Contributions of standing-wave components to dynamic stresses in a beam

Vibration modes of a beam have been identified previously as superposed standing waves, each of which consisting of two components, denoted by ka and kb, respectively. In this paper, individual contributions of these two components to dynamic normal and shear stresses in a beam are investigated, and their roles in the development of cracks in a beam are briefly addressed. A cantilever beam subject to an impulse tip force of different time durations is chosen as an example to illustrate their respective effects. The contributions of the kb-component are studied in particular, since it changes its wave characteristics from evanescent to propagating when its frequency exceeds the critical frequency. For an impulse force of long duration, it is shown that the kb-component is dominant in the normal stress while the ka-component is dominant in the shear stress. They would play dominant roles for Mode I and Mode II of crack propagation, respectively. For an impulse force of short duration, both normal and shear stresses are dominated by the ka-component. Therefore, the ka-component would play a dominant role for both Mode I and Mode II of crack propagation. It should be noted, however, ka- and kb-components above the critical frequency contribute almost equally to both normal and shear stresses, regardless of the duration of the force.