direct analysis and design using amplified first-order analysis
TRANSCRIPT
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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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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)
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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)
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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)
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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.
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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
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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-
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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.
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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)
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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)
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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
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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
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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