CHAP. XH.] METHODS COMBINED. 175 



Now in any framed structure of the above kind, the counter 

 ties are inserted to prevent the deforming action of the rolling 

 load only. For the dead load but one system of triangulation 

 is required, and the strains in every piece due to this dead load 

 can therefore easily be determined. 



We have then only to determine the strains in the same pieces 

 due to the rolling load also. If now in any diagonal the strain 

 due to this rolling load exceeds the constant strain due to the 

 dead load, and is of opposite character, and if the diagonal is 

 to be so constructed as to take but one kind of strain, then a 

 counter diagonal must be inserted in that panel, and propor- 

 tioned to this excess of strain only. For instance, if a diagonal 

 takes only the compressive strain (a condition which is easily 

 secured in practice) due to the dead load, and the live load 

 would cause in that diagonal a tensile strain, then the excess of 

 this tensile strain over the constant compressive strain due to 

 the dead load must be resisted by a counter diagonal, which 

 also takes compressive strain only. The method is precisely 

 the same as by calculation (see Stoney and other authors on 

 the subject), and we only notice the point here, as in all our 

 examples we have taken a single system of triangulation only 

 a system which, we may here remark in passing, has many ad- 

 vantages, and is worthy of more general attention * than it has 

 hitherto obtained. 



[See also on this point Art. 10 of Appendix.] 



116. Beam fixed horizontally at one end, supported at 

 the other Supports on Level. In this case, equation (1), 

 Art. 112, becomes for left end fixed, since T& = 1, ^5 = 0, * = 0, 



*i a I 



x = 



, 1 ft) " *1 <* 



But for this case the distance of the point of inflection from 

 the fixed end is 



. _ (2 I a) I a , 



* See " A Treatise on Bracing." By S. H. Bow. D. Van Nostrand, pub- 

 lisher. 



f The values of the distance of the inflection points which we assume above 

 as known, may easily be deduced by the theory of elasticity. See Supplement 

 to Chap. VII., Arts. 16 and 19. See Wood, Strength of Materials; 

 Mecanique Appliquee ; or other treatises on the subject. 



