Nonlocal Formulation for Numerical Analysis of Post-Blast Behavior of RC Columns

As known, the volumetric and deviatoric responses of concrete are decoupled in the KCC model, in which a user defined equation of state (EOS) defines volumetric strain versus pressure relationship and a movable failure surface limits the deviatoric stress. In order to govern the evolution of the failure surface, three fixed independent failure surfaces termed as yield, maximum and residual are defined. For hardening behavior, the failure surface is interpolated between the yield and maximum failure surface based on a damage variable, and for softening behavior, a similar interpolation is performed between the maximum and residual failure surfaces. The strain rate dependent behavior is considered through a user defined dynamic increase factor (DIF) curve (Hallquist 2007).

Three parameters are used in the KCC model to control the damage evolution, they are b ₁, b ₂ and \( \omega \) for compression damage, tensile damage and volume expansion, respectively. It is a common sense that the numerical results will not be objective upon mesh refinement if the strain softening behavior of the constitutive model is not associated with a localization limiter or characteristic length. To eliminate the pathological mesh size dependent phenomenon, the crack band method is incorporated into this model by forcing the area under the stress–strain curve to be a constant value G _f /h, where G _f is the fracture energy and h the element characteristic size. G _f can be obtained through experiment (Lee and Lopez 2014). However, this method is only valid for tensile softening and becomes invalid when the element size exceeds 250 mm or the element size is smaller than the localization width (Magallanes et al. 2010).

The CSC model is originally developed for use in roadside safety simulations, and is also frequently used to model concrete materials subjected to impact and blast loads. Its yield surface combines a shear surface with a hardening compaction surface smoothly and continuously through the multiplicative formulation, which avoids the numerical complexity of dealing with “corner” regions. As softening is a remarkable behavior of concrete in tension and low confinement compression, the strain-based damage formulations are incorporated into this model, and brittle damage and ductile damage are distinguished. In order to consider the strain rate effect for impact and blast analysis, the viscoplastic formulation is adopted. Details of the description of the model can be referred to the reports (Murray et al. 2007; Murray 2007).

In all the HFPB simulations, concrete is modeled by 8-noded solid element with single integration point. The reinforcing steel is modeled with 2-noded beam element. And perfect bond between concrete and reinforcement is assumed by sharing nodes. The KCC or CSC model is used for concrete material and an elasto-plastic material (*MAT_PICEWISE_LINEAR_PLASTICITY) is used to model steel. The erosion technique is implemented to consider the severely damaged concrete elements.

In order to validate the reliability of the numerical model, two numerical analyses of the CTEST20 column (Crawford et al. 2012) and B40 beam (Magnusson et al. 2010a) under blast loading are conducted and the numerical results are compared with experimental data.

The KCC and CSC models have been frequently used in dynamic response analysis of RC columns under blast loading, and it is generally considered to be converged when the displacement response does not vary a lot upon mesh refinement. However, there is limited information on the studies that considered the mesh size influence on the post-blast residual axial capacity to date.

A specific RC column is taken as an example here to calculate the post-blast residual axial capacity by varying the element sizes. Figure 6 shows the scheme of the RC column, the clear height of which is 2750 mm. And the section width and height are both 300 mm with a concrete cover of 40 mm. A total of four Φ20 mm longitudinal rebar are uniformly distributed along the perimeter of the section, and the stirrup reinforcement with a diameter of 10 mm is spaced 200 mm along the length. The uniaxial compressive strength for the concrete is 30 MPa. The yield and ultimate strength of steel reinforcement are 335 and 480 MPa, respectively. The loading scheme consists of three stages, as shown in Fig. 6. In stage 1, an initial axial load of 821 kN is applied on the top end to produce an initial stress state in the column. After a steady state is reached, an idealized blast load with peak overpressure of 10 MPa and duration time of 1 ms is applied to the column along the full height and certain damages will be generated till the dynamic response is damped out in stage 2. Finally, the displacement control loading scheme is used to calculate the post-blast residual axial capacity in stage 3.

Four different element sizes, i.e., 50, 25, 12.5 and 6.25 mm, are used in this study. Both KCC and CSC models are adopted for the concrete material. And the piecewise linear isotropic plasticity model is employed to model the steel reinforcement. The mid-span transverse displacement time history curves are described in Fig. 7 for different mesh sizes. As can be observed, for the results calculated by both KCC and CSC models, the 50 mm element size seems a little large for the results to achieve convergence, while the displacement time history curves are almost identical for cases with element sizes of 25, 12.5 and 6.25 mm before loading stage 3. In stage 3, the displacement curves diverge a lot, which indicates different post-blast response of the RC column for analysis results with different mesh sizes. The post-blast residual axial capacity can be easily obtained from the axial load time history curves as shown in Fig. 8. It can be observed that the residual axial capacity has a pathological dependence on the mesh size. This is mainly because that the concrete material models are not equipped with a complete regularization formulation, hence the damages generated in the local areas still have a mesh dependent character even if the constant fracture energy method is introduced. Figure 9 shows the damage distribution of the RC column at the end of stage 2, the local damages generated during blast loading are obviously very different, and leading to different residual axial capacity. It is no doubt that the different residual axial capacity will affect the dynamic response of RC structures under blast loading. Therefore, it is essential to take measures eliminating the mesh dependent issue in order to ensure reliable analysis results for RC structures under blast loading.

Damage mechanics has been proved to be efficient for modelling concrete behavior in recent years (Heo and Kunnath 2013; Ren et al. 2015). Mazars damage model belongs to one of the first models based on this framework and has been widely used in simulating the damage behavior of concrete (Mazars and Pijaudier-Cabot 1989). It is described in Fig. 10 by the relation

where \( {\varvec{\upsigma}} \) is the stress tensor, \( {\varvec{\upvarepsilon}} \) is the strain tensor, \( {\mathbf{C}} \) is the stiffness matrix of material and D is the scalar damage which varies from 0 to 1 with 0 represents the intact material and 1 for fully failure of the material. And the damage criterion inspired from the St. Venant criterion of maximum principal strain is adopted, its equation is:

where \( \kappa \) represents the current damage value which is the maximum value of equivalent strain ever reached in time history, and the initial value of \( \kappa \) is \( \kappa_{0} \), which indicates the initiation of damage.

As extensions play a significant role in the damage behavior of concrete, Mazars (1986) suggested the equivalent strain evaluating the local intensity of extensions, which is defined as:

where \( \varepsilon_{i} \) are the principal strains and \( \left\langle \cdot \right\rangle \) is the Macaulay bracket.

In order to describe the different behaviors of tension and compression, two separated damage variables denoted as \( D_{t} \) and \( D_{c} \) are defined, and the total scalar damage D is calculated from the weighted combination of them, namely:

With

where, \( A_{i} \) and \( B_{i} \) are characteristic parameters of the concrete material identified from test data. In Eq. (4), the weighting coefficients \( \alpha_{t} \) and \( \alpha_{c} \) are functions of the strain state (\( \alpha_{t} = 0 \) for pure compression and \( \alpha_{c} = 0 \) for pure tension, \( \alpha_{t} \) and \( \alpha_{c} \) vary from 0 to 1 in other cases).

The post-blast residual axial capacity of RC columns is closely related to the blast damages of concrete material. As stated in Sect. 2, the blast damages show pathological sensitivity to the mesh size, which directly results in the mesh-dependent residual axial capacity. The integral type nonlocal model provides a fully regularization for the strain softening problem through introducing a characteristic length in the local constitutive laws (Pijaudier-Cabot and Bažant 1987), which is achieved by weighted spatial averaging of a suitable state variable. Generally, state variables such as the damage energy release rate, damage value, total strain and equivalent strain are chosen for applying the nonlocal formulation. In this paper, the equivalent strain in Mazars damage model is taken as the nonlocal state variable, and with its nonlocal counterpart replacing \( \varepsilon_{eq} \) in Eq. (5), which is defined as

where \( \bar{\varepsilon } \) and \( \tilde{\varepsilon } \) are the nonlocal and local equivalent strains, respectively. V is the spatial domain of interest, which is decided by the characteristic length. And \( \alpha (x,\xi ) \) is the nonlocal weight function that defines the interaction coefficients between point x and \( \xi \). The weight function is typically normalized by \( \int_{V} {\alpha (x,\xi )} {\text{ d}}\xi = 1 \) for all x, then the normalized weight function can be obtained:

where \( \alpha_{0} (s) \) is a function which depends on the distance s between point x and \( \xi \). In this study, the Bell-type weight function is adopted (Jirásek and Bazant 2002):

where, R is the characteristic length that limits the interaction zone, which makes the function exactly zero for \( s \ge R \).

LS-DYNA is a general finite element code which provides abundant interfaces for users developing their own subroutines to achieve research objectives (Hallquist 2007). In this study, before invoking the local constitutive model, the local equivalent strain must be replaced by the nonlocal counterpart of the current time step. The scheme of the beam element formulation is shown in Fig. 11, only one integration point is used for one element to save computational time in explicit analysis. The values of the nonlocal equivalent strain must be traced at every individual integration point, whose coordinates are denoted as \( x_{k} \) (\( {\text{k}} = 1, 2, \ldots ,{\text{N}}, \) N represents the total number of integration points). The numerical algorithm can be determined as

with

where \( w_{l} \) is the integration weight of the integration point number l, and the value of which is 1 as single integration point is used for all the beam elements. It is worth noting that \( \alpha_{kl} \) would be zero when the distance between k and l is larger than the interaction radius R (characteristic length). It could be calculated only once before the first step and then stored as it does not vary during the simulation.

Figure 12 shows the numerical procedure of the nonlocal formulation. The transformation matrix from local to nonlocal equivalent strain is defined as \( {\varvec{\upbeta}} \), in which \( \beta_{kl} = w_{l} \alpha_{kl} \). Then, before invoking the local constitutive model at every new time step, the local equivalent strain is transformed to the nonlocal one, and stresses of each integration point are updated, the corresponding history variables are stored for next step.

One of the tests on RC columns conducted by Tanaka and Park (Tanaka 1990), labeled specimen 7, is analyzed in this example due to the considerable axial load applied, which makes the overall behavior entering post-peak region. The geometry and section details are shown in Fig. 13. The unconfined and confined compressive strength of the concrete are \( f_{c}^{'} = 32 \) MPa and \( f_{c} = 39 \) MPa, respectively. A bilinear stress–strain relationship is used for the reinforcement with yield strength of 510 MPa and 1% strain hardening ratio.

In order to have a deep insight into the influence of concrete strain softening behavior on post-peak response of the RC column with mesh refinement, the analysis is carried out with 10, 20 and 40 elements and both the local and nonlocal damage models are adopted for concrete. The main material parameters for concrete are \( A_{c} = 0.501 \), \( B_{c} = 707.1 \), \( A_{t} = 0.0005 \) and R = 1100. The displacement control scheme is applied on the top node of the column. The computed force–displacement curves are shown in Fig. 14. As can be seen in the figure, the post-peak stage of the curves become more and more brittle as the number of elements increases for the local damage model. This is mainly because that the inelastic deformation localizes in one element, resulting in the increasing strains in the extreme element as the column enters the post-peak stage. However, the post-peak results computed by the nonlocal damage model are almost the same with mesh refinement, and match well with the envelope of the cyclic response from the experiment.

In Sect. 2.4, the post-blast residual axial capacity of the RC column is pathologically sensitive to the mesh size as the constant fracture energy method used in the concrete constitutive models fails to capture objective damage values under blast loading. In this section, the RC column is used as an example to demonstrate the feasibility of the nonlocal damage model in calculating the post-blast residual axial capacity. Finite element models with 20, 40 and 80 elements are analyzed with the same load in Sect. 2.4. The main material parameters for concrete are \( A_{c} = 1.5154 \), \( B_{c} = 1767.77 \), \( A_{t} = 0.1 \) and R = 600. And the strain rate effect of concrete is accounted by the dynamic increase factor (DIF) calculated from Malvar (Malvar and Ross 1998).

The computed axial force versus time history curves are presented in Fig. 15. It is obvious that the curve calculated by the finite element model with 20 elements shows a minor difference with the rest two curves, which means that the finite element model with 20 elements does not guarantee the convergence of the numerical results. This is consistent with the consensus that a sufficiently fine mesh is generally required for nonlocal model. The results computed from the finite element models with 40 and 80 elements are almost the same, indicating that the proposed numerical model can not only capture identical damage values during blast loading but also ensure objective post-blast residual axial capacity of the RC column.

Furthermore, the finite element model with 40 elements is herein taken for a series of calculations by varying the characteristic length R in the nonlocal model. According to Bažant (Pijaudier-Cabot and Bažant 1987), the material characteristic length cannot be less than the beam depth h, which is 300 mm in this example. As can be observed in Fig. 16, when the characteristic length R equals to 200 mm, the RC column loses its loading capacity under blast loading. And it is obvious that adopting a smaller characteristic length will lead to underestimate the dissipated energy within the damage zones as well as the post-blast residual axial capacity. The proper calibration of the nonlocal internal length is not a trivial task. Although this quantity is usually related to the theoretical width of the fracture process zone (material characteristic length), its actual definition is not completely clear. In this study, the simulation results seem to match the experiment well when twice of the column height is used for R. However, some specifically-designated experiments are still needed to calibrate this important parameter in the future.