OF IRON BEAMS. 41^^ 



= Xf any double ordinate on the surface of 

 compression = Y, its distance from AB = y; 

 call moreover the weight sustained by the 

 stretched fibres in a unity of section at the 

 top of the beam = ^, and the cotemporary 

 resistance of the same quantity of fibres at the 

 bottom = r. 



We have then a : a; :: / : — i= force of stretched 

 fibres in a unity of section at distance x from 

 AB, whence — x Xdx = force in X, when the 

 thickness of X is dx. Integrating the force 

 in X gives --fXxdx = sum of the forces of 



tension = F- ; whence— J^ X x^dx = sum of the 



forces of tension multiplied by their distances 

 from AB = F X P F. 



In like manner we should have for the forces 



of compression, ^ jYydy—f\ ^n^~jYy^dy 



=/' X P/. But to obtain the value of r in 

 terms of 5, suppose the forces necessary to 

 produce an equal extension and compression 

 in a fibre to be as 1 to « ; then, if the extension 

 from t be unity, nt would be the force to 

 produce the same compression in the same 

 fibres* But the extension at the top of the 

 beam is to the compression at the bottom as 



