Engineering Structures 102 (2015) 120–131

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

RC shear walls: Full-scale cyclic test, insights and derived analyticalmodel

http://dx.doi.org/10.1016/j.engstruct.2015.07.0530141-0296/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: beko@vce.at (A. Bek}o), peter.rosko@tuwien.ac.at (P. Rosko),

pierre.pegon@jrc.ec.europa.eu (P. Pegon), damijan.markovic@edf.fr (D. Markovic).

Adrián Bek}o a,⇑, Peter Rosko b, Helmut Wenzel a, Pierre Pegon c, Damijan Markovic d,Francisco Javier Molina c

a Vienna Consulting Engineers ZT GmbH, Hadikgasse 60, A-1130 Vienna, Austriab Vienna University of Technology, Center of Mechanics and Structural Dynamics, Karlsplatz 13, A-1040 Vienna, Austriac European Commission, Joint Research Centre, Institute for the Protection and Security of the Citizen, ELSA Unit, Via E. Fermi 2749, I-21027 Ispra, VA, Italyd Électricité de France SEPTEN, 12–14 Avenue Dutriévoz, 69628 Villeurbanne, France

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

Article history:Received 19 March 2015Revised 28 July 2015Accepted 30 July 2015Available online 24 August 2015

Keywords:Shear wallReinforced concreteCyclic testingHysteresisStrengthDampingAnalytical model

The unpredictable consequences of earthquakes have proven the necessity to study the shear mechanismof low-rise reinforced concrete walls further. Experimental testing remains the preferred approach togather insight into the workings of the material and structural elements as the complex behavior of rein-forced concrete does not lend itself to widely applicable generalized solutions. The contribution dealswith cyclic testing of full-scale low aspect ratio reinforced concrete walls in a principally uniform shearstate. An approach with various novel aspects in testing massive specimens is introduced. The obtaineddata was analyzed resulting in well-defined hysteresis curves. Characteristics of ultimate shear capacity,energy dissipation, damping and nonlinear effects are discussed. It was found that the sequence of loadcycle amplitudes at the quasi-static rate does not significantly influence ultimate strength or overallbehavior of the wall. Yet, hysteretic damping ratios depend on loading history and range from values con-siderably higher than commonly assumed in design to values that are lower. An advanced shear strengthcalculation is presented to relate the findings to previous research. A nonlinear mathematical model isdescribed which is capable of simulating the hysteresis of the tested shear walls. The reported develop-ments are applicable in design of structures.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Excessive horizontal forces may lead to vast structural damagesif structures are not properly equipped to deal with them. In case ofcivil engineering shear walls, commonly of reinforced concrete, aredesigned to transfer horizontal loading. Drifts due to ground shak-ing must be kept within reasonable limits by the stiffness of theshear walls [1]. Earthquakes around the world repeatedly testour understanding of the involved physics and the procedures ofdesign we employ, either with unexpected collapses or surpriseresistances in certain cases. These varied consequences still urgeresearchers to look into the working of structural elements andmaterials at a base level. The scientific research in the area of shearwalls, which started growing in the 1950s [2–4] is still intensivenowadays [5].

In the case of reinforced concrete the parameter variability islarge and the influence of quantitatively different factors is pre-sent, therefore each experimental research is but a contributionto the knowledge base. The applicability of results is limited bythe type of wall and the whole parameter subspace. A number offailure modes can occur depending on parameters such as the typeof the cross-section, reinforcement detailing and quantities, prop-erties of reinforcing steel [6,7], concrete compressive strengthand loading [8,9] and support conditions. The phenomena of struc-tural failure are complex and include concrete cracking, interactioneffects between steel and concrete, steel yielding and concretecrushing in compression [10,11]. For all of the reasons mentionedit is rather challenging to generalize any given experimental ornumerical result [12]. When a test studies a specific type of struc-ture with well-defined loading the outcome is of appreciable value.

Shear walls can, based on their geometry, be divided into twogroups of high-rise and low-rise shear walls. High-rise shear wallsare governed by flexural behavior similar to that of a cantileverbeam [13]. The flexural behavior of reinforced concrete walls hasbeen examined and is theoretically described. Low-rise shear walls

A. Bek}o et al. / Engineering Structures 102 (2015) 120–131 121

on the other hand are governed by shear behavior. There has beenconsiderable interest directed towards low-rise shear walls overthe years. Nonetheless, research in this field is continuing alsostimulated by the common use of shear walls in nuclear powerplants. To accurately predict or model the behavior of shear wallsis an important competence to the engineering community.

Experimental testing remains the preferred method to accu-rately assess capacity and behavior of structural elements.Consequently laboratory tests were envisaged considering severalobjectives. The walls were required to have comparatively largerthickness and a low aspect ratio. A demand had arisen to foregoscale effects, thus specimens were to be full thickness using stan-dard diameter reinforcing bars and aggregate size. Informationabout ultimate strength, hysteretic behavior, hysteretic dampingand failure mode was to be gathered under distinct yet comparableload histories. This resulted in the preparation of a complete failuretest for a relatively stiff low-rise wall with a thickness of 0.4 m.Full-scale testing of concrete shear walls is expensive, especiallyso for more massive specimens, and only a small number of testshave been carried out. Four walls have been tested in two groupsof load regimens. The joint research effort resulted in an innovativetesting method at the European Laboratory for StructuralAssessment. A special loading device was designed and con-structed to facilitate the cyclic loading of the strong wall in a closedsystem setup. Elaborate boundary conditions were applied at thetop to simulate the element as if it was part of a larger structure.This was done in accordance with boundary conditions of wallspresent in nuclear power plants (NPP), which prevent wall rotationat the top. The whole setup was instrumented to gather data and tocontrol the experiment. Cyclic uniplanar loading in two oppositedirections was gradually increased until complete failure of thespecimen.

The test data is used to characterize the response of the shearwalls. As the bearing capacity of the wall was one of the main con-cerns an analytical assessment of the wall strength based on plastictheory is compared to the test results. For possible simulation ofwall response under different loading conditions an analyticalmodel of the reinforced concrete shear wall is developed on thebase of Takeda’s model [14] and tuned by utilizing the experimen-tal results.

2. Laboratory testing of RC walls in shear

The devised tests had two main distinctive features which were,first the thickness of the wall which was robust enough not to letthe wall fail in buckling and second, the top of the wall was kepthorizontal, with a corresponding axial load, to keep the wallresponse predominantly in shear as much as it was achievable.

These tests were carried out within the scope of the IRIS projectwhich targeted industrial risk assessment and management as

Fig. 1. Designed reinforcement (spec

basic research. The tests were a result of the collaboration of mul-tiple parties. Based on the incentive of Électricité de France thespecimens and loading apparatus were designed jointly by theJoint Research Centre (JRC) of the European Commission andVienna Consulting Engineers who also commissioned the hard-ware. The lab tests were carried out by the European Laboratoryfor Structural Assessment (ELSA) of JRC at their facilities in Ispra,Italy. The specimens were loaded by cyclic quasi-static loadregimens.

2.1. Wall specimens

The reinforced concrete specimen comprised three main parts.The actual wall part which was 3 m long, 1.2 m high and 0.4 mthick was lined with two beams at the bottom and the top witha cross-section of 1.25 � 0.8 m. The beams were cantilevered0.5 m on either side of the wall which resulted in the 4.0 m totallength of the specimen. The design of the reinforcements and prin-cipal dimensions are shown in Fig. 1. The middle part of the spec-imen classifies as a rectangular wall without boundary stiffeningelements. With an aspect ratio of 0.4 it also classifies as alow-rise or short wall. The thickness of the wall was full-scale,characteristic e.g. of NPP electric buildings, which allowed for theuse of realistic reinforcement diameters and sizes of aggregates.The choice of the aspect ratio was dictated by the failure modeto be provoked, namely shear. Actual sizing in terms of the lengthand height of the specimen was based on laboratorycircumstances.

The reinforcement was characteristic of a wall to undergo shearas a primary mode of loading. The edges of the wall were strength-ened with concentrated reinforcement to take the bendingmoment occurring in the wall. The mesh of vertical and horizontal(shear) reinforcement was designed based on structural principlesand suggestions by EDF to be representative of walls in NPPs. Asthe wall is short direct load transfer is possible and engagementof shear reinforcement for force transfer was not foreseen.Vertical reinforcement runs from the bottom of the bottom beamthrough the wall to the top of the top beam creating a solid connec-tion of the segments.

The specimens were concreted in a rotated vertical position toreduce inhomogeneities along the height of the wall. The concretemix used was of 54 MPa cubic strength and the reinforcementyield strength was 500 MPa. The planned loading frequency wassmall and no notable increase in the strength of the materialswas expected.

An important aspect of the specimen design and manufacturewas the load transfer from the loading apparatus to the concretewall. The wall was not loaded directly but through the upper andlower containment beams. The solution used, was a pair of 3 cmthick steel plates (see Fig. 1) on both sides of the containment

imen dimensions given in cm).

Fig. 2. Force transfer mechanism layout (framed circles are hi-tensile prestressing bars, unframed circles are connection studs in the concrete, left), detail of prestressing-barheads with a safety beam running across (right).

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beams which were connected to the concrete through connectingstuds and later casting. The force transfer between the loadingframe and the steel plates of the specimen was achieved by steelto steel friction. A large enough normal force was required to allowfor the transmission of the total shear force. The necessary frictionwas obtained by using prestressing bars running through the con-tainment beams of the specimen anchored on the outer surfaces ofthe loading frame. The solution is shown in Fig. 2.

2.2. Apparatus and instrumentation

Due to the robustness of the tested wall the forces required tocarry the specimen to failure could have damaged the reaction wallat the ELSA laboratory. For this reason the shear testing was con-ceived in a self-equilibrating manner. The complete loading equip-ment consisted of three systems: the main loading system, asystem controlling the rotations at the end of the actuators and arestraining system that controlled the out-of-plane motion. Themain loading system, a four part jaw-like steel frame pictured inFig. 3, was designed to be placed around the lower and upper con-tainment beams. The frame had been conceived so that the hori-zontal load was applied at the height of the centerline of thewall as not to introduce any bending at that section of the speci-men. The horizontal load was anti-symmetrically transmitted tothe lower and upper beams. This type of loading would in principleresult in zero bending moment acting on the system as a whole.But for lack of symmetry in the setup and finite stiffness of theloading frame a small rotation could be induced and so the lowerframe was anchored.

The horizontal actuators were fixed to the lower part of themain loading frame and pushed against the vertical plates of theupper part of the system. The four 300 ton actuators were uniaxialdevices capable of pushing in positive and negative direction. Allfour actuators worked in a synchronized fashion to exert therequired loads. A smooth evolution of the forces and displacementswas obtained by adopting the control of the horizontal loading inthe form Fi þ auTi ¼ ramp, with i ¼ 5;6;7;8 (refer to Fig. 4); whereFi are the forces on the horizontal actuators, uTi are the displace-ments on the Temposonics sensors on the actuators and a is a con-stant chosen in relation to the performance and safety of thecontrol [15].

The shearing of the wall also translates into vertical displace-ments and rotations at the point of load application to the framewhich might exceed the shear or bending capacity of the acting

rod. In order to protect the actuators a sliding and a rotation mit-igating mechanism has been devised (see Fig. 3). The sliding mech-anism was based on the ‘‘Xlide’’ antifriction disk between twocontact plates. The rotation mitigating device was a round boxaround the load cell closed by a gouged steel plate, which allowedfor rotation over the head of the load cell. Lastly, to stabilize thewall laterally and to prevent spurious distortions out of plane thetop section of the main loading system was braced on one sideagainst the reaction wall by means of two steel bars capable ofresisting 550 kN buckling load.

To force the specimen to deform in shear mode as much as pos-sible, considerable effort was devoted to keeping the top of thespecimen horizontal without introducing an additional verticalforce overall. To compensate the effects of the lack of symmetryfour additional actuators were placed around the specimen verti-cally. These actuators operated in a push–pull mode, thus beingable to exert forces in both directions. These jacks were also usedto simulate the additional vertical load on the structure. The con-trol of these actuators was implemented in such a way that equi-librium of forces was to be maintained diagonally, i.e. assumingnotation according to Fig. 4: F1 þ F4 ¼ 0 and F2 þ F3 ¼ 0 or actuallyfor nonzero vertical forces the sum of each couple had to equal halfof the required vertical load. Also, in-plane and out-of-plane rota-tion was controlled by the condition posed upon displacementsu1 ¼ u4 and u2 ¼ u3 [15]. The vertical forces were being continu-ously adjusted during the entire duration of the test by aclosed-loop control system.

The objective was to follow the behavior of the stiff wall incyclic shear loading up to its failure. To measure the sheardisplacement four transducers were mounted at the top and bot-tom of the wall on each end. These Heidenhain sensors (H5–H8)were actually fixed to the containment beams close to the wallsince failure of the wall could render them useless. To measurethe displacement of the wall in the vertical direction therewere four Gefran potentiometer transducers (G1–G4) mountedat the four corners of the wall between the upper and lowercontainment beams. Load cells were installed as part of therotation system at the end of the action rods of horizontal actu-ators. There were eight 5 MN load cells in the horizontaldirection.

For the control of the horizontal motion four Temposonicstransducers were mounted on the horizontal actuators. The verti-cal control was serviced by four Heidenhain transducers, each nextto the vertical actuator, and standard 1 MN load cells at the heads

Fig. 3. Prepared and instrumented set up (top). Exploded-view of the loading frame (bottom).

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of these actuators. It was assumed that the deformations of theloading system were small and there was no sliding between theloading system and the wall. Nonetheless, there were extra sensorsmounted to check the relative sliding of the main loading framewith respect to the specimen.

Finally, an optical measurement system was set up to monitorthe motion and degradation of the wall. Both faces of the wall werekept void of conventional testing devices not to hinder the applica-tion of high resolution cameras. Two stereo pco.edge cameras(with scale of 1.2 mm/pixel) were placed on both sides of wall.

Fig. 4. Main instrumentation of the specimen.

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These cameras were used to monitor displacements, crack initia-tion, evolution, opening and sliding. Except for the standardGaussian square targets the surface of the specimen had a treat-ment of a black and white natural–artificial pattern that was notrepetitive but had repeating features.

2.3. Testing and output

Four specimens were tested in a comparable fashion. Each wallwas subjected to a combination of constant vertical load of 600 kNapplied at the top of the shear wall and variably increasing hori-zontal load. Generally on all tested walls ten initial cycles of500 kN were performed. For the ‘‘monotonically increasing targetload’’ regimen (walls 3 and 4) the load was increased in steps of500–1000 kN and two cycles were performed at each load leveluntil failure. A load cycle was defined as reaching the target loadin one direction, retracting, reaching the target load in the oppositedirection and centering the wall. On the second cycle only the max-imum displacement recorded at the target load on the first cycle

Table 1Load histories for the tested walls.

Wall 1 Wall 2

0.5MN/2c 02/10/2012 0.5MN/2c 20/11/20120.5MN/7c 04/10/2012 0.5MN/7c 22/11/20120.5MN/1c 0.5MN/1c7.0MN/2c 08/10/2012 7.0MN/2c 23/11/20124.0MN/2c 4.0MN/2c3.0MN/2c 3.0MN/2c 26/11/20126.0MN/2c 08/10/2012a 6.0MN/2c5.0MN/4c 5.0MN/2c 27/11/20125.5MN/1/2c 09/10/2012 5.5MN/2c�5.5MN/0.5c 10/10/2012 6.0MN/2c6.0MN/2c 6.5MN/2c 28/11/20126.5MN/2c 10/10/2012a 7.0MN/2c7.0MN/2c 7.5MN/2c 29/11/20127.5MN/2c 11/10/2012 1.0MN/2c 04/12/20128.0MN/2c 8.0MN/2c8.0MN/1/2c 12/10/2012 Final cycle 05/12/2012Final cycle

Notes:a Continuous measurements without power down.b Absent from the processed data set.

was followed up. The second type of loading history had anunevenly changing target load and was characterized by a largecycle at the beginning, followed by lesser cycles before aimingfor the ultimate load level (walls 1 and 2). The detailed load histo-ries of these tests are summed up in Table 1. The ‘‘c’’ characterdenotes cycles.

Testing at different load levels was treated as separate stages.The loading stage of a specimen followed this basic sequence:increase global vertical force from 0 to 600 kN – equilibrate forcesin the vertical actuators by offsets in vertical displacement – startthe motion of horizontal actuators – carry out all cycles of the stage– bring horizontal actuators back to middle position – decreasevertical force to 0 – switch off pressure system. In special casescontinuous measurements were done for wall 1 from one stageto another, marked by an asterisk at the date in Table 1. In thesecases the vertical load was maintained and the pumps were notshut down.

2.3.1. Load–displacement curvesAs representative cases wall 2 and wall 4 were selected, based

on preliminary data analysis, to represent both loading regimens.This enables comparison of the behavior of a wall with increasingload levels (wall 4) and one which experiences a major load cyclein the beginning and lower cycles afterwards, before increasing tothe ultimate load. The plots in Fig. 5 show the performance of thewalls in terms of the main parameters, i.e. horizontal load, sheardisplacement, vertical displacement. Shear displacement is definedas the differential horizontal displacement between the top andthe bottom of the wall. Generally, wall 4 is presented as the firstbased on its simpler load history. Major points of interest havebeen marked in the data. The complete work diagrams are pre-sented in Fig. 6.

The walls had increased in height considerably over the courseof the tests. Vertical displacements, plotted against shear displace-ments in Fig. 7, show that wall 4 was some 16 mm higher at thepoint of collapse and remained 11 mm higher upon itsre-centering after the failure in the initial direction (‘‘+’’). The timeof the first major crack appearing is defined as the point when thevertical displacements start to accumulate. This occurs when thetwo surfaces slide enough with respect to each other and aggregateinterlocking prevents perfect closure of the crack. The onset ofyielding in the reinforcement is difficult to accurately determine

Wall 3 Wall 4

0.5MN/9c 25/05/2012 0.5MN/2cb 11/07/20121.0MN/2c 0.5MN/7c 12/07/20122.0MN/2c 05/06/2012 0.5MN/1c3.0MN/2c 1.0MN/2c4.0MN/2c 2.0MN/2c 16/07/20125.0MN/2c 06/06/2012 3.0MN/2c6.0MN/2c 4.0MN/2c6.5MN/1c 07/06/2012 5.0MN/2c 17/07/20126.5MN/1c 08/06/2012 6.0MN/2c7.0MN/2c 6.5MN/2c 18/07/20127.5MN/2c 7.0MN/2c 24/07/20128.0MN/1c 11/06/2012 7.5MN/2cFinal cycle 12/06/2012 8.0MN/2c

Final cycle 26/07/2012

Fig. 5. Horizontal load, shear displacement and vertical displacements over pseudo-time wall 4 (left) and wall 2 (right), ‘‘c’’ denotes cycle.

Fig. 6. Load–displacement curve for wall 4 (left) and wall 2 (right) separated into cycles.

A. Bek}o et al. / Engineering Structures 102 (2015) 120–131 125

for lack of direct instrumentation of the reinforcing bars.Nonetheless, the position after which the effect of steel yieldingis apparent as a significant stepwise increase in the residual

Fig. 7. Shear displacement vs. vertical displacement for sensors G1–G4 (wall 4).

vertical displacements can be identified (for detailed view seeFig. 8). This corresponds to the cycle in which wall 4 reaches about5.1 mm or 4.25‰ vertical elongation and 6.6 mm or 2.3‰ sheardistortion, and wall 2 reaches 4.1 mm or 3.4‰ vertical and5.3 mm or 1.8‰ shear deformation respectively. Also the pinchingaround the origin becomes more pronounced after this point. Toclearly discriminate which bars are under yielding is not possible;nonetheless judging from the deformation of the wall, the horizon-tal reinforcement in the direction of shearing is not the one underexcessive strain. The point of collapse is well defined by the suddendrop of the horizontal load, which in this case also indicates thefailure was due to crushing of the concrete. By the end of the testthe specimens had developed, provoked by load reversals, awell-defined pattern of parallel diagonal cracks in both directions,shown in Fig. 9. The images from the optical system present a qual-itative view of the cracking, in terms of absolute vertical shifts, byevaluating the optical flow on a distorted image.

3. Results and discussion

The behavior of a reinforced concrete shear wall cannot be sep-arated into distinct linear and nonlinear regions because concretehas an inherently damage behavior. Although steel has a defined

Fig. 9. Photographs of the crack pattern at the end of the test of wall 2 (above). Optical flow as absolute vertical shifts in the penultimate cycle in one direction and theterminal cycle in the opposite direction (below, courtesy of Ph. Capéran, ELSA, JRC).

Fig. 8. Partial detail of shear displacement vs. vertical displacement for sensors G1–G4 (wall 4 left, wall 2 right). Effect of the reinforcement yielding is visible after unloadingin the last cycle plotted.

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yield point, concrete is nonlinear from initial loading and so thespecimen as a whole shows a composite behavior. The hystereses,presented in the previous section, show well the cyclic strengthdegradation as the second cycle on a damaged specimen doesnot reach the set load level at an equal horizontal displacement,which was the controlled variable. Progressive degrading phenom-ena occur, starting with concrete cracking, through reinforcementyielding until the specimen fails by concrete crushing in the middleregion of the wall. The different load histories of the two selectedwalls did not result in noteworthy differences in the global behav-ior of the wall. Judging from the points of interest in the work dia-grams the behavior of the wall does not significantly depend on theload history, meaning the succession of smaller or larger hystereticexcursions. Load levels for each feature are very similar includingthe ultimate load and the failure mechanism is the same. Theseresults can be explained by the quasi-static rate of loading andimportantly by the symmetrical loading cycles.

According to damage mechanics the behavior of the shear wallspecimen can be characterized as unidirectional. During the cyclicloading of the wall to specific directions of the shearing two dis-tinctive sets of shear cracks form in the wall (see Fig. 9). Whenthe shear force changes sign one set of cracks tends to closeand its presence has a reduced effect in the wall behavior whilethe other set of cracks tends to open and becomes the dominantstiffness reduction phenomenon. During opening and closing ofexisting cracks the reinforcement in the disturbed areas isactivated.

In the load–displacement curves of the tests in Fig. 6 pinchingaround the origin can be seen. The phenomenon of pinching isdue to sliding between the already cracked surfaces of concretebefore they come in full contact. This can be observed in the hys-teresis curves where the sign of the shear force changes upon loadreversal. This action of sliding across a shear crack can be explainedin terms of Coulomb friction. Under extensive shear load a crack

A. Bek}o et al. / Engineering Structures 102 (2015) 120–131 127

has been formed in the wall. As the load is reduced to zero thecrack remains partially open. Once the load is applied in the oppo-site direction friction across the crack is small initially, but as thecrack begins to close, friction gradually increases which translatesinto increased resistance to sliding. Furthermore, if reinforcementyielding has occurred upon the crack opening, it is evident thatin order to close the crack completely the reinforcement must yieldin compression. An interaction between two phenomena existshere: sliding across the shear cracks and yielding of the reinforce-ment. Both mechanisms contribute to the plastic behavior of thewall. The features related to these phenomena are observable inthe load displacement plots in Figs. 6 and 7. Next, some propertiesfor the selected specimens are calculated and discussed.

3.1. Hysteretic damping

Damping is a fundamental property of any structure and a cru-cial one for dynamic applications. It also can be difficult to assessrealistically. Raggett [16] proposed a practical way of assessingdamping for real structures. Essentially, it is based on fundamentalmechanics of energy dissipation by viscous damping of a SDOF sys-tem. Considering a SDOF system in a steady-state motion with har-monic excitation, defined by angular frequency x, the energydissipated by viscous damping can be expressed as

ED ¼ 2pf xxn

kD2 where k is the stiffness, f is the damping ratio, D

is the maximum displacement in the cycle and xn is the naturalangular frequency of the system. The dissipated energy is propor-tional to the square of the amplitude of motion. The elastic energyof the system is related to the maximum amplitude of the cycle by

the relation ES ¼ 12 kD2. Equivalent viscous damping is commonly

defined by equating the energy dissipated in a vibration cycle of

Fig. 11. Equivalent damping estimates for each test cycle

Fig. 10. Method to evaluate equivalent damping according to Muroi.

an actual structure with that of the equivalent SDOF viscous sys-tem. The energy dissipated in the actual structure in a hysteresisloop is the area enclosed by this loop and for discretized data canbe computed using the trapezoidal rule. Placing it equal to theenergy dissipated by damping in a viscous system yields

feq ¼1

4p1

x=xn

ED

ESor feq ¼

14p

ED

ESfor x ¼ xn ð1Þ

The equivalent stiffness k required to compute the elasticenergy may be calculated by doing a linear regression over theloop.

As an alternative, for the sake of comparison, Muroi’s modifica-tion of Jacobsen’s approach presented by Igarashi [17] was alsoimplemented. The simplest explanation of the method is a graph-ical one depicted in Fig. 10. The equivalent damping ratio is com-puted as the area of a half-loop over 2p times the area of atriangle defined by the point of maximum displacement (pointD) and the middle point of the loop width at zero force level(point F).

Using these two methods the equivalent damping ratios foreach test cycle were computed and are presented in Fig. 11. Theultimate ‘‘cycle’’ in which failure occurred is omitted. The resultingdamping ratios are taken to represent material damping. For wall 2the first ten cycles (at 0.5 MN) show a more or less constant damp-ing. The same is true for the first eight cycles of wall 4. Then, as theload gradually increases the damping ratio decreases on average oftwo cycles to equal target load. Reviewing the energy plots inFig. 12, one can see that the proportion of the dissipated and elasticenergies changes in favor of the elastic energy, which means longand narrow cycles, mostly as an effect of pinching. This means thatin general the damping ratio decreases with increased pinching inlarge deformation cycles. For such cycles the dissipated energy canget lower in proportion to the spring energy. It shows, that withstrong pinching present, damping ratios are in proportion to‘‘new’’ damage in the cycle rather than to the total cumulativedeterioration of the specimen.

In case of wall 4 it should be noted that the first cycle to adefined load level always has a higher damping ratio than the sub-sequent cycle which reached a lower maximum load level. Lookingat the result for wall 2 a similar trend is not visible for the lowerlevel load cycles after the first major cycle to 7.0 MN (No. 11). Asloads are later raised beyond the 7.0 MN level (cycle 31) the damp-ing ratio increases again and it does show the drop in value on thesecond repetition cycle for the same load level. The notable excep-tion are cycles 33, 34 which can be classified as small displacementcycles, meaning a low amount of pinching. Based on these out-comes we may state that damping in the specimens is highly

by both methods for wall 4 (left) and wall 2 (right).

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dependent on the current state of the composite material and itshistory. Given a cycle when a higher load is reached than the pre-vious maximum, that is to say the material is pushed beyond itsprevious limit, an increased damping is realized e.g. wall 4 in cycles15 or 19. The element vibrating within its previously establishedlimits can be expected to exhibit comparatively lower amountsof damping, like wall 2 in cycles 12–32. Moreover, more pinchingyields lower damping. Generally, equivalent damping is an equiv-alent to all mechanisms of energy dissipation involved. In our casethe damping ratios computed are assumed equivalent to the mate-rial dissipation in the hysteresis loops, since no significant energyloss was possible through the support boundary conditions andalso due to the slow rate of the tests. Since reinforced concrete isa composite material and it deteriorates gradually, slippage andfriction bonding along the cracks occur. As bonds are being loston the macro level the overall damping exhibits besides usualmaterial damping a strong character of structural damping.

3.2. Shear capacity

To assess the strength of a shear wall realistically, an elaborateanalysis was undertaken based on the work of Liu [18]. The author

Fig. 13. Shear with strut and homogeneous stress fields for lower bound solution (left), f

Fig. 12. Energy dissipated in each cycle and elastic energy by Raggett and Muroi for wacycles.

describes two distinct approaches for estimating the load bearingcapacity in shear of a reinforced concrete wall. These are an upperbound solution based on the plasticity theory and a lower boundsolution as an extension of the known strut-and-tie approach.The wall is subjected to transverse force and a vertical normalforce. The basic assumptions are that the transverse load P is trans-ferred to the wall by a top beam or slab and the wall channels theforce to the bottom beam or slab; the wall acts in plane stress andshould the wall have boundary stiffening elements they act asstringers carrying a pair of tension and compression forces.

By postulating the yield force as a force that produces a stressfield lying on the yield surface, then any load that corresponds toa statically admissible stress fields will be smaller or equal to theyield load. This constitutes the lower bound solution of whichthe maximum must be sought to find the best lower bound esti-mate of the bearing capacity. Generally a shear wall is suppliedwith a mesh of minimum reinforcement. The lower bound solutionfor a wall with stirrup reinforcement takes the horizontal bars intoaccount by a combination of compressive strut and homogenousstress fields (Fig. 13, left). In a wall which is wider than its heighta compression strut develops to the base of the wall and theremaining area is assumed to be divisible into three triangular

ailure mechanism of shear wall for upper bound solution (right, adapted from [18]).

ll 4 (left) and wall 2 (right). Average values are given for the initial low load-level

A. Bek}o et al. / Engineering Structures 102 (2015) 120–131 129

fields with the same stress levels. Based on the geometric relationsof the stresses for the concrete and reinforcement, Liu derives a setof nonlinear equations which represents the maximum solution ofthe lower bound approximation and requires methods of mathe-matical optimization to be solved.

The upper bound solution is based on the work equation whichequates the work of the external load to the work absorbed by thebody. Any load derived from the work equation for a geometricallyadmissible failure mechanism will be greater than the yield load. Aminimization of the resulting set of equations provides the bestupper bound for the bearing capacity of the wall. A modifiedCoulomb material model with a sliding resistance and a separationresistance had been assumed. To determine the work absorbed in afailure mechanism, a flow rule for perfectly plastic materials hadbeen selected. Consequently, a kinematical discontinuity had beenpresumed in form of a yield line separating the body in rigid parts(Fig. 13, right). Such discontinuity is a mathematical representationof a highly deforming narrow band. Employing the normality con-dition, the rate of internal work dissipated per unit area of the yieldline is derived. Equating this to the external work yields the workequation which is to be minimized to retrieve the upper boundsolution. Liu [18] provides this extremal solution in a closed form.

In both cases, the variable to solve for appears in the form of smf c

,

where s is the shear stress at the base of the wall, f c is the concretestrength and m is the shear effectiveness factor. It is noteworthy todiscuss this coefficient. Any accordance between test and theorycan only be achieved by the introduction of an effective compres-sive concrete strength for shear f ce ¼ mf c . As a qualitative explana-tion is lacking, empirical formulas from tests must be adopted. Theavailable literature provides several formulae to compute m as

m ¼ 0:7� f c=200 for f c < 50 MPa and m ¼ 1:9=f 0:34c for

f c > 50 MPa while m must be less than or equal 1. These formulashad been updated by Liu [18], to include the effect of the normalforce on the change of the value of the effectiveness factor asm ¼ 0:8� f c=200� 0:725N=ðAfcÞ for f c < 70 MPa and

m ¼ 1:9=f 0:34c � 0:725N=ðAfcÞ for f c > 70 MPa. Upon evaluation for

the tested walls the effectiveness factor for shear can range from0.43 to 0.54. The results of the upper bound and lower bound solu-tions for two selected values of m are given in Table 2 along withthe ultimate test loads for each specimen. A value of m ¼ 0:41,lower than the values given by the provided formulae, was foundto provide a better match to the test results. This essentially lowershear strength is thought to be a consequence of the hysteretic

Table 3Shear wall strength based on design codes.

Design code Formula (unified for SI units)

EN 1992 clause 6.2.5 [19] vR ¼ cfct þ lrn þ qf yðl sin aþ cos aÞ 6 0:5mf c

ACI 349 clause 11.10 [20] v f ¼ 0:219ffiffiffiffiffif c

pþ 0:2ru þ 0:8qt f y

� �� /

ACI 349 clause 21.6 [20] v f ¼ 0:25ffiffiffiffiffif c

pþ qt f y

� �� / (for ac ¼ 3)

Hirosawa’s formula [21]

Note: f c – compressive strength of concrete; f ct – tensile strength of concrete; f y – yield stratio resisting shear; a – angle between the shear friction reinforcement and shear planeconcrete cracked in shear (0.468); / – behavioral factor of safety (0.75).

Table 2Shear capacity of wall specimen based on Liu [18] compared to test results.

LB-lower bound; UB-upper bound; Wx-specimen number

m ¼ 0:43 PLB ¼ 8500 kN; PUB ¼ 8866 kNm ¼ 0:41 PLB ¼ 7998 kN; PUB ¼ 8500 kNTest results

PW1 ¼ 8053 kN; PW2 ¼ 8279 kN; PW3 ¼ 8117 kN; PW4 ¼ 7997 kN

response evoked by the cyclic nature of the test loading. The thick-ness of the wall could be another contributing factor, since a planestress state had been assumed for the analysis.

Finally, it is informative to compare the shear capacity of thetested walls with design provisions. A concise overview is givenin Table 3. Particularities to the evaluation and limits of the appliedformulae can be found in the respective codes. In addition thewell-known Hirosawa’s formula was evaluated. The code basedestimates of the wall strength are significantly lower than the val-ues reached in the tests. The reason for this is the failure mecha-nism that was triggered by the tests. All walls failed in shearcompression, i.e. by crushing concrete in the center region afterdeveloping a series of diagonal cracks. The code treatment of wallsunder shear is by way of a shearing cut of the cross-section.

3.3. Hysteretic model

A numerical model has been developed based on the work dia-grams of the tested specimens. As a base for model development,the modified Takeda hysteresis rule was chosen. This is a discretepiecewise linear model. The original set of rules was developedby Takeda in [14] and it was based on a trilinear backbone curve,featured degrading unloading stiffness and distinguished betweensmall and large cycles. There have been two major modifications ofthis model by Otani [22] and by Litton [23]. The latter simplifica-tion was chosen for further development to have liberty in adopt-ing the basic rules to a shear specific case, while also maintaining arelative simplicity of the model. Here we describe the final numer-ical model in short.

Strength w/o safety factors [kN] Strength factored for safety [kN]

5805 56555814 4360

6849 5137

5628

rength of reinforcement; rn; ru – normal stress per unit area; q; qt – reinforcement; c, l – factors depending on the roughness of the interface; m – reduction factor for

Fig. 14. Nonlinear hysteresis model with pinching (shown with trilinear backbone).Model status can be: YI – yielding, LO – loading, UN – unloading, PI – pinching, QE –quasi-elastic (possible to occur once only to the direction opposite of the firstyielding /1/ until yielding is reached /5/).

Fig. 15. Response of the numerical model overlain over test work diagram in terms of dimensionless force over ductility tuned for wall 2 (left) and validated by data of wall 4(right).

130 A. Bek}o et al. / Engineering Structures 102 (2015) 120–131

Two separate sets of governing parameters were adopted forthe positive and negative quadrants each with four variablesa; b; c; d. The governing equations of the model are defined as fol-lows (for definition of the symbols see Fig. 14)

- Unloading stiffness ku ¼ k0umax

uy

� ��a,

- Reloading point Yþ ¼ ½uþ; fþ�; uþ ¼ umaxuy� b umax�uy

uy,

fþ ¼ f y þ kiðuþ � uyÞ,

- Pinching stiffness by Kato et al. [24] ks ¼ f maxumax�uq

uminuy

� ��c,

- Pinching point P ¼ ½uP ; f P �; uP ¼ �ksuqþkuux

ku�ksd, f P ¼ ksðuP � uqÞ.

The equation of motion was adopted in the dimensionless form of

€lþ 2fxn _lþx2nef sðl; _lÞ ¼ �x2

n

€ugðtÞay

; ð2Þ

where l is the ductility, f is the viscous damping ratio, xn is theangular natural frequency of the inelastic system vibrating within

its elastic range, ef s is the dimensionless nonlinear force–deforma-

tion function, €ug is the ground acceleration and ay ¼f y

m is the accel-eration of the mass needed to produce the yield force [25]. The rulesthat govern the advancement of the hysteresis are summed up here

for the positive quadrant i.e. for ef s > 0 with examples based onFig. 14 (c stands for cycle):

– If in terms of ef s the curve is advanced beyond a yield point (1, 2,5, 6, 13), or a pinching point (9, 12, 16), or a previous maximumlesser than the backbone (10, after a small cycle); Newton iter-ations are used to refine the transition of the stiffness. Theseoptions are also conditioned by a positive velocity, no reloadingon the unloading path and active pinching for crossing a pinch-ing point.

– A set of rules, analogous to the first, is applied with the differ-ence that if in the very same time step there is change of direc-tion in the velocities, a peak is found by Newton iterations andthe model goes into unloading.

– If there is sign reversal of ef s , i.e. it was negative in the previousstep (8, 15) the stiffness is directed towards Yþ ¼ ½uþ; fþ� (17)via a pinching point P ¼ ½uP ; f P � (9, 16) except:– If uq > 0 and uq < up;c�1 advance via the previous pinching

point ½uP;c�1; f P;c�1�.– If uq > 0 and uq > up;c�1 point towards Yþ (17) directly,

unless an unfinished path to Yþ exists defined by point

Ymax ¼ ½lmax; f ðlmaxÞ� (10), then progress via Ymax (10) toYþ (17).

– If unloaded from pinching (120) and uq > 0 progress to Ymax

(3) or via Ymax (10) to Yþ (17) if applicable.– Upon simple velocity change the model goes into unloading

/UN/.– Velocity change on the unloading path leads to reloading along

the same path until the yield point, at which unloadingoccurred, is reached.

It was observed that the steering parameters would benefitfrom a dependence on the amount of displacement, most visiblyin the unloading stiffness parameter a. In a final refinement ofthe model the steering variables a; b; c; d have been extended to afunctional form written for a in the positive (P) quadrant as:

aPðlÞ ¼ aP1ð1� sPðlÞÞ þ aP

2sPðlÞ; ð3Þ

sPðlÞ ¼ 0:5þ 0:5 tanhl� 0:5 lP

1 þ lP2

� �lP

2=lP1

; ð4Þ

where aP1; aP

2 are the tuned values of the steering parameter a atpositions lP

1; lP2, the limits of the transition interval. The hyperbolic

tangent function provides a smooth change between the values andreturns a near constant value below and above the transition inter-val. Also the backbone curve was extended to quadrilinear as ithelped to accurately follow the shape of the test data.

The implementation of the model has been developed with anunconditionally stable version of the Newmark time integrationscheme. It was observed that for the test input the simulationhas a low sensitivity to the value of the damping ratio and to thesize of the times step, provided the latter is small enough to assureconverge. The conversion of the forces applied in the tests topseudo-accelerations is governed by the value of the assumedmass, which had to be tuned to match the quasi-static testresponse. The response of the model to the input loading of wall2 is shown in Fig. 15. A good agreement has been achieved in gen-eral. A discrepancy shows in the last computed cycle, where thereis a large shift (b) of reloading in the positive quadrant and a minis-cule shift in the negative quadrant, which in fact seems ratherincompatible with the rest of the hysteresis. We put forth that tocapture this feature at nearing failure further extension and thuscomplication of the model would be required. Nonetheless, for fur-ther numerical applications the model is accepted as satisfactory.The model was validated by the test data of wall 4 (see Fig. 15).

A. Bek}o et al. / Engineering Structures 102 (2015) 120–131 131

4. Conclusions

Strong short thick shear walls have been tested in a novel waywhere shearing was the dominant mode of behavior all the way tofailure. The shear wall had a thickness of 40 cm to be a close rep-resentation of a structural part existent in nuclear facilities. A spe-cial loading device was designed and constructed allowing asymmetrical closed system loading. An elaborate displacementand rotation control was needed at the load transfer point.Additionally, the top of the specimen was held horizontal to pro-mote shear behavior. The quasi-static loading was administeredin cycles of preconceived load levels. The unilateral response ofthe wall was monitored by a group of horizontal and vertical trans-ducers. On the resulting load displacement curves features ofdegradation are identifiable. Wall 2 and wall 4 were selected asrepresentative for primary data processing. Points of interest havebeen identified in the data and it was established that the load his-tory does not affect yield force significantly at the rate of speed thetest were conducted. It does however influence the displacementat which the yield force is reached. A greater number of graduallyincreasing smaller cycles leaves the concrete more cracked andyield force is reached at a higher displacement in a major cycle(wall 4) as opposed to the case when a major cycle was adminis-tered right away (wall 2).

The data was analyzed to assess hysteretic damping. The result-ing damping ratios were found to depend on the cycle amplitudeand damage history. Estimates of damping, calculated by twomethods, range from values significantly higher than thoseassumed by design practice to about half of those, yet remain rel-atively high on average. This should be viewed positively as thedamping values essentially reflect available material dampingonly, which is however related to the geometry of the reinforcedconcrete element and its loading.

The failure loads in the initial loading direction were closelymatched around the 8.1 MN average. The mechanism of the failurewas identical in all cases specifically the concrete was crushed inthe middle of the wall. A check of the strength of the wall was car-ried out with a method that reflects the failure mechanism of theshear wall. The detailed mathematical analysis resulted in a veryaccurate estimate of its strength. The shear effectiveness factorthough, was shown to be slightly lower than suggested by earlierresearch, for which the cyclic nature of the loading is consideredto be the major contributing factor. In this light it is adequate toconclude that the strength reduction of the shear wall due to cyclicloading is minor as compared to its static strength. When com-pared to code based design checks of the shear wall its experimen-tal strength showed great reserves.

Exploiting the test data a mathematical hysteretic model wasdeveloped based on Litton’s modification of the original Takedamodel. The final development is a single degree of freedom nonlin-ear material model with the important feature of pinchingincluded. The model is tuned by two sets of four steering parame-ters for the two opposite quadrants. Upon the observation that themodel would require different steering parameters for small andlarge displacement a functional relation was introduced for them.With such a refined model a good agreement with test data wasreached. The derived analytical model based on measurementresults enables its application in structural design and analysisfor a wide engineering community.

Acknowledgements

The authors wish to express their gratitude to the EuropeanCommission which under Grant Agreement Number CP-IP213968-2 funded the IRIS research project within the scope ofwhich the present development was carried out. We would liketo warmly thank Ms. Bianca Mick for the unified graphic designof all figures in the article.

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