Engineering Journal AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC. Page 1: Lawrence G. Griffis Servicability Limit States Under Wind Load Page 17: A. Zureick Design Strength of Concentrically Loaded Single Angle Struts Page 31: Tony Lue and Duane S. Ellifritt The Warping Constant for the W-Section with a Channel Cap Page 34: Charles J. Carter and Louis F. Geschwindner The Economic Impact of Overspecifying Simple Connections Page 37: Discussion—William E. Moore II Simple Equations for Effective Length Factors—Pierre Dumonteil Page 38: Correction—Pierre Dumonteil Simple Equations for Effective Length Factors Page 39: Correction—Lewis B. Burgett Fast Check for Block Shear 1st Quarter 1993/Volume 30, No. 1
Transcript
EngineeringJournalAMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Page 1: Lawrence G. GriffisServicability Limit States Under Wind Load
Page 17: A. ZureickDesign Strength of Concentrically Loaded SingleAngle Struts
Page 31: Tony Lue and Duane S. EllifrittThe Warping Constant for the W-Section with aChannel Cap
Page 34: Charles J. Carter and Louis F. GeschwindnerThe Economic Impact of Overspecifying SimpleConnections
Page 37: Discussion—William E. Moore IISimple Equations for Effective Length Factors—Pierre Dumonteil
Page 38: Correction—Pierre DumonteilSimple Equations for Effective Length Factors
Page 39: Correction—Lewis B. BurgettFast Check for Block Shear
1st Quarter 1993/Volume 30, No. 1
INTRODUCTION
The increasing use and reliance on probability based limitstates design methods, such as the recently adopted AISCLRFD Specification,1 has focused new attention on the prob-lems of serviceability in steel buildings. These methods,along with the development of higher-strength steels andconcretes and the use of lighter and less rigid building mate-rials, have led to more flexible and lightly damped structuresthan ever before, making serviceability problems moreprevalent.
The purpose of this paper is to focus attention on twoimportant serviceability limit states under wind loads;namely, deformation (including deflection, curvature, anddrift) and motion perception (acceleration). These issues areparticularly important for tall and/or slender steel and com-posite structures. A brief review of available information onthese subjects will be presented followed by a discussion ofcurrent standards of practice, particularly in the United States.Finally, proposed standards will be presented that, hopefully,will focus attention, debate, and perhaps new research effortson these very important issues in design.
IMPORTANCE OF SERVICEABILITYLIMIT STATES 12,31
Every building or other structure must satisfy a strength limitstate, in which each member is proportioned to carry thedesign loads to resist buckling, yielding, instability, fracture,etc.; and serviceability limit states which define functionalperformance and behavior under load and include such itemsas deflection, vibration, and corrosion. In the United States,strength limit states have traditionally been specified in build-ing codes because they control the safety of the structure.Serviceability limit states, on the other hand, are usuallynoncatastrophic, define a level of quality of the structure orelement, and are a matter of judgment as to their application.Serviceability limit states involve the perceptions and expec-tations of the owner or user and are a contractual matterbetween the owner or user and the designer and builder. It isfor these reasons, and because the benefits themselves areoften subjective and difficult to define or quantify, that ser-
viceability limit states for the most part are not includedwithin U.S. building codes. The fact that serviceability limitstates are usually not codified should not diminish theirimportance. Exceeding a serviceability limit state in a build-ing or other structure usually means that its function is dis-rupted or impaired because of local minor damage, deterio-rations, or because of occupant discomfort or annoyance.While safety is usually not at issue, the economic conse-quences can be substantial. Interestingly, there are someserviceability items that can also be safety related. For in-stance, excessive building drift can influence frame stabilitybecause of the P-∆ effect. Excessive building drift can alsocause portions of the building cladding to fall and potentiallyinjure pedestrians below.
Serviceability limit states can be grouped into three cate-gories as follows:
1. Deformation (deflection, curvature, drift). Common ex-amples include local damage to nonstructural elements(e.g., ceilings, cladding, partitions) due to deflectionsunder dead, live, wind, or seismic load; and damagefrom temperature change, moisture, shrinkage, or creep.
2. Motion perception (vibration, acceleration). Commonexamples include human discomfort caused by wind ormachinery, particularly if resonance occurs. Floor vibra-tions from people or machinery and acceleration in tallbuildings under wind load are usual areas of concern inthis category.
3. Deterioration. Included are such items as corrosion,weathering, efflorescence, discoloration, rotting, andfatigue.
The focus on this paper will be items one and two.
CURRENT TREATMENT OF SERVICEABILITYISSUES IN U.S. CODES
A review of the three model building codes3,29,35 in the UnitedStates reveals a somewhat inconsistent and haphazard ap-proach to serviceability issues. For instance, it is implied thatthe codes exist strictly to protect life safety of the generalpublic. Yet, traditionally they have contained provisions fordeflection control of floor members while ignoring provisionsfor other member types (columns, walls, mullions, etc.). Nomention is made of limits for wind drift, vibration, expansionand contraction (expansion joint guidelines), or corrosion.
The author’s work in professional committees and codebodies, coupled with a review of recent surveys of the profes-
Lawrence G. Griffis is Senior Vice President and Director ofStructural Engineering for Walter P. Moore and Associates,Houston, TX.
Serviceability Limit States Under Wind LoadLAWRENCE G. GRIFFIS
FIRST QUARTER / 1993 1
sion36 seem to reveal a reluctance of engineers to codifyserviceability issues. This reluctance probably stems in parton differences of opinion as to the purpose of building codes(i.e., protection for life safety exclusively or establishment ofcomplete minimum design standards including strength andserviceability), but also a genuine concern for restrictingdesign options, stifling creativity, and removing the all-important concept of “engineering judgment” from the solu-tion to the problem. There is also the belief, rightly so, thattoo little hard data exists to justify rigid standards on mostserviceability issues.
It is important that engineers recognize these problems andbegin to focus on the solution of serviceability related designissues. The reason for doing so is the large economic impactthat serviceability items are having on the operational costsof buildings.
MEAN RECURRENCE INTERVAL WIND LOADSFOR SERVICEABILITY DESIGN
The first step in establishing a serviceability design criterionis to define the load under which it is to be checked. Windloading criteria for strength limit states in the United Statesare normally based on a 50-year mean recurrence interval fornormal buildings and a 100-year mean recurrence interval forcritical structures. There seems to be a general consensus thatbasing serviceability criteria on such a severe loading thatmay occur only once, on the average, during the lifetime ofthe structure is unrealistic and too stringent a standard toapply. The average tenant occupancy in office buildings hasbeen defined as eight years.26 It seems reasonable to baseserviceability criteria on a mean recurrence interval more inthis range of time because the consequences of exceeding aserviceability limit state are usually not safety related. Variousresearchers have suggested mean recurrence intervals of fromfive to ten years for serviceability issues.10,11,12,14,17,18,19,20,33,36
If no permanent damage results from exceeding the service-ability limit, some researchers have also suggested selectingserviceability criteria (such as floor deflection) on an annualbasis.14
A wind load for a mean recurrence interval of 10 years isrecommended for checking the two wind serviceability limitstates defined herein (deformation and motion perception).This corresponds to a 10 percent probability of being ex-ceeded in any given year. While it has become standardpractice to base building accelerations under wind load on thismean recurrence interval, drift criteria typically have beenformulated around the same mean recurrence interval (50years or 100 years) as the strength limit state.36
The proposed 10-year mean recurrence interval comparesto five years as proposed in ISO Standard 6897-1984, 10 yearsas proposed by the National Building Code of Canada (1990),20 years in the Australian Standard AS 1170.2-1989 and 0.1years as proposed by the Japanese.28
BUILDING DRIFT—STANDARD OF PRACTICE
Serviceability of buildings under wind loads has traditionallybeen checked in the design office by evaluation of the lateralframe deflection calculated on the basis of a statically appliedwind load obtained from the local building code. The magni-tude of the wind load is usually the same as that used inproportioning the frame for strength and typically is based ona 50-year or 100-year mean recurrence interval load. Some-times, an arbitrary wind load (i.e., 20 PSF above 100 ft, 0(zero) PSF below 100 ft as has been used in New York Cityon the design of some buildings15) is used in the serviceabilitycheck. This serviceability check, for all but the tallest andmost slender of buildings (where wind tunnel studies areutilized), has been used to prevent damage to collateral build-ing materials, such as cladding and partitions, and also tocontrol the perception of building motion. None of the threenational building codes in the United States specify a limit tolateral frame deflection under wind load. The degree of thisserviceability check is left to the judgment of the designengineer. Lateral frame deflection is usually evaluated for thebuilding as a whole, where the applicable parameter is totalbuilding drift, defined as the lateral frame deflection at thetop-most occupied floor divided by the height from grade tothe uppermost floor (∆ / H); and for each floor of the building,where the applicable parameter is interstory drift, defined asthe lateral deflection of a floor relative to the one immediatelybelow it divided by the distance between floors ((δn − δn−1) /h). Typical values of this parameter (commonly called driftindex) used in this serviceability check are H / 100 to H / 600for total building drift and h / 200 to h / 600 for interstory driftdepending on building type and materials used. The mostwidely used values are 1 / 400 to 1 / 500.36 Lateral framedeflections have historically been based on a first order analysis.
DRIFT—A REVISED DEFINITION 7
Drift Measurement Index (DMI)
If the goal in defining a drift limit is limited to only the controlof damage to collateral building elements, such as claddingand partitions, and is separated from the problem of buildingmotion, then frame racking or shear distortion (strain) is thelogical parameter to evaluate.
Mathematically, if the local x, y displacements are knownat each corner of an element or panel, then the overall averageshear distortion for rectangular panel ABCD as shown inFigure 1 may be termed the drift measurement index (DMI)and defined as follows:
Drift measurement index (DMI) = average shear distortion
DMI = 0.5 × [(XA − XC) / H + (XB − XD) / H + (YD − YC ) / L + (YB − YA ) / L]
DMI = 0.5 × (D1 + D2 + D3 + D4)
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION2
where,
Xi = vertical displacement of point i
Yi = lateral displacement of point i
D1 = (XA − XC) / H, horizontal component of racking driftD2 = (XB − XD) / H, horizontal component of racking driftD3 = (YD − YC ) / L, vertical component of racking driftD4 = (YB − YA ) / L, vertical component of racking drift
It is to be noted that terms D1 and D2 are the horizontalcomponents of the shear distortion or frame racking and arethe familiar terms commonly referred to as interstory drift.The terms D3 and D4 are the vertical components of the sheardistortion or frame racking caused by axial deformation ofadjacent columns.
If it can be accepted that the DMI is the true measure ofpotential damage, then it becomes readily apparent that theevaluation of interstory drift alone can be misleading inobtaining a true picture of potential damage. Interstory driftalone does not account for the vertical component of frameracking in the rectangular panel that also contributes to thepotential damage, nor does it exclude rigid body rotation ofthe rectangular panel which, in itself, does not contribute todamage. It can be shown that evaluation of the commonlyused interstory drift can significantly underestimate the dam-age potential in a combined shear wall/frame type buildingwhere the vertical component of frame racking can be impor-tant; and significantly overestimate the damage potential in ashear wall or braced frame building where large rigid bodyrotation of a story can occur due to axial shortening ofcolumns.7
Consider for example, the eight-story building shown inFigure 2. This frame represents a typical windframe that maybe found in any office building with 36-ft lease depths (build-ing perimeter to center core) and a central core containingelevator, stairs, etc. The frame shown consists of a combinedmoment frame and X-braced frame. Figure 3 shows a plot
(exaggerated) of the deflected shape of the top level underwind loads. Table 1 shows calculations for the traditionalstory drift and the revised drift definition DMI. The signifi-cant thing to note is that the potential damaging deformations,as represented by the DMIs, are more severe in the externalbays (panels 1, 3) and much less severe in the internal bay(panel 2) than predicted by the traditional story drift calcula-tion. Most of the deformation in the center bay (panel 2) issimply rigid body rotation that, by itself, is not damaging topartitions.
Drift Measurement Zone (DMZ)
It is logical to identify the rectangular panel ABCD in Fig-ure 1 as the zone in which the damage potential is to beevaluated and define it the drift measurement zone (DMZ).From a practical standpoint, these zones will typically repre-sent column bays within a building and would be incorporatedas part of the building frame analysis.
Drift Damage Index (DDI)
Once the determination of the shear distortion or drift meas-urement index (DMI) is made for different column bays ordrift measurement zones (DMZs), it must be compared to adamage threshold value for the element being protected.These damage threshold limits can be defined as the sheardistortion or racking that produces the maximum amount ofcracking or distress that can be accepted, on the average, once
Fig. 1. Drift measurement index (DMI). Figure 3
Figure 2
FIRST QUARTER / 1993 3
every 10 years. It is logical to define these damage thresholdshear distortions as the drift damage index (DDI). From thestandpoint of serviceability limit states it is necessary toobserve the following inequality:
drift measurement index ≤ dr ift damage index
DMI ≤ DDI
A significant body of information is available from rackingtests for different building materials that may be utilized todefine DDIs.2 This is discussed further below in conjunctionwith Figure 4.
Calculation of Building Frame Deflection
If drift measurement indices (DMIs) are to be effective incontrolling collateral building material damage, there must bea consistency and accuracy in the method of calculation. Arecent survey36 on drift clearly pointed out the problems thatexist in the structural engineering community on controllingdamage by excessive drift. There appears to be a wide vari-ation in the methods of structural analysis performed tocalculate building frame deflection. Ideally, if DMIs are to bean effective parameter in controlling damage caused by build-ing deflection, then the structural analysis employed mustreasonably capture the significant response of the buildingframe under load. As previously stated, it is suggested that thewind load be defined by the 10-year mean recurrence intervalstorm. The designer should recognize that the wind loads used
in the structural analysis are “static equivalent” wind loadsthat are intended to estimate the peak load effect (mean plusdynamic) caused by the vibratory nature of the buildingmotion. The structural analysis must then capture all signifi-cant components of potential frame deflection as follows:
1. Flexural deformation of beams and columns.2. Axial deformation of columns.3. Shear deformation of beams and columns.4. Beam-column joint deformation.5. Effect of member joint size.6. P-∆ effect.
The behavioral knowledge of each of the above effects onframe deflection is sufficiently understood to permit a reason-ably accurate prediction of the contribution to the total re-sponse. Computer programs and analytical models are nowwithin reach of most engineers to afford consideration of allof the above effects.
Depending on the height, slenderness, and column baygeometry, each of these effects can have a significant influ-ence on building deflection. A recent study8 on the sources ofelastic deformation for different height (10 to 50 stories) andnumber-of-bay (5 to 13 bays) frames showed the following:
1. Axial deformations in columns can be very significantfor tall slender frames, amounting to 26 percent to 59percent of the total deflection, depending on bay widths.
2. Shear deformations, as a percentage of the total framedeflection, tend to increase with the number of bays andalso as the bay size (beam span) reduces. Shear defor-mation can account for as much as 26 percent of the totaldeflection. For slender “tube” structures (10- to 15-ftbays and 40 to 50 stories tall) shear deformation cancontribute as much as flexural deformation to the totalbuilding deflection. Shear deformations should never beignored in frame deflection if an accurate response pre-diction is expected.
3. Beam-column joint deformations, particularly for steelstructures, constitute a significant portion of the totaldeflection for all frames studied and should never byignored. As with shear deformations, there is a generaltrend for deformations to increase as the number of baysincreases and the size of the bay decreases. Participation
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION4
as a percentage of the total varied from 16 percent to 41percent.
4. The P-∆ effect can easily increase total frame dis-placement by 10 to 15 percent depending on frameslenderness.
Errors in the determination of frame stiffness can alsoaffect proper design for strength. For example, the P-∆ effectis a function of frame stiffness; the magnitude of wind forcesin tall buildings is affected by building period; and the mag-nitude of seismic forces is also affected by building period.
DAMAGE THRESHOLDS FORBUILDING MATERIALS
General guidelines to the behavior that might be expectedfrom different building elements and materials at various driftindices may be obtained from a review of the literature.2,13,31
A summary of behavior, taken from a recent study on ser-viceability research needs31 is shown in Table 2. Anothersource of information may be found in seismic racking testsof exterior cladding systems for buildings sometimes per-formed during routine testing of mock-ups at testing labora-tories. One of the most comprehensive studies of damageintensity as a function of shear distortion can be found inReference 2 which contains a summary of over 700 rackingtests on various nonstructural partitions taken from more than30 different sources. Partition types surveyed included tileand hollow brick, concrete block, brick and “veneer”; wallswhich consisted of gypsum wall board, plaster, and plywood.Veneer walls are often referred to as “drywall” in engineeringpractice. Damage intensity was defined on a scale from 0.0
to 1.0 with 0.1 to 0.3 defined as minor damage, 0.4 to 0.5defined as moderate damage, 0.6 to 0.7 defined as substantialdamage, and 0.8 to 1.0 defined as major damage. A damageintensity of 1.0 is defined as complete or intolerable. Figure 4shows a plot of damage intensity versus shear distortion forthe partition groups discussed. If the upper limit of the “minordamage” range is selected as the maximum acceptable dam-age to occur in a 10-year design period, then the deflectionlimit of 0.25 percent (1 / 400) is obtained for veneer ordrywall in Figure 4. This number correlates reasonably wellwith the first damage threshold limit of 1⁄4-in. displacementfor an eight foot tall test panel as described in Reference 13for gypsum wallboard. The 0.3 damage intensity has beenused as the maximum acceptable shear distortion for thevarious partition types in Table 3.
Once a wind load (mean recurrence interval) has been definedfor use in the serviceability check, the appropriate deforma-tion to measure it has been defined (drift measurement index(DMI)) and damage thresholds are determined from tests orestimated, it remains only to establish an appropriate limit fordifferent building components. Table 3 is a compilation ofmost common building elements with recommended defor-mation limits. The building elements considered include roof,exterior cladding, interior partitions, elevators, and cranes.Most of the more common building cladding and partitiontypes are considered. Deformation types addressed includedeflection perpendicular to the plane of the building elementand shear deformation (racking) in the plane of the element.
Table 2.Serviceability Problems at Various Deflection or Drift Indices 14,31
Deformation as aFraction of Span or
HeightVisibility of
Deformation Typical Behavior
≤ 1 / 1000 Not Visible Cracking of brickwork
1 / 500 Not Visible Cracking of partition walls
1 / 300 Visible General architectural damage Cracking in reinforced walls Cracking in secondary members Damage to ceiling and flooring Facade damage Cladding leakage Visual annoyance
1 / 200 – 1 / 300 Visible Improper drainage
1 / 100 – 1 / 200 Visible Damage to lightweight partitions, windows, finishes Impaired operation of removable components such as doors, windows, sliding partitions
Pendant Operated Wind Frame Shear Strain (DMI) H / 100 Note 23
Notes to Table C.H = story height L = span length of supporting member DMI = drift measurement index 1. Metal roofs include standing seam and thru fastener type roofs.12
2. Deflection limit shown is relative support movement measured perpendicular to a line drawn between skylight support points. Racking movements in the planeof the glass should be limited to 1⁄4-in. for gasketed mullions and 1⁄8-in. for flush (butt) glazing.12
3. Deflection limits recommended by the Brick Institute of America34 are L/600 – L/720. 4. L/600 is recommended for the case when predominant flexural stress in masonry is perpendicular to bed joint. L/300 may be used for the case when
predominant flexural stress in masonry is parallel to bed joint. 5. H/400 limit applies if brick is supported on relief angles at each floor with 3⁄8-in. soft joint and 3⁄8-in. control joints are used in each column bay. 6. Reinforced concrete masonry implies vertical reinforcing bars in grouted cells and/or horizontal reinforcing bars in bond beams. 7. Assumes only windframe designed to carry lateral loads and flexible connections used between wall and parallel windframe. H/600 limit also protects wall
perpendicular to plane of windframe from excessive flexural cracking. A horizontal crack control joint at base of wall is recommended. Limit crack widthunder wind load to 1⁄16-in. if no joint is used and 1⁄8-in. if control joint is used.12
8. Assumes only windframe designed to carry lateral loads and flexible connections used between wall and parallel windframe. H/400 applies only if in-fill wallshave 3⁄8-in. soft joints against structural frame.
9. Assumes only windframe designed to carry lateral loads and flexible connections used between wall and parallel windframe. Stricter limit should be consideredif required to protect other building elements. If walls designed as shear walls, then design DMI should be based on damage control of other building elements.H/200 limit also protects wall perpendicular to plane of windframe from excessive flexural cracking. If a horizontal control joint at base of wall is used, thenlimit may be changed to H/100.12
10. H/400 limit applies to reinforced masonry walls designed as shear walls unless stricter limit is required to protect other more critical building elements. Reinforcedmasonry walls infilled “hard” against structural windframe should not be used without assessing their stiffness in a compatibility analysis with windframe, unlessisolation joints are provided between wall and building frame.
11. In cases where wall support is indeterminate, differential support deflection should be considered in design of wall panel.12. Assumes only windframe designed to carry lateral loads and flexible connections are used between wall and parallel windframe. Stricter limit should be
considered if required to protect other building elements. If panels designed as shear walls then H/400 is recommended limit with minimum 3⁄4-in. panel joints.13. Control joints are recommended to limit cracking from shrinkage, thermal, and building movement.14. H/400 limit applies if wall is panelized with 3⁄8-in. control joints and relief joints are used between floors and at each column bay. If plaster applied to unreinforced
masonry, then limits should be same as masonry.15. H/400 applies if panel connection to frame is determinate, flexible connections are used between panel and parallel windframe and minimum 3⁄4-in. panel joints
are used. Panels with indeterminate support to frame should be designed for differential support movement.16. Consult metal panel manufacturer for possible stricter requirements.17. L/100 limit applies for metal panel only. Other building components may warrant stricter limit.18. L/175 recommended by American Architectural Manufacturers Association.27 Recommended limit changes to L/360 when a plastered surface or dry wall
subjected to bending is affected. At roof parapet or other overhangs recommended limit is 2L/175 except that the deflection of a member overhanging ananchor joint with sealed joint (such as for roof flashing, parapet cover, soffit) shall be limited to no more than one half the sealant joint depth between theframing member and fixed building element.
19. H/400 limit is to protect connections to building frame and also sealants between panels. More liberal limits may be applicable for custom designedcurtain/window walls where racking can be accounted for in design and where wall will be tested in a labortory mock-up. Consult manufacturer for rackinglimits of off-the-shelf systems.
20. Recommended limits shown assume partition is constructed “hard” against structural frame. More liberal limits may be appropriate if isolation (“soft”) joints aredesigned between partition edge and structural frame. Design of structural frame for DMI limits stricter than H/600 is probably not practical or cost effective.
21. In addition to the static deflection limit shown, proper elevator performance requires consideration of building dynamic behavior. Design of elevator systems(guide rails, cables, sheaves) will require knowledge of predominant building frequencies and amplitude of dynamic motion. This information should be furnishedon the drawings or in the specifications.
22. Limit shown applies to wind loads or crane forces, either lateral or longitudinal to crane runway. Deflection limit specified is to be measured at the elevation ofcrane runways.12
23. Buildings designed to H/100 limit will exhibit observable movements during crane operation. Stricter limits may be appropriate to control this and/or to protectother building components.12
FIRST QUARTER / 1993 7
Notes are included at the end of the table to explain or clarifya recommendation.
It should be pointed out that the recommended limitsshown are guidelines based on past successful performance.The degree of distress in any of the building elements (clad-ding, partitions) under the action of wind loads is highlydependent upon the nature and design of the attachments orjoints to the building frame. If specific attention is paid to thisaspect then oftentimes any reasonable deformation can beaccommodated without damage. Indeed, it may be moreprudent and cost effective to detail joints to accommodate ahigher deflection than to design a higher level of stiffness intothe building wind frame.
SERVICEABILITY LIMIT STATE—MOTION PERCEPTION
Motion Perception Parameter—Acceleration28,30
Perception to building motion under the action of wind maybe described by various physical quantities including maxi-mum values of velocity, acceleration, and rate of change ofacceleration, sometimes called jerk. Since wind induced mo-tion of tall buildings is composed of sinusoids having a nearlyconstant frequency f but varying phase, each quantity isrelated by the constant 2πf where f is the frequency of motion(V = 2 π f )D; A = (2 π f )2D; J = (2 π f )3D where D, V, A, andJ are maximum displacement, velocity, acceleration, and jerkrespectively). Human response to motion in buildings is acomplex phenomenon involving many psychological andphysiological factors. It is believed that human beings are notdirectly sensitive to velocity if isolated from visual effectsbecause, once in motion at any constant velocity, no forcesoperate upon the body to keep it in such motion. Acceleration,on the other hand, requires a force to act which stimulatesvarious body organs and senses. Some researchers believethat the human body can adapt to a constant force acting uponit whereas with changing acceleration (jerk) a continuouslychanging bodily adjustment is required. This changing accel-eration may be an important component of motion perceptionin tall buildings. It appears that acceleration has become thestandard for evaluation of motion perception in buildingsbecause it is the best compromise of the various parameters.It also is readily measurable in the field with available equip-ment and has become a standard for comparison and estab-lishment of motion perception guidelines among various re-searchers around the world.
Factors Affecting Human Response to Building Motion25
Perception and tolerance thresholds of acceleration as a meas-ure of building motion are known to depend on various factorsas described below. These factors have been determined frommotion simulators that have attempted to model the action ofbuildings subjected to wind loads.
1. Frequency or Period of Building. Field tests haveshown that perception and tolerance to accelerationtend to increase as the building period increases (fre-quency decreases) within the range of frequency com-monly occurring in tall buildings.
2. Sex. The general trend of response between men andwomen is the same although women are slightly moresensitive than men.
3. Age. The sensitivity of humans to motion is an inversefunction of age, with children being more sensitivethan adults.
4. Body Posture. The sensitivity of humans to motion isproportional to the distance of the persons head fromthe floor; the higher the person’s head, the greater thesensitivity. Thus, a person’s perception increases as hegoes from sitting on the floor, to sitting in a chair, tostanding. However, since freedom of the head may beimportant to motion sensitivity, a person sitting in achair may be more sensitive than a standing personbecause of the body hitting the back of the chair.
5. Body Orientation. Humans tend to be more sensitive tofore-and-aft motion than to side-to-side motion be-cause the head can move more freely in the fore-and-aftdirection.
6. Expectancy of Motion. Perception threshold decreasesif a person has prior knowledge that motion will occur.Threshold acceleration for the case of no knowledge isapproximately twice that for the case of prior knowledge.
7. Body Movement. Perception thresholds are higher forwalking subjects than standing subjects, particularly ifthe subject has prior knowledge that the motion willoccur. The perception threshold is more than twice asmuch between the walking and standing case if thereis prior knowledge of the event, but only slightlygreater if there is no knowledge of the event.
8. Visual Cues. Visual cues play an important part inconfirming a person’s perception to motion. The eyescan perceive the motion of objects in a building suchas hanging lights, blinds, and furniture. People are alsovery sensitive to rotation of the building relative tofixed landmarks outside.
9. Acoustic Cues. Buildings make sounds as a result ofswaying from rubbing of contact surfaces in framejoints, cladding, partitions, and other building ele-ments. These sounds and the sound of the wind whis-tling outside or through the building are known to focusattention on building motion even before subjects areable to perceive the motion, and thus lower their per-ception threshold.
10. Type of Motion. Under the influences of dynamic windloads, occupants of tall buildings can be subjected totranslational acceleration in the x and y direction andtorsional acceleration as a result of building oscillationin the along-wind, across-wind, and torsional direc-
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION8
tions, respectively. While all three components con-tribute to the response, angular motion appears to bemore noticeable to occupants, probably caused by anincreased awareness of the motion from the aforemen-tioned visual cues. Also, torsional motions are oftenperceived by a visual-vestibular mechanism at motionthresholds which are an order of magnitude smallerthan those for lateral translatory motion.24
Root-Mean-Square (RMS) Versus Peak Acceleration
A review of the literature on the subject of motion perceptionas measured by acceleration shows a difference in the pres-entation of the results. Some researchers report maximum orpeak acceleration and some report root-mean-square or RMSaccelerations. This dual definition has extended into estab-lishing standards for motion perception.
Most of the research conducted on motion perception hasbeen with motion simulators subjected to sinusoidal motionwith varying frequency and amplitude. In these tests it hasbeen common to report the results in maximum or peakacceleration since that was the quantity directly measured. Itshould be pointed out that for sinusoidal acceleration, thepeak is equal to √2 times the RMS value. It appears that windtunnel research has tended to report peak acceleration or bothpeak and RMS in order to correlate the wind tunnel studieswith these motion simulation tests. Many researchers believethat, when the vibration persists for an extended period oftime (10 to 20 minutes) as is common with windstorms havinga 10-year mean recurrence interval, that RMS acceleration isa better indicator of objectionable motion in the minds ofbuilding occupants than isolated peak accelerations that maybe dampened out within a few cycles.17,18,22,33 Also, the RMSstatistic is easier to deal with during the process of temporaland spatial averaging because the 20-minute averaging periodfor a storm represents a time interval over which the meanvelocity fluctuates very little. The relationship between peakand RMS accelerations in tall buildings subjected to thedynamic action of wind loads has been defined by the peakfactor which varies with building frequency, but which isoftentimes taken as 3.5. Correlation between peak and RMSaccelerations in tall building motion may be made using thispeak factor.
Relationship Between Building Drift andMotion Perception
Engineers of tall buildings have long recognized the need forcontrolling annoying vibrations to protect the psychologicalwell being of the occupants. Prior to the advent of wind tunnelstudies this need was addressed using rule-of-thumb driftratios of approximately 1/400 to 1/600 and code specifiedloads. Recent research,22 based on measurement of windforces in the wind tunnel, has clearly shown that adherenceto commonly accepted lateral drift criteria, per se, does notexplicitly ensure a satisfactory performance with regard to
motion perception. The results of one such study22 are plottedin Figure 5 for two square buildings having height/widthratios of 6/1 and 8/1 where each is designed to varying driftratios. Plots are shown of combined transitional and torsionalacceleration as a function of design drift ratio. At drift ratiosof 1/400 and 1/500 neither building conforms to acceptablestandards for acceleration limits. The reason that drift ratiosby themselves do not adequately control motion perceptionis because they only address stiffness and do not recognizethe important contribution of mass and damping, which to-gether with stiffness, are the predominant parameters affect-ing acceleration in tall buildings. This is discussed furtherlater in the paper.
Human Response to Acceleration
Considerable research in the last 20 years has been conductedon the subject of determining perception threshold values foracceleration caused by building motion.9,25,28 Much of thiswork has also attempted to formulate design guidelines fortolerance thresholds to be used in the design of tall and slenderbuildings.
Some of the earliest attempts to quantify the problem wereperformed by Chang5,6 who proposed peak acceleration limitsfor different comfort levels that were extrapolated from datain the aircraft industry. Chang’s proposed limits, plotted inFigure 6 as a function of building period, are stated as follows:
Peak Acceleration Comfort Limit <0.5% g Not Perceptible0.5% to 1.5% g Threshold of Perceptibility1.5% to 5.0% g Annoying5% to 15.0% g Very Annoying
>15% g Intolerable
Additional data has been reported by researchers whoutilized motion simulators to define perception levels.9,28 Asummary of this work is shown in Figures 7 and 8 showingplots of perception thresholds for both peak and RMS accel-eration as a function of building period.
Perhaps the most comprehensive studies of the problemhave been performed in Japan28 for a wide range of variables.
Fig. 5. RMS acceleration vs drift index.
FIRST QUARTER / 1993 9
This work is summarized in Figure 8 where peak accelerationis plotted as a function of building period. Each curve andzone between curves is identified in the figure. The discussionbelow is keyed to the letters and numbers in the figure and istaken from Reference 28:
1. Zone A, below Curve 1, identifies peak accelerationless than about 0.5 percent g. In this zone, a humancannot perceive motion at all. No evidence of motionexists except for possible rubbing of building compo-nent surfaces in contact. Curve 1 defines the limit ofperception threshold for an average population.
2. Curve 2 (0.5 percent g) defines the point where somebuilding objects (furniture, hanging lights, water) be-gin to move.
3. Curve 3 separates zones between “very normal walk-ing” and “nearly normal walking.”
4. Zone B (between 0.5 percent g and 1.0 percent g)identifies a zone where some people can perceivemotion. Some building fixtures and objects will beginto move slightly, but these movements are generallynot observable except to a person who looks directlyat them.
5. Curve 4 (1 percent g) separates the zones where peoplecan be affected by working at a desk.
6. Curve 5 defines the threshold where people can start to
become subjected to motion sickness when exposed tothis level of motion for extended periods.
7. Zone C (between 1.0 percent g and 2.5 percent g) iswhere most people are able to perceive motion andbecome affected by desk work. Generally, in this zone,people can be subjected to motion sickness if exposedfor extended periods but can walk without hindrance.
8. Curve 6 defines the limit between normal and hinderedwalking.
9. Zone D (between 2.5 percent g and 4.0 percent g)defines the acceleration range where desk work be-comes difficult and at times impossible. Most peoplecan walk and go up and down stairs without too muchdifficulty.
10. Curve 7 (3.5 percent g) defines the point where workingat a desk is difficult.
11. Curve 8 (4.0 percent g) defines the acceleration wherefurniture and fixtures begin to make sounds, whichmay evoke a strong concern or alarm among somepeople.
12. In Zone E people strongly perceive motion and stand-ing people lose their balance and find it hard to walknaturally.
13. Curve 9 marks the point where people are unable towalk.
14. Curve 10 defines the maximum tolerance for motion.15. In Zones F and G (above 5.0 percent g) most people
cannot tolerate the motion and are unable to walk.These zones are considered to be at the limit of walkingability.
16. In Zone H people cannot walk. Motion is intolerable.
Design of Tall Buildings for Acceleration
The design of most tall buildings is controlled by lateraldeflection and most often by perception to motion. Indeed,this characteristic is often proposed as one definition of a“tall” building.
While the problem of designing for motion perception intall buildings is usually solved by conducting a scale model
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION10
force-balance or aeroelastic test in the wind tunnel, certaincriteria have been established to aid the designer. Empiricalexpressions now exist21,22,32 that allow approximate evalu-ation of the susceptibility of a building to excessive motion.This can be very helpful in the early design stages particularlywhere geometry, site orientation, or floor plan are not yetfixed.
The following simple expressions22 for along-wind (drag),across-wind (lift), and torsional RMS acceleration have beenderived for square, symmetric (coincident centers of mass,rigidity, and geometry), tall buildings in an urban environ-ment:
Along-wind:
AD(Z) = CD(Z) UH
2.74
KD0.37 × ζ0.5 × MD
0.63 (1)
Across-wind:
AL(Z) = CL(Z) UH
3.54
KL0.77 × ζ0.5 × ML
0.23 (2)
Torsional:
AΘ(Z) = CΘ(Z) UH
1.88
KΘ −0.06 × ζ0.5 × MΘ
1.06 ,
NΘBUH
≤ 0.25 (3a)
AΘ(Z) = CΘ(Z) UH
2.76
KΘ 0.38 × ζ0.5 × MΘ
0.62 , NΘBUH
> 0.25 (3b)
The proportionality constants CD(Z), CL(Z), and CΘ(Z) aredefined as follows:
CD(Z) = 0.0116 × B0.26 × Z
CL(Z) = 0.0263 × B−0.54 × Z
CΘ(Z) = 0.00341 × B2.12 × Z, NΘBUH
≤ 0.25
CΘ(Z) = 0.00510 × B1.24 × Z, NΘBUH
> 0.25
The definition of terms in the above expressions are listedbelow:
UH = mean hourly wind speed at the top of the building(meters/sec.)
H = building height (meters)B = plan dimension of square building (meters)M = generalized mass of the building (kilogram)
= ∑mi
i = 1
n
φi2 mi is mass of floor i and φi is modal
coordinate at floor i, normalized so that φ = 1 at(Z) = H
N = frequency (hertz)K = generalized stiffness (newton/meters)
= (2πN)2 × M
ζ = damping ratio
For rectangular buildings, B may be taken as the squareroot of the plan area. The resultant RMS acceleration at thecorner of the building, AR, is calculated as follows:
AR = (AD2 + AL
2 + (B / √2 × AΘ)2)0.5 (4)
These expressions were used in a parametric study of a150-ft square building having slenderness ratios (H / B) offive through ten (building heights varying from 754 feet to1,495 feet). The buildings were subjected to basic wind speedof 70 mph in an Exposure B (suburban) environment asdefined in ASCE 7-88. The buildings were assumed to beall-steel with steel weights typical of tall buildings of theseheights, varying from 25 psf to 44 psf. Building densities wereassumed to vary from 7.77 pcf to 9.23 pcf, typical for officebuildings having lightweight concrete metal deck floors andcurtain wall cladding. Translational building periods werecalculated using the well-known Rayleigh formula,35 whichfor uniform prismatic buildings with a linear deflected shapecan be approximated by the following expression:
T = 0.904HρDR
pR
0.5
(5)
In this expression, T is the building period in seconds, H isthe building height (feet), ρ is the density (PCF), DR is thedesign drift ratio (∆ / H), p is the equivalent uniform pressure(PSF) and R is the aspect ratio H / B. Torsional periods weretaken as 85 percent of the translational periods. For this study,the drift ratio under design wind load as defined by ASCE7-88 is set at 1/400 or 0.0025. This practice is typical of theprocedure used in many building designs.
Along-wind, across-wind, and torsional RMS accelera-tions were calculated at the building top corner using 10-yearmean recurrence interval wind loads. Complete building datais shown in Table 4 and the accelerations are plotted inFigure 9. Also shown in Figure 9 is the design limit as definedlater in this paper. The results clearly show that controllingdrift limits to the traditional design value of 0.0025 does notensure satisfactory performance from the standpoint of mo-tion perception. In examining Figure 9, it is interesting to notethat for the common aspect ratios of 5-6, torsional accelera-tion is comparable to across-wind acceleration and both aresignificantly larger than the along-wind acceleration.
Generally, for most tall buildings without eccentric massor stiffness, the across-wind response will predominate if(WD)0.5 / H < 0.33 where W and D are the across-wind and
FIRST QUARTER / 1993 11
along-wind plan dimensions respectively and H is the build-ing height.32
In examining the across-wind proportionality, which often-times is the predominant response, it is possible to make thefollowing deductions:
1. If stiffness is added without a change in mass, accelera-tion will be reduced in proportion to 1 / N1.54, which isproportional to 1 / K0.77, where K is the stiffness.
2. If mass is added throughout the building without chang-ing the stiffness, acceleration will be reduced in propor-tion to 1 / M0.23.
3. If mass is added with a proportionate increase in stiffness
Notation:T = period (seconds)f = frequency (hertz)gp = peak factor
NOTE:1. RMS and peak accelerations listed in this table are the traditional “unofficial” standard applied in U.S. practice
based on the author’s experience.
Fig. 9. Parametric study—150 -ft sq bldg.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION12
so that N does not change, then the acceleration will bereduced in proportion 1 / M or 1 / K.
4. If additional damping is added, then the acceleration willbe reduced in proportion to 1 / ζ0.5.
It should be pointed out that torsional response can beimportant even for symmetrical buildings with uniform stiff-ness. This is because a torsional wind loading can occur fromunbalance in the instantaneous pressure distribution on thebuilding surface.
Oftentimes, in very slender buildings, it is not possible toobtain satisfactory performance, given building geometry andsite constraints, by adding stiffness and/or mass alone. Theoptions available to the engineer in such a case involve addingadditional artificial damping and/or designing mass or pen-dulum dampers to counteract the sway.16
Standards of Motion Perception
Numerous high-rise buildings have been designed and areperforming successfully all over the world. Many have beendesigned according to an “unofficial” standard observed inthe author’s practice as defined in Table 5. Both peak accel-eration and RMS accelerations are used, their relationshipgenerally defined by the use of a peak factor, gp, approxi-mately 3.5–4.0. The true peak factor for a building whichrelates the RMS loading or response to the peak, can bedetermined in a wind tunnel aeroelastic model study.32 Targetpeak accelerations of 21 milli-g’s and 15 milli-g’s are oftenused for commercial and residential buildings respectively.Corresponding RMS values are ratioed accordingly using theappropriate peak factor. A stricter standard is often applied toresidential buildings for the following reasons:4
1. Residential buildings are occupied for more hours of theday and week and are therefore more likely to experiencethe design storm event.
2. People are less sensitive to motion when at work thanwhen in the home at leisure.
3. People are more tolerant of their work environment thanof their home environment.
4. Occupancy turnover rates are higher in office buildingsthan in residential buildings.
5. Office buildings are more easily evacuated in the eventof a peak storm event.
The apparent shortcoming in the standard defined byTable 5 is the fact that the tolerance levels are not related tobuilding frequency. Research has clearly shown a relationshipbetween acceptable acceleration levels and building fre-quency. Generally higher acceleration levels can be toleratedfor lower frequencies (see Curves 1, 4 and 5 in Figure 7 andCurves 5 and 6 in Figure 8).
The International Organization of Standardization has es-tablished a design standard for occupant comfort in fixedstructures subjected to low frequency horizontal motion—ISO Standard 6897-1984.17 This standard is based on a five-year mean recurrence interval and seems to agree quite wellwith the experimental work described in Figures 7 and 8. Themean threshold curve from this standard is plotted for com-parison to the research of Reference 9 in Figure 7. The ISOStandard 6897 design curves are plotted in Figure 10. Theinteresting feature of the ISO approach is that accelerationlimits increase as the building period increases and thereforeit represents a better correlation to available research. Theacceleration limits defined by the “General Purpose Build-
ings” curve of Figure 10 agree very well with U.S. practicefor commercial buildings if an upward adjustment of approxi-mately 10 percent is used to account for the difference in meanrecurrence intervals for U.S. practice (10-year versus the ISOfive-year mean recurrence interval). The 10 percent adjust-ment seems reasonable in light of the author’s experience inwind engineering studies performed on office buildings.
The author’s observation and experience with U.S. prac-tice, combined with a study of the available research pre-viously described and also the ISO Standard 6897-1984, formthe basis of a proposed new standard defined in Figures 11and 12. Design limits are proposed for both peak and RMSacceleration using a 10-year mean recurrence interval windas customarily used in U.S. practice. The logic used in theformulation of these curves is described below:
1. Design curves are established for residential buildingsand for commercial buildings. Residential buildings de-mand a separate stricter standard for the reasons pre-viously stated. Target values are given for each buildingtype centered between an upper and lower bound. Theupper bound values are 12.5 percent above and the lowerbound values 12.5 percent below the target values. Theconcept of a design range seems reasonable consideringthe limited available research and the uncertainty in thepresent state-of-the-art.
2. The ISO 6897 curve for mean threshold acceleration(middle curve of Figure 10) is taken as a lower boundfor the residential building curves shown in Figure 11.
3. The target and the upper bound values are establishedconsidering the design range defined in Item 1.
4. The commercial building target curve is defined byusing the ISO Standard “General Purpose Building”curve of Figure 10, increased by 10 percent to reflect thechange in mean recurrence intervals. The upper andlower bounds are defined 12.5 percent above and belowthe target curve respectively.
5. The peak acceleration curves defined in Figure 12 arebased on the corresponding RMS acceleration curves ofFigure 11 multiplied by a peak factor as defined in Table 5.
Additional research and experience will be required toconfirm the validity of this proposed new standard. Theacceleration levels relate reasonably well (slightly higher)with the successful experience of Table 5 and the new stand-ard has the advantage of frequency dependency that seems tobe confirmed by research.
CONCLUSIONS
This paper has focused on two serviceability limit states forbuildings (particularly tall and/or slender buildings); namely,deformation (deflection, curvature, and drift) and motionperception as measured by acceleration.
The conclusions reached in this paper are summarizedbelow:
1. The current practice of using 50-year or 100-year meanrecurrence interval wind loads to evaluate buildingdrift with currently accepted drift limits is overly con-servative. Wind drift and acceleration are proposed tobe based on a mean recurrence interval of 10 years.
2. A revised definition of building drift is proposed tobetter reflect the potential for damage to building ele-ments. The new definition, termed herein as the driftmeasurement index (DMI) is a mathematical formula-tion of shear deformation or racking that occurs in abuilding element. It includes the vertical component ofracking and filters out the effect of rigid body rotation,both of which are shortcomings in the present defini-tion of building drift. The term given to the rectangularpanel forming the zone over which shear deformationis to be measured is drift measurement zone (DMZ).The threshold damage distortion that represents thelimit of shear deformation that causes distress is termedthe drift damage index (DDI). The drift limit state maythen be stated as DMI ≤ DDI under 10-year wind loads.
3. If rational drift limits are to be effective, the calculationof building drift must capture all significant compo-nents of frame deflection including flexural deforma-tion of beams and columns, axial deformation of col-umns, shear deformation of beams and columns,beam-column joint deformation (panel zone deforma-tion), effect of member joint size, and the P-∆ effect.
4. A review of available racking distortion data for differ-ent partition types is made. Based on this information,and past successful experience, guidelines (Table 3)are proposed for different building elements (roofs,cladding, partitions, elevators, and cranes) subjected to10-year wind loads.
5. Factors affecting human response to building motionare reviewed and include building frequency, sex, age,body posture, body orientation, expectancy of motion,and body movement of the occupants; visual cues,acoustic cues, and type of motion.
6. Acceleration appears to be the best indicator of build-ing motion at present.
7. Both RMS (root-mean-square) and peak accelerationvalues are commonly used to represent building mo-tion. There appears to be a difference of opinion amongengineers and researchers as to the relative importanceand merits of each. This issue should be resolved toavoid confusion in the development of designstandards.
8. Contrary to early attempts by engineers to controlannoying lateral vibrations in buildings, building stiff-ness, represented by drift ratios, by itself is not a goodindicator of occupant susceptibility to building motion(Figure 5). Perception of building motion is influencedby available damping and also building mass as wellas building stiffness.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION14
9. Research seems to indicate that human perception toacceleration begins at about 0.5 percent (peak accel-eration) and appears to increase as the building periodincreases (Figure 8).
10. Human tolerance to acceleration tends to increase withbuilding period above about three to four seconds(Figure 8).
11. Current practice in tall building design has targeteddesign values for acceleration at 21 milli-g’s peakacceleration (6 milli-g’s RMS) for office buildings and15 milli-g’s peak acceleration (4.3 milli-g’s RMS) forresidential buildings. These limits do not recognize theapparent trend for the dependence of acceleration lim-its on building period.
12. Humans appear to be particularly sensitive to torsionalacceleration and so this component should be mini-mized in the assignment of building mass and stiffnessas much as possible during the design stage.
13. The factors affecting building acceleration are bestevaluated in a wind tunnel study. Acceleration involvesthe complex inter-relationship of the variables of mass,stiffness, and damping and also the influence of build-ing orientation on the site and the surrounding windenvironment.
14. Most tall slender building motion is controlled byacross-wind effects (vortex shedding). Generallyspeaking, this component of acceleration is propor-tional to the wind velocity to a power of about 3.5 andthe period of the building to a power of about 1.5; andinversely proportional to mass and the square root ofdamping.
15. Proposed standards for building perception (commer-cial and residential buildings) are shown in Figures 11and 12 which show acceleration limits increasing withbuilding period. This seems to follow the results of pastresearch and is an improvement over current standards.
16. The current approach to serviceability design is incon-sistent in U.S. building codes and seems to reflect ageneral reluctance by practicing engineers to codifyserviceability standards.
17. Structural engineers must begin to address service-ability issues in design and establish rational standardsbecause of the increasing economic impact service-ability issues are having on construction.
REFERENCES
1. American Institute of Steel Construction, Load and Resis-tance Factor Design, AISC, Chicago, 1986.
2. Algan, B., “Drift and Damage Considerations in Earth-quake Resistant Design of Reinforced Concrete Build-ings,” Ph.D. Dissertation, Department of Civil Engineer-ing, University of Illinois at Urbana, 1982.
3. BOCA National Building Code/1990, Building Officials
and Code Administrators International, Inc., EleventhEdition, 1990.
4. Cermak J., Boggs, D., Paper to be published. 5. Chang, F., “Wind and Movement in Tall Buildings,” Civil
Engineering Magazine, ASCE, August 1967. 6. Chang, F. K., “Human Response to Motions in Tall Build-
ings,” Paper presented at ASCE National EnvironmentalEngineering Meeting, Houston, Texas, October 16–22.
7. Charney, F. A., “Wind Drift Serviceability Limit StateDesign of Multi-story Buildings,” Journal of Wind Engi-neering and Industrial Aerodynamics, 36, 1990.
8. Charney, F. A., “Sources of Elastic Deformation in Later-ally Loaded Steel Frame and Tube Structures,” Councilon Tall Buildings and Urban Habitat, Fourth World Con-gress, Tall Buildings: 2000 and Beyond, Hong Kong,November 5–9, 1990.
9. Chen, P. W., Robertson, L. E., “Human PerceptionThresholds of Horizontal Motion,” Journal of StructuralEngineering, ASCE, Vol. 98, No. ST8, August 1972.
10. Davenport, A. G., “Tall Buildings—An Anatomy of WindRisks,” Construction in South Africa, December 1975.
11. Ellingwood, B., “Serviceability Guidelines for SteelStructures,” Engineering Journal, AISC, Volume 26, No.1, First Quarter, 1989.
12. Fisher, J. M., West, M. A., “Serviceability Design Consid-erations for Low Rise Buildings,” Steel Design GuideSeries No. 3, AISC, 1990.
13. Freeman, S. A., “Racking Tests of High Rise BuildingPartitions,” Journal of Structural Engineering, ASCE,Volume 103, No. ST8, August 1977.
14. Galambos, T. V., Ellingwood, B., “Serviceability LimitState: Deflection,” Journal of Structural Engineering,ASCE, Volume 12, No. 1, January 1986.
15. Gaylord, Edwin H. Jr., Gaylord, Charles N., StructuralEngineering Handbook, McGraw Hill Book Company,New York, 1979.
17. “Guidelines for the Evaluation of the Response of Occu-pants of Fixed Structures to Low Frequency HorizontalMotion (0.063 to 1 Hz.),” ISO Standard 6897-1984, In-ternational Organization of Standardization, 1984.
18. Hansen, R. J., Reed, J. W., Vanmarcke, E. H., “HumanResponse to Wind-Induced Motion of Buildings,” Jour-nal of Structural Engineering, ASCE, Volume 99, No.ST7, July 1973.
19. Irwin, A. W., “Human Response to Dynamic Motion ofStructures,” Structural Engineer, Volume 56A, No. 9,September 1978.
20. Irwin, A. W., “Motion in Tall Buildings,” Paper presentedat Second Century of the Skyscrapers, Chicago, Illinois,January 6–10, 1986.
21. Irwin, P. A., Ferraro, V., Stone, G. K., “Wind Induced
FIRST QUARTER / 1993 15
Motions of Buildings,” Proceedings Symposium/Work-shop on Serviceability of Buildings (Movements, Defor-mations, Vibrations) Volume I, University of Ottawa, May16–18, 1988.
22. Islam, M. S., Ellingwood, B., Corotis, R. B., “DynamicResponse of Tall Buildings to Stochastic Wind Load,”Journal of Structural Engineering, ASCE, Volume 116,No. 11, November 1990.
23. Isyumov, N., Poole, M., “Wind Induced Torque on Squareand Rectangular Building Shapes,” Journal of Wind En-gineering and Industrial Aerodyamics, 13, 1983.
24. Kareem, A., “Lateral Torsional Motion of Tall Buildingsto Wind Loads,” Journal of Structural Engineering,ASCE, Volume 111, No. 11, November 1985.
25. Khan, F., Parmelee, R., “Service Criteria for Tall Build-ings for Wind Loadings,” Proceedings 3rd InternationalConference on Wind Effects on Buildings and Structures,Tokyo, Japan, 1971.
26. Minimum Design Loads for Buildings and Other Struc-tures, ASCE 7-88, American Society of Civil Engineers.
27. Metal Curtain Wall Manual, American ArchitecturalManufacturers Association, Des Plaines, Illinois, 1989.
28. Planning and Environmental Criteria for Tall Buildings,A Monograph on Planning and Design of Tall Buildings,ASCE, Volume PC, Chapter PC-13, 1981.
29. Standard Building Code, Southern Building Code Con-gress, 1991 Edition.
30. Structural Design of Tall Steel Buildings, A Monographon Planning and Design of Tall Buildings, ASCE, VolumeSB, Chapter SB-5, 1979.
31. “Structural Serviceability: A Critical Appraisal of Re-search Needs,” Journal of Structural Engineering, ASCE,Volume 112, No. 12, December 1986.
32. Supplement to the National Building Code of Canada—1990, National Research Council of Canada, Ottawa,1990.
33. Tallin, A., Ellingwood, B., “Serviceability Limit States:Wind Induced Vibrations,” Journal of Structural Engi-neering, ASCE, Volume 110, No. 10, October 1984.
34. “Brick Veneer Steel Stud Panel Walls,” Technical Noteson Brick Construction No. 28B (revised February 1987),Brick Institute of America.
35. Uniform Building Code, International Conference ofBuilding Officials, 1991 Edition.
36. “Wind Drift Design of Steel-Framed Buildings: State-of-the-Art Report,” Journal of Structural Engineering,ASCE, Volume 114, No. 9, September 1988.
37. Vickery, B. J., Isyumov, N., Davenport, A. G., “The Roleof Damping, Mass, and Stiffness in the Reduction of WindEffects on Structures,” Journal of Wind Engineering andIndustrial Aerodynamics, 11, 1983.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION16
INTRODUCTION
In practice, the majority of single angle struts are eccentri-cally loaded, and an attempt to calculate exactly the strengthof such members is a task of formidable complexity. There-fore, the designer often relies on approximate methods andguidelines, in which the key ingredient is the value of thedesign strength when the strut is loaded concentrically (AISC1986, ASCE 1988, and Adluri and Madugula 1992).
In this brief note, a step-by-step solution and load tablesare presented for the determination of the design strength ofconcentrically loaded single angle struts according to theAISC 1986 LRFD Specification. Equation A-E3-7, of Appen-dix E of the LRFD Specification requires the computation ofthe torsional-flexural buckling stress as the smallest of thethree roots of a cubic equation.
GEOMETRY AND COORDINATE SYSTEMS
Consider the unsymmetrical single angle section, shown inFigure 1, with geomatric axes x and y and principal axes uand v, each passing through the centroid C. Let α be the angleof inclination between the horizonal axis x and the principalaxis u; and b1, b2, and t be the dimensions of the vertical leg,horizontal leg, and thickness of the angle, respectively. Theshear center of the section SC is located at the intersection ofthe center lines of the two legs and measures from the centroidxo and yo, in the (x, y) system, and uo and vo, in the (u, v)system, as shown in Figure 1. They are defined as follows:
xo = x_ −
t2
yo = y_ −
t2
(1)
uo= yo sin α + xo cos αvo = yo cos α − xo sin α
x_, y_, and Tan α tabulated in the AISC Manual (AISC 1986).
FUNDAMENTAL FORMULAE
For an unequal-angle member under concentric loading theelastic flexural-torsional buckling stress is defined as the
lowest root of the following cubic equation (Timoshenko andGere 1961):
In the above equations, Feu Fev, and Fez are the elastic flexuralbuckling stresses about the u and v axes and the elastictorsional buckling stress about the z-axis respectively. KuL,KvL are the effective lengths for bending about the principalaxes u and v. KzL is the effective length for twisting about thez-axis, which passes through the centroid of the section and
A. Zureick is Associate Professor, School of Civil Engineering,Georgia Institute of Technology, Atlanta, GA.
Design Strength of Concentrically LoadedSingle Angle StrutsA. ZUREICK
Figure 1
FIRST QUARTER / 1993 17
coincides with the angle’s longitudinal axis. The cross-sec-tional area of the angle is A; Cw and J are the warping constantand the St. Venant torsion constant, respectively. E and G arethe modulus of elasticity and the shear modulus of the steelwith values of 29,000 ksi and 11,200 ksi respectively. ro, thepolar radius of gyration of the cross section about the shearcenter, is defined as
ro = √uo2 + vo
2 + Iu + Iv
A(4)
Equation 2 is identical to AISC LRFD Equation A-E3-7and can be written conveniently in the form
Fe 3 + a2Fe
2 + a1Fe + a0 = 0 (5)
where
a2 = −1H
1 −
vo2
ro2
Feu +
1 −
uo2
ro2
Fev + Fez
,
a1 = 1H
(Feu Fev + Feu Fez + Fev Fez ) , a0 = −1H
(Feu Fev Fez ) (6)
in which H is defined as
H = 1 − uo
2 + vo2
ro2 (7)
SOLUTION OF THE CUBIC EQUATION
In general, a cubic equation has either three real roots or onereal and two complex conjugate roots. In the case of bucklingof an unsymmetrical section, however, it can be shown thatthe three roots of the cubic equation are always real andpositive (See, for example, the proof in texts by Timoshenkoand Gere, 1961 or Galambos, 1968). In such a case thesolution of the cubic equation can most conveniently beobtained as follows: Let
M = 3a1 − a2
2
9 , N =
9a2a1 − 2a23 − 27a0
54 ,
θ = arccosN
√−M3 (8)
then the three real roots are given by
Fe1 = 2√−Mcos θ3
− a2
3 , Fe2 = 2√−Mcos
θ3 +
2π3
−
a2
3 ,
Fe3 = 2√−Mcos θ3
+ 4π3
−
a2
3(9)
The above solution was first presented in 1615 by theFrench mathematician François Viète (Kline 1972) and isused in most mathematical handbooks nowadays (Tuma1970, Beyer 1982). In the above equation the quantity M isalways negative.
DESIGN STRENGTH OF CONCENTRICALLYLOADED STRUTS
Upon solving the cubic equation, the critical flexural-tor-sional buckling stress Fc for the angle strut is defined as thelowest of the three roots obtained previously. Thus,
Fe = min [Fe1 , Fe2 , Fe3 ]
from which the slenderness parameter λc, the nominal criticalstress φcFcr, and the design strength φcPn, can be calculatedaccording to AISC LRFD Appendix E. A schematic showingall design steps for calculating the design strength of concen-trically loaded angle struts is shown in Figure 2.Figure 2
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION18
Example
Calculate design strength for a 9 ft long concentrically loaded5×3×1⁄4 single angle made of A36 steel.
Solution
Given:
Ku L = Kv L = Kz L = 9 ft, Fy = 36 ksi, E = 29,000 ksi, G = 11,200 ksi
For L5×3×1⁄4:
A = 1.94 in.2, rx = 1.62 in., ry = 0.861 in., rv = 0.663 in.,
Tan α = 0.371 or ( α = 0.355 rad. = 20.35°)Cw ≈ 0.0606 in.6, J ≈ 0.0438 in.4,
Ix = 5.11 in.4
Iy = 1.44 in.4
Properties
Iv = Arv2 = (1.94)(0.663)2 = 0.852 in.4
Iu = Ix + Iy − Iv = 5.11 + 1.44 − 0.852 = 5.698 in.4
ru = √ Iu
A = √5.698
1.94 = 1.713 in.
xo = x_ −
t2
= 0.657 − 0.25
2 = 0.532 in.
yo = y_ −
t2
= 1.66 − 0.25
2 = 1.535 in.
uo = yo sin α + xo cos α = 1.535 sin (20.35°) + 0.532 cos (20.35°) = 1.033 in.
vo = yo cos α − xo sin α = 1.535 cos (20.35°) − 0.532 sin (20.35°) = 1.254 in.
ro = √uo2 + vo
2 + Iu + Iv
A
= √(1.033)2 + (1.254)2 + 5.698 + 0.852
1.94 = 2.453 in.
H = 1 − uo
2 + vo2
ro2 = 1 −
(1.033)2 + (1.254)2
(2.452)2 = 0.561
Check local buckling
bt =
51⁄4
= 20 > 76
√36 = 12.67 <
155√36
= 25.83
Thus, use AISC LRFD Eq. A-B5-1
Qs = 1.34 − 0.00447bt
√Fy
= 1.34 − 0.00447(20) √36 = 0.804
Q = Qs = 0.804
Check global buckling
The slenderness ratios about the principal axes are
KuLru
= 9(12)1.713
= 63.05 , KvLrv
= 9(12)0.663
= 162.9
The elastic buckling stresses about the u, v, and z axes are
Feu = π2E
KuLru
2 =
π2(29,000)(63.05)2 = 72 ksi,
Fev = π2E
KvLrv
2 =
π2(29,000)(162.9)2 = 10.78 ksi
Fez = π2ECw
(KzL)2 + GJ
1Ar
_o2
= π2(29,000)(0.0606)
(108)2 +(11,200)(0.0438)
1(1.94)(2.453)2
= (1.48 + 490)
11.67 = 42.1 ksi
Now it is required to solve the cubic equation Fe
3 + a2Fe 2 + a1Fe + a0 = 0, where
a2 = −1H
1 −
vo2
ro2
Feu +
1 −
uo2
ro2
Fev + Fez
− 1
0.561
1−
(1.254)2
(2.453)2
(72)+
1−
(1.033)2
(2.453)2
(10.78)+(42.1)
= −186
a1 = 1H
(Feu Fev + Feu Fez + Fev Fez ) = 1
0.561 [(72)(10.78)
+ (72)(42.1) + (10.78)(42.1)] = 7,596
a0 = − 1H
(Feu Fev Fez ) = − 1
0.561(72)(10.78)(42.1) = −58,247
M = 3a1 − a2
2
9 =
3(7,596) − (−186)2
9 = −1,312
N = 9a1a2 − 27a0 − 2a2
3
54
FIRST QUARTER / 1993 19
= 9(7,596)(−186) − 27(−58,247) − 2(−186)3
54 = 31,976
θ = arccos N
√−M3 = arccos
31,976√−(−1,312)3
= arccos (0.6729) = 0.832 rad.
Fe1 = 2√−M cos θ3
− a2
3
= 2√−(−1,312) cos 0.832
3 −
(−186)3
= 131.6 ksi
Fe2 = 2√−M cos θ3 +
2π3
−
a2
3
= 2√−(−1,312) cos 0.832
3 +
2π3
−
(−186)3
= 9.98 ksi
Fe3 = 2√−M cosθ3
+ 4π3
−
a2
3
= 2√−(−1,312) cos0.832
3 +
4π3
−
(−186)3
= 44.4 ksi
Therefore
Fe = min[131.6, 9.98, 44.4] = 9.98 ksi
(please note how close the flexural-torsional buckling stress(Fe = 9.98 ksi) is to that calculated from the flexural bucklingabout the minor axis (Fev = 10.78 ksi). The difference here isonly eight percent.)
λe = √ Fy
Fe = √36
9.98 = 1.9
λe√Q = 1.9√0.804 = 1.704 > 1.5
Fcr = 0.877
λe2
Fy =
0.877(1.9)2
(36) = 8.74 ksi
φPn = φFcr Ag = 0.85(8.74)(1.94) = 14.4 kips
Of some note, if the flexural buckling stress with respect tothe minor axis (Fev = 10.78 ksi) were used instead of theflexural-torsional buckling stress (Fe = 9.98 ksi), the designstrength of the angle member would be 15.6 kips. Such anincrease (8.3 percent) may not be regarded as significant. Itshould be emphasized, however, that for a strut with a smallslenderness ratio the difference between the elastic flexuraland the elastic flexural-torsional buckling stresses can bequite large. A procedure related to the design of axially loaded
compressed angles by the method of minor axis buckling waspresented by Galambos (1991).
DESIGN TABLES
The procedure outlined in this note was implemented togenerate angle load tables for 139 equal and unequal leg anglesections made of either A36 or Grade 50 steel. Tabulated loadsare for, KuL = KvL = L, and are terminated when the slender-ness ratio with respect to the minor axis of the angle exceeds200. Numerical values are rounded to the nearest wholenumber. It should be noted that it is, of course, not necessaryto use the solution of the cubic equation to calculate the designstrength of a concentrically loaded equal leg angle strut.Equal leg angle struts can be regarded as singly symmetricsections (symmetry about the u-axis) for which eitherflexural-torsional buckling about the axis of symmetry (u-axis) or flexural buckling about an axis perpendicular to theaxis of symmetry (v-axis) may occur. In such a case the cubicequation is reduced to a quadratic equation, the solution ofwhich is given as Equation A-E3-6 in the AISC LRFD Speci-fication. Solutions from either AISC LRFD Equation A-E3-6or A-E3-7 yield the same results when applied to equal legangles.
REFERENCES
Adluri, S. M. R., and Madugula, M. K. S. (1992). “Eccentri-cally Loaded Steel Single Angle Struts,” AISC EngineeringJournal, 2nd Quarter 1992, pp. 59–66.
AISC (1986), Manual of Steel Construction—Load and Re-sistance Factor Design, First Edition, Chicago, IL.
ASCE (1988), Guide for Design of Steel Transmission Tow-ers, Manuals and Reports on Engineering Practice No. 52,Second Edition, New York.
Beyer, William H. (1982). CRC Mathematical Tables, 26thEdition, CRC Press, Boca Raton, FL.
Galambos, T. V. (1968). Structural Members and Frames,Prentice Hall, Englewood Cliffs, NJ.
Galambos, T. V. (1991), “Design of Axially Loaded Com-pressed Angles,” Proceedings of the SSRC Annual TechnicalMeeting, April 15–17, Chicago, pp. 353–359.
Kline, Morris, (1972). Mathematical Thought from Ancientto Modern Times, Oxford University Press, New York.
Timoshenko, S. P., and Gere, J. M. (1961). Theory of ElasticStability, Second Edition, McGraw-Hill Book Company.
Tuma, Jan J. (1970). Engineering Mathematics Handbook,McGraw-Hill Book Company.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION20
Fy = 36 ksi
Fy = 50 ksi
CONCENTRICALLY LOADEDCOLUMNS
Single AnglesDesign axial strength in kips ( φ = 0.85)
Single AnglesDesign axial strength in kips ( φ = 0.85)
Size 21⁄⁄2 ×× 21⁄⁄2
Thickness 1⁄⁄2 3⁄⁄8 5⁄⁄16 1⁄⁄4 3⁄⁄16
Wt. / ft 7.7 5.9 5.0 4.1 3.07
Fy 36 50 36 50 36 50 36 50 36 50
Effe
ctiv
e le
ngth
in fe
et, k
L 0 69 96 53 74 45 62 36 51 27 35
12345
6751524131
9180644732
5047403224
6962493624
4239342720
5652423121
3232272217
4342342517
2322211713
2827251913
678
221612
221612
1712 9
1712 9
1411 8
1411 8
12 9 7
12 9 7
9 7 5
9 7 5
Size 2 ×× 2
Thickness 3⁄⁄8 5⁄⁄16 1⁄⁄4 3⁄⁄16 1⁄⁄8
Wt. / ft 4.7 3.92 3.19 2.44 1.65
Fy 36 50 36 50 36 50 36 50 36 50
Effe
ctiv
e le
ngth
in fe
et,
kL
0 42 58 35 49 29 40 22 30 13 17
12345
4034271912
5444311912
3329221610
4637261610
27241813 8
36302113 8
19181410 7
25231710 7
1010 9 7 5
121210 7 5
6 8 8 7 7 6 6 5 5 3 3
Size 13⁄⁄4 ×× 13⁄⁄4 11⁄⁄2 ×× 11⁄⁄2
Thickness 1⁄⁄4 3⁄⁄16 1⁄⁄4 3⁄⁄16
Wt. / ft 2.77 2.12 2.34 1.8
Fy 36 50 36 50 36 50 36 50
Effe
ctiv
e le
ngth
infe
et, k
L
0 25 35 19 26 21 29 16 22
12345
231914 9 6
322415 9 6
171511 7 4
231912 7 4
191510 5
261810 5
1511 7 4
2014 7 4
Size 11⁄⁄4 ×× 11⁄⁄4 11⁄⁄8 ×× 11⁄⁄8 1 ×× 1
Thickness 1⁄⁄4 3⁄⁄16 1⁄⁄8 1⁄⁄8
Wt. / ft 1.92 1.48 0.9 0.8
Fy 36 50 36 50 36 50 36 50
Effe
ctiv
e le
ngth
in fe
et, k
L
0 17 24 13 19 8 11 7 10
1234
1510 6 3
2012 6 3
12 8 4 2
16 9 4 2
742
9 5 2
631
8 3 1
x
u y v
x
uy
v
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION30
I t is common practice in crane runway beams to place achannel, open-side down, over the top flange of a W-section,as shown in Figure 1, to increase its lateral stability. This isdone because it is not always convenient to brace the com-pression flange between columns.
The lateral-torsional buckling capacity of a singly symmet-ric section may be determined by the formulas (Footnote c onpage 6-96 of the current LRFD manual1) without knowing thewarping section constant. However, these formulas weredeveloped from research on three-plate monosymmetric sec-tions.2,3,4,5,6,7 If one is to develop a similar equation for the caseof the W-section with a channel cap, it is necessary to returnto the basic theory of lateral-torsional buckling and thatmeans that one must calculate the warping constant (Cwc) forthe cross-section.
It is not a difficult matter to do this with a computer, butsince the program may not be available to everyone, it wouldbe useful to develop a simple empirical equation for prelimi-nary design, making use of the section properties alreadygiven in the AISC steel manual.
According to Kitipornchai and Trahair,3 Equation 1, whichprovides for calculating the warping section constant, “isexact for an unlipped section and approximate for a lippedsection.” Their “lipped” section of Figure 2 closely approxi-mates the subject of this paper, the W-section with a channelcap.
Iw = a2Iyc + b2Iyt (1)
where
a = (1 − ρ)h b = ρhh = hc + e ρ = Iyc / Iy e = (DL
2Bc2TL ) / (4ρIy)
Iw = warping section constantIy = moment of inertia of the combined section about the
axis parallel to webIyc = moment of inertia of the compression flange about
the axis parallel to web
Iyt = moment of inertia of the tension flange about the axisparallel to web
hc = distance from the centroid of the compression flangeto the centroid of the tension flange
h = distance from the shear center of the compressionflange to the centroid of the tension flange
e = distance from the shear center of the compressionflange to the centroid of the compression flange
DL = the depth of the lipBc = the width of the lipped flangeTL = the thickness of the lip
Tony Lue is Graduate Assistant, University of Florida, Gaines-ville, FL.
Duane S. Ellifritt is Crom Professor, University of Forida,Gainesville, FL.
The Warping Constant for the W-Sectionwith a Channel CapTONY LUE AND DUANE S. ELLIFRITT
Fig. 2. W-section with lipped flange.
Fig. 1. Crane runway beam.
FIRST QUARTER / 1993 31
The authors have enhanced a computer program, whichwas originally written in BASIC language by Professor Theo-dore V. Galambos of the University of Minnesota in Minnea-polis and converted to FORTRAN language by Dr. ThomasSputo of the University of Florida, to calculate the exactvalues of warping section constants (Cwc) of the W-sectionwith a channel cap. The user need only input the W-sectionand channel dimensions. The 28 combined sections shown onpages 1-98 and 1-99 of the AISC LRFD steel manual areshown in Table 1 with their warping section constants (Cwc)as calculated by the program described.
The ratios of Cwc / Cw are plotted against the ratios ofAc / Aw, as shown in Figure 3 for the 28 combined sections,where Cw, Ac, and Aw are explained in Table 1.
When Equation 1 was applied to the case of the W-sectionwith a channel cap and the results Iw / Cw were compared withthe Cwc / Cw values obtained from the above-mentioned pro-
gram, as shown in Figure 4, it can be observed that Equation 1gives a very conservative estimate (−7 percent to −17 percent)for this case (the W-section with a channel cap). The equationalso has the added disadvantage of requiring the user tocalculate certain section properties and parameters (Iy, Iyt, Iyc,ρ, hc, h, e, a, and b) first.
The purpose of this paper is, therefore, to present a reason-ably accurate method of calculating the warping sectionconstant (Cwc), using a simple model which can be expressedby Equations 2, 3, or 4 and known section properties that canbe found in the AISC steel manual.
The proposed method which seemed to offer the mostpromise was to use the ratio of the channel area (Ac) to theW-section area (Aw) as the independent variable and plot itagainst the ratio of the warping section constant (Cwc) for thecombined section to the warping section constant (Cw) for theW-section alone.
W—W-section; C—Channel; MC—Miscellaneous ChannelAc —Area of channel, in.2; Aw —Area of W-section, in.2
Cwc —Warping section constant (in.6) for the W-section with a channel cap, which is obtained by using the program as mentioned in this paperCw —Warping section constant (in.6) for the W-section alone, which can be found in the AISC steel manual
*MCMetric Conversion:To convert to Multiply byinches(in.) millimeters(mm) 25.4
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION32
Several curves (or models) were then fit to these data whichare shown in Figure 3. These curves are represented by thefollowing equations.
Cwc = Cw 1.31 + 2.55
Ac
Aw
− 1.31
Ac
Aw
2
+ 0.29Ac
Aw
3
(2)
Cwc = Cw 1.25 + 2.55
Ac
Aw
− 1.31
Ac
Aw
2
+ 0.29Ac
Aw
3
(3)
Cwc = Cw 1.7
Ac
Aw + 1
√ Ac
Aw(4)
Equations 2, 3, and 4 were superimposed on the data ofFigure 3, and plotted in Figure 5. Equation 2 results in anestimated error of −2.9 percent to +2.9 percent, and Equa-tion 3 with an estimated error of −5.9 percent to 0. Equation4 is a simplified model with fewer terms involved, with anestimated error of −7.1 percent to +1.5 percent. Equations 2and 3 were derived based on the multiple linear regressiontechnique of statistics using the data of Figure 3. Equation 4was also based on the same statistical technique with somefurther modifications.
Equation 2 is the best fit curve for the data of Figure 3, butdoes overestimate the warping section constant (Cwc) by asmuch as 2.9 percent. Equation 3 is simply Equation 2 shifteddown until all the data points fall above the curve and may beup to 5.9 percent conservative. However, for those who wanta formula that they can easily memorize and still get aconservative result within 7.1 percent, Equation 4 is offered.
The Equations 2, 3, and 4 proposed by the authors requireno calculation of certain section properties and parameters,and are close to an exact solution. The required parametersfor these equations are the channel area (Ac), the area of theW-section (Aw), and its warping section constant (Cw), all ofwhich can be found in the AISC steel manual.
REFERENCES
1. American Institute of Steel Construction, Load and Resis-tance Factor Design, First Edition, Chicago, Illinois, 1986.
2. Anderson, J. M., Trahair, N. S., “Stability of Monosymmet-ric Beams and Cantilevers,” Journal of the StructuralDivision, ASCE, Vol. 98, No. ST1, January 1972.
3. Kitipornchai, S., Trahair, N. S., “Buckling Properties ofMonosymmetric I-Beams,” Journal of the Structural Divi-sion, ASCE, Vol. 106, No. ST5, May 1980.
4. Kitipornchai, S., Wang, C. M., Trahair, N. S., “Buckling ofMonosymmetric I-Beams under Moment Gradient,” Jour-nal of the Structural Division, ASCE, Vol. 112, No. 4, April1986.
5. Bradford, M. A., Cuk, P. E., “Elastic Buckling of TaperedMonosymmetric I-Beams,” Journal of the Structural Divi-sion, ASCE, Vol. 114, No. 5, May 1988.
6. Galambos, T. V., Guide to Stability Design Criteria forMetal Structures (SSRC), Wiley, New York, 1988.
7. Trahair, N. S., Bradford, M. A., The Behaviour and Designof Steel Structures, Chapman and Hall, London, 1988.
Fig. 3. Plot of Cwc / Cw vs. Ac / Aw. Fig. 5. Equations 2, 3, and 4 fit to the data of Figure 3.
Fig. 4. Iw / Cw compared with the data of Figure 3Cwc is exact for the W-section with a channel cap.Iw is an approximation from Reference 3 for theW-section with a channel cap (see Equation 1).
FIRST QUARTER / 1993 33
INTRODUCTION
An accurate and complete design will result in an economi-cal and safe connection. Yet it is entirely common for theengineer of record to withhold, either intentionally or unin-tentionally, the information necessary to the fabricator ordetailer to perform a design which is both accurate andcomplete. Specifically, actual reactions are seldom shown onthe contract drawings from which the connections must bedesigned.1
AISC states, “For economical connections, beam reactionsshould be shown on the contract drawings. If these reactionsare not shown, connections must be selected to support one-half the total uniform load capacity…for the given beam,span, and grade of steel specified.”2,3 No quantification isgiven, however, of the actual difference in economy betweenthe two cases. In fact, this difference is somewhat elusive asit may vary greatly among specific examples. For the generalcase, however, it is possible to determine a reasonable esti-mate of the economic sacrifice incurred when a larger con-nection than required is used. The focus of this paper, then, isthis economic sacrifice. For simplicity, a standard configura-tion4 of the double angle connection will be considered inwhich only n, the number of bolt rows (and consequently, thelength of the angles), varies. Based on values of n from 2 to10, the cost of these standard connections will be estimated.Ranges of n compatible with each beam size group will beidentified and the percent increase in cost which results whena larger connection than required is used will then be deter-mined over these ranges.
Note that the practices which routinely result in uneconom-ical connection designs are not specifically addressed in thispaper. For a discussion of these practices, the reader is re-ferred to “Eliminating the Guesswork in Connection De-sign—Communication of Design Requirements BetweenFabricator and Engineer is Crucial for a Safe and EconomicStructure” by W.A. Thornton, in the June 1992 issue ofModern Steel Construction. Also not addressed is the effect
of standardization on the detailing costs, ease of fabrication,and overall quality of the constructed product. While ingeneral, standardization will reduce detailing costs, increasethe ease of fabrication, and lead to improvements in qualitybecause of decreased variability, these considerations aremore project related than connection specific. Thus, it wouldnot be feasible to consider their effect in this paper.
THE STANDARD CONFIGURATION
The standard parameters of the double angle connection to beconsidered are as follows. The shop and field bolts will be3⁄4-in. diameter A325-N at 3-in. spacing with 11⁄4-in. edgedistance. The holes will be short-slotted in the outstandingangle legs (those connected to the supporting member) andstandard otherwise. The angles will be 2L 4×31⁄2×5⁄16 (SLBB).This standard configuration produces nine connections withthe number of bolts rows n ranging from 2 to 10. While theseconnections will not satisfy every case, they will be adequatefor the typical case and, therefore, will be used in this costcomparison.
CONNECTION COSTS
The costs which will be considered in this paper can bedivided into three categories: material, shop labor, and fieldlabor. The material costs include the cost of the bolts, washersand nuts, and the framing angles. The shop labor cost includesshearing and punching the angles, punching the supportedand supporting members, and installing the shop bolts. Thefield labor cost is comprised of installing the field bolts. Whilematerial costs are readily available, labor costs are seldom amatter of public knowledge. Furthermore, labor costs willvary from fabricator to fabricator and from region to region.Consequently, those presented in this paper should be re-garded as an average estimate and should by no means beconstrued to be universal. The fabricators’ costs which willbe used in this paper are as follows:
A325 tension control (TC) bolts $90.00 per 100 lbs.L4×31⁄2×5⁄16 $14.75 per 100 lbs.Shop Labor $20.00 per hourField Labor $30.00 per hour
These are base costs; selling costs, which would includeoverhead and profit, would be higher.
The material and labor costs for double angle connections
Charles J. Carter is staff engineer-structures, The AmericanInstitute of Steel Construction, Inc., Chicago, IL.
Louis F. Geschwindner is professor of architectural engineer-ing, The Pennsylvania State University, University Park, PA.
The Economic Impact of OverspecifyingSimple ConnectionsCHARLES J. CARTER AND LOUIS F. GESCHWINDNER
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION34
in the standard configuration are summarized in Table 1. Thebolt material cost is based on a bolt length of 3-in. with onewasher and nut each; about 83 pounds per 100 count. Theangle material cost is based on an angle size of 4×31⁄2×5⁄16
which weighs 7.7 pounds per foot. The labor costs are basedon the labor time estimates4 summarized in the same table.Note that, in each row of bolts, there are three bolts: one shopbolt and two field bolts. Total costs have been rounded to thenearest whole dollar.
COMPATIBILITY WITH BEAM SHAPES
The deepest compatible standard connection must fit withinthe T-dimension of the beam as listed in Part 1 of the ASDand LRFD Manuals. As recommended in Part 4 of the ASDManual and Part 5 of the LRFD Manual, the depth of theminimum standard connection should be greater than T/2.Given these limits, the compatibility of the nine standardconnections with W-shapes is summarized in Table 2. Notethat limitations such as coping, which may further restrict themaximum value of n are not considered.
PERCENT INCREASE IN CONNECTION COST
Given the allowable variations in n of Table 2, percent in-creases in connection cost per unnecessary row of boltsprovided are listed in Figure 1. Cells below the heavy line falloutside the spacially permissible variations in n given inTable 2. As an example of the use of Figure 1, consider aW18×50 and assume an end reaction which would requirefour rows of bolts. Using five rows of bolts instead, the largestn possible given the T-dimension of a W18, would increasethe connection cost by 26 percent. Similarly, using six rows
of bolts in a W21×44 requiring only five rows would resultin a cost increase of 17 percent.
Some general observations may be made from Figure 1.The predicted range of economic sacrifice when all beams
Table 1.Material, Shop Labor, Field Labor, and Total Cost Estimates
of Double Angle Connections in Standard Configuration
Table 2.Compatibility of W-shape Beamsand Standard Connection Depths
Shape Weight nmin nmax
W36 359-230256-135
66
1010
W33 354-118 6 10
W30 235-99 5 9
W27 217-84 5 8
W24 176-55 4 7
W21 166-44 4 6
W18 143-35 3 5
W16 100-26 3 4
W14 132-9082-4338-22
322
334
W12 87-4035-14
22
33
W10 112-3330-12
22
23
W8 67-2421-10
22
22
FIRST QUARTER / 1993 35
and ranges of n are considered is from 11 percent to 85percent. As the size of the beam being connected decreases,the percent change in cost increases. Additionally, and obvi-ously, as the number of unneccesary bolt rows increases, sodoes the percent change in cost increase.
Focusing on the range of typical simple beam sizes, thevariation in percent change can be narrowed. First, do notconsider beams larger than a W24; from Table 2, this wouldeliminate the 8, 9, and 10 row connections. Additionally,consider only uniformly loaded cases, the designs of whichare usually controlled by moment. The actual end reactionswill likely be close to, but still less than, the end reactionscalculated from the Design Uniform Load Tables.4 Thus, inmost cases, the number of unnecessary rows of bolts will beone. Accordingly, the cost increase given these limitations isbetween 13 percent and 41 percent.
It should not be forgotten, however, that there are manycases which cannot be classified as typical. Shorter spanbeams, often sized for convenience or for similarity to otherbeams, and in-fill beams, which may serve no other purposethan to reduce the unbraced length of another member, maysustain actual reactions which are significantly lower thanone-half the the total uniform load capacity of the beam. Asimilar situation is found in beams controlled by deflectionconsiderations. In these cases, the percent increase in econ-omy can be much higher than the range identified as typical.
CONCLUSIONS
A generalized and simplified approach has been taken toestimate the added cost of providing more rows of bolts in a
simple connection than were necessary. This approach wasintended to estimate the possible economic implicationswhen the engineer of record does not indicate the actualreactions for which the connections must be designed on thecontract drawings. While this approach centered on the shopand field bolted double angle connection, this information islikely similar to that which might be obtained when othertypes of simple shear connections are considered. Given thepotential for unnecessary increase in connection cost, theengineer of record should always indicate the actual reactionson the contract drawings. In this manner, the best opportunityfor safe and economical connections will be realized.
REFERENCES
1. Thornton, W. A., “Eliminating the Guesswork in Connec-tion Design—Communication of Design RequirementsBetween Fabricator and Engineer is Crucial for a Safe andEconomic Structure,” Modern Steel Construction, June1992, p. 27.
2. American Institute of Steel Construction, Manual of SteelConstruction, Allowable Stress Design, 9th Ed., Chicago,IL, 1989, pp. 4–9.
3. American Institute of Steel Construction, Manual of SteelConstruction, Load and Resistance Factor Design, 1st Ed.,Chicago, IL, 1986, pp. 5–15.
4. Carter, Charles J., “Standardizing Simple Shear Connec-tions in Steel”, Master of Science Thesis, PennsylvaniaState University, University Park, PA, 1991.
Fig.1. Percent increase in connection cost.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION36
The approximate solutions for K-factor in Pierre Dumonteil’spaper will assist engineers in programming both spreadsheetand general language column solutions.
While the general equation for K-factor is rapidly solvedby Newton’s Method of Successive Approximations, the dis-continuity caused by Tan(π / K) being infinite when K = 2interferes with the solution if the initial estimate of K is on thewrong side of 2. This engineer derived a simple method ofestimating K which works but lacks the accuracy or simplicityof the solution in the Engineering Journal.
For those interested, Newton’s Method for the sway caseis:
xi = x − f(x)f ′(x)
= x − [Ax2 − 36]Tan (x) − 6Bx
[Ax2 − 36]Sec2(x) + 2AxTan(x) − 6B
and for the non-sway case is:
xi = x − [Ax3 − 2Bx2Cot(x) + 2Bx − 4x + 8Tan(x / 2
3Ax2 − 4BxCot(x) + 2Bx2Csc2(x) + Tan2(x / 2)
where
xi = Improved value of x = π / K
x = Estmated value of x = π / KA= Gt × Gb = (Stiffness Top) × (Stiffness Bottom)B= Gt + Gb = (Stiffness Top) + (Stiffness Bottom)
The iteration will close to any reasonable degree of accu-racy very rapidly.
CLOSURE BY PIERRE DUMONTEIL
The author wishes to thank Mr. Moore for pointing out that,should a mathematically exact solution be required, it can becalculated by Newton’s method. The author has found thatusing the approximate K factor as a starting value gives a veryrapid convergence.
The approximate formulae are obviously more convenientfor spreadsheet and programmable calculator use.
DISCUSSIONSimple Equations for Effective Length FactorsPaper by PIERRE DUMONTEIL(3rd Quarter, 1992)
Discussion by William E. Moore II
FIRST QUARTER / 1993 37
CORRECTIONS
Simple Equations for Effective Length Factors
Paper by PIERRE DUMONTEIL(AISC Engineering Journal, Third Quarter, 1992)
p. III, Equation 4 should read:
K = 3GA + 1.4
3GA + 2.0
p. 112, Table 2 should read:
Table 2.Comparison of Eqs. 7 and 8—Unbraced frames
GA
GB
K exactK approx.Error, percent
0.1000.4001.0831.0901.000
0.2500.2501.0831.1001.200
0.1000.9001.1591.1701.000
0.2500.7501.1621.1801.400
0.5000.5001.1641.1801.700
0.1001.9001.2861.2900.300
0.2501.7501.2951.3100.800
0.5001.5001.3071.3301.400
GA
GB
K exactK approx.Error, percent
1.0001.0001.3171.3401.800
0.5004.5001.5751.5800.200
1.0004.0001.6341.6500.800
2.5002.5001.7111.7301.200
0.5009.5001.7771.770
−0.200
1.0009.0001.8741.8800.400
2.5007.5002.0922.1000.600
5.0005.0002.2282.2400.400
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION38
p. 125: Second Column, Line 12 should read:
dht = diameter of hole in tension plane(bolt diameter + 1⁄8-in.)
p. 125: Second Column, Line 13 should read:
dhs = diameter of hole in shear plane(bolt diamter + 1⁄16-in.)
p. 125: Second Column, Line 17 should read:
lh = N(1.8 0.6dhs) + 0.3dns + 0.9d + 0.5dht − 1.8
p. 126: Figure 2 should read:
CORRECTIONS
Fast Check for Block Shear
Paper by LEWIS B. BURGETT(4th Quarter 1992)
Figure 2
FIRST QUARTER / 1993 39
EngineeringJournalAMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Page 41: Thomas SputoDesign of Pipe Column Base Plates Under GravityLoad
Page 44: W. Samuel Easterling, David R. Gibbings,and Thomas M. MurrayStrength of Shear Studs in Steel Deck on CompositeBeams and Joists
Page 56: James W. MarshEarthquakes: Steel Structures Performance andDesign Code Developments
Page 66: American Institute of Steel Construction, Inc.SI Units for Structural Steel Design
Page 68: Neil WexlerComposite Girders with Partial Restraints: A NewApproach
2nd Quarter 1993/Volume 30, No. 2
INTRODUCTION
Round pipe columns are often used in large, low buildingssuch as warehouses and department stores. No guidanceregarding the design of the base plates for these columns is tobe found in the Manual of Steel Construction,1 and no defini-tive guidance may be found in the literature. In lieu of havinga rational method for sizing the base plate, designers haveoften resorted to rules of thumb to determine the plate thick-ness. This paper provides a design procedure for determiningthe thickness of these base plates under gravity loads, appliedto Allowable Stress Design criteria. This method is not appli-cable for uplift conditions where the column is in net tensionnor does it consider erection criteria which is within thejudgement of the detailer and erector.
EFFECTIVE PLATE AREA
For a square base plate under a round column, it is reasonableto assume the effective bearing area for this analysis to bewithin an inscribed circle of the same diameter as the platedimensions as shown in Figure 1. Neglecting the area outsidethe inscribed circle will have little effect on the design. If around base plate is used, the entire area may be consideredeffective in the design. In any case, the value of D should belimited to no greater than 2R.
Therefore:
fp = P / πD2
BASE PLATE DESIGN
The base plate can be visualized as being overstressed in twoareas, inside the column diameter and outside the columndiameter. These areas may be designed by applying a yieldline analysis to the base plate.
Yield line analysis of situations similar to this can be foundin many references on yield lines and plastic plate analysis.2
The simplest method of applying this analysis is that of virtualwork.
Considering first the case of bending within the area en-closed by the column (Figure 2):
Wi = Work of the perimeter yielding + work of the radiallines yielding
= 2πM + 2πM = 4πMWe = Bearing pressure × volume of cone shown in Figure 2
= fp(1⁄3πR2)
Setting internal work equal to external work:
4πM = fp(πR2 / 3)
Allowing M to be equal to the elastic moment capacity ofa rectangle, to be consistent with the 9th Edition method:
M = t2Fb / 6
setting Fb = 0.75Fy and solving for the plate thicknessprovides:
t = R(2fp / 3Fy )1⁄2 (1)
Now consider the case of bending of the base plate outsidethe column as shown in Figure 3.
Wi = Work of the radial lines yielding + work of theperimeter yielding
= 2πM[(D − R) / D] + 2πM(R / D)= 2πM
Thomas Sputo is consulting structural engineer, Gainesville,FL.
Design of Pipe Column Base Plates Under Gravity LoadTHOMAS SPUTO
Figure 1
SECOND QUARTER / 1993 41
We = Bearing pressure × volume under the deflected areaoutside the column perimeter as shown in Figure 3.
= fp[πD2 − πD2 / 3 − π(D2 − R2)(R / D) +
(πR3 / 3)(R / D)]= fp[2D2 / 3 − RD + R3 / 3D]
Setting internal work equal to external work and insertingM = t2Fb / 6 yields:
t = [4fp / 3Fy (2D2 − 3RD + R3 / D)]1⁄2 (2)
LIGHTLY LOADED BASE PLATES
Reference 3 recommends that the loaded “H” section methoddescribed in the Manual of Steel Construction be applied tothe design of these base plates. Referring to Figure 4, andagain applying a yield line analysis to the plate:
A = P / Fp = π(Ro2 − Rc
2) therefore:Rc =[Ro
2 − (P / πFp )]1⁄2
Applying a yield line analysis to the base plate,
Wi = 2πM + 2πM = 4πM, as for the first case
Figure 2
Figure 3
Figure 4
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION42
We = Bearing pressure under loaded portion of cone inFigure 4
= Fp [πR2 / 3 − πRc2((R − Rc) / R) − (π / 3)Rc
2(Rc / R)]= Fp π[R2 / 3 − Rc
2 + 2Rc2 / 3R]
Setting internal work equal to external work, and substitut-ing for the value of M and Fb:
t = [(2Fp / 3Fy )(R2 − 3Rc2 + 2Rc
2 / R)]1⁄2 (3)
It should be noted that if Fp = fp and Rc = 0, Equation 3reduces to Equation 1.
If the result of Equation 3 is a plate thickness greater thanthat given by Equation 1, Equation 3 does not apply. Thethickness of a lightly loaded base plate should not be greaterthan the thickness of one loaded over its entire area.4
The required base plate thickness is the greater of Equa-tions 1 or 3 and 2.
EXAMPLE
Given:
Pipe Column, Standard Weight, 4 in. nominal diameter (OD = 4.500 in., ID = 4.026 in.)Load = 12 kipsBase Plate = 7 in. × 7 in.Steel = A36Concrete = 3,000 psi
t = {[2(1.05) / 3(36)][2.132 − 3(1.19)2 + 2(1.19)3 / 2.13]}1/2 = 0.191 in. (3)
This column requires a base plate 0.237 inches thick. Usea plate 7×7×1⁄4-in.
REFERENCES
1. American Institute of Steel Construction, Manual of SteelConstruction, 9th Edition, 1989, Chicago, Ill.
2. Save, M. A. and C. E. Massonnet, Plastic Analysis andDesign of Plates, Shells, and Disks, 1967, Elsevier, NewYork.
3. DeWolf, John T., and David T. Ricker, Column Base Plates,1990, AISC, Chicago, Ill.
4. Ahmed, Salahuddin, and Robert R. Kreps, “Inconsisten-cies in Column Base Plate Design in the New AISC ASDManual,” Engineering Journal, 3rd Qtr. 1990.
NOMENCLATURE
B = Base plate widthD = Radius of loaded area of base plate (D ≤ 2R)Fb = Allowable bending stress in base plate = 0.75Fy
Fp = Allowable concrete bearing pressure as defined inmanual
fp = Actual bearing pressure as defined in manualM = Base plate internal resisting moment per unit lengthP = Total column loadR = Average radius of pipe column
= (Ri + Ro) / 2Rc = Inside radius of loaded area for lightly loaded columnRi = Pipe column outside radiusRo = Pipe column inside radiust = Thickness of base plateWe = External work done by bearingWi = Internal work done by plate bending
SECOND QUARTER / 1993 43
INTRODUCTION
Composite beam or joist and slab systems typically providethe most efficient design alternative in steel frame construc-tion, and indeed it is one of these systems that make steel aneconomically attractive alternative to concrete framed struc-tures. Composite beam specification requirements and designaids are given in the American Institute of Steel Construction(AISC) Load and Resistance Factor Design (LRFD) Manual.1
The LRFD composite beam design procedure results in de-signs that are typically 10–15 percent more economical thanthose obtained using the AISC allowable stress design (ASD)procedure. The efficiency of composite beam design usingLRFD procedures has, in the authors’ opinions, been theprimary motivating factor for the use of the LRFD specifica-tion2 to date.
The design strength and stiffness of composite beamsdepends on the shear connection behavior. The strength of theshear connectors may be reduced because of the influence ofthe steel deck geometry. An empirical expression for thisreduction was developed by evaluating results of compositebeam tests in which the deck ribs were oriented perpendicularto the steel beam.3 A reduced stud strength is obtained bymultiplying the stud reduction factor, SRF, by the nominalstrength of a shear stud, Qn. The expression for the nominalstud strength,4 which has been incorporated in the AISCLRFD specification and is the basis for the tabular valuesgiven in the AISC ASD specification,5 is given by:
Qn = 0.5Asc√fc′Ec ≤ AscFu (1)
where
Asc= cross-sectional area of a stud shear connector
fc′ = specified compressive strength of concrete
Ec = modulus of elasticity of concreteFu = minimum specified tensile stress of the stud shear
connector
This equation was developed based on results from elementalpush-out tests.4 The stud reduction factor is given by:
SRF = 0.85√Nr
wr
hr
Hs
hr − 1.0
≤ 1.0 (2)
where
Nr = number of studs in one rib at a beam intersectionwr = average width of concrete ribhr = nominal rib heightHs = length of shear stud after welding
This reduction factor applies to cases in which the deck ribsare perpendicular to the steel beam and is used in both theAISC LRFD and ASD specifications.
These equations, or similar forms, have been used in sev-eral design specifications, both in the United States andabroad. However, in recent years several researchers6–11 haveshown that Equation 2 is unconservative for certain configu-rations. The studies have considered numerous parameters,including depth of steel deck shear stud height, concrete unitweight, position of shear stud in the deck rib relative to thebottom flange stiffener, number of shear studs in a given deckrib, and the amount and position of reinforcement in the slab.The studies reported results from push-out tests alone6,10,11 ora combination of push-out tests and beam tests.7–9 A conclu-sion common to all of the studies is that a modified, orcompletely different, stud reduction factor is needed. Modi-fied calculation procedures have been developed and reportedin the recent research studies. However, none of the studieshave reported reasons for the discrepancy between the experi-mental data and Equations 1 and 2.
The reason for the discrepancy between recent experimen-tal results with those predicted using Equations 1 and 2 is notclear. However, it is clear that a significant base of data existsto substantiate the procedures.3,12,13 A proper resolution of thisdilemma will require careful consideration of all the data.
A review of the data reported by Grant, et al.,3 along withrelated studies conducted by Henderson12 and Klyce13 reveal
W. Samuel Easterling is assistant professor in the Charles E.Via, Jr. Department of Civil Engineering, Virginia PolytechnicInstitute and State University, Blacksburg, VA.
David R. Gibbings is graduate research assistant in theCharles E. Via, Jr. Department of Civil Engineering, VirginiaPolytechnic Institute and State University, Blacksburg, VA.
Thomas M. Murray is Montague-Betts professor of structuralsteel design in the Charles E. Via, Jr. Department of CivilEngineering, Virginia Polytechnic, Blacksburg, VA.
Strength of Shear Studs in Steel Deck onComposite Beams and JoistsW. SAMUEL EASTERLING, DAVID R. GIBBINGS and THOMAS M. MURRAY
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION44
two important characteristics that relate directly to the dis-crepancy. The majority, but not all, of the tests reported byGrant, et al. and all the tests reported by Henderson weredetailed such that the studs were placed in pairs within a givenrib. The single test reported by Klyce had two-thirds of thestuds placed in pairs. Also, the deck used in the studiesreported by Grant, et al. did not have a stiffener in the bottomflange. Both of these details make the position of the shearstud relative to the stiffener in the bottom flange of the deck,which is described in greater detail in the following para-graph, of less concern.
One of the important parameters identified in some of therecent studies was the position of the shear stud relative to thestiffener in the bottom flange of the deck. Most deck profilesmanufactured in the United States have a stiffener in themiddle of the bottom flange, thus making it necessary to weldshear studs off center. Tests have shown differences in shearstud strengths for the two choices. A stud placed on the sideof the stiffener nearest the end of the span is in the “strong”position and one placed on the side of the stiffener nearest thelocation of maximum moment is in the “weak” position. Aschematic of both strong and weak position stud locations isshown in Figure 1. The difference in strength is partly attrib-utable to the differences in the amount of concrete betweenthe stud and the web of the deck that is nearest to mid-spanfor the two positions. This detail will be considered further insubsequent sections of this paper.
A characteristic of partial composite beam design must bekept in mind when one evaluates results of beam tests andpush-out tests. The relationship between the percentage ofshear connection and the moment capacity is shown in Fig-ure 2 for a W16×31 A36 section. The curves shown in Figure 2were developed using the calculation procedure in the Com-mentary to the LRFD specification.2 The nominal momentcapacity, Mn, is shown normalized with respect to the fullycomposite moment, Mfc. The percent shear connection isgiven by ΣQn / AsFy, where ΣQn is the sum of the shearconnector strength between the points of maximum and zero
moment, As is area of steel cross section, and Fy is yield stressof the steel cross section. Curves are shown for three valuesof Y2, which is the distance from the top of the steel sectionto the center of the effective concrete flange. Although thecurves were generated for a W16×31, they are representativeof a wide range of cross sections because of the normalizationprocedure. A value of Mn / Mfc of about 0.9 is obtained froma partial shear connection value of 0.7. This relation can beextended to evaluating test results, in that if a measured topredicted moment capacity of 0.9 is obtained, then the meas-ured to predicted shear connector capacity is 0.7. Because ofthis relationship, one can argue that an accurate evaluation ofthe shear connector strength must be made using carefullycontrolled elemental push-out tests, as opposed to evaluatingstud strengths using only beam tests. The sensitivity of thestud strength to various parameters is difficult to discern if thestrength is back calculated from beam test results. The bestapproach is to use a combination of the two test configura-tions, with the push-out tests being used to evaluate a widerange of parameters and formulate strength relationships, andwith the beam tests used as confirmatory tests.
The remaining sections of this paper describe a researchproject conducted at Virginia Tech to evaluate the strong vs.weak shear stud position issue.14 Results from a series of fourcomposite beam tests are presented. Additionally, the resultsfrom a series of push-out tests are described. The push-outtests were part of another research project conducted prior tothe beam tests.15 An analysis of the results is presented whichcompares the experimental beam strengths with calculatedvalues based on Equations 1 and 2, as well as values based onthe push-out tests.
Fig. 1. Strong and weak position shear stud locations. Fig. 2. Normalized moment versus percent shear connection.
SECOND QUARTER / 1993 45
STRENGTH AND STIFFNESS CALCULATIONPROCEDURES
Test results were compared to calculated strength and stiff-ness values. The calculated shear stud strengths were deter-mined using the LRFD Specification Equations I5-1 and I3-1(Equations 1 and 2 in this paper). The flexural strengthcalculations were made using the equations given in theCommentary to the LRFD Specification. The elastic stiffnessvalues were calculated using the lower bound moment ofinertia defined in Part 4 of the LRFD Manual. Measuredmaterial properties were used in all calculations. The steelsection properties that were measured (depth, flange thick-ness, flange width, and web thickness) were nearly identicalto the tabular values given in Part 1 of the LRFD Manual.Therefore, tabulated cross-section properties for the steelshape were used in the calculations.
The flexural strength calculation procedure gives threeequations for the nominal moment capacity, with the govern-ing one determined based on the location of the plastic neutralaxis (PNA). Yield stresses were determined separately for theweb and flanges, thus the hybrid section idealization wasused. All the specimens in this study were designed approxi-mately 40 percent composite and the PNA was located in theweb for all tests. The calculated moment capacity, Mc, usingEquation C-I3-5,2 is given by:
Mn = Mp −
CPyw
2
Mpw + Ce (3)
where
Mp = steel section plastic momentC = compressive force in the concrete slabPyw = web yield forceMpw = web plastic momente = distance from center of steel section to the center of
the compressive stress block in the slab
The force C is given by:
C =
min
AswFyw + 2Asf Fyf
0.85fc′Ac
ΣQn
(4)
where
Asw = area of steel webFyw = yield stress of web steelAsf = area of steel flangeFyf = yield stress of flange steelAc = area of concrete slab within effective width
The distance e is given by:
e = 0.5d + hr + tc − 0.5a (5)
where
d = depth of steel sectiontc = slab thickness above the steel decka = depth of compression stress block
The lower bound moment of inertia was calculated using themoment of inertia of the steel beam plus an equivalent areaof concrete, which is a function of the quantity of shearconnection provided. The lower bound moment of inertia,ILB, is given by
ILB = Ix + As
YENA −
d2
2
+ ΣQn
Fy
(d + Y2 + YENA)2 (6)
where
Ix = moment of inertia about x-axis of structural steelsection
YENA = the distance from bottom of beam to elastic neutralaxis (ENA) and is given by:
YENA =
Asd2
+ ΣQn
Fy
(d + Y2)
As +
ΣQn
Fy
(7)
TEST PROGRAM
Beam Test Specimens
The four composite beam tests were similarly constructed.Each specimen consisted of a single W16×31 A36 sectionwith a composite slab attached. The span of each specimenwas 30 ft and the total beam length was 32 ft because of a 1ft cantilever at each end. The composite slab used for thebeam tests was constructed using a 20 gage (0.036 in.), 3 in.deep, composite deck with a total of 6 in. of normal weight(145 pcf) concrete. The steel deck profile is shown inFigure 3. A single layer of welded wire fabric (WWF 6×6-W1.4×W1.4) was placed directly on the top of the deck. Atotal of 12 headed shear studs, 3⁄4-in. × 5 in. after welding, wasused in each test. The studs were welded directly through thesteel deck. The deck was placed with the ribs perpendicularto the beam span and the slab width was 81 in. A self-drillingscrew was placed in each rib that did not have a shear stud init, thus satisfying the requirement of having one fastenerevery 12 inches.16 Deck seams were crimped (button-punched) twice on either side of centerline, resulting in anapproximately 14-in. spacing. The only nominal difference inthe specimens was the position of the shear studs. However,the material properties varied for each test.
All of the studs were placed in the strong position for Test 1and the weak position for Test 2. In Tests 3 and 4 the studpositions were alternated, thus there were 3 in the strongposition and 3 in the weak position along each half span. The
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION46
stud nearest the support was placed in the strong position andthe stud placement was alternated toward midspan. Thisresulted in a symmetric stud pattern in the two half-spans.(Test 4 was a repeat of the configuration used in Test 3 andwas conducted due to the low concrete strengths obtained inTest 3.) The ribs in which shear studs were placed are shownin Figure 4. Note that all of the studs appear in the center ofthe deck ribs in Figure 4, however the studs were placed asdescribed above.
The concrete slabs were formed using 6-in. cold-formedpour-stop material, resulting in three inches of cover on the3-in. steel deck. A detail of the deck and slab is shown inFigure 5. After the concrete was placed, the slab was coveredwith plastic and cured for seven days. During this curing timethe slab was kept moist. After seven days, the plastic and thepour-stop on the sides of the specimen were removed and theslab was allowed to cure for at least 21 additional days priorto testing. Concrete cylinders (4 in. × 8 in.) were cast at thesame time as the concrete slab. The cylinders were keptadjacent to the slab, thus were covered with plastic and keptmoist for the initial seven days.
Each specimen was partially supported during construc-tion. Timber supports were used to prop the steel deck alongthe sides of the slab at the quarter points during concreteplacement. This bracing prevented the slab from warpingduring the placement of the concrete and was not intended toshore the beam. The timber props were cut to allow for thedeflection of the beam under the weight of the fresh concreteand were removed along with the pour-stop after seven days.Additional support was provided by concrete blocks placedunder the four corners of each specimen to prevent rockingof the slab during construction and testing.
Beam Instrumentation
A standard instrumentation arrangement for strain, deflec-tion, end rotation and slip measurement was used for all beamtests. All of the instruments were monitored using a computercontrolled data acquisition system.
Eight strain gages were used to measure the strain throughthe beam cross-section at three different locations, resultingin a total of 24 strain gages per specimen. Two gages wereplaced at each of the following locations: the bottom of thetop flange, the center of the web, the top of the bottom flangeand the bottom of the bottom flange, as indicated in Figure 6.Gages were placed near one end support, at one quarter pointand at the centerline.
Vertical deflections were measured at the centerline and thequarter points. Measurements were taken using linear wiretransducers.
Slip measurements were made using potentiometers at-tached to the top flange of the beam. The potentiometersmeasured the relative movement between the top flange ofthe beam and a screw embedded in the concrete slab througha hole in the steel deck. A total of 12 potentiometers were usedin each test, except Test 1, with one placed adjacent to eachshear stud. Slip was not measured adjacent to the two studsnearest to midspan in Test 1. The slip measurement detail isshown in Figure 7.
End rotations were measured using two different tech-niques. Transducers were used to measure the upward deflec-tion of the ends of the specimen and the support beam. The1 ft overhang was assumed to rotate rigidly about the support,
Fig. 3. Composite deck profile.
Fig. 4. Shear stud locations for composite beam specimens.
Fig. 5. Deck/slab detail.
Fig. 6. Strain gage locations for composite beam specimens.
SECOND QUARTER / 1993 47
thus using the net upward deflection and the distance betweenthe measurement and the support, the end rotation was calcu-lated. Additionally, a digital level was used to measure theangle of the slab relative to horizontal, over the support, tothe nearest 0.1 degrees.
In addition to the strain measurements already described,axial strain was measured in a select number of studs inTests 2–4. This measurement was made using an innovativeapproach, adapted from bolt strain measurement techniques.However, due to problems with the gage installation tech-nique, only a limited amount of usable data was obtained. Forthe benefit of those involved with similar research in thefuture, the instrumentation technique is presented here.
A cylindrical uniaxial strain gage, referred to as a bolt gageby the manufacturer, was inserted in the stud into a pre-drilledhole (approximately 0.1-in. diameter) after it had been weldedto the beam. Lead wires were attached and electrical shrinktubing was placed over the lead wires to protect them duringconcrete placement. The end of the shrink tubing was embed-ded in a small amount of protective coating that was appliedto the top of the stud. Subsequently the tubing was heated toconform to the general shape of the lead wire bundle. The leadwires were brought from the gage straight up through the
concrete to prevent interference with the bonding between theconcrete and the shear stud. A detail of the strain-gaged shearstud is shown in Figure 8.
The problems with the installation technique were attrib-uted to the method used to insert the glue in the pre-drilledhole. The viscosity of the glue was such that the glue had tobe worked into the hole using a blunt probe. Once the gagewas inserted, it was worked back and forth to eliminate anyair bubbles. A different technique, which utilizes a syringe tofill the hole from the bottom, has been used in other tests oncomposite members since the completion of the beam tests.The change in installation procedures appears to have cor-rected the problem.
Beam Load Apparatus and Test Procedure
A four-point loading system was used for all tests, with theloads spaced seven feet apart. The load was applied with asingle hydraulic ram and distributed to the slab by a two-tierdistribution system, as shown in Figure 9.
The load program was similar for all tests. An initial load,equal to approximately 15 percent of the calculated strength,was applied to seat the specimen and was then removed. Theinstrumentation was then re-initialized. Load incrementswere applied to the specimen until the load vs. centerlinedisplacement response became non-linear. The specimen wasthen unloaded and then reloaded to the previous peak in three,approximately equal, increments. Displacement increments,based on the mid-span vertical deflection, were subsequentlyused to complete the test. The specimen was unloaded during
Fig. 7. Slip measurement detail.
Fig. 8. Detail of strain gage in a shear stud. Fig. 9. Loading frame for composite beam specimens.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION48
the displacement controlled phase if it was necessary to adjustthe loading apparatus.
Push-Out Test Specimens
A total of eight push-out specimens were fabricated, four withstuds in the weak position and four with studs in the strongposition. These tests were performed as part of another studyreported by Sublett, et al.15 The push-out tests were con-structed using the same deck profile and shear stud size thatwere used in the beam tests. Each half of a push-out specimenwas constructed by attaching a piece of 3-in. deep compositesteel deck to a W5×11. The ribs of the deck were orientedperpendicular to the length of the WT section. One or twoshear studs (3⁄4-in. × 5 in. after welding) were welded throughthe deck to the structural tee. Two each of the strong and weakposition groups had one stud per specimen half. The other twospecimens in each group had two studs, spaced 12 inchesapart along the length of the WT, on each specimen half. Anormal weight concrete slab, 6-in. thick by 24-in. wide by36-in., was cast on the deck. Welded wire fabric (WWF6×6-W1.4×W1.4) was placed on top of the deck prior tocasting the concrete. The specimens were covered and keptmoist for seven days, at which time the forms were removed.Concrete test cylinders (4 in. × 8 in.) were cast along with thepush-out specimens and cured in a similar manner.
After the slabs had cured, two halves were bolted throughthe stems of the structural tees to form a complete specimen.This manner of casting permitted the slabs to be cast horizon-tally and from the same batch of concrete. By doing this theconcrete curing problems associated with either casting thespecimens vertically or from different mixes were avoided.Overlapping the stems of the tees induced an eccentricity inthe built-up steel section, as compared to using a rolledH-shape. The effect due to this eccentricity was deemednegligible.
Push-Out Test Instrumentation
A standard instrumentation arrangement for measurement ofslip, shear load, and normal load was used for all tests. Slipbetween the steel deck and steel section was measured at twolocations on each half of the push-out specimen using me-chanical dial gages. The applied shear load was measuredusing a load cell that was part of the universal test machine.A normal force was applied to the slab, as described in thenext section of the paper, and monitored using a electronicload cell.
Push-Out Load Apparatus and Test Procedure
To prevent premature separation between the slab and steeldeck, in a direction normal to the slab surface, a yoke devicewas placed on the specimen. This manner of loading simu-lated the gravity load placed on a slab in a compositebeam/slab arrangement. A load cell and hydraulic ram were
part of the yoke assembly. The specimen configuration withthe yoke in place is shown in Figure 10.
Specimens were placed in a universal testing machine onan elastomeric bearing pad, which minimized the effectscaused by any unevenness in the bottom of the specimen.Shear load was applied with the universal testing machine inload increments equal to approximately 10 percent of theexpected specimen capacity. Displacement control was usedonce the load levels reached approximately 80 percent of theexpected capacity.
Load normal to the slab surface was applied using the yokeassembly. The load was monitored using a load cell andcontrolled with a hydraulic hand pump and ram. The normalload was increased along with the applied shear load. Thenormal load was approximately 10 percent of the appliedshear load throughout a test.
Material Tests
Standard material tests were conducted on the concrete andsteel components. The concrete cylinders were tested to de-termine compressive strength on the days of the various beamand push-out tests. Tensile coupons (0.5 in. width, 2 in. gage)were cut and machined from both the web and one flange ofeach structural steel shape, as well as from flat widths of thesteel deck profile. The ultimate tensile stress for the shear
Fig. 10. Push-out specimen schematic.
SECOND QUARTER / 1993 49
studs was reported by the manufacturer. Material propertiesare given in Table 1.
TEST RESULTS
Beam Test Results
The observed behavior was similar for all beam tests, butnotable differences exist. A normalized moment versus de-flection plot of the four tests is shown in Figure 11. Theexperimental moments, Me, were normalized with calculatedmoment strengths using measured material properties and theprocedure described previously. Note that the plots in Fig-ure 11 include the non-composite load and correspondingdeflection. The vertical mid-span deflection, ∆, was normal-ized with ∆H. The deflection corresponding to the point wherethe elastic stiffness, calculated using the lower bound momentof inertia, intersects the calculated moment strength is definedas ∆H.
As indicated in Figure 11, all tests exhibited a ductileresponse. The moment versus deflection response in Tests 1, 3,and 4 (strong and alternating stud position tests) remainedelastic up to a normalized moment of approximately 0.6.Test 2 (weak stud position test) remained elastic up to anormalized moment of approximately 0.4.
The behavior of the shear studs was distinctly different forthe strong and weak position studs. Strong position studs
exhibited failure by developing concrete shear cones or byshearing off in the shank. Weak position studs exhibitedfailure by punching through the deck rib without developinga significant shear cone in the concrete or shearing in the studshank. In Tests 1, 3 and 4, one or two of the strong positionstuds closest to one of the specimen supports sheared off inthe shank. However, the weak position stud between the twostrong position studs in Tests 3 and 4 did not shear off, butpunched through the deck web and remained attached to thebeam.
Push-Out Test Results
An average strength of 13.55 kips per stud was obtained fromthe four push-out tests in which the studs were in the weakposition. The concrete compressive strength was similar foreach of the tests, with an average for the four tests of 4.27 ksi.There was no significant difference between the strengths(load per stud) obtained from the tests with one stud perspecimen half and the tests with two studs per specimen half.In all of the weak position tests, failure occurred by the studspunching through the adjacent web of the steel deck. A smallwedge of concrete between the stud and the deck web wascrushed or broken out in each of the tests. The deck wasnoticeably bulged out adjacent to the stud prior to reachingthe maximum applied shear load. This behavior was an indi-cation that the load was being primarily resisted by the deck.
An average of 18.82 kips per stud was obtained from thethree push-out tests in which the studs were in the strongposition. The average concrete strength was 4.57 ksi. Theresults for the fourth specimen were inexplicably low and arenot included in the average. The decision to omit this test wasbased on the other three tests plus an additional 11 tests,similarly constructed, that were part of a proprietary study inwhich double angle sections were used as the base membersinstead of structural tees. There was no significant differencebetween the strengths (load per stud) obtained from the testwith one stud per specimen half and the tests with two studsper specimen half. In all of the strong position tests, thestrength was limited by the development of a failure surface
Fig. 11. Normalized midspan moment versusdisplacement for composite beam specimens.
Table 1.Material Properties for Composite Beam Specimens
Test
FlangeFyf
(ksi)
FlangeFuf
(ksi)
WebFyw
(ksi)
WebFuw
(ksi)
Slabfc′
(ksi)
1234
42.041.942.543.6
68.870.470.163.4
47.045.447.049.1
71.973.875.762.9
4.813.202.284.99
Shear Studs: Fu = 64.8 ksiSteel Deck: Fy = 40.3 ksi Fu = 53.6 ksi
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION50
in the concrete. None of the shear studs exhibited a shearfailure in the shank.
The response of the studs in the weak position, in terms ofload versus slip, was more ductile than that of the studs in thestrong position. This difference is attributed to the way inwhich the load appeared to be resisted, based on the observedfailure modes. The failure mode for the strong position testswas brittle; concrete shear, and the failure mode for the weakposition tests was more ductile; bearing and eventual tearingof the steel deck web. A typical plot of load versus slipbehavior for strong and weak position shear studs is illus-trated in Figure 12.
ANALYSIS OF RESULTS
The results of the beam and push-out tests were comparedwith calculated values. Several comparisons have been madeand are presented in this section. The calculated momentvalues were based on the expressions described previously inthis paper, using measured material properties and values ofshear connector strength that were calculated using the LRFDspecification or taken from normalized push-out test results.Shear connector strength was also back calculated using theexperimental moment values obtained from the beam tests.The results of each of these calculations and comparisons aregiven in Table 2.
The values Qc given in Table 2 are calculated stud strengths.These were determined using Equations 1 and 2 with meas-ured material properties. Stud strengths Qcb, were back-calculated using the experimental moment from the beamtests, measured material properties and the calculation proce-dure described previously.
Because the shear studs in the weak position, in both thepush-out and beam tests, failed by punching through the webof the deck it was hypothesized that their strength was not
primarily a function of concrete strength. Rather, the studstrength is primarily a function of the steel deck strength (i.e.,the yield stress of the steel deck). Certainly some interactionbetween the concrete and the deck occurred, but the dominantcomponent was the steel deck. Based on this hypothesis, theweak position push-out test strengths were averaged and usedfor all the weak position stud strengths in the calculations forthe beam tests. No adjustment was made to account forvariable concrete strengths.
The strength of the shear studs in the strong position wastaken as a function of the concrete strength. The strong
Table 2.Experimental and Calculated Results
TestQc
(kips)Qpo
(kips)Qcb
(kips)Mc
(ft-kips)Mpo
(ft-kips)Me
(ft-kips) Qcb / Qc Qpo / Qc Qcb / Qpo Me / Mc Me / Mpo
All values based on measured material propertiesQc = calculated stud strength using Equations 1 and 2.Qpo = calculated stud strength using Equation 8 and concrete strength from beam test for strong position studs and a constant value of 13.55 kips
for the weak position studs.Qcb = calculated stud strength using Equation 3 with Me in place of Mn.Mc = calculated moment strength using Equation 3 and Qc.Mpo = calculated moment strength using Equation 3 and Qpo.Me = maximum applied experimental moment including weight of specimen, load beams, and applied ram load.
Fig. 12. Load vs. slip for strong and weakposition shear studs for push-out tests.
SECOND QUARTER / 1993 51
position stud strengths in the beam tests were calculated bynormalizing the push-out test results with the concretestrengths as given by:
Qpo = 18.82 kips√fc′4.57 ksi(8)
where fc′ is the concrete compressive strength for the compos-ite beam test, 18.82 kips is the average stud strength from thepush-out tests, and 4.57 ksi is the concrete compressivestrength from the push-out tests. The Qpo values represent studstrengths for the beam tests based on push-out test results.
Equation 8 was used to calculate the values for Qpo in theTest 1, and the constant value reported in the push-out resultssection was used for Test 2. The strong and weak positionvalues were averaged in determining the values for Opo inTests 3 and 4.
Three values of moment are shown in Table 2, Mc, Mpo, andMe. The first, Mc, was calculated using Qc, Mpo was calculatedusing Qpo, and Me represents the maximum experimentalmoment from the beam tests. Various ratios of stud strengthsand moment strengths are also given in Table 2.
Two trends are clearly indicated by the results in Table 2.One of these is that the stud strengths predicted by Equa-tions 1 and 2 do not compare favorably to the values from thepush-out tests or the beam tests. This is indicated by the ratiosQcb / Qc and Qpo / Qc. The second trend that is evident is thatthe results from the push-out tests and beam tests comparevery well, as indicated by the ratio Qcb / Qpo.
Additionally, while a comparison between strong and weakposition shear stud strengths indicates some difference, themore pronounced and significant difference is between thepredicted values and the beam and push-out test results. Theratios Qcb / Qc or Qpo / Qc indicate the strong position valuesare approximately 70 percent of the predicted and the weakposition values are approximately 60 percent of predicted.
The sensitivity of the moment strength to the shear studstrength is also illustrated in the results. Values of experimen-tal to calculated shear stud strengths varied between 0.59 and0.83, while the experimental to calculated moment values,indicated by Me / Mc, varied between 0.85 and 0.94. Therelationship between shear connection and moment strengthis illustrated for the W16×31 used in this study by the nor-malized moment versus shear connection relationship in Fig-ure 2. Although as previously indicated, this relationship isgenerally presented in the context of partial composite design,it can also be used to consider the reduction in momentstrength due to a reduction in shear connector strength.
The strain data collected from the beam tests also indicatethe difference between strong, weak, and alternating positionstuds. The relationship between the position of neutral axisand the applied moment is illustrated in Figure 13. A linearregression analysis was performed using the eight strainreadings located at midspan in the steel section to determinethe neutral axis location. As noted in Figure 13, the strongposition studs resulted in the neutral axis being higher in thesteel than for the weak or alternating tests. Further, the posi-tion for the alternating tests fell between the strong and weakvalues.
Using Figure 13, the plastic neutral axis position can beestablished by visually locating the point at which the slopeof line is approximately vertical. These values are given inTable 3. Also shown in Table 3 are calculated values of theplastic neutral axis based on Qc, Qpo, and Qcb. Note that thecalculated values using either Qpo or Qcb correspond moreclosely to the experimental values than do the positionscalculated using Qc, in all but Test 3.
DESIGN IMPLICATIONS
The implications of the study described here, as well asprevious studies, on composite beam design merit considera-
Fig. 13. Applied moment versus position of neutralaxis for composite beam specimens.
Table 3.Experimental and Calculated Neutral Axis Positions
TestPNAe
(in.)PNAc
(in.)PNApo
(in.)PNAcb
(in.)
1234
2.74.23.93.6
0.782.323.880.88
3.134.564.643.85
3.154.614.573.76
All values of PNA are measured from top of steel section.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION52
tion at this point. Based on the test results presented in theprevious sections, it is evident that Equation 2 is not conser-vative in all cases. Specifically, if single shear studs are used,as opposed to pairs of studs, the equation over-predicts thestrength of the stud. Based on a review of previous stud-ies3,12,13 the authors believe that Equation 1 is conservative fordesigns in which two studs per rib are utilized. No generalmodifications to the form of the equation are proposed at thistime. Until such modifications are formulated, the followingrecommendations are offered:
1. The stud reduction factor should not exceed 0.75 forcases in which there is one stud in a rib.
2. Detail all single studs in the strong position. The imple-mentation of this detail requires coordination betweenthe structural engineer and the stud contractor to effec-tively relay the objective of the detail.
3. Use 50 percent composite action as a minimum, i.e.,keep ΣQn / AsFy greater than or equal to 0.50. This willminimize the adverse effect of under-strength studs onthe design moment strength, as reflected by the trend ofthe curves in Figure 2.
The result of implementing the above recommendations isan increase in the number of shear studs for designs utilizingone stud per rib. This will obviously result in a small increasein the cost, however the percentage increase in the in-placecost of the composite beam for these situations will be minor.Certainly in view of the questions that have been raisedregarding the strength of the studs, the increase is warranted.
A consideration in future composite beam studies andmodifications to the specification procedures should be theapplication of a strength reduction factor, φ, to the shear studs.In the current AISC LRFD specification2 a single strengthreduction factor is applied to the nominal moment strengthfor the composite beam system, which includes the variableeffects of the shear connectors. However, the flexural strengthof the beam and the shear strength needed at the steel concreteinterface are associated with different modes of behavior andlimit states and therefore merit separate consideration. If thisapproach were pursued, one would expect that the value ofφ for the flexural limit state may increase above the presentvalue of 0.85, thus making more efficient use of the steelshape which is the dominant component in the cost of thecomposite bearn. At the same time the variability that existsin the shear stud strength would be reflected in a φ value forshear studs.
The flexural and shear stud limit states are treated inde-pendently in other limit states design specifications.17,18 Thenominal strengths, as well as the stud reduction factors, varybetween the three specifications. A graphical comparison ofthe three specifications for the 3-in. deep composite deckshown in Figure 3 is given in Figure 14. The differencesillustrated in Figure 14 in part reflect the uncertainty thatexists at the present time regarding shear connector strength.
SUMMARY AND CONCLUSIONS
Results were described for a recent study conducted at Vir-ginia Tech in which a series of push-out tests and composite
Fig. 14. Shear strength comparison for AISC, CSA, and Eurocode specifications.
SECOND QUARTER / 1993 53
beam tests were conducted. The results were consistent withother recent studies reported in the literature, in that thestrength of shear studs placed in the ribs of steel deck orientedtransverse to the beam span, calculated using Equation 2,were higher than measured values. Review of the test dataused to develop Equation 2 indicated that the majority of thetests were conducted with the shear studs placed in pairs.Equation 2, when combined with Equation 1, accuratelyreflects the stud strength for these cases.
Specific modifications to Equation 2 were not proposed, asfurther evaluation of existing procedures is required. Thehypothesis regarding the influence of the steel deck materialproperties on the stud strength must be evaluated at the sametime and perhaps included as a modification to one of theexisting methods. This hypothesis, while not conclusivelyverified, was supported by the results of the Virginia Techresearch program.
ACKNOWLEDGMENTS
Graduate research assistant support for the project was pro-vided by the American Institute of Steel Construction. Thefollowing organizations generously supplied material andequipment for the project: Virginia-Carolinas Structural SteelFabricators Association (structural steel), Vulcraft Divisionof Nucor (steel deck pour-stop and welded wire fabric), andNelson Stud Welding Division of TRW (shear studs and studwelding equipment). The remaining project costs were pro-vided by Virginia Tech. The project from which the push-outtest results were taken was sponsored by Nucor Research andDevelopment.
REFERENCES
1. American Institute of Steel Construction, Manual of SteelConstruction—Load and Resistance Factor Design, FirstEdition, Chicago, Illinois, 1986.
2. American Institute of Steel Construction, Load and Re-sistance Factor Design Specification for Structural SteelBuildings, Chicago, Illinois, September 1986.
3. Grant, J. A., Fisher, J. W. and Slutter, R. G, “CompositeBeams with Formed Steel Deck,” Engineering Journal,AISC, 14(1), 1977, pp. 24–43.
4. Ollgaard, J. G., Slutter, R. G. and Fisher, J. W., “ShearStrength of Stud Connectors in Lightweight and NormalWeight Concrete,” Engineering Journal, AISC, 8(2),1971, pp. 55–64.
5. American Institute of Steel Construction, Specificationsfor Structural Steel Buildings: Allowable Stress Designand Plastic Design, Chicago, Illinois, June 1989.
6. Hawkins, N. M. and Mitchell, D., “Seismic Response ofComposite Shear Connections,” Journal Structural Engi-neering, ASCE, 110(9),1984, pp. 2120–2136.
7. Jayes, B. S. and Hosain, M. U., “Behaviour of HeadedStuds in Composite Beams: Push-out Tests,” CanadianJournal of Civil Engineering, 15, 1988, pp. 240–253.
8. Jayes, B. S. and Hosain, M. U., “Behaviour of HeadedStuds in Composite Beams: Full-Size Tests,” CanadianJournal of Civil Engineering, 16, 1989, pp. 712–724.
9. Robinson, H., “Multiple Stud Shear Connections in DeepRibbed Metal Deck,” Canadian Journal of Civil Engi-neering, 15, 1988, pp. 553–569.
10. Mottram, J. T. and Johnson, R. P., “Push Tests on StudsWelded Through Profiled Steel Sheeting,” The StructuralEngineer, 68(10), (1990), pp. 187–193.
11. Lloyd, R. M. and Wright, H. D., “Shear Connectionbetween Composite Slabs and Steel Beams,” Journal ofConstruction Steel Research, 15, 1990, pp. 255–285.
12. Henderson, W. D., “Effects of Stud Height on ShearConnector Strength in Composite Beams with Light-weight Concrete in Three-Inch Metal Deck,” Master ofScience Thesis, The University of Texas, Austin, TX,1976.
13. Klyce, D. C., “Shear Connector Spacing in CompositeMembers with Formed Steel Deck,” Master of ScienceThesis, Lehigh University, Bethlehem, PA, 1988.
14. Gibbings, D. R., Easterling, W. S. and Murray, T. M.,“Composite Beam Strength as Influenced by the ShearStud Position Relative to the Stiffener in the Steel DeckBottom Flange,” Report No. CE/VPI-ST 92/07. VirginiaPolytechnic Institute and State University, Blacksburg,VA, 1992.
15. Sublett, C. N., Easterling, W. S. and Murray, T. M.,“Strength of Welded Headed Studs in Ribbed Metal Deckon Composite Joists,” Report No. CE/VPI-ST 92/03, Vir-ginia Polytechnic Institute and State University,Blacksburg, VA, 1992.
16. American Society of Civil Engineers, Specifications forthe Design and Construction of Composite Slabs,ANSI/ASCE 3-84, New York, 1984.
17. Commission of the European Communities, Eurocode 4:Common Unified Rules for Composite Steel and ConcreteStructures, Rep. EUR 9886, 1992.
18. Canadian Standards Association, Limit States Design ofSteel Structures, CAN/CSA-S16.1-M89, Rexdale, On-tario, 1989.
NOMENCLATURE
Ac = area of concrete slab within effective widthAs = area of steel cross sectionAsc = cross sectional area of a stud shear connectorAsf = area of steel flangeAsw = area of steel weba = depth of compression stress blockC = compressive force in concrete slabd = depth of steel sectionEc = modulus of elasticity of concretee = distance from center of steel section to the center of
the compressive stress block in the slab
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION54
Fu = minimum specified tensile stress of stud shearconnector
Fy = yield stress of steel cross sectionFyf = yield stress of steel webFyw = yield stress of steel web
fc′ = specified compressive strength of concrete
Hs = length of shear stud after weldinghr = nominal rib heightILB = lower bound moment of inertiaIx = moment of inertia about x-axis of structural steel
sectionMc = moment strength calculated using Qc
Me = maximum experimental momentMfc = fully composite moment strengthMn = nominal moment strengthMp = steel section plastic moment strength
Mpo = moment strength calculated using Qpo
Mpw = web plastic moment
Nr = number of studs in one rib at a beam intersection
Pyw = web yield force
Qc = calculated stud strength using Equations 1 and 2
Qcb = stud strength calculated using Me and Equation 3.
Qpo = stud strength calculated using push-out test results
Qn = nominal strength of a shear stud
tc = slab thickness above the steel deck
wr = average width of concrete rib
Ycon = distance from top of steel beam to top of concrete
YENA = distance from bottom of beam to elastic neutral axis
Y2 = Ycon − a / 2
ΣQn = sum of strengths of shear connectors
SECOND QUARTER / 1993 55
INTRODUCTION
Major earthquakes occur several times each year through-out the world with heavy loss of life and property. Recentexamples are the 1992 Cairo, Egypt earthquake with the lossof over 500 lives, and the Mexico earthquake of 1985 withthe loss of 8,000 lives and the collapse of over 400 buildings.
The United States has experienced many large earth-quakes, with the most seismic activity to date being locatedin California, i.e., Loma Prieta, California, 1989, 7.1 Richtermagnitude and Landers, California, 1992, 7.5 Richter magni-tude. It is evident from past occurrences of earthquakes thatthe highly seismic regions of the United States have a seriousearthquake problem, and the less serious regions in the centraland eastern parts of the country now realize that they have anearthquake problem which is being addressed through adop-tion of the latest seismic design provisions into the BOCA andSBCCI building codes. Some of these newly acquired seismicprovisions are taken from the Building Seismic Safety Coun-cil program on improved seismic safety.
The Building Seismic Safety Council (BSSC) was estab-lished in 1979 under the auspices of the National Institute ofBuilding Sciences (NIBS) as an entirely new type of instru-ment to develop and promulgate building earthquake hazardmitigation regulatory provisions that are national in scope. Itsfundamental purpose is to enhance public safety by providinga national form that fosters improved seismic safety provi-sions for use by the building community in the planning,design and construction of buildings. To fulfill its purpose,the BSSC promotes the development of seismic safety provi-sions suitable for use throughout the United States. The BSSCbelieves that the regional and local differences in the natureand magnitude of potentially hazardous earthquake eventsrequire a flexible approach to seismic safety that allows forconsideration of the relative risk, resources and capabilitiesof each community. The BSSC itself assumes no standards-making and promulgating role; rather, it advocates that codeand standards formulation organizations consider BSSCrecommendations for inclusion into their documents and stand-ards. A recommendation that is taking place today in code writing.
The basic problem of earthquake design is to synthesize thestructural configuration; the size, shape, and material of the
structural elements along with the methods of fabrication, sothat the structure will safely and economically withstand theaction of earthquake ground motions. This of course requiresa broad knowledge of the behavior of structures during earth-quakes, and the final evaluation of the design will be madeby a future earthquake. It is this ultimate test that has shownthat steel-framed buildings and bridges have an excellentrecord of protecting life safety, as well as minimizing eco-nomic loss and business interruption.1
STRUCTURES PERFORMANCE
Mexico Earthquake
On September 19, 1985, a magnitude 8.1 earthquake struckMexico, followed by a magnitude 7.5 aftershock 36 hourslater.2 Data compiled by the Institute of Engineering of theNational Autonomous University of Mexico revealed that atotal of 330 buildings collapsed in central Mexico City. Ofthese, 12 were steel frame, and 318 were reinforced concreteor masonry. The majority of steel buildings that collapsedwere built before 1957, while most concrete and masonrybuildings were built between 1957 and 1976. Both years markmajor revisions in the building code, adopted in response topast damaging earthquakes.
Steel structures constructed after 1957 fared much betterthan the norm, with only 6.8 percent of such buildings expe-riencing severe damage or collapse. Steel structures con-structed after 1976 performed excellently; no cases of severedamage or collapse were noted in this group, and only foursuch buildings sustained any structural damage.2
Some steel buildings constructed from 1920 through the1940s experienced severe damage. These structures werebuilt prior to the adoption of earthquake codes, and usedconstruction types that were abandoned following the 1957earthquake. The most common type of steel building con-struction used over the last three decades has been highlyredundant moment frames where almost every beam-columnjoint in these structures is moment resisting. The second mostcommon lateral system for steel structures was moment-resisting frames with braced bays. The Pino Suarez complexaccounted for all the reported failures of this system in the1985 earthquake.
Analyses performed after the earthquake have provided anexplanation of the Pino Suarez failures.3 Very large axial
Earthquakes: Steel Structures Performance andDesign Code DevelopmentsJAMES W. MARSH
James W. Marsh is a professional engineer in El Monte, CA.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION56
loads, due to gravity and seismic overturning, overstressedthe exterior columns in the braced bays. The bracing systembeing capable of resisting story shears several times higherthan the code design level produced unanticipated large axialforces in the columns. As the result of these findings, provi-sions have been added to the 1988 edition of the UniformBuilding Code to prevent overload of columns from overturn-ing forces that exceed those calculated from the basic seismicprovisions of the Code.
Structural steel was successfully used to strengthen rein-forced concrete buildings prior to the 1985 earthquake.4 The12-story Durango Building is located in the heaviest damagedregion of the city. After being heavily damaged in the 1979earthquake, the building was retrofitted with steel frames,which added ductility as well as strength to the structure. Inthe 1985 earthquake the building performed excellently, sus-taining no damage. The steel frames are believed to havecarried over 80 percent of the total lateral force.
Although steel construction in Mexico differs substantiallyfrom practice in the United States, dozens of modern steelbuildings located in the badly shaken lake bed area of MexicoCity received no damage. A good example is the 44-storyTorre Latinoamericana, constructed in the early 1950s anddesigned for earthquake loads, which performed excellentlyin 1985 as it had in three previous earthquakes in 1957, 1978and 1979.
Whittier-Narrows, California Earthquake
The October 1, 1987 Whittier-Narrows earthquake of magni-tude 5.9 (Richter Scale) was considered a moderate earth-quake. Several aftershocks caused a few structures that werebadly damaged on October 1 to collapse in an October 4aftershock of 5.5 magnitude. USGS records show unusualhigh ground accelerations of 0.40g to 0.60g, and grounddisplacements of 1 to 2 inches.5 According to the NationalCenter for Earthquake Engineering Research (NCEER), mostearthquake damage occurred in unreinforced masonry build-ings, older homes and modern buildings in construction typeslacking in ductility.6
Several reinforced concrete and shear wall buildings,bridges constructed according to pre-1971 engineering prac-tice sustained heavy damage. Major shear damage was expe-rienced by the supporting concrete columns of the overpasslocated at the junction of the I-5 and I-605 Freeways, 15 milesEast of downtown Los Angeles. Whereas bridge abutmentsexperienced moderate to minor damage by spauling of con-crete underneath the supporting pads, no damage was notedin abutments, columns and piers of bridges that were retrofit-ted by cable restrainers.6
A two-story concrete parking structure built in 1964 andlocated in the Whittier Quad shopping center collapsed aftershear failure of its columns. Large girders had much strongersections than the supporting columns, thereby creating astrong-beam–weak-column situation. The requirement for a
strong-column vs. a weak-beam design was first required forconcrete structures in the 1985 Uniform Building Code. TheCode requires that the sum of the column moments at abeam-column joint be a minimum of 20 percent greater thanthe sum of the girder moments. A similar design provisionbecame a requirement for structural steel seismic design in1988.7
The four-story steel-framed California Federal SavingsService Center relied upon braced (chevron) frames for lateralresistance and was designed in accordance with the 1979Uniform Building Code. During the earthquake the buildingexperienced a peak ground motion several times higher thanthe working stress design levels, Structural damage was lim-ited to the buckling of a single wide flange bracing memberon each of the second, third and fourth floors.8 In spite of thesevere ground motions that the building experienced, it wasrestored to service within a week. In contrast, an adjacenttwo-story precast concrete structure built in 1980 experiencedsuch extensive damage that repair to the building took ninemonths.
Loma Prieta Earthquake
On October 17, 1989 an earthquake of 7.1 Richter magnitudeoccurred that was centered approximately 60 miles south ofSan Francisco. Among the seismic-induced events were thecollapse of the elevated Cypress Street section of Interstate880 in Oakland; the collapse of a section of the San Francisco-Oakland Bay Bridge; and major structural damage to modernbuildings in Oakland, San Francisco and Burlinghame. Over62 people died.
Some of the heaviest concentration of damage occurred inthe city of Oakland, 60 miles north of the earthquake epi-center, where peak ground accelerations were only 0.2g to0.26g.9 A 15-story concrete shear wall structure in downtownOakland suffered extensive damage when its lightweightconcrete shear walls shattered at the first story. The presenceof a redundant steel frame within the building saved thestructure.2
The Hyatt Regency Hotel located in Burlingame, a mid-
Table 1.Statistical Summary of Damage to Buildings
1985 Earthquake
TypeStructure
Extent ofDamage
Year When Built
Pre-1957 ’57–’76 Post 1976 Total
SteelFrame
CollapseSevere
7 1
3 1
0 0
10 2
RC Frame CollapseSevere
2716
5123
4 6
8245
WaffleSlab
CollapseSevere
8 4
6222
2118
9144
SECOND QUARTER / 1993 57
rise reinforced concrete structure, sustained extensive dam-age to its shear walls and floor slab around the elevator core.The structure was designed to the 1985 Uniform BuildingCode and construction completed just prior to the earthquake.Repair of damage resulted in closure of the hotel for morethan eight months. In contrast, modern steel-framed buildingsperformed excellently in the Loma Prieta earthquake, as theyhave in the past.
Damage to steel structures was typically limited to crack-ing of cladding and interior partitions with wide-spread dis-array of building contents. The nonstructural damage sus-tained by steel frame buildings may largely be attributed totheir flexibility, which results in large displacements.10
Major transportation routes were affected by the LomaPrieta earthquake. Immediately after the earthquake 11 majorhighways and freeways were closed due to landslides, struc-tural damage or bridge collapse. The collapse of the CypressStreet elevated section of I-880 (near downtown Oakland)was responsible for the majority of earthquake deaths. Thedouble-deck highway system consists of box girder deckssupported by concrete frames. The failure occurred at theconnection of the support columns and the transverse beams,at the lower road level.
Redesign of the Cypress Street roadway was completed inOctober of 1992, with reconstruction scheduled to begin inearly 1993. Five sections of the new design of I-880 will beconstructed of structural steel.
While a mile-long section of the Cypress Street overpassstructure of I-880 collapsed, buildings of various types andvintage right next to the collapsed freeway experienced verylittle or no damage. It is of further interest that the collapsedportion of I-880 is located on man-made ground, whereas thesurviving elevated section is located on a competent sandformation.10
The collapse of a section of the San Francisco-Oakland BayBridge greatly impacted bay area commuting. The BayBridge carries an average of 250,000 vehicles per day be-tween San Francisco and the cities of the East Bay. The BayBridge is a double-decked steel bridge about 8.5 miles long.Its west bay crossing is a suspension span, while the east baycrossing consists of deck trusses and through trusses. Abouttwo miles west of the Oakland toll plaza, 50-foot horizontalspans, situated along the top and bottom decks and locatedabove a main pier, serve to link the bridge’s deck-truss sectionto the east with its through-truss section to the west. Theanchor bolts that attached the bridge’s deck-truss section tothe pier were the only constraint that prevented the twodeck-spans from displacing longitudinally with the bridge’sdeck-truss section to the east. During the earthquake, largelongitudinal and lateral seismic forces caused these bolts tofail in shear. Following this failure, the earthquake-inducedlongitudinal deformations along the length of the deck-trusssection were sufficiently large (7 in.) to result in collapse ofthe upper and lower spans. The cause of the bridge failure was
easily understood, and the 50-foot section was repaired in onemonth and opened to traffic again.11
Landers, California Earthquake
On June 28, 1992 a magnitude 7.5 earthquake, epicenterednear Landers, California in the Southern Mojave Desert,occurred at 4:58 a.m. At 8:04 a.m. a second earthquake, of 6.5magnitude and centered near Big Bear Lake 20 miles to theWest of Landers in the San Bernardino mountains, occurred.
Both earthquakes occurred near the so-called “Big Bend”of the San Andreas Fault, causing scientists to speculate aboutthe possibility of a larger earthquake on this conspicuouslyquiet stretch of the longest fault in California.
Accelerations of as much as 1.0g were recorded in LucerneValley, and 0.55g in Big Bear, although most epicentralstations reported peak accelerations of 0.3g or less.12 Caltranshad instrumented one of the tall (70 ft.) bridge columns onInterstate 10, near the city of Colton, after retrofitting theconcrete column with a steel plate jacket as a result of the1989 earthquake. Although the Landers earthquake showed aground acceleration of only 0.1g, acceleration at the top ofthe column was 0.8g in the longitudinal direction and 1.02gin the transverse direction. The column experienced no dam-age which can be attributed in part to the retrofit method ofwrapping the concrete column in steel.
The Landers earthquake sequence appears to have oc-curred in a northerly northwest direction, striking the CampRocks, Emerson and Johnson Valley faults. It appears that theBig Bear seismic event was initiated by movement on theCamp Rock-Emerson Fault. Two sets of 500kV and two setsof 220kV transmission lines crossed the Camp Rock-Emer-son Fault near the north end of the rupture. The fault passeddirectly between the legs of a bolted steel frame 220kV tower,moving two of the legs approximately 9 feet. This movementresulted in substantial deformation of the steel tower andfailure of several braces. No damage was sustained by thelines or ceramic insulators and the tower continued to provideadequate support until repaired.13
EARTHQUAKE LEGISLATION
Earthquake Hazard Reduction
A review of the history of seismic code development in theUnited States14 helps to more fully understand the lethargy inbringing modern seismic code requirements into the buildingcodes. Much like other areas of the world the early seismicdesign codes in the United States were the result of disastrousearthquakes, primarily in California. Major milestones inseismic code development closely follow many of our signifi-cant earthquakes. California’s Earthquake Reduction Act of1986 was signed into law shortly after the 1985 MexicoEarthquake.15
This Act is sponsored by the Seismic Safety Commissionwhich has the responsibility of preparing and administering
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION58
the California Earthquake Hazard Reduction Program. It is amultidisciplinary commission consisting of 17 commission-ers and 12 staff. The commission’s goal is to significantlyreduce earthquake risk in California by the year 2000. Theresponsibility for meeting this goal must be shared by State,City, and County agencies as well as the private sector.
The first step is an advisory document which is based oninitiatives to improve seismic safety. Between 1987 and 1992there were 72 initiatives passed by the legislature, with an-other 42 initiatives scheduled for the 1992–96 period. Theadvisory document contains 150 milestones to measure pro-gress and record accomplishments.
The hazard reduction program is based on five criteria: (1)lives saved, (2) damage reduction, (3) socioeconomic conti-nuity, (4) opportunity (ease of implementation), (5) cost.These priorities must pass the common sense test of will thedecision maker and the general public consider the initiativeas being practical, sensible, and feasible? The program has 42initiatives integrating actions needed in the public and privatesector.
The size of the earthquake and the amount of damagegreatly influence safety legislation. For instance, during the1987–88 session of the California legislature the Whittier,California earthquake, M 5.9, occurred with 23 seismic safetybills being introduced and 11 of the bills passed by thelegislature but only six bills finally being signed into law bythe Governor.
Two years later during the 1989–90 legislative session theLoma Prieta earthquake of M 7.1 occurred with 443 billsbeing introduced. Of these bills, 164 passed the legislatureand 137 of those were signed into law.
If a historical comparison of legislative action is made forthe 1906–1989 period, 112 seismic safety bills were signedinto law during that 83-year period. But from 1990 to present,a short two-year period, 206 seismic safety bills were madelaw. Obviously the impact of the Loma Prieta earthquake.
How seismic safety translates into dollars is shown inTable 3. Out of $670 million for fiscal year 1991–92 the bulkof the money, 63 percent, went to the California Departmentof Transportation (Caltrans), again as the result of the majorroad and bridge damage inflicted by the Loma Prieta earth-quake.
A review of the 12,500 California State highway bridgesafter the 1971 Sylmar California earthquake (6.6 Richtermagnitude) showed that ten percent of the bridges constructedprior to 1971 would need to be strengthened. The initialportion of the 1973 program was aimed at retrofitting bridgehinges. Inexpensive joint restraining devices were developedand installed. This portion of the program focused on 1,249bridges statewide and was scheduled to be completed in 1988.Despite these safeguards, the possibility of bridge damagewas not eliminated. The 1987 earthquake on the WhittierFault verified the action Caltrans started in 1973 becausealthough there was the expected damage during the earth-
quake no bridges collapsed. The program is currently focusedon wrapping a steel reinforcement shield around the concretecolumn in bridges with single-column designs.
Seismic Design Provisions
It is impossible to predict the location and magnitude ofearthquakes accurately. It is therefore essential to adopt apreventive design philosophy in order to avoid repeatingerrors in rebuilding after an earthquake and in planning newconstruction. This goal is best achieved through code adop-tion where state-of-the-art seismic design criteria is specified.
One such specification is the AISC Seismic Provisions forStructural Steel Buildings. First published in 1990 for Load& Resistance Factor Design, it has been updated in a 1992version to encompass both LRFD and ASD design proce-dures.16 A significant change in the 1992 edition of the seismicprovisions is the conversion to the loads and design formatrecommended by the 1991 National Earthquake HazardsReduction Program (NEHRP) document.1
Whereas the provisions contained in the AISC seismicdocument are to be used in conjunction with the AISC Load& Resistance Factor Design (LRFD) Specification, the loadprovisions have been modified from those in the LRFD inorder to be consistent with the load provisions contained inthe BOCA, SBCCI Codes, and the ASCE 7-93 MinimumDesign Loads for Buildings and Other Structures.17 All thesenew seismic load provisions are modeled after the 1991NEHRP earthquake provisions.
Table 2.California at Risk 1992–1996
CategoryNumber ofInitiatives Focus
12345
20 5 9 5 3
Existing FacilitiesNew FacilitiesEmergency ManagementDisaster RecoveryResearch and Education
Table 3.California Seismic Safety Activities
FY 1991–1992, $670 Million Total
Agency $ Millions
Department of ConservationOffice of Emergency ServicesGeneral ServicesPublic Utilities CommissionSeismic Safety CommissionDepartment of Water ResourcesUniversity of CaliforniaOffice Statewide Health Planning & DevelopmentDepartment of Transportation (Caltrans)
6.6192.6 17.0 0.6 1.4 1.5 3.4 16.6420.3
SECOND QUARTER / 1993 59
The requirements for analysis and design of buildingsunder the 1991 NEHRP and the 1992 AISC Seismic Provi-sions are based on a seismic hazard criteria, Table 4, thatreflects the relationship between the use of the building andthe level of earthquake to which it may be exposed. Thisrelationship primarily reflects concern for life safety and,therefore, the degree of exposure of the public to the hazardbased on a measure of risk.
The purpose of the NEHRP seismic ground accelerationmaps and corresponding seismic hazard exposure groups isto provide the means for establishing the measure of seismicrisk/performance for a building of any use group, and in anyarea of the United States, Table 5.
Seismic performance design requirements get progres-sively more stringent as the categories proceed from Athrough E. The seismic hazard exposure groups listed in
Table 4 are defined in detail, with examples of buildings ineach type, in ASCE 7-93.17
The most frequently used load combinations given in theLRFD Specification are repeated in the AISC Seismic Provi-sions publication in order to reduce the amount of cross-referencing by the engineer. The load combinations, Table 6,have been modified to be consistent with the anticipatedASCE 7-93 document.
The most notable modification is the reduction of the loadfactor applied to the earthquake load, E, to 1.0. This resultsfrom the limit states load model used in ASCE 7-93. Theearthquake load and load effects E in the ASCE 7-93 arecomposed of two parts. E is the sum of the seismic horizontalload effects and one half of A times the dead load effects. Thesecond part adds an effect simulating vertical accelerationsconcurrent to the usual horizontal earthquake effects. Anamplification factor to earthquake load E of 0.4R is pre-scribed. The amount of this amplification was assumed to betwo times the deflections generated by forces specified for abuilding with R = 5. This amplification factor is thus 2R / 5or 0.4R. The added complication that would be required toconsider orthogonal effects with the amplified force is notdeemed necessary.
Base Shear and the R Factor
The equivalent lateral force procedure for a Special MomentResisting Frame is greatly influenced by the R or Rw factor, anumerical coefficient commonly referred to as a responsemodification factor.
NEHRP UBCV = CsW V = ZICW / Rw
Cs = 1.2AvS
RT2⁄3
C = 1.25S
T2⁄3
For Map Area 7, Av = 0.4 Seismic Zone 4, Z = 0.4Soil/Site Coefficient Importance Factor I = 1.0 S = 1.0 Site Coefficient S = 1.0
V = 1.2(0.4)SW
8T2⁄3
V = 0.4(I)1.25(S)W
(12)T 2⁄3
V = 0.06W V = 0.04W (for equal T values)
Basically, the 50 percent difference in the base shear valuesis due to the different response modification factor, R, usedby the Uniform Building Code and NEHRP. The R valuedepends on the degree to which the system can be allowed togo beyond the elastic range, its energy dissipation in so doing,and the stability of the vertical load carrying system duringinelastic response due to maximum expected ground motion.It is recognized that the assigned R values must be peri-odically reviewed as earthquake performance is observed andmore data on material and system performance becomesavailable.18
Under NEHRP design provisions, the design of a structure
Table 4.Seismic Hazard Exposure Groups
Group III Buildings having essential facilities that are necessary for post-earthquake recovery and requiring special requirements for access and functionality.
Group II Buildings that constitute a substantial public hazardbecause of occupancy or use.
Group I All buildings not classified in Groups II and III.
Table 5.Seismic Performance Categories
Value of Av
Seismic Hazard Exposure Group
I II III
0.20 ≤ Av < 0.200.15 ≤ Av < 0.200.10 ≤ Av < 0.150.05 ≤ Av < 0.100.20 ≤ Av < 0.05
DCCBA
DDCBA
EDCCA
Table 6.Load Combinations
1.4D (3-1)
1.2D + 1.6L + 0.5(Lr or S or R′) (3-2)
1.2D + 1.6(Lr or S or R′) + (0.5L or 0.8W) (3-3)
1.2D + 1.3W + 0.5L + 0.5(Lr or S or R′) (3-4)
1.2D ± 1.0E + 0.5L + 0.2S (3-5)
0.9D ± (1.0E or 1.3W) (3-6)
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION60
(sizing of members, connections, etc.) is based on the internalforces resulting from a linear elastic analysis using the pre-scribed forces. It assumes that the structure as a whole, underthe prescribed forces, will not deform beyond a point ofsignificant yield. The elastic deformations then are amplifiedto estimate the real deformations in response to the designground motion.1
Earthquake load combinations in the AISC Provision16 are:
1.2D + 0.5L + 0.2S ± 0.4R × E (3-7)
0.9D ± 0.4R × E (3-8)
The amplification factor (3Rw / 8) was derived by using thesimilar assumptions that were used in deriving the factor forASCE 7-93. The same building type with R = 5 in ASCE 7-93has a Structural System Coefficient Rw = 8 in the 1991Uniform Building Code. The deflection determined by thisRw was used as the value to be amplified by 3. Thus (3Rw /8)E.
Drift Limits
Model Codes and resource documents such as NEHRP con-
tain specific seismic drift limits, but there are major differ-ences among them, i.e., UBC drift allowable is 1⁄3 greater thanthat allowed by NEHRP for a Special Moment Frame in steel,Seismic Hazard Exposure Group I for “All other buildings”category, Table 7.19
There are many reasons for controlling story drift in abuilding. Stability considerations dictate that flexibility becontrolled. The stability problem is resolved by limiting thedrift of the building columns and the resulting secondarymoments commonly referred to as P-∆ effects. Buildingssubject to earthquakes also need drift control in order to limitdamage to partitions, emergency stair towers, exterior curtainwalls and other fragile nonstructural elements. The designstory drift limits of NEHRP take into account these needs, andin order to provide a higher performance standard for essen-tial facilities the drift limit for Seismic Hazard ExposureGroup III is more stringent than that for Groups I and II, Table4 and Table 8.
The story drift limitations of ASCE 7-93 and NEHRPprovisions are applied to an amplified story drift that esti-mates the story drift that would occur during a large earth-quake. For determining the story drift the deflection deter-
Table 7.Comparison of 1991 NEHRP and 1991 UBC Drift Limits
Single Story Buildings (Assumed to have a C = 2.75 [UBC] and a Cs = 2.5 Aa / R [NEHRP])[Z = Av]
Framing System
NEHRP UBCForce
Amplifier
UBC Drift UBC Drift NEHRP Allowable Elastic Drift Ratio of NEHRP to UBC
(0.005h or[0.04 / Rw]h)
Scaled to NEHRPby 0.91Rw / R (Delta / Cd ) SHEG I SHEG II SHEG III
Cd R Rw 0.91Rw / R I = 1.0 I = 1.25 I = 1.0 I = 1.250.025Hsx
0.020Hsx
0.015Hsx
0.010Hsx NA
0.020Hsx
0.015Hsx
Bearing Wall System Light framed w / sp CBF
4 3.5
6.54
8 6
1.121.37
0.00500.0050
0.00400.0040
0.00560.0068
0.00450.0055
0.00630.0071
0.00500.0057
0.00380.0043
0.00250.0029
1.121.05
0.840.78
Building Frame System EBF Light Framed w / sp CBF
4 4.54.5
8 7 5
10 9 8
1.141.171.46
0.00400.00440.0050
0.00320.00360.0040
0.00460.00520.0073
0.00360.00420.0058
0.00630.00560.0056
0.00500.00440.0044
0.00380.00330.0033
0.00250.00220.0022
1.371.070.76
1.030.800.57
Moment ResistingFrame System SMF Steel OMF Steel SMF Conc. IMF Conc. OMF Conc.
5.54 5.53.52
8 4.58 4 2
12 612 8 5
1.371.211.371.822.28
0.00330.00500.00330.00500.0050
0.00270.00400.00270.00400.0040
0.00460.00610.00460.00910.0114
0.00360.00490.00360.00730.0091
0.00450.00630.00450.00710.0125
0.00360.00500.00360.00570.0100
0.00270.00380.00270.00430.0075
0.00180.00250.00180.00290.0050
1.001.031.000.781.10
0.750.770.750.590.82
Average 1.03 0.77
Avg. for all moment frames 0.98 0.74
SECOND QUARTER / 1993 61
mined using the earthquake forces E is amplified by a deflec-tion amplification factor, Cd (51⁄2 for a SMF of steel) which isdependent on the type of building system.
The 1991 Uniform Building Code7 drift provisions arenumerically specific and require that story drift shall becalculated including the translational and torsional deflectionresulting from the application of unfactored lateral forces.There are no drift limits on single-story steel-framed struc-tures with low occupancies.
The AISC Seismic Provisions do not specify specific driftlimits but defer to the governing design code, stating that thestory drift shall be calculated using the appropriate loadeffects consistent with the structural system and method ofanalysis.
Ordinary Moment Frames
Ordinary moment frames (OMF) of structural steel are momentframes which do not meet the requirements for special designand detailing required of the Special Moment Frame. OMF ofstructural steel exist in all areas of seismic activity throughoutthe country, and experience has shown that this type of buildinghas responded without significant structural damage.
The 1992 AISC Seismic Provisions for OMF have beam-to-column joint requirements that allow the use of either fullyrestrained (FR) or partially restrained (PR) connections, con-trary to the Uniform Building Code. But the beam-to-columnconnection must meet one of three criteria depending onwhether it is a fully restrained (FR) or partially restrained (PR)connection:
1. If fully restrained then the connection may conform tothe requirements for SMF except that the requiredflexural strength of a column-to-beam joint is not re-quired to exceed the nominal plastic flexural strength ofthe connection
2. If fully restrained with a connection design strengthmeeting the requirements of Load Combinations 3-1through 3-8
3. If either FR or PR connections meeting all the following:
a. The design strengths of the members and connectionsshall have a design strength to resist Load Combina-tions 3-1 through 3-6.
b. The connections have been demonstrated by cyclictests to have adequate rotation capacity at a story driftcalculated at a horizontal load of 0.4R × E.
c. The additional drift due to PR connections shall beconsidered in design.
The provision requiring a demonstration of rotation capac-ity is included to permit the use of connections not permittedunder the provisions for SMF, such as top and bottom anglejoints, in areas where the additional drift is acceptable.
Column Strength
As the result of the reduction in the actual lateral forces foruse in a code elastic analysis of the structure, overturningforces are underestimated and are amplified by unaccounted-for concurrent vertical accelerations. These two load combi-nations account for these effects:
Axial compression loads:
1.2PD + 0.5PL + 0.2PS + 0.4R × PE ≤ φcPn (6-1)
where the term 0.4R is greater or equal to 1.0.
Axial tension loads:
0.9PD − 0.4R × PE ≤ φtPn (6-2)
where the term 0.4R is greater or equal to 1.0.
Table 8.Tentative Allowable Story Drift
Building
Seismic Hazard Exposure Group
I II III
Single-story buildings without equipment attached to the structural resisting system and with interior walls, partitions, ceilings, and exterior wall system that have been designed to accommodate the story drifts.
No limit 0.020hsx 0.015hsx
Buildings with four stories or less with interior walls, partitions, ceilings, and exterior wall system that have been designed to accommodate the story drifts.
0.025hsx 0.020hsx 0.015hsx
All other buildings. 0.020hsx 0.015hsx 0.010hsx
Where hsx is the story height of the story drift calculated.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION62
These load combinations are to be applied without considera-tion of any concurrent flexure forces on the member.
Column Splices
Column splices, as a minimum, must be able to transmit theprescribed design code forces, but more stringent provisionsare required for column splices in frames that due to seismicforces are required to transmit net tension forces.
The AISC Seismic Provisions require partial penetrationwelded joints that are subject to net tension to be designed forforces in excess of the code forces (150 percent of the requiredstrength) and that the column splice be located three feet fromthe beam-to-column connection.
For column splices in seismic design, using either completeor partial penetration welded joints, beveled transitions asgiven in AWS D1.1, Section 9.20,20 are not required whenchanges in thickness and width of flanges/webs occur.
The possibility of developing high tensile stresses in partialpenetration welded column splices during a maximum prob-able seismic event is real and the use of splice plates weldedto the lower part of the column and bolted to the upper partshould be considered.
The designer should always review the conditions found incolumns in tall stories, large changes in column sizes at thesplice, or where the possibility of a single curvature exists ona column over multiple stories to determine if special designstrength or special detailing is necessary at the splice.
Panel Zone Design
Cyclic tests of beam-to-column joints has shown the ductilityof shear yielding in column panel zones.21 The usual VonMises shear limit of Fy / √3 did not accurately predict theactual panel zone behavior. Tests have shown that strainhardening and other phenomena have enabled panel zoneshear strengths in excess of 1.0Fy dt to be developed.
In calculating the required panel zone shear strength forAISC LRFD Seismic Provisions, the typical Load Combina-tions 3-5 and 3-6 are used with the nominal web shear strengthdefined as 0.6Fy dt. In order to provide the same level of safetyas determined by tests and as contained in the 1991 UniformBuilding Code, a lower resistance factor of 0.75 was selected:
φvVn = 0.6φvFy dctp 1 +
3bcf tcf2
dbdctp
(8-1)
where for this case φv = 0.75
where:
tp = Total thickness of panel zone including doublerplates, in.
dc = Overall column section depth, in.bcf = Width of the column flange, in.tcf = Thickness of the column flange, in.
db = Overall beam depth, in.Fy = Specified yield strength of the panel zone steel, ksi.
Eccentrically Braced Frames (EBF)
Whereas concentrically braced frames (CBF) are braced sys-tems whose worklines essentially intersect at points with noeccentricities, the EBF is composed of columns, beams, andbraces in which at least one end of each bracing memberconnects to a beam at a short distance (eccentricity) from abeam-to-column connection.
Research22 has shown that buildings using the EBF systempossess the ability to combine high stiffness in the elasticrange together with excellent ductility and energy dissipationcapacity in the inelastic range. In the elastic range, the lateralstiffness of an EBF system is comparable to that of a CBFsystem, particularly when short link lengths are used.
In the inelastic range, EBF systems provide stable, ductilebehavior under severe cyclic loading, comparable to that of aSMF system. The design purpose of an EBF system creates asystem that will yield primarily in the links. The specialprovisions for EBF systems are intended to satisfy this crite-rion and to ensure that cyclic yielding of the links can occurin a stable manner.
Upon publication of the first research report22 on EBF,several important applications of this design concept wereemployed in the design of major buildings. Ten years later theStructural Engineers Association of California developedrecommended seismic design provisions for the EBF whichwere accepted for inclusion into the 1988 Uniform BuildingCode. It is to be noted that the SEAOC and UBC designprovisions for EBF are essentially identical and are based onthe allowable stress design approach, whereas the NEHRPand AISC Provisions are based on the strength design ap-proach.
Eccentrically braced frames are designed so that underearthquake loading, yielding will occur primarily in the links.The diagonal braces, the columns, and the beam segmentsoutside of the links are designed to remain essentially elasticunder the maximum forces that can be generated by the fullyyielded and strain hardened links.
EBF have become a well established structural steel systemfor seismic resistant construction. Sustained research since1975 combined with experience from many buildings thatemployed the system has provided the database for properdesign of eccentrically braced frames. The most recent EBFcode provisions are contained in the 1992 AISC SeismicProvisions.14 This document represents the most up-to-dateand comprehensive code requirements for EBFs currentlyavailable in the United States.
Conclusion
Several years ago, it was not uncommon for local jurisdictionto each have their own building code. However, this did little
SECOND QUARTER / 1993 63
to promote efficient construction, nor to promote uniformsafety. Today the system has evolved to where most citiesadopt one of three model codes: the Uniform Building Code,promulgated by the International Conference of BuildingOfficials (ICBO) and used throughout the western UnitedStates; the National Building Code, promulgated by theBuilding Officials and Code Administrators International(BOCA) and used in the northeastern United States; theStandard Building Code, promulgated by the Southern Build-ing Code Congress International (SBCCI) and used in thesoutheastern United States.
Since 1957 the Seismology Committee of the StructuralEngineers Association of California has published its seismicdesign recommendations. They have also acted as an effectivebridge between seismic research and the application of theirrecommendations by assuring that the provisions wereadopted into the UBC in a timely manner. The last majorrewrite of the SEAOC recommendations occurred in 1988,which formed the basis for the seismic provisions in the 1988UBC. The SEAOC Seismology Committee is beginning thepreparation of a code change to convert the seismic provisionsin the UBC to a limit state design basis. Their goal is for acompletion time to allow the changes to be incorporated inthe 1997 UBC.
Whenever possible BOCA and SBCCI prefer adoptingdesign standards by reference.
Unfortunately, the national seismic standard adopted wasANSI A58.1 / ASCE 7, which was made up of UBC criteriathat was developed by SEAOC. The delay that results fromthis technology transfer resulted in the 1987 NBC and 1988SBC being based on 18 and 14 year old SEAOC recommen-dations respectively.
But in 1991 BOCA approved seismic code changes basedon NEHRP provisions from its 1988 publication and updatedthat to the 1991 NEHRP provisions in 1992. The SBCCIfollowed a similar path to code update and the 1993 StandardBuilding Code Supplement will contain seismic provisionsbased on the 1991 NEHRP.
Within a few months of publication of the June 1992 AISCSeismic Provisions for Structural Steel Buildings,16 bothBOCA and SBCCI approved the provisions which will bereferenced in the 1993 NBC, and will appear in the 1993Amendments to the SBC. Could uniformity in code seismicdesign criteria be just around the corner for the United States?
The provisions contained in the AISC Seismic Provisionsfor Structural Steel Buildings,14 are to be used in conjunctionwith the AISC LRFD Specification in the design of buildingsin the areas of moderate, high seismicity. The First Edition ofthe LRFD Specification was published in 1986. It did notcontain seismic design criteria.
Load and Resistance Factor Design (LRFD) is an improvedapproach to the design of structural steel for buildings. Themethod involves explicit consideration of limit states, multi-ple load and resistance factors, and implicit probabilistic
determination of reliability. The designation LRFD reflectsthe concept of factoring both loads and resistance. The LRFDmethod was devised to offer the designer greater flexibility,more rationality of design, and possible overall economy.
REFERENCES
1. BSSC, NEHRP (National Earthquake Hazards ReductionProgram), Recommended Provisions for the Developmentof Seismic Regulations for Buildings, Building SeismicSafety Council, Federal Emergency ManagementAgency, Washington, DC, 1992.
2. EQE Engineering, Inc., The Performance of Steel Build-ings in Past Earthquakes, American Iron and Steel Insti-tute, Washington, DC, 1991.
3. Krawinkler, H., and E. Martinez-Romero, 1989, ‘‘Per-formance Evaluation of Steel Structures in Mexico City,’’Lessons Learned from the 1985 Mexico Earthquake,Earthquake Engineering Research Institute.
4. Valle-Calderon, E. D. A. Foutch, and D. K. Hejelmstad,1989, ‘‘Investigation of Two Buildings Shaken During the19 September 1985 Mexico Earthquake,’’ LessonsLearned from the 1985 Mexico Earthquake, EarthquakeEngineering Research Institute.
5. Etheredge E. and Porcella, R., Strong Motion Data fromthe October 1, 1987 Whittier Narrows Earthquake, Open-file report 87-616, US Geological Survey, October 1987.
6. Pantelic, J. and Reinhorn, A., Report on the Whittier-Nar-rows, California Earthquake of October 1, 1987, Techni-cal Report NCEER-87-0026, National Center for Earth-quake Engineering Research, Buffalo, New York,November 1987.
7. ICBO, Uniform Building Code, International Conferenceof Building Officials, Whittier, CA, 1988 and 1991,
8. Hamburger, R. O., D. L. McCormick, and S. Hom, Sep-tember-October 1990. Building for Earthquake Survival,A Historic Perspective, Modern Steel Construction,AISC, Chicago, IL.
9. U.S. Geological Survey, November 1988, PreliminaryReport of Strong Ground Motion Data, October 17, 1989Loma Prieta Earthquake, Menlo Park, CA.
10. EQE Engineering, October 1989, The October 17, 1989Loma Prieta Earthquake: A Quick Look Report, SanFrancisco, CA.
11. Dames & Moore, 1989, A Special Report on the October17, 1989 Loma Prieta Earthquake, Los Angeles, CA.
13. EQE International, The Landers and Big Bear Earth-quakes of June 28, 1992, San Francisco, CA.
14. Martin, H. W. 1993, Recent Changes to Seismic Codesand Standards: Are They Coordinated or RandomEvents?, US National Earthquake Conference, Memphis,TN, May 1993.
15. Cluff, L. S. 1992, California Earthquake Hazard Reduc-
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION64
21. Slutter, R., Tests of Panel Zone Behavior in Beam Col-umn Connections, Lehigh University, Report No.200.81.403.1, Bethlehem, PA, 1981.
22. Roeder, C. W. and Popov, E. P., ‘‘Eccentrically BracedFrames for Earthquakes,’’ Journal of the Structural Divi-sion, Vol. 104, No. 3, American Society of Civil Engi-neers, March 1978.
SECOND QUARTER / 1993 65
Although there are seven metric base units in the SI system,only four are currently used by AISC in structural steel design.These base units are listed in the following table.
Quantity Unit Symbol
lengthmasstime
temperature
meterkilogramsecondcelcius
mkgs
°C
Similarly, of the numerous decimal prefixes included in theSI system, only three are used in steel design.
Prefix SymbolOrder of
Magnitude Expression
megakilomilli
Mkm
106
103
10−3
1,000,000 (one million)1,000 (one thousand)
0.001 (one thousandth)
In addition, three derived units are applicable to the presentconversion.
Quantity Name Symbol Expression
forcestressenergy
newtonpascaljoule
NPaJ
N = kg × m / s2
Pa = N / m2
J = N × m
Although specified in SI, the pascal is not universallyaccepted as the unit of stress. Because section properties areexpressed in millimeters, it is more convenient to expressstress in newtons per square millimeter (1N / mm2 = 1 MPa).This is the practice followed in recent international structuraldesign standards, including the International Standards Or-ganization (ISO), Draft International Standard for Steel De-sign,1 as well as the April 1990 draft of Eurocode 3, Designof Steel Structures, Part 1—General Rules and Rules forBuildings. It should be noted that the joule, as the unit ofenergy, is used to express energy absorption requirements forimpact tests. Moments are expressed in terms of N × m.
The following conversion factors relate traditional U.S.units of measurement to the corresponding SI units:
Multiply by: to obtain:
inch (in.)foot (ft)
pound-mass (lb)pound-force (lbf)
ksift-lbf
25.4305
0.4544.4486.8951.356
millimeters (mm)millimeters (mm)
kilogram (kg)newton (N)
N / mm2
joule (J)
Note that fractions resulting from metric conversion shouldbe rounded to whole millimeters. Following are commonfractions of inches and their metric equivalent.
Fraction, in. Exact conversion, mm Rounded to: (mm)
Bolt diameters are taken directly from the ASTM Specifi-cation A325M and A490M rather than converting the diame-ters of bolts dimensioned in inches. The metric bolt designa-tions are as follows:
Designation Diameter, mm Diameter, in.
M16M20M22M24M27M30M36
16202224273036
0.630.790.870.941.061.181.42
The yield strengths of structural steels covered in the metricLRFD Specification are taken from the metric ASTM Speci-fications. It should be noted that the yield points are slightlydifferent from the traditional values.
SI Units for Structural Steel DesignAMERICAN INSTITUTE OF STEEL CONTRUCTION, INC.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION66
ASTM DesignationYield stress,
N / mm2Yield stress,
ksi
A36M 250 36.26
A572M Gr. 345A588M
345 50.04
A852M 485 70.34
A514M 690 100.07
On the basis of the above selection of units and conversionfactors, the 1986 LRFD Specification has been translated intothe SI system. When necessary, formulas were revised tomake all coefficients nondimensional. In most instances, thiscould be achieved by explicitly showing the modulus ofelasticity, E, in the formulation.
The converted LRFD Specification is offered to the federalagencies and consultants as an interim document to facilitatedesign of metric demonstration projects. It will also serve asan introduction of the SI units of measurement to the generaldesign profession and fabricating industry. More completeinformation is available in the Metric Guide for FederalConstruction, First Edition, prepared by the ConstructionSubcommittee of the Metrication Operating Committee. Theguide is available from the National Institute of BuildingSciences in Washington D.C.*
Most designs for buildings with steel frames are based ongirders with simple connections. To eliminate the problemsassociated with traditional construction (such as deep andheavy girders and large deflections during construction)structural engineers have been searching for a new designsystem for a long time. The stub girder system is one exampleof such efforts. However, the stub girder system proved to beuneconomical for most common buildings. A new and differ-ent approach to composite steel/concrete designs was under-taken by the writer resulting in light building frames and costsavings. This new design system is called partial RestraintGirder System (“RGS”) (Figure 1). (A composite section isobtained in buildings with metal deck and concrete floors bywelding steel studs to the top flange.) With RGS two types ofrestraint are possible: the first makes use of moment connec-tions to columns; the second includes concrete reinforcement.
In buildings utilizing composite girders, deflections werecontrolled by either shoring, camber, or further increase ingirder size. RGS has arisen as a viable and cost effectivealternative.
In traditional designs, the engineer determined the build-ing’s moment diagram from a moment distribution or stressanalysis. In the RGS method, the Structural Engineer cancontrol the maximum and minimum values of moment on themoment diagram (the governing design values) from theoutset to fit his design, by establishing the amount of restraint.
Although composite girders with partial restraints improvethe moment resistance of composite girders significantly,such design is commonly ignored and the codes of practicegive no guidance as to procedures that might take advantageof the improved properties.
PAST RESEARCH
Extensive knowledge is available on non-composite girder-to-column moment connections. Reference 3 provides a goodsummary with design examples of various non-compositegirder-to-column moment connections. Composite girders
with girder-to-column moment connections is a new topic andless covered in past research.
Karl Van Dalen (Reference 4), Ammerman, and Leon(Reference 6) and others, realized the significance of com-posite girders with negative concrete reinforcement (com-posite connections). They tested specimens to determinestrength, stiffness, and ductility. Reference 6 provides a goodsummary of past research on composite girders with negativereinforcement (semi-rigid connections).
Also, Ammerman recognized the significance of semi-rigid composite connections (Reference 9) and suggested amethod for the design of frames incorporating such connec-tions. By considering the construction phase, RGS improveson the previous work. This paper describes RGS, provides themathematical formulae which describe the system and pro-vides examples of buildings designed and built with RGS.
INTRODUCTION
The traditional design for buildings with steel frames is basedon composite girders with simple connections. The disadvan-tage of this traditional design is that the entire moment re-quirement is at one portion of the girder resulting in large sizegirders. Also, girders have large mid-span deflections duringconstruction when the concrete is wet. In order to eliminatethese disadvantages, the designer specified camber or tempo-rary shoring. However, since both of these methods are costlyand difficult to implement, contractors often preferred to dowithout them and instead, increased girder sizes even further.With partial restraints, girder sizes can be decreased and thedeflections reduced.
Two different restraint types are possible, as follows:
a. Girder-to-column moment connection.b. Negative concrete reinforcement.
Neil Wexler is president, N. Wexler, P.E., P.C. ConsultingStructural Engineers.
Composite Girders with Partial Restraints:A New ApproachNEIL WEXLER
Fig. 1. Composite girder with partial restraint.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION68
The two different types of restraint are best utilized basedon the following rule of thumb: when deflections duringconstruction are large, and/or the girder sizes are governed byconstruction loads, girder-to-column connections are pre-ferred; when deflections do not govern, and the girder size isgoverned by superimposed loads, negative concrete rein-forcement bars are preferred; however excellent results areachieved when a combination of both restraint types is used.
Just like a composite girder with simple connections, thedesign of a Restraint Girder System is also done in twophases—construction phase, when the concrete is wet and thefinal phase, after the concrete hardens.
CONSTRUCTION PHASE
At this phase, the steel girder alone supports all the loads.Some steel girders with simple connections have significantmid-span deflections at this phase. Introducing end momentconnections results in reduced mid-span deflections. For ex-ample, for a beam with fixed connections, Figure 2 shows thatmid-span deflections can be reduced by as much as 58 percentif only one end is fixed and by 80 percent if both ends arefixed. These are very significant reductions. Considering thatthe end moment connections also have the added benefit ofreducing the mid-span moment, it becomes quite clear whata powerful design tool has been created.
To provide the rigidity required for this phase, the endmoment connection must be designed as a rigid connection(AISC Type 1 construction) for this stage (Figure 3). Itprovides just enough strength and rigidity to hold the originalangles between the members unchanged. During this phaseall connection components are stressed elastically. The con-
nection is strong enough to hold the original angles betweenmembers unchanged, reducing the center moment and mid-span deflections.
FINAL PHASE
At this phase, the steel girder acts compositely with theconcrete. The girder now is both strong and rigid.
Once the concrete hardens, superimposed loads such aspartitions, mechanical, ceiling, and live loads are applied. Atthis time, the moment at the girder end wants to increase,however since the connection has reached its elastic capacity,it will deform plastically. This now corresponds to AISC Type3 construction; the connection now becomes semi-rigid. Allexcess moment “shaken-off” by the semi-rigid moment con-nection is now transferred to the middle section of the girder.Since this middle section is composite with the concrete, it isboth strong and rigid. Therefore, any deflections associatedwith the final phase are small.
The design described above results in smaller girders andreduced costs. Where before A36 steel was used, governed bydeflection considerations, A572 steel now often becomesmore economical.
The sensitivity of the RGS system to deflections when theconcrete is wet and unevenly placed ought to be investigatedby the design engineer for each individual project. In somecases the RGS system should be specified with recommenda-
Figure 2 Figure 3
SECOND QUARTER / 1993 69
tions for the concrete pour sequence and the acceptablelocations of construction joints.
END-MOMENT CONNECTIONS
A cost efficient end-moment connection is an end-plate con-nection (for moderate size moments) (Figure 3a and 3b). Itperforms well as a rigid connection during the constructionphase, and a semi-rigid connection at the final phase. An endplate is a particularly good choice because not only does itdeliver forces to the column but it also reinforces the columnby spreading compression forces over larger areas, just like abearing plate, thus reducing the need for compression columnstiffeners. It is especially economical when full penetrationwelds are not required. However, other connections can alsobe used. Figure 3d shows a girder to column moment connec-tion with angles. Angles are also a good choice because thebottom flange is reinforced against local buckling by thehorizontal leg of the angle. Reference 3 and others providedesign guidelines for the design of such connections.
Increasing the connection size beyond that which providesfull fixity during the construction phase is usually not neces-sary and proves to be uneconomical. Therefore, for besteconomy, the end moment connection need not be over-de-signed. Details A and B in Figure 3 show a relatively inex-pensive moment connection. The connection shown in detailC is more expensive.
One way to evaluate a girder with moment connections isby making use of the connection moment rotation curve(Figure 4). A composite girder with partial restraint behavesjust like a steel girder with full restraint when the concrete iswet (line A). After the concrete hardens, and additional loadsare superimposed, the connection provides additional re-straint until yielding; then the girder behaves just like a simplesupported composite girder. Figure 4 shows a moment rota-tion curve with concrete reinforcement added at the joint. Theconnection curves shown are diagrammatic and in realityyielding may occur sequentially.
In order to determine the various points on the momentrotation curve, Figure 5 is reproduced herein from Refer-ence 3.
ADDITIONAL RESTRAINT
Research done by Professors Karl Van Dalen and HernanGodoy at Queen’s University, Kingston, Ontario (Refer-ence 4) revealed that additional moment strength can beachieved at the beam-column connection if only 0.46 percentof the concrete slab area is provided as slab reinforcement.This additional strength is at least equal to the ultimatemoment capacity of the composite beam and is not influencedby the type of connection between the steel elements. Therotational capacity of the composite beam—column connec-tion is also at least equal to that of a conventional, non-com-posite rigid steel connection.
The AISC specifications for Structural Steel for Buildings(Reference 2) allows the calculations of the negative designmoment strength based on the plastic stress distribution of thecomposite section, provided that the following are met:
a. Shear connectors are located in the negative momentregion.
Fig. 4. Girder-to-column connection + reinforcement.Fig. 5. Moments and end rotation for various
load/beam conditions.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION70
b. The slab reinforcement is adequately developed.c. Steel beam is compact and braced.
The designer can use this additional strength to reduce thegirder size further. Only additional studs and negative con-crete reinforcement are needed. However, in order to ensurea uniform cracking pattern in the slab in the vicinity of thecolumn, Karl Van Dalen (Reference 4) recommends that atleast twice the minimum area of steel reinforcement be ex-tended on each side of the column centerline (Figure 6).
UNBALANCED LOADS
For a long time engineers assumed that unbalanced loadsmight overstress non-composite steel girders designed withpartial restraint and therefore avoided the use of such restraintin steel buildings. RGS however is not very sensitive tounbalanced loads for two reasons:
a. Traditionally, structures have been analyzed on the basisof “frame action”—meaning beams with joints whichare allowed to rotate when subjected to unbalancedmoments. For building structures with concrete floors anadditional horizontal restraint exists. This restraint isprovided by the concrete diaphragm and is usually ig-nored by engineers. Unbalanced loads create a jointrotation; any joint rotation is associated with horizontaland vertical translations. However, in certain buildingswith girders connected to concrete floor diaphragmshorizontal translations are restrained. Therefore unbal-anced loads in buildings with concrete floors are gener-ally less able to generate joint rotation.
b. If the load is increased the reinforcement might yield.
Tests indicate that reinforced concrete has some ductilityto transfer of moments from one section to another afterfirst yielding of reinforcement. When tensile reinforce-ment reaches yield at one section it will continue to yieldwhile the section rotates. For a continuous member, theload will increase until all sections reach yield or untilthe concrete reached ultimate strain. As a result for mostconcrete structures moment redistribution is possibleand accepted by the Building Codes.
Because of the above reasons, for most common buildingsusing RGS, adequate strength is assured with unbalancedloads present if adequate concrete reinforcement is provided.
DUCTILITY
Ductility is associated with the ability of the joint to rotateafter yielding. Joint rotation can be prevented by prematurelocal or overall buckling of the bottom flange and bucklingof the web. The use of under-reinforced sections and class 1steel shapes assures adequate post-yielding rotations.
COMPOSITE STUDS
Stud-design criteria is similar to composite girders withoutrestraint, with the exception that if top reinforcement is usedfor restraint, then additional studs are required between thepoint of maximum negative moment and point of zero mo-ment. The number of such studs shall be selected to developthe negative moment (Figure 7).
ALTERNATIVES
Alternatives to composite girders with partial restraint in-clude the stub girder system, haunch girders, and simplecomposite girders. These alternatives however, are expen-
sive, complicated, deep, and might require duct openings,camber, shoring.
ADVANTAGES
The advantages of composite girders with partial restraint aremany. First, the design usually results in shallow and smallgirders with no duct penetrations. The lighter weight girdersalso have small mid-span deflections. Camber or shoring arenot required. Wind loads can be incorporated as part of the33 percent increase in allowable stresses at no additionalcosts. Medium to large spans can be accommodated. Longterm creep deflections associated with shored constructionare eliminated. The engineering analysis and design is simpleand suitable for hand calculations or computer use. All engi-neering principles involved are based on the AISC specifica-tions and in accordance with standard practice.
It is important to point out what is new about RGS. Girder-to-column moment connections, rigid or semi-rigid, are notnew; negative concrete reinforcement with composite con-struction is not new for wind loads, but it is new for gravityloads when used to reduce the girder size. (Some reportsindicate that negative reinforcement with composite con-struction has been used with the stub girder system). The useof rigid and semi-rigid construction in the same frame at thesame location is new. The use of such construction in con-junction with negative concrete reinforcement is also new.The traditional use of moment connections alone or thetraditional use of negative reinforcement alone cannot pro-vide the benefits which are created by the RGS system. Forthis reason, RGS has arisen as a viable new alternative systemfor building construction. The final result, which is a smallgirder system with controlled deflections, and without shor-ing or camber, is new.
This is a 28-story office tower with a triangular floor plan.The bay size is 30×30. The floor construction consists of 2-in.deep metal deck and 3.25-in. light weight concrete. The fillerbeams are W14×22 at 10 feet on center. The girders arecomposite, restrained, hunched girders. Both hunches andrestraints were used in order to develop the maximum possi-ble negative moment at the columns and reduce the mid-spanmoment. The straight portion of the girders was W14×30.Hunches were fabricated from 16-in.-deep sections, cutdiagonally. Restraint was obtained with end plates. This con-tinuous girder system was also used to provide additionallateral load resistance.
Contractor: Turner Construction Company, Columbus, Ohio.Steel Fabricator: Ohio Steel Fabricators
2. Minolta Office Building—Ramsey N.J. (Figure 8)
A 4-story office building with a steel frame on a concretefoundation. The building has a horseshoe footprint and120,000 sq. ft of space. The bay size is 25×30 with girdersframing the long direction. 3.25 inches of light-weight con-crete were poured over 2-in. composite metal deck. 14-in.deep composite filler beams are spaced at 10 feet on centerand span the 25-ft dimension. 14-in. composite girders withpartial restraint at columns support all gravity and wind loads.Simply supported girders, 50 ksi steel, would have beenW14×53 with a 1-in. camber and 57 studs. Using partialrestraint, only W14×43 girders were used with no camber and24 studs. Many mechanical units and a continuous roof screencreated “heavy congestion” on an open web joist-framed roof.Despite the roof congestion, the total steel weight was under7.5 lbs. per sq. ft. No floor deflections or vibrations werereported. The contractor and steel fabricator reported easyfabrication and construction details.
Steel fabricator: Mulach Steel, Pennsylvania
Fig. 8. Part plan—Minolta Office Building.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION72
3. Parking/Retail Building—New York, N.Y. (Figure 10)
This is a 45,000 sq. ft building with retail on the groundfloor, parking in the basement, and a playground/communityarea on the roof. The bay size varies; filler beams are 18 to 22feet long and the girders 17 to 35.5 feet long.
A traditional design with simply supported composite gird-ers would have resulted in W18×76 or W24×62 camberedgirders. Using Partially Restraint Girders resulted in W16×45(Fy = 50) for the first exterior girder and W16×36 (Fy = 50) forinterior girders without camber. The end-moment connectionis an end plate. The shallower girders were necessary for alower overall building height. Figure 11 shows a comparison
Fig. 9. Part plan—Capitol Square Office Building.
Fig. 10. Part plan—Parking/Retail Building.
Fig. 11. Summary.
Figure 12
SECOND QUARTER / 1993 73
of different design schemes using RGS alternatives. SchemeC which includes a girder-to-column moment connection andnegative concrete reinforcement is the best design alternative.It requires no camber, yet the result is small girder sizes,economical connections, and light concrete reinforcement.
ANALYTICAL EVALUATIONS
Figure 12 shows formulas which govern the design of RGSusing LRFD. Equations 6-1 and 6-2 are based on Van Dalen(Reference 4). While Van Dalen does not address the case ofa girder-to-column moment connection alone, an increasedamount of minimum reinforcement is recommended by thewriter to control cracking and for unbalanced loads.
Equation 6-3 is based upon partial-partial restraint whenthe amount of concrete negative reinforcement required,based on loads, is less than the maximum that can be provided.
Equation 6-4 represents the maximum amount of concretenegative reinforcement that can be provided and still ensureits yielding.
A ductility factor K is provided to ensure that the concretereinforcement will yield first. This factor can vary dependingupon certain factors such as the steel girder size, the type of
girder-to-column moment connection and/or the type of bot-tom flange restraint.
The number of composite studs should be selected basedon the formulas shown in Figure 7.
Figure 13 shows tables prepared by the author for a quickdesign of RGS in a engineering office.
ADDITIONAL RESEARCH
It is recommended that resources be allocated for researchinto the following and other topics for better prediction of thebehavior of the RGS system in building structures:
1. Requirements for column stiffeners.2. Behavior under reversed loading. Wind and seismic
loading.3. Short and long term effects of shrinkage, creep, relaxa-
tion.4. Non-linear behavior of steel connections and concrete
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION74
7. Shored versus unshored construction.8. Local buckling.
CONCLUSIONS
A new girder system was presented. The system resolves theproblems associated with traditional steel construction. Re-straint Girder System (“RGS”) provides a powerful tool forreducing girder sizes and mid-span deflections. End restraintis achieved with end-moment connections and/or concretetop reinforcement. The design is especially economical whenthe end-moment connection is detailed with economy inmind. The shape of the moment diagram can be controlled bythe designer, resulting in cost savings. The new designmethod can be used with significant savings for most steelbuildings with metal deck concrete and composite action. Theresulting analysis and design method will most likely becomea favorite for steel design into the next century, creating a newclass of building structures. In the future engineers might findit practical to apply RGS knowledge to other building struc-tural components such as filler beams and also to bridges.
ACKNOWLEDGMENTS
The author would like to thank Professor Karl Van Dalen,Queens University, Kingston, Ontario for his contribution onthe subject of composite connections.
2. Manual of Steel Construction, Load and Resistance FactorDesign, First Edition, AISC, 1986.
3. Blodgett, Omer W., Design of Welded Structures, TheJames F. Lincoln Ave Welding Foundation, ClevelandOhio, 1966.
4. Van Dalen, Karl and Hernan Godoy, Strength and Rota-tional Behavior of Composite Beams—Column Connec-tions, Queen’s University, Kingston, Ontario, October 5,1982.
5. McCormac, Jack C., Structural Steel Design, LRFDMethod, Harper & Row, New York, 1989.
6. Ammerman, Douglas J. and Roberto T. Leon, Behavior ofSemi-Rigid Composite Connections, University of Minne-sota, AISC Engineering Journal, Volume 24, No. 2, 1987.
7. Fling, Russel S., Practical Design of Reinforced ConcreteStructures, John Wiley & Sons.
8. Building Code Requirements for Reinforced Concrete, ACI318-89.
9. Ammerman, Douglas James, Behavior and Design ofFrames with Semi-Rigid Composite Connections, Thesis—Degree of Doctor of Philosophy, University of Minnesota.
Figure 13c
SECOND QUARTER / 1993 75
EngineeringJournalAMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Page 77: W. Samuel Easterling and Lisa Gonzalez GirouxShear Lag Effects in Steel Tension Members
Page 90: N. Kishi, W. F. Chen, Y. Goto, and K. G. MatsuokaDesign Aid of Semi-rigid Connections for FrameAnalysis
Page 108: Atorod Azizinamini and Bangalore PrakashA Tentative Design Guideline for a New Steel BeamConnection Detail to Composite Tube Columns
3rd Quarter 1993/Volume 30, No. 3
INTRODUCTION
The non-uniform stress distribution that occurs in a tensionmember adjacent to a connection, in which all elements of thecross section are not directly connected, is commonly referredto as the shear lag effect. This effect reduces the designstrength of the member because the entire cross section is notfully effective at the critical section location. Shear lag effectsin bolted tension members have been accounted for in theAmerican Institute of Steel Construction (AISC) allowablestress design specification1 (ASD) since 1978. The 1986 loadand resistance factor design specification2 (LRFD) and the1989 ASD specification3 stipulate that the shear lag effectsare applicable to welded, as well as bolted, tension members.
Past research on the subject of shear lag has focusedprimarily on bolted tension members. Recently, more atten-tion has been given to welded members, evident by theirinclusion in the AISC specifications. Shear lag provisions forwelded members were introduced into the specifications pri-marily because of a large welded hanger plate failure.8 Tomaintain a uniform approach to both welded and boltedmembers, the same provisions for shear lag in bolted mem-bers were applied to welded members. Additional require-ments for welded plates were added. However, the applica-tion of the shear lag requirements to welded members hasraised several questions.
This paper examines shear lag in steel tension members inthe following context. First, the background for the currentAISC specification provisions is reviewed. Second, the re-sults of an experimental research program in which 27 weldedtension members were loaded to failure is presented. Third,based on the first two parts of the paper, recommendedchanges to the AISC specifications are presented.
BACKGROUND FOR CURRENTDESIGN PROVISIONS
Bolted Connections
The shear lag provisions in the current AISC specifications2,3
are based on work reported by Chesson and Munse.6,11 This
work included experimental tests of riveted and bolted ten-sion members conducted by Chesson and Munse and a reviewof experimental tests by other researchers. Chesson andMunse6 defined test efficiency as the ratio, in percentage, ofthe ultimate test load to the product of the material tensilestress and the gross area of the specimen, and used this ratioto evaluate the test results. Several factors influence the testefficiency of connections failing through a net section: the netsection area, a geometrical efficiency factor, a bearing factor,a shear lag factor, and a ductility factor.
The data base Chesson and Munse gathered included teststhat failed in a variety of ways, including rupture of the netsection, rivet or bolt shear, and gusset plate shear or tear-out.However, only tests exhibiting a net section rupture, approxi-mately 200, were included in the validation of the tensionmember reduction coefficients. Munse and Chesson seldomobserved efficiencies greater than 90 percent and thereforerecommended, for design use, an upper limit efficiency of 85percent.11 Chesson5 reported on two additional studies thatrecommended maximum efficiencies of 0.75 and 0.85.
Fourteen of the 30 tests conducted by Chesson and Munse6
failed by net section rupture. Nine of the 14 tests failed at loadlevels exceeding the gross cross section yield load. Testsreported by Davis and Boomslitter7 were used in the overalldata base and also exhibited net section failures at load levelsexceeding gross section yield. References to other tests aregiven by Chesson and Munse.
Research reported prior to 1963 indicated that shear lagwas a function of the connection length5 and the eccentricityof the connected parts.7 Combining previous research resultswith their own investigation of structural joints, Munse andChesson11 developed empirical expressions to account forvarious factors influencing the section efficiency. The twomost dominant parts of their formulation were the net sectioncalculation, which accounts for stagger of the fasteners, andthe shear lag effect. The shear lag expression is given by
U = 1 − x_
l(1)
where
U = shear lag coefficientx_
= connection eccentricityl = connection length
An AISC Task Committee concluded from a review ofMunse and Chesson’s results that the recommended design
W. Samuel Easterling is associate professor in the Charles E.Via, Jr. Department of Civil Engineering, Virginia PolytechnicInstitute and State University, Blacksburg, VA.
Lisa Gonzalez Giroux is staff engineer, Hazen and Sawyer,P.C., Raleigh, NC.
Shear Lag Effects in Steel Tension MembersW. SAMUEL EASTERLING and LISA GONZALEZ GIROUX
THIRD QUARTER / 1993 77
procedure could be simplified.10 The simplification is in theform of coefficients given in the AISC Specifications.2,3 Al-though the work of Chesson and Munse included the effectsof several factors on the net section efficiency, the AISCspecifications only account for the two dominant factors, netarea and shear lag. The commentaries of both specificationsinclude Equation 1 as an alternate approach for determiningthe shear lag coefficients. The calculation of the effective netarea, Ae, incorporates the shear lag coefficient and is given by
Ae = UAn (2)
where
An = net area
Welded Connections
In 1931 the American Bureau of Welding published the resultsof an extensive study in which safe working stresses for weldswere determined. The American Bureau of Welding was anadvisory board for welding research and standardization ofthe American Welding Society (AWS) and the National Re-search Council Division of Engineering.4 The study was acollaborative effort between three steel mills, 39 fabricators,61 welders, 18 inspectors, and 24 testing laboratories. Severalspecimen configurations were used in the test program andwere assigned a series designation, e.g. 2400, 2500, etc.,based on the configuration. Those directly applicable to thisdiscussion consist of flat plate specimens, welded eitherlongitudinally or both longitudinally and transversely. Bothsingle and double plate tension specimens, as shown in Fig-ure 1, were tested in the research program.
Most of the tests in the AWS program failed through thethroat of the weld; but several of the specimens rupturedthrough the plate. The tests that ruptured are the ones appli-cable to the study described here. Key results from these testshave been taken from the report and are presented in Table 1.Figure 2 is a plot of the results in terms of plate thickness vs.experimental shear lag coefficient (efficiency), Ue.
Several trends are apparent in Figure 2. First, as the platethickness increases, the scatter in the data tends to increase,with the average experimental shear lag coefficient increasingslightly. This trend appears to hold except for the 5⁄8-in. group,which shows the least scatter, although this is the group withthe smallest number of tests.
Second, the amount of scatter in the 3⁄4-in. group is unexpect-edly high. There are groups of tests in which specimens havevirtually identical details, yet the results vary by as much as 30percent. For instance, consider the two 3⁄4-in. specimens in series2200. The specimen details are nearly the same, yet the experi-mental efficiency varies from 0.69 to 1.03. Likewise, the 3⁄4-in.specimens of series 2400 had very similar details, but the experi-mental efficiencies varied from 0.65 to 0.94.
A number of factors may have caused the scatter, includingvariation in the quality of the welds. An interesting observa-
tion pertaining to the issue of weld quality was made whilereviewing the AWS report. The last column of Table 1 is acode used in the report to indicate the welding process (arc orgas), fabricating shop, welder, and mill that supplied the steel.Nine specimens, all of which were arc-welded, failed at anefficiency less that 0.80. Most of these specimens had com-panion specimens, which had similar fabrication details, yetthey exhibited test efficiencies well above 0.80. A hypothesisthat welding techniques, which may have created gouges ornotches in the base material, caused the scatter in the data wasformed by the authors of this paper. This seems plausiblebecause the nine tests with efficiencies below 0.80 werefabricated in two shops, by three welders, using steel fromtwo mills (3 heats), and seven of those were welded in thesame shop by two welders. Unfortunately, this hypothesiscannot be confirmed for tests conducted more than 60 yearsago.
The results of the AWS research were considered in thedevelopment of the AISC specification provisions accountingfor shear lag in welded members. However, as will be pre-
Fig. 1. AWS test specimen configuration.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION78
sented later in this paper, questions have arisen regarding theapplication of the provisions to welded members. A researchprogram was initiated to address the questions. The remainderof this paper presents the results of the research program.
RESEARCH PROGRAM FOR WELDEDTENSION MEMBERS
This section of the paper summarizes a research projectconducted at Virginia Tech focusing on the application ofshear lag specification provisions to welded tension mem-bers, presenting both experimental and analytical results. Theexperimental program included tests of 27 welded tensionmembers, along with the associated tensile coupon tests.Analytical studies included elastic finite element analyses of
the experimental specimens, as well as a review of the AISCspecification provisions pertaining to shear lag.
Description of Experimental Specimens
Each test specimen consisted of two members welded back-to-back to gusset plates, as shown in Figure 3. The gussetplates were then gripped in a universal testing machine andpulled until failure. Use of double members minimized thedistortion due to the out-of-plane eccentricity, however,eccentric effects were ignored in the design of the testspecimens.
Three types of member were tested: plates, angles, andchannels. Fillet weld configurations used for each membertype, except the plates, were longitudinal, transverse, and a
a. All welds nominally 3⁄8-in.; measured variation between 3⁄8 and 1⁄2-in.
b. Proc—welding process; A = arc welding G = gas weldingFab—fabricator designationWeld—welder designation (within particular fabricating shop)Mill—mill designation for steel supply
THIRD QUARTER / 1993 79
combination of both longitudinal and transverse. For theplates, two different lengths of longitudinal weld and a com-bination of longitudinal and transverse welds were used.
For a given specimen configuration, three nominally iden-tical tests were conducted; specimens with only transversewelds were the exception. Calculations indicate that tensionmembers connected with only transverse fillet welds willalways fail through the welds. For the purpose of confirmingthe calculations three specimens were fabricated with onlytransverse welds. Details of the specimens are given inTable 2. Test designations in Table 2 indicate the type ofmember (P = plate, L = angle, C = channel), weld configura-tion (L = longitudinal, T = transverse, B = longitudinal and
transverse) and specimen number for a given member typeand weld configuration. For instance, test designation P-B-2is a plate specimen with both longitudinal and transversewelds and is the second test in that particular group. Anadditional number appears in the weld designation for someof the plate specimens (e.g. P-L2-3). This is because thelongitudinal weld lengths were varied in some of the platespecimens that were fabricated with only longitudinal welds.
In an attempt to ensure net section failures in the members,all welds, except the transverse welds, were designed to have10–15 percent greater strength than the gross section tensilestrength of the member. The width and thickness of theconnected member elements prevented oversizing of thetransverse welds. Welds were balanced by size for all anglespecimens, except L-B-1a, with the longitudinal weld lengthsbeing equal on each specimen. Specimen L-B-1a was unbal-anced with the two longitudinal welds being the same sizeand length.
Strain gages were used in one of the tests for each membertype to study the stress distribution near the critical section ofthe member and the distribution of stress in the member alongthe length of the connected region. A displacement transducerwas used to monitor the overall cross head movement. Thismeasurement is only of qualitative value since it includes anyslip between the specimen and the testing machine grips.Each specimen was whitewashed before testing to permit theobservance of qualitative yield pattern formation. Completespecimen details are reported by Gonzalez and Easterling.9
Two aspects of the authors’ research program should bekept in mind while reviewing the following results. The firstis that the number of tests was limited, compared to the manytests available for consideration when the shear lag provisionswere developed by Munse and Chesson. Second, the membersizes used to fabricate the test specimens were small. Thecapacity of the testing equipment available at the time thetests were conducted limited the member sizes. There areundoubtedly size effects that the results of this study do notreflect. However, the same can be said of the data base thatforms the basis for the current shear lag specification provi-sions, and thus the results of this study can be consideredsimilar to the bolted and riveted test results.
Description of Analytical Models
Linear elastic finite element analyses were performed forexperimental test specimens using ANSYS, a commercialfinite element analysis package.12 None of the transversewelded members were analyzed.
A two-dimensional, four node, isoparametric plane stresselement was used to model the plate specimens. The anglesand channels were modeled using three-dimensional, fournode, quadrilateral shell elements. Linear elastic spring ele-ments simulated the welds. The spring force constant for theweld elements was determined using a calibration procedure.Only the members and the welds were modeled elastically;Fig. 3. Test specimen configuration.
Fig. 2. Plate thickness vs. experimental shear lagcoefficient for AWS tests.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION80
the gusset plates were considered to be rigid. Interface, or gap,elements were used to prevent the member from displacinginto the gusset plate. A typical finite element mesh andboundary conditions for a model of an angle specimen areillustrated in Figure 4. Plate and channel models were con-structed in a similar manner. Only results from the platemodels are presented in this paper. Results from other analy-ses are reported by Gonzalez and Easterling.9
The stiffness for the weld elements was determined bycalibrating a model of a plate with 5-in. longitudinal welds tothe corresponding experimental specimen. In the calibration
process, the model was analyzed with eight different weldstiffness values, ranging from 100 to 5,000 k/in. The com-pletely rigid case was also considered. The remaining analyti-cal stresses and displacements were then compared to thoseobserved experimentally. A spring constant of 350 k/in. pro-vided the best correlation between the analytical and experi-mental stresses. The calibration weld size was 3⁄16-in. Thespring constant for other weld sizes were determined assum-ing a linear relationship between the shear stiffness of theweld and the spring constant. All models contained the samenumber of spring elements per linear inch of weld.
a. See Figure 3.b. Gross area based on measured cross section dimensions.
THIRD QUARTER / 1993 81
General Results
In all tests with cross section ruptures, the failure occurredafter the cross section yielded. The yielding was qualitativelyobserved by flaking of whitewash and quantitatively ob-served in the instrumented specimens from strain readingsand in all specimens from load cell readings, which exceededthe yield load. Ideally, specimens used to determine shear lagcoefficients would rupture at the critical section prior toyielding on the gross cross section. Shear lag coefficientsdetermined from tests in which yielding occurs on the grosscross section prior to rupture on the net cross section maydiffer from those determined from tests that do not yield priorto rupture. This hypothesis has not been verified in the studyreported here, nor in past studies. As indicated in the reviewof past research, this limitation was also present in most ofthe tests conducted as part of the research reported by Ches-son and Munse.6
Yield lines, indicated by flaking of the whitewash, gener-ally were not observed within the directly connected portionof the members. In some instances, the portion of the crosssection that was not connected, e.g. outstanding angle leg,showed indications of yielding. Yielding mostly occurred inthe portion of the member between the welded ends.
Experimental results are given in Table 3. The experimen-tal shear lag coefficients, Ue, were calculated as the ratio ofthe failure load to the rupture strength (gross area × tensilestress). The shear lag coefficients for the specimens that didnot exhibit rupture at the critical cross section can be taken atleast equal to those shown in Table 3. The values for thesetests do not explicitly represent shear lag coefficients becauserupture was not the controlling limit state. Calculated shearlag coefficients, Ut, were determined using Equation 1, exceptfor the plate specimens and the transversely welded speci-mens. Coefficients for the plate specimens were determined
from the current AISC specifications,2,3 which give the coef-ficients according to:
a. If l ≥ 2w . . . . . . . . U = 1.0b. If 2w > l ≥ 1.5w . . . . U = 0.87c. If 1.5w > l ≥ w . . . . U = 0.75
where
w = plate width (distance between welds)
The shear lag coefficient for longitudinally welded plates canalso be calculated using Equation 1 with each half of the platetreated independently. Therefore, x
_ would be one-fourth of
the plate width. These values are not shown in Table 3.Coefficients for the transversely welded members were cal-culated as the ratio of the area of the directly connectedelements to the gross area. This is also an AISC specificationprovision. The calculation procedure for the shear lag coeffi-cients is deemed acceptable if the ratios of experimental tocalculated shear lag coefficients, given in Table 3, fall withina 10 percent scatter band, i.e. 0.9 to 1.1. A similar evaluationwas made for bolted and riveted tests reported by Munse andChesson.11
Plate Specimens
Results are summarized in Table 3 and as indicated, the platetests can be divided into three groups according to the speci-fication shear lag coefficients of 0.75, 0.87, and 1.0. (Valuescomputed using Equation 1, as described in the previousparagraph, are 0.82, 0.85, and 1.0.) Two of the groups haveonly longitudinal welds and one has both transverse andlongitudinal welds.
The plate specimens exhibited tearing across the memberat the critical section, which was at the end of the welds.Yielding in the plates was first observed at the critical crosssection at the end of the welds. None of the plate specimensdisplayed significant out-of-plane effects. For all of the platespecimens the ratio of Ue / Ut was greater than 0.9. Six of thenine tests have values of Ue / Ut greater than or equal to 1.1.Note that a failure load was not obtained for Test 1 becausethe testing machine capacity was exceeded. However a shearlag coefficient of 0.92 is reported. This represents the maxi-mum load applied, and the true coefficient would have beengreater.
Longitudinal strains were recorded across the width of theinstrumented plate specimens near the end of the welds. Thestrain gage locations for Test 3 are shown in Figure 5. Strainswere converted to stresses and distributions plotted along thecritical section. The stress distribution at various load levelswithin the elastic range of material behavior for Test 3 isillustrated in Figure 6.
Note the unsymmetrical stress distributions shown in Fig-ure 6(a), most likely caused by imbalance in the longitudinalwelds or eccentrically applied load. The welds were detailedfor a balanced configuration, but given the stress distribu-Fig. 4. Finite element model of typical angle.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION82
tions, they apparently were not fabricated symmetrically. Onewould expect the distributions to be symmetric if the weldswere balanced and the load applied concentrically.
Figure 7 is a plot of the stress distributions at approximatelythe critical section for Tests 3, 6, and 8. The strain gages werewithin 0.5 in. of the critical section. Note that both specimenswelded only longitudinally exhibited unsymmetric stress dis-tributions, while the specimen that was welded with bothlongitudinal and transverse welds exhibited essentially a sym-metric distribution. Assuming that the stress distributionwould be symmetric if the welds were balanced, then theexperimental stresses may be modified to permit an evalu-ation of the influence of the longitudinal weld length. Figure 8
is a plot of the distributions in which the symmetric strainreadings, e.g. gages 1 and 5 and gages 2 and 4, were averagedprior to converting the values to stresses.
In Figure 8, the three tests show similar distribution pat-terns, but with varying magnitudes of stress. The higheststresses occurred for Test 8, which had 3-in. longitudinalwelds along with a transverse weld. Test 3, which had 41⁄4-in.longitudinal welds, exhibited lower stresses than Test 6,which had 5-in. longitudinal welds. The variations in thestresses for Tests 3 and 6 ranged between 3 and 7 percent,while the stresses for Test 8 were 5–7 percent higher thanthose for Test 6.
The analytical, based on finite element analyses, and ex-
Table 3.Experimental Results
Test No.Test
DesignationFy
(ksi)Fu
(ksi)
Failure LoadPer Member,
Pu / 2 (k)
CalculatedShear Lag
Coefficient, Ue
TheoreticalShear Lag
Coefficient, Ut Ue / Ut
1 P-L1-1a 48.4 73.2 99.0a 0.92 0.75 1.23
2 P-L1-1b 51.9 73.0 53.7 0.94 0.75 1.25
3 P-L1-2 51.9 73.0 56.0 0.98 0.75 1.31
4 P-L1-3 51.9 73.0 57.5 1.00 0.75 1.33
5 P-L2-1 51.9 73.0 55.9 0.98 0.87 1.13
6 P-L2-2 51.9 73.0 55.8 0.98 0.87 1.13
7 P-L2-3 51.9 73.0 54.4 0.96 0.87 1.10
8 P-B-1 51.9 73.0 51.2 0.90 1.00 0.9
9 P-B-2 51.9 73.0 56.1 0.99 1.00 0.99
10 P-B-3 51.9 73.0 55.7 0.97 1.00 0.97
11 L-L-1 54.1 81.1 50.0 0.81 0.87 0.93
12 L-L-2 54.1 81.1 50.5 0.82 0.87 0.94
13 L-L-3 54.1 81.1 50.4 0.82 0.87 0.94
14 L-B-1a 47.8 71.3 98.7 0.82 0.80 1.03
15 L-B-1b 54.1 81.1 49.5b — 0.81 —
16 L-B-1c 54.1 81.1 50.0 0.80 0.81 0.99
17 L-B-2 54.1 81.1 46.2 0.75 0.81 0.93
18 L-B-3 54.1 81.1 48.8 0.80 0.81 0.99
19 L T-1 47.8 71.3 55.8b — 0.59 —
20 C-L-1 57.0 75.5 87.0c 0.89 0.91 0.98
21 C-L-2 57.0 75.5 86.7c 0.90 0.91 0.99
22 C-L-3 57.0 75.5 86.9c 0.91 0.91 1.00
23 C-B-1 55.7 76.6 85.1c 0.92 0.91 1.01
24 C-B-2 55.7 76.6 84.0c 0.92 0.91 1.01
25 C-B-3 56.0 77.1 83.1 0.88 0.91 0.97
26 C-T-1 58.5 77.6 60.0b — 0.44 —
27 C-T-2 51.1 73.8 32.3b — 0.49 —
a. Testing machine capacity exceeded.b. Weld failure.c. Gross cross section failure away from welds.
THIRD QUARTER / 1993 83
perimental stresses for Tests 3, 6, and 8 can be compared withFigures 9–11. The trend for each of the three cases is similar.The experimental stresses near the center of the plate areapproximately the same, or somewhat less, than the calcu-lated stresses. Experimental stresses nearer the edge of theplate are greater than the calculated stresses. This trend mayhave been caused by the stopping or starting of the weldingprocess, causing imperfections at the critical section in theform of gouges or notches due to blow out. These imperfec-tions would result in stress concentrations adjacent to the edgeof the plate, which would in turn cause yielding earlier in theloading process. Stress concentrations caused by the imper-fections at the critical cross section were not considered in thefinite element model.
The experimental results for the plate specimens indicatedthat the longitudinal weld length appears to not influence therupture strength based on shear lag effects. This observationwas reinforced by the elastic finite element results whichshowed virtually the same stress distribution at the criticalsection for models in which the weld length is 3, 41⁄4 and 5 in.Neither the stress distribution at the critical section nor theexperimental shear lag coefficient were significantly affectedby the addition of the transverse weld, as compared to thespecimens with only longitudinal welds. However, note thatthe differences in longitudinal weld length were relativelysmall.
Angle Specimens
All but two of the angle specimens exhibited a tearingfailure, with the tearing initiating at the welded toe. Thewelds sheared in Tests 15 and 19. The outstanding legs ofthe specimens generally exhibited more signs of yieldingat the critical section, evident by whitewash flaking, thanthe area of the angles directly connected to the gusset
Fig. 5. Strain gage locations for Test 3.
Fig. 6. Experimental stresses for Test 3.Fig. 7. Experimental stresses at the critical section
for Tests 3, 6, and 8.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION84
plates. This behavior was observed in both longitudinallywelded specimens and specimens with a combination oflongitudinal and transverse welds, and was attributed to com-bined stress caused by out-of-plane eccentricity. Yieldinggenerally was first visible at the heel of the angle in theconnected leg.
Significant out-of-plane bending occurred in the speci-mens fabricated with 2L4×3×1⁄4. Bending in the plane of theconnected leg also occurred in Test 14. The welds for thisspecimen were not balanced, nor was the centerline of theangle coincident with the centerline of the end plates, i.e. theline of load application. Negligible out-of-plane bending wasobserved in the specimens fabricated with 2L2×2×3⁄16.
The ratios of Ue / Ut, given in Table 3, for the angle speci-mens vary from 0.93 to 1.03, the majority of the values being
less than or equal to 0.99. These results indicate that thecalculated shear lag coefficients compare well to the experi-mental results for this group of tests. However, it is interestingto note that the experimental values for all but one test werebetween 0.80 and 0.82, while the calculated values rangedbetween 0.80 and 0.87. The increased length of the welds forTests 11–13 did not affect the shear lag coefficient as ex-pected. As with the plate tests, the addition of the transversewelds did not affect the maximum loads, or shear lag coeffi-cients, for the angle tests.
Channel Specimens
The predominant limit state observed in the channel tests wasrupture in the cross section away from the welded region.Each specimen in the series of longitudinally welded chan-
Fig. 8. Modified experimental stresses at the critical sectionfor Tests 3, 6, and 8.
Fig. 11. Analytical vs. experimental stresses for Test 8.
Fig. 10. Analytical vs. experimental stresses for Test 6.
Fig. 9. Analytical vs. experimental stresses for Test 3.
THIRD QUARTER / 1993 85
nels, and all but one of the longitudinally and transverselywelded channels, failed in the center of the specimen. Thiswas attributed to the combined state of stress induced in themembers by the out-of-plane load eccentricity.
Initial yielding was generally concentrated in the channelflanges and near the web-flange intersection. In the weldedarea, as with the angles, yielding was visible in the outstand-ing flanges while none was present in the directly connectedweb. The propagation of yielding into the channel flanges wasattributed to the combined axial stress and bending stress dueto out-of-plane eccentricity.
Note in Table 3 that all the experimental shear lag coeffi-cients for the channels ranged between 0.88 and 0.92. How-ever, the predominant limit state was rupture of the gross crosssection approximately halfway between the two ends of thespecimen, and not rupture of the net section. These resultsagree with the observed practical upper limit of 0.9 thatChesson and Munse11 identified from their studies. Due toeccentricities and fabrication imperfections in welded speci-mens, an upper limit of 0.9 for the shear lag coefficientappears prudent. As with the plate and angle specimens, themaximum loads and experimental shear lag coefficients werenot affected by the addition of a transverse weld.
SPECIFICATION REVIEW ANDRECOMMENDED REVISIONS
The AISC design specifications for shear lag pertained onlyto bolted or riveted connections prior to the inclusion ofwelded members in the 1986 LRFD Specification,2 and sub-sequently in the 1989 ASD Specification.3 Welded membersare treated similar to bolted members to maintain continuityin the specifications. However, the provisions for weldedmembers are not clear in all instances and have thereforeraised questions regarding their application. A review of thequestions and related issues, along with recommendedchanges to the specification, are presented in this section ofthe paper.
Although the specification indicates that welded membersare subject to shear lag reductions, there is no minimum weldlength criterion to distinguish between different coefficientvalues in Chapter B3. The first set of subparagraphs a, b, andc in section B3 identify a minimum fastener length indirectly,by specifying a minimum number of fasteners, in the directionof stress for bolted or riveted connections but not for weldedconnections. In fact, because welding is not mentioned insubparagraphs a, b, or c, while bolting and riveting are, it isunclear that the definitions apply to welded members. Nev-ertheless, these sections are intended to be applicable towelded specimens.
Another unclear portion of the specification pertains to theuse of members with both transverse and longitudinal welds.A specification provision is given for members connected byonly transverse welds. If the first group of subparagraphs a,b, and c are assumed to apply only to members with longitu-
dinal welds, then no provisions exist for cases in which acombination of longitudinal and transverse welds are used.Results of this study indicate that the addition of a transverseweld does not significantly affect the rupture strength com-pared to a specimen with only longitudinal welds.
The shear lag provision for members welded only withtransverse welds specifies that the effective area shall be thearea of the connected element. Reviewing the limit states ofweld shear and shear lag, summarized in the Appendix of thispaper, indicates that weld shear will always control thestrength if fillet welds are used. If partial- or full-penetrationwelds are used then the present specification provision isappropriate.
According to the specification commentary, previous re-search4 determined that plates welded only longitudinally canfail prematurely due to shear lag if the distance between thewelds is too great. Thus, a minimum weld length equal to theplate width or distance between the welds, w, is required.Currently, the specification does not consider shear lag alimiting factor as long as the weld length is greater than twicethe plate width. Two shear lag coefficients are specified forintermediate ranges of longitudinal weld length between wand 2w. Results from this study indicate that the weld lengthsgreater than the distance between the welds have little influ-ence on the shear lag coefficient. However, due to the limitednumber of tests conducted and to the small size of the mem-bers, no modifications are recommended to the shear lagcoefficients for longitudinally welded plates.
Reviewing the AWS4 results, along with the statement byMunse and Chesson11 that efficiencies greater than 90 percentare seldom observed, an upper limit for U of 0.9 is deemedappropriate. This is also consistent with the upper limit thatappears in the current specifications2,3 in section B3 subpara-graph a. The strength of welded tension members is reduceddue to the coupled effects of shear lag, stress concentrations,and eccentricities. The stress concentrations are due to thesudden change in stiffness caused by the presence of the weld,or to notches or gouges created at the critical section by thewelding process. Although all play a role in reducing thestrength, it is difficult to determine the relative participationof each component. Using an empirical approach, such as theshear lag coefficient, is an approximate way to account for allthe effects.
Recommended Revisions to the Specification and Commentary
Recommended revisions to the specification were developedjointly by the authors of this paper and the AISC Task Com-mittee 108—Connections and Force Introduction. The rec-ommended changes address all of the issues identified in theprevious section. All of the recommended changes apply tosection B3 of the AISC Specifications.2,3
Subparagraphs a, b, and c that follow the line “Unless alarger coefficient can be justified by tests or other rational
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION86
criteria....” should be replaced with a single equation for U,given by:
U = 1 − x_
l ≤ 0.9 (3)
The specific values of U, given for certain groups of sectionsin subparagraphs a, b, and c, are acceptable for use in lieu ofthe values calculated from Equation 3 and may be retained inthe commentary for continued use by designers.
The section that addresses sections only connected withtransverse welds should be modified to include all shapes, notjust W, M, or S shapes and structural tees cut from theseshapes. A provision should be added to indicate that thissection is only applicable if partial- or full-penetration weldsare used, and is not applicable if fillet welds are used as thetransverse weld type.
In addition to changing the commentary to incorporate theinformation of subparagraphs a, b, and c, several otherchanges would help to clarify specification revisions. Severalof these are indicated in the following paragraph. Althoughthe primary focus of the research project reported in this paperwas welded tension members, literature and specificationprovisions for bolted members were reviewed. The commentsmade in the following paragraphs pertaining to bolted mem-bers are the authors’ judgment based on that review.
For any given profile and connected elements, x_ is a geo-
metric property. It is defined as the distance from the connec-tion plane, or face of the member, to the member centroid, asindicated in Figure 12. Note that the “member” may be aportion of the cross section for particular cases. Connectionlength, l, is dependent upon the number of fasteners or length
of weld required to develop the given tensile force, and thisin turn is dependent upon the mechanical properties of themember and the strength of the fasteners or weld used. Thelength l is defined as the distance, parallel to the line of force,between the first and last fasteners in a line for bolted con-nections. The number of bolts in a line, for the purpose ofdetermining l, is determined by the line with the maximumnumber of bolts in the connection. For staggered bolts, usethe out-to-out dimension for l (See Figure 13). For weldedconnections, l is the length of the member parallel to the lineof force that is welded. For combinations of longitudinal andtransverse welds (see Figure 14), l is the length of longitudinalweld because the transverse weld does not significantly affectthe rupture strength based on shear lag. The presence of thetransverse weld does little to get the load into the unattachedportions of the member.
SUMMARY AND CONCLUSIONS
The purpose of this investigation was to review the shear lagprovisions for welded tension members relative to those forbolted members, and to make recommendations for pertinentspecification changes. Experimentally, three different mem-ber types and three different weld configurations were con-sidered. Results of 27 tests were reported. Longitudinalstresses were determined analytically in a finite elementstudy. Experimental strains were determined directly from
Fig. 12. Definition of x_ for various members.
Fig. 13. Definition of l for bolted members with staggered holes.
Fig. 14. Definition of l for welded members.
THIRD QUARTER / 1993 87
tensile tests. The analytical and experimental stress patternsin the elastic region were compared. Shear lag criteria arerecommended based on the experimental results. The currentAISC provisions have been reviewed and revisions recom-mended.
The recommended revisions to the specification are basedon the results of the experimental and analytical studiesreported here, a review of the specification and judgment ofthe authors. In particular the definitions of x
_ in Figure 14 and
l in Figures 15 and 16 are based on the authors’ judgment.Further, the hypothesis that the net section failure is due to acombination of the shear lag effect and stress concentrationcaused by welding induced imperfections is also based on theauthors’ judgment, given the insight gained from the researchprogram. Each of these topics would require further study tobe proven explicitly.
The following conclusions have been drawn from theexperimental and analytical investigations:
1. Shear lag controlled the strength of the angle and platespecimens.
2. For plates connected only by longitudinal welds, con-nection length had little influence on the experimentalshear lag coefficient.
3. The transverse weld in the angle members welded bothlongitudinally and transversely did not increase theshear lag coefficient as expected. The experimentalshear lag coefficients of the longitudinally welded an-gles and the angles with both longitudinal and transversewelds were equivalent.
4. Shear lag will not control the strength of tension mem-bers connected only by transverse fillet welds. Weldshear will be the controlling limit state, regardless ofelectrode strength or fillet weld size. This conclusiondoes not apply to partial- or full-penetration welds.
5. Due to the small size of the experimental specimens inthis study as well as past studies, caution should beexercised when applying the design provisions to muchlarger tension members. There is a need for some limitedconfirmatory testing on large tension members designedso that shear lag effects control the strength.
6. The recommended upper limit for the shear lag coeffi-cient is 0.9.
7. The implementation of the recommended changes toAISC specifications and commentaries would result ina simpler, more uniform approach to the application ofshear lag provisions to bolted and welded tension mem-bers. The changes should result in fewer questions re-garding the application of the provisions.
ACKNOWLEDGMENTS
Financial support for this project was provided by the Ameri-can Institute of Steel Construction and Virginia Tech. Valu-able assistance was provided throughout the project by Nestor
Iwankiw of AISC. Steel for the test specimens was providedby Montague-Betts Co., Inc. The technical input provided bythe AISC Task Committee 108, particularly comments by J.W. Fisher, T. M. Murray and W. A. Thornton, was verybeneficial to the authors. Additionally, comments provided byD. R. Sherman proved very useful. The authors are gratefulto all of the above. Many of the specimens were tested in thematerials testing laboratory at Virginia Military Institute. Thewillingness of C. D. Buckner and D. K. Jamison to permitaccess to the laboratory made the completion of the experi-mental parts of this project possible, and for this the authorsare grateful.
APPENDIX
The strength of members welded only with transverse filletwelds will not be controlled by the rupture based on shear lageffects, but rather will be controlled by weld shear. This is trueregardless of the steel or electrode strength. This can beshown by considering the following parameters:
Ae = bt = area of connected elementE70XX electrodes (fillet welds)A36 steel
The strength of the tension member based on rupture is givenby:
φPn = φFu Ae = 0.75(58 ksi)bt (A1)
where φ for tension rupture is 0.75.
The weld strength is given by:
φRw = φ0.6FEXX Aw (A2)
where φ for weld shear is 0.75 and Rw = nominal weldresistance.
If the weld area, Aw, is taken as (0.707t)b (the maximumpossible dimension for a fillet weld made along the edge ofplate element), then φRw becomes
Comparing Equations A1 and A3, one observes that the weldstrength, Equation A3, is less and will therefore control thestrength. The same conclusion will be reached for any prac-tical combination of weld electrode and steel. If the sub-merged arc process were used, the weld strength result, Equa-tion A3, would increase, but it would still remain less thanEquation A1.
REFERENCES
1. American Institute of Steel Construction, Specificationfor the Design, Fabrication, and Erection of StructuralSteel for Buildings with Commentary, Chicago: AISC,1978.
2. American Institute of Steel Construction, Load and Re-
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION88
sistance Factor Design Specification for Structural SteelBuildings, Chicago: AISC, 1986.
3. American Institute of Steel Construction, Specificationfor Structural Steel Buildings, Allowable Stress Designand Plastic Design, Chicago: AISC, 1989.
4. American Bureau of Welding, “Report of Structural SteelWelding Committee,” American Welding Society, NewYork, 1931.
5. Chesson, E., Jr. “Behavior of Large Riveted and BoltedStructural Connections.” Thesis presented to the Univer-sity of Illinois in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy. Urbana, 1959.
6. Chesson, E., Jr. and Munse, W. H. “Riveted and BoltedJoints: Truss-Type Tensile Connections,” Journal of theStructural Division, ASCE, Vol. 89(ST1), 1963, pp. 67–106.
7. Davis, R. P. and Boomslitter, G. P. “Tensile Tests of
Welded and Riveted Structural Members,” Journal of theAmerican Welding Society, 13(4), 1934, 21–27.
8. Fisher, J. W., personal communication, ATLSS, LehighUniversity, Bethlehem, PA, 1990.
9. Gonzalez, L. and Easterling, W. S. “Investigation of theShear Lag Coefficient for Welded Tension Members,” Re-port No. CE/VPI-ST 89/13. Virginia Polytechnic Instituteand State University, Blacksburg, VA, December, 1989.
10. Kulak, G. L., Fisher, J. W. and Struik, J. H. A., Guide toDesign Criteria for Bolted and Riveted Joints, 2nd Edi-tion, John Wiley and Sons, New York, 1987.
11. Munse, W. H. and Chesson, E., Jr. “Riveted and BoltedJoints: Net Section Design,” Journal of the StructuralDivision, ASCE, Vol. 89(ST1), 1963, pp. 107–126.
12. Swanson Analysis Systems, Inc., ANSYS EngineeringAnalysis System User’s Manual, Vol. I and II. Houston,PA, 1989.
THIRD QUARTER / 1993 89
ABSTRACT
In this paper, a useful design aid for determining the valuesof the initial connection stiffness Rki, the ultimate momentcapacity Mu, and the shape parameter n of a three-parameterpower model describing the moment-rotation curve (M-θr) ofsemi-rigid connections with angles is prepared for its use inthe practical design of flexibly jointed frames with angles. Aset of nomographs allows the engineer to rapidly determinethe M-θr curve for a given connection.
Applying the design aid, numerical simulations on drift andcolumn moment of a flexibly jointed frame with angles areillustrated.
1. INTRODUCTION
The aim of the work described in this paper is to provide apractical procedure for the analysis and design of semi-rigidframes with angles. To this end, a set of nomographs allowsthe engineer to determine rapidly the values of the initialconnection stiffness Rki, the ultimate moment capacity Mu,and the shape parameter n of a three-parameter power modeldescribing the M-θr curve of connections. In this develop-ment, a data base of steel beam-to-column connections wasfirst built and simple procedures to enable engineers to assessthis M-θr behavior were then formulated. Using this data base,extensive comparisons were made with the results of tests onactual connections providing final confirmation of the valid-ity of the three-parameter power model. The model is recom-mended for general use in semi-rigid frame analysis.
In this paper, we have established a design procedure forconnections with angles. The three-parameter power modelis adopted to represent the nonlinear M-θr curve proposed
previously by Richard and Abbott (1975). The three parame-ters in this model are the initial connection stiffness Rki, theultimate moment capacity of connection Mu, and the shapeparameter n. The values of Rki and Mu can be determined bya simple mechanical procedure with an assumed failuremechanism (Kishi and Chen, 1990). Herein, we prepared auseful design aid for the values of these parameters corre-sponding to given angles and beam, or the main parametersof connection angles for given values of Rki and Mu. The shapeparameter n can be determined as a linear function oflog10θ0 (Kishi and Chen et al. 1991) which is an empiricalequation based on experimental data installed in the ProgramSCDB (Chen and Kishi 1989), where θ0 = Mu / Rki.
Using the nomographs for Rki and Mu and the empiricalequation for n, we can determine rapidly the nonlinear M-θr
curve of connections with angles. Then, using the second-or-der elastic analysis program FRAME formulated by Goto andChen (1987), we can analyze the flexibly jointed frame in asimple manner (Chen and Lui, 1991). As an illustrative ex-ample, studies of a four-bay, two-story frame with variablebeam section and/or length or thickness of connection anglesmade using this analysis are presented.
2. ASSUMPTIONS AND NOTATIONS
In this paper, four types of connections with angles areconsidered: single/double web-angle connections and top-and seat-angle with/without double web-angle connections asshown in Figures 1 to 3. To prepare the design charts for theinitial connection stiffness Rki and the ultimate moment ca-pacity Mu, the dimensions used for angle as shown in Figure 4are defined as:
t = angle thicknessk = gauge distance from heel to the top of filletl = angle lengthg = distance between heel to the center of fastener closest
to web or flange of beamW = nut widthI0 = t3 / 12 = geometrical moment of inertiaM0 = σy t
2 / 4 = pure plastic bending moment
where σy is the yielding stress of steel, and I0 and M0 the valuesper unit length of plate element of angle. We assume that topangle and seat angle have the same dimensions.
Design Aid of Semi-rigid Connectionsfor Frame AnalysisN. KISHI, W. F. CHEN, Y. GOTO, and K. G. MATSUOKA
N. Kishi is associate professor, civil engineering, MuroranInstitute of Technology, Muroran, Japan 050.
W. F. Chen is George E. Goodwin distinguished professor ofcivil engineering and head of structural engineering, School ofCivil Engineering, Purdue University, West Lafayette, IN.
Y. Goto is professor, civil engineering, Nagoya Institute ofTechnology, Nagoya, Japan 466.
K. G. Matsuoka is professor, civil engineering, Muroran Insti-tute of Technology, Muroran, Japan 050.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION90
Furthermore, we shall introduce the following non-dimen-sional parameters:
β = gl, γ =
lt, δ =
dt, κ =
kt, ω =
Wt
, ρ = twtt
in which d is the height of beam and subscripts “t” and “w”denote top angle and web angle respectively.
3. CHARTS FOR Rki and Mu INCONNECTIONS WITH ANGLES
To prepare the charts for the initial connection stiffness Rki
and the ultimate moment capacity Mu, the equations devel-oped previously by Kishi and Chen (1990) are used. All chartsare basically related to the parameter β. Extensive compari-
sons of the analytical solutions with experimental test resultscan be found in the paper by Kishi and Chen (1990).
3.1 Single/Double Web-Angle Connections
Using a simple mechanical procedure described in the papercited above, the values of Rkiw and Mu for single web-angleconnections are formulated as:
Rkiw
EI0w =
12α cos h(αβw′)
7.8 (αβw′) cos h (αβw′) − sin h (αβw′)
(1)
Muw
M0wtw =
(2ξw + 1)3
γw2 (2)
Fig. 1. Single web-angle connections.
Fig. 3. Top and seat angle with/without double web-angle connections.
(a) Top and seat angle with double web-angle connection (b) Top and seat angle without double web-angle connection
Fig. 2. Double web-angle connections.
THIRD QUARTER / 1993 91
in which α = 4.2967 and βw′ is defined as
βw′ = βw − 1γw
κw +
ωw
2
(3)
ξw in Equation 2 is obtained by solving Equation 4 whichis derived by combining the Drucker interaction equationbetween bending moment and shearing force with the Trescayielding criterion as
ξw4 + (βwγw − κw)ξw − 1 = 0 (4)
The non-dimensional initial connection stiffness as a func-tion of β′w for single web-angle connections is shown inFigure 5. The non-dimensional ultimate connection momentis shown in Figure 6 in which βw is taken as abscissa and γw
is varied from 5 to 20 with an increment of 5 for κw = 1.5and 2.0.
The case of double web-angle connections can be obtainedsimply by doubling the values found from these charts.
3.2 Top and Seat Angle Without DoubleWeb-Angle Connection
Assuming that the center of rotation is located at the angle legadjacent to the compression beam flange and the top angle actsas a cantilever beam to resist surcharged moment, the initialconnection stiffness Rkits is obtained as (Kishi and Chen, 1990).
Rkits
EI0t = (1 + δt)2Dts (5)
in which Dts is a function of βt′ and γt and those Dts and βt′ aregiven by
Dts = 3
βt′(γt2βt′2 + 0.78)
(6)
βt′ = βt − (1 + ωt)
2γt
(7)
The ultimate moment capacity Muts is obtained by assuminga simple failure mechanism. The equation for Muts is given by
Muts
M0ttt = γt
1 + ξt (1 + βt
∗ + 2(κt + δt)
where the variable ξt is a non-dimensional ultimate shearingforce acting at the plastic hinge. Here, as in the case of single
Fig. 4. Main parameters for an angle.Fig. 6. Ultimate moment capacity for
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION92
web-angle connections, it is obtained by solving Equation 9as
ξt4 + βt
∗ξt − 1 = 0 (9)
in which βt∗ is defined as
βt∗ = βt′γt − κt (10)
The distributions of the coefficient Dts for Rkits with respectto βt′ are shown in Figure 7 in which γt is varied from 5 to 40with an increment of 5. Figure 8 is the chart for the non-dimensional ultimate moment capacity Muts. Figure 8(a)shows the results with the variation of γt for δt = 40 and 80,while Figure 8(b) shows the variation of δt (10 to 80) for γt =5 and 40.
3.3 Top and Seat Angle With DoubleWeb-Angle Connection
In this type of connection, the initial connection stiffness Rki
and the ultimate moment capacity Mu can be evaluated byseparating the top- and seat-angle part and the double web-angle part as
Rki
EI0t =
Rkits
EI0t +
Rkiw
EI0t ,
Mu
M0ttt =
Muts
M0ttt +
Muw
M0ttt(11,12)
Since the top- and seat-angle part of the equations derivedabove are also applicable for the case of the top- and seat-an-gle connections, Figure 7 can be used for Rkits / EI0t and Figure8 for Muts / M0ttt in this type of connections.
As for the web angle, it acts as a cantilever beam similar tothe behavior of the top angle, the initial connection stiffness
Rkiw is related to the double web-angle connection part as(Kishi and Chen, 1990)
Rkiw
EI0t = (1 + δt)2ρDw (13)
in which Dw is
Dw = 3
2βw′(γw2βw′2 + 0.78)
(14)
where βw′ is defined the same as βt′ in Equation 7.In the limit state, choosing a simple failure mechanism of
web angle and taking moment about the center of rotation atthe angle leg adjacent to the compression beam flange, theultimate moment capacity Muw is
Muw
M0ttt = γw(1 + ξw)
ξw − 13(ξw + 1)
γw + δw + 1ρ
ρ3 (15)
in which ξw is obtained by solving Equation 4, since themechanism assumed here is the same as in the single web-an-gle connections.
The distributions of the coefficient Dw for Rkiw / EI0t aregiven in Figure 9. Muw has a total of five variables: βw, δw, γw,ρ, and κw. Taking βw as abscissa, two types of charts areprepared in this study. In the first case, γw is varied from 5 to35 and/or 40 with an increment of 5 while the values of δw,ρ, and κw are kept constant (Figure 10). In the second case,δw instead of γw is varied in a similar manner (Figure 11).Though it is easy to obtain these curves for arbitrary valuesof these parameters, we consider here only two cases ρ = 0.5or 1.0 and κw = 1.5 or 2.0.
4. DETERMINATION OF M-θθr CURVEOF CONNECTION
It is a simple matter to obtain the values of Rki and Mu for givendimensions of angle or to determine the angle dimensions forgiven values of Rki and Mu. Moreover, we must also determinethe nonlinear characteristics of connection behavior for astructural analysis (Chen, 1987). The three-parameter powermodel is adopted here to represent these characteristics ofsemi-rigid connections which is a simplification of the four-parameter power model proposed previously by Richard andAbbott (1975).
Assuming m = M / Mu, θ = θr / θ0 and θ0 = Mu / Rki andintroducing the shape parameter n, the power model used herehas the simple form
m = θ
(1 + θn)1⁄n (16)
Figure 12 shows the M-θr curves of a connection withseveral values of shape parameter n. In one extreme, if theshape parameter n is taken to be infinity, the model reducesto a bilinear curve with the initial connection stiffness Rki and
Fig. 7. Coefficient Dts for Rkits of top and seat anglewithout double web-angle connections.
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Fig. 8. Ultimate moment capacity for top and seat angle without double web-angle connections.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION94
the ultimate moment capacity Mu. The principal merit of thepresent model is a significant saving of computing time in anon-linear structural analysis program, since the connectionmoment M can be represented as a function of relative rota-tion θr. Furthermore, the tangent connection stiffness Rk canbe directly obtained without iteration.
As for the shape parameter n, we use the following proce-dure for its determination (Kishi and Chen et al. 1991):
1. The shape parameter n for each experimental test data isnumerically determined first by the least mean squaretechnique of the test data with Equation 16.
2. The shape parameter is assumed to be a linear functionof log10θ0. Using a statistical technique for n valuesobtained from the above procedure, empirical equationfor n for each connection type are determined.
Figure 13 shows comparisons of the distributions of nvalues of the empirical equation with the experimental testdata installed in the program SCDB (Kishi and Chen, 1989).From these numerical considerations, we conclude thatwithin the current practice of the range of the connectionvariables, the three-parameter power model with the shapeparameter n obtained from the empirical equation can beapplied in practical design (Kishi and Chen et al. 1991). Inthis study, we set the shape parameter n to be constant for theregion of θ0 less than the smallest one obtained from experi-mental test data. The equation refined for each connectiontype is listed in Table 1.
5. NUMERICAL EXAMPLE OF STRUCTURALANALYSIS OF FLEXIBLY JOINTED FRAME
In this study, a four-bay, two-story frame used by Lindsey(1987) is taken as the frame of basic skeleton for the presentnumerical analysis (Figure 14).
W8×24 and W8×31 sections are used as the external andinternal columns respectively and the frames are placed 25 ftcenter to center. The loads are: floor dead load: 65 psf, roofdead load: 20 psf, reduced floor live load: 40 psf, and rooflive load: 12 psf. Wind loads are assumed to be 15 psf with ashape factor of 1.3. Two types of load combination are con-sidered referring to AISC-LRFD specification (1986). One isthe unfactored loads (D+L+W) to check the drift under serviceload. Another is the factored loads (1.2D + 0.5L + 1.3W) tocheck the frame stability. Load intensities are WR = 0.80 k/ft,WF = 2.70 k/ft, PR = 2.925 kip, PF = 6.581 kip for the unfactoredloads and WR = 0.75 k/ft, WF = 2.54 k/ft, PR = 3.803 kip, PF =8.556 kip for the factored loads.
In the present study, we adopt the top and seat angle withdouble web-angle connections as a part of the beam-to-col-umn connection. The combinations of beam, column, andconnection angle for several cases are listed in Table 2 inwhich the size of the columns is constant for each case. Beamsused in Cases 1 and 2 are stronger than those in Cases 3 and 4.
Fig. 9. Coefficient Dw for Rkiw / EI0t of top and seat anglewith double web-angle connections.
Table 1.Empirical Equations for Shape Parameter n
Type No. Connection Type n
I Single web-angle connection 0.520 log10 θ0 + 2.2910.695
… log10 θ0 > −3.073≤ −3.073
II Double web-anlge connection 1.322 log10 θ0 + 3.9520.573
… log10 θ0 > −2.582≤ −2.582
III Top- and seat-angle connection(with double web angle)
1.398 log10 θ0 + 4.6310.827
… log10 θ0 > −2.721≤ −2.721
IV Top- and seat-angle connection(with double web angle)
2.003 log10 θ0 + 6.0700.302
… log10 θ0 > −2.880≤ −2.880
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Fig. 10. Ultimate moment for the variation of γw for top and seat angle with double web-angle connections.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION96
Fig. 11. Ultimate moment for the variation of δw for top and seat angle with double web-angle connections.
THIRD QUARTER / 1993 97
Heavy hex structural bolts with 1-in. nominal size are used asfasteners for all cases.
6. NUMERICAL RESULTS
6.1 Characteristics of M-θθr Curve of Connections
The M-θr curves of connections used are shown in Figure 15in which the dimensions of beams and angles are specified inCase 1 and the length of top and seat angles is six inches.
6.2 Drift of Frame in Case Surcharging Unfactored Loads
The general configurations of deformation of flexibly jointedframe under the service loads are shown in Figure 16 com-paring with the results of rigid connections, in which thesections of members are taken as lt = 6 in. and tt = 1⁄4-in. and/or1⁄2-in. as in Cases 2 and 4. Though Rki and Mu in the case oftt = 1⁄2-in. may be twice than that of tt = 1⁄4-in. as we can see inFigure 15, the drift of roof in the case of tt = 1⁄4-in. is less thantwice that of tt = 1⁄2-in. The drifts for tt = 1⁄2-in. and 1⁄4-in. are
almost two to three times than that of the result of rigidconnections, respectively.
Distributions of the non-dimensional roof drift (∆ / H) foreach case are shown in Figure 17 taking lt or gt as abscissa inwhich ∆ and H are roof drift and height of frame respectively.From this figure, we can select some dimensions for top andseat angles for a given drift. For example, if the maximumdrift is set to be ∆ / H = 1⁄300, we can choose three types ofangles to meet this requirement as
For Cases 1 and 2:
tt = 1⁄2-in. gt = 2.75 in. lt = 6 in.
and
tt = 1⁄2-in. gt = 2.50 in. lt = 5 in.
For Cases 3 and 4:
tt = 1⁄2-in. gt = 2.50 in. lt = 6 in.
6.3 Frame Stability in Case Surcharging Factored Loads
Bending moment diagrams of the frame in Cases 2 and 4 withlt = 6 in. and tt = 1⁄4-in. and/or 1⁄2-in. under the factored loadsare shown in Figure 18 together with the results of rigidconnections. The bending moments on floor beams show alarge difference between the semi-rigid and rigid connections.On the other hand, the differences on other members aresmaller than those of the floor beams.
The non-dimensional end moments of columns of Cases 1to 4 are tabulated in Table 3 together with the results of rigidconnections and the B1, B2 method as given in AISC-LRFDspecification (Chen and Lui, 1987). The reference values ineach case are obtained from a first order elastic analysis withrigid connections. In these tables, all values at the fixed pointsof flexibly jointed frame in all cases are greater than theFig. 12. M-θr curves for the three-parameter power model.
Table 2.Combinations of Beam, Column, and Connection Angles
Beam and Column Sizes:
Case 1, 2 Case 3, 4
Floor beam W18×50 W18×46
Roof Beam W14×22 W12×19
External column W8×24 W8×24
Internal column W8×31 W8×31
Top- and Seat-Angle Sizes:
Size l g l g
T & S Angles L4×31⁄2×tt 6 in. var. var. 2.5 in.
Web Angles L3×21⁄2×1⁄4 8.5 in. for floor5.5 in. for roof
1.75 in. 8.5 in. for floor5.5 in. for roof
1.75 in.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION98
Fig. 13. Comparison of the shape parameter n of the empirical equation with experimental test data.
THIRD QUARTER / 1993 99
reference value, and its maximum value is almost 3.5 timesgreater than the reference value. Alternatively, almost all ofthe bending moments at the top of the 1st floor columns areless than the reference ones and the values at Node No. 2 ofElement No. 1 in Figure 14 for some cases have an oppositesign for the reference one. The results of the flexibly jointedframe in all cases are substantially different from the results(1, 2, and 3) of rigid connections.
On the other hand, referring to the results of rigid connec-
tions, it is clear that the values (1) obtained from the secondorder elastic analysis are almost the same with the values (2)obtained from the B1, B2 method with Equation H1-5 in theAISC LRFD specification (1986).
7. SUMMARY AND CONCLUSIONS
A considerable amount of test data on semi-rigid connectionswith angles has been collected and analyzed and simplemodels developed in the past years. Against the background
Fig. 14. General view of a four-bay, two-story frame.
Fig. 15. M-θr curves of connections in Case 1.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION100
of this information, design aids for a three-parameter connec-tion model put forward recently by Kishi and Chen is pre-pared here. A set of nomographs allows the engineer to rapidlydetermine the values of initial connection stiffness Rki and theultimate moment capacity Mu for a given connection, or thebasic dimensions of angles for given values of Rki and Mu. Thegeneral validity of the design procedures based on thesedeveloped design charts is demonstrated by comparisons withcomputed displacements and moments at service loads andfactored loads of a four-bay, two-story frame with semi-rigidconnections using a second-order elastic analysis program.With the aid of these design charts, the present analysisprocedure for the design of semi-rigid frames with angles hasachieved both simplicity in use and, as far as possible, arealistic representation of actual behavior. Taking this pointin conjunction with the demonstrated validity of the ap-proach, it is recommended for general use.
REFERENCES
1. American Institute of Steel Construction, Load and Re-sistance Factor Design Specification for StructuralBuildings, Chicago, IL, 1986.
2. Chen, W. F., editor, “Joint Flexibility in Steel Frames”,Journal of Construction Steel Research, Special Issue,Vol. 8, Elsevier Applied Science, London, 290 pp, 1987.
3. Chen, W. F. and Lui, E. M., Structural Stability: Theoryand Implementation, Elsevier, New York, 486 pp, 1987.
4. Chen, W. F. and Lui, E. M., Stability Design of SteelFrames, CRC Press, Boca Raton, Florida, 380 pp, 1991.
5. Chen, W. F. and Kishi, N., “Semi-Rigid Steel Beam-to-Column Connections: Data Base and Modeling,” ASCEJournal Structural Engineering, 115(ST1), pp. 105–119,1989.
6. Goto, Y. and Chen, W. F, “On the Computer-Based DesignAnalysis for Flexibly Jointed Frames,” Journal of Con-struction Steel Research, 8, pp. 203–231, 1987.
7. Kishi, N. and Chen, W. F., “Data Based of Steel Beam-to-Column Connections,” Structural Engineering ReportNo. CE-STR-80-26, School of Civil Engineering, PurdueUniversity, West Lafayette, IN, 653 pp, 1986.
8. Kishi, N. and Chen, W. F., “Moment-Rotation Relationsof Semi-Rigid Connections with Angles,” ASCE JournalStructural Engineering, 116(ST7), pp. 1813–1834, 1990.
9. Kishi, N., Chen, W. F., Goto, Y. and Matsuoka, K. G.“Applicability of Three-Parameter Power Model to Struc-tural Analysis of Flexibly Jointed Frames,” ASCE Me-chanics Computing in 1990s and Beyond, pp. 238–242,1991.
10. Lindsey, S. D. “Design of Frames with PR Connections,”Journal of Construction Steel Research, 8, pp. 251–260,1987.
11. Richard, R. M. and Abbott, B. J., “Versatile Elastic-PlasticStress-Strain Formula,” ASCE Journal Engineering Me-chanical Division, Vol. 101, No. EM4, pp. 511–515, 1975.
NOMENCLATURE
d beam depthDts defined in Equation 6Dw defined in Equation 14g distance between heel to the center of fastener
closest to web or flange of beam (Figure 4)Io t3 / 12 = moment of inertia per unit lengthk gauge distance from heel to the top of fillet (Figure
4)l length of the angleM connection momentM0 σyt
2 / 4 = plastic bending moment capacity per unitlength
m M / Mu
Mu ultimate connection moment capacityn shape parameter of the three-parameter power
model defined in Equation 16Rk tangent connection stiffnessRki initial connection stiffnessV shear forceVo δyt / 2 = plastic shear capacity per unit lengtht angle thickness (Figure 4), subscripts w and t may
be used to refer to web angle and top anglerespectively
W nut diameter (Figure 4)Fig. 16. General deformations of frame under service loads.
THIRD QUARTER / 1993 101
Fig. 17. Distributions of roof drift under service loads.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION102
θ θr / θo
θr relative rotation of connection
θo Mu / Rki
ξw V / Vo = non-dimensional ultimate shear forceparameter in web angle
ξt non-dimensional ultimate shear force parameter atthe plastic hinge
σy yield stress of steel
β g / l
βw′ defined in Equation 3
βt∗ defined in Equation 10
βt′ defined in Equation 7
γ l / t
δ d / t
κ k / t
ω W / t
ρ tw / tt
Fig. 18. Bending moment diagrams of frameunder factored loads.
THIRD QUARTER / 1993 103
Table 3.Non-dimensional End Moments of Columns
gt = 2.5 in. of Case 1
T&S Angle with W Angle Connection Rigid Connection
Steel tubes of relatively thin wall thickness filled with high-strength concrete have been used in building construction inthe U.S. and Far East Asian countries. This structural systemallows the designer to maintain manageable column sizeswhile obtaining increased stiffness and ductility for wind andseismic loads. Column shapes can take the form of tubes orpipes as required by architectural restrictions. Additionally,shop fabrication of steel shapes helps insure quality control.
In this type of construction, in general, at each floor levelheavy steel beam is framed to these composite columns.Often, these connections are required to develop shear yieldand plastic moment capacity of the beam simultaneously.
This paper summarizes results and recommendations froma pilot study conducted to develop a moment-resisting steelconnection detail for connecting steel beams to compositecolumns of the type described above. The focus of this pilotstudy was on composite columns having a square or rectan-gular cross section.
CURRENT PRACTICE
Beam-column connections in concrete-filled steel tubes areusually constructed by directly welding the steel beam to thetube when connections are required to develop plastic mo-ment capacity of the beam. Current design practices for theseconnections rely heavily on the judgment and experience ofindividual designers, with little research and testing informa-tion available.
When beams are welded or attached to steel tubes throughconnection elements, complicated stiffener assemblies arerequired in the joint area within the column. However, weld-ing of the steel beam or connecting element directly to thesteel tube of composite columns should be avoided for thefollowing reasons:
1. Transfer of tensile forces to the steel tube can result inseparation of the tube from the concrete core, thereby
overstressing the steel tube. In addition, the deformationof the steel tube will increase connection rotation, de-creasing its stiffness.
2. Welding of the thin steel tube results in large residualstresses because of the restraint provided by other con-nection elements.
3. The steel tube is designed primarily to provide lateralconfinement for the concrete which could be compromisedby the additional stress due to the welded connection.
POSSIBLE CONNECTION DETAIL
With these considerations in mind, attempts should be madeto prevent direct transfer of beam forces to the steel tube. Twogeneral types of connection details were envisioned, types Aand B.
Type A Connection Detail
Figure 1 shows one alternative in which forces are transmittedto the core concrete via anchor bolts connecting the steelelements to the steel tube. In this alternative, all elementscould be pre-connected to the steel tube in the shop. The nutinside the steel tube is designed to accomplish this task. Thecapacity of this type of connection would be limited with thepull-out capacity of the anchor bolts and local capacity of thetube.
Another variation of the same idea is shown in Figure 2,where connecting elements would be embedded in the coreconcrete via slots cut in the steel tube. In this variation slotsmust be welded to connection elements after beam assemblyfor concrete confinement. The ultimate capacity of this detailalso would be limited to the pull-out capacity of the connec-tion elements and the concrete in the tube.
Type B Connection Detail
Another option is to pass the beam completely through thecolumn (see Figure 3). This type of connection is believed tobe the most suitable. In this type of detail a certain height ofcolumn tube, together with a short beam stub passing throughthe column and welded to the tube, could be shop fabricatedto form a “tree column.” The beam portion of the “treecolumn” could then be bolted to girders in the field. A com-bination of analytical and experimental investigations wasundertaken to comprehend and identify the force transfer
Atorod Azizinamini is an assistant professor, Department of CivilEngineering, University of Nebraska-Lincoln, Lincoln, NE.
Bangalore Prakash, structural engineer with Nabih Youssefand Associates, Los Angeles, California, formerly a graduatestudent, Department of Civil Engineering, University of Ne-braska, Lincoln, NE.
A Tentative Design Guideline for a New Steel BeamConnection Detail to Composite Tube ColumnsATOROD AZIZINAMINI and BANGALORE PRAKASH
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION108
mechanism and suggest a tentative design procedure for thistype of connection.
ANALYTICAL INVESTIGATION
To investigate the performance of the connection detail inwhich the beam completely passes through the column (here-after referred to as a through connection), detailed finiteelement analyses were conducted. The finite element modelused in these analyses consisted of a three-dimensional modelof the column with a small portion of the beam extending fromthe column. In these analyses concrete cracking and non-lin-ear behavior of the steel elements were modeled. In addition,the interface between steel and concrete elements was care-fully modeled.
Results of the analyses were used to identify the forcetransfer mechanism between the steel beam and compositecolumn in the joint region, and to identify the effects of someof the connection details on its performance. Major conclu-sions from the analytical investigation associated with thethrough beam connection detail are discussed in the followingsection.
Figure 4 shows the force transfer mechanism observedfrom the analyses. The portion of the steel tube between thebeam flanges acts as a stiffener, resulting in a concrete com-pression strut which assists the beam web within the joint in
carrying shear. The effectiveness of the compression strut wasshown to be increased to a limit by increasing the thicknessof the steel plate between the beam flanges. The width of theconcrete compression strut on each side of the beam web inthe direction normal to the beam web is approximately equalto half the beam flange width.
A compressive force block is created when beam flangesare compressed against the upper and lower columns (Figure4). The width of this compression block is approximatelyequal to the width of the beam flange. In the upper and lowercolumns shown in Figure 4 the compressive force, C, is shownto be balanced by the tensile force provided by an embeddedrod in the concrete and possibly welded to the beam flanges.This rod was not modeled in the finite element model, forcingthe steel liner plate to carry this tensile force.
Since one of the objectives of this phase of the study is todevise means to improve connection performance, it is bene-ficial to require rods be attached to beam flanges as shown inFigure 4. The presence of such rods is believed to make thebeam web within the joints stiffer and reduce the stress levelin the steel tube.
EXPERIMENTAL INVESTIGATION
To gain additional insight of the behavior of the through beamconnection detail, one test specimen representing approxi-mately a one-half scale model of a prototype column used inhigh-rise building construction in seismic zones was con-
Fig. 1. Type A connection detail using anchor bolt. Fig. 2. Type A connection detail using embedded elements.
THIRD QUARTER / 1993 109
structed and tested. The prototype column consists of a 4-ft(1.22-m) square hollow tube with a 2-in. (50.8-mm) wallthickness, 10′-9 (3.28-m) story height, and W30x99 beamsection framing to the column. In this particular building theW30x99 beams were welded directly to the steel tube. Toprevent overstressing of the steel tube a complicated schemeof stiffener assemblies was placed inside the hollow tubedirectly behind the beam section.
Figure 5 shows the general configuration of the test speci-men. The height of the column from the beam’s top andbottom flanges to the support point is 3111⁄16 inches (0.8 m)and represents the distance from the floor to the inflectionpoint in the upper and lower stories of a building framesubjected to lateral loading (assuming the inflection point tobe located at mid-height of the column). The length of thebeam extending from each side of the column is 27 inches(0.69 m). This length was selected such that the beam’scross-section shear yield and plastic moment capacitieswould develop simultaneously.
Figure 6 shows the different components of the test speci-men. The test specimen consisted of three major components:
a. hollow steel tube made of A36 steelb. hybrid built-up beam sectionc. four #11 grade 60 reinforcing bars with anchor plates
welded to each end of the reinforcing bars
The hollow steel tube is 24 inches (0.6 m) square with 1⁄2-in.(12.7-mm) wall thickness. A half-scale model of the prototypecolumn (which has a 2-in. (50.8-mm) wall thickness) wouldhave required using 1-in. (25.4-mm) wall thickness in the testspecimen. However, only 1⁄2-in. (12.7-mm) wall thickness isused.
As shown in Figure 6, two slots in the shape of the beamcross section were prepared on two faces of the steel tube.These slots were used to pass the beam through the column.
Four holes were drilled on each flange of the beam withinthe column as shown in Figure 6. These holes were used topass four #11 grade 60 (414 MPa) reinforcing bars throughthe beam flanges. Reinforcing bars were then welded to thebeam flanges. As discussed earlier, these reinforcing barswere provided to resist tensile forces in the lower and uppercolumns arising from applied beam loads. The 4×2×l-in.(102×50.8×25.4-mm) plates welded to each end of the rein-forcing rods were intended to reduce the amount of slip in therebars. “Excessive” slip of the rebars could transfer largetensile forces to the steel tube. It may be possible to achievethis same objective by using longer rebars (develop the re-bars) or by using a hook at the end of the rebars, particularlysince it has been reported that the use of steel plates at the endof anchor bolts could potentially reduce their capacity.1
The specimen was cast and cured in the vertical position.The concrete compressive strength at time of testing was14,000 psi (99 MPa).
TEST RESULTS
In this section the general behavior of the test specimen interms of function of the beam web within the joint aredescribed briefly. Further details are given elsewhere.2
Figure 7 shows the location and orientation of six gagesattached to the beam web within the column. Also shown inthis figure is the direction of the applied beam loads. Datafrom these gages, as shown in Figure 8, indicate that the beamweb within the joint is subjected primarily to compressive andtensile strains along the lines GG and HH, respectively. This
Fig. 3. Through connection detail.Fig. 4. Force transfer mechanism for through
beam connection detail.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION110
type of deformation indicates that the beam web experiencesshear type deformation.
Closer examination of data from gages shown in Figure 7indicate that tensile strains along lines parallel to HH aresignificantly larger than compressive strains parallel to lineGG. This observation can be explained as follows. The typeof shear deformation imposed on the beam web within thejoint results in the creation of a concrete compressive strutparallel to line GG in Figure 7. This compressive strut acts asa stiffener along the diagonal GG, consequently reducing thecompressive strain in the beam web in that direction. How-ever, in the other direction (along line HH) tensile strains inthe web increase since concrete is not effective. This obser-vation verified the force transfer mechanism deduced fromthe analytical investigation and explained earlier.
BEHAVIORAL MODEL
Based on results of the finite-element analysis and experi-mental results, a behavioral model in the form of equationsrelating the applied external forces to the connection’s inter-nal forces was developed. These equations are then used tosuggest a tentative design criteria for through-beam connec-tion detail.
In developing the behavioral model the following assump-tions were made:
a. Externally applied shear forces and moments at the jointsare known.
b. Failure is defined as the point at which the beam webwithin the joint reaches its shear stress limit when exter-nally applied forces are at their ultimate values.
c. At failure the concrete stress distribution is linear andmaximum concrete compressive stress is below its lim-iting value.
The joint forces implied in assumption (a.) could be ob-tained from analysis and requires the knowledge of applied
shear and moment at the joint at failure. These quantities areassumed to be related as follows:
Vc = αVb
Mb = l1Vb
Mc = l2Vc
where Vb and Mb are ultimate beam shear and moment, respec-tively, while Vc and Mc are ultimate column shear and moment,respectively. Figure 9 shows these forces for an isolatedportion of a structure subjected to lateral loads.
The validity of assumption (c.) above could be justified forthe following reasons:
1. Column sizes for the type of construction considered inthis paper are generally much larger than the beam sizes.
2. The concrete type used in these columns is generallyhigh-strength concrete with compressive strength wellabove 10,000 psi. The uniaxial stress-strain charac-teristics of high-strength concrete exhibit a linear behav-ior up to maximum strength, followed by a sharp de-scending portion.
Derivation of Behavioral Model
The type of joint is shown in Figure 9. Figure 10 shows theFree Body Diagram (FBD) of the beam web within the joint
Fig. 5. General configuration of test specimen.
Fig. 6. Different components of the test specimen:a) hollow steel tube, b) hybrid built-up beam section,
c) four #11 reinforcing bars with anchor plateswelded to each end.
THIRD QUARTER / 1993 111
and upper column at ultimate load. With reference to Figure10, the following additional assumptions are made in derivingthe Behavioral Model:
1. The concrete stress distribution is assumed to be linear.The width of the concrete stress block is assumed toequal bf, beam flange width.
2. As shown in Figure 10, strain distribution over the uppercolumn is assumed to be linear.
3. The steel tube and concrete act compositely.4. The portion of the upper column shear, Vc, transferred to
the steel beam is assumed to be βCc, where Cc is theresultant concrete compressive force bearing against thebeam flange and β is the coefficient of friction.
5. Applied beam moments are resolved into couples con-centrated at beam flanges.
6. Resultant of concrete compression strut is along a diago-nal as shown in Figure 10.
Considering the above assumptions and strain distributionshown for the upper column in Figure 10, strain for differentconnection elements could be related to ε1, steel tube strainin tension.
εc = a
dc − a ε1 (1)
εsc = a − d1
dc − a ε1 (2)
εst = dc − d1 − a
dc − a ε1 (3)
where
εc = maximum compressive strain in steel tube andconcrete in compression
εsc= strain in steel rod in compression
εst= strain in steel rod in tension
Next, maximum stress in concrete and stresses in the steelrod and steel tube could be calculated as follows:
fc = Ecεc (4)
fsc = Esεsc (5)
flc = Esεc (6)
fst = Esεst (7)
flt = Esε1 (8)
where fc, fsc, flc, fst, and flt are maximum concrete concretecompressive stress, stress in rod in compression, stress in steeltube in compression, stress in rod in tension, and stress in steeltube in tension, respectively.
Substituting Equations 1 through 3 in Equations 4 through8 and multiplying Equations 4 through 8 by correspondingarea, the resultant forces for different connection elementscould be calculated as follows:
Cc = 1⁄2η′ξbf a2
dc − a fy1 (9)
C1 = γξbf t1 a
dc − a fy1 (10)
Cs = Asξ a − d1
dc − a fy1 (11)
Ts = Asξ dc − d1 − a
dc − a fy1 (12)
T1 = ξγbf t1 fy1 (13)
Using the FBD of the upper column shown in Figure 10,
Fig. 7. Location and orientation of gagesattached to beam web within the column.
Fig. 8. Strain data from gages attached tobeam web within the column.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION112
Equations 9 through 13, and satisfying vertical force equilib-rium, the following equation could be obtained.
As = 1
dc − 2a [1⁄2η′bf a
2 − A1(dc − 2a)] (14)
where
bf = beam flange widthdc = depth of the columna = depth of the concrete compression blockη′ = ratio of modulus of elasticity for concrete over
modulus of elasticity of steelA1 = effective area of steel tube = 2bf t1As = area of steel rod at each corner of the beamt1 = thickness of steel tube
In defining A1 it is assumed that a steel tube width equal totwo times the beam flange width is effective in carryingtension and compression. This value was estimated fromexperimental results.
Next, considering the moment equilibrium of the FBD ofthe upper column shown in Figure 10 the following expres-sion can be derived.
a3
A1adc + As(adc − 2d1dc + 2d12) +
1⁄2η′bf a2 dc −
a3
ξfy1
αl2(dc − a) = Vb (15)
where
d1 = distance between steel rod and steel tubefy1 = yield strength of steel tube
In Equation 15 ξ fy1 is the stress level the steel tube isallowed to approach at ultimate condition. ξ fy1 could also beviewed as the portion of the steel tube strength utilized toresist the forces transferred by the connection. Based on thelimited experimental data obtained from this investigation itis suggested that a value of 0.35 be used for ξ.
Equations 14 and 15 relate the externally applied force,Vb, directly and the externally applied forces Vc and Mc indi-
rectly (through the coefficients α and l2) to different connec-tion parameters such as As, A1, and a.
DESIGN APPROACH
Before proceeding with the steps necessary in designing thethrough-beam connection detail, additional equations will bederived to relate the shear stress in the beam web within thejoint to the compressive force in the concrete compressionstrut and externally applied forces.
Considering the FBD of a portion of the beam web within
Fig. 9. Assumed forces on an interior jointin a frame subjected to lateral loads.
Fig. 10. FBD of the upper column and beamweb within the joint area.
THIRD QUARTER / 1993 113
the joint area as shown in Figure 11 and satisfying the hori-zontal force equilibrium, the following equation could bederived:
Vw + Cst cos θ + βCc − 2Mb
db = 0 (16)
where
Vw = shear force in the beam web at ultimate condition
θ = arctan db
dc
Equations 14, 15, and 16 could be used to proportion thethrough-beam connection detail.
Until further research is conducted the following steps aresuggested for designing the through-beam connection detailfollowing the LRFD format.
Step 1. From analysis, obtain factored joint forces.
Step 2. Select the following quantities: t1, bf, db, dc, d1, fy1
Step 3. Solving Equations 14 and 15 simultaneously, obtainAs and a. This could be achieved using the trial and errorapproach.
Step 4. Check stress in different connection elements.
Step 5. Assume the beam web yields at ultimate load. Withthis assumption Vw could be calculated as follows:
Vw = 0.6Fyw tw dc (17)
where
Fyw = beam web yield stresstw = thickness of the beam web
Step 6. Using Equation 16 calculate Cst, compressive force inthe concrete compressive strut, and applied shear force toconcrete in the joint area.
Step 7. Check shear stress in concrete in the joint area. Thelimiting shear force could be assumed to be as suggested byACI 352 [2]:
Vu = φR√fc′Ae (18)
where
φ = 0.85
R = 20, 15, and 12 for interior, exterior, and corner joints,respectively
fc′ = concrete compressive strength
It is suggested that the value of √fc′ be limited to 100 psi,implying that in the case of 15,000 psi concrete, for instance,√fc′ be taken as 100 rather than 122 as would be obtained fromVu calculations.
Until further research is conducted it is suggested that Ae
be calculated as follows:
Ae = 2bf × dc
DESIGN EXAMPLE
Design a through-beam connection detail with the followinggeometry and properties.
Substitute As = 1.03 in.2 in Equation 15 and calculate Vb. Ifthe result is approximately equal to 79 kips the assumed valueof a is o.k. Equation 15 yields:
Step 7: The shear force carried by concrete within the jointbetween the beam flanges is assumed to be the horizontalcomponent, Cst.
Vc = Cstcos(θ)Vc = 90.9 cos(31.1) = 77.8k
For the interior joint the shear capacity is
Vu = φ(20)fc′(2bf)(dc)Vu = 0.85(20)100×[(2×5.5)(24)]/1,000 = 449k >
77.8k o.k.
SUMMARY AND CONCLUSIONS
The use of composite columns of the type described in thispaper is proven to be economical. This paper has summarizeda suggested connection detail (a through-beam connectiondetail) for connecting steel beams to these columns as well astentative design guidelines. The information presented in thispaper is based on a pilot study and, therefore, it is suggestedthat this information be viewed as a general guideline untilfurther research is carried out. It should also be noted that theeffect of axial load in the column on performance of theconnection was not considered. The intent of the paper is tosuggest an economical connection detail and outline a proce-dure to comprehend its behavior through the behavioralmodel presented.
ACKNOWLEDGMENT
The authors greatly appreciate the support provided by theAmerican Institute of Steel Construction which providedpartial funding of a graduate student. Valmont Industries ofOmaha, Nebraska is greatly appreciated for constructing thetest specimen. Special thanks are also due the Center forInfrastructure Research at the University of Nebraska-Lincoln for supporting this research. The authors thank Dr. J.P. Colaco of CBM Engineers, Inc., Houston, Texas for hishelpful suggestions and great encouragement while pursuingthis research.
REFERENCES
1. Shipp, G. John and Haninger, R. Edward, “Design ofHeaded Anchor Bolts,” AISC Engineering Journal, Sec-ond Quarter, 1983, Vol. 20, No. 2 pp. 58–69.
2. Prakash, A. Bangalore, “Development of Connection De-tail for Connecting Steel Beams to Composite Columns,”M.S. Thesis, Civil Engineering Department, University ofNebraska-Lincoln, 1992.
3. ACI-ASCE Committee 352, “Recommendations for De-sign of Beam-Column Joints in Monolithic ReinforcedConcrete Structures,” ACI Journal, Proceedings, Vol. 82,No. 3, May–June 1985, pp. 266–283.
THIRD QUARTER / 1993 115
EngineeringJournalAMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Page 117: D. E. Allen and T. M. MurrayDesign Criterion for Vibrations Due to Walking
Page 130: Won-Sun King and Wai-Fah ChenLRFD Analysis for Semi-Rigid Frame Design
Page 141: Lai-Choon Ting, Nandivaram E.Shanmugam, and Seng-Lip LeeDesign of I-Beam to Box-Column Connection StiffenedExternally
Page 150: Discussion—Thomas Sputo and Nestor R. IwankiwDesign of Pipe Column Base Plates Under Gravity Load—Thomas Sputo
A design criterion for walking vibrations of broader appli-cation than previous criteria is proposed for steel floor orfootbridge structures. The criterion is based on the dynamicresponse of steel structures to walking forces, as well as thesensitivity of occupants to vibration motion. The criterion isapplicable to structures with natural frequencies below 9 Hz,where resonance can occur with a harmonic of the stepfrequency, but is extended beyond 9 Hz where footstep im-pulse response becomes important.
INTRODUCTION
Walking, good for body and soul, sometimes produces vibra-tions which are annoying to others. This is not a new problem.Tredgold (1828) wrote that girders over long spans should be“made deep to avoid the inconvenience of not being able tomove on the floor without shaking everything in the room.”It also became common practice for soldiers to break stepwhen marching across bridges to avoid large and potentiallydangerous resonance vibrations. Both stiffness and resonanceare therefore important considerations in the design of steelfloor structures and footbridges for walking vibrations.
Stiffness has been taken into account for many years in thedesign of floor structures using criteria dating fromTredgold’s time. A traditional stiffness criterion for residentialfloors is to limit the deflection under 2 kPa (42 psf) to lessthan span/360. This criterion is restricted to traditional woodfloor construction with high transverse stiffness. The Ameri-can Institute of Steel Construction Allowable Stress DesignSpecification (AISC, 1989) limits the live load deflection ofbeams and girders supporting “plastered ceilings” tospan/360, a limitation which has also been widely applied tosteel floor systems in an attempt to control vibrations. A betterstiffness criterion applicable to all floor construction is to
limit the deflection due to 1 kN (225 lb.) concentrated load toless than approximately 1 mm (0.04 in.).
Resonance, however, has been ignored in the design offloors and footbridges for walking vibrations until recently.Approximately 30 years ago, problems arose with walkingvibrations for steel-joist floors that satisfied code stiffnesscriteria. Lenzen (1966) determined that damping and mass,not stiffness, were the most important factors in preventingunacceptable walking vibrations for these floors. To takedamping and mass into account, a simple dynamic designcriterion based on heel-impact response was developed (Al-len and Rainer, 1976) and introduced 18 years ago into anAppendix to the Canadian design standard for steel structures(Canadian Standards Association, 1989), In 1981, Murrayrecommended a similar dynamic design criterion based ondata from 91 floor measurements (Murray, 1981). More re-cently a design criterion for footbridges has been introducedinto British and Canadian bridge standards based on reso-nance response to a sinusoidal force (BSI, 1978; OHBDC,1983).
Since these criteria were introduced, more has been learnedabout the loading function due to walking, in particular thatresonance can occur at a harmonic multiple of the step fre-quency. This has been verified by reviewing past cases ofvibration problems with steel joist and beam floors, most ofwhich corresponded to third harmonic resonance of the stepfrequency (6 Hz floors approximately), but more recently alsoto second harmonic resonance (4 Hz floors approximately).Also the Canadian CSA criterion has recently been found notto correctly predict the vibration behavior of two-way joistgirder systems.
In this paper a simple yet rational design criterion forwalking vibration is proposed based on harmonic resonance.The criterion is calibrated to floor experience. It is similar toone recently recommended by Wyatt (1989). The criterion isextended to floor frequencies beyond 9 Hz to control impulsevibration from footsteps.
VIBRATION LIMIT STATE—ACCELERATION LIMITS
International Standards Association (ISO, 1989; ISO, 1992)recommends vibration limits below which the probability ofadverse reaction is low. Limits for different occupancies aregiven in terms of rms acceleration as a multiple of the baselinecurve shown in Figure 1. For offices, ISO recommends a
D. E. Allen is senior research officer, Institute for Research inConstruction, National Research Council Canada, Ottawa, On-tario, Canada.
T. M. Murray is Montague-Betts Professor of Structural SteelDesign, The Charles E. Via Department of Civil Engineering,Virginia Polytechnic Institute and State University, Blacksburg,VA.
Design Criterion for Vibrations Due to WalkingD. E. ALLEN AND T. M. MURRAY
FOURTH QUARTER / 1993 117
multiplier of 4 for continuous or intermittent vibrations and60 to 128 for transient vibrations. Intermittent vibration isdefined as a string of vibration incidents such as those causedby a pile driver, whereas transient vibration is caused rarely,for example by blasting. Walking vibration is intermittent innature but not as frequent and repetitive as vibration causedby a pile driver. It is therefore estimated that the multiplier forwalking vibration in offices is in the range of 5 to 8, whichcorresponds to an rms acceleration in the range 0.25 to 0.4percent g for the critical frequency range 4 to 8 Hz shown inFigure 1. Based on an estimated ratio of peak to rms accel-eration of approximately 1.7 for typical walking vibration, theannoyance criterion for peak acceleration is estimated to bein the range 0.4 to 0.7 percent g. From experience (Allen andRainer, 1976), a value of 0.5 percent g is recommended forthe frequency range 4–8 Hz. The resulting acceleration limitfor offices is shown in Figure 1.
For footbridges, ISO (1992) recommends a multiplier of60 which, combined with an estimated ratio of peak to rmsacceleration of 1.7, results in a criterion of approximately ten
times the vibration limit for offices. People in shoppingcentres will accept something in between, depending onwhether they are standing or sitting down. Suggested peakacceleration limits for these occupancies are given inFigure 1.
LOADING FUNCTION
Walking across a floor or footbridge produces a movingrepetitive force. Figure 2 shows the dynamic reaction atmid-support of a footbridge due to a person walking acrossit: the Fourier spectrum of the reaction clearly indicates thepresence of sinusoidal loading components at the first, sec-ond, and third harmonic multiples of the step frequency. Theforce, F, can therefore be represented in time by a Fourierseries
F = P (1 + Σαi cos 2πift) (1)
where P is the person’s weight, taken as 0.7 kN (160 lbs) fordesign, f the step frequency, i the harmonic multiple, αi is adynamic coefficient for the harmonic, and t is time. Table 1recommends design values for these parameters based on testinformation on dynamic coefficient (Rainer, et al, 1988) and
Fig. 2. Center support reaction produced by walking along afootbridge on three supports (Rainer, et al, 1988).
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION118
observations of step frequencies which are in the range 1.9± 0.3 Hz for offices.
Jogging, or more than one person walking in step, is a moresevere dynamic loading, but only for the first two harmonics.Generally such cases are rare enough not to be a problem inpractice. Similarly a large group of people walking in an areaproduces a greater dynamic loading at the step frequency(2 Hz approximately), but lack of coherence at the higherharmonics plus the damping effect of people has meant that,except for footbridges close to entertainment events (Bach-mann, 1992) such loading has not been a problem in practice.
RESPONSE
Walking across a footbridge or floor causes a complex dy-namic response, involving different natural modes of vibra-tion, as well as motion due to time variation of static deflec-tion. The problem can be simplified by considering a personstepping up and down at mid-span of a simply supportedbeam which has only one mode of vibration—the fundamen-tal mode. Maximum dynamic response will occur when thenatural frequency corresponds to one of the harmonic forcingfrequencies. The steady-state acceleration, a, due to harmonicresonance is given by (Rainer, et al, 1988),
ag
= αiP
0.5W ×
R2β
× cos 2πift = RαiP
βW × cos 2πift (2)
where W is the weight of the beam, β is the damping ratio, gis the acceleration due to gravity, and R is a reduction factordiscussed later. The factor 1 / (2β) is the familiar dynamicamplification factor for steady-state resonance and 0.5W / gis the mass of an SDOF oscillator which is dynamicallyequivalent to the simply supported beam of weight W vibrat-ing in its fundamental mode. The other harmonics will alsoproduce steady-state vibrations at their forcing frequencies,but the level of vibration is generally much smaller. For floorstructures, an exception occurs when there is resonance of twomodes of vibration at two multiples of the step frequency;floor experience indicates, however, that only one resonantmode whose frequency is near to the fundamental frequencyneed be considered for design.
The reduction factor R is introduced into Equation 2 to takeinto account (a) that full steady-state resonance is notachieved when someone steps along the beam instead of upand down at mid-span and (b) that the walker and the personannoyed are not simultaneously at the location of maximummodal displacement. Figure 3 shows test results for a personwalking across two simply supported footbridges which ver-ify the harmonic resonance response model, Equation 2. Thevalue R = 0.56 in Figure 3a was determined by dynamicanalysis of a person walking across the footbridge (Rainer, etal, 1988). It is recommended that for design R be taken as 0.7for footbridges and 0.5 for floor structures having two-waymodal configurations.
PROPOSED DESIGN CRITERIA
Equation 2 predicts peak acceleration due to harmonic reso-nance, RαiP / βW, which can be compared to the accelerationlimit, ao / g shown in Figure 1. It is useful to express this interms of a minimum value of damping ratio times equivalentmass weight (βW):
βW ≥ RαiPao / g
(3)
Table 2 contains specific minimum values of βW for thevalues of dynamic loading (αiP) from Table 1, accelerationlimit (ao / g) from Figure 1 and reduction factor (R) recom-mended above.
As shown in Figure 4 the results of Table 2 can be approxi-mated by the following criterion for walking vibrations:
βW ≥ K exp (−0.35fo) (4a)
where fo is the fundamental natural frequency (Hz) and K isa constant given in Table 3 which depends on the accelerationlimit for the occupancy. Equation 4a can be inverted toexpress the criterion for walking vibrations in terms of mini-mum fundamental natural frequency:
fo ≥ 2.86 ln
KβW
(4b)
The following section provides guidance for estimating therequired floor properties for application of Equations 4.
DAMPING RATIO ββThe damping ratio depends primarily on non-structural com-ponents and furnishings. The Canadian steel structures speci-fication (CSA, 1989) recommends damping ratios of 0.03 fora bare floor; 0.06 for a finished floor with ceiling, ducts,flooring, and furniture; and 0.12 for a finished floor withpartitions. Murray (1991) recommends damping ratios of0.01 to 0.03 for a bare floor, 0.01 to 0.03 for ceilings, 0.01 to0.10 for mechanical ducts, and 0.10 to 0.20 for partitions.These damping ratios, however, are based on vibration decayresulting from heel impact and include a component for
Table 1.Loading Function for Walking (See Equation 1)
Harmonici
Frequency Rangei ×× f
Dynamic Load Factorααi
1 1.6 to 2.2 0.5
2 3.2 to 4.4 0.2
3 4.8 to 6.6 0.1
4 6.4 to 8.8 0.05
FOURTH QUARTER / 1993 119
geometric dispersion of vibration as well as frictional andmaterial damping. More recent testing of modal dampingratios shows that the frictional and material damping ratiosare approximately half of the values determined from heelimpact tests. Based on available information (Wyatt, 1989;ISO, 1992), Table 3 recommends damping values for use inthe proposed criterion, Equation 4.
NATURAL FREQUENCY, fo, AND EQUIVALENTMASS WEIGHT, W
In the case of a simply supported panel such as a footbridge,the natural frequency is equal to the fundamental beam fre-quency of the panel and the equivalent mass weight is equalto the panel weight. Floors of steel construction, however, aretwo-way systems with many vibration modes having closely
spaced frequencies. Natural frequency and equivalent massweight of a critical mode in resonance with a harmonic of stepfrequency is therefore difficult to assess. A dynamic modalanalysis of the floor structure can be used to determine thecritical modal properties, but there are factors that are difficultto incorporate in the structural model. Composite action anddiscontinuity conditions are two such factors, but more diffi-cult to assess is the effect of partitions and other non-structuralcomponents. An unfinished floor with uniform bays can havea variety of modal pattern configurations extending over thewhole floor area, but partitions and other non-structural com-ponents tend to constrain the modal configurations to localareas in such a way that the floor vibrates locally like a singletwo-way panel. The following simplified procedure is recom-mended to estimate the properties of such a panel. Some ofthe recommendations are based on judgment guided by floortest experience. Further research is needed to obtain betterestimates, particularly for W.
The floor is assumed to consist of a concrete slab (or deck)supported on steel joists or beams (open-web or rolled sec-tions) which, in turn are supported on walls or on steel girdersbetween columns. The fundamental natural frequency, fo, andequivalent mass weight, W, for a critical mode is estimatedby first considering a “joist panel” mode and a “girder panel”mode separately and then combining them. If the joist span isless than half the girder span, however, both the joist panelmode and the combined mode should be checked against thecriterion, Equations 4.
Table 2.Minimum Value of ββW determined from Equation 3
for Satisfactory Performance
FloorFrequency
fo (Hz)
OfficeFloors
kN (kips)
ShoppingMalls
kN (kips)Footbridges
kN (kips)
1.6 to 2.2 28 (6.3) 9.3 (2.10) 4 (0.50)
3.2 to 4.4 14 (3.2) 4.7 (1.05) 2 (0.45)
4.8 to 6.6 7 (1.6) 2.3 ( 0.52) 1 (0.22)
6.4 to 8.8 3.5 (0.8) 1.1 (0.26) 0.5 (0.11)
Fig. 3. Peak response of two footbridge spans to a person walking across at different step frequencies (Rainer, et al, 1988).
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION120
In the following, the concrete modulus of elasticity isassumed equal to 1.35 times that assumed in current structuralstandards, the increase being due to the greater stiffness ofconcrete under dynamic, as compared to static, loading. Alsofor determining composite moment of inertia, the width ofconcrete slab is equal to the member spacing but not morethan 0.4 times the member span. For edge members, it is halfof this value plus the projection of the slab beyond themember center line.
Also the floor weight per unit area, w, should include thesustained component of live load (approximately 0.5 kPa(11 psf) for offices).
JOIST PANEL MODE
The joist panel mode is associated with the natural frequencyof the joist or beam alone. The natural frequency of this modecan be estimated from the simple beam formula
fj = 0.18 √g / ∆j (5)
where ∆j is the deflection of a beam or joist relative to itssupports due to the weight supported by the individual beamor joist. Composite action is normally assumed provided thejoists are directly connected to the concrete slab by welds tosteel deck. Normally the joists or beams are assumed to besimply supported unless dynamic restraint is verified by adynamic analysis or experiment. For open-web joists, sheardeformations should be included in the calculations for ∆j.
The mass weight of the joist panel mode can be estimatedfrom
Wj = wBj Lj (6)
where w is the floor weight per unit area, Lj the joist or beamspan, and Bj the effective joist panel width determined from
Bj = 2(Ds / Dj )1⁄4Lj (7)
where Dj is the flexural rigidity per unit width in the joistdirection and Ds the flexural rigidity per unit width in the slabdirection (including a correction for shear in open-web joists)based on the moment of inertia of the uncracked concrete(assume an average thickness tc for ribbed decks). The formof Equation 7 is based on orthotropic plate action and thefactor 2 was determined by calibration to floor data as de-scribed later. The effective panel width, Bj, determined byEquation 7 should be assumed to have an upper limit oftwo-thirds of the total width of the floor perpendicular to thejoists or beams.
Where the beams or joists are continuous over their sup-ports (including rolled sections shear connected to girderwebs), and an adjacent span is 0.7Lj or greater, the effectivejoist panel weight, Wj, can be increased by 50 percent. Thereason for this increase is that continuity over supports en-gages participation of adjacent floor panels in the fundamen-tal mode of vibration. (Wyatt (1988) recommends an increase
of 70 percent where the adjacent span is 0.8Lj or greater, 100percent when it is 1.0Lj.)
GIRDER PANEL MODE
The girder panel mode is associated with the natural fre-quency of the girder alone. The natural frequency of this modecan be estimated from
fg = 0.18 √g / ∆g (8)
Table 3.Values of K and ββ for use in Equation (4)
KkN (kips) ββ
Offices, residences, churches 58 (13.0) 0.03*
Shopping Malls 20 (4.5) 0.02
Footbridges 8 (1.8) 0.01
*0.05 for full-height partitions, 0.02 for floors with few non-structural com-ponents (ceilings, ducts, partitions, etc.) as can occur in churches
Fig. 4. Proposed criterion for walking vibrations.
FOURTH QUARTER / 1993 121
where ∆g is the deflection of individual girders relative to theirsupports due to the weight supported. Composite action canbe assumed when the girders are directly connected to theconcrete slab, for example by welds to the steel deck. Whenthe girders are separated from the concrete slab by beams orjoist seats (shoes), they act as Vierendeel girders, i.e. partiallycomposite. It is recommended that the moment of inertia ofgirders supporting joist seats be determined from:
Ig = Inc + (Ic − Inc) / 2 (9a)
for seat heights 75 mm (3 in.) or less, and
Ig = Inc + (Ic − Inc) / 4 (9b)
for seat heights 100 mm (4 in.) or more, where Inc and Ic arenon-composite and fully composite moments of inertia re-spectively. (These recommendations are subject to changedepending on the results of current research.) Normally thegirders are assumed to be simply supported unless dynamicrestraint is verified by analysis or experiment.
The mass weight of the girder panel mode can be estimatedfrom
Wg = wBg Lg (10)
where Lg is the girder span and Bg is the effective girder panelwidth determined from
Bg = 1.6 (Dj / Dg )1⁄4Lg (11)
where Dg is the flexural rigidity per unit width in the girderdirection and Dj the flexural rigidity per unit width in the joistdirection. Equation 11 is the same as Equation 7 except thatthe factor 2 is reduced to 1.6 to take into account discontinuityof joist systems over supports; if the joists consist of rolledbeams shear connected to girder webs the factor 1.6 can beincreased to 1.8. Bg determined by Equation 11 should beassumed to have a lower limit equal to the tributary panelwidth supported by the girder and an upper limit of two-thirdsof the total floor width perpendicular to the girders.
Where the girders are continuous over their supports, andan adjacent span is 0.7Lg or greater, the mass weight, Wg, canbe increased by 50 percent. This is due to participation ofadjacent floor panels, as discussed above for the joist panelmode.
COMBINED MODE
Combined flexibilities of the joists and girders reduces thenatural frequency and makes the floor more susceptible tonoticeable walking vibration. For design purposes this can betaken into account by a “combined” mode whose propertiesmay be estimated using the following interaction equations:
(i) The fundamental natural frequency can be approxi-mated by the Dunkerly relationship:
fo = 0.18 √g / (∆j + ∆g) (12)
(ii) The equivalent mass weight can be approximated bythe interaction formula:
W = ∆j
∆j + ∆g
Wj + ∆g
∆j + ∆g
Wg (13)
If the girder span, Lg, is less than the joist panel width, Bj,the combined mode is restricted and the system is effectivelystiffened. This can be accounted for by reducing the deflec-tion, ∆g, used in Equations 12 and 13 to
∆g′ = Lg
Bj (∆g) (14)
where
0.5 ≤ Lg / Bj ≤ 1.0
EXAMPLE
Determine if the framing system for the typical interior bayshown in Figure 5 satisfies the proposed criterion for walkingvibration. The structural system supports the office floorswithout full-height partitions. For ease in reading, this exam-ple will be carried out using Imperial units.
Concrete: 110 pcf, fc′ = 4,000 psi; n = Es / 1.35 Ec = 9.3Deck thickness = 3.25 in. + 2 in. ribs = 5.25 in.Deck weight = 42 psf
Beam Mode Properties
With an effective concrete slab width of 120 in. < 0.4 Lj =0.4 × 35 × 12 = 168 in., and considering only the concreteabove the steel form deck, the transformed moment of inertiaIj = 2,105 in.4 For each beam
wj = 10(11 + 42 + 4 + 40 / 10) = 610 plf
which includes 11 psf live load and 4 psf for mechanical/ceil-ing, and
∆j = 5wjLj
4
384EIj =
5 × 610 × 354 × 1,728384 × 29 × 106 × 2,105
= 0.337 in.
The beam mode natural frequency from Equation 5 is:
fj = 0.18 √3860.337
= 6.09 Hz
Using an average concrete thickness, 4.25 in., the transformedmoment of inertia per unit width in the slab direction is
Ds = 12 × 4.253 / 12 × 9.3 = 8.25 in.4/ft
The transformed moment of inertia per unit width in the beamdirection is (beam spacing is 10 ft)
Dj = 2,105 / 10 = 210.5 in.4/ft
The effective beam panel width from Equation 7 is:
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION122
Bj = 2(8.25 / 210.5)1⁄4(35) = 31.3 ft
Since this is a typical interior bay, the actual floor width is atleast 3 × 30 = 90 ft, and 2⁄3 × 90 = 60 ft > 31.3 ft. Therefore,the effective beam panel width is 31.3 ft.
The mass weight of the beam panel is from Equation 6,adjusted by a factor of 1.5 to account for continuity:
With an effective slab width of 0.4 × 30 × 12 = 144 in. andconsidering the concrete in the form of deck ribs, the trans-formed moment of inertia Ig = 3,279 in.4 For each girder
In this case the girder span (30 ft) is less than the beam panelwidth (31.3 ft) and the girder deflection, ∆g, is thereforereduced according to 0.350 × 30 / 31.3 = 0.334 in. FromEquation 12,
fo = 0.18 √386 / (0.337 + 0.334) = 4.32 Hz
and from Equation 13
W = 0.337
0.337 + 0.334(100) +
0.3340.337 + 0.334
(104) = 102 kips
For office occupancy without full-height partitions, β = 0.03from Table 3, thus
βW = 0.03 × 102 = 3.06 kips
Evaluation
Application of Equations 4 for offices (see Table 3) results in
Unsupported edges of floors can cause a special problembecause of low-mass weight and sometimes decreased damp-ing. Normally this is not a problem for exterior floor edges,because of stiffening by exterior cladding or because walk-ways are not located near exterior walls. Problems haveoccurred, however, at interior floor edges adjacent to atria.These edge members should often be made stiffer than currentpractice suggests by use of the following assumptions in theproposal criterion.
Where an interior edge is supported by a joist, the equiva-lent mass weight of the joist panel can be estimated usingEquation 6 by replacing the coefficient 2 with 1 in Equation7. Where an interior edge is supported by a girder, the equiva-lent mass weight of the girder panel should be estimated onthe basis of the tributary weight supported by the girder. Theseedge panels are then combined with their orthogonal panelsas recommended above.
CALIBRATION OF PROPOSED CRITERIONTO EXPERIENCE
The factor 2 in Equation (7) was determined by calibration todata on one-way joist floor systems in Table 1 of Allen andRainer (1976). The results of applying the proposed criterion,including recommended design parameters, to floors thathave been evaluated and tested is given in Tables 4 and 5.Table 4 confirms application of the proposed criterion forone-way systems, two-way systems, and interior edge panels.Application of the CSA criterion (CSA, 1989) to the two-wayfloor systems in Table 4, on the other hand, predicts that allare satisfactory when in fact floors 12 and 13 are definitelyunsatisfactory. Table 5 confirms application of the proposedcriterion to two-way systems except for floor 3, a heavy floor(3.6 kPa) with continuity in both directions. Two factors forFig. 5. Floor framing system—typical interior bay.
FOURTH QUARTER / 1993 123
unsatisfactory performance of this floor are low damping(criterion just met for β = 0.015) and vibration transmissiondue to girder continuity. Floors 7 and 10 are predicted to bemarginal.
The proposed criterion can also be compared to existingcriteria. Table 6 makes this comparison on the basis of mini-mum values of βWj for one-way beam or joist systems. Thebasis for the values shown in Table 6 is given in Appendix III.For office floors, Table 6 shows that all criteria are similar forresonance with the third harmonic of the step frequency (5 to7 Hz). This is not surprising because existing design criteriaare based to a large extent on experience with floors in thefrequency range 5 to 8 Hz.
The criteria, however, differ at other floor frequencies. TheCSA criterion is insufficient for frequencies less than 5 Hzand conservative for frequencies beyond 7 Hz. The Murraycriterion has tendencies similar to the CSA criterion, but thediscrepancy with the proposed criterion is less severe. TheWyatt criterion is close to the proposed criterion within a
broad frequency range, 3 to 8 Hz, but is more conservativebeyond 8 Hz.
For footbridges the proposed criterion is apparently a littlemore conservative than the OHBDC (1983) criterion, but thisis offset by the difference in recommended values of β (0.01vs. 0.005 to 0.008 in the OHBDC). Third and fourth harmonicresonance is not adequately considered by the OHBDC butthis is not serious in practice because footbridges with thesefrequencies generally have sufficient mass to satisfy the pro-posed criterion, Equation 4a.
Information on shopping centers is scarce. Application ofEquation 4a for shopping centers to the floor data in Cases 16and 19 of Table 4, however, indicates agreement with userreaction.
Tables 4–6, as well as Figure 3, therefore confirm theapplicability of the proposed criterion for walking vibrationto a wide variety of structures and occupancies.
Table 4.Application of Proposed Design Criterion to Tested Floors
Quebec CityQuebec CityQuebec CityMatthews, et al (1982)Pernica and Allen (1982)
4.55.47.26.25.2
(7.6, 7.6)3
(7.6, 7.6)(7.6, 7.6)(9, 12.5)
(7.6, 12.2)
(9.1, 11.9)3
(9.1, 8.6)(7.4, 10.7)(9.7, 11.3)(8.1, 15.0)
0.030.030.030.030.02
6.2 5.4 5.2 9.511.8
12.0 8.8 4.7 6.6 3.2
Very UUSSS
Interior Edge Panels
171819
Quebec CityEdmontonPernica and Allen (1982)
5.15.15.6
13.717.512.2
2.34.63.3
0.030.030.02
2.48.42.5
9.79.72.8
Very UUU
Notes:1K = 58 for all cases except #16 and #19, where K = 20 applies2U = unsatisfactory, S = satisfactory, B = borderline3The first entry inside the brackets refers to the joist panel, the second refers to the girder panel
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION124
NATURAL FREQUENCIES GREATER THAN 9 HZ
When the natural frequency is greater than 9 Hz, harmonicresonance does not occur, but walking vibration can still be aproblem. Because the natural frequencies are high comparedto the main loading frequencies, the floor response is gov-erned primarily by stiffness relative to a concentrated load.Experience indicates a minimum stiffness of approximately1 kN per mm (5.7 kips per in.) deflection for office andresidential occupancies.
For light floors with natural frequencies in the range 9 to18 Hz there may also be adverse reaction to floor motioncaused by step-impulse forces. Experience indicates that ad-verse reaction to step impulses depends primarily on mass(initial floor velocity equals impulse divided by mass) andvibration decay time, the shorter the decay time the better.The decay time decreases in proportion to clamping ratiotimes floor frequency. Wyatt (1989) recommends an impulsecriterion beyond 7 Hz floor frequency, but beyond approxi-mately 9 Hz the criterion becomes overly conservative be-cause it ignores the benefits of decreased decay time. Ohlsson(1988) recommends an impulse criterion which takes decaytime into account, but the criterion is complex for design. Theresonance criterion, Equation 4a, is in a form that correctly
reflects impulse discomfort except that the right-hand side hasnot been correctly determined. If, however, Equation 4a withK = 58 for office floors is extended beyond 9 Hz, it decreasesrapidly until approximately 18 Hz when the stiffness criterionof 1 kN/mm (5.7 k/in.) starts to control the design of the floor.Application of Equation 4a to the examples in Ohlsson (1988)also indicates that it gives a reasonable evaluation for floorsbetween 9 and 18 Hz.
To ensure satisfactory performance of office and residen-tial floors with frequencies greater than 9 Hz it is recom-mended that Equations 4 be used in conjunction with thestiffness criterion of 1 kN/mm (5.7 k/in.).
CONCLUSION
Walking forces produce motions which are related to reso-nance, impulse response, and static stiffness. Resonance con-trols the design of floors and footbridges with natural frequen-cies less than approximately 9 Hz, static stiffness controls thedesign of floors with frequencies greater than approximately18 Hz, and impulse response controls the design of floors withfrequencies in between.
A simple criterion for resonance vibration of floor andfootbridge structures, Equations 4, is proposed for design,along with a recommended procedure for determining the
Table 5.Application of Proposed Design Criterion to Floors Investigated by Murray (1981) 1
Notes:1All open web joist on girder systems except #3 and #10 (beams shear connected to girders)2Members continuous over supports (Wj or Wg increased by 1.5)3Joist systems supported on stiff girders, frequency fo estimated from fj
FOURTH QUARTER / 1993 125
required floor properties. The proposed criterion, based onacceptable vibration for human reaction, compares well withexisting criteria and is confirmed by experience with testedfloors. Recommended values of the criterion parameters,however, are expected to be improved by further experienceand research.
Floors of offices and residential occupancies with frequen-cies greater than 9 Hz should also be checked both for aminimum static stress under concentrated load of 1 kN/mm(5.7 kips/in.) and for impulse response by means of Equa-tions 4.
APPENDIX I: REFERENCES
1. American Institute of Steel Construction, Specificationfor Structural Steel Buildings—Allowable Stress Designand Plastic Design, AISC, Chicago, 1989.
2. Allen, D. E. and Rainer, J. H., “Vibration Criteria forLong-Span Floors,” Canadian Journal of Civil Engineer-ing, 3(2), June, 1976, pp. 165–171.
3. Bachmann H., “Case Studies of Structures withMan-Induced Vibrations,” Journal of Structural Engi-neering, ASCE, Vol. 118, No. 3, 1992, 631–647.
4. British Standard BS5400, Part 2: Steel, Concrete andComposite Bridges: Specification for Loads, Appendix C,British Standards Institution, 1978.
5. Canadian Standard CAN3-S16. 1-M89: Steel Structuresfor Buildings—Limit States Design, Appendix G: Guidefor Floor Vibrations, Canadian Standards Association,Rexdale, Ontario, 1989.
6. International Standard ISO 2631-2, Evaluation of HumanExposure to Whole-Body Vibration—Part 2: Human Ex-posure to Continuous and Shock-Induced Vibrations inBuildings (1 to 80 Hz), International Standards Organiza-tion, 1989.
7. International Standards ISO 10137, Basis for the Designof Structures—Serviceability of Buildings Against Vibra-tion, International Standards Organization, 1992.
8. Lenzen, K. H., “Vibration of Steel Joists,” EngineeringJournal 3(3), 1966, pp. 133–136.
9. Matthews, C. M., Montgomery, C. J., and Murray, D. W.,“Designing Floor Systems for Dynamic Response,”Structural Engineering Report No. 106, Department ofCivil Engineering, University of Alberta, Edmonton, Al-berta, 1982.
10. Murray, T. M., “Acceptability Criterion for Occupant-In-duced Floor Vibrations,” Engineering Journal, 18(2),1981, 62–70.
11. Murray, T. M., “Building Floor Vibrations,” EngineeringJournal, Third Quarter, 1991, 102–109.
13. Ohlsson, S. V., “Ten Years of Floor Vibration Research—A Review of Aspects and Some Results,” Proceedings ofthe Symposium/Workshop on Serviceability of Buildings.Vol. I, Ottawa, 1988, pp. 435–450.
14. Pernica, G., and Allen, D. E., “Floor Vibration Measure-ments in a Shopping Centre,” Canadian Journal of CivilEngineering, 9(2), 1982, pp. 149–155.
15. Rainer, J. H., Pernica, G., and Allen, D. E., “DynamicLoading and Response of Footbridges,” Canadian Jour-nal of Civil Engineering, 15(1), 1988, pp. 66–71.
17. Wyatt, T. A., “Design Guide on the Vibration of Floors,”Steel Construction Institute Publication 076, London,1989.
APPENDIX II: NOTATION
The following symbols are used in this paper:a = accelerationao = acceleration limitB = effective width of a panelD = flexural rigidity or transformed moment of inertia
per unit width of a panel
Table 6.Comparison of Various Design Criteria for Walking Vibrations
NaturalFrequency
fo (Hz)
Minimum Value of Damping Ratio Times Effective Mass Weight, ββWj (kN)
Note:1Results are given for a standard case of finished floor without full-height partitions (β = 0.03)
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION126
Es = modulus of elasticity for steelf = step frequencyfj = natural frequency of joist or beam panelfg = natural frequency of girder panelfo = fundamental natural frequency of floor structureg = acceleration due to gravity; subscript indicating
girderi = ith harmonic of step frequencyj = subscript indicating joint or beamK = factor in Equation 4 taking into account occupant
sensitivity to vibrationL = span of joint, beam, or girder (with subscript j
or g)P = weight of a person (0.7 kN assumed)R = reduction factor in Equation 2W = effective mass weight of floor vibrating in the
fundamental modew = unit weight of floor panel, including acting live
loadwj or wg = unit weight of joist or girder per unit lengthαi = dynamic load factor for ith harmonic of step fre-
quencyβ = damping ratio∆ = deflection of member under weight supported
APPENDIX III: BASIS FOR COMPARISON OFVIBRATION CRITERIA
Existing design criteria for walking vibration can be com-pared with the proposed criterion by considering a standardjoist or beam panel on stiff supports. To make a valid com-parison, each criterion must be considered as a total package.This requires adjustments to the criteria to take account ofdifferences in the form of the design equations and in therecommended values of design parameters. To make a com-parison, all criteria will be transformed to a common measureβWj as defined for the proposed criterion.
The following frequency relationship for a simply sup-ported joist panel will be used to transform all criteria to thecommon measure, βWj:
fo = π2
√gDj wLj
4 (A1)
where w is the unit weight of the panel
Canadian Standards Association (CSA, 1989)
This criterion has been used in Canada since 1975, with minormodifications in 1984. For the standard joist panel, the CSAcriterion can be expressed as follows:
w(40tc)Lj (kN) > 0.6fo / (ao / g) (A2)
where tc is the effective concrete thickness, 40tc is the effec-tive slab width, and ao / g is a limiting heel-impact accelera-
tion determined from Figure 6. Equation A2 can be expressedin terms of Wj if a correction is made for the effective panelwidth. For a typical case of a 5.5 Hz floor, span Lj = 12 m andconcrete thickness tc = 75 mm, application of Equation 7results in an effective width of 8.3 m or 110tc compared to40tc in Equation A2. If Equation A2 is multiplied by110β / 40 it becomes
βWj (kN) > β1.65fo / (ao / g) (A3)
Minimum values of βWj for the CSA criterion in Table 6were determined from Equation A3 using the criterion forfinished floors in Figure 6 and β = 0.03 from Table 3.
Murray (1981)
On the basis of a review of field data from 91 floors, Murray(1981) recommended the following criterion, presentlywidely used in the U.S.:
β > 0.35Ao fo + 0.025 (A4)
where Ao is the initial amplitude of vibration (inches) due to
Fig. 6. Annoyance criteria for floor vibrations in residential,school, and office occupanices (CSA, 1989).
FOURTH QUARTER / 1993 127
a standard heel impact. Equation A4 is plotted in Figure 7along with the floor data. To determine Ao Murray providesthe following expression for a simply supported one-wayfloor system:
Ao = DLF × ∆s (A5)
where ∆s is the static deflection of the joist panel under aconcentrated load of 600 lb. and DLF is a dynamic load factorto obtain the maximum amplitude of vibration for a standardheel impact. DLF ranges from 0.15fo at fo = 4 Hz to 0.12fo atfo = 10 Hz, and can therefore be approximated by 0.14fo, itsvalue at fo = 6 Hz. Thus,
Ao = (1.14fo ) 600Lj
3
48Dj BM (A6)
where BM is the effective joist panel width as defined later.Substitution of Equation A6 in Equation A4 after eliminationof Dj by means of Equation A1 results in the followingcriterion:
β > 584
wBM Lj + 0.025 (A7)
For the standard case of finished office floor without full-height partitions, β = 0.03 according to Table 3 and β = 0.045according to Murray. For this case Equation A7 becomes
wBM Lj = 584 / (0.045 − 0.025) = 29,200 (A8)
Murray (1991) provides expressions for determining BM interms of beam or joist spacing times the number of effectivejoists. Two expressions are used, one for normal hot-rolledbeam (spacing more than 30 in.); the other for closely spaced
joists (30 in. or less). The expression for narrow spacing isequivalent to
BM = 3√2
π
DS Dj
1⁄4
Lj = 1.35
DS Dj
1⁄4
Lj = 0.675Bj (A9)
where Bj is defined according to Equation 7, and the expres-sion for wide spacing can be approximated by
Bm = 1.03
DS Dj
1⁄4
Lj = 0.515 Bj (A10)
Substitution of Equations A9 or A10 in Equation A8 resultsin minimum values of Wj equal to 43,260 lb for narrowspacing and 56,700 lb for wide spacing. For the standard case,β = 0.03, the corresponding minimum values of βWj includedin Table 6 are 1,300 lb (5.8 kN) and 1,700 lb (7.6 kN).
Wyatt (1989)
Wyatt (1989) proposed two design criteria for office floors,one a resonance criterion for floor frequencies up to 7 Hz, theother an impulse response criterion for floor frequenciesgreater than 7 Hz. For the one-way beam or joist system, theresonance criterion can be expressed (with rearrangement andchange of symbols) as
β(wBw Lj) > 667Cf / F (A11)
where Cf is a loading coefficient (0.4 for second harmonicloading and 0.2 for third harmonic loading), F is a ratingfactor which depends on the office environment (12 for a busyoffice, 8 for a general office, and 4 for a special office) andBw is the joist panel width. For the one-way system Wyattrecommends
BW = 4.5
gDS fo
2 w
1⁄4
(A12)
which can be expressed in the same form as Equation 7 byuse of Equation A1. After substitution, Bw in Equation A12becomes equal to 1.8Bj, where Bj is defined by Equation 7.Wyatt, however, recommends a concrete modulus elasticity25 percent higher than recommended for Ds in Equation 7.With this correction Bw, becomes equal to 1.9Bj. Equation A11can therefore be expressed as
βWj > 351Cf / F (A13)
for floor frequencies below 7 Hz. Table 6 contains minimumvalues of βWj assuming F = 8 for a general office.
For floor frequencies greater than 7 Hz Wyatt recommendsthe following impulse criterion:
wSLj > 294 / F (A14)
where S is the member spacing. Equation A14 may be ex-pressed in terms of βWj if it is multiplied by βBj / S. Based on
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION128
an assumed beam spacing of 2.5 m used in Wyatt’s examplesand a typical value Bj = 6.8 m for an 8 Hz floor of span 9 mand concrete thickness of 75 mm. Equation A14 can beapproximated by
βWj > 800β / F (A15)
Table 6 contains a minimum value of Equation A15 atfo = 8 Hz for a general office floor for which F = 8 and β =0.03.
The OHBDC (1983) design criterion for footbridges is basedon a pedestrian or jogger exerting a dynamic force ofαP cos 2π ft where P is 0.7 kN, α = 0.257 and f, the stepfrequency, takes on any value between 1 and 4 Hz. Thefootbridge is modeled as an SDOF beam which vibrates at the
first flexural frequency, fo. For a simply supported footbridge,the resonance response for flexural frequencies up to 4 Hz canbe determined from Equation 2 with a value of R which isdetermined by the length of the footbridge. If, for a typicalcase R is assumed equal to 0.7, the maximum acceleration isdetermined from
amax / g = 0.7(0.257)0.7 / βWj = 0.126 βWj (A17)
where Wj is the weight of the footbridge. The OHBDC rec-ommends limiting values of amax / g equal to 0.042 at fo = 2 Hzand 0.072 at fo = 4 Hz. Thus Equation A17 can be inverted toa criterion for minimum value of βWj equal to 0.126 / 0.042= 3 kN at fo = 2 Hz and 0.126 / 0.072 = 1.8 kN at fo = 4 Hz.
For a flexural frequency beyond 4 Hz, the OHBDC givesan incorrect assessment because it neglects resonance withthe higher harmonics of the walking and jogging forces.
FOURTH QUARTER / 1993 129
ABSTRACT
A practical LRFD-based analysis method for the design ofsemi-rigid frames is proposed. The proposed method usesfirst-order elastic analysis with a notional lateral load for thesecond-order effects. In the proposed method, a simplifiedthree-parameter model describing the tangent rotational stiff-ness of semi-rigid connections is used.
1. INTRODUCTION
Although partially restrained (PR) construction is permittedby the AISC Specification for Structural Steel Buildings—Load and Resistance Factor Design, no specific analysis ordesign guidance is given in the current LRFD and ASDspecifications for these partially restrained frames.
Recently, a simplified procedure for the analysis and de-sign of semi-rigid frames was proposed by Barakat andChen,1 using the B1 and B2 amplification factors together withthe beam-line concept. However, the beam-line method cannot adequately predict the drift of unbraced frames and thecalculation of effective length factor is cumbersome andtime-consuming.
A simplified procedure to improve these drawbacks isintroduced in this paper. Here, as in the Barakat method, theproposed method is based on first-order linear elastic analy-sis, but the second-order effect will be included with the useof notional lateral loads.
2. MODELING OF SEMI-RIGID CONNECTIONS
2.1 Connection Models
Most existing connection models express the moment interms of rotation from which the tangent stiffness can bederived. This paper proposes a direct tangent-stiffness expres-sion for flexible connections. This proposed tangent-stiffnessmodel is based on the concept that connection stiffness de-grades gradually from an initial stiffness, Ki, to zero followinga nonlinear relationship of the simple form:
dMdθr
= Kt = Ki 1 −
MMu
c
(1)
where
Kt = tangent stiffnessKi = initial connection stiffnessMu = ultimate bending moment capacityM = connection momentC = shape factor account for decay rate of Kt, C > 0
The moment-rotation (M − θr) behavior of bolted extendedend-plate beam-to-column connections tested by Yee andMelchers2 is compared with the proposed model in Figure 1and a good agreement is observed with C = 1.6. In Figure 1,the initial stiffness, Ki = 546,666 in-kip/rad, is the tangent tothe starting point of the curve. The ultimate moment capacity,Mu = 3,539 in-kip, is determined by test. The value C is usedto control the shape of a convex curve. If C is equal to 1, Kt
decreases linearly. When C is less than 1, Kt decreases morerapidly. If C is greater than 1, Kt decreases much slower. Thisis illustrated in Figure 1 with C = 1.0, 1.6, and 2.2 respectively.
In the following, the proposed tangent stiffness connectionmodel will be applied to several types of connections, includ-ing the extended end-plate, top and seat angle with doubleweb angles, framing angles, and single-plate connections.
LRFD Analysis for Semi-Rigid Frame DesignWON-SUN KING AND WAI-FAH CHEN
Won-Sun King is associate professor, Department of CivilEngineering, Chung Cheng Institute of Technology, Ta-Hsi,Tao-Yuan, Taiwan.
Wai-Fah Chen is professor and head of Structural Engineer-ing, School of Civil Engineering, Purdue University.
Fig. 1. Moment-rotation curves of Yee connection (1986).
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION130
The connections ranged from very stiff to rather soft connec-tions. The moment-rotation curve is obtained by numericalintegration of tangent-stiffness Equation 1.
(a) Jenkins Bolted Extended End Plate
A comparison of the proposed model with one of the Jenkins,Tong, and Prescott extended end-plate connection test3 isshown in Figure 2. In this test, the beam is 305×165 UB54,the column is 254×254 UC132 (stiffened), bolts are M20grade 8.8, thickness of end plate is 20 mm and Ki = 786,732in.-kip/rad and Mu = 1,989 in.-kip. Good agreement is ob-served with C = 0.555.
(b) Bolted Top and Bottom Angles with Web Angles
Azizinamini, Bradburn, and Radziminski4 reported test re-sults on bolted semi-rigid steel beam-to-column connections.These connections are comprised of top and bottom anglesconnected to the flanges along with web angles. ASTM A36steel was used for the members and the connection elements.Eighteen specimens were tested. The beam tested was aW14×38, the bolt diameter is 22.2 mm, and the web anglesare 2L4×31⁄2×1⁄4. The thickness of flange angles is 15.9 mm,and the length of the test beam is 203.2 mm. The test number14S8 with Ki = 677,025 in.-kip/rad and Mu = 1,707 in.-kipcompares well with that of the proposed model in Figure 3with C = 0.34.
(c) Bolted Framing Angles
Bolted double-web angles were tested by Lewitt, Chesson,and Munse at the University of Illinois. In 1987, Richard, etal5 proposed a four-parameter formula to describe these full-scale tests. Figure 4 compares the results of the proposedmodel with one of these tests using a five-bolt design withrivets in the angle-to-beam web connection with Ki = 206,667in.-kip/rad and Mu = 761 in.-kip.
(d) Single Plate
A total of seven tests were made by Richard6 on single-plateconnections. The first set of two-, three-, five-, and seven-bolttests were run with the framing connection plate welded to aflange plate which was in turn bolted to the support column.A second set of tests was run on the two-, three-, and five-boltconnections with the framing connection plate welded to thesupport column. In these tests, three bolts were used toconnect the beam and the single plate. The bolts are A3253⁄4-in. diameter, and the plate thickness is 3⁄8-in. The moment-rotation curve of the proposed model compares well with oneof the tests as illustrated in Figure 5 with Ki = 51,000 in.-kip/rad, Mu = 137 in.-kip, and C = 0.22.
2.2 Initial Stiffness
For simplicity, researchers7,8,9 have been using the initialconnection stiffness, Ki, for their semi-rigid frames analysis.The use of initial stiffness throughout the flexible frameanalysis results in a frame behavior that is generally too stiffwhen the frame is subjected to a normal loading condition.
Extensive studies of frames by Ackroyd10 with nonlinearconnections indicate that the secant stiffness of beam-to-col-umn connections near ultimate frame capacity was typically20 percent of the initial stiffness, Ki, at leeward ends of girdersand 80 percent of Ki at the windward ends of girders, whenthe frame is subjected to combined gravity and wind loading.It seems, therefore, reasonable to use an average connectionstiffness of 0.5Ki when computing the design moments. Thisis adopted in the present analysis.
3. DESIGN FORMULA IN AISC-LRFD
The equation for the maximum strength of beam-columns isgiven by AISC-LRFD as
Fig. 2. Moment-rotation curves of Jenkins connection (1986).Fig. 3. Moment-rotation curves of Azizinamini
connection (1987).
FOURTH QUARTER / 1993 131
for Pu
φcPn ≥ 0.2,
Pu
φc Pn +
89
Mux
φb Mnx
+ Muy
φb Mny
≤ 1.0 (2)
for Pu
φcPn < 0.2,
Pu
2φc Pn +
Mux
φb Mnx
+ Muy
φb Mny
≤ 1.0 (3)
where
Pn = ultimate compression capacity of an axiallyloaded column
Mnx, Mny = ultimate moment-resisting capacity of alaterally unsupported beam about x and y axes,respectively
φc = column resistance factor (= 0.85)
φb = beam resistance factor (= 0.9)
Pu = design axial forceMux, Muy = member design moment about x and y axes,
respectively, with:
Mu = B1Mnt + B2Mlt (4)
Mnt = first-order moment in the member assuming nolateral translation in the frame
Mlt = first-order moment in the member as a result oflateral translation of the frame
B1 = P-δ moment amplification factor
B1 = Cm
1 − Pu Pe
≥ 1 (5)
B2 = P-∆ moment amplification factor
B2 = 1
1 − ΣPu ∆o
ΣHL
(6)
Cm = 0.6 − 0.4M1 / M2, where M1 / M2 is the ratio of thesmaller to the larger end moment of a member
Pe = π2EI / (KL)2
ΣPu = axial loads on all columns in a story
∆o = first-order translational deflection of the story underconsideration
ΣH = sum of all story horizontal forces producing ∆o
L = story heightK = effective length factor determined from the
alignment chart
The second-order effects are taken into account approxi-mately by the moment amplification factors B1 and B2 on thenonsway and sway moments obtained from first-order elasticanalyses, respectively. It usually leads to conservative results.
For second-order elastic analysis, we use the usual elementgeometric stiffness matrix combined with the update of theelement geometry during the analysis. The first three terms inthe Taylor series expansion of the elastic stability functionsare retained for the axial compressive force P to increase theaccuracy of the element stiffness. The corresponding terms instiffness matrix were obtained by Goto and Chen11 as
Kii = 4EIL
+ 2PL15
+ 44P2L3
25,000EI(7)
Kij = 2EIL
− PL30
− 26P2L3
25,000EI(8)
Fig. 4. Moment-rotation curves of Lewittconnection (Richard, 1987). Fig. 5. Moment-rotation curves of Richard connection (1980).
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION132
Kji = 2EIL
− PL30
− 26P2L3
25,000EI(9)
Kjj = 4EIL
+ 2PL15
+ 44P2L3
25,000EI(10)
where
P = axial force in memberA = area of a cross sectionri = internal reactions at both ends of a member = 1,6
di = displacements at both ends of a member = 1,6
The beam stiffness matrix in Equation 11 can be simplifiedby recognizing that the axial force in beams of rectangularframes is usually negligible. That is, by setting Kii = Kjj =4EI / L, Kij = Kji = 2EI / L, and P = 0.
The stiffness matrix of a beam-column can be modified toinclude the effect of semi-rigid connections by combining themember stiffness with the connection stiffness using a staticcondensation. Details of this procedure are given in Chen andLui,12 and the resulting member stiffness matrix has the form:
=
r1
r2
r3
r4
r5
r6
AEL
0 0−AE
L0 0
d1
d2
d3
d4
d5
d6
(11)
0(Kii + 2Kij + Kjj)
L2 + PL
(Kii + Kji)L
0− (Kii + 2Kij + Kjj)
L2 − PL
(Kij + Kjj)L
0(Kii + Kij)
LKii 0
− (Kii + Kij)L
Kij
−AEL
0 0AEL
0 0
0− (Kii + 2Kij + Kjj)
L2 − PL
− (Kii + Kji)L
0(Kii + 2Kij + Kjj)
L2 + PL
− (Kij + Kjj)L
0(Kji + Kjj)
LKji 0
− (Kji + Kjj)L
Kjj
=
r1
r2
r3
r4
r5
r6
AEL
0 0−AE
L0 0
d1
d2
d3
d4
d5
d6
(12)
0(Kii ′ + 2Kij ′ + Kjj′)
L2 + PL
(Kii ′ + Kji ′)L
0− (Kii ′ + 2Kij′ + Kjj′)
L2 − PL
(Kij ′ + Kjj′)L
0(Kii ′ + Kij ′)
LKii ′ 0
− (Kii ′ + Kij′)L
Kij′
−AEL
0 0AEL
0 0
0− (Kii ′ + 2Kij ′ + Kjj′)
L2 − PL
− (Kii ′ + Kji′)L
0(Kii ′ + 2Kij ′ + Kjj′)
L2 + PL
− (Kij ′ + Kjj′)L
0(Kji ′ + Kjj ′)
LKji ′ 0
− (Kji ′ + Kjj′)L
Kjj′
FOURTH QUARTER / 1993 133
where
Kii′ = Sii′ +
Sii′Sjj′Rj
− Sij′Sij′
Rj
1R∗
(13)
Kjj′ = Sjj′ +
Sii′Sjj′Ri
− Sij′Sij′
Ri
1R∗
(14)
Kij′ = Kji′ = Sij′R∗
(15)
The coefficients Ri and Rj in Equations 13 and 14 are theinstantaneous tangent stiffness coefficients of the connectionsat ends i and j of the member respectively. These coefficientsare obtained from Equation 1 when the connection is in thestate of loading, and are set equal to Ki when the connectionis in the state of unloading. Also, the parameter R* is givenby
R∗ = 1 +
Sii′Ri
1 +
Sjj′Rj
−
Sij′Sij′Ri Rj
(16)
Sii′ = Sjj′ = Kii = Kjj (17)
Sij′ = Sji′ = Kij = Kji (18)
in which P is negative for compressive force, and is small orzero for beam elements and can be neglected.
5. THE PROPOSED METHOD
Several simplifications are made in the present formulation.The moments of beam-column joints must be less than theultimate moment Mu of semi-rigid connections or the plasticmoment capacity Mpc of beam-columns. The combined axialload and end moments in any member must satisfy the AISC-LRFD bilinear interaction equations.
5.1 Rigid Frame Analysis
1. Perform the first-order elastic rigid frame analysis.2. Compute notional lateral loads, ΣH′, using the relation-
ship
SF = ΣH
∆o
= ΣH + ΣPu ∆ / L
∆ =
ΣH ′∆
(19)
where ∆o is the first-order translational deflection of the story,and ∆ is the second-order translational deflection of the storyunder consideration. From Equation 19, we have
∆ = (ΣH + ΣPu ∆ / L)∆o
ΣH =
1 +
ΣPu ∆ΣHL
∆o (20)
from which we obtain
∆ 1 −
ΣPu ∆o
ΣHL
= ∆o (21)
or
∆ = ∆o
1 −
ΣPu ∆o
ΣHL
= B2∆o (22)
The ∆ is the second-order lateral deflection due to P-∆effect, and the notional lateral load is defined as
ΣH′ = ΣH + ΣPu ∆ / L (23)
3. Use ΣH′ and original gravity loads to perform first-orderelastic rigid frame analysis. The results of this stepinclude the second-order effect.
4. Calculate B1 factor with the effective length factor K =1.0 for each column and multiply the corresponding endmoments.
5. Check the AISC-LRFD bilinear interaction equations.
5.2 Semi-Rigid Frame Analysis
1. Select connections from the maximum beam-columnjoint moments in rigid frame analysis.
2. Determine the initial stiffness, Ki, of connections fromtest results or any other available methods.
3. Substitute 0.5Ki of connection stiffness for the semi-rigid joint. The average connection stiffness 0.5Ki assuggested by Ackroyd10 is adopted here.
4. Use 0.5Ki for semi-rigid connection stiffness with thenotional lateral loads ΣH′ to carry out the first-orderelastic analysis.
5. Calculate B1 factor with effective length factor K = 1.0for Pe of each column and multiply the correspondinglarger end moments.
6. Check the AISC-LRFD bilinear interaction equations.The effective length factor, K, for the column strength,Pn, has to be modified in the case of semi-rigid frames.For beams connected to columns with semi-rigid con-nections rotational stiffness, Ki, at both ends, a simplemodification of the relative stiffness, G, factors with themodified moment inertia of beam is (Chen and Lui13):
I′ = I
1 + 2EIKiL
(24)
The I′ is used in G factors for the determination of theeffective length factor, K, for the value of Fcr which is thecritical column stress.
6. NUMERICAL EXAMPLES
The proposed method will now be illustrated by numericalexamples. Comparisons are made between results using di-rect second-order elastic analysis, Barakat’s method,1 and theproposed method. The semi-rigid frame examples includesingle-story and multi-story frames. All examples are ana-
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION134
lyzed with a personal computer. All beams subjected to uni-formly distributed loads are divided into two equal elements.
6.1 Two-Story One-Bay Frame with ConcentratedLoads
The two-story one-bay frame as shown in Figure 6 is analyzedwith both rigid and semi-rigid connections. Two lateral loads,H, and four constant concentrated gravity loads, P, of 100 kipsare applied at the beam-column joints of the frame. Theflexible connections used are shown in Figure 2 where Mu isless than the plastic moment Mp of beams and columns. The0.5Ki of Jenkins connection is 393,366 in-kip/rad. The sec-ond-order lateral displacement at Joint 5 is
All moments of beams and columns predicted by theproposed method are normalized with respect to that ofsecond-order elastic analysis and are summarized in Tables 1and 2. The lateral displacements at windward beam-columnjoints are shown in Table 3. The mean values are the sum ofnormalized values of each member divided by the number oftotal members. All the results predicted by the proposedmethod are close to the exact solutions. It is found that themaximum moment and lateral displacement can be predictedwell by the proposed method.
6.2 Two-Story One-Bay Frame with UniformlyDistributed Loads
A two-story one-bay frame used by Barakat,1 et al as shownin Figure 7(a) is employed here for comparison of the maxi-mum moments in members. The semi-rigid connection la-beled III-17 is shown in Figure 8 and compared with theproposed connection model. The moments predicted by sec-ond-order elastic analysis the Barakat method, and the pro-posed method are compared in Table 4. The average value ofColumn 3 in Table 4 is 0.98, while the average value ofColumn 5 is 0.97. The Barakat method is slightly less conser-vative in this example.
To verify the validity of the proposed method for softsemi-rigid connections, the bolted framing angles tested byLewitt5 are used. The lateral loads and uniformly distributedFig. 6. Two-story one-bay frame with concentrated loads.
loads are reduced as shown in Figure 7(b), so the maximummoments in members of the two-story one-bay frame are lessthan the ultimate moment, Mu, semi-rigid connections. Theresults predicted by the proposed method are compared withthat of second-order elastic analysis in Table 4(b) and thelateral displacements are shown in Table 4(c). It can be
concluded that the proposed method is valid for soft connec-tions, although the second-order lateral loads in the proposedmethod are determined from a rigid frame.
6.3 Three-Story One-Bay Frame with UniformlyDistributed Loads
The three-story one-bay frame shown in Figure 9 is analyzedwith semi-rigid connections labeled III-17. Three beam-col-umn joints are subjected to concentrated lateral loads. All thebeams are subjected to uniformly distributed gravity loads.The results by the Barakat method1 are compared with thoseresults of the proposed method (Table 5). The average valueof Column 3 in Table 5 is 1.00, while the average value of
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION136
Column 5 is 0.97. It can be seen that the Barakat method isless conservative in this case.
6.4 Four-Story Two-Bay Frame with UniformlyDistributed Loads
A four-story two-bay frame as shown in Figure 10 is investi-gated here for the maximum column moments in both rigidand semi-rigid frames. The semi-rigid connection of Jenkinsis utilized. The average value of Column 3 in Table 6 is 1.06,while the average value of Column 6 in Table 6 is 1.01. The
proposed method represents reasonably well the second-or-der effect for rigid and semi-rigid frames. The lateral dis-placements at windward beam-column joints are summarizedin Table 7. The lateral displacements predicted by the pro-posed method are less than those of the second-order elasticsemi-rigid frame analysis. However, the lateral displacementspredicted by the proposed method are larger than that of thesecond-order elastic rigid frame analysis.
7. SUMMARY AND CONCLUSIONS
Several conclusions can be drawn from the present studies:
1. The moment-rotation relationships of semi-rigid con-nections as represented by a simple tangent stiffnessFig. 8. Experimental III-17 connection curves (Barakat, 1991)
expression are convenient and can lead to a close mo-ment-rotation curve by numerical integration whencompared with test result. Note that only the tangentstiffness is needed in an incremental nonlinear frameanalysis.
2. The proposed method gives close results to that of sec-ond-order elastic analysis. It can handle both the uni-formly distributed gravity loads and concentrated loads,and predicts well the drift of unbraced frames.
3. All mean values of the normalized moment ratios arefound close to or slightly greater than one in the pro-posed method. This shows that the proposed method ismore accurate when compared with that of the Barakatmethod. The proposed method gives a reasonable pro-cedure for estimating the approximate P-∆ column mo-ments for both rigid and semi-rigid frames.
4. The notional lateral loads calculation is relatively simpleand straightforward because the tedious determination
of the effective length factor, K, can be avoided. It is asimple and practical method for semi-rigid frame design.
REFERENCES
1. Barakat, M. and Chen, W. F., “Design Analysis of Semi-Rigid Frames: Evaluation and Implementation,” AISC,Engineering Journal, 2nd Qtr., 1991, pp. 55–64.
2. Yee, Y. L. and Melchers, R. E., “Moment-Rotation Curvesfor Bolted Connections,” ASCE, J Struct. Eng., 112(3),1986, pp. 615–634
3. Jenkins, W. M., Tong, C. S. and Prescott, A. T., “Moment-Transmitting End-Plate Connections in Steel Construc-
Fig. 10. Four-story two-bay frame.Fig. 9. Three-story one-bay frame with
uniformly distributed loads.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION138
tion, and a Proposed Basis for Flush End-Plate Design,”Struct. Engrg., 64A(5), 1986, pp. 121–132.
4. Azizinamini, A., Bradburn, J. H., and Radziminski, J. B.,“Initial Stiffness of Semi-Rigid Steel Beam-to-ColumnConnections,” J. Construct. Steel Research 8, 1987, pp.71–90
5. Richard, R. M., Hsia, W. K. and Chmielowiec, M., “Mo-ment Rotation Curves for Double Framing Angles,” Ma-terials and Member Behavior, 1987, 107–121.
6. Richard, R. M., Gillett, P. E., Kriegh, J. D. and Lewis, B.A., “The Analysis and Design of Single-Plate FramingConnections,” AISC, Engineering Journal, 2nd Qtr.,1980, pp. 38–52.
7. Frye, M. J. and Morris, G. A., “Analysis of FlexiblyConnected Steel Frames,” Can. J. Civ. Eng., 1975, 2(3),pp. 280–291.
8. Ang, K. M. and Morris, G. A., “Analysis of Three-Dimen-sional Frames with Flexible Beam-Column Connec-tions,” Can. J. Civ. Eng., 11, 1984, pp. 245–254.
9. Romstad, K. M. and Subramanian, C. V., “Analysis ofFrames with Partial Connection Rigidity,” ASCE, J.Struct. Div., 96(11), 1970, pp. 2283–2300.
10. Ackroyd, M. H., “Simplified Frame Design of Type PRConstruction,” AISC, Engineering Journal, 4th Qtr.,1987, pp. 141–46.
11. Goto, Y. and Chen, W. F., “Second-Order Elastic Analysis
for Frame Design,” ASCE, Journal of Structural Engi-neering, Vol. 113, No. 7, 1987, pp. 1501–1519.
12. Chen, W. F. and Lui, E. M., Structural Stability: Theoryand Implementation, Elsevier, New York, 1987.
13. Chen, W. F. and Lui, E. M., “Stability Design Criteria forSteel Members and Frames in the United States,” J. ofConstr. Steel Research, 5, Great Britain, 1985, pp. 31–74.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION140
ABSTRACT
This paper is concerned with I-beam to box-column connec-tions stiffened externally. A design method to determine thedimensions of T-stiffeners is proposed. Connections of I-beams and box-columns for a wide range of dimensions werestudied by the finite-element method and found to satisfy thebasic design criteria for a moment connection. The Ramberg-Osgood function was used to curve-fit the moment-rotationcurves based on the geometric parameters of the connectionsand the results are found to agree well with those from thefinite-element analyses. Finally, the design procedure and thecurve-fitting parameters were compared with the experimen-tal results of a four-way connection tested to failure.
INTRODUCTION
It is a well known fact that the behavior of beams and columnsat their connection is one of the most important factorsconsidered in the analysis of steel frames. A vast number ofdifferent types of connections are used, and the rigidity ofconnections range from one that is extremely flexible, behav-ing more like a pin joint, to one that is almost rigid. Re-searchers have carried out studies on the effect of the semi-rigid connection on frame behavior. A state-of-the-art paperwas presented by Jones, et al1 on the analysis of frames withsemi-rigid joints. A modified stiffness matrix method, incor-porating the partial rigidity of joints to find the elastic buck-ling load of semi-rigid frames was presented by Yu andShanmugam.2 Gerstle3 noted that the effect of the connectionflexibility on frames can be two-fold: (a) connection rotationcontributes to the overall frame stability and (b) it affects thedistribution of internal forces and moments in the girders andcolumns. The effects of connections on columns was consid-ered by Nethercot and Chen4 and Jones, et al5 while Kato, etal6 carried out a study on the effect of joint flexibility due tojoint-panel shear deformation on frames. Barakat and Chen7
used idealized connection models in the analysis of frames
and subsequently implemented the method on personal com-puter.
Analytical models incorporating semi-rigid connectionswill result in efficient design. In such design the frame mem-bers will be utilized more efficiently, resulting in a lower cost.Therefore, there is an important need to accurately determinethe moment-rotation (M-φ) characteristics of various types ofconnections and to define them in a convenient way suitablefor incorporating them in frame analyses. Many researchershave carried out both analytical and experimental investiga-tions on the behavior of different types of connections. Vari-ous types of models defining the M-φ relationship, rangingfrom the simple linear model in the 1930s to the present daycomplicated cubic B-spline curve-fitting model have beenreported by Jones, et al1 The curve fitting technique was alsoused by Attiogbe and Morris.8 Experimental data was fittedto the Richard-Abbott function while Ang and Morris9 usedthe Ramberg-Osgood function for the curve fitting process.Due to the diversity of the behavior of the connections,researchers10,11 have proposed a classification system in anattempt to present the behavior of connections consistently.An alternative approach was to build up a data base for thevarious types of connections12,13 so that designers can obtainthe necessary data for their specific use.
However, most of the past work was carried out on connec-tions between I-beams and I-columns. Limited work is avail-able on the behavior of I-beam to box-column connec-tions.14,15,16 The authors have carried out an investigation onsuch connections stiffened externally using the finite elementmethod.17 Experimental and analytical results from tests car-ried out on a series of specimens stiffened both internally aswell as externally have been reported.18 It has been found thatthe T-section provides an efficient external stiffener for theconnection. In this paper, a simple design procedure to deter-mine the dimension of external T-stiffeners is proposed. Acurve fitting method using the Ramberg-Osgood function todefine the M-φ characteristic of such connections is alsodiscussed.
BASIC DESIGN PHILOSOPHY
The basic design criteria for rigid or moment connectionsare:19
1. sufficient strength
Lai-Choon Ting, Nandivaram E. Shanmugam and Seng-Lip Leeare research assistant, associate professor and emeritus profes-sor, respectively, of the Department of Civil Engineering, NationalUniversity of Singapore, Singapore.
Design of I-Beam to Box-Column ConnectionsStiffened ExternallyLAI-CHOON TING, NANDIVARAM E. SHANMUGAM AND SENG-LIP LEE
FOURTH QUARTER / 1993 141
2. sufficient rotation capacity3. adequate stiffness4. ease of erection and economical fabrication
Whether a particular connection satisfies the first threeconditions can be determined by observing its moment-rota-tion curve. The last criterion is a matter of practical applica-tion which has to take into account both the material cost ofthe various components of the connection and the labor costin fabricating it.
Figure 1 shows four moment-rotation curves exhibitingdifferent characteristics of connections. Connection A is con-sidered to be properly designed as it satisfies criteria (1) to(3), i.e., it can attain sufficient strength in excess of the plasticmoment of the beam as well as having adequate stiffness androtation capacity before failure. Connection B, however, hasinsufficient rotation capacity although it is adequate in termsof stiffness and strength. As for connection C, it only hasrotation capacity but not stiffness and strength while connec-tion D has neither sufficient strength nor rotation capacity. Itis the objective of this paper to present a design method forI-beam to box-column connections stiffened externally withT-stiffeners (Figure 2) which can exhibit the property of theconnection A in Figure 1.
CONNECTIONS WITH EXTERNAL T-STIFFENERS
When an I-beam frames into a box-column, the width of thebeam flange is normally less than the column width, as a resultthe connection will be weak if it is not stiffened. In order toachieve the conditions of an acceptable moment connection(connection A in Figure 1), the traditional method is to stiffenthe connection by welding internal continuity plates at thelevels of the beam flanges inside the box-columns. This,however, is a difficult and expensive process. The authorshave carried out a study17,18 to investigate the possibility of
using external stiffeners in place of internal continuity platessuch that the basic design criteria are still satisfied.
The investigations showed that by using external T-stiffen-ers, all the basic design criteria can be satisfied. Two series ofexperiments have been carried out18 on connection specimensstiffened by internal continuity plates or by two differenttypes of external stiffeners namely angle and T-stiffeners. Inthe first series, the specimens were subjected to a monotoni-cally increasing load while in the second series, the specimenswere subjected to cyclic loads. The specimens consisted oftwo 1.5 m long beams welded to opposite sides of a box-col-umn of 1.0 m height. The dimensions of the external stiffenerswere designed based on a preliminary design method by usingfinite element analysis. The length of the stiffeners were sochosen that the normal stress distribution is uniform acrossthe stiffener. This would prevent any premature failure of thestiffeners due to stress concentration. However, it was foundthat this method gives rise to overdesign of the stiffeners andan alternative method of design for the T-stiffener is, there-fore, presented in this paper.
Due to the complexity of the connection involved, the finiteelement method was used to analyze the connections.MSC/NASTRAN,20 which can carry out both material andgeometrical non-linear analyses was used to analyze all theconnections. Due to the symmetry of both the model andloading, only a quarter of the model was analyzed and atypical finite-element mesh is shown in Figure 3. The resultsobtained are compared with those obtained from the experi-ments.
T-STIFFENER DESIGN
Minimum stiffener length for stiffness
From the finite-element analyses, it was observed that aminimum length for the T-stiffener is required to transfer theforces from the beam flanges to the column webs effectively.It was also found that for the stiffener to be effective, its webthickness must be at least equal to half that of the beam-flange
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION142
thickness. Otherwise, the stiffener web may yield prema-turely, resulting in a weak connection.
Figure 4 shows the load-deflection curves of a typicalconnection with external T-stiffeners of various lengths.These curves were obtained by the elasto-plastic finite-ele-ment method using the MSC/NASTRAN package. The plas-tic capacity of the beam, Pp is also shown. The lengths weredefined by the angle θ as shown in the figure. The curvescorresponding to θ = 15° and 20° have sufficient stiffness andstrength while the other two curves are more flexible. Thecurve corresponding to θ = 15° show marginal increase inultimate strength capacity over the curve with θ = 20°. It wasthus decided to adopt θ = 20° as the design criterion becausethis would result in a shorter stiffener length and hence moreeconomical design. The stiffener length l can thus be writtenas
l = (B − b) / (2 tan 20°)
where
B = column widthb = beam-flange width
The other factor affecting the stiffness of the connection isthe stiffener-flange width which is connected to the edge ofthe column web. This stiffener flange serves two purposes: 1)it increases the moment of inertia of the beam cross-sectionat the connection significantly, thus increasing the stiffness of
the connection, and 2) it transfers the stresses from the beamto the column web more evenly, minimizing the possibility ofstress concentration.
Minimum Stiffener Length for Strength
The length of the stiffener, therefore, depends upon the ratioof beam-flange width to column-flange width (b / B). Whenthis ratio reaches a value close to one, the stiffener length willbecome so short that it will result in premature failure at thestiffener web. For such cases, a check has to be made on theminimum length based on the strength criteria of the stiffenerweb. To determine the minimum length based on this type offailure of the web, the following assumptions are made. Themoment developed at the connection should be at least equalto the plastic moment capacity Mp of the beam and it is carriedby the beam flanges such that Tp = Mp / db (Figure 5); the stressdistribution at failure on the beam flanges and stiffeners areas shown in Figure 6(a) with stiffener flanges and stiffenerweb between the flange and K-line reaching yield; the flangeforces, Tp are transferred to the column webs through thestiffeners as shown in Figure 6(b). It can be seen from Figure6(b) that
Tp 2
= T1 + T2 (1)
where
T1 = (Af + Aw) (2)T2 = ltswτy (3)Af = stiffener flange areaAw = area of stiffener web between the flange and K-linel = stiffener lengthtsw = thickness of stiffener web
Fig. 3. Typical finite element mesh.Fig. 4. Load-deflection curves of specimen
with various stiffener lengths.
FOURTH QUARTER / 1993 143
τy = σyt √3σyt = tensile strength
The stiffener length l can be calculated for Equations 1–3.
DESIGN PROCEDURE
Given the dimensions of a beam and a column, the followingsimple procedure can thus be adopted to determine the suit-able size and length of the T-stiffener required. From thesection table, an I-beam or T-section having web thicknessequal to at least half the beam-flange thickness is chosen forthe stiffener. Assuming θ = 20° (Figure 4), the stiffener lengthis determined; the stiffener length based on the strengthcriteria is calculated from Equations 1–3. The larger stiffenerlength is finally chosen. All welds between the various com-ponents at the connection are assumed to be full penetrationwelds. Two examples based on the above design proceduresare shown in Appendix II.
LOAD-DEFLECTION CURVES FORTYPICAL SPECIMENS
Since there is no closed-form solution to define the behaviorof connections, the finite-element method has been used toanalyze these connections. This method has been shown topredict the load-deflection characteristic of specimens withreasonable accuracy. As such, it was decided to use thismethod to test the validity of the design procedure proposed.Two series of specimens were designed based on the aboveprocedure. One series consisted of specimens with two beamsframing into the box column on opposite sides while the otherseries consists of connections with four beams framing intothe column on all four sides. The same design procedure wasused for both series, resulting in the same stiffener size forconnections between a particular beam and column dimen-sions. Both the beam and box-column sections were obtainedfrom section tables.22 Beams and column sizes were takensuch that the whole range of sections in the table can berepresented. The external T-stiffeners were then designedaccordingly and the specimens were analyzed usingMSC/NASTRAN.
Figure 7 shows some typical load-deflection curves of one
of the specimens. For comparison, the results obtained usingthe simple elastic-plastic method are also plotted. It can beseen from the figure that the connections are able to developstrength well in excess of the plastic capacity of the beams.In addition, the initial stiffness of the connections satisfied thebasic criteria for a moment connection. The slight differencein the initial stiffness between the two- and four-way connec-tions is expected since the column web for the two-wayconnection is unrestrained in one direction while that of thefour-way connection is restrained all round.
It is commonly known that the behavior of connectionsbetween beams and columns is one of the most uncertainparameters in the design of frames at present. Analyses arecarried out assuming the connections to be either fixed orpinned but in practice it is never the case. An accurate mo-ment-rotation (M-φ) characteristic for different types of con-nection is therefore essential if any work is to be carried outto incorporate the semi-rigid nature of these connections.Many researchers have tried to standardize the M-φ relation-ships for various types of connections so that they can beincorporated into the computer programs during the analysisof frames. One of the most common method of standardizingthe M-φ curves is by curve-fitting the available test data forthe different types of connections.
Fig. 5. Internal forces at connection under symmetrical load. Fig. 6. Stress distribution at failure.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION144
STANDARDIZED MOMENT-ROTATIONFUNCTIONS
In the present study, a series of 15 M-φ curves has beengenerated for each of the two cases, namely, two-way andfour-way connections using the finite-element method. Thesespecimens cover the combination of full range of beams andbox columns available in the section table.22 The Ramberg-Osgood function was used to curve-fit the data. This functioncan be expressed in terms of moment, M, and rotation, φ, asfollows:
φφο
= MMo
1 +
MMo
(n − 1)
(4)
where Mo, φo, and n are the independent parameters of thefunction. Mo and φo are the reference moment and rotationrespectively, while n defines the sharpness of the curve. Theseindependent parameters can be expressed in terms of thegeometric properties of the connection as follows:
Mo = ∏ i = 1
m
piai (5)
φo = ∏ i = 1
m
pibi (6)
n = ∏ i = 1
m
pici (7)
where pi represents the ith geometric parameter of the con-nection and ai, bi, and ci are the exponents that indicate theeffect of the ith geometric parameter; m is the number ofgeometric parameters considered. Taking the logarithms ofboth sides of the above equations, the Ramberg-Osgoodparameters can be expressed as:
log Mo = a1 log p1 + a2 log p2+...+ am log pm (8)
log φo = b1 log p1 + b2 log p2+...+ bm log pm (9)
log n = c1 log p1 + c2 log p2+...+ cm log pm (10)
Multiple linear regression analysis was then carried out todetermine the coefficients a, b, and c.
A total of six terms, representing the various geometries ofthe connection, has been used to determine the coefficients,and the relationships thus obtained are given as follows:
αM = 5.395 × 10−6 and 5.935 × 10−6, αφ = 0.0324 and 0.0308and αn = 0.019 and 0.0285 for the two-way and four-wayconnections respectively. All dimensions are in millimeters.Units for M and Mo are in kNm, φ and φo are in radians ,whilethe geometrical parameters were measured in millimeters.
Figure 8 shows the comparison between results obtainedfrom the Ramberg-Osgood function by using the standardizedconnection parameters and the corresponding results from thefinite element analysis. Curves for typical specimens (Exam-ple 1 in Appendix II) are shown for both the two-way andfour-way connections. It can be seen that the correlationbetween the curves is very good. Similar observation has beenmade for all the other specimens.
Figures 9(a) and (b) show plots of normalized moment-ro-tation relationships for all the 15 specimens of two-way andfour-way connections, respectively. It can be seen that almostall the curves in each case lie very close, except for twospecimens. These two specimens, which are the same for bothcases, consist of specimens with beam and column of extremesizes obtained from the section table i.e., one specimen con-sists of the smallest column and very small beams while theother specimen consists of the largest column size with verylarge beams. It is suggested that a single curve, as shown in
Fig. 7. Typical load-deflection curves forthree-way and four-way connections.
FOURTH QUARTER / 1993 145
the respective figures, can be used for the prediction of themoment-rotation characteristic of the connections betweenall beams and columns of practical dimensions.
EXPERIMENTAL VERIFICATION
An experimental investigation was carried out to study thebehavior of a connection in which four I-beams frame into abox column. This is the typical connection which could occurat an interior column of a building. The connection was testedto failure, and strain and displacement measurements weremade to obtain the stress distribution and moment-rotationcharacteristics. Details of the experimental program togetherwith the results have been reported elsewhere.23 For compari-son, results from one of the specimens are shown here.
Figure 10 is typical specimen which was designed in
accordance with the proposed method. It was supported at theends of the four beams and subjected to a load appliedvertically on the column until failure. Figures 11 and 12 showthe load-deflection and the dimensionless moment-rotationcurves, respectively. It can be seen that the initial stiffness ofthe connection is good and the ultimate strength capacityexceeds the plastic capacity of the beam. The large rotationcapacity of the specimen also indicates that the connection isductile. Good correlation is observed between the resultsfrom the experiment and the finite element method. Also, itcan be seen that the fitted Ramberg-Osgood curve using thestandardized moment-rotation function agrees with the ex-perimental results.
CONCLUSIONS
A simple design procedure has been proposed to determinethe size of the T-section to be used as the external stiffener for
Fig. 8. Normalized moment-rotationcurves for typical speciments. Fig. 9. Normalized moment-rotation curves for 15 specimens.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION146
an I-beam to box-column connection. The design method isapplicable for both the two-way and four-way connections.Results from connections, formed by a wide range of I-beamsand box columns obtained from section tables, show theaccuracy of the proposed design method.
It is common practice in modern design codes to proposea capacity reduction factor φ for ultimate-strength design. Asimilar factor may be adopted for the calculation of T1 andT2 to account for various uncertainties in the connectiondesign and performance. A value of 0.9 given in the AISC-LRFD Specification for both shear and tension yield limitstates may be used in this case also.
A curve-fitting procedure using the Ramberg-Osgoodfunction has been used to obtain the moment rotation rela-tionship for the connections. The independent parameters ofthe function was expressed in terms of the geometrical prop-erty of the connection. The curves obtained by using the fittedparameters compare well with those obtained from the finite-element analysis for all the specimens considered. Resultsindicate that a single moment-rotation curve can be used forthe design of connections formed by a wide range of beamand column sections. Finally, both the design procedure andthe Ramberg-Osgood function obtained have been found toagree well with experimental results. The results presentedare with reference to two-way and four-way connections.Further research is in progress to study other types of configu-rations and to investigate the suitability of this connection forseismic design.
NOTATION
B column widthM moment imposed on the connectionMo reference momentMp plastic moment of beamTp beam-flange force corresponding to plastic
moment of beam
ai, bi, ci exponents indicating effect of ith geometricparameter
b beam flange widthdb beam depthh stiffener flange widthl stiffener lengthm number of geometric parameters consideredn “sharpness” of Ramberg-Osgood curvepi ith geometric parameter of connectiontbf beam flange thicknesstc column wall thicknesstsf stiffener flange thicknesstsw stiffener web thicknessφ rotation at the connectionφo reference rotation
ACKNOWLEDGEMENT
The investigation presented in this paper is part of a programof research on box columns being carried out in the Depart-ment of Civil Engineering at the National University ofSingapore. The work is funded by research grant RP94/85made available by the National University of Singapore.
APPENDIX I—REFERENCES
1. Jones, W. S., Kirby, P. A., and Nethercot, D. A., “TheAnalysis of Frames with Semi-Rigid Connections—AState-of-the-Art Report,” Journal of Constructional SteelResearch, Vol. 3, No. 2, 1983, pp. 2–13.
2. Yu, C. H. and Shanmugam, N. E., “Stability of Frameswith Semi-Rigid Joints,” Computers and Structures, Vol.23, No. 5, 1986, pp. 639–648.
3. Gerstle, K. H., “Effect of Connection on Frames,” Journalof Constructional Steel Research, Vol. 10, 1988, pp. 241–267.
4. Nethercot, D. A., and Chen, W. F., “Effects of Connectionson Columns,” Journal of Constructional Steel Research,Vol. 10, 1988, pp. 201–239.
5. Jones, W. S., Kirby, P. A., and Nethercot, D. A., “Effect of
Fig. 10. 4-way connection test specimen. Fig. 11. Load-deflection curve for 4-way test specimen.
FOURTH QUARTER / 1993 147
Semi-Rigid Connections on Steel Column Strength,”Journal of Constructional Steel Research, Vol. 1, 1980,pp. 38–46.
6. Kato, B., Chen, W. F., and Nakao, M., “Effects of Joint-Panel Shear Deformation on Frames,” Journal of Con-structional Steel Research, Vol. 10, 1988, pp. 269–320.
7. Barakat, M. and Chen, W. F., “Design Analysis of Semi-Rigid Frames: Evaluation and Implementation,” Engi-neering Journal, AISC, 2nd Quarter, 1991, pp. 55–64.
8. Attiogbe, E. and Morris, G., “Moment-Rotation Func-tions for Steel Connections, ” Journal of Structural Engi-neering Division, ASCE, Vol. 117, ST6, 1991, pp. 1703–1718.
9. Ang. K. M. and Morris, G. A., “Analysis of Three-Dimen-sional Frames with Flexible Beam-Column Connec-tions,” Canadian Journal of Civil Engineering, Vol. 11No. 2, 1984, pp. 245–254.
10. Bjorhovde, R., Colson, A., and Brozzetti, J., “Classifica-tion System for Beam-to-Column Connections,” Journalof Structural Division, ASCE, Vol. 116, No. 11, Nov. l990,pp. 3059–3076.
11. Maquoi, R., “Semi-Rigid Joints: from Research to DesignPractice,” International Conference on Steel and Alu-minium Structures, Singapore, 22–24 May, 1991, pp.32–43.
12. Chen, W. F. and Kishi, N., “Semi-Rigid Steel Beam-to-Column Connections: Data Base and Modelling,” Jour-nal of Structural Division, ASCE, Vol. 107, ST 9, 1981,pp. 105–119.
13. Nethercot, D. A., “Utilization of Experimentally Ob-tained Connection Data in Assessing the Performance ofSteel Frames,” Connection Flexibility and Steel Frames,W. F. Chen, Ed., 1985, pp. 13–37.
14. Chen, S. J. and Lin, H. Y., “Experimental Study of SteelI-Beam to Box-Column Moment Connections,” 4th Inter-national Conference on Steel Structures and SpaceFrames, Feb. 15–16, 1990, Singapore, pp. 41–47.
15. Dawe, J. L. and Grondin, G. Y., “W-Shape Beam to RHSColumn Connections,” Canadian Journal of Civil Engi-neering, Vol. 17, Oct., 1990, pp. 788–797.
16. White, R. N. and Fang, P. J., “Framing Connections forSquare Structural Tubing,” AISC National EngineeringConference, Memphis, Tenn, April, 1965, pp. 74–102.
17. Ting, L. C., Shanmugam, N. E., and Lee, S. L., “Box-Col-umn to I-Beam Connections Stiffened Externally,” Jour-nal of Constructional Steel Research, Vol. 18, No. 3, 1991,pp. 209–226.
18. Ting, L. C., Shanmugam, N. E., and Lee, S. L, “ExternallyStiffened Steel Beam to Box-Column Connections,” Pro-ceedings, International Conference on Steel and Alu-minium Structures, Singapore, May, 1991.
19. Shanmugam, N. E., Ting, L. C., and Lee, S. L., “Behaviorof I-Beam to Box-Column Connections Stiffened Exter-nally and Subjected to Fluctuating Loads,” Journal ofConstructional Steel Research, Vol. 20, No. 2, 1991, pp.129–148.
20. Chen, W. F. and Lui, E. M., “Static Flange MomentConnections,” Journal of Constructional Steel Research,Vol. 10, 1988, pp. 38–88.
21. MSC/NASTRAN Application Manual, Volumes I and II,The MacNeal-Schwendler Corporation, April, 1983.
22. Steelwork Design Guide to BS5950: Part 1, Vol. 1, SectionProperties, Member Capacities, 2nd ed., The Steel Con-struction Institute, 1985.
23. Lee, S. L., Ting, L. C., and Shanmugam, N. E., “Use ofExternal T-Stiffeners in Box-Column to I-Beam,” Journalof Constructional Steel Research, Vol. 26, Nos. 2&3,1993, pp. 77–98.
APPENDIX II—DESIGN EXAMPLES
Design examples to determine the size of the T-section whichcan be used as the external stiffener for I-beam to box-columnconnections. Example 1 illustrates the case when the stiffenerlength is governed by the angle θ =2° while Example 2illustrates the case when the length is governed by the shearcapacity of the stiffener web.
Example 1
Column size: 200×200×16 mm
Beam size: 305×165×40 kg/m
Grade 43 steel is assumed for column, beam, and stiffeners.
1. Choose a T- or I-section with a web thickness of at leasthalf the beam-flange thickness.
Try T-section 102×102×12 kg/m
2. For θ = 20°, stiffener length l = (200 − 165) / (2 × tan20°) = 50 mm (2 in.)
3. Check minimum stiffener length based on strength criteria.
Plastic moment capacity of the beam,Fig. 12. Moment-rotation curve for four-way test specimen.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION148
The allowable shear stress according to von Mises yieldcriteria,
τy = 275 / 1.732 = 159 N/mm2 (23.1 ksi)
Therefore, the minimum length
l = 164,000 / (159 × 6.4) = 161 mm (6.34 in.)
4. From Steps 2 and 3, the longer length is chosen.
Therefore, use section 102×127×14 kg/m of length 161 mm(6.34 in.) as the external stiffener.
FOURTH QUARTER / 1993 149
The author has provided useful suggestions for this particulardesign problem based on yield-line analysis. The purpose ofthis discussion is to clarify some aspects of these solutionsand to re-organize them in concise decision form for ease ofcalculations.
While the AISC Manuals have only exclusively addressedbase plates for wide-flange column shapes, the AISC DesignGuide No. 1 Column Base Plates (Ref. 3) does briefly covertubular and pipe columns. It suggests that the usual cantileverplate model employed under wide-flange columns can beextended to such closed sections: the critical overhang dimen-sion (m or n) for determining plate thickness becomes 0.95times the outside column dimension for rectangular tubes and0.80 times the outside dimension for round pipes.
Also, similar to base plates with wide-flange columns, animportant consideration is the limiting case when the columnapproaches the size of its base plate. For this so-called “small”plate condition, the cantilever overhang distance m can becomerather short and almost zero, thereby rendering this simple modeluseless for design. The proposed yield-line Equation 2 possessesthis same characteristic since for R / D = 1, the required thicknessreduces to zero. Fortunately, Equation 1 does provide a rationaldesign answer for R / D = 1.
While Equations 1 and 2 in combination offer valid andcomplementary design solutions, the presented application oflightly loaded Equation 3 in this context is confusing. Thepaper states that Equation 3 is a special case of Equation 1.The required plate thickness is limited to no more than givenby Equation 1 and, finally, the greater of Equations 1 or 3 and2. Based on this rationale, it is never necessary to checkEquation 3. In order to parallel the logic of the revised AISCsmall-column base-plate procedure for wide-flange shapes, itappears that the real intent should be for the required thick-ness to be the greater of:
a. the lesser of Equations 1 and 3 or, conservatively,Equation 1
b. Equation 2
The author’s Equations 1 and 2 may be interpreted suchthat the former applies to “small” plates (R / D near 1.0)whereas the latter covers the “larger” plates (smaller R / D).Use of Equation 3, in my opinion, is optional in conjunctionwith Equation 1 for lightly loaded conditions. The recom-mended practical limit of R / D ≥ 0.5 gives an actual designrange of 0.5 ≤ R / D ≤ 1.0. One may easily compute that theintersection of Equations 1 and 2 occurs at about R / D = 0.7,hence, the following general design criteria can be formulatedto minimize calculations:
If 0.5 ≤ R / D ≤ 0.7, (large base plate case) use Equation 2..
If 0.7 < R / D ≤ 1.0, (small base plate case) use the lesserof Equations 1 and 3, or, conservatively, Equation 1.
In the original example, R / D = 0.61 < 0.7, and Equation2 governs, as expected.
As alluded to previously, the cantilever bending modelcould also be utilized for the large base-plate case. Becauseits solution is slightly more conservative than given by Equa-tion 2, the R / D = 0.7 limit should be increased to 0.8 for itsrange of applicability. Applying this procedure to the author’sexample problem results in:
R / D = 0.61 < 0.8 o.k.
m = D − 0.8R = 3.5 − 0.8 (2.13) = 1.796 in.
fp = P
4D2 = 12
4(3.5)2 = 0.245 ksi
t = 2m √ fpFy
= 2(1.796) √.24536
= 0.296 in.
This solution requires an extra one-sixteenth base-plate thick-ness compared to the yield-line based Equation 2. However,it can also serve to demonstrate the reasonableness of bothmethods.
Easy conversions can be made for LRFD design equivalentto the proposed ASD Equations 1, 2, and 3:
1. replace M by φMp = 0.9t2Fy / 42. replace P by Pu (factored loads) in fp3. solve appropriate work balance expressions for required
thickness
DISCUSSIONDesign of Pipe Column Base Plates Under Gravity LoadPaper by THOMAS SPUTO(2nd Quarter, 1993)
Discussion by Nestor R. Iwankiw
Nestor R. Iwankiw is Director of Research & Codes, AISC,Chicago, IL.
ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION150
LRFD:
t = RD
√Pu / (2.7πFy ) (1)
t = √0.74Pu
πFy 2 − 3
RD
+
RD
3
(2)
Rc = √Ro2 − Pu / πFp (3)
t = R √Fp 2.7Fy
1 − 3
Rc
R
+ 2
Rc
R
3
In summary, this paper contributes new and useful designcriteria for column base plates under gravity loads which,with additional reflection, can be further simplified and gen-eralized for applications.
CLOSURE BY THOMAS SPUTO
The discussor has provided some interesting suggestions forsimplifying the content of this paper. His clarification of theapplicability of each design equation is especially welcome.The author thanks him for his interest in this topic.
ERRATA
1. π should be included as multiplier in the final externalwork expression above Equation 2.
2. Equation 3 and We equation above it: change exponentin last term on Rc from 2 to 3.
3. Examplet by Equation 2, last term should be changed from(2.13)2 / 3.15 to (2.13)3 / 3.5
4. Page 42, upper left, external work expression shouldread
= fp πD2 −
πD2
3 − π (D2 − R2)
RD
− πR2
RD
+
πR2
3 RD
= π fp 2D2
3 − RD +
R3
3D
FOURTH QUARTER / 1993 151
CORRECTION
Shear Tab Design Tables
ASD/LRFD Volume II—Connections
The following are corrected tables for pages C-11 and C-12 of the AISC Manual of Steel Construction Volume II—Connections.
