TIMBER PHYSICS — RELATION OF CRUSHING TO BENDING. 



375 



from the similar triangle OLT and OeZ (fig. 100), 



<yc 



Ee 



(j& r — Mp) C 





Ee 



2 ' 



whence, 



(3) d p =VWxE i 

 and after d p is found, T can be obtained: 



Now, when the vertical line LU is assumed to lepresent the real depth of the beam in inches = d, every verti- 



d 

 cal measure will be changed in the latio E ~ (see Jig. 102) ; whence, 



u 



d A 



(5) Ar- Kr . 



(real distance of neutral plane from tension side). 



(6) d ( , 



c 



J 7? ^o 



(I because E e total distortion, while d< is the distance 

 on one side of the neutral plane). 



The area OLT would then become : 



(7) T a =^ r , and the area OUCl^ 



(8) C a = (^d r )C-(2XC) 



(C a must equal T a ). 



The distance of centers of gravity would be : 



(9) *t = t<2„ 



(10) d c = — 2~~ r +4 P? 

 and the sum of internal moments. 







. 



i & 





1 * 





i 





P 



P 

 i 





i 





i 





f 



1 



a 



^^ °P A/EUTf?AL AAME 



^\ i * 



^^^- i 1 







^^v^-^ V 





Fig. 102.— Position of neutral plane at rupture. 



T 



(11) Mr = (Ca^c + T a <#t)&, and since C ft = T a , hence M T = C & (d c + d t )I>. 



But since the sum of internal moments equals the sum of external moments : 



- 4 - = Jf r = C a (^H-^)&. 



And since P r is the breaking load of the beam, and C a involves only the compression endwise strength and lineal 

 dimensions, we haAe a formula directly connecting the breaking load of a beam with the compression strength. 1 



Application of these formula, — Unfortunately no tests have been made to study the application, of these formulae 

 directly and in particular. The tests on beams published in this circular were made for a different purpose. For 

 the purpose of ascertaining the correctness of the formulae only the tests made on large beams have been utilized, 

 since m these the deflections were specially accurately measured. In addition to the quantities to be calculated 

 already given in this discussion, the fiber stress at the true elastic limit is also calculated, and called S e , to be 

 compared with C, and the load producing it, P e , is also set down as an observed quantity. If the modulus of 



rupture, R, has already been calculated by the "ordinary formula/' S e can be obtained from the relation^ =|T and 



P> 

 (12) Sc=p~It. 



The modulus of elasticity at true elastic limit E c is recomputed as a check, and of course is : 



S 



(13) E e -=j£. 



Since P t is an arbitrary quantity within certain limits, and can not be determined with any degree of accuracy, 

 S e will be found to differ more or less from C. For these reasons, however, C is a more reliable value for the true 

 elastic limit than S e itself, and in the formulae is used as such; for instance, K e is the fiber distortion produced by the 

 same load which produces a fiber stress ==C, not by the load which produces S e . 



The following table exhibits the results of applying the formulae to the data from these tests: 



pThe factors d c ~\~d t , within such limits as the cross-bending strength is constant, are constants; they will have 

 to be ascertained by actual experiment for each species and quality, and might then be expressed as a proportion of 

 the depth. In the material used, pine as well as oak, it appears to be about 3/5. The material on which this rela- 

 tionship has been mainly studied was green wood, and it may be questioned whether the factors d Q and dt would 

 remain the same in material of all moisture conditions. There is no logic which would, lead us to expect a difference 

 greater than the limits of "maximum uniformity," i. e., 10 per cent. A few comparisons of data obtained from 

 material of otuer species witlj varvfng jnoisture percentage indict© that a difference, 4o,§s, pot exist?— B. $, $\;j 



