34 



STRENGTH OF MATERIALS 



direction of the axis, called the axial loading, and the other perpen- 

 dicular to the axis, called the shear. 



38. Common theory of flexure. In the majority of practical cases 

 of flexure (or bending) of beams, the external forces acting on the 

 beam all lie in one plane through its axis and are perpendicular to 

 this axis. The single force through the center of gravity .f any cross 

 section is then perpendicular to the axis of the beam, and tin- plane of 

 the moment passes through this axis. The theory based on the assuiui- 

 tion of this condition of strain is called tin- common theory of flexure. 



39. Bernoulli's assumption. In order to obtain a starting point 

 for the analysis of stress in beams, the arbitrary assuinj 



tltat a cross section of the beam which 

 plane before flexure rein 

 flexure. This assumption was first made 

 by Bernoulli, and since his time has formed 

 the basis for all investigations in the theory 

 of iKiams.* 



40. Curvature due to bending moment. 

 The effect of the external mon,. 

 beam originally straight is to cause its 

 to become bent into a curve, called the elas- 

 tic curve. Si nee, by Bernoulli's ass u 1 1 1 ;, 

 any cross sect i'ii of the beam remains iden- 

 tical with itself during deformat i< n, 

 FIG. 19 consecutive cross sections of the beam \\ 



were perpendicular to its axis before flexure will remain perjenli -u- 

 lar to it after flexure, and will therefore intersect in a o 

 vature of the elastic curve, as shown in Fig. 19. 



The fibers of the beam between these two cross sections were 

 originally of the same length. After flexure, however, it will U found 

 that the fibers on the eonvex side have been lengthened by a certain 

 amount AB y while those on the concave side have been 1 by 



an amount C7).f Between these two there must lie a strip of 1 



* St. Venant has shown that Bernoulli's assumption is rigorously true only for certain 

 forms of cross section. For materials hi-h r,.nf..rm to Hooke's law, however, it is 

 sufficiently exact to assure results approximately corr 



t This can be shown experimentally by placing two thin steel strips In : 

 grooves in a wooden beam, one on the upper side and the other on the lower side, so that 



