184 



STRENGTH OF MATERIALS 



force diagram. A third condition is therefore necessaiy in order to 

 make the problem determinate. 



The problem can be solved in various ways, depending on the 

 choice of the third condition. The first solution that will be given is 

 that found by applying the principle of least work, that is, by apply- 

 ing Castigliano's condition that the work of deformation shall be a 

 minimum. 



Consider a two-hinged arched rib supporting a system of vertical 

 loads, as shown in Fig. 133. Then the moment at any point A is 

 equal to the moment of the forces on the left of the section mn 

 through A, minus the moment of P h about A y where P k is the unknown 



I n 



Fio. 188 



horizontal reaction, or pole distance of the force diagram, which is to 

 be determined. Consequently, if M denotes the moment at-1. .J/ tin- 

 moment of the forces on the left of A, and z the perpendicular distance 



of P h from A, we have 



M = M p P k z. 



Since the work of deformation due to the shear and axial l>a<l is small, 

 it may be neglected in comparison with that due to the bending i mo- 

 ment. Under this assumption the work of deformation is 



El 



in which the integral is to be extended over the entire length f 

 the rib. Applying the principle of least work to this expression, the 

 partial derivative of W with respect to the unknown quantity /',_ 

 must be zero. Hence 



