Reinforced concrete squat walls are common in low-rise constructionand as wall segments formed by window and door openings inperimeter walls. Existing approaches used to model the lateral forceversus deformation responses of wall segments typically assumeuncoupled axial/flexural and shear responses. A more comprehensivemodeling approach, which incorporates flexure-shear interaction, isimplemented, validated, and improved upon using test results. Theexperimental program consisted of reversed cyclic lateral loadtesting of heavily instrumented wall segments dominated by shearbehavior. Model results indicate that variation in the assumedtransverse normal stress or strain distribution produces importantresponse variations. The use of the average experimentallyrecorded transverse normal strain data or a calibrated analyticalexpression resulted in better predictions of shear strength andlateral load-displacement behavior, as did incorporating a rotationalspring at wall ends to model extension of longitudinal reinforcing barswithin the pedestals.
INTRODUCTIONSquat walls (with aspect ratios typically less than 1.5) are
very common in low-rise construction and at lower levels oftall buildings (for example, parking level walls or basementwalls). They can also be found in long walls with perforationsdue to window and door openings, resulting in wall segmentsbetween openings. The design of wall elements is usuallyoriented toward supplying sufficient shear strength topromote flexural yielding (for example, ACI 318-081
Section 21.9); therefore, a model that appropriately accountsfor nonlinear flexural behavior is required. For low aspect-ratio walls or wall segments, behavior is often dominated bynonlinear shear responses, and the modeling parametersselected for shear stiffness and strength can have a significantimpact on the predicted distribution of member forces and onbuilding lateral drift. For example, FEMA 3562 recommendsuse of 0.4Ec to model the effective preyield shear stiffnessdespite the lack of test data to support the use of this valueand the impact of shear cracking on effective stiffness.Additionally, the impact of flexural deformations on theoverall load-deformation relation for wall segments withshear-dominant behavior (for example, with failure modecontrolled by crushing along a diagonal compression strutand lateral displacements dominated by shear deformationassociated with diagonal tension and cracking) may beimportant, although the FEMA 356 backbone relationsappear to imply that nonlinear load-versus-deformationresponse is either due to flexure or shear, but not both.
According to experimental evidence, flexural and sheardeformation interaction exists even for relatively slenderwalls with an aspect ratio of 3 to 4, with shear deformationscontributing approximately 30% and 10% of the first-storyand roof level lateral displacements, respectively.3 Thedegree of interaction for smaller aspect ratios, and particularlyfor squat walls with an aspect ratio of less than 1, is unclear.
There is a need for relatively simple modeling approachesthat consider interaction between flexure and shearresponses, and capture important response features for awide range of wall geometries and reinforcing details.Although a relatively large number of wall tests are reportedin the literature, the primary focus for most of these tests isthe assessment of wall shear strength and overall loadversus lateral displacement response, as opposed to assessmentof relative contributions of flexural, shear, and anchoragedeformations to wall lateral displacements, which is necessaryfor validating existing and developing new modelingapproaches. Therefore, experimental studies that incorporatevery detailed instrumentation layouts are needed to allowdevelopment and verification of new approaches that focuson robust and mechanical and/or behavioral models torepresent the load-deformation responses of squat wallscontrolled by shear.
Based on the preceding discussion, a modeling approachcapable of incorporating flexure-shear interaction isimplemented and evaluated using experimental results.Model improvements needed to capture the overall lateralload versus lateral displacement response, as well as the relativecontribution of flexural and shear deformation responses, areimplemented and validated using data from an experimentalprogram conducted on lightly reinforced squat walls withshear-dominant behavior. Test results on heavily instrumentedtest specimens enabled a range of modeling parameters andassumptions to be investigated, ultimately yielding improvedagreement between experimental and model results.
RESEARCH SIGNIFICANCELow-aspect-ratio reinforced concrete walls, with height-
to-length ratios less than 1.5, are common in low-rise andperforated-wall-type construction. Accurate modeling of theload-versus-deformation response and the lateral-load stiffnessof such walls is essential to estimate building lateraldisplacements as well as the distribution of forces amongelements of the lateral-load-resisting system. A model thatreasonably captures measured responses, includingcoupling of flexure-shear behavior, is proposed and modelresults are validated by comparison with test results fromheavily instrumented wall segments. Based on the study,recommendations for modeling the load-deformationresponses of squat walls are provided.
MODEL DESCRIPTION AND BACKGROUNDAn analytical model that couples wall flexural and shear
responses was proposed by Massone4 and Massone et al.5
Title no. 106-S60
Modeling of Squat Structural Walls Controlled by Shearby Leonardo M. Massone, Kutay Orakcal, and John W. Wallace
647ACI Structural Journal/September-October 2009
based on framework proposed by Petrangeli et al.6 Themodel incorporates reinforced concrete (RC) panel behaviorinto a two-dimensional macroscopic fiber model (multiplevertical line element model [MVLEM])7-9 (Fig. 1(a)) tocapture the experimentally observed shear-flexure interactionin RC walls.3 The model formulation involves modifying theMVLEM by assigning a shear spring to each macrofiber ofthe MVLEM element (Fig. 1(b)). Each macrofiber is thentreated as an RC panel element subjected to membraneactions, that is, in-plane uniform normal and shear stresses(Fig. 1(b)). Therefore, the interaction between flexure andshear is incorporated at the fiber level. It has been shown thatthis modeling approach adequately captures overall load-versus-deformation responses for relatively slender walls(aspect ratio of 3) and walls with aspect ratios (or moment-to-shear ratio divided by wall length) greater than approximately1.5 This study identifies significant model variables that influenceresponse predictions for very squat walls (moment-to-shear ratiodivided by wall length equal to 0.5). To represent constitutivepanel behavior, a rotating-angle modeling approach, such asthe modified compression field theory (MCFT)10 or therotating-angle softened truss model (RA-STM),11 can beused, among other models. For each constitutive RC panelelement, a uniaxial constitutive stress-strain model forconcrete is applied along the principal directions to obtainthe stress field associated with the principal directions,assuming that the principal stress and strain directionscoincide.10,11 For reinforcing steel, a uniaxial constitutivestress-strain model is applied in the directions of the reinforcingbars, based on the assumption of perfect bond betweenreinforcing steel and concrete. Dowel action on reinforcementis neglected; only the uniaxial response of the reinforcing barsis included in the model. Accordingly, the axial and shearresponses of each fiber (panel) element are coupled, whichenables coupling of flexural and shear responses of theMVLEM, because the combination of axial response of theuniaxial elements constitute the overall flexural response ofeach MVLE. Although the cyclic constitutive stress-strain models may be implemented for concrete and steelreinforcement, the present model was formulated toconsider only monotonic loading.5 Therefore, the presentstudy focuses on analytical prediction of envelope responsesobtained from cyclic tests.
As described by Massone et al.,5 the deformations orstrains within the components of each panel element aredetermined from the six prescribed degrees of freedom (ux ,uy , and at both ends of the model element), as shown inθ
⋅
Fig. 1(b). Assuming that the shear strain is uniform along thesection and that plane sections remain plane, the longitudinalnormal (axial) strain εy and shear distortion γxy componentsof the strain field are calculated for the entire section (for allthe strips (i)) based on the prescribed degrees of freedom forthe current analysis step. The transverse normal strain withineach strip (εx) is initially estimated to complete the definitionof the strain field, allowing stresses and forces to be determinedfrom the constitutive material relationships and geometricproperties (dimensions and reinforcement and concrete areasfor each strip). A numerical solution procedure (for example,the Newton-Raphson method) can be employed to linearizethe equilibrium equation and iterate on the unknown quantityεx (transverse normal strain in each strip i), to achieve horizontalequilibrium for a given resultant transverse normal stress σx(resultant of transverse normal stress components inconcrete and reinforcing steel) within each strip. In the casewhere the transverse normal strains are known, this iterationis not required. As an initial approximation in developmentof the model, the transverse normal stress σx within eachstrip was assumed to be equal to zero, which is consistentwith the boundary conditions at the sides of a wall with notransverse loads applied over its height.
Cheng et al.12 revealed that transverse normal strainsexperienced along the length of the wall are significantlyreduced for low-aspect-ratio walls (for example, h/l = 0.5),especially in regions close to the top and bottom of the walldue to the constraining effect of pedestal used at the wallends required for testing. Thus, using an assumption of zerotransverse normal strain (εx = 0), especially in regions closeto the top and bottom of the wall, may be more appropriatethan assuming zero resultant transverse normal stress (σx = 0)along the entire height of a wall with a low shear span-depthratio. To investigate this idea, an alternative model formulationwas developed to represent an extreme case of assuming zerotransverse normal strains (εx = 0) along wall length over theentire wall height. Additional details on model formulationare presented elsewhere.4,5
Comparing predictions of the two alternative modelformulations (σx = 0 and εx = 0), studies by Massone et al.5
revealed that neither model formulation is capable ofcorrectly reproducing the experimental responses observedin walls with low shear span-depth ratios (lower than 0.5).Therefore, a more detailed description of the distribution oftransverse normal strains or stresses, or possibly variation of
Leonardo M. Massone is an Assistant Professor in the Department of CivilEngineering at the University of Chile, Santiago, Chile. He received his BS fromthe University of Chile in 1999 and his MS and PhD from the University of California-Los Angeles (UCLA), Los Angeles, CA, in 2003 and 2006, respectively. His researchinterests include analytical and experimental studies of reinforced concrete systems,with an emphasis on seismic response.
Kutay Orakcal is an Assistant Professor in the Department of Civil Engineering atBogaziçi University, Istanbul, Turkey. He received his BS from Middle East TechnicalUniversity, Ankara, Turkey, in 1998, and his MS and PhD from UCLA in 2001 and 2004,respectively. His research interests include structural and earthquake engineering, withan emphasis on response assessment for structural elements and systems subjected toearthquake actions through laboratory testing and multi-scale analytical modeling.
John W. Wallace, FACI, is a Professor of civil engineering at UCLA. He is a memberof ACI Committees 318-H, Seismic Provisions (Concrete Building Code), and 374,Performance-Based Seismic Design of Concrete Buildings, and Joint ACI-ASCE Committee352, Joints and Connections in Monolithic Concrete Structures. His research interestsinclude seismic design of buildings and bridges, laboratory and field testing of structuralcomponents and systems, and seismic monitoring using sensor networks.
Fig. 1—Element models: (a) MVLEM element; and (b) coupledmodel element.
648 ACI Structural Journal/September-October 2009
other model parameters or assumptions, was deemed necessaryto accurately predict the responses of such squat walls. Mosttests reported in the literature, however, incorporate relativelysparse instrumentation, primarily to measure the lateraldisplacement over wall height. Well instrumented test specimensare required to provide the detailed information needed to assessthe importance of various modeling assumptions. Thefollowing section describes a test program with heavilyinstrumented specimens that provided the data needed togain insight into the validity of various model assumptions.
TEST PROGRAMSpecimen description
The experimental program involved testing of 14 wall pier(WP) and spandrel specimens (WS), with dimensions,reinforcement configurations, and material propertiesselected to be representative of perimeter wall segmentsconstructed in California between approximately 1940 and1970. The wall segments tested were 3/4-scale replicas ofsegments from an actual hospital building. The spandrelspecimens were 1520 mm (60 in.) tall, 1520 mm (60 in.)long, and 150 mm (6 in.) thick, and the piers were 1220 mm(48 in.) tall, 1370 mm (54 in.) long, and 150 mm (6 in.) thickwith a shear span-depth ratio (M/(Vl)) of 0.5 and 0.44,respectively. Relatively low shear span-depth ratios(corresponding to one-half of the aspect ratio for eachspecimen) were achieved during testing of these specimensvia fixing the base of the walls, restraining rotations at thetop of the walls, and by having the line-of-action of theapplied lateral load pass through the specimen midheight.The distributed reinforcing steel ratios of the specimens inlongitudinal and transverse directions (ρl and ρt), thecorresponding boundary reinforcement ratio ρb, and theaxial load levels applied on the specimens during the tests, aswell as other specimen characteristics, are presented in Table 1.
The reinforcing steel ratios in the table were calculated basedon the total area of boundary (Asb) or distributed webreinforcement per the total tributary area of concrete(boundary or web) over which the reinforcement is located.The tributary area of the boundary zones was calculated asspecified in ACI 318-08,1 Fig. R21.9.6.5.
The test specimens included some specific featuresobserved in older buildings, such as a weakened plane joint(WPJ) in the spandrels, where the concrete thickness isreduced and web reinforcement is cut to provide a crackinitiator along the span between window openings either atmidspan or adjacent to the end of the span, and a lack ofhooks on horizontal web reinforcement for piers and selectedspandrels. Four test groups, with two identical (companion)specimens in each group, were tested for the spandrel (WS)specimen configurations. Repeating tests for a givenconfiguration enabled an assessment of variability of thetest results. The first two WS types were differentiatedprimarily by the amount of boundary reinforcementprovided, whereas for the last two groups, which had nohooks on horizontal reinforcement (similar to pier configuration),incorporated less web and boundary reinforcement, and thelocation of the WPJ was varied (Table 1). The other sixspecimens were identical piers (WP) tested under threedifferent levels of axial load (0%, 5% fc′Ag, and 10% fc′Ag).A detailed description of the experimental program andresults can be found elsewhere.4
InstrumentationEach test specimen was provided with a very detailed set
of instrumentation to enable post-test studies focused onmodel development and validation. DC-LVDTs (DC-excitedlinear variable differential transducer, referred to as DCDTs)were mounted on the specimens to provide measurements of
Table 1—Properties of wall spandrel (WS) and wall pier (WP) specimens
Specimen
Test no.
tw,cm
lw,cm
hw,cm M/(Vlw)*
Transverse web reinforcement
Longitudinal web reinforcement
Boundary reinforcement
Axial load
Materialproperties, MPa
ID No. TypeReinforcing
bar† ρt, % HooksReinforcing
bar† ρt ,%Cut bars
Reinforcing bar†
Asb,
cm2ρb, %
N/Ag f c′, % f c′
fy, φ13
fy,φ16
WS-T1-S11
Test 1 15.2 152 152 0.50 φ13 @ 33 cm 0.278 Yes φ13@ 23 cm 0.428 4 of 6‡ 4 φ16 8 3.12 0 25.5 424.0 448.2
WS-T1-S2 Test 4 15.2 152 152 0.50 φ13 @ 33 cm 0.278 Yes φ13@ 23 cm 0.428 4 of 6‡ 4 φ16 8 3.12 0 43.7 424.0 448.2
WS-T2-S12
Test 2 15.2 152 152 0.50 φ13 @ 33 cm 0.278 Yes φ13@ 23 cm 0.400 4 of 6‡ 1 φ13 +1 φ16 3.29 1.70 0 31.4 424.0 448.2
WS-T2-S2 Test 3 15.2 152 152 0.50 φ13 @ 33 cm 0.278 Yes φ13@ 23 cm 0.400 4 of 6‡ 1 φ13 +1 φ16 3.29 1.70 0 31.0 424.0 448.2
WS-T3-S13
Test 11 15.2 152 152 0.50 φ13 @ 28 cm 0.278 No φ13@ 28 cm 0.256 2 of 4‡ 2 φ13 2.58 1.33 0 31.7 351.6 —
WS-T3-S2 Test 14 15.2 152 152 0.50 φ13 @ 28 cm 0.278 No φ13@ 28 cm 0.256 2 of 4‡ 2 φ13 2.58 1.33 0 33.6 351.6 —
WS-T4-S14
Test 12 15.2 152 152 0.50 φ13 @ 28 cm 0.278 No φ13@ 28 cm 0.256 2 of 4§ 2 φ13 2.58 1.33 0 31.9 351.6 —
WS-T4-S2 Test 13 15.2 152 152 0.50 φ13 @ 28 cm 0.278 No φ13@ 28 cm 0.256 2 of 4§ 2 φ13 2.58 1.33 0 33.0 351.6 —
WP-T5-N0-S1
5
Test 9 15.2 137 122 0.44 φ13 @ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 0 29.9 424.0 —
WP-T5-N0-S2 Test 10 15.2 137 122 0.44 φ13 @ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 0 31.0 424.0 —
WP-T5-N5-S1 Test 7 15.2 137 122 0.44 φ13@ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 5 31.9 424.0 —
WP-T5-N5-S2 Test 8 15.2 137 122 0.44 φ13@ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 5 32.0 424.0 —
WP-T5-N10-S1 Test 5 15.2 137 122 0.44 φ13@ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 10 28.3 424.0 —
WP-T5-N10-S2 Test 6 15.2 137 122 0.44 φ13@ 30.5 cm 0.278 No φ13@ 33 cm 0.227 — 2 φ13 2.58 1.33 10 31.4 424.0 —*Shear span-depth ratio.†Ribbed φ13 (13 mm diameter) = U.S. No. 4; ribbed φ16 (16 mm diameter) = U.S. No. 5.‡Weakened plane joint at wall midheight.§Weakened plane joint at wall-pedestal interface.Note: 1 cm = 0.394 in.; 1 cm2 = 0.155 in.2; 1 MPa = 0.145 ksi.
649ACI Structural Journal/September-October 2009
average deformations at specified locations. DCDTs werelocated to determine overall deformations as well as localdeformations to assess, for example, the contribution ofshear and flexural deformations to the relative lateraldisplacement over the specimen height. Additional DCDTsalso were mounted on the specimen to obtain average shear,transverse normal, and longitudinal normal strains. Straingauges also were used to measure local strains at selectedlocations on the reinforcing steel bars. Figure 2 shows ascheme of the DCDT sensor distribution for Tests 5 to 8.Additional details on the wall instrumentation used can befound in the report by Massone and Wallace.13
ANALYTICAL MODELING STUDIESAn overview of the analytical modeling studies and
comparisons with experimental results for four spandrelspecimens (Tests 1 to 4) and five pier specimens (Tests 5 to 9)are presented. Results for the remaining spandrel specimens(Tests 11 to 14) are not included herein because they did notincorporate sufficient instrumentation to provide the dataneeded for the comparisons included in this paper. Results ofTest 10 are not discussed because the pier specimen WP-T5-N0-S2 was accidentally damaged (cracked) prior to testing.
The shear-flexure interaction model previously describedis initially used to predict the response of each wall specimen.The analysis considers monotonic lateral loading with a zero-rotation kinematic boundary condition enforced at the topand bottom of the wall, whereas the vertical displacement atthe top of the wall is not restrained and the lateral displacementat the top of the wall is prescribed. A displacement-controllednonlinear analysis solution strategy is used, where the lateraldisplacement at the top of each wall model is increasedduring the analysis. The axial load applied at the top of eachwall model corresponds to the resultant of the forces appliedby the vertical actuators and the self-weight of the steel reactionframe (approximately 53 kN [12 kips]).
The constitutive material models were calibrated to matchthe as-tested material properties (that is, compressivestrength of concrete, strain at peak stress of concretecompressive strength, yield strength of reinforcing steel, andpost-yield steel stiffness) used in the construction of the wall
specimens, obtained through uniaxial stress-strain testsconducted on 152 x 304 mm (6 x 12 in.) concrete cylindersand reinforcing bar coupons.
The constitutive relationship implemented in the analyticalmodel for concrete considered the effects of biaxial compressionsoftening (reduction in principal compressive stresses in concretedue to cracking under tensile strains in the orthogonal direction),and tension stiffening (average postpeak tensile stresses inconcrete due to the bonding of concrete and reinforcing steelbetween cracks). To incorporate the tension stiffening effectin the stress-strain behavior of concrete in tension, theaverage (smeared) stress-strain relationship proposed byBelarbi and Hsu14 is implemented. To describe the stress-strain behavior of concrete in compression, the Thorenfeldtbase curve is used, calibrated by Collins and Porasz15 andCarreira and Kuang-Han,16 and updated via the introductionof the compression softening parameter proposed byVecchio and Collins.17
The monotonic stress-strain relationship implemented inthe wall model for reinforcing steel is the well-knownuniaxial constitutive model of Menegotto and Pinto.18 Therelationship is in the form of a curved transition, from astraight-line asymptote with slope E0 (modulus of elasticity)to another asymptote with slope E1= bE0, where parameter bis the strain hardening ratio (b = 0.008). To consider theeffects of tension stiffening on the reinforcement, that is,softening (and weakening) of the average (smeared) stress-strain relationship of reinforcing steel bars embedded inconcrete (due to concentration of strains in steel at the cracklocations), Belarbi and Hsu14 proposed two empiricalrelationships, the first of which enforces smoothing of thetransition curve between the elastic and yield asymptotes, andthe second of which accounts for the reduction of the effectiveaverage yield strength for steel bars embedded in concrete.These relationships (the so-called single-curve model) wereincorporated directly into the Menegotto and Pinto equationused in the present wall model. According to the Belarbi andHsu model, the effective yield stress and strain (at the inter-section of the elastic and yield asymptotes) for barsembedded in concrete correspond to approximately 91% ofthe yield stresses and strain for bare bars.
Fig. 2—Specimen sensor configuration: Tests 5 to 8. (Note: 1 mm = 0.0394 in.)
650 ACI Structural Journal/September-October 2009
In the analytical models used to represent the test specimens,seven model elements were stacked along the height of thewall, where each model element consisted of eight strips(panel elements) along wall length. Two strips were used todiscretize each boundary element of a wall specimen,whereas the remaining four strips were used to define theweb. Steel reinforcement was assumed to be distributeduniformly within the tributary area of each particular strip.
The presence of the WPJ and the unfavorable anchoragecondition due to discontinuity of the longitudinal webreinforcement at the WPJ on the spandrel specimens(Fig. 3) were not considered in the analysis. Based on testobservations, it is noted that diagonal cracks formed acrossthe WPJ, with only slight cracking along the WPJ for a fewcentimeters near the center of the spandrel. In addition, thelack of hooks on the transverse web reinforcement of the pierspecimens was not considered in the analysis.
Model results: zero transverse normal stressor strain
The experimental results demonstrated that the endpedestal at the top and bottom of the walls restrained thetransverse normal strains. The model formulation describedpreviously, which assumes zero transverse normal stressresultant (σx = 0), does not account for this effect, producinga softer and weaker analytical load-versus-displacementresponse than measured. A stiffer and stronger response isexpected for the condition where the constraint of the endblocks is considered (εx = 0). Response predictions wereobtained for the wall specimens using both model formulations,and the results are discussed in the following subsections.
Lateral load versus top displacement response—Figure 4compares the measured and predicted lateral load versus topdisplacement responses of selected wall specimens. In Fig. 4,analytical results obtained using three different modelformulations—shear-flexure interaction model with zero
Fig. 4—Load versus top displacement (full response): (a) Test 1, WS-T1-S1; and (b) Test 6,WP-T5-N10-S2. (Note: 1 kN = 0.225 kips; 1 cm = 0.394 in.)
651ACI Structural Journal/September-October 2009
transverse normal strain, shear-flexure interaction model withzero transverse normal stress, and flexural model (withoutshear deformation)—are compared with the experimentallymeasured responses. Results of the shear-flexure interactionmodels are plotted until significant degradation of the walllateral load capacity occurs, as postpeak (degrading) responseis sensitive to model discretization (element size).
For all specimens, the experimentally measured load-displacement response falls between the upper and lowerbound analytical responses predicted by the shear-flexureinteraction models with zero transverse normal strain andzero transverse normal stress assumptions, respectively. Inaddition, the flexural model predictions for the wall lateralload capacity and stiffness are significantly larger than thatfor the shear-flexure interaction model predictions and thetest results (Fig. 4). This is consistent with the observed shearfailure (diagonal tension and web crushing) of the test specimens.
The lateral load capacities predicted by the interactionmodels are compared with the experimentally observedcapacities in Fig. 5. As expected, the two extreme interactionmodel formulations (with the εx = 0 and σx = 0 assumptions)provide upper and lower bound strength predictions. Thecapacity prediction using the model with the zero transversenormal strain assumption is better, that is, closer to a ratio of 1,for piers (shorter specimens) compared to spandrels, and abetter prediction is obtained for spandrels (higher aspect ratio)using the model with zero transverse normal stress assumption.Thus, as expected, the constraint provided by the end blocks ismore pronounced for the shorter specimens and the betterresponse prediction is obtained using the model based on the
assumption of zero transverse normal strain. These resultssuggest that the model could be improved by allowing themagnitude of the transverse normal stresses or strains to varyalong the height and across the length of a wall.
Flexural and shear deformation components—With eithermodel formulation, significant inconsistencies are notedbetween model and experimental results for the relativecontribution of flexural and shear deformations to the overalllateral displacement. For example, the analytical model forzero transverse normal strains (εx = 0) produces flexuraldeformations that generally exceed the experimental results,whereas the model with zero transverse normal stresses (σx = 0)substantially underestimates the relative contribution of lateralflexural displacements at low drift levels (Fig. 6). Clearly amore sophisticated modeling approach is needed to producemore consistent results for low-aspect-ratio walls over abroad range of applied drift ratios.
Model results: measured average transverse normal strains
To assess whether incorporating a more realistic transversenormal strain distribution would improve the accuracy of theanalytical model, the ability to incorporate the distribution ofthe measured transverse normal strains obtained from testresults was implemented into the model formulation. Thedistribution of transverse normal strains was obtained fromthe wall segment tests using data measured from DCDTsmounted at seven levels over the wall height (for example,Fig. 2, Face A); seven elements also were used in the modelof the test specimens. For the top three sensors (51, 52, 53 in
Fig. 6—Load versus top displacement: Test 6, WP-T5-N10-S2: (a) shear; and (b) flexure.(Note: 1 kN = 0.225 kips; 1 cm = 0.394 in.)
652 ACI Structural Journal/September-October 2009
Fig. 2, Face A), a single horizontal transducer (DCDT) wasused to measure the average transverse normal strain acrossthe entire length of the wall. Although Fig. 2 shows singletransducers (57, 58, 59, and 60) on Face A (which was thecase for Tests 5 to 8), for Tests 1 to 4 and Test 9, four horizontaltransducers were used at these levels to measure the distributionof transverse normal strains across the wall length. For the testswith multiple DCDTs, the average strain value was taken asthe average of the measurements of all four sensors. Modelresults obtained using the measured average transversenormal strain distributions for the peak (load reversal) pointsfor each positive and negative loading cycle (that is, pointsthat define the load-displacement envelope) are denoted asεx
exp in Fig. 5 to 8.Lateral load versus top displacement response—Figure 7
presents lateral load versus top displacement responsecomparisons for selected wall specimens up to the pointwhere reliable experimental horizontal normal strain data areavailable. The model incorporating the experimental transversenormal strain profiles produces improved results for lateralload versus top displacement behavior, with the predictedload-displacement responses falling between the upper andlower bound model results (εx = 0 and σx = 0; refer to Fig. 4).The ratio of the model to the experimental lateral load capacities,with the experimental value taken as the average value for thepositive and negative cycles, are compared in Fig. 5 for nine
tests. As observed in Fig. 5, the model with the measuredhorizontal normal strains provides a more accurate capacityprediction than the models with upper and lower boundassumptions (εx = 0 and σx = 0, respectively). The modelwith the measured horizontal normal strains overestimatesthe capacity by 14% (average of all nine specimens),whereas the models with upper and lower bounds assumptionsresult in overestimations of 34% and underestimations of 29%of the capacity, respectively. Although the model results areimproved, the analytical model tends to over estimate both thelateral stiffness and the lateral load capacity of the test walls atall lateral displacement levels (for example, Fig. 7).
Flexural and shear deformation components—Figure 6compares model and experimentally derived shear and flexuraldeformation components of the lateral load versus topdisplacement response for a representative test specimen(Test 6 of Table 1). The model results are significantlydifferent than results obtained using the model that assumeszero transverse normal strains (εx = 0), which was observedto produce inconsistent results over a broad range ofdisplacements. As shown in Fig. 6, the model results indicatethat the flexural deformation component contributes significantlyless to the top displacement and better matches the experimentalresults. Although the initial shear stiffness for the model isclose to the experimental stiffness (Fig. 6(a)), for most cases,the flexural stiffness is still overpredicted (Fig. 6(b)).
Fig. 7—Load versus top displacement (selected displacement range): (a) Test 1, WS-T1-S1;and (b) Test 6, WP-T5-N10-S2. (Note: 1 kN = 0.225 kips; 1 cm = 0.394 in.)
Fig. 8—Deformation component of top displacement: (a) spandrels; and (b) piers.
ACI Structural Journal/September-October 2009 653
Model results: wall-pedestal interfacerotational spring
The experimental lateral load versus top displacementresponses tend to be softer than the model predictions thatincorporate measured average transverse normal strainsεx
exp, especially for the flexural deformations (Fig. 6(b) and 7).A review of the experimental results reveals that the contributionof flexural deformations to the top displacement is concentratedwithin the first pair of gauges (67, 68, 70, and 71 in Fig. 2)located at the boundaries of the test specimens, whichaccount for approximately 60% of flexural top displacementat drift levels lower than or equal to 0.2%. The gauge lengthfor these sensors is 7.6 mm (3 in.), with one end anchored(fixed) to the upper and lower support pedestals. Given thatthese sensors span the wall-pedestal interface, the potentialcontribution of the extension of the longitudinal reinforcingbars within the pedestals to lateral displacements measuredover the wall height was investigated.
To model the potential impact of reinforcing bar extensionwithin the pedestals, an initial moment-curvature analysiswas conducted at the wall-pedestal interfaces, assuming thata crack forms along the entire length of the interface betweenthe specimen and the pedestals (Fig. 9). Interface cracks,formed during post-tensioning of anchor bars or micro-cracking caused by differential shrinkage of concrete at theinterfaces, were observed in several specimens. These interfacecracks are expected to reduce the flexural rigidity of the wallcross section and soften the moment-rotation response at theinterface, particularly at low lateral-load levels. For all specimens,the wall cross section was assumed to be cracked at the interface,but contact of concrete was considered within the compressivezone. A linear strain distribution was assumed along theembedment length of the longitudinal bars within the top andbottom pedestals, with maximum strains developed at theinterface (per results of the moment-curvature analysis for agiven moment at the interface) and zero strain at a distanceequal to the development length of the bar from the interface(Fig. 9). The axial strains in the longitudinal bars within thepedestals were integrated to obtain cumulative displacements(bar extension) at the interface, which were converted intointerface rotations (via dividing by the neutral axis depth)and used to calibrate the linear elastic stiffness (moment/rotation) of the interface rotational springs. For the wall pierspecimens with axial load, the influence of axial load wasconsidered on the moment-curvature analysis. The interfacesprings were implemented into the model as zero-lengthrotational springs. All other model parameters remainidentical to those used in the previous model formulations.
Lateral load versus top displacement response—As seenin Fig. 7, implementation of the interface springs (denoted inthe figure as εx
exp and Rot.) improves the agreement betweenthe model response prediction and the experimental resultsfor both lateral stiffness and lateral load capacity. Regardingthe lateral load capacity (Fig. 5), peak strength predictedusing the model with the interface rotational springs isapproximately 5 to 10% lower than the model without theinterface rotational springs, and better represents the experimentalresults. The results indicate that cracking at the wall-pedestalinterfaces and extension of the reinforcing bars within thepedestals has a modest impact on the lateral strength, but amore pronounced effect on the lateral stiffness of the testspecimens, especially during the initial cycles. These interfacecracks are less likely to exist in real buildings, as the wall crosssection does not change abruptly as it does in the test specimens,
and the wall is not subjected to the post-tensioning process thatis required in the laboratory to affix the specimens to the strongfloor. In addition, in an actual building, a floor diaphragmlikely exists to restrain axial growth and limit interface crackwidths. The rotational springs, however, were incorporatedinto the model to provide a robust definition of the test conditionsand a consistent comparison between model and test results.
Flexural and shear deformation components—As expected,the model formulations that include the interface rotationalsprings provide significantly improved correlation (Fig. 6).The initial shear and flexural stiffness predicted by themodel that incorporates interface rotational springs (εx
exp
and Rot.) is in good agreement with experimental results.Figure 8 compares the model predictions and measurements
for the flexural deformation contribution to top displacementfor all wall specimens (with the spandrels and pierscompared separately), for three distinct ranges of lateral drift(0 to 0.2%, 0.2 to 0.4%, and 0.4% to the maximum recordeddrift level) by averaging the readings obtained within eachrange for the envelope (peak) points for positive and negativeloading for each specimen, and then averaging obtained valuesfor all spandrel and pier specimens. A similar approach is usedto represent the model results. The combined contribution ofmeasured flexural and shear deformations is also includedto verify data consistency. The combined contribution of themeasured flexural and shear deformations determined usinginstrumentation mounted directly on the test specimen(referred to as internal instrumentation) generally accounts for80 to 95% of the lateral displacement over the specimen heightmeasured with lateral displacement sensors (referred toexternal instrumentation; sensors that are connected to thereference frame) indicating that the experimental data areconsistent. The greatest discrepancy between internal andexternal instrumentation readings exists for the lowest driftrange, indicating that the experimental data for the internalinstrumentation are less reliable in this range, likely due to theimpact of sensor noise when results are combined for multiplesensors in series. As noted in the figure, when interfacerotational springs are included, the model results are closerto the experimental results for the spandrels (Fig. 8). Thissituation is not true, however, for the piers at low drift, whichcan be due to the fact that the internal instrumentationmeasurements underestimate the top displacement byapproximately 20% at low drift levels, which means the flexuraldeformation measurements of the internal instrumentation maybe biased (low) by as much as 20% at those drifts.
Model improvements: empirical strain distribution functions
In the previous section, the model formulation wasupdated to include rotational interface springs and to allowthe use of the average experimental horizontal normal strainsmeasured during the test (versus the extreme conditions of
Fig. 9—Interface crack and rotational spring model.
654 ACI Structural Journal/September-October 2009
either εx = 0 or σx = 0). In this section, a curve-fittingapproach is used to develop a general model formulation thatreasonably captures the distribution of measured averagetransverse normal strains. It is noted that this approach canonly be validated using the limited geometries of the testprogram described herein; therefore, the objective was toassess the potential of the approach. The formulation of themodel is maintained; however, empirical relationshipscalibrated using the results of these specific tests are usedfor the transverse normal strain distribution. The formulationof the empirical relationships considers constant averagetransverse normal strains along wall length, which vary onlyalong the height of the wall.
Average transverse normal strains—The experimentalresults for the spandrel and pier specimens indicated that theaverage transverse normal strains generally reach amaximum value at wall midheight and progressivelydiminish to near zero at the top and bottom boundaries of thewalls (top beam and bottom pedestal). In addition, themagnitude of the average transverse normal strains increasewith the lateral drift applied on the walls. Based on theseobservations, a function was used to account for the shape ofthe transverse normal strains over the wall height, andanother function was used to relate the maximum value oftransverse normal strain at wall midheight to the topdisplacement or drift of the wall.
Based on the test data, the function to account for theaverage transverse normal strains along the height of a wallwas characterized as a geometric function with the maximumstrain value at wall midheight. To define the relative variationof the strain values over the wall height, transverse normalstrain measurements corresponding to the seven instrumentedlevels on the pier and spandrel specimens (level one being thebottom level and level seven the top) were compared to thestrain values measured at midheight, and the correlation wasrepresented by a best-fit relationship for all of the pier orspandrel specimens to establish multipliers for all levels.Although symmetry is expected to occur between the upperand lower halves of the specimens due to the symmetricalloading conditions during testing, some asymmetry isobserved in the transverse normal strain distribution(Fig. 10(a)) possibly due to non-uniform cracking andvariation in material properties.
The distribution of the average transverse normal straincoefficients at the seven levels along the height of the wall iswell represented by the following expression
(1)
where εx is the horizontal normal strain at a specific position(level) along the height of the wall, y is the distance from thatspecific position to the bottom boundary of the wall, h is thewall height, and α is a parameter to be calibrated. Based onresults of best-fit analyses, α-values were determined to be0.7 and 0.4 for the spandrel and pier specimens, respectively.The average transverse normal strain coefficients (averagetransverse normal strain along wall height to midheightaverage transverse normal strain) are plotted together withthe derived model strain distribution functions for both arepresentative case in Fig. 10(a).
The average transverse normal strain measurements atwall midheight are plotted against the total lateral topdisplacement of the spandrel and pier specimens in Fig. 10(b).The data from all specimens show a similar relationshipbetween midheight strain and top displacement, with nosignificant variation between the spandrels and the piers.Thus, a single expression (function) was selected to relate theaverage normal strain at the midheight to the top displacementfor all specimens. The relationship was represented by thefollowing expression
εx(h/2, Δ) = 0.0045Δ1.6 (2)
where εx(h/2) is the horizontal normal strain at wallmidheight, and Δ is the lateral top displacement (cm) of thewall. In terms of drift (δ = Δ/h), the previous expression,although with larger dispersion, can be rewritten as
εx(h/2, δ) = 12δ1.6 (3)
Lateral load-displacement response—The model with theempirical average transverse normal strains and rotationalinterface springs (denoted as εx
model and Rot. in Fig. 7)predicts, with reasonable accuracy, the experimental load-
εx y( )εx h 2⁄( )------------------- sinα y
h---π⎝ ⎠⎛ ⎞=
Fig. 10—Average horizontal normal strains: (a) horizontal normal strain distribution:experimental average distribution (exp) and calibrated distribution function (model); and(b) average horizontal normal strains versus top displacement. (Note: 1 cm = 0.394 in.)
655ACI Structural Journal/September-October 2009
displacement response, providing a reasonably goodrepresentation of not only the lateral load capacity (see alsoFig. 5) but also the lateral stiffness of the walls.
Flexural and shear deformation components—As seen inFig. 6, the initial shear stiffness predicted by all models is ingood agreement with the experimental results. The flexuralstiffness predictions also compare favorably with theexperimental results, provided the rotational interfacesprings are incorporated. According to Fig. 8, the modelresults capture the flexural deformation component contributionwith reasonable accuracy. The model overestimates the flexuraldeformation component of the piers within the low drift range(0 to 0.2%); however, as discussed previously, theexperimental data are less reliable (for example, due tonoise) for this drift range.
SUMMARY AND CONCLUSIONSThis study investigated and verified experimentally a
modeling approach that integrates flexure and shear interactionto reasonably predict the inelastic response of reinforcedconcrete squat walls. The model incorporates RC panelbehavior described by a rotating-angle approach, similar tothe RA-STM into the fiber-based MVLEM. The experimentalprogram carried out as part of this study was used to improvemodel predictions by proving test measurements to enableassessment of various model assumptions by comparinganalytically predicted and experimentally observed behavior.
The experimental results demonstrated that the wall ends(pedestal and beam) constrained the transverse normal strainat such locations, which was not considered in the originalinteraction model. The model with a zero resultant horizontalstress (σx = 0) assumption resulted in a softer load-displacementrelation and a lower bound strength estimate model (lower by13 to 40%). On the other hand, a model with a zero horizontalstrain (εx = 0) assumption (the other extreme case, where theconstraint is highly effective, representative of very shortwalls) produced a stiffer and stronger model result (higherby 18 to 50%). The overall load-displacement responsesobtained from the model were improved, resulting in anoverestimation of the experimental shear capacity by 3 to28% by using the experimentally measured average transversenormal strain. Further improvement of predictions for stiffness,strength, and the relative contributions of flexural and sheardeformations was achieved by modeling the additional rotationalflexibility at the wall-pedestal interfaces due to cracking.
Based on the findings from this specific experimentalprogram, an average transverse normal strain equation (anddistribution function) was calibrated with the experimentaldata to investigate the potential of improving the shear-flexure interaction model. The approach adopted involvedusing a shape function that relates values of the averagehorizontal strain along the wall height, with the maximumaverage horizontal strain (at wall midheight) for both spandrelsand piers, which was also correlated to the top displacement ordrift. A good correlation was obtained for shear strength and thecontribution of flexural deformations to the top displacement,which are 30 to 50% of the total lateral displacement for low driftlevels, but only 15 to 20% of the total lateral displacementfor large drift levels. The experimental shear capacitywas estimated with an error of approximately 10% forindividual specimens.
Overall, the results indicate that the proposed model, afterthe improvements, captures with reasonable accuracy andsimplicity the lateral load-displacement response and monotonic
deformation modes of wall pier and spandrel specimenssubjected to the boundary conditions imposed during testing.Potential model improvements considered as future workinclude consideration of broader set of test results (aspectratios, axial load, and quantities of reinforcement), refinementof the transverse normal strain distribution for generalizedboundary conditions at the pier-spandrel connection regionsin an actual building (for example, the potential difference inend restraint provided by the large end blocks in the testsrelative to connection regions in actual buildings with thicknessequal to the pier or spandrel), and cyclic response predictions.Efforts are underway to incorporate these refinements intothe model formulation to enable building system studies.
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