Abstract
The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.
Highlights
Deflection of a cantilever beam (CB) has been the subject of numerous engineering problems nowadays which have very attractive civilian applications, e.g., shipbuilding, forestry, roofed structures, cranes, heavy bridges, flexible manipulator, etc
Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam
The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam boundary conditions (Dado & Al-Sadder, 2005)
Summary
Deflection of a cantilever beam (CB) has been the subject of numerous engineering problems nowadays which have very attractive civilian applications, e.g., shipbuilding, forestry, roofed structures, cranes, heavy bridges, flexible manipulator, etc. A uniform CB under the action of a combined load consisting of a uniformly distributed load and an external vertical concentrated load applied at the free end were analyzed (Belendez et al, 2005; Belendez et al, 2003; Belendez et al, 2002) In these articles the numerical solution based on the Runge-Kutta-Felhberg method was compared with experimental results. Exact and numerical solutions of non-prismatic, nonlinear bi-modulus CB subjected to a tip moment by applying a power series approach to analytically solve highly nonlinear simultaneous first-order differential equations (Shatnawi & Al-Sadder, 2007; Baykara et al, 2005) In these references, the stress–strain relationship of the nonlinear material was represented by the Ludwick constitutive law. ADM needs too many algebraic computations to achieve the required accuracy for more complicated loading conditions
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