Keywords: stiffened plates, plate buckling, optimal spacing, in-plane forces, in-plane bending, large deflection, optimal design.

In this paper, the buckling of stiffened plates with longitudinal, transverse, and combined orthogonal stiffeners is investigated. The plate is simply supported along all its edges and is subjected to a combination of in-plane forces and bending moments. The virtual work principle is implemented in order to obtain the critical load of the structure in terms of the displacement field coefficients and the stiffener position. It is then minimized with respect to the displacement field coefficients which results in the actual buckling load, together with the mode shape of the buckled plate. Finally, in order to find the most efficient spacing of the stiffeners, the spacing corresponding to the maximum buckling load is obtained.

Stiffened plates are in extensive use in a variety of applications like box girders, plate girders, ship hulls and wing structures. The interest in using stiffened plate construction has been widespread in recent years due to its economic and structural benefits. The main reason is the high strength to weight ratio that is observed in stiffened plates. While the inclusion of stiffeners slightly increases the weight of the overall structure, their influence on strength and stability is enormous. By the proper choice of the stiffener location, they can still become much more efficient. There are excellent textbooks on the general behaviour of plates like Ugural [1]. Narrowing down the attention to stiffened plates, Troitsky [2] may be considered as a comprehensive reference book on the bending, vibration and buckling of stiffened plates.

One of the earliest works on the subject of buckling of stiffened plates has been that of Cox and Riddel [3] where the stain energy equations were implemented in the analysis. Later, Seide [4] investigated the effect of the eccentricity of the stiffeners on the critical buckling load of stiffened plates.

Timoshenko and Gere [5] have used the energy method in order to obtain critical load of stiffened plates with orthogonal longitudinal and transverse stiffeners. In recent years, Bedair has presented several studies on the subject of stiffened plate buckling, [6,7,8]. He has mainly used the Sequential Quadratic Programming (SQP) method in order to obtain the buckling load of stiffened plates.

In the present paper, the virtual work principle has been applied in order to calculate the buckling coefficient and the critical load factor of the stiffened plates. It is observed that the buckling coefficient is a function of the displacement coefficients. Therefore, in order to find its real value (corresponding to the actual instability of the plate) the buckling coefficient has been minimized with respect to these coefficients. In this way, not only the buckling load, but also the mode shape of the buckled plate has been obtained.

The critical load obtained in this way, is a function of the stiffener spacing. Hence, at a subsequent step, search is made in order to obtain the best stiffener spacing for maximum buckling load. In this way, the design would be most efficient and the best configuration of structural material has been obtained in order to yield the maximum strength at a constant material consumption level.

1

A.C. Ugural, "Stresses in Plates and Shells, 2nd Edition", McGraw-Hill, 1999.

2

M.S. Troitsky, "Stiffened Plates, Bending, Stability and Vibrations", Elsevier, 1976.

3

H.L. Cox and J.R. Riddel, "Buckling of a Longitudinally Stiffened Flat Panel", Aeronautical Quarterly Journal, 225-244, 1949.

4

P. Seide, "The Effect of Longitudinal Stiffeners Located on one Side of a Plate on the Compressive Buckling Stress of the Plate-Stiffener Combination", Technical Report No.2873, NACA, 1953.

5

S.P. Timoshenko and J.M. Gere, "Theory of Elastic Stability", McGraw-Hill, 1961.

6

O.K. Bedair and A.N. Sherbourne, "Unified Approach to Local Stability of Plate/Stiffened Assemblies", ASCE Journal of Engineering Mechanics, 121, 214- 229, 1995. doi:10.1061/(ASCE)0733-9399(1995)121:2(214)

7

O.K. Bedair, "The Elastic Behaviour of Multi-Stiffened Plates Under Uniform Compression", Thin-Walled Structures, 27, 311-335, 1997. doi:10.1016/S0263-8231(96)00032-8

8

O.K. Bedair, "Influence of Stiffener Location on the Stability of Stiffened Plates Under Compression and In-plane Bending", International Journal of Mechanical Sciences, 39, 33-49, 1997. doi:10.1016/0020-7403(96)00017-3

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