一种新的集成非线性杆件单元刚度矩阵的方法

对于非线性杆件单元,本文提出一种新的简便有效的集成单元刚度矩阵的算法。该方法直接从结构力学中的位移法的概念出发,通过解析积分或数值积分求解积分算子,由积分算子线性组合,能快速求解考虑弯、剪、扭、轴压等各种非线形刚度的杆件单元的刚度矩阵。该方法具有广泛的普适性,能适用于所有多项式、插值多项式、解析式、离散点描述的变刚度、变截面直杆的单元刚度矩阵集成计算。文中通过求解线性直杆单元刚度矩阵验证了该方法的正确性。     

ACCURATE CALCULATION OF ELASTIC BUCKLING LOADS FOR SPACE FRAMES BUILT UP OF UNIFORM BEAMS WITH OPEN THIN-WALLED CROSS-SECTION

A so-called exact static stiffness matrix for a uniform beam element with open thin-walled cross-section carrying an axial compressive load is derived. This stiffness matrix is useful in an accurate calculation of bifurcation loads and corresponding buckling modes of space frames built up of such beam elements. One may also calculate displacements and sectional forces caused by external joint loads taking into account the second-order effect of the axial beam loads. The exact stiffness matrix is derived by use of the general solution to a set of three coupled differential equations. This means that no preselected shape functions need be introduced and that discretization errors are avoided. The differential equations model coupled Euler–Bernoulli bending in the two principal planes and Saint-Venant/Vlasov torsion and warping with respect to the shear centre axis. No cross-sectional symmetries are assumed. Numerical examples are given. One application will be to loaded pallet racks. The ‘effective length’ for a rack column is calculated.     

联肢剪力墙的刚度、稳定性以及二阶效应

该文采用连续化模型,对双肢剪力墙结构平面内稳定性进行了研究,求得了顶部作用集中压力时临界荷载的精确显式表达式和显式屈曲波形。这个临界荷载公式表明,联肢剪力墙是一种双重抗侧力结构,并且可以采用串并联电路模型来表示两者之间的相互作用。串并联模型推广到线性分析的情况,得到顶部抗侧刚度的显式表达式,与精确解进行了比较。推导了顶部作用竖向集中荷载时,在不同水平荷载作用下结构的侧移、墙肢弯矩、墙肢轴力和连梁弯矩放大系数,并提供了近似计算公式。     

Free vibrations of shear‐flexible and compressible arches by FEM

The purpose of this paper is to analyse free vibrations of arches with influence of shear and axial forces taken into account. Arches with various depth of cross‐section and various types of supports are considered. In the calculations, the curved finite element elaborated by the authors is adopted. It is the plane two‐node, six‐degree‐of‐freedom arch element with constant curvature. Its application to the static analysis yields the exact results, coinciding with the analytical ones. This feature results from the use of the exact shape functions in derivation of the element stiffness matrix. In the free vibration analysis the consistent mass matrix is used. It is obtained on the base of the same functions. Their coefficients contain the influences of shear flexibility and compressibility of the arch. The numerical results are compared with the results obtained for the simple diagonal mass matrix representing the lumped mass model. The natural frequencies are also compared with the ones for the continuous arches for which the analytically determined frequencies are known. The advantage of the paper is a thorough analysis of selected examples, where the influences of shear forces, axial forces as well as the rotary and tangential inertia on the natural frequencies are examined. Copyright © 2001 John Wiley & Sons, Ltd.     

Effects of geometrical and loading parameters on wave propagation in a circular ring

A theoretical investigation into the transient elastic response of a circular ring under radial impact is undertaken. The effects of ring curvature, cross-section geometry and impact duration on the resulting shear force, axial force and bending moment distributions and ring deformation are examined. This extends an earlier effort (Shim, Quah, J. Appl. Mech. 65 (1998) 569–579) which employed classical Timoshenko beam theory, coupled with the method of characteristics, to analyse propagation of flexural waves in rings. The present study shows that curvature and ring cross-sectional geometry do not significantly affect the development of bending moments and shear forces but influence axial forces induced and the deformed shape. It is found that for a given impulse, propagation of generalised forces and the deformation geometry depend on impact duration. Wave speeds relating to different deformation modes are governed primarily by curvature and cross-sectional geometry.     

Exact solutions of axial vibration problems of elastic bars

While exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to obtain exact solutions for free vibration and harmonic analyses for non‐trivial cases. This paper presents a study on new finite element formulation and algorithms for exact solutions of undamped axial vibration problems of elastic bars. Appropriate shape functions are constructed by using the homogeneous governing equations, and based on the new shape functions, a novel element is formulated. An iterative procedure is proposed for determining both the exact natural frequency values and the corresponding vibration mode shapes. Exact solutions can also be obtained for undamped harmonic response analyses by using the new element, as its stiffness and mass matrices are exact for a specified frequency. Illustrative examples are presented to demonstrate the effectiveness of the proposed element and algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

An inelastic Timoshenko beam element with axial–shear–flexural interaction

An enhanced beam element is proposed for the nonlinear dynamic analysis of skeletal structures. The formulation extends the displacement based elastic Timoshenko beam element. Shear-locking effects are eliminated using exact shape functions. A variant of the Bouc–Wen model is implemented to incorporate plasticity due to combined axial, shear and bending deformation components. Interaction is introduced through the implementation of yield functions, expressed in the stress resultant space. Three additional hysteretic degrees of freedom are introduced to account for the hysteretic part of the deformation components. Numerical results are presented that demonstrate the advantages of the proposed element in simulating cyclic phenomena, in which shear deformations are significant.     

AN EFFICIENT CURVED BEAM FINITE ELEMENT

The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio. © 1997 John Wiley & Sons, Ltd.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

不同截面梁构件的刚度和稳定性优化设计

本文运用有限元分析与优化设计软件JIFEX，对五种常用截面梁结构的尺寸和形状进行了抗剪、抗弯、抗扭的刚度优化设计和在轴力、剪力作用下的结构稳定性优化设计。通过对优化设计的计算结果分析，得到了对工程设计有意义的若干结论。然后通过飞机结构中一种波形梁构件的优化，进一步讨论了波形梁的波数对结构稳定性和刚度的影响。     

On the accuracy of p-version elements for the Reissner–Mindlin plate problem

This paper addresses the question of accuracy of p-version finite element formulations for Reissner–Mindlin plate problems. Three model problems, a circular arc, a rhombic plate and a geometrically complex structure are investigated. Whereas displacements and bending moments turn out to be very accurate without any post-processing even for very coarse meshes, the quality of shear forces computed from constitutive equations is poor. It is shown that significantly improved results can be obtained, if shear forces are computed from equilibrium equations instead. A consistent computation of second derivatives of the shape functions is derived. © 1998 John Wiley & Sons, Ltd.     

Analysis of dowelled timber to timber moment-resisting joints

Dowelled joints, widely used in timber structures, are designed to transfer shear forces and bending moments between timber members. The anisotropic non-linear behaviour of the timber beneath the fasteners controls the stiffness of these joints. At the ultimate load-carrying capacity, the failure modes result from the shear stresses induced by the load distribution among the fasteners. The paper presents the experimental results obtained for beam to column joint with or without reinforcement using glued plywood plates. Based on these results, a two-dimensional finite element model was developed in two stages to describe the three-dimensional behaviour of the joint. At the single fastener scale, the model considers the non-linearity induced by the timber embedding and the fastener bending. At the structural scale, the modelling approach considers the timber as an elastic orthotropic material whereas each fastener is modelled by two non-linear springs. The elastic-plastic behaviour of each spring element is defined by the local scale model defined in two perpendicular directions. The load distribution among the fasteners is compared to the analytical results according to design rules. Considering the global load displacement curves, the results show that the modelling approach provides a good estimation of the structural response.     

SPATIAL STABILITY ANALYSIS OF THIN-WALLED SPACE FRAMES

A clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption. The improved displacement field for unsymmetric thin-walled cross-sections is introduced based on inclusion of second-order terms of finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. In the present formulation, displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis, and all bending-torsional coupling effects due to unsymmetric cross-sections are taken into account. For finite element analysis, cubic Hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of Hermitian space frame element. Also, load correction stiffness matrices for off-axis point loadings are derived based on the second-order rotation terms. Finite element solutions for the spatial buckling analysis of thin-walled space frames are compared with available solutions and other researcher's results.     

An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics

An equivalent single-layer model for the dynamic analysis of magnetoelectroelastic laminated plates is presented. The electric and magnetic fields are assumed to be quasi-static and the first-order shear deformation theory is used. The formulation of the model provides for a preliminary fulfillment of the electro-magnetic governing equations, which allows to determine the electric and magnetic potential as functions of the mechanical variables. Then, by using this result, the equations of motion are written leading to the problem governing equations. They involve the same terms of the elastic dynamic problem weighted by effective stiffness coefficients, which take the magneto-electro-mechanical couplings into account. Additional terms, exclusively arising in force of the piezoelectric and piezomagnetic behavior, appear. The electromagnetic inputs are treated as equivalent external distributed axial forces and bending moments. Free and forced vibrations solutions for simply-supported plates are presented to validate the model by comparing the present results with benchmark solutions found in the literature.     

A boundary element method applied to the elastostatic bending problem of beam-stiffened plates

The present paper proposes a basic formulation for the static bending problem of beam-stiffened elastic plates. This problem has been so far analyzed using the Timoshenko theory in which the equivalent shear force and bending moments are assumed to act on the beam stiffener. Since fourth-order derivatives of unknown displacements are included in the formulation, in its numerical implementation fourth-order polynomials must be used as the interpolation functions.

In this paper, the interactive forces and moments between the plate and the stiffener are treated as line distributed unknown loads. In the numerical implementation of the formulation, these forces can be approximated using a suitable family of interpolation functions. The formulation is presented in detail and a computer code is developed. The numerical results obtained by the computer code are discussed, whereby the usefulness of the proposed solution procedure is demonstrated.     

液化场地中Timoshenko桩自由振动与屈曲的精确数值解

基于单桩的Timoshenko梁模型和桩-土相互作用的Winkler模型,建立考虑轴力效应的具有分布参数的Timoshenko梁模型微分控制方程,确定对应的齐次方程的通解,并以此作为有限单元的基函数。推导得精确形函数矩阵,建立分布参数Timoshenko梁的精确有限单元,根据拉格朗日方程得到有限元离散方程和单元刚度矩阵、几何刚度矩阵和一致质量矩阵。应用建立的精确Timoshenko梁单元于分层液化土中单桩-土-结构系统的自由振动与屈曲模态分析,通过与对应解析解以及常规有限元解的对比,表明精确Timoshenko桩基础单元的可靠性与较常规有限元法的优势。     

PFVIBAT—a computer program for plane frame vibration analysis by an exact method

The described program PFVIBAT uses the exact displacement method to perform free and forced vibration analysis entirely within the differential equation theory of beams thus avoiding assumed modes and lumped masses. The frame may contain rigid bodies. Clamped, hinged, guided and rolling connections are allowed for. Consideration of rotatory inertia, shear deformation and second-order bending moments and shear forces as caused by static axial load is optional. Eigenfrequencies and modal masses are calculated with an accuracy that may be specified. Displacement and moment modes are plotted. Transient vibrations are studied.     

The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes

The statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non-circular helicoidal axis and cross-section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first-order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non-circular axis and arbitrary cross-section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran-77, which has been applied to various example problems available in the relevant literature, and the results have been compared.     

Stress analysis of flat plates with shear using explicit stiffness matrix

An explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation. The element has twelve nodes on the sides and four nodes internal to it. The formulation is displacement type and the use of area co-ordinates makes it possible to obtain the shape functions explicitly. Separate polynomials are assumed for transverse displacement and rotations. To obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary. The element, which is initially of thirty five degrees of freedom, can be reduced to a thirty degrees of freedom one by condensation of the internal nodes. An interesting feature of the element developed is that the values of nodal moments computed at a node point, considering different elements surrounding the node, do not vary significantly. Thus the nodal moments can be obtained directly at node points. Also, the element does not give rise to any inconvenience like locking, even for very thin plates. The straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less CPU time, thus making the use of the element well suited to low capacity computers. A number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results. Good accuracy has been observed in all cases, even for a small number of elements.     

A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.