THE PLASTIC STATE. 39 



In the foregoing Table I. are the formulae, which are 

 most likely to be of use in testing work. 

 Here let W = the whole load on the beam. 



,, I =the length of the beam. 



., E=the modulus of elasticity of the material. 



,, 8 = the deflection caused by the load W. 



I =the moment of inertia of the section. 



,, M = the maximum bending moment. 

 It should not be forgotten that the linear dimensions must 

 all be taken in the same units, and the same remark applies 

 to the forces. 



Table II. contains the moduli and moments of inertia 

 of the most commonly occurring beam sections will also be 

 found useful for reference. More complicated sections 

 should be treated graphically. 



A most important thing to remember is that all the 

 relations as to stress and strain which have just been 

 described are only true so long as the beam is perfectly 

 elastic. When the elastic limit has been passed these 

 laws no longer hold. 



15. Stresses and Strains in the Plastic State. It has 

 already been stated that if a strain following a stress is 

 in any way permanent, then the material under considera- 

 tion is in a more or less plastic state. 



In the case of most of the metals stresses of a certain 

 magnitude produce almost perfectly elastic strains ; if these 

 stresses are increased beyond a fixed point, a condition of 

 semi-plasticity is reached, and the plastic condition becomes 

 more pronounced as the stresses are further increased. 



Thus if the loads upon a bar of mild steel be small, they 

 will produce elastic extensions and the bar will return to 

 its original length on the removal of the load ; the 

 extensions during this elastic stage are very small, and are 

 only capable of being measured by means of delicate 

 appliances. So long as the stress does not exceed about 

 15 tons per square inch the elastic condition will be 

 maintained ; but when the stress is increased to much 

 beyond this point, the semi-plastic extensions begin, and 

 what is called permanent set takes place. The point where 

 elasticity ceases and plasticity begins is called the elastic 

 limit. When this point is passed, the plasticity is only 

 partial and some elasticity remains. As the load is 

 increased, however, the plasticity becomes more and more 

 perfect, and when the point of fracture is nearly reached, a 

 state very nearly approaching to true plasticity is arrived at. 



