Science Letters:Analytical solution for fixed-end beam subjected to uniform load

INTRODUCTION The plane stress problem of beams is very clas-sical in elasticity theory and is encountered frequently in practical cases. Timoshenko and Goodier (1970) investigated isotropic beams for different cases, such as tension, shearing, pure bending, bending of a can-tilever by transverse load at the end, bending of a simply supported beam by uniform load and other cases of continuously loaded beams. Lekhnitskii (1968) studied the anisotropic beams problem including tension, shear…     

Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to thermal and uniformly dis-tributed load is investigated,with material parameters being arbitrary functions of the thickness coordinate. The heat conduction problem is treated as a 1D problem through the thickness. Based on the elementary formulations for plane stress problem,the stress function is assumed to be in the form of polynomial of the longitudinal coordinate variable,from which the stresses can be derived. The stress function is then determined completely with the compatibility equation and boundary conditions. A practical example is presented to show the application of the method.     

Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads

In this paper, the axisymmetric general solutions of transversely isotropic magnetoelectroelastic media are expressed with four harmonic displacement functions at first. Then, based on the solutions, the analytical three-dimensional solutions are provided for a simply supported magnetoelectroelastic circular plate subjected to uniform loads. Finally, the example of circular plate is presented. Project supported by the National Natural Science Foundation of China (No. 10172075) and Ningbo Natural Science Foundation for Young Scientist (No. 02J20102-13)     

Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads

INTRODUCTIONInelasticity ,theproblemofasimplysup portedcircularplatesubjectedtouniformloadsisaclassicone .Timoshenkoetal.(1970 )pre sentedasolutionforanisotropiccircularplate .Dingetal.(2 0 0 0 ;2 0 0 1)obtainedtheanalyti calthree dimensionalsolutionforat…     

Analytical study on the dynamic displacement response of a curved track subjected to moving loads

A closed-form out-of-plane dynamic displacement response of a curved track subjected to moving loads was pro- posed. The track structure was modeled as a planar curved Timoshenko beam periodically supported by the double-layer spring-damping elements. The general dynamic displacement response induced by the moving loads along the curve on the elastic semi-infinite space was firstly obtained in the frequency domain, according to the Duhamel integral and the dynamic reciprocity theorem. In the case of the periodic curved track structure subjected to moving loads, the dynamic displacement equation was simplified into a form of summation within the basic track cell instead of the integral. The transfer function for the curved track was expressed in the form of a transfer matrix. Single and series moving loads were involved in the calculation program. For the verification of the analytical model, the mid-span vertical deflection of a simply support curved beam subjected to moving load was recalculated and compared with the same case in the reference. The research results indicate that： under the same moving loads, the displacement response of the curved track decreases slightly with the increasing track radius, and the displacement response of the curved track with the radius greater than or equal to 600 m is almost equivalent to the displacement response of the straight track; the frequency spectrum of the curved track is more abundant than that of the straight track, which may result in more wheel-rail resonance and rail corrugation in the curved lines.     

Analytical solution for 1D consolidation of unsaturated soil with mixed boundary condition

Based on consolidation equations proposed for unsaturated soil, an analytical solution for 1D consolidation of an unsaturated single-layer soil with nonhomogeneous mixed boundary condition is developed. The mixed boundary condition can be used for special applications, such as tests occur in laboratory. The analytical solution is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the nonhomogeneous boundary condition is first transformed into a homogeneous boundary condition. Then, the eigenfunction and eigenvalue are derived according to the consolidation equations and the new boundary condition. Finally, using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution for the new boundary condition is obtained. The present method is applicable to various types of boundary conditions. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with mixed boundary condition.     

均布载荷作用下悬臂不可压饱和多孔梁的非线性变形

Nonlinear bending of cantilever incompressible poroelastic beams subjected to a uniform load is investigated with the constraint that fluid flow is only in the axial direction. The governing equations for large deflection of the poroelastic beam are derived from theory of incompressible saturated porous media. Then, nonlinear responses of a cantilever beam with impermeable fixed end and permeable free end are examined with the Galerkin truncation method. The deflections and bending moments of the poroelastie beam and the equivalent couples of the pore fluid pressures are shown in figures. The differences of the results between the large deflection and the small deflection theories are analyzed. It is shown that the results of the large deflection theory are smaller than those of the small deflection theory, and the time needed to approach their stationary states for the large deflection theory is shorter than that for the small deflection theory.     

估测湿样品空气渗透系数的近似解析解

利用文献[1]的自然降压法测量非饱和材料的空气渗透系数,推出一个估测湿样品空气渗透系数的近似解析解.通过解析方法和数值方法验证解析解的理论值和实验数据吻合.若忽略可预计的误差,在给定条件下,运用自然降压法得到的近似解析解可求湿样品的空气渗透系数.     

深埋圆形富水隧道应力与位移的弹塑性解

研究目的：推导获得深埋圆形富水隧道应力场与位移场分布的弹塑性解析解。创新要点：皋于弹性力学厚壁圆筒受均布压力的拉姆解答和Mohr—Coulomb屈服条件,推导了考虑渗流作用和应力释放时含衬砌深埋隧道的弹塑性解,并采用FLAC3D有限差分程序验证了其正确性。研究方法：通过理论分析建立考虑渗流作用和应力释放的含衬砌深埋圆形隧道的解析模型（图1）,然后推导了基于Mohr-Coulomb屈服准则的隧道同岩与衬砌渗流场、位移场和应力场计算的弹塑性解析公式,并通过数值模拟程序（图2）验证了推导公式的正确性。重要结论：基于前人研究皋础,推导获得了深埋富水隧道应力与位移场分布的理论解析解,并通过数值模拟方法验证了其正确性。该解析解一方面可以用于深埋高水头隧道的预设计,如预测隧道周围的水压力、应力场和位移场分布规律,另一方面可用于校核大型复杂数值模型的正确性。     

3D analytical solution for a rotating transversely isotropic annular plate of functionally graded materials

The analytical solution for an annular plate rotating at a constant angular velocity is derived by means of direct displacement method from the elasticity equations for axisymmetric problems of functionally graded transversely isotropic media. The displacement components are assumed as a linear combination of certain explicit functions of the radial coordinate, with seven undetermined coefficients being functions of the axial coordinate z. Seven equations governing these z-dependent functions are derived and solved by a progressive integrating scheme. The present solution can be degenerated into the solution of a rotating isotropic functionally graded annular plate. The solution also can be degenerated into that for transversely isotropic or isotropic homogeneous materials. Finally, a special case is considered and the effect of the material gradient index on the elastic field is illustrated numerically.     

Exact solution of an external circular crack in a piezoelectric solid subjected to shear loading

A three-dimensional, exact analysis is presented in this paper for the problem of an external circular crack in a transversely isotropic piezoelectric medium subjected to arbitrary antisymmetric shear loading. A recently proposed general solution of three-dimensional piezoelectricity is employed. It is shown that four quasi harmonic functions involved in the general solution can be respresented by just one complex potential. Previous results in potential theory are then used to obtain the exact solution of the problem. For point shear loading, Green’s functions for the elastoelectric field are derived in terms of elementary functions. Project supported by Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry and partly by National Natural Science Foundation of China (NSFC) (10002016).     

Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading

INTRODUCTION Conventional consolidation analysis based on Terzaghi’s one-dimensional theory neglects the non-linearity of soil for the sake of mathematical expediency, which is considered to be one of the theory’s shortcomings. Many researchers have pro-posed one-dimensional (1D) consolidation models where the coefficient of volume compressibility mv, coefficient of permeability kv, and coefficient of consolidation cv, vary with either depth or time. Schiffman (1958), Davis and Raymon…     

含海底泉的含水层系统中水头波动的解析解

考察了含有海底泉的半承压含水层系统中,在潮汐效应与越流共同作用下地下水水头的波动.该系统由潜水含水层、承压含水层和介于其间的弱透水层构成.在文献[1]的基础上,综合考虑弱透水层的越流及海潮载荷效率,利用地下水动力学、复变函数及数学物理方程的基本原理和方法,推出了承压含水层水头波动的近似解析解.     

用Microsoft Excel求解线性方程组

本文将计算机运用于数学教学中 ,以具体例子说明用 Excel中内含的规划求解软件环境求出一般线性方程组的一组解。而且给出具体步骤和要点 ,有一定的实用价值。     

加强分析化学实验操作考核提高实验教学质量

本文提出了分析化学实验操作考核的具体内容和方法．强调正确掌握定量分析仪器操作的重要性．全面、客观、合理考核学生实验操作，有利于提高分析化学实验教学质量．     

Dynamic response of a beam on a Pasternak foundation and under a moving load

The dynamic response of an infinite beam placed on a Pasternak foundation when the system was subjected to a moving load was investigated. We used the double Fourier transform and its inversion to solve the formulations of the problem. A closed form analytic solution of the beam was obtained by the theorem of residues. We selected a numerical example to illustrate the dynamic response of the beam on Pasternak and Winkler foundations, respectively. We discuss the effect of the moving load velocity on the dynamic displacement response of the beam. The maximum deflection of the beam increases slightly with increased load velocity but increases significantly with reduced shear modulus of subgrade at a given velocity. The maximum deflection of a beam resting on a Pasternak foundation is much smaller than that of a beam on a Winkler foundation.     

冲击荷载作用下钢框架的动态响应分析

在冲击荷载作用下,钢框架结构的动态响应是一个复杂的非线性过程。运用ANSYS/LS-DYNA建立了钢框架的三维模型,对冲击荷载作用下钢框架的动态响应和破坏形态进行研究。采取多点积分算法,有效避免沙漏问题,保证计算结果精确。以冲击块与钢框架的碰撞冲击为例,研究了冲击速度、冲击块质量、冲击位置等参数对钢框架在横向冲击作用下动力响应的影响,并分析冲击荷载下钢框架的破坏形态。结果表明:冲击块的质量、速度、冲击位置的增加都会不同程度地加剧钢框架的动态响应;在冲击作用下,高应变主要出现在被冲击柱子的冲击区域、柱脚处和梁柱节点处;冲击荷载下钢框架的破坏形态为整个框架发生侧移,受冲击柱在冲击处翼缘出现局部屈曲,受冲击区域的截面发生弯扭,未受冲击柱出现倾斜。     

温度场中斜梁受简谐激励的1/2次亚谐-主参数共振

以温度场中斜梁受简谐激励的非线性振动方程为基础,应用非线性振动的多尺度法,求得非线性振动系统1/2亚谐-主参数共振的一次近似解,并进行定常解稳定性分析和数值计算.分析了温度、激励、几何尺寸对1/2亚谐-主参数共振幅频响应曲线的影响.     

The analytical solutions for orthotropic cantilever beams （Ⅰ）： Subjected to surface forces

This paper first gives the general solution of two-dimensional orthotropic media expressed with two harmonic displacement functions by using the governing equations. Then, based on the general solution in the case of distinct eigenvalues, a series of beam problems, including the problem of cantilever beam under uniform loads, cantilever beam with axial load and bending moment at the free end, cantilever beam under the first, second, third and fourth power ofx tangential loads, is solved by the superposition principle and the trial-and-error methods.     

Analytical layer-element solutions for a multi-layered transversely isotropic elastic medium subjected to axisymmetric loading

This paper presents an analytical layer-element method used to analyze the displacement of a multi-layered transversely isotropic elastic medium of arbitrary depth subjected to axisymmetric loading. Based on the basic constitutive equations and the HU Hai-chang’s solutions for transversely isotropic elastic media, the state vectors of a multi-layered transversely isotropic medium were deduced. From the state vectors, an analytical layer element for a single layer (i.e., a symmetric and exact stiffness matrix) was acquired in the Hankel transformed domain, which not only simplified the calculation but also improved the numerical efficiency and stability due to the absence of positive exponential functions. The global stiffness matrix was obtained by assembling the interrelated layer elements based on the principle of the finite layer method. By solving the algebraic equations of the global stiffness matrix which satisfy the boundary conditions, the solutions for multi-layered transversely isotropic media in the Hankel transformed domain were obtained. The actual solutions of this problem in the physical domain were acquired by inverting the Hankel transform. This paper presents numerical examples to verify the proposed solutions and investigate the influence of the properties of the multi-layered medium on the load-displacement response.