ES-PIM applied to buckling of variable angle tow laminates

Aeroelastic behavior of stiffened composite laminated panel with embedded SMA wire using the hierarchical Rayleigh–Ritz method

Koiter's asymptotic method enables the calculation and deep understanding of the initial post-buckling behaviour of thin-walled structures. For the single-mode asymptotic analysis, Budiansky (1974) presented a clear and general formulation for Koiter's method, based on the expansion of the total potential energy function. The formulation from Budiansky is herein revisited and expanded for the multi-modal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. Given the admittedly difficult implementation of Koiter's method, especially for multi-modal analysis and during the evaluation of the third– and fourth–order tensors involved in Koiter's analysis; the presented study proposes a formulation and notation with close correspondence with the implemented algorithms. The implementation is based on state-of-the-art collaborative tools: Python, NumPy and Cython. The kinematic relations are specialized using von Karman shell kinematics, and the displacement field variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) finite element, modified to reach third-order interpolation also for the in-plane displacements, using only 4 nodes per element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order fields. The formulation and implementation are verified by comparing results for isotropic and composite plates against established literature. Finally, results for multi-modal displacement fields with up to 5 modes and corresponding post-buckling factors are reported for future reference.

Displacement-based formulation of Koiter's method: Application to multi-modal post-buckling finite element analysis of plates

The present study investigates how to apply continuous tow shearing (CTS) in a manufacturable design parameterization to obtain reduced imperfection sensitivity in lightweight, cylindrical shell designs. The asymptotic nonlinear method developed by Koiter is applied to predict the post-buckled stiffness, whose index is constrained to be positive in the optimal design, together with a minimum design load. The performance of three machine learning methods, namely, Support Vector Machine, Kriging, and Random Forest, are compared as drivers to the optimization towards lightweight designs. The new methodology consists of contributions in the areas of problem modeling, the selection of machine learning strategies, and an optimization formulation that results in optimal designs around the compromise frontier between mass and stiffness. The proposed ML-based framework proved to be able to solve the inverse problem for which a target design load is given as input, returning as output lightweight designs with reduced imperfection sensitivity. The results obtained are compatible with the existing literature where hoop-oriented reinforcements were added to obtain reduced imperfection sensitivity in composite cylinders.

/pdf/lightweight-design-of-variable-stiffness-cylinders-with-9j3uriew.pdf

Lightweight Design of Variable-Stiffness Cylinders with Reduced Imperfection Sensitivity Enabled by Continuous Tow Shearing and Machine Learning

Variable Angle Tow designs have shown to improve structural performance by providing a better stiffness distribution. However, additional manufacturing constraints are involved compared to straight fibre laminates: mainly the maximum fibre steering curvature that can be achieved. Moreover, if a gap free laminate is constructed, overlaps will be formed depending on the relative fibre orientation when the continuous tows are bent to follow a reference path in Automated Fibre Placement and a one-sided irregular thickness profile is created. To counter the latter effect and still obtain uniform thickness laminates, a cut and restart strategy of the tows is often used in industry, leaving the final product with small gap areas. However, previous research has hinted both numerically and experimentally on further buckling improvements of variable stiffness laminates incorporating overlaps, resulting in a variable thickness profile. To investigate these possible benefits and their extent, the thickness build-up and overlap locations are specially considered in this proposed framework on Variable Angle Tows. In first instance, a virtual manufacturing surrogate of the laminate is produced to represent the discrete thickness profile due to the overlaps. This is performed by plotting each tow graphically and subsequently retracing the total thickness based on the superimposed opacity over the laminate. The tow paths are obtained from the interpolated fibre orientation, which simultaneously incorporates the steering limit of the manufacturing process. The surrogate information is then compared to a smeared approximation of the thickness build-up, based solely on the steering angle. The linear buckling simulation is performed by means of a semi-analytical model on a plate formulation, where the one-sided thickness profile variation is modelled by incorporating an offset from the geometrical varying midplane to a common one, around which the laminate stiffnesses are calculated. These adapted stiffnesses are then used in solving the neutral equilibrium buckling problem, where the displacements are approximated by means of the Ritz method, with a set of Legendre polynomial shape functions and numerical integration of the stiffness matrices. The semi-analytical smeared model correlates to within ± 5% of the discrete thickness Finite Element Model for a range of geometries, loading and boundary conditions. This verifies the thickness approximation for linear buckling of small tow widths compared to the laminate's dimensions, yet was inconclusive for larger tow width ratios. Finally, laminates with a varying thickness profile are optimised alongside the variable stiffness for different cases and compared to uniform thickness counterparts at isomass. The results show small improvements in the symmetric problem, but a higher gain is obtained for unsymmetrical conditions with an unrestricted layup sequence, as the thickness increase can be highly tailored together with the variable fibre orientation to locations requiring higher stiffness.

/pdf/semi-analytical-buckling-and-optimisation-of-variable-7dr12hied6.pdf

Semi-Analytical Buckling and Optimisation of Variable Stiffness, Variable Thickness Laminates

The non-linear buckling of unstiffened laminated composite cones and cylinders will be investigated and new semi-analytical models capable to predict the static and the instability response of these shells under various loads and boundary conditions will be proposed. An introduction is given to the reader in order to present and discuss some of the main deterministic approaches currently used for the design of imperfection sensitive structures. From this introduction it will become clear the need for non-linear tools that can consider both geometric and load imperfections, which are recognized to be among the main factors affecting the load carrying capacity of the shells under discussion.
The complete non-linear strain equations are derived using two Equivalent Single-Layer Theories: the Classical Laminated Plate Theory and the First-order Shear Deformation Theory. The non-linear terms will be identified corresponding to Donnell’s, Sanders’ and Timoshenko and Gere’ assumptions, but the discussion will focus on Donnell’s and Sanders’ equations. The resulting strain-displacement equations will then be applied to the stationary conditions of the total potential energy in order to obtain the non-linear static equations and the eigenvalue problem that can be used to predict the instability behavior, the latter using the neutral equilibrium criterion.
The Ritz method is selected to solve the non-linear set of equations and a new set of appropriate approximation functions for the displacement field is proposed, in order to simulate axial compression, torsion, pressure, load asymmetry, any arbitrary surface or concentrated loads, and any load case combining these loads. Elastic constraints are used to produce a wide range of boundary conditions, covering the four types of boundary conditions commonly used in the literature.
Two methods to solve the non-linear static equations are discussed: Newton-Raphson with line search and Riks; both presented in the full form, where the tangent stiffness matrix is calculated at every iteration, and in the modified form, where the tangent stiffness matrix is updated at the beginning of each load increment (or arc-length increment) and kept constant along the iterations.
An analytical integration scheme is proposed for the linear stiffness matrices and a numerical integration scheme proposed for the non-linear stiffness matrices. The analytical integration schemes assume constant laminate properties over the whole conical/cylindrical surface. For conical shells a novel approximation is proposed in order to efficiently perform the analytical integration of the linear stiffness matrices.
Detailed convergence analyses are presented and the proposed models are verified with finite element results, models available from the literature and test results from the literature. All the developed tools and algorithms are presented in detail and made available to the reader online.
Keywords:
Ritz method, Linear, Non-Linear, Static, Buckling, Composite, Conical, Cylindrical, Pressure, Torsion, Axial Compression, Donnell, Sanders, Geometric Imperfection, Load Imperfection

Castro, Saullo, Giovani, Pereira

Papers

Semi-analytical tools for the analysis of laminated composite cylindrical and conical imperfect shells under various loading and boundary conditions