# Simplified Procedure For The Free Vibration Analysis Of Rectangular Plate Structures With Holes And Stiffeners

Open access

## Abstract

Thin and thick plates, plates with holes, stiffened panels and stiffened panels with holes are primary structural members in almost all fields of engineering: civil, mechanical, aerospace, naval, ocean etc. In this paper, a simple and efficient procedure for the free vibration analysis of such elements is presented. It is based on the assumed mode method and can handle different plate thickness, various shapes and sizes of holes, different framing sizes and types as well as different combinations of boundary conditions. Natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations. Mindlin theory is applied for a plate and Timoshenko beam theory for stiffeners. The applicability of the method in the design procedure is illustrated with several numerical examples obtained by the in-house developed code VAPS. Very good agreement with standard commercial finite element software is achieved.

1. [1]Reissner E.: The effect of transverse shear deformation on the bending of elastic plate. Transactions of ASME Journal of Applied Mechanics, 12, (1945), pp. 69-77

2. Mindlin R. D.: Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics, 18, 1(1951), pp. 31-38

3. Liew K. M., Xiang Y., Kitipornchai S.: Research on thick plate vibration: a literature survey. Journal of Sound and Vibration, 180, (1995), pp. 163-176

4. Senjanović I., Vladimir N., Tomić M.: An advanced theory of moderately thick plate vibrations. Journal of Sound and Vibration, 332, (2013), pp. 1868-1880

5. Senjanović I., Tomić M., Vladimir N.: Cho D. S.: Analytical solution for free vibrations of a moderately thick rectangular plate. Mathematical Problems in Engineering, 2013, (2013), Article ID 207460

6. Liew K. M., Xiang Y., Kitipornchai S.: Transverse vibration of thick plates – I. Comprehensive sets of boundary conditions. Computers and Structures, 49, (1993), pp. 1-29

7. Dawe D. J., Roufaeil O. L.: Rayleigh-Ritz vibration analysis of Mindlin plates. Journal of Sound and Vibration, 69, 3(1980), pp. 345-359

8. Kim K.H., Kim B.H., Choi T.M., Cho D.S.: Free vibration analysis of rectangular plate with arbitrary edge constraints using characteristic orthogonal polynomials in assumed mode method. International Journal of Naval Architecture and Ocean Engineering, 4, (2012), pp. 267-280

9. Chung J.H., Chung T.Y., Kim K.C.: Vibration analysis of orthotropic Mindlin plates with edges elastically restrained against rotation. Journal of Sound and Vibration, 163, (1993), pp. 151-163

10. Auricchio F., Taylor R.L.: A triangular thick plate finite element with an exact thin limit. Finite Elements in Analysis and Design, 19, (1995), pp. 57-68

11. Lovadina C.: Analysis of a mixed finite element method for the Reissner-Mindlin plate problems. Computer Methods in Applied Mechanics and Engineering, 163, (1998), pp. 71-85

12. Hughes T.J.R., Tezduyar T.: Finite elements based upon Mindlin plate theory with particular reference to the four-node isoparametric element. Journal of Applied Mechanics, 48, (1981), pp. 587-596

13. Bletzinger K., Bischoff M., Ramm E.: A unified approach for shear-locking free triangular and rectangular shell finite elements. Computers and Structures, 75, (2000), pp. 321-334

14. Nguyen-Xuan H., Liu G. R., Thai-Hong C.: Nguyen-Thoi T. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 199, (2010), pp. 471-489

15. Senjanović I., Vladimir N., Hadžić N. Modified Mindlin plate theory and shear locking-free finite element formulation. Mechanics Research Communications, 55, (2014), pp. 95-104

16. Cho D.S., Vladimir N., Choi T.M.: Approximate natural vibration analysis of rectangular plates with openings using assumed mode method. International Journal of Naval Architecture and Ocean Engineering, 5, 3(2013), pp. 478-491

17. Paramasivam P.: Free vibration of square plates with openings. Journal of Sound of Vibration, 30, (1973), pp. 173-178

18. Kwak M.K., Han S.: Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method. Journal of Sound of Vibration, 306, (2007), pp. 12-30

19. Grossi R.O., del V. Arenas B., Laura P.A.A.: Free vibration of rectangular plates with circular openings. Ocean Engineering, 24, 1(1997), pp. 19-24

20. Monahan L.J., Nemergut P.J., Maddux G.E.: Natural frequencies and mode shapes of plates with interior cut-outs. The Shock and Vibration Bulletin, 41, (1970), pp. 37-49

21. Cho D.S., Vladimir N., Choi T.M.: Natural vibration analysis of stiffened panels with arbitrary edge constraints using the assumed mode method. Proceedings of the IMechE, Part M: Journal of Engineering for the Maritime Environment, (2014), DOI: 10.1177/1475090214521179, published online

22. Samanta A., Mukhopadhyay M.: Free vibration analysis of stiffened shells by the finite element technique. European Journal of Mechanics, A Solids, 23, (2004), pp. 159-179

23. Sivasubramonian B., Kulkarni A. M., Rao G.V.; Krishnan A.: Free vibration of curved panels with cutouts. Journal of Sound and Vibration, 200, (1997), pp. 227-234

24. Sivasubramonian B., Rao G.V., Krishnan A.: Free vibration of longitudinally stiffened curved panels with cutout. Journal of Sound and Vibration, 226, 1(1999), pp. 41-55

25. Srivastava A.K.L.: Vibration of stiffened plates with cutout subjected to partial edge loading. Journal of the Institution of Engineers (India) Series A, 93, 2(2012), pp. 129-135

26. MSC. MD Nastran 2010 Dynamic analysis user’s guide. MSC Software, 2010

# Polish Maritime Research

## The Journal of Gdansk University of Technology

### Journal Information

IMPACT FACTOR 2017: 0.763
5-year IMPACT FACTOR: 0.816

CiteScore 2017: 0.99

SCImago Journal Rank (SJR) 2017: 0.280
Source Normalized Impact per Paper (SNIP) 2017: 0.788

### Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 264 234 10