In this work, a generalization of the Timoshenko beam theory is introduced, which is based on fractal continuum calculus. The mapping of the bending problem onto a non-differentiable self-similar beam into a corresponding problem for a fractal continuum is derived using local fractional differential operators. Consequently, the functions defined in the fractal continua beam are differentiable in the ordinary calculus sense. Therefore, the non-conventional local derivatives defined in the fractal continua beam can be expressed in terms of the ordinary derivatives, which are solved theoretically and numerically. Lastly, examples of classical beams with different boundary conditions are shown in order to check some details of the physical phenomenon under study.

In this paper a new generalized fractal equation for studying the behaviour of self-similar beams using the Timoshenko beam theory is introduced. This equation is established in fractal dimensions by applying the concept of fractal continuum calculus $F^{\alpha }$-CC introduced recently by Balankin and Elizarraraz in order to study engineering phenomena in complex bodies. Ultimately, the achieved formulation is a fourth-order fractal single equation generated by superposing a shear deformation on an Euler–Bernoulli beam. A mapping of the Timoshenko principle onto self-similar beams in the integer space into a corresponding principle for fractal continuum space is formulated employing local fractional differential operators. Consequently, the single equation that describes the stress/strain of a fractal Timoshenko beam is solved, which is simple, exact, and algorithmic as an alternative description of the fractal bending of beams. Therefore, the elastic curve function and rotation function can be described. Illustrative examples of classical beams are presented and show both the benefits and the efficiency of the suggested model.