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

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

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

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

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

A cohesive FE model for simulating the cracking/debonding pattern of composite NSC-HPFRC/UHPFRC members

The aim of this work is to propose to practitioners a simple cohesive Finite-Element model able to simulate the cracking/debonding pattern of retrofitted concrete elements, in particular Normal-Strength-Concrete members (slabs, bridge decks, pavements) rehabilitated by applying a layer of High-Performance or Ultra-High-Performance Fiber-Reinforced-Concrete as overlay. The interface was modeled with a proper nonlinear cohesive law which couples mode I (tension-crack) with mode II (shear-slip) behaviors. The input parameters of the FE simulation were provided by a new bond test which reproduces a realistic condition of cracking/debonding pattern. The FE simulations were accomplished by varying the overlay materials and the moisture levels of the substrate surface prior to overlay, since findings about their influence on the bond performances are still controversial. The proposed FE model proved to effectively predict the bond failure of composite NSC-HPFRC/UHPFRC members

Thin film bonded to elastic orthotropic substrate under thermal loading

The problem of thin elastic films bonded on an elastic orthotropic substrate under thermal load is investigated in this work. Differently from past studies on the same topic, the effects induced by anisotropic behavior of the elastic substrate will be taken into account. Particular attention will also be paid to the determination of the displacement and stress fields induced by thermal loading. In particular, it is assumed that the thin films are deposed on the substrate at high temperature, and then the mismatch occurring during the cooling process, due to the difference between the thermal expansion coefficients of the two materials, is responsible for the permanent deformation assumed by the system. This phenomenon can be exploited for realizing a crystalline undulator. To this aim, the permanent deformation must be optimized in order to encourage the channeling phenomenon. By imposing equilibrium conditions and perfect adhesion between the film and the substrate, a singular integral equation is derived. A closed-form solution is achieved by expanding the interfacial shear stress fields in Chebyshev series. The unknown coefficients in the series expansion are then determined by transforming the integral equation into an infinite algebraic system

Analytical and experimental study of snap-through instability in truss structures

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