Vibration analysis of cracked beam based on Reddy beam theory by finite element method

Abstract

The lateral vibration of cracked isotropic thick beams is investigated. Generally, the analysis of thick beam based on line elements can be undertaken using either Timoshenko beam theory or a third order beam theory (TSDT) such as Reddy beam theory. TSDT is superior to Timoshenko beam theory, as it eliminates the need for a shear correction coefficient. However, there is no available solution for a cracked beam based on Reddy beam theory which is the main focus of this paper. The investigated beam is divided into several elements where their stiffness and mass matrices are derived using Reddy beam theory. Each element has two nodes with 3 degrees of freedom (1 lateral and 2 rotational) at each node. The crack was modeled using two rotating springs with stiffnesses that varied with crack depth. These elements are used to connect the adjacent beam elements. The impact of boundary conditions, slenderness ratio, crack location, and crack depth are investigated. In addition, experimental and finite element analyses of cracked steel beams is undertaken to validate the results of the present model. The results show that the maximum deviation between the analytical and the experimental results is less than 3% up to the third mode shape.

Free vibration Analysis of Beams with Functional Gradient Materials with Cracks

“…They derived the control equations for cracked beams under both symmetric and asymmetric boundary conditions and analytically solved these equations using basic standard trigonometric and hyperbolic functions, analyzing the free vibration characteristics of the system and investigating the effects of different system conditions on its dynamic behavior. Taima MS and El-Sayed TA et al [4] studied the lateral vibration of isotropic and thick beams with cracks, comparing the analysis of Timoshenko beam theory and Reddy beam theory for beams with cracks. By dividing the beam into multiple elements and deriving stiffness and mass matrices for each beam element based on Reddy beam theory equations, they simulated the impact of cracks and analyzed the effects of boundary conditions, slenderness ratio, crack position, and depth on lateral vibration.…”

Comparative study on free vibration analysis of rotating bi-directional functionally graded beams using multiple beam theories with uncertainty considerations

“…Third-order shear deformation theory (TSDT), also known as Reddy beam theory, goes a step further, and accounts for the fact that the cross-section will no longer remains straight or perpendicular to the beam axis after deformation. In TSDT, the transverse shear strain and stress are assumed to have a parabolic distribution with respect to the thickness coordinate 4 , 8 , 40 .…”

Dynamic Analysis of Double-Tapered Rotating Beams Composed of Bidirectional Functionally Graded Materials (BFGMs)

“…To overcome these limitations and accurately predict vibrational characteristics, higher-order theories such as TSDT, have been developed. TSDT assumes a parabolic distribution of transverse shear strain and stress along the beam thickness, as shown in figure 1(b), eliminating the need for a shear coefficient [7].…”