Direct Analysis and Design Using Amplified First-Order Analysis

7/28/2019 Direct Analysis and Design Using Amplified First-Order Analysis

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ENGINEERING JOURNAL / FOURTH QUARTER / 2007 / 305

Spec/Manual Reference

In 1976 and 1977, LeMessurier published two landmark

papers on practical methods o calculating second-order e-

ects in rame structures. LeMessurier addressed the proper

calculation o second-order displacements and internal

orces in general rectangular raming systems based on rst-

order elastic analysis. He also addressed the calculation o

column buckling loads or eective length actors using theresults rom rst-order analysis. Several important acts are

emphasized in LeMessuriers papers:

1. In certain situations, braced-rame structures can have

substantial second-order eects.

2. The design o girders in moment-rame systems must ac-

count or second-order moment amplication.

3. Control o drit does not necessarily prevent large second-

order eects.

4. In general, second-order eects should be consideredboth in the assessment o service drit as well as maxi-

mum strength.

Furthermore, LeMessurier discussed the infuence o

nominal out-o-plumbness in rectangular rames as well as

inelastic stiness reduction in members subjected to large

axial loads, although the handling o these actors has ma-

tured in the time since his seminal work.

Direct Analysis and Design Using AmplifedFirst-Order AnalysisPart 1: Combined Braced and Gravity

Framing Systems

In spite o the signicant contributions rom LeMessurier

and others during the past 30 years, there is still a great deal

o conusion regarding the proper consideration o second-

order eects in rame design. Engineers can easily misinter-

pret and incorrectly apply analysis and/or design approxi-

mations due to an incomplete understanding o their origins

and limitations. For instance, in braced rames, it is commonto neglect second-order eects altogether. Although this

practice is acceptable or certain structures, it can lead to

unconservative results in some cases. LeMessurier (1976)

presents an example that provides an excellent illustration

o this issue.

The 2005 AISC Specifcation or Structural Steel Build-

ings (AISC, 2005a), hereater reerred to as the 2005 AISC

Specifcation, provides a new method o analysis and design,

termed the Direct Analysis Method (or DM). This approach

is attractive in that:

It does not require any Kactor calculations,

It provides an improved representation o the internal

orces throughout the structure at the ultimate strength

limit state,

It applies in a logical and consistent ashion or all

types o rames including braced rames, moment

rames and combined raming systems.

The DM involves the use o a second-order elastic analy-

sis that includes a nominally reduced stiness and an ini-

tial out-o-plumbness o the structure. The 1999 AISCLoad

and Resistance Factor Design Specifcation or Structural

Steel Buildings (AISC, 1999), hereater reerred to as the1999 AISC Specifcation and the 2005 AISC Specifcation

permit this type o analysis as a undamental alternative to

their base provisions or design o stability bracing. In act,

the base AISC (1999) and AISC (2005a) stability bracing

requirements are obtained rom this type o analysis.

This paper demonstrates how a orm o LeMessuriers

(1976) simplied second-order analysis equations can be

combined with the AISC (2005a) DM to achieve a particular-

ly powerul analysis-design procedure. In this approach, P

Donald W. White is proessor, structural engineering, me-

chanics and materials, Georgia Institute o Technology, At-

lanta, GA.

Andrea Surovek is assistant proessor, civil and environmen-

tal engineering, South Dakota School o Mines and Technol-

ogy, Rapid City, SD.

Sang-Cheol Kim is structural engineer, AMEC Americas,

Tucker, GA.

DONALD W. WHITE, ANDREA SUROVEK, and SANG-CHEOL KIM


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306 / ENGINEERING JOURNAL / FOURTH QUARTER / 2007


shears associated with the amplied sidesway displacements

are applied to the structure using a rst-order analysis. This

removes the need or separate analyses or no lateral trans-

lation (NT) and with lateral translation (LT), which are

required in general or accurate amplication o the internal

orces in the AISC (2005a)B1-B2 (or NT-LT) approach. The

combination o the DM with the proposed orm o LeMes-suriers equations or the underlying second-order analysis

streamlines the analysis and design process while ocusing

on the ollowing important system-related attributes:

Ensuring adequate overall sidesway stiness.

Accounting or second-order P eects on all the lat-eral load-resisting components in the structure, when

these eects are signicant, including the infuence o

reductions in stiness and increases in displacements

as the structure approaches its maximum strength.

The rst part o the paper gives an overview o the AISC

(2005a) DM in the context o braced rames, or combinedbraced and gravity raming systems. This is ollowed by a

step-by-step outline o the combined use o the DM with

LeMessuriers (1976) second-order analysis approach. The

paper closes by presenting analysis results and load and re-

sistance actor design (LRFD) checks or a basic braced col-

umn subjected to concentric axial compression, and or an

example long-span braced rame rom LeMessurier (1976). A

companion paper, Part 2 (White, Surovek, and Chang, 2007),

discusses an extension o the above integrated approach to

general raming systems including moment rames and mo-

ment rames combined with gravity and braced raming.

OVERVIEW OF THE DIRECT ANALYSIS METHOD

FOR BRACED-FRAME SYSTEMS

For simply-connected braced structures, the DM requires

two modications to a conventional elastic analysis:

1. A uniorm nominal out-o-plumbness o o = L/500 isincluded in the analysis, to account or the infuence o

initial geometric imperections, incidental load eccen-

tricities and other related eects on the internal orces

under ultimate strength loadings. This out-o-plumbness

eect may be modeled by applying an equivalent notional

lateral load o

Ni = 0.002Yi

at each level in the structure, where Yi is the actored grav-

ity load acting at the i th level. Alternatively, the nonverti-

cality may be modeled explicitly by altering the rame

geometry. The above nominal out-o-plumbness is equal

to the maximum tolerance specied in the AISC Code o

Standard Practice (AISC, 2005b).

2. The nominal stinesses o all the components in the

structure are reduced by a uniorm actor o 0.8. This ac-

tor accounts or the infuence o partial yielding o the

most critically loaded component(s), as well as uncertain-

ties with respect to the overall displacements and stiness

o the structure at the strength limit states.

These adjustments to the elastic analysis model, combinedwith an accurate calculation o the second-order eects, pro-

vide an improved representation o the second-order inelas-

tic orces in the structure at the ultimate strength limit. Due

to this improvement, the AISC (2005a) DM bases the mem-

ber axial resistance, Pn, on the actual unsupported length not

only or braced and gravity rames, but also or all types o

moment rames and combined raming systems.

The above modications are or the assessment o

strength. In contrast, serviceability limits are checked using

the ideal geometry and the nominal (unreduced) elastic sti-

ness. Also, it should be noted that the uniorm actor o 0.8,

applied to all the stiness contributions, infuences only the

second-order eects in the system. That is, or structures in

which the second-order eects are small, the stiness reduc-

tion has a negligible eect on the magnitude and distribu-

tion o the system internal orces. The rationale or the above

modications is discussed in detail by White, Surovek-Ma-

leck, and Kim (2003a), White, Surovek-Maleck, and Chang

(2003b), Surovek-Maleck and White (2004), and White,

Surovek, Alemdar, Chang, Kim, and Kuchenbecker (2006).

The reader is reerred to Maleck (2001), Martinez-Garcia

(2002), Deierlein (2003, 2004), Surovek-Maleck, White,

and Ziemian (2003), Maleck and White (2003), Nair (2005),

and Martinez-Garcia and Ziemian (2006) or other detailed

discussions as well as or validation and demonstration othe DM concepts.

STEP-BY-STEP APPLICATION: COMBINATION

OF THE DIRECT ANALYSIS METHOD WITH

LEMESSURIERS SECOND-ORDER

ANALYSIS PROCEDURE

The proposed second-order analysis and design procedure

involves a combination o the DM with a specic orm o

the approach or determining second-order orces in braced-

rame systems originally presented by LeMessurier (1976).

A succinct derivation o the key equations is provided in Ap-pendix A. For a given load combination, the basic steps o

the proposed combined procedure are as ollows:

1. Perorm a rst-order elastic analysis o the structure.

2. Obtain the total rst-order story lateral displacement(s).

3. Calculate the story sidesway displacement amplication

actor(s).

(1)


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4. Calculate the story P shears based on the amplied storysidesway displacement(s).

5. Apply the P shears in a separate rst-order analysisto determine the second-order portion o the internal

orces.

6. Calculate the required orces by summing the appropriate

rst- and second-order contributions and check against

the corresponding design resistances.

In the context o the DM, the above P shears include theeects o a nominal reduction in the structure stiness and

a nominal initial out-o-plumbness, o. However, or check-ing o service defection limits, or or conventional strength

analysis and design using the Eective Length Method

(ELM), the same approach can be applied using o = 0 andzero stiness reduction, in other words, with a stiness re-

duction actor o 1.0.

The ollowing is a more detailed description o the steps:

1. Perorm a rst-order analysis to obtain the rst-order in-

ternal member orces and story sidesway displacements

or each o the load types that need to be considered. The

authors recommend the use o separate analyses or each

load type (D, L, W, Lr, E, etc.) at the nominal (unactored)

load levels. By arranging the analyses in this way, the

results can be actored and combined using superposition

or each o the required load combinations.

2. Obtain 1, the total rst-order story lateral displacement(s)or a given load combination, by summing the analysis

results rom step 1 multiplied by the appropriate load ac-

tors. Note that throughout this paper, the over bar on avariable means that it is obtained by applying a stiness

reduction actor o 0.8 whenever the DM is used.

3. Calculate the story sidesway amplication actor(s) asso-

ciated with a given load combination using the equation

where

Blt

i

=

1

1

(2)

irP

L= (3)

White et al. (2003a) discuss an appropriate reduction in the o values, or the corresponding notional loads,Ni, rom Equation 1,or tall multi-story rames. The reader should note that the resulting lateral loads shown in Figure 1 include the eect o both the

amplied initial out-o-plumbness, o, as well as the amplied sidesway defections due to the applied loads, .Blt Blt1

is reerred to as the ideal stiness (Galambos, 1998) and

is the actual story sidesway stiness o the lateral load

resisting system in the rst-order analysis model. The

term Pr in Equation 3 is the total actored vertical loadsupported by the story andL is the story height. The term

Hin Equation 4 is the total story shear orce due to theapplied loads and 1His the rst-order horizontal displace-ment due to H. IH= 0 (and thereore 1H= 0), maybe determined by applying any raction oPras an equaland opposite set o shear orces at the top and bottom o

all the stories and then dividing by the corresponding 1H.For combined braced and gravity rames, the sidesway

amplication actorBltis the same as the termB2 in AISC

(2005a). The symbol Blt is used in this paper, since the

subscripts 0, 1, and 2 are reserved to denote the initial,

rst-order and second-order orces and/or displacements.The notation lt stands or lateral translation.

4. For each o the load combinations, calculate the story Pshears,HP, using the equation

and apply to each story o the rame in a separate rst-

order analysis to determine the second-order component

o the internal orces. Figure 1 illustrates the application

o these orces in a multi-story rame.

5. Add the second-order orces rom step 4 to the rst-order

orces rom step 1 times the load actors or the strength

load combination under consideration. This gives the re-

quired strengths throughout the structure.

6. For axially loaded members subjected to transverse loads,

ampliy the internal moments using the traditional NT

amplication actor, denoted byB1 in AISC (2005a). The

calculation o internal moments in general raming sys-

tems is addressed by White, Surovek, and Chang (2007).

7. Check the required orces rom steps 5 and 6 versus thecorresponding design resistances.

=H

H1

(4)

H P BL

P r lt

o

=+

1

(5)


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It should be noted that the above procedure may be ap-

plied or design by either LRFD or ASD. However, i ap-

plied or ASD, AISC (2005a) requires that the applied loads

must be multiplied by a parameter = 1.6 to account or thesecond-order eects at the ultimate load level. The resulting

internal orces are subsequently divided by to obtain the

required ASD orces.To use the above procedure or strength analysis and de-

sign by the DM, the nominal initial out-o-plumbness

and the reduced stiness

are employed. Also, since the stiness is reduced, the cor-

responding rst-order story sidesway displacements are

where 1 is the rst-order story displacement, obtained us-ing the nominal (unreduced) stiness values. The out-o-

plumbness, o, may be modeled explicitly by canting therame geometry or it may be represented by the equivalent

notional lateral loads given by Equation 1. I a notional lat-

eral load is used, this load is handled the same as any other

applied lateral load in the above procedure. Alternately, orservice load analysis, or or strength analysis and design by

the conventional Eective Length Method (ELM), where

the analysis is conducted on the idealized nominally-elastic

initially-perect structure, the above terms are

and

The above analysis approach gives an exact solution or

the second-order DM, ELM, or service level orces and dis-

placements within the limits o:

the idealization o the lateral load resisting system as a

truss,

the approximation cos

the assumption that the stiness

or any story is the

same value or both o the loadings HandH

P, and

the approximation o equal sidesway displacements

throughout each foor or roo level.

That is, with these qualications, LeMessuriers procedure

is an exact noniterative second-order analysis (Vandepitte,

1982; Gaiotti and Smith, 1989; White and Hajjar, 1991) or

combined gravity and braced-rame systems. The assump-

tion o cos is certainly a reasonable one, since

any structure that violates this limit will likely have objec-

tionable sidesway defections under service loads. For multi-

story structures, the interactions between the stories, or ex-

ample, the rotational restraint provided at the top and bottomo the columns in a given story as well as the accumulation

o story lateral displacements due to overall cantilever bend-

ing deormations o the structure, strictly are dierent or

dierent loadings and load eects. However, the dierences

in the values or the dierent loadings are typically small.The handling o unequal lateral displacements at a given

story level, or example, due to thermal expansion or due to

fexible foor or roo diaphragms, is addressed by White et

al. (2003a).Fig. 1. Application o the P shear orces

in a multi-story rame.HP

o L= 0 002.

= 0.8 .

11

0 8

=.

(6a)

(6b)

(6c)

o = 0

=

1 1=

(7a)

(7b)

(7c)

tot L/( ) 1, where

tot L/( ) 1

tot lt oB ,= +( )1


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Since the infuence o member axial deormations, in-

cluding dierential column shortening, as well as other

contributions to the displacements can be included in the

rst-order analysis, the above approach is applicable to all

types o combined gravity and braced building rames. For

a multi-story rame, sidesway due to column elongation and

shortening in the stories below the one under consideration,in other words, sidesway due to cantilever deormations o

the building, is addressed by analyzing the ull structure. In

general, it is not appropriate to determine

= H /

1H by

just applying equal and opposite lateral loads at the top and

bottom o a single story.

The proposed approach is particularly amenable to pre-

liminary analysis and design. Typically, it is desired or

structures to satisy a certain drit limit under service load

conditions. The story P shear orces, , can be calculatedor preliminary design by scaling the target service load drit

limit to obtain the corresponding /L or use in Equation

5. Part 2 o the paper illustrates this process (White et al.,

2007).For structures with a large number o stories and/or com-

plex three-dimensional geometry, the proposed analysis ap-

proach can be programmed to avoid excessive manual calcu-

lations. The above approach can be implemented or general

3D analysis o building rames by incorporating the concepts

discussed by Wilson and Habibullah (1987) and White and

Hajjar (1991) to include the P eects associated with over-all torsion o the structural system. However, in the view o

the authors, the use o general-purpose second-order analy-

sis sotware is oten preerable or complex 3D rames. The

most important benet o the proposed approach is that it

acilitates preliminary analysis and design (using an estimat-ed based on target service drit limits). Also, it is a useul

aid or understanding second-order responses and checking

o computer results. For instance, i rom Equation 5 is

smaller than a certain raction o the story shear due to the

applied lateral loads H(say 5%), the Engineer may chooseto exercise his or her judgment and assume that the second-

order sidesway eects are negligible.

BRACED COLUMN EXAMPLE

Figure 2 shows the results obtained using the proposed com-

bination o the DM with LeMessuriers second-order analy-

sis equations or one o the most basic analysis and designsolutions addressed in the AISC (2005a) Specifcation

determination o the required bracing orces or simply-

supported columns. For this problem, the stability bracing

provisions o AISC (2005a) also apply. These provisions

require a minimum brace stiness o

in LRFD, where = 0.75 and Pr(LRFD) = Pu is the required axialcompressive strength o the column obtained rom the LRFD

load combinations, such as 1.2D + 1.6L. They require

in ASD, where = 2.0 and Pr(ASD) is the required axial com-pressive strength o the column obtained using the ASD load

combinations, or example,D +L.

I one assumes a live-to-dead load ratio,L /D = 3, then the

ratio, Pr(LRFD)/Pr(ASD) is 1.5 or the above dead and live load

combinations. By substituting Pr(ASD) = Pu /1.5 into Equa-

tion 8b, one can observe that this equation is equivalent to

Equation 8a atL /D = 3. However, or other live-to-dead load

ratios and/or other load combinations, the required minimum

brace stiness is slightly dierent in ASD and LRFD.

At the br limit given by Equation 8a, the sidesway am-plication obtained rom an explicit second-order analysis

without any reduction in the stiness is Blt = 1.6. Whenthe analysis o the system shown in Figure 2 is conducted

using the DM, = 0.8br and the sidesway amplication

Fig. 2. Braced column example.

HP

HP

1

1

br

r LRFD u

i

P

L

P

L= = =

22 67 2 67

( ). . (8a)

br

r ASDP

L= 2

( )(8b)


6/18



is = 1.88. As a result, the brace orce induced by the axial

orce, Pr, acting through the amplied nominal initial out-o-

plumbness o o = (0.002L) is

Pbr= Pu (0.002) = 0.00376Pu

rom Equation 5 (note that in this problem). The corre-

sponding base AISC (2005a) Appendix 6 brace orce re-quirement is

Pbr= 0.004Pu

in LRFD (6% higher). The above 6% reduction in the brace

orce may be considered as a benet allowed by AISC

(2005a) or the explicit use o a second-order analysis or

LRFD. Figure 2 shows the brace orce requirements or the

above basic column case as a unction o increasing values

oact/i, where act is the nominal (unreduced) brace sti-ness. The requirements are plotted using the AISC (2005a)

DM with LeMessuriers second-order analysis approach as

well as the using the equation

which is specied as a renement o the brace orce require-

ment in the AISC (2005a) Appendix 6 Commentary. One

can observe that the orce requirements rom the DM and

rom Equation 10 are increasingly close to one another or

larger values oact/i. At act/i = 10, the DM requires Pbr=0.00228Pu whereas Equation 10 gives Pbr= 0.00231Pu, only

57% and 58% o the base AISC (2005a) brace orce require-

ment, respectively. It should also be noted that i the above

analyses are conducted using the AISC (2005a) ASD provi-

sions, the results rom the corresponding orm o Equation10 and the corresponding DM solution match exactly.

LONG-SPAN BRACED FRAME EXAMPLE

Figure 3 shows a long-span roo structure originally consid-

ered by LeMessurier (1976). The rame consists o 165-t-

long trusses at 20 t on center, supporting a total (unactored)

gravity load o 100 ps. A live-to-dead load ratio o 3 is as-

sumed or the purposes o checking this rame by LRFD.

The structure is 18 t high and is required to resist a nominal

wind load o 15 ps. These nominal loadings are the same as

in (LeMessurier, 1976); however, the LRFD actored load-

ings obtained using the above assumptions are dierent thanLeMessuriers actored loads. The nominal column axial

loads are 0.5(165 t)(2 kips/t) = 165 kips. A W848 with Fy= 50 ksi is selected or the column size based on the LRFD

load combinations. LeMessurier selected an ASTM A572

Grade 50 W1448 section in the original design. A braceis provided on the let side o the structure; or architectural

reasons this member is an HSS 4.54.5x with Fy = 46 ksi.The original brace selected by LeMessurier was an

Blt

Blt Blt

Blt(9a)

(9b)

1

P Pbr

br

act

u=

0 004

2

.

(10)

Fig. 3. LeMessuriers (1976) example rame, designed by LRFD.


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HSS 3.53.5x with the same yield strength. The increasein size o the brace is predominantly a result o the LRFD

wind load actor in SEI/ASCE 7-05 (ASCE, 2005) com-

pared to that originally assumed by LeMessurier. This brace

is sized to act in both tension and compression.

The load combinations considered in this example include

the service cases (ASCE, 2005),

D +Lrand

D + 0.5Lr+ 0.7W

where

D = dead load

Lr = roo live load

W = wind load

as well as the strength combinations,

1.2D + 1.6Lr+ 0.8W,

0.9D + 1.6W

and

1.2D + 0.5Lr+ 1.6W

The size o column bc and the tension orce requirement

in the brace ab are governed by the strength load combina-

tion 1.2D + 1.6Lr + 0.8Wwith the wind acting to the right.

However, the size o the diagonal ab is governed by its com-

pressive strength under the load combination 0.9D + 1.6W

with the wind acting to the let.

In the ollowing, separate rst-order analyses o the abovestructure under the nominal (unactored) gravity and wind

loads are presented rst. Then the calculation o the required

axial strengths by the DM is illustrated or the load combina-

tion 1.2D + 1.6Lr + 0.8Wwith the wind acting to the right.

The DM results or the other load combinations are summa-

rized at the end o this presentation. Finally, the results rom

the ollowing analysis models are compared and contrasted

with the DM analysis results:

1. First- and second-order analysis using the nominal (un-

reduced) elastic stiness and o = 0. These solutions areappropriate or analysis o service load conditions, but

are not always appropriate or calculation o the orce re-quirements at strength load levels. The proposed orm o

LeMessuriers (1976) equations is used or these second-

order analyses with zero elastic stiness reduction and

zero initial out-o-plumbness. Since all the strength load

combinations considered here involve a lateral wind load,

these second-order analyses satisy all the requirements

o the Eective Length Method (ELM) o the 2005 AISC

Specication.

2. Summation o the stability bracing orces obtained rom

the rened equations specied in the AISC (2005a) Ap-

pendix 6 Commentary with the bracing orces obtained

rom a rst-order structural analysis. This is the approach

specied in AISC (1999) or calculation o the required

strength o braced-rame systems. The AISC (2005a)

Specication no longer uses this approach. Instead, itspecies the use o either the DM (in its Appendix 7) or

the use o a second-order elastic analysis with nominal

stiness and perect rame geometry, but with a minimum

lateral load included in gravity-only load combinations

(in the ELM procedure o its Chapter C). The stability

bracing provisions o the AISC (2005a) Appendix 6 are

specied solely to handle bracing intended to stabilize

individual members, in other words, cases where bracing

orces due to the applied loads on the structure are not

calculated. However, it is useul to understand the close

relationship between the DM and the AISC (1999) and

AISC (2005a) Appendix 6 stability bracing provisions.

3. Distributed Plasticity Analysis. This method o analysis

is useul or evaluation o all o the analysis and design

methods, since it accounts rigorously or the eects o

nominal geometric imperections and member internal

residual stresses. The details o the Distributed Plasticity

Analysis solutions are explained in Appendix B.

Base First-Order Analysis, Nominal (Unfactored) Loads

LeMessurier (1976) derived base rst-order elastic analysis

equations or the structure in Figure 3. These equations are

summarized here or purposes o continuity. As noted above,

the nominal (D +L) column load in this example is P = 165

kips. The corresponding total story vertical load is P = 330kips. The axial compression in column bc causes it to short-

en. Consequently, compatibility between the diagonal ab and

the top o the column at b requires a story drit ratio o

where

L = story height

B = horizontal distance between the base o col-umn bc and the base o the diagonal ab

Ac = column area = 14.1 in.2

E = 29,000 ksi

The nominal elastic sidesway stiness o the structure is

given by

1 0 002421

413

P

cL

PL

B

A E=

= =. (11)


8/18



Lab = length o the diagonal = L B2 2+ = 18.25 t

Aab = area o the brace = 2.93 in.2

Equations 11 and 12 are rather simple to derive or the ex-

ample rame. In cases where the raming is more complex, it

is oten more straightorward to determine the story stiness

as H/1H. Based on Equation 12, the drit ratio due to thenominal horizontal load H= 2.7 kips is

1 0 001431

699

H

L

H

L

=. (14)

It is important to note that the story drit ratio under the

nominal gravity load alone (1P/L) is signicantly largerthan 1/500. Furthermore, the drit due to the nominal gravity

load is 1.7 times that due to the nominal wind load. This

attribute o the response, as well as the magnitude o the

total vertical load, makes this example a severe test o any

stability analysis and design procedure or braced rames.

The rst-order drit under the nominal wind load itsel is

rather modest.

Strength Analysis Under 1.2D + 1.6Lr + 0.8W, with the Wind

Acting to the Right, Using the Direct Analysis Method

The ideal stiness o the structure or the load combination

(1.2D + 1.6Lr+ 0.8W) is

i D L

P

L( . . )

..1 2 1 6

1 52 292+ = =

kipsin.

(15)

where the actor 1.5 is obtained rom [1.2D +1.6(3D)] / [D +

(3D)]. Furthermore, the sidesway amplication based on the

reduced stiness model is

Blt D L

i D L

( . . )

( . . )

.1 2 1 6

1 2 1 6

1

10.8

1 487+

+

=

= (16)

The total story drit at the maximum strength limit is thereore

tot

D L W

lt D L

P

LB

L

=

+ ++ +

+( . . . )

( . . )

. .

1 2 1 6 0 8

1 2 1 6

10 002 1 5

00 81

0 8

0 01201

83

1..

.

H

L

= =

(17)

This results in a story P shear o

a tension orce requirement in the diagonal ab o

and a maximum compressive strength requirement in col-

umn bc o

Synthesis of Results

Table 1 summarizes the key results rom the service and

the LRFD strength load combinations considered or the

example long-span braced rame. The results or the ser-

vice load combinations are presented rst, ollowed by the

results or the strength load combinations. The procedures

utilized or the analysis calculations are listed in the second

column o the table. The third through sixth columns in-clude the ollowing inormation: the orces in the brace ab

(Fab based on the nominal stiness or Fab based on the

reduced stiness, as applicable); the comparable orces in

column bc (Fbc or Fab ); the drit values (tot/L based on thenominal stiness or tot/L based on the reduced stiness, asapplicable); and the sidesway amplication actor (Bltbased

on the nominal stiness orBltbased on the reduced stiness,

as applicable).

The second-order sidesway amplication is 1.21 and

1.12 under the service load combinations, (D + Lr) and

(D + 0.5Lr + 0.7W), respectively. Thereore, although the

structure is braced, its second-order eects are signicanteven under service loading conditions. Engineers oten as-

sume that second-order eects are small in braced struc-

tures. This is true in many situations, but this assumption

can lead to unacceptable service load perormance in some

cases. The maximum total drit (including the second-order

P eects) or the two service load combinations consid-ered here is 1/341. This drit is acceptable or many types

o structures (West, Fisher and Gris, 2003). However, the

corresponding rst-order drit is 1/413, 21% smaller than

=

+( )

+

=A E

LL

B

c

1

8 7432

.kips

in.

=L A

A L

ab c

ab

(12)

(13)

H P

L

P D L W

tot

D L W

( . . . )

( . . . )

.

.

1 2 1 6 0 8

1 2 1 6 0 8

1 5

5 87

+ +

+ +

=

=

kkips

F H H

L

B

ab D L W P D L W

ab

( . . . ) ( . . . ).1 2 1 6 0 8 1 2 1 6 0 80 8+ + + += +( )

== 48 8. kips

F P H H

L

B

bc D L W P D L W ( . . . ) ( . . . ). .1 2 1 6 0 8 1 2 1 6 0 81 5 0 8+ + + += + +( )

= 296 kips

(18)

(19)

(20)


9/18



the actual drit calculated including the P eects under theservice loading,D +Lr.

The infuence o unavoidable geometric imperections

tends to be relatively small at service load levels, where

these eects tend to be oset somewhat by incidental con-

tributions to the stiness rom connection rotational sti-

nesses, cladding, etc. Also, yielding eects are usually neg-ligible at service load levels. Thereore, the reduction in the

elastic stiness and the out-o-plumbness utilized in the DM

solution at strength load levels are not necessary and should

not be employed or service load analysis. However, as dis-

cussed by LeMessurier (1976 and 1977), the second-order

eects at service (or working) load levels generally should

not be neglected. Neglecting the second-order eects can

lead to a violation o the service defection limits and inad-

equate structural perormance in cases where these eects

are signicant.

As noted previously, the strength load combination (1.2D

+ 1.6Lr + 0.8W) gives the largest required tension orce in

the brace ab and the largest required compression orce in

the column bc o the example rame. Thereore, most o the

ollowing discussions are ocused on the results or this loadcombination. A conventional rst-order analysis or this load

combination gives a tension orce in the brace ab o only

13.1 kips. Equilibrium o the defected geometry requires a

brace orce o Fab = 32.6 kips based on the AISC (2005a)

ELM model, which assumes no initial geometric imperec-

tions and no reduction in the eective stiness o the struc-

ture at the strength limit state. This is due largely to the sig-

nicant drit o the structure under the gravity loads, 1.2D +

1.6Lr, in other words, 1.51P, plus the signicant sidesway

Table 1. Summary of Analysis Results

Load

CombinationAnalysis Procedure

Fab or

Fab (kips)

Fbc or

Fbc (kips)

tot/L ortot/L

Blt or

Blt

Service

D + Lr

First-order, nominal stiness, o = 0 1/413

Second-order, nominal stiness, o = 0 1/341 1.212

ServiceD + 0.5Lr + 0.7W

Wacting to right

First-order, nominal stiness, o = 0 1/398

Second-order, nominal stiness, o = 0 1/354 1.123

Strength

1.2D + 1.6Lr + 0.8W

Wacting to right

First-order, nominal stiness, o = 0 13.1 260 1/209

AISC (2005a) ELM(a) 32.6 280 1/154 1.355

AISC (2005a) Appendix 6 44.5 291 1/96 1.537

AISC (2005a) DM(b) 48.8 296 1/83 1.487

Distributed Plasticity Analysis(c) 40.8 272 1/98

Distributed Plasticity Analysis(d) 44.2 291 1/97

Strength

0.9D + 1.6WWacting to let

First-order, nominal stiness, o = 0 26.3 11.2 1/574

AISC (2005a) ELM 27.1 10.4 1/551 1.041

AISC (2005a) Appendix 6 28.1 9.4 1/253 1.055

AISC (2005a) DM 28.3 9.2 1/228 1.052

Distributed Plasticity Analysis 28.1 9.3 1/243

Strength

1.2D + 0.5Lr + 1.6W

Wacting to right

First-order, nominal stiness, o = 0 26.3 137 1/255

AISC (2005a) ELM 32.3 143 1/225 1.134

AISC (2005a) Appendix 6 35.8 147 1/142 1.187

AISC (2005a) DM 37.2 148 1/124 1.173

Distributed Plasticity Analysis 36.2 147 1/137

(a)The ELM analysis is based on the nominal (unreduced) elastic stiness and o = 0.(b)DM analysis is based on a reduced elastic stiness o 0.8 o the nominal stiness, and o = 0.002L.(c)Maximum load capacity reached due to a compression ailure o column bc at P = 272 kips at 0.936 o 1.2D + 1.6Lr + 0.8W. The AISC

(2005a) LRFD column strength is cPn = 288 kips.(d)Analysis conduced up to 1.2D + 1.6Lr + 0.8Wwith distirbuted yielding neglected such that the structure remains elastic at this load

level.


10/18



amplication oBlt = 1.355. LeMessurier (1976) notes a

similar signicant increase in the diagonal brace tension in

his discussions (but in the context o allowable stress de-

sign). The column bc axial compression is also increased

rom 260 to 280 kips due to the P eects.I one assumes a nominal o = 0.002L in the direction o

the sidesway and conducts a Distributed Plasticity Analysisto rigorously account or the eects o early yielding due to

residual stresses (see Appendix B), a diagonal brace tension

o 40.8 kips is obtained when the rame reaches its maximum

strength. In this example, the structures maximum strength

determined rom this type o analysis occurs at 0.936 o

(1.2D + 1.6Lr + 0.8W). The maximum strength is governed

by a ailure o column bc at P = 272 kips [6% smaller than

the AISC (2005a) LRFD column design strength]. I yielding

is delayed such that the rame remains ully elastic at (1.2D

+ 1.6Lr + 0.8W) (or example, i the actual Fy is suciently

larger than the specied minimum value such that column bc

remains elastic), the Distributed Plasticity Analysis model

gives Fab = 44.2 kips and Fbc = 291 kips at this load level.This is shown as a second entry in Table 1 or the Distributed

Plasticity Analysis [also see ootnote (d)]. The reader should

note that none o the Distributed Plasticity results shown in

Table 1 account or attributes such as connection slip, con-

nection elastic deormations or yielding, or oundation fex-

ibility. These attributes can increase the diagonal brace ten-

sion urther. In addition, the infuence o axial elongation in

the long-span roo system, causing a larger sidesway o the

leaning column on the right-hand side o the rame, is not

considered here. White et al. (2003a) address the handling

o this eect, including the eect o axial deormation in

member bd due to changes in temperature. The DM providesa reasonable estimate o the rame defections and internal

orces rom the above Distributed Plasticity solutions, giv-

ing = 48.8 kips (tension) and = 296 kips (compres-

sion) as illustrated in the previous section.

Interestingly, or this example the calculation using the

stability bracing equations in AISC (1999) and the AISC

(2005a) Commentary to Appendix 6 gives the closest esti-

mate o the second Distributed Plasticity results or the load

combination 1.2D + 1.6Lr + 0.8W. These calculations are

summarized as ollows:

1. The designer must recognize that the drit o the rame

under the above gravity plus wind load combination sub-stantially violates the assumption o o.total = o + 1 =0.002L in the base Appendix 6 equation or the stability

bracing orce, in other words, the horizontal orce com-

ponent in the bracing system due to second-order eects.

Based on an assumed nominal initial out-o-plumbness

oo = 0.002L and the rst-order analysis calculationssummarized in Table 1, o.total/L = (o + 1)/L = 0.002 +1/209 = 0.00678 = 1/148.

2. The Commentary to Appendix 6 o (AISC, 2005a) and the

Commentary to Chapter C o (AISC, 1999) indicate, or

other o and o values, use direct proportion to modiy thebrace strength requirements. Note that the above Com-

mentaries use the term o to represent o.total. The notationo.total is used here to clariy the two sources o relative

transverse displacement between the brace points. There-ore, the appropriately modied base equation or the

stability bracing orce is

P P Pbr u u=

=0 0040 00678

0 0020 0136.

.

.. (21)

3. In addition to the above modication, the AISC (2005a

and 1999) Commentaries speciy, I the brace stiness

provided, act, is dierent rom the requirement, then thebrace orce or brace moment can be multiplied by the

ollowing actor:

1

2

br

act

" (22)

where

bruP

L=

2 (23)

and = 0.75. By applying this actor to the above calcula-tion o the stability bracing orce, one obtains

P Pbr u=

0 01361

22 1 5 330

0 75 216 8 743

. ( . )( )

. ( ) .

kips

in.kips

in.

= = =0 0104 0 0104 1 5 330 5 16. . ( . )( ) .Pu

kips kips

(24)

It is important to note that in this example, the above

coecient o 0.0104 is substantially larger than the base

Appendix 6 coecient o 0.004.

4. Finally, the above stability bracing orce must be applied

to the bracing system along with the horizontal orce de-

termined rom a rst-order elastic analysis or the subjectapplied load combination. That is, the bracing system

must be designed or a total horizontal orce o

Pbr+ 0.8H= 5.16 kips + 2.16 kips = 7.32 kips

in addition to the vertical load o 1.5P = 1.5(165 kips) =

247.5 kips applied to the let-hand column. This is specied

Fab Fbc


11/18



only load combinations. The approach is reerred to as

the Eective Length Method (ELM) in Table 2-1 o AISC

(2005c). For the load combinations considered in Table 1,

this approach amounts to the use o a conventional second-

order analysis with the nominal elastic stiness and o =0 (since all the load combinations considered in the table

include wind load). As a result, the AISC (2005a) ChapterC provisions give required strengths or members ab and bc

o only 32.6 kips and 280 kips, respectively. The relatively

large dierence between Fab = 32.6 kips and the DM value

oFab = 48.8 kips is due to: (1) the small angle o the brace

relative to the vertical orientation; (2) the large gravity load

supported by the structure; (3) the relatively small wind

load; and (4) the correspondingly small lateral stiness o

the bracing system. I a symmetrical conguration o the

bracing were introduced, the second-order P eects in thisrame are dramatically reduced. Also, Figure 2 shows that

the second-order internal story shears approach 0.002Prwhen the bracing system has substantial stiness relative

to the ideal stiness i. In many cases, particularly i Pris relatively small, these internal shear orces are only a

small raction o the lateral load resistance o the bracing

system. However, in sensitive stability critical rames such

as LeMessuriers example in Figure 3, the application o the

AISC (2005a) DM provisions is considered prudent.

The required strengths or the other load combinations

shown in Table 1 are less sensitive to the design approach.

For example, the governing axial compression in brace ab is

determined as 28.3 kips using the DM or the load combi-

nation 0.9D + 1.6Wwith the wind acting to the let. A ba-

sic rst-order analysis or this load combination gives -26.3

kips, only 7.1% smaller. The AISC (2005a) Chapter C ap-proach gives a compressive orce o 27.1 kips, only 3.0%

smaller.

SUMMARY

This paper presents an analysis-design approach based on a

combination o the AISC (2005a) Direct Analysis Method

(DM) with a orm o LeMessuriers (1976) simplied

second-order analysis equations. The results rom several

DM analysis solutions are compared and contrasted with the

results rom other analysis solutions including second-order

service drit calculations, AISC (2005) Eective Length

Method (ELM) solutions, rened calculations based on theCommentaries to AISC (1999) Chapter C and AISC (2005a)

Appendix 6, and benchmark Distributed Plasticity Analysis

solutions. The DM is attractive in that:

It does not require any Kactor calculations,

It provides an improved representation o the internal

orces throughout the structure at the ultimate strength

limit state,

in Section C3.2 o AISC (1999) by the clause, These

story stability requirements shall be combined with the

lateral orces and drit requirements rom other sources,

such as wind or seismic loading. AISC (2005a) no longer

includes this clause in its Appendix 6, since Appendix 6 is

intended only or cases where the bracing is not subjected

to any orces determined rom a structural analysis.The resulting calculations are

F P HL

Bab D L W br

ab

( . . . ) ( . )

.

1 2 1 6 0 8 0 8

44 5

+ + = +

=

kips

(25)

and

F P P HL

Bbc D L W br( . . . ) . ( . )1 2 1 6 0 8 1 5 0 8

291

+ + = + +

=

kips

(26)

Note that the result rom Equation 24 can be obtained more

directly by applying LeMessuriers (1976) approach as ol-

lows:

P PL

br ui

act

o=+

=

1

10 75

1 5 3301

11 5 330

1

.

. ( ). (

kipskipss

in.

kips

in.

kips

)

. ( ) .

.

.

0 75 216 8 743

0 0021

209

5 16

+

=

(27)

Equation 27 is simply Equation 5 but with a stiness reduc-

tion actor o 0.75 used in the sidesway amplier and with

zero stiness reduction assumed in the calculation o1. Thecorresponding sidesway amplication and total drit ratio

are shown in the last two columns o Table 1. Obviously, the

use o a reduced stiness in the sidesway amplier and the

use o a nominal (unreduced) stiness in the calculation othe defections due to the applied loads is inconsistent. The

DM provides a consistent second-order analysis calculation

based on o = 0.002L and an elastic stiness reduction actoro 0.8.

As noted previously, AISC (2005a) Chapter C also allows

the use o a second-order elastic analysis with the nominal

elastic stiness and idealized perect geometry, as long as

notional lateral loads o 0.002Yi are included in all gravity-


12/18



It applies in a logical and consistent ashion or all types

o rames including braced rames, moment rames and

combined raming systems.

LeMessuriers (1976) second-order analysis equations are

particularly useul in that they capture the second-order e-

ects in rectangular rame structures by explicitly applying

the P shears associated with the amplied sidesway dis-placements in a rst-order analysis. LeMessuriers approach

also can be used or analysis o service defections and or

conventional strength analysis and design by the Eective

Length Method (ELM) using the nominal elastic stiness

and the idealized perect structure geometry. However, the

combination o the DM with LeMessuriers equations or

the underlying second-order analysis streamlines the analy-

sis and design process, while also ocusing the Engineers

attention on the importance o:

Ensuring adequate overall sidesway stiness, and

Accounting or second-order

P eects on all the lateralload-resisting components in the structural system at thestrength load levels.

ACKNOWLEDGMENTS

The concepts in this paper have beneted rom many discus-

sions with the members o AISC Task Committee 10 and the

ormer AISC-SSRC Ad hoc Committee on Frame Stability in

the development o the Direct Analysis Method. The authors

express their appreciation to the members o these commit-

tees or their many contributions. The opinions, ndings and

conclusions expressed in this paper are those o the authors

and do not necessarily refect the views o the individuals inthe above groups.

REFERENCES

AISC (2005a), Specifcation or Structural Steel Buildings,

ANSI/AISC 360-05, American Institute o Steel Con-

struction, Inc., Chicago, IL.

AISC (2005b), Code o Standard Practice or Steel Build-

ings and Bridges, American Institute o Steel Construc-

tion, Inc., Chicago, IL.

AISC (2005c), Steel Construction Manual, 13th Ed., Ameri-

can Institute o Steel Construction, Inc., Chicago, IL.AISC (1999),Load and Resistance Factor Design Specifca-

tion or Steel Buildings, American Institute o Steel Con-

struction, Inc., Chicago, IL.

Alemdar, B.N. (2001), Distributed Plasticity Analysis o

Steel Building Structural Systems, Doctoral dissertation,

School o Civil and Environmental Engineering, Georgia

Institute o Technology, Atlanta, GA.

ASCE (1997), Eective Length and Notional Load Ap-

proaches or Assessing Frame Stability: Implications or

American Steel Design, Task Committee on Eective

Length, Technical Committee on Load and Resistance

Factor Design, Structural Engineering Institute, American

Society o Civil Engineers, p. 442.

ASCE (2005), Minimum Design Loads or Buildings andOther Structures, ASCE/SEI 7-05, American Society o

Civil Engineers, Reston, VA.

Deierlein, G. (2003), Background and Illustrative Examples

on Proposed Direct Analysis Method or Stability Design

o Moment Frames, AISC Task Committee 10 and AISC-

SSRC Ad Hoc Committee on Frame Stability, Department

o Civil and Environmental Engineering, Stanord Uni-

versity, p. 17.

Deierlein, G. (2004), Stable Improvements: Direct Analysis

Method or Stability Design o Steel-Framed Buildings,

Structural Engineer, November, pp. 2428.

Galambos, T.V. (1998), Guide to Stability Design Criteriaor Metal Structures, 5th E, T.V. Galambos (ed.), Struc-

tural Stability Research Council, Wiley.

Galambos, T.V. and Ketter, R.L. (1959), "Columns Under

Combined Bending and Thrust,"Journal o the Engineer-

ing Mechanics Division, ASCE, Vol. 85, No. EM2, pp.

135152.

Giaotti, R. and Smith, B.S. (1989), P-Delta Analysis o

Building Structures, Journal o Structural Engineering,

ASCE, Vol. 115, No. 1, pp. 755770.

LeMessurier, W.J. (1976), A Practical Method or Second

Order Analysis. Part 1: Pin Jointed Systems,Engineering

Journal, AISC, Vol. 12, No. 4, pp. 8996.

LeMessurier, W.J. (1977), A Practical Method or Second

Order Analysis. Part 2: Rigid Frames,Engineering Jour-

nal, AISC, Vol. 14, No. 2, pp. 4967.

Maleck, A. (2001), Second-Order Inelastic and Modied

Elastic Analysis and Design Evaluation o Planar Steel

Frames, Doctoral dissertation, School o Civil and Envi-

ronmental Engineering, Georgia Institute o Technology,

Atlanta, GA, p. 579.

Martinez-Garcia, J.M. and Ziemian, R.D. (2006), Bench-

mark Studies to Compare Frame Stability Provisions,

Proceedings, SSRC Annual Technical Sessions, pp. 425442.

Martinez-Garcia, J.M. (2002), Benchmark Studies to Eval-

uate New Provisions or Frame Stability Using Second-

Order Analysis, M.S. Thesis, School o Civil Engineer-

ing, Bucknell University, Lewisburg, PA, p. 757.


13/18



Maleck, A. and White, D.W. (2003), Direct Analysis Ap-

proach or the Assessment o Frame Stability: Verication

Studies, Proceedings, SSRC Annual Technical Sessions,

pp. 423441.

Nair, R.S. (2005), Stability and Analysis Provisions o the

2005 AISC Specication or Steel Buildings, Proceed-

ings, Structures Congress 2005, SEI/ASCE, p. 3.Surovek-Maleck, A. and White, D.W. (2004), Alternative

Approaches or Elastic Analysis and Design o Steel

Frames. I: Overview,Journal o Structural Engineering,

ASCE, Vol. 130, No. 8, pp. 11861196.

Surovek-Maleck, A., White, D.W., and Ziemian, R.D. (2003),

Validation o the Direct Analysis Method, Structural

Engineering, Mechanics and Materials Report No. 35,

School o Civil and Environmental Engineering, Georgia

Institute o Technology, Atlanta, GA.

Vandepitte, D. (1982), Non-Iterative Analysis o Frames

Including the P- Eect,Journal o Constructional SteelResearch, Vol. 2, No. 2, pp. 310.

West, M., Fisher, J., and Gris, L.G. (2003), Serviceability

Design Considerations or Low-Rise Buildings, Steel De-

sign Guide Series 3, 2nd Ed., American Institute o Steel

Construction, Inc., Chicago, IL, p. 42.

Wilson, E.L. and Habibullah, A. (1987), Static and Dynam-

ic Analysis o Multi-Story Buildings, Including P-Delta

Eects,Earthquake Spectra, Vol. 3, No. 2, pp. 289298.

White, D.W. and Hajjar, J.F. (1991), "Application o Second-

Order Elastic Analysis in LRFD: Research To Practice,"

Engineering Journal, AISC, Vol. 28, No. 4, pp. 133148.

White, D.W., Surovek-Maleck, A.E., and Kim, S.-C. (2003a),

Direct Analysis and Design Using Amplied First-Order

Analysis, Part 1: Combined Braced and Gravity Framing

Systems, Structural Engineering, Mechanics and Mate-rials Report No. 42, School o Civil and Environmental

Engineering, Georgia Institute o Technology, Atlanta,

GA., p. 34.

White, D.W., Surovek-Maleck, A.E., and Chang, C.-J.

(2003b), Direct Analysis and Design Using Amplied

First-Order Analysis, Part 2: Moment Frames and Gener-

al Framing Systems, Structural Engineering, Mechanics

and Materials Report No. 43, School o Civil and Envi-

ronmental Engineering, Georgia Institute o Technology,

Atlanta, GA., p. 39.

White, D.W., Surovek, A.E., Alemdar, B.N., Chang, C.-J.,

Kim, Y.D., and Kuchenbecker, G.H. (2006), StabilityAnalysis and Design o Steel Building Frames Using the

2005 AISC Specication,International Journal o Steel

Structures, Vol. 6, No. 2, pp. 7191.

White, D.W., Surovek, A.E., and Chang, C-J. (2007), Di-

rect Analysis and Design Using Amplied First-Order

Analysis, Part 2: Moment Frames and General Framing

Systems,Engineering Journal, AISC, Vol. 44, No. 4, pp.

323340.


14/18



APPENDIX A

DERIVATION OF AMPLIFIED FIRST-ORDER

ELASTIC ANALYSIS EQUATIONS

This appendix derives a specic orm o the method or

determining second-order orces in braced-rame systems

originally presented by LeMessurier (1976). The heavilyidealized model shown in Figure 4 represents the essen-

tial attributes o a story in a rectangular rame composed

o gravity raming combined with a braced-rame system.

Gravity raming is dened by AISC (2005a) as a portion

o the raming system not included in the lateral load re-

sisting system. This type o raming typically has simple

connections between the beams and columns and assumed

to provide zero lateral load resistance. The braced-rame

system is designed to transer lateral loads and some portion

o the vertical loads to the base o the structure, as well as to

provide lateral stability or the ull structure. The structures

sidesway stiness is assumed to be provided solely by thebraced-rame system, which AISC (2005a) allows to be ide-

alized as a vertically-cantilevered simply-connected truss.

The vertical load carrying members in the physical struc-

ture are represented by a single column in the Figure 4 mod-

el. The bracing system is represented by a spring at the top o

this column. This spring controls the relative displacement

between the top and bottom o the story. An axial load oPris applied to the model, where Pris the total requiredverti-cal load supported by the story. The model has a nominal

initial out-o-plumbness oo, taken equal to a base valueo 0.002L in the DM but taken equal to zero in conventional

analysis and design by the Eective Length Method (ELM).

This nominal out-o-plumbness represents the eects o un-avoidable sidesway imperections. A horizontal load, H,is also applied to the model, where Hrepresents the storyshear due to the applied loads on the structure. The rst-or-

der shear orce in the bracing system, (F1), is equal to H.Figure 4 may be considered as a representation o a single-

story braced-rame structure, or as an idealized ree-body

diagram o one level in a multi-story system. In the latter

case, a portion oHand Pr is transerred rom the storyabove the level under consideration.

Based on a rst-order elastic analysis o the above model,

the load Hproduces a story drit o = H/ , where isthe reduced lateral stiness o the bracing system as speci-

ed or the DM. In general, the vertical loads also produce a

drit o the story whenever the loadings and/or the rame ge-ometry are not symmetric. This rst-order lateral displace-

ment is denoted by 1P. The net lateral orce in the bracing

system is zero due to this displacement. The total rst-order

inter-story drit is denoted by 1 1 1= +H P . Summationo moments about point A gives

where the displacement 2 is the additional drit due to

second-order (P) eects. Since

the terms HL and 1HL cancel on the let and right-handsides o Equation A1, and thereore this equation becomes

I the columns are initially plumb (in other words, o = 0)and the horizontal load His such that is also equal tozero, then one nds that either 2 must be zero, or the system

is in equilibrium under an arbitrary sidesway displacement

2 at a vertical load

where Pcr is the sidesway buckling load o the combinedsystem. One can observe that this load is proportional to the

bracing system stiness, . The minimum bracing stinessrequired to prevent sidesway buckling under the load Pr(with o and Hequal to zero or a symmetrical structure

1H

1

The symbol is selected or the lateral stiness o the structural system in this paper and in White et al. (2007), consistent withthe use o this term or the lateral stiness o a bracing system in AISC (1999 & 2005a). This is dierent rom the denition o

in LeMessurier (1976). Also, in this paper, an over bar is shown on all quantities that are infuenced by the stiness reduc-tion employed within the DM. The equations presented are equally valid or a service load analysis or a conventional strength

load analysis with zero stiness reduction, in other words, with a stiness reduction actor o 1.0.

In the context o the bar-spring model o Figure 4, the displacement 1P is equivalent to a lateral movement o the horizontal

springs support. See the example o Figure 3 or an illustration o the source o this displacement.

HL P Lr o H

+ + +( ) = +( )1 2 1 2

H H= 1

P Lr o + +( ) =1 2 2

Pr = =P Lcr

(A1)

(A2)

(A3)

(A4)


15/18



subjected to symmetrical vertical load) is dened as the

ideal stiness (Galambos, 1998)

whereas the sidesway stinessprovidedby the bracing sys-

tem is denoted by the symbol . Based on Equation A4, theprovided stiness can be written in terms o the sidesway

buckling load as

By solving or the displacement 2 o the imperect laterallyloaded system rom Equation A3, one obtains the ollowing

ater some algebraic manipulation,

This displacement induces a second-order lateral orce in the

bracing system o

where

is the sidesway displacement amplication actor. The last

two orms shown in Equation A9 are the same as the expres-

sions provided or the sidesway amplication actor, B2, in

AISC (2005a) or braced-rame systems. The symbol Blt is

used in this paper, since the subscripts 0, 1 and 2 are reserved

to denote the initial, rst-order and second-order orce and/

or displacement quantities in this work. The notation lt

stands or lateral translation.

Summation o all the contributions to the total drit in Fig-ure 4 gives

or in other words, the total drit o the story tot = (o + 1+ 2) is equal to the total rst order displacement (o + 1)

multiplied by the sidesway displacement amplication ac-tor, Blt. The total horizontal shear orce developed in the

Fig. 4. Idealized model o a story in a general rectangular rame composed o gravity raming combined with a braced-rame system.

irP

L=

=PL

cr

2

1 1 1

1 1

=+( )

=+( )

=+( )P

L P P

P

r o

r

o

cr

r

o

i

FP

Li

r o

i

2 2

1

1

1

1

= =+( )

=

ii

o

r

lt

o

r

LP

BL

P

+( )

=+( )

1

1

(A5)

(A6)

(A7)

(A8)

BP L

H

P

P

lti

i

i r

H

r

cr

=

=

=

=

1

1

1

1

1

1

1

1

/

/

(A9)

tot o

i

ltB

= + +( )

= +( ) ++( )

= +( )

1 2

0 1

0 1

0 1

1(A10)


16/18



bracing system is in turn

where

or

From Equation A13, it is apparent that the total horizontal

orce in the bracing system is equal to the sidesway amplier

Blt times the sum o the ollowing internal shear orces: (1)

the rst order orce due to the applied lateral loads, H; (2)the P shear orce due to the initial out-o-plumbness, o;and (3) the P shear orce due to the lateral defection causedby the vertical loads on the story. In the traditional AISC

(2005a) no translation-lateral translation (NT-LT) analysis

approach, the last o the above horizontal orces is captured

by articially restraining the structure against sidesway in an

NT analysis. The reverse o the articial reactions is then ap-plied to the structure along with any other horizontal orces

in an LT analysis. An estimate o the corresponding total

second-order internal orces is then obtained by multiply-

ing the results rom the LT analysis by the corresponding

storyBlt amplier. White et al. (2003a) show the derivation

o this procedure in the context o the above undamental

equations. However, one can see rom Equation A11 that the

total internal shear orce in the lateral load resisting system

is also simply equal to the primary (or rst-order) applied

load eect Hplus the eect oPracting through the totalamplied story drit

Stated most directly, the total story internal shear orce is

simply equal to Hplus the P shear orce HP given byEquation A12. Thereore, Equation A11 points to a more

straightorward procedure than the traditional B1-B2 or NT-

LT analysis approach. This equation shows that the second-

order shear orces may be obtained by a rst-order analysis

in which equal and opposite P shearsHP (Equation A12)are applied at the top and bottom o each story o the struc-

ture. The Engineer never needs to consider the subdivision

o the analyses into articial NT and LT parts. Additional

considerations associated with P amplication o internal

moments in moment-rame systems are addressed by Whiteet al. (2007).

APPENDIX B

DISTRIBUTED PLASTICITY ANALYSIS RESULTS

FOR LEMESSURIERS (1976) EXAMPLE FRAME

It is inormative to compare the elastic analysis and design

solutions presented in the paper or LeMessuriers example

rame to the results rom a Distributed Plasticity Analysis.

Distributed Plasticity Analysis is a useul metric or evalu-

ation o all o the analysis and design methods, since it ac-

counts rigorously or the eects o nominal geometric im-perections and member internal residual stresses. The load

combination (1.2D + 1.6Lr + 0.8W) with the wind applied

to the right is considered here (reer to Figure 3), since this

combination gives the governing axial orce requirement

in the most critically loaded member, column bc. As noted

previously, this load combination also produces the largest

tension in brace ab. For the Distributed Plasticity Analysis, a

nominal out-o-plumbness o 0.002L to the right is assumed

throughout the rame and a nominal out-o-straightness o

0.001L is assumed in column bc. Also, the Lehigh residual

stress pattern (Galambos and Ketter, 1959), which has a

maximum residual compression o 0.3Fy at the fange tips

and a linear variation over the hal-fange width to a constantsel-equilibrating residual tension in the web, is taken as

the nominal residual stress distribution or the wide-fange

columns. These are established parameters or calculation

o benchmark design strengths in LRFD using a Distributed

Plasticity Analysis (ASCE, 1997; Martinez-Garcia, 2002;

Deierlein, 2003; Surovek-Maleck et al., 2003; Surovek-

Maleck and White, 2004; White et al., 2006). Columns bc

and de are assumed to have their webs oriented in the di-

rection normal to the plane o the rame. The tension at the

maximum load level in brace ab is signicantly less than its

yield load; thereore, the residual stresses in the brace are not

a consideration in the Distributed Plasticity Analysis or theabove load combination. A resistance actor o = 0.90 is ap-plied to both the yield strength, Fy, and the elastic modulus,

E, including the occurrence o Fy in the above description

o the nominal residual stresses. The steel is assumed to be

elastic-plastic, with a small inelastic modulus o 0.0009Eor

numerical purposes. The gravity and the lateral loads are ap-

plied proportionally to the rame in the Distributed Plasticity

solution. Two mixed elements (Alemdar, 2001), which are

capable o accurately capturing the inelastic P moments in

F F H BL

P

H H

lto

r

P

1 21+ = +

+

= +

H P BL

P r lt o

=+

1

F F H B

H BH

B H

lt o P H i

lt o P i

lt

1 2 1 1

1

+ = + +

+( )

= + + +

= + P Lro P+

1

(A11)

(A12)

(A13)

tot

lt

o o

LB

L L=

+( )=

+ +( )1 1 2


17/18



column bc as this member approaches its maximum strength,

are employed to model all the members. All the members are

modeled as ideally pin-connected. The roo system is mod-

eled by a strut between the two columns, and the gravity

loads rom the roo system are applied as concentrated verti-

cal loads at the top o each o the columns. The Engineer

should note that it is essential to include column de in anysecond-order analysis model. Otherwise, the second-order

eects caused by the leaning o this column on the lateral

load resisting system are missed.

The Distributed Plasticity solution predicts a maximum

load capacity o the rame at 0.936 o the maximum roo live

load combination. The predominant ailure mode is the fex-

ural buckling o column bc at an axial compression o 272

kips. This load is 0.944 o the AISC (1999) column strength

ocPn = 288 kips based on c = 0.90. There is some yieldingin the middle o the column unbraced length at the predicted

maximum load level, but the amount o yielding is relatively

minor and the rame deormations are still predominantly

elastic. Approximately 10% o the column area is yieldedresulting in a reduction in the eective elastic weak-axis mo-

ment o inertia o 29% at the mid-length o the column. Col-

umn bc is still ully elastic over a length o 4.5 t at each o

its ends. The above solution or the column strength is within

the expected scatter band or the actual-to-predicted column

strengths based on the single AISC (1999) column curve or-

mula. The total drit o the rame at the maximum load level

is tot/L = 0.0102, including the initial out-o-plumbness oo/L = 0.002. It should be noted that this drit is only slightlylarger than tot/L = 0.00958 obtained by a second-order elas-tic analysis o the structure at 0.936 o (1.2D + 1.6Lr+ 0.8W)

using 0.9 o the nominal elastic stiness. The tension orcein brace ab is 40.8 kips at the maximum strength limit in the

Distributed Plasticity Analysis, versus 38.8 kips in the above

corresponding second-order elastic analysis.

I the maximum load capacity o column bc is assumed to

be greater than or equal to that required to reach the design

load level o (1.2D + 1.6Lr+ 0.8W), and i the eects o mi-

nor yielding at this load level are assumed to be negligible,

the above inelastic analysis gives the second-order elastic

solution (based on 0.9 o the structure nominal elastic sti-

ness) oFbc = 291 kips, Fab = 44.2 kips and tot/L = 0.0103.The Engineer should note that this solution is only slightly

less conservative than the recommended DM values oFbc

= 296 kips rom Equation 20, Fab = 48.8 kips rom Equation19, and tot /L = 0.0120 rom Equation 17. The DM valuesaccount approximately or the potential additional sidesway

defections and P eects associated with yielding at themaximum strength limit. The reader is reerred to Martinez-

Garcia (2002) or other DM and Distributed Plasticity Anal-

ysis examples involving truss raming and using the estab-

lished parameters employed in the above study.

The results rom the Distributed Plasticity Analysis solu-

tions or the load combinations 0.9D + 1.6W and 1.2D +

0.5Lr+ 1.6Ware summarized in Table 1. The member orces

or these load combinations are suciently small such that

no yielding occurs at the strength load levels (including the

consideration o residual stress eects). Thereore, these so-

lutions are the same as obtained using a second-order elasticanalysis with a nominal stiness reduction actor o 0.9 and

o = 0.002L.

APPENDIX C

NOMENCLATURE

B1 = Nonsway moment amplication actor in

AISC (1999)

Blt,Blt = Sidesway displacement amplication actor

given by Equation 2 or Equation A9

E = Modulus o elasticity

F1 = First-order story shear orce in the bracing

system, equal to H

F2 = Second-order shear orce in the bracing

system; second-order contribution to the orce

in a component o the lateral load resisting

system

Fy = Yield stress

HP,HP = Story shear due to P eects

L = Story height

Ni,Ni = Notional load at ith level in the structure

P = Column axial load

Pbr = Stability bracing shear orce required by AISC

(1999) & (2005a)

Pn = Nominal axial load resistance

Pr = Required axial load resistance

Yi = Total actored gravity load acting on the ith

level

, = Total story sidesway stiness o the lateralload resisting system

i = Story sidesway destabilizing eect, or idealstory stiness, given by Equation 3

= Resistance actor

o = Initial story out-o-plumbness

1, 1 = First-order interstory sidesway displacementdue to applied loads = 1H+ 1P or 1H+ 1P


18/18



2, 2 = Additional interstory sidesway displacementdue to second-order (P) eects

1H, 1H = First-order interstory sidesway displacementdue to H

1P, 1P = First-order interstory sidesway displacement

due to vertical loadstot, tot = Total interstory sidesway displacement

= Summation

H = Story shear due to the applied loads on thestructure

P = Total story vertical load

Pr = Total required story vertical load

Pcr = Story sidesway buckling load, given by Equa-tion A4

(over bar) = Indicates quantities that are infuenced by the

stiness reduction employed in the DirectAnalysis Method

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Direct Analysis and Design Using Amplified First-Order Analysis

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