Composite beam–columns with interlayer slip—Approximate analysis

An approximate second order analysis procedure for composite beam–columns with interlayer slip subjected to transverse loading and axial compressive loads is developed. The magnification factors to be applied to the first order solutions in order to estimate the deflections and internal forces obtained by the second order analysis approach are presented. The method of applying magnification factors to internal axial forces is discussed. The approximate second order analysis procedure is developed for the four Euler cases with various transverse load conditions. The procedure is applied to and the accuracy is illustrated for simply supported partially beam–columns of steel and concrete, and timber and concrete with different bending stiffness and interlayer slip properties. The deflections and internal forces obtained by the approximate method compared extremely well, except for slip forces in case of very flexible shear connectors, with those obtained by the more rigorous second order analysis approach for different composite action (partial interaction) parameters (shear connector stiffness values). The study also shows that the magnification factor associated with the deflections can be utilized to estimate also the internal actions, except shear forces in case of very flexible shear connectors, in the second order case with minimal error for simply supported beam–columns. Thus, for members with shear connector stiffness of structural significance the proposed approximate method can be used in general for simply supported beam–columns. For other boundary and loading conditions, the approximate method needs to be re-evaluated. The approach of using one magnification factor greatly simplifies the analysis task for those components.     

Exact dynamic analysis of composite beams with partial interaction

The partial differential equations and general solutions for the deflection and internal actions and the pertaining consistent boundary conditions are presented for composite Euler-Bernoulli members with interlayer slip subjected to general dynamic loading. Both free and forced vibrations are treated. The solutions are shown to be unique and complete under certain conditions, and valid for all so-called restricted admissible boundary conditions. Specifically, the exact eigenmode length coefficients are derived for the four Euler BC. They differ from those valid for ordinary, fully composite (solid) beams, except for the pinned-pinned case. The maximum deviation for beams with the other three Euler BC is shown to be less than 2-6% with respect to the eigenmode length coefficient and 3-10% with respect to the eigenfrequency, respectively, depending on the two non-dimensional parameters, composite action or shear connector stiffness and relative bending stiffness parameters. However, these deviations occur in a rather narrow range of the determining parameters, so for most practical cases the eigenmode length coefficients given for solid (fully composite) beams can approximately be used also for partially composite beams. The procedures of analysing beam vibrations are applied to a specific case. These solutions illustrate the effect of interlayer connection on the peak velocity of the beam vibrations. The proposed analytical theory is verified by tests and finite element calculations.     

A simplified analysis method for composite beams with interlayer slip

A simplified static procedure is proposed for analysing and designing composite beams with interlayer slip. The method is parallel to the Eurocode 5 method, but it is general in nature and can be applied to arbitrary boundary and loading conditions. In contrast with Eurocode 5, a general and correct way of choosing the effective beam length of the problem is given by the present procedure, which is that the effective beam length equals the buckling length that is found in the corresponding column buckling problem. The procedure predicts the deflections and internal actions and stresses, in principle by replacing the fully composite bending stiffness (EI_∞) with the effective (partially) composite bending stiffness (EI_eff) in the expressions for these quantities in the corresponding fully composite beam. This effective bending stiffness depends on two non-dimensional parameters: the composite action parameter (shear connection stiffness) and the relative bending stiffness parameter. The method is applied to a number of simple practical cases and the results obtained have been compared with the exact values. The applicability of the simplified analysis procedure was found to be very good, except for interlayer shear stresses. The error in the Eurocode 5 procedure, as compared with the method proposed in this paper, can in some cases be up to almost 30% depending on the boundary conditions.     

Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading

In this paper, the function for describing the distribution of flexural stiffness K(x) of a non-uniform column is arbitrary, and the distribution of axial distributed loading N(x) acting on the column is expressed as a function of K(x) and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of variable transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 14 cases. The analytical buckling solutions of one-step non-uniform columns are thus found. Then the obtained analytical solutions are used to derive the eigenvalue equation for buckling of a multi-step non-uniform column for several boundary supports by using the transfer matrix method. A numerical example shows that the proposed procedure is an efficient method for buckling analysis of multi-step non-uniform columns.     

Nonconservative dynamic axial–torsional buckling of structural frames using power series

Axial deformation is not involved in the formulation of linear buckling caused by axial force. Likewise, twisting is not present in linear buckling caused by axial torque. The dynamic axial–torsional buckling of structural frames in the presence of follower axial force will be solved by means of dynamic stiffness using power series. Variationally consistent natural boundary conditions are given so that the resulting dynamic stiffness is symmetrical for conservative loading. Some parts of the boundary forces disappeared for follower axial forces due to consistent tangency to the neutral axis. The deficiency of the power series method to deal with non-uniform sections is highlighted. New instability phenomena for a simple column are studied in detail. It is shown that columns can buckle under direct follower tension. Follower tension decreases the natural frequency initially and then increases it rapidly after a turning point. The first pair of modes about the major axis and that about the minor axis of a rectangular section column meet at one crossing point. A very small axial torque will change the crossing into flutter-like tongues. These tongues are common in compressive follower force. These tongues caused by axial torque are reported here for the first time.     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

Inelastic buckling of two-layer composite columns with non-linear interface compliance

A mathematical model for a slip-buckling problem has been proposed and its exact solution has been found for the analysis of materially inelastic two-layer composite columns with non-linear interface compliance. The mathematical model has been carried out to evaluate exact critical buckling loads. It has been demonstrated mathematically exactly, that exact critical buckling loads are influenced by the initial stiffness, and hence on linear portion of the interface force-slip relationship. Besides, it has been shown that material inelasticity can reduce the critical buckling loads significantly and that the interlayer stiffness has an important effect on the transition between the elastic and inelastic buckling.     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

轴向预压缩双晶片动力学模型与特性分析

基于欧拉梁的假设,将双晶片简化为3层压电层合梁,由Hamilton原理建立系统的动力学模型,得到描述轴向预压缩压电双晶片的偏微分方程及特定边界条件,通过求解上述微分方程的边值问题得到双晶片在不同轴向力下的输出特性的解析表达式。利用有限元软件对双晶片进行仿真,验证了理论分析的正确性,增加理论分析结果的可信度。最后通过实验的方法研究双晶片的静、动力学特性,证明了施加轴向力可以降低双晶片的弯曲刚度和固有频率,显著增加其机电耦合效率,提高力和位移输出能力。轴向力虽然会影响双晶片启动时的位移峰值,但是对响应时间和带宽影响较小,大轴向力下双晶片仍具有带宽高与响应快的优势。     

Parametric studies on buckling loads and critical speeds of microdrill bits

Transverse bending vibrations of the spinning microdrill bit subjected to a compressive axial load are developed based on the Timoshenko beam theory. The system equations of motion are discretized into the form of time-dependent ordinary differential equations by the finite element method. Two types of eigenvalue problems are formulated and utilized to study the effects of the drill helix angle, flute length and diameter on the buckling load and critical speed of microdrill bits with different supported ends. Equivalent formulae similar to those of untwisted Euler beams are established to predict critical buckling loads and critical speeds for microdrills and provide results with sufficient accuracy. The effect of rotational speed on the buckling load, and the influence of thrust force on critical speed are also investigated. A Galef-type equation associated with critical speed, thrust force and buckling load is formulated.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.     

Horizontal cylinder-in-cylinder buckling under compression and torsion: Review and application to composite drill pipe

Available analytical results and experiments on the structural behavior of constrained horizontal cylinders subjected to axial compression, torsion, and gravitational loads are reviewed. Such configurations are of interest to the oil-drilling field and provide static design expressions for steel tubulars. The buckling problem is similar to restrained railroad tracks and submerged/underwater pipelines under thermal expansion. Due to outer cylinder constraint and gravitational loads, analysis has shown that long cylinders initiate buckling at loads significantly higher than classical Euler buckling loads. For these constrained long cylinders, buckling initiates in a sinusoidal mode that snakes along the lower surface of the constraining cylinder. Classic analytical expressions hold that as the axial load increases, the cylinder achieves an overall helically buckled state in which the buckled cylinder forms a helix spiraling around the inner surface of the constraining cylinder. Torsion is shown to have little effect on either buckling load but controls the sense/direction of the helical buckling. Little experimental data exist on constrained cylinder buckling, and it is unclear how the initiating sinusoidal mode transitions to the helical mode. Implications of the buckling progression for composite cylinder applications are described including the finding that composites perform poorly relative to steel on the metric of buckling due to lower density and axial stiffness; composites perform well on the metric of lock-up length when friction is considered. Based on this review and findings for composite cylinders, recommendations are made for further work.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

基于Mathcad求弹性梁弯曲问题的数值解

提出一种求解受复杂载荷作用的弹性梁弯曲问题数值解的新方法。该方法利用奇异函数表示弹性梁的弯曲内力和弯曲变形方程,与梁的边界条件和连续条件联立,应用数学软件Mathcad中相应的求解块和绘图工具,可以快速获得弹性梁的支座约束力、任意截面的剪力和弯矩、挠度和转角等参数以及相应的曲线图。该方法无需编程,既适用于静定或超静定梁,也适用于等截面或变截面梁,为梁的强度和刚度校核以及优化设计等提供了新途径。     

Zigzag theory for buckling of hybrid piezoelectric beams under electromechanical loads

A new efficient coupled one-dimensional (1D) geometrically nonlinear zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. The potential field is approximated layerwise as piecewise linear. The deflection is approximated to account for the normal strain due to electric field. The axial displacement is approximated as a combination of a global third-order variation and layerwise linear variation. It is expressed in terms of three primary displacement variables and a set of electric potential variables by enforcing exactly the conditions of zero transverse shear stress at the top and bottom and the conditions of its continuity at the layer interfaces. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of simply supported beams under electromechanical loads are presented. Comparisons with the exact 2D piezoelasticity solution establish that the present zigzag theory is very accurate for buckling analysis.     

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.     

复合材料中厚层合板在轴压位移和剪切作用下的稳定性研究

采用线性屈曲分析方法和弧长法对单向压缩和剪切两种载荷作用下复合材料中厚板的稳定性问题进行研究,给出两种载荷作用下的一阶失稳模态以及失稳平衡路径,并得出在单向压缩载荷作用下,过分叉点后载荷的微小增加将引起板的横向大挠度变形,而不能继续承受更大的载荷,而板在剪切载荷的作用下不发生失稳的结论。     

大挠度后屈曲倾斜梁结构的非线性力学特性

基于弹性梁的几何非线性大挠度屈曲理论,建立两端固定对称倾斜支撑梁结构的大挠度后屈曲控制微分方程,采用几何非线性隐式变形协调关系来表达强非线性超静定边值问题,得到描述倾斜梁大挠度后屈曲行为的精确解析解.采用数值方法求解含有第一、二类椭圆积分的强非线性微分方程,给出不同倾角梁结构从初始屈曲到后屈曲并发生两态跳转过程中的位形曲线及非线性刚度.根据最小能量原理和挠曲线拐点个数,分析对称屈曲模态与非对称屈曲模态之间相互跳转的内在联系及其对结构非线性刚度突变的影响,得到了屈曲模态之间的转换条件.跳转过程的数值仿真表明,倾斜支撑梁结构发生大挠度后屈曲时具有明显的双稳态特性且只出现低阶(1、2阶)屈曲模态,仿真计算结果与试验结果相一致.     

The buckling and post-buckling behaviour of composite plates under biaxial loading

The response to applied load of general composite plates exhibits a coupling between the bending and extensional modes of deformation which may be significant when shear or compressive loads are applied in-plane. The additional deformation modes may affect the nature of the buckling behaviour, reduce the buckling load or change the post-buckled stiffness. This paper considers the stiffness immediately after buckling of a rectangular panel of angle-ply type in which coupling effects occur and which undergoes bifurcational buckling when biaxial load is applied in directions parallel to the panel edges. Expressions are derived for the buckling loads and for the in-plane stiffness of the panel immediately after the instant of buckling, it being found that the coupling terms affect the stiffness at buckling mainly via the associated change in buckling mode shape.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.