Abstract
In the previous chapter we were able to find solutions to the beam dynamics problem using the separation of variables technique. The process went very much like it did for the axial bar problem. But the beam differential equation involves fourth order derivatives with respect to x, and consequently the spatial part of the solution was more complicated. While the classical techniques provide a good route to solving wave propagation problems in beams, we want to solve more general problems (e.g., problems with time-dependent forcing functions and possibly non-prismatic cross sections). In this chapter we will approach the problem in a different way. We will start by introducing the principle of virtual work and develop an approximate numerical approach using the Ritz method with finite element base functions.
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- 1.
It is possible for the real displacement to be prescribed but not equal to zero, e.g., w(0, t) = w o. That is a non-homogeneous essential boundary condition. The reason the essential boundary conditions must be homogeneous for the virtual displacements is precisely because we want the terms that result from integration by parts to vanish.
- 2.
The notion of completeness is essentially a measure of the ability of the Ritz base functions to represent certain basic functions. For example, if the Ritz functions are based on polynomials, you would not leave out the x 2 term because that would hamper the ability to represent low-order polynomials. One commonly-used strategy is the patch test, which tests the ability of the base functions to represent a constant.
- 3.
More precisely, the second derivatives are zero almost everywhere with infinite jumps at the locations where the first derivative is discontinuous. These functions fail to be square integrable, which is the mathematical requirement for functions to be admissible in the virtual work setting.
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Hjelmstad, K.D. (2022). Finite Element Analysis of Linear Planar Beams. In: Fundamentals of Structural Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-89944-8_12
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DOI: https://doi.org/10.1007/978-3-030-89944-8_12
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