The bending problem of a cantilever beam is schematically defined in Fig. 1. Under the assumption that the material of beam remains linearly elastic, the relationship of bending moment and beam deformation reads ()
(1)
where ? is the angle of rotation of the deflection curve, s is the distance measured along the beam, M is the bending moment, E is the module of elasticity of the material and I is the moment of inertia of the cross-sectional area of the beam about the axis of bending.
Fig. 1. Bending of a cantilever beam.
Dado and Al-sadder summarized several major approaches used for large deflection problems: the elliptic integral approach developed by Bisshopp and Drucker and is still widely used at present (e.g., and ); the numerical integration approach with iterative shooting techniques (e.g., and ); and the incremental finite element or finite difference method with Newton-Rhapson iteration techniques (e.g., and ). The drawbacks of these approaches, mainly including applicability, complexity and stability, were analyzed in the same article. Dado and Al-sadder developed an approach that approximates the angle of rotation by a polynomial function and minimizes the residual of the governing equation caused by the approximation. This approach is effective for complex load and non-prismatic cantilever beam with very large deflection. However, the formulation of this approach is also complicated which increases the difficulty of application in practical problems.
Ang et al. proposed a numerical method based on the form of Eq. (1) in Cartesian coordinates, which provided the basis of this study. The left-hand side of Eq. (1) is the curvature of the beam curve. Thus the equation can be reformed to
(2)
where x and y are coordinates in which y is parallel to the original beam. Ang et al applied a search procedure to solve the cantilever beam bending problem defined by Eq. (2). Define
(3)
thus the curve length of the beam can be calculated with
(4)
And we also have
(5)
Hence Eq. (2) can be converted to
(6)
Eqs. (3), (5) and (6) can be solved numerically for a given projective length l with boundary conditions z(0)=x(0)=s(0)=0at the fixed end. The problem will then be solved by searching the projective length l until
(7)
s(l)=L
is satisfied. Ang et al. proposed an approach using Runge-Kutta method to solve the Eqs. (6), (3) and (5).
Experiment 2
We propose a new approach to solve the cantilever beam bending problem based on the formulation by Ang et al. .
Rewrite Eq. (6) into
(8)
This equation can be integrated and we obtain
(9)
Note that the boundary condition z(0)=0 is used here. Also it is worthy to point out that the left-hand side of the above equation is actually sin ?. Using Eq. (5), the above equation can be converted to
(10)
From Eq. (3), another equation for variable x can be obtained following the same procedure, which reads
(11)
Eqs. (10) and (11) become new governing equations for cantilever beam bending problems. For simple loads and uniform beam properties, they are regular first order ODEs in which dependent variables do not appear in the right-hand side thus can be integrated ...