A unified approach to nonlinear buckling optimization of composite structures

Abstract

A unified approach to nonlinear buckling fiber angle optimization of laminated composite shell structures is presented. The method includes loss of stability due to bifurcation and limiting behaviour. The optimization formulation is formulated as a mathematical programming problem and solved using gradient-based techniques. Buckling of a well-known cylindrical shell benchmark problem is studied and the solutions found in literature are proved to be incorrect. The nonlinear buckling optimization formulation is benchmarked against the traditional linear buckling optimization formulation through several numerical optimization cases of a composite cylindrical shell panel which clearly illustrates the advantage and potential of the presented approach.

Introduction

The use of fibre-reinforced polymers has gained an ever-increasing popularity due to their superior mechanical properties. Designing structures made out of composite material represents a challenging task, since both thicknesses, number of plies in the laminate and their relative orientation must be selected. The best use of the capabilities of the material can only be gained through a careful selection of the layup. This work focuses on optimal design of laminated composite shell structures i.e. the optimal fiber orientations within the laminate which is a complicated problem. One of the most significant advances of optimal design of laminate composites is the ability of tailoring the material to meet particular structural requirements with little waste of material capability. Perfect tailoring of a composite material yields only the stiffness and strength required in each direction. A survey of optimal design of laminated plates and shells can be found in [1].

Stability is one of the most important objectives/constraints in structural optimization and this also holds for many laminated composite structures, e.g., a wind turbine blade. Traditionally, stability is regarded as the linear buckling load, but for structures exhibiting a nonlinear response when loaded, and especially for shell like structures, the traditional approach can lead to unreliable predictions of the buckling load. In the case where nonlinear effects cannot be ignored nonlinear path tracing analysis is necessary. For limit point instability, several standard finite element procedures allow the nonlinear equilibrium path to be traced until a point just before the limit point. The traditional Newton like methods will probably fail in the vicinity of the limit point and the post-critical path cannot be traced. More sophisticated techniques, as the arc-length methods suggested by [2] and subsequently modified by [3], [4] are among some of the techniques available today for path tracing analysis in the post-buckling regime. Despite such sophisticated techniques exist, buckling analysis of shell like structures is today still a difficult task which consequently makes it difficult to optimize shell structure w.r.t. stability.

For many years a common shell buckling problem, first introduced by [5] and later appeared in numerous journal articles, has been a classical example for describing buckling behaviour of cylindrical shell panels. The example has been used as a benchmark to investigate advances in numerical finite elements methods for handling load and/or deflection reversals in nonlinear buckling problems. Furthermore, it is used to demonstrate the capability of finite element procedures to traverse such complicated load paths.

Lately, [6], [7] noticed that the solution by [5] and re-produced by many other authors through several decades was incorrect. The incorrect solution only involves symmetric deformation modes and makes the assumption that limit point buckling occurs. [6], [7] discovered through numerical studies and related experiments that the former symmetric solution is incorrect and the existence of bifurcation and asymmetric buckling mode at a lower load level. Furthermore, [6], [7] concludes that the bifurcation point is stable which means that the structure is able to carry more load after bifurcation until, according to [6], [7], a load limit point instability is encountered. The results by [6], [7] is also included and discussed in the book by [8]. Their conclusion about stability of the bifurcation point turns out to be incorrect, i.e. the bifurcation point is not stable but unstable, which demonstrates that buckling analysis of relatively simple structures still represent a challenging task. The entire solution of the buckling benchmark problem is shown in Section 4 where new features of the buckling problem are revealed.

Research on the subject of structural optimization of composite structures considering stability has been reported by many investigators. The first work to appear concerned simple composite laminated plates and circular cylindrical shells where stability was determined by solution of buckling differential equations, see [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Later, buckling optimization of composite structures was considered in a finite element framework where the buckling load was determined by the solution to the linearized discretized matrix eigenvalue problem at an initial prebuckling point. Optimization of laminated composite plates has been studied by [19], [20], [21], [22], [23], while others considered more complex composite structures as curved shell panels and circular cylindrical shells, see [24], [25], [26], [27], [28], [29]. Applications of optimization methods to stability analysis including nonlinear prebuckling effects have been very limited. To the best knowledge of the authors only the papers by [30], [31], [32] report on nonlinear gradient based buckling optimization of composite laminated plates and shells where buckling is considered in terms of the limit load of the structure. Thus there is a lack of optimization procedures that handles bifurcation instability including nonlinear prebuckling effects but also optimization procedures that simultaneously handles bifurcation and limit point instability. Despite bifurcation points, if unstable, in many cases may be transformed into limit points by introducing imperfections into the system, see e.g. [33], [34], [35], whereby only limit points may be concerned in the optimization formulation in order to optimize the buckling load, a general optimization formulation that handles both types of instability may prove to be important. In cases of stable bifurcation points the method of introducing imperfections will not work since the stability point simply vanish, i.e. the bifurcation point is not converted into a limit point but vanish and the load response keeps rising stably. Also in cases of unstable bifurcation points the method of introducing imperfections may not be without difficulties since a proper choice of imperfections can be difficult. The latter is shown in Section 4. Furthermore, the type of stability may also change during buckling optimization, i.e. from one optimization iteration to another the stability type may change from e.g., a bifurcation point to a limit point. An optimization formulation that operates on the initial structure without imperfections and handles a general type of stability is needed.

This paper presents an integrated and reliable method for doing optimization of composite structures w.r.t. a general type stability, i.e. bifurcation instability and limit point instability, depending on what to appear first on the equilibrium path. Features for detecting bifurcation points and limit points during nonlinear path tracing analysis is developed. The nonlinear buckling formulation described in [31] is utilized, i.e. optimization w.r.t. stability is accomplished by including the nonlinear response by a path tracing analysis, after the arc-length method, in the optimization formulation, using the Total Lagrangian formulation. The nonlinear path tracing analysis is stopped when a stability point is encountered and the critical load is approximated at a precritical load step according to the “one-point” approach, i.e. the stiffness information is extrapolated from one precritical equilibrium point until a singular tangent stiffness is obtained. Design sensitivities of the critical load factor are obtained semi-analytically by the direct differentiation approach on the approximate eigenvalue problem described by discretized finite element matrix equations. A number of the lowest buckling load factors are considered in the optimization formulation in order to avoid problems related to “mode switching”. The proposed method is benchmarked against a formulation based on linear buckling analysis on a shell buckling problem and helps to clarify the importance of including nonlinear prebuckling effects in structural design optimization w.r.t. stability.

In this work only Continuous Fiber Angle Optimization (CFAO) is considered, thus fiber orientations in laminate layers with preselected thickness and material are chosen as design variables in the laminate optimization. Despite fiber angle optimization is known to be associated with a non-convex design space with many local minima it has been applied since the laminate parametrization has not been the focus in this work, i.e. the presented method in this paper is generic and can easily be used with other parametrizations.

The proposed procedure regarding nonlinear buckling analysis is described in Section 2 together with detection features applied for discovering stability points during geometrically nonlinear analysis. Derivations of design sensitivities, using the direct approach, of the nonlinear buckling load are presented along with the general type nonlinear buckling optimization formulation in Section 3. The benchmark shell buckling problem is treated in Section 4 where it is shown that the solutions found in literature still are not correct and new features of the problem are revealed. Buckling optimization of a composite laminated curved shell panel is considered in Section 5. Conclusions are outlined in Section 6.