A triangular finite difference scheme for large deflection problems

A lumped triangular element formulation is developed based on a finite difference approach for the large deflection analysis of plates and shallow shells. The presented formulation is independent of the boundary condition (unlike the finite difference formulation) and uses energy principles to derive a set of nonlinear algebraic equations which are solved by using an incremental Newton-Raphson iterative procedure. A study of the large deflection behaviour of thin plates is made for various edge conditions and aspect ratios, and the results obtained are compared with those using a finite element scheme. Representative nondimensional solutions for deflections and stresses are presented in the form of graphs.     

Nonlinear dynamic analysis of stiffened plates

A new finite strip formulation for the nonlinear analysis of stiffened plate structures subjected to transient pressure loadings is presented. The effects of large deflections, and strain rate sensitive yielding material properties are included. An explicit central difference/diagonal mass matrix time stepping method is adopted. Example results are presented for an I-beam, an isotropic plate and a five-bay stiffened panel and compared with other predictions and/or experimental results. It is observed that design level accuracy can be obtained for practical structures for a fraction of the cost of full finite element analyses.     

Elasto-plastic analysis of orthogonally stiffened plates with initial imperfections under uniaxial compression

In general, the stiffened plates consisting of steel plate elements are unavoidably accompanied by initial imperfections such as residual stresses and initial deflections, which have considerable effects on their ultimate strength. Therefore, it is needed for designing them to develop more rational method taking the ultimate strength influenced by initial imperfections into account rather than the conventional design method being on the basis of the linear elastic buckling theory.From this point of view, this study aims to evaluate rigorously the ultimate strength of orthogonally stiffened plate with initial imperfections under uniaxial in-plane compression. The elasto-plastic finite element method is applied to attain this purpose. By a happy combination of modal analytical technique and conventional finite element method, much reduction of the degree of freedom can be expected to be realized herewith. Some numerical calculations are performed by means of this rigorous method to examine the exactness of the analysis. Moreover, the numerical results are compared with the experimental ones.     

Application of a trigonometric finite difference procedure to numerical analysis of compressive and shear buckling of orthotropic panels

A numerical analysis developed for the buckling of rectangular orthotropic layered panels under combined shear and compression is described. This analysis uses a central finite difference procedure based on trigonometric functions instead of using the conventional finite differences which are based on polynomial functions. Inasmuch as the buckle mode shape is usually trigonometric in nature, the analysis using trigonometric finite differences can be made to exhibit a much faster convergence rate than that using conventional differences. Also, the trigonometric finite difference procedure leads to difference equations having the same form as conventional finite differences; thereby allowing available conventional finite difference formulations to be converted readily to trigonometric form. For two-dimensional problems, the procedure introduces two numerical parameters into the analysis. Engineering approaches for the selection of these parameters are presented and the analysis procedure is demonstrated by application to several isotropic and orthotropic panel buckling problems. Among these problems is the shear buckling of stiffened isotropic and filamentary composite panels in which the stiffener is broken. Results indicate that a break may degrade the effect of the stiffener to the extent that the panel will not carry much more load than if the stiffener were absent.     

Optimization of cable-stayed bridges with box-girder decks

Box-girder decks are very effective solutions for long span cable-stayed bridges, due to their high torsional stiffness and streamlined profile, which usually lead to a good aerodynamic behaviour. A study on the optimization of such structural system is presented in this paper. The deck is modelled through the assembly of planes of plate-membrane elements. A multicriteria approach is considered for the optimization itself, with constraints on maximum stresses, minimum stresses in stays and deflections under dead load condition. Two illustrative examples are shown.     

Weight optimization of stiffened cylinders under axial compression

A procedure is developed for the design of a stiffened cylinder under a given uniform axial compression with minimum weight. The approach allows the consideration of various shapes of stiffening members. The effective stiffness of the skin in its post-buckled state is taken into account in the basic analysis. The buckling analyses are accomplished as a minimum problem in the buckling mode shape parameters space using the variable metric method. A mixed procedure which combines the exterior penalty function concept and random search is used to minimize the weight of the stiffened cylinders. The design examples demonstrate the validity of the present approach.     

Optimization of composite stiffened panels subject to compression and lateral pressure using a bi-level approach

Composite stiffened panel optimization is typically a mixed discrete-continuous design problem constrained by buckling and material strength. Previous work applied a bi-level optimization strategy to the problem by decomposing the mixed problem to continuous and discrete levels to reduce the optimization search space and satisfy manufacturing constraints. A fast-running optimization package, VICONOPT, was used at the continuous optimization level where the buckling analysis was accurately and effectively performed. However, the discrete level was manually adjusted to satisfy laminate design rules. This paper develops the strategy to application on continuously long aircraft wing panels subjected to compression and lateral pressure loading. The beam-column approach used to account for lateral loading for analysis during optimization is reported. A genetic algorithm is newly developed and applied to the discrete level for automated selection of laminated designs. The results that are presented show at least 13% weight saving compared with an existing datum design.     

Optimal composite structures by deflection-variable programming

This paper deals with the problem of optimising multilaminar fibre-reinforced continua. A formulation is proposed in which primary optimisation variables are the nodal deflections of the finite element model. This leads to a decomposition of the problem into an inner and an outer subproblem; algorithms for solving both are described and numerical results given. It is shown that this approach is convenient for solving the nonlinear mixed integer program involved and allows the optimal number of layers in each element, the layer thicknesses and the fibre angles are to be determined simultaneously for very general inequality constraints on stresses and deflections. The case of structures of maximum stiffness is particularly considered.     

Bending and extension of cross-ply laminates with different moduli in tension and compression

Composite materials often exhibit different stiffnesses or moduli under tension loading than under compression loading. This behavior is modeled with a bilinear stress-strain curve having a modulus E_t in tension and E_c in compression as an approximation to the real nonlinear behavior. Under bending loads, laminated composites have both tensile and compressive stresses and hence are not subject to the same behavioral rules as ordinary single modulus materials. The resulting transcendental equilibrium equation is dependent upon the unknown neutral surface. This neutral surface is found and, hence, the equilibrium problem is solved with an iteration technique. The approach is applied to laminates ordinarily thought to be symmetric, antisymmetric, and unsymmetric about the middle surface. All laminates are found to exhibit coupling between bending and extension under bending in contrast to the usual concepts of symmetry and antisymmetry for single modulus laminates. Several approximate approaches are investigated for treating the multimodulus laminate problem. The effect of coupling due to different moduli in tension and compression on stresses and deflections is found to be generally significant for common composite materials such as boron/epoxy and graphitc/epoxy as well as carbon-carbon.     

Plastic behaviour and stability constraints in the shakedown analysis and optimal design of trusses

A shakedown analysis and optimum shakedown design of elasto-plastic trusses under multi-parameter static loading are presented. To control the plastic behaviour of the truss, bounds on the complementary strain energy of the residual forces and on the residual displacements are applied and for the bars under compression critical stresses updated during the iteration are taken into consideration. The formulation of problems is suitable for nonlinear mathematical programming which is solved by the use of an iterative procedure. The application of the method is illustrated by three test examples.     

Aircraft wing panel buckling analysis: efficiency by approximations

A comparison of ‘exact’ and approximate methods for the determination of critical buckling loads of prismatic benchmark metal and composite panels is presented. The panels are stiffened by either J-, blade- or hat-stiffeners and are representative of typical aircraft wing panel configurations, with in-plane shear and compression load combinations. Buckling design curves and modes are illustrated, and associated CPU times are given to demonstrate the accuracy and efficiency of the approximations adopted.Initial results for the benchmarks, which are rectangular in plan-form, are compared with rigorous finite element solutions. Thereafter, attention is focused on results for the same panels but with parallelogram plan-form.Two analysis methods based on Classical Plate Theory are used as follows: an existing, ‘exact’ method, incorporating Lagrangian multipliers to constrain the transverse (or skew) boundary conditions; and a recently developed approximate infinite width technique, based on the previous one but analysing only a repeating portion of the plate assembly.     

Experiments on interactive buckling in optimized stiffened panels

Five aluminium blade stiffened panels of three different geometries were tested in compression in a displacement controlled loading apparatus. Panel designs were achieved using VICONOPT, a fast-running optimization package based on linear eigenvalue buckling theory, and embrace two different design philosophies. The panels were loaded beyond initial buckling to collapse, and the effects of initial overall imperfections were monitored. In all cases the final failure showed evidence of significant interaction between buckling modes. The tests draw particular attention to a violently unstable and unpredictable form of failure, involving a combination of overall and stiffener buckling, which can occur even when initial buckling in the skin has the effect of pushing the panel in the opposite sense. Received November 14, 2000     

Stability analysis of ring stiffened shells of revolution

A finite element formulation is presented for the general instability of ring stiffened shells of revolution subjected to external pressure. Linear bifurcation buckling theory is used. A rigorous derivation for the potential due to the hydrostatic loading including follower force effect is presented. Comparison with results obtained by earlier research workers in this field is given. Substantial reduction in buckling pressures due to follower force effect is reported.     

Two-stage size-layout optimization of axially compressed stiffened panels

In this study, a two-stage optimization framework is proposed for cylindrical or flat stiffened panels under uniform or non-uniform axial compression, which are extensively used in the aerospace industry. In the first stage, traditional sizing optimization is performed. Based on the buckling or collapse-like deformed shape evaluated for the optimized design, the panel can be divided in sub-regions each of which shows characteristic deformations along axial and circumferential directions. Layout optimization is then performed using a stiffener spacing distribution function to represent the location of each stiffener. A layout coefficient is assigned to each sub-region and the overall layout of the panel is optimized. Three test problems are solved in order to demonstrate the validity of the proposed optimization framework: remarkably, the load-carrying capacity improves by 17.4 %, 66.2 % and 102.2 % with respect to the initial design.     

Optimum design of plano-milling machine structure using finite element analysis

The problem of optimum design of plano-milling machine structure is formulated as a nonlinear mathematical programming problem with the objective of minimizing the structural weight. The plano-milling machine structure is idealized with triangular plate elements and three dimensional frame elements based on finite element displacement method. Constraints are placed on static deflections and principal stresses in the problem formulation. The optimization problem is solved by using an interior penalty function method in which the Davidon-Fletcher-Powell variable metric unconstrained minimization technique and cubic interpolation method of one dimensional search are employed. A numerical example is presented for demonstrating the effectiveness of the procedure outlined. The results of sensitivity analysis conducted with respect to design variables and fixed parameters about the optimum point are also reported.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Sensitivity of buckling loads of anisotropic shells of revolution to geometric imperfections and design changes

Buckling load sensitivity calculations in the shell-of-revolution program FASOR are discussed. This development is based on Koiter's initial postbuckling theory, which has been generalized to include the effect of stiffness changes, as well as geometric imperfections. The implementation in FASOR is valid for anisotropic, as well as orthotropic, shells. Examples are presented for cylindrical panels under axial compression, complete cylindrical shells in torsion, and antisymmetric angle-ply cylindrical panels under edge shear.     

Deflection and stress analysis of orthotropic plates in flexure

A finite difference (FD) method is developed for computing deflections and stresses in rectangular orthotropic plates subjected to transverse flexural loadings. The plate material is elastic with symmetry axes parallel to the plate edges, and plate deflections are assumed to be small. The FD equations are solved iteratively using successive over relaxation techniques. The numerical procedure converges rapidly and is simple enough to be implemented in a minicomputer program for the design analysis of composite plates. As an example the program is applied to the analysis of simply supported and clamped square plates under uniform pressure or point loads, and the effect of material anisotropy on plate behaviour is discussed.     

水下爆炸对船板冲击作用仿真

根据DAA1近似建立了一种简单实用的用于分析船板水下冲击响应模型。模型基于流-固作用原理，在不考虑结构低频响应的情况下，得到了可直接作用于结构湿面的总的压力载荷表达式。利用有限元代码LS—DYNA的用户定义的分布载荷子程序实现了这一方法。该方法适用于求解在早期冲击波作用下，结构的高频响应和局部变形。采用该方法对水下冲击波对船板和加筋船板(气背条件)的作用进行了仿真计算。计算机模拟结果表明采用该方法，在保证较高精度的前提下可以大大地降低CPU时间，提高仿真计算效率。