Analysis of Elastic Beams on Linear and Nonlinear Foundations Using Finite Difference Method

An approximate method is developed to analyze the deflection in beams and beam-column by solving the differential equation for the elastic deformation of beam and beam-column. The analysis is performed using the central difference of finite difference method for the Euler-Bernoulli beam and beam-column supported on an elastic, nonlinear foundation with rigid or elastic discrete supports. To make a verification of the results, Laplace Transformation method was used to solve the elastic differential equation of beam and beam-column based on linear elastic supports and the results were compared with the finite difference method. Two types of beams were selected, simply supported and fixed-fixed with five elastic supports of an idealized soil. In the nonlinear idealization, the division of force into many levels were assumed and based on these forces, the equivalent displacements were obtained from an assumed power law equation by using the finite difference method. Central finite difference scheme, which has a second order, was used throughout the numerical analysis with five nonlinear behavior of springs separated by an equal distance between them

Steady states analysis and exponential stability of an extensible thermoelastic system

In this work we consider a nonlinear model for the vibrations of a thermoelastic beam with fixed ends resting on an elastic foundation. The behavior of the related dissipative system accounts for both the midplane stretching of the beam and the Fourier heat conduction. The nonlinear term enters the motion equation, only, while the dissipation is entirely contributed by the heat equation. Under stationary axial load and uniform external temperature the problem uncouples and the bending equilibria of the beam satisfy a semilinear equation. For a general axial load $p$, the existence of a finite/infinite set of steady states is proved and buckling occurrence is discussed. Finally, long-term dynamics of solutions and exponential stability of the straight position are scrutinized

The role of damage-softened material behavior in the fracture of composites and adhesives

Failure mechanisms of materials under very high strains experienced at and ahead of the crack tip such as formation, growth, and interaction of microvoids in ductile materials, microcracks in brittle solids or crazes in polymers and adhesives are represented by one-dimensional, nonlinear stress-strain relations possessing different ways by which the material loses capacity to carry load up to fracture or total separation. A double cantilever beam (DCB) type specimen is considered. The nonlinear material is confined to a thin strip between the two elastic beams loaded by a wedge. The problem is first modeled as a beam on a nonlinear foundation. The pertinent equation is solved numerically as a two-point boundary value problem for both the stationary and the quasi-stationay propagating crack. A finite element model is then used to model the problem in more detail in order to assess the adequacy of the beam model for the reduction of experimental data to determine in-situ properties of the thin interlayer

HOMOTOPY ANALYSIS OF A FORCED NONLINEAR BEAM MODEL WITH QUADRATIC AND CUBIC NONLINEARITIES

This study investigates forced nonlinear vibrations of a simply supported Euler-Bernoulli beam on a nonlinear elastic foundation with quadratic and cubic nonlinearities. Applying the homotopy analysis method (HAM) to the spatially discretized governing equation, we derive novel analytical solutions and discuss their convergence to present nonlinear frequency responses with varying contributions of the nonlinearity coefficients. A comparison with numerical solutions is conducted and nonlinear time responses and phase planes are compared to the results from linear beam theory. The study demonstrates that apart from nonlinear problems of free vibrations, HAM is equally capable of solving strongly nonlinear problems of forced vibrations

Rasio Modulus Penampang Elastik Balok Kayu Laminasi-Baut

. Laminated beam can be an alternative for solid timber, because it provides the advantage that it can be fabricated with a needed-span and a bigger cross section. The purpose of this research is to obtain an empirical equation of the bolt-laminated timber beam elastic section modulus ratio. Elastic section modulus ratio is elastic section modulus ratio between laminated and solid beams. Scope of this research are horizontally laminated system, Indonesian timber with specific grafity ranged 0.4-0.8 which are red meranti (shorea spp), keruing (dipterocarpus spp), and acacia mangium, prismatic beam section, experimental test in laboratorium and numerical simulation using nonlinear finite element method. The parameters discussed are timber type, bolt diameter, number of row, and spacing. Beam has a 3-meter span and arranged by 4 laminae. Timber stress-strain model for numerical simulation based on Hill plasticity, bolt stress-strain model is elasto-plastic. Results obtained are beam load-displacement curve trend is bilinear, the elastic section modulus ratio equation are the fuction of timber type, bolt diameter, and number of row against bolt spacing ratio. The elastic section modulus ratio can be used to predict the bending strength at the proportional limit

An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model

An efficient numerical iterative method is constructed for the static deflection of an infinite beam on a nonlinear elastic foundation. The proposed iterative scheme consists of quasilinear method (QLM) and Green’s function technique. The QLM translates the nonlinear ordinary differential equation into iterative linear ordinary differential equation. The successive iterations of quasilinear form of ordinary differential equation (ODE) show the quadratic convergence if an initial guess is chosen in the neighbourhood of true solution. The Green’s function technique converts the differential operator into an integral operator and the integral operator is approximated by discrete summation which finally gives us an iterative formula for the resulting set of algebraic equations.The numerical validity and efficiency are proved by simulating some nonlinear problems.Peer ReviewedPostprint (published version

Eigenvalue problem for nonlinear elastic beam equation of fractional order

In this study, under some suitable assumptions, we determine an explicit eigenvalue interval for the existence of positive solution of singular fractional-order nonlinear elastic beam equation with bending term. Our analysis rely on cone theoretic techniques. Moreover, we consider some special cases and an example to affirm the applicability of the main result

Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core

The nonlinear free vibration behavior of shear deformable sandwich porous beam is investigated in this paper within the context of Timoshenko beam theory. The proposed beam is composed of two face layers and a functionally graded porous core which contains internal pores following different porosity distributions. Two non-uniform functionally graded distributions are considered in this paper based on the equivalent beam mass, associated with a uniform distribution for purpose of comparison. The elastic moduli and mass density are assumed to vary along the thickness direction in terms of the coefficients of porosity and mass density, whose relationship" is determined by employing the typical mechanical characteristic of an open-cell metal foam. The Ritz method and von Karman type nonlinear strain-displacement relationships are applied to derive the equation system, which governs the nonlinear vibration behavior of sandwich porous beams under hinged or clamped end supports. A direct iterative algorithm is then used to solve the governing equation system to predict the linear and nonlinear frequencies which are presented by a detailed numerical study to discuss the effects of porosity coefficient, slenderness ratio, thickness ratio and to compare the varying porosity distributions and boundary conditions, providing a feasible way to improve the vibration behavior of sandwich porous beams