Stress-driven nonlocal integral model with discontinuities for transverse vibration of multi-cracked non-uniform Timoshenko beams with general boundary conditions

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We present a formulation for the size-affected vibration study of multi-cracked non-uniform Timoshenko beams based on the well-posed stress-driven nonlocal elastic theory with discontinuities. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. The presence of cracks divides the beam into segments connected by translational and rotational springs, and compatibility conditions are established to address the geometric discontinuities introduced by these cracks. The stress-driven constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at the two ends of the entire structure and multi-sets of constitutive continuity conditions at the junctions of the sub-structures. To solve the equations of motion, the constraint conditions and the integrals involved, we employ the differential quadrature method (DQM) alongside an interpolation quadrature formula, which allows us to efficiently compute the frequencies of the cracked beams across various boundary types. After validating our approach against results in the existing literature, we present numerical studies that examine the effects of the nonlocal parameter, the slope of the beam’s thickness variation, crack location, severity, number, and the stiffness of the springs on the vibrational behavior of the beams.