102 TOWERS AND TANKS FOR WATER-WORKS. 



quantity termed the "moment of inertia," /, which, referred 

 to the neutral axis of the beam, is the product of the square 

 of the distance from that axis to all the elementary areas of 

 the cross-section, and its value is determined by summing up 

 the product of the elementary areas, multiplied by the square 

 of their distances from the neutral axis, or solving 2az* where 

 2 represents the summation, a the elementary area, and z its 

 distance from the neutral axis. 



Without demonstration, the resisting moment, R, of a 

 beam is determined by dividing the moment of inertia, /, 

 by the distance, as c, from the neutral axis to the extreme 



fibres; therefore the formula, R = -. 



c 



Modulus of Elasticity. As has been said, not only does 

 the cross-section of the beam, representing the arrangement 

 of the fibres, have to be taken into consideration in determin- 

 ing the resistance offered by a given form to an external force, 

 but the tenacity of those fibres or their cohesive force, and 

 this last consideration deals with the relative ability to resist 

 " elastic deformation " to the point of " ultimate elongation " 

 and rupture. Provided none of the stresses exceed the " elastic 

 limit " of the material, the elongation and deflection of beams 

 can be computed. 



The letter E is generally taken to represent the "mod- 

 ulus of elasticity" or the "coefficient of elasticity," rep- 

 resentative terms expressing the ratio of " unit stress" to 

 li unit deformation," and to be found by dividing the unit 

 stress, as 5, representing say, the stress in pounds per 

 square inch, by the unit of elongation which, by experiment, 

 has been found to follow the application of stress on different 



S 



materials, as s\ hence, E = '. 



s 



Under tension, and compression, experiment has deter- 

 mined that the coefficient or modulus E is practically the 



