Analytical solution for functionally graded anisotropic cantilever beam subjected to linearly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated.The analysis is based on the exact elasticity equations for the plane stress problem.The stress function is introduced and assumed in the form of a polynomial of the longitudinal coordinate.The expressions for stress components are then educed from the stress function by simple differentiation. The stress function is determined from the compatibility equation as well as the bound- ary conditions by a skilful deduction.The analytical solution is compared with FEM calculation,indicating a good agreement.     

The transient fracture behavior for a functionally graded layered structure subjected to an in-plane impact load

The transient fracture behavior of a functionally graded layered structure subjected to an in-plane impact load is investigated. The studied structure is composed of two homogeneous layers and a functionally graded interlayer with a crack perpendicular to the boundaries. The impact load is applied on the face of the crack. Fourier transform and Laplace transform methods are used to formulate the present problem in terms of a singular integral equation in Laplace transform domain. Considering variations of parameters such as the nonhomogeneity constant, the thickness ratio and the crack length, the dynamic stress intensity factors (DSIFs) in time domain are studied and some meaningful conclusions are obtained.The project supported by the National Science Foundation for Excellent Young Investigators (10325208), the National Natural Science Foundation of China (10432030) and the China Postdoctoral Science Foundation (2004036018)The English text was polished by Ron Marshall.     

Functionally graded piezoelectric cantilever beam under load

Summary In the present paper, the problem of a functionally graded piezoelectric cantilever beam subjected to different loadings is studied. The piezoelectric beam is characterized by continuously graded properties for one elastic parameter and the material density. A pair of stress and induction functions in the form of polynomials is proposed and determined. Based on these functions, a set of analytical solutions for the beam subjected to different loadings is obtained. As particular cases, series of solutions for some canonical problems can be directly obtained from the solutions of the present paper, such as for the problems of a piezoelectric cantilever beam with constant body force or without body forces, etc.This research work is supported by the National Natural Science Foundation of China (50272003). Support was also given by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.     

TiNi相变悬臂梁的横向冲击特性实验研究

利用改装的霍普金森压杆装置对TiNi形状记忆合金圆截面悬臂梁进行了横向冲击实验研究,并与45#钢悬臂梁的实验结果进行了对比,目的是探索相变对结构动态响应的影响。结果表明:在同样冲击条件下,TiNi悬臂梁的吸能效率优于钢悬臂梁;发现冲击过程中,TiNi试件根部内侧可能形成局部相变铰,使阻力曲线的斜率发生变化;卸载后相变铰消失,TiNi悬臂梁试件基本回复原状,钢试件则留下显著的残余变形。TiNi悬臂梁的冲击特性受热弹性马氏体相变和逆相变的支配,不同于传统的弹塑性机制。     

Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

This study applies two analytical approaches, Laplace transform and normal mode methods, to investigate the dynamic transient response of a cantilever Timoshenko beam subjected to impact forces. Explicit solutions for the normal mode method and the Laplace transform method are presented. The Durbin method is used to perform the Laplace inverse transformation, and numerical results based on these two approaches are compared. The comparison indicates that the normal mode method is more efficient than the Laplace transform method in the transient response analysis of a cantilever Timoshenko beam, whereas the Laplace transform method is more appropriate than the normal mode method when analyzing the complicated multi-span Timoshenko beam. Furthermore, a three-dimensional finite element cantilever beam model is implemented. The results are compared with the transient responses for displacement, normal stress, shear stress, and the resonant frequencies of a Timoshenko beam and Bernoulli–Euler beam theories. The transient displacement response for a cantilever beam can be appropriately evaluated using the Timoshenko beam theory if the slender ratio is greater than 10 or using the Bernoulli–Euler beam theory if the slender ratio is greater than 100. Moreover, the resonant frequency of a cantilever beam can be accurately determined by the Timoshenko beam theory if the slender ratio is greater than 100 or by the Bernoulli–Euler beam theory if the slender ratio is greater than 400.     

Fracture behavior of brittle circular rings subjected to concentrated impulsive loading

The influence of impact velocity and ring geometry in the fracture patterns produced by in-plane concentrated impulsive loading on brittle circular rings has been studied both experimentally and theoretically. The experiments were performed by impacting circular rings made of plaster with a steel plate. The fracture behavior was photographed by a camera with a flash, and the crack-initiation time was measured using a memoriscope. The fracture behavior is explained using dynamic photoelasticity and the theory of flexural motion of a circular ring. With the addition of an impact-fracture criterion to this theory, the fracture patterns of brittle circular rings are predicted.     

On the parametric stability of a Timoshenko beam subjected to a periodic axial load

Summary The dynamic buckling of a hinged bar under harmonic axial load is studied using Timoshenko's beam theory and considering the effects of longitudinal vibrations. Several new types of parametric resonances were found. The influences of the more accurate Timoshenko beam theory on the instability regions found with the Bernoulli-Euler theory is also discussed.

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Übersicht Die dynamische Stabilität eines gelenkig gelagerten Balkens mit harmonischer Axiallast wird mit Hilfe der Timoshenkoschen Balkentheorie und unter Berücksichtigung der Längsschwingungen untersucht. Verschiedene neue Arten von parametrischen Instabilitäten konnten gefunden werden. Der Einfluß der genaueren Timoshenkoschen Balkentheorie auf die sich aus der Bernoulli-Eulerschen Theorie ergebenden Instabilitätsbereiche wird diskutiert.

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Visiting Professors at COPPE, Universidade Federal do Rio de Janeiro, Brazil.     

Direct method for analysis of flexible cantilever beam subjected to two follower forces

The static analysis of the flexible non-uniform cantilever beams under a tip-concentrated and intermediate follower forces is considered. The angles of inclination of the concentrated forces with respect to the deformed axis of the beam remain unchanged during deformation. The governing non-linear boundary-value problem is reduced to an initial-value problem by change of variables. The resulting problem can be solved without iterations. It is shown that there are no critical loads in the Euler sense (divergence) for any flexural-stiffness distribution and angles of inclination of the follower forces. In particular, if the follower forces are tangential, the rectilinear shape of the non-uniform cantilever beam is the only possible equilibrium configuration. In this paper some equilibrium configurations of the uniform cantilever under normal or tangential follower forces are presented using direct method.     

Nonlinear static behavior of a cantilever subjected to an inclined follower force

The nonlinear static behavior of a linearly elastic cantilever subjected to a nonconservative force of the follower type is formulated and examined. The formulation allows for finite rotations with small strains (the elastica). Exact solutions are found. The investigation is greatly facilitated by means of a phase plane analysis in which the phase plane variables are related to slope angle and bending moment. Some of the interesting and unusual effects occurring in this system are discussed and illustrated with a set of deflection curves for a typical case.Nomenclature x, y coordinates of a point on the deformed elastic axis - slope angle - s arc length - L length of beam (assumed constant) - x _L , y _L , _L values of x, y, at s=L - P applied force - constant angle between P and end tangent - angle between P and the horizontal - EI beam stiffness (assumed constant) - u, dimensionless variables defined by (7) and (8) - c ² load parameter defined by (10) - k, transformation parameters defined by (21) - F(, k), E(, k) elliptic integrals of the first and second kind - argument of elliptic integrals - ₀, ₁ values of at u=0 and u=1 - m, n positive integers - N mode number     

阶跃载荷作用下弹塑性悬臂梁的变形机制与能量耗散

基于大挠度动力控制方程,应用有限差分离散求解,研究了阶跃载荷作用下弹塑性悬臂梁的动力行为。通过对动力响应早期内力、变形以及能量分布规律的分析,考察了悬臂梁的弹塑性响应模式和变形机制,并与已有的刚塑性分析进行了系统的比较。数值计算表明,阶跃载荷的不同幅值使得梁的响应模式存在较大差异,弹塑性分析肯定了刚塑性理论在处理中载情形的准确性,同时也指出了其在处理低载和高载情形时的缺陷。通过与小变形理论计算结果的比较,指出了考虑大变形效应的必要性,为今后的大变形刚塑性动力分析提出了建议。     

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.     

Fracture behavior of brittle plates and cylindrical shells subjected to concentrated impulse loading

The influence of impact velocity and geometry in the fracture patterns produced by a concentrated impulse loading on brittle plates and cylindrical shells has been studied both experimentally and theoretically. The experiments were performed by impacting plates and cylindrical shells made of plaster with a steel ball. The fracture behavior was photographed by a camera with a flash. The crack-initiation time was measured using a memoriscope. The fracture behavior is explained using the theory of flexural motion of a plate and a cylindrical shell. With the addition of impact-fracture criteria to these theories, the fracture patterns of brittle plates and cylindrical shells are predicted and the resemblance is discussed.     

On possible causes of brittle fracture

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 137–141, May–June, 1988.     

Deflection of a partially supported elastic beam subjected to non uniform load

Summary The bending of a partially supported, nonhomogeneous elastic beam is studied by means of variational inequalities. Properties of the contact set are obtained along with the continuous dependence of the displacement on the load. Eventually, for a wide class of external forces we give an explicit formula for the contact set and prove further properties for the solution.

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Sommario E' qui studiata mediante disequazioni variazionali la flessione di una trave elastica non omogenea in parte appoggiata. Vengono ottenute proprietà dell'insieme di contatto e della soluzione (spostamento dalla configurazione di riferimento), quali ad esempio la dipendenza continua dal carico e, per un'ampia classe di forze esterne, una formula esplicita per la zona di contatto e la concavità della configurazione di equilibrio.

     

Large deflection of a circular cantilever beam with uniformly distributed load

Summary In this paper, the bending of thin circular cantilever beam, convex downward, under a uniformly distributed load is discussedby the use of the Bernoulli-Euler equation. The solutions are obtained in the form of power series. Numerical results are also presented.The author wishes to thank Dr. S. Higuchi, Professor Emeritus of the Thoku University, for his suggestion and encouragement in the work. The author also wishes to thank Professor F. Numachi and Professor O. Tamate for their kind advice.     

On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load

The problem of a uniform cantilever beam under a tip-concentrated load, which rotates in relation with the tip-rotation of the beam, is studied in this paper. The formulation of the problem results in non-linear ordinary differential equations amenable to numerical integration. A relation is obtained for the applied tip-concentrated load in terms of the tip-angle of the beam. When the tip-concentrated load acts always normal to the undeformed axis of the beam (the rotation parameter, β=0) there is a possibility of obtaining non-unique solution for the applied load. This phenomenon is also observed for other rotation parameters less than unity. When the tip-concentrated load is acting normal to the deformed axis of the beam (β=1), many load parameters are obtained for a tip-angle with different deformed configurations of the beam. However, each load parameter corresponds to a tip-angle, which confirms the uniqueness on the solution of non-linear differential equations.     

Nonlinear dynamic modeling and periodic vibration of a cantilever beam subjected to axial movement of basement

Dynamic modeling of a cantilever beam under an axial movement of its basement is presented. The dynamic equation of motion for the cantilever beam is established by using Kane's equation first and then simplified through the Rayleigh-Ritz method. Compared with the older modeling method, which linearizes the generalized inertia forces and the generalized active forces, the present modeling takes the coupled cubic nonlinearities of geometrical and inertial types into consideration. The method of multiple scales is used to directly solve the nonlinear differential equations and to derive the nonlinear modulation equation for the principal parametric resonance. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling is clarified through the comparisons of its coefficients with those experimentally verified in previous studies. Project supported by the Fundamental Fund of National Defense of China (No. 10172005).