STRESS AND STRAIN. I5 



next step in the development was perhaps the realization of 

 the fact that the actual maximum stresses in the beam were 

 not horizontal (flexural) stresses together with horizontal and 

 vertical equal shears, but that the maximum stresses are usu- 

 ally in some other direction. That is, the horizontal stresses 

 and horizontal and vertical shears xnust be resolved in dififer- 

 ent directions and the direction of their maximum resultants 

 for tension, compression and shear ascertained. The forces 

 to be resolved are indicated in Figure i. 



If we plot the directions of the maximum stresses as 

 found from the above resolutions, curves somewhat similar- 

 to those of Figure 2 are obtained, the heavy lines being ten- 

 sion and compression and the dotted lines representing shears. 

 The two direct stresses are at right angles to each other and 

 also the shears are mutually at right angles, while the two 

 systems of lines make angles of 45° with each other. 



This was a long stride in the development of the theory. 

 Yet it must be borne in mind that up to this point the theory 

 does not take into consideration the question of strain. 

 Hence in the actual beam where deformation does take place, 

 it is evident that some modification of the theory must be made. 



The lines and calculated stresses of Figure 2 are based 

 upon statically determined horizontal stresses and vertical 

 and horizontal shears unaffected by lateral stresses. The hor- 

 izontal stresses themselves produce 45° shears which, however, 

 appear in the general resolution when the horizontal stresses 

 are combined with the applied vertical and horizontal shears in 

 order to find the amount and direction of maximum shears. 

 So much for statical conditions without lateral deformation. 

 When, however, lateral strain is considered, vertical and hori- 

 zontal forces at right angles to the axis come into play. 

 Hence, limiting ourselves to the vertical-longitudinal plane, 

 the forces to be resolved are those of Figure i, together with 

 vertical compression or tension, as the case may be. We 

 might resolve the new combination of forces as before and 

 find corrected directions and values for maximum stresses 



