Analytical and Numerical Methods for Estimating the Deformation Capacity of RC Shear Walls

Abstract

The present research aims to the evaluation of the deformation capacity of existing reinforced concrete shear walls designed with past non-conforming seismic regulations. A refined analytical model (referred to as the Proposed Model) is presented for generating Load–displacement (P-d) curves for RC shear walls. The model is applicable to medium-rise walls designed with or without modern seismic provisions and incorporates shear effects in both deformation and strength capacity. The application of the Proposed Model is assessed through comparison with numerical models implemented in the widely accepted OpenSees platform. Specifically, two types of elements are examined: the widely used flexural element Force-Based Beam-Column Element (FBE) and the Flexure-Shear Interaction Displacement-Based Beam-Column Element (FSI), which accounts for the interaction between flexure and shear. The results of both analytical and numerical approaches are compared with experimental data from four RC shear wall specimens reported in previous studies.

1. Introduction

Modern seismic design principles rely on ensuring the ductile response of the structure under earthquake actions. In this context, current codes emphasize the adequate detailing of reinforced concrete (RC) shear walls, trying to secure their flexural behavior and achieve acceptable ductility levels. Specifically in RC shear walls, the above provisions are summarized in two main design requirements. The first requirement is the configuration of confined boundary elements, which enhance local ductility in critical regions, and the second requirement is a capacity design procedure to prevent shear brittle failures, which typically leads to high demand of shear reinforcement. However, in a lot of countries with high seismic exposure, such as Greece, a large number of existing structures are designed according to outdated regulations and include shear walls that do not conform to current standards. These non-conforming shear walls lack confined boundary elements and typically have relatively low ratios of shear reinforcement.

In most cases, the scientific approach for estimating strength and deformation capacity of RC members is based on enhanced numerical analyses, which involve various levels of complexity (e.g., beam, shell, or 3D finite elements, linear vs. non-linear analysis, etc.). On the contrary, code-based design and assessment provisions of codes require simplified analytical models that minimize complexity and computational cost, making them suitable for practical engineering applications.

The present paper aims to present a refined analytical procedure (referred to as the Proposed Model) to form Load–displacement (P-d) curves in reinforced concrete shear walls. The Proposed Model finds application to medium-rise shear walls designed with and without modern seismic provisions. A key feature of the Proposed Model is the incorporation of the shear effects in both deformation and capacity. The Proposed Model is compared to existing numerical models (implemented in the widely accepted Opensees platform). Both analytical and numerical approach are compared with the experimental results of four RC shear walls originally reported to previous studies.

2. Literature Review

The extensive and detailed experimental investigation of the behavior of RC shear walls started taking place in the 1970s and 1980s [1,2,3,4]. However, in the majority of cases, the studies aimed at revising the applicable regulations at that time, and thus the resulting conclusions and models primarily addressed enhanced design provisions rather than assessment procedures. In the following decades, research on the behavior of RC members placed increased emphasis not only on the design, but also on the assessment and redesign of existing members. The assessment methodologies have their basis on a displacement analysis approach, which gives particular attention in estimating both the strength and the deformation capacity of each individual member. The estimation of the deformation capacity lies not only in calculating the total deformation, but in separating the contribution of each individual mechanism (flexure, shear, slip, etc.), as well. These issues are often addressed through the application of advanced numerical techniques [5,6,7,8].

The computational cost of these methods has led to the development of easy-applied analytical models which have their scientific foundation on the so-called plastic hinge analysis. According to the plastic hinge analysis, for the case of a cantilever member, the plastic deformations after yielding are concentrated in an area near the member base, called the plastic zone. The plastic zone length may be replaced by an equivalent length, the plastic hinge length, L_pl ≈ 0.5L_pz, where the plastic deformation is assumed to be constant (Figure 1). Thus, the plastic hinge length, L_pl, can be calculated as follows:

$${\mathrm{L}}_{\mathrm{pl}}=0.5{\mathrm{L}}_{\mathrm{pz}}=0.5{\mathrm{L}}_{\mathrm{V}}\left(1-{\displaystyle \frac{{\mathrm{M}}_{\mathrm{y}}}{{\mathrm{M}}_{\mathrm{u}}}}\right)$$

(1)

where M_y is the moment corresponding to yielding point; M_max is the moment capacity and L_V is the length of the shear span (L_V = M/V = moment/shear at the end section).

Several empirical equations for the estimation of the plastic hinge length can be found in the international literature (i.e., [9,10]). One of the most commonly used expressions is the one proposed by Paulay and Priestley [11]:

$${\mathrm{L}}_{\mathrm{pl}}=0.08{\mathrm{L}}_{\mathrm{V}}+0.022{\mathrm{d}}_{\mathrm{b}}{\mathrm{f}}_{\mathrm{y}}$$

(2)

where d_b is the rebar diameter and f_y the reinforcement yield stress.

EC8-3 [12] proposes Equation (3) which, however, has application only in members including confined concrete and typically in accordance with the relevant confinement model (for model see EC8-3 [12]).

$${\mathrm{L}}_{\mathrm{pl}}={\displaystyle \frac{{\mathrm{L}}_{\mathrm{V}}}{30}}+0.20\mathrm{h}+0.11{\displaystyle \frac{{\mathrm{d}}_{\mathrm{b}}{\mathrm{f}}_{\mathrm{y}}}{\sqrt{{\mathrm{f}}_{\mathrm{c}}}}}$$

(3)

where h is the cross-section height and f_c the concrete compressive strength.

Assuming a pure flexural behavior, which means neglecting shear and bond slip effects within the member, the ultimate displacement at failure, d_u, can be expressed by Equation (4) as the sum of two separated terms: the displacement at yielding, d_y, and the plastic part of the displacement, d_pl [11].

$${\mathrm{d}}_{\mathrm{u}}={\mathrm{d}}_{\mathrm{y}}+{\mathrm{d}}_{\mathrm{pl}}={ \mathsf{\phi }}_{\mathrm{y}}{\displaystyle \frac{{{\mathrm{L}}_{\mathrm{V}}}^2}{3}}+\left({\mathsf{\phi }}_{\mathrm{u}}-{\mathsf{\phi }}_{\mathrm{y}}\right){\mathrm{L}}_{\mathrm{pl}}\left(1-{\displaystyle \frac{0.5{\mathrm{L}}_{\mathrm{pl}}}{{\mathrm{L}}_{\mathrm{V}}}}\right){\mathrm{L}}_{\mathrm{V}}$$

(4)

where φ_y is the section curvature at yielding stage and φ_u is the ultimate curvature at failure (ultimate limit state).

Based on the plastic hinge analysis theory, several enhanced models have been developed trying to estimate the envelope of the cyclic response of an RC member including shear and bond slip effects and expressions for calculating the plastic hinge length taking, at the same time, into account the shear strength of the member assumed to be degraded with inelastic cyclic displacements [13,14,15,16,17,18,19,20,21]. Several of these models have been incorporated, with some modifications, in modern seismic regulations. However, in most cases the above models include empirical expressions for estimating the plastic hinge length trying to fit Equation (3) (or refined expressions) to the results from experimental databases. Thus, each empirical expression for predicting the plastic hinge length can be applied only in accordance with a specific model for calculating the ultimate curvature; therefore, with a specific model for calculating the concrete compressive strength and the ultimate concrete strain. In addition, especially for RC shear walls, the experimental research of the past decades consists mainly of walls designed according to modern detailing provisions rather than existing non-conforming ones. Consequently, the fact that the databases, on which the above models were calibrated, consist primarily of members with seismic detailing, leading to the conclusion that these expressions are applicable mainly—if not exclusively—to members with seismic detailing. However, recent references (even relatively fewer) regarding inadequate reinforced walls can be found addressing issues such as the importance and the efficiency of confining reinforcement, the influence of low ratios of shear reinforcement, the buckling of compressive reinforcement, etc., leading to refined empirical and analytical models [22,23,24,25,26,27,28,29,30,31,32,33,34,35].

3. Experimental Study

The present paper focuses on the numerical and analytical investigation of the behavior of a series of four specimens—W₇, W₉, W₁₁, and W₁₃—whose experimental study has been presented and discussed in detail in Christidis [36] and Christidis and Trezos [37]. Thus, a brief discussion of the specimens’ characteristics and experimental results is presented herein. The geometrical and the reinforcement configuration characteristics are depicted in Figure 2. The specimens represent typical medium-rise walls with aspect ratio equal to L_V/h = 2.0, tested as cantilevers under static cyclic loading.

Walls W₇ and W₁₃ serve as benchmarks for two distinct types of reinforced concrete (RC) shear walls. W₇ represents a modern design compliant with Eurocode standards, while W₁₃ represents a typical older, non-conforming wall. Specimens W₉ and W₁₁ also represent non-conforming walls. However, they were modified by incorporating open stirrups to prevent buckling of compressive reinforcement; in this way the influence of shear reinforcement was directly evaluated. Details regarding material properties and the reinforcement ratios are provided in

Table 1. Reinforcement ratios and material properties of the specimens considered.

Specimen (Wall)	Concrete Compression Strength fc (MPa) (CV%)	Yield/Failure Stress fy/fu (MPa) (CV%)	Hardening/Failure Strain esh/esu (‰)	Stirrups Yield/Failure Stress fy/fu (MPa) (CV%)	Longitudinal Reinforcement Ratio ρtot a (‰)	Transverse (Shear) Reinforcement Ratio ρw b (‰)
W7	31.12 (3.33)	D10:604/705 (2.25/2.51) D8:588/681 (2.58/2.51)	D10:26.2/100.2 D8:28.1/88.2	588/681 (2.58/2.51)	14.33	6.69
W9	31.12 (3.33)	580/670 (3.34/2.55)	26.3/107.9	588/681 (2.58/2.51)	12.06	2.01
W11	31.12 (3.33)	580/670 (3.34/2.55)	26.3/107.9	568/654 (9.23/7.24)	12.06	1.13
W13	25.37 (1,50)	580/670 (3.34/2.55)	26.3/107.9	568/654 (9.23/7.24)	12.06	1.13

Note: CV = Coefficient of Variation. ^a ρ_tot = Area sum of longitudinal reinforcement/cross-section area = ΣA_s,L/(b_w*h). ^b ρ_w = Area sum of stirrup/(wall width*stirrups distance) = ΣA_s,w/(b_w*s).

.

As indicated from the experimental results [36,37], low ratios of shear reinforcement appear to have a limited impact on both the bearing capacity and the top deformation capacity of the specimens. All the four walls reached a maximum measured load close to their flexural capacity, while in most cases (except W₁₃) exhibited a flexural post-yield behavior, followed by significant values of ductility. In all examined cases, the observed loss of bearing capacity was primarily associated with the deterioration of the compressive zone. This conclusion seems to be applicable even in the case of W₁₃, considering that W₁₁ and W₁₃ include the same shear reinforcement, with the only difference being the presence of open stirrups in W₁₁ (configured primary to prevent rebar buckling). Note that, although it cannot quantitatively evaluated, the presence of edge stirrups in W₁₁ may at some point prevent the initiation of shear inclined cracks. However, the type of post yield behavior for W₁₃ is mainly attributed to the buckling of compressive rebars. On the contrary, low ratios of shear reinforcement seem to influence the cracking pattern of the walls, characterized by the formation of significant inclined X cracks; nevertheless, the shear cracks did not lead to a direct loss of bearing capacity.

The modified Kent and Park model [38] has been adopted to represent the behavior of both unconfined and confined concrete, with the corresponding stress–strain curves illustrated in Figure 3. Regarding the reinforcing steel, the Chang and Mander [39] model has been employed, while the buckling behavior of compressive longitudinal rebar was incorporated based on the Dhakal and Maekawa [40] model. The buckling of compressive rebars is primary influenced by the slenderness ratio L/D, where L denotes the unsupported bar length and D denotes its diameter. Three different cases occur—L/D = 5.0 for W₇, L/D = 8.33 for walls W₉ and W₁₁ and, finally, L/D = 33.33 for W₁₃ as shown in Figure 4.

4. Analytical Approach—Proposed Model

The proposed model is based on Christidis and Karagiannaki work [31]. Christidis and Karagiannaki, based on experimental observation and results, tried to point out the significant influence of shear deformation proportion to the total deformation capacity of shear walls with low ratios of shear reinforcement. The output of this effort was a modified model to estimate the shear-to-flexural deformation ratio (d_sh/d_fl) within the inelastic deformation range, based on Beyer et al. [41] model. A complete procedure is presented herein to construct a P-d curve.

Step-1

A moment curvature analysis is conducted considering material properties for unconfined concrete, confined concrete (if any) and steel reinforcement. For steel reinforcement, buckling models for compressive reinforcement may be adopted (i.e., [30,40]). Moment curvature curve (M-φ) is derived, together with the mean axial strain of the cross-section.

Step-2

Bilinearization of M-φ curve in order to define conventional curvature at yielding, φ_y. The initial elastic stiffness is defined from the first yielding point (Μ_y,_1st, φ_y,_1st). The conventional yield point (Μ_y, φ_y) may be derived by equating the area (energy) of the two curves. The ultimate curvature, φ_u, is defined as the minimum curvature value from the one corresponding to a percentage 20% loss of the bearing capacity of the cross-section and the one corresponding to the tension steel failure strain e_su = 60‰ (as proposed in EC8-3 [12] to take into account the degradation due to cyclic loading).

Step-3

The flexural deformation capacity at yielding state may be calculated as follows:

$${\mathrm{d}}_{\mathrm{fl},\mathrm{y}}={\iint }_0^{{\mathrm{L}}_{\mathrm{V}}}{\mathsf{\phi }}_{\mathrm{y}}\mathrm{dx}={\displaystyle \frac{1}{3}}{\mathsf{\phi }}_{\mathrm{y}}{{\mathrm{L}}_{\mathrm{V}} }^2$$

(5)

It should be noted that the total displacement at yielding may include the contribution of mechanisms further than the flexural one. For example, as pointed out in EC8-3 [12] (whose scientific background is based on the statistical analysis of an extensive experimental database), the total displacement at yielding includes the contribution of the following:

• The tension shift effect (if shear cracking precedes flexural yielding);
• The shear deformation;
• The slippage of longitudinal rebars.

Since, during the experimental results, neither (significant) shear cracking before flexural cracking nor rebar slippage was observed, and taking into consideration that the examined numerical model (see Section 4) takes no account for these mechanisms, these two terms are neglected.

For shear deformation, EC8-3 [12] based on the aforementioned statistical analysis models adopts a value equal to 0.0013L_V. However, more recent refined (statistical based) models [9] propose that shear deformation at yielding may be estimated as

$${\mathrm{d}}_{\mathrm{sh},\mathrm{y}}={ \mathsf{\lambda }}_{\mathrm{sh}}(1+{\mathsf{\mu }}_{\mathrm{sh}}{\displaystyle \frac{\mathrm{h}}{{\mathrm{L}}_{\mathrm{V}}}}){\mathrm{L}}_{\mathrm{V}}$$

(6)

where h is the cross-section height and for shear walls λ_sh = 0.00105 and μ_sh = 0.475 may be adopted.

Note that for h/L_V = 0.75/1.5 = 0.5 (values h and L_V of the shear wall specimens examined within the present paper), Equation (6) results in 0.001299L_V ≈ 0.0013L_V. Assuming a constant d_sh,y/d_fl,y ratio within the elastic deformation range, the total deformation in every ductility step (curvature step) until yielding can be estimated as

$${\mathrm{d}}_{\mathrm{tot}}={\displaystyle \frac{1}{3}}\mathsf{\phi }{{\mathrm{L}}_{\mathrm{V}} }^2+{\displaystyle \frac{{\mathrm{d}}_{\mathrm{s}\mathrm{h},\mathrm{y}}}{{\mathrm{d}}_{\mathrm{fl},\mathrm{y}}}} {\displaystyle \frac{1}{3}}\mathsf{\phi }{{\mathrm{L}}_{\mathrm{V}} }^2$$

(7)

Step-4

The top flexural deformation capacity at every curvature step at inelastic range (φ > φ_y) may be calculated as

$${\mathrm{d}}_{\mathrm{fl}}= {\mathrm{B}}_2{\mathsf{\phi }}_{\mathrm{y}}{{\mathrm{L}}_{\mathrm{V}} }^2$$

(8)

where

$${\mathrm{B}}_2={\displaystyle \frac{1}{6}}\left(-\mathrm{k}{\mathrm{n}}^2+3\mathrm{nk}-\mathrm{n}+2\right)$$

(9)

and k and n are defined in Figure 1. For detailed proof of Equations (8) and (9) see Christidis and Karagiannaki [31].

Step-5

The shear-to-flexural deformation at inelastic deformation range may be calculated as

$${\displaystyle \frac{{\mathrm{d}}_{\mathrm{sh}}}{{\mathrm{d}}_{\mathrm{fl}}}}={\displaystyle \frac{(\frac{{\mathsf{\epsilon }}_{\mathrm{m}1}}{\tan \mathsf{\theta }}+{\mathsf{\epsilon }}_{\mathrm{s}1}\tan \mathsf{\theta }){\mathrm{L}}_{\mathrm{pl}}+(\frac{{\mathsf{\epsilon }}_{\mathrm{m}2}}{\tan \mathsf{\theta }}+{\mathsf{\epsilon }}_{\mathrm{s}2}\tan \mathsf{\theta })({\mathrm{L}}_{\mathrm{V}}-{\mathrm{L}}_{\mathrm{pl}})}{{{\mathrm{B}}_2\mathsf{\phi }}_{\mathrm{y}}{{\mathrm{L}}_{\mathrm{V}}}^2}}$$

(10)

where

ε_m1 is the mean axial strain of the cross-section assumed constant within the plastic hinge length, L_pl.

ε_m2 is the mean axial strain of the cross-section of the upper part of the wall L_V-L_pl, estimated as ε_m2 = 0.5ε_my.

ε_s1 is the shear reinforcement strain within the plastic hinge length, L_pl

ε_s2 is the shear reinforcement strain within the length L_V-L_pl

θ is the inclined strut angle, and it refers to the angle outside the plastic area where the cracks are approximately parallel.

The strain value ε_s,i at each displacement step, i, can be estimated as

$${\mathsf{\epsilon }}_{\mathrm{s},\mathrm{i}} =\mathrm{A}{\left({\displaystyle \frac{{\mathrm{d}}_{\mathrm{fl},\mathrm{i}}}{{\mathrm{L}}_{\mathrm{V}}}}\right)}^{\mathrm{B}}$$

(11)

Factors A and B can be estimated as

-

for V_R/P_f ≤ 0.85

Bottom panel (plastic hinge length):

$$\mathrm{A}=2233{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-22.31}$$

(12)

$$\mathrm{B} =1.80{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-1.37}$$

(13)

Top panel:

$$\mathrm{A} =143{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-25.80}$$

(14)

$$\mathrm{B} =1.42{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-1.78}$$

(15)

-

for V_R/P_f > 0.85

Bottom panel (plastic hinge length):

$$\mathrm{A} =\mathrm{20,016}{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-2.05}$$

(16)

$$\mathrm{B} =2.07{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-0.13}$$

(17)

Top panel:

$$\mathrm{A} =2170{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-5.87}$$

(18)

$$\mathrm{B} =1.65{\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{R}}}{{\mathrm{P}}_{\mathrm{f}}}}\right)}^{-0.62}$$

(19)

where V_R is the shear capacity before degradation (see also Equation (20)) and P_f is the flexural capacity in load terms.

For detailed proof of Equations (10)–(19) see Christidis and Karagiannaki [31].

The above flexural capacity can be attained provided that the structural member has not previously reached its shear strength capacity. For assessment purposes, experimental and analytical results have shown that the shear capacity degrades with the inelastic cyclic displacements (e.g., [16,34]). EC8-3 [12], adopting the aforementioned assumption, proposes Equation (13) for estimating the shear strength as controlled by the stirrups, V_R.

$${\mathrm{V}}_{\mathrm{R}}={\displaystyle \frac{1}{{\mathsf{\gamma }}_{\mathrm{el}}}}\left[\begin{array}{c}{\displaystyle \frac{\mathrm{h}-\mathrm{x}}{2{\mathrm{L}}_{\mathrm{V}}}}\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{N};0.55{\mathrm{A}}_{\mathrm{c}}{\mathrm{f}}_{\mathrm{c}}\right)+\left(1-0.05\min \left(5;{{\mathsf{\mu }}_{\Delta }}^{\mathrm{pl}}\right)\right)\ast \\ \ast \left[0.16\max \left(0.5;100{\mathsf{\rho }}_{\mathrm{tot}}\right)\left(1-0.16\min \left(5;{\displaystyle \frac{{\mathrm{L}}_{\mathrm{V}}}{\mathrm{h}}}\right)\right)\sqrt{{\mathrm{f}}_{\mathrm{c}}}{\mathrm{A}}_{\mathrm{c}}+{\mathrm{V}}_{\mathrm{w}}\right]\end{array}\right]$$

(20)

where γ_el = 1.15 for primary seismic elements, γ_el = 1.00 for secondary seismic elements; x is the compression zone depth; N is the compressive axial force (positive, taken as being zero for tension); A_c is the cross-section area equal to b_wd for a cross-section with a rectangular web of thickness b_w and structural depth d; ρ_tot is the total longitudinal reinforcement ratio; V_w = ρ_wb_wzf_yw is the contribution of transverse reinforcement to shear resistance (ρ_w = A_sw/b_ws is the transverse reinforcement ratio and A_sw is the transverse reinforcement area); ductility factor μ_Δ^pl = μ_Δ − 1 = d_tot/d_y − 1.

The final P-d curve may be derived as the lower envelope of shear and flexural capacity, as shown in Figure 5.

In order to apply the analytical procedure, a curvature analysis was preceded in the Opensees platform [42]. The moment-curvature diagrams are depicted in Figure 6 together with the equivalent bilinear curves.

The proposed model was applied for all specimens and the resulting P-d curves are depicted in Figure 7. The curves are plotted up to maximum total deformation of 50 mm. For comparison reasons, in the same figure, the P-d curves model neglecting the shear components in both deformation and capacity are also included, representing a pure flexural element model. It should be noted that in order to enable a direct comparison between the two models, the curves of the model without shear are plotted only up to the (flexural) deformation which corresponds to the same ductility level as the ones with shear effects (the difference between the two displacement values is attributed to the shear deformation). Finally, it should be noted that in the case of W₇, shear strength is greater than the flexural capacity in all the displacement range; thus, it does not influence the P-d curve in terms of capacity.

5. Numerical Analysis Approach—Opensees

In addition to the previous analytical approach, numerical simulations were carried out in the OpenSees platform [42] using two alternative elements available in the OpenSees library—the widely used Force-Based Beam-Column Element (FBE) and a Flexure-Shear Interaction Displacement-Based Beam-Column Element (FSI), which takes into account the interaction between flexure and shear.

5.1. Force-Based Beam-Column Element (FBE)

The Force-Based Beam-Column Element (FBE) is a flexural element formulation that accounts for the spread of plasticity along the height of the element. The internal force distributions are expressed as functions of the nodal forces and the response is evaluated solely based on the flexural characteristics of the cross-section (i.e., geometry, longitudinal reinforcement, material constitutive laws) for the response evaluation. As such, the model does not incorporate shear contributions in the evaluation of either total strength or overall displacement. Consequently, the FBE allows for the assessment of differences in the total P-d response of a wall arising solely from flexural effects, i.e., the influence of confinement and the buckling of compressive longitudinal reinforcement, while neglecting the effects associated with varying shear reinforcement ratios. While the FBE offers clear advantages over the traditional Displacement-Based Element (DBE), particularly in capturing the descending branch of the post-yield response, it also involves challenges related to the solution uniqueness, commonly referred to as the ‘size-effect’ [43,44]. Specifically, for a predefined top displacement the peak curvature demand at the cantilever base is influenced by the number of integration points (IPs). The inelastic curvature tends to localize at the finite length of the first IP, L_IP₁ = w₁L_V, while the other IPs remain in the linear elastic range and contribute minimally to the overall curvature or moment distribution. As the number of IPs increases, the effective length of the first IP decreases, resulting in a higher curvature demand to achieve the same displacement at the top.

When modeling a RC cantilever using a single FBΕ and adopting the commonly used Gauss-Lobatto integration method, the number of the IPs can be decided such that the finite element length of the first (base) IP aligns with the estimated the plastic zone length. However, since the Gauss-Lobatto weights, w, are fixed for each integration scheme, an exact match is not always possible. Consequently, within this paper, the examined walls were modeled using two elements. The length, L₁, of the first element using IP = 2 (bottom and top), is defined as such that the finite length of the first node L_IP₁ = w₁L₁ = 0.5L₁ precisely corresponds the estimated plastic hinge length. The second element remains in the elastic range, making the choice of its integration scheme (number of IPs) less critical. IP = 3 is selected which leads to x₁ = −1, x₂ = 0, x₃ = 1 namely x₁ = 0, x₂ = 0.5L2, x₃ = L₂ for points position and w₁ = 0.333, w₂ = 1.333, w₃ = 0.333 namely w₁ = 0.1665L₂, w₂ = 0.6665L₂, w₃ = 0.1665L₂ for points weight. Although it is difficult to set objective rules to, numerically, define the “correct” P-d solution (an energy approach is included in [44]), it is possible to set a lower bound in plastic hinge length such that the curvature demand leads to acceptable results. This is clearly shown in Figure 8a which includes the curvature profile of wall W₇ for two different cases—for L_pl = 0.32 m (according to EC8-3 from Equation (3)) using the aforementioned 2-element modeling and for a typical 5-point Gauss-Lobatto integration using one element, which leads to a plastic hinge length, L_pl = L_IP₁ = 0.05L_V = 0.075 m (w₁ = 0.1 = 0.05L). Both cases lead to similar P-d results (Figure 8b) but in the case of L_pl = 0.075 m the tension reinforcement strain output (Figure 8c) is significantly higher than the tension failure steel strain derived from uniaxial tension tests (see also Table 1).

The influence of the plastic hinge length is shown in Figure 9 which includes the numerical results of wall W₇ for three different lengths using Equations (1)–(3). Considering that for all the three cases the P-d curve, the steel stress–strain diagram and the curvature demand do not exhibit significant differences and that for the non-conforming walls EC8-3 does not include any expression for estimating the plastic hinge length, the Paulay and Priestley expression (Equation (2)) was finally used.

As also discussed earlier in Section 3, one of the main factors that determines the wall behavior is the buckling of the longitudinal rebars, which has been taken account as a diminution of the compression steel stresses according to the Dhakal and Maekawa [40] model. In the case of wall W₁₃ (with L/D = 33.33), the buckling effect leads to a more descending post-yield M-φ diagram, to lower ultimate curvature value (corresponding to 0.80 M_max, lower in comparison to the one corresponding to the tension failure steel strain) and, therefore, to lower ultimate displacement value. The influence of buckling is shown in Figure 10.

5.2. Flexure-Shear Interaction Displacement-Based Beam-Column Element (FSI)

An alternative numerical model which specially concentrates on the simulation of RC shear walls is the one proposed in Massone et al. [6]. The foundational basis of the model is a macroscopic fiber-based model, known as Multiple-Vertical-Line-Element-Model (MVLEM), as originally introduced by Vulcano et al. [45]. According to this approach, an RC shear wall is modeled as a stack of Multiple-Vertical-Line-Elements (MVLEs), placed one upon the other. Each MVLE is represented by a series of uniaxial elements connected to rigid beams, to model the flexural component, complemented by a horizontal spring that simulates the shear behavior. However, this simulation does not consider the shear–flexure interaction. In view of this limitation, Massone et al. [6] and Orakcal et al. [46] suggested an improved model, by assigning a shear spring for each uniaxial element, achieving an RC membrane behavior for each element. This model is incorporated in the OpenSees platform in a Displacement-Based Beam-Column Element (DBE) known as the Flexure–Shear Interaction Displacement-Based Beam-Column Element (FSI). The specific model is sensitive to three modeling factors: the discretization of the cross-section in n strips, the number of subelements, m, along the wall height and the value of the parameter c, which is the center of rotation (or the center of gravity of the curvature distribution) in each subelement.

The discretization of the cross-section in n strips for each element was performed to achieve the best simulation of the actual cross-section configuration, i.e., the exact place of the rebars and the transitions between confined and unconfined concrete (Figure 11).

Contrary to the FBE, when modeling an RC member using a DBE, the analysis results are influenced not by the number of the IPs but by the number of elements along the height and especially on the length of the first element, while the number of remaining elements does not have a significant role in the response. Accordingly, in the present study, the walls were modeled using two elements where the length L₁ of the first element was defined equal to the plastic hinge length L₁ = L_pl. The implementation of the FSI additionally demands the specification of shear reinforcement. Consequently, each of the two elements was assigned to such an area of shear reinforcement that, when divided by the product of the element height and the wall width, results in the actual shear reinforcement ratio, ρ_w, of the corresponding shear wall.

Finally, regarding the location of the center of rotation, a value c = 0.4 was recommended by Vulcano et al. [45], although using a large number of MVLEs along wall height diminishes the influence of this parameter on the predicted response [46]. Figure 12 shows the analysis results using two elements with c = 0.5 and c = 0.4. Applying the value c = 0.4 in wall W₇, although leading to slightly higher values of curvature, φ, and shear deformation (rotation), γ, demand (Figure 12a,b), it did not seem to significantly affect the overall P-d response (Figure 12c). Thus, a value of c = 0.5 was adopted in this study without; however, this value is being considered as determinant for the analysis results.

6. Comparison of Proposed Model with Numerical and Experimental Results

The results of the application of the proposed model and the numerical analysis are summarized in Figure 13, together with the experimental results (in the form of envelope in the positive direction). As shown in Figure 13, numerical and analytical results show satisfactory agreement, especially for specimens W₇ and W₉, which include higher ratios of shear reinforcement. However, in specimens W₁₁ and W₁₃, FSI seems to lead to more descending post-yield branches especially for large displacement values (note that for wall W₁₃ the FBE analysis stops converging in 23.5 cm).

An insightful comparison can be drawn between the shear-to-flexural deformation ratio (d_sh/d_fl) for the inelastic range of displacement. The results are illustrated in Figure 14. The procedure for estimating the experimental ratio can be found in detail in Christidis and Karagiannaki [31]. As shown in Figure 13 the results of the proposed model show close agreement with the numerical results. In both cases, d_sh/d_fl tends to exhibit higher values as the total displacement increases. This assumption is more pronounced in the cases of W₁₁ and W₁₃ which include lower ratios of shear reinforcement.

On the contrary, the results of FSI model demonstrate a different behavior. Although, as expected, d_sh/d_fl generally exhibits higher values in cases of low ratios of shear reinforcement, a different pattern emerges between walls with and without adequate shear reinforcement. In the first case, the ratio clearly has the tendency to decrease after yielding. Conversely, in walls with inadequate shear reinforcement, the ratio initially decreases but then begins to increase beyond a certain displacement value.

7. Conclusions

The present study aims to investigate the deformation behavior, in both deformation and capacity terms, of existing reinforced concrete (RC) shear walls designed with and without modern seismic provisions, through the application of an analytical model. The Proposed Model was compared with numerical models (FBE and FSI elements implemented in OpenSees) and experimental data from four RC wall specimens (see also Christidis [36] and Christidis and Trezos [37]).

As concluded from the experimental results, low ratios of shear reinforcement tend to affect the cracking pattern of the reinforced concrete walls, characterized by the formation of significant inclined cracks, which, nevertheless, do not necessarily cause a direct reduction in bearing capacity. On the contrary, the ratio of shear reinforcement, and therefore the shear-related inclined cracks affect the development of shear deformations and their proportion to the total displacement. In all cases examined, the loss of bearing capacity appears to be linked mainly to the deterioration of the compressive zone, which is directly associated with the effect of buckling of the compressive longitudinal reinforcement.

Based on the aforementioned experimental observations, the Proposed Model, applied to medium-rise shear walls (normally shear walls with aspect ratio equal to L_V/h = 2.0–2.5), focuses on estimating the shear-to-flexural ratio within the inelastic deformation range. In addition, the Proposed Model includes a full analytical procedure to construct the Load–displacement (P-d) curve, based only on an initial moment-curvature analysis.

The experimental results and Proposed Model were also compared with the results derived from non-linear numerical analysis using beam elements in OpenSees platform. Two alternative models, available in OpenSees library, were used—the Force-Based Beam-Column Element (FBE) and the Flexure-Shear Interaction Displacement-Based Beam-Column Element (FSI), which considers the interaction between flexure and shear. The use of FBE (which does not consider the shear component) seems to lead to realistic predictions for both strength and deformation capacity, even in cases where the modern seismic detailing is not fully applied (wall W₉). Considering that the FBE model is based on the assumption of the concentration of the curvature demand near the base, it leads to similar results with the Proposed Model, if shear mechanism neglected. In the case of walls sufficiently reinforced against shear (walls W₇ and W₉), the use of FSI and FBE yields to similar results. On the contrary, in the case of walls with low shear reinforcement (walls W₁₁ and W₁₃) combined with the absence of confining reinforcement, the use of FSI seems to lead to lower values of bearing capacity followed by descending post-peak branches and poor deformation capacity. These results are, to some extent, in accordance with the experimental observations where, although the low ratio of shear reinforcement did not seem to be determinant for the wall behavior in strength capacity terms, the severe cracking mode and generally the interaction between flexure and shear definitely accelerated the wall failure and led to lower displacement values than estimated using a pure flexural approach.

The comparison of load–displacement responses demonstrated satisfactory agreement between the Proposed Model and numerical simulations, particularly for specimens with adequate shear reinforcement (W₇ and W₉). In contrast, for walls with lower shear reinforcement (W₁₁ and W₁₃), the FSI model exhibited a more significant post-yield degradation. However, comparing the shear-to-flexural deformation ratio in the inelastic deformation range, the FSI model showed different trends depending on the level of shear reinforcement. While both the Proposed Model and numerical results exhibited an increasing trend of this ratio with displacement, the ratio in FSI results tended to decrease after yielding in walls with sufficient shear reinforcement, whereas in walls with inadequate reinforcement, the ratio initially decreased and subsequently increased beyond a certain displacement value.

Overall, the results highlight the validity of the Proposed Model in capturing the non-linear behavior of RC shear walls, particularly in reflecting the influence of shear deformation. The Proposed Model can be applied either as a standalone methodology or integrated into larger analytical frameworks to evaluate the performance of both new and existing reinforced concrete shear wall structures.