For given values of dc and de, h and fe are obtained proportionate to the square root of M.

It frequently occurs that in the cross-section also the compressive zone is reinforced, for the purpose of obtaining a higher compressive strength, or to have a sufficiently strong reinforcement for changing moments.

THRUST.

Apart from ordinary deflection, the deflection by thrust can come into operation, as it usually occurs with bridge arches. If there are only compressive strains on the whole cross section, then the calculation can be made in the same way as in a cross-section of a homogeneous material, if the cross-section of the reinforcement is multiplied by

The case is different if the compressive force shows such an eccentricity that tensile strains appear on the opposite side of the cross-section. We then obtain for the calculation of the distance of the neutral axis by a given cross-section and reinforcement, an equation of the third degree, or the small tensile strength of the concrete should exceptionally be taken into account, whereby the calculation for the homogeneous cross-section could be made.

In reinforced concrete arches, or, as they were formerly called, the Monier-Arches, there occur in most cases no tensile strains, provided that the arch is given a serviceable shape. The employment of the reinforcement only serves to ensure a greater safety and to lend a greater compressive strength to the concrete. Should any tensile strains still occur, the following method may be applied provided one does not want a very intricate calculation. The strains of the pure concrete cross-section are first ascertained, and then the total of all tensile strains of the concrete put to the reinforcement on the side of tension.

The above-mentioned methods of calculation are meant for the rectangular cross-section only, and are therefore essential for the calculation of the ceilings-plates and arches. It is evident that the plates can either rest free or be fixed on two or more supports, and that accordingly special types for the shape of the reinforcement must be chosen. In these works the reinforcement is placed in the direction of the tension in the lower part of the plate, and in such a manner that the iron is still sufficiently covered by the concrete. The case is a different one with fixed plates, or with plates resting continually over several supports. The moments are in this case negative near the supports, and the moments of the supports proper are rather greater than the positive maximum moments in the centres of the plates. The reinforcement must therefore be placed over and near the supports.

DEFLECTION.

As a rule bent iron reinforcements do not suffice, as the load can change its position, thereby causing a change of the moments. With continual ceilings we have a positive and a negative line of maximum moments, according to which the reinforcement has to be selected. Frequently a reinforcement running right through the upper part of the plate is necessitated, especially if a short and wide span are interchanging. Reinforced ceiling constructions have arrived at such a pitch that it is impossible to refer to them in detail; there are close on 300 different methods of construction in existence, and nearly every week a new system appears, which, however, in most cases does not show any material improvement. Different ceiling systems have the technical fault, that above the supports the reinforcement of the ceilings are placed in the lower part, and not in the upper part of the plate as required by calculation. A progress is noticeable in connection with those ceiling constructions, in which the distance between the tensile and compressive zone is made as great as possible, without, however, considerably increasing the dead weight of the ceilings. This aim can be obtained through groins; so we then obtain groined ceilings consisting of interlocked T girders of concrete, with reinforcement in the lower part of the groins. If these reinforced groins are set apart in greater intervals, which naturally necessitates a stronger construction; the upper part forming the compressive zone, should be reinforced as a plain concrete plate, fixed between the groins, according to the theory just mentioned. Whereby we obtain a groined ceiling construction in which the ceilings with the reinforced concrete form a T profile. From a theoretical point of view a plate strengthened by groins offers a more economical method of using the materials than a plate of a constant thickness. Up to a certain span, however, the greater cost for the centering of the groins will compensate for the saving of material, so that the groined ceilings can only be constructed with advantage with a span of 10-12 feet or over.

In the groined ceilings construction, the ceilings always statically co-operate to a certain extent with the groin. As, however, the bending moments are negative as in the case with fixed girders or with unsupported girders running right through, and thus again the tensile strength of the concrete is not taken into consideration, so the calculation will be found to be the same as if the ceilings were not in existence, that means that you have to proceed the same way shown in the foregoing for the rectangular cross-section, with this difference, however, that the tension zone with the reinforcement is to be looked for in the upper part, while the compressive zone acts in the lower part of the beam. Assuming that the reinforcement of the girder is evenly distributed on the operative width of the plate, then the circulation can also be made with positive bending moments for the rectangular cross-section if the neutral axis lies within the ceiling plate, or lies horizontally with the lower part of the plate.

In reality the neutral axis always lies near the lower part of the plate; the hatched part of the girder, Fig. 4, in which smal! compressive forces are still active, can simply be omitted without

substantially affecting its strength.

If we only want to determine the reinforcement, the smallest possible value, namely, the distance between the reinforcement and the centre of the ceiling plate can be

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 which 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 consideration, one obtains, assuming a constant modulus of elasticity Ec for the compressed concrete, the distance of the neutral axis :2 x axkx fe + d2

= X

2 x (a x fe + 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:

:

2 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 formulæ :

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 calculation 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 underneath the neutral axis will remain constant.

The value in rectangular cross-section can be calculated as follows:

shows also the sum of the adhesion strains acting on the outside of the reinforcement, so that the same can also be calculated in a simple way. With the cross-section of the plate beam the distance of the tensile and compressive centre point is substituted in place of

It may also be mentioned that the simultaneous presence of a great bending moment the mentioned value for the shearing strain is not altered even if the tensile strength of the concrete is taken into consideration. For in concrete, subjected to great tensions, the difference between neighbouring cross-sections is zero. An increase can therefore not take place for the shearing strains in the part of the cross-section under tension.

Out of the adhesion strain the number of iron bars is calculated which must still be present on the bearer, and it is therefore evident that it will not do to assume the dimensions by the line of the maximum moments solely. If sufficient reinforcement is not given to the supports the destruction of the girder occurs through the irons being pulled out of the concrete at the bearer ends. The distance and the thickness of the stirrups are calculated out of the shearing strain; it is a rule that 43 to 72 lbs. per square inch can be taken up by the concrete and the rest falls to the stirrups. If we know that concrete is capable of following the extension of the iron, and we do not take into consideration the tensile strains in the calculations of the dimensions. This is, in my opinion, a method of calculation giving the greatest protection against the occurrence of cracks, quite apart from the absolute safety and strength which is obtained thereby. In reality the tensile strength of the concrete is naturally still present, which fact will show only slight deflections of reinforced concrete constructions under loading tests.

Further, it must be considered that in consequence of the strong and close interlocking of all parts of a ferro-concrete structure more factors take part in the bearing of the load than generally appears in the calculations, and any of the other building systems manifest it. The slight deflection can also be easily accounted for and determined beforehand.