Probabilistic free vibration analysis of beams subjected to axial loads

A stochastic finite-element-based algorithm for the probabilistic free vibration analysis of beams subjected to axial forces is proposed in this paper through combination of the advantages of the response surface method, finite element method and Monte Carlo simulation. Uncertainties in the structural parameters can be taken into account in this algorithm. Three response surface models are proposed. Model I: star experiment design using a quadratic polynomial without cross-terms; Model II: minimum experiment design using a quadratic polynomial with cross-terms; Model III: composite experiment design using a quadratic polynomial with cross-terms.A separate set of finite element data is generated to verify the models. The results show that the Model II is the most promising one in view of its accuracy and efficiency. Probabilistic free vibration analysis of a simply supported beam is performed to investigate the effects of various parameters on the statistical moments of the frequency response of beams. It is found that the geometric properties of beams have significant effects on the variation of frequency response.     

Maximum stiffness and minimum weight optimization of laminated composite beams using continuous fiber angles

This paper deals with identification of optimal fiber orientations and laminate thicknesses in maximum stiffness and minimum weight design of laminated composite beams. The structural response is evaluated using beam finite elements which correctly account for the influence of the fiber orientation and cross section geometry. The resulting finite element matrices are significantly smaller than those obtained using equivalent finite element models. This modeling approach is therefore an attractive alternative in computationally intensive applications at the conceptual design stage where the focus is on the global structural response. An optimization strategy is presented which aims at enabling the use of fiber angles as continuous design variables albeit the problems may have many local minima. A sequence of closely related problems with an increasing number of design variables is treated. The design found for a problem in the sequence is projected to generate the starting point for the next problem in the sequence. Numerical results are presented for cantilever beams with different geometries and load cases. The results indicate that the devised strategy is well suited for finding optimal fiber orientations and laminate thicknesses in the design of slender laminated composite structures.     

基于微悬臂梁的化学传感器的灵敏度研究

基于探测的原理,在动态模式下,首先对不同结构梁的本征频率偏移量进行了有限元分析,结果表明三角形梁的灵敏度最高。然后重点针对不同结构参数的三角形梁进行了动态计算,获得了其频率偏移量随结构参数变化的关系曲线。另外,通过静态模式下的有限元计算,发现各种梁自由端的偏移量相同,表明它们的静态灵敏度与形状无关。最后,对三角形梁在不同测量模式下结构参数的优化设计进行了分析。     

Topology optimization of frame structures with flexible joints

A method for structural topology optimization of frame structures with flexible joints is presented. A typical frame structure is a set of beams and joints assembled to carry an applied load. The problem considered in this paper is to find the stiffest frame for a given mass. By introducing design variables for beams and joints, a mass distribution for optimal structural stiffness can be found. Each beam can have several design variables connected to its cross section. One of these is an area-type design variable which is used to represent the global size of the beam. The other design variables are of length ratio type, controlling the cross section of the beam. Joints are flexible elements connecting the beams in the structure. Each joint has stiffness properties and a mass. A framework for modelling these stiffnesses is presented and design variables for joints are introduced. We prove a theorem which can be interpreted as the fact that the removal of structural elements, e.g. joints or beams, can be modelled by a small strictly positive material amount assigned to the element. This is needed for the computations of sensitivities used in the applied gradient based iterative method. Both two and three dimensional problems, as well as multiple load cases and multiple mass constraints, are treated.     

Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials

In this paper, two different methods for modelling of functionally graded material (FGM) beam with continuous spatially varying material properties will be presented and compared, namely the multilayering method and the direct integration method. Both the methods are related to homogenization of spatially varying material properties of real FGM beam and to calculation of the secondary variables of the FGM beams. The multilayering method is based on the laminate theory, which is very often used by modelling of the multilayer composite beams. The direct integration method transform spatial continuous varying material properties to the effective ones by direct integration of derived homogenization rules. In next part of the paper, new multiphysical beam finite element will be presented, which in conjunction with the proposed homogenization methods can be used for very effective analysis of the FGM beam structures. The numerical experiment will be presented concerning the multiphysical (electro–thermal–structural) analysis of the chosen FGM beams with spatial continuous variation of material properties.     

Weight optimization for flexural reinforced concrete beams with static nonlinear response

The weight optimization of reinforced concrete (RC) beams with material nonlinear response is formulated as a general nonlinear optimization problem. Incremental finite element procedures are used to integrate the structural response analysis and design sensitivity analysis in a consistent manner. In the finite element discretization, the concrete is modelled by plane stress elements and steel reinforcement is modelled by discrete truss elements. The cross-sectional areas of the steel and the thickness of the concrete are chosen as design variables, and design constraints can include the displacement, stress and sizing constraints. The objective function is the weight of the RC beams. The optimal design is performed by using the sequential linear programming algorithm for the changing process of design variables, and the gradient projection method for the calculations of the search direction. Three example problems are considered. The first two are demonstrated to show the stability and accuracy of the approaches by comparing previous results for truss and plane stress elements, separately. The last one is an example of an RC beam. Comparative cost objective functions are presented to prove the validity of the approach.     

Micromechanics-based FEM simulation of fiber-reinforced cementitious composite components

Fiber bridging along cracks is an important mechanism governing the fracture toughness and the pseudo-ductility of fiber-reinforced brittle materials and structures. This paper attempts to predict structural behavior of fiber-reinforced cementitious composite (FRCC) components using the finite-element procedure with micromechanics-based constitutive modeling of the stress-displacement relation along the crack. The tensile stress-displacement relation along a Mode I (opening) crack is established based on fiber pullout curves derived from a micromechanical model. A statistical model is used to account for random fiber distribution. Two-dimensional finite-element simulations of beam behavior are performed with the finite-element package ADINA. Using the discrete crack approach, strain softening truss elements are placed along the crack to simulate the fiber bridging effect. Experiments of beams under four-point bending are performed with specimens containing different fiber volume fractions (up to 1.5%). The numerical results for the load vs deformation behavior of the beams agree well with the experimental results. The FEM procedure for micromechanics-based design and analysis of FRCC components is therefore established. Simulation of component behavior to identify the most cost-effective design can, hence, be carried out.     

Static, dynamic and stability analysis of structures composed of tapered beams

A new numerical method is proposed for the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth. It is a finite element method based on an exact flexural and axial stiffness matrix and approximate consistent mass and geometric stiffness matrices for a linearly tapered beam element with constant width. Use of this method provides the exact solution of the static problem with just one element per member of a structure with linearly tapered beams and excellent approximate solutions of the dynamic and stability problems with very few elements per member of the structure in a computationally very efficient way. Very detailed comparison studies of the proposed method against a number of other known finite element methods with respect to accuracy and computational efficiency for cantilever tapered beams of rectangular and I cross section clearly favor the proposed method. A continuous beam, a gable frame and a portal frame consisting of tapered members are analyzed by the proposed method as well as by other known methods to illustrate the use of the method to structures composed of tapered beams.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

Modelling of continuous steel–concrete composite beams: computational aspects

Steel–concrete composite members are an interesting option for structural designers, but the reliability of design procedures both in the case of gravity and seismic loads is in continuous development. The issue is very complex, since behaviour of continuous composite beams results from local phenomena of interaction such as partial shear connection and bond.Furthermore, composite beams in buildings generally are not characterised by a full continuity due to the beam to column connections; thus the analysis and the detailing of such parts have a key role in the development of suitable design procedures.In the present paper, some computational aspects related to the modelling of composite flexural members are discussed with reference to continuous and semi-continuous structural systems widely used in practice.     

Optimal cross-section and configuration design of elastic-plastic structures subject to dynamic cyclic loading

This paper shows an optimal design problem with continuum variational formulation, applied to nonlinear elasticplastic structures subject to dynamic loading. The total Lagrangian procedure is used to describe the response of the structure. The direct differentiation method is used to obtain the sensitivities of the structural response that are needed to solve the optimization problem. Since unloading and reloading of the structure are allowed, the structural response is path-dependent and an additional step is needed to integrate the constitutive equations. It can be shown, consequently, that design sensitivity analysis is also path-dependent. A finite element method with implicit time integration is used to discretize the state and sensitivity equations.A mathematical programming approach is used for the optimization process. Numerical applications are performed on a 3-D truss structure, where cross-sectional areas and nodal point coordinates are treated as design variables. Optimal designs have been obtained and compared by using two different strategies: a twolevel strategy where the levels are defined according to the type of design variables, cross sectional areas or node coordinates, and optimizing simultaneously with respect to both types of design variables. Comparisons have also been made between optimal designs obtained by considering or not considering the inertial term of the structural equilibrium.     

Optimum design of structures composed of metal and composite materials

This paper describes a general program system in the optimum design of structures. The element library in the program system consists of a bar (BAR), a shear trapezoidal panel (STP), a constant strain triangle (CST), a plate linear isoparametric element (PLIE), and a beam element in a plane (BEAM). The structural design procedure is performed by combining finite element analysis and hybrid approximation technique with dual solutions. It is suitable for structures with various materials (metal, composite, etc.), especially the structures of the aircraft (wing, tail, fuselage, etc.). Five examples show that the computer program system is capable of generalized applicability.     

Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.     

Isogeometric configuration design optimization of shape memory polymer curved beam structures for extremal negative Poisson’s ratio

Using a continuum-based design sensitivity analysis (DSA) method, a configuration design optimization method is developed for curved Kirchhoff beams with shape memory polymers (SMP), from which we systematically synthesize lattice structures achieving target negative Poisson’s ratio. A SMP phenomenological constitutive model for small strains is utilized. A Jaumann strain, based on the geometrically exact beam theory, is additively decomposed into elastic, stored, and thermal parts. Non-homogeneous displacement boundary conditions are employed to impose mechanical loadings. At each equilibrium configuration, an additional nonlinear analysis is performed to calculate the Poisson’s ratio and its design sensitivity of the SMP material. The design objectives are twofold: for purely elastic materials, lattice structures are designed to achieve prescribed Poisson’s ratios under finite compressive deformations. Also, SMP-based lattice structures are synthesized to possess target Poisson’s ratios in specified temperature ranges. The analytical design sensitivity of the Poisson’s ratio is verified through comparison with finite difference sensitivity. Several configuration design optimization examples are demonstrated.     

Topology optimization of frame structures—joint penalty and material selection

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

Sensitivity analysis and optimization of thin laminated structures with a nonsmooth eigenvalue based criterion

This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.     

Algebraic reduction of beams for CAD-integrated analysis

Beams are high aspect ratio structural members that are used extensively in civil, automotive, aerospace, and MEMS applications. In all such applications, one must typically analyze and optimize the beams through computer simulations. Standard 3D finite element analysis (FEA) of beams can be used in such simulations; it is however prone to errors, and is computationally expensive for thin structures. Therefore, a common strategy is to carry out a dimensionally reduced 1D beam analysis. Unfortunately, 1D beam analysis is hard to automate and integrate with 3D CAD.In this paper, we propose an alternate “algebraic reduction” method that combines the generality of 3D FEA, and the computational efficiency of 1D beam analysis. This is achieved via a dual-representation framework where the geometry of the beam is captured via a 3D finite element mesh, while the physics is captured via a 1D beam model. The proposed method is formally established, and supported through numerical experiments.     

Cost optimization of composite beams using genetic algorithms

This paper presents a genetic algorithm model for the cost optimization of composite beams based on the load and resistance factor design (LRFD) specifications of the AISC. The model formulation includes the cost of concrete, steel beam, and shear studs. Two design examples taken from the literature were analyzed in order to validate the proposed model, to illustrate its use, and to demonstrate its capabilities in optimizing composite beam designs. The results obtained show that the model is capable of achieving substantial cost savings. Hence, it can be of practical value to structural designers. A parametric study was also conducted to investigate the effects of beam spans and loadings on the cost optimization of composite beams.