INTRODUCTION 3 2 THEORY OF ELASTICITY 2.Stress.Let Fig.1 represent a body in equilibrium.Under the stress is inclined to the area 6A on which it acts and we usually resolve action of external forces P...,P,internal forces will be produced it into two components:a normal stress perpendicular to the area,and between the parts of the body.To study the magnitude of these forces a shearing stress acting in the plane of the area 5A. at any point O,let us imagine the body divided into two parts A and B 3.Notation for Forces and Stresses.There are two kinds of exter- nal forces which may act on bodies.Forces distributed over the sur- by a cross section mm through this point. Considering one of these parts,for instance,A,it can be face of the body,such as the pressure of one body on another,or hydro- stated that it is in equilibrium static pressure,are called surface forces.Forces distributed over the under the action of external volume of a body,such as gravitational forees,magnetic forces,or in forces Pi,...,Prand the inner the ease of a body in motion,inertia forces,are called body forces.The forces distributed over the cross surface force per unit area we shall usually resolve into three compo- section mm and representing the nents parallel to the coordinate axes and use for these components the actions of the material of the notation X,F,2.Weshall also resolve the body force per unit volume part B on the material of the part into three components and denote A.It will be assumed that these components by X,Y,Z. these forees are continuously dis- We shall use the letter o for de- tributed over the area mm in the noting normal stress and the letter same way that hydrostatic pres- r for shearing stress.To indicate sure or wind pressure is contin- the direction of the plane on which Fro.1. uously distributed over the sur- the stress is acting,subscripts to face on which it acts.The magnitudes of such forces are usually these letters are used.If we take a defined by their intensity,i.e.,by the amount of force per unit area of very small cubic element at a point the surface on which they act.In discussing internal forees this 0,Fig.1,with sides parallel to the coordinate axes,the notations for intensity is called stress. In the simplest case of a prismatical bar submitted to tension by the components of stress acting on f1a.3. forces uniformly distributed over the ends (Fig.2),the internal forces the sides of this element and the directions taken as positive are as indicated in Fig.3.For the sides of the element perpendicular to the are also uniformly distributed over any croes section mm.Hence the intensity of this distribution,i.e.,the y-axis,for instance,the normal components of stress acting on these stress,can be obtained by dividing the total tensile sides are denoted by The subscript y indicates that the stress is acting on a plane normal to the y-axis.The normal stress is taken force P by the cross-sectional area A. positive when it produces tension and negative when it produces In the case just considered the stress was uniformly compression. distributed over the cross section.In the general case of Fig.I the stress is not uniformly distributed over The shearing stress is resolved into two components parallel to the coordinate axes.Two subscript letters are used in this case,the first mm.To obtain the magnitude of stress acting on a indicating the direction of the normal to the plane under consideration small area 6A,cut out from the croes section mm at any and the second indicating the direction of the component of the stress. point O,we assume that the forces acting across this 1a.2. elemental area,due to the action of material of the part For instance,if we again consider the sides perpendicular to the y-axis, B on the material of the part A,can be reduced to a resultant op.If the component in the r-direction is denoted by and that in the we now continuously contract the elemental area 8A,the limiting value -direction byThe positive directions of the components of shear- of the ratio p/6A gives us the magnitude of the stress acting on the ing stress on any side of the cubic element are taken as the positive eross section mm at the point O.Thelimiting direction of the resultant directions of the coordinate axes if a tensile stress on the same side 6P is the direction of the stress.In the general case the direction of would have the positive direction of the corresponding axis.If the