372 FORESTRY INVESTIGATIONS U. S. DEPARTMENT OF AGRICULTURE, 



and, thougu little more was accomplished than to find proper ways, the study of these results, 

 amplified by the large ordinary series, led to several interesting discoveries, the most important 

 of which is the discovery of the relation between the strength in cross bending at elastic limit 

 and the compression endwise, this latter being equal to the fiber stress of the former. Though 

 still requiring special experiments to become convincing, it is fair to state at this point that a 

 great deal of useless testing will be saved in the future, since the test in compression is by all 

 means the simplest, the selection and treatment of the material for it the easiest, and the result 

 the most satisfactory. The importance of this discovery by Mr. S. T. Neely is such that a reprint 

 of Mr. Eeely's discussion here will be found justified. 



Relation op Compression-endwise Strength to Breaking Load of Beam. 



In testing timber to obtain its various coefficients of strength, the test which is at once the simplest, most 

 expedient, satisfactory, and reliable is the " compression-endwise test," which is made by crushing a specimen 

 parallel to the fibers. All other tests are either mechanically less easily performed, or else, as in the case of cross- 

 bending, the stresses are complex, and the unit coefficient can be expressed only by reliance upon a theoretical 

 foimula, the correctness of which is in doubt. It would, therefore, be of great practical value to find a relation 

 between the cross-bending strength, the most important coefficient for the practitioner, and the compression strength, 

 when the study of wood would not only be greatly simplified and cheapened, but the data could be applied with 

 much greater satisfaction and safety. 



The consideration of such a relation resolves itself naturally into two parts, namely, a study of the relation o± 

 the internal stresses in a beam to the external load which produces them, and a study of the relation of the internal 

 stresses in a beam to the compression-endwise strength of the material of which the beam is made. 



The first relation has been a subject of study for more than two centuries, and from the time of Galileo down to 

 the present day the theory of beams has been gradually evolved. Within recent years several eminent physicists 

 and engineers have given a true analysis of both the elastic and ultimate strength of a beam, a clear exposition of 

 which is made by Prof. J. B. Johnson in his work on Modern Framed Structures. He points out that the " ordinary 

 ( quation" for obtaining the extreme fiber stresses, when the external load and dimensions of the beam aie given, is 

 not applicable to a beam strained beyond its elastic limit ; and he follows this statement with a discussion of the true 

 distribution of internal stresses in a beam at time of rupture, and with a " Kational equation for the moment of 

 resistance at rupture/ 7 devised by M. Saint- Venant, which really does connect the extreme fiber stress in a bent beam 

 with the compression-endwise strength and also with the tension strength. Professor Johnson's final conclusion, 

 however, is that for practical use the " ordinary formula" may be applied to a beam at luptuie, providing the fiber 

 stress involved is obtained from cross-bending tests ; and this is the present practice among engineers. 



relation op internal stresses. 



Assume for the discussion of the relation of internal stresses to external load the simple conditions of a beam 

 of rectangular cross section loaded at the middle. 



Regarding the distribution of internal stresses, it must be agreed that the neutral plane lies in the center of the 

 beam so long as the beam is loaded within the elastic limit; this follows from the fact that the modulus of elasticity 

 is the same whether derived from compression tests or from tension tests (i. e., E c = E t ), as proved by experiments 

 of Nordlinger, Bauschmger, Tetmayer, and others. 



Since the distortion of any given fiber in the beam is proportional to its distance fiom the neutral plane, the 

 distribution of stresses in a longitudinal section of a beam loaded up to its elastic limit may be represented by the 

 following diagram, in which the vertical scale represents increments of distortion and the horizontal scale the fiber 

 stresses. 



In this diagram the angle a = angle &, since E c = E t ; and furthermore, since these latter quantities are each 

 equal to the modulus of elasticity obtained from cross-bending tests (according to the same authorities), this angle 

 a (or b) can be obtained by platting the results of the cross-bending test itself. 



It is a well-established fact that the tension strength of wood is much greater than the compression strength, 

 and also, as shown by the German experimenters quoted, that the elastic limit in either case is not reached until 

 shortly bofore the ultimate strength. Furthermore, it seems reasonable to suppose, and is essential to the construc- 

 tion of the above diagram, that the true elastic limit of the beam (shown on the strain diagram of a beam at the 

 point where it ceases to be a straight line) is reached at the same instant that the elastic limit of the extreme com- 

 pression fiber is reached; for when the loading is continued beyond this latter condition the line OC must begin to 

 curve upward (since the proportion of load to distortion on that side begins to increase more rapidly), while the line 

 OT continues in its original direction. Therefore, in order to maintain the equilibrium, the whole distribution of 

 stresses will necessarily be changed, the position of the neutral axis will be lowered, and these changes will, of 

 course, show an effect on the deflection of the beam. 



Now, even at rupture the proportionality of fiber distortion to distance from neutral axis is maintained (because 

 a plane cross section will always remain a plane), and therefore the distribution of internal stresses just at the point 

 of rupture can be represented by a diagram similar to fig 99, in which, as before, the vertical scale represents incre. 

 ments of distortion and the horizontal scale fiber stresses. The fibers on either side of the neutral plane are under 

 stresses which vary from zero at the neutral plane to the maximum stress in the extreme fiber, changing in proportion 



