Essentials in the theory of framed structures . Beams with Varying Cross-section 150. General Expressions.—The moment of inertia of beamshaving uniform cross-section is constant, and for this reason Iappears outside the integral sign in Eqs. (i) and (2). Whenthe cross-section is not uniform the moment of inertia varies,and Eqs. (i) and (2) become In order to perform the integration, / as well as M must beexpressed as a function of x. This is relatively a simple matterwhen the beam has a rectangular cross-section varying uni-formly in breadth or depth; but this method often results in 236 THEOR
Essentials in the theory of framed structures . Beams with Varying Cross-section 150. General Expressions.—The moment of inertia of beamshaving uniform cross-section is constant, and for this reason Iappears outside the integral sign in Eqs. (i) and (2). Whenthe cross-section is not uniform the moment of inertia varies,and Eqs. (i) and (2) become In order to perform the integration, / as well as M must beexpressed as a function of x. This is relatively a simple matterwhen the beam has a rectangular cross-section varying uni-formly in breadth or depth; but this method often results in 236 THEORY OF FRAMED STRUCTURES Chap. V long and cumbersome expressions when applied to structuralsteel sections. In all such instances the geometric method ispreferable. 151. Beams with Vaiying Depth.—The beam in Fig. 147a isa plate girder. The % in. web plate is 24 in. wide at the ends,and the width increases uniformly to 36 in. at the flange is composed of two angles 5 by 3 H by % with the ?0,000* 20,000* innnn* --4« 5--->H---5-. 3-iii. leg against the web. The distance back to back of angles is the width of the web plus ^ in. Ordinates in the M-diagram (Fig. 1476) are given in inch-pounds every 3 ft. An I-diagram is shown in Fig. 147c. The ordinates represent the moment of inertia in inches* at 3-ft. intervals. Each ordinate in the M-diagram has been divided by the corresponding ordinate Min the I-diagram, and the quotient recorded in the Y-diagram (Fig. i47</). The ordinates in this diagram are expressed inpounds/inches.^ Since the girder and the loads are symme- Mtrical, the -:j^-diagram is symmetrical about the vertical ordinate Sec. V DEFLECTION OF BEAMS 237 through the center of the span; the maximum deflection is atthe center, and the tangent to the elastic curve (not drawn)at the center is horizontal; hence the tangential deviation t atthe left support equals the area-moment A BCD about Adivided by about A X 36 X 36 = S75)Ooo X
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