Second order approximation solution of nonlinear large deflection problems of Yongjiang Railway Bridge in Ningbo

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A periodic solution to a nonlinear diffusion equation

An analytic solution to the one dimensional heat diffusion equation is presented where the diffusion coefficient varies as a power of temperature. The discussion is motivated by the transmission of heat through the strongly nonlinear medium of soil. Under boundary conditions representing the daily, or seasonal, sinusoidal fluctuation in temperature it is seen that, despite the nonlinearity, the period of the oscillation is preserved on passage through the medium. The nonlinearity acts to accelerate the heating phase and retard the cooling phase within a period which itself remains stable. These effects are calculable from a second harmonic arising in the analysis.     

A new technique for large deflection analysis of non-prismatic cantilever beams

This paper studies the very large deflection behavior of prismatic and non-prismatic cantilever beams subjected to various types of loadings. The formulation is based on representing the angle of rotation of the beam by a polynomial on the position variable along the deflected beam axis. The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam’s boundary conditions. Several numerical examples are presented covering prismatic and non-prismatic cantilever beams subjected to uniform, non-uniform distributed loads and tip concentrated loadings in vertical and horizontal directions. The loads considered in this study are restricted to the non-follower type loads. Cases with different loadings and geometries are compared with MSC/NASTRAN computer package. However, for some very large deflection case, the MSC/NASTRAN failed to predict the deflected shape due to divergence problems.     

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.     

A solution of the diffusion equation with nonlinear right-hand side

The division of a rectangular pile of subsoil water between two channels with different water levels is considered. Evaporation is allowed for as a function of the depth of the subsoil stream. The problem reduces to solution of a nonlinear integral equation and is solved approximately by a method developed from the method of successive replacement of stationary states. A numerical example is given.     

有摩擦时悬臂细长大杆大挠度弯曲的解

在专用汽车开发中,遇到有磨擦时细长杆受集中载荷时大挠度弯曲变形件的设计问题,本文利用数值积分法对此进行讨论,提出有摩擦时细长杆悬臂梁所受最大弯矩计算方法,同时就磨擦力对变形的影响进行了分析,为此类杆件的强度计算提供依据。     

A new method for finding special solutions of nonlinear diffusion equation

A straightforward and concise method is proposed to construct special solutions of nonlinear diffusion equation. Many new special solutions obtained. We believe this method is effective on solving other nonlinear differential equations.     

环形薄板轴对称非线性屈曲的样条函数解法

环形薄板的大挠度计算因为边界条件复杂,仅有少数特殊情形的数值解答。均布边缘径向力作用下环形薄板非线性屈曲迄今尚未有研究成果。作者以三次B样条函数为试函数,用配点法计算环形薄板的大挠度。在12种不同的边界条件下,首次计算了环形薄板的压曲临界荷载及超临界荷载作用的变形。在所有的算例中均取得了收敛的数值结果。结果表明样条配点法具有收敛范围大、精度高和计算时间少的优点。     

A nonlinear mathematical model for large deflection of incompressible saturated poroelastic beams

Nonlinear governing equations are established for large deflection of incom- pressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams.Then,the nonlinear bend- ing of a saturated poroelastic cantilever beam with fixed end impermeable and free end permeable,subjected to a suddenly applied constant concentrated transverse load at its free end,is examined with the Gaierkin truncation method.The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure are shown in figures.The results of the large deflection and the small deflection theories of the cantilever poroelastic beam are compared,and the differences between them are revealed.It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory,and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.     

A periodic solution of the diffusion equation

Under some circumstances, autooscillations can arise in an electrochemical system with decreasing characteristics [1–3]. A method for finding the polarization curve P=P() (here is the electrode potential, P=i I c (0, t), where i is the current density, and c(0, t) is the mass concentration at the electrode surface), if the distribut ion in time of the current density is given, is proposed in [1]. In the numerical solution of this problem, which is considered below, considerable computational difficulties were encountered.     

悬臂梁大挠度变形的近似估计法

研究了悬臂梁自由端受集中力作用时的大挠度变形问题,对大挠度的界定方法做出了一些讨论,并从计算数据分析和理论推导两方面归纳出一种不通过复杂计算就能对大挠度变形进行定量估计的方法. 分析表明,由挠曲线近似微分方程得出的自由端挠度值与梁长度之比值的平方,可以近似表示小挠度法计算挠度值偏离精确挠度值的误差,并由此得出大挠度变形的估计值. 该方法避免了复杂的微分方程求解和数值计算,有一定的工程实际意义.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

A new approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat plates

In ref. [1], Lin obtained an approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat flates.In this paper, the author shows that one may obtain the result coinciding with that obtained in ref. [1] by making use of the Shvez's method^[2]. Morever, we can obtain a more accurate result than that obtained in ref. [1], if we make use of the improved Shvez's method^[2]. Some calculating examples are presented.     

The complete process of large elastic-plastic deflection of a cantilever

An extension of the Elastica theory is developed to study the large deflection of an elastic-perfectly plastic horizontal cantilever beam subjected to a vertical concentrated force at its tip. The entire process is divided into four stages: I.elastic in the whole cantilever; II.loading and developing of the plastic region; III.unloading in the plastic region; and IV.reverse loading. Solutions for stages I and II are presented in a closed form. A combination of closed-form solution and numerical integration is presented for stage III. Finally, stage IV is qualitatively studied. Computed results are given and compared with those from small-deflection theory and from the Elastica theory.     

Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force

A realistic beam structure often exhibits material and geometrical non-linearity, in particular for those made of metals. The mechanical behaviors of a non-linear functionally graded-material (FGM) cantilever beam subjected to an end force are investigated by using large and small deformation theories. Young's modulus is assumed to be depth-dependent. For an FGM beam of power-law hardening, the location of the neutral axis is determined. The effects of depth-dependent Young's modulus and non-linearity parameter on the deflections and rotations of the FGM beams are analyzed. Our results show that different gradient indexes may change the bending stiffness of the beam so that an FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.     

海洋管线在有摩擦的弹性地基上铺设时的大变形分析

本文在[1]文献的基础上,进一步考虑了地基的弹性变形和摩擦的影响,分析和计算了海洋管线在铺设工况下的大变形.文中采用奇异摄动技术和幂级数解相结合,并针对几种不同的地基模型,提出了含二参数和四参数两类非线性方程组的迭代格式,成功地求得了问题的解.     

A new approach to on-line vibration monitoring and fault diagnosis

This paper presents a finite element method by deriving the stiffness matrix of cracked element from the stress intensity factor. The method can be used to evaluate the strain response of the cantilever beam with a single edge crack. By using the method of identification in time domain, the modal parameters are identified and at the same time the effect of crack closure is taken into consideration. Analysis has also been made of the sensitivity of the strain to the faults. Test results show that the strain response is more sensitive to faults than the displacement response. Then, a distinguishing function is introduced, which is more effective than any other existing ones in failure detection. Finally a new approach to on-line vibration monitoring and fault diagnosis is given.