An extended piecewise functions formalism for computing the internal forces and deflections of beams

Abstract

Beams are slender bodies vastly used to build many kinds of structures. In sit- uations where a beam can be accurately modeled using a set of governing equations given by linear differential equations, the solution for a loading case that is the combination of sev- eral loading cases, is given by the superposition of the solutions for each particular loading case. In structural mechanics this result is called the superposition principle. The most usual loading cases appearing in practice impose discontinuities in the derivatives of the internal forces and the deflection (displacements) of a beam. Therefore, when solving the governing equations, the domain must be partitioned in regions where all the derivatives are defined, then a solution is found in each region, and finally the boundary conditions and a set of compatibility conditions at the points of discontinuity are applied. Conse- quently the solutions are piecewise functions and the solution process becomes lengthy. Several techniques have been proposed to abbreviate this procedure, all supported in the validity of the superposition of solutions. One very straightforward technique involving a compact and intuitive notation, sometimes designated Macaulay’s method, is the use of the so called ”singularity functions” [1], ”step functions” [2] or Macaulay brackets, a sort of generalization of the Heaviside function, an idea introduced by Macaulay in 1919 [3], and further refined by other authors, as explained in [4]. This technique considers loadings involving only point forces, point couples or distributed forces of polynomial type which are active from some starting point until de end of the beam. When a distributed force is nonzero in only an interior segment of the beam, it must be modeled as two dis- tributed loads that are nonzero until the end of the beam, but which cancel each other in the portion of the beam where the original loading is zero. Besides possible mistakes with finding the fictitious cancellation load, this extra load doubles the computations when evaluating the solution. We consider that this technique can be extended to any loading case and avoid the need for fictitious loads, using therefore less computations, by adding another piecewise term to the formalism, at the expense of an arguably less expressive notation. Basically the superposition of solutions given by piecewise functions, with each function representing the complete solution for a single load, is generated and applied in a very systematic way. This communication is devoted to present this extension, as well as the corresponding au- tomation of the solution procedure using symbolic computation. Several illustrative exam- ples of application, which can appear in a context of teaching as well as in real structural design, are considered. With the availability of free computer algebraic systems, a few lines of code can provide solutions for any beam and frame problem that are accurately modeled by linear differential equations. Therefore symbolic computational tools should be introduced in the curricula and used when teaching these subjects.