Minimum cost design of a welded stiffened square plate loaded by biaxial compression

The optimized plate structure consists of a simply supported square base plate stiffened with an orthogonal grid of flat stiffeners welded to the base plate by fillet welds. The uniformly distributed compressive load acts biaxially in the plane determined by the centre of gravity of T-sections, which consist of a part of the base plate and of a stiffener. In the optimization process the number of stiffeners as well as the thicknesses of the base plate and flat stiffeners, which minimize the cost function and fulfil the design constraints, as sought. The cost function includes the cost of material, assembly, welding and painting. Constraints relate to the global buckling, local buckling of base plate parts and stiffeners as well as to the deflection due to shrinkage of welds. To illustrate the effectiveness of the mathematical methods, the problem is solved by the Rosenbrocks hill-climb algorithm as well as by entropy-based unconstrained minimization.     

Buckling of prismatic structures under biaxial loading

This work presents an analysis and numerical results for the buckling of longitudinally stiffened prismatic structures that consist of an assemblage of flat plate and cylindrical panel components. The applied loading is assumed to induce uniform in-plane biaxial stresses, and all types of buckling modes (general, local, and coupled) are automatically accounted for in the analysis. Previous analyses of such problems by the exact (Wittrick) and approximate (Cheung) finite strip methods are based on the assumption that the structure is simply supported on the transverse ends by diaphragms (SS3 classical simple supports). The present work extends these analyses to include the case of completely clamped transverse ends (CC4), as well as simply supported ends. A one term Galerkin method is used to replace the governing partial differential equations of equilibrium for each component by ordinary differential equations in terms of the transverse coordinate, and the ordinary equations are solved exactly. The principle of virtual work is employed to obtain the stiffness matrix for each component. Numerial results are presented for a wide variety of CC4 and SS3 prismatic structures. These results compare favorably with available solutions.     

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.     

铝合金受压板件局部屈曲承载力设计方法

为考察现有铝合金设计规范预测非焊接受压板件承载力的准确性,采用壳单元建立单向受压四边简支板、方形截面管柱和十字形截面轴压柱有限元模型,使用ANSYS计算得到铝合金板件弹塑性屈曲临界应力和极限承载力。数值计算结果与理论分析结果对比发现：现有板件弹塑性屈曲应力理论能够给出铝合金板件屈曲临界应力的下限值,误差较小。中国规范非加劲板件有效厚度预测公式偏保守,因此给出修正的有效厚度预测公式。修正后的非加劲板件有效厚度公式可以提高板件利用率,增强经济性。     

Bifurcation, pre- and post-buckling analysis of frame structures

A procedure is developed for investigating the stability of complex structures that consist of an assembly of stiffened rectangular panels and three-dimensional beam elements. Such panels often form one of the basic structural components of an aircraft or ship structure. In the present study, the stiffeners are treated as beam elements, and the panels between them as thin rectangular plate elements, which may be subject to membrane and/or bending and twisting actions.

The main objective of the study is the determination of the critical buckling loads and the generation of the complete force-deformation behavior of such structures within a specified load range, based on the use of a computer program developed for this purpose. The present formulation can trace through the postbuckling or post limit behavior whether it is of an ascending or descending type. A limit load extrapolation technique is automatically initiated within the computer program, when the stability analysis of an imperfect or laterally loaded structure is being carried out.

The general approach to the solution of the problem is based on the finite element method and incremental numerical solution techniques. Initially, nonlinear strain-displacement relations together with the assumed displacement functions are utilized to generate the geometric stiffness matrices for the beam and plate elements. Based on energy methods and variational principles, the basic expressions governing the behavior of the structure are then obtained. In the incremental solution process, the stiffness properties of the structure are continuously updated in order to properly account for large changes in the geometry of the structure.

The computer program developed during the course of this study is referred at as GWU-SAP, or the George Washington University Stability Analysis Program.     

Lateral buckling of locally buckled beams using finite element techniques

This paper deals with lateral-torsional buckling of beams which have already buckled locally before the occurrence of overall buckling. Due to the weakening effects of local buckling, the stiffness of the beam is reduced. As a result, overall lateral buckling takes place at a lower load than the member would carry in the absence of local buckling. The effective width concept is used in this investigation to account for the post-buckling strength in the buckled compression plate elements of the beam section. A finite element formulation in conjunction with effective width concept is presented. Due to the nonlinearity involved because of local buckling, an iterative procedure is necessary. Search techniques are used to find the load factor. The method combined with an analysis on nonlinear bending moment distribution can be used to analyze the lateral stability problem of locally buckled continuous structure. In this case, both elastic stiffness matrix and geometric stiffness matrix must be revised at each load level. A computer program has been prepared for an IBM 370/165 computer.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Finite element analysis of eccentrically stiffened plates in free vibration

A compound finite element model is developed to investigate eccentrically stiffened plates in free vibration. The plate elements and beam elements are treated as integral parts of a compound section, and not as independent bending components. The derivation is based on the assumptions of small deflection theory. In the orthogonally stiffened directions of the compound section, the neutral surfaces may not coincide. They lie between the middle surface of the plate and the centroidal axes of the stiffeners. The results of this study are compared with existing ones and with those of the orthotropic plate approximation. Modifications to the existing equivalent orthotropic rigidities are proposed.     

Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.     

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.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

Nonlinear transient response of stiffened plates to air blast loading by a superelement approach

In this paper, new plate and stiffener beam elements are developed for the nonlinear dynamic analysis of stiffened plate structures subject to blast-type pressure waves. The displacement fields for the elements are represented by polynomial and analytical functions in both in-plane directions and they have been constructed so that only one element per bay or span is required to model the response. Geometric and material nonlinearities are included and the temporal equations are solved by the implicit Newmark beta method with Newton-Raphson sub-iteration. The new formulation has been tested on several numerical examples and the results obtained are compared with other available solutions. The present model is simple, requires much reduced storage and computing time and yet gives results suitable for practical design purposes.     

Critical buckling of delaminated composite plates using exact stiffness analysis

The critical buckling of composite plates with through-the-length delaminations is studied using exact stiffness analysis and the Wittrick–Williams algorithm. Computational efficiency is achieved by avoiding discretisation into elements, making the method suitable for preliminary aircraft design. Numerical results for longitudinal, transverse and shear loading show a transition from global to local buckling modes as the delamination width is increased. The critical buckling strength is dramatically reduced as the delamination is moved towards the plate surface, but is relatively insensitive to its widthwise location or to the edge conditions. The results are compared with finite element analysis.     

Buckling of thin-walled members by finite elements

We present a general two-dimensional thin plate/shell theory for the study of the elastic stability of sections formed by an assemblage of flat elements. These types of sections are generally analyzed by a one-dimensional model, but when, for instance, the width of the flanges is large or when there are openings in the section, the one-dimensional model cannot correctly predict the buckling load. We have developed a new finite element called DKM for the plate/shell model. We compare the result obtained with the present model with those of known analytical theories. The new model proves to be efficient and reliable. We also demonstrate that for wide plates, the analytical solution based on the one-dimensional model gives results slightly different from those of the proposed model.     

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.     

Two approaches for truss topology optimization: a comparison for practical use

This paper discusses ground structure approaches for topology optimization of trusses. These topology optimization methods select an optimal subset of bars from the set of all possible bars defined on a discrete grid. The objectives used are based either on minimum compliance or on minimum volume. Advantages and disadvantages are discussed and it is shown that constraints exist where the formulations become equivalent. The incorporation of stability constraints (buckling) into topology design is important. The influence of buckling on the optimal layout is demonstrated by a bridge design example. A second example shows the applicability of truss topology optimization to a real engineering stiffened membrane problem.     

On the design optimization of built-up stiffened steel beams with buckling and frequency constraints

The literature on the structural design optimization of steel-plate girders indicates a need for more refined research studies to obtain optimal designs by formulating and solving the design problem that combines structural sizing and shape parameters in one unified, constrained problem. For this purpose, the structural optimization design problem of stiffened steel-plate girders is formulated with specified loading conditions and constraints on strength and serviceability considerations including limits on fundamental frequency and buckling modes. The finite-element method-based model is used to define the objective function and the structural/geometric response functions, while the geometric domain elements are used to systematically perturb the structural shape during the search for an optimal shape of the structure. The mathematical statement of the gradient-based-design problem is solved for an optimal structural size and shape with buckling and frequency constraints in addition to the traditional strength constraints. The numerical results obtained are compared with results obtained from a less formal ad hoc design procedure, and some conclusions are drawn to emphasize the design benefits obtained from solving the design problem for optimal structural size and shape.     

Finite element analysis of out-of-plane buckling of reinforced concrete walls

In this paper a FE model for the study of the out-of-plane buckling of reinforced concrete walls is derived. The concrete is modelled using non-linear orthotropic 16 d.f. plate bending elements; the reinforcing steel using elasto-plastic beam elements. In the plane of the structure stresses are calculated using either four or eight-node membrane elements with bar elements used for the reinforcing steel. The buckling load is calculated by determining when the determinant of the out-of-plane tangent stiffness becomes zero. Comparison of the FE model with available experimental results shows good agreement.     

Ultimate strength of stiffened symmetrical welded steel beam-to-column flange connections

Several studies have confirmed the importance, in beam-column connections, of reinforcing the column web with horizontal stiffeners opposite the beam flanges welded to the column web and flanges. Present design codes, besides being conservative, do not adequately reflect the role of the stiffeners as lateral support to the column web aginst buckling. Non-linear finite element formulations, employing an incremental version of Stowell's deformation theory and a scaled version of the inverse power method, allowed accurate predictions of the bifurcation load of general concentrically stiffened steel plates in the elasto-plastic range. In this paper, the analytical procedure developed has been used to provide qualitative and quantitative design information on the strength of this type of connection by conducting a parameteric study, the parameters considered being the effect of stiffener size on column web buckling capacity and the effect of column axial load on the stiffening requirements of a connection. It has been demonstrated that proper reinforcing of the column web can, in practice, diminish the harmful effects of high column axial loads on connections.     

Optimization of light-gage cold-formed steel shapes by parametric bandwidth constriction

Optimization of structural shapes governed by the American Iron and Steel Institute Specification for the Design of Cold-Formed Steel Structural Members presents particular problems. Not only must a variety of buckling formulas be considered for flexural-torsional buckling of the shape as a whole as well as for buckling of the separate plate elements of the cross-section, but also the section properties used to evaluate buckling stress constraints are themselves stress dependent.A simplicial technique is detailed for forcing convergence to competitive designs in spite of the iterative computational oscillations caused by the nonlinear complexity and discontinuities that are present in the mathematical programming problem which must be solved. The set of constraints includes all applicable and unmodified design formulas appearing in the AISI specification. The technique is a two-phase process which gradually constricts the range limits of the design parameters as observed over a set number of iterations and adaptively relieves cyclic infeasibilities caused by the constriction. Convergence is also aided by the use of a character string manipulation language to generate machine readable expressions for the exact differentials of all nonlinear functions.The behavior of the convergence forcing technique is demonstrated for a lipped Z shape in flexure and for a two-channel lipped wide flange beam column.