A macro-element model for inelastic building analysis

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2000; 29:1725–1757

A macro-element model for inelastic building analysis

Juan C. de la Llera∗;1, Jorge V�asquez1, Anil K. Chopra2 and Jos�e L. Almaz�an1

1 Department of Structural Engineering; Ponti�cia Universidad Cat�olica de Chile; Casilla 306; Correo 22;Santiago; Chile

2 Department of Civil Engineering; University of California; Berkeley; CA 94720; U.S.A.

SUMMARY

A three-dimensional model for approximate inelastic analysis of buildings is presented herein. The model isbased on a single macro-element per building storey. The inelastic properties of the model are characterizedby the so-called ultimate storey shear and torque (USST) surfaces. Di�erent algorithms for the constructionof these surfaces, as well as their applications in building modelling, are presented and discussed. Twoalternative procedures are developed to integrate the force-deformation constitutive relationship of the macro-elements. The �rst one follows the exact trajectory of the load path of the structure on the USST, andthe second uses linear programming without ever forming the USST surface. The accuracy of the modeland integration procedure is evaluated by means of the earthquake response of single-storey systems. Themodel and integration procedure developed is �nally used to compute the inelastic response of a seven-storeyR=C building. The results of this investigation show that the model proposed, although approximate, canbe e�ective in estimating the inelastic deformation demand of a building. It also enables the engineer tocapture and interpret important features of the three-dimensional inelastic response of a structure even beforeperforming any inelastic dynamic analysis. Copyright ? 2000 John Wiley & Sons, Ltd.

KEY WORDS: inelastic building analysis; ultimate storey-shear and torque surface; storey mechanisms;single-element model; lateral-torsional coupling

INTRODUCTION

The current philosophy for earthquake resisting design of conventional buildings formulated al-most forty years ago admits, motivated mainly by economical reasons, that most buildings couldexperience inelastic behaviour during a severe earthquake but should not collapse [1]. Mostbuilding codes, however, prescribe a linear elastic model for building analysis, thus leaving alarge uncertainty about the expected inelastic performance of the structure during a severe earth-quake.The task of checking such a philosophy is not simple, and our wise predecessors left a di�cult

task in front of us. Inelastic building models are usually tedious to build, check, and interpret.

∗ Correspondence to: Juan C. de la Llera, Department of Structural Engineering, Ponti�cia Universidad Cat�olica de Chile,Casilla 306, Correo 22, Santiago, Chile.

Received 23 October 1999Revised 30 March 2000

Copyright ? 2000 John Wiley & Sons, Ltd. Accepted 10 April 2000

1726 J. C. DE LA LLERA ET AL.

Further, complete inelastic models for dynamic analysis of buildings used to be essentially impos-sible to implement in computers ten years ago due to the large numerical e�ort involved. Althoughcomputers have improved tremendously during this last decade, even today a three-dimensionalinelastic building model is cumbersome and few software packages are available to construct andanalyse such models reliably. There is little doubt, however, that the time when the design andperformance of a building are veri�ed for a group of earthquakes using an inelastic model isclose.Meanwhile, simpli�ed inelastic models can be used to predict with su�cient accuracy the

true earthquake response of a building. The model presented next has proven to be e�ectivein predicting the inelastic performance of an instrumented structure during the Northridge earth-quake [2]. Simpli�ed non-linear models are a reasonable compromise between the most sophis-ticated three-dimensional inelastic models and the elastic models currently used inpractice.A two-dimensional version of such a simpli�ed model, devised for the study of structures with

a single axis of sti�ness symmetry in plan (mono-symmetric), was developed earlier [3]. Themodel presented herein is intended for full three-dimensional inelastic analysis and it constitutesan important extension of the previous 2D-model. More important, its construction leads to anumber of properties that are quite useful in interpreting aspects of the inelastic plan behaviour ofa structure, such as inelastic torsion. These properties, for instance, may prove useful in retro�ttingor re-designing a torsionally unbalanced structure. An attempt to incorporate these ideas into acapacity design format has been presented recently by Pauley [4].The structural model presented consists of a single column-like macro-element per building

storey and is denoted hereafter as single element model (SEM). In general, a macro-element iscapable of representing approximately the elastic and inelastic properties of a storey. Its inelas-tic properties are based strongly on the concept of an ultimate storey-shear and torque surface(USST) [3]. These surfaces provide a convenient and simple framework for the engineer to un-derstand the main trends of the inelastic performance of a structure. Since no combination ofstorey shears and torque can go beyond these surfaces, they bound the structural response inthis force space in the same way as the P–M interaction curves do for an R=C column section.The properties used in the construction of these surfaces may help the engineer to assess thequality of a three-dimensional design, in height as well as in plan. Once the model is imple-mented computationally, many di�erent structural layouts for the building can be tested due tothe small number of inelastic elements—a twenty-storey building requires only twenty macro-elements.To better understand the three-dimensional SEM, the construction and properties of the two-

dimensional USST surfaces will be summarized �rst using a newly developed procedure. Thisprocedure will then be extended to the 3D-case and used to formulate the SEM. Two algorithmswill be developed to perform the time integration in the force space along the USST surface:(1) an exact algorithm that follows the trajectory of the structure along all possible load paths onthe di�erent faces, edges, and vertices of the USST surfaces, and (2) an approximate model basedon linear programming that uses a central correction strategy and is computationally very e�cient.Because such integration procedures are useful and their implementation is not trivial, pseudo-algorithms have been provided in the appendix. The accuracy of both integration procedures will beevaluated by using examples of single-storey buildings. Finally, the hypothetical three-dimensionalinelastic earthquake response of a real seven-storey R=C building subjected to the Sylmar (1994)record will be predicted using the simpli�ed model.

Copyright ? 2000 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2000; 29:1725–1757

MACRO-ELEMENT MODEL FOR INELASTIC BUILDING ANALYSIS 1727

Figure 1. Typical building considered and single element model (SEM).

BUILDING MODEL AND SEM

The buildings considered are three-dimensional structures with in-plane rigid oor diaphragmswhose position in space is described by three degrees of freedom, two horizontal translationsand one rotation (Figure 1). The building may have any number of resisting planes that neednot be arranged in an orthogonal grid; an orthogonal grid has been used exclusively to ease thepresentation. The procedure developed to construct the SEM is general and could be extendedto the case of resisting planes or structural elements with shear interaction in the two horizontaldirections.As it was mentioned before, the structural model developed assumes a single macro-element in

each storey of the building (Figure 1). The macro-element responds to the relative displacementsand rotations of two consecutive oors. The forces developed by the macro-elements are the shearsand torque of a storey; they may result from the elastic or fully plastic behaviour of the storey.Because one of the main goals of the model is to remain as simple as possible, the transitionbetween the elastic and plastic behaviour of the structure has not been included since the resultspresented later do not warrant the corresponding increase in complexity, especially for pre-designpurposes.The elastic and inelastic properties of the SEM are obtained from the elastic and inelastic

properties of the individual resisting planes. In the analysis that follows, the force-deformationrelationship of each resisting plane at each storey is assumed to be adequately represented by anequivalent elasto-plastic constitutive law that could be obtained from di�erent procedures, such aslateral pushover analysis, plastic analysis, or a family of storey mechanisms [5]. In any case, theinelastic properties of the resisting plane should be computed from the individual shear capacitiesof structural members and realistic values of vertical load. Consequently, each SEM macro-elementis de�ned by a set of elastic (sti�ness matrix) and inelastic properties (USST surface).



The equations of motion that govern the inelastic response of the structure shown in Figure 1subjected to a ground motion acceleration vector �ug can be expressed as

M �u+ Cu̇+ R(q̇; q; t)= −Mr �ug (1)

where M and C are the well-known mass and damping matrices; R(q̇; q; t) represents the vector ofrestoring forces of the structure acting on the degrees of freedom u (Figure 1); q̇=Lu̇ and q=Lurepresent the vectors of deformation rates and deformations of the resisting planes, respectively,where L is the kinematic transformation matrix between u and q; and r is the in uence vector ofthe input. Each component of the restoring force vector R(q̇; q; t) is associated with a oor andmay be interpreted as the di�erence between the shears and torques of the stories above and belowthat oor. The restoring force vector R(q̇; q; t) can be computed from the forces f(q̇; q; t) in theresisting planes by

R(q̇; q; t)=LTf(q̇; q; t) (2)

The integration in time of Equation (1) leads to the true inelastic response of the structure u. Theonly di�culty in performing such integration is the computation of R(q̇; q; t) that requires techniquesand software that is not readily available in most engineering o�ces. Therefore, an alternative isto construct an approximation R̃(q̇; q; t) for the restoring force vector by using the SEM. ThusEquation (1) may be integrated by any of the well-known numerical integration procedures ofsecond order di�erential equations, such as Newmark’s method.The approximation R̃(q̇; q; t) is constructed from an assemblage of the restoring forces of the

macro-elements. For the linear response of the SEM, R̃(q̇; q; t)= (L̃Tk̃ L̃)u, where k̃ is the composite

block diagonal sti�ness matrix formed by the sti�ness matrices k̃ i of the individual macro-elements,and L̃ is the kinematic transformation between the degrees of freedom of the structure u and thedeformations q of the macro-elements. On the other hand, when some macro-elements are in theinelastic range, the general expression R̃(q̇; q; t)= L̃

Tf̃ for the restoring force vector is used, where

f̃ T = [f1 : : : fi : : : fn] is the macro-element force vector and fi= [−Vxi;−Vyi;−Ti; Vxi; Vyi; Ti]T is therestoring force vector of the ‘ith’ macro-element. It is the computation of these inelastic forcesthat requires the construction of the USST surfaces.

ULTIMATE STOREY-SHEAR AND TORQUE SURFACE

Before attempting to describe the more complex three-dimensional USST surface, a review of themost relevant properties of the two-dimensional surface for mono-symmetric structures is presentedby using the example described in Figure 2. The USST surface is de�ned as the locus of allstorey-shear and torque combinations that applied statically lead to a collapse mechanism of thestructure. Given the assumption of perfect plasticity in the resisting planes, the actual behaviourof the structure in the storey-shear and torque space is bounded by this surface. Because theimplementation of the SEM is based on the properties of these surfaces, they need to be studiedin detail.

Two-dimensional surface

Consider a single-storey system with three resisting planes in the Y-direction and two resistingplanes in the X-direction (Figure 2). Since the structure is mono-symmetric, its X-direction motion



Figure 2. Generic USST for a single-storey system with 2 and 3 resisting planes in the X- and Y-direction.

is uncoupled from the motions in the Y and �-directions. Hence, for a ground excitation in theY-direction, the USST surface may be represented in the two-dimensional force space (Vy; T ), i.e.Vx =0. The USST surface in this �gure has been constructed using the guidelines presented byde la Llera and Chopra [6]. To avoid an unnecessary duplication, the description presented nexthas been oriented toward an understanding of the most relevant features of the surface and itsproperties. This section will lead to a new, generally applicable, procedure to construct the surfacein either two or three dimensions.



In the two-dimensional case, the USST surface is formed by a number of linear segmentsforming a convex polygon. Each linear segment is associated with a speci�c collapse mechanism,for which typically one resisting plane remains elastic while all others are plastic. The in�nitepossible values of the shear force in this resisting plane within plus and minus its shear capacityf(o)y , lead to the di�erent (Vy; T ) pairs that de�ne the linear segment. This property can be derivedby noting that any variation in shear Vy and torque T results from the variation in shear forceof the only elastic plane. Thus, dVy =dfyi and dT =dfyixi, and thus the slope dT=dVy = xi. Thisresult implies, for instance, that linear segment 1-2 in Figure 2 has a slope dT=dVy = −b indicatingthat plane 1 remains elastic for the corresponding mechanism.Since the co-ordinate xi of the elastic plane is a geometric quantity, this condition is very

strong and has small uncertainty in practice. Therefore, when important yielding in the structure isobserved, the response histories of storey-shear and torque combinations will tend to follow suchlinear segments. An empirical result in a real instrumented structure that suggests this behaviourwas presented earlier [2].Another useful property derived from this result is that each segment has a projection on the

shear axis equal to the sum of the strengths of the resisting plane in both positive and negativeloading directions. This can be proved by noticing that the vertices that de�ne a linear segmentcorrespond to the case when the elastic plane just reaches yielding in either loading direction.Therefore, the variation in shear produced by a resisting plane is �Vy =f

(o)+yi −f(o)−yi = |f(o)+yi |+

|f(o)−yi |, where f(o)+yi and f(o)−yi are the ‘ith’ plane shear capacities in the positive and negativeloading directions, respectively. For instance, the projection along the shear axis of segment 7-8 is�Vy = |f(o)+y3 | + |f(o)−y3 |=2f(o)y3 , which implies that resisting plane 3 is stronger than the restof the planes.Two singularities of the USST surface, the vertical segments and the vertices, require further

interpretation. The vertical segments 8-1 and 4-5 in the example (Figure 2) represent mecha-nisms for which all Y-direction planes are yielding in the same direction. As an example, point 0(Figure 2) represents a purely translational mechanism with yielding in all the Y-direction planesand no force in the X-direction planes, leading to Vy0 =f

(o)y1 +f

(o)y2 +f

(o)y3 and T0 = b(f

(o)y3 −f(o)y1 ).

The quotient between this torque and shear is the well-known strength eccentricity of the plan, ep.As the shear and torque pairs travel from 0 to 1, the X-direction planes A and B develop elasticshear forces in opposite directions. At point 1 planes A and B reach yielding in opposite directionsand the torque resisted by the structure increases to T1 = b(f

(o)y3 − f(o)y1 ) + 2af

(o)xA . Because planes

A and B remain elastic for any base shear and torque combination between points 8 and 1, therotation of the plan ‘freezes’ at the onset of the mechanism and the structure displaces in puretranslation into the inelastic range.The kinematics of the plan at the onset of collapse can be better understood by using the concept

of instantaneous centre of plastic rotation (CPR). The CPR is de�ned as the point relative to whichthe inelastic rotation of the plan takes place once a mechanism has occurred. In the two dimensionalcase the CPR lies at a �nite number of points along the X-axis corresponding to the location of theY-direction planes. Indeed, the location of the CPR is the same as that of the resisting plane that re-mains elastic for a mechanism. In the particular case of the translational mechanism associated withpoint 0, it corresponds to a CPR located at x=−∞, since a translational mechanism of the plane inthe positive Y-direction corresponds to a positive rotation about minus in�nity. This same locationof the CPR is valid for all other shear and torque combinations corresponding to segment 0-1 (Fig-ure 2). For segment 1-2 the CPR jumps from x=−∞ to x= x1, i.e., the location of plane 1; it then



jumps to plane 2 for segment 2-3, and so on. It �nally reaches x=∞ for segment 4-0′. Thus, trav-elling along the surface from 0 to 0′ in a counterclockwise (positive) sense, is equivalent as shiftingthe CPR from x=−∞ to x=∞; the rest of the segments, 0′-5, 5-6, 6-7, 7-8, and 8-0 can beobtained by an identical shifting of the CPR but considering a clockwise (negative) plane rotation.It will be shown next that vertices correspond to a shift of the CPR from one location to

another. Suppose that in between resisting planes 1 and 2 there existed a large number of veryweak imaginary resisting planes. As it was shown above, segment 1-2 corresponds to a CPRlocated at plane 1 (x=−b) and segment 2-3 to a CPR located at plane 2 (x=0). As the CPRmoves away from plane 1 toward plane 2, it starts jumping sequentially into the imaginary resistingplanes. However, since the intermediate planes are very weak their contribution to the shear andtorque resisted by the structure is negligible and, hence, the same shear and torque as that of point2 is obtained. This enables us to interpret the vertex as a sequence of in�nitesimal linear segmentsthat lead to a transition of the slope of the USST surface at vertex 2 from −b (segment 1-2) to 0(segment 2-3). Therefore, if instead of having a sharp corner at the vertex of the USST we hada smooth round corner—corresponding to an in�nite number of resisting planes with negligiblestrength in between planes 1 and 2, the CPR could be precisely de�ned at any location in betweenthese two planes. Although a conceptual point, this interpretation of the vertex helps understandthe construction procedure of the USST developed next. Finally, it is important to note also thatat a vertex all resisting planes in the plan are yielding and unrestricted plastic ow occurs.By using the CPR, a new conceptual procedure to generate the two-dimensional USST surface,

di�erent from the pivot strategy followed earlier [6], is developed. It begins by using the trivialobservation that each linear segment is uniquely determined by two consecutive vertices and, hence,the need to compute only the vertices of the surface. As an example, consider the computationof linear segment 1-2 (Figure 2). A simpli�ed structural model of the building is presented inFigure 3. Because the linear segment 1-2 corresponds to mechanisms for which plane 1 is elastic,the two vertices de�ning this segment may be obtained as the CPR approaches plane 1 from theleft and right, respectively.To compute the shear and torque values at vertices 1 and 2, an in�nitesimal circle is drawn

around the intersection between plane 1 and a parallel to the X-axis. An in�nitesimal counterclock-wise rotation � of the plane relative to the CPR when it is located at the left of plane 1, largeenough to produce yielding in all planes, is indicated in Figure 3a. In this case, planes 1–3 yield inthe positive Y-direction (Figure 2) while planes A and B in opposite directions. The equilibrium atthe CM of the plan for this con�guration leads to Vy1 =f

(o)y1 +f

(o)y2 +f

(o)y3 and T1 = b(f

(o)y3 −f(o)y1 )+

2af(o)xA , which correspond to the co-ordinates of vertex 1. If the equilibrium is examined for allother possible locations of the CPR in the left hemisphere, the same result is obtained. Similarly,an in�nitesimal positive rotation of the plane when the CPR is at the right of plane 1 is shownin Figure 3b. The equilibrium in this case shows that Vy2 =−f(o)y1 +f

(o)y2 +f

(o)y3 and T2 = b(f

(o)y3 +

f(o)y1 ) + 2af(o)xA , are the co-ordinates of vertex 2. Notice that the only change occurs in the shear

force resisted by plane 1 (elastic plane) that goes from f(o)y1 to −f(o)y1 as the CPR shifts from theleft to the right hemisphere of the circle; the rest of the USST vertices are obtained analogously.The idea of drawing a circle around the intersection of plane 1 and the X-axis is to account

for all possible directions as the CPR approaches the elastic plane that de�nes the linear segmentof interest. In the two dimensional case it is intuitive that only the positions of the CPR at theleft and right of the elastic plane are of interest; in the 3D-case, however, four di�erent positionsof the CPR will be required to determine each face of the surface.



Figure 3. Computation of a USST surface for monosymmetric buildings using the CPR.

THREE-DIMENSIONAL SURFACE

Let us turn now into a more algebraic approach to investigate the properties of the 3D-USSTsurfaces; these properties will lead to a set of construction rules. The starting points are thethree equilibrium conditions that need to be satis�ed everywhere on the surface. If the externallyapplied lateral forces and torque are identi�ed by Vx, Vy, and T , respectively, the equations of



equilibrium are

Vx =nx∑i=1fxi

Vy =ny∑j=1fyj (3)

T =−nx∑i=1fxiyi +

ny∑j=1fyjxj

where fxi and fyj are the forces in the ‘ith’ and ‘jth’ resisting planes acting in the X andY-directions, respectively; xj and yi are the X and Y-co-ordinates of these planes, respectively;and nx and ny are the number of planes in the corresponding directions. Because of the assump-tion of perfect plasticity, and assuming for simplicity in the presentation that the capacity of theresisting planes in the positive and negative directions of loading is the same, the shear forcein each plane is limited by −f(o)xi 6fxi6f(o)xi , and a similar relation holds for the Y-directionplanes.In the three-dimensional case, a storey mechanism is formed with at most two resisting planes

remaining elastic. These planes will be denoted by p and q in the X- and Y-direction, respectively.Since all the other planes must be yielding, their forces are known in magnitude and can beintroduced in Equation (3). Their sign, however, depends on the kinematics of the mechanismat collapse that corresponds to a rotation about the CPR. Substitution of these force values inEquation (3) leads to

Vx =nx∑

i=1; 6=psgn(�xi)f

(o)xi + fxp

Vy =ny∑

j=1; 6=qsgn(�yj)f

(o)yj + fyq (4)

T =−nx∑

i=1; 6=psgn(�xi)f

(o)xi yi +

ny∑j=1; 6=q

sgn(�yj)f(o)yj xj − fxpyp + fyqxq

where the last equation may be recast in order to observe the functional dependence of torque andshear,

T =Vyxq − Vxyp −nx∑

i=1; 6=psgn(�xi)f

(o)xi (yi − yp) +

ny∑j=1; 6=q

sgn(�yj)f(o)yj (xj−xq) (5)

From Equation (5) it is concluded that the mechanisms associated with the two elastic planesdescribe a planar surface in the (Vx; Vy; T ) space. This property is identical to that of linear segmentsin the 2D-case. Moreover, the gradient of this �pq plane surface is

gpq= [yp − xq 1]T (6)



which is a function of the relevant distances between the two planes that remain elastic during themechanism and the CM. Because the inelastic rotation at imminent collapse takes place about thepoint of intersection of the two elastic planes, this intersection point coincides with the locationof the CPR. Further, the variation of the base torque T relative to the storey shears Vx and Vy canbe directly computed from Equation (5) as

@T@Vx

= − yp and@T@Vy

= xq (7)

This implies that the slopes of the planar face of the surface containing these mechanismscoincide, with the appropriate sign, with the co-ordinates of the point of intersection of the twoelastic resisting planes, or analogously, with the position of the CPR. Again, this is the extensionof the equivalent property stated earlier for the 2D-case. Equation (7) also implies that the steepest�pq planes will be those corresponding to yielding in the resisting planes farther from the CM.The plane �pq will be limited by linear edges and vertices forming the di�erent faces of

the surface. As in the 2D-case, these limits can be obtained now by taking all combinations ofpossible values for fxp and fyq in Equation (4). Since each resisting plane has two possible statesof yielding (positive or negative), there will always be four vertices in the face �pq, correspondingto all possible combinations (f(o)xp ; f

(o)yq ); (−f(o)xp ; f(o)yq ); (−f(o)xp ;−f(o)yq ), and (f(o)xp ;−f(o)yq ) for the

pair of forces fxp and fyq; denoted from now on as (+;+), (−;+), (−;−), (+;−).Let us compute the actual three-dimensional USST surface for the building example shown in

Figure 4(a). For this example, the face of the surface corresponding to the elastic behaviour ofplanes 1 and A has to be determined. Note that the independent action of these two planes maybe thought of as a double spring element located at their intersection, having a square shear-forceinteraction as shown in Figure 4(a). The four corners of this interaction surface correspond tothe force combinations (f;f), (−f;f), (−f;−f), and (f;−f). As it was done for the 2D-case,only the vertices are needed to de�ne the planar face �A1—four in this case. The vertices will bede�ned by simultaneous yielding of all resisting planes in the building.Among the many alternatives to compute these vertices, the richest in concept and interpretation

is the one that uses the CPR. According to the procedure already presented, a circle with in�nites-imal radius is drawn around the point of intersection of the two planes that remain elastic, A and1. Inscribed in the circle is the square interaction surface indicated above. As the CPR approachesthe point of intersection of planes A and 1 from any possible direction in the plan, di�erent planemechanisms may be generated. It turns out that for elements with no shear interaction as thosein this study, it is su�cient to consider the CPR located anywhere in the �rst, second, third, andfourth quadrants. This is so because for any position of the CPR in a speci�c quadrant, the sameyielding pattern of the resisting planes occurs. In particular, the same combination of yield forcesoccurs in resisting planes A and 1 and, hence, the same values are obtained for the resisted shearsand torque according to Equation (4).To verify this, shown in Figure 4(b) are arbitrary locations of the CPR in each quadrant and

the associated kinematic conditions of the building plan that lead to the four vertices of �A1.Indicated in the �gure are the in�nitesimal rotations of the plane about each location of the CPR.The positive displacement direction of the intersection point of the two elastic planes A and 1 isshown by the arrows. Thus, if the CPR is located in the �rst quadrant, a counterclockwise rotationof the plane will lead to a displacement that always has a component pointing toward the (f;−f)vertex of the square shear interaction. This implies that for this kinematic condition yielding occurs



Figure 4. Computation of a planar face of a 3D-USST surface using the concept of CPR.

in the fourth quadrant of this square interaction, and is associated with the combination (f;−f) offorces in planes A and 1, respectively. Similarly, when the CPR is located in the second quadrant,a counterclockwise rotation of the plane leads to positive yielding of both the planes A and 1, i.e.to the force combination (f;f), and so forth.Also shown in Figure 4(b) are the resulting vertices of face �A1. The shear forces and torque

corresponding to each of the vertices are computed directly by replacing in Equation (4) the pair



of values (fxp; fyq) by the force combinations (f;f); (−f;f); (−f;−f); (f;−f), while the shearvalues in the rest of the resisting planes remain invariant and consistent with a positive rotationof the plane about the CPR located in the intersection of planes A and 1. The other four-vertexleaning faces of the surface shown in the �gure are obtained by repeating this procedure for allother possible intersections of the resisting planes. The USST surface also has four vertical clos-ing faces parallel to the coordinate planes (Vx; T ) and (Vy; T ), denoted hereafter as vertical faces,that correspond to full yielding of all resisting planes in one direction; these vertical faces mayhave in general more than four vertices. They need not be obtained explicitly because they areformed by the edges of the already computed four-vertex faces. Nevertheless, they may be obtainedexplicitly as two-dimensional surfaces by considering that the planes in one direction are fullyyielding, either in the positive or negative direction. Note also that the general two-dimensionalUSST surface presented in Figure 2 corresponds to the section Vx =0 of the surface shown inFigure 3.Also shown in Figure 4 is the projection of �A1 on the (Vx; Vy) plane. The projection has two

interesting aspects. First, the projection of the edges of �A1 are parallel to the coordinate shearaxes, and second, the length of the projection of the edges is twice the shear capacity of theresisting planes. The latter observation coincides with the property described earlier for the two-dimensional surface. These two observations are just examples of the general properties of thethree-dimensional USST surfaces, which can be summarized as follows:

(1) The surface is convex and formed by planar faces bounded by linear edges. The surfacehas two types of faces according to their spatial orientation: leaning and vertical. Leaning faceshave four-vertices and correspond to building mechanisms with two resisting planes remainingelastic. On the other hand, vertical faces correspond to full yielding of all planes in either theX- or Y-direction—although not a necessary condition, these faces usually have more than fourvertices (multi-vertex). Furthermore, an edge of the surface corresponds to mechanisms for whicha single resisting plane is elastic and vertices to building mechanisms with all resisting planesyielding.(2) If the positive and negative shear capacity for each resisting plane is the same, the USST

surface is point symmetric with respect to the origin. The positive faces of the surface, torqueaxis coming out of the faces, correspond to equilibrium conditions of the plane associated withthe counterclockwise rotations about all possible locations of the CPR in the plane; its negativeface corresponds to clockwise rotations about the same CPRs.(3) The location of the CPR in the plane is the point of intersection of the resisting planes that

remain elastic for a given mechanism. For the co-ordinate system chosen, the change in torquerelative to a change in shear Vy is given by the position of the resisting plane that remains elasticin the Y-direction; similarly, the change in torque relative to a change in shear Vx corresponds tominus the position of the elastic plane in the X-direction.(4) Contiguous faces of the USST surface have a common elastic plane. Thus the connectivity

of four-vertex faces follows exactly the same pattern as the intersection of the resisting planes.(5) The edges of the leaning faces have projections on the (Vx; Vy) plane that are parallel to the

co-ordinate axes as a result of the lack of shear interaction and orthogonal arrangement of resistingplanes. Besides, faces may not have vertical edges parallel to the torque axis.(6) The projection length of the surface edges on the (Vx; Vy) plane is twice the capacity of the

element that remains elastic for the corresponding edge. This property can be used in conjunctionwith Equation (5) to generate from a known vertex to the other vertices of a face.



COMPUTATIONAL FORMULATION OF THE SEM

In this section the computational implementation of the force–deformation constitutive relationshipof the USST surface is discussed. A summary of the actual numerical construction procedure of thesurface is presented �rst. Then the integration of the elasto-plastic force–deformation constitutiverelationship is considered. Because the latter implies a rather cumbersome procedure that keepstrack of the load path along the USST surface, a pseudo-code algorithm has been included in theappendix in order to facilitate its understanding.Although, the conceptual aspects of the construction of the surface have been explained in detail

already, the algorithmic implementation presented next deserves special attention. The algorithmstarts by assuming that the shear capacities of all resisting planes have been already determinedby any procedure, say, pushover analysis or plastic analysis. It is �rst convenient to identify theresisting planes by numbers and the generated four-vertex faces by the notation �pq, where pand q represent as before the planes that remain elastic for that mechanism. Starting then at anyof the four vertices, the three other vertices are computed by simply adding or subtracting to thecorresponding shears of a vertex twice the element capacity, or the peak-to-peak strength in casethe yielding is not symmetric in the element (property 6 above). At each vertex Equation (5) isused to compute the associated torque. For instance, to compute the shear Vx at the (−;+) vertexstarting from the (+;+) vertex, twice the shear capacity of plane p is subtracted from the shearVx at vertex (+;+). This procedure is then repeated to compute the other vertices of the face, andfor the rest of the p and q planes.Certain bookkeeping is required to keep track of the spatial connectivity of the faces. The

connectivity pattern of planar faces becomes trivial when using the fourth property of the USSTsurfaces stated above. To see this, consider for instance the faces generated by the intersection ofplane A with all planes in the Y-direction, i.e. �A1;�A2; and �A3 (Figure 4(a)). Shown in Figure 5is the complete USST surface as an example. According to Equation (7), �A1 has to be the farthestplane from the origin towards the positive X- and Y-direction. Since the intersection of planes Aand 2 is the next to come as the CPR shifts to the right, face �A2 will be below �A1; similarly,�A2 will be above �A3. The easiest way to understand the pattern is to realize that by rotatingcounterclockwise ninety degrees the surface projections on the (Vx; Vy) plane, the indices pq forthe positive faces �pq are exactly in the same position as given by the intersections of resistingplanes in the building plan. Analogously, the pattern for the negative faces of the USST surface(torque axis coming into the surface) is such that if rotated ninety degrees clockwise, the sameresult as above is obtained. Finally, the four vertical faces of the USST surface are automaticallyde�ned by the edges of the leaning faces.The USST surface represents the inelastic force–deformation constitutive relationship of a macro-

element in a SEM. Once the surface is known, the problem is to compute the shear and torquevariation associated with a deformation increment of the building. This force variation is used inthe integration of the equations of motion of the building. If the macro-element is yielding, theresulting variation in shear and torque corresponding to a change in the element deformation fromuk to uk+1 may be expressed as

�Q=

�Vx�Vy�T

=

∫ uk+1

ukKep(Q) du (8)



Figure 5. Complete USST surface and projections for the building sample of Figure 4(a).

where Kep(Q) is the elasto-plastic operator, which depends on the load path. At the interval[k; k + 1], Equation (8) may be evaluated exactly due to the linearity of the faces and their �nitenumber. To accomplish such integration, the elasto-plastic operator Kep(Q) needs to be computedall over the surface, i.e. on an arbitrary planar face, along an edge, or in a vertex. On the other



hand, if the change in deformation imposed to the macro-element produces unloading, or themacro-element is elastic, the force variation is simply �Q=K(uk+1−uk), where K is the sti�nessmatrix of the macro-element.While the element is yielding, the elasto-plastic operator Kep(Q) is such that it imposes the

so-called consistency condition, or equivalently, that the computed shears and torques, Q + �Q;are on the USST surface. It is a well known fact from perfect plasticity theory with an associated ow rule that this operator adopts the following form [7]:

K(�pq)ep =K − K∇�pq∇�TpqK∇�TpqK∇�pq

(9)

where ∇�pq is in this case the gradient of face �pq of the USST surface (Figure 6(a)) and thesupraindex �pq is used to identify that the operator is related to yielding within this face. Theelasto-plastic operator K(�pq)ep in Equation (9) is rank de�cient since it satis�es the consistencycondition on �. Notice that ∇�pq is an eigenvector belonging to the null-space of K(�pq)ep , i.e.K(�pq)ep ∇�pq=0 (Equation (9)) and K(�pq)ep is of rank 2. Shown schematically in Figure 6(a) isthe action of K(�pq)ep on an arbitrary deformation vector �u.For the macro-element to have loading along the edge lq, generated by the intersection of planes

�pq and �qr of the USST surface (Figure 6(b)), the elasto-plastic operator K(‘q)ep must be of rank

1, with two eigenvectors in the null-space of K(‘q)ep corresponding to the gradients of the faces�pq and �qr . Since the operator K

(�pq)ep already has an eigenvector parallel to the gradient ∇�pq

of face �pq, the elasto-plastic operator for the edge lq may be computed by using recursivelyEquation (9) with K(�pq)ep in place of K, i.e.

K(lq)ep =K(�pq)ep − K(�pq)ep ∇�qr∇�TqrK(�pq)ep

∇�TqrK(�pq)ep ∇�qr(10)

Alternatively, K(lq)ep could have been obtained from the elasto-plastic operator K(�qr)ep and Equation(10) by switching the roles of ∇�pq and ∇�qr . As it should, both procedures lead to the identicaloperator K(lq)ep . That the operator K

(lq)ep satis�es the conditions K(lq)ep ∇�pq=K(lq)ep ∇�qr =0 may be

seen easily from the de�nition of Equations (9) and (10). Thus, ∇�pq and ∇�qr are eigenvectorsof the null space of K(lq)ep and, hence, this matrix is of rank 1. The load increment �Q fora given deformation increment �u is then �Q=K(lq)ep �u. This load increment is valid whenthe deformation increment �u has a non-zero component along the edge lq and the orthogonalcomponent is in between the normal of planes �pq and �qr . Otherwise, either loading occursalong these planes or the load increment �Q is zero. In the latter case, the whole deformationincrement is plastic.This last situation is similar to that occurring at the vertices. To better understand what happens

on a vertex of the USST surface, the simpler two-dimensional case is considered �rst. In thiscase, if the direction of the deformation increment is in between the sector described by thenormal vectors n̂1 and n̂2 (Figure 6(c), left), the loading increment �Q= 0, and the deformationincrement is entirely taken as a plastic deformation. This is consistent with the fact that at a vertexall resisting planes are yielding and the elasto-plastic operator Kep must be the zero matrix. Ananalogous situation occurs in the three-dimensional case. The elasto-plastic operator for the edge



Figure 6. De�nition of the elasto-plastic operators for a face, edge, and vertex of the USST surface.

K(lq)ep could now be used to derive the elasto-plastic operator for the vertex. To eliminate anyloading along the edge lq, this �nal eigenvector of K

(lq)ep is sent to its null space, thus leading to

the zero matrix. If the deformation increment is in between the pyramid formed by the normal ofthe planar faces shown in Figure 6(c) (right), yielding will occur at the vertex and �Q= 0. Assoon as the deformation increment �u contains a component outside this pyramid, loading willoccur along an edge or along any of the di�erent faces converging to the vertex.



As said before, the computational implementation of this constitutive relationship is a bit tricky.Because of that the algorithm used in this research is described in pseudo-code in the appendix.The code is self-explanatory and is based in the equations already described. The di�culty of thealgorithm is due to the veri�cation of all possible conditions for loading and unloading. However,once it works, such complexity is transparent to the user and the SEM model is extremely e�cientcomputationally. Such is the case because the number of inelastic elements is equal to the numberof the storeys of the building, which is usually very small compared to the true number of inelasticstructural members that would be used in a complete three-dimensional analysis of the building.

ACCURACY OF THE SEM MODEL

Numerical experimentation with the two-dimensional monosymmetric model developed earliershowed that for a large number of inelastic analyses the typical accuracy of the SEM modelis generally within 20 per cent error in peak displacements relative to the exact inelastic buildingresponse considering all structural elements [3]. Although depending on the problem considered,this accuracy of the SEM model may be considerably better, it is important to insist that themodel is not intended for �ne tuning of the inelastic behaviour of a structure—if such tuningwere possible given the large uncertainties present in a real-life structure subjected to earthquakemotions. Further, the trends of the inelastic response, which are of primary importance to identifyproblems in the planwise distribution of strength in the structure, are always well captured by theSEM. The results presented next show that a similar accuracy as for the 2D-case is obtained withthe three-dimensional model.Just as an example of the large number of validations generated to evaluate the accuracy of the

model, two single-storey structures with 3 and 5 resisting planes in the X- and Y-directions arepresented (Figure 7). The �rst building is mono-symmetric with a sti�ness eccentricity esx = 0:125band strength eccentricity epx = 0:214b for the earthquake motion in the Y-direction (Figure 7(a)).The second building is asymmetric in both principal directions of analysis with esx = 0:125b,epx = 0:214b; esy = 0:142b, and epy = 0:228b. To enable the comparison of both structures, the spa-tial distribution of resisting planes and their global sti�ness and strength is the same in both cases.The uncoupled lateral frequencies of both systems are 2Hz; the uncoupled torsional frequenciesare 2.60 and 2.63Hz, respectively.Shown in Figure 8 are the USST surfaces corresponding to the two buildings and their projec-

tions on the (Vx; T ); (Vy; T ), and (Vx; Vy) planes. The faces have been identi�ed in the plan as �pqfor a better understanding of the surface. As shown in the �gure, the only di�erence among thesurfaces is in the X-direction as a result of the strength eccentricity epy present in the asymmetricmodel. Several topological aspects of these surfaces can be checked using the USST propertiesdescribed earlier. For instance, the number of faces on either side is 15, i.e. the same as thenumber of possible locations of the CPR, or analogously, the number of intersections of resistingplanes. More important, the pattern of faces in the plan may be checked by following the simplerule described in the previous section.The inelastic response of the two buildings subjected to the two horizontal components of the

Sylmar record (1994) and El Centro record (1940) is shown in Figures 9 and 10, respectively. The�rst row of plots shows the history of displacements at the sti� and exible edges in both directionsof analysis. Solid and dotted lines correspond to the true and approximate response, respectively.In general, the trends of the inelastic displacement response of the buildings is followed reasonably



Figure 7. Building examples with lateral sti�ness KY = KX = 25:75 ton=cm, lateral capacitiesFY = FX = 0:262W = 42 ton, sides 2b = 16m, and 2a = 10m, and sti�ness and strength eccentricities:(a) esx = 0:125b, epx = 0:214b, esy = epy = 0, monosymmetric-plane; and (b) esx = 0:125b, epx = 0:214b,

esy = 0:1428a, epy = 0:228a, asymmetric-plane.

well by the SEM. The larger errors occur at the instant of the response when the building is intransition from the purely elastic to the purely plastic stage. Although not the subject of this study,the di�erence between the peak displacement demand of the sti� and exible edges of the plan inthis example is signi�cant, even though the buildings do not have a large eccentricity.Shown in the second row of plots in Figures 9 and 10 are the histories of base shear for the two

buildings and both records. Again, the results obtained with the SEM seem to follow quite wellthe true inelastic response of the structure. It is interesting to note that the episodes of yieldingare considerably longer for the Sylmar record and coincide with the instant at which deformationo�sets occur at the edges of the structure. The error involved in these shear histories is in general



Figure8.USSTsurfacesforbuildingexamplesofFigure7



Figure9.InelasticearthquakeresponseofthebuildingexamplessubjectedtoSylmargroundmotion(1994).



Figure10.InelasticearthquakeresponseofthebuildingexamplessubjectedtotheElCentrogroundmotion(1940).



smaller than the errors in the deformation demand. The reason is that the former is a globalresponse quantity of the structure which can be better approximated by the response of the SEMat the CM of the structure; the latter instead, is a local response that is not explicitly computedby the model.Finally, the histories of storey shear and torque are presented in the third row of plots of the

�gures. To better visualize the response, the positive and vertical faces of the USST surfaces havebeen made transparent. The exact and approximated response histories are presented on the left-and right-hand columns for each structure. In this �gure, the storey shears and torque combinationshave been identi�ed by di�erent markers corresponding to di�erent states of the structure. For theexact building model, the elastic, partially plastic, and fully plastic states are identi�ed, and forthe SEM, only the elastic and plastic state. The latter is divided into yielding along the leaningand vertical faces, respectively. A comparison of the traces of shear and torque pairs between theexact and SEM shows important similarities. Only a single comment about the inelastic buildingresponse. It is clearer for the Sylmar record, for which yielding is larger, that the inelastic responseof the building tends to occur at the vertical faces of the USST. This implies that all resisting planesyield simultaneously in one direction, and the torque resisted by the plane comes only from theresisting planes in the other direction. This result is important and states that the inelastic rotationsof the plane will tend to increase as a result of the uniform yielding of planes in one direction [6].

SEM USING LINEAR PROGRAMMING

It turns out that the force-deformation constitutive relationship of the USST surface can be eval-uated without ever constructing such a surface. This can be done exclusively by using linearprogramming techniques as brie y discussed in this section.The algorithm will be introduced by �rst assuming that the structure is elastic at instant tk = kT

and that the deformation increment �u(k + 1)= u(k + 1) − u(k) is enough to take the buildinginto a plastic state. Otherwise, the solution would be �Q=K(uk+1 − uk), which is independentof the USST surface. The problem that comes up immediately is how to compute the fractionof the deformation �u(k + 1) that will just take the structure to the USST surface without everconstructing the surface? The solution to this problem is equivalent to �nding the intersectionbetween the vector K(uk+1 − uk) and the USST surface. It can be stated as the following linearprogramming problemMaximize �; subjected to

�l̂ −Q = 0

Vx −nx∑i=1fxi = 0

Vy −ny∑j=1fyj = 0 (11)

T +nx∑i=1fxiyi −

ny∑j=1fyjxj = 0

−f(o)xi 6fxi6f(o)xi ; i=1; : : : ; nx

−f(o)yj 6fyj6f(o)yj ; j=1; : : : ; ny



where � represents the scalar factor (event factor) such that �K�u(k + 1) is just on the USSTsurface; l̂T = [�1 �2 �3] is a vector in the direction of K�u(k + 1); and QT = [Vx Vy T ] representsthe vector of shears and torque. The �rst equation in Equation (11) imposes the condition that� must satisfy in order to reach the USST surface, i.e. �nd the intersection. The following threeequations are nothing but the equilibrium stated before in Equation (3); the last nx + ny equationsare the shear capacity restrictions for the resisting planes. Although neither necessary nor e�cient,the USST surface could be generated point to point by blindly repeating the above optimizationfor all possible directions l̂ pointing outward from the origin at Q= 0 toward the surface.Equation (11) is standard and can be solved by any linear programming software such as the

one contained in the optimization tool box of Matlab [8]. Indeed, the solution of this equation isa very simple and robust step. For the algorithm described next, the simplex step (Equation (11))can be repeated several times during the same time step. To avoid confusion from now on, thevalue of � corresponding to the kth solution of Equation (11) will be denoted as �(k).The simplex solution of Equation (11) not only provides the values of � but also whether shear

capacity restrictions are active or not. This is done by looking at the vector of Lagrange multipliers.For instance, if the combinations of shear and torque lie on a face, two shear capacity restrictionsare not active; they indicate the planes that remain elastic for that mechanism. The intersectionof these two planes p and q de�nes the location of the CPR. Therefore, the simplex method tellswhat face �pq of the USST surface is active. By using Equation (6), the corresponding gradientof the face can be computed and used to evaluate the elasto-plastic operator K(�pq)ep (Equation (9)).From there on, the integration proceeds as described in the previous section and the �nal point inthe force space is Q(1)(k + 1)=Q(k) + �(0)K�u(k + 1) + (1− �(0)) K(�pq)ep �u(k + 1), where �(0)

represents the event factor computed by the �rst simplex step.If Q(k +1) is on the �pq face of the USST surface, the process stops and the time integration

of Equation (1) advances one step. To check if Q(k+1) is on the surface, Equation (11) is solvedagain using the direction l̂=Q(k + 1)=|Q(k + 1)|. If the new simplex solution indicates �(1) = 1,then the force vector Q(k +1) is on the surface and the same face �pq. Otherwise, Q(k +1) willbe outside the USST surface (�(1)¡1) and it will not correspond to a statically admissible forcecombination.If a central correction algorithm is employed [9], the simplex solution of Equation (11) in

the direction l̂=Q(k + 1)=|Q(k + 1)| will provide an estimate for the searched value, Q̃(k +1)= �(1)Q(1)(k + 1), and the time integration will proceed to the next step. Thus, a single stepin the evaluation of the force–deformation constitutive relationship of Equation (8) for a macro-element requires solving Equation (11) twice. This is an extremely inexpensive computationalprocedure; a pseudo-code of the linear programming algorithm with central correction developedin this study is presented in the appendix.At this point, the reader may wonder if the exact evaluation of Equation (8) can be done

by using linear programming. The answer is yes, but the computational cost increases consider-ably. Since the force Q(1)(k + 1) lies outside the USST surface, Equation (11) is solved againstarting from Q(0)(k + 1)=Q(k) + �(0)K�u(k + 1) and seeking the intersection of the direction

l̂=K(�pq)ep �u(k+1) with any of the edges of the �pq face; a more robust numerical alternative is to�nd the intersection with the edge by any iteration scheme and monitor the point were one of thetwo inactive restrictions for the face �pq becomes active. Any of these procedures will de-�ne a new simplex solution �(1) and it will indicate the shear capacity restriction that remainsinactive—the elastic resisting plane. Thus, the new force vector at the edge of �pq is given by



Q(1)(k + 1)=Q(0)(k + 1) + �(1)(1− �(0))K(�pq)ep �u(k + 1). Once on the edge, the integration maycontinue into the next face or go along the edge.Since �(1) will be greater than one, there is still a deformation increment, (1 − �(1))(1 −

�(0))�u(k+1), that has not been accounted for. In order to continue one more step with the inte-gration, the direction l̂=Q(1)(k +1) is slightly perturbed in order to identify the faces that de�nethe edge found. With that information, the three elasto-plastic operators K(�pq)ep ; K(�qr)ep , and K(lq)ep

are known. The �nal force vector will be Q(2)(k+1)=Q(1)(k+1)+(1−�(1))(1−�(0))K(∗)ep v(k+1),where K(∗)ep will be either K(�qr)ep or K(lq)ep depending on the load path followed. Usually, the newforce Q(2)(k + 1) will be on the USST surface. Although it is possible to continue with a newsimplex step for a new increment �(2), such a situation would be indicative of a large deforma-tion increment resulting probably from an excessively large time step used in the integration ofEquation (1). Although not presented here for brevity, it is important to mention that if the exactintegration of Equation (8) is implemented with linear programming, the same results are obtainedas those by using the USST surfaces.The accuracy of the linear programming algorithm with a central correction strategy is evaluated

in Figure 11. This �gure shows the earthquake response of the two structures presented earlierin Figure 7 subjected to the two horizontal ground motion components of the Corralitos (1990)and Sylmar (1994) records. The solid traces correspond to the earthquake response computed bythe exact integration of Equation (8) at each time step; the dashed traces are the result of linearprogramming with central correction. The di�erences between the responses for both structures andearthquakes are quite small, showing that the simpler linear programming algorithm with centralcorrection could be used instead of the exact integration procedure. Indeed, the relative error inpeak response using the central correction strategy is usually within 15 per cent.

BUILDING EXAMPLE

In this section a seven-storey reinforced concrete building is analysed using the SEM. The exactintegration procedure for the force–deformation constitutive relationship of the macro-elements isconsidered in the analysis. The building plan is presented schematically in Figure 12; material andstructural properties for the members may be found elsewhere [2]. With the only purpose of betterillustrating the performance of the SEM model, the building was subjected to the two horizontalcomponents of the Sylmar (1994) record instead of the actual motions recorded in the structureduring Northridge. The main objective of the example is to evaluate the computational performanceof the SEM for a real-life structure, not to interpret in detail its interesting inelastic response.The construction of the model followed similar steps to those presented for this building in an

earlier work [2] and is brie y discussed next. The �rst step was to determine according to ACI-95,the shear capacity of all laterally resistant structural members. This shear capacity was selectedas the minimum between the shear force necessary to develop a ductile exural-type mechanismor a brittle shear-type mechanism in the columns. The evaluation of the nominal bending andshear strength of columns was based on vertical loads obtained from a gravitational analysis ofthe structure. Because of the low shear strength in columns and the large sti�ness of beams, thestructure was assumed to behave as a shear building. As mentioned before, this is not essentialto the model and the designer could use, for instance, the shears associated with a strong-columnweak-girder design, those from a pushover analysis, or those from any other mechanism that is



Figure 11. Comparison between the inelastic building response computed by the exact and linear programingalgorithm with central correction: (a) monosymmetric structure subjected to the Corralitos ground motion;

and (b) asymmetric structure subjected to the Sylmar ground motion.



Figure 12. Schematic structural plan of the seven-storey R=C building considered.

kinematically possible. One of the most attractive features of the SEM, is that it can constructdi�erent structural models associated with di�erent types of earthquake-behaviour of the structure.The computed shear strengths in columns, f(o)xi and f(o)yj , were used to construct the USST

surfaces for each storey of the building. The resulting three-dimensional USST surfaces are shownin Figure 13. Because the structure is essentially symmetric, it is assumed, in order to make theplan behaviour of the structure more interesting, that resisting plane A is not present at the fourthand �fth storeys. Although this assumption is consistent with the true behaviour experienced by thecolumns in this plane during the Northridge earthquake, its purpose for the analysis is only that ofcreating an eccentricity in plan. The generated strength eccentricity is observed in the asymmetryof the (Vx,T ) projection of the USST surfaces at these storeys. The asymmetry is also veri�edby counting the number of faces of the USST surfaces in the (Vx,T ) and (Vy; T ) projections.Since the structure has 4 and 9 resisting planes in the X- and Y-directions, respectively, the samenumber of faces should be observed in the surface projections. As shown in the �gure, the facecorresponding to the mechanisms in plane A is missing from the fourth- and �fth-storey surfaces.The strength and sti�ness eccentricity of the structure at these levels will be enough to couple thelateral and rotational motions of the structure.Also shown in the third column of plots of Figure 13 is the earthquake response of the structure

in the shear and torque space. The results of the analysis show that the inelastic excursions ofthe structure are larger at the intermediate stories, four and �ve, as they should be, due to thelower strength of the structure in these storeys. Thus if plane A is damaged, or non-existent,damage will further concentrate on these storeys. Another observation obtained from these plots isthat most of the inelastic action in the building occurs on the vertical faces of the USST surface[6]; this is consistent with the inelastic behaviour observed in single-storey structures shown inFigures 9 and 10. This implies that all resisting planes in one direction are yielding in the samedirection and cannot resist torque. Consequently, if the structure is analysed for the e�ect of oneground motion component, the resisting planes in the direction orthogonal to this ground motioncomponent should not be included in the model.



Figure 13. Analytical USST surfaces, storey-shears, and torque histories for the buildingsubjected to the Sylmar ground motion.



Figure 14. Earthquake deformation histories for resisting planes A and D.

Finally, the deformation histories at the edge of the building plan are presented in Figure 14.Note the larger di�erences in deformation demand of planes A and D at the fourth and �fth storeys.This is due to the strength eccentricity present in these storeys. The deformation traces also showthe large deformation o�set that results in the �fth storey for this ground motion. Such o�set couldbe reduced essentially to zero if the USST surfaces in the fourth and �fth storeys are modi�edappropriately. It is concluded that, although an approximate procedure, the SEM provides a numer-ically robust structural model for estimating the main trends of the inelastic response of a structure.

OTHER APPLICATIONS

It seems interesting to mention brie y some other possible applications of the techniques developedhere. One case of interest is the analysis of the earthquake response of structures with frictionaldampers. The use of a concept such as the USST surface provides a practical tool for optimalplacement of the dampers in plan. Since the dampers may be ideally thought of as rigid-plastic,the usual assumption of perfect plasticity behind the USST surfaces is particularly appropriate in



this case. Similarly, the SEM may be used for analysis and pre-design of structures with lead-plugrubber and frictional isolators. In this case, a USST surface is constructed from the shear capacitiesof the lead-plugs or the mean frictional forces of all frictional isolators. In addition, since theearthquake behaviour of an isolated structure is controlled mainly by the behaviour of the isolationsystem, a model of the structure would typically include a single macro-element at the isolationinterface. The SEM provides a very useful tool for pre-design, selection, and placement of theisolators in plan, especially for structures that have torsional unbalance. Finally, due to the fractalnature of structural problems, the concept of the USST surfaces and the algorithms developed couldbe directly applied down one level to structural members. An important application is the modellingof the biaxial behaviour of columns and other related structural problems with force interactions.

SUMMARY AND CONCLUSIONS

The SEM proposed for approximate inelastic analysis of buildings leads to estimates of the buildingresponse that lie in most cases within 20% of the exact inelastic response of the structure. TheSEM is constructed from the elastic and inelastic properties of a building storey. The inelasticproperties are represented by the so-called USST surfaces that determine an upper bound forthe inelastic histories of shears and torque developed by the structure. The USST surfaces areformed by planar faces, each corresponding to a di�erent collapse mechanism of the storey. Thesemechanisms control the inelastic behaviour of the building plan. Results of the accuracy of themodel also show that the estimated histories of storey shears are usually more accurate than theestimated histories of deformation of the resisting planes. The numerical e�ciency of the SEMdeveloped was tested by computing the inelastic response of a seven-storey R=C building subjectedto the Sylmar ground motion record. These results, together with those of single-storey structures,suggest that the inelastic behaviour of a building will tend to occur on the four vertical faces ofthe USST surfaces, which are characterized by full yielding of the resisting planes in one direction.Under this condition of yielding, the structure has a reduced torsional strength. Although the SEMis capable of estimating the true inelastic behaviour of a building, it also provides, by means of theproperties of the USST surfaces, useful guidelines for locating optimally the strength in structures,especially in those with torsional unbalance.

ACKNOWLEDGEMENTS

This research has been sponsored by the Chilean National Research Fund for Science and Technology,Fondecyt, under Grant No. 1960358, the National Fund for the Foment of Science and Technology underGrant No. D96I1008, and the Fondecyt fund for the support of doctoral dissertations under Grant No. 2990069.The authors are grateful for this support. The authors would also like to acknowledge the useful commentsand improvements suggested by the anonymous reviewers of the paper.

APPENDIX A

This appendix describes the algorithms used for the integration of the force-deformation constitu-tive relationship of the USST. Two algorithms are presented: (1) exact integration using the USST,and (2) using linear programming and a central correction strategy. Each algorithm is presented



as a function whose input and output are the deformation increment �u and restoring force Qa,respectively.

A.1. Exact USST integration algorithm

[Qa; new state] = USST(Qo, old state, �u)state=‘old state’while ‖�u‖¿0

if state= ‘elastic’ {Qo⊂USST}Qa = Qo + K�uif Qa⊂USSTnew state= state; return % element remains elastic

else{compute intersection with USST: Qc ∈�pq where � = [�A1 ∪ · · · ∪�pq : : :]}state= ‘plastic face’

{event factor} ef= ‖Qc −Qo‖‖Qa −Qo‖ ; �u=�u(1− ef); Qo = Qc

{compute elasto-plastic operator}K(�pq)ep = K − K∇�pq∇�TpqK

∇�TpqK∇�pq% Equation (9)

end

elseif state= ‘plastic face’ {Qo ∈�pq}{check elastic unloading} if (Qo + K�u)⊂USSTstate= ‘elastic’

elseQa = Qo + K

(�pq)ep �u

if Qa ∈�pqstate new= state; return % element stays in �pq

else{compute intersection with edge: Qc ∈ lq = �pq ∩�qr}state= ‘plastic edge’

ef =‖Qc −Qo‖‖Qa −Qo‖ ; �u = �u(1− ef); Qo = Qc

{compute elasto-plastic operator}

K(lq)ep = K(�pq)ep − K(�pq)ep ∇�qr∇�TqrK(�pq)ep

∇�TqrK(�pq)ep ∇�qr% Equation (10)

endend

elseif state= ‘plastic edge’ {Qo ∈ lq}{check elastic unloading} if (Qo + K�u)⊂USSTstate= ‘elastic’



else{decide if loading is on face}[�vw; resp] = inquire face(Qo;�u;K

(�pq)ep ;K(�qr)ep )

if resp= ‘true’state= ‘plastic face’; K(�pq)ep = K(�vw)ep

elseQa= Qo + K

(lq)ep �u

if Qa ∈ lqstate new= state; return % element stays in lq

else{stop in vertex, K(v)ep = 0}Qc ≡ vk , where vk = lq ∩ lr ∩ · · ·state= ‘plastic vertex’

ef =‖Qc −Qo‖‖Qa −Qo‖ ; �u=�u(1− ef); Qo = Qc

endend

end

elseif state= ‘plastic vertex’ {Qo ≡ vk}{check elastic unloading} if (Qo + K�u)⊂USSTstate= ‘elastic’

else{decide if loading is on a face}[�vw; resp] = inquire face(Qo;�u;K

(�pq)ep ; : : : ;K(�tu)ep )

if resp= ‘true’state= ‘plastic face’; K(�pq)ep = K(�vw)ep

else{decide if loading is on an edge}[lv; resp] = inquire edge(Qo;�u;K

(lq)ep ; : : : ;K

(lu)ep )

if resp= ‘true’state= ‘plastic edge’; K(lq)ep =K

(lv)ep

elseQa= Qo; state new= state; return % element stays in vertex vk

endend

end

endend while

return

A.2. Linear programming and central correction strategy

[Qa, new state] =USST LP (Qo, old state, �u)



state= ‘old state’while ‖�u‖¿0

if state= ‘elastic’ {Qo⊂USST}Qa = Qo + K�u

l̂ =K�u‖K�u‖ {compute loading direction l̂}

[Qc;�x;�y] = lp(Qint = Qo; l̂; f(o)xi ; f

(o)yj ; yi; xj) {solve simplex Eq. (11)}

{where �x = [�x1 ; : : : ; �xnx ], and �y = [�y1 ; : : : ; �yny ] are the Lagrangians}ef=

‖Qc −Qo‖‖Qa −Qo‖ {compute event factor ⇒ � = ef}

if ef¿1new state= state; return% element remains elastic

elsestate= ‘plastic’�u = �u(1− evf); Qo = Qc{compute gradient using lagrangian information}p =�nd(�x 6= 0), q =�nd(�y 6= 0)if p = ∅ & q 6= ∅; {full yielding in X-direction (vertical face)}

∇�pq = ±[1; 0; 0]Telseif p 6= ∅ & q = ∅ ; {full yielding in Y-direction (vertical face)}

∇�pq = ±[0; 1; 0]Telse

∇�pq = [yp;−xq; 1]T % Equation (6)end{compute elasto-plastic operator}K(�pq)ep = K − K∇�pq∇�TpqK

∇�TpqK∇�pq% Equation (9)

end

elseif state= ‘plastic’ {Qo ∈�pq}{check elastic unloading using ow rule} if d� = ∇�TpqK�u

∇�TpqK∇�pq¡0

state= ‘elastic’elseQa=Qo + K

(�pq)ep �u

l̂o =Qa

‖Qa‖ {compute central correction direction l̂o}[Qa] = lp(Qint = 0; l̂o; f

(o)xi ; f

(o)yj ; yi; xj) {solve simplex Eq. (11) for central correction}

new state= state; return % element remains plasticend

endend whilereturn



REFERENCES

1. Blume J, Newmark N, Corning L. Design of Multistorey Reinforced Concrete Buildings for Earthquake Motions.Portland Cement Association: Skokie, Illinois, 1961.

2. de la Llera JC, Chopra AK, Evaluating seismic code provisions using strong-motion building records from the 1994Northridge earthquake. EERC Report No. 97/16, Earthquake Engineering Research Center, University of California atBerkeley, CA, 1998.

3. de la Llera JC, Chopra AK. A simpli�ed model for analysis and design of asymmetric-plan buildings. EarthquakeEngineering and Structural Dynamics, 1995; 24: 573–594.

4. Paulay T. Torsional mechanisms in ductile building systems. Earthquake Engineering and Structural Dynamics, 1998;27: 1101–1121.

5. de la Llera JC, Some fundamental aspects of torsionally coupled structures. Proceedings of the 12WCEE, Auckland,New Zealand (2000). To be printed.

6. de la Llera JC, Chopra AK, Accidental and natural torsion in earthquake response and design of buildings. Report No.94/07, Earthquake Engineering Research Center, University of California at Berkeley, CA, 1994.

7. Lubliner J. Plasticity Theory. Macmillan: New York, 1990.8. MATLAB, Reference Guide. The Mathworks, Inc: Natick, MA.9. Wilkins ML. Calculation of plastic ow. Methods of Computational Physics, vol. 3. Academic Press: New York, 1964.


A macro-element model for inelastic building analysis

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