Large deflections of nonlinearly elastic functionally graded composite beams (original) (raw)
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This paper presents an exact solution for large deflection behavior of nonprismatic cantilever beams of nonlinear bimodulus type material and subjected to tip concentrated moment. The highly nonlinear simultaneous first-order differential equations were solved analytically using the power series approach. Several numerical examples are carried out to investigate the effect of bimodulus properties, material parameter, n, and the applied tip moment on the large deflection behavior of nonprismatic cantilever beams. The numerical results show that the interaction (i.e., coupling) between the bimodulus material properties, the material constant, n, and the applied tip moment, plays a significant role on the large deflection behavior of cantilever beams. A comparative study with ADINA has been made to verify the accuracy of the presented analytical solution, and excellent agreement has been obtained.
This paper presents an exact solution for large deflection behavior of nonprismatic cantilever beams of nonlinear bimodulus type material and subjected to tip concentrated moment. The highly nonlinear simultaneous first-order differential equations were solved analytically using the power series approach. Several numerical examples are carried out to investigate the effect of bimodulus properties, material parameter, n, and the applied tip moment on the large deflection behavior of nonprismatic cantilever beams. The numerical results show that the interaction (i.e., coupling) between the bimodulus material properties, the material constant, n, and the applied tip moment, plays a significant role on the large deflection behavior of cantilever beams. A comparative study with ADINA has been made to verify the accuracy of the presented analytical solution, and excellent agreement has been obtained.
Meccanica, 2014
The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler-Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper. Keywords Large deflections Á Asymmetric Ludwick constitutive law Á Material nonlinearity Á Geometrical non-linearity Á Cantilever beam List of symbols L Initial length of the beam and length of the beam neutral curve, (m) b Width of rectangular cross-section, (m) h Height of rectangular cross-section, (m) F V Constant vertical force at the free end of the beam, (N) F H Constant horizontal force at the free end of the beam, (N) T Constant bending torque at the free end of the beam, (Nm) Oxyz Coordinate system of reference configuration O 0 x 0 y 0 z 0 Coordinate system defined for each crosssection h 1 , h 2 Quotes individuating the neutral axis of cross-section, (m)
Mathematical Problems in Engineering, 2012
This paper presents a large deflection of variable-arc-length beams, which are made from nonlinear elastic materials, subjected to its uniform self-weight. The stress-strain relation of materials obeys the Ludwick constitutive law. The governing equations of this problem, which are the nonlinear differential equations, are derived by considering the equilibrium of a differential beam element and geometric relations of a beam segment. The model formulation presented herein can be applied to several types of nonlinear elastica problems. With presence of geometric and material nonlinearities, the system of nonlinear differential equations becomes complicated. Consequently, the numerical method plays an important role in finding solutions of the presented problem. In this study, the shooting optimization technique is employed to compute the numerical solutions. From the results, it is found that there is a critical self-weight of the beam for each value of a material constant n. Two possible equilibrium configurations i.e., stable and unstable configurations can be found when the uniform self-weight is less than its critical value. The relationship between the material constant n and the critical self-weight of the beam is also presented.
Large Deflection of Composite Beams
In this study, finite element method is used to compute deflections of composite beams undergoing geometric nonlinearity. The beams analyzed are laminates with different lay – ups and different end conditions, subjected to uniformly distributed loads. Integrals encountered in the analysis are performed by hand, and therefore considered more accurate than numerical integration. The results obtained showed excellent agreement with those found in literature. Extra results are generated to serve as bench marks for further investigations. As expected, large deflections have resulted in stiffer beams. The stiffness increase is more pronounced in beams which are less restrained.
Experimental Static Analysis of a Cantilever Beam with Nonlinear Parameters
The beam-like structures are typically subjected to dynamic loads. In this paper classical problem of deflection of a cantilever beam of linear elastic material, under the action of a uniformly distributed load along its length (its own weight), is experimentally and numerically analyzed. Paper presents the differential equation governing the behavior of this system and shows that these equations are difficult to solve due to the presence of nonlinear term. The experiment described in this paper is an easy way to introduce the concept of geometric nonlinearity in mechanics of material. Finally numerical result is carried out by ANSYS program and comparedwith the experimental results.