i.iNi-:s oi' inti-;knai. stkicss. 79 



shades insensibly into the other in passing- the central point of 

 the l)eani. In this connection, however, attention will be 

 called to the fact that at the central point of the beam and the 

 four corners the curves must satisfy the condition of pointing 

 in two directions, making an angle of 45° with each other, at 

 the same time. This indicates no fault in the theory, never- 

 theless, for it is precisely at these points that the formulas show 

 the stresses to become zero and the directions indeterminate. 



As mentioned before, the two right-angled shears have the 

 same value at any point. This is a necessaiy condition of inter- 

 nal equilibrium, as otherwise the element under consideration 

 would suffer rotation. On the contrary, the direct stresses usually 

 have different values, there being no condition of equilibrium 

 requiring their equality. The diagram shows how wonderfully 

 intricate is the interplay of stresses throughout the beam. 



In what has preceded, the subject has been treated under 

 static conditions, that is, as though the beam were perfectly 

 rigid and loaded and supported as mentioned above. Under 

 ordinaiy circumstances, however, the beam bends slightly, 

 shortening and broadening in its upper half, and lengthening 

 and narrowing in the lower half. This change of configiira- 

 tion of the particles, as before stated, calls into play certain 

 definite stresses other than those necessary for external equili- 

 brium. For strains within the elastic limit of the material, 

 the value of the true maximum stresses will, therefore, be 

 slightly different from the theoretical stresses, although the 

 direction of the lines of maximum stress will be the same. 

 That is, the lines of the figure, supposing them to have been 

 plotted to scale, are still valid. When, on the other hand, the 

 strain exceeds the elastic limit so that we have to deal with 

 internal friction and the flow of solids, then l)oth the values 

 and direction of the maximum stresses materially differ from 

 those just found. In other words, while the lines in the figure 

 give the direction of the maximum stresses in the beam when 

 normally loaded, they by no means give the lines along which 

 ruptirre will take place. 



