Art. 80. STRESSES BEYOND THE ELASTIC LIMIT. 



121 



girders whose discussion would be foreign to the purpose of this 

 article. 



80. Beams Stressed Beyond the Elastic Limit. When the 

 extreme fiber stress in a beam is greater than the elastic limit 

 of the material, the theory of flexure is no longer applicable, 

 and the equations based upon it are not applicable. The discrep- 

 ancies between the theory of flexure and the facts in the case 

 are negligible within the elastic limit, but not beyond it. The 

 stresses do not increase uniformly from the neutral axis out- 

 ward, even though the deformations may sensibly do so ; Hooke 's 

 law does not hold. Hence experiments on the ultimate strength 

 of beams can not confirm or disprove the theory of flexure. 



If the extreme fiber unit stress at rupture is calculated by 

 equation (10), it will be found to be much greater than the 

 ultimate strength in tension or compression, unless there is con- 

 siderable difference in these, in which case it will lie between 

 them; its value will depend upon the form of the cross section 

 to a large extent. Timber being stronger -in tension than in 

 compression, fails on the compression side of a beam ; the reverse 

 is true with cast iron. For a rectangular cross section s u in a 

 cast iron beam will be nearly twice the ultimate strength in 

 tension. 



The discrepancy between the ultimate strength in bending 

 as determined by equation (10), and the ultimate strength in 

 tension or compression, is explained as follows : 



1. The stresses do not increase outward from the neutral 

 axis as rapidly as for stresses within the elastic limit. 



2. The neutral axis moves toward the stronger side as the 

 stresses increase, since equilibrium requires the total tension to 

 be equal to the total compression. 



3. The interaction of the fibers, preventing transverse 

 deformation as explained in the third assumption under the 

 theory of flexure (68), has a much greater influence for stresses 

 beyond the elastic limit than for stresses within it ; in both cases 

 this influence is such as to increase the strength of the beam in 

 bending. 



4. The influence of the shearing stresses is greater beyond 

 the elastic limit than within it. 



5. The usual ultimate strength in tension is not the real 



