Computational Technology Resources - CCP

Keywords: cracked plate, hierarchical trigonometric functions, buckling, free vibration, plate, crack.

Summary

Ageing steel structures are prone to suffer several damages such as corrosion, cracks and dents. These damages influence the behaviour of the structure significantly and it is essential to study the buckling and free vibration characteristics of plates in damaged condition.

The finite element method provides solutions to such complex problems. However in the conventional method, the discretization of the global structure in the presence of cracks is a laborious task involving a large number of elements for achieving acceptable levels of accuracy. Moreover, the size of the matrices increases enormously and requires huge amount of computational resources to estimate the behaviour of cracked structures.

**Figure 39.1:** Discretisation of the plate with vertical edge crack into four elements (b) ith plate element in $(\xi-\eta)$ coordinate system

Therefore, in the present paper, the method based on hierarchical trigonometric functions [1] is proposed to analyse the cracked plates with possible minimum number of elements. The plate is discretised into several elements based on the disposition of the crack as shown in Figure 39.1. The hierarchical trigonometric functions are defined separately for each element based on the boundary conditions of the element. The local displacement of the ith element is expressed using hierarchical trigonometric functions as

-->

$\displaystyle w^i(\xi^i,\eta^i)=\sum_{m=1}^{M_i}\sum_{n=1}^{N_i}q_{mn}^i \phi_m (\xi^i)\phi_n(\eta^i)$

(39.1)

where $\phi_m(\xi^i)$ and $\phi_n(\eta^i)$ are trial functions and qmn is amplitude of displacement,

and

are the number of functions in

and

directions respectively. The trigonometric set $\{\phi_m(\xi^i)\}$ is defined as

$\displaystyle \phi_m(\xi)=\sin(a_m\xi+b_m)\sin(c_m\xi+d_m)$

(39.2)

Coefficients , , and ensure appropriate boundary conditions for the element. The displacement and slope compatibility between the elements is established by equating the amplitudes of the respective displacement functions of each element. Therefore, using the present method, the analysis of discontinuous structures such as cracked plates can be carried out with minimum discretisation, minimum computational efforts and with more accuracy.

The present method is applied to estimate the buckling and free vibration characteristics of plates with various types of cracks, such as edge crack and central crack and the results are compared with those obtained using finite element method.

References

1: O. Beslin, J. Nicolas, "A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions", Journal of Sound and Vibration, 202(5), 633-655, 1997. doi:10.1006/jsvi.1996.0797

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