On evaluation of shape sensitivities of non‐linear critical loads

Abstract

SUMMARYThe present paper focuses on the evaluation of the shape sensitivities of the limit and bifurcation loads of geometrically non-linear structures. The analytical approach is applied for isoparametric elements, leading to exact results for a given mesh. Since this approach is di cult to apply to other element types, the semi-analytical method has been widely used for shape sensitivity computation. This method combines ease of implementation with computational e ciency, but presents severe accuracy problems. Thus, a general procedure to improve the semi-analytical sensitivities of the non-linear critical loads is presented. The numerical examples show that this procedure leads to sensitivities with su cient accuracy for shape optimization applications.

Tracing nonlinear equilibrium paths of structures subjected to thermal loading

“…It should be noted that the numerical differentiation has been successfully used in different areas of computational mechanics, as design sensitivity analysis and critical point computation [22]. Equation (40) is efficient since g(u, λ) was already computed for the evaluation of the residual forces.…”

Direct calculation of critical points in parameter sensitive systems

“…One way to define G for imperfect structures is based on the existence of (a few) eigenvectors corresponding to the null space of the tangent stiffness matrix [13,37]. Subsequently, for simple critical points, the critical condition can be written as [38,39]:…”

Corotational non‐linear analysis of thin plates and shells using a new shell element

“…The shell element formulated in this study is a triangular flat element with 18 degrees of freedom: (27) We rearrange this vector to separate membrane and bending degrees of freedom:…”

“…A limit point arises when the load-displacement curve reaches a local extremum, as the points L1 and L2 shown in Figure 5(b), while a bifurcation occurs when different equilibrium paths meet at a certain point, as the point B in Figure 5(b). Detection and classification of the critical points are usually performed by checking the null eigenvector condition, and orthogonality between the load vector and the buckling mode [27]. The non-linear analysis approach explained in this section may also be extended to materially non-linear problems as long as the pure strains remain small.…”