Bending of Timoshenko beam with effect of crack gap based on equivalent spring model

Considering the effect of crack gap, the bending deformation of the Timoshenko beam with switching cracks is studied. To represent a crack with gap as a nonlinear unidirectional rotational spring, the equivalent flexural rigidity of the cracked beam is derived with the generalized Dirac delta function. A closed-form general solution is obtained for bending of a Timoshenko beam with an arbitrary number of switching cracks. Three examples of bending of the Timoshenko beam are presented. The influence of the beam’s slenderness ratio, the crack’s depth, and the external load on the crack state and bending performances of the cracked beam is analyzed. It is revealed that a cusp exists on the deflection curve, and a jump on the rotation angle curve occurs at a crack location. The relation between the beam’s deflection and load is bilinear, each part corresponding to an open or closed state of crack, respectively. When the crack is open, flexibility of the cracked beam decreases with the increase of the beam’s slenderness ratio and the decrease of the crack depth. The results are useful in identifying non-destructive cracks on a beam.