Influence Surfaces for Stresses in Slabs

Abstract

Abstract The investigation reported herein presents a formal analytical solution of some problems involving influence surfaces for stresses in slabs. It deals primarily with the determination of influence diagrams for stresses in homogeneous elastic slabs and is based upon the extension of Müller-Breslau’s principle established in structural mechanics. The analysis is based upon the classical procedure of obtaining a solution of Lagrange’s differential equation of the deflected middle surface of the slab, satisfying at the same time, all of the boundary conditions at the various edges. Systematic use of a stress function, introduced by Nádai in the theory of slabs, is made in forming the expressions for bending moments, twisting moments, and shears at any point of a semi-infinite slab cantilevered from a single fixed edge. The problem is meant only as an illustration in the use of Nádai’s stress function when a fixed edge condition is desired. The influence surface for a bending moment at a point on the fixed edge of an infinitely long rectangular slab, with a fixed edge transverse to two parallel simply supported edges, is defined by a function in finite form. Contour lines of the influence surfaces for the bending moments at the quarter points and at the mid-point of the fixed edge are given. The solution for the infinitely long rectangular slab is considered as a basic solution to which a correction function is added to obtain any desired condition along a line parallel to the fixed edge. The correction functions required to obtain either a simple or fixed support along a line parallel to the given fixed edge are stated in the form of an infinite series. A few illustrative figures of the resultant diagrams are given. Approximate formulas for the correction functions are also stated. Influence functions are stated for the moments at the mid-point and at a point on the fixed edge of a circular slab with a fixed edge. Contour lines of the influence surfaces are given. Each solution defining an influence surface for a stress at a given point of a homogeneous elastic slab may also be used to solve the related two-dimensional problem of stresses in a slice. A slab is converted into a slice when the loads applied transverse to the slab are replaced by forces that are parallel to the middle plane of the slice and produce no bending or transverse shears.