Keywords: cracked beam, dynamic stiffness method, Wittrick-Williams algorithm, Timoshenko-Ehrenfest beam.

The dynamic stiffness matrix of a cracked Timoshenko-Ehrenfest beam is developed to investigate its free vibration characteristics. The cracked beam is modelled by connecting two intact Timoshenko-Ehrenfest beam elements and an infinitesimal small length cracked element. For the cracked element, the flexibility matrix and subsequent stiffness matrix are established by applying fracture mechanics. The governing differential equations of motion and natural boundary conditions are obtained by applying Hamilton’s principle. For harmonic oscillation the equations are solved for displacements and bending rotation. The shear force and bending moment are obtained from the natural boundary conditions. The dynamic stiffness matrix of the intact beam is then derived by relating the amplitudes of loads to those of the displacements. Next, the compliance properties of the crack element are derived using facture mechanics theory. The dynamic stiffness matrices of the three components, namely the two intact elements and one crack element, are assembled to form the overall dynamic stiffness matrix for the cracked beam. The formulation leads to a nonlinear eigenvalue problem. The natural frequencies and mode shapes are extracted by applying the Wittrick-Williams algorithm. Results for the cantilever boundary conditions of the cracked beam are presented for illustrative purposes, and the effects of crack location and crack depth on the natural frequencies and mode shapes are examined. Some results are compared with published literature to confirm the validity and accuracy of the proposed method. The theory developed can be extended to include frameworks and other structures.

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