VALUES OF n. Beam. Cantilever. 12 End load (a) Central load (b) Evenly distributed load 9'6 4 (c) Two equal symmetrically placed loads dividing 93 beam into three equal parts (d) Irregular loading (approx.) This table shows the relation that must be observed between the span and the depth of the section for a given stiffness. The stress can be found direct from the deflection of a given beam if the modulus of elasticity be known; as this does not vary much for any given material, a fairly accurate estimate of the stress can be made. We have above The system of loading being known, the value of n can be found from the table above. The value of E must be assumed for the material in the beam. The depth of the section d can readily be measured, also & and L. The above method is extremely convenient for finding approximately the stress in any given beam. The error cannot well exceed 10 per cent., and usually will not amount to more than 5 per cent. CHAPTER XII. COMBINED BENDING AND DIRECT STRESSES. IN the figure, let a weight W be supported by two bars, 1 and 2, whose sectional areas are respectively A, and A, and the corresponding loads on the bars R, R, น FIG. 438. and R; then, in order that the stress may be the same in each, W must be so placed that R, and R, are proPportional to the sectional areas of the R1 Αν 1 bars, or or A1u = But R1u = R, Az; hence W passes through the centre of gravity of the two bars when the stress is equal on all parts of the section. This relation holds, how ever many bars may be taken, even if taken so close together as to form a solid section; hence, in order to obtain a direct stress of uniform intensity all over a section, the external force must be so applied that it passes through the centre of gravity of the section. If W be not placed at the centre of gravity of the section, but at a distance x from it, we shall have FIG. 439. Thus when W is not placed at the centre of gravity of the section, the section is subjected to a bending moment Wx in addition to the direct force W. Thus If an external force W acts on a section at a distance x from its centre of gravity, it will be subjected to a direct force W acting The uniformly all over the section and a bending moment Wx. intensity of stress on any part of the section will be the sum of the direct stress and the stress due to bending, tension and compression being regarded as stresses of opposite sign. The Loaded In the figure let the bar be subjected to both a direct stress (+), say tension, and bending stresses. direct stress acting uniformly all over the section may be represented by the diagram abcd, where ab or cd is the intensity of the tensile stress (+); Unloaded side FIG. 440. then if the intensity of tensile stress due to bending be represented by be (+), and the compressive stress (-) by fc, we shall have The total tensile stress on the outer skin = ab + be = ae compressive = defe = df If the bending moment had been still greater, as shown in Stresses on Bars loaded out of the Centre.— Let W the load on the bar producing either direct tensile or compressive stresses; A the sectional area of the bar; Z = the modulus of the section in bending; x f'. 1= the eccentricity of the load, i.e. the distance of the point of application of the load from the centre of gravity of the section; the direct tensile stress acting evenly over the section; f' the direct compressive stress acting evenly over = the section; f = the tensile stress due to bending ; = f the compressive stress due to bending; M = Ꮓ Ꮓ Then the maximum stress on the skin) = f+f= W Wx of the section on the loaded side + A Z Then the maximum stress on the f' of the section on the unloaded side} = ƒ' - ƒ = W(A − 2) In order that the stress on the unloaded side may not be of opposite sign to the direct stress, the quantity must be greater x I A than When they are equal, the stress will be zero on the Z unloaded side, and of twice the intensity of the direct stress on the loaded side; then we have = I x Ꮓ A Z' A or = x. Hence, in order that the stress may not change sign or that there may be no reversal of stress in a section, the line of action of the Ꮓ external force must not be situated at a greater distance than from the neutral axis. A For convenience of reference, we give various values of in the following table :- Ꮓ A |