Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

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.     

Optimal design of shells against buckling subjected to combined loadings

In this paper, the problem of optimal design of shells against instability is considered. A thin-walled shell is loaded, in general, by overall bending moment, constant or varying along an axis of a shell, by the appropriate shearing force and by an axial force and a constant torsional moment. We look for the shape of middle surface as well as the thickness of a shell, which ensures the maximum critical value of the loading parameter. The volume of material and the capacity of a shell are considered as equality constraints. The concept of a shell of uniform stability is applied.     

受轴向激励弹性支承梁的稳定性分析

对于广泛存在的弹性支撑梁,首次呈现支承弹簧刚度对轴向激励下梁横向振动稳定性的影响.应用Hamilton原理,建立了两端由线性弹簧支撑的受轴向激励梁的动力学控制方程.通过解析方法计算了受轴向压力梁的固有频率,得到了支撑弹簧刚度与系统固有频率和临界轴力的关系.Galerkin截断后,通过多尺度法和Runge-Kutta法,计算得到了梁参激振动稳态响应的半解析与数值解.讨论了激励幅值、支撑弹簧刚度、平均轴力对系统非线性响应幅值及软硬特性的影响.利用Routh-Hurwitz稳定性判据,求得系统的参激稳定边界,着重讨论了支撑弹簧刚度、阻尼系数的影响.研究发现,边界支撑弹簧的刚度可以显著改变受轴向激励梁的参激稳定边界.因此,研究结果将为广泛存在受到轴向激励结构的设计提供指导.     

Optimization of geometry described by curves

In CAD, the shape of a design is often described by an interpolating curve. In this paper, an automated method for finding the optimal geometry of such a curve is used to study the blade geometry of vertical axis wind turbines. To minimize bending moments due to centrifugal loads during rotation the blades are often curved in the shape of the troposkien (Greek for ‘turning rope’). From a fabrication point of view a simpler shape is preferred. An optimal number of interpolating points along the curve are used as design variables and their movements controlled by an optimization algorithm. During the optimization, the shape of the curve is successively recomputed and current objective and constraint functions are analysed. Stresses along the beam are computed using the finite element method by regarding the curve as a plane elastic beam.     

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations.

An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.     

变截面铁木辛柯梁振动特性快速计算方法

提出了一种快速计算变截面铁木辛柯梁横向振动特性的方法.基于铁木辛柯梁理论建立的变截面梁的横向振动方程,其梁的截面参数如有效剪切面积、密度、弯曲刚度、转动惯量等沿梁轴线连续或非连续变化；首先将变截面梁等效为多段均匀阶梯梁；然后基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)连续条件,建立相邻两段模态函数间相互关系,并递推出首段段与末段模态函数相互关系,利用边界条件得到相应特征方程,使用Newton-Raphson方法计算其固有频率；最后针对梁常见边界条件,得到计算变截面铁木辛柯梁横向振动固有频率特征     

The interaction of plane frames with elastic foundation having normal and shear moduli of subgrade reactions

The interaction of plane frames with an elastic foundation, of the Winkler type, having normal and shear moduli of subgrade reactions was studied.

An exact stiffness matrix for a beam element on an elastic foundation having only a normal modulus of subgrade reaction was modified to include the shear modulus of subgrade reaction of the foundation as well as the axial force in the beam. A computer program was written and used in two case studies. In the first case study the convergence of the modified element was tested: in the second it was employed to study the effect of the foundation normal and shear moduli of subgrade reactions on the bending moments in a selected plane frame. The results indicated that bending moments might be considerably affected according to the type of frame and loading.     

Modelling of double chord rectangular hollow section T-joints by finite element method

A linear elastic T-joint comprised of double chord RHS has been modelled by treating the mated flanges as thin plates supported by coupled linear springs thus simulating the action of the side walls and connecting bottom flanges. A rigid rectangular inclusion is presumed for the branch member. Two loading conditions are analyzed—branch member axial force and branch member bending moment. The finite element formulation that is proposed incorporates rectangular plate and edge boundary spring elements. The model is then used to determine the punching shear and rotational stiffnesses of both double chord T-joints and single chord T-joints, thus demonstrating its versatility. The numerical values obtained are in good agreement with the experimental results available in the literature.     

Bending of beams of arbitrary cross sections - optimal design by analytical formulae

The paper deals with the conceptual design of a beam under bending. The common problem of designing a beam in a state of pure bending is discussed in the framework of Pareto-optimality theory. The analytical formulation of the Pareto-optimal set is derived by using a procedure based on the reformulation of the Fritz John Pareto-optimality conditions. The shape of the cross section of the beam is defined by a number of design variables pertaining to the optimization process by means of efficiency factors. Such efficiency factors are able to describe the bending properties of any beam cross section and can be used to derive analytical formulae. Design performance is determined by the combination of cross section shape, material and process. Simple expressions for the Pareto-optimal set of a beam of arbitrary cross section shape under bending are derived. This expression can be used at the very early stage of the design to choose a possible cross section shape and material for the beam among optimal solutions.     

The finite element solution of a class of elastica problems

A finite element procedure for the analysis of an inextensional elastica bent through frictionless supports is presented. Element displacements are expressed in terms of cubic Hermite polynomials with nodal displacements and derivatives being determined to minimise the strain energy. It is shown that the axial force at any point is proportional to the square of the bending moment, which enables member incremental stiffness matrices to be expressed in a similar form to those used in stability analysis. The iterative procedure proposed for solving the nonlinear stiffness equations may readily be incorporated into any continuous beam program by the modification and addition of very few statements. An example is considered which indicates that accurate solutions may be obtained to a wide range of practical problems using the proposed technique.     

A general iterative numerical approach to the finite deflection analysis of beams

The nonlinear bending analysis of a beam with immovable ends and of any edge conditions (hinged or clamped) is studied here. In addition to the nonlinearity arising from the stretching of the neutral axis, the exact expression for curvature is also considered. The governing equation is formulated using the inplane and transverse displacements of the neutral axis of the beam. A simple, numerically exact iterative method is used to solve the resulting equations for evaluating the deflections and stresses for various loadings. Numerical results are compared with available ones for both immovable and movable end conditions. It is observed that the hardening effect due to axial force predominates over that resulting from the use of an actual nonlinear expression for curvature, which leads to the conclusion that it is enough to consider only the nonlinearity arising from the axial force in the nonlinear analysis of beams with immovable ends.     

Dynamic finite element analysis of laminated beams with delamination cracks using contact-impact conditions

A new finite element modeling technique is presented to investigate the static and dynamic behavior of laminated composite beams with partial delamination. In this study, a recently developed rectangular beam element is used. The element has lateral and axial displacements as degrees of freedom, but not rotation. For simplicity, linear shape functions are used for the beam element. As a result, the element has six degrees of freedom, four of which are the axial nodal displacements at the corner points and the other two are the lateral displacments at the ends. In addition, contact-impact conditions are applied to the finite element modeling to avoid overlapping of the upper and lower portions of a delaminated section. The numerical study shows that, depending on existence of an embedded delamination crack and its size, the response is different for a beam with a crack and subjected to a short impulse load. Hence, the present modeling technique may be used for detection of an embedded delamination crack.     

Simultaneous optimization of photostrictive actuator locations,numbers and light intensities for structural shape control using hierarchical genetic algorithm

The present paper introduces an investigation into simultaneous optimization of the PbLaZrTi-based actuator configuration and corresponding applied light intensity for morphing beam structural shapes. A finite element formulation for multiphysics analysis of coupled opto-electro-thermo-mechanical fields in PbLaZrTi ceramics is derived and verified with the theoretical solution and the commercial software ANSYS. This element is then used to simulate beam bending shape control using the orthotropic PbLaZrTi actuators and the simultaneous optimization. In this procedure, the controlling and geometrical variables are simultaneously optimized via a hierarchical genetic algorithm. A bi-coded chromosome is proposed in a hierarchical mode, which consists of some control genes (i.e. actuator location and number) and parametric genes (i.e. applied light intensity). Whether the parametric gene is activated or not is managed by the value of the first-grade control genes. The numerical results demonstrate that the achieved beam bending shapes correlate remarkably well with the expected ones and the simultaneous optimization of photostrictive actuator locations, numbers and light intensities can result in optimal actuator layout with less PbLaZrTi actuators and irradiated light energy. The simulation results also show that the hierarchical genetic algorithm has more superior performance over the conventional real-coded genetic algorithm.     

Arch shape optimization using force approximation methods

The objective of this paper is to provide a method of optimizing areas of the members as well as the shape of both two-hinged and fixed arches. The design process includes satisfaction of combined stress constraints under the assumption that the arch ribs can be approximated by a finite number of straight members.In order to reduce the number of detailed finite element analyses, the Force Approximization Method is used. A finite element analysis of the initial structure is performed and the gradients of the member end forces (axial, bending moment) are calculated with respect to the areas and nodal coordinates. The gradients are used to form an approximate structural analysis based on first order Taylor series expansions of the member end forces. Using move limits, a numerical optimizer minimizes the volume of the arch with information from the approximate structural analysis.Numerical examples are presented to demonstrate the efficiency and reliablity of the proposed method for shape optimization. It is shown that the number of finite element analysis is minimal and the procedure provides a highly efficient method of arch shape optimization.     

不同脱层位置下脱层屈曲梁振动实验研究

通过实验对一端固定一端夹支脱层屈曲梁在轴向周期激励作用下的非线性动力响应进行了实验研究.利用位移时间历程图,相图和频谱图,对多组不同脱层位置下脱层屈曲梁的非线性动力响应进行了分析.实验表明脱层梁结构存在倍周期以及混沌运动等非线性动力学行为.同时实验还表明,在相同的脱层长度下,脱层位置对脱层梁的动力学特性有明显影响,即脱层区域中心越靠近梁结构的中心位置,脱层梁的一阶自然频率越低,且越容易在较低的激励频率和激励荷载下发生周期分叉和混沌等行为.     

Blocked force measurement of electro-active paper actuator by micro-balance

This paper presents a blocked force measurement of electro-active paper (EAPap) with a simple configuration and high accuracy. Blocked force is a force at the tip of a bending actuator when the tip is fixed. A micro-balance with a fixture is used to measure the blocked force of EAPap bending actuator. The measurement is conducted with dc electric fields. A theoretical blocked force is calculated by using the cantilever beam model, and the measured blocked force is compared with the calculated result. Comparing the measured blocked force with the calculated result shows a good agreement. This force measurement method is simple and accurate, and can be applied for measuring blocked forces of EAP actuators.     

On plotting of section force components diagrams for framed structures

A procedure for the automatic choice of the side of drawing of the section force components of two- and three-dimensional framed structures is presented in this paper. These components are the shear and axial forces and the bending and twisting moments and their signs are usually defined taking into account the local reference system of the structural member. In structural analysis, a beam or structural element may be divided into two or more members or elements with different local reference systems and discontinuities with no physical meaning in the diagrams of those components may be obtained if only these systems are considered for the sign convention. For each element it is necessary unambiguously to choose the sign and the side of drawing of each one of these force components. The literature does not present a general procedure for this choice and the solution used in the SALT system is presented here. It takes into account the position of each local reference system in relation to the global reference system and solves the problem.     

Working equations for piezoelectric actuators and sensors

A new solution to the force, displacements, and charges developed in piezoelectric beams is derived. Differing from previous solutions, this development determines the neutral axis where the bending strains are zero and results in a closed form solution (without matrix inversion). With the closed form, simplifications become evident which increase understanding and facilitate calculations. These equations are than expanded to account for axial, built-in strains in the beam. A design example where axial forces exerted by the piezoelectric layer are important is presented