Designing cutouts for optimum residual strength in plane structural elements

Abstract

The paper presents a new approach for shape optimisation of structures with residual strength as the design objective. It must be emphasized that flaws are inevitably present in most structures, and hence the influence of cracks on optimised shapes needs to be investigated. Numerical simulation of cracks using the finite element method requires a very fine mesh to model the singularity at crack tip. This makes fracture calculations computationally intensive. Furthermore, for a damage tolerance based optimisation numerous cracks are to be considered along the structural boundary, and fracture analysis needs to be repeated for each crack at every iteration, thus making the whole process extremely computationally expensive for practical purpose. Moreover, the lack of information concerning crack size, orientation, and location makes the formulation of the optimisation problem difficult. As a result, little attention has been paid to date to consider fracture parameters in the optimisation objective. To address this, the paper presents a methodology for the shape optimisation of structures with strength and durability as the design objectives. In particular, the damage tolerance optimisation is illustrated via the problem of optimal design of a ‘cutout in a rectangular block under biaxial loading’. A parametric shape representation has been used to describe the problem geometry. Damage tolerance based optimisation was performed using nonlinear programming algorithms, and NE-NASTRAN was used for finite element analysis. The first order mathematical programming algorithms, viz: the Broydon–Fletcher–Goldfarb–Shanno (BFGS) and the Fletcher–Reeves (Conjugate Direction) Methods were evaluated as the potential algorithms for optimising the residual strength in the presence of flaws. Another recently developed Sequential Unconstrained Minimisation Technique (SUMT), based on an exterior penalty function method and especially suited for large problems, was also investigated for fracture based optimisation. The effects of the orientation and the number of boundary cracks on the optimal solutions were also studied. It has been shown that the residual strength optimised shapes can be different from the corresponding and commonly adopted stress optimised solution. This emphasises the need to explicitly consider residual strength as the design objective. In all cases a significant reduction in the maximum stress intensity factor was achieved with the generation of a ‘near’ uniform fracture critical surface. The design space near the optimal region was found to be relatively flat. This is beneficial as a significant structural performance enhancement is important rather than precise identification of the local/global optimum solution. The optimal solutions obtained using the nonlinear programming algorithms were compared against those obtained in the literature using a heuristic optimisation method (Biological algorithm). The results obtained using the two methods, employing inherently different (gradient-based and gradient-less) algorithms, were found to agree very well.