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Timoshenko梁通过假设截面的剪切刚度和附加平均剪切转角变形的方式来近似修正初等梁中未考虑剪切变形能的问题,这与梁剪应力沿梁高变化的实际不符。本文基于材料力学剪应力计算式和相应的剪切变形理论,从剪切变形与梁的位移关系入手,导出矩形梁考虑剪切变形时的纵向位移沿梁高方向的函数关系式,证明该位移可分解为纯弯曲引起的位移和剪力引起的剪力滞翘曲位移之和。应用剪力滞广义坐标与广义力的概念,基于能量变分原理得到等截面梁剪力滞控制微分方程组及其通解形式。对均布荷载作用下矩形简支梁的算例分析表明,本文算法与弹性力学精确解对比,两者的应力和挠度剪力滞系数求解结果非常接近,本文算法有足够的精度,且比弹性力学简单。 相似文献
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本以一种新的思路和做法对变截面曲杆进行单元分析,直接把变截面曲杆作为单元,采用弹性中心法导出了单元刚度矩阵通用公式,确定了截面按余弦规律变化时,轴线分别为圆弧形、抛物线形、半椭圆形平面曲杆的单元刚度矩阵精确式,供结构设计参考使用。 相似文献
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George C. Tsiatas 《Archive of Applied Mechanics (Ingenieur Archiv)》2014,84(5):615-623
In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature. 相似文献
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变截面梁横向振动固有频率数值计算 总被引:1,自引:0,他引:1
根据边界条件对变截面梁横向振动四阶变系数微分方程降阶, 形成关于挠度和弯矩的二
阶非显式递推变系数微分方程组; 利用有限差分法, 研究了变截面简支梁横向振动固有频率
的数值计算方法及其精度. 理论分析和正交计算的算例表明: 数值计算算法简单, 计算精度
取决于计算步长的数目和梁横截面竖向渐变率, 与梁宽和梁长无关; 对于给定的计算步长或
数目, 可以估算数值计算的精度; 对于给定的精度要求, 可以确定合理的计算步长或数目. 相似文献
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The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions
is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without
axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature
so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable
cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse
displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential
equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory
inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial
compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies.
At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations
of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical
technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies
of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several
tables and figures and are compared with the results of the analytical solution where a very good agreement is observed. 相似文献
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对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法。文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解。引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系。当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数。求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解。对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达。当取Boltzmann粘弹模型时,进行不同边值问题的分析。分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同。本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析。此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解。 相似文献
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为了改进变截面连续箱梁桥的扭转分析理论,将截面总扭转角分解为自由翘曲扭转角和约束剪切扭转角,选取自由翘曲转角扭率作为广义位移,提出一个2节点8自由度的扭转梁段单元。从约束扭转控制微分方程出发,推导单元刚度矩阵及等效节点荷载列阵。引入应力增大系数,以反映约束扭转对初等梁应力的增大效应。数值算例验证了本文梁段单元的可靠性。最后对一个三跨变截面连续箱梁桥进行分析,结果表明,双力矩影响线与弯矩影响线较为类似,按双力矩影响线进行最不利荷载加载时最大应力值偏小;应力增大系数在集中荷载作用截面出现极值,均发生在腹板与顶板交点处;利用偏载放大系数来考虑扭转附加效应时,不宜考虑弯曲正应力较小及翘曲正应力出现极值的梁段区域。 相似文献
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基于结构力学中的矩阵位移法,提出了一种利用节点位移参数来反推拱肋区段刚度的计算方法。算例验证该方法在只利用部分节点位移参数时依然具有较高的精度。然后考虑不同的加载模式和损伤模式,采用该方法对借助非线性有限元分析技术和试验手段获得的既有钢筋混凝土拱肋加载过程中的节点位移值进行了分析,证实了既有拱肋的区段刚度具有较大幅度的波动变化特性。基于拱肋区段的变化规律,研究了既有拱肋破坏过程中节点竖向刚度值的变化趋势。结果表明,既有拱肋的节点竖向刚度变化规律与区段刚度较为一致。在此研究基础上,提出了一种适用于既有钢筋混凝土拱肋的承载能力评估方法。实例分析表明,该评估方法具有较高的精度。 相似文献
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运用变截面梁单元有限元法和MATLAB高级编程语言对全波纹腹板H型钢在轴心压力下的屈曲性能及其参数对临界荷载的影响进行了分析,并且与等截面H型钢进行对比,用实例说明了全波纹腹板H 型钢的优越性,为这种H型钢在工程中的应用提供了理论依据. 相似文献
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《International Journal of Solids and Structures》2005,42(1):239-253
A beam theory for the stability analysis of short beam that includes shear deformation and warping of the cross-section is developed. The warping of the cross-section is taken to be an independent kinematics quantity and corresponding force resultants are defined. For the beam subjected to the external loading only at the ends of the beam, equilibrium equations have been obtained by the principle of virtual work. The variations of lateral displacement, rotational angle of the cross-section and the multiplier of the warping shape along the beam axis are solved in closed form and expressed in terms of deformation quantities at the ends of the beam. Based on this beam theory, the lateral stiffness of the beam sustained an axial compression force and a lateral shear force at one end is explicitly derived, from which the equation of the buckling load is established and the buckling load can be solved. When the effect of cross-section warping is neglected, the derived lateral stiffness and buckling load converge to the solutions of the Haringx theory. 相似文献
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Afsin Saritas 《Mechanics Research Communications》2011,38(2):126-130
In this paper, general closest point projection algorithm is derived for the elastoplastic behavior of a cross-section of a beam finite element. For given section deformations, the section forces (stress resultants) and the section tangent stiffness matrix are obtained as the response for the cross-section. Backward Euler time integration rule is used for the solution of the nonlinear evolution equations. The solution yields the general closest projection algorithm for stress resultants plasticity model. Algorithmic consistent tangent stiffness matrix for the section is derived. Numerical verification of the algorithms in a mixed formulation beam finite element proves the accuracy and robustness of the approach in simulating nonlinear behavior. 相似文献
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A new class of beam finite elements is proposed in a three-dimensional fully parameterized absolute nodal coordinate formulation, in which the distortion of the beam cross section can be characterized. The linear, second-order, third-order, and fourth-order models of beam cross section are proposed based on the Pascal triangle polynomials. It is shown that Poisson locking can be eliminated with the proposed higher-order beam models, and the warping displacement of a square beam is well described in the fourth-order beam model. The accuracy of the proposed beam elements and the influence of cross-section distortion on structure deformation and dynamics are examined through several numerical examples. We find that the proposed higher-order models can capture more accurately the structure deformation such as cross-section distortion including warping, compared to the existing beam models in the absolute nodal coordinate formulation. 相似文献