12 LIVE-LOAD STRESSES 



value when the wheel is to the left of the point. A change, 



dS 

 therefore, of -r- from + to would not result. Similarly, 



it may be shown to be unnecessary to seek a numerically 

 maximum negative stress by trying wheels at any salient 

 point which has a negative coefficient. 



Formulas (7) and (8) are the general formulas for the 

 solution of the sum of the ordinate-load products of an 

 influence line and the rate of change of this sum, and are 

 applicable to any form of influence line. They give at once 

 a definite solution of the position of a set of loads produc- 

 ing maximum positive and negative stresses in any member 

 of any truss or girder for which an influence line can be 

 drawn and the values of such stresses. The method is 

 particularly advantageous in the case of statically indeter- 

 minate structures, such as two-hinged and no-hinged an h< - 

 swing bridges, continuous girders, etc., where general ana- 

 lytical criteria for the positions of loads producing maxi- 

 mum stresses cannot readily be expressed and where such 

 maximum stresses have had to be found by assuming P<M- 

 tions of loadings and scaling the influence-line ordinates 

 under all the loads, a laborious process and one open to 

 much liability of mechanical inaccuracy. 



In applying the present method to the simple forms of 

 girders and trusses (viz., the statically determinate struc- 

 tures where the ordinates of the influence lines are readily 

 expressible algebraically) it will generally be more conve- 

 nient to transform formulas (7) and (8) in each case whereby 

 the coefficients C may be expressed in terms of the geo- 

 metric proportions of the truss or girder. This, in the fol- 

 lowing articles (4 to 7 inclusive), we shall proceed to do for 

 the case of girder bridges (with and without panels), pier 

 reactions, and through Pratt trusses with curved or hori- 

 zontal chords. The general method will, however, be ap- 

 plied directly to the case of the three-hinged arch in Art. 

 8, which will serve as a typical example of the application 

 of the method to any influence line. 



