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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) some BMs change their signs as the frames change their proportions. This will be appreciated by examining the charts. For simple frames, i.e. for single-storey frames, the formulae for reactions immediately follow the formulae for BMs for each load. Considering the simple frames only, the type of formula depends on the degree of indeterminacy and the shape of the frame.Auxiliary coefficients X are introduced whenever the direct expressions become complicated or for other reasons of expediency. No hard and fast rules can be laid down for the notation and it must be noted that each set of symbols and constants applies only to the particular frame under consideration, although, of course, an attempt has been made to produce similarity in the types of symbols. 11.1.3 Sign conventions All computations must be carried out algebraically, hence every quantity must be given its correct sign. The results will then be automatically correct in sign and magnitude. The direction of the load or applied moment shown in the left-hand diagram for each load condition is considered to be positive. If the direction of the load or moment is reversed, the signs of all the results obtained from the formulae as printed must be reversed. For simple frames, the moments causing tension on the inside faces of the frame are considered to be positive. Upward vertical reactions and inward horizontal reactions are also positive. For multi-storey or multi-bay frames the same general rules apply to moments and vertical reactions. 11.1.4 Checking calculations for indeterminate frames Formulae for rigid frames 345 Calculations for indeterminate frames may be checked by using some other method of analysis, but it is also possible to check any frame or portion of a frame, such as that above the line AB in Fig. 11.2, by ensuring that the following rules are obeyed: (1) the three fundamental statical equations, i.e. SH = 0, SV = 0 and SM = 0, have been satisfied, and, in addition, either that (2) the sum of the areas of the M/EI diagram above any line, such as AB, is zero if A and B are fully fixed; or (3) the sum of the moments, with respect to the base AB, of the areas of the M/EI diagram above the line AB is zero if A and B are partially restrained (as shown in Fig. 11.2) or are hinged.
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) 346 Plane frame analysis -— side boding A B I Fig. 11.2 Checking an indeterminate frame I L I — 1 '1 I + -7 I / -7 I I / The underlying principles in rules (2) and (3) above are those used in the application of the ‘column analogy’ method of analysis. As an example of rule (2), consider the frame in Fig. 11.3 where EI is constant. Then the sum of the areas of the M/EI diagram, considering the legs first, is 2 È( + 0 0736 - 0 0826)¥ 6 0 ˘ 2 0 0826 0 0893 8 078 EI ÎÍ 2 ˚˙ EI 2 0 054 0 054 0 EI + È( - + )¥ ˘ ÎÍ ˚˙ = - + . . . . . . . . = thus demonstrating that the moments calculated are correct. Now consider the frame in Fig. 11.4 as an example for rule (3). Then the sum of the moments of the areas of the M/EI diagram, working from A round to D, is 1 È11. 25¥ 4. 8 2¥ 4. 8˘ 6 4. 8 2 48 . ¥ EI ÎÍ 2 3 ˚˙ 3 2 11. 25 12. 75 12. 75 4. 8 2 4. 8 96 . 48 . 0 2 2 3 2 1 4. 8 2 11. 25 6 EI 2 3 + Ï È ¥ ¥ ˘ Ì ¥ Ó ÎÍ ˚˙ ÈÊ - ˆ ˘ È- ¥ ¥ ˘ + ¥ + ¥ ÎÍ Ë ¯ ˚˙ ÎÍ ˚˙ ¸ ˝ = ˛ È ¥ ˘ ( + - - ) Î Í ¥ ˚ ˙ = 4. 5 12. 75 0 demonstrating again that the calculations are correct. 11.2 Portal frame analysis The design process explained here assumes that the reader is familiar with the basic concepts of limit state steel design, and the basic analysis and design of continuous
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