Initial post-buckling of variable-stiffness curved panels

White, Simon; Raju, Gangadharan; Weaver, Paul M.

doi:10.1016/j.jmps.2014.07.003

Cited by 64 publications

(36 citation statements)

References 37 publications

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“…The sensitivities are determined by differentiation of Equation (21). Assuming design independent loads the sensitivities of the load vector are zero.…”

Section: Sensitivity Analysis Of the Pre-buckling Displacement Fieldmentioning

confidence: 99%

Post‐buckling optimization of composite structures using Koiter's method

Henrichsen

Lindgaard

Lund

2016

Numerical Meth Engineering

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SUMMARYThin-walled structures, when compressed, are prone to buckling. To fully utilize the capabilities of such structures, the post-buckling response should be considered and optimized in the design process. This work presents a novel method for gradient based design optimization of the post-buckling performance of structures. The post-buckling analysis is based on Koiter's asymptotic method. To perform gradient based optimization, the design sensitivities of the Koiter factors are derived and new design optimization formulations based on the Koiter factors are presented. The proposed optimization formulations are demonstrated on a composite square plate and a curved panel where the post-buckling stability is optimized. Copyright c 0000 John Wiley & Sons, Ltd. Thin-walled structures are often designed such that buckling does not occur during service. To ensure a structure does not buckle, the specified buckling load is often much greater than the design load. If a structure can be allowed to buckle during operation, thus operating in the post-buckling regime, it enables the possibility to design lighter and more efficient structures. To enable such an approach, the engineer must optimize the post-buckling response of the structure. Fiber reinforced polymers are ideally suited for such design tasks, as these materials allow a high degree of tailoring of the considered structure, and thus applied here for the post-buckling design optimization of structures. When optimizing structures, robustness of the resulting structure is of major importance, as the imperfection sensitivity can increase during the optimization process [1]. One method is to collect these into an equivalent geometric imperfection and use that to evaluate the knockdown in performance. Refs. [2,3] demonstrate robust design optimization by combining "worst" shape imperfection optimization and laminate optimization, thereby efficiently decreasing the imperfection sensitivity of laminated composite structures. A different approach is to handle all imperfections simultaneously by modeling the uncertainties using statistics, therefore quantifying * Correspondence to: Department of Mechanical and Manufacturing Engineering, Aalborg University (AAU), the imperfections arising from material, geometry, load etc., and perform robust buckling optimization based on the uncertainties, see e.g., [4] for a review of different approaches. Many textbooks describe the coupling between imperfections, buckling load factor, and post-buckling stability, see e.g., [5]. Focusing on simple i.e., distinct buckling load factors, the sensitivity towards imperfections relates to the stability of the post-buckling response. Generally speaking, a stable post-buckled structure is less sensitive towards imperfections than an unstable post-buckled structure. Because of these considerations, the post-buckling stability can be utilized to design robust structures.Post-buckling analysis of plates and shells has been subject to much research. Driven by the aerospace industry, large re...

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“…The sensitivities are determined by differentiation of Equation (21). Assuming design independent loads the sensitivities of the load vector are zero.…”

Section: Sensitivity Analysis Of the Pre-buckling Displacement Fieldmentioning

confidence: 99%

Post‐buckling optimization of composite structures using Koiter's method

Henrichsen

Lindgaard

Lund

2016

Numerical Meth Engineering

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“…Even though this method considerably reduces the condition number of the stiffness matrix, the authors are aware that damping the diagonal in this manner perturbs the underlying numerical problem and more elegant solutions may be possible. Second, the matrix inversion is not computed using the backslash operator in MATLAB, but rather with the Moore-Penrose pseudoinverse as detailed in reference [48]. One possible solution is to discretise the plate into multiple DQM elements using the method proposed by Tornabene et al [43].…”

Section: Model Implementationmentioning

confidence: 99%

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part II: Model validation

Groh

2016

Composite Structures

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. AbstractThe higher-order, equivalent single-layer model developed in Part I is applied to the stretching and bending of exemplar multilayered flat plates, where the results are compared with different 3D models, and trends and insights are subsequently drawn. The present mixed displacement/stress-based model is derived from inherently equilibrated 3D stress fields that satisfy the interlaminar and surface traction equilibrium conditions. A new set of governing equations is derived from a contracted Hellinger-Reissner functional that only enforces the classical membrane and bending equations via Lagrange multipliers. Combined with the fact that the same set of stress resultants is used for all stress fields, the number of unknown variables of the theory reduces, while maintaining sufficient fidelity to capture higher-order transverse shearing and zig-zag effects. A wide range of stacking sequences are considered ranging from orthotropic straight-fibre laminates to sandwich panels with variable-stiffness face sheets, i.e. composite plies in which the reinforcing fibres describe curvilinear paths. Hence, the model is used to study laminated plates with 3D heterogeneity, that is laminates comprising layers with material properties that can differ by multiple orders of magnitude and that vary continuously in-plane. The governing equations are solved both analytically using trigonometric expansions and numerically using the pseudo-spectral differential quadrature method. The 3D stress fields predictions correlate closely with 3D elasticity and 3D finite element solutions and are accurate to within a few percent for thick plates with characteristic length to thickness ratios as small as 5:1. In fact, the results suggest that 3D stress fields from our model satisfy Cauchy's 3D equilibrium equations more accurately, and at a three-order degree of freedom reduction in computational cost, compared to high-fidelity 3D FEM models.

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“…Koiter's approach was applied by Stein 33 for rectangular isotropic plates, Chandra & Raju 34 for symmetrically laminated orthotropic plates, and was extended in an optimization framework by Wu et al 35 Furthermore, Koiter's perturbation technique lends itself to robust implementation in numerical solution techniques such as the Finite Element Method (FEM) 36 and the Differential Quadrature Method (DQM). 37 Recently, Koiter's perturbation scheme was coupled with a Newton-Raphson iterative solver in order to combine the merits of both asymptotic and path-following techniques, thereby allowing efficient modeling of nonlinear prebuckling paths and limit-point buckling.…”

Section: Iib Koiter's Perturbation Approachmentioning

confidence: 99%

“…DQM is a numerical technique for solving differential boundary value problems developed in the 1970s by Bellmann et al 40 Since then DQM has been shown to give robust and efficient solutions to many problems in structural mechanics. 41 Details on implementing Koiter's perturbation approach in DQM are presented by White et al 37 …”

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confidence: 99%

Mass Optimisation of Variable Angle Tow, Variable Thickness Panels with Static Failure and Buckling Constraints

Groh

2015

56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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By taking advantage of curved fiber paths, Variable Angle Tow (VAT) laminates increase the design space for tailoring the structural behavior of thin-walled aerospace structures. In recent years, advancements in Automated Fiber Placement (AFP) and Continuous Tow Shearing (CTS) have facilitated the manufacture of these laminates. The CTS technique holds the advantage of reducing many of the manufacturing defects characteristic of the AFP process such as fiber wrinkling, tow gaps and tow overlaps, while also allowing for tighter steering radii. On the other hand, the CTS process features added complexity due to the coupling of fiber steering angle with tow thickness. In this study, a minimummass design of a typical aircraft wing panel under end-compression subject to pre-defined manufacturing, static failure and buckling load constraints is sought. The geometric effects of the asymmetric thickness distribution of the CTS panel on the critical buckling loads, postbuckling paths and static failure behavior are captured for the first time. A hybrid optimization scheme that couples a genetic algorithm with a pattern-search algorithm is used to define a VAT laminate that reduces the mass of both square and rectangular aircraft panels by 31% compared to a baseline straight fiber design. The optimization of the fiber paths is driven by two distinct requirements, namely local and global stiffness tailoring that influence the buckling performance and static strength, respectively. Finally, the initial postbuckling behavior of the optimized designs is investigated using Koiter's perturbation approach, which reveals that postbuckling stability should be considered when optimising VAT panels manufactured by the CTS technique.

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Initial post-buckling of variable-stiffness curved panels

Cited by 64 publications

References 37 publications

Post‐buckling optimization of composite structures using Koiter's method

Post‐buckling optimization of composite structures using Koiter's method

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part II: Model validation

Mass Optimisation of Variable Angle Tow, Variable Thickness Panels with Static Failure and Buckling Constraints

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