i8o.] GRAPHICAL METHODS. IO / 



the stress diagram for half the frame. At the vertex F, a difficulty 

 arises, there being apparently three members whose stresses are not 



I 



Fig. 52. 



known from the previous construction ; but on account of the symmetry 

 with respect to EF, the members FG and FH must have equal stresses. 



180. Shearing Force and Bending Moment. Consider a hori- 

 zontal beam fixed at one end A (Fig. 53), and acted upon at the 

 other end B by a vertical force F. If the beam be cut at any 

 point C of its length, and the equilibrium of the portion AC be 

 considered, the action on AC of the 

 portion removed must be replaced 



by its equivalent. Now the force B .C . 



F at B is equivalent, by Art. 136, 



to an equal and parallel force F at I 



C in connection with a couple whose Fig. 53. 



moment is F- BC. 



The force F at C is called the shearing force of the cross- 

 section C, and the moment F-BC the bending moment at C. 

 Both are of great importance in engineering, as their combined 

 effect represents what must be overcome by the resistance of 

 the material of the beam, i.e. by the internal forces holding 

 together its fibres. 



These definitions are readily generalised. Let any beam or 

 girder, supported in any manner, and acted upon by any number 

 of vertical forces, be divided by a vertical cross-section into two 

 portions A and B. For the portion A the shearing force at the 

 cross-section is the sum of all the external forces acting on B ; 

 and the bending moment is the sum of the moments of all these 

 forces with respect to some point in the cross-section. 



