ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 603 



In structures acted on by weights in which we wish to determine the 

 deflection at any point, we may regard the points of support as the extremities 

 of pieces connecting the structure with the centre of the earth ; and if the 

 supports are capable of resisting a horizontal thrust, we must suppose them 

 connected by a piece of equivalent elasticity. The deflection is then the 

 shortening of a piece extending from the given point to the centre of the 

 earth. 



EXAMPLE. Thus in a triangular or Warren girder of length I, depth d, 

 with a load W placed at a distance a from one end, ; to find the deflection 

 at a point distant b from the same end, due to the yielding of a piece of 

 the boom whose extensibility is e, distant x from the sam6 end. 



The pressure of the support at = W j ; and if a; is less than a, the 



W 



force at x will be ~jr x(l a), or 

 al 



If x is greater than a, 

 Similarly, if x is less than b, 

 but if x is greater than b, 



x(l a) 



\ _ ' 



dl 



('-&). 



~ dl ' 



The deflection due to x is therefore Wepq, where the proper values of p 

 and q must be taken according to the relative position of a, b, and x. 



If a, b, I, x represent the number of the respective pieces, reckoning from 

 the beginning and calling the first joint 0, the second joint and the piece 

 opposite 1, &c., and if L be the length of each piece, and the extensibility of 

 each piece =e, then the deflection of b due to W at a will be, by summation 

 of series, 



762 



