4 RESISTANCE OF MATERIALS 



3. Strain diagram. In the case of tension or compression it is 

 easy to show graphically the chief features of the strain. Thus, 

 suppose that a test bar is placed in a testing machine, and that the 

 total load on the bar at any instant is read on the scale beam of 

 the machine, and its corresponding length in inches is measured 

 with an extensometer. Assuming that the stress is uniformly dis- 

 tributed over any cross section of the bar, the unit stress is obtained 

 by dividing the total load in pounds acting on the bar by the area 

 of its cross section in square inches. That is, 



total load in pounds 



(1) p = unit stress = . 



area of cross section in square inches 



Also, the total deformation, or change in length, is divided by the 

 original unstrained length of the bar, giving the unit deformation 

 in inches per inch. Let this be denoted by s ; that is, let 



., , - change in length 



(2) s = unit deformation = - ^ . 



original length 



The unit deformation is therefore an abstract number. Moreover, 

 both the unit stress and the unit deformation are independent of the 

 actual dimensions of the test bar and depend only on the physical 

 properties of the material. 



If, now, the unit stresses are plotted as ordinates and the corre- 

 sponding unit deformations as abscissas, a strain diagram is obtained, 

 as shown in Fig. 2. Such a diagram shows at a glance the physical 

 properties of the material it represents, as explained in what follows. 



4. Hooke's law. By inspection of the curves in Fig. 2 it is 

 evident that the strain diagram for each material has certain char- 

 acteristic features. For instance, in the case of wrought iron the 

 strain diagram from to A is a straight line ; this means that for 

 points between and A the stress is proportional to the correspond- 

 ing deformation. That is to say, within certain limits the ratio of 

 p to is constant, or 



(3) = Jf, 



where the constant E denotes the slope of the initial line. This 

 important property is known as Hooke's law, and the constant E 

 is called Young's modulus of elasticity. 



