154 



STRENGTH OF MATERIALS 



Hence the expression for p becomes 



M(y 



FIG. 107 



(100) j 



The maximum stress is the value of p for y = - Consequently, 



M (h + 2 d) 



From this formula, it is evident that the stress is greatest where the 

 radius of curvature is least, a result which is 

 amply verified by experience, as mentioned 

 in the following article. 



Problem 151. A boat's davits are compos- 



two wrought-iron bars '2( in. squaiv. lnit to a radius 

 of 2 ft, as shown in Fii:. 1<>7. If the boat \\ 

 500 Ib. and is hung 8J ft from the vertical axis of 

 the davits, find the maximum stress in the davits and 

 the factor of sat 



127. Effect of sharp curvature on bending 

 strength. Consider a sharply curved pris- 

 matic piece which is subjected to bending strain, l-'mni the above 

 discussion, it is known that for aseeti<>n taken in the neighborhood of 

 the bend, the neutral axis does not coincide with the gravity axis but 

 approaches the center of curvature. The neutral 

 fiber is therefore separated from the mean liber, 

 or axis of the piece, and takes some such posi- 

 tion as that shown by the broken line in Fig. 108. 

 Consequently the inner fiber through A must \ \\ 

 endure a far greater stress than that deduced 

 from formulas for the straight portion. Engi- 

 neers and constructors have learned by experi- 

 ence that sharp curvature produces weakness 

 of this kind, and that it is necessary to reenforce a piece at a bend 

 either by increasing its diameter or by adding a brace. 



As an illustration of the effect of sharp curvature <>n bending 

 strength, suppose that a bar of rectangular cross section is bent into a 

 right angle, as shown in Fig. 109. In this case the center of curvature 



. 



FI... 



