398 Report S.A.A. Advancement of Science. 



chosen as a lever arm between tension and compression. The strain 

 in the upper part of the concrete of the ceiling plate does not move 

 within such narrow limits as this lever arm of Z and D and this 

 strain should be calculated, for w-hich purpose the following more 

 exact course might be adopted. The neutral axis lies within the 

 distance x from the upper part of the plate in the groin, k to be 

 the distance of the reinforcement from the same part, fe means the 

 cross-section of the reinforcement reduced to the unit of the operative 

 width of the plate. The calculation is then easily made, the small 

 compressive forces in the upper part of the girder left out of con- 

 sideration, one obtains, assuming a constant modulus of elasticity 

 Ec for the compressed concrete, the distance of the neutral axis : — 



2 X a X k X f e -I- d^ 



= X 



2 X (a X fe -f d) 



The distance of the centre of the compressive strains or the 

 distance of the point of gravity from the neutral part, forming a 

 trapezium, can be calculated as follows : — 



d , d'^ 



y = x - — + 



6 X (2x - d) 



If the centre point of the pressure is known, the compressive 

 force D = Z as well as the strain de can be ascertained by means 

 of the following formulae : — 



de X X 



dc = 



(k -xY 



In the case of plate beams the correct calculation for their 

 cross-sections as regards the shearing forces is just as important as 

 the one regarding the tensile strains, and the construction of the 

 plate beams became only possible when it was recognised that the 

 concrete on the one part could take up a considerable shearing stress 

 in itself, and that on the other part with a suitable reinforcement 

 it could counteract the shearing strains. With the plates the calcu- 

 lation shows such small values for the shearing stress, that they 

 can safely be taken up by the concrete itself. 



With plate beams, however, special reinforcements for the 

 shearing forces are to be added, and the calculation for the shearing 

 strains of the stirrups and the adhesion strains becomes necessary. 

 The calculation of the shearing strains is done in the following 

 way : — The shearing forces appearing in the area CC Fig. 5 

 between two neighbouring cross-sections are equal to the difference 

 of the normal forces in AC and A'C\ If we, therefore, draw the 

 line t of the shearing strains, the shearing strains in the upper part 

 will be equal to O and will increase up to the value T in the direction 

 of the neutral axis. Under the assumption hitherto observed in 

 connection with all strain calculations, namely, that the concrete 

 should not take up any tensile strains, the shearing strains under- 

 neath the neutral axis will remain constant. 



