6 STRENGTH OF MATERIALS 



to take place. At this stage of the experiment, indicated by C on the 

 diagram, the material in the neighborhood of the place where rupture 

 is to occur begins to draw out very rapidly, and in consequence the 

 cross section of the piece diminishes at this point until rupture occurs. 



Within the portion OA of the strain diagram the stress is pro- 

 portional to the deformation produced, and the material may be con- 

 sidered to be perfectly elastic. For this reason the point A, which is 

 the limit of proportionality of stress to deformation, is called the 

 elastic limit The point B, at which the first signs of weakening occur, 

 is called the yield point 



In commercial testing the tests are usually conducted so hurriedly 

 that the position of the point A is not noted, and consequently the 

 yield point is often called the elastic limit. The yield point, however, 

 is not the true elastic limit, because plastic deformation begins to he 

 manifested before this point is reached, namely, as soon as the stress 

 passes A. 



At C the tangent to the strain curve is horizontal. Therefore the 

 ordinate at this point indicates the maximum stress preceding rup- 

 ture, which is called the ultimate strength of the material. 



8. Hooke's law and Young's modulus. The fact that within the 

 elastic limit the deformation of a body is proportional to the stress 

 producing it was discovered in 1678 by Robert Hooke, and is there- 

 fore known as Hooke's law. It can be stated by saying that the ratio 

 of the unit stress to the unit deformation is a constant; or, expressed 

 as a formula, 



where E is a constant called the modulus of elasticity. E is also called 

 Young's modulus, from the name of the first scientist who made any 

 practical use of it. 



Since s is an abstract number, E has the same dimensions as p 

 and is therefore expressed in Ib./in. 2 Geometrically^ is the slnjie 

 of the line OA in Fig. 2. 



The answers given to the following problems were obtained by 

 using the average values of Young's modulus given in Article L'L'. 



Problem 6. A steel cable 500 ft. long and 1 in. in diameter is pulled by a force 

 of 25 tons. How much does it stretch, uiul what is its unit elongation ? 



