Mechanics of high-flexible beams under live loads

In this paper the mathematical formulation of the equilibrium problem of high-flexible beams in the framework of fully nonlinear structural mechanics is presented. The analysis is based on the recent model proposed by L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. In this model the complete three-dimensional kinematics of the beam is taken into account, both deformations and displacements are considered large and a nonlinear constitutive law in assumed. After having illustrated and discussed the peculiar mechanical aspects of this special class of structures, the criteria and methods of analysis have been addressed. A classification of the structures based on the degree of kinematic constraints has been proposed, distinguishing between isogeometric and hypergeometric structures. External static loads dependent on deformation (live loads) are also considered. The governing equations are derived on the basis of a moment-curvature relationship obtained in L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. The governing equations take the form of a highly nonlinear coupled system of equations in integral form, which is solved through an iterative numerical procedure. Finally, the proposed analysis is applied to some popular structural systems subjected to dead and live loads. The results are compared and discussed

Countermeasures assessment of liquefaction-induced lateral deformation in a slope ground system

Liquefaction-induced lateral spreading may resul in significant damage and disruption of functionality for structures and slope groung system

Bending of beams in finite elasticity and some applications

The 2D Rivlin solution concerning the finite bending of a prismatic solid has been recently extended by accounting for the complete 3D displacement field [1]. In particular, the relationship between the principal and transverse (anticlastic) deformation of a bent solid has been investigated, founding the coupling relationships among three kinematic parameters which govern the problem. Later, based on the formulation reported in [1], and making reference to a (hyper)elastic material, the formulation has been extended to slender beams by introducing some simplifying assumptions [2]. This leads to a challenging relation between the external bending moment m and the curvature R0\uf02d1 of the longitudinal axis, which involves both the constitutive and geometric parameters of the beam. This relation can be viewed as a generalization of the Elastica [3]. However, such a relationship can be simplified through a series expansion, thus obtaining a reliable moment-curvature relation as follows [4], being a, b, c the constitutive parameters involved in the stored energy function according to a compressible Mooney-Rivlin material, whereas r denotes the anticlastic radius of the cross section [1]. In eqn (1)1 the radius of curvature R0 depends on the curvilinear abscissa s describing the beam axis in its deformed configuration. The rotation \uf071 of the beam cross section follows from the derivative of the curvature with respect abscissa s, i.e. \uf071\u2019(s) = R0\uf02d1(s). Thus, the axial and vertical components of the displacement field and the rotation of the beam cross section are found to be coupled in a set of three equations in integral form, which is handled in an iterative procedure in order to analyse elastic structures exhibiting deformations and displacements both large. Some basic structural schemes under both dead and live loads are here investigated, thus assessing the deformed configuration and the arising internal forces into the beam. It is found that the magnitude of the external loads strongly affects the qualitative distribution of the axial and shear forces and the bending moment in the inflexed beam, giving rise to a solution which completely differs to that corresponding to infinitesimal strains and small displacements

Large nonuniform bending of beams with compressible stored energy functions of polynomial-type

The large bending of beams made with complex materials finds application in many emerging fields. To describe the nonlinear behavior of these complex materials such as rubbers, polymers and biological tissues, stored energy functions of polynomial-type are commonly used. Using polyconvex and compressible stored energy functions of polynomial-type, in the present paper the equilibrium problem of slender beams in the fully nonlinear context of finite elasticity is formulated. In the analysis, the bending is considered nonuniform, the complete three-dimensional kinematics of the beam is taken into account and both deformation and displacement fields are deemed large. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Explicit formulae for displacements, stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. By way of example, a complete analysis has been performed for the Euler beam

FE Analyses of Hyperelastic Solids under Large Bending: The Role of the Searle Parameter and Eulerian Slenderness

A theoretical model concerning the finite bending of a prismatic hyperelastic solid has been recently proposed. Such a model provides the 3D kinematics and the stress field, taking into account the anticlastic effects arising in the transverse cross sections also. That model has been used later to extend the Elastica in the framework of finite elasticity. In the present work, Finite Element (FE) analyses of some basic structural systems subjected to finite bending have been carried out and the results have been compared with those provided by the theoretical model performed previously. In the theoretical formulation, the governing equation is the nonlinear local relationship between the bending moment and the curvature of the longitudinal axis of the bent beam. Such a relation has been provided in dimensionless form as a function of the Mooney–Rivlin constitutive constants and two kinematic dimensionless parameters termed Eulerian slenderness and compactness index of the cross section. Such parameters take relevance as they are involved in the well-known Searle parameter for bent solids. Two significant study cases have been investigated in detail. The results point out that the theoretical model leads to reliable results provided that the Eulerian slenderness and the compactness index of the cross sections do not exceed fixed threshold values

Finite bending of non-slender beams and the limitations of the Elastica theory

The problem of slender solids under finite bending has been addressed recently in Lanzoni and Tarantino (2018). In the present work, such a model is extended to short solids by improving the background formulation. In particular, the model is refined by imposing the vanishing of the axial force over the cross sections. The geometrical neutral loci, corresponding to unstretched and unstressed surfaces, are provided in a closed form. Two approximations of the models are obtained linearising both kinematics and constitutive law and kinematics only. It is shown that the approximations of the model, corresponding to the Euler Elastica formulation, can lead to significant values of the axial stress resultants despite pure bending conditions. For a generic form of compressible energy function, a nonlinear moment–curvature relation accounting for both material and geometric nonlinearities is provided and then specialised for a Mooney–Rivlin material. The obtained results are compared with simulations of 3D finite element models providing negligible errors. The normalisation of the moment–curvature relation provides the dimensionless bending moment as a function of the Eulerian slenderness of the solid. This dimensionless relation is shown to be valid for any aspect ratio of the bent solid and, in turn, it highlights the limitations of the Elastica arising in case of large deformations of solids

Lateral buckling of the compressed edge of a beam under finite bending

This paper investigates the critical condition whereby the compressed edge of a beam subjected to large bending exhibits a sudden lateral heeling. This instability phenomenon occurs through a mechanism different from that usually studied in linear theory and known as flexural-torsional buckling. An experimental test device was specifically designed and built to perform pure bending tests on soft materials. Thus, the experimental campaign provides not only the moment-curvature behavior of beams of narrow rectangular cross section, but also information regarding the post-critical lateral buckling behavior. To study the local bifurcation phenomenon, an analytical model is proposed in which a field of small transversal displacements, typical of the linear stability of thin plates, is superimposed on the large vertical displacement field of an inflexed beam in the nonlinear elasticity theory. Furthermore, numerous numerical simulations through nonlinear FE analysis have been performed. Finally, the results provided by the different methods applied were compared and discussed

Lateral buckling of a hyperleastic solid under finite bending

The problem of a beam that laterally buckles when subjected to flexure in the plane of greatest bending stiffness has been investigated first in the pioneering works by Prandtl and Michell in 1899. Those studies, and many others appeared in Literature afterwards, are based on the classical beam theory, which predicts that the cross sections experience a rigid rotation maintaining their original shape after deformation. However, experimental investigations highlight that a more challenging scenario takes place when, instead of beam-like solids, plate-like bodies are bent in their stiffest plane. In such a situation, an elastic deformation takes place also in the planes of the cross sections. This holds true particularly if large bending is needed to reach the onset of flexural–torsional buckling. The present works addresses the problem of the lateral buckling of a hyperelastic prismatic body under finite bending accounting for the deformation of the cross sections also. The stored energy function for compressible Mooney-Rivlin materials is considered, accounting for the material and geometric nonlinearities. The problem is handled through the energy method. Starting from the bent configuration of the prism, an out-of-plane displacement field is superposed as a small perturbation. Then, the vanishing of the variation of the total potential energy allows assessing the critical angle (or, equivalently, the critical bending moment) for which a deflected equilibrium configuration adjacent to the purely bent one becomes possible. According to the energetic approach, the method provides upper bounds of the critical loads. However, it is shown that the accuracy of the solution may be conveniently improved by enriching the general expression of the perturbation