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2014 Elsevier Ltd. All rights reserved.

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Journal of Constructional Steel Research 99 (2014) 7284

Contents lists available at ScienceDirect

Journal of Constructiomore recent studies have worked on new types of SPSW systems, past, particularly before the existence of codied design requirementsfor SPSWs, dealt with systems not meeting the capacity design require-ments and therefore not necessarily having a desirable mode of failure.shear panels as dampers [1012], as alternatives to conventionalSPSWs to improve the dissipation capacity of the system. However,

design level earthquakes will be satised [21]. Noperimental and analytical research studies perfoload resisting framing.With these advantages, this systemhas attractedmany research activities throughout the world. Many researchers havefocused their research on the discovery of the behavior of SPSWs[15], while others have proposed the use of light-gauge [6] or lowyield point (LYP) SPSWs [7,8], SPSWs with slits [9] or special metal

pendence of the angle of the tension elds on the cross-sectional prop-erties of the surroundingmembers and the inll plate thickness), resultsof recent research have shown that if the SPSWs are designed accordingto the code recommendations, the desired sequence of yielding will beachieved [20] and maximum interstory drift requirements considering Corresponding author. Tel.: +98 171 444 1002; fax:E-mail addresses: ahosseinzade@gmail.com (S.A.A. Ho

(M. Tehranizadeh).1 Tel.: +98 21 6454 3030; fax: +98 21 6454 3037.

http://dx.doi.org/10.1016/j.jcsr.2014.04.0040143-974X/ 2014 Elsevier Ltd. All rights reserved.ting frames, SPSWs havete fast construction, andfor fewer bays of lateral

resulting from both tension eld yielding of the inll plates and exuralyielding of the HBE ends. Although, the suggested design procedurescan be an iterative and time-consuming process (due in part to the de-fewer costly detailing requirements, facilitahave high strength and ductility that allow1. Introduction

Steel plate shearwall (SPSW) systemlateral load resisting systems currentlyThe SPSW system comprises a steel frthat are allowed to buckle in shear anunder lateral loading. Past studies haveexemplary seismic performance. In comeral load resisting systems, such as reinious types of braced frames and momene of themost promisingle to structural engineers.ith thin steel inll plateselop tension eld actionn that SPSWs can exhibitn with conventional lat-concrete shear walls, var-

such as semi-supported [13,14] and self-centering [15] steel shearwalls, to further increase the efciency of the system.

Design clauses for design of SPSWs were provided rst in CAN/CSA16-01 [16] and then in FEMA 450 [17], AISC 341 [18] and AISC DesignGuide 20 [19]. To ensure a ductile and desirable behavior, the currentcodes require capacity design of SPSWs. Capacity design implies thatthe horizontal boundary elements (HBEs)must be designed to resist de-mands resulting from tension eld yielding of the inll plates, and thevertical boundary elements (VBEs) must be designed to resist demandsBehavioral characteristics of code designed

S.A.A. Hosseinzadeh a,, Mohsen Tehranizadeh b,1

a Department of Civil Engineering, Faculty of Engineering, Golestan University, Al-Ghadir Ave, Gb Department of Civil Engineering, Amirkabir University of Technology, 424 Hafez Ave, Tehran

a b s t r a c ta r t i c l e i n f o

Article history:Received 10 June 2013Accepted 2 April 2014Available online 4 May 2014

Keywords:Shear wallsWallframe contributionInteractionStrengthDuctilityStiffness

A series of code designed stenumerically analyzed to invethe wall/frame contributionfew of lower stories and in80% of the compressive columore effective in resisting bshear force of the frame coSPSWs due to early bucklingboundary frames of differentof the current design procedutility of SPSWs having almos+98 171 444 1003.sseinzadeh), dtehz@yahoo.comteel plate shear wall systems

stan 49188-88369, Iran75-4413, Iran

late shear walls (SPSWs) with different aspect ratios and number of stories areate different aspects of the behavior of such SPSWs, particularly with regard toesults show that frames contribute effectively in resisting story shear only at alates absorb substantial part of story shear at the remaining stories. About 70axial force comes from plate tension elds. The tensile column is found to beshear than the compressive one and it contributes about 5595% of the totaln bases at the ultimate state. Up to 32% reduction in the overall stiffness oftheir inll plates is observed. The rst yield points in the inll walls and in theWs occur at about 2545% and 7085% of their strength, respectively. As a resultthat neglects the boundary framemoment resisting action, the stiffness and duc-e same design lateral loads but different aspect ratios can be quite different.

nal Steel ResearchHence, the ndings from those studies, although valuable, are not nec-essarily valid for all SPSW systems, especially for those designed per de-sign code.

In general, for a given panel geometry, inll plate thickness andma-terial properties, the behavior of a SPSW system depending on the

cross-sectional properties of its boundarymembers can be controlled bytwo general types of failure mechanisms, namely brittle and ductile. Alist of possible failure mechanisms of typical SPSWs in the order oftheir desirability was provided by Astaneh-Asl [5]. It is generally accept-ed that the tension yielding in the inll plate, occurring under the actionof story shear, should be considered as the primarymode of energy dis-sipation of SPSWs. Again, for the full-tension yielding of the inll plates,the boundary frame members should have adequate strength and stiff-ness. However, if the frame member sections are selected from weakerproles, the system behavior is primarily governed by an undesirable orless desirable failuremode rather than a ductile or desirable one and thefull-tension yielding of inll plates would not be realized even at theultimate state. Several experimental investigations have conrmed theabove discussion [3,2224]. In some of those tests excessive deforma-tion, premature yielding and/or buckling of boundary elements limitedthe strength and ductility of the SPSW systems.

The overall behavior of a SPSW comprises the contributions of theinll wall and boundary frame actions. Hence, separation of the contri-butions of inll wall tension eld action and boundary frame momentresisting action to the overall behavior, in addition to studying the over-all behavior of the system, provides a better insight into the systembehavior. It is to be noted that there is an interaction effect betweenthe inll wall and the boundary frame, which is too complicated to bedened by a closed form solution. Nevertheless, in order to accuratelypredict the overall behavior of the SPSWbased on the discrete behaviorsof the inll wall and the frame, or to separate the wall and frame

interstory drift ratios for different SPSWs. The results show that in all

73S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284responses from the overall response of the SPSW, the effect of the inter-action must be taken into account somehow.

The purpose of this research is to investigate different aspects of thebehavior of code designed SPSWs, particularly with regard to the rela-tive or respective contributions of their inll walls and boundary framesto the overall behavior. To accomplish this, a series of SPSWswith differ-ent aspect ratios and number of stories, designed per design code [18,19], are analyzed using the nite element method and the obtained re-sults are utilized to investigate: (a) wallframe contribution shares ofstory shears, (b) wallframe contribution shares of the VBE axial forces,(c) comparison of the VBE axial and shear forces, (d) overall stiffnessand ductility, and the contributions from inll walls and frames,(e) base shear levels associated with the rst yielding of walls andFig. 1. Typical plan and considered SPSW.designs, the maximum interstory drifts are relatively low (lower than0.01 h). This indicates that all designs are governed by strength andframes, and (f) inuence of the SPSW aspect ratio and number ofstory on the above.

2. Method of the study

2.1. Design of models

A number of SPSW systems having different aspect ratios and num-ber of stories are considered for this research. SPSWs are designed for atypical building plan (Fig. 1). The buildings considered are 1, 2, 4, 6, 8,10, 12 and 15 stories tall and have uniform story heights of 3.40 m.SPSWs are designed according to the recommendations given in AISCSeismic Provisions [18] and AISC Design Guide 20 [19]. The perimetergravity frames without shear walls are assumed to have pinned beamto column connections and therefore, they are not incorporated in de-sign and analysis. However, gravity loads transmitted by perimeterframe beams to SPSW beamcolumn connections, are considered inthe design and analysis.

All models have beam-to-column connection details that include re-duced beam sections (RBS) at each end, as recommended by the AISCDesign Guide 20 [19], to ensure inelastic beam action at the desired lo-cations. Also, the use of the RBS is reasonable considering the followingtwo properties. First, the exural force in the VBE due to HBE hinging istypically greater than that due to plate tension. In such cases, the exureaway from the connection does not govern the design of the VBE. Sec-ond, the required HBE exural strength is governed by exure in themid-span due to plate tension (in combinationwith gravity load effects,if any), not at the ends. Based on these two properties, it is convenient touse a RBS in the HBE to limit the required exural strength of the VBE.Moreover, the RBS reduces the demand on the VBE when applying thestrong columnweak beam requirement. However, special concernmust be paid to the design of HBEs, particularly for intermediate oneshaving RBS connections, since recent research [25] has shown that thecurrent design approach does not necessarily lead to a HBEwith the ex-pected performance. To reliably achieve capacity design, analyticalmodels for estimating the design forces for intermediate HBEs havebeen proposed by researchers [26].

A dead load of 4.6 kPa is used for each oor and 3.2 kPa for the roof.Live loads are taken equal to 2.4 kPa for each oor and 0.96 kPa for theroof. According to the code-compliant range of aspect ratios, the baywidths (L), measured from center to center of VBEs, are assumed tovary from 2.9 to 8.5 m (i.e. L/h = 0.85, 1.4, 2 and 2.5). Inll plate thick-nesses are designed to resist the entire story shear per [18]. Plate thick-nesses are selected from those available in ASTM A36 steel [19]. Duringthe design, the minimum practical inll plate thickness requiredfor handling and welding considerations is considered to be 3.18 mm(1/8 in.). Note that the inll plate, however, will often have someoverstrength (i.e. the specied plate thickness will be thicker than re-quired by design) not only due to the consideration of the minimumpractical plate thickness but also due the fact that steel plates are avail-able in discrete thicknesses on the market. The boundary frame mem-bers are designed using the capacity design principles to resist theforces from inll plate yielding. The resulting plate thicknesses andmember sizes are shown in Table 1. Table 2 presents the RBS connectiondimensions (see Fig. 2) for different HBE proles per AISC 358-05 [27].Throughout the article, each model will be identied by the value ofSPSW aspect ratio (L/h) and number of story (n).

Finally, in order to check that drift requirements are met, the de-signed SPSW systems in Table 1 are numerically analyzed under the de-sign level earthquake forces as determined by ASCE 7-05 [28]. Table 3presents the design base shears and the corresponding maximumnot by drift limitations. However, as can be inferred from the results

Table 1Inll plate thicknesses and frame member sizes of different SPSW models at different stories.

Case # of stories,n

Bay width, L(m)

Aspect ratio,L/h

Plate thickness, tw (mm) HBE size VBE size

Intermediate Top

1 1 2.9 0.85 3.18 (1st) W14 132 W14 2112 1 4.8 1.4 3.18 (1st) W14 233 W14 3113 1 6.8 2.0 3.18 (1st) W27 194 W14 3984 1 8.5 2.5 3.18 (1st) W40 211 W36 3305 2 2.9 0.85 4.76 (1st), 3.18 (2nd) W14 132 W14 132 W14 3116 2 4.8 1.4 3.18 (1st, 2nd) W14 132 W14 233 W14 3117 2 6.8 2.0 3.18 (1st, 2nd) W14 132 W27 194 W14 3988 2 8.5 2.5 3.18 (1st, 2nd) W14 132 W40 211 W36 3309 4 2.9 0.85 4.76 (1st, 2nd), 3.18 (3rd, 4th) W14 132 W14 132 W14 37010 4 4.8 1.4 3.18 (1st4th) W14 132 W14 233 W14 31111 4 6.8 2.0 3.18 (1st4th) W14 132 W27 194 W14 39812 4 8.5 2.5 3.18 (1st4th) W14 132 W40 211 W36 33013 6 2.9 0.85 7.94 (1st, 2nd), 6.35 (3rd, 4th), 4.76 (5th), 3.18 (6th) W14 132 W14 132 W14 605 (1st4th), W14 311 (5th, 6th)14 6 4.8 1.4 4.76 (1st4th), 3.18 (5th, 6th) W14 132 W14 233 W14 500 (1st4th), W14 311 (5th, 6th)15 6 6.8 2.0 3.42 (1st, 2nd), 3.18 (3rd6th) W14 176 W27 194 W14 500 (1st4th), W14 398 (5th, 6th)16 6 8.5 2.5 3.18 (1st6th) W14 132 W40 211 W36 361 (1st4th), W36 330 (5th, 6th)17 8 2.9 0.85 12.7 (1st, 2nd), 11.1 (3rd), 9.53 (4th), 7.94 (5th), 6.35 (6th), 4.76 (7th), 3.18 (8th) W14 132 W14 132 BUILT UP1a (1st4th), W14 455 (5th8th)18 8 4.8 1.4 6.35 (1st4th), 4.76 (5th, 6th), 3.18 (7th, 8th) W14 132 W14 233 W14 730 (1st4th), W14 398 (5th8th)19 8 6.8 2.0 4.76 (1st4th), 3.18 (5th8th) W14 176 W27 194 W14 730 (1st4th), W14 398 (5th8th)20 8 8.5 2.5 4.76 (1st4th), 3.18 (5th8th) W14 233 W40 211 W36 652 (1st4th), W36 330 (5th8th)21 10 2.9 0.85 12.7 + 3.18 (1st3rd), 12.7 (4th, 5th), 11.1 (6th), 9.53 (7th), 7.94 (8th), 4.76 (9th), 3.18 (10th) W14 176 W14 176 BUILT UP2b (1st5th), W14 730 (6th10th)22 10 4.8 1.4 9.53 (1st, 2nd), 7.94 (3rd5th), 6.35 (6th, 7th), 4.76 (8th), 3.18 (9th, 10th) W14 132 W14 233 W36 800 (1st5th), W36 487 (6th10th)23 10 6.8 2.0 7.94 (1st, 2nd), 6.35 (3rd6th), 4.76 (7th, 8th), 3.18 (9th, 10th) W14 176 W27 194 W36 800 (1st5th), W36 487 (6th10th)24 10 8.5 2.5 6.35 (1st4th), 4.76 (5th7th), 3.18 (8th10th) W14 233 W40 211 W36 652 (1st5th), W36 487 (6th10th)25 12 4.8 1.4 11.1 (1st4th), 9.53 (5th, 6th), 7.94 (7th, 8th), 6.35 (9th), 4.76 (10th), 3.42 (11th), 3.18 (12th) W14 132 W14 233 BUILT UP3c (1st6th), W36 652 (7th12th)26 12 6.8 2.0 7.94 (1st5th), 6.35 (6th8th), 4.76 (9th, 10th), 3.18 (11th, 12th) W14 176 W27 194 BUILT UP4d (1st6th), W36 652 (7th12th)27 15 4.8 1.4 12.7 + 1.59 (1st5th), 12.7 (6th, 7th), 11.1 (8th, 9th), 9.53 (10th, 11th), 7.94 (12th), 6.35 (13th),

4.76 (14th), 3.18 (15th)W14 132 W14 233 BUILT UP5e (1st8th), W36 800 (9th15th)

28 15 6.8 2.0 11.1 (1st4th), 9.53 (5th7th), 7.94 (8th10th), 6.35 (11th), 4.76 (12th, 13th), 3.18 (14th, 15th) W14 176 W27 194 BUILT UP6f (1st8th), W36 652 (9th15th)

a BUILT UP1: section depth = 550 mm, ange width = 500 mm, ange thickness = 160 mm, web thickness = 100 mm.b BUILT UP2: section depth = 650 mm, ange width = 600 mm, ange thickness = 180 mm, web thickness = 120 mm.c BUILT UP3: section depth = 1150 mm, ange width = 550 mm, ange thickness = 140 mm, web thickness = 80 mm.d BUILT UP4: section depth = 1150 mm, ange width = 500 mm, ange thickness = 120 mm, web thickness = 70 mm.e BUILT UP5: section depth = 1150 mm, ange width = 700 mm, ange thickness = 180 mm, web thickness = 100 mm.f BUILT UP6: section depth = 1150 mm, ange width = 600 mm, ange thickness = 160 mm, web thickness = 90 mm.

74S.A.A.H

osseinzadeh,M.Tehranizadeh

/JournalofConstructionalSteelResearch99

(2014)7284

in Table 3, for taller (over 15-story) SPSWs, especially for those havinglow aspect ratios, the drift may control the design.

2.2. Description of nite element models

Using a detailed FE modeling approach similar to that used by theauthors in the previous work [29], SPSWs are modeled and analyzed.The FEmodeling approach's adequacy for representing the pushover re-sponse of SPSWs has already been veried through comparisons withexperimental results [29] and thus, is not repeated here. However, forcompleteness and for convenience, a brief description of the nite ele-ment modeling is presented herein.

The ASTM-A36 and ASTM-A572 conventional structural steel stan-dards are, respectively, selected for inll wall and framemember mate-rials. In order to study the nonlinear behaviors of inll walls and frame

Table 2RBS connection dimensions for different HBE proles per AISC 358-05.

RBSdimensions

W14 132 W14 176 W14 233 W27 194 W40 211

a (mm) 200 200 220 200 150b (mm) 300 300 330 600 800c (mm) 90 95 100 85 70

75S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284members, respective stressstrain diagrams are adapted from [20].The nite element program ABAQUS/Standard [30] is utilized for all

eigen-value and incremental nonlinear pushover analyses. All framemembers and inll plates are modeled with a reasonably ne meshusing the four-noded S4R element, a general purpose shell elementwith reduced integration. The element size is selected from a mesh re-nement study. The geometric nonlinearity phenomenon is includedas a result of large displacements with small strains. An isotropic hard-ening model is used for all nonlinear pushover analyses.

The inll plates are considered to be connected directly to theboundary members. The rst-story inll panel is assumed to be an-chored to the ground rather than to an anchor beam at that level andits nodes at the base are xed in displacement. Similarly, to simulatethe x support conditions at the VBE bases, the bottom nodes of bothFig. 2. Typical SPSW system.VBEanges andwebs are restrained fromdisplacement in all directions.In order to replicate the effects of the concrete slab of the oors, all HBEwebs are also restrained against movement in the out-of-planedirection.

Initial imperfections need to be considered in the nite elementmodels. To ensure satisfactory results, using imperfections based onthe rst eigenmode together with the maximum anticipated amplitudevalue based on the consideration of fabrication procedures was recom-mended by some FE codes [31]. However, in the case of typical slenderplates, the choice of imperfection amplitude and shape does not inu-ence signicantly the overall behavior of the system [32,33]. Based onthe above discussion, an initial imperfection pattern corresponding tothe rst buckling mode of each inll plate with peak magnitude of 1mm is applied in themodels. Preliminary analyses veried the sufcien-cy of the considered value. Gravity loads are applied to SPSW beamcolumn connections in order to include P-delta effects, although P-delta effects are not important for SPSWs that are stiff enough to meetthe code drift limit [34]. Lateral loads, as shown in Fig. 2, are equally ap-plied to the exterior nodes of panel zones on either side of each storybeam and are gradually increased from zero to a magnitude beyondthe system's capacity. The ultimate state of SPSWs, per ASCE 7-05 [28],is considered to occur when the drift ratio at least at one of the storiesof the system reaches a value of 2.5%.

3. Discussion of results

3.1. General behavior

In this section, characteristics of the general behavior of SPSWs arediscussed. As such, SPSWs are classied into two groups, namelylow-rise and high-rise, according to the behaviors of their inllwalls and frames. Only SPSWs having practical dimensions (i.e. 1 b L/h 2) are considered here. However, such classication, in general,depends on both the SPSW aspect ratio and height.

Typical and idealized behaviors of low-rise (1- or 2-story) and high-rise (10-story or taller) SPSWs are depicted in Fig. 3. The behavior of amedium-rise SPSW may be closer to a low-rise SPSW or a high-riseSPSW, depending on the number of story and aspect ratio, and thus isnot discussed here separately. In Fig. 3(a) and (b), the absorbed baseshears by the inll wall and frame columnswere individually calculatedby summation of the reaction forces at the base nodes of each one in thein-plane direction. In thisway, the effect of thewallframe interaction isimplicitly considered in the behaviors of the inll wall and the frameand therefore, the overall base sheardisplacement curve of the systemis equal to the summation of the inll wall and frame base sheardisplacement curves (that is, the principle of superposition applies). InFig. 3(c) and (d), it was assumed that the idealized behaviors of the inllwall and the frame are elasticperfectly plastic and that the principle ofsuperposition is applicable.

Fig. 3(c) shows that in a low-rise SPSW, the inll wall has a relativelyhigher stiffness than the frame in the early stages of loading and absorbsa higher portion of story shear. As the inll wall reaches its yieldedstrength (point (yw, Pw)), system stiffness and additional strength areonly provided by moment frame action. Thereafter, at = yf, theframe reaches its yielded strength (Pf). At this point, the frame losesits stiffness and does not absorb any additional shear force up to theultimate state. Eventually, the frame exhibits greater strength than theinll wall at the ultimate state. Comparing Fig. 3(c) and (d), it is ob-served that the frame of a high-rise SPSW acts like the inll wall of alow-rise SPSW and vice versa. This can be attributed to the increasedbending effects in a high-rise SPSW, which in turn affect the effective-ness of the inll wall and cause signicant yielding of the inll wall tohappen after signicant yielding of the frame (i.e. yw N yf). On the con-trary, in a low-rise SPSW where shear deformations are dominant, sig-nicant yielding of the inll wall occurs prior to signicant yielding of

the frame (i.e. yw b yf). From the results in Fig. 3(d), it is also noted

that in a high-rise SPSW, the inll wall and the frame contribute almostequally to the overall strength at the ultimate state.

3.2. Wallframe contribution shares of story shear

An effective method to better understand the behavior of SPSWs isto measure the portion of the total shear force contribution providedby the inll plates at different levels. It has been shown from the push-over analyses of single-story SPSWs that the inll plate is very effectivein the initial stages of loading and it absorbs substantial part of storyshear (at the base level) before development of diagonal yield zones

[20]. However, according to the work by the authors [35], in a typicalsingle-story SPSW, the inll plate is less effective in resisting lateralloads at the lower level than at upper levels in the initial stages of load-ing. Once yielding occurs across the inll plate, the percentage contribu-tion from the inll plate at different levels becomes similar and after theframe reaches its yielded strength, it remains almost constant up to theultimate state (see Fig. 4).

Table 4 presents the percentage base shear contribution provided byinll walls in different SPSWs at the ultimate state. As shown, the per-centage strength provided by the inll wall increases slightly with theheight of the system and generally varies between 32 and 55% of the

Table 3Design base shears and the corresponding maximum interstory drift ratios for different SPSWs.

# of stories(n)

SPSW aspect ratio

L/h = 0.85 L/h = 1.4 L/h = 2.0 L/h = 2.5

Design baseshear, (KN)

Maximum interstorydrift, (%)

Design baseshear, (KN)

Maximum interstorydrift, (%)

Design baseshear, (KN)

Maximum interstorydrift, (%)

Design baseshear, (KN)

Maximum interstorydrift, (%)

1 770a 0.11 1360a 0.12 1945a 0.12 2320a 0.082 1125a 0.16 1360a 0.12 1945a 0.12 2320a 0.094 1125 0.19 1360 0.15 1945 0.15 2320a 0.136 1810 0.30 2000 0.19 2085 0.17 2320 0.158 2830 0.40 2620 0.26 2865 0.21 3440 0.1910 3410 0.56 3450 0.40 4383 0.30 4590 0.2712 3935 0.46 4365 0.33 15 5010 0.68 6030 0.48

a The design base shears for these cases are modied for the overstrength due to the consideration of the minimum practical plate thickness.

76 S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284Fig. 3.Typical and idealizedbehaviors of low-rise andhigh-rise SPSWs: (a) typical shear forcedrise SPSW, (c) idealized behavior of a low-rise SPSW, and (d) idealized behavior of a high-riseisplacement curve of a low-rise SPSW, (b) typical shear forcedisplacement curveof a high-SPSW.

l sin

77S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284overall strength. Table 5, on the other hand, compares the percentageshear forces resisted by inll plates at different stories of typical four-

Fig. 4. Percentage share of story shear by the inll walls at different levels of typicat = 3.18 mm, (c) L/h = 2.0, t = 3.18 mm, and (d) L/h = 2.5, t = 3.18 mm.and six-story SPSWs having various aspect ratios at the ultimate state.Fairly similar results are obtained for other multi-story SPSWs, andthus will not be presented for brevity. It is also to be noted that the re-sults in Table 5 may be somewhat affected by the fact that in the designphase, as can be seen in Table 1 and as mentioned in Section 2.1, somethicknesses were homogenized due to the availability of steel platesand/or the consideration of the minimum practical plate thickness. Infact, the additional stiffness and overstrength provided by the use ofthe thicker inll plate (relative to the demand) would increase theplate contribution to the story shear to some extent.

From the results in Table 5, it is shown that, generally, the effective-ness of inll plates in resisting story shear increases at upper stories.This is related to the decreased moment to story shear ratio at higherlevels of the system, which tends to dominate the shear-type behaviorat those levels compared to exural deformation. Based on the resultsgiven in Table 5, inll plates absorb substantial part of story shear atlevels above the second oor. This implies that VBEs are effective in

Table 4Percentage ultimate base shear strength provided by inll walls in different SPSWs.

# of stories (n) SPSW aspect ratio (L/h)

0.85 1.4 2.0 2.5

1 37 40 41 322 41 41 42 364 42 48 50 416 46 49 50 468 49 52 52 4710 49 51 51 5512 46 50 15 48 52 resisting story shear only at a few of lower stories and they do not con-tribute much in resisting story shear at the remaining stories. It is also

gle-story SPSWs, adapted from [35]: (a) L/h = 0.85, t = 3.18 mm, (b) L/h = 1.4,observed from the results in Table 5 that at some upper stories (espe-cially at the top), the shear forces in the inll plates are greater thanstory shear. The reason for this observation, as illustrated in Fig. 5 (themoments at the end of columns are not shown in the gure for clarity),is that the total shear force of the two VBEs due to inll plate tensionforces (i.e. VLwall VRwall ), that acts in the direction of story shear, isgreater than the total shear force of the two VBEs due to the frame ac-tion (i.e. VLframe VRframe ), that acts in the opposite direction of storyshear. Thus, the shear force of the inll plate should be greater thanstory shear to satisfy the equilibrium condition in the horizontal direc-tion. This is discussed further below.

Consider two facts. First, on account of the small shear contributionof the frame and limited story drift at upper stories, the total shearforce of the VBEs due tomoment frame sway is relatively small. Second,the upper story inll plates, which are usually thicker than required bydesign, do not yield completely even at the ultimate state (yield zones,

Table 5Percentage shear forces resisted by inll plates at different stories of typical four- and six-story SPSWs having various aspect ratios at the ultimate state.

Four-story SPSWs (n = 4) Six-story SPSWs (n = 6)

Story SPSW aspect ratio (L/h) Story SPSW aspect ratio (L/h)

0.85 1.4 2.0 2.5 0.85 1.4 2.0 2.5

1 42 48 50 41 1 46 46 50 462 58 65 67 54 2 64 79 83 743 59 98 99 81 3 64 85 96 914 122 163 176 158 4 87 96 98 99

5 99 105 108 1166 115 165 154 174

as shown in Fig. 5, develop only in one side of the inll plates between inuential. As a result, the total shear force of the two VBEs acts in the

Fig. 5.Mises stress distribution of a typical four-story SPSW at the ultimate state and the illustration of frame reaction forces for a partially yielded inll plate.

78 S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284the compressive (right) columns and the lower beams),while the stresslevel in other areas (i.e. near the tensile (left) columns and the topbeams) remains low. These cause the shear force of the compressiveVBE due to plate tension forces, that acts in the direction of storyshear, in comparison to the shear force of the tensile VBE due to platetension forces, that acts in the opposite direction of story shear, to beFig. 6. Illustration of wall/frame contributidirection of story shear.

3.3. Wallframe contribution shares of the VBE axial forces

Axial forces in the VBEs come from two sources: plate tension eldsand moment frame sway (see Fig. 6). Plate tension elds impose axialons to the VBE axial and shear forces.

forces on the VBEs in two ways. First, the vertical components of platetension forces on interfaces with the VBEs produce tensile PL

Fig. 7(a), it is also noted that in single-story SPSWs, and to some extentin two-story SPSWs, the system aspect ratio does affect the contribu-

Fig. 7. Percentage contribution of inll wall tension elds to the VBE axial forces at the base of different SPSWs at the ultimate state: (a) tensile and (b) compressive VBEs.

79S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284wall 1 and compressive PRwall 1 axial forces in the left and right VBEs, respec-tively, when the lateral loads act from left to right on the SPSW. Second,the vertical components of plate tension forces on interfaces with theHBEs indirectly cause axial forces in the VBEs (PLwall 2 and PRwall 2 ).These axial forces are normally compressive. This is mainly due to thefact that the net vertical components of plate tension elds on HBEs atstory levels where plate transitions occur and especially at the top,where the inll plate is present on only one side of theHBE, act in down-ward direction (note that the net vertical forces acting on HBEs thathave equal thickness plates above and below are negligible). The axialforces of the left and right VBEs due to moment frame sway (i.e. PLframeand PRframe ) are respectively tensile and compressive, when the lateralloads act from left to right on the SPSW.

Fig. 7 compares the percentage contribution of inll plate tensionelds to the tensile and compressive VBE axial forces at the base of dif-ferent SPSWs at the ultimate state. The curves in Fig. 7(a) show a sharpdifference between the percentage contribution of plate tension eldsto the tensile (left) VBE axial force in single-story SPSWs and in multi-story ones. Whereas the tensile VBE axial forces in single-story SPSWs(especially those having high aspect ratios) are governed by momentframe actions rather than by plate tension eld actions, a high percent-age of the tensile VBE axial forces inmulti-story SPSWs (especially thosewith n 4) comes from plate tension elds. From the results inFig. 8. Typical percentage contributions of the inll wall and frame actions to the VBEtions of plate tension elds to the tensile VBE axial forces. However, intaller SPSWs (n 4), the system aspect ratio does not play an importantrole in the contribution of plate tension elds to the tensile VBE axialforces (about 15% difference between curves for different aspect ratios).Notably, the axial force of the left (so-called tensile) VBE of a single-story SPSW due to plate tension elds can be compressive if the SPSWlength is adequately greater than its height (i.e. the case with n = 1and L/h = 2.5). This is because of the greater compressive force of thetop beam on the left VBE due to plate tension elds compared tothe tension from plate tension forces along the height of the left VBE(i.e. PLwall 2

NPLwall 1 ).

The results in Fig. 7(b) demonstrate that neither the system aspectratio nor the system height has any effect on the contribution of platetension elds to the compressive (right) VBE axial force at the baselevel. Based on the results, about 7080% of the ultimate axial force ofthe compressive VBE base is resulted from plate tension elds, regard-less of the SPSW aspect ratio and height.

Fig. 8 is used to illustrate the percentage contribution shares of theinll wall and frame actions on the VBE axial forces during the loadinghistory of typical low-rise and high-rise SPSWs. The gures show thatsimilarly, a high percentage (more than 80%) of the tensile and com-pressive VBE axial forces at the early stage of loading is due to inllwall tension forces in different SPSWs. In a low-rise SPSW, with theaxial forces during the loading history of (a) low-rise and (b) high-rise SPSWs.

The ratios of the compressive (right) to the tensile (left) VBE axial

Fig. 9. Ratios of the compressive to the tensile VBE axial forces at the base of differentSPSWs at the ultimate state.

80 S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284forces (i.e. PR/PL) at the base level of different SPSWs at the ultimatestate are presented in Fig. 9. Generally, the amount of the tensionforce resisted by the inll wall (i.e. Pp) relative to the total tensionforce in the system (i.e. PL + Pp), which is resisted by both the inllwall and the left VBE, tends to become smaller as the system aspectratio decreases or the height increases. Therefore, the difference be-tween the tensile and compressive VBE axial forces generally decreasesas the systemaspect ratio decreases or the height increases; however, inincrease of lateral displacement, the percentage contribution of inllwall tension forces to the tensile VBE axial force decreases, while thepercentage contribution of inll wall tension forces to the compressiveVBE axial force remains nearly constant. In a high-rise SPSW, however,the percentage contributions of inll wall tension forces to the tensileand compressive VBE axial forces are nearly the same and remainalmost constant during the loading history of the system.

3.4. The tensile and compressive VBE axial and shear forces

Asmentioned before, the forces from inll plate tension elds on theHBEs reduce the tension in the VBE. Said in another way, at each level ofthe SPSW, the total tension force in the system is resisted by both theinll wall and the tensile VBE, while the total compression force, dueto the plate high slenderness ratio, is resisted essentially by the com-pressive VBE. Thus, the axial force in the compressive VBE is greaterthan that in the tensile VBE and the difference between the axial forcesof the two VBEs at each level is equal to the amount of the vertical com-ponent of the total tension force resisted by the inll wall at that level.Fig. 10. Percentage frame base shear intakes by the tensile VBEs of different SPSWs at theultimate state.taller SPSWs, regardless of the aspect ratio, the ratios of the compressiveto the tensile VBE axial forces are very close to unity.

The components of the VBE shear forces at the base of a typicalsingle-story SPSW regarding the inll wall tension eld action and theboundary frame moment resisting action are illustrated in Fig. 6. Asshown, whereas the shear forces of the tensile, VLframe , and compressive,VRframe , VBEs due tomoment frame sway similarly act against story shearand are therefore additive, the shear forces of the tensile,VLwall, and com-pressive, VRwall , VBEs due to plate tension elds oppose each other (theshear force of the tensile VBE, VLwall , acts against story shear and that ofthe compressive VBE, VRwall , acts in the direction of story shear, whenthe lateral loads act from left to right on the SPSW). Therefore, in a typ-ical SPSW system, the shear force in the tensile VBE base would begreater than that in the compressive VBE base.

Fig. 10 presents the percentage of frame base shears resisted bythe tensile (left) VBEs of different SPSWs at the ultimate state (i.e. [VL/(VL + VR) 100]). Note that the VBEs at the lower levels, especially atthe base, would be more active in resisting story shear than at upperlevels, as mentioned before. Fig. 10 shows that generally, the contribu-tion of the tensile VBE to the frame base shear resistance increaseswith the height of the system, while it decreases with the aspect ratio.As shown, the tensile VBEs especially in taller SPSWswith low aspect ra-tios contribute signicantly to the frame base shear resistance.

Typical variation of the percentage frame base shear resisted byeither the tensile or compressive VBE during the loading history oflow-rise and high-rise SPSWs is drawn in Fig. 11. As noted, in bothlow-rise and high-rise SPSWs, each of the tensile and compressiveVBEs absorbs approximately half the frame base shear at the earlystage of loading. In the case of a low-rise SPSW, the contribution curvefor the tensile VBE rises with increasing lateral displacement until theformation of full yield lines across the inll wall. Afterwards, the contri-bution curve for the tensile VBE declines until the frame reaches itsyielded strength and then becomes almost constant up to the ultimatestate. In the case of a high-rise SPSW, however, due to the increasedbending effects that tend to delay the full-tension yielding of the inllwall, the percentage shear contribution from the tensile VBE always in-creases with an almost constant slope with increasing lateraldisplacement.

3.5. SPSW ductility and stiffness

The measured ductility () of different SPSWs is presented inTable 6. The ductility was calculated as the ratio of the maximum dis-placement to the yield displacement (i.e. = max/y). The maximumdisplacement (max) was dened as the top story displacement at adrift ratio of 2.5% at least at one of the stories of the system. The yielddisplacement (y) was measured through the concept of equal plasticenergy, so that the area enclosed by the idealized elasto-plastic curvewas equal to that of the actual pushover curve, as depicted in Fig. 12.For comparison purposes, the ductility ratios of the inll wall and theboundary frame calculated from the respective shear forceroof dis-placement curves using the procedure stated above are also presentedin Table 6. It is observed from the results that generally, the ductilityvalues of inll walls, frames and consequently SPSWs decrease as theheight of the systems increases. In shorter SPSWs, inll walls are signif-icantlymore ductile than frames. As the height of the systems increases,exural deformations dominate over the shear deformations; with thiseffect the effectiveness of inll walls decreases with the height of thesystems and frames behave in a more ductile manner than inll walls.The results also show that although the ductility of shorter SPSWs isnot sensitive to the system aspect ratio, the ductility of taller ones issomewhat dependent on the system aspect ratio; that is, the greaterthe aspect ratio, the higher the ductility.

Initial stiffness of different SPSWs, calculated at very early stages ofnonlinear pushover analyses, is presented in Table 7. For comparison

purposes, the wall/frame contribution shares on the overall stiffness

sults show that the reductions in the stiffness due to early buckling ofinll plates can be considerable (up to 32%) especially for shorter

ve V

81S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284are also presented in Table 7. The respective contributions of the inllwall and the moment frame to the overall stiffness were calculated bydividing the absorbed base shear by each one by the roof displacement.From the results in Table 7, it can be seen that for a given number ofstories, the stiffness values of inll walls, frames and consequentlySPSWs increase signicantly with the increase of the aspect ratio, al-though all the SPSWs have been designed for the same plan area andtherefore the same design lateral loads (neglecting the difference inthe plan area or design forces resulted fromdifferent SPSWbaywidths).Further, due to the increased bending effects, the stiffness of SPSWs de-creases noticeably with the height of the systems. According to the re-sults given in Table 7, for a given system height (i.e. number ofstories), the initial stiffness of a SPSW with L/h = 2.5 is about 3 timesgreater than that of a SPSW with L/h = 0.85, and for a given aspectratio, the initial stiffness of a ten-story SPSW is about 10% of that of asingle-story SPSW. The results also show that both inll walls and mo-ment frames are effective in providing stiffness to the SPSWs. As a gen-eral trend, however, it can be seen that the effectiveness of inll wallsdecreases with the height of the systems so that in taller SPSWs, framesbecome more effective than inll walls.

Typical SPSWs due to early buckling of their inll plates at very early

Fig. 11. Percentage frame base shear intakes by the tensile/compressistages of loading experience a signicant loss of stiffness. However, thisis not the case for SPSWs with stiffened large rectangular openings,where the inll plates are divided into separate parts with relativelylower slenderness ratios by the introduction of the openings and localboundary elements around the openings [29]. In typical SPSWs, buck-ling of the inll plates due to gravity or fabrication process occurseven before the application of lateral loads. Therefore, a realistic esti-mate of the system stiffness (termed effective stiffness here) is found

Table 6Ductility ratios of SPSWs and their inll walls and boundary frames.

# of stories (n) SPSW aspect ratio

L/h = 0.85 L/h = 1.4

SPSW Wall Frame SPSW Wall F

1 7.4 9.7 6.0 7.5 10.4 52 7.2 8.4 6.4 7.7 10.0 54 4.5 4.7 4.4 5.7 6.5 56 3.2 2.7 3.5 4.4 4.6 48 2.5 2.2 2.7 3.5 3.5 310 2.0 1.4 2.4 2.9 1.9 312 2.6 1.6 315 2.2 1.3 2SPSWs with high aspect ratios, and generally, the amount of the reduc-tions decreases with the height of the systems. In fact, as the systemheight increases, the contribution of the inll wall to the overall stiffnessand consequently the effect of inll wall buckling on the overall stiffnessdecreases.

3.6. Lateral load levels associated with the rst yielding of walls and frames

According to recent research studies [20,29], the general behavior ofa SPSW system can be divided by three stages (see Fig. 14). The rststage (OA) is considered from the onset of applying shear load (pointO) to the occurrence of the rst yield point in the inll plate (point A).by considering the effect of inll plate buckling. Fig. 13 depicts typicalstiffnessdrift ratio curves of four typical SPSW systems of differentnumber of stories and aspect ratios, obtained from pushover analyses.As shown, a similar trend is observed in the stiffness curves of differentSPSW systems. Table 8 is used to quantitatively demonstrate that effectfor SPSWs having different number of stories and aspect ratios. The re-

BE versus lateral displacement, (a) low-rise and (b) high-rise SPSWs.During this stage, the inll plate is geometrically nonlinear but materi-ally elastic. Note that the inll plate after installation is already in abuckled shape due to fabrication tolerances, welding distortion andgravity. During the second stage (AC), tension zones spread across theinll plate, while frame members remain essentially elastic. Finally, inthe third stage (CD), that is considered from the occurrence of therst yield point in frame members (point C) to the ultimate state(point D), partial or fully plastic hinges form in frame members.

L/h = 2.0 L/h = 2.5

rame SPSW Wall Frame SPSW Wall Frame

.4 7.6 11.5 4.8 7.9 12.2 5.8

.8 8.1 11.1 5.5 8.3 11.8 6.2

.0 6.8 8.2 5.4 7.3 8.8 6.1

.3 5.1 5.8 4.4 5.8 6.3 5.3

.6 4.7 5.0 4.3 5.4 5.3 5.4

.6 3.6 3.0 4.1 4.1 3.6 4.5

.3 3.3 2.8 3.7

.8 2.7 2.0 3.2

The contribution of plate tension elds to the tensile VBE axial forcesis dependent on both the system aspect ratio and height; however,

Fig. 12. Denition of a yield point, adapted from [29]. Fig. 13. Stiffnessdrift ratio curves of typical SPSWs with n = 1, 4 and L/h = 1.4, 2.

82 S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284The knowledge of the load magnitude associated with each of theabove stages can be useful not only in ensuring the efciency of designfor different earthquake hazard levels but also in reducing post-earthquake rehabilitation costs. From the seismic design point of view,minor earthquake loads can be controlled by the system during therst stage of the behavior. In moderate earthquakes, the inll plates ex-perience extensive yielding, ensuring a relatively large global energydissipation capability, while frame members remain essentially elastic(i.e. the second stage of the behavior). Consequently, if necessary, dam-aged inll plates due to earthquake loading can be simply replacedwithoutmajor involvement of the primary structure. Under severe seis-mic events, widespread yielding of the inll plates and plastic hinging offramemembers ensure an extremely large energy dissipation capabilityand also a stable response (i.e. the third stage of the behavior).

The relative base shears (normalized by the ultimate base shearstrength) associated with the rst yielding of inll walls and boundaryframes for different SPSWs are presented in Fig. 15. For comparison pur-poses, the normalized design base shears for different SPSWs are alsopresented in the gure. As shown, the normalized base shears associat-ed with the rst yielding of inll walls and frames increase slightly withthe height of systems. Based on the result, the rst yield points in theinll walls and in the frame members of different SPSWs occur at baseshear levels of about 2545% and 7085% of the ultimate strength, re-spectively. In addition, the design base shears for the different systemslie between about 20 and 30% of their ultimate strength. This indicatesthat the current design approach, which assumes that 100% of storyshear is resisted by each inll panel, can lead to a SPSW system withan overstrength factor of between about 3 and 5 (based on the obtainedresults), although ASCE 7-05 [28] suggests an overstrength factor of 2.5for SPSWs. From the results in Fig. 15, it can also be implied that unlikethe assumption in design codes (i.e. uniform yielding of inll plates onall stories under design level earthquake forces), no signicant yieldingwould occur in the inll wall of a code designed SPSW under designlevel earthquake forces. This is mainly due to the fact that theTable 7Initial stiffness of different SPSWs and contribution shares of their inll walls and frames.

# of stories (n) SPSW aspect ratio

L/h = 0.85 L/h = 1.4

SPSW Wall Frame SPSW Wall F

1 273 130 143 461 263 12 188 87 102 267 150 14 75 31 44 117 646 53 20 33 93 428 41 17 24 68 3210 33 11 22 50 1612 45 1315 30 9contribution of the boundary frame in resisting story shear, which canbe considerable especially at the lower stories as mentioned before, isnot considered by design codes.

4. Conclusions

A series of code designed SPSWswith different aspect ratios (L/h=0.85, 1.4, 2 and 2.5) and number of stories (n = 1, 2, 4, 6, 8, 10, 12 and15) were numerically analyzed using the nite element method andthe obtained results were utilized to investigate characteristics of thebehavior of code designed SPSWs and to evaluate the effectiveness oftheir inll walls and boundary frames. The following can be concludedfrom this study:

- In a low-rise SPSWwhere the behavior is shear-type, the signicantyielding of the inll wall occurs prior to the signicant yielding of theframe, while in a high-rise SPSW the increased bending effects delaythe full-tension yielding of the inll plates.

- VBEs are found to be effective in resisting story shear only at a few oflower stories where the exural deformations are dominant, andinll walls absorb substantial part of story shear at the remainingstories where the behavior mode is mainly shear-type.

- At upper stories,where the plate thicknesses are usually thicker thanrequired by design and drifts are limited, inll plates do not yieldcompletely. As a result, the total shear force of the VBEs will act inthe direction of story shear and the shear force of the inll platewill be therefore greater than story shear to satisfy the equilibriumcondition in the horizontal direction.

- Regardless of the system aspect ratio and height, about 7080% ofthe compressive VBE axial forces come from plate tension elds.L/h = 2.0 L/h = 2.5

rame SPSW Wall Frame SPSW Wall Frame

98 666 415 252 1084 562 52217 389 240 149 554 290 26452 189 116 73 256 135 12251 120 69 51 147 74 7336 99 54 45 135 62 7334 82 34 48 100 48 5232 64 24 40 21 53 19 33

Table 8Effective stiffness and loss of stiffness due to early buckling of inll plates for different SPSWs.

# of stories (n) Effective stiffness (KN/mm)

L/h = 0.85 L/h = 1.4 L/h = 2.0 L/h =

1 187 317 456 7352 140 190 271 3844 63 89 141 1866 51 77 95 1148 40 59 83 10910 33 48 72 8412 45 57 15 30 49

d (b

83S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284Fig. 14. (a) Typical lateral loaddisplacement anin taller SPSWs (i.e. with n 4), regardless of the system aspect ratioand height, about 6080% of the tensile VBE axial forces at the ulti-mate state come from plate tension elds.

- In a typical SPSW, the inll wall has a contribution in resisting thetotal tension in the system, but a small or no contribution in resistingthe total compression. Therefore, the axial force in the tensile VBE issmaller than that in the compressive one. Generally, the ratio of thetension force resisted by the inll wall to the total tension force inthe SPSW and consequently the difference in the axial forces of thetwo VBEs tend to become smaller as the system aspect ratio de-creases or the height increases.

- The shear forces of the tensile VBE base due to moment frame swayand due to inll panel tension elds act in the same direction, whilethose of the compressive one act in opposite directions. As a result,the tensile VBE is more effective in absorbing base shear than the

Fig. 15. Normalized base shears associatedwith different stages of the behavior of SPSWs.Loss of stiffness due to inll buckling (%)

2.5 L/h = 0.85 L/h = 1.4 L/h = 2.0 L/h = 2.5

31 31 32 3226 29 30 3116 23 26 275 17 21 232 13 16 190 4 12 16 0 11 0 6

) stiffnessdrift ratio curves, adapted from [29].compressive one. Based on the results obtained, the tensile VBEs ab-sorb about 5595% of the total base shear in the frame columns atthe ultimate state.

- As a result of the current design procedures that neglect the contri-bution from the boundary frames, the stiffness and ductility ofSPSWs that are designed for the same plan and consequently thesame design lateral loads can vary signicantly depending on the as-pect ratio; that is, the greater the aspect ratio, the higher the stiffnessand the ductility. In turn, these variations are not only due to thechange in the stiffness and ductility of frames, but also due to thechange in the stiffness and ductility of inll walls.

- Typical SPSWs experience a signicant loss of stiffness at the earlystage of loading due to buckling of their inll plates (up to 32% re-duction). The percentage reduction in the stiffness generally in-creases with the system aspect ratio, while it decreases with theheight.

- The rst yield points in the inll walls and in the boundary framemembers of different SPSWs occur at base shear levels of about2545% and 7085% of the ultimate strength, respectively.

- Unlike the assumption in design codes (i.e. uniform yielding of inllplates on all stories under design lateral loads), no signicant yield-ingwould occur in the inll wall of a code designed SPSW under de-sign level earthquake forces. This is mainly due to the fact that thecontribution of the boundary frame in resisting story shear, whichcan be considerable especially at lower stories, is not considered bydesign codes.

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84 S.A.A. Hosseinzadeh, M. Tehranizadeh / Journal of Constructional Steel Research 99 (2014) 7284

Behavioral characteristics of code designed steel plate shear wall systems1. Introduction2. Method of the study2.1. Design of models2.2. Description of finite element models

3. Discussion of results3.1. General behavior3.2. Wallframe contribution shares of story shear3.3. Wallframe contribution shares of the VBE axial forces3.4. The tensile and compressive VBE axial and shear forces3.5. SPSW ductility and stiffness3.6. Lateral load levels associated with the first yielding of walls and frames

4. ConclusionsReferences