Author Topic: Example: Cantilevered beam  (Read 458 times)

Mahabubur rahman

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Example: Cantilevered beam
« on: June 15, 2013, 01:04:06 PM »
The boundary conditions for a cantilevered beam of length L (fixed at x = 0) are

    \begin{align} &\hat{w}_n = 0 ~,~~ \frac{d\hat{w}_n}{dx} = 0 \quad \text{at} ~~ x = 0 \\ &\frac{d^2\hat{w}_n}{dx^2} = 0 ~,~~ \frac{d^3\hat{w}_n}{dx^3} = 0 \quad \text{at} ~~ x = L \,. \end{align}

If we apply these conditions, non-trivial solutions are found to exist only if \cosh(\beta_n L)\,\cos(\beta_n L) + 1 = 0 \,. This nonlinear equation can be solved numerically. The first few roots are β1 L = 1.8751, β2 L = 4.69409, β3 L = 7.85476, β4 L = 10.9955 , ...

The corresponding natural frequencies of vibration are

    \omega_1 = \beta_1^2 \sqrt{\frac{EI}{\mu}} = \frac{3.515}{L^2}\sqrt{\frac{EI}{\mu}} ~,~~ \dots

The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:

    \hat{w}_n = A_1 \Bigl[\cosh\beta_n x - \cos\beta_n x + \frac{(\cos\beta_n L + \cosh\beta_n L)(\sin\beta_n x - \sinh\beta_n x)}{\sin\beta_n L + \sinh\beta_n L}\Bigr]

For a non-trivial value of the displacement, A_1 has to remain arbitrary, and the magnitude of the displacement is unknown for free vibrations. Typically a value of A_1 = 1 is used when plotting mode shapes. However, the arbitrary nature of the displacement amplitude implies that under the right conditions very large displacements can be experienced by the beam, i.e., the beam can resonate. The natural frequencies of a beam therefore correspond to the frequencies at which resonance can occur.

Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.

Both the bending moment and the shear force cause stresses in the beam. The stress due to shear force is maximum along the neutral axis of the beam (when the width of the beam, t, is constant along the cross section of the beam; otherwise an integral involving the first moment and the beam's width needs to be evaluated for the particular cross section), and the maximum tensile stress is at either the top or bottom surfaces. Thus the maximum principal stress in the beam may be neither at the surface nor at the center but in some general area. However, shear force stresses are negligible in comparison to bending moment stresses in all but the stockiest of beams as well as the fact that stress concentrations commonly occur at surfaces, meaning that the maximum stress in a beam is likely to be at the surface.