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Vibration Analysis of Laminated Plates and Shells with Improved Degenerated Shell Element

Hyo-Gyoung Kwak1 and Hyuk-Chun Noh2

[1]Department of Civil and Environmental Engineering, KAIST, Daejeon 305-701, Korea

[2] Smart Infra-Structure Technology Center, KAIST, Daejeon 305-701, Korea

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ABSTRACT

In this paper, 4 node degenerated shell element adopting the assumed strain fields to eliminate the shear and membrane locking in the thin shell is applied in the nonlinear dynamic analysis of composite materials. The nonlinearity in the isotropic and anisotropic material is considered through the flow theory of plasticity. The influence of lamination on the structural behavior in static and dynamic is also discussed.

INTRODUCTION

One of the recent research interests in the structural analysis is in the area of plates and shells, especially in the laminated composite form. Structural composites are widely adopted due to their high specific stiffness and strengths. Obtaining the analytical solution for the laminated composite shells with variety of geometrical configurations such as saddle, cylindrical, spherical quadratic and hypar are very difficult if it is not impossible in some cases.

The analysis of general shells is performed based on the various analytical and numerical techniques (Dipankar et al. 1998, Swaddiwudhipong 1996). The frequency characteristics of the eccentrically hollow shells are given by Chakravorty (Dipankar et al. 1998) and the investigation of the doubly cantilevered shell of saddle, cylindrical and spherical shells are presented by Qatu using Rits method. The material nonlinear behavior of anisotropic plates and shells are delivered by Huang.

EQUATIONS OF MOTION

The stiffness matrix is approximated by the layered approach, where the through thickness integration is performed by the summation of stiffness of each layers. In this case, the volume integration is divided into area integration plus the summation through thickness. The integration in the parenthesis is replaced by the summation of contribution from each layer, which are transformed in accordance with the skewed material axis.

Anisotropic Material

The yield criterion used in this study is the generalized Huber-Mises law. The yield function f for the anisotropic material can be defined in the similar way to that of the isotropic material. The anisotropic parameters etc. can be determined experimentally by the six independent yield tests. In the case of the work-hardening material, the change in the anisotropic parameters due to the increase of the yield stress is taken into account using the concept of plastic work.

Flow rule for the concrete material Using the flow theory of plasticity, the material nonlinearity is represented according to the following elasto-plastic constitutive matrix.

(1)

where a is the flow vector and A denotes the hardening parameter.

NUMERICAL EXAMPLES

A cylindrical shell with the dimension as depicted in Fig. 3 is subjected to an impulsive patch load of an initial velocity of 143.5´103mm/s (5650 in/s) over an area of 259mm´78.2mm along the crown line is analyzed.

Fig. 1. Cylindrical shell subjected to impulsive patch load

A half is modeled taking into account of the symmetries of structure and of load. The time history at the point y1=159.5mm (6.28in) and y2=239mm (9.42in) along the crown line of the shell shows reasonable agreement with the experiment as given in Fig. 3 (b). In this analysis, no damping is assumed.

REFERENCES

Dipankar Chkravorty, P. K. Sinha and J. N. Bandyoopadhyay, “Application of FEM on free and forced vibration of laminated shells”, Journal of Engineering Mechanics, ASCE, Vol. 124, No. 1, January, 1998, pp. 1-8.

S. Swaddiwudhipong and Z. S. Liu, “Dynamic response of large strain elasto-plastic plate and shell structures”, Thin Walled Structures, Vol. 26, No. 4, 1996, pp. 223-239.

Madenci, E and Barut, A. “A free-formulation-based flat shell element for non-linear analysis of thin composite structures”, International Journal for Numerical Methods in Engineering, Vol. 37, 1994, pp. 3825-3842.

[1] Professor

[2] Post Doctoral Researcher