Implicit differentiation-based reliability analysis for shallow shell structures with the Boundary Element Method

Abstract

A novel methodology for evaluating the response sensitivities of shallow shell structures using the Boundary Element Method (BEM) is presented in this work. The implicit derivatives of the BEM formulations for shallow shell structures, with respect to the geometrical variables, such as curvature and thickness, have been derived for the first time and incorporated into an Implicit Differentiation Method (IDM). The IDM is employed in conjunction with the First Order Reliability Method (FORM) to evaluate the reliability of shallow shell structures. The accuracy of the IDM formulation is first validated against an analytical solution, with results showing a maximum difference of only 2.61%. The IDM was later validated against the Finite Difference Method (FDM), with results showing a maximum difference of only 0.11%. The IDM was also found to be significantly more efficient than the FDM, requiring 35% less CPU time when calculating sensitivities. This is further compounded by the fact that, unlike the FDM, the IDM does not require a step size. A numerical example featuring a circular shallow shell is used to demonstrate the application of the IDM-based FORM for assessing structural reliability. The uncertainty in curvature is set as a variable for the purpose of investigating its impact on reliability. The results of the reliability index obtained from the IDM-FORM are compared to the results obtained from FDM-FORM and were found to be very similar. An analysis of sensitivity is conducted to identify the most significant variables affecting reliability. It is found that uncertainties in curvature, thickness, and applied pressure distribution parameters have the largest impact on structural reliability. To demonstrate how the IDM could be used in practice, it was employed as gradient-based optimisation procedure featuring shallow-shell structures. The IDM was found to be a very efficient and accurate alternative to existing methods for calculating structural response sensitivities.