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.

An integral approach for large deflection cantilever beams

Large deflection of constant curvature cantilever beam under follower load

An electromechanical energy harvesting system with a fractional order current–voltage relationship for the electrical circuit and fractional power law in the restoring force of its mechanical part is considered to act as an energy harvester. Our results show that, under a single-well potential configuration, for a small amplitude of the perturbation, as the order of derivative increases, the resonant amplitude of mechanical vibration decreases while the bending degree (hardening case) remains fairly constant. For a large amplitude of the perturbation, the output power is increased due to the hardening effects. Under a double-well configuration, the fractional power stiffness $$\alpha $$
 strongly affects the crossing well dynamics (large amplitude motion) and consequently the output electrical power. The harvested electric power appears to be maximal for deterministic and random excitation for small $$\alpha $$
 . High-level noise intensity is found to reduce the output power in the region of resonance and surprisingly increases the output in other regions of $$\alpha $$
 . For sufficiently large amplitude of harmonic excitation, this effect is realized in a stochastic resonance.

Nonlinear analysis of energy harvesting systems with fractional order physical properties

In this paper, a theoretical and numerical model based on the power series method is investigated for the lateral buckling stability of tapered thin-walled beams with arbitrary cross-sections and boundary conditions. Total potential energy is derived for an elastic behavior from strain energy and work of the applied loads. The effects of the initial stresses and load eccentricities are also considered in the study. The lateral-torsional equilibrium equations and the associated boundary conditions are obtained from the stationary condition. In presence of tapering, all stiffness coefficients are not constant. The power series approximation is then used to solve the fourth-order differential equations of tapered thin-walled beam with variable geometric parameters having generalized end conditions. Displacement components and cross-section properties are expanded in terms of power series of a known degree. The lateral buckling loads are determined by solving the eigenvalue problem of the obtained algebraic system. Several numerical examples of tapered thin-walled beams are presented to investigate the accuracy and the efficiency of the method. The obtained results are compared with finite element solutions using Ansys software and other available numerical or analytical approaches. It is observed that suggested method can be applied to stability of beams with constant cross-sections as well as tapered beams.

Lateral-torsional buckling of tapered thin-walled beams with arbitrary cross-sections

The analysis of the large deformation of a non-linear cantilever functionally graded material (FGM) beam is made. When subjected to an end moment, explicit expressions for deflection and rotation are derived for a functionally graded beam with work hardening of power law. The effects of the gradient distribution of Young’s modulus and the material non-linearity parameter on the deflections of the FGM beam are analyzed. Our results show that depth-dependent Young’s modulus and material non-linearity have a significant influence on the deflections of the beam, and a FGM beam can bear larger applied load than a homogeneous beam. Moreover, to determine an optimal gradient distribution, an optimum design of a beam of a lighter weight and larger stiffness is given. The influence of the geometric non-linearity of the beam is also studied. Large and small deformation theories predict nearly the same deflections with 5% error when rotation is less than 45°, and the predictions based on the small deformation theory...

Large Deflections of a Non-linear Cantilever Functionally Graded Beam

A new technique for large deflection analysis of non-prismatic cantilever beams

Exact expressions for stability functions of a general non-prismatic beam–column member

An efficient scheme, called quasi-linearization finite differences, is developed for large-deflection analysis of prismatic and non-prismatic slender cantilever beams subjected to various types of continuous and discontinuous external variable distributed and concentrated loads in horizontal and vertical global directions. Simultaneous equations of highly nonlinear and linear terms are obtained when casting the derived exact highly nonlinear governing differential equation using central finite differences on the nodes along the beam. A quasi-linearization scheme is used to solve these equations based on successive corrections of the nonlinear terms in the simultaneous equations. The nonlinear terms in the simultaneous equations are assumed constant during each correction (iteration). Several representative numerical examples of prismatic and non-prismatic slender cantilever beams with different loading conditions are analyzed to illustrate the merits of the adopted numerical scheme as well as its validity, accuracy and efficiency. The results of the present scheme are checked using large-displacement finite element analysis by the MSC/NASTRAN program. A comparison between the present secheme, MSC/NASTRAN and available results from the literature reveals excellent agreement. The advantage of the new scheme is that the load can be applied in one step with few iterations (3–6 iterations).

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

The post-buckling problem of a fixed-free column composed of two segments connected by a linear rotational spring with different flexural stiffness and lengths is studied. A semi-analytical solution is derived and the results are checked using two different numerical approaches. In the first approach, the problem is solved using numerical integration technique, while in the second approach, the problem is solved using large displacement finite element analysis of NASTRAN program. Excellent agreements have been obtained between the semi-analytical and the numerical approaches. Furthermore, a regression analysis has been made for the parameters covered in this study and a regression function has been obtained which relates to the tip axial horizontal displacement ( δ ) as a function of load parameter ( α ) and fixity ratio ( f ). From this function, the stored strain energy ( U ) is obtained and plotted to be used as a design chart for engineers to obtain the required fixity ratio for a given stored strain energy and load parameter.

Post-buckling behavior of two elastica columns linked with a rotational spring

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.

Samir Al-Sadder

Papers

A new technique for large deflection analysis of non-prismatic cantilever beams

Exact expressions for stability functions of a general non-prismatic beam–column member

Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings

Post-buckling behavior of two elastica columns linked with a rotational spring

Exact Large Deflection Analysis of Nonprismatic Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment