Geometric modelling, surface interaction, boundary conditions and solution scheme
The FE model of base restrained reinforced concrete (RC) RW has been developed using the FE software Abaqus [1]. Figure 7a shows the geometrical details of the base restrained RW model. Plane strain conditions have been assumed for the development of 2D RC RW model. In order to simplify the boundary value problem, it was assumed that the RW and backfill are resting over a firm rock outcrop. The RW, backfill and base rock have been modelled using the 2D solid sections, and the steel reinforcement has been modelled using the wire element option available in Abaqus. Figure 7b shows the reinforcement details of the base restrained RW. The 25 mm diameter steel bar has been used as the primary rebar (backfill side) with 250 mm c/c spacing. The 16 mm diameter steel bar has been used as secondary rebar with 250 mm c/c spacing. The base of the RW has also been reinforced using the 16 mm diameter rebars with 250 mm c/c spacing. The rebar frame has been embedded inside the RW using the embedded region option available in Abaqus. The base slab of the RW has been restrained with the rock by modelling fixity between the base slab of the RW and the top surface of the base rock. Frictional contact has been modelled between (i) the backfill and the base rock, (ii) the RW stem and the backfill and (iii) the RW heel slab and the backfill. A friction coefficient of \(\frac{2}{3}\mathrm{tan\phi }\) has been specified for modelling the frictional surface interaction, where \(\phi\) is the angle of internal friction of the backfill. In order to prevent the mesh penetration and formulate pressure overclosure relation between different frictional contact surfaces, a hard contact has been modelled in the normal direction for all the contact surfaces.
The boundaries of the FE model have been modelled using the acceleration and displacement controlled boundary option available in Abaqus. The base of the FE model has been modelled as pinned support type which was free to translate in “x” direction (lateral direction) and restrained in “y” direction (vertical direction). The sign conventions used in the FE investigation are shown in Fig. 7. The spring and dashpot system has also been adopted to model the vertical (viscous) boundaries of the FE model as shown in Fig. 7c. The viscous boundaries have proven effective for reducing the boundary effects and the computational time [8, 22]. Details of domain size analyses and viscous boundaries are presented in Sect. 4.3 of the present manuscript.
Geostatic stresses have also been defined for the backfill and the base rock domain. The prime objective to define the geostatic stresses was to ensure equilibrium of forces and accuracy of the FE results. Gravitational loading has also been applied to the entire FE model. The earthquake loading has been applied at the base of the FE model using “x” direction acceleration.
The nonlinear dynamic time history FE analyses have been carried out in the dynamic explicit module of Abaqus. The dynamic explicit solution scheme of the FE software Abaqus is popular for the large deformation numerical analysis [1]. The dynamic explicit analyses of Abaqus use an explicit central difference integration rule and adopt several small time increments for solving the boundary value problems. Low sampling frequency has been kept for capturing the FE analyses results in order to reduce the data noise.
Constitutive modelling of the materials
Constitutive modelling of the backfill
To understand the role of backfill into the seismic performance of the base restrained RW, a detailed parametric investigation has been carried out for three different backfill types. The dune sand [11], Fontainebleau sand [12] and crushed rocks (present study) have been considered as backfill material for the parametric FE investigations. The constitutive behaviour of soils observed during the consolidated drained (CD) triaxial test is shown in Fig. 8.
The Mohr Coulomb (MC) material model has been used for modelling the constitutive behaviour of backfill. Several studies used the MC material model for simulating the pre and post yield behaviour of backfill [2, 48]. It should be noted herein that the post yield behaviour of soil could also be simulated by providing an extension to the MC material model [43, 48]. Therefore, in the present study the MC model has been calibrated using the available triaxial test results of all three backfills considered for the FE investigations. Details of the MC material model and calibrations with the triaxial test data have been presented by Song [48]. The triaxial test results obtained from the calibrated MC material model (hardening and softening) have been compared with the laboratory triaxial test results of different backfills. The details of MC material modelling, calibrations of post yield response of backfill with triaxial test data and modelling of Rayleigh damping of backfill have been presented by the authors in a separate investigation [53]. Figure 8 shows the comparison of constitutive behaviour of different backfills observed during the triaxial testings and simulated using the calibrated MC model. Figure 9 shows the particle size distribution curve for the crushed rock used in the backfill construction.
Constitutive modelling of the concrete
The concrete has been modelled using the concrete damaged plasticity (CDP) model which is built into the FE software Abaqus. The CDP model is popular in modelling the constitutive behaviour of concrete [27, 52]. The CDP model simulates the constitutive behaviour of concrete in compression and tension using the following formulation:
$${\upsigma }_{{\text{t}}} = \left( {1 - d_{{\text{t}}} } \right)E_{0}^{{{\text{el}}}} :\left( {{\upvarepsilon } - {\upvarepsilon }_{{\text{t}}}^{{{\text{pl}}}} } \right)$$
(2)
$${\upsigma }_{{\text{c}}} = \left( {1 - d_{{\text{c}}} } \right)E_{0}^{{{\text{el}}}} :\left( {{\upvarepsilon } - {\upvarepsilon }_{{\text{c}}}^{{{\text{pl}}}} } \right)$$
(3)
where, \({\upsigma }_{\mathrm{t}}\) and \({\upsigma }_{\mathrm{c}}\) represent the stress vectors for the compressive and tensile stresses, respectively. The \({\upvarepsilon }_{\mathrm{t}}^{\mathrm{pl}}\) and \({\upvarepsilon }_{\mathrm{c}}^{\mathrm{pl}}\) represent the equivalent plastic strains in tension and compression, respectively. The initial undamaged elastic modulus \(\left({E}_{0}^{\mathrm{el}}\right)\) has been calculated from the stress–strain response of uniaxial compressive strength test of concrete (as shown in Fig. 10a). Damage variables \({d}_{\mathrm{t}}\) and \({d}_{\mathrm{c}}\) are functions of the plastic strains [1].
The yield function of the CDP model is initially developed by Lubliner et al. [29] and later modified by Lee and Fenves [25]. Details of the CDP yield function could be found in the Abaqus/Explicit User’s Manual [1]. The CDP model follows a non-associative flow rule. The plastic potential function is controlled by the dilation angle (at the deviatoric stress plane) and the eccentricity.
Table 3 shows the engineering properties considered for modelling the concrete using the CDP model. The stress–strain response of concrete with a characteristic strength (fcu) of 40 MPa has been generated using the procedure suggested by Carreira and Chu [9] and shown in Fig. 10b. It was assumed that the inelastic behaviour of the concrete (in compression) starts when the stresses in the concrete reaches to 0.4 fcu.
Table 3 The engineering properties used to model concrete and steel The tensile behaviour of the concrete against the uniaxial tension has been modelled using the fracture energy approach. The tensile failure of concrete has been simulated by a linear softening model. Equations 4 and 5 were used to calculate the tensile strength of concrete (ft) and the fracture energy (Gf) respectively.
The ft and Gf have been calculated for the maximum compressive strength of concrete (fcu) and the maximum aggregate size (da), [1, 27].
$$f_{{\text{t}}} = 1.4\left( {\frac{{f_{{{\text{cu}}}} - 8}}{10}} \right)^{\frac{2}{3}}$$
(4)
$$G_{{\text{f}}} = \left( {0.0469d_{a}^{2} - 0.5d_{a} + 26} \right)\left( {\frac{{f_{{{\text{cu}}}} }}{10}} \right)^{0.7}$$
(5)
Constitutive modelling of the reinforcement steel
The bilinear stress–strain response of the steel has been used to simulate the constitutive behaviour of steel. The details of steel are shown in Table 3. Figure 10c shows the stress–strain response of the steel considered in the present study [14]. The von Mises plasticity model has been used to model the plastic behaviour of steel. In the von Mises plasticity model, an isotropic hardening of the material has been defined with the help of the uniaxial yield stress and equivalent plastic strain. It should be noted herein that the size of the yield surface in the isotropic hardening changes uniformly in all directions [1].
Effect of the soil domain size on the seismic behaviour of the base restrained RW
Boundary effects may significantly affect the results of the FE simulations [5, 28]. Therefore, a detailed parametric investigation has been performed for different backfill domain lengths behind the plain concrete RW for estimating an optimum backfill domain length. Three RW models with 25 m, 50 m and 100 m backfill domain length have been considered for the domain size study. The crushed rock has been used as the backfill type for all three RW models. The MC model (Sect. 4.2.1) has been used to model the backfill. The Northridge accelerogram (090 CDMG Station 24278) has been used as the input base excitation for the nonlinear time history FE analyses for backfill domain size study.
Figure 11a shows the relative displacement time history at the top of the RW; observed from different backfill domain lengths. Higher active state displacement of the RW's with 25 m and 50 m long backfill domains was observed than the RW with 100 m long backfill domain which is due to the higher reflection of stress waves from the boundaries of 25 m and 50 m long backfill domain, respectively. However, the computational time for the RW with 100 m long backfill domain was significantly higher than the 50 m and 25 m long backfill domains. Therefore, for reducing the boundary effects and minimizing the computational time the vertical boundaries of the FE model have been modelled using the viscous boundaries (spring and dashpot system) as shown in Fig. 7.
A typical spring and dashpot system available in the FE software Abaqus is shown in the inset of Fig. 11a [1]. The stiffness and damping for the spring and dashpot system have been estimated for 80 m long backfill domain. The vertical spacing between the consecutive spring and dashpot sets has been kept as 200 mm. It was assumed that the external side of each spring and dashpot set is connected to the high stiffness rock located at the left and right hand sides of the FE model. The Northridge accelerogram (090 CDMG Station 24278) has been used as the input base excitation for the nonlinear time history FE analyses with spring and dashpot boundaries.
Figure 11a also shows the relative displacement at the top of the RW (with spring and dashpot boundaries). It was observed that the RW with the spring and dashpot boundaries shows lesser displacement than other considered backfill domain lengths. Moreover, the relative displacement of the RW with spring and dashpot boundaries was close to the relative displacement of RW with 100 m long backfill domain length. Apart from this, the computational time for the RW with spring and dashpot boundaries was significantly lesser than the computational time for the RW with 50 m and 100 m long backfill domains.
Mesh sensitivity analysis for the base restrained RW
The mesh sensitivity analyses have been carried out to study the effects of mesh sizes on the seismic response of the base restrained RW. Plane strain elements with reduced integration and hourglass control (CPE4R) have been used to model the FE model except the steel reinforcement. The steel reinforcement (rebar) has been meshed using beam element (B31). Several researchers have studied the effects of the mesh size on the structural response and observed that the FE analyses results are sensitive to mesh size. Moreover, an optimum mesh size could provide accurate FE results with lesser computational time [8, 22].
It was observed during the shaking table experiments on scaled down RW model and the FE investigations performed by Tiwari and Lam [53] that the backfill near the RW stem and the heel slab highly influence seismic response of the base restrained RW. Therefore, the mesh sensitivity analyses have been performed for different mesh sizes at the RW and the backfill contact locations.
The mesh sensitivity analyses were performed by varying the mesh sizes of the backfill near the RW stem and heel. A medium dense mesh was used for the RW and the base rock for minimizing the shear locking effects. Four different backfill mesh sizes (15 mm, 25 mm, 50 mm and 200 mm) have been used for the mesh sensitivity analyses. The FE model (with spring and dashpot boundaries) used for the domain size study (Sect. 2.3) has been used for the mesh sensitivity analyses.
Figure 11b shows the relative displacement at the top of the RW observed form different backfill mesh sizes. A minor difference has been observed between the results from different backfill mesh sizes. Based on the mesh sensitivity analyses, 25 mm mesh size of the backfill has been selected for the detailed FE investigations.
Validation of FE modelling approach
The capability of present FE modelling approach has been verified against the shaking table experiment results performed by the authors. A 2D plane strain FE model of the scaled down RW has been developed using the FE modelling approached defined in Sects. 4.1–4.4. The aluminium RW has been modelled using the elastic material properties, and the backfill has been modelled using the MC material model. The post-yield behaviour of the scaled down backfill has also been simulated for 7 kPa confinement pressure. Tables 2 and 3 show properties of different materials considered for the FE modelling of the scaled down RW. The earthquake response of scaled down RW model has been evaluated for two different base excitations as shown in the inset of Fig. 4. The nonlinear time history analyses have been performed in the dynamic explicit scheme of the FE software Abaqus. It should be noted herein that the captured displacement time history of the shaking table base during the shaking table experiment has been used as input base excitation for the FE models. Figure 4 also shows the comparison of relative displacement at the RW top observed during the shaking table experiment and obtained from the FE simulations. A good agreement has been observed between the shaking table experiment and the FE simulation results. The authors have also performed numerical investigations for evaluating the capabilities of the scaled down RW models for replication of seismic response of prototype RW's. Good agreement has been observed between the seismic response of scaled down and prototype RW’s (Tiwari and Lam; [53]). Therefore, it can be concluded that the present FE modelling approach can effectively replicate the seismic response of base restrained RW’s.