50 



ON THE EQUILIBRIUM AND STRENGTH OF ELASTIC SUBSTANCES. 



in their intimate constitution to be perfectly elastic, yet 

 it often happens that their toughness with respect to exten- 

 sion and compression differs very materially. In general, 

 bodies are said to have less toughness in resisting extension 

 _?han compression. 



335. Theore.m. The transverse strengtli 

 of a beam is directly as the breadth and as 

 the square of the depth, and inversely as the 

 length. 



The strength U limited by the extension or compression 

 wliich the substance will bear without failing ; the curva- 

 ture at the instant of fracture must therefore be inversely as 

 the depth, and the radius of curvature as the depth, or 



Hm hm 



■ — -, as b, consequently bm must be as a/*, and/ as — , or, 



bhh 

 since m is as Ih, as — . 

 a 



SciioMUM. If one of the siufaces of a beam wrerc in- 

 compressible, and the cohesive force of all its strata collect- 

 ed in the other, its strength would be six times as great as 

 in the natural state ; for the radius of curvature would be 



—— , which could not be less than twice as great as in the 

 oj 



natural state, because the strata would be twice as much 



extended, with the same curvature, as when the neutrd 



point is in the axis ; and/ would then be six times as great. 



3.36. Definition. The resilience of a 

 beam jnay be considered as proportional to 

 the lieight from which a given body must 

 fall to break it 



337. Theorem. The resilience of pris- 

 matic beams is simply as their bulk. 



The space through which the force or stiffness of a beam 

 acts, in generating or destroying motion, is determined by the 

 curvature that it will bear without breaking ; and this cur- 

 vature is inversely as the depth-, ■consequently, the depres- 

 sion will be as the square of the length directly, and as the 

 depth inversely : but the force in similar parts of the spaces 

 to be described is every where as the strength, ex as the 

 square of the depth directly, and as the length inversely t 

 . therefore the joint ratio of the spaces and the forces is the 

 ratio of the products of the length by the depth ; but this 

 ratio is that of the squares of the velocities generated or des- 

 troyed, or of the heights from which a body must fall to 

 acquire these velocities. And if the breadth vary, the force 

 Will obviously vary in the same ratio ; therefore the resili- 

 ence will be in the joint ratio of the length, breadth, and 

 ilcpth, ^ 



338. Theorem, The stiffest beam that 

 can be cut out of a given cylinder is that of 

 which the depth is to the breadth as the 

 square root of 3 to 1, and the strongest as the 

 square root of 2 to ] ; but the most resilient 

 will be that which has its depth and breadth 

 equal. 



Let the diameter or diagonal be o, and the breadth x ; 

 then the depth being ^/■ {aa — xx),the stiffness is {aa — xxfx, 

 and the strength aax — i*, which must be maximvims ; 

 and [aa — xxYxx must be a maximum ; so that 3(oa — xt)'. 

 ( — 2ir).xx-t-(aa — Tr)'(2.ri)— 0, aa — rir:3xx; and the 

 squares of the breadth and depth are as 1 to 3 ; also aaizz 

 3x'i,x^^ia, and the depth v' jo, for the strongest form. It 

 is evident that the bulk, and consequently the resilience; 

 will be greatest when the depth and breadth are equal. 



33Q. Theorem. Supposing a tube of 

 evanescent thickness to be expanded into a 

 similar tube of greater diameter, but of 

 equal length, the quantity of matter remain- 

 ing the same, the strength will be increased 

 in the ratio of the diameter, and the stiffness 

 in the ratio of the square of the diameter, but 

 the resilience will remain unaltered. 



For the quantity of matter remaining the same, its actioa 

 is in both cases simply as its distance from the fulcrum, or 

 from the axis of motion, and this distance is simply as tho 

 diameter, since the section remains similar in all its parts : 

 the tension at a given angular flexure being also increased 

 with the distance, the stiffness will be as the square of the dis- 

 tance, and the force in similar parts of the space described 

 being always inversely as the space, the square gf the velo- 

 city produced or destroyed will remain unaltered. 



Scholium. When a beam of finite thickness is made 

 hollow, retaining the same quantity of matter, the strength 

 is increased in a ratio somewhat greater than that of the 

 diameter, because the tension of the internal fibres at the 

 instant of breaking is increased. 



340. Theorem. If a column, subjected 

 to a longitudinal force, be cut out of a plank 

 or slab of equable depth, in order that the 

 extension and compression of the surfaces 

 may be initially every where equal, its out- 

 line must be a circular arc. 

 Neglecting the distance of the neutral point from the sxis. 



