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Introduction
Reinforced concrete (RC) slender walls, when appropriately designed and detailed, provide adequate ductility and inelastic displacement capacity to building structures. This behaviour is achieved by using boundary elements (confinement reinforcement), and by limiting the axial load and the concrete compressive strains in the wall. Currently, the Chilean provisions that are contained in DS60 (2011), limit the maximum concrete strain in the fibre of extreme compression (εc), as shown in equations (1) and (2).
Regarding the displacement capacity of RC walls, according to DS60 (2011), at the critical section of slender walls with an aspect ratio (Hw/Lw) greater than 3 the curvature capacity (ϕ) should be greater than the curvature demand (ϕu), in order to obtain a ductile behaviour and displacement capacity higher than the displacement demand (δu). The curvature demand can be estimated using the equations (1) or (2):
where Lw is the wall length, Lp is the plastic hinge length, δu is the design displacement according to DS61 (2011), δy is the elastic displacement capacity (or yield displacement), ϕy is the yield curvature, C is the distance from the extreme compression fibre to the neutral axis and Hw is the wall height measured from the critical section. To estimate the curvature capacity, the maximum axial load on the wall must be considered; moreover, it is necessary to justify the values for ϕy and δy used in the analysis (DS60, 2011). However, Chilean regulations do not provide recommendations to compute these values. In addition, if equation (2) is analyzed, it is observed that the yield displacement is a key parameter in order to design RC slender walls, therefore, a reliable estimation of this value is required.
Previous studies have proposed closed-form expressions to estimate the elastic displacement capacity. However, most of these studies are based on pushover analyses, where the dynamic phenomenon is simplified to a fixed or variable lateral load pattern. An exception is presented by Quintana (2018), who proposes a relationship between the elastic displacement and a dynamic amplification factor, that is used in the capacity design for shear of cantilever walls (Paulay and Priestley, 1992), and that is a function only of the number of stories in the building. However, more recent studies have demonstrated that the dynamic amplification factor depends on the ductility demand too (Priestley et al., 2007; Morales, 2017; Jiménez et al., 2019; Morales et al., 2019).
Another common assumption, to estimate the elastic displacement capacity (or yield displacement) for slender RC walls, is that the flexural stiffness above the critical section is constant along the wall height. This assumption is not appropriate since the flexural stiffness of a wall varies during the dynamic response depending on its level of deformation (Beyer et al., 2014; Moehle, 2015). Additionally, due to the fact that for RC members the effective flexural stiffness is proportional to the nominal flexural strength (Priestley, 2003; Priestley et al., 2007), the axial force and its variation along the wall height has a relevant influence on the flexural stiffness that cannot be ignored. Furthermore, in the upper regions of the wall, the moment demands may be less than the cracking moment, therefore, the flexural stiffness should be larger than that of the lower stories (Priestley et al., 2007; Adebar et al., 2007).
In some cases, assumptions and simplifications considered in previous studies could produce errors in the estimation of the yield displacement and, as a consequence, inadequate design of wall boundary elements (i.e., underestimation of confinement reinforcement or wall thickness).
Considering the previous ideas, in this paper, a parametric study of the yield displacement for slender cantilever RC walls is presented. The results were obtained from nonlinear response history analyses (NRHA) for a set of cantilever RC walls, representative of real wall buildings. The building prototypes were created by varying the wall aspect ratio, height, thickness and longitudinal reinforcement, and were modelled using a distributed inelasticity fibre model. To carry out NRHA, natural and artificial records with different features are used. The displacement at which first yielding occurs is recorded in order to correlate it with features of the wall such as aspect ratio, amount of reinforcement, and the accelerograms characteristics (intensity, frequency content). The final discussion is focused on the influence of higher mode effects on the yield displacement and its variability, in order to provide design recommendations.
Description of the analytical model
The analysis model was implemented in SeismoStruct (Seismosoft, 2018) using a classical forced-based fibre element formulation in which the wall cross-section is discretized into uniaxial fibres representing the reinforcing steel and the confined (concrete core) and unconfined concrete (concrete cover). This formulation assumes that plane sections remain plane during the deformation history (the Bernoulli hypothesis); hence shear deformations are not accounted for in the analysis. On the other hand, a relevant aspect of this modelling approach is that calibration of the input parameters is not required and that axial-flexural interaction can be explicitly captured. Therefore, the model is able to represent the elastic and inelastic stiffness of the members with a non-critical shear or flexure-shear response (Sedgh et al., 2015).
In this study single cantilever walls are used as a simplification to represent a building (see Figure 1). This approach has been used in the past by other authors (Priestley and Amaris, 2002; Pennucci et al., 2013); however, some improvements are included in this work. In order to make the analysis as general as possible, 10-, 15-, 20-, 25- and 30-story buildings are considered and, for each building height, three different rectangular cross-sections are defined (see Figure 2 and Table 1). Note that the axial load ratios (ALR) were selected considering typical values found in the design practice of RC wall buildings in Chile.
Table 1 Characteristics of the building prototypes
| Prototype | ρl, % | lw, m | lc, m | tw, m | hn, m | hs, m | mi, ton | ALR |
|---|---|---|---|---|---|---|---|---|
| W. 1 | 0.31 | 8.0 | 1.2 | 0.25 | 27.0 | 2.7 | 60.0 | 0.20 |
| W. 2 | 0.30 | 6.0 | 0.8 | 0.25 | 27.0 | 2.7 | 25.0 | 0.15 |
| W. 3 | 0.32 | 4.0 | 0.8 | 0.25 | 27.0 | 2.7 | 5.0 | 0.15 |
| W. 4 | 0.31 | 12.0 | 1.2 | 0.25 | 40.5 | 2.7 | 70.0 | 0.20 |
| W. 5 | 0.31 | 8.0 | 0.8 | 0.25 | 40.5 | 2.7 | 20.0 | 0.15 |
| W. 6 | 0.33 | 6.0 | 0.8 | 0.25 | 40.5 | 2.7 | 60.0 | 0.35 |
| W. 7 | 0.36 | 15.0 | 2.0 | 0.30 | 54.0 | 2.7 | 80.0 | 0.20 |
| W. 8 | 0.38 | 10.0 | 1.6 | 0.30 | 54.0 | 2.7 | 20.0 | 0.15 |
| W. 9 | 0.37 | 8.0 | 1.2 | 0.30 | 54 .0 | 2.7 | 60.0 | 0.30 |
| W.10 | 0.41 | 15.0 | 2.0 | 0.30 | 67.5 | 2.7 | 60.0 | 0.20 |
| W. 1 1 | 0.44 | 10.0 | 1.6 | 0.30 | 67.5 | 2.7 | 6.0 | 0.15 |
| W.12 | 0.43 | 8.0 | 1.2 | 0.30 | 67.5 | 2.7 | 6.0 | 0.15 |
| W.13 | 0.41 | 15.0 | 2.0 | 0.35 | 81.0 | 2.7 | 26.0 | 0.15 |
| W.14 | 0.43 | 10.0 | 1.6 | 0.35 | 81.0 | 2.7 | 6.0 | 0.15 |
| W.15 | 0.42 | 8.0 | 1.2 | 0.35 | 81.0 | 2.7 | 10.0 | 0.20 |
Story masses showed in Table 1 were adjusted to obtain first-mode elastic periods representative of real wall structures. This consideration is not trivial since, as shown by Morales (2017) and Morales et al. (2019), previous studies with similar modelling approaches have resulted in building models with first-mode elastic periods larger than what is expected from real structures. The periods of the buildings from this study are compared against the expressions proposed in ASCE 7-10 (2010) and in Massone et al. (2012), the latter based on the values measured by Wood et al. (1987). The expressions are reproduced below as equations (3) and (4), respectively.
In equation (3)hn is the building height and the coefficients Ct and x are 0.0488 and 0.75, respectively. On the other hand, in equation (4)N represents the number of stories of the building. Figure 3 compares the periods of the buildings considered in this study with the periods predicted using equations (3) and (4). It is observed a good correlation between the values obtained in this study and the empirical expressions.
Materials: concrete and reinforcement steel
A trilinear concrete model (named con_tl in SeismoStruct) was used for the confined and unconfined regions of the wall. The con_tl is a simplified uniaxial model that assumes no resistance in tension and features a residual strength plateau in compression. Five parameters must be defined in order to fully describe the mechanical characteristics of the material: mean compression strength (fc1), initial stiffness (E1), post-peak stiffness (E2), residual strength (fc2) and specific weight (yc). The parameters used in this study are as shown in Table 2.
Table 2 Concrete model parameters
| Parameter | Unconfined | Confined |
|---|---|---|
| fc1, MPa | 25 | 29 |
| E1, MPa | 13000 | 15600 |
| E2, MPa | -5500 | -6600 |
| fc2, MPa | 5 | 6 |
| yc, kN/m3 | 24 | 24 |
To represent the cyclic behaviour of the reinforcement steel the model of Menegotto-Pinto (1973), included in SeismoStruct, was selected. This is a uniaxial model with the isotropic hardening rules as proposed by Filippou et al. (1983). An additional memory rule proposed by Fragiadakis et al. (2008) is also introduced for higher numerical stability and accuracy under transient seismic loading (Seismosoft, 2018). Table 3 shows the ten parameters used in this study to calibrate the Menegotto-Pinto model.
Table 3 Reinforcement model parameters
| Parameter | Considered value |
|---|---|
| Modulus of elasticity Es, MPa | 200000 |
| Yield strength fy, MPa | 420 |
| Strain hardening parameter μ | 0.01 |
| Transition curve initial shape parameter Ro | 20 |
| Transition curve shape calibrating coefficient A1 | 18.5 |
| Transition curve shape calibrating coefficient A2 | 0.15 |
| Isotropic hardening calibrating coefficient A3 | 0.0 |
| Fracture/buckling strain | 0.1 |
| Specific weight ys, kN/m3 | 78 |
The material models for concrete and reinforcement steel were selected after a trial and error procedure in which the performance of four different material models was compared against the experimental results of Thomsen and Wallace (1995; 2004) for a RC rectangular wall specimen (specimen RW2). For this purpose, an analytical model of the specimen was developed in SeismoStruct (SeismoSoft, 2018) and was subjected to the same loading protocol implemented by the authors. The analytical results are shown in Figure 4, for the selected con_ctl and Menegoto-Pinto material models, and are compared against the experimental ones. It is noticeable the close match between the simulation and the experimental data which furthers validate the modelling approach implemented here. For more details about the analysis model, its definition and calibration, the reader is referred to the work of Rubina (2020).
Nonlinear response history analysis (NRHA) and results
To carry out the NRHA of the simplified buildings a set of six accelerograms was considered: three artificial (Figure 5a, 5c and 5e) and three natural records (Figure 5b, 5d and 5f).
Note that the duration of the natural accelerograms presented in the Figure 5 was reduced to an effective duration in order to obtain shorter analysis times. The concept of effective duration refers to the duration of the strong motion phase of an earthquake recording measured from an Arias Intensity (AI) versus time plot (or Husid plot). Bommer and Martinez-Pereira (1999) proposed to define the beginning of the strong motion phase as the instant when AI is 0.01 m/s and pointed out that AI less than 0.135 m/s do not have an effective duration since they are not considered to be strong motions.
The artificial accelerograms were generated with SeismoArtif v2016 (Seismosoft, 2016) to be compatible with the elastic pseudo-acceleration spectra defined in DS61 (2011) for soil types B, C and D in seismic zone III. Response spectra of the artificial accelerograms are shown as dotted lines in Figure 6b and, as expected, they match the code spectra (solid lines) in a wide period range with comparatively small dispersion. Regarding the natural accelerograms, three recorded ground motions in Chile from 2010 are considered in this study: Angol NS (soil type D and seismic zone III), Maipu NS (soil type D and seismic zone II) and Talca NS (soil type B and seismic zone III). Figure 6a shows the elastic pseudo-acceleration spectra for 5% damping, obtained from the natural recordings (dotted line), and their comparison with the elastic spectra from NCh433 (2009) (solid lines).
Yield displacement
In this section, the results of the NRHA are presented and discussed in terms of the yield displacement of the case study buildings (see Figure 1). In order to estimate the yield displacement, the yield curvature at the base of the walls is considered according to equation (5) proposed by Priestley (1998). In this case the yield curvature is defined as the lowest between the curvature at the first yielding of tension reinforcement, or the curvature when the extreme fibre reaches a compressive strain of 0.002, extrapolated to the nominal flexural strength (Priestley, 1998).
In the equation above, Lw is the wall length and εy is the yield strain of the longitudinal reinforcement. As shown in equation (5), the yield curvature (ϕy) is defined with a lower and upper limit due to the dispersion observed in experimental tests. In order to consider the variability observed in laboratory tests and achieve representative results, this work used three different values of ϕy for each wall prototype. In this manner it is possible to account for the uncertainty in the definition of the yield curvature, and in the estimation of the yield displacement of the wall prototypes.
Commonly, the yield displacement δy in slender walls is presented in terms of the yield curvature ϕy, the wall height hw and a dimensionless yield constant ky, as shown in equation (6). This constant is a function of lateral load pattern applied to the wall and, according to the literature, could take values between 0.15 and 0.33.
Using the NRHA results and based on the equation (6), this paper proposes values for ky that includes features of the earthquake ground motions considered. To accomplish this goal, the following procedure is applied:
For each wall section upper
From NRHA the curvature time history response at the base of walls is obtained, and the time instant at which the upper, lower and average yield curvatures occur is identified. Then, using the roof displacement time history and the time instants identified previously, three values for the yield displacement (
Finally, applying the equation (7), three values for the dimensionless yield constant Ky are estimated for each nonlinear response history analysis carried out.
In the following paragraphs, the results of the NRHA campaign carried out in this study are presented in terms of the dimensionless constant Ky and the stiffness index H/T (Lagos et al., 2012, 2020), which is defined as the ratio between the total building height above the ground level and the first translational natural period in the direction of analysis. Figures 7a, 7b and 7c show the values of the dimensionless yield constants (
Figure 7 Dimensionless yield constants for artificial accelerograms: (a) soil type B, (b) soil type C, (c) soil type D and (d) summary results
Similarly, Figure 8 presents the results obtained from NRHA using natural accelerograms. In comparison with the artificial accelerograms, the same tendency is observed: lack of a clear correlation between the dimensionless yield constant and the soil type, mainly for soils type C and D (Figures 8b and 8c). However, in this case, Ky takes values between 0 and 0.25 showcasing a dispersion larger than that observed from the artificial accelerograms (see Figure 8d).
Figure 8 Dimensionless yield constants for natural accelerograms: (a) soil type B, (b) soil type C, (c) soil type D and (d) summary results
Even though an evident correlation between the dimensionless yield constant, the soil type and the building stiffness index is not shown in Figures 7d and 8d, an interesting result is observed: a good definition of the minimum values of Ky as a function of the building stiffness index. Based on this finding, a simple and practical expression to estimate the yield displacement is proposed, as shown below.
Influence of higher-mode effects
The pattern of lateral inertial forces in wall buildings changes during the dynamic response. At some instants of the response, displacement and accelerations may be strongly influenced by the second and third modes of vibration (Paulay and Priestley, 1992). If the wall is designed to have a single critical section (or plastic hinge) for moment and axial force, moments at critical section are bounded by the provided moment strength. However, moment and shear elsewhere are not well limited due to the changes in the lateral inertial forces (Moehle, 2015).
Analytical and experimental results show that the base shear demand is a function of two factors. Firstly, the moment strength at the base of the wall and, secondly, the distribution of lateral inertial forces. If the pattern of the lateral forces is close or approximately proportional to the first mode shape, the centroid of lateral forces (hv) is typically located around 0.7 (see Figure 9a). However, if the dynamic response is strongly influenced by higher modes, the resultant force is located much lower than in the previous case with an increment of the base shear, as shown in Figure 9b.
According to previous ideas, the influence of higher-mode effects can be visualized studying the position of the centroid of lateral forces. If the value of hv/hw is less than 0.7, the dynamic response is influenced by higher-modes. In fact, the value of hv/hw decreases as the higher-mode effects increase.
Figure 10 shows the dimensionless yield constant Ky as a function of the centroid of lateral forces, it is noted that most of the values of hv/hw are less than 0.7. However, a tendency is observed: the influence of higher-mode effects increase as the values of Ky decrease, mainly for the results obtained with artificial accelerograms.
Proposed expression to estimate the yield displacement
As mentioned above, a clear relationship or correlation between the dimensionless yield constant and the soil types and building stiffness indices is not detected. Figure 11 summarizes all the results obtained from NRHA where, despite the scatter of the data, upper and lower limits of the Ky values are identified. These limits define a band of points which decreases as the building stiffness index increases. The main objective of this study is to provide a simple and conservative expression to estimate the yield displacement in slender cantilever reinforced concrete walls. Therefore, the expression proposed makes use of the lower limit of the data showed in Figure 11 (dashed black line).
The yield displacement of a cantilever wall building is estimated as per equation (6), which for convenience is rewritten here as equation (8). As stated above, using only the lower limit of the scatter plot in Figure 11, the dimensionless yield constant can be estimated as a function of the building index H/T as proposed in equation (9). Note that for building stiffness indexes larger than 55 m/s the yield displacement of slender cantilever walls is neglected.
with
Regarding the applicability of the equation (9), it shall be limited to the range of case study buildings considered here, and to wall buildings with stiffness indexes between 35 m/s and 55 m/s. To increase this range, further analyzes are required.
Conclusions and future research
In this paper, a simple and useful expression to compute yield displacement of slender cantilever RC walls was presented. This expression is more conservative than previous approaches presented in the literature and only depends on the building stiffness index, which is a common and available parameter for structural designers. Several analytical improvements were introduced in this study; for instance, the wall flexural stiffness was defined as a function of the moment capacity, the use of distributed inelasticity fibre elements along the wall height, explicit seismic demands (accelerograms) and a good representation of the reinforced concrete wall buildings using their stiffness indices. Furthermore, the axial load ratios and natural periods of the prototypes analyzed are representative of real wall buildings.
With regard to the effects of higher-modes, it should be noted that their influence increases with the increase of the building period; nonetheless, further analyses are required to obtain reliable results.
During the development of this research, a certain number of approximations and assumptions were necessary. Some of these assumptions may introduce uncertainty to the findings presented. Future research is focused to give answers to the potential sources of uncertainty. Firstly, this study only considered the analysis of one configuration of buildings, represented as single cantilever rectangular wall with different features (thickness, aspect ratio and reinforcement amount). In order to extrapolate the results of this study, non-rectangular walls (e.g. C- and T-shaped wall) could be considered in future works. Finally, analysis models (or prototypes) with two or more walls acting in parallel should be used to account for their interaction and possible coupling within the building.
