On the use of differential quadrature-three-term conjugate finite-step length methods for reliability analysis of steel fiber-reinforced sinusoidal rupture beams

Abstract

The reliability of a complex mathematical problem is presented in this work. The problem is a rupture concrete beam reinforced by steel fibers with various orientation angle. The structure is simulated by sinusoidal shear deformation theory and energy method. Harmonic differential quadrature method is applied for the solution of the problem under the buckling load. The reliability analysis of the mentioned system is studied utilizing a three-term conjugate finite-step length (TCFS) approach. The TCFS is formulated utilizing the conjugate gradient method with a limited scalar parameter to reach the numerical stabilization. The implicit buckling limit state function of the structure includes different parameters of length to thickness ratio of the beam, volume percent and orientation angle of steel fibers, rupture constant and boundary conditions. The results show that the failure probabilities of the studied structure may be decreased by increasing the rupture constant. In addition, the orientation angle of zero is the best choice for steel fiber in the concrete beam.

Appendix 1

Step 0	Define the performance function $g({\varvec{U}}_{{}}^{{}} ) = 0$, Give the statistical charactristic of random parameters (${\varvec{\mu}}$, ${\varvec{\sigma}}$), Set $k = 0$, ${\varvec{X}}_{0}^{{}} = {\varvec{\mu}}$, ${\varvec{d}}_{0}^{{}} = {\varvec{0}}$, and $\eta_{0}^{{}} = 0$, Choose step length $\lambda > > 0$ and stopping criterion $\varepsilon < < 1$
Step 1	Transfer the random parameters form X-space to U-space
Step 2	Calculate the gradient vector and LSF at point ${\varvec{U}}_{k}^{{}}$ Calculate the conjugate scalar factor as $\theta_{k}^{{}} = - {{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } {\left\| {\nabla g({\varvec{U}}_{k - 1}^{{}} )} \right\|^{2} }}} \right. \kern-\nulldelimiterspace} {\left\| {\nabla g({\varvec{U}}_{k - 1}^{{}} )} \right\|^{2} }}$ Calculate the adjusted scalar factor as $\eta_{k}^{{}} = \min \{ 0.9{{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } {\left\| {\nabla g({\varvec{U}}_{k - 1}^{{}} )} \right\|^{2} }}} \right. \kern-\nulldelimiterspace} {\left\| {\nabla g({\varvec{U}}_{k - 1}^{{}} )} \right\|^{2} }},{{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } \mathord{\left/ {\vphantom {{\left| {\left| {\nabla g({\varvec{U}}_{k}^{{}} )} \right|} \right|^{2} } {\left\| {{\varvec{d}}_{k - 1}^{{}} } \right\|^{2} }}} \right. \kern-\nulldelimiterspace} {\left\| {{\varvec{d}}_{k - 1}^{{}} } \right\|^{2} }}\}$ Calculate the conjugate search direction utilizing the nonlinear discrete three-term map as: ${\varvec{d}}_{k} = - \nabla g({\varvec{U}}_{k}^{{}} ) + \theta_{k}^{{}} {\varvec{d}}_{k - 1} \, + \eta_{k}^{{}} \nabla g({\varvec{U}}_{k - 1}^{{}} )$ Calculate the point along beta-circle direction as $\tilde{\user2{U}}_{k}^{{}} = {\varvec{U}}_{k}^{{}} + \lambda_{{}} {\varvec{d}}_{k}^{{}}$ Calculate the normalized conjugate search direction ${\varvec{\alpha}}_{k}^{{}} = \frac{{\tilde{\user2{U}}_{k}^{{}} }}{{\left\| {\tilde{\user2{U}}_{k}^{{}} } \right\|}}$ Calculate the reliability index as $\beta_{k} = \frac{{\nabla^{T} g({\varvec{U}}_{k}^{{}} )\,{\varvec{U}}_{k}^{{}} - g({\varvec{U}}_{k}^{{}} )}}{{\nabla^{T} g({\varvec{U}}_{k}^{{}} ){\varvec{\alpha}}_{k}^{{}} }}\,\,$ Calculate the new point as follows: ${\varvec{U}}_{k + 1}^{{}} = \beta_{k} {\varvec{\alpha}}_{k}^{{}} \,\,$
Step 3	If $\left\| {{\varvec{U}}_{k}^{{}} - {\varvec{U}}_{k - 1}^{{}} } \right\| \ge \varepsilon$ then $k = k + 1$ and Go to Step 1, else, print ${\varvec{X}}_{{}}^{*} = {\varvec{X}}_{k + 1}^{{}}$, ${\varvec{U}}_{{}}^{*} = {\varvec{U}}_{k + 1}^{{}}$, $\beta_{k}^{{}}$, and $P_{f} \approx \Phi ( - \beta_{k} )$