Damage induced instability in beam bending

The effect of internal damage creation on the load carrying capacity of a beam is studied. Time independent relations are postulated for the development of strain and damage with increasing net stress. The resulting relations between load and deformation for a Bernoulli-Navier beam lamina with rectangular cross-section are then derived. A state of instability is shown to exist, characterized by unlimited rate of increase in deformation and damage with load. An instability locus in the plane of bending moment and normal force is defined. The shape of this locus is studied for varying parameters in the deformation and damage laws.     

Large deflections of a beam subject to three-point bending

In this paper, a solution for the equilibrium configuration of an elastic beam subject to three-point bending is given in terms of Jacobi elliptical functions. General equations are derived, and the domain of the solution is established. Several examples that illustrate a use of the solution are discussed. The obtained numerical results are compared with the results of other authors. An approximation formula by which the beam load is given as a polynomial function of beam deflection is also derived. The range of applicability of the approximation is illustrated by numerical examples.     

Thrust and slip line analysis on a swamp shoe

This paper attempts to develop an analytical method which can evaluate the thrust and the slip line exerted by continuos swamp shoes. In this analysis, all the slip lines in the ground and the adhesion between a swamp shoe and the ground are considered. An equation was found through the analysis and calculated values of the thrust and the slip line are examined through experiments with a swamp shoe.     

In-plane bending of a beam resting on a rigid rough foundation

Summary The bending of a finite-length beam that lies on a rigid, rough, flat foundation and interacts with it in accordance to the dry friction law is considered. Loading by bending moments applied at the ends of the beam is studied in detail. The problem is found to be a self-similar one. For small moments, the central part of the beam remains undeflected, and the problem reduces to the solution of an infinite system of algebraic equations. Large moments deflect the entire length of the beam, and the problem partly loses its self-similarity. In this case, the problem reduces to the solution of a successively decreasing number of ordinary differential equations along with some algebraical equations. The solution for the latter case provides initial conditions for the former one. This permits to obtain a solution for any value of the moment. Received 5 November 1996; accepted for publication 27 January 1997     

Free rigid/plastic plane bending of a slender beam

Summary Large rigid/plastic plane bending of a slender structural element as e.g., a beam or a sheet metal strip is examined, in view of its applications to metal forming. Loads, i.e. forces and moments, are admitted only at the ends of the element; the loads or the corresponding displacements and rotations may be prescribed. Using the normal force, the transversal force and the bending moment as generalized stresses acting at the cross sections the differential equations of the problem are set up for an arbitrary, strain-hardening and/or rate-sensitive material. As an example a homogeneous, ideally plastic beam is considered to be plastified by means of the bending moment only. It is shown that it can be brought into any shape provided the end conditions are adequately controlled. Circular bending as a special case becomes possible in two different ways i.e., by pure bending (without end forces) or by localized bending (generated by a moving yield hinge).

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Ebenes, freies biegen von schlanken, starrplastischen trägern

Übersicht Es wird die große, ebene, starrplastische Biegung schlanker Träger, also beispielsweise von Balken oder Blechen, im Hinblick auf Anwendungen in der Umformtechnik von Metallen untersucht. Lediglich an den Enden der Träger greifen Lasten (Kräfte, Momente) an; diese Lasten oder die entsprechenden Verschiebungen und Neigungen kann man vorgeben. Mit der Normalkraft, der Querkraft und dem Biegemoment als generalisierte Spannungen in den Querschnitten werden die Differentialgleichungen des Problems für ein beliebig verfestigendes und/oder geschwindigkeitsabhängiges Material formuliert und speziell auf einen homogenen, idealplastischen Träger angewendet, dessen Plastifizierung nur vom Biegemoment abhängt. Es wird gezeigt, daß man ihn in jede beliebige Gestalt bringen kann, vorausgesetzt, die Bedingungen an den Enden werden angemessen gesteuert. Eine kreisförmige Biegung erreicht man zum Beispiel auf zwei verschiedenen Wegen: Durch reine Biegung (ohne Kräfte an den Enden) oder durch lokalisierte Biegung infolge eines wandernden Fließgelenkes).

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Herrn Prof. Dr. Dr. h.c. H.-P. Stüwe, Montan-Universität Leoben, zum 60. Geburtstag am 14. Sept. 1990 gewidmet.     

On the shear and bending of a degrading polymer beam

In this paper we develop a model, within a general framework that has been developed to describe the response of dissipative systems, for the strain induced degradation of polymeric solids, due to scission. The theory can be generalized to include degradation due to ultraviolet radiation, oxygen diffusion etc., by incorporating an appropriate form for the rate of dissipation associated with these processes. We study the simple shear and pure bending of such degrading polymer beams.     

Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.     

An engineering theory for beam bending

Summary A beam bending theory is proposed that is similar to the Timoshenko and Reissner theories but uses different kinematic variables. Under regular end conditions the theory is shown to predict stresses having a relative mean square error proportional to the depth cubed compared with plane stress elasticity solutions. For a cantilever beam the error from irregular end constraints is found to be smaller in the present theory than in those of Timoshenko and Reissner.

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Eine technische Theorie der Balkenbiegung

übersicht Es wird eine den Theorien von Timoshenko und Reissner analoge Balkentheorie vorgeschlagen, die aber andersartige kinematische Variable enthÄlt. Unter der Voraussetzung regulÄren Randbedingungen ergeben sich aus dieser Theorie Spannungen, deren relativer mittlerer quadratischer Fehler im Vergleich zu Lösungen ebener ElastizitÄtstheorie proportional zur 3. Potenz der Tragerhöhe ist. Für Kragbalken wird der Fehler von nichtregulÄren Randbedingungen in der dargestellten Theorie kleiner als in den Theorien von Timoshenko und Reissner.

     

Optimal design of a beam subject to bending: a basic application

The minimisation of both the mass and deflection of a beam in bending is addressed in the paper. To solve the minimisation problem, a multi-objective approach is adopted by imposing the Fritz John conditions for Pareto-optimality. Constraints on the maximum stress and elastic stability (buckling) of the structure are taken into account. Additional constraints are set on the beam cross section dimensions. Three different cross sections of the beam are analysed and compared, namely the hollow square, the I-shaped and the hollow rectangular cross sections. The analytical expressions of the Pareto-optimal sets are derived. As expected, the I-shaped beam exhibits the best compromise in structural performance, which is related on the particular loading considered.     

Shear and bending response of a rigid-plastic beam to blast-type loading

Summary An analysis is presented of the dynamic behavior and permanent deformation of a rigid-perfectly plastic beam subjected to blast-type loading. The beam is simply supported at the ends, and the load is uniformly distributed over the span. The anaylsis is based on an approximate yield curve relating limiting values of shear force and bending moment. Various patterns of deformation are found to occur with combination of plastic bending and shear sliding depending upon the load magnitude and beam characteristics. The effects of shear and of the shape of the load pulse are examined, and the paper is concluded with the results expressed in general terms for application to similar cases.

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Übersicht Das dynamische Verhalten und die Dauer-Verformung eines ideal starr-plastischen Balkens unter dem Einfluß von Stoßbelastungen wird untersucht. Die Belastung ist gleichmäßig über die Länge des an den Enden einfach gelagerten Balkens verteilt. Den Untersuchungen wird eine genäherte Fließkurve zugrunde gelegt, bei der die Grenzwerte für Schubkraft und Biegemoment in Beziehung gesetzt werden. Abhängig von der Belastungsgröße und von den Balkenparametern werden verschiedenartige Deformationsmuster gefunden, bei denen plastisches Biegen und Schubgleiten kombiniert auftreten. Der Einfluß der Form des Belastungsimpulses und des Schubes wird untersucht. Abschließend werden die Ergebnisse in allgemeiner Form ausgedrückt um sie so für ähnliche Fälle anwendbar zu machen.

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The essential part of the material presented in this paper was prepared while the author was engaged in graduate study at Brown University, under the general guidance of Prof. P. S. Symonds, to whom the author wishes to express his gratitude.     

Transverse vibrations of a single-span beam with the motion of a bending moment

Conclusions Analysis of the dynamic action of a moving bending moment on a single-span beam-type system showed that, with v=(0.2–0.8v ₁ ⁰ , taking account of the inertial forces of the load does not enter into the margin of strength of the construction, and these forces must be taken into consideration in dynamic calculations. The greatest deflections of the beam, when the mass of the load M=0.5ml, exceed the static deformations, taking account of the inertial forces of the load, by 2.5 times. The value of the velocity here is v=0.6v ₁ ⁰ .The maximal coefficient of the dynamics, calculated without taking account of the weight of the load, is equal to 1.95 and occurs with v=0.8v ₁ ⁰ . We note that, with the motion of a vertical force along the beam, the maximal value of the dynamic coefficient is equal to 1.77 and is observed with v=0.6v ₁ ⁰ [3].If v<0.6v ₁ ⁰ , where the mass of the load is not introduced into the calculations, and v<0.4v ₁ ⁰ , where account is taken of the inertial forces of the load, then the maximal deformations of the beam take place during the process of forced vibrations at the moment that the load is located in the construction. With large values of v, the greatest deflections are observed after passage of the load, during the period of free vibrations of the system.In distinction from the solution of the problem of the vibrations of a beam under the action of a moving force (load), where a sufficient degree of exactness of the computations assures taking account of the first form of the vibrations of the construction, with an analysis of dynamic deformations of a beam, brought about by the action of a moving bending moment, the higher forms of the vibrations of the system must be taken into consideration.Leningrad Institute of Railroad Engineers. Translated from Prikladnaya Mekhanika, Vol. 14, No. 1, pp. 111–115, January, 1978.     

Free and forced nonlinear oscillations of a two-layer composite beam with interface slip

Free and forced flexural nonlinear vibrations of a two-layer beam are investigated. Each beam is assumed to have Euler?CBernoulli kinematics and free-free boundary conditions. The interface allows only nonlinear elastic slip between adjacent sides of the beams, so that the transversal displacement is unique. Free vibrations are considered first by the multiple time scale method, which allows to determine the amplitude dependent nonlinear natural frequencies of the system. It is shown that the nonlinear coefficient of the backbone curve is positive, so that hardening/softening behavior of the interface generates hardening/softening behavior of the whole structure. The modifications of the linear normal modes for moderate excitation amplitudes have been computed. Forced and damped nonlinear oscillations are then considered by the same mathematical method, and the nonlinear frequency response curves are obtained.     

直梁弯曲切应力的讨论

针对矩形截面直梁横力弯曲,讨论了弯曲切应力推导的假设条件,给出了弯曲切应力公式的适用范围.并通过图像的方式展示了不同截面形状下弯曲切应力的分布模式,为学生全面认识与理解弯曲切应力分布提供课堂教学补充材料.     

The non-linear bending of a cantilever beam with shear and longitudinal deformations

A non-linear bending theory for beams is constructed which accommodates shear and longitudinal deformations. Using the theory, an analytical solution for the cantilever beam subject to a compressive load (the elastica) and a series solution for the horizontal cantilever under weight loading are derived. The effects of including shear and longitudinal deformations are found to be negligible for configurations in which the two deformations tend to offset one another—such as at the onset of buckling under a compressive load—but are shown to be significant for certain configurations in which the two deformations are additive—as in some instances of weight loading.