behaviourofasymmetricbuildingsystemsunderamonotonicload—ii

1、<p>　　Engineering Structures, Vol. 18, No. 2, pp. 133-141, 1996 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved</p><p>　　0141-0296(95)00073-90141-0296/96 $15.00

2、 + 0.00</p><p>　　Behaviour of asymmetric building systems under a monotonic load—I</p><p>　　A. Ferhi and K. Z. Truman</p><p>　　Department of Civil Engineering, Washington University

3、, St Louis, MO 63130, USA (Received May 1994; revised version accepted October 1994)</p><p>　　Buildings subjected to a large intensity ground motion deform well into the inelastic range. In particular asymme

4、tric buildings undergo coupled inelastic lateral and torsional deformations that could be the governing factors in their design. The relationship between the system's asymmetry and its inelastic deformations is yet t

5、o be determined. This study defines the asymmetric building system and its key components namely the stiffness and strength eccentricities. The solution is mathematically</p><p>　　Keywords: asymmetric build

6、ings, inelastic behaviour, lateral- torsional coupling, stiffness eccentricity, strength eccentricity, biaxial bending</p><p>　　Introduction</p><p>　　It would be ideal if all buildings had tneir

7、 lateral-load resisting elements symmetrically arranged and earthquake ground motions would strike in known directions. But most buildings have some degree or irregularity in the geometric configuration or the distributi

8、on of mass, stiffness, and/or strength and earthquakes; do not strike in a predetermined direction. Due to one or more of these irregularities, the structure’s lateral resistance to the ground motion is usually torsional

9、ly unbalanced cr</p><p>　　A survey of previous studies of asymmetric building systems reveals a wide range of differences and contradictions in their conclusions. Typical differences are in the assessment

10、of the peak ductility demands1*3 and edge displacements1 -3 b^9. Most researchers agree that inelastic torsional coupling induces torsion and reduces lateral response but they differ in their assessments of the amount of

11、 reduction14 7-810. These differences point to the difficulties in studying the inelastic response</p><p>　　tribution is always constant. It was not until the late 1980s that the strength eccentricity was f

12、ormulated as a parameter that affects the behaviour of an inelastic system. Much of the research done before that did not consider the strength eccentricity as a parameter. Comparisons oi inelastic systems without a com

13、mon strength eccentricity are meaningless. In a dynamic study, the conclusions often reflect the choice of the loading function making it difficult to conduct direct comparisons. Of</p><p>　　As most researc

14、hers indicate, the inelastic response of asymmetric building systems is highly model dependent. In addition to other parameters, the inelastic response depends on the location, the number, and the type of elements. Howe

15、ver, most of the studies reviewed have used simpimed models that rail to preserve the inelastic characteristics of asymmetric buildings. These studies typically reduced the number of elements. Typically they considered e

16、lements that provide resistance only in one d</p><p>　　In this study of the behaviour of asymmetric building systems, the analytical models are a step closer to realistic structures. The lateral load resisti

17、ng elements are made representative of either wall-columns neglecting biaxial bending effects or columns exhibiting significant biaxial bending effects. The force-deformation relationship of the columns is trilinear w

18、ith distinct first yield and plastic limit points. This is a step beyond current practice which typically assumes an EPP force-</p><p>　　Objective and scope</p><p>　　Buildings subjected to a la

19、rge intensity ground motion deform well into the inelastic range. In particular asymmetric buildings undergo coupled inelastic lateral and torsional deformations that could be the governing factors in their design. The

20、relationship between the system’s asymmetry and its inelastic deformations is yet to be determined. This study is a detailed examination of the inelastic behaviour of two asymmetric building models loaded well into the i

21、nelastic range. The inelastic def</p><p>　　The asymmetric shear building is represented by two simplified yet general analytical models. They both consist of a homogeneous rigid diaphragm and four lateral lo

22、ad resisting elements located at the comers. For reference, the two models are called the wall-column model and the column model. As the names suggest, the wall-column model is laterally supported on wall-columns that p

23、rovide resistance in one principal direction where each wall-column is replaced by a single spring with an EPP forc</p><p>　　The asymmetric building problem</p><p>　　2.1. Definition</p>

24、<p>　　A structure is asymmetric if when subjected to a lateral force at its CM, it undergoes rotational as well as translational deformations. This behaviour is best known as the lateral-torsional coupling and is i

25、nfluenced by the stiffness eccentricity</p><p><b>　　n</b></p><p><b>　　2 ^hxi</b></p><p><b>　　=氣—⑴</b></p><p><b>　　/=i</b&

26、gt;</p><p>　　and by the strength eccentricity</p><p>　　EPy = -?——(2)</p><p><b>　　i=\</b></p><p>　　where, n represents the number of lateral load resisting

27、elements used in this study; kxi is the stiffness of member i in the x-direction; is the y location of member i measured with reference to the CM; Vpxi is the ultimate strength of member i in the ^-direction; Esy, also

28、known as rigidity eccentricity or simply eccentricity, is a measure of the offset between the CM and the centre of stiffness or CR; and Epy is a measure of the offset between the CM and the centre of strength or centre

29、 of</p><p>　　Other parameters that affect the severity of the torsionalcoupling are the torsional to lateral frequency ratio, the soil-structure interaction, and the torsional ground motion. In a static anal

30、ysis, the torsional to lateral frequency ratio is represented by the torsional to lateral stiffness ratio</p><p><b>　　n</b></p><p>　　E (取,■ + nkj</p><p>　　^ = ^ (3)<

31、/p><p><b>　　i=l</b></p><p>　　where, Ke is the system’s torsional stiffness formulated with respect to the CM, Kx is the system’s lateral stiffness in the ズ-direction, and X,- is the x l

32、ocation of member i measured with respect to the CM. It is assumed that there is no torsional ground motion and that the structure is fully attached to its rigid foundation.</p><p>　　2.2. Modelling</p>

33、<p>　　For simplicity, the two asymmetric building models studied are one-bay, one-storey frames. The diaphragm is assumed rigid and homogeneous with its CM coincident with its geometric centroid. The number n of

34、lateral-load resisting elements is equal to four with an element at each comer of the diaphragm. As shown in Figure 1，the models areanalytically represented by an equivalent spring model. The springs replacing the latera

35、l-load resisting elements are nonlinear with force-deformation relationsh</p><p>　　Linearly, the wall-column and column models are equivalent. Nonlinearly, the elements have identical uniaxial strength and

36、postyield stiffness. The purpose of this equivalency between the two models is not to make a direct comparison of their behaviours but to take advantage of the wall-column’s simplicity and use it to: (1) study the beha

37、viour of building systems supported on EPP wall columns; (2) relate the conclusions of this study to those of others that have used EPP wall-column element</p><p>　　(a) Wall-column model</p><p>

38、;　　(c) Equivalent spring model</p><p><b>　　し</b></p><p>　　Pu=(aab2)/(2L) : K0=(ab3E)/(L3) : 5j=(oL2)/{2bE)</p><p>　　Figure 2 Elastic-perfectly plastic and trilinear forc

39、e- deformation relationships for a column in shear</p><p>　　plify and explain the behaviour of the column model by making use of the more general closed form solution.</p><p>　　Parametric variat

40、ion</p><p>　　Desired levels of stiffness and/or strength eccentricities are achieved by adjusting the springs’ elastic stiffness and ultimate strength without changing the system’s remaining properties. The

41、 springs oriented along the loading direction have stiffnesses</p><p>　　kx, = ~(l ± r)(4)</p><p>　　and strengths</p><p>　　vpxi=^f{\ ±p)(5)</p><p>　　The spri

42、ngs perpendicular to the loading direction have stiffnessesin the loading direction, Vpv = 2j ルw is the total strength</p><p><b>　　/= 1</b></p><p>　　perpendicular to the loading dire

43、ction, r is the normalized stiffness eccentricity and is equal to the ratio of the stiffness eccentricity to the distance d (d is half the frame size measured perpendicular to the loading direction, see Figure 7), and p

44、 is the normalized strength eccentricity and is equal to the ratio of the strength eccentricity to the distance d. A plus sign is used for all members located on the positive y-axis side, known as the least-loaded-elemen

45、t (LLE) side, and a minus s</p><p>　　Analysis procedure</p><p>　　The wall-column asymmetric model with EPP springs is often simple enough that a closed form solution is obtainable. This is the

46、case of wall-column models having a zero stiffness eccentricity, zero strength eccentricity, or equal stiffness and strength eccentricities. Closed-form solutions were also obtainable for limit states namely the first yi

47、eld and the plastic limits. In order to provide solutions for the remainder of the asymmetric cases with any choice of stiffness and strength eccentri</p><p>　　The solution formulation assumes small displac

48、ements (i.e., sin 6 ~ 0 and cos 6 — 1.0) and uses the CM as a point of reference. The asymmetric problem in general has three in-plane degrees of freedom, two lateral displacements ux and uy and a rigid body rotation 6.

49、 The asymmetric models studied in this paper are asymmetric with respect to the x-axis and symmetric about the j-axis. They are subjected to a single lateral load in the ズ-direction. With these assumptions, the lateral

50、displacement o</p><p>　　Kt AD - A(2(8)</p><p><b>　　where</b></p><p>　　^, = ^(1 ± r)(6)</p><p>　　and strengths</p><p>　　Vpyi = ^f{\ ±

51、;p)(7)</p><p><b>　　4</b></p><p>　　Where, Kx=ム kxiis the total lateral stiffness in the loading</p><p><b>　　i=i</b></p><p><b>　　4</b>

52、;</p><p>　　direction, Ky = kyi is the total lateral stiffness perpendicu-</p><p><b>　　i=\</b></p><p><b>　　4</b></p><p>　　lar to the loading dire

53、ction, Vpx = 2j ^Pxi is the total strength</p><p><b>　　<=i</b></p><p>　　is the incremental applied load vector at the CM</p><p>　　is the incremental displacement vect

54、or at the CM</p><p>　　and Kt is the updated stiffness matrix of the structure with respect to the CM. Having computed the displacement vector at the CM, the displacements at the comer elements are obtained

55、using in-plane transformations. Using the updated local moment components and the axial force, the elements are checked for yielding. The load step is readjusted if necessary and this procedure is terminated when all ele

56、ments have reached their plastic limit.</p><p>　　The aforedescribed solution procedure is carried out in a computer program SIRABS. A specifically developed</p><p>　　program proved more effectiv

57、e than a commercially available computer program as the former is best suited for parametric studies. However, SIRABS was compared to a finite element solution using ABAQUS13. This comparison was used to verify the sim

58、plilied model and the computations of SIRABS. ABAQUS is capable of tracing the load path of the structure up to the full plasticity of the members. The lateral load resisting dements are full columns that are descretized

59、 to a finite element mesh which </p><p><b>　　i-iフ1</b></p><p>　　0.200:5\1:522.53</p><p>　　Lateral displ. at CM</p><p>　　Figure 5 CM lateral displac

60、ement and rigid body rotation for a system with r~ 0.4, p-0</p><p><b>　　1</b></p><p><b>　　0.8-</b></p><p>　　close. Not shown in these results are the drawbac

61、ks of using a commercial program like ABAQUS to do a specific research task. A postprocessing routine was developed in order to assist in the interpretation of ABAQUS’ results like tracking the degradation of stiffness a

62、nd computing the instantaneous location of the CR in the inelastic range. The CPU time required to solve one asymmetric case was on the order of several hours for an ABAQUS solution whereas it takes SIRABS only a few sec

63、onds.</p><p><b>　　Examples</b></p><p>　　In order to show the fundamental characteristics of the inelastic behaviour of asymmetric building systems, four distinct cases are studied. O

64、ne system has both its initial stiffness and strength eccentricities equal to zero and is called a symmetric system. A second system has no initial stiffness eccentricity but a strength eccentricity larger than zero ana

65、is called a stiffness symmetric system. A third system has no strength eccentricity but a stiffness eccentricity larger than zero and is </p><p>　　In these examples, the deformations and behaviours of the a

66、forementioned asymmetric building systems are compared. Loading history plots or rigid body rotations, lateral displacements at the CM, and the instantaneous location of the CR are presented. The oDjective of these exam

67、ples is to give a basic understanding to the various forms of building's asymmetry namely the stiffness asymmetry and the strength asymmetry. Also these examples will show in detail the path of travel of the CR as t

68、he el</p><p>　　Figure 7 shows the rigid body rotation versus the applied load for the four cases identmed previously using both models. The rotation is characterized by an elastic region, an inelastic region

69、, a first yield point, and a plastic limit point. The elastic deformations, as expected, are dependent on only the stiffness eccentricity and not on the strength eccentricity. In the inelastic regions, the two strength e

70、ccentric systems (r = 0, p = 0.4, and r = 0.4, p = 0.4) undergo</p><p>　　(b) Elements are wall-columns neglecting biaxial bending effects</p><p>　　Figure 7 Rigid body rotations for four asymmet

71、ric systems: r= 0, p = 0; r=0, p = 0.4; r= 0.4, p = 0.4, and r=0.4, p= 0</p><p>　　large inelastic rotations that are higher than the linear elastic level and increasing with applied loads. Whereas, the stre

72、ngth symmetric system (r = 0.4, p = 0) undergoes inelastic rotations decreasing with the applied loads. This preliminary example shows that the inelastic rotation depends a great deal on both strength and stiffness eccen

73、tricities. From a serviceability criterion, these examples show that the system with the lowest strength eccentricity is the most favourable because its i</p><p>　　A trace of the path of travel of the CR ex

74、plains the changes in the inelastic rotation. Figure 8 shows the instantaneous position of the CR for the four cases under study using the column and wall-column models. For a system with a 40% normalized initial stiffn

75、ess eccentricity (r = 0.4)，at first yield, the CR moves toward the LLE side if the normalized strength eccentricity is equal to 40% (p = 0.4)，and it moves toward the WLE side if the strength eccentricity is equal to zero

76、 (p = 0). In the f</p><p>　　(b) Elements arewal-columns neglecting biaxial bending effects</p><p>　　Figure8 Instantaneous centre of rigidity location for: r=0, p= 0; r=0, p = 0.4; r=Q.4f p= 0.4;

77、 r= 0.4, p = 0</p><p>　　actual or instantaneous stiffness eccentricity increases give rise to higher inelastic rotations whereas in the latter, the instantaneous stiffness eccentricity decreases and changes

78、sign resulting in reduced inelastic rotations. The knowledge of the instantaneous location of the CR is useful in these studies because it makes use of the elastic intuition in explaining some of the sudden and complex v

79、ariations in the inelastic deformations. As an example, the difference in ultimate rotations betw</p><p>　　The knowledge of the first-yield-loading capacity and its relationship to the stiffness and strength

80、 eccentricities is useful to the designer because, along with the plastic limit, it provides mm/her with information on the ductility of the system. The first yield is defined as the point where yield is initiated wherea

81、s the plastic limit is defined as the point at which the building model loses all of its lateral stiffness. The plots of Figure 7 show differences in the first yield deformations, </p><p>　　Figure 9 shows th

82、e CM lateral displacement as a function of the applied load. The two stiffness eccentric systems (r-0.4, p = 0 and r = 0.4, p = 0.4) have the same elastic lateral displacement, whereas the two stiffness symmetric system

83、s (r = 0, p = 0 and r = 0, /? = 0.4) undergo a slightly lower deformation. This difference is explained by recalling that the linear elastic lateral displacement at the CM is the superposition of a pure translation and a

84、 lateral displacement component resulting </p><p>　　(b) Elements are wad-columns neglecting biaxial bending effects</p><p>　　Figure 9 Lateral displacements at CM for four asymmetric systems: r

85、=0, p = 0; r=0r p = 0.4; r=0.4, p=0.4; r= 0.4, p = 0</p><p>　　placement of the stiffness eccentric systems. In the inelastic range, the four systems undergo displacements much larger than the linear elastic

86、levels. The plastic limit points show ultimate-load capacities that are mostly influenced by the magnitude of the strength eccentricity. For example, the two systems with a zero strength ecentricity have the ultimate-lo

87、ad capacities at or near the ultimate-load capacity of an equivalent symmetric system (r = p = 0). In both models, the plastic limit po</p><p>　　The wall-column and the column models are made equivalent. Th

88、ey differ only in their force—deformation relationships and the fact that the biaxial bending effects are significant in the column model and are negligible in the wall- column model. With this in mind, the wall-column

89、 model will serve to explain some of the column model’s complex behaviour and simplify its results. Figure 9b shows the CM lateral displacement versus the applied load for a wall-column model. The lateral stiffness of&l

90、t;/p><p>　　Conclusions</p><p>　　To summarize, the results illustrated the key effects of stiffness and strength eccentricities by comparing the behaviour of a symmetric system, an initial-stiffnes

91、s symmetric system, a strength symmetric system, and a system with equal eccentricities. As a verification, the elastic deformations are dependent on only the stiffness eccentricity. The inelastic aeformations, on the o

92、ther hand, were strongly dependent on both stiffness and strength eccentricities. This is shown in the wide range o</p><p>　　References</p><p>　　Tso, W. K. and Sadek, A. W. ‘Inelastic seismic re

93、sponse of simple eccentric structures’，Earthquake Engng Struct. Dyn. 1985, 13, 255-269</p><p>　　Irvine, H. M. and Kountouris, G. E. ‘Inelastic seismic response of a torsionally unbalanced single-story buildi

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100、th eccentricity concept for inelastic analysis of asymmetric structures% Proc. Ninth World Conf. on Earthquake Engineering, Vol. 7，2-9 August 1988，Tokyo-Kyoto, Japan</p><p>　　Chandler, A. M. and Duan, X. N.