(2) ~Cantilever beam~--a bar resting upon one support or fulcrum, or that portion of any beam projecting out of a wall or beyond a support. (See Fig. 16, No. 2.) (3) ~Continuous beam~--a bar resting upon more than two supports. (See Fig. 16, No. 3.) [Illustration: FIG. 16.--Three common forms of beams. 1. Simple. 2. Cantilever. 3. Continuous.] _Stiffness of Beams_ The two main requirements of a beam are stiffness and strength. The formulæ for the _modulus of elasticity (E)_ or measure of stiffness of a rectangular prismatic simple beam loaded at the centre and resting freely on supports at either end is:[10] [Footnote 10: Only this form of beam is considered since it is the simplest. For cantilever and continuous beams, and beams rigidly fixed at one or both ends, as well as for different methods of loading, different forms of cross section, etc., other formulæ are required. See any book on mechanics.] P' l^{3} E = ------------- 4 D b h^{3} b = breadth or width of beam, inches. h = height or depth of beam, inches. l = span (length between points of supports) of beam, inches. D = deflection produced by load P', inches. P' = load at or below elastic limit, pounds. From this formulæ it is evident that for rectangular beams of the same material, mode of support, and loading, the deflection is affected as follows: (1) It is inversely proportional to the width for beams of the same length and depth. If the width is tripled the deflection is one-third as great. (2) It is inversely proportional to the cube of the depth for beams of the same length and breadth. If the depth is tripled the deflection is one twenty-seventh as great. (3) It is directly proportional to the cube of the span for beams of the same breadth and depth. Tripling the span gives twenty-seven times the deflection. The number of pounds which concentrated at the centre will deflect a rectangular prismatic simple beam one inch may be found from the preceding formulæ by substituting D = 1" and solving for P'. The formulæ then becomes: 4 E b h^{3} Necessary weight (P') = ------------- l^{3} In this case the values for E are read from tables prepared from data obtained by experimentation on the given material. _Strength of Beams_ The measure of the breaking strength of a beam is expressed in terms of unit stress by a _modulus of rupture_, which is a purely hypothetical expression for points beyond the elastic limit. The formulæ used in computing this modulus is as follows: 1.5 P l R = --------- b h{^2} b, h, l = breadth, height, and span, respectively, as in preceding formulæ. R = modulus of rupture, pounds per square inch. P = maximum load, pounds. In calculating the fibre stress at the elastic limit the same formulæ is used except that the load at elastic limit (P_{1}) is substituted for the maximum load (P).