Keywords: plates, limit analysis, collapse load, rigid-perfectly plastic media, reinforced concrete, orthotropic reinforcement.

The definition of the collapse load of a reinforced concrete (RC) plate requires the determination of the solution of the upper bound (by applying the kinematic theorem) and that of the lower bound (by applying the static theorem). The two solutions reveal the interval within which we find the exact collapse load value. If the two solutions coincide, the common value represents the exact collapse value (by the mixed theorem of Limit analysis).

Application of the kinematic theorem normally facilitates the finding of the upper bound of the collapse load. Many results are to be found in the works of Johansen (for example, [1]) who was the first to elaborate the yield-line theory. This work was later continued by other authors (for example, [2,3]). On the contrary, research on the finding of lower bounds and consequently the application of the static theorem are less numerous and more episodic. In general, when automatic calculation methods are not employed (see for example [4]) it is necessary to use "intuition" in assigning a statistically admissible field of moments and from this to deduct the lower bound of the collapse load (see for example the solutions for the lower bound in [5,6,7,8,9,10,11]). In the works just cited, only [9] and [10] concern models of plates with free edges.

The paucity of lower bound solutions is unfortunate because lower bound solutions furnish information on the distribution of bending and torsional moments throughout the plate and the reactions on the supporting system, and ways of minimizing the reinforcement, including bar cutoff points, can be found [11].

In this note we address the problem of defining the collapse load for a RC rectangular plate clamped at one edge and supported at the two opposite corners of the boundary under a uniformly distributed load. The most general conditions of the ratio between the edges and the orthotropism of the reinforcement are considered.

In accordance with the fundamental theorems of limit analysis of rigid-perfectly plastic media, herein the formulation used to find the lower and upper bounds of the collapse load for reinforced concrete plates is discussed..

It will be shown by means of a comparison of the upper and lower bounds that in most cases the collapse load is found exactly, and in the other cases the lower bound agrees very favourably with the corresponding upper bound (and the collapse load is defined with sufficient precision for technical purposes). The specified moment fields could be used in determining the required distribution of positive and negative reinforcement.

1

K.W. Johansen, "Brudlinieteorie", English Translation: "Yield-Line Theory", Cement and Concrete Association, London, UK, 1952.

2

R.W. Wood, "Plastic and elastic design of slabs and plates", Thames and Hudson, London, UK, 1961.

3

C.E.B., "Annexes aux Recommandations internationales pour le calcul et l'exécution des ouvrages en Béton. Dalles et structures planes - Structures hyperstatiques", AITEC, Rome, Italy, 1972.

4

J. Munro, A.M.A. Da Fonseca, "Yield-line method by finite elements and linear programming", The Structural Engineer, 568(2), 37-44, 1978.

5

C. Massonnet, M. Save, "Calcolo a rottura delle strutture, Vol II, Strutture spaziali", Zanichelli, Bologna, Italy, 1969.

6

F.M. Mazzolani, M. Pignataro, "Limiti di collasso rigido-plastico per piastre in cemento armato incastrate a forma di parallelogramma", Giornale del Genio Civile, n. 5-6, 251-275, 1969.

7

B.V. Rangan, "Lower bound solution for continuous orthotropic slabs", Journal of the Structural Division, ASCE, 99(3), 443-452, 1973.

8

K.S. Subba Rao, N. Rangaiah, B.V. Ranganatham, "Lower Bound Limit Analysis of Rectangular Slabs", Journal of the Structural Division, ASCE, 103(11), 2193-2209, 1977.

9

K.S. Subba Rao, N. Rangaiah, B.V. Ranganatham, "Limit Analysis of Slabs with Free Edge", Journal of the Structural Division, ASCE, 105(11), 2419-2431, 1979.

10

Mura I., "Calcolo a rottura della piastra rettangolare ortotropa in C.A. appoggiata negli angoli e soggetta a carico uniformemente distribuito sulla superficie", Atti della Facoltà di Ingegneria, Università di Cagliari, Vol. 13, 31-52, 1980.

11

R. Park, W.L. Gamble, "Reinforced Concrete Slabs", 2nd ed., John Wiley & Sons Inc., New York, 2000.

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