Experimental Evaluation of Inelastic Higher-Mode Effects on the Seismic Behavior of RC Structural Walls

Reinforced concrete structural walls are widely used in midrise and high-rise concrete structures as the seismic force resisting system (SFRS). Ductile reinforced concrete (RC) structural walls designed according to modern building codes are typically detailed to undergo plastic hinging at their base. Both the design moment and design shear force envelopes for the remaining portion of the RC wall are evaluated based on the probable flexural resistance of the RC wall in the plastic hinge region according to the capacity design procedure. It has been found that so-designed structural walls can be subjected to shear forces in excess of the design values. Plastic hinging can also develop in the upper portion of the walls. These effects are primarily attributed to higher mode responses, and hence are more severe in taller or slender walls with long fundamental periods. The contribution of higher vibration modes to the elastic response of such RC walls produces base shear force demands that are significantly larger than those resulting from the traditional static procedure based on the inverted triangular distribution of the lateral seismic forces. The contribution of higher modes in the structural response of an RC wall becomes more critical when the RC wall starts to respond inelastically. This effect is primarily due to the elongation of the fundamental mode period due to yielding of steel reinforcement and development of wide cracks in the concrete.

Since the 1970s, researchers have studied higher mode effects on the flexural moment and shear force demands through various research projects. Currently, a long list of research publications about higher mode effects and amplification factors are available, but it appears that a unique consensus is not available yet (Rutenberg 2013). Table 1 presents all proposed shear amplification factors reviewed in this paper. For the sake of conciseness, all variables in the equations reported in Table 1 are defined in the list of symbols. Blakeley et al. (1975) performed the very early research on the higher mode effects on RC structural walls. This pioneering work yielded some important results, including amplified base shear force demand and possible flexural hinging at the upper levels when using capacity design procedures. The authors proposed an equation (Table 1) to amplify the design base shear force accounting for the inelastic effects of the higher modes based on the number of stories $N$ only and indirectly on the fundamental period of vibration. Kabeyasawa and Ogata (1984) proposed an equation that directly modifies the base shear corresponding to the base flexural capacity instead of the shear force obtained from analysis. This formulation, as shown in Table 1, considers the effect of base flexural overstrength as well as the peak design acceleration, while the effects of the fundamental period of the structure are neglected. Chapdelaine (1987) studied a ductile designed reinforced concrete wall-frame building through nonlinear dynamic analyses. He reported that the shear force demand at the base of the RC wall is significantly larger than the nominal shear resistance calculated based on the Canadian codes that were in practice at the time and could lead to a diagonal shear failure. Eibl and Keintzel (1988) developed the concept of modal limit force to calculate shear demands of RC walls. As presented in Table 1, the authors proposed a function that was later adopted by the Eurocode to calculate the shear force demand by amplifying the design shear force. Based on a numerical study on isolated cantilever structural walls with 10, 20, 30, and 40 stories, Ghosh and Markevicius (1990) simplified the equation proposed by Kabeyasawa and Ogata (1984) (Table 1). Note that the fundamental periods used by Ghosh and Markevicius correspond to buildings with high lateral stiffness in comparison with the well-known practical period formulation employed in most design codes. Priestley (2003) proposed two different methods for taking into account the effects of higher modes in cantilever RC walls. The first is based on a simple modification of the modal response spectrum method that extended the basic method of Eibl and Keintzel (1988) for shear forces to the full height of cantilever RC structural walls. In the second method, the author proposed a simple amplification factor for practical design of cantilever RC structural walls, incorporating both the design ductility and the fundamental period of the structure. Formulation of both methods are presented in Table 1. Renaud (2004) studied the seismic behavior of a shear wall designed with a force modification factor of 3.5 and reported that under the design-level seismic excitations, the shear force demand at the base of the RC wall is higher than the factored resistance of the section required by the then in force Canadian standard for design of concrete structures. In light of a parametric study, Rutenberg and Nsieri (2006) demonstrated that seismic provisions for RC walls in the medium- and high-ductility classes addressed in Eurocode 8 (EC8), and the amplification factor addressed in the New Zealand Concrete Structures Standard [NZS 4203 (NZS 1992)], are not conservative. The authors proposed an equation that considers the effects of the fundamental period and the design ductility level while the base flexural overstrength factor is neglected (Table 1). Sullivan et al. (2008) observed that flexural yielding at the base of RC walls cannot limit base shear force demands from increasing. The authors proposed the concept of transitory inelastic modes to calculate the base shear forces (Table 1). Pennucci et al. (2010) proposed that the internal force (shear and moment) distribution in cantilever RC shear walls could be expressed as a combination of fixed and pinned base models as shown in Table 1. The fixed base model represents the elastic response before base yielding of the wall and the pinned base model represents the additional forces because of postyield higher mode response. Based on this assumption, Yathon (2011) proposed a function to predict the shear force demand at the base of RC walls (Table 1). Boivin and Paultre (2012a) conducted a parametric study and showed that the most effective characteristics affecting the dynamic shear amplification and seismic force demands are the fundamental period and the base flexural overstrength. Based on this parametric study, the authors proposed a function to consider the inelastic amplification of the base shear force demand, presented in Table 1, and a new capacity design method accounting for the higher mode amplification effects all over the height of RC walls with a rectangular section (Boivin and Paultre 2012b). Through a parametric study, Rejec et al. (2012) examined the shear force amplification factor addressed in EC8. Rejec et al. (2012) showed that the current procedure used in EC8 for moderately ductile (DCM) structures (using a constant shear magnification factor of 1.5 for all walls) is nonconservative and recommended a new equation shown in Table 1. Nonlinear time history analysis (NLTHA) was used by Dezhdar and Adebar (2012) to study the base shear force demand in 13 different cantilever shear wall buildings. The authors showed that base shear force demand can be amplified by a factor of 2 due to the inelastic effects of higher modes. Ambroise et al. (2013) extended the parametric study carried out by Boivin and Paultre (2012a) to eastern Canada. The authors proposed a dynamic shear amplification function ${\omega }_v$ for the base and over all stories of RC structural walls based on the fundamental period $T$ and flexural overstrength ${\gamma }_w$ for Vancouver and Rivière-du-Loup cities as representatives of western and eastern Canada (Fig. 1). Same conclusions have been reached by Luu et al. (2014). Pennucci et al. (2015) explained that where the response of higher modes is significant, the use of a single reduction factor $R$ across all modes to account for ductility development is not appropriate. To circumvent this problem, the authors proposed two different methods. The first method requires only an elastic modal analysis, using a substitute structure, and is suitable for traditional design purposes. The second method is based on structural dynamics closed-form solutions that do not require finite-element analysis and is satisfactory for simplified capacity design purposes (Pennucci et al. 2015). Recent works investigated RC coupled walls (Pennucci et al. 2011; Fox et al. 2014). Although they are out of the scope of this paper, note that they are among the very early research studies that considered the influence of the shape of the acceleration spectrum on the higher mode effects.

In the literature, compared to analytical studies, much fewer laboratory tests on shear walls under ground motion excitations are available to study this phenomenon. Abrams and Sozen (1979) performed shake table tests on four 10-story small-scale RC wall-frame systems. Each small-scale structure consisted of two three-bay RC frames and one central cantilever RC wall. The tests program consisted in applying a series of three earthquake excitations of progressively increasing intensity to each of the four structures. The authors reported that unlike displacement and acceleration responses, force demands are sensitive to participation of higher modes that are highly dependent on the fundamental period of the structure and characteristics of the earthquake. Lybas (1981) performed shake table tests on six one-twelfth scale 6-story RC coupled structural walls mainly to investigate the effect of the coupling level on the structural response. The author reported that under the seismic loading, the second vibration mode has a major effect on the high shear force demand at the base of RC walls tested. Ghorbanirenani et al. (2012) conducted an experimental study on two dimensionally identical RC shear walls to investigate the inelastic responses and interaction of shear, flexure, and axial loads in the plastic hinge zones through shaking table tests. The authors reported that under the design-level ground motion, the peak base shear force was amplified by a factor of 1.82. They observed that, despite a high shear demand at the base, no shear failure mechanism took place at the base of the wall, and they explained that this higher resistance is essentially due to a greater contribution of concrete in the shear strength and by the fact that the maximum base shear force demands did not occur simultaneously with the maximum curvatures at the base of the RC walls. In another experimental study, a full-scale 4-story RC wall-frame building designed in accordance with the Japanese seismic design code and adjusted with seismic provisions in the ACI Code and ASCE Standard was tested under multidirectional accelerations on the E-Defence shake table facilities (Nagae et al. 2015). Two moment resisting frame systems were used in the longer direction of the floor plan and two walls were incorporated in the exterior frames in the shorter plan direction (Nagae et al. 2011). During the test, the measured base shear demands were 40% larger than the calculated shear capacity of the wall-foundation interface. The authors concluded that the effects of higher modes and localized damage on the vertical distribution of lateral forces should be considered in the design process of the RC shear walls.

The dynamic amplification factors for the base shear in the New Zealand Code of Practice for General Structural Design and Design Loading for Buildings [NZS 4203 (NZS 1992)] are based on the early inelastic time history analyses carried out by Blakeley et al. (1975). The New Zealand code was the first building code that considered higher mode amplification factors to evaluate the design base shear force. In addition to the dynamic amplification factor of the base shear, the New Zealand code [NZS 4203 (NZS 1992)] has proposed that the lateral force distribution extracted from equivalent lateral force procedure should be amplified by the base flexural overstrength factor ${\varphi }^o$. The Comité Européen du Béton (CEB) adopted the New Zealand standard equation with a minor modification in their model code (CEB 1985).

Eurocode 8 (CEN 2004) uses the Eibl and Keintzel (1988) shear force amplification factor since the 2004 edition for walls designed for high ductility. The equation in the Eurocode (Table 1) has a minor difference with Eibl and Keintzel’s equation. The equation accounts for the overstrength due to the strain hardening of the reinforcement through ${\gamma }_{Rd}$ that may be taken as 1.2. Eurocode 8 gives larger amplification factors for shear force in comparison with other available design codes in the world (Rutenberg 2013).

In the last edition of the CSA standard A23.3 (CSA 2014), two amplification factors were presented to increase the factored shear force in order to take into account higher mode effects. One of the factors is to consider the effect of the flexural overstrength at the base of the RC wall, and the other one is to account for the inelastic effects of higher modes. As shown in Table 1, the CSA standard limited the second factor to 1.5. Additionally, the National Building Code of Canada NBCC2015 (NRCC 2015) presents amplification factors to increase design shear force in order to consider higher mode effects when using the equivalent lateral force method.

A pseudodynamic hybrid test designed to investigate inelastic shear amplification due to inelastic effects of higher modes has been developed at the University of Sherbrooke. Hybrid testing involves splitting the studied structure into two parts: the physical substructure that contains a key region of interest and is experimentally tested, and a numerical substructure that contains the remainder of the structure that is simulated numerically (Fig. 2). The numerical substructure was modeled using nonlinear force-based multilayer beam-column elements. The nonlinear beam-column element considers the spread of plasticity along the element and accounts for the stiffness deterioration due to concrete cracking. The mass of each story was lumped at each floor level, and the effects of the vertical components of ground motions were ignored. As the shear wall was modeled as an isolated structural element, the three-dimensional (3D) effects such as slab coupling as explained in Sullivan (2010) on shear force demands was not considered. Results of NLTHA of the complete structure showed that using reinforcement yield strength up to 800 MPa above the plastic hinge zone of the wall had no significant effect on the base shear force demand as well as on the structural behavior. Therefore, to ensure linear behavior in the upper stories, in the hybrid tests, 800 MPa was used as the reinforcement yield strength in the numerical subassembly. During the tests, the two substructures feedback displacement or force to each other as shown in Fig. 3. In Step 1, using the AlphaOS numerical integration method in OpenSees version 2.5.0 (Schellenberg et al. 2009), the displacement vector is calculated from the numerical model in the computer controlling the experimental substructure for time step $\mathrm{\Delta }t$. In the second step, the calculated displacements are imposed by the control system to the experimental substructure using servohydraulic actuators through a mixed displacement-force control method. In Step 3, the restoring force vector, associated with the applied displacements, is measured and fed back to the numerical substructure in the control computer. In Step 4, time history responses of the structure are updated. The same procedure is started again from Step 1 for the next time step and repeated until the end of the seismic excitation. A combination of OpenSees version 2.5.0 (McKenna et al. 2006), OpenFresco version 2.7.1 (Schellenberg et al. 2010), LabView version 12 (National Instruments 2004), and FlexTest version 60 (MTS 2004) programs was used to carry out the hybrid tests.

The reference reinforced concrete structural wall tested in this research is the SFRS of an 8-story RC building structure. The structure was designed according to the acceleration response spectra corresponding to the city of Rivière-du-Loup (Canada), on soil Type D. The fundamental period of the building is $T=1.2\mathrm{s}$. The height of the first story, corresponding to the experimental part, was chosen as 6.05 m and is larger than the base plastic hinge region length defined as $h_p=0.5{\ell }_w+0.1h_w$, where ${\ell }_w$ is the length of the wall, and $h_w$ is the total height of the wall (CSA 2014). This length is larger than the plastic hinge length, which is used to relate curvatures to displacements and is assumed to be equal to $1.0{\ell }_w$ or $0.5{\ell }_w$ in the CSA A23.3 standard, depending if the flexural demand or capacity is being calculated. The plastic hinge region is where significant flexural and shear cracking is expected to occur and where special reinforcement must be provided according to the CSA A23.3 standard (CSA 2014). Nonlinear force-based beam-column elements were used in the numerical subassembly in order to account for the effects of any significant cracking outside of the plastic hinge zone. All other stories were set at 3.15 m in height. The reference RC wall was designed and detailed according to the 2014 edition of the CSA A23.3 Standard Design of Concrete Structures and the 2015 edition of the National Building Code of Canada (NRCC 2015). Note that in calculating the design shear force, the amplification factor to consider the nonlinear effects of the higher modes presented in the CSA A23.3 standard was not considered in order to measure it experimentally. Regarding the CSA A23.3 standard, this amplification factor for the given RC wall is 1.5. The first story of the RC wall was constructed in the structural laboratory while the remainder of the RC wall structure was modeled numerically. Taking into account the capacity of the equipment available in the structural laboratory, a scale factor of 2.75 was used to reduce the size of the test specimen. The numerical subassembly was constructed in full-scale and the Cauchy method was used to scale down the displacement of the control point calculated by the numerical part and to scale up the experimental forces and moments. Experimental forces were multiplied by a factor of ${2.75}^2$ and experimental flexural moments by ${2.75}^3$ before they were fed back to the numerical model. More information about Cauchy method could be found in Kumar et al. (1997). The RC shear wall was designed using a concrete with a specified compression strength of $f_c^{\prime }=30\mathrm{MPa}$ and steel reinforcements with a specified yield strength of $f_y=400\mathrm{MPa}$. Table 2 summarizes concrete material properties measured on the day of testing. The concrete compressive strength, ranging from 28.5 to 57.1 MPa, was determined from standard compressive tests on at least three $150\times 300\mathrm{mm}$ cylinders. The postpeak strain at 50% of maximum stress, ${ϵ}_{c50u}$, was determined from the complete stress-strain curves obtained from the standard compressive tests. This strain ranged from 0.00452 to 0.00527. The cracking strength of the concrete, $f_r$, obtained from modulus of rupture tests on at least three $100\times 100\times 400\mathrm{mm}$ prisms for each specimen, ranged from 4.35 to 5.62 MPa. The higher strength concrete was used to construct the top slab in order to reduce cracking and stiffness degradation in the slab transmitting the seismic force to the wall. Complete stress-strain curves were obtained from test coupons on each bar diameter used. The averages of at least three steel coupons for each bar diameter are shown in Table 3 where $f_y$ is the yield strength, ${ϵ}_{sh}$ is the strain at the commencement of strain hardening, and ${ϵ}_{su}$ is the ultimate strain corresponding to the ultimate stress $f_{su}$. Fig. 4 presents the dimensions of the reference RC wall and the specimen.

A total of 49 strain gauges with a gauge factor of $2.125\pm 0.5\%$ and grid resistance of $120\pm 0.6\%\mathrm{\Omega }$ were used on vertical and horizontal reinforcements as internal instruments. A total of 50 linear voltage differential transducers (LVDTs) and potentiometers were used to measure deformations externally. Potentiometers were employed in the locations where displacement demands were high, while LVDTs were installed to measure sliding of the foundation and displacements at lower levels of the wall as shown in Fig. 5. In this figure, the letter H stands for horizontal potentiometers, V for vertical potentiometers, and D for diagonal potentiometers. The letter L denotes LVDTs.

The control point of the test specimen is located halfway along the length of the wall, just below the slab (Fig. 5). The potentiometers H7, H8, and H9, as shown in Fig. 5, are placed in order to control the horizontal displacement and rotation at the control point. The average displacement of two potentiometers mounted on each side of the wall, H7 and H8, was used to control the horizontal displacement. The rotation was controlled using two different measurement points, the top of the wall middle just below the slab (average of H7 and H8) and the potentiometer H9, which are located 2,300 mm apart from each other.

The peak shear force demand observed during the hybrid testing was 1,020 kN under the amplified seismic excitation, GM2. By taking into account the scale factor, the corresponding peak shear force for the full-scale wall is 7,715 kN. The maximum nominal and factored shear strength were calculated using code equations in CSA A23.3 (CSA 2014). The calculated nominal and factored peak shear strengths of the reference RC wall were 4,230 and 3,400 kN, respectively. Code equations proposed by the ACI 2014 [ACI 318 (ACI 2014)] standard and the New Zealand Concrete Structures Standard [NZS 3101 (NZS 2006)] predicted almost the same shear strength for the reference RC wall. However, using the detailed method proposed by the New Zealand Code NZS 3101 (NZS 2006), the nominal and factored shear resistance of the reference wall were 8,100 and 6,080 kN, respectively. It appears that the CSA and ACI standards are more conservative in calculating the shear resistance of ductile RC shear walls, while the detailed method of NSZ 3101 (NSZ 2006) predicts a nominal shear resistance in agreement with the test results. A numerical model was created in Response2000 version 1.0.5 (Bentz 1999) software to evaluate the shear force capacity of the RC wall. This model yielded a 850 kN nominal shear strength corresponding to a 6,400 kN resistance for the reference RC wall.

The seismic demands are presented in Table 4. The amplification factor corresponding to the shear force demand at the base of the RC wall measured during the GM1 ground motion is 2.27.

In the case of the very strong ground motion, GM2, the dynamic shear amplification due to the effect of higher modes measured during the test increased to 3.01, while due to period shifting and smaller displacement and rotational demands, the maximum flexural moment observed under the GM2 excitation was less than the one measured during the design-level ground motion test (GM1). The dynamic shear amplification factors measured during the hybrid tests are reported on the shear amplification function (Ambroise et al. 2013) presented in Fig. 11. The test results showed a very good correlation with the predicted dynamic amplification factors by Ambroise et al. (2013) at the base of the RC wall.

When higher modes govern the response of the RC wall, the moments at the top and bottom of the first story bend the sections in opposite directions as can be seen in Fig. 12. Fig. 13(a) displays the displacement at each story of the reference RC wall when the maximum horizontal displacement at the roof level of the reference RC wall (broken line) and the maximum shear force demand (continuous line) are reached. It is clear that at the time the maximum shear force demand is reached, the second mode of vibration dominates the structural behavior.

When the maximum lateral displacement is reached at the top level of the wall, the first mode of vibration is governing the structural response as expected. The test results showed that the maximum shear force demand occurred in four different peak points during two successive cycles under the GM1 seismic excitation. The shear force demands in these four consecutive peak points were 708 ($t=8.579\mathrm{s}$), $-770$ ($t=8.876\mathrm{s}$), 726 ($t=9.116\mathrm{s}$), and $-720\mathrm{kN}$ ($t=9.404\mathrm{s}$). Fig. 13(b) shows the deformation modes of the reference RC wall at the aforementioned times. At all four points, the moments at top and bottom of the test specimen are bending the sections in opposite direction, implying that the higher modes govern the response of the RC wall structure at those points.

Despite a high shear demand and considerable yielding of the horizontal reinforcements, no sudden significant deterioration of the lateral stiffness was observed under the design-level earthquake (GM1). In other words, under the design level excitation a ductile shear failure occurred. The maximum shear force and maximum flexural moment demands occurred at different times. Note that at time $t=9.116\mathrm{s}$, the 726 kN shear force demand (5% less than the peak shear force) was accompanied by a base flexural moment equal to $\mathrm{1,382}\mathrm{kN}·\mathrm{m}$ (10% less than the peak flexural moment).

Under the amplified seismic excitation GM2, the maximum demands occurred almost simultaneously. Based on the test results under the GM1 and GM2 excitations, it appears that the ACI and the CSA standards are conservative in calculating the shear capacity of the RC walls.

Note that during the 16-s-long ground motion (GM1), higher modes (primarily the second mode) governed the response during at least two cycles. A long-duration ground motion could have imposed much more shear amplified cycles. For the sake of comparison, the Chile 2010 earthquake had a magnitude of 8.8 on the moment magnitude scale (USGS 2010), with intense shaking lasting for approximately 3 min. This is a good example of a very long and strong ground motion. Such a long and intense seismic event would be capable of inducing a large number of inelastic deformation reversals, thus creating several peak points with a high shear force demand. Such repetitive high shear force peak demands could lead to a brittle shear failure. This consideration is important for a large amount of American and Canadian structures because the literature indicates that earthquakes generated by the Cascadia subduction zone that stretch from northern Vancouver Island to northern California generally produce high-intensity long-duration ground motions (Tremblay 1998).

In order to compare the contribution of the different deformation mechanisms to the total displacement of the physical subassembly, the drift of the specimen was decomposed into four displacement components coming from (1) rigid body motion due to sliding of the system at the foundation–wall base interface, ${\delta }_{bs}$; (2) concentrated rotation at the foundation–wall base interface, ${\delta }_{br}$; (3) shear deformations, ${\delta }_s$; and (4) flexural deformations, ${\delta }_f$. Test results showed that sliding of the foundation block with respect to the strong floor of the laboratory is negligible and thus not considered in this analysis. Note that only 1,445 mm out of the total 2,200 mm height of the RC wall specimen was instrumented externally to provide data for extracting shear deformations. Thus, displacement analysis was performed at a height of 1,445 mm above the foundation. Components of displacement are presented at selected time. During the GM1 excitation, the selected times are 2.697, 3.027, 3.873 (maximum shear demands), 4.431, and 4.986 s (maximum flexural moment demands) when the actuators are pushing in the positive direction. When the actuators are pulling in the negative direction, the selected times are 3.582, 4.122, 6.747, and 7.402 s. Displacement history of the control point under the GM1 and GM2 seismic excitations are presented in Figs. 14(a and d), respectively. Figs. 14(b and c) present the contribution of each source of deformation to the total drift under the design level seismic excitation GM1. As can be seen from Fig. 14(b), the contribution of the shear deformations is initially marginal (less than 5%). At about the time that the maximum shear demand takes place (3.876 s from the starting point of the GM1 motion) shear deformations govern the response in contributing over 50% of the total drift. When the maximum displacement response at the top of the specimen was reached, flexural deformations dominated the response, accounting for more than 60% of its value. This caused the flexural crack openings to increase and the vertical reinforcements to yield, forming a plastic hinge at the base of the wall. At that time during the test, the concentrated rotation at the wall-foundation interface increased significantly, contributing a larger portion of the total displacement. At the time corresponding to the maximum shear force demand under the GM1 motion, the flexural contribution to the total drift is slightly more than 40%. In the negative direction, as presented in Fig. 14(c), the flexural deformations govern the response of the RC wall from the beginning until the end. Between the selected times, the relative contributions are obtained by linear interpolations. These linear interpolations do not represent the exact contributions but provide a reasonable estimation (Whyte and Stojadinovic 2013). Figs. 14(e and f) show the components of the lateral deformations under the GM2 excitation in the positive and negative directions, respectively. It is obvious that in the positive and negative directions, the shear deformation has an important contribution in the response of the RC wall. Because the wall experienced severe yielding accompanied by wide crack openings during the GM1 excitation, it exhibited a very soft behavior, governed primarily by shear mechanisms in the positive and negative directions under the GM2 excitation.

Figs. 15(a and b) show the base shear force versus horizontal displacement of the control point and base flexural moment versus rotation at the control point under the GM1 excitation, respectively. The corresponding hysteretic behavior of the specimen under the GM2 excitation is presented in Figs. 15(c and d). The spikes on the hysteretic curves corresponding to the GM1 excitation are due to the high stiffness of the RC wall before significant cracking of the concrete and longitudinal reinforcement yielding at the base. Stiff experimental specimens tend to oscillate in interaction with actuation systems. This oscillation tendency due to high stiffness makes the control process more challenging. The base shear force hysteresis curve under the GM2 excitation, Fig. 15(c), begins with a very low rigidity because the specimen was heavily damaged under the GM1 excitation. The RC wall recovers its rigidity as the base sliding at the foundation-wall base interface is complete and cracks close on the compressive side, thus increasing aggregate interlock and shear friction along the cracks. As shown in Fig. 15(c), the shear force-displacement hysteresis displays a pinched curve due to shear deformations. Contrary to the shear hysteresis curve, the base flexural moment as shown in Fig. 15(d), exhibits a higher rigidity because no buckling of the longitudinal reinforcements of the RC wall was observed under the GM1 excitation.

Shear force amplification due to inelastic higher mode effects was identified in the early 1970s. Even though failures of shear wall structures have been rare under past earthquakes, it is nevertheless a concern to avoid shear failures in walls designed according to the capacity design principles.

Interaction between the shear and flexural deformations plays an important role In how RC shear walls respond to dynamic loads. Thus, it is important to quantify the shear-flexure interaction and the dynamic amplification of the shear force due to the higher-mode effects in order to clearly define the design shear forces and thereby ensure that a given RC shear wall will fail in a ductile mechanism when submitted to a severe ground motion. Recent studies have shown that the shear amplification can be significantly larger than the value of the shear force corresponding to the development of the probable moment capacity at the base of the wall. The 2014 edition of the CSA A23.3 standard introduced for the first time in North America expressions for the shear force amplification factor that are limited to a maximum value of 1.5.

Three successive pseudodynamic hybrid tests were performed on an 8-story RC shear wall to investigate the effect of higher modes on the nonlinear dynamic response at the base of the RC wall. The displacement and rotation demands imposed on the test specimen during the hybrid tests were successfully applied with sufficient precision, thereby validating the adequacy of the test setup and control schemes. As expected, under the design ground motion (GM1), yielding that occurred in the longitudinal reinforcements at the base of the wall spread out over a length larger than ${\ell }_w/2$, thereby forming a plastic hinge. However, several shear horizontal reinforcements yielded due to shear force amplification induced by the response of higher modes. The yielding of the bars was accompanied by major horizontal and diagonal cracks. The shear demand observed during the GM1 hybrid test was 2.27 times greater than the expected shear capacity as evaluated using the capacity design procedure in CSA A23.3. The test results showed that larger shear force demands took place in four points during two successive cycles. The results revealed that the dynamic amplification factor for the design shear force presented in the capacity design procedure of the CSA A23-3 is lower than that observed in reality for this particular test specimen. Despite the higher amplification of the shear force, no sudden significant deterioration of the lateral stiffness was observed. Nevertheless, the CSA A23.3 standard underestimated the shear force capacity of the RC walls. Under the GM2 ground motion, which had a PGA twice as high as GM1, more yielding was observed, especially in the horizontal reinforcements. The yielding of the bars was also accompanied by severe horizontal and diagonal cracks. The shear amplification observed during the GM2 test was 3.01 times greater than the capacity design value obtained using the CSA A23.3 design code. Nevertheless, the structural integrity of the wall (and building structure) was maintained after the three hybrid tests. The test results, as well as some test results reported in the literature, seems to suggest that the design codes, such as CSA A23 and ACI 318, are conservative in calculating the shear force capacity of RC structural walls. Thus, formulas for calculating the shear capacity of RC walls might need to be investigated further in future research studies. Note that a long and intense seismic event would be capable of inducing a large number of inelastic deformation reversals, giving rise to several peak points with a high shear force demand that can lead to a brittle shear failure.

The following symbols are used in this paper:

$A_g$

peak acceleration ratio normalized by gravitational acceleration g;

$A_{gw}$

gross cross-sectional area of RC wall;

$D_m$

a coefficient dependent on the number of stories;

$E_c$

secant modulus of elasticity of concrete;

$E{I}_g$

gross flexural stiffness of shear wall;

$F_u$

ultimate strength of reinforcement steel;

$F_y$

yield strength of reinforcement steel;

$f_c^{\prime }$

maximum compressive strength of concrete measured at the age of 28 days;

$f_r$

concrete modulus of rupture;

$H$

wall’s total height;

$H_f$

fraction of pinned base shear;

$h_p$

plastic hinge length;

$K$

behavior factor;

$k_1,k_2,k_3$

characteristics of buildings and earthquakes in Japanese standard;

$l_w$

RC wall width;

$M_{Ed}$

design flexural bending moment at the base;

$M_f$

factored design moment;

$M_I$

design bending moment according to simplified dynamic analysis;

$M_{nb}$

nominal base moment capacity;

$M_{Rd}$

flexural resistance at the wall base;

$M_y$

yielding moment of the wall structure;

$N$

number of stories of building;

$P$

behavior factor (equivalent to $R$ in Canadian code);

$Q$

seismic shear force at the base of wall;

$Q_I$

design shear force according to simplified dynamic analysis;

$q$

behavior factor (equivalent to $R$ in Canadian code);

$R_d$

ductility related force reduction factor;

$R_o$

overstrength related force reduction factor;

$R_{pf}$

correction factor for pinned to fixed base shear ratio;

$R_{0.5E{I}_g}$

force reduction factor corresponding to $0.5E{I}_g$ effective stiffness;

$S_a(T)$

acceleration response spectrum corresponding to period $T$;

$S_e(T)$

ordinate of the elastic acceleration response spectrum at period $T$;

$T$

period of vibration;

$T_c$

upper limit period of the constant spectral acceleration region;

$T_1$

fundamental mode period;

$T_2$

second mode period;

$V_a$, $V_{am}$

amplified base shear force;

$V_d$

base shear force from inverted triangular distribution;

$V_E$

base shear force from equivalent lateral force approach;

$V_e$

base shear from response spectrum analysis;

$V_f$

factored shear force;

$V_{hm}$

amplified shear force corresponding to higher modes;

$V_i$

shear force envelope at level $i$;

$V_{\max }$

maximum dynamic base shear in isolated walls;

$V_{pb}$

probable shear strength of RC wall base section;

$V_R$

probable shear strength of RC wall base section;

$V_w$

shear resistance of RC wall section;

$V_y$

amplified shear force corresponding to first mode;

$V_1,V_2,\dots$

modal shears;

$W$

total weight of the building;

$\gamma$

correction factor;

${\gamma }_{Rd}$

overstrength factor due to strain hardening of steel reinforcements;

${\gamma }_w$

shear wall overstrength factor;

${\delta }_{br}$

concentrated rotation within the base sliding zone of RC wall;

${\delta }_{bs}$

rigid sliding motion;

${\delta }_f$

flexural deformation;

${\delta }_s$

shear deformation;

$ϵ$

dynamic shear amplification factor;

${ϵ}_a$

dynamic base shear amplification factor;

${\epsilon }_c^{\prime }$

concrete axial strain corresponding to $f_c^{\prime }$;

${\epsilon }_{c50u}$

concrete axial strain corresponding to $0.5f_c^{\prime }$ after peak;

${\epsilon }_{sh}$

strain corresponding to beginning of steel hardening;

${\epsilon }_u$

ultimate strain of reinforcing steel;

${\epsilon }_y$

yield strain of reinforcing steel;

${\eta }_{vw}$

shear force amplification factor;

$\mu$

design displacement ductility level (New Zealand standard);

${\varphi }_o$, ${\varphi }^o$

flexural overstrength factor;

${\omega }^{*}$, ${\mathrm{\Omega }}_v$,

base shear amplification factor;

${\omega }_{mv}$

dynamic base shear amplification factor;

${\omega }_v$

shear force amplification factor; and

${\omega }_v$, ${\overline{\omega }}_v$

dynamic shear amplification factor.

The authors gratefully acknowledge the financial support of the Natural Science and Engineering Research Council of Canada and the Fond de recherche du Québec—Nature et technologie through the Centre d’étude interuniversitaire des structures sous charges extrêmes (CEISCE) (Interuniversity Center for the Study of Structures under Extreme Loading). Support from the Centre de recherche en génie parasismique et en dynamique des structures (CRGP) (Earthquake Engineering and Dynamics of Stuctures Research Center) of the University of Sherbrooke is also acknowledged. The authors wish to thank Claude Aubé, Jeason Desmarais, Éric Beaudoin, and Raphaël Prévost, technicians at the University of Sherbrooke, who participated in the testing program. Gabriel Rivard and Wilsonne François helped during the tests.