374 FORESTRY INVESTIGATIONS U. S. DEPARTMENT OF AGRICULTURE. 



KELATION OF CRUSHING- ENDWISE STRENGTH. 



The second relation (that of crushing-endwise strength to internal stresses) was touched upon in discussing the 

 first, when it -was stated: (1) That the true elastic limit of the beam is probably leached at the same instant that 

 the extreme fibers on the compression side leach their elastic limit in compression. (2) That this latter limit lies 

 close to the ultimate compression-end wise strength (so close that former experimenters have been unable satisfactorily 

 to separate them). (3) That a piece of gieen wood will stand a great deal of distortion after the ultimate load is 

 applied before actually failing. And to these statements may be added the evident fact (4) that the stress on any 

 fiber on the compression side can not exceed the compression-endwise strength of the material. (5) Finally and 

 most important it appears from (1) and (2), but especially irom an examination of the several thousand test results 

 on the several species of conifers made by the Division of Forestry, that the extreme fiber stress at the true elastic 

 limit of a beam is practically identical with the compression-endwise strength of the material. (This last observa- 

 tion, which was forced upon the writei by its continual repetition in the large series of tests under review, lies at 

 the basis of this discussion.) The observation of this identity makes the distribution of internal stresses appear 

 more simple than was hitherto assumed, and the desired relation between compression and cross-bending strength 

 capable of mathematical expression. 



DE\ELOPMENT OF FORMULAE. 



From these considerations the distance UC in fig. 100, which represents the ultimate compression-endwise 

 strength of the material, becomes practically equal to the distance el, which represents the compression strength at 

 the true elastic limit, and hence the line IC straight and veitical ; and if OT is taken as straight, the diagram will 

 be made up of simple geometric figures, as in fig. 100. 



The line LU will represent the total fiber distortion at time of rupture, and is equal to the sum of the amounts 

 by which the extreme compression fibers shorten and the extreme tension fibers elongate. 



Let a test in which the following quantities have been observed and recorded be considered: 



Let P r = the external load at rupture (pounds). 



z/ r = the corresponding deflection of the beam (inches). 

 C = compression-endwise strength of the material (pounds). 

 E = modulus of elasticity (pounds). 

 d .= depth of beam (inches). 

 b = breadth of beam (inches). 

 l=z length of beam (inches). 

 jd L = deflection at true elastic limit. 



Then, based upon the above statements, by means of formulas dei ived from the geometric relations of the diagram 

 and the fundamental equations of equilibrium, the following quantities can be calculated : 



Let E c = total fiber distortion due to bending at true elastic limit (inches). 

 2£ r = total fiber distortion due to bending at rupture — LU (inches). 



d p = distortion in extreme tension fiber at rupture=LO (inches); also the proportional dis- 

 tance of neutral plane from tension side of beam. 

 tf r = reaJ distance of neutral plane at rupture from tension side of beam (inches). 

 d L =real distance of neutial plane at ruptuie from that fiber on compression side which has 



just reached the elastic limit, in inches™ Oe. 

 T = stress in extreme tension liber (pounds). 

 T a = sum of forces on tension side — aiea OLT (pounds) 

 Ca = sum of forces on compression side — area OUC/ (pounds). 

 #t = distance of center of gravity of tension area from neutral plane (inches). 

 d L ~ distance of center of gravity of compression area fiom neutral plane (inches). 

 M r = sum of the internal moments about the point (inch-pounds). 



The formulas connecting these quantities are derived as follows: 



To find E c let fig. 101 represent a portion of the beam one unit in length bent to its elastic 

 limit; then, 



Pig. 101.— Fiber dis- 1 r" 



tortion m unit 



length of beam, at where r i& the radius of curvature, but from fundamental formulas true at elastic limit 

 elastic limit. 



> ~EI~~1ET~ j2~-''W^c — —jr ' 



Since this involves only geometric relations, it is true also at rupture (since the beam preserves its original form). 



12J r d 



(2) E 



P 



To find d» and T : 



Since the sum of stresses on the tension side = sum of stresses on compression side, 



the area OLT = area OUCZ.\| p T==(E r — d p )C — ~p and T=|^ 



