# Composite Action between Beams and Slabs Under Transverse Load (1) A Slab, with Identical Spanwise Beams Repeated at Equal Intervals, supported over several Transverse Supports

*The analysis of a structure composed of beams integral with a slab is properly a three dimensional problem. In this paper, the need for a three dimensional treatment is obviated by an analysis of the slab according to thin plate theory and of the beams,
treated separately from the slab, by the theory of simple bending. The fact that the beams are actually integral with the slab is then recognised by enforcing displacement and strain compatibility at the slab-beam junctions. Both the stress-function, describing forces in the plane of the slab, and the transverse deflection of the slab are governed by a biharmonic equation; series solutions for these two quantities are obtained subject to typical, specified, boundary conditions and to conditions of compatibility at the slab-beam junctions. The calculation of the coefficients in the series solutions would be extremely laborious if carried out by hand-computation so that a digital computer programme has been written (for the Ferranti Pegasus Computer) by use of which the complete computation can be carried out in a few minutes; thus the effect of any prescribed variation in the parameters describing either the geometry of the structure or its loading can be quickly followed through. Details of the computer programme are given.
D. N. de G. Allen and R. T. Severn*