CHAPTER III 



ANALYSIS OF STRESS IN BEAMS 



37. System of equivalent forces. The theory of beams deals, in 

 general, with the stresses produced in a prismatic body by a set of 

 external forces in static 'equilibrium. Ordinarily these forces all lie 

 in one plane ; in this case it is proved in mechanics that they can be 

 replaced by a single force acting at any given point in this plane, 

 and a moment. To balance this equivalent system of external forces, 

 the stresses acting on any cross section of the beam must also consist 

 of a single force and a moment, the point of application of this single 

 force being conveniently chosen as the 

 center of gravity of the cross section. 



The fulluwing special cases are of fre- 

 qiirnt <ccurr- : 



It the moment is zero and the single 

 force through the center of gravity of a 

 cross section acts in the direction of the 

 axis of the beam,the strain is simple tension 

 or compression ; if it is perpendicular to the axis ..f the beam, the strain 

 is simple shear. 



If the single force is zero and the plane of the moment passes 

 through the axis ..f the beam, pure bending strain occurs; if the single 

 force is zero and the plane of the moment is perpendicular to the 

 axis of the beam, a twisting strain called torsion is produced. These 

 two cases are illustrated in Fig. 18, A and B. 



If the plane of the moment forms an arbitrary angle with the axis 

 of the beam, the moment can l>e ivsnlved int<> two components whose 

 I >la nes are parallel and perpendicular respectively to the axis of the 

 beam. In this case the strain < -(insists of combined bending and torsion. 



If the single f<>n-e thr<>iii;h the center of gravity is inclined to the 

 f the brain, iu can be ivsulvnl intntxvo components, one in the 



