ELASTIC ARCHES 421. Technical Meaning. All of the previous demonstrations in arches have been made on the basis that the arch is made up of voussoirs, which are acted on only by compressive forces. The demonstration would still remain the same, even if the arches were monolithic rather than composed of voussoirs; but in the case of an arch composed of voussoirs, it is essential that the line of pressure shall pass within the middle third of each joint, in order to avoid a tendency for the joint to open. If the line of pressure passes very far outside of the middle third of the joint, the arch will certainly collapse. An elastic arch is one which is capable of withstanding tension, which practically means that the line of pressure may pass outside of the middle third and even outside of the arch rib itself. In such a case, transverse stresses will be developed in the arch at such a section, and the stability of the arch will depend upon the ability of the arch rib to withstand the transverse stresses developed at that section. A voussoir arch is, of course, incapable of withstanding any such stresses. A monolithic arch of plain concrete could withstand a con siderable variation of the line of pressure from the middle third of the arch rib; but since its tensile strength is comparatively low, this variation is very small compared with the variation that would be possible with a steel arch rib. A reinforced-concrete arch rib can be designed to stand a very considerable variation of the line of pressure from the center of the arch rib.
422. Advantages and Economy. The durability of concrete, and the perfect protection that it affords to the reinforcing steel which is buried in it, give a great advantage to these materials in the con struction of arch ribs. Although the theoretical economy is not so great as might be expected, there are some very practical features which render the method economical. It is always found that, before any considerable transverse stresses can be developed in a reinforced concrete arch bridge, the concrete will be compressed to the maximum safe limit while the unit-stress in the steel is still comparatively low. Since a variation in the dead load often changes the line of pressure from one side of the arch rib to the other, and thus ,changes the direction of the transverse bending, it becomes necessary to place steel near both faces of the arch rib, in order to withstand the tension which will be alternately on either side of the rib. Of course the steel
which is (for the moment) on the compressive side of the rib will assist the concrete in withstanding compression, but this is not an economi cal use of the steel. There is, however, the practical economy and advantage, that the reinforcement of the concrete makes it far more reliable, even from the compression standpoint. It prevents cracks in the concrete, and it also permits the use of a much higher unit pressure than would be considered good practice in the use of plain concrete. This advantage becomes especially great in the con struction of arches of long span, since in such a case the dead load is generally several times as great as the live load. Therefore the maximum variation in the line of pressure produced by any possible change in loading is not very great; and any method which will per mit the use of a higher unit-pressure in the concrete is fully justified by the use of such an amount of steel as is required in this case.
423.• Elements of Integral Calculus. It has been found im practicable to develop the theory of elastic arches without employing some of the fundamental principles of integral calculus; but an effort will be made to explain each one of the equations which are used, in such a way that the application of calculus to this particular ease may be understood. To facilitate this demonstration, a few of the fundamental principles of integral calculus will be briefly demon strated. All of the calculus equations which are used are similar to Equation 48. In this, a character f , somewhat similar to the letter S (which may be considered to stand for the word summation), is placed in front of some mathematical quantities. The equation generally reads that this summation equals zero. The general meaning of the equation is that there is a group of quantities all of which are in general similar, but which have a variation in magnitude. In general, some of these quantities arc positive, and some are negative, and the equation reads that the summation or the algebraic addition of all these positive and negative quantities just equals zero; or, in other words, that the sum of all the positive quantities is just equal to the sum of all the negative quantities.