AISC Engg Journal 92

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EngineeringJ ournal

AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.

Page 1: Ronald D. Ziemian and William McguireA Method for Incorporating Live Load ReductionProvisions in Frame Analysis

Page 4: Krishna K. Verma and Fred R. Beckmann

High-Strength Bolts for Bridges

Page 12: Thomas R. Rauscher and Kurt H. GerstleReliability of Rotational Behavior of FramingConnections

Page 20: Patrick D. ZuraskiThe Significance and Application of Cb inBeam Design

Page 26: J ack D. Bakos, J r. and J ames A. OLeary

An Equivalent Radius of Gyration Approach toFlexural-Torsional Buckling for SinglySymmetric Sections

Page 45: R. Shankar NairForces on Bracing Systems

1st Quarter 1992/ Volume 29, No. 1


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INTRODUCTION

The effects of live load are often reduced to reflect the lowprobability of all live load existing simultaneously through-

out a substantial portion of a structure. Subject to certain

limitations, ASCE 7-881 provides the following permissible

reduction

L=0.25 + 15

Al

LoLo (1)

where:

L = reduced design live load

Al = member influence area in square feet

(Al 400 ft2)Lo = unreduced design live load

= 0.5 for members supporting one floor and 0.4otherwise

In the analysis of entire structural systems or substantial

portions thereof, methods for incorporating live load reduc-

tion are essential. They can have a significant influence on

a structures response. Not to include live load reduction

provisions may be overly conservative. For example, reduced

live loads may produce smaller second-order effects. In some

cases, however, use of full live load may be unconservative.

For example, full live load may not be in place to resist an

overturning moment produced by lateral load.

The incorporation of live load reduction provisions of the

type in ASCE 7-88 requires careful consideration when

analyzing structural systems. This is because (i) the influ-

ence area for beams and columns are generally different,

and (ii) Eq. 1 is a nonlinear function of this area. Several

methods for including live load reduction in system analyses

have been suggested.2,3,4 These methods, however, have

only treated reduction of member forces for the purpose of

member proportioning. Also, they may produce memberforces that are not consistent with the calculated deflections

of the frame. With this in mind, a more comprehensive

method for incorporating live load reduction in system ana-

lyses has been developed.5

OUTLINE OF APPROACH

The method is based on the use of compensating forces

calculated by: (a) applying beam live load reduction factors

to the column connected beams; (b) applying column live

load reduction factors to the columns; and (c) determining

any out of balance at the beam-to-column intersections.

Because columns typically have a larger influence area than

beams (providing for a larger reduction), the compensating

forces are generally upwardly directed (opposite of gravity).

All structural system analyses which include live load are

then performed by applying a combination of the reduced

beam live loads and the calculated compensating forces. By

applying this combination of live load, the resulting forces

A Method for Incorporating Live LoadReduction Provisions in Frame Analysis

RONALD D. ZIEMIAN and WILLIAM McGUIRE

Ronald D. Ziemian is assistant professor of civil engineering,

Bucknell University, Lewisburg, PA.

William McGuire is professor of civil engineering, emeritus,

Cornell University, Ithaca, NY.Fig. 1. Description of example frame.

FIRST QUARTER / 1992 1


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in both the main girders and the columns will reflect the

ASCE-7 live load reduction provisions.

The frame shown inFig. 1 will be used to illustrate the

determination of compensating forces. Each of the relevant

structural components in the frame is assigned a two part

identifier. The first part, a beam, column, or area number,

is defined in the plan view ofFig. 1. The second part, the

level (for beams and areas) or story (for columns), is pro-

vided in the corresponding elevation view. For example, the

member designation B1-3 refers to Beam 1 of level 3, and

C2-1 refers to Column 2 of story 1.

The following steps outline how the live load compensat-

ing forces could be calculated:

1. Based on tributary area, estimate the axial force in each

column without applying any reduction factors. For col-

umn C2-1 (seeFig. 2), an estimate of the unreduced

axial force is

FC21 =1

2i= 13

[(BliLBli)+(B2iLB2i)+(B9iLB9i)] (2)

where:

Bji = unreduced uniform live load along beam Bj-i LBji = length of beam Bj-i

(In lieu of assuming one-half of the beam loads con-

tributing to each of the column forces, a structural anal-

ysis that accounts for the actual continuity of the sys-

tem could be performed to obtain a more accurate

estimate of the column axial force distribution).

2. Based on each columns influence area, reduce the

above axial force by the ASCE 7-88 live load reduc-tion factor (Eq. 1). For column C2-1, the reduced axial

force is

FC21 =

0.25 + 15

FC21 (3)

i= 1

3

(Area1i+ Area2i)

where:

i= 1

3

(Area1i+ Area2i) = total influence area for col-umn C2-1.

Note that FC21 should not be less than 0.4FC21.3. Based on tributary area, estimate the axial force in each

column by applying only beam live load reduction fac-

tors. For column C2-1, this axial force is approximately

FC21 =12i= 1

3

[(B1i LB1i)+(B2i LB2i)

+(B9i LB9i)] (4)

where:

Bji = reduced uniform live load along beam Bj-i LBji = length of beam Bj-i

As in step 1, a separate structural analysis could be per-

formed to obtain a more accurate estimate of these col-

umn axial forces.4. Determine the difference in axial forces calculated in

steps 2 and 3. For column C2-1, this force is

FC21 =FC21 FC21 (5)

5. Determine the additional upward axial force, compen-

sating force, to be applied at the top of each column

segment. For column C2-1, this force is

fC21 =FC21 i= 2

3

fC21 (6)

Fig. 2. Components used in live load reduction example.Fig. 3. Description of applied live load to be used in

frame analysis.

2 ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION


4/174

A summary of typical forces used in this frames live load

calculations are provided inTables 1(a) and1(b).Figure 3

shows the net applied live load distribution.Table 1(c) shows

distributions obtained by calculating the forces for steps 1

and 3 by a three-dimensional linear elastic analysis of the

rigidly jointed system.

In all cases where factored load combinations are inves-

tigated, both the beam live loads and the compensating forcesshould be multiplied by the appropriate live load factors.

SUMMARY

An approach for incorporating live load reduction provisions

within system analyses is presented. By determining an

appropriate applied live load, the resulting forces in both

the beams and the columns will include the ASCE 7-88 live

load reduction provisions. In applying this live load, any dis-

placements calculated by a structural analysis will be con-

sistent with the reduced internal member force distribution.

Joint equilibrium will be maintained. Because the procedure

does not rely on applying the principle of superposition, it

may be used with either linear or nonlinear structuralanalyses.

The concept of compensating forces has been illustrated

by applying them at beam-to-column intersections only. The

same idea can be extended to accommodate any desired

degree of modeling of interior floor framing.

ACKNOWLEDGMENTS

This research was supported by the National Science Foun-

dation under Grant Number MSM-8608803, the American

Institute of Steel Construction, and the School of Civil and

Environmental Engineering at Cornell University. The

authors wish to thank Dr. Jerome F. Hajjar of Skidmore,

Owings and Merrill for his comments and suggestions.

REFERENCES

1.American Society of Civil Engineers Minimum DesignLoads for Buildings and Other Structures, ASCE 7-88,

American Society of Civil Engineers, New York, 1990

(formally,American National Standard Minimum Design

Loads for Buildings and Other Structures, ANSI A58.1,

American National Standards Institute, New York, March

1982).

2. Parikh, B. P., Elastic-Plastic Analysis and Design of

Unbraced Multi-Story Steel Frames, Ph.D. Thesis,

Lehigh University, June 1966.

3. Pesquera, C. I., Integrated Analysis and Design of Steel

Frames with Interactive Computer Graphics, Ph.D. The-

sis, Cornell University, Ithaca, New York, March 1984.

4. White, D. W. and Hajjar, J. F., Application of Second-Order Elastic Analysis in DesignResearch to Practice,

AISC, National Steel Construction Conference, Kansas

City, Missouri, March 1990, pp. 11.111.22.

5. Ziemian, R. D., Advanced Methods of Inelastic Analy-

sis in the Limit States Design of Steel Structures, Ph.D.

Thesis, Cornell University, Ithaca, New York, August

1990.

Table 1.Example of Reduced Live Load Calculations

(a) Beams

Member

Length

ft

Tributary

Area

ft2

, UnreducedUniform L.L.

k/ft

Influence

Area

ft2

Permissible

Reduction

Factor

, ReducedUniform L.L.

k/ft

B1-i, B2-iB9-i

for i = 1 to 3

3020 200200 0.5000.750 6001200 0.8620.683 0.4310.512

(b) Columns with Force Distribution Estimated

Member

F, Unreduced

Axial L.L.

kips

Influence

Area

ft2

Permissible

Reduction

Factor

F, reducedAxial L.L.

kips

F, ReducedAxial L.L.

kips

F==FFkips

f, Compensating

Force (Upward)

kips

C2-1

C2-2

C2-3

67.50

45.00

22.50

3600

2400

1200

0.500

0.556

0.683

33.75

25.02

15.37

54.15

36.10

18.05

20.40

11.08

2.68

9.32

8.40

2.68

(c) Columns with Force Distribution Determined by Linear Elastic Analysis

Member

F, Unreduced

Axial L.L.

kips

Influence

Area

ft2

Permissible

Reduction

Factor

F, reducedAxial L.L.

kips

F, ReducedAxial L.L.

kips

F==FF

kips

f, Compensating

Force (Upward)

kips

C2-1

C2-2

C2-3

69.48

46.45

23.37

3600

2400

1200

0.500

0.556

0.683

34.74

25.83

15.96

55.71

37.21

18.78

20.97

11.38

2.82

9.59

8.56

2.82



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ABSTRACT

The use of substandard and mismatched bolts continues to

be a major concern to bridge owners in the United States.

Based on FHWA-sponsored research at the University of

Texas, supplemental specifications were developed and issued

modifying fastener manufacturing, testing, and installation

procedures.

Nearly all bridge bolts are designed for dynamic loading.

They are designed to resist either tension forces and/or shear

forces. Fatigue concerns govern bolts designed for cyclic ten-

sion forces. Cyclic shear forces require slip critical connec-

tions. Both loading conditions require bolts to be installed

to a minimum preload.

The FHWA recommendations were developed in order to

assure the ability of bolts to achieve this preload. Minimum

nut strength is increased, maximum bolt strength is reduced,

thread fit tolerance is reduced, additional rotational-capacity

testing is required, and additional testing, documentation,

handling and shipping requirements are imposed. The ration-

ale for these new FHWA provisions are discussed.

Finally, slip critical joints depend upon friction between

faying surfaces to develop strength. Values of slip resistance

or coefficient of friction for various paints and coatings must

be determined by testing. Bolt design parameters depend

upon minimum values of tested coatings.

INTRODUCTION

The behavior of bolted joints depends on a large number of

variables many of which are rather difficult to predict.

Depending on the usage, and concerns for protection from

the environment, different materials and acceptance require-

ments have been specified by the users depending on their

current knowledge. In spite of over 30 years of experience

with high-strength fasteners, there continue to be problems

in ensuring that fasteners are of adequate quality and are

installed properly. There are concerns that bolted connec-

tions in many bridges built over the past 10 years or so might

not meet acceptance criteria if they were subjected to testrequirements of today.

These concerns can be eliminated when fasteners are

manufactured to code requirements and subsequent quality

control testing is done by the fastener manufacturers, accept-

able installation procedures are practiced by the installers,

followed by a reliable quality assurance (QA) and traceabil-

ity program by the owner.

FASTENER REQUIREMENTS AND RATIONALE

Researchers, owners, code writing organizations, and the fas-

tener industry have been attempting to constantly improve

the quality of fasteners and fastener installation practices to

produce a better end product. To ensure that only those

fasteners which meet the minimum quality standards areused, the Federal Highway Administration (FHWA) initiated

an extensive experimental research program with the Depart-

ment of Civil Engineering of the University of Texas at Austin

to evaluate the performance of both black and galvanized

high-strength bolts for steel bridge structures. The study was

done using ASTM A325 hot dipped or mechanically gal-

vanized bolts and A325/A490 black bolts. Only normal size

fasteners commonly used in steel bridge superstructures were

tested. Research findings were reported in the FHWA pub-

lication FHWA/RD-87/088 High-Strength Bolts for

Bridges. Recognizing the need to underscore the various

recommendations made in the report and to implement them,

the recommendations were compiled, modified in consulta-tion with the researcher and the fastener industry, and later

distributed to the field offices via an FHWA memorandum.

The objective of the FHWA memorandum was to allow the

AASHTO (American Association of State Highway and

Transportation Officials) bridge owners to incorporate these

high-strength bolt specifications in the state standard speci-

fications or contract documents without duplicating the effort

of sorting out the recommendations from the report. A copy

of the FHWA supplemental specifications contained in the

memorandum is included in the appendix. The rationale

behind the pertinent specifications is discussed in this paper.

The supplemental specifications were written for AASHTO

M164 (ASTM A325) bolts but it is recognized that similarspecifications are needed for A490 bolts and other alternate

fasteners. The supplemental specifications for A325 bolts

were written first because those bolts are used most com-

monly for bolted connections of bridge members.

The following background information should be helpful

in understanding the rationale for the various requirements

in the memorandum.

Essentially, a clamping force is needed to prevent fatigue

High-Strength Bolts for Bridges

KRISHNA K. VERMA and FRED R. BECKMANN

Krishna K. Verma is welding engineer, Federal Highway Ad-

ministration, Washington, D.C.

Fred R. Beckmann is Director of Bridges, American Institute

of Steel Construction, Chicago, IL.



6/174

failure of bolts subjected to cyclic tension and to prevent slip

and increase fatigue strength in shear connections. Fatigue

failure of threaded fasteners is well known. It can be traced

to points of stress concentrations such as those locations

where there are abrupt geometric changes, a notch or a nick,

or locations where the material may have poor fracture

toughness. The torque applied to the fastener assembly is

not uniformly distributed over the engaged length because

fastener materials are not inelastic and there are manufac-

turing tolerances resulting in less than perfect matching of

bolt and nut threads over the engaged length. However, fail-

ure in threaded fasteners is often located at the washer face

of the nut, at the thread runout, or at the junction of the bolt

head and shank. This is primarily because of probable high

stress concentrations at these locations, although the aver-

age stress levels in the body of the bolt may remain well

below the endurance limit of the material. Furthermore,

cyclic external forces applied to the bolt can reduce the life

of the fasteners by fatigue.

As the torque is applied to the nut, a portion of it is resistedby friction between the nut and the gripped material; the

remainder is resisted by friction at the thread interface result-

ing in torsional stresses in the bolt shank. The bolt is thus

subjected to a combined torque-tension stress condition. Load

deformation characteristics of bolts subject to direct tension

compared to torque-tension reveal that specimens subject to

torque-tension are less ductile2 and have strength levels

reduced between five and 25 percent.

Clamping force is an important consideration if a bolted

joint must function as a slip resistant joint. In such a joint

the external load component parallel to the faying surface(s)

is resisted by the frictional resistance which is dependent on

the clamping force of the bolt and the coefficient of frictionat the faying surface. In a bearing type connection, slip is

allowed and movement stops as the material bears against

the bolt. In such joints the critical factors are the permissi-

ble bearing stress on the connection material, the axial stress

on the net section and the shear stress of the fastenersnot

the initial preload of the bolt. Comparative studies of bolts

subject to shear stresses under tension or compression show

that shear stress deformation characteristic of A325 bolts and

A490 bolts are similar; however, A490 bolts have a lesser

ability to deform than A325 bolts under similar conditions,

and the maximum shear stress experienced by A490 bolts

(of higher strength material) is greater than that in A325

bolts. The research also suggests that when the same typeof bolt (A325 or A490) is subjected to shear test in tension

or compression jigs, samples in tension jigs show lower shear

strength (a tension jig is preferred for testing shear strength

of bolts because it produces the lower range of the shear

value). Available data also demonstrate that the shear strength

of A325 or A490 bolts is approximately 62 percent of the

tensile strength. It is significant to note that unlike bolts sub-

ject to tensile loads, the clamping force has no significant

effect on the ultimate shear strength of the bolt. Thus, for

slip critical joints subjected to dynamic loads, it is apparent

that not only should initial preloads as high as practicable

be applied to fasteners, but it is also critical that the desired

preload is indeed in the bolt after it is installed.

Until 1985, the practice in North America had been to pro-

vide as high a preload as practical regardless of whether or

not the joint was slip critical and whether or not tensile forces

were applied. Though the apparent objective was to achieve

uniformity and simplicity in bolt installation, there were

inherent economic disadvantages in attempting to accurately

preload bolts where preloading was not even necessary. Since

the introduction of high-strength bolts the requirement has

been that high-strength fasteners in slip critical joints and

connections subject to direct tension or reversible loads need

to be preloaded to a predetermined level. Since 1985, snug

tightening has been permitted in many situations where it

is adequate in buildings for bearing type fasteners though

generally not used for bridges.

Obviously, an adequate preload is essential within certaintolerances for dynamically loaded structures such as steel

bridges. Proper preloading of fasteners in such structures is

an important and critical task faced by bridge engineers and

inspectors. There are, however, numerous related problems

and issues and hence the need to specify adequate control.

Material specifications, e.g., ASTM Specifications,

AASHTO Materials Specifications and other specifications

provide necessary controls during the manufacturing pro-

cess. Installation of fasteners for bridges is addressed by

AASHTO, in Division II of the AASHTO Standard Speci-

fications for Highway Bridges. In addition, AASHTO bridge

owners may have their own special requirements and pre-

ferred practices.The FHWA memorandum cited earlier supplements to

AASHTO Specifications based on the research findings

reported in Ref. 1, High-Strength Bolts for Bridges. It

should be understood that except for the proposed sup-

plemental specifications, other ASTM Specifications and

AASHTO Material Specifications remain valid. The memo-

randum amends or revises AASHTO Material Specifications

but does not replace them. These modifications also ensure

the strength of the bolts, nuts, and washers during manufac-

turing and cover issues pertaining to testing of fasteners and

fastener assemblies, needed documentation, shipping, and

installation at the job site. As an example, the FHWA sup-

plemental specifications take some exceptions to AASHTOMaterial Specifications for tensile strength and hardness

requirements and modify related specifications. Some of

these are:

1. A325 bolts are available as Type 1, 2 and 3 fasteners.

These require a minimum strength of 105 ksi for 118-in.

to 112-in. diameter bolts and 120 ksi minimum strength

for 12-in. to 1-in. diameter bolts. Though A325 bolt



7/174

specifications provide a range of hardness, the upper

bound of tensile strength is not included in the ASTM

or AASHTO Material Specifications. The hardness can

generally be converted to an equivalent tensile strength

using conversion tables such as those in ASTM Speci-

fications (A370) or other references. Current AASHTO

Material Specifications and ASTM Specifications

require matching nuts for A325 bolts. These include

heat treated nuts as well as non-heat treated nuts with

hardness values as low as 78 HRB (Hardness, Rock-

well B). Similarly, A490 bolts are available as Type 1,

2, and 3. These bolts have required material strength

ranges from 150 ksi to 170 ksi with matching nuts of

hardness greater than 24 Rc (Rockwell C) which is

much greater than 89 HRB. For A490 bolts non-heat

treated nuts are not permitted by either ASTM Speci-

fications or AASHTO Material Specifications. An

examination of these two specifications reveals an

inconsistency in fastener specifications. As noted

above, current specifications allow manufacturing A490bolts with a minimum tensile strength 150 ksi and hard-

ness value of approximately 33 Rc, but these A490 bolts

are not permitted to be galvanized. However, using cur-

rent ASTM Specification or AASHTO Material Spec-

ifications, A325 bolts can be manufactured with hard-

ness as high as 35 Rc which is equivalent to 156 ksi

tensile strength, well into the A490 strength range. The

current AASHTO Material Specifications and ASTM

Specifications do allow galvanizing A325 (M164) bolts.

Thus comparing the two situations it does not seem log-

ical to allow galvanizing A325 bolts of 35 Rc hardness

when galvanizing A490 bolts of 33 Rc hardness is pro-

hibited. The FHWA supplemental specifications includemodified requirements to correct this inconsistency.

2. Thread stripping is controlled by (a) bolt and nut

strength and (b) fit of threads at the interface. Preven-

tion of stripping requires proper fit of bolt-nut assem-

blies and often requires that heat treated nuts be speci-

fied. Non-heat treated nuts with lower hardness values

have potential for nut stripping. In previous years,

AASHTO had been allowing the use of non-heat treated

nuts which could have a minimum hardness as low as

78 HRB. The FHWA supplemental specifications re-

quire that the minimum hardness of the nut should be

89 HRB to prevent possible stripping of nuts. The need

for this minimum hardness can be explained by Alex-anders model1 which was developed based on

experimental data. It is illustrated inFig. 1.

Curves have been plotted for 78-in. diameter bolts

of tensile strength 156 ksi (equivalent to 35 Rc hard-

ness). InFig. 1, the ratio of the stripping strength of

nut (or stripping strength of bolt) to the tensile strength

of the bolt has been plotted against the nut strength.

The dotted horizontal line represents those assemblies

which have stripping strength equal to the tensile

strength of the bolt. Points on the curve which are below

this horizontal dotted line are subject to possible fail-

ure by thread stripping only. Those above the dotted

line will fail by tension in the bolt rather than strip-

ping of threads. FromFig. 1, it is evident that for those

assemblies which have nut strength greater than 87 ksi,

neither the bolts nor the nuts will strip since the corre-

sponding points lie above the horizontal dotted line.

Since 87 ksi tensile strength is approximately equiva-

lent to 89 HRB hardness, the FHWA supplemental

specification requires hardness of nuts not less than 89

HRB. On the abscissa inFig. 1, nut strength and vari-

ous nut designations have been shown. These nut

representations indicate lowest permissible strength (or

hardness) as permitted by the current ASTM/AASHTO

Material Specifications. From this figure, it is possi-

ble to infer that heat treated nuts, 2H, DH, and DH3,

have minimum hardness well above 89 HRB, the sug-

gested minimum hardness to prevent nut stripping.However, non-heat treated nuts, if manufactured with

minimum hardness as permitted by ASTM and

AASHTO Material Specifications, will be prone to nut

stripping. The suggested minimum hardness 89 HRB

is within the upper and the lower limits of hardness

permitted in those specifications. Nut stripping in non-

heat treated nuts can be prevented if such nuts are

manufactured to a hardness not less than 89 HRB.

A limited study1 of comparable fasteners produced

in accordance with ASTM specifications using tradi-

Fig. 1. Effect of nut strength on bolt and nut stripping.(Reproduced from Ref. 1.)



8/174

tional U.S. units of measurement with fasteners pro-

duced in accordance with ASTM specifications using

metric units of measurement seems to suggest that met-

ric fasteners with loose fit and minimum hardness of

89 HRB are less prone to stripping, whereas other

fasteners with tighter thread fit tolerances and mini-

mum hardness of 78 HRB are prone to stripping. The

study revealed that fasteners made using the metric

standard with slightly greater nut strength (approxi-

mately two percent), as evidenced by hardness num-

bers, are more forgiving, even with a loose fit. It is

important to recognize that failures resulting from

thread stripping must be avoided because such failures

could go undetected during the service life of the

bridge, resulting in possible failure of bridge members

and related consequences to the travelling public. How-

ever, it may be noted that even though the minimum

hardness requirement of 89 HRB for non-heat treated

nuts 2, C, C3 and D is specified in the FHWA sup-

plemental specifications, stripping failure can still occurif there are only a few threads in the grip. For that rea-

son it is desirable to ensure that a minimum three to

five complete threads are in the grip. Bolts with more

threads in the grip have greater ductility and lower

apparent tensile strength.

3. Some of the test requirements for bolts, nuts, washers

and fastener assemblies have also been modified by the

FHWA supplemental specifications. Proof load testing

of bolts and nuts is required. Proof load is the tension

applied load which the fasteners must resist without

evidence of any permanent deformation. This test pro-

vides a check on the yielding behavior of the material

since the elongation is measured during testing. If gal-vanized fasteners are used, proof load testing is required

after galvanizing. Wedge testing of bolts and hardness

testing of washers is also required, but in the case of

galvanized fasteners these tests are required after gal-

vanizing. For galvanized fasteners, zinc thickness meas-

urements are also needed. Zinc thickness measurements

on bolts and nuts are important for proper fit and to

control overtapping. Performance capability of these

fasteners together in an assembly is checked via

rotational-capacity testing for either black or galvanized

units. Rotational-capacity testing is required prior to

shipping as well as at the job site. Job site testing is

important but only a minimal amount is needed.Rotational-capacity testing prior to shipping can be

done either by the manufacturer or the distributor, as

appropriate.

The purpose of the rotational-capacity testing is to

verify the torque tension relationship in order to ensure

(a) efficiency of lubrication, (b) adequate installation

ductility and (c) adequate resistance to stripping. Essen-

tially the rotational-capacity test requires measurement

of the bolt tension at the specified minimum rotation

(twice the amount of the required installation rotation)

from a snug tight condition; and also torque tension

values in a Skidmore-Wilhelm Calibrator, at any point

above installation rotation, to satisfy the following

requirement:

Torque (foot-pounds) 0.25 P (bolt tension-pounds) D (bolt dia. feet)

The FHWA supplemental specification does not allow

rotational-capacity testing of long bolts in a steel joint

as currently permitted by both ASTM Specifications

and ASHTO Material Specifications. Testing in a steel

joint does not allow direct measurements of bolt ten-

sion during rotational-capacity testing. A Skidmore-

Wilhelm Calibrator or similar device is required by the

FHWA supplemental specification because such a

device allows direct measurement of bolt tension as the

rotational-capacity test is performed. The torque-

tension relationship curves for these two situations havedifferent slopes at the lower levels of bolt tensioning,

but then the curves level out, merge and form a hori-

zontal plateau prior to sloping downwards as the bolt

tension is increased. Because the values of tension and

torque from this somewhat horizontal portion of the

curve are used for acceptance or rejection of the

rotational-capacity test, and for determination of the

maximum tension in the bolt, the values obtained using

a steel joint or a Skidmore-Wilhelm Calibrator will be

the same for all practical purposes.

In the case of short bolts which cannot be installed in

a Skidmore-Wilhelm Calibrator, the FHWA sup-

plemental specification does not require measurementof the actual maximum tension for the turn test. Antic-

ipated turn test tension as tabulated in the FHWA sup-

plemental specifications is used to calculate torque

using the equation noted above. This calculated torque

can then be compared with the measured torque.

4. In addition to job site rotational-capacity tests, calibra-

tion tests are also required. This is because for a given

tension there can be large variation in bolt torque as

measured in the laboratory prior to shipping to the job

site and that obtained in the field. Hence, it is required

that calibration tests be performed after fasteners are

received at the job site using a Skidmore-Wilhelm

Calibrator or an acceptable equivalent tension meas-uring device to ensure compliance with the minimum

installation pretension.

SLIP RESISTANCE OF FAYING SURFACES

As previously noted, the intent of the FHWA supplemental

specification is to ensure that the washer/nut/bolt combina-

tion functions as a matched unit. It is appropriate to con-

sider the influence of surface preparations and coatings on



9/174

the faying surfaces in achieving slip critical joints using high-

strength fastener assemblies.

The design of a bolted connection may be governed by

bearing on the connected material, shear in the shank, or

thread plane of the fastener or the slip resistance of the con-

tact surfaces of the connection. In nearly all bridge design,

because of dynamic loading, slip resistance of the joint is

the critical criterion. Bolts are seldom used in tension in

bridge structures.

Slip resistance of the contact of faying surfaces is a func-

tion of the surface condition. The design specification recog-

nizes three classes of surface conditions:

Class AClean mill scale surfaces and surfaces coated

with a Class A coating.

Class BBlasted surfaces and surfaces coated with a

Class B coating.

Class CGalvanized and roughened surfaces.

The most economical joint design generally occurs using

Class B surfaces. These are either uncoated blasted surfaces

or surfaces coated with a Class B coating. Where the struc-

ture is to be unpainted, it makes sense to specify uncoated

blasted surfaces. Where the structure is to be painted, the

structure should be designed with painted faying surfaces

using Class B coatings.

Coatings are classified as Class A or B based on slip coeffi-

cient testing performed in accordance with Appendix A of

the Specification for Structural Joints Using ASTM A325

or A490 Bolts. The essential variables for the test are paint

formulation, cure time, dry film thickness, and thinner used.

Actual coating application procedures that deviate from the

essential variables beyond certain limits require retesting.

Because there are many combinations of essential variables,choosing the proper values when performing the test is very

important.

Part of the test lasts 42 days; to retest is costly and can

delay a project.

As of the summer of 1990, very little testing of candidate

Class B coatings has been performed. Since bridges are cur-

rently being designed using the Class B coatings, it is impor-

tant that testing proceed at a faster rate. Steps are currently

underway to increase the number of paints that have been

tested. Hopefully, by the spring of 1991, the situation rela-

tive to the testing will improve and designers will be using

the higher slip values with the full knowledge that there are

an adequate number of paints available to meet the need.

REFERENCES

1. J. A. Yura, K. H. Frank, D. Polyzois. High-Strength Bolts

for Bridges. Publication No. FHWA/RD-87/088. U.S.

Department of Transportation. Federal Highway Adminis-

tration.

2. G. L. Kulak, J. W. Fisher, and J. H. A. Struik. Guide to

Design Criteria for Bolted and Riveted Joints. A Wiley-

Interscience Publication. John Wiley and Sons, New

York.

3. J. H. Bickford. An Introduction to the Design and

Behavior of Bolted Joints. Marcel Deckker Inc., New

York.

4. J. A. MacDonald. For Want of Bolt. Civil Engineering,

October 1988.

5. FHWA Memorandum. High-Strength Bolts, November

1989.

APPENDIX

November 1989

SUPPLEMENTAL CONTRACT SPECIFICATIONS

FOR PROJECTS WITH AASHTO M164 (ASTM A325)

HIGH-STRENGTH BOLTS

A. Scope

A1. All AASHTO M164 (ASTM A325) high-strength

bolts, nuts and washers shall be furnished in accor-

dance with the appropriate AASHTO MaterialsSpecifications as amended and revised herein.

Additional requirements for field or shop instal-

lation of AASHTO M164 (ASTM A325) high-

strength bolts are also included. These additional

requirements supplement AASHTO Division II,

Section 10.

B. Specifications

B1. All bolts shall meet the requirements of AASHTO

M164 (ASTM A325) and these revisions.

B2. All nuts shall meet the requirements of AASHTO

M292 (ASTM A194) as applicable or AASHTO

M291 (ASTM A563) and these revisions.

B3. All washers shall meet the requirements of

AASHTO M293 (ASTM F436) and these revisions.

C. Manufacturing

C1. Bolts

1. Hardness for bolt diameters 12-in. to 1-in. inclu-

sive shall be as noted below:

Hardness Number

Bolt Size, In. Brinell Rockwell C

Min. Max. Min. Max.

12- to 1-in. 248 311 24 33

C2. Nuts

1. Nuts to be galvanized (hot dip or mechanicallygalvanized) shall be heat treated grade 2H, DH,

or DH3.

2. Plain (ungalvanized) nuts shall be grades 2, C,

D, or C3 with a minimum Rockwell hardness

of 89 HRB (or Brinell hardness 180 HB), or heat

treated grades 2H, DH, or DH3. (The hardness

requirements for grades 2, C, D, and C3 exceed

the current AASHTO/ASTM requirements.)



10/174

3. Nuts that are to be galvanized shall be tapped

oversize the minimum amount required for

proper assembly. The amount of overtap in the

nut shall be such that the nut will assemble freely

on the bolt in the coated condition and shall meet

the mechanical requirements of AASHTO M291

(ASTM A563) and the rotational-capacity test

herein (the overtapping requirements of

AASHTO M291 (ASTM A563) paragraph 7.4

shall be considered maximum values instead of

minimum, as currently shown).

4. Galvanized nuts shall be lubricated with a lubri-

cant containing a dye of any color that contrasts

with the color of the galvanizing.

C3. MarkingAll bolts, nuts and washers shall be

marked in accordance with the appropriate

AASHTO/ASTM Specifications.

D. Testing

D1. Bolts

1. Proof load tests (ASTM F606 Method 1) are

required. Minimum frequency of tests shall be

as specified in AASHTO M164 (ASTM A325)

paragraph 9.2.4.

2. Wedge tests on full size bolts (ASTM F606 para-

graph 3.5) are required. If bolts are to be gal-

vanized, tests shall be performed after galvaniz-

ing. Minimum frequency of tests shall be as

specified in AASHTO M164 (ASTM A325)

paragraph 9.2.4.

3. If galvanized bolts are supplied, the thickness of

the zinc coating shall be measured. Measure-

ments shall be taken on the wrench flats or topof bolt head.

D2. Nuts

1. Proof load tests (ASTM F606 paragraph 4.2) are

required. Minimum frequency of tests shall be

as specified in AASHTO M291 (ASTM A563)

paragraph 9.3 or AASHTO M292 (ASTM A194)

paragraph 7.1.2.1. If nuts are to be galvanized,

tests shall be performed after galvanizing, over-

tapping and lubricating.

2. If galvanized nuts are supplied, the thickness of

the zinc coating shall be measured. Measure-

ments shall be taken on the wrench flats.

D3. Washers1. If galvanized washers are supplied, hardness

testing shall be performed after galvanizing.

(Coating shall be removed prior to taking hard-

ness measurements).

2. If galvanized washers are supplied, the thickness

of the zinc coating shall be measured.

D4. Assemblies

1. Rotational-capacity tests are required and shall

be performed on all black or galvanized (after

galvanizing) bolt, nut and washer assemblies by

the manufacturer or distributor prior to shipping.

Washers are required as part of the test even

though they may not be required as part of the

installation procedure.

The following shall apply:

a. Except as modified herein, the rotational-

capacity test shall be performed in accor-

dance with the requirements of AASHTO

M164 (ASTM A325).

b. Each combination of bolt production lot, nut

lot and washer lot shall be tested as an assem-

bly. Where washers are not required by the

installation procedures, they need not be

included in the lot identification.

c. A rotational-capacity lot number shall be

assigned to each combination of lots tested.

d. The minimum frequency of testing shall be

two assemblies per rotational-capacity lot.e. The bolt, nut and washer assembly shall be

assembled in a Skidmore-Wilhelm Calibra-

tor or an acceptable equivalent device (note:

this requirement supersedes the current

AASHTO M164 (ASTM A325) requirement

that the test be performed in a steel joint).

For short bolts which are too short to be

assembled in the Skidmore-Wilhelm Calibra-

tor, see Section D4.1i.

f. The minimum rotation, from a snug tight

condition (10% of the specified proof load),

shall be:

240 (23 turn) for bolt lengths < 4 diameters

360 (1 turn) for bolt lengths > 4 diameters

and < 8 diameters

480 (113 turn) for bolt lengths > 8 diameters

(Note that these values differ from the

AASHTO M164 Table 8/ASTM A325 Table

6 Specifications.)

g. The tension reached at the above rotation

shall be equal to or greater than 1.15 times

the required installation tension. The instal-

lation tension and the tension for the turn testare shown below:

Diameter (in.) 12 58 34 78 1 118 114 138 112

Req. installation

tension (kips) 12 19 28 39 51 56 71 85 103

Turn test

tension (kips) 14 22 32 45 59 64 82 98 118



11/174

h. After the required installation tension listed

above has been exceeded, one reading of

tension and torque shall be taken and

recorded. The torque value shall conform to

the following:

Torque 0.25 PD

WhereTorque = measured torque (foot-pounds)

P = measured bolt tension (pounds)

D = bolt diameter (feet).

i. Bolts that are too short to test in a Skidmore-

Wilhelm Calibrator may be tested in a steel

joint. The tension requirement of Section

D4.1g need not apply. The maximum torque

requirement of Section D4.1h shall be com-

puted using a value of P equal to the turn test

tension shown in the table in Section D4.1g.

D5. Reporting

1. The results of all tests (including zinc coatingthickness) required herein and in the appropri-

ate AASHTO specifications shall be recorded on

the appropriate document.

2. Location where tests are performed and date of

tests shall be reported on the appropriate

document.

D6. Witnessing

1. The tests need not be witnessed by an inspec-

tion agency; however, the manufacturer or dis-

tributor that performs the tests shall certify that

the results recorded are accurate.

E. DocumentationE1. Mill Test Report(s) (MTR)

1. MTR shall be furnished for all mill steel used

in the manufacture of the bolts, nuts, or washers.

2. MTR shall indicate the place where the mate-

rial was melted and manufactured.

E2. Manufacturer Certified Test Report(s) (MCTR)

1. The manufacturer of the bolts, nuts and washers

shall furnish test reports (MCTR) for the item

furnished.

2. Each MCTR shall show the relevant informa-

tion required in accordance with Section D5.

3. The manufacturer performing the rotational-

capacity test shall include on the MCTR:a. The lot number of each of the items tested.

b. The rotational-capacity lot number as

required in Section D4.1c.

c. The results of the tests required in Section D4.

d. The pertinent information required in Sec-

tion D5.2.

e. A statement that MCTR for the items are in

conformance to this specification and the

appropriate AASHTO specifications.

f. The location where the bolt assembly com-

ponents were manufactured.

E3. Distributor Certified Test Report(s) (DCTR)

1. The DCTR shall include MCTR above for the

various bolt assembly components.

2. The rotational-capacity test may be performed

by a distributor (in lieu of a manufacturer) and

reported on the DCTR.

3. The DCTR shall show the results of the tests

required in Section D4.

4. The DCTR shall also show the pertinent infor-

mation required in Section D5.2.

5. The DCTR shall show the rotational-capacity lot

number as required in Section D4.1c.

6. The DCTR shall certify that the MCTR are in

conformance to this specification and the

appropriate AASHTO specifications.

F. ShippingF1. Bolts, nuts and washers (where required) from each

rotational-capacity lot shall be shipped in the same

container. If there is only one production lot num-

ber for each size of nut and washer, the nuts and

washers may be shipped in separate containers. Each

container shall be permanently marked with the

rotational-capacity lot number such that identifica-

tion will be possible at any stage prior to installation.

F2. The appropriate MTR, MCTR or DCTR shall be

supplied to the contractor or owner as required by

the Contract Documents.

G. InstallationThe following requirements for installation apply in addi-

tion to the specifications in AASHTO Division II, Sec-

tion 10 when high-strength bolts are installed in the field

or shop.

G1. Bolts shall be installed in accordance with AASHTO

Division II Article 10.17.4. During installation,

regardless of the tightening method used, particu-

lar care should be exercised so that the snug tight

condition as defined in Article 10.17.4 is achieved.

G2. The rotational-capacity test described in Section D4

above shall be performed on each rotational-

capacity lot prior to the start of bolt installation.

Hardened steel washers are required as part of thetest although they may not be required in the actual

installation procedures.

G3. A Skidmore-Wilhelm Calibrator or an acceptable

equivalent tension measuring device shall be

required at each job site during erection. Periodic

testing (at least once each working day when the

calibrated wrench method is used) shall be per-

formed to assure compliance with the installation



12/174

test procedures required in AASHTO Division II,

Article 10.17.4.1 for Turn-of-Nut Tightening,

Calibrated Wrench Tightening, Installation of Alter-

nate Design Bolts and Direct Tension Indicator

Tightening. Bolts that are too short for the

Skidmore-Wilhelm Calibrator may be tested using

direct tension indicators (DTIs). The DTIs must be

calibrated in the Skidmore-Wilhelm Calibrator using

longer bolts.

G4. Lubrication

1. Galvanized nuts shall be checked to verify that

a visible lubricant is on the threads.

2. Black bolts shall be oily to the touch when

delivered and installed.

3. Weathered or rusted bolts or nuts not satisfying

the requirements of G2 or G3 above shall be

cleaned and relubricated prior to installation.

Recleaned or relubricated bolt, nut and washer

assemblies shall be retested in accordance with

G2 above prior to installation.

G5. Bolt, nut and washer (when required) combinations

as installed shall be from the same rotational-

capacity lot.



13/174

INTRODUCTION

Recent studies have pointed to the behavior of beam-columnconnections as having an important effect on stiffness and

strength of steel frames,1,2 and considerable work has been

done to develop analysis methods intended to include not

only member, but also connection behavior.3,4

Design methods as outlined in the AISC Allowable

Stress5 and LRFD6 Specifications authorize inclusion of

connection effects under the heading of Type 3 in the

former, and Partially Restrained (PR) in the latter.In both analysis and design including connection effects,

connection behavior must be known. For typical beam-to-

column connections of building frames, voluminous, if frag-

mentary, data are available.7,8,9 Attempts at rational predic-

tion of connection behavior have been less than successful,

but empirical expressions, based on test data, of the relation

between the applied moment Mand the resulting connec-

tion rotation are available. Among these, the most com-monly used are those of Frye and Morris,10 shown inFig. 1.

The deterministic moment-rotation curves shown inFig. 1,

and others similar, are often based on one single test, and

do not account for the scatter which may inevitably be

expected of connection behavior, specially if field-bolted.Little is available in the way of replicate tests which might

provide a database necessary for statistical prediction of con-

nection behavior. Until such information about reliability of

connection behavior is provided, its inclusion in design or

analysis rests, at best, on a shaky basis.

This paper reports a study the aim of which is to provide a

statistical database for the purpose of establishing the

degree of reliability of strength and stiffness for one con-

nection type. To this end, nominally identical framing con-

nection specimens from different sources were tested under

identical conditions. The individual moment-rotation curves

obtained from these tests form the database for probabil-

istic determination of the reliability with which specified

behavior of these connections can be expected.

TEST PROGRAM

Specimens

Six fabricators volunteered to provide double-web angle con-

nection specimens fabricated according to the drawing and

specifications shown inFig. 2. Two identical specimens were

provided with untensioned bearing-type bolts (B-bolts), and

two with friction-type bolts (F-bolts) tensioned according toshop practice of the individual fabricator, for a total of 12

specimens for each bolt type. Since each specimen contained

two web-angle connections, we had in fact a sample of 24

Reliability of Rotational Behaviorof Framing Connections

THOMAS R. RAUSCHER and KURT H. GERSTLE

Thomas R. Rauscher is a master degree candidate in the

Civil, Environmental and Architectural Engineering Depart-

ment at the University of Colorado, Boulder, CO.

Kurt H. Gerstle is a professor in the Civil, Environmental and

Architectural Engineering Department at the University of

Colorado, Boulder, CO.Fig. 2. Test specimen.

Fig. 1. Connection moment-rotation curves.10



14/174

of each connection.

In addition, one fabricator supplied us with a set of six

specimens with 38-in. thick web angles with F-bolts, attached

to previously tested members.Table 1 summarizes the test

specimens. This program gave us the opportunity to assess

the following factors:

Scatter of connection behavior Comparison of B-bolt versus F-bolt behavior

Influence of connection stiffness Effect of applied load history.

The ratio of moment to shear transmitted by the connec-

tion might have considerable influence on its behavior, but

was not a variable in our study. It was held constant at the

value of shear span shown inFig. 2.

It should be noted that these double web-angle connec-

tions are commonly used as shear connections. Our discus-

sion only concerns their rotational characteristics and there-

fore none of the conclusions should be interpreted as

addressing their reliability in transmitting shear. We are here

only concerned with the way in which they can be expected

to rotate under applied moment.The test configuration used in this study, consisting of

beams and column stub as shown inFigs. 2 and3, should

not be construed as suggesting that web angles should be used

to provide lateral resistance for unbraced frames. This speci-

men type was used here only to provide a simple connection

test setup.

Test Method and Instrumentation

The specimens were mounted as shown inFig. 3 in a 1000 kip

MTS universal testing machine with load and displacement

control. Instrumentation consisted of rotation meters and

strain-gaged links to determine applied moments. The former,

also shown inFig. 3, consisted of an aluminum frame mountedon the beam, with linear variable differential transducers

(LVDTs) bearing against the column flange. Each link support

shown inFig. 3 was instrumented for measurement of reac-

tions in order to determine the connection moment.

Test Procedure

All tests were carried out under load control. Two types of

load history were applied: A cyclic regime (C-Type) con-

sisting of three cyclic reversals each up to moments of 80,

160, and 240 kip-inches for Test Series 1 and 2, and 160,

320, and 480 kip-inches for Series 3, followed by load

increase up to a rotation of about 0.06 radians which would

entail contact between beam and column flanges. For com-

parison, some of the specimens were subjected to a mono-

tonic load increase (M-Type) up to maximum connection

rotation.

During tests, data were collected by a ten-channel data

acquisition system at specified time intervals, and signifi-cant events were recorded. In some tests, the shock caused

by sudden bolt slip was sufficient to cause displacement of

the LVDTs; corrections were made to the readings in such

cases.

TEST RESULTS

All test results will be presented in the form of monotoni-

cally increasing moment-rotation curves. These were

obtained from the cyclic tests by drawing envelope, or spline,

curves circumscribing the cyclic response. Comparison with

curves from monotonic tests, described in greater detail in

Ref. 11, was in general good.

Test results will be described separately for the different

series specified inTable 1.

Table 1.Test Program

Test

Series

No. of

Fabricators

Connection

Type

No. of

Specimens

No. of

Connections

Angle

Thickness

1 6 B-Bolt 12 24 14

2 6 F-Bolt 12 241

4

3 1 F-Bolt 6 12 38

Fig. 3. Test setup.



15/174

Series 1

Figure 4 shows monotonic moment-rotation curves obtained

from 24 connections in 12 specimens obtained from six dif-

ferent fabricators. As might be expected of connections with

non-tensioned bolts of random location within 116-in. over-

size holes, the range of rotational behavior is vast. These

results are sufficiently unpredictable that no reliance what-ever can be placed on the rotational resistance of such web-

angle connections with bearing bolts. No further reference

will be made to the results of Test Series 1.

Series 2

Moment-rotation curves from 22 connections of 11 speci-

mens of Series 2, obtained either from monotonic, or as enve-

lope curves from cyclic tests, are shown inFig. 5. Although

showing considerable variation, a systematic random pattern

is seen here for both stiffness and strength. Non-linearity

is mainly due to yielding of the outstanding angle legs, and

bolt slip occurs only under rotations well in excess of admis-

sible values.

Series 3

The 12 moment-rotation curves for these 38-in. web angle

connections furnished by one fabricator are shown inFig. 6,

indicating consistency in the initial stiffness, but considera-

ble scatter in the occurrence of bolt slip which accounted

for the onset of softening of these connections.

Descriptive Parameters of Connection Response

The parameters used to describe the connection response in

the statistical analysis which follows were the secant modu-

lus Ksec, the elastic limit moment Mel, and the moment

under permissible rotationMs, as shown inFig. 7.

The secant modulus Ksec was based on the moment cor-

responding to a rotation of 0.002 radians, well within the elas-tic range. Mel was obtained visually as the moment corre-

sponding to the onset of softening of theM- curve.Ms wasthe moment corresponding to the end rotation of a uniformly

loaded simple beam under allowable midspan deflection

L/360, computed as 0.009 radians.

Tables 2 and 3 show the values of these parameters for

the right and left connection of each of the 12 specimens

of Series 2, and of the six specimens of Series 3. In these

tables, fabricator, test number, and loading type, parameter

values, and tension control are shown. These values furnish

the database for the statistical study of the next section.

STATISTICAL ANALYSISThe purpose of our study is to assess the reliability with

which strength and stiffness of these web angle connections

can be predicted. To this end, we will subject the strength

parameters Mel and the stiffness parameter Ksec, defined in

Fig. 7, to statistical analysis with the aim of predicting their

minimum values which may be expected with specified prob-

ability, or confidence level. In addition, we will try to extract

Fig. 4. Bearing-type bolt connection response. Fig. 5. Friction-type connection response, Series 2.



16/174

information about systematic differences between products

of different fabricators in order to obtain insight into prob-lems of quality control.

Statistical Methods

The value of any characteristic will vary among the speci-

mens tested. The total of these specimens is called the sam-

ple. The individual values can be plotted in the form of a

histogram. We assume that this histogram can be matched

under increasing sample size by a continuous bell-shaped

curve containing an area of value unity, as shown inFig. 8,

representing a normal distribution. This curve displays the

character of the population of an infinite number of such

specimens, of which the sample is assumed to be a part. The

shape of this curve can be defined by just two parameters,the meanXand the standard deviation S, defined inFig. 8.

The coefficient of variation S/Xindicates the degree of scat-

ter of results among nominally identical specimens.

The probability P of exceeding any particular value of the

parameterx is given by the area under the bell curve (shown

shaded in Fig. 8) which is to the right of that value, and which

can range from zero to unity.

The probability P can be found for a distribution with given

Xand Sfor any value ofx by integration, or from available

tables.12

In this way, we will determine the minimumstrength and stiffness which can be expected at a specified

level of confidencesay, 95 times out of the next 100 speci-

mens, as will be assumed in what follows.

The methods just described depend on the premise that

all specimens belong to the same population. However, the

techniques of different fabricators could be so different that

their products might not belong to one population. Such con-

ditions are determined by an analysis of variance

(ANOVA).12 An occurrence of this type will be discussed

below in connection with the stiffnesses of Series 2.

These techniques were applied to the test data in the

following sequence: the strengths Mel and Ms, and the

stiffness Ksec of Series 2 and 3 were first subjected to ananalysis of variance to determine the likelihood of their

belonging to one or more populations to within the 95

percent level of confidence, using the F-Test described in

Ref. 12.

For each population, the valuesXand Sof the normal dis-

tribution were computed, and the minimum value of each

parameter which might be expected within 95 percent con-

fidence level was calculated.

Table 2.Sample Data for Series 2

Fabricator

Test

No.

Loading

Type

Ksec (kip-in./radian) Mel (kip-in.) Ms (kip-in.)

Tension Control Methodos rs is rs is rs

1 1

15

C

C

30,000

46,500

33,000

52,000

168

100

182

167

183

110

200

185

Calibrated wrench

Specified tension

2 1114

CM

80,00059,500

72,50039,500

175160

240218

172200

212220

not available

3 23

24

C

M

69,000

75,000

74,000

66,000

140

150

179

195

135

133

179

195Twist-off

4 13

8

C

M

40,000

50,000

168

200

210

225

Turn-of-nut

(no data recorded)

5 3

4

M

C

35,000

37,000

34,000

32,500

120

140

135

142

172

155

175

160Twist-off

6 6

16

M

M

47,500

24,242

35,000

25,806

175

130

185

165

150

161

118

183Twist-off

Table 3.

Sample Data Test Series 3

Fabricator

Test

No.

Loading

Type

Ksec (kip-in./radian) Mel (kip-in.) Ms (kip-in.)

Tension Control Methodis rs is rs is rs

3

25

26

27

28

29

30

M

C

C

M

C

M

95,000

135,000

89,000

95,000

105,000

99,000

115,000

89,000

112,500

115,000

130,000

100,000

338

265

395

265

360

370

338

270

350

230

325

365

345

280

360

243

370

370

345

280

340

280

335

378

Twist-off



17/174

Series 2

Strength

The strengthsMel andMs, defined in Fig. 7, were subjected

to the statistical treatment outlined, and the results are sum-

marized inTable 4. An ANOVA showed to within a 95 per-

cent confidence level that the strength of all 22 specimens

belonged to one population, whose characteristic values Xand Sare shown in Table 4, and that one might expect 95

out of the next 100 specimens to have strengths in excess

ofMel = 89 kip-inches andMs = 99 kip-inches.

Stiffness

The observed stiffnesses Ksec listed in Table 2 showed a

great deal of scatter, indicated by the coefficient of varia-

tion shown inTable 4 and the dashed curve ofFig. 9. The

ANOVA showed two distinct populations: Population A, con-

sisting of 14 specimens from Fabricators 1, 4, 5, and 6, andPopulation B, of eight specimens from Fabricators 2 and 3.

The statistical characteristics of each of these populations,

as well as those of the composite sample of 22 specimens,

are presented inFig. 9 andTable 5. These results show that

of the next 100 specimens from the first set of fabricators,

95 can be expected to have a stiffness Ksec in excess of

14,486 kip-in./radian, and of those from the second set of

fabricators, 95 can be expected to have stiffnesses in excess

of 26,438 kip-in./radian. If all 22 specimens are lumped

together, then a minimum stiffness of only 6,475 kip-in./

Fig. 7. Descriptive parameters of connection response.

Fig. 6. Friction-type bolt connection response, Series 3.

Fig. 9. Assumed distribution for Ksec.

Fig. 8. Assumed population distribution.

X__

=1

ni= 1

n

xi ; n= Sample Size

S=1ni= 1

n

(xiX__)2



18/174

radian can be assumed at the 95 percent confidence level,

a value so low as to be negligible.

The expected stiffness of specimens from Fabricators 2

and 3 is about twice that of specimens from Fabricators 1,

4, 5, or 6. One might look for obvious manufacturing differ-

ences among these fabricators. The last column ofTable 2

gives little clue as to causes: Three different bolt tension con-

trol methods were used by the fabricators of Population A,

among whom two used the same method as one of the fabri-

cators of Population B. The reason for these seemingly sys-

tematic differences remains unknown.

Series 3

The 12 38-in. web angle specimens constituting Series 3

came from one Fabricator (No. 3). In fact, the M- curvesofFig. 6show much less scatter prior to bolt slip than those

of Fig. 5 for Series 2. The strength of these connections,

defined by the onset of softening, was determined by bolt

slip; this is in contrast to the softening of the 14-in. angle

connections which was caused by yielding of the outstand-

ing angle legs. The uncertainty of this event seems to be about

the same, no matter what the cause, as evidenced by com-

parison of the coefficients of variation for the strength mea-

sures of Series 2 and 3.

The statistical analysis summarized inTable 6 indicates

that at the 95 percent confidence level both strength and stiff-

ness belong to one population. Values of strength and stiff-

ness which may be expected to be exceeded in 95 out of the

next 100 specimens from Fabricator 3 are also shown in

Table 6.

The coefficient of variation for the stiffness Ksec of

the specimens of Series 3 is less than half of that of

Series 2, indicating good quality control within one fabrica-

tor. For strength, Series 2 and 3 have similar scatter,

Table 4.Composite Sample Statistics Test Series 2

Series 2

Sample Size: n= 22

Stiffness Strength

Ksec (kip-in./radian) Ms (kip-in.) Mel (kip-in.)

Sample mean

Standard deviation

Coefficient of variation

48,093

17,710

36.8%

174

32

18.4%

165

35

21.2%

Stat. minimum P= 95%

C= 95% Confidence Interval

min Ksec= 6,475 min Ms= 99 min Mel= 89

Table 5.Population Dependent Statistics Test Series 2

Series 2

Population A Population B

Ksec (kip-in./radian) Ksec (kip-in./radian)

Fabricator

Sample size

1, 4, 5, 6

n= 14

2, 3

n= 8

Sample mean

Standard deviation


37,325

8,737

23.4%

66,938

12,704

18.9%



min Ksec= 14,487 min Ksec= 26,438

Table 6.Statistics Test Series 3

Series 3

Sample Size: n= 12

Stiffness Strength

Ksec (kip-in./radian) Ms (kip-in.) Mel (kip-in.)

Sample mean

Standard deviation


107,667

15,091

14.2%

328

44

13.4%

323

52

16.1%



min Ksec= 65,377 min Ms= 208 min Mel= 180



19/174

indicating the difficulty of predicting bolt slip even within

one shop.

DISCUSSION OF RESULTS

How will these results affect the designer who might wish

to include connection restraint as provided by Type 3 Con-

struction in the ASD, and PR Design in the LRFD Specifi-

cations? An example of this approach has been given by

Lindsey13 in an effort to optimize purlin size. In such a

case, the engineers likely recourse for the determination of

connection stiffness and strength is to rely on analytical for-

mulations such as that of Frye and Morris, which, as stated

earlier, are deterministic and have in some cases8 been

found at variance with test data.

For the 14-in. web angle connections of Series 2, the curve

predicted by Frye and Morris is shown inFig. 10, along with

the range of the M- curves from our tests. The Frye andMorris curve is somewhat on the high side. Its initial stiff-

ness is also shown, and the connection strength can readily

be extrapolated.If for safetys sake it is specified that these connection prop-

erties should be at the 95 percent level of confidence, then

our statistical calculations would permit a serviceability

moment and stiffnesses as also shown inFig. 10, of values

greatly below those given by deterministic formulation, or

by any one of the test curves.

Figure 11shows similar comparisons for Series 3: The Frye

and Morris prediction is much too high (a fact which veri-

fies the findings of Ref. 8). Because of the low scatter of

the observed initial stiffnesses, the stiffness at the 95 per-

cent confidence level is close to the measured values, but

the strength under serviceability is much lower than any

observed value.

It is clear that in any case the choice of either a determinis-

tic formulation such as that of Frye and Morris, or a single

test case, may lead to connection strength and stiffness

grossly on the unsafe side of values in the actual structure.

CONCLUSIONS

Based on the test results and analyses which have been

presented, we can draw the following conclusions for ro-

tational behavior of the web angle connections under

consideration:

1. The bearing-bolt connections showed unpredictable

behavior; they are not recommended for joints intended

to offer rotational constraint.

2. The friction-bolt connections exhibited a systematic pat-

tern of behavior, whose non-linearity was caused

largely by yielding for thin web angles, and by bolt slipfor thicker angles.

3. The scatter of stiffness is much less for the stronger

than for the weaker connections; on this basis, it may

be expected that the statistical variation of joints

designed as moment-resistant may be more favorable

than that of the web-angle connections.

4. The strength of the connections, while showing con-

siderable scatter, varied insignificantly among fabri-

Fig. 10. Properties of Test Series 2. Fig. 11. Properties of Test Series 3.



20/174

cators. Statistical minimum values can be determined

with a reasonable level of confidence.

5. Initial stiffness varied significantly among fabricators

for the thin web-angle connections, although no physi-

cal reasons could be identified. It was not possible to

assign meaningful statistical stiffness values for these

specimens based on the totality of our test data. The

thicker web-angle connections, from one fabricator,

showed much more consistent response.

6. Deterministic predictions of connection behavior, based

on either empirical formulations or single test data, are

likely to overestimate reliable values of strength and

stiffness. Statistically designed replicate test series are

needed to establish these characteristics.

REFERENCES

1. Gerstle, K. H., Flexibly Connected Steel Frames, in

Steel Framed Structures, R. Narayanan, Ed., Elsevier,

London and N.Y., 1985.

2. Bjorhovde, R., Brozzetti, J., and Colson, A., Connec-tions in Steel Structures, Elsevier, London and N.Y.,

1988.

3. Gerstle, K. H., Effects of Connections on Frames,

Jnl. Construct. Steel Res., Vol. 10, 1988, p. 241.

4. Ackroyd, M. H., and Gerstle, K. H., Strength of

Flexibly-Connected Steel Franes, Engineering Struc-

tures, Vol. 5, 1983, p. 31.

5. AISC,Allowable Stress Specifications, Sec.1.2, 1978.

6. AISC,LRFD Specifications, Sec. A2, 1986.

7. Goverdhan, A., A Collection of Experimental Moment-

Rotation Curves for Semi-Rigid Connections, M.S.

Thesis, Dept. of Civil Engineering, Vanderbilt Univer-

sity, Nashville, TN, 1983.8. Kishi, N., and Chen, W. F., Data Base of Beam-

Column ConnectionsA Review of Test Data, Report,

School of Civil Engineering, Purdue University, West

Lafayette, IN, 1986.

9. Nethercot, D. A., Steel Beam to Column Connec-

tionsA Review of Test Data and its Applicability to

the Evaluation of Joint Behavior on the Performance of

Steel Frames, CIRIA Project Record 338, Sept. 1985.

10. Frye, M. J., and Morris, G. A., Analysis of Flexibly-

Connected Steel Frames, Can. Jnl. of C.E., Vol. 2,

1975, p. 280.

11. Rauscher, T. R., Reliability of Double-Web Angle Con-

nection Behavior, M.S. Thesis, C.E.A.E. Dept., Uni-

versity of Colorado, Boulder, 1989.

12. Lipson, C., and Sheth, N. J., Statistical Design and

Analysis of Engineering Experiments, McGraw-Hill,

N.Y., 1973.

13. Lindsey, S. D., Ioannides, S. A., and Goverdhan, A.,

LRFD Analysis and Design of Beams, AISC Engi-

neering Journal, Fourth Quarter 1985, p. 157.

ACKNOWLEDGMENT

This study was carried out as an M.S. thesis by the first-

named author in the C.E.A.E. Department of the Univer-

sity of Colorado in Boulder. The following colleagues and

fabricators made this work possible by furnishing specimens,

equipment, funds, and, most importantly of all, enthusiasm

and expert advice, for which we thank them sincerely: Wil-

liam Ashton, Egger Steel, Sioux City, IA; Michael Milot,

Boulder Steel, Boulder, CO; Jim Roscoe, Roscoe Steel,

Boise, ID; Ron Singleton, Stanley Structures, Denver, CO;

Maynard Trostel, Platte River Steel, Greeley, CO; George

Zimmerman, Zimmerman Metals, Denver, CO; and Wil-

liam Zimmerman, Zimkor Industries, Littleton, CO.



21/174

INTRODUCTION

The basic provisions related to design and evaluation ofbending members in the structural steel specifications, either

according to Load and Resistance Factor Design (LRFD)1 or

Allowable Stress Design (ASD),2 typically are first presented

from the point of view that the magnitude of bending moment

is constant throughout the entire distance between points of

lateral support for the compression flange. Then, to account

for variations in moment, one multiplies the expression asso-

ciated with constant moment by a correction factor Cb to

arrive at a result which predicts the actual bending strength

(or allowable stress) for a specific moment gradient. What one

accomplishes is to account for changes that occur in the force

within the compression flange of the beam throughout the

unbraced length.

A procedure for selecting beams in situations involving

non-uniform moment is suggested within the prelude to the

charts of design moments in the LRFD manual, but only in

extremely brief fashion. The purpose of this paper is to review

the principles associated with the application ofCb, and to

elaborate on the procedure briefly suggested in the LRFD

manual for selecting beams which experience non-uniform

moment (Cb 1).BENDING STABILITY

Basic notions of column strength apply to stability-related

issues in the strength of sections in bending. With a beam,

however, only a portion of the cross section resists the com-

pression. The key issues are still the restraint provided at the

boundaries of the element resisting the compression and the

distance between the locations of lateral support.

The magnitude of the compressive force within a beam

cross section, which will nearly always vary with position

along the span, may be determined by inspection of the

moment diagram. Since resistance to bending is composed of

the internal C(compressive force) and T(tensile force) cou-ple, the magnitude ofCat any location along a span equals

the applied bending moment divided by the internal moment

arm (Fig. 1). Thus, the variation in force within the compres-

sion flange has the same shape as the moment diagram.

Analogies to Single Columns

Three single columns with different variations in axial load

are shown inFig. 2. Each column experiences the same axial

compression (equal to the applied force P) within the upper-

most segment. There is a difference, however, in the maxi-

mum force P that could be applied to each column because

the magnitude of axial compression is reduced along the

length of the columns in parts (b) and (c). Intuition dictates

that the greatest load P may be sustained by the column in

Fig. 2(c). By considering free-body diagrams at various po-

sitions along the length of the columns, one observes that

substantial segments of columns (b) and (c) experience re-

duced compression, compared to column (a). Furthermore,

Patrick D. Zuraski is assistant professor of civil engineering,

Louisiana State University, Baton Rouge, LA.

The Significance and Application ofCb in Beam Design

PATRICK D. ZURASKI

Fig. 1. Internal bending resistance.

Fig. 2. Columns with varying axial compression.

ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION20


22/174

(c) will compare favorably to (b) because the lowermost

segments of (c) experience less compression and, notably, are

in tension for the last 25 percent of the column.

The moment diagrams in parts (a), (b), and (c) ofFig. 3

may be associated with the individual columns in the respec-

tive parts ofFig. 2. The moment diagrams inFigs. 3(b) and

3(c)would exactly correspond to the respective columns in

Fig. 2 only if the opposing mid-length forces were applied as

a continuous distribution, but the basic analogy is the same.

In part (b) of bothFigs. 2 and3, the force in the compression-

carrying element is high at one end of the element and

decreases by 60 percent at the other end. Similar behavior is

demonstrated in part (c) ofFigs. 2 and3, but the change is

more dramatic. At the far end, that part of the element which

had been experiencing compression actually changes and

becomes a tensile element.

In Figs. 2 and 3, for a given length of compression member,

part (c) exhibits the least vulnerability to instability and may

be assigned the greatest magnitude of compression/bending.

The function ofCb is to take these aspects of behavior intoaccount.

BASIC DESIGN EXPRESSIONS

As mentioned previously, design equations pertaining to

beams are first developed from the standpoint of constant

bending moment over the unbraced length. In the LRFD

specification, bending strength is controlled by either of two

equations, Eq. F1-3 or F1-13, depending on whether the yield

stress will have appeared within the cross section at the instant

a loss in load carrying capacity occurs. The presence of

residual stress is taken into account. For the ASD specifica-

tion, allowable bending stress is most frequently controlled

by either Eq. F1-7 or Eq. F1-8, depending on whether lateralor torsional moment strength, respectively, is the more domi-

nant component of bending strength for a given cross section

at the instant that instability occurs.

Only the basic form (and not specific terms) of these design

equations is relevant to discussing the significance ofCb. For

LRFD the basic equation is

Mn=Cb {LRFD Eq. F1-3 or F1-13, for constant moment}

Mp (1)

and for ASD the following usually applies (unless Eq. F1-6

controls)

Fb=Cb {ASD Eq. F1-7 or F1-8, for constant moment}

0.6Fy (2)

where

Cb= 1.75 + 1.05

M1

M2

+ 0.3

M1

M2

2

2.3 (3)

In Eq. 3, the end moments are considered on an absolute value

basis, withM1 equal to the smaller of the moments at the ends

of the unbraced length. Should the moment anywhere within

the unbraced length exceed that which occurs at either end,

the above expression is disregarded and Cb is assigned a value

of one. The matter of the plus/minus sign is considered in thefollowing.

When moment decreases from a value of any magnitude at

one end of the unbraced length to zero at the other end, the

ratioM1/M2 = 0 and Cb = 1.75. (Thus, bending strength [or

allowable stress] is 75 percent greater than that which could

have been achieved had the moment been uniform over the

unbraced length, but limited to a result that does not exceed

Mp [LRFD] or 0.6Fy [ASD].) When the smaller end moment

is non-zero, one must decide on a proper sign for the second

term in Eq. 3. Situations that provide increased strength

compared to zero moment at one end must lead to Cb > 1.75,

and situations with less strength (more closely resembling

uniform moment) should reduce Cb below 1.75, backtoward1.00. Recalling previous discussion associated withFigs. 2

and 3, the decision regarding the proper sign is very straight-

forward.

For situations of single curvature (moment diagram on only

one side of the baseline for the entire unbraced length), the

same flange is always in compression. If there is little change

in moment over the unbraced length, the compressive force

in the flange will be maintained at a fairly constant level. Such

a condition is more susceptible to instability, and bending

strength should not be increased appreciably from that exhib-

ited when moment is constant. Thus, for single curvature, the

sign ofM1/M2 is negative.

For double (reversed) curvature (moment diagram changesfrom one side of the baseline to the other), the flange that

experiences compression eventually changes to tension.

There is greater stability in such a situation, and the moment

that may be applied at one end may be significantly increased

from that which could be applied as constant moment (Cb =

1), and beyond that for zero moment at one end (Cb = 1.75).

Thus, for double curvature, the sign ofM1/M2 is positive.

One may note the same expression is used for Cb in bothFig. 3. Segments with varying bending moment.



23/174

the LRFD and ASD specifications. Since it is the purpose of

this factor to reflect structural behavior and account for the

shape of the moment diagram, it is expected that there would

be no difference in Cb equations between the specifications.

BEAM DESIGN WHEN Cb 1Both the LRFD and ASD manuals contain a series of beam

curves that provide invaluable assistance in selecting a sec-

tion that is suitable for a given combination of bending

moment and unbraced length. These curves, designated Beam

Design Moments (LRFD) and Allowable Moments in Beams

(ASD), apply directly when moment is constant throughout

the unbraced length (Cb = 1). The curves can also provide

significant assistance when Cb 1, after properly accountingfor increased strength resulting from non-uniform moment.

ASD Procedures

Designers have extensive experience with the beam curves in

the ASD manual and have developed a methodology for

selecting beams for situations of non-uniform moment that isconsistent with the design equations governing allowable

stress. Observing ASD Eq. F1-8 (which usually controls

allowable stress, especially for moderate-to-large unbraced

lengthsLb)

Fb =12 103Cb

Lbd

Af

0.60Fy (4)

it is appropriate to useLb/Cb and the applied service moment

as an entry point to the curves because Fb is linear in that

parameter. Should one anticipate that Eq. F1-6 will apply,

experience has shown thatLb/Cb and applied service mo-ment are an appropriate entry point in the curves for finding

an acceptable section. These well-established procedures for

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