Surface-Loading on the Flexure of Beams. 501 



at D and at A. If we suppose it to vary uniformly between, 

 we are not likely to be far wrong. 



" This leads to the following expression for the vertical 

 pressure in A D : — 



it \y 6V* 



"Now for the horizontal. We know that the complete 

 system of external forces must satisfy the conditions of equi- 

 librium of a rigid body. The direction in each element of the 

 fan passes through A, about which therefore the fan has no 

 moment. Hence the moment of the horizontal forces along 

 AD taken about A must equal ^Pa. Again, the resultant of 

 the semi-fan is a force passing through A, and its vertical 

 component is \ P. Its horizontal component is the integral 

 of 



2P6 2 xdn 



IT '(b 2 +xY 

 P 



taken from to infinity, or — • 



" Hence of the horizontal forces along A D we know these 

 two things : — 



p 



(1) The sum must equal — > 



IT 



(2) The moment round A must equal \Ya. 



" In default of a knowledge of the law according to which 

 the force varies with y, it is natural to take it, for a more or 

 less close approximation, to be expressed by the linear func- 

 tion A + By, or say Y. To determine the arbitrary constants 



p 

 A, B, we have only to equate the integral of Y . dy to — , and 



7T 



that of Yy . dy to J Pa, the limits being to b. We thus get 

 for the expression for the tension at any point of A D, 



b\lT b) + b \b ir) 



" At neutral points the vertical pressure equals minus the 

 horizontal tension, giving 



or, putting for shortness — '- !=■ m, 



