Towards practical multiscale approach for analysis of reinforced concrete structures

Abstract

We present a novel multiscale approach for analysis of reinforced concrete structural elements that overcomes two major hurdles in utilization of multiscale technologies in practice: (1) coupling between material and structural scales due to consideration of large representative volume elements (RVE), and (2) computational complexity of solving complex nonlinear multiscale problems. The former is accomplished using a variant of computational continua framework that accounts for sizeable reinforced concrete RVEs by adjusting the location of quadrature points. The latter is accomplished by means of reduced order homogenization customized for structural elements. The proposed multiscale approach has been verified against direct numerical simulations and validated against experimental results.

Appendix: Coarse-scale discretization

The shape functions that satisfy the displacement field described by Eqs. (25) through (27) are defined as

$$\begin{aligned} N^{c}=\left[ {{\begin{array}{ccc} {1-x_1 /L}&{} 0&{} 0 \\ {\frac{6\left( {L-x_1 } \right) x_1 x_2 }{L^{3}}}&{} {\frac{\left( {L-x_1 } \right) ^{2}\left( {L+2x_1 } \right) }{L^{3}}}&{} 0 \\ {\frac{6\left( {L-x_1 } \right) x_1 x_3 }{L^{3}}}&{} 0&{} {\frac{\left( {L-x_1 } \right) ^{2}\left( {L+2x_1 } \right) }{L^{3}}} \\ 0&{} {\left( {-1+\frac{x_1 }{L}} \right) x_3 }&{} {\frac{\left( {L-x_1 } \right) x_2 }{L}} \\ {\frac{\left( {L-3x_1 } \right) \left( {L-x_1 } \right) x_3 }{L^{2}}}&{} 0&{} {-\frac{\left( {L-x_1 } \right) ^{2}x_1 }{L^{2}}} \\ {-\frac{\left( {L-3x_1 } \right) \left( {L-x_1 } \right) x_2 }{L^{2}}}&{} {\frac{\left( {L-x_1 } \right) ^{2}x_1 }{L^{2}}}&{} 0 \\ {\frac{\left( {L-2x_1 } \right) \left( {L-x_1 } \right) x_3 \left( {1-\frac{4x_3^2 }{3h^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ {\frac{\left( {L-2x_1 } \right) \left( {L-x_1 } \right) x_2 \left( {1-\frac{4x_2^2 }{3b^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ {x_1 /L}&{} 0&{} 0 \\ {\frac{6x_1 \left( {-L+x_1 } \right) x_2 }{L^{3}}}&{} {\frac{\left( {3L-2x_1 } \right) x_1^2 }{L^{3}}}&{} 0 \\ {\frac{6x_1 \left( {-L+x_1 } \right) x_3 }{L^{3}}}&{} 0&{} {\frac{\left( {3L-2x_1 } \right) x_1^2 }{L^{3}}} \\ 0&{} {-\frac{x_1 x_3 }{L}}&{} {\frac{x_1 x_2 }{L}} \\ {\frac{x_1 \left( {-2L+3x_1 } \right) x_3 }{L^{2}}}&{} 0&{} {\frac{\left( {L-x_1 } \right) x_1^2 }{L^{2}}} \\ {\frac{\left( {2L-3x_1 } \right) x_1 x_2 }{L^{2}}}&{} {\frac{x_1^2 \left( {-L+x_1 } \right) }{L^{2}}}&{} 0 \\ {-\frac{\left( {L-2x_1 } \right) x_1 x_3 \left( {1-\frac{4x_3^2 }{3h^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ {-\frac{\left( {L-2x_1 } \right) x_1 x_2 \left( {1-\frac{4x_2^2 }{3b^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ {\frac{4\left( {L-x_1 } \right) x_1 x_3 \left( {1-\frac{4x_3^2 }{3h^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ {\frac{4\left( {L-x_1 } \right) x_1 x_2 \left( {1-\frac{4x_2^2 }{3b^{2}}} \right) }{L^{2}}}&{} 0&{} 0 \\ \end{array} }} \right] ^{T} \end{aligned}$$

(59)

The family of $g(\mathbf{x})$ functions corresponding to the Reddy [38] beam formulation are obtained by equating Eqs. (29) and (30), which yields

$$\begin{aligned}&g_1 =0 \end{aligned}$$

(60)

$$\begin{aligned}&g_2 =x_2 \end{aligned}$$

(61)

$$\begin{aligned}&g_3 =x_3 \end{aligned}$$

(62)

$$\begin{aligned}&g_{11} =0 \end{aligned}$$

(63)

$$\begin{aligned}&g_{12} =x_1 x_2 \end{aligned}$$

(64)

$$\begin{aligned}&g_{13} =x_1 x_3 \end{aligned}$$

(65)

$$\begin{aligned}&g_{22} =-\frac{h_2^{2}}{24}+\frac{x_2^{2}}{2} \end{aligned}$$

(66)

$$\begin{aligned}&g_{33} =-\,\frac{h_3^{2}}{24}+\frac{x_3^{2}}{2} \end{aligned}$$

(67)

$$\begin{aligned}&g_{122} =-\,\frac{1}{8}x_1 \left( {h_2^{2}-4x_2^{2}} \right) \end{aligned}$$

(68)

$$\begin{aligned}&g_{133} =-\,\frac{1}{8}x_1 \left( {h_3^{2}-4x_3^{2}} \right) \end{aligned}$$

(69)

$$\begin{aligned}&g_{222} =-\,\frac{1}{24}x_2 \left( {h_2^{2}-4x_2^{2}} \right) \end{aligned}$$

(70)

$$\begin{aligned}&g_{333} =-\,\frac{1}{24}x_3 \left( {h_3^{2}-4x_3^{2}} \right) \end{aligned}$$

(71)

$$\begin{aligned}&g_{1122} =\frac{1}{192}\left( {h_2^{2}-4x_2^{2}} \right) \left( {\left| \Theta \right| _1^{2}-12x_1^{2}} \right) \end{aligned}$$

(72)

$$\begin{aligned}&g_{1133} =\frac{1}{192}\left( {h_3^{2}-4x_3^{2}} \right) \left( {\left| \Theta \right| _1^{2}-12x_1^{2}} \right) \end{aligned}$$

(73)

$$\begin{aligned}&g_{1222} =\frac{1}{24}x_2 \left( {-h_2^{2}+4x_2^{2}} \right) x_1 \end{aligned}$$

(74)

$$\begin{aligned}&g_{1333} =\frac{1}{24}x_3 \left( {-h_3^{2}+4x_3^{2}} \right) x_1 \end{aligned}$$

(75)