s Truc Tres p Vertical Comp

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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



12 Conoga Park, Santa Susana 613 573 1.07


A.R. Salazar, A. Haldar / Computers and Structures 74 (2000) 131145132


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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.

A.R. Salazar, A. Haldar / Computers and Structures 74 (2000) 131145 133


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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



9/15

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



10/15

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



11/15

Fig. 2. Error in the axial load on the interior column of frame 1, elastic case.




12/15





13/15

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.



14/15

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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