A high-fidelity first-order reliability analysis for shear deformable laminated composite plates
Introduction
The growth in application of composite materials reflects the increasing importance of being able to design material properties consistent with mechanics performance metrics. Simulation is a key component of the engineering design process, at both material and component levels, and normally requires experimental validation. It is in this final stage that many deterministic studies have failed to simulate the mechanical behaviour of composite materials and components, with considerable observed differences between theoretical predictions and experimental measurements [1]. Variations in fibre volume fractions, matrix-fibre voids, damage, fibre misalignment, residual stresses, etc. (e.g. [2], [3], [4], [5]) introduce uncertainty at a local level that then propagates to a larger scale and is reflected in the variability of stiffness and strength descriptors characterising material or component scale structural performance. In addition to the aleatoric uncertainty representing natural or intrinsic variability, epistemic uncertainty describes knowledge or information that is missing because, for example, quantities may not have been measured or may have been measured with insufficient accuracy, loading and boundary conditions have been inadequately represented, the numerical representation contains assumptions that results in certain phenomena being omitted or misrepresented, or the analysis method at both simulation and reliability levels are inappropriate or inadequate [6], [7]. The significance of these uncertainties is reflected in the use of high safety factors in deterministic structural analysis, and is particularly manifested as engineering conservatism in the presence of modelling or simulation uncertainty.
The complexity of laminated composites is partly reflected in the different approaches that are available to study these materials and structures. Single layer and discrete layer theories have been proposed in which the laminated structure is treated respectively as either a type a homogenised whole or a combination of individual layers (layer-wise). Plate theories are similarly divided into stress-based and displacement-based theories. Shear deformation may also contribute significantly to the behaviour of a composite plate or shell. Shear deformation theories are often considered to be those that are represented in an equivalent single layer formulation, whereas the theories that are “layer-wise” are not normally included in this category, even though shear effects are considered in these models. Transverse shear stress components are absent in classical laminated plate theory which may lead to errors for thick plates, especially where the transverse shear stiffness is low, as often found for advanced composites, making the inclusion of shear deformation a normal prerequisite for the analysis of composite plate and shell structures. We consider an equivalent single layer formulation in this paper (the sine approach of Touratier) in contrast to a layer-wise approach.
The reliability analysis of composites is challenging because it combines uncertainty quantification with the numerical estimation of behaviour and performance criteria that are themselves complex. Fundamental reliability analysis techniques have been developed and applied to a number of fields. The need for a framework into which to set this work has been identified [7]. A required component of this framework will be the identification of appropriate solid mechanics formulations that inherently minimise the epistemic uncertainty associated with the simulation. Recent research demonstrates the combination of reliability analysis with classical lamination theory [9], Mindlin theory [8], and higher-order (cubic) shear deformation theory [10]. In this paper we present a high-fidelity first-order reliability analysis for laminated composite plates.
Section snippets
Selection
It is well known that with a ratio of elastic modulus to shear modulus of the order of 25–40, compared with a value of 2.6 for a typical isotropic material, classical Kirchhoff theory (CLT) is unable to simulate the behaviour of an advanced composite plate. First-order shear deformation plate theory (FSDT) assumes transverse shear strains that are, along with the through-thickness shear stresses, constant through the thickness, contradicting the physical behaviour. Whilst the shear stresses
Principles
Reliability analysis evaluates the probability of structural failure by determining whether the limit-state functions are exceeded. The structure will be considered unreliable if the failure probability of the structure limit-state exceeds the required value. For most structures, ultimate (related to collapse of part or all of the structure) and serviceability (related to disruption of the normal use – e.g. deflection, vibration, etc.) limit states are defined.
Generally, the limit-state
Conclusions
The analytical derivation of linear and non-linear sensitivities of a complex shear-deformable composite laminate plate model has been demonstrated in this paper. Displacement, strain, and stress components have been differentiated with respect to fibre orientation and ply thickness as examples and verified using finite difference approximations. The non-linear components of these sensitivities make a significant contribution for moderately thick laminates when considering changes in
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