Large deflection of curved elastic beams made of Ludwick type material

The large deflection of an axially extensible curved beam with a rectangular cross-section is investigated. The elastic beam is assumed to satisfy the Euler-Bernoulli postulation and be made of the Ludwick type material. Through reasonably simplified integration, the strain and curvature of the axis of the beam are presented in implicit formulations. The governing equations involving both geometric and material nonlinearities of the curved beam are derived and solved by the shooting method. When the initial curvature of the beam is zero, the curved beam is degenerated into a straight beam,and the predicted results obtained by the present model are consistent with those in the open literature. Numerical examples are further given for curved cantilever and simply supported beams, and the couplings between elongation and bending are found for the curved beams.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

Finite elastic straining

Summary Finite elastic straining is analysed with all quantities referred consistently to the deformed body taken as the function domain. The straining-displacement of a typical point is relative to a set of axes imbedded in the body at one arbitrary point and rotating in fixed space with that neighborhood if necessary in a particular problem. The resulting |plane stress' equations have precisely the same form as in the classical theory but relate to |true' quantities in the deformed body.The solution of a circular hole in a deformed sheet under simple tension is given and checks closely with experiment on rubber. Cauchy strains of order 65% and local rotation of order 30° are found to occur at the hole boundary.The solution of a deformed quadrantal cantilever is given. Cauchy strains of several hundred percent and local rotation of order 90° occur.Any boundary value problem already solved for the classical infinitesimal strains theory can be applied directly as a finite strains solution for the deformed body.Notation x, y, z, r, , z Cartesian and polar co-ordinates respectively - , Normal and shear true stresses respectively - , Normal and shear true strains respectively - r Position vector - Airy stress function - S Simple tensile stress applied to sheet - a Radius of circular hole in deformed sheet - a, b Inner and outer radii of quadrantal cantilever - u Straining-displacement vector - u, v Straining-displacement scalar components - E, True Young's modulus and Poisson's ratio respectively - c ₁, c ₂ Local unit vectors in principal normal strains directions - i, j Cartesian axes constant unit vectors - Stress dyadic or tensor - First stress invariant - I Idemfactor or spherical tensor - P Shear load per unit thickness applied to quadrantal cantilever - A, B, D, N, H, K, L Arbitrary constants of integration     

An integral approach for large deflection cantilever beams

A new integral approach is proposed to solve the large deflection cantilever beam problems. By using the moment integral treatment, this approach can be applied to problems of complex load and varying beam properties. This versatile approach generally requires only simple numerical techniques thus is easy for application. Treatment for typical loading and beam property conditions are presented to demonstrate the capability of this approach.     

Finite twist of composite beams

Summary A pretwisted composite beam with straight centroidal axis, subjected to axial tension and torsion is proved to achieve a better structural efficiency if the reinforcing fibers are located along helicoidal generators in the pretwisted configuration, than in the Hodges' problem where those fibers are parallel to the beam axis.

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Sommario Si trova che una trave svergolata in materiale composito, con asse baricentrico diritto, soggetto a trazione e torsione, raggiunge una maggiore efficienza strutturale se le fibre di rinforzo sono dispose già inizialmente secondo linee generatrici elicoidali piuttosto che in direzione assiale come supposto in un precedente lavoro di Hodges.

     

Asymptotic analysis of layered elastic beams

We consider an elastic beam formed by three layers, fixed at one end and loaded at the free end. We call adherents the upper and lower layers ${\Omega }_{+}^{?}$ and ${\Omega }_{?}^{?}$ and an adhesive layer ${\Omega }_m^{?}$. We denote by $?h_{\pm ,m}$ the thickness of each layer and we suppose that the stiffness of the adhesive layer is ${?}^2$, with respect to that of the adherents. By an asymptotic analysis we obtain the zeroth order limit problem and the form of the second order displacements. To cite this article: M. Serpilli, C. R. Mecanique 333 (2005).     

Bending of beams on three-parameter elastic foundation

The bending of a Timoshenko beam resting on a Kerr-type three-parameter elastic foundation is introduced, its governing differential equations are formulated and analytically solved, and the solutions are discussed and applied to particular problems. Parametric analyses of elastically supported beams of infinite and finite length are carried out and comparisons are made between one, two or three-parameter foundation models and more accurate 2D finite element models. In order to estimate the necessary soil parameters, an analytical procedure based on the modified Vlasov model is proposed. The presented solutions and applications show the superiority of the Kerr-type foundation model compared to one or two-parameter models.     

Asymptotic theories of elastic beams and rods

Summary Through the asymptotic approach to elastic theory of beam-like structural elements, various categories of problems are distinguished with different first approximation formulations, showing different degrees of non-linearity.

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Sommario Per la teoria elastica di elementi strutturali tipo trave, il metodo asintotico porta a individuare varie categorie di problemi, retti da diverse formulazioni con diversi gradi di non-linearità, qui considerate in forma di prima approssimazione. Il metodo, oltre a fornire giustificazione di formulazioni classiche, offre la possibilità di approfondimenti sistematici.

     

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.     

An integral equation approach to finite deflection of elastic plates

The present paper concerns an approximate integral equation approach to finite deflection of elastic plates with aribitrary plan form. With the combined help of the Berger equation governing non-linear bending and a weighted residual technique for boundary-value problem, a boundary integral equation is formulated for immovable edge conditions. We here clarify the formulation and show that the calculation can be performed, with a slight modification, through a procedure similar to that conventionally employed in the linear bending analysis. Availability of the derived integral equation and the solution scheme is shown by way of simple numerical examples.     

Continuous elastic beams with unilateral rigid supports

Summary We analyze the problem of the continuous elastic beam on unilateral rigid supports, by means of a mathematical model based on classical derivatives.

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Sommario Si esegue una analisi qualitativa del problema dell'equilibrio elastico della trave continua su appoggi rigidi unilaterali, mediante un modello matematico proposto in termini classici.

     

Finite displacements of annular elastic membranes

Axisymmetric deformations of annular membranes subjected to normal surface loads and radial edge loads or displacements are considered within the Föppl nonlinear membrane theory. When the inner edger=a is free of radial traction, the solution of the annular membrane problem is shown to reduce to the solution for the circular membrane (a=0). For nonvanishing traction atr=a, the problem is reduced to a circular pseudo-membrane problem. For both cases, existence and uniqueness of tensile solutions of the annular membrane problem are proved, including a rigorous derivation of a stress concentration factor originally found by Schwerin by formal methods.     

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.     

Dynamic response of beams on generalised elastic foundations

Dynamic responses of beams on generalised elastic foundations is studied using the method of Initial parameters. The foundation model proposed by Vlasov and Leontev is modified by incorporating in the analysis the horizontal displacements in the elastic foundation thus making it more general and physically close to the actual situation. Results are compared with those reported by Rades, using Pasternak's foundation model and Winkler's model. The insufficiency of the Winkler's model in the study of dynamic responses (mainly the bending moments) is emphasized. Solutions presented are quite general for application to beams on generalised elastic foundations subjected to arbitrary external dynamic loads and (or) moments.     

Bending and stability analysis of gradient elastic beams

The problems of bending and stability of Bernoulli–Euler beams are solved analytically on the basis of a simple linear theory of gradient elasticity with surface energy. The governing equations of equilibrium are obtained by both a combination of the basic equations and a variational statement. The additional boundary conditions are obtained by both variational and weighted residual approaches. Two boundary value problems (one for bending and one for stability) are solved and the gradient elasticity effect on the beam bending response and its critical (buckling) load is assessed for both cases. It is found that beam deflections decrease and buckling load increases for increasing values of the gradient coefficient, while the surface energy effect is small and insignificant for bending and buckling, respectively.     

The exact deflection of an elastic beam over a soft obstacle

Summary We consider the problem of an elastic beam which is in unilateral contact with a soft obstacle. More precisely, we calculate the bending moments in the beam and the positions of the points where the beam leaves the obstacle.

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Sommario Si considera il problema di una trave elastica a contatto unilaterale con un ostacolo soffice. Più precisamente, si calcolano i momenti flettenti nella trave e le posizioni dei punti dove la trave abbandona l'ostacolo.

     

Finite Fracture Mechanics at elastic interfaces

In the present paper we provide a method to determine the load causing delamination along an interface in a composite structure. The method is based on the elastic interface model, according to which the interface is equivalent to a bed of linear elastic springs, and on Finite Fracture Mechanics, a crack propagation criterion recently proposed for homogeneous structures. The procedure outlined is general. Details are given for the pull–push shear test. For such geometry, the failure load is obtained and compared with the estimates provided by stress concentration analysis and Linear Elastic Fracture Mechanics. It is seen that Finite Fracture Mechanics provides intermediate values. Furthermore, it is shown that the predictions provided by Finite Fracture Mechanics are almost coincident with the ones provided by the Cohesive Crack Model. As far as we are concerned with the determination of the failure load, the advantage of using Finite Fracture Mechanics with respect to the Cohesive Crack Model is evident, since a troublesome analysis of the softening taking place in the fracture process zone is not necessary. A final comparison with classical fracture criteria based on critical distances, such as the average stress criterion, concludes the paper.