414 Mr. R. C. Nichols on_ the Resistances 



centre of gravity is 



The foregoing equations are sufficient to determine the con- 

 ditions under which the facts relating to transverse strength 

 may be reconciled with those of tensile strength without 

 recourse to the supposition of any kind of resistance other 

 than the known resistances to extension and compression. 



The truth of the theory and its adequacy to account for the 

 facts shown by experiment may be tested by comparing the 

 amounts of deflexion calculated to exist according to the 

 theory with those experimentally determined. 



For this purpose it is necessary to find the curve assumed 

 by a beam, and its consequent deflexion, under the supposed 

 condition of overstrain. 



Now it is evident that the area CLHMD of maximum 

 tensile stress is the same as that of an equal stress would be 

 if the stress continued to vary with the strain down to the 

 lower side of the section, arid every horizontal dimension of 

 the beam between L C and M D were reduced in the propor- 

 tion which the actual stress at any horizontal line bears to 

 that which would then exist at that line, that is as c d to ef. 



(cd) 2 

 Make tv= — -~, and draw lines from L and M through all 



such points t, v, to meet the base in T and Y. Then the 

 actual strength of the section A C D B is the same as would 

 be that of the section ALTVMB if the stress continued to 

 increase with the strain -to its utmost limit. And its de- 

 flexion under any given load must likewise be the same. 



The centre of gravity of the section ALTYMB is the 

 point H, which has been already determined to be the neutral 

 axis of the section A D B as subjected to overstrain. And 

 the neutral axis of ALTYMB will coincide with its centre 

 of gravity. 



The moment of inertia of this section about that point is 

 readily found to be 



T bd 3 /(2>/o— <r) 8 <j z 0/1 , w., , A 



= gcr(3-2^cr)-T|(3-0 2 ? ....... (7) 



(where I is the moment of inertia of the section A C D B 

 about its centre of gravity), which is therefore the moment 

 of inertia of the reduced section at the centre of the length of 

 the beam. 



