Beams on elastic foundation
General Theory. — The subject of this chapter grew out of the practical problem of railroad track. A long rail is a beam of small bending stiffness, and in order to sustain the large wheel loads placed on it, the rail must be supported almost along its entire length, by closely spaced crossties. The investigation of this problem led (about 1880) to a theory of interaction between a beam of moderate bending stiffness and an "elastic" foundation which imposes reaction forces on the beam that are proportional to the deflection of the foundation. This theory then was of great importance to civil engineers only, but later it was found that the fundamental theory applied not only to railroad track but to many other situations as well. An example is a bridge deck or floor structure consisting of a "grillage", or rectangular network of beams, closely spaced. The many beams crossing it at right angles support each individual beam of this network, and these crossbeams assert reactions on the first beam proportional to the local deflection. Each individual beam in the network thus is placed on an elastic foundation consisting of all the crossbeams. This line of thinking has proved to be very useful in the design of ship's bottoms and similar structures.
A second example is a thin-walled cylindrical shell loaded by pressures which vary with the longitudinal coordinate z only and which are constant with в, circumferentially. If we cut out of this shell a longitudinal strip of width rd, then this strip is a "beam", subjected to some radial loading along the length z. The beam then finds its reaction forces from the remaining part (2 — rd) of the shell in the form of hoop stresses on the two sides, having the small angle d between them and thus having a resultant in the radial direction, i. e., in the direction of the load.
Returning to the railroad track, the assumption made regarding the behavior of the elastic foundation is
q = -ky (34)
where у is the local downward Reflection of the foundation under the rail; q is the downward (and — q the upward) force from the foundation on the rail per unit length of rail, and k is the "foundation modulus", measured in units of q/y = lb/in./in. = lb/sq in. For the usual railroad track this constant has a value of the order of k= 1,500 lb/sq in., which means that if the long rail is uniformly loaded with g= 1,500 lb per running inch, then the whole rail is pushed uniformly 1 in. into the foundation. The assumption (34) has the great advantage of being mathematically as simple as can be; it also is in fairly good agreement with the facts, although it can be criticized on two points. The first and most important is that an actual soil behaves non-linearly, becoming gradually stiffer for greater deflections. Therefore the q=f(y) relation is represented by a curve rather than a
straight line, and the slope к depends on the deflection y, becoming larger I with increasing y. The mathematics of such non-linear phenomena is extremely
Fig. 28.
complicated and unsatisfactory so that here as well as in other cases we work out a linear theory, use it as far as it goes , and discuss deviations from it in a qualitative manner only. The second objection to eq. (34) is illustrated in Fig. 23. The assumption (34) describes a soil entirely without continuity; the deflection at any point is caused by the load on that point only and is completely independent of other loads nearby. This, of course, is not in agreement with the actual behavior of most soils, but the objection is not as serious as it would seem at first sight. We do not consider cases of loads placed directly on the soil; there always is a rail in between, and if we place a rectangular discontinuous loading p (Fig. 23a) on the rail, then the deflection of the rail be quite smooth and the reaction from the ground is also smoothly distributed over a comparatively great length.
Now we are ready to set up the differential equation of the rail. If p is the downward loading per unit length on the rail and q is the downward reaction force from the foundation, then the rail will obey the classical beam equation (which the reader has to look up in some elementary text)
EI(d4 y/d x4)=p+q (35)
where El is the bending stiffness of the rail. Substituting the assumption (34)