(a) That the beam is symmetrical with respect to a certain plane. 



(b) That the material of the beam is homogeneous. 



(c) That sections which are pin no before bending remain plane after 

 bending. 



(d~) That the ratio of longitudinal stress to the corresponding strain 

 is the ordinary (i. e. Young's) modulus of elasticity, notwithstanding 

 the lateral connection of the elementary layers. 



(e) That these elementary layers expand and contract freely under 

 tensile and compressive forces. 



In each case, the skin stress at the point of fracture in Ibs. per sq. in 

 has been determined by means of the formula, 



_s I (2W 1 + W,) 

 J bd* 



W r lbs. being the weight at an end, Wo -Ibs. half the weight of the beam 

 Z-ins. the length of the beam between the two end centres of pressure, 

 6-ins. the breadth and d-ins. the depth at the section of fracture. 



In practice, the breaking weight, \V\ + | W-,, is usually determined 



from the formula. 



b d- 

 W, + \ W 2 = C , 



C being the co-efficient of rupture. Hence,/ = 3 C. 



It may perhaps be well to point out that a very small error in esti- 

 mating the depth of a beam may lead to a considerable error in the 

 calculated skin stress. Thus from the formula just given it appears 

 that if A/be the change in the skin stress corresponding to a change 

 Ad in the depth, then 



A / = _ 2./A d , 

 d 



and the skin stress will be increased or diminished by this amount, 

 according as the estimated depth is too small or too great by the 

 amount "Arf. 



For instance, in the case of the Spruce Beam No. L, the calculated 

 skin stress, disregarding the diminution of depth due to compression, 

 is 5123 Ibs. The initial depth (d) of the beam was 17.5 ins., and the 

 amount of the compression (&d) 2 ins. Thus the error (A/) in the -skin 



stress is 



5123 

 A/= 2^ry^-2 1 171 Ibs. per sq. in., 



and the actual stress becomes 5123 + 1171:= 6294 Ibs. per sq. In., 

 showing an increase of 22.8 per cent. 



Now, in every example of transverse t< sting, the material is more or 

 less compressed at the central support. The central support in the 

 following examples was a hardwood block of 20 ins. diameter. The 

 amount of the compression at this support depends not only upon the 

 nature of the material of the beam and upon the character of the sup- 

 port, but also very especially upon the ratio of the length of the beam to 

 its depth. In calculating the skin stress corresponding to the breaking 

 weight, therefore, three assumptions may be made : 



1st. That the compression at the support may be disregarded. 



2nd. That the effective depth of the beam may be taken as equal to 

 the initial depth minus the amount of the compression, and that the usual 

 law may be assumed to hold good for the whole of this effective depth. 



3rd. That the compression portion of the beam is alone affected, so 

 that the so called neutral plane remains in the same position relatively 

 to the tension face of the beam from the common cement of the test to 

 the end. 



Calculations based upon these three assumptions have been made in 

 several of the following cases, an d it will be observed that in all cases 

 the skin stress calculated upon the first assumption is invariably less than 

 the skin stress determined upon either of the remaining assumptions. 



Tims any error is on the safe side. 



It should be remembered, however, that it is possible, and even pro- 

 bable, that neither of these assumptions is even approximately correct, 

 at all events, beyond the limit of elasticity, w Inch in the case of timber 

 still remains indefinite. The portion in compression doubtless acquires 



4 



