A layer-by-layer limit analysis approach for composite laminates

Limit analysis, among the theoretical and numerical methods aimed at predicting the load bearing capacity of structures and structural elements, plays an eminent role either because it gives a rapid answer to the above problem, or because it becomes very effective and attractive for the design of many modern industrial prototypes often manufactured with materials whose constitutive behaviour does not have a well defined mathematical description. In the field of composite laminates in facts, many constitutive models are affected by values of material parameters hardly identifiable via experimental tests, so that in this context results obtained by a step-by-step post-elastic analysis may be useless for applications of engineering interest. The classical approaches of limit analysis rely upon mathematical programming procedures which have progressed significantly in recent years, see e.g. [1]. Nevertheless, grounding on classical theorems of plasticity, many of these procedures are effective and computationally competitive only within the realm of perfect plasticity. A number of recent contributions (see e.g.[1-3]) adopt a different approach whose basic assumption is that limit state solutions may be developed from iterative FE linear solutions. The extension to non-standard limit analysis (in the sense of [4]) is in this case quite easy. In this context a numerical FE-based approach, known as Linear Matching Method [2], has been recently applied by the authors to a peculiar problem of orthotropic composite laminates; namely the evaluation of the load bearing capacity of a pinned-joint composite plate [3]. Assuming a Tsai-Wu type yield surface and a non-associate flow rule, the value of the collapse load is bounded there with an upper bound to the collapse load multiplier. The results given in [3], though very encouraging, were all affected by a number of factors related to the through-thickness effects like, for example, stacking sequence of the laminate, the latter not taken into account in the above quoted contribution where an equivalent single layer laminate is analysed. On this base grounds the present study in which the whole procedure is enhanced and rephrased at the laminate layer, so capturing the different mechanical behaviours exhibited by the analysed prototypes for different thickness and, consequently, different stacking sequence of the laminate layers. At layer laminate level is also applied here and rephrased a technique known as Elastic Compensation Method, [5], used to evaluate a lower bound to the collapse load multiplier via a stress redistribution procedure. The proposed methodology seems to be very promising as witnessed by the comparison between the numerical findings and a great number of experimental data detected on real prototypes and available in the relevant literature.