194 On ike Strength of Beams. 



part of the beam which is neither extended nor compressed : 

 then the curvature at any point, is as the extension of the inde- 

 hnitely small part cd, is to its original .^^^ ^^ 



dimension alj, which varies as the strain, -— -j — r — '^ 



that is as the length and weight ; but A-- «l >^^ •■» 



the deflexion is as the curvature and as -...,,^^^ f I ^^^^-^^ 

 the sum of the extensions: therefore it '~~^ — ^ 



is as L*W. 



Suppose a beam, of a given length L, to be supported in the 

 middle, and loaded at each end, and let the deflexion be E; then 

 conceive a part of each end of the beam to be taken away, and 

 at the same time let the ends of the remaining beam be loaded 

 so that the strain in every part may be exactly the same it was 

 before the ends were taken away. Let / be the length of the 



remaining beam ; then the deflexion will be as L^ : Z^ : : E : — - . 



It is obvious the curvature is exactly the same in both cases. If 

 I do not mistake, this imaginary experiment is the same with 

 the uniform curvature of your correspondent. But it is evident, 

 that when we say the deflexion is as the square of the length, 

 we do not suppose the weight to have been varied, but it was 

 varied as the length, that the strain might remain the same : 

 therefore, the deflexion is as the length and weight. Hence the 

 deflexion is as the length and weight when the curvature is the 



same ; and the curvature will vary as the strain : therefore, 



is as tlse deflexion. 



The deflexion of beams at the time of fracture, when the 

 dej'th is very small in proportion to the length, is subject to se- 

 veral irregularities : there is one which I do jio recollect having 

 seen noticed, that is, the sliding of the ends of the beams on 

 ti\e supports: this motion sometimes gives such a mon.entum to 

 tlie weight, that the beam breaks with a much less weight than 

 it would have done had ihe deflexion been gradually increased. 



In practice, where we wish the deviation from the natural po- 

 fiuion to be as small as possible, the rule given for the stiffness 

 of beams will be found as correct as the nature of the subject 

 will admit of. 



I am, sir. 



Your obedient servant, 



Sept. 5, 1815. T. T. 



XXXV. Oi- 



