BD.

Assuming that the assumptions are correct, the line AB shows the cross-section after the change of form, and the line CD represents the same cross-section before the change of form. The compressive or tensive strains in the concrete above and below the neutral axis respectively, are values depending upon the changes of form of the body equally increasing from O to AC and from 0 to These are represented by two curves ОЕ and OF which in a body of rectangular cross-section are nothing else than the law of the change of form of the concrete through compression or tension, and the latter was constructed by accepting on the one hand O a and OA, on the other hand O ẞ and OB as co-ordinate lines for the strains and changes of form respectively. As the reinforcement takes part in the changes of form of the concrete, it is accepted that the cross-sections K and I of the iron bar are removed to H and G in the line CD, which enables the strain to be determined as a value depending on the modulus of elasticity. As all relations connecting the strains with the changes of form are known, one can succeed through two conditions to determine the elastic powers in the horizontal direction, which are transmitted between the two intercepts divided by the line AB, one being the moment, the other the projection. The problem is therfore solved as soon as it is possible to determine the equation of the curves OE and OF.

In the following I will refer to the different strains appearing in different structures, in order to prove the probability of the assumptions above mentioned. The most simple of all strains is resulting from compression, and this has to be specially dealt with in the statical calculation of columns.

COMPRESSION.

The compressive strength of concrete depends entirely upon the proportion of the mixture, the shape and the height of the test bodies. With small concrete bodies the strength is very great, it diminishes with the increasing proportion of the height to the width; the strength of the cubic bodies is known as the cubiform strength of the concrete. In the high test bodies, the rupture occurs through removing the sliding resistance in inclined planes, and the compressive strength, which generally cannot be taken into account, then appears very small, when dividing the crushing load by the sectional contents. The purpose of the reinforcement in the columns is thus to prevent that sliding to inclined planes. Round iron vertical bars. interlocked with horizontal iron hoops, are usually employed for the reinforcement of columns. This arrangement affords the advantage that by the eccentrical loading of the columns, it can still sustain tensile strains. For the compressive strain on the axis, the calculation is made under the supposition that the concrete and the iron are equally compressed. Thus if fc means the sectional contents of the concrete, fe that of the iron, dc and de the corresponding strains of both materials, the load P will be::

P fcx dc+fe x de

dc and de are thus tensions in the concrete and the iron, which correspond with the same compressions. We must therefore apply the law of the elastic strain for concrete.

The latest investigations by Considere led to the discovery of quite a new system for the reinforcement of columns, which system is especially recommended for heavily loaded columns, of which, for certain reasons, the diameter must be chosen as small as possible. His publication on Le beton frette," which is best translated by encircled concrete, appeared some time ago in the French Journal, Genie Civil. They are the results of experiments with concrete cylinders, the reinforcement of which consists of spirally wound iron bars. This system allows a more effective use of the material than the ordinary vertical reinforcement; it is 2.4 times stronger than the latter. The strength of the concrete can thus be increased up to 11,500 lbs. per square inch, and is consequently practically quadrupled. The calculation for reinforced concrete columns should be done in the following manner :

The cross-section of the iron is multiplied by

a=

=

=

Ee elasticity of iron 3000000
Ec elasticity of concrete 300000

= 10,

Then

and the result added to the sectional area of the concrete. the strain of the concrete is equal to the loading force, divided by the cross-section of the column, and the strain in the iron consequently is ten times the strain in the concrete. On the basis of Euler's breaking formula

and the formula which in bridge construction is known as Rankine's breaking formula, namely:

and further on the basis of the formula for the strain of the concrete (and of the cast iron)

d = K (1 - e - 1000o)

the calculation of reinforced concrete bodies under compression can easily be made.

DEFLECTION.

In demonstrating the deflection in ferro-concrete constructions, I will first refer to the ordinary bending :

The fluxure equations with the homogeneous bodies with constant coefficients of strain are obtained as mentioned before, on the assumption that the vertical sections, straight before the bending, also remain straight after the bending, although that assumption is incompatible with the presence of the shearing strains, as the latter cause

an S-shaped curve of the sections. With equal force such an assumption can be applied to the deflection of reinforced concrete bodies. The question whether the tensile strength of the concrete shall be taken into account with the deflection caused some diversity of opinion. By practical engineers the point at issue was settled from the beginning, with the result that the tensile strength of the concrete is not considered at all, and the full permissible tensile strength of the reinforcement at the side of the tension is used. On that sound basis the theory since then has still further been developed, and greatly assisted by the elasticity experiments. This theory, in which the tensile strength of the concrete remains unconsidered, is supported by such authorities as Emperger, Christophe, and Considere.

It is mainly owing to the endeavour to theoretically explain the favourable adaptability of ferro-concrete that many methods of calculation were brought into existence, mostly by theorists. The oldest methods take the tensile and compressive elasticity of the concrete as equally great, subsequently the modulus of elasticity for concrete was taken smaller, then parabolic curves were taken for the elongation curves, and at last, through Considere's experiments, the line of tension of the concrete under strain was represented by a straight line being parallel to the cross-section. It must be observed that with such assumptions formulæ must be obtained, the length of which may be regarded by the authors to reflect the most minute accuracy and reliability; for the constructing engineer, however, these intricate formulæ offer little inducement for their employment. Further, the substitution of the elongation curve is less correct, than the substitution by a straight line, because for the law of power the exponent m is nearer to 1 than to 2, and the elongation curves would have to be forced into the shape of a parabolic curve. But apart from all that, these methods of calculation do not afford the desired degree of safety; and they can become somewhat dangerous if the percentage of reinforcement in the concrete chosen is too small. Considering that the concrete is liable to crack, caused either through faulty preparation or interruption during the concreting or by drying too quickly, one cannot reckon on a uniform tensile strength of the concrete. Under the circumstances, there is, therefore, no protection against cracks in the concrete under strain, and, for instance, in loading tests small cracks occur frequently at an early stage, originating from the tension, in the body, the cause whereof is as yet unknown. At any rate, the exact time when these cracks appear on the side of tension cannot be predicted with certainty. Further, considering that the object of every statical calculation is less the precise demonstration of the strains in a structure under load than it is proof of a sufficiently great factor of safety in the structure, the tensile strains of concrete must not be considered, for the reason that its tensile strength cannot surpass the limit of elasticity of the iron, thus already giving way before the point of rupture is reached.

For a rectangular cross-section we can deduct formulæ for the calculations of the dimensions, if the elongation curve of the concrete is known. In Fig. 2 the line represents the line of the strain, which is identical with the elongation curve of the concrete; for the compressions are proportionate to the ordinates, while the abscisses represent the corresponding compressive strains. On the side of tension the tensile forces of the reinforcement only act, and will be reduced to the unit of the width. The line from the central axis O to the centre of the reinforcement is defined through strain, which corresponds with the permissible strain of the iron. The area of strain of the iron is a rectangle, the basis of which is of smaller dimensions than the height. Assuming the permissible strain for the concrete to be 750 lbs. per square inch, the maximum area of the concrete is thereby limited above the neutral axis. Under

ordinary deflection no external compound forces are present in the longitudinal section; therefore the tensile and compressive forces must equalise themselves in the cross-section or the contents of the area of compression is equal to the rectangle of the tensile force. If the distance from the centre of gravity of the reinforcement to the top edge of the concrete body equal h, then fe is calculated by means of the function of h and dc, and the moment M, which is equal to the contents of the area of compression multiplied by the distance of the same from the centre of gravity of the. reinforcement, will be found as the function of hz or h, and fe is found to be proportionate to the square root of M. By this method the dimensions of a construction can easily be determined, while through circumstantial test calculations it is only possible to find the strain of an existing construction..

The same method can be applied purely analytically by the use of the law of power, whereby the thickness of the concrete and the reinforcement for given strains is obtained proportionally to the

square root of the moment M. For the law of power can then also be substituted the proportion between the elongations and the strains, so that we obtain the distribution of the strain as shown in Fig.

ᄑ

No. 3. Through the equality of the tensile and compressive force, and on condition that the elongations of the concrete and iron are proportionate to the distance from the neutral axis, we obtain for the calculation of the distance x of the neutral axis, the quadratic equation :

Having obtained x, it is possible if Z represents tension, and D compression, to find that

This method thus enables the experimenter to determine in a simple manner the strain of a given construction, but formulæ can also be given for calculating the dimensions of a construction to be erected, namely :