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Structural responses considering the vertical component ofearthquakes
Alfredo Reyes Salazara, Achintya Haldarb,*aFacultad de Ingeniera, Universidad Autonoma de Sinaloa (UAS), Culiancan, Sinaloa, Mexico
bDepartment of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721, USA
Received 31 December 1997; accepted 18 November 1998
Abstract
The guidelines in the NEHRP Provisions and the Mexican Code regarding the eects of the vertical component
of earthquakes on the response of frames are re-evaluated. Using a time domain nonlinear nite element program
developed by the authors, the seismic responses of frames are evaluated realistically by simultaneously applying the
horizontal and vertical components of earthquake motion. Three steel frames and 13 recorded earthquake motions
are considered. The same response parameters are then estimated using the two codes, and their error is evaluated.
It is found that, if the frames remain elastic, the NEHRP Provisions estimate the maximum horizontal deection at
the top of the frames and the bending moment in the columns very accurately; the Mexican Code overestimates
them. If the frames develop plastic hinges, the Mexican Code conservatively overestimates them, but the NEHRP
Provisions underestimate them in some cases. Both codes signicantly underestimate the axial loads in columns. The
underestimation increases as the frames develop plastic hinges. The underestimation is more for interior columns
than for exterior columns. If the ratio R of the PGA of the vertical and horizontal components of an earthquake is
higher than normal, the underestimation increases as R increases. The underestimation is not correlated with frame
height. The vertical component may increase the axial load signicantly. Since they are designed as beamcolumns,
the increase in the axial load will have a very detrimental eect on the performance of the columns. In light of the
results obtained in this study, the design requirements for the vertical components need modication. At the veryleast, further study is required. # 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Seismic response; Seismic design; Steel frames; Load combinations; Design criteria; Lateral deection; Vertical accelera-
tion
1. Introduction
The inuence of the vertical component of an earth-
quake on the overall seismic response of structures has
long been of considerable interest to the profession.
Several design codes tried to address the issue in many
dierent and, it is hoped, conservative ways. Despite
this, many steel structures suered a considerable
amount of damage during the Northridge earthquake
of January 1994. Severe cracks developed in many
structures during the earthquake. Researchers mostlyattribute this damage to defects in welding and ma-
terial, and to design-related causes. Several recorded
ground motions during the Northridge earthquake in-
dicate that the vertical component was much larger
Computers and Structures 74 (2000) 131145
0045-7949/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
P I I : S 0 0 4 5 -7 9 4 9 (9 9 )0 0 0 3 1 -0
www.elsevier.com/locate/compstruc
* Corresponding author. Tel.: +1-520-621-2142; fax: +1-
520-621-2550.
E-mail address: [email protected] (A. Haldar)
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than is usually considered normal in design. This ob-
servation prompted a discussion about whether the
excessive vertical acceleration may have caused the
damage since, in the past, similar steel structures
behaved well when the vertical component of the
earthquakes was not so strong. Although extensive stu-
dies are now being conducted in related areas, it is im-
portant for the profession to reconsider the adequacy
of the design provisions outlined in the model building
codes to consider the eect of the vertical component
of earthquakes.
For numerical evaluation, earthquake motions are
generally represented by three components: two hori-
zontal and one vertical. The peak ground acceleration
of the vertical component is usually smaller than those
of the two horizontal components. Since the horizontal
motion of the ground has the most signicant eect on
the structural response, it is that motion which is
usually thought of as earthquake load. Therefore, most
building codes with earthquake provisions require that
an equivalent lateral load as a result of the horizontal
ground motion be used in simplied empirical
approaches [1]. The eect of the vertical component is
considered indirectly. Obviously, if the vertical com-
ponent is much stronger than is usually considered
normal, then the simplied code approaches may
underestimate the seismic load, and the structure willnot perform as intended.
The ratio of the peak ground acceleration of the ver-
tical component (PGAV) to the maximum horizontal
peak ground acceleration (PGAH), denoted hereafter
as R, can be used to study the inuence of the vertical
component on the overall seismic response behavior of
structures. For normal earthquakes, this ratio is
expected to be around 2/3. For the widely used earth-
quake time histories recorded during the El Centro
earthquake of 1940, this ratio is 0.60, as shown in
Table 1. For the 12 earthquake time histories recorded
during the Northridge earthquake listed in Table 1,
this ratio varies between 0.27 and 1.11, and ve of
them have a ratio greater than 2/3. Any one of these
12 earthquake time histories can be used to represent
the Northridge earthquake in future designs. Thus, the
date collected during the Northridge earthquake gives
the profession an opportunity to re-evaluate the ade-
quacy of the provisions suggested in design codes on
how to consider the eect of the vertical component in
design.
This study specically addresses two major seismic
design guidelines for buildings, namely, the National
Earthquake Hazard Reduction Program (NEHRP)
Recommended Provisions for Seismic Regulations for
New Buildings [2], hereafter denoted as the NEHRP
Provisions, and the Mexico City Seismic Code [3]. The
design requirements in other codes are expected to be
similar. In the 1994 edition of the NEHRP Provisions,
a new requirement was added to consider the com-
bined eects of the horizontal and vertical components
on the structural response. It is addressed indirectly in
the section on `Combination of load eects'. It is
suggested that the eect of gravity loads and seismicforces be combined in accordance with the factored
load combinations as presented in the American
Society of Civil Engineers Minimum Design Loads for
Buildings and Other Structures (ASCE 7-95) [4], except
that the eect of seismic loads, E, shall be dened as
E=QE+0.5 CaD to consider the eect of both the
horizontal and vertical components of an earthquake,
Table 1
Strong motion earthquakes
Earthquake Station Acceleration (cm/s2)
PGAV PGAH PGAV/PGAH
1 El Centro 206 342 0.60
Northridge earthquakes2 Los Angeles, 1526 Edgemont Ave 225 825 0.27
3 Los Angeles, Wadsworth V.A. 135 359 0.38
4 Los Angeles, 10660 Wilshire Blvd 441 998 0.44
5 Los Angeles, Grith Observatory 142 276 0.51
6 Jenson Filtration Plant 369 615 0.60
7 Los Angeles, Wadsworth V.A. 152 250 0.61
8 Topanga Fire Station 201 326 0.62
9 Sherman Oaks, 1525 Ventura Blvd 377 551 0.68
10 Los Angeles, 4929 Wilshire Blvd 285 389 0.73
11 Los Angeles, 10751 Wilshire Blvd 326 380 0.86
12 Conoga Park, Santa Susana 613 573 1.07
13 Los Angeles, 4929 Wilshire Blvd 526 472 1.11
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where QE is the eect of horizontal seismic forces, Cais the seismic coecient based upon the soil prole
type and the value of Aa as determined from Section
1.4.2.3 or Table 1.4.2.4a of the NEHRP Provisions,
and D is the eect of the dead load. The commentary
of the Provisions further adds that `0.5 Ca was placed
on the dead load to account for the eects of vertical
acceleration. The 0.5 Ca factor on dead load is notintended to represent the total vertical response. The
concurrent maximum response of vertical acceleration
and horizontal accelerations, direct and orthogonal, is
unlikely and, therefore, the direct addition of responses
was not considered appropriate.'
In the Mexico City Seismic Code, the eect of the
vertical component is considered to be a fraction of
the eect of the horizontal component. It states that
`For the buildings located in seismic zones C and D
the eect of the vertical component should be con-
sidered. This eect shall be taken as 2/3 of that of the
largest horizontal component. This eect while com-
bined with gravity and horizontal component eectsshould be taken as 0.3 of the above equivalent vertical
eect'. Eectively, the code recommends that the eect
of the vertical component should be estimated as 20%
(the product of 2/3 and 0.3) of the eect of the largest
horizontal component. This criterion can be inter-
preted another way. If the horizonal maximum re-
sponse is H, than the vertical maximum response will
be 2/3 H. Assuming that both maxima occur at the
same time and using the square root of the sum of
squares (SRSS) rule, the total response considering
both components can be calculated as
H2 2a3H2q 1X2H. Obviously, the requirements
in the Mexico City Seismic Code appear to be much
more conservative than those of the Provisions because
of this assumption.
In light of the extensive damage suered by steel
structures during the Northridge earthquake and the
signicant amount of information collected during the
earthquake enabling detailed analytical studies, it is
very desirable to compare the accuracy of the two
codes in estimating the eect of the vertical com-
ponent. The main objectives of this paper are: (1) to
evaluate the eect of the vertical component analyti-
cally for several recorded earthquakes for several steel
frames representing dierent dynamic properties in
terms of their maximum lateral displacements and the
maximum axial loads and bending moments in the
members; (2) to evaluate the eect of the vertical com-
ponent according to the NEHRP Provisions and the
Mexican Code; and (3) to compare the analytical
results with the codes' recommendations, in order to
evaluate the adequacy of current design practices.
2. Analysis procedure
In order to meet the objectives of this study, the
nonlinear seismic responses of structures subjected to
both horizontal and vertical components of an earth-
quake need to be evaluated as realistically as possible.
The authors, with the help of other research team
members, developed a highly ecient time domain
nite element-based algorithm to estimate the non-
linear seismic responses of steel frames considering
geometric and material nonlinearities. This sophisti-
cated algorithm can also be used to estimate the seis-
mic response of structures, instead of using simplied
approaches such as the equivalent lateral load pro-
cedure and the modal analysis procedure suggested in
the NEHRP Provisions. This type of elaborate analyti-
cal procedure is not expected to be used routinely by
the design profession; however, it can be used to study
the adequacy of the simplied methods suggested in
the design codes.
The fundamentals of the analytical procedure are
available in the literature [5,6], but cannot be describedhere due to lack of space. Therefore, only the essential
features required for the purposes of this paper are dis-
cussed brie y below. Nonlinear bahavior of a frame
can be produced by changes in the geometry, including
the PD eect and/or material properties. The eects
of geometric nonlinearity are changes in the member
lateral stiness due to the eect of axial force, the
change in member length due to the bowing eect and
axial force, and the nite rigid body deformation of a
member with small to moderate relative rotation. Most
of the currently available nite element-based non-
linear analysis techniques for frames are based on an
assumed displacement eld. In order to capture the
eects of change in the axial length of an element due
to large deformation, several elements are needed to
model each member. The necessity for a large number
of elements coupled with the use of a numerical inte-
gration scheme to obtain the tangent stiness matrix
for each element several times during the analysis
makes this approach uneconomical. Alternatively, the
assumed stress-based nite element method [79] can
be used to derive an explicit form of the tangent sti-
ness. In this approach, the stresses on an element can
be obtained directly instead of using the less accurate
method of taking the derivatives of the displacement
functions as in the assumed displacement eld
approach. The method is very ecient and economicalbecause of this feature and the use of fewer elements
in describing a large deformation conguration, and
because it needs no integration to obtain the tangent
stiness. This procedure is particularly applicable to
steel structures. It gives very accurate results and is
very ecient compared to the displacement-based
approach [6,8,9]. This method is used in this study.
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The other major source of nonlinearity in frames is
material nonlinearity. Material nonlinearity occurs as a
result of the nonlinear constitutive relationship of the
material. In the analysis of steel structures, the three
most common assumptions for the material behavior
are the elasticperfectly plastic, isotropic strain harden-
ing and kinematic strain hardening models [10].
Considering the complexity of the problem under con-
sideration and the usual practice in the profession, the
material nonlinearity of steel will be considered to be
elasto-perfectly plastic in this study. The von Mises cri-
terion [10] is very appropriate, and is used in this
study.
The development of the static governing equations
using the assumed stress method is not described here
due to lack of space. Only the dynamic governing
equations required for the nonlinear seismic analysis of
frames and the solution strategy are presented very
briey below. The equation of motion of a linear sys-
tem under dynamic and seismic loadings can be
expressed as [11]:
MD C D KD FMDg 1
where M, C and K are the mass, damping and stiness
matrices of the frame, respectively; D , D.
and D are the
acceleration, velocity and the relative displacement vec-
tors, respectively; D g is the ground acceleration vector,
and F is the external dynamic force vector, if present.
For the nonlinear case the dynamic and seismic gov-
erning equations of motion can be expressed in incre-
mental form as [12]:
MtDt D
kt CtDt D
kt KtDtDDk
tDt Fk tDt Rk1 MtDt D kg 2
where tK(k ) is the tangent stiness matrix of the system
of the kth iteration at time t; (t+Dt )D(k ) and (t+Dt )F(k )
are the incremental displacement vector and external
load vector of the kth iteration at time t+Dt, respect-
ively; and (t+Dt )R(k 1) is the internal force vector of
the (k 1)th iteration at time t+Dt. All other par-
ameters were dened earlier. The step-by-step direct in-
tegration numerical analysis procedure using the
Newmark b method is used to solve Eq. (2).
Explicit expressions for the tangent stiness matrix
consisting of geometric and material nonlinearities and
the internal force vector are developed for each beamcolumn element using the assumed stress method for
each iteration at a given time t. The mathematical
details of the derivation are not shown here, but can
be found in the literature [79].
As stated earlier, in this study the material is con-
sidered to be linear elastic except at plastic hinges.
Concentrated plasticity behavior is assumed at plastic
hinge locations. In the past, several analytical pro-
cedures have been proposed to predict the deformation
of elasto-plastic frames under increasing seismic and
static loads. However, most of these formulations were
based on small deformation theory. In this study, each
elasto-plastic beamcolumn element can experience
arbitrary large rigid deformations and small relative
deformations.
In addition to the elastic stressstrain relationships,
the plastic stressstrain relationships need to be incor-
porated into the constitutive equations if the yield con-
dition is satised. Several yield criteria have been
proposed in the literature in terms of stress com-
ponents or nodal forces. Since the nodal forces can be
obtained directly from the proposed methods, the yield
used in this study is expressed in terms of nodal forces.
When the combined action of axial force and bending
moment (this is for plane structures only) satisfy a pre-
scribed yield function at a given node of an element, a
plastic hinge is assumed to occur instantaneously at
that location. Plastic hinges are considered to form at
the ends of the beamcolumns elements. The yieldfunction (or interaction equation) depends on the type
of section and loading acting on the beamcolumn el-
ement [13]. The yield function for the two-dimensional
beamcolumn element has the following general form:
fP,M,sy 0 atX lp 3
where P is the axial force; M is the bending moment;
sy is the yield stress, and lp is the location of the plas-
tic hinge. For the W-type sections used in this study,
this equation has the following form:
P
Pn
M
Mnx 1 0 4
where Pn and Mnx are the axial strength and the ex-
ural strength with respect to the major axis, respect-
ively.
The presence of a plastic hinge in the structure will
produce additional axial deformations and relative ro-
tations in a particular element. This is considered in
the stiness matrix and the internal force vector of the
plastic stage. Explicit expressions for the elasto-plastic
tangent stiness matrix and the elasto-plastic internal
force vector are also developed. The mathematical
derivations can be found in the literature [79].
Depending on the level of earthquake excitation, ina typical structure all the elements may remain elastic,
or some of the elements will remain elastic and the rest
will yield. The structural stiness matrix and the in-
ternal force vector can be explicitly developed by con-
sidering individual elements and the particular state
they are in.
Since actual earthquake time histories are used in
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this study, the inertia and applied forces are available.
However, further discussion of damping is necessary at
this stage. In a realistic seismic analysis of steel frames,
the amount of damping energy that will be generated
will depend on the nonyielding and yielding state of
the material and on the hysteretic behavior if the ma-
terial yields. For mathematical simplicity, the eect of
nonyielding energy dissipation is usually represented
by equivalent viscous damping varying between 0.1
and 7% of the critical damping. The damping is often
increased in linear analysis to approximate energy
losses due to anticipated inelastic behavior [14]. In a
rigorous seismic analysis this practice is not appropri-
ate, since the energy losses due to inelastic behavior
would be counted twice. Based on an extensive litera-
ture review, it is observed that the following Rayleigh-
type damping is very commonly used in the profession:
t
C aM gt
K 5
where a and g are the proportional constants. The use
of both the tangent stiness and the mass matrices is a
very rational approach to estimate the energy dissi-
pated by viscous damping in a nonlinear seismic analy-
sis. The constants a and g can be determined from
specied damping ratios xi and xj for the ith and jth
modes, respectively. Then the following algebraic
equation system is solved for a and g [11]:
1
2
1
o io i
1
ojoj
a
g
xixj
6
where oi and oj are the natural frequencies of the ith
and jth mode, respectively, and are calculated using
the Stodola method in this study. Usually the ith mode
is selected as the rst mode, and the jth mode as the
higher mode that contributes signicantly to the struc-
tural response.
A computer program has been developed to im-
plement the algorithm. The program was extensively
veried using information available in the literature.
The structural response behavior and the members'
forces in terms of axial load, shear force and bendingmoment can be estimated using the computer pro-
gram.
3. Description of structures and earthquakes
Three steel frame structures representing dierent
Fig. 1. Three steel frames. (a) Frame 1; (b) frame 2; (c) frame 3.
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ENEHRP 1X2D H 0X5CaD 1X2D HV
1X2D H 0X5CaD8
where the term 1.2D+H+ 0.5CaD represents the
combined eect of dead load, horizontal seismic load
and vertical seismic load according to the NEHRP
Provisions; the term 1.2D+HV represents the com-
bined eect of dead load, horizontal and vertical seis-
mic loads according to analytical results obtained by
the algorithm discussed in Section 2, H is the eect of
the horizontal component containing the maximum
PGA acting alone (case 1), 0.5CaD represents the eect
of the vertical component, and HV represents the eect
of both the horizontal and vertical components acting
simultaneously.
Similarly, for the Mexican Code, Eq. (7) can be
expressed as:
EMEX 1X2D H 0X2H 1X2D HV
1X2D H 0X2H9
where the term 1.2D+H+ 0.2H represents the com-
bined eect of deal load, horizontal seismic load and
the vertical seismic load according to the Mexican
Code, and 0.2H represents the eect of the vertical
component. All other terms in Eq. (9) were dened
earlier.
A positive error in Eqs. (8) and (9) implies that the
codes overestimate the load eect due to the vertical
component; in other words, the codes' recommen-
dations are conservative. A negative error indicates
that the codes underestimate the load eect, and thus
are unconservative. The responses of the three frames
can be compared in light of the error terms just dis-
cussed
4.1. Eect of the vertical component of DMAX
Frame 1 is considered rst. The DMAX values for
the two damping ratios and all 13 earthquakes areshown in Table 3. Column 3 contains analytical
DMAX values for excitation by the horizontal com-
ponent only. Column 4 contains the same information
when the frame is subjected to 1.2D plus both the hori-
zontal and vertical components. Columns 5 and 6 con-
tain the combined eect for the DMAX values
according to the NEHRP Provisions and the Mexican
Table 3
Maximum top displacements (DMAX) for frame 1
EAR x H (cm) 1.2D+HV (cm) NEHRP (cm) MEX (cm) ENEHRP (%), Eq. (8) EMEX (%), Eq. (9)
(1) (2) (3) (4) (5) (6) (7) (8)
1 2 2.88 2.86 2.88 3.46 1 17
5 2.05 2.06 2.05 2.46 0 16
2 2 6.92 6.92 6.92 8.30 1 175 5.38 5.37 5.38 6.46 0 17
3 2 1.82 1.82 1.82 2.18 0 17
5 1.72 1.72 1.71 2.05 1 16
4 2 6.39 6.40 6.39 7.67 0 17
5 4.54 4.54 4.54 5.45 0 17
5 2 1.79 1.79 1.79 2.15 0 17
5 1.49 1.48 1.49 1.79 1 17
6 2 3.79 3.78 3.79 4.55 0 17
5 3.08 3.08 3.08 3.70 0 17
7 2 1.08 1.08 1.08 1.30 0 17
5 0.81 0.82 0.81 0.97 1 15
8 2 4.26 4.26 4.26 5.11 0 17
5 2.65 2.66 2.65 3.18 0 16
9 2 7.44 7.46 7.44 8.93 0 16
5 4.74 4.74 4.74 5.69 0 1710 2 2.32 2.32 2.32 2.78 0 17
5 1.78 1.78 1.78 2.14 0 17
11 2 1.96 1.96 1.96 2.35 0 17
5 1.52 1.52 1.52 1.82 0 16
12 2 4.48 4.47 4.47 5.36 0 16
5 3.59 3.59 3.59 4.31 0 17
13 2 2.74 2.74 2.74 3.29 0 17
5 2.29 2.28 2.29 2.75 0 17
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Code, respectively. Using Eqs. (8) and (9), the corre-
sponding error terms are calculated and are shown in
columns 7 and 8, respectively. As stated earlier, theframe is designed so that it did not develop any plastic
hinges when excited by any of the 13 earthquakes.
From the results given in Table 3, several important
observations can be made. The maximum analytical
horizontal deections of the frame are observed to be
almost the same for excitation by the horizontal com-
ponent alone or by both the horizontal and vertical
components. This is expected. Since the frame is sym-
metric and did not develop any plastic hinges, the
eect of the vertical component in the estimation of
the horizontal deection is expected to be small. The
DMAX values estimated according to the NEHRP
Provisions (column 5) are very similar to the analyticalresults. This is also expected, since the eect of the
dead load on the DMAX calculation is negligible. The
corresponding error according to Eq. (8) is also negli-
gible (column 7). However, the situation is quite dier-
ent for the Mexican Code. The results in column 6
indicate that the Mexican Code overestimates the
DMAX values, and this overestimation is about 17%
(column 8). Thus, for frame 1, the NEHRP Provisions
estimate DMAX very accurately, but the Mexican
Code overestimates it by about 17%.In order to study the behavior of the same frame
subjected to stronger earthquakes, all the earthquake
time histories are scaled up so that six to eight plastic
hinges develop in the frame. The frame is reanalyzed,
and the results in term of DMAX are given in Table 4.
For 2% damping, the frame developed six to eight
plastic hinges, but remained elastic for 5% damping.
For 2% damping, when the structure lost its symmetry
due to the development of the plastic hinges in the
frame, the NEHRP Provisions underestimated DMAX
by more than 11% in some cases, and overestimated
by over 13% in other cases. This underestimation or
overestimation cannot be correlated with the R par-ameter. It appears to be problem-specic. As before,
the Mexican Code overestimates the DMAX values in
this case too; however, the amount of overestimation
is, in some cases, smaller than that of Table 3, and
could be as small as 7%. For 5% damping when the
frame remains elastic, the eect of the vertical com-
ponent on the NEHRP calculation remains negligible
Table 4
Maximum top displacements (DMAX) for frame 1 (plastic case)
EAR x Case 1 (cm) 1.2D+HV (cm) NEHRP (cm) MEX (cm) ENEHRP (%), Eq. (8) EMEX (%), Eq. (9)
(1) (2) (3) (4) (5) (6) (7) (8)
1 2 12.59 13.73 12.59 15.11 9 9
5 8.20 8.2 8.20 9.84 0 17
2 2 12.26 12.45 12.26 14.71 2 155 8.08 8.08 8.08 9.70 0 17
3 2 9.09 9.10 9.09 10.91 0 17
5 8.54 8.54 8.54 10.25 0 17
4 2 13.75 13.77 13.75 16.50 0 17
5 11.72 11.72 11.72 14.06 0 17
5 2 8.94 8.94 8.94 10.73 0 17
5 7.44 7.44 7.44 8.93 0 17
6 2 13.08 11.35 13.08 15.70 13 28
5 9.25 9.25 9.25 11.10 0 17
7 2 8.66 8.66 8.66 10.39 0 17
5 6.51 6.51 6.51 7.81 0 17
8 2 11.32 11.31 11.32 13.58 0 17
5 6.62 6.62 6.62 7.94 0 17
9 2 14.40 14.23 14.40 17.28 1 18
5 7.12 7.12 7.12 8.54 0 1710 2 10.26 10.24 10.26 12.31 0 17
5 7.10 7.10 7.10 8.52 0 17
11 2 9.83 9.79 9.83 11.80 0 17
5 7.63 7.61 7.63 9.16 0 17
12 2 12.67 14.11 12.67 15.20 11 7
5 12.38 12.36 12.38 14.86 0 17
13 2 10.66 10.66 10.66 12.79 0 17
5 8.03 8.07 8.03 9.64 1 16
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for the NEHRP Provisions, but the Mexican Code
overestimates it by about 17%, as before.
In summary, whether the frame remains elastic or
develops plastic hinges, the Mexican Code always over-
estimates the DMAX values; however, the NEHRP
Provisions could unconservatively underestimate
DMAX if plastic hinges develop in the frame.
The benecial eect in terms of the reduction inDMAX as a function of damping can also be noted
from Tables 3 and 4. However, the amount of re-
duction varies from earthquake to earthquake and
depends on the degree of yielding occurring in the
structure. If no plastic hinge develops in the structure,
the reduction could be around 20% (Table 3), and if
plastic hinges develop, the reductions could be larger
than 40% (Table 4).
Frames 2 and 3 are considered next. Results similar
to Tables 3 and 4 for frame 1 were estimated for
frames 2 and 3. They cannot be shown here due to
lack of space. The major conclusions made for frame 1
are also valid for these frames. If frames 2 and 3
remain elastic, the error according to the NEHRP
Provisions is almost zero, but according to the
Mexican Code, the conservative error is about 17%, as
before. If plastic hinges develop, the error according to
the NEHRP Provisions could be on the unconservative
side by about 13% for frame 2, and about 9.5%
for frame 3. Thus, the trends are very similar for all
three frames. The heights of the frames cannot be cor-
related with the corresponding errors, particularly
when the errors are negative or unconservative.
4.2. Eect of the vertical component on bending
moments in columns
The eect of the vertical component on the evalu-
ation of the bending moments for the interior and
exterior columns at the ground level of frame 1 is
considered next. Results for recorded time histories
and scaled up time histories, similar to Tables 3 and
4, are shown in Tables 5 and 6, respectively. The
Table 5
Maximum moments at ground level columns for frame 1
EAR x H (kN m) 1.2D+HV(kN m) NEHRP (kN m) MEX (kN m) ENEHRP (%), Eq. (8) EMEX (%), Eq. (9)
Interior Exterior Interior Exterior Interior Exterior Interior Exterior Interior Exterior Interior Exterior
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
1 2 767 496 769 513 769 516 922 612 0 1 17 16
5 545 353 547 369 547 373 656 440 0 1 17 16
2 2 1766 1141 1769 1158 1768 1161 2121 1386 0 0 17 16
5 1360 879 1362 895 1362 899 1634 1073 0 0 17 173 2 486 315 488 332 488 335 585 395 0 1 17 16
5 452 293 454 310 454 313 544 368 0 1 17 16
4 2 1526 985 1528 1002 1528 1005 1833 1199 0 0 17 16
5 1083 700 1086 717 1085 720 1301 857 0 0 17 16
5 2 461 298 463 315 463 318 555 374 0 1 17 16
5 383 248 385 165 385 268 461 314 0 1 16 16
6 2 989 640 991 656 991 660 1188 785 0 1 17 16
5 801 518 803 534 803 538 963 638 0 1 17 16
7 2 261 167 264 184 263 187 315 217 0 2 16 15
5 196 125 198 142 198 145 237 167 0 2 16 15
8 2 1004 648 1006 666 1006 668 1206 794 0 0 17 16
5 625 403 626 420 627 423 752 500 0 1 17 16
9 2 1860 1202 1863 1218 2862 1222 2234 1459 0 0 17 17
5 1190 769 1193 786 1192 789 1430 940 0 0 17 16
10 2 571 369 572 386 573 389 687 460 0 1 17 165 440 284 441 301 442 304 530 358 0 1 17 16
11 2 480 313 486 332 482 333 578 392 1 0 16 15
5 372 243 377 261 374 263 448 308 1 1 16 15
12 2 1195 773 1197 790 1197 793 1436 944 0 0 17 16
5 960 621 961 637 962 641 1154 762 0 1 17 16
13 2 700 453 703 470 702 473 842 560 0 1 17 16
5 585 378 587 395 587 398 704 470 0 1 17 16
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major observations made for the DMAX calculations
are also valid for the estimation of moments at the
ground level of columns. If the frame remains elas-
tic, the errors in the bending moment calculations
according to the NEHRP Provisions are almost zero
for both interior and exterior columns. However,
when plastic hinges develop, the underestimation
could be about 5%. The Mexican Code always
overestimates the bending moments; the correspond-
ing overestimation errors are about 17% when the
frame remains elastic, and as low as 12% when plas-
tic hinges develop.
Frames 2 and 3 were similarly analyzed, and the cor-
responding errors are almost identical to Frame 1. The
results are not shown due to lack of space. As in the
DMAX evaluation, the Mexican Code is conservative
in the estimation of bending moments, but the
NEHRP Provisions could underestimate this eect if
plastic hinges develop. These errors have no corre-
lation with the R parameter or with the height of the
frame.
4.3. Eect of the vertical component on axial loads in
columns
The eect of the vertical component on the evalu-
ation of the maximum axial loads at interior and ex-
terior ground level columns for all the frames is
considered next. The estimation errors according to
both codes are calculated using Eqs. (8) and (9) for
both interior and exterior columns. For ease of discus-
sion, the errors versus R are plotted. Only underesti-
mation of the axial load with errors larger than 25%,
which occurs for the interior column only, is empha-
sized in the following discussion. Other results cannot
be shown due to lack of space. The results for the in-
terior column of frame 1 are shown in Fig. 2 for theelastic case and in Fig. 3 when plastic hinges develop
in the frame. Unlike the DMAX and the bending
moment evaluation cases, the eect of the vertical
component on the axial estimation is observed to be
signicant, even when the frame remains elastic. The
underestimation error could be very large, on the
order of 50% for 2% damping and elastic behavior
Table 6
Maximum moments at columns for frame 1 (plastic case)
EAR x H (kN m) 1.2D+HV(kN m) NEHRP (kN m) MEX (kN m) ENEHRP (%), Eq. (8) EMEX (%), Eq. (9)
Interior Exterior Interior Exterior Interior Exterior Interior Exterior Interior Exterior Interior Exterior
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
1 2 3293 2109 3394 2098 3295 2129 3953 2548 3 1 14 185 2178 1409 2180 1425 2180 1429 2615 1708 0 0 17 17
2 2 2834 1902 2833 1917 2836 1922 3402 2299 0 0 17 17
5 2040 1318 2043 1335 2042 1338 2450 1598 0 0 17 16
3 2 2437 1575 2441 1593 2439 1595 2926 1907 0 0 17 16
5 2259 1464 2263 1481 2261 1484 2712 1774 0 0 17 17
4 2 3128 2166 3139 2124 3130 2186 3755 2616 0 3 16 19
5 1950 1259 1952 1275 1952 1279 2342 1528 0 0 17 17
5 2 2305 1492 2298 1507 2307 1512 2768 1807 0 0 17 17
5 1914 1238 1916 1255 1916 1258 2298 1502 0 0 17 16
6 2 3180 1996 2929 1948 3182 2016 3818 2412 8 3 23 19
5 2043 1312 2045 1337 2045 1341 2453 1602 0 0 17 17
7 2 2072 1338 2075 1356 2074 1358 2488 1622 0 0 17 16
5 1566 1004 1572 1020 1568 1024 1881 1222 0 0 16 17
8 2 2605 1694 2618 1719 2607 1714 3128 2050 0 0 16 16
5 1561 1008 1563 1025 1563 1028 1875 1226 0 0 17 169 2 3097 2046 3262 2148 3099 2066 3718 2472 5 4 12 13
5 1785 1154 1789 1171 1787 1174 2144 1402 0 0 17 16
10 2 2431 1577 2428 1591 2433 1597 2919 1909 0 0 17 17
5 1578 1136 1759 1152 1760 1156 2111 1380 0 0 17 17
11 2 2523 1634 2545 1661 2525 1654 3029 1978 1 0 16 16
5 1858 1214 1879 1236 1860 1234 2231 1474 1 0 16 16
12 2 3305 2017 3467 2078 3307 2037 3968 2437 5 2 13 15
5 2303 1490 2304 1507 2305 1510 2765 1805 0 0 17 17
13 2 2535 1637 2347 1658 2537 1657 3044 1981 0 0 16 16
5 2046 1323 2051 1341 2048 1343 2457 1604 0 0 17 16
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Fig. 2. Error in the axial load on the interior column of frame 1, elastic case.
Fig. 3. Error in the axial load on the interior column of frame 1, elastic case.
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Fig. 4. Error in the axial load on the interior column of frame 2, elastic case.
Fig. 5. Error in the axial load on the interior column of frame 2, elastic case.
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according to the NEHRP Provisions, and about 70%
according to the Mexican Code. If inelastic behavior is
considered, the corresponding errors increase to about
150% for both codes. It is also observed from the
above gures that for a given code, the error is always
larger for 2% damping than for 5% damping.
However, this observation is not valid if the structure
develops plastic hinges (Fig. 3). It is interesting to note
that the magnitude of the unconservative error
increases as the R value increases.
Frame 2 is considered next. The underestimation
error for the interior column is shown in Figs. 4 and 5
for the elastic and plastic cases, respectively. The
major observations made for frame 1 apply to frame 2.
If the frame remain elastic, the unconservative error
associated with the Mexican Code is greater than that
of the NEHRP Provisions. This observation is not
valid if plastic hinges develop. Although the NEHRP
Provisions are better than the Mexican Code for the
elastic case, the unconservative error associated with it
may not be acceptable. it is also observed from Figs. 4
and 5 that the magnitude of the unconservative error
is not a function of the height of the frame; however,
it increases as the R value increases.
Frame 3 is analyzed last. The results for the interior
column are shown in Fig. 6, when plastic hinges
develop in the structure. The major observations made
for frames 1 and 2 are valid for this frame too. The
only additional observation is that, unlike frames 1
and 2, both codes are conservative for the interior col-
umn when R values are smaller than about 0.6. For
larger values of R, however, they again considerably
underestimate the axial force.
4.3.1. Design implications
The eect of the vertical component on the axial
load evaluation is signicant for all three frames con-
sidered in this study. All these members are expected
to be designed as a beamcolumn, and the exact form
of Eq. (4), according to the AISC LRFD code, is:
Pu
fPn
8
9
Mux
fbMnx1X0; if
Pu
fPnr0X2 10
Fig. 6. Error in the axial load on the interior column of frame 3, plastic case.
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Pu
2fPn
Mux
fbMnx1X0; if
Pu
fPn` 0X2 11
where Pu is the required axial strength, Pn the nominal
axial strength, Mux and Mnx are the required exural
and the nominal exural strength with respect to the
major axis, respectively, f is the resistance factor for
compression (or tension) and fb is the resistance factor
for exure.Although the eect of the vertical component on the
moment calculation is negligible, it may increase the
axial load signicantly. Since both the axial load and
moment are considered in the interaction equations,
the increase in the axial load will have a very detrimen-
tal eect on the performance of the columns. This ob-
servation indicates the need for modication of the
way the eect of the vertical component is considered
in design codes, or at least indicates the need for
further study. If the R values are greater than the
value usually considered to be normal, say 2/3, the
underestimation increases as the R value increases. The
underestimation error is observed to be more for in-
terior columns than for exterior columns, indicating
that the location of the columns may also be import-
ant. The underestimation error also depends on the
elastic or plastic state of the frames, but no correlation
is observed between the underestimation error and the
height of the frames.
5. Conclusions
The eect of the vertical component on the seismic
responses of frames, as outlined in the NEHRP
Provisions and the Mexican Codes, is reevaluated.
Using a time domain nonlinear nite element program
developed by the authors, the seismic responses of
frames, in terms of the maximum lateral displacement
at the top of the frame and the maximum axial and
bending moments in columns, are evaluated as realisti-
cally as possible by applying the horizontal and verti-
cal components of earthquake motion simultaneously.
Three steel frames and 13 recorded earthquake
motions are considered in the study. The same re-
sponse parameters are then estimated using the two
codes, and the error associated with their recommen-
dations is evaluated. Several important observations
are made. If the frames remain elastic, the NEHRP
provisions estimate DMAX and the bending momentsvery accurately; however, the Mexican Code overesti-
mates them. If the frames develop plastic hinges, the
Mexican Code still conservatively overestimates them,
but the NEHRP Provisions underestimate them in
some cases. Both codes signicantly underestimate the
axial loads in columns. The underestimation increases
as the frames develop plastic hinges. Also, the underes-
timation is more for interior columns than for exterior
columns. If the ratio R of the PGA of the vertical and
horizontal components of an earthquake is more than
is usually considered to be normal, the underestimation
increases as R increases. The underestimation can not
be correlated with the height of the frames.
Although the eect of the vertical component in the
moment calculation of columns is negligible or conser-
vative in most cases, it may increase the axial load sig-
nicantly. Since they are designed as beamcolumns,
the increase in the axial load will have a very detrimen-
tal eect on the performance of the columns. In light
of the results obtained in this study, the design require-
ments for the vertical components, as outlined in the
NEHRP Provisions and the Mexican Code, need
modication. At the very least, further study is
required.
Acknowledgements
This paper is based on work partly supported by the
National Science Foundation under grant nos MSM-
8896267 and CMS-9526809. The nancial support
received from the American Institute of Steel
Construction (AISC), Chicago, is appreciated. The
work is also partially supported by El Consejo Nacional
de Ciencia y Tecnologia (CONACYT), Mexico, and La
Universidad Autonoma de Sinaloa (UAS), Mexico. Any
opinions, ndings, conclusions, or recommendations
expressed in this publication are those of the authors
and do not necessarily reect the views of the sponsors.
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