热效应对静不定梁热弯曲的影响1）

建立了静不定梁在温度场中热弯曲的微分方程,推导出了在小挠度变形条件下静不定梁热弯曲的挠曲线表达式.研究结果表明：当温度沿梁高呈线性分布时,梁的温度使静不定梁受到轴向热力作用,梁底与梁顶的温度差使静不定梁发生热弯曲.在小挠度变形条件下：考虑轴向热力的作用时,静不定梁的热弯曲是非线性问题;忽略轴向热力的作用时,静不定梁的热弯曲是线性问题.Timoshenko的名著《材料力学》,在研究两端固支梁热弯曲问题时,得到了“两端固支梁热弯曲挠曲线表达式有时是意想不到的”结论,即两端固支梁热弯曲挠曲线表达式为零的结论.因此在考虑轴向热力对静不定梁热弯曲影响的基础上,研究了静不定梁热弯曲问题,把两端固支梁热弯曲问题与其他静不定梁热弯曲问题进行对比,对两端固支梁热弯曲挠曲线表达式为零的结论进行了理论解释,可知两端固支梁在热状态下的变形是一个弹性稳定问题.     

热效应对静不定梁热弯曲的影响

建立了静不定梁在温度场中热弯曲的微分方程,推导出了在小挠度变形条件下静不定梁热弯曲的挠曲线表达式.研究结果表明:当温度沿梁高呈线性分布时,梁的温度使静不定梁受到轴向热力作用,梁底与梁顶的温度差使静不定梁发生热弯曲.在小挠度变形条件下:考虑轴向热力的作用时,静不定梁的热弯曲是非线性问题;忽略轴向热力的作用时,静不定梁的热弯曲是线性问题.Timoshenko的名著《材料力学》,在研究两端固支梁热弯曲问题时,得到了"两端固支梁热弯曲挠曲线表达式有时是意想不到的"结论,即两端固支梁热弯曲挠曲线表达式为零的结论.因此在考虑轴向热力对静不定梁热弯曲影响的基础上,研究了静不定梁热弯曲问题,把两端固支梁热弯曲问题与其他静不定梁热弯曲问题进行对比,对两端固支梁热弯曲挠曲线表达式为零的结论进行了理论解释,可知两端固支梁在热状态下的变形是一个弹性稳定问题.     

双参数地基圆(环)形板的边界剪力及其力学响应

针对目前考量双参数地基上圆形或环形板边界剪力中存在的一些不足,应用Hankel变换法求得双参数地基的解析解,研究了任意环状线荷载作用下双参数地基的挠度、转角及地基反力间的关系。指出双参数地基在线荷载处的转角差与表示横向联系的地基参数之乘积等于线荷载集度;环内的分布荷载仅影响环内侧转角,而环外分布荷载也仅影响环外侧转角;进而又给出了适用于双参数地基圆形或环形板的内/外边界剪力的简明表达式。最后分别给出了双参数地基上自由边界圆形薄板中点承受集中力、圆形均布荷载作用时边界剪力对板中点挠度、弯矩的影响规律;双参数地基边界剪力对板挠度、弯矩有较明显影响,尤其是板的相对半径ρ_04或是荷载距板边界较近时这种影响更为明显。     

不均匀土中有限长桩的横向瞬态波动

将Timoshenko梁模型引入到弹性地基粱中,土体对桩的作用用连续分布的弹簧来模拟,基于回传射线矩阵法,研究了桩顶受到横向冲击荷载作用下有限长桩的横向瞬态波动．经参数分析表明：土体刚度比对上软下硬、上硬下软两层地基中的有限长桩剪力波和弯矩波影响很大;从挠曲波曲线中难以辨别有无软硬夹层．     

考虑轴向力的双参数弹性地基有限长梁的静力问题

探讨轴向荷载对双参数地基梁弯曲的影响,以最小势能原理为基础,采用变分法推导了双参数地基上承受轴向力的梁的控制微分方程及边界条件,并明确了衰减参数γ需要满足的方程。对地基梁的参数γ进行了迭代,给出了双参数弹性地基上承受轴向力的有限长梁的内力及变形的求解方法。结果表明:轴向力的存在,使得地基梁的跨中挠度、最大弯矩、转角均有所增大;轴向力对地基梁的剪力有所影响,但影响程度并不大。本文计算方法准确可行,为双参数弹性地基模型的推广应用奠定了基础,具有广阔的应用前景。     

基于物理中面与高阶剪切变形理论双参数弹性地基上FGM梁的非线性弯曲分析

在本文中,基于物理中面概念与高阶剪切变形理论,用Ritz法给出双参数弹性地基上FGM梁非线性弯曲的近似解答,并且讨论不同温度场、地基参数、不同边界条件、以及体积分数变化对FGM梁力学行为的影响。值得进一步指出的是：在基准温度场中,Winkler地基FGM梁的挠度介于Pasternak型与无地基梁之间;在热传导场中固支FGM梁的挠度介于均匀热场与基准温度场之间,而简支FGM梁由于有初始热挠度的影响,并非总是如此。     

考虑接缝传荷的文克勒地基上矩形薄板的级数解

应用双重正弦级数展开的方法,推演得到了文克勒地基上计入两块矩形板间接缝传荷效应的级数解。假设接缝传递剪力与板间挠度差成正比、传递弯矩与板间转角差成正比,进而分析了单轮和单轴荷载作用在纵缝边缘中部时,接缝传荷效应对板边最大挠度和最大应力的影响规律;通过引入接缝传荷效应系数和接缝极限传荷的两个挠度比及两个应力比,建立了计入接缝传荷效应的板边最大挠度和最大应力的一般式,总结了不同板尺寸、荷载面积尺寸和类型及板材料泊松比对四个接缝影响系数和四个接缝极限传荷的挠度比及应力比的影响规律。结果表明:不同荷载面积下,受荷板接缝边缘最大挠度、最大应力均随弯矩或剪力传荷刚度系数的增大而减小,且应力的变化幅度相较挠度要小。影响系数fV^w(ξV)、fM^w(ξM)、fM^σ(ξM)与荷载圆相对半径(a/l)、相对板长(L/l)和相对板宽(B/l)无关,且单轮荷载与双轮荷载规律相同;而影响系数fV^σ(ξV)与荷载圆相对半径(a/l)有关,与相对板长(L/l)和相对板宽(B/l)无关;挠度比λV^w与荷载圆相对半径、板尺寸(L/l,B/l)及泊松比v无关,恒等于0.5,而λM^w、λV^σ、λM^σ均与荷载相对半径(a/l)、板尺寸(L/l,B/l)及泊松比v有关,且影响因素中荷载面积尺寸的影响最为显著。     

考虑地基水平摩阻的Winkler地基上板模型解析

在Winkler地基模型基础上,采用板与地基之间的水平摩阻应力正比于板与地基接触点间水平位移假设,推演了包含板弯曲和轴向拉压效应的非线性地基板微分方程;应用摄动法分析竖向集中力作用下,板与地基水平摩阻应力对板内力（弯曲力、轴向力和剪应力）、位移（挠度、水平位移）的影响规律,实证了板与地基水平摩阻应力对板剪应力的非线性影响可以忽略不计,从而得到了轴对称条件下计入地基水平摩阻的Winkler地基上薄板的线性微分方程．随后通过汉克尔变换得到了轴对称无限大板的解析解,以及包含四项复宗量贝塞尔函数的轴对称圆形板的解析解,给出了板内力、位移的计算式;最后,通过算例分析了板与地基间水平摩阻状况对板挠度、截面弯矩的影响规律．结果表明：地基板挠度与弯矩随着板与地基间水平摩阻的增大而减少;当地基板水平摩阻参数大于0.01时,地基水平摩阻力对板挠度和弯矩影响有可能超过2%,应予考虑;无限大板作用圆形均布荷载时,水平摩阻的存在最大可使板最大挠度（即荷载圆中心点处挠度）下降约50%,板最大截面弯矩（即荷载圆中心点处弯矩）下降约70%.     

超静定梁的挠曲线初参数方程

本文建立了超静定梁的挠曲线初参数方程，利用超静定梁的边界条件和支座处的约束条件以及静力学平衡条件定出了方程中的所有未知参数．可通过研究梁的初参数方程，求出整个梁中挠度，转角的最大值．     

弹性地基双层梁理论下的混凝土路面力学分析

为建立混凝土路面结构受力分析计算模型,以Winkler弹性地基梁模型为基础,推导出了弹性地基双层梁理论的表达式;给定边界条件,利用MATLAB软件获得了无限长弹性地基梁在集中力作用下的挠度表达式。将混凝土路面结构简化为弹性地基上的双层梁,当车辆荷载作用于混凝土路面时,在集中载荷的作用下,建立了面层与基层的微分平衡方程。应用广义“初参数”法,得到了双层梁位移和应力的解析解。通过算例,对面层及基层的变形和应力进行了分析,结果表明:增大面层、基层的轴惯性矩和地基的弹性常数,可以有效地减少面层和基层的变形量,降低最大应力数值,但抗弯刚度对基层和面层的弯矩受力影响不大。最后将结果与ANSYS分析结果进行了比较,佐证了解的可靠性,研究结果可为混凝土路面结构设计提供依据。     

变刚度梁挠度曲线的Green函数法求解

在变刚度梁的线弹性问题中,求解梁受静力荷载的挠度曲线常用解法有积分法与单位荷载法.本文从变刚度梁挠度曲线的微分方程出发,给出了变刚度梁挠度曲线的Green函数法解答,并分析了该解法的优点.从推导结果可以看到,本文提出的公式具有统一、精确、简洁、适合电算的特点,在编制杆系结构计算软件中将具有重要应用价值.     

Parametrically Excited Vibration of a Timoshenko Beam on Random Viscoelastic Foundation jected to a Harmonic Moving Load

The vibration response of a Timoshenko beam supported by a viscoelastic foundation with randomly distributed parameters along the beam length and jected to a harmonic moving load, is studied. By means of the first-order two-dimensional regular perturbation method and employing appropriate Green's functions, the dynamic response of the beam consisting of the mean and variance of the deflection and of the bending moment are obtained analytically in integral forms. Results of a field measurement for a test track are utilized to model the uncertainty of the foundation parameters. A frequency analysis is carried out and the effect of the load speed on the response is studied. It is found that the covariance functions of the stiffness and the loss factor both have the shape of an exponential function multiplied by a cosine function. Furthermore, it is shown that in each frequency response there is a peak value for the frequency, which changes inversely with the load speed. It is also found that the peak value of the mean and also standard deviation of the deflection and bending moment can be a decreasing or increasing function of the load speed depending on its frequency. An erratum to this article is available at .     

Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force.     

Response of a finite beam in contact with a tensionless foundation under symmetric and asymmetric loading

In this work a general analytical model is developed for the static response of a beam resting on a tensionless elastic foundation subjected to a lateral point load. This load may either be located at the center of the beam or may be offset. An analytical/numerical solution is obtained to the governing equations; this solution makes no assumption about either the contact area or the kinematics associated with the transverse deflection of the beam. This is in contrast to previous work in which, for an infinite beam (where the load is symmetric by definition), implicit assumptions about the contact area and the response kinematics were made. Because these assumptions are dropped, the contact behavior differs in several fundamental ways from its infinite counterpart. Specifically, it is shown that (i) the contact area is a sensitive function of the beam length and that this function may change nonmonotonically, (ii) the contact area may depend on the magnitude of the load, (iii) asymmetric loads, which cannot exist in the infinite problem, have a dramatic influence the contact area for the finite system. These features are demonstrated with specific examples and explained in terms of the fundamental physics of the system. The implications for these behaviors are also discussed.     

Effects of thermo-magneto-electro nonlinearity characteristics on the stability of functionally graded piezoelectric beam

Due to the increasing interests in using functionally graded piezoelectric materials(FGPMs) in the design of advanced micro-electro-mechanical systems, it is important to understand the stability behaviors of the FGPM beams. In this study, considering the effects of geometrical nonlinearity, temperature, and electricity in the constitutive relations and the effect of the magnetic field on the FGPM beam, the Euler-Bernoulli beam model is adopted, and the nonlinear governing equation of motion is derived via Hamilton's principle. A perturbation method, which can decompose the deflection into static and dynamic components, is utilized to linearize the nonlinear governing equation. Then,a dynamic stability analysis is carried out, and the approximate analytical solutions for the nonlinear frequency and boundary frequencies of the unstable region are obtained.Numerical examples are performed to verify the present analysis. The effects of the static deflection, the static load factor, the temperature change, and the magnetic field flux on the stability behaviors of the FGPM beam are discussed. From the proposed analytical solutions and numerical results, one can easily and clearly find the effects of various controlled parameters, such as geometric and physical properties of the system, on the mechanical behaviors of structures, and the conclusions are very important and useful for the design of micro-devices.     

DEFLECTION OF FOUR TYPES OF CANTILEVER BEAMS WITH VARIABLE CROSS SECTION UNDER LATERAL TRIANGULAR DISTRIBUTED LOADS1)

研究了圆柱、圆锥、抛物型和双曲型回转变截面悬臂梁在侧向三角形分布载荷下的挠度。基于四类回转悬臂梁的惯性矩沿长度方向的分布规律,得到其任意侧向分布载荷下的挠曲线方程。基于三角形分布载荷下的挠曲线方程,得到其端部挠度值。在等长度和等体积假设下,通过比较端部挠度值找到四类悬臂梁中挠度最小者。研究表明三角形分布载荷下,特征参数在特定范围内,母线为双曲线的悬臂梁挠度最小。     

Low-velocity impact response of sandwich beams with functionally graded core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young’s modulus is represented by a polynomial in the thickness coordinate, but the Poisson’s ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel—the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.     

Non-linear studies of orthotropic shallow spherical shells on elastic foundation

Governing non-linear integro-differential equations for cylindrically orthotropic shallow spherical shells resting on linear Winkler-Pasternak elastic foundations, undergoing moderately large deformations are presented. Three problems, namely, non-linear static deflection response, non-linear dynamic deflection response and dynamic snap-through buckling of orthotropic shells have been investigated. The influences of material orthotropy, foundation parameters and shell-material damping on the deflection response are determined for the clamped and the simply- supported immovable edge conditions accurately. Orthotropy, foundation interaction and material damping play significant roles in improving the load carrying capacity of the shell structures.     

ADOMIAN POLYNOMIALS FOR NONLINEAR RESPONSE OF SUPPORTED TIMOSHENKO BEAMS SUBJECTED TO A MOVING HARMONIC LOAD

This paper investigates the steady-state responses of a Timoshenko beam of infinite length supported by a nonlinear viscoelastic Pasternak foundation subjected to a moving harmonic load. The nonlinear viscoelastic foundation is assumed to be a Pasternak foundation with linear-plus-cubic stiffness and viscous damping. Based on Timoshenko beam theory, the nonlinear equations of motion are derived by considering the effects of the shear deformable beams and the shear modulus of foundations at the same time. For the first time, the modified Adomian decomposition method（ADM） is used for solving the response of the beam resting on a nonlinear foundation. By employing the standard ADM and the modified ADM, the nonlinear term is decomposed, respectively. Based on the Green＇s function and the theorem of residues presented,the closed form solutions for those linear iterative equations have been determined via complex Fourier transform. Numerical results indicate that two kinds of ADM predict qualitatively identical tendencies of the dynamic response with variable parameters, but the deflection of beam predicted by the modified ADM is smaller than that by the standard ADM. The influence of the shear modulus of beams and foundation is investigated. The numerical results show that the deflection of Timoshenko beams decrease with an increase of the shear modulus of beams and that of foundations.     

Determination of the microscale stress–strain curve and strain gradient effect from the micro-bend of ultra-thin beams

A simple method is established to determine the microscale uniaxial stress–strain curve from the load and deflection data for a doubly clamped beam. The method is based on the fact that, for beam deflection much larger than the beam thickness, the axial stretching dominates the deformation in the doubly clamped beam and the doubly clamped beam behaves like a simple plastic hinge. The microscale uniaxial stress–strain curve, together with the cantilever beam experiments, is used to determine the strain gradient effect in Au thin beams. The effect of finite rotation is also discussed.