Composite laminate optimization program suitable for microcomputers

The design of a composite panel requires some way of finding the minimum thickness laminate which will withstand the load requirements without failure. The mathematical complexity of this problem dictates the use of nonlinear optimization techniques. Although there are sophisticated optimization programs available capable of solving for the ply ratios, these programs are not often used in preliminary design because they require a large computer and some knowledge of the program's operation. As an alternative, specialized laminate optimization programs were developed which are compact and efficient enough to run on microcomputers. Only stresses at a point and inplane loads and deflections are considered. The programs are simple to use and require no knowledge of optimization. Techniques are developed in this paper that find minimum thickness laminates with either ply ratios or ply angles as design variables. Many test cases were run with these programs to demonstrate the weight savings possible over quasi-isotropic laminates. Particular interest is directed toward performance of the laminates under multiple independent loads. Initial orientations for the programs to operate on were studied, and 0/90/45/-45 laminates were found to be an effective starting point for design.     

Coupled thermal-structural problems in the optimization of laminated plates

The behaviour of a laminated plate with given boundary temperatures and displacement constraints may be tailored by varying the orientation of the reinforcement in the different layers. Because the material parameters in a thermal conductivity problem, as also in a structural problem, depend on the orientations of the layers, there is a coupled-field problem to be solved. FEM is applied here to the analysis of such problems, which now consists of two phases in each iteration cycle: first solution of the temperature distribution over the structure and then computation of the displacements, stresses and strains. Strain energy and the sum of selected displacements for the structure are minimized with respect to the fibre orientations in the layers. Only mid-plane symmetric laminates with constant temperature over the thickness are considered, i.e. the response of the laminate is restricted to in-plane behaviour. Mathematically, the problem is a nonlinear one, and thus the minimum point can be either a local or a global one. The gradients needed during minimization are computed analytically. Examples with different numbers of design variables are given.     

Stacking sequence optimization for constant stiffness laminates based on a continuous optimization approach

In this paper, an optimization procedure based on multi-phase topology optimization is developed to determine the optimal stacking sequence of laminates made up of conventional plies oriented at ?45°, 0°, 45 and 90°. The formulation relies on the SFP (Shape Functions with Penalization) parameterization, in which the discrete optimization problem is replaced by a continuous approach with a penalty to exclude intermediate values of the design variables. In this approach, the material stiffness of each physical ply is expressed as a weighted sum over the stiffness of the candidate plies corresponding to ?45°, 0°, 45 and 90° orientations. In SFP, two design variables are needed for each physical ply in the laminate to parameterize the problem with respect to the 4 candidate orientations. Even if only constant stiffness laminates of constant thickness are considered in this paper, specific design rules used in aeronautics for composite panels (i.e., no more than a maximum number of consecutive plies with the same orientation in the stacking sequence) are however formulated and taken into account in the optimization problem. The methodology is demonstrated on an application. It is discussed how the different design rules can affect the solution.     

Topology optimization to minimize the dynamic compliance of a bi-material plate in a thermal environment

Topology optimization to minimize the structural dynamic compliance in a thermal environment is carried out in this paper. A bi-material plate subjected to a uniform temperature rise is investigated. The structure is driven by a time-harmonic surface loading with prescribed excitation frequency and amplitude. The stress induced by the equivalent thermal force which is known as design-dependent load could reduce the stiffness of the structure, thus altering the optimal topology design. A way to carry out the optimization in the thermal environments is presented here. The thermal stress is first evaluated, and then considered as pre-stress in the subsequent dynamic analysis with the introduction of the geometric stiffness matrix. The sensitivity analysis is carried out through adjoint method which can save significant computational resources by avoiding the derivatives of the thermal displacement (or thermal stress) on the design variables. The cost to obtain these derivatives can be very high since each design variable affects all the nodal displacements. The structural damping is neglected. Several pre-buckling cases are investigated.     

An integrated optimization for laminate design and manufacturing of a CFRP wheel hub based on structural performance

An integrated optimization that comprehensively considers design and manufacturing factors such as the geometric appearance, laminate constitutions, laminate distribution, laminate thickness and stacking sequence, is proposed for designing a carbon fiber reinforced polymer wheel hub of a racecar. First, the driving conditions of the racecar are analyzed to determine the performance requirements. Then, under the condition that the geometric design regions are partitioned and the constitutions of fiber plies with different directions are defined, laminate design and manufacturing model is established. A multi-objective optimization is then performed to achieve a lightweight, high-stiffness laminate structure in different design regions. Next, number of plies in each region is obtained from the thickness of laminate, and then, the stacking sequence is optimized to improve the stiffness of the laminate structure. Finally, laminate transitions for different regions are investigated. The results showed that laminate design and manufacturing optimization can reduce the weight of the wheel hub and improve the performance of the wheel hub under static, dynamic and impact conditions. The proposed optimization approach provides a feasible solution for a performance-based design of composite structures.     

A general approach forcing convexity of ply angle optimization in composite laminates

This paper deals with optimization of laminated composite structures in which the ply angles are taken as design variables. One of the major problems when using ply-angles as design variables, is the lack of convexity of the objective function and thus the existence of local optima, which implies that usual gradient based optimization procedures may not be effective. Therefore, a new general approach that avoids the abovementioned problems of nonconvexity when ply-angles are used as design variables is proposed. The methodology is based upon the fact that the design space for an optimization problem formulated in lamination parameters [introduced by Tsai and Pagano (1968)] is proven to be convex, because the laminate stiffnesses are expressed linearly in terms of the lamination parameters. However, lamination parameters have at least two major shortcomings: as yet, for the general case involving membrane-bending coupling, the constraints between the lamination parameters are not completely defined; also, for a prescribed set of lamination parameters physically realizable composite laminates (e.g. laminates with equal thickness plies) may not exist. The approach here, uses both lamination parameters and ply-angles and thereby uses the advantages of both and eliminates the shortcomings of both.In order to illustrate this approach, several stiffness optimization examples are provided.     

Controlling failure using structural fuses for predictable progressive failure of composite laminates

Structural fuses have been used to bias and control failures in structural applications where predictability of the progressive failure or collapse response is important. Tailoring structural fuses by trial and error in large structures that have numerous possible load and failure paths is not possible because the optimum failure sequence is not known a priori. Using nondeterministic methods to tailor structural fuses is computationally expensive. A procedure for developing deterministic measures to optimize structural fuses is presented here. The progressive failure of composite laminates is used for demonstration. Structural fuses are optimized using a reliability optimization. The failure response characteristics of the laminate with optimum structural fuses are used to identify deterministic measures that correlate with high progressive failure predictability. The deterministic measures are validated by using them as surrogate design criteria in a deterministic optimization to optimize structural fuses that control failure and improve progressive failure predictability. The improvement in predictability of the deterministic optimum design achieved by using optimized structural fuses is better than that obtained by optimizing the ply angles of the laminate explicitly for predictability.     

Orthotropic material orientation optimization method in composite laminates

This paper proposes a optimization method that is capable of simultaneous design of multiple layers in a composite laminate with respect to multiple objective functions. The optimization process obtains a continuous orientation of an orthotropic material for each layer of the laminate. Each layer by itself is a single design domain, which allows multiple domains to be stacked in various orientations. Multiple optimization objectives are considered resulting in layers that perform different functions. The layers are modeled within a three-dimensional structure and by discretizing the structure using three-dimensional elements, the interaction between individual layers can be modeled. This also allows the optimization method to obtain a three-dimensional orientation vector. In this study, the individual layers are assumed to be thin, limiting the orientation vector to the mid-plane of the layer. The optimization model is tested on a two-layer laminate in which one layer is optimized for thermal control by directing heat toward specified sections while shielding other sections and the second layer is optimized to reduce the total deformation of the laminate structure that results from the thermal load. The results of simultaneous optimization for both layers are shown for several different configurations of boundary conditions.     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

Optimization of coupled thermal-structural problems of laminated plates with lamination parameters

The behaviour of a laminated plate with given boundary temperatures and displacement constraints is optimized and the optimization problem is expressed in terms of lamination parameters. Because the thermal conductivity and structural properties of a laminate depend on the lamination parameters of the laminate, the analysis of the plate consists of solving a coupled-field problem. The strain energy, or certain displacements of the laminated plate due to given boundary temperatures and displacement boundary conditions, is optimized with respect to in-plane lamination parameters, and also buckling of the plate is considered. The buckling factors for thermal loading are expressed as a function of four in-plane and four bending lamination parameters, and the smallest factor is maximized with respect to these parameters. In addition to these thermal problems, the natural frequencies of the laminated plate are studied. Since transverse shear deformations are taken into account,the natural frequencies can be expressed as functions of two in-plane and four bending lamination parameters, with respect to which the lowest natural frequency of the plate is maximized. The lay-up for the laminate, corresponding to four optimal in-plane or bending lamination parameters, consists of three layers at most and can be determined using explicit equations. Explicit equations are derived for creating a lay-up having optimal bending lamination parameters. Received May 12, 1999     

A lagrange parametrization for the design of variable stiffness laminates

A Lagrange parameterization of lamination parameters which are used for optimization of variable stiffness laminates is presented. The advantages of the approach are: a) the design variables become independent of the finite element mesh and, b) the smoothness of the solution is inherently guaranteed. Due to independency of design variables of the finite element mesh, the reduction in the number of design variables is drastic, once variable stiffness laminates usually demand a fine mesh. The lamination parameters formulation allows a more precise and concise approach to laminate design, removing difficulties related to fiber direction and stacking sequence, increasing the chances to find a global optimal solution. In this paper, the Lagrange parameterization is used for the maximization of the buckling load of a variable stiffness composite plate.     

Benchmark solution of imperfect angle-ply laminated rectangular plates in cylindrical bending with surface piezoelectric layers as actuator and sensor

The static and dynamic responses of simply supported adaptive angle-ply laminates in cylindrical bending are considered. The interlaminar bonding of the host elastic laminate is assumed to be imperfect, described by a spring-layer model, while the bonding between the host elastic laminate and the surface piezoelectric actuator and sensor layers is perfect. The state-space approach, which is directly based on the three-dimensional exact elasticity (piezoelasticity) equations and very effective in analyzing laminated structures, is employed. The numerical results should provide a useful means of comparison in the development of simplified analyses or numerical methods.     

Aerothermoelastic topology optimization with flutter and buckling metrics

This work develops a framework for SIMP-based topology optimization of a metallic panel structure subjected to design-dependent aerodynamic, inertial, elastic, and thermal loads. Multi-physics eigenvalue-based design metrics such as thermal buckling and dynamic flutter are derived, along with their adjoint-based design derivatives. Locating the flutter point (Hopf-bifurcation) in a precise and efficient manner is a particular challenge, as is outfitting the optimization problem with sufficient constraints such that the critical flutter mode does not switch during the design process. Results are presented for flutter-optimal topologies of an unheated panel, thermal buckling-optimal topologies, and flutter-optimality of a heated panel (where the latter case presents a topological compromise between the former two). The effect of various constraint boundaries, temperature gradients, and (for the flutter of the heated panel) thermal load magnitude are assessed. Off-design flutter and thermal buckling boundaries are given as well.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

复合材料整体成型过程分层缺陷动态扩展研究

分层是复合材料层合板最主要的缺陷/损伤形式,当前研究多集中于复合材料使用过程中的分层损伤,对制造过程中的分层缺陷研究较少.复合材料层合板的分层缺陷在整体成型过程中因不均匀温度场、非对称结构等原因易发生扩展,采用预埋隔离纸模拟分层缺陷,研究分层缺陷在整体成型过程中的扩展行为,对分层扩展的驱动力进行了计算分析,并通过分析热循环对T300/QY8911复合材料层合板界面性能的影响以及计算裂纹尖端能量释放率,从实验研究和有限元计算两方面证明了分层扩展是一个动态过程.研究结果表明,热残余应力是T300/QY8911复合材料层合板整体成型过程中发生分层缺陷扩展的主要驱动力;增加热循环次数和提高热循环温度会显著降低T300/QY8911复合材料层合板的层间剪切强度,当能量释放率降低到低于层间断裂韧性值时,分层动态扩展过程则停止.     

Note on singular optima in laminate design problems

This paper studies the design of laminates subject to restrictions on the ply strength. The minimum weight design is considered. It is shown that this formulation includes singular optima, which are similar to the ones observed in topology optimization including local stress constraints. In laminate design, these singular optima are linked to the removal of ‘zero thickness’ plies from the stacking sequence. It is shown how the fiber orientation variables can circumvent the singularity by relaxing the strength constraints related to such vanishing plies. This demonstrates the key role of fiber orientations in the optimization of laminates and the need for their efficient treatment as design variables.     

A fortran program for the design of laminates with required mechanical properties

A FORTRAN 77 program for designing laminates with required mechanical properties is presented. This user-friendly program is developed for interactive use by users with either micro or mainframe computers to provide maximum flexibility and reduce the amount of inputting. The inputs required to the program are layer orthotropic unidirectional (UD) material properties and thickness and the desired laminate mechanical properties. The output is the optimum lamination that will provide these properties. This optimum is obtained by minimizing the difference between the calculated properties of trial laminates and the desired properties. The program allows such designed laminates to be unsymmetric in either geometry or material about the middle surface of a plate and consist of a specified number of different fibre-reinforced orthotropic layers. This program can either be used on its own for the purpose of designing laminated plates or be considered as a package-subroutine within a complex structural design package. As a demonstration of the use of the program, the design of a symmetric laminate is conducted to show how an optimal lay-up sequence can provide the required laminate mechanical properties determined by the design. A full listing of the program is given as an Appendix.     

Optimal design of structural steel frameworks

This paper reviews work conducted at the University of Waterloo during the 1980s concerning the computer-automated design of least-weight structural steel frameworks. First, design under static loads is considered whereby the members of the structure are automatically sized using commercial steel sections in full conformance with design standard provisions for elastic strength/stability and stiffness. This problem is illustrated for the least-weight design of a steel mill crane framework comprised of a variety of member types and subject to a number of load effects. Then, the design methodology is extended to the least-weight design of structural steel frameworks under both service and ultimate loading conditions. Here, acceptable elastic stresses and displacements are ensured at the service-load level while, simultaneously, adequate safety against plastic collapse is ensured at the ultimate-load level. This design problem is illustrated for the least-weight design of an industrial steel mill framework for which plastic behaviour is governed by conservative piecewise linear yield conditions. Finally, the computer-based design methodology is extended to the least-weight design of structural steel frameworks subjected to dynamic loading. Constraints are placed on dynamic displacements, dynamic stresses, natural frequencies and member sizes. The design problem is illustrated for the least-weight design of a steel trussed arch subjected to non-structural masses and an impulse force.     

Problems of structural optimization for post-buckling behaviour

A proposal of a new approach to the optimal design of structures under stability constraints is presented. It is shown that the standard problem of maximization of the instability load may be modified so as to obtain a specified post-critical behaviour of the designed structure. The modified optimal structure represents stable post-buckling behaviour either locally, that is, in the vicinity of the critical point, or for a specified range of generalized displacements. First, some rigid–elastic finite-degree-of-freedom models are optimized, giving an insight into the modified design problems. Then a classification of the new optimization problems is presented. Various forms of instability are taken into account and a broad selection of objective as well as constraint functions is proposed. Based on the presented classification and following the proposed optimization concept, detailed formulations of nonlinear problems of design for post-buckling behaviour are given.     

Variable-stiffness composite panels: Buckling and first-ply failure improvements over straight-fibre laminates

One of the primary advantages of using fibre-reinforced laminated composites in structural design is the ability to change the stiffness and strength properties of the laminate by designing the laminate stacking sequence in order to improve its performance. This procedure is typically referred to as laminate tailoring. Traditionally, tailoring is done by keeping the fibre orientation angle within each layer constant throughout a structural component. Allowing the fibres to follow curvilinear paths within the plane of the laminates constitutes an advanced tailoring option that can lead to modification of load paths within the laminate to result in more favourable stress distributions and improve the laminate performance.Based on numerical simulations, the present work demonstrates the advantages of variable-stiffness over straight-fibre laminates in terms of compressive buckling and first-ply failure. A physically based set of failure criteria, able to predict the various modes of failure of a composite laminated structure, is implemented in finite element models of straight and variable-stiffness panels under compression. Non-linear analyses are carried out to simulate first-ply failure in the postbuckling regime.