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This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) 1L I I I I I (a) BL, IA4- (b) 328 Beam analysis (c) -± 4RB pIIIUUWWI (a) + (c) (a) + (b) Fig. 10.2 Bending moment diagram for propped cantilever with sinking support 10.3.1 Beams with supports at the same level The two conditions for solution, derived by Mohr, are: (1) the area of the fixing or negative BM diagram is equal to that of the free or positive BM diagram (2) the centres of gravity of the two diagrams lie in the same vertical line, i.e. are equidistant from a given end of the beam. Figure 10.3 shows a typical BM diagram for a built-in beam. ACDB is the diagram of the free moment Ms and the trapezium AEFB is the diagram of the fixing moment Mi, the portions shaded representing the final diagram. Let As = the area of the free BM diagram and Ai = the area of the fixing moment diagram.
This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) Fig. 10.3 Bending moment diagram for fixed-end beam load I I Then from condition (1) above, As = Ai, while from condition (2) their centres of gravity lie in the same vertical line, say, distance x from the left-hand support A. Now AE = the fixing moment MA and BE = the fixing moment MB. Therefore Ai and MA + MB = L (10.1) Divide the trapezium AEFB by drawing the diagonal EB and take area moments about the support A. 2 Then i Ê MA ¥ L Lˆ MB L L Ax = ¥ Ë ¯ L MA MB Ax i MA MB L + Ê ¥ 2 ˆ ¥ 2 3 Ë 2 3 ¯ 2 = ( + 2 ) 6 6 + 2 = 2 B B (10.2) But Ai MA + MB = L (10.3) Also As = Ai 2 Subtracting Equation (10.1) from Equation (10.2) and substituting A s for A i gives M Similarly M MA + MB ¥ L = Ai 2 B A 6Ax s 2As = - 2 L L 4As 6Ax s = - 2 L L Fixed, built-in or encastré beams 329
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