CHAP. V.] MOMENT OF ROTATION PARALLEL FORCES. 59 



outer polygon sides is at a 6 and the nearest polygon angle is 5. 

 Half-way between a 6 and 5 we must then take the centre, while 

 before it lay at a, z (nearly). If we take then M 2 half-way be- 

 tween 02 and this new centre, we find precisely as before the 

 span S 2 S 2 with centre M 2 , and right end at P 7 , as the limiting 

 span for the six wheels P t to P 6 . The same holds good for the 

 span S 8 S 3 with centre M 3 , for the seven wheels P L to P 7 , and 

 so on. If, according to supposition, P 3 P 4 P 5 are 3 ft. 6 in. 

 apart, P 2 and P 3 4 ft., and P : and P 2 5 ft. 6 in. apart; then for 

 spans up to s 81 = say 8 ft., the two \vheels P 4 P 5 will give the 

 greatest maximum moment, and their place upon the beam is 

 given by the position of the centre (half-way between a^ and 4). 

 From about 8 ft. to 15 ft. span, or s 2 s 2 the three wheels P 3 to P 5 

 give the greatest maximum moment, anc^the centre of the span 

 is located at a%. For spans from 1 5 ft. to 19 ft. span, or s' s s\, 

 the four wheels P 2 to P 5 give the maximum moment, and the 

 centre is at St ; and so on. Thus for a span of any given length 

 we have at once the weights and their position, in order to 

 cause the greatest maximum moment, as also the place of this 

 moment, viz., the point vertically over that angle of the equi- 

 librium polygon nearest the centre of the span. The ordinate 

 through this point included by the equilibrium polygon, and 

 the closing line for the given span, taken to the moment scale 

 gives this moment at once ; or this ordinate taken to the scale 

 of force must be multiplied by the previously assumed pole 

 distance. 



51. Greatest moment of Rupture caused by a System of 

 moving Loads at a given Cross-Section of a Beam of given 

 Span. For beams or trusses of long span, which are as a rule 

 caused to vary in cross-section, it is not sufficient merely to find 

 the greatest maximum moment which a given system of con- 

 centrated forces can cause ; we must also know for a number 

 of individual cross-sections, the maximum moments which can 

 ever occur. 



For this purpose the force and equilibrium polygons being 

 first constructed, we shift as above the given span along a 

 horizontal line, and draw for each successive position of the 

 span the corresponding closing line in the equilibrium polygon, 

 marking the point where each closing line is intersected by a 

 vertical through the given cross-section, which of course moves 

 with the span, keeping always the same position with reference 



