Deformation Limits of L-Section RC Shear Walls

3.1. Analyzed Walls

Numerical models shall take account of main variables, and shall be representative in geometry. To achieve this, FE models of 216 L-section RC shear walls designed to meet the Chinese codes are set up. Sizes of these specimens are shown in Fig. (2). Main parameters and their values are: shear-span ratio (λ=2, 2.5 and 3.0), axial load ratio (n=0.071, 0.21, 0.36 and 0.50, 0.50 is beyond the code's regulation), longitudinal reinforcement ratio of the web and flange end columns as well as corner column (ρ=1.15%, 2.36% and 3.68%), the confinement reinforcement(stirrups) ratio of the web and flange end columns and corner column (ρ_v1=Ф8@200, Ф10@200, Ф12@200) and the horizontal shear reinforcement ratio(ρ_v2=1.14% and 1.55%). Shear span ratio is defined as λ=M/(V·h) where M and V are moment and shear force at a given cross section, and h is the length of the section. For cantilever walls, λ equals to L/ h where L is the height of the walls. Axial load ratio is n=F/(f_c·A) where F is the applied axial load, f_c is the compressive strength of concrete and A is the gross area of wall section. Moment-shear ratio refers to m=M_u/(V_u·L) where M_u is the ultimate sectional moment capacity, V_u is the shear resistant capacity. Note that the value of moment-shear ratio is influenced by reinforcement ratio, axial load ratio, horizontal reinforcement ratio and the height of the wall. It is a comprehensive index that characterizes the shear level in the web and is not treated explicitly as an independent variable herein. However, given the wide range of the influencing variables, moment-shear ratio will has a representative distribution.

3.2. Numerical Modeling of the Analyzed Walls

FE software ABAQUS is adopted. The Concrete Damaged Plasticity constitution model is used to simulate material behavior of concrete. A general plastic model is used to model steel bars. According to [26], dilation angle of concrete is between 36° to 40°, an optimal value based on material test is 38° which is adopted herein. K_c is the ratio of the distances between the hydrostatic axis and respectively the compression and tension meridian in the deviatoric section [27]. It is taken as recommended value 2/3 which conforms to test results [27]. Eccentricity describes the rate of approach of the plastic potential hyperbola to the asymptote. The recommended value of 0.1 is used. The parameter f_bo / f_co refers to the ratio of strength in the biaxial state to that in the uniaxial state. It is taken as the default value 1.16 which is also suggested by material test [27]. With respect to steel, the simple plastic model considering strengthening is adopted. Yield stress is 400 MPa for class HRB400 steel, ultimate strength is 570 MPa with corresponding ultimate strain of 0.075. These values are based on Code for design of concrete structures (GB 50010-2010) [28].

The concrete body uses 3D cubic elements and steel bars are simulated by truss elements. Truss elements can capture the steel bars' axial behavior. Moments from rotation is not considered since it is generally negligible comparing with axial force. Connection between concrete and steel bars is realized by embedded element technique. Nodal displacement of truss elements is interpolated by the shape function of the surrounding concrete element. Such restraint can well reveal the behavior of reinforced concrete members when steel bars are gripped tightly by concrete. In real specimens, slipping only concentrates in limited areas undergoing severe nonlinearity, such as plastic hinges. This study mainly focuses on the macro deformation capacities to which slipping is not as influential in terms of numerical analysis. Moreover, taking account of slipping is too costly in computation for 216 models. Therefore, effect of slipping is not considered herein. FE models are firstly loaded with axial force. Monotonic lateral displacement loading is carried out with constant axial load. Monotonic curves are the upper bound of the hysteretic curves, and they are usually close [18]. According to Shen Ju-min et al. [29], error of ultimate resistance is below 10%, which is acceptable. Also, cyclic loading is extremely costly in computation. Thus, the push-over loading is considered rational.

3.3. Comparison Between FE Models and Test Results

To verify the parameters of the FE analysis, comparison between numerical models and test specimens is necessary. Data of 49 tests of L-section RC shear walls and L-section RC columns from literature [10 -12, 30 -39] are collected. Finite element analysis is performed to 9 of them. Parameters of the 9 specimens are given in Table 2. These walls are representative in geometry, and include a wide range of axial load ratio, shear-span ratio and reinforcement ratio. Values of parameters are determined according to the discussion in last section. Comparison between numerical load-displacement curves and backbone curves attained from tested specimens are shown in Fig. (3).

It is seen that the FE analysis and tests results show general agreement on backbone curves, maximum resistance and deformation abilities. Error of maximum resistance is limited within 15%, which guarantees the accuracy of FE models. With respect to failure modes, analysis and tests also display very similar features. Take the specimen LSW-6.5 as an illustrative example. Failure modes of both the tested specimen and numerical model are given in (Fig. 4). As described in the literature, lateral cracks were first seen, concrete crush occurred at the eighth cycle, spalling was observed at the tenth cycle together with the buckling of longitudinal bars (Fig. 4a). The model's concrete compressive damage, tensile damage and yielding of reinforcement are respectively presented in Fig. (4b-d) The concrete's damage represents the comprehensive extent of material nonlinearity. The results of the FE model reveal satisfactory accordance with the tested specimen.

3.4. Failure Modes Categorization

Failure modes of the collected 49 specimens show strong relevance with shear-span ratio λ and moment-shear ratio m Fig. (5). Specimens' moment and shear capacities are calculated according to formulas and regulations in technical specification for concrete structures of tall building (JGJ3-2010) [39]. Relation between failure modes and shear-span ratio and end column reinforcement ratio are given in Fig. (6). As can be seen, shear failure specimens are mostly distributed in the range of λ<1.5 and m>0.9 while flexure failure dominates the phase of λ>3.5 and m<0.5. This indicates an increasing involvement of shear effect with the ascendance of moment-shear ratio. Positive correlation between flexure-shear failure and end column reinforcement ratio can be observed when shear-span ratio λ is below 3.0. The correlation weakens when λ is over 3.5.

To compensate for the limit amount of specimens and the unavoidable divergence of test data, FE models are analyzed to do further study and gain quantified evaluation of the connection between failure modes and the above mentioned factors. 3 categories of failure modes are defined: flexure failure, flexure-shear failure and shear failure. Typical characteristics of the 3 modes in ABAQUS are shown through (Fig. 7-9). The sign of a flexure failure mode (Fig. 8) includes: yielding of the tensile longitudinal reinforcement, horizontal shear reinforcements' remaining elastic, damage concentration in the bottom confined area. The sign of a flexure-shear failure mode (Fig. 8) includes: yielding of the tensile longitudinal reinforcement, following small amount of yielding horizontal shear reinforcement, the damage's incline stretching out to web area from the bottom. The shear failure mode's indication (Fig. 9) includes: large amount of yielding shear reinforcement and saturation of concrete tensile damage in the web. The state of failure, with reference to literature [40], is defined as one of the following: 1). strain of the longitudinal reinforcement reaches the ultimate value, 2). strain of the compressive concrete surpasses the ultimate value, and 3). resistance of lateral load drops by 15%.

With reference to the collected 49 specimens and 216 FE models, further study of the influence of shear-span ratio, moment-shear ratio, reinforcement ratio and stirrup ratio of end columns on failure modes is carried out. Given the potential difference in two reversed directions, a judgment criterion for discerning failure modes of the overall shear wall is defined herein. The criterion is based on damage features of the flange submitted to both tension (negative direction) and compression (positive direction) and is listed in Table 3.

To verify the above method, FE analysis is performed to several specimens. Failure modes prediction is shown in Table 4 where satisfactory coincidence between of test specimens and FE models is seen. Prediction for the 216 FE models is also conducted. The relation between the anticipated failure modes and factors including shear-span ratio, moment-shear ratio, reinforcement ratio of end columns is given in Fig. (10).

According to ASCE/SEI 41-13 [41], failure is controlled by flexure when λ≥3, and by shear when λ<1.5, and by both flexure and shear when λ is in between. In addition to the geometric dimensions, the mechanical state under various loads is also as influential. But this factor has been omitted by ASCE/SEI 41-13. Thus, a more detailed criterion for failure modes judgment of L-section RC shear wall is given in Table 5. It is summarized on the basis of the numerical models, and has further involved the impact of flexure-shear ratio and reinforcement ratio of end columns. In Table 5, λ is the shear-span ratio, m is the moment-shear ratio and ρ is the reinforcement ratio of end columns.

The above standard is proved with satisfactory accuracy by comparing the predicted failure modes with the actual failure modes observed in the 49 specimens and 216 FE models. The rate of correct predictions is given in Table 6.

4.1. Selection of Deformation Index

The index of deformation shall represent the actual state of member performance, that is, to have high correlation with the extent of damage. The drift angle, as indicated by this study, is qualified to serve as reliable indicator of damage states. Calculation of drift angle is illustrated in Fig. (11). Total drift angle is the sum of the elastic and plastic drift angle. The elastic limits (the first limits) use elastic drift (also total drift) while other limits use plastic drift. This could give direct clues of the nonlinear deformation at each level, and is more convenient for practical use in most software.

The derivation of the deformation limits in the loading direction that imposes tension to the flange is analyzed. Determination of the limits in the opposite direction follows exactly the same process and will not be presented herein for the sake of conciseness.

4.2. Factors Influencing the Deformation Limits

Based on the 7 damage levels of the L-section RC shear wall, 6 deformation limits represented by drift angle δ_1e, δ₂, δ₃, δ₄, δ₅ and δ₆ are put forward. Correlation analysis is carried out to study the relevance between deformation limits and factors including shear-span ratio, axial load ratio, end columns' longitudinal reinforcement ratio, end columns' stirrup ratio and moment-shear ratio, respectively. Relevance is considered strong when the p value of F-statistic is lower than 0.05.

According to the results given in Table 7 and Table 8, the axial load ratio and flexure-shear ratio are the two dominant factors over the deformation limits of flexure failure and flexure-shear failure. Since shear failure is brittle, large ductility is not expected. The deformation of the peak resistance δ_ultimate is defined as ultimate deformation limit for shear failure. Moreover, data show high divergence of the limits δ₂, δ₃, δ₄, δ₅ defined in Table 1 when shear failure occurs. Some of the intermediate limits are even greater than δ_ultimate. Thus, performance evaluation and control of shear-failure-walls only relies on the elastic limit δ_1e and the ultimate limit δ_ultimate. As can be seen in Table 9, shear-span ratio and longitudinal reinforcement ratio are main influential factors of δ_ultimate.

4.3. Fitted Formulas of Deformation Limits

Deformation limits at different performance levels are fitted into linear formulas. Only the 2 most dominant variables are considered as variables. The fitted formulas are given in Table 10.

4.4. Criteria of Deformation Limits

The criteria of deformation limits are established based on axial load ratio and moment-shear ratio. Limits of shear failure mode are determined conservatively as a lower bound value. The limits criteria are given in Table 11 value of the limits is determined as mean minus one time standard deviation.