A Seismic Design Lateral Force
Distribution Based on Inelastic State
of Structures
Shih-Ho Chao,a) M.EERI, Subhash C. Goel,b) M.EERI, and Soon-Sik Leec)
It is well recognized that structures designed by current codes undergo
large inelastic deformations during major earthquakes. However, lateral force
distributions given in the seismic design codes are typically based on results of
elastic-response studies. In this paper, lateral force distributions used in the
current seismic codes are reviewed and the results obtained from nonlinear
dynamic analyses of a number of example structures are presented and
discussed. It is concluded that code lateral force distributions do not represent
the maximum force distributions that may be induced during nonlinear
response, which may lead to inaccurate predictions of deformation and force
demands, causing structures to behave in a rather unpredictable and
undesirable manner. A new lateral force distribution based on study of inelastic
behavior is developed by using relative distribution of maximum story shears
of the example structures subjected to a wide variety of earthquake ground
motions. The results show that the suggested lateral force distribution,
especially for the types of framed structures investigated in this study, is more
rational and gives a much better prediction of inelastic seismic demands at
global as well as at element levels. 关DOI: 10.1193/1.2753549兴
INTRODUCTION
The current lateral seismic-force distributions in building codes are generally based
on first-mode dynamic solution of lumped multiple-degree-of-freedom (MDOF) elastic
systems (ATC 1978, Clough and Penzien 1993, Chopra 2000, BSSC 2003), which can
be expressed as:
fi1 = V1
冉
wi␾i1
n
兺j=1wj␾j1
冊
共1兲
where fi1 is the lateral force at level i; V1 is the first-mode base shear; wj is lumped seismic weight at jth level; and ␾j is amplitude of the first mode at jth level. The code expression assumes that the lateral force distribution can be expressed using the first-mode
a)
Postdoctoral Research Fellow, Dept. of Civil and Environmental Engineering, University of Michigan, Ann
Arbor, MI; E-mail: shchao@umich.edu
b)
Professor, Dept. of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI; E-mail:
subhash@engin.umich.edu
c)
Senior Bridge Engineer, URS Corporation, Roseville, CA; E-mail: Soon_Sik_Lee@URSCorp.com
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Earthquake Spectra, Volume 23, No. 3, pages 547–569, August 2007; © 2007, Earthquake Engineering Research Institute
548
S. H. CHAO, S. C. GOEL, AND S. S. LEE
deflected shape of an elastic lumped-mass system subjected to dynamic loading, which
can be written as:
␾i1 = hki /L
共2兲
where L is the total height of the structure; hki is the height of level i above the base with
exponent k being related to the building’s fundamental period 共T兲. Observations of the
elastic response of buildings suggest that the first-mode shape is close to a straight line
共k = 1兲 when the fundamental period is 0.5 s or less; and is close to a parabola 共k = 2兲
when the fundamental period is 2.5 s or longer. This leads to the following code lateral
force distribution (BSSC 2003, ICC 2006):
Fi = CviV
共3兲
where
Cvi =
wihki
n
兺j=1
wjhkj
共4兲
and Fi is the lateral force applied at level i; Cvi is the vertical distribution factor; V is the
total design base shear, which replaces V1 in Equation 1 since V1 is the dominant part of
the total force V; wi and wj are the total effective seismic weights at levels i and j, respectively; hi and hj are the heights of levels i and j from the ground, respectively; and
n is the number of stories. For structures with natural period between 0.5 and 2.5 s, k is
determined by linear interpolation between 1 and 2.
Equations 3 and 4 were originally adopted by ATC 3-06 Provisions (ATC 1978),
which served as the basis of the NEHRP Provisions (BSSC 2003). These expressions are
also used by the current International Building Code (ICC 2006). Similarly, the UBC 97
(ICBO 1997) used distribution of design lateral forces based on a linear mode shape,
together with an additional concentrated lateral force at the top level to account for
higher mode effects (SEAOC 1999):
Fi = 共V − Ft兲
w ih i
n
兺j=1wjhj
共5兲
The force at the top level computed from Equation 5 is increased by an additional
force:
Ft = 0.07TV
Ft = 0
if T ⬎ 0.7 s
if T ⱕ 0.7 s
共6a兲
共6b兲
where Ft is the additional concentrated force applied at the top level of the structure, in
order to account for higher mode effects. It is noted that the IBC/NEHRP and UBC distributions are essentially the same when the fundamental period of the structure is less
than 0.5 s.
Since most building structures (especially those with large R values) designed ac-
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549
cording to current code procedures are expected to undergo large deformations in the
inelastic range when subjected to major earthquakes, lateral force distributions can be
quite different from those given by the code formulas such as Equation 4. In order to
achieve the main goal of performance-based seismic design, i.e., a desirable and predictable structural response, it is important to consider inelastic behavior of structures directly in the design process. The commonly used elastic analysis and design procedures
in current practice, together with elastic-design lateral force distributions, may not be
well suited to fulfill this goal in a realistic manner.
One of the essential elements of performance-based seismic design of structures
should be to use more realistic design lateral force distribution, which represents peak
lateral force distribution in a structure in the inelastic state and includes the higher mode
effects. This paper presents a new lateral force distribution based on the study of inelastic responses of various types of structural systems, using extensive nonlinear dynamic
analysis results.
NEW LATERAL FORCE DISTRIBUTION
The format for this new design lateral force distribution based on inelastic state of a
structure was originally proposed by Lee and Goel (2001) by using shear distribution
factor derived from the relative distribution of maximum story shears of a large number
of steel moment frames subjected to selected earthquake records. The suggested expression has a format similar to that of the IBC/NEHRP expression:
⬘V
Fi = Cvi
共7兲
where
冉
wh
⬘ = 共␤i − ␤i+1兲 n n n
Cvi
兺j=1wjhj
冉
冊
␣T−0.2
n
兺j=i
w jh j
Vi
␤i =
=
Vn
w nh n
when i = n, ␤n+1 = 0
冊
␣T−0.2
共8a兲
共8b兲
where ␤i is the shear distribution factor at level i; Vi and Vn, respectively, are the story
shear forces at level i and at the top (nth) level; wj is the seismic weight at level j; hj is
the height of level j from the base; wn is the weight at the top level; hn is the height of
roof level from the base; T is the fundamental period; Fi is the lateral force at level i; and
V is the total design base shear. The value of parameter ␣ was originally proposed as 0.5
by Lee and Goel (2001), which was later revised to 0.75 based on more extensive nonlinear dynamic analyses on eccentrically braced frames (EBFs) and special truss moment frames (STMFs) by Chao and Goel (2005 and 2006a) and is briefly presented in
this paper. It is interesting to note that, by using a similar approach (story shear distribution), Park and Medina (2006) proposed a design lateral force distribution for moment
frame structures exposed to pulse-type near-fault ground motions.
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 1. (a) Floor plan of nine-story building and (b) nine-story moment frame.
It should be mentioned that the lateral force distribution as presented in this study is
based on story shear distribution Vi / Vn, because it facilitates calculation of lateral forces
starting from the top (level n).
JUSTIFICATION OF THE NEW LATERAL FORCE DISTRIBUTION
Four types of steel frames were evaluated in this study: moment frames (MFs),
EBFs, STMFs, and concentrically braced frames (CBFs). The floor plan and elevation
views are shown in Figures 1–5. The nine-story MF is the same that was studied by
Gupta and Krawinkler (1999), and was designed based on UBC lateral force distribu-
Figure 2. (a) Floor plan of three-story building and (b) three-story eccentrically braced frame.
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
551
Figure 3. (a) Floor plan of 10-story building and (b) 10-story eccentrically braced frame.
tion. The three- and 10-story EBFs were designed according to both the IBC and the
suggested lateral force distributions (Chao and Goel 2005). The nine-story STMF as
shown in Figure 4 was designed by using the suggested expression (Chao and Goel
2006a). The six-story CBF (Sabelli 2000), which was originally designed using NEHRP
lateral force distribution and 1997 AISC Seismic Provisions (AISC 1997), was rede-
Figure 4. (a) Floor plan of nine-story building and (b) nine-story special truss moment frame.
552
S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 5. (a) Floor plan of six-story building and (b) six-story concentrically braced frame.
signed for this study according to the 2005 AISC Seismic Provisions (AISC 2005). The
member sizes and other details for the study frames can be found in the related references.
Nonlinear dynamic analyses were carried out to evaluate the validity of the suggested design lateral force distribution. A total of 21 SAC Los Angeles–region ground
motions records (Somerville et al. 1997; fifteen at 10% and six at 2% probability of exceedance in 50 years) were used. The characteristics of the selected SAC ground motions are given in Table 1. Analyses for MFs and CBFs were carried out by using the
SNAP-2DX program (Rai et al. 1996), while analyses for EBFs and STMFs were conducted by using Perform-3D (RAM 2003) and Perform-2D (RAM 2003), respectively.
Details regarding modeling work can be found elsewhere (Lee and Goel 2001, Chao and
Goel 2005, 2006a, 2006b). Table 2 summarizes the fundamental periods of all the study
frames. The merits of the suggested expression over the code distributions are discussed
below.
RELATIVE STORY SHEAR DISTRIBUTIONS
The suggested new lateral force distribution was justified by investigating the relative story shear distributions (defined as the ratio of maximum story shear force at level
i to that at top level n; i.e.,Vi / Vn) as obtained from the nonlinear dynamic analyses. Note
that story shear distribution and lateral force distribution have a direct relationship. The
story shear, Vx, in any story is the sum of the lateral forces above that story:
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
553
Table 1. Characteristics of ground motions used in this study
SAC
Name
Record
Earthquake
Magnitude, Mw
Distance
(km)
PGA
共cm/ sec2兲
LA01
LA02
LA04
LA06
LA07
LA09
LA11
LA12
LA13
LA14
LA16
LA17
LA18
LA19
LA20
LA21
LA23
LA24
LA26
LA27
LA30
Imperial Valley, 1940, El Centro
Imperial Valley, 1940, El Centro
Imperial Valley, 1979, Array #05
Imperial Valley, 1979, Array #06
Landers, 1992, Barstow
Landers, 1992, Yermo
Loma Prieta, 1989, Gilroy
Loma Prieta, 1989, Gilroy
Northridge, 1994, Newhall
Northridge, 1994, Newhall
Northridge, 1994, Rinaldi RS
Northridge, 1994, Sylmar
Northridge, 1994, Sylmar
North Palm Springs, 1986
North Palm Springs, 1986
1995 Kobe
1989 Loma Prieta
1989 Loma Prieta
1994 Northridge
1994 Northridge
1974 Tabas
6.9
6.9
6.5
6.5
7.3
7.3
7
7
6.7
6.7
6.7
6.7
6.7
6
6
6.9
7
7
6.7
6.7
7.4
10
10
4.1
1.2
36
25
12
12
6.7
6.7
7.5
6.4
6.4
6.7
6.7
3.4
3.5
3.5
7.5
6.4
1.2
452.03
662.88
478.65
230.08
412.98
509.70
652.49
950.93
664.93
644.49
568.58
558.43
801.44
999.43
967.61
1258.00
409.95
463.76
925.29
908.70
972.58
Probability
of Exceedence
10%/50yrs
2%/50yrs
n
Vx = 兺 Fi
共9兲
i=x
As can be seen in Figure 6 for the nine-story MF, IBC/NEHRP lateral force distribution
significantly deviates from the nonlinear dynamic analysis results. This leads to relatively smaller design story shear forces at upper levels while nonlinear dynamic analyses
show that the upper floors are subjected to larger forces than those given by the code
formulas. The relatively smaller lateral design forces at upper floors generally result in
Table 2. Fundamental period of the study frames
Frame
Nine-story MF
Three-story EBF
Ten-story EBF
Nine-story STMF
Six-story CBF
Fundamental Period, T (s)
1.29
0.42
1.42
1.93
0.55
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 6. Relative story shear distributions from nonlinear dynamic analyses, code expressions,
and suggested expression for nine-story moment frame designed based on NEHRP expression.
smaller member sizes and therefore larger story drifts and member deformations at those
levels. This trend was noticed in all types of study frames.
On the other hand, the suggested lateral force distribution (thus, the relative story
shear distribution) is closer to the results obtained from nonlinear dynamic analyses. It is
noted, as shown in Figure 6, that relative story shear distribution using ␣ = 0.5 generally
represents a lower bound of the nonlinear dynamic analysis results. This would normally
lead to larger design forces at upper floors, which may result in concentration of inelastic deformation at the lower levels. Further analyses by Chao and Goel (2005 and 2006a)
show that relative story shear distribution using ␣ = 0.75 represents an upper bound of
the nonlinear dynamic analysis results (Figure 6) and generally leads to more uniform
deformations of elements as well as stories over the height of the structure, which will
be discussed later.
It is also noticed that, by applying an additional lateral force at the top level, the
UBC equation gives better prediction than the IBC/NEHRP equation. The reason that
the ATC 3-06 (thus IBC and NEHRP) does not include the additional top force can be
attributed to the concern that an underestimation of story shears in the lower stories
might be more serious than those in the top stories (Newmark and Rosenblueth 1971).
However, based on nonlinear dynamic analysis results, it is obvious that the IBC/
NEHRP equation can result in significant underdesign of the top stories with consequent
adverse results.
Figures 7–10 show the nonlinear dynamic analysis results for three- and 10-story
EBFs designed by using both the IBC and the suggested expressions. It is seen that, generally, the difference between the analysis results and the code predictions becomes
larger when the building height (or fundamental period) increases (see Table 2). It is also
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
555
Figure 7. Relative story shear distributions from nonlinear dynamic analyses, code expressions,
and suggested expression for three-story eccentrically braced frame designed based on IBC
expression.
noticed from Figures 9 and 10 that the relative story shear distributions obtained from
nonlinear dynamic analyses have the same trend (i.e., are closer to the suggested distribution) regardless of whether the frame was designed according to the code or the suggested formula. These observations are further confirmed through nonlinear dynamic
Figure 8. Relative story shear distributions from nonlinear dynamic analyses, code expressions,
and suggested expression for three-story eccentrically braced frame designed based on suggested expression.
556
S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 9. Relative story shear distributions from nonlinear dynamic analyses, code expressions,
and suggested expression for 10-story eccentrically braced frame designed based on IBC
expression.
analysis results of the nine-story STMF and six-story CBF as shown in Figures 11 and
12, respectively. It should be noted that, because the six-story CBF has a relatively short
fundamental period (see Table 2), some analysis results are closer to the code predictions
as also observed in the three-story EBFs with even shorter period.
Figure 10. Relative story shear distributions from nonlinear dynamic analyses, code expressions, and suggested expression for 10-story eccentrically braced frame designed based on suggested expression.
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
557
Figure 11. Relative story shear distributions from nonlinear dynamic analyses, code expressions, and suggested expression for nine-story special truss moment frame designed based on
suggested expression.
The suggested design lateral force distribution was further verified by using structural wall systems from the analytical results carried out by other researchers (Iqbal and
Derecho 1980). By using the DRAIN-2D program, Iqbal and Derecho conducted nonlinear dynamic analyses of a series of isolated reinforced-concrete structural walls hav-
Figure 12. Relative story shear distributions from nonlinear dynamic analyses, code expressions, and suggested expression for six-story concentrically braced frame designed based on
NEHRP (IBC) expression.
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Table 3. Structural parameters and ground motions used for analyses of the 10-, 20-, 30-, and
40-story isolated structural walls (Iqbal and Derecho 1980)
Wall
First-Story
Height (ft.)
Other Story
Height (ft.)
Fundamental
Period, T (s)
Stiffness and
Mass
Earthquake Ground
Motions
10-Story
12
8.75
0.5
1971 Pacoima Dam, S16E
20-Story
12
8.75
1.4
30-Story
12
8.75
2.0
40-Story
12
8.75
3.0
Uniform over
wall height
Uniform over
wall height
Uniform over
wall height
Uniform over
wall height
1971 Holiday Orion, E-W
1952 Taft, S69E
1940 El Centro, E-W
S1 (Artificial Acc.)
1940 El Centro, N-S
ing 10, 20, 30, and 40 stories. These individual cantilever walls represent interior elements of a building and provide the entire lateral load resistance for their tributary floor
area. The corresponding parameters and the six ground motions used are given in Table
3. Based on the analysis results, they concluded that the UBC and ATC 3-06 formulas
tend to underestimate the shears at the top levels for all the study structural walls. A
modified UBC force distribution was proposed based on their observations (Iqbal and
Derecho 1980):
(a) Normalized shear forces in the top 25% of the wall height should be increased
by a factor of:
T
共10兲
where 1.0 ⱕ ␤1 ⱕ 1.5
3
Use of ␤1 should not result in a normalized shear force greater than that allocated to any portion in the lower 75% of the wall height using UBC
distribution.
␤1 = 2 −
(b)
Relative story shear distributions for the 10- and 20-story structural walls based on
their proposed formula, along with the code distributions and expression suggested in
this study, are plotted and shown in Figure 13. As can be seen, the distribution proposed
by Iqbal and Derecho for reinforced-concrete structural walls generally agrees with the
findings in this study for frame systems, in that the current code formulas can significantly underestimate the story shears at the upper levels when a structural wall deforms
into the inelastic range. Note that the straight lines in Iqbal and Derecho’s distribution at
upper levels in Figure 13 are due to the constraint (b) in their proposed method. Based
on Figure 13, it appears that the suggested formula in this study can also be applied to
structural wall systems. In any case, further research, such as using more ground motions as well as coupled wall systems, should be undertaken to determine if a slightly
smaller value of ␣ would be more appropriate for structural wall systems.
In order to examine the effects of inelastic and elastic behavior on the lateral force
distributions, elastic dynamic analyses were carried out for the nine-story STMF. As in-
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
559
Figure 13. Comparison between relative story shear distributions for 10- and 20-story structural walls.
dicated in Figure 14, the elastic relative story shear distributions have a tendency to shift
toward the code distribution a bit more as opposed to the inelastic analysis results as
shown in Figure 11. However, the code expression still does not predict the elastic responses well enough, even though it was developed based on elastic behavior. Figure 15
shows the relative story shear distributions for a 10-story EBF, as obtained from elastic
and inelastic dynamic response analyses. As can be seen, the elastic relative story shear
distributions are not unique and can vary quite a bit with the ground motion used. On the
other hand, the inelastic relative story shear distributions were closer to the values given
by the suggested expression (Equation 8; ␣ = 0.75). This is quite understandable because
Figure 14. Relative story shear distributions from elastic dynamic analyses, code expressions,
and suggested expression for nine-story special truss moment frame.
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 15. Selected results of relative story shear distributions from elastic and inelastic dynamic analyses for 10-story eccentrically braced frame.
the story shears in the inelastic state are limited by the strength of the system. It is interesting to note that a similar trend was also observed by Goel (1967) in his early study
of inelastic seismic behavior of multistory steel moment frames, as shown in Figure 16.
MAXIMUM INTERSTORY DRIFT DISTRIBUTIONS
Since the suggested lateral force distribution is based on inelastic response, the
structures designed by using such distribution tend to be better proportioned. In other
words, the possibility of overdesign or underdesign in certain regions is greatly reduced.
Figure 17 shows that an EBF designed by using the suggested lateral force distribution
Figure 16. Elastic and inelastic story shear distributions of a 10-story moment frame (Goel
1967).
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
561
Figure 17. Comparison of maximum interstory drift distribution between EBFs designed by
using the suggested and code design lateral force distributions (all ground motions have a probability of exceedence of 10% in 50 years): (a) suggested, (b) IBC 2006.
exhibited more uniform distribution of maximum interstory drift than the EBF designed
by IBC lateral force distribution. A similar trend was also observed in the nine-story
STMF designed according to the suggested distribution, as shown in Figure 18. A uniform interstory drift distribution also implies that intended “fuse” elements (such as
beams in moment frames, shear links in EBFs, and chord members in STMFs) at all
Figure 18. Maximum interstory drift distribution of nine-story STMF designed by the suggested lateral force distribution (all ground motions have a probability of exceedence of 10% in
50 years).
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 19. Free body diagram of an exterior “column tree.”
levels can be mobilized to dissipate earthquake energy, thus minimizing the possibility
of concentration of excessive inelastic deformation and damage in some stories or localized regions.
COLUMN DESIGN MOMENTS
It has been pointed out by several investigators in the past that when a structure is
subjected to seismic loading, especially in the inelastic state, large moments can occur in
the columns, which can be quite different from those calculated by elastic analysis (eg,
Paulay and Priestley 1992). Conventional design approaches usually do not accurately
estimate the maximum column moments and their location (Bondy 1996, Medina and
Krawinkler 2005). In fact, the column moments are quite often underestimated because
the columns are subjected to moments not only from those delivered from the beams or
other members framing into the columns (conventional capacity design approach), but
also from their own deformation (Bondy 1996).
In view of the above-mentioned shortcoming in conventional design approach, an alternative design method was proposed by Leelataviwat et al. (1999) in considering the
equilibrium of an entire “column tree” in the extreme limit state. Figure 19 shows the
free-body diagram of an exterior “column tree” of a moment frame for calculating the
design column moments in each story. In order to ensure the formation of strong-column
weak-beam mechanism, columns should be designed for maximum expected forces by
considering a reasonable extent of strain-hardening in the beam plastic hinges and material overstrength. The moment at a strain-hardened beam plastic hinge can be obtained
by multiplying its nominal plastic moment 共Mpb兲 by an overstrength factor 共␰兲, which
accounts for the effect of strain hardening and material overstrength. The column moments and shear forces can be calculated by applying the expected beam end moments
and equivalent lateral (inertia) forces applied at each level 共Fiu兲 necessary to keep the
“column tree” in equilibrium. Through this approach the moments due to column elastic
deformation are also accounted for. It is worth mentioning that conventional capacity
design approaches only consider individual joints, in which the column moment capacity
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
563
Figure 20. Design column moments and maximum column moment envelopes from nonlinear
dynamic analyses for a nine-story moment frame (Lee and Goel 2001).
must be greater than the moment demands coming from beam ends. The approach used
herein and illustrated in Figure 19 considers the entire column tree under all applicable
forces needed for equilibrium. An appropriate moment at the column base must also be
applied. It is in the calculation of column design moments and shears that the importance of using a realistic lateral force distribution becomes more critical.
Figure 20 shows a comparison of the column moments calculated by the abovementioned procedure with the new lateral force distribution, the UBC lateral force distribution, and maximum column moment envelopes obtained from nonlinear dynamic
analyses for the exterior and interior columns of the nine-story moment frame (Lee and
Goel 2001). The column design moments calculated by using the new lateral force distribution agree very well with the maximum column moment envelopes at most levels.
On the other hand, the design column moments calculated by using the UBC lateral
force distribution significantly deviate from the nonlinear dynamic analysis results. The
overestimation becomes even greater for taller frames (Figure 21). It should be mentioned that the overestimated column design moments due to code lateral force distribution occur only when the proposed column design method (i.e., “column tree” approach)
is used. As mentioned earlier, the column design moments can be underestimated when
conventional elastic analysis and design methods are employed along with the code distribution of lateral forces.
HIGHER MODE EFFECTS
The new lateral force distribution increases the forces in the upper stories, thereby
better representing the higher mode effects. Figure 22 shows the shapes of IBC (or NEHRP) and the suggested lateral force distributions for the example nine-story moment
frame; the corresponding parameters are summarized in Tables 4 and 5. It can be seen
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S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 21. Design column moments and maximum column moment envelopes from nonlinear
dynamic analyses for a 20-story moment frame (Lee and Goel 2001).
that the new lateral force distribution results in larger force at the upper level than that
given by the IBC lateral force distribution. This implies that the additional design force
at the top level as given by the IBC (intended to reflect the higher mode effects) may not
be adequate, especially when the structures are taller and respond inelastically.
Dynamic analyses conducted on moment frames with various beam-to-column stiffness ratios have shown that, as the beam-to-column stiffness decreases, the higher mode
Figure 22. The suggested lateral force distribution and IBC 2006 lateral force distribution for
the nine-story moment frame.
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565
Table 4. IBC/NEHRP design lateral forces and design story shears for the nine-story moment
frame
Floor
hi (ft.)
9
8
7
6
5
4
3
2
1
122
109
96
83
70
57
44
31
18
wi (kips)
1177.0
1092.5
1092.5
1092.5
1092.5
1092.5
1092.5
1092.5
1111.5
wihki (kip-ft)
Fi (kips)
Story Shear (kips)
0.239V
0.190V
0.159V
0.130V
0.102V
0.077V
0.054V
0.033V
0.016V
945870
750484
628844
513514
405082
304304
212212
130316
62195
0.239V
0.429V
0.588V
0.718V
0.821V
0.898V
0.951V
0.984V
1.000V
response becomes an increasing percentage of the total response (Chopra and Cruz
1986, Chopra 2005). For the extreme case when the beam stiffness approaches zero, the
lateral forces at the upper levels of a moment frame become much larger than those calculated by the code expression, as shown in Figure 23. A moment frame with negligible
beam stiffness resembles a moment frame in which plastic hinges have occurred at the
beam ends. Therefore, the response as shown in Figure 23b supports the observations in
this study that the influence of higher mode effects increases in the inelastic state when
the plastic hinges form at the beam ends during strong shaking.
Nonlinear dynamic analyses carried out by Villaverde (1991,1997) also showed that
using the linear elastic first-mode distribution of lateral forces (without accounting for
the fact that a structure would enter inelastic state during a major earthquake) could be
the primary reason for numerous upper story collapses during the 1985 Mexico City
earthquake. The modal periods of a structure elongate as plastic deformation occurs.
That leads to increased contribution from the higher modes, hence columns designed
Table 5. Suggested design lateral forces and design story shears for the nine-story moment
frame
Floor
9
8
7
6
5
4
3
2
1
hj (ft.)
122
109
96
83
70
57
44
31
18
wj (kips)
wjhj (kip-ft)
1177.0
1092.5
1092.5
1092.5
1092.5
1092.5
1092.5
1092.5
1111.5
143594
119083
104880
90678
76475
62273
48070
33868
20007
兺wjhj
143594
262677
367557
458234
534709
596982
645052
678919
698926
␤i
Fi (kips)
Story Shear (kips)
1.000
1.538
1.955
2.288
2.554
2.763
2.920
3.029
3.092
0.323V
0.174V
0.135V
0.108V
0.086V
0.068V
0.051V
0.035V
0.021V
0.323V
0.498V
0.632V
0.740V
0.826V
0.894V
0.944V
0.979V
1.000V
566
S. H. CHAO, S. C. GOEL, AND S. S. LEE
Figure 23. (a) Moment frame with small beam-to-column stiffness ratio, and (b) equivalent
static lateral force distribution from dynamic analysis and IBC expression (Chopra 2005).
based on elastic first-mode distribution of story shears are subjected to significantly
higher moments when a structure deforms in the inelastic range, especially those in the
upper stories (Villaverde 1991). Villaverde’s conclusions further justify the importance
of using a lateral force distribution, such as the one suggested herein.
SUMMARY AND CONCLUSIONS
This paper presents a study aimed at formulating a more realistic design lateral force
distribution that accounts for inelastic behavior of structures when subjected to major
earthquakes. Use of a realistic force distribution based on inelastic response is one of the
important steps in a comprehensive seismic design methodology if accurate representation of expected structural response is to be realized. In this study, extensive nonlinear
dynamic analyses were carried out on different types of frames to verify the suggested
distribution, and the results showed that the suggested lateral force distribution, especially for the types of framed structures investigated in this study, is more rational and
gives a much better prediction of inelastic seismic demands at global as well as at element levels. It should be mentioned that the suggested lateral force distribution can be
easily further refined and modified by using more ground motions, such as near-field
A SEISMIC DESIGN LATERAL FORCE DISTRIBUTION BASED ON INELASTIC STATE OF STRUCTURES
567
records and other structural systems, than those used in this study. It is also noted that
the current codes generally use one distribution (derived based on first-mode elastic dynamic response) for all types of structures. The suggested lateral force distribution can
be applied to most of the commonly used frame types, and it can also be modified for
other structures by changing the value of the ␣ factor.
The following conclusions can be drawn from the results of this study:
1. Steel frames (MFs, EBFs, STMFs, and CBFs) designed by using the suggested
lateral force distribution resulted in maximum story shears that agreed well with
those obtained from nonlinear dynamic analyses. Maximum story shear distributions as given in the codes, which are based on first-mode elastic behavior,
deviate significantly from the time-history dynamic analysis results regardless
of whether the structures respond in the elastic or inelastic range.
2. Frames designed by using the suggested lateral force distribution experienced
more uniform maximum interstory drifts along the heights than the frames designed based on current code distributions.
3. The proposed column design procedure, using the “column tree” concept and
the new design lateral force distribution, gives a very good estimation of maximum column moment demands when the structures respond to severe ground
motions and deform into their inelastic state.
Higher mode effects are also well reflected in the suggested design lateral force distribution.
ACKNOWLEDGMENTS
The financial support for this study was provided by NUCOR Research & Development and the G. S. Agarwal Fellowship Fund in the Department of Civil & Environmental Engineering at the University of Michigan. The opinions and views expressed in this
paper are solely those of the authors and do not necessarily reflect those of the sponsors.
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(Received 23 August 2006; accepted 13 February 2007兲