EEINFORCED CONCRETK STRENGTH OF BEAMS. 645 



and he hopes that the Httle he has done may help to advance this 

 branch of engineering science. He would especially invite criticism 

 as to the assumption of inextensibility and incompressibility, i.e., as 

 to whether or not the stresses calculated upon this assumption are 

 sufficiently accurate for the determination of the dimensions of the 

 members. In very shallow structures, an estimate may be made 

 of the deflection of the whole by reason of extensions and compi-es- 

 sicns throughout, using, for this purpose, average values of the 

 nioduli of elasticity. Tlien, in the case, say, of the double-post 

 trussed beam, instead of equating the drop at one post to the rise 

 at the other, we would introduce a term for the drop due to the said 

 extensions and compressions. 



2.— REIXFOKCED CONCRETE— THE STEEXGTH OF BEAMS. 

 By W. J. DOAK, B.E., Assoc. M. Inst. C.E. 



I propose to consider the ordinary elastic theory of beams as 

 modified for reinforced concrete. 



The well-established formula M ==- /' - for beams of all sections, 



or M = I bd^fioY beams ot rectangular section, depends upon two 

 principal assumptions — 



1. Navier's hypothesis that a section of a beam normal to the 



neutral axis plane before bending remains plane after 

 bending. 



2. Hooke's Law — that stress is proportional to strain within 



the elastic limit. 



From the first it follows that strains are proportional to distance 

 from neutral axis, and from the second that stresses are also propor- 

 tional to distance from the neutral axis. 



Experiments made by Talbot, Schule, and others show that 

 Navier's hypothesis does not hold absolutely for concrete beams, and 

 Professor Warren's tests at the Sydney University in 1906 also show- 

 that plane sections become slightly curved. 



It is generally conceded, however, that for purposes of calculation 

 Navier's hy[)othesis may be accepted. 



As regards Hooke's Law, it may safely be said that it is not 

 perfectly true for any known material. Even with steel, an experi- 

 ment in bending, say, a piece of rail, with careful observation of 

 deflections, will show that the stress strain line has a small curvature 

 well within the elastic limit on the first application of the load ; on 

 gradually unloading and reloading several times it will be found that 

 the stress strain diagram ultimately becomes straight. 



Concrete exhibits something of the same phenomenon, but in a 

 much more marked degree. Tests made to determine the elasticity of 

 concrete show that the elastic limit as ordinarily understood is either 

 non-existent or else very small ; that is to say, the stress strain dia- 

 gram is curved from the beginning. 



Nnw there is no obvious reason why the only elastic law should 

 be Hooke's .Straig'ht Line Law. 



A material would deserve to be classed as elastic if it always 

 within limits followed a law that stress was proportional to square of 



