On the Calculation of Beams and Plates Resting on an Elastic Foundation

P. B. Antonevich

1. Statement of the Problem

At the present time the calculation of foundation beams and plates is carried out mainly by two methods.

The first method—the method of the coefficient of subgrade reaction—is based on the Fuss–Winkler assumption that the settlement of the surface of an elastic foundation at a given point is proportional to the intensity of the normal pressure acting at that point. In other words, according to this hypothesis the properties of an elastic foundation are identified with the properties of a heavy fluid devoid of surface tension.

The method of the coefficient of subgrade reaction gives results that do not always agree satisfactorily enough with practical data. Its principal shortcoming is that the entire set of questions arising in the calculation of beams and plates resting on an elastic foundation is characterized by only one parameter—the coefficient of subgrade reaction \(K_z\). As experiments have shown, the magnitude of this coefficient is determined not only by the properties of the foundation, but also by a number of other factors, among which the most important are the dimensions and shape of the bases of foundation beams and plates, as well as the character of the loads applied to them.

At the present time the method of the coefficient of subgrade reaction is customarily applied only for calculating structures in whose foundations the development of large regions of plastic deformation is expected. It is also applicable to the solution of a number of problems that have no relation whatever to soil mechanics.

The second, more modern method is based on representing the foundation as an ideally elastic half-space. Its application to the calculation of foundation beams and plates is valid only in those cases where the regions of plastic deformation in their foundations are negligibly small.

It follows from what has been said that, in cases of a sufficiently moderate development of regions of plastic deformation, a justified choice of applying one or the other of the two named methods becomes difficult. Under these conditions, for the calculation of structures resting on an elastic foundation, one ought to use a method intermediate between the method of the coefficient of subgrade reaction and the method of the elastic half-space. An attempt to develop such a method is the subject of our work. Its purpose is to construct such a theory of an elastic foundation which, possessing a certain universality, would, on the one hand, make it possible to solve problems on the calculation of foundation beams and plates by comparatively simple means of analysis, and, on the other hand, would be free from the principal shortcomings of the method of the coefficient of subgrade reaction and would open possibilities for solving such problems as cannot be solved at all by means of the Fuss–Winkler hypothesis. This can be achieved if the construction of the theory is based on a new approximate model of an elastic foundation,

properties of which are characterized by parameters that make it possible to approximate satisfactorily the dependence, occurring in real soils, between the settlements of their surface and the reaction thereby arising.

We shall now proceed to the construction of such a model.

2. Construction of an approximate model of an elastic foundation

The basic requirements which the new approximate model of an elastic foundation must satisfy can be clarified by comparing the results of solutions of problems on the settlement of a rigid punch resting on an elastic half-space and on a Fuss–Winkler elastic foundation.

As is known, the settlement of a punch resting on an elastic half-space may be computed by Schleicher’s formula

\[ w=\frac{P}{A\sqrt{F}}, \]

in which \(P\) is the external load applied to the punch, balanced by the reaction of the elastic foundation \(R\), \(F\) is the area of the punch, and \(A\) is a coefficient depending on the elastic properties of the half-space and the shape of the punch.

From this formula it is easy to see that, other conditions being equal, the total reaction of the foundation under the punch proves proportional to its perimeter:

\[ P=R_1=wA\sqrt{F}. \]

On the other hand, the settlement of a punch placed on a Fuss–Winkler elastic foundation is determined by the formula

\[ w=\frac{P}{K_zF}, \]

from which it follows that the reaction of the Winkler foundation is proportional to the area of the punch:

\[ P=R_2=wK_zF. \]

Experiments show that the settlements of foundations resting on real soils, for equal values of the average stresses in their bases, vary with changes in the magnitude of their bearing area. This variation is, in most cases, characterized by a dependence between the dimensions of the foundations and their settlements that is close to linear. Most often the settlements increase more slowly than follows from calculations by Schleicher’s formula. However, there are cases when the settlements of foundations increase faster than is given by the formula just mentioned.

A rationally constructed model of an elastic foundation must reflect these experimentally established facts. This can be achieved by dividing the supporting forces of the soil into two systems of reactive forces: forces distributed over the area of the foundation, and forces distributed along its perimeter.

Taking into account that the reactions both of the Fuss–Winkler foundation and of the elastic half-space are proportional to the settlements of the foundations, let us admit the existence of the same proportionality in the proposed model of an elastic foundation. Then the total reaction of such a foundation will be expressed by the formula:

\[ R=R_1+R_2=(kF+mS)w, \tag{1} \]

in which \(k\) is the coefficient of proportionality of the reactive forces \(R_1\) distributed over the area of the foundation \(F\); \(m\) is the coefficient of proportionality of the reactive forces \(R_2\) distributed along its perimeter.

Thus, the properties of the proposed model of an elastic foundation, as well as the properties of an elastic half-space, are characterized by two parameters instead of one, in the manner of the coefficient of subgrade reaction. These parameters \(k\) and \(m\) will henceforth be called elastic constants.

With the aid of the substitution

\[ R = pF \]

we transform formula (1) into the form:

\[ w = \frac{p}{k+\dfrac{ms}{F}} . \tag{2} \]

For \(k=0\) this formula becomes Schleicher’s formula; for \(m=0\), the Fuss-Winkler formula, but in itself it is essentially empirical. Its merits and shortcomings can be clarified by comparing the settlements computed from it with the data of the corresponding experiments. Some results of such a comparison are given in Section 4.

3. Physical interpretation of the properties of the model

Let us give a physical interpretation of the coefficients \(k\) and \(m\) entering into formula (2). From this formula it is clear that the constant \(k\), in its physical meaning, is entirely analogous to the Fuss-Winkler coefficient of subgrade reaction. The constant \(m\) determines, for a unit settlement of the punch, a certain system of specific forces distributed along the perimeter of the punch—edge reactive forces. These forces schematically reflect the concentration of normal stresses along the edges of the punch and the influence of shear stresses actually existing along the cylindrical surface passing through the contour of the base of the punch. It may be assumed that they arise as a result of a break in the surface of the foundation at the boundary of the punch and are caused by forces similar to surface-tension forces, analogous to those that arise on the surface of any heavy liquid.

Thus, we arrive at a model of an elastic foundation representing something like a heavy liquid with surface tension.

The constant \(k\) represents the specific weight of this liquid; the constant \(m\) is simply expressed through its surface tension.

A model of an elastic foundation close to the one described was proposed by Prof. M. M. Filonenko-Borodich [3]. Among other things, he proposed a model in which the elastic foundation is represented by a system of infinitely often arranged springs connected on top by a membrane stretched in all directions.

It is easy to see that this model of Prof. M. M. Filonenko-Borodich can be wholly likened to a heavy liquid with surface tension. In the model of an elastic foundation proposed by us, such an analogy cannot be carried through to the end.

Thus, in particular, it does not exclude the supposition that the constant \(k\) may assume different values at different points of the foundation. As the analysis of experimental data carried out by us has shown, the need for such a supposition sometimes arises. Moreover, our model makes it possible to eliminate superfluous complications, not following from the essence of the problem, which may arise if, in solving all problems, one strictly follows the membrane model of Prof. M. M. Filonenko-Borodich.

In general, our model of an elastic foundation is more flexible than the membrane model. To the constants \(k\) and \(m\) that characterize it we may assign arbitrary values, being guided only by experimental data.

At the same time, as a convenient working apparatus we shall sometimes use the analogy between our model and a heavy fluid with surface tension. But this analogy plays for us a purely auxiliary role, to some extent resembling the role of the membrane analogy in the theory of torsion of prismatic bars.

4. Some results of an experimental verification of the elastic-foundation model

In order to clarify the merits and shortcomings of the proposed approximate model of an elastic foundation, we analyzed data, borrowed from various sources, from tests of soils by rigid stamps, whose dimensions in different series of experiments varied from \(0.02\) to \(15\) square meters.

This analysis was carried out according to the following general plan.

From the data of two tests with one stamp, or from the averaged data of several tests, the constants \(k\) and \(m\) were determined; then, by formula (2), the settlements of the remaining stamps of the experimental series were determined. These theoretically calculated settlement values were compared with the settlements obtained directly in the experiments.

Another type of analysis of experimental data was proposed by Eng. O. A. Savinov [2]. It consists in comparing the bed coefficients calculated from the experimental data for uniform (\(k_z\)) and nonuniform (\(k_\varphi\)) compression with their reduced values, calculated by special formulas relating the magnitude of these coefficients to the elastic constants of the foundation and to the dimensions of the stamps. These formulas, transformed by us for determining the constants \(k\) and \(m\) introduced above, have the form:

\[ \left. \begin{aligned} k_z &= k + \frac{2(a+b)}{a \cdot b}\,m,\\ k_\varphi &= k + \frac{2(3a+b)}{a \cdot b}\,m, \end{aligned} \right\} \tag{3} \]

where \(a\) and \(b\) are the sides of the rectangular base of the stamp, with side \(a\) parallel to, and \(b\) perpendicular to, the axis of rotation of the stamp.

In Fig. 1 are given the results of such processing of the VIOS experiments, the data of which we borrowed from the work of Eng. Kh. R. Khakimov [4]. The ascending curves express the dependence of the change in settlements of stamps of different dimensions at identical average stresses at their bases. The descending curves represent analogous dependences constructed for the reduced bed coefficients. The points mark the values obtained experimentally.

In Figs. 2–4 are presented graphs of an analogous processing of experimental data borrowed by us from the work of Eng. O. A. Savinov [2], in which he considered the problem of equilibrium of a rigid rectangular stamp resting on the membrane model of Prof. M. M. Filonenko-Borodich. In doing so, Savinov made a number of serious simplifications of the membrane model, as a result of which its properties can be described by formula (1).

In Figs. 5–6 are presented graphs illustrating our processing of the results of Press’s experiments [5] on testing by trial loads—

Fig. 1. Experiments of VIOS on fine-grained sand: \(\gamma = 1.87\ t/m^3\).

Fig. 2. Kerner’s experiments on medium-grained sand. Curve 1 is plotted by formula (3); curve 2—by Schleicher’s formula.

Fig. 3. Experiments of Barkan, Smolikov, and Sanchev (Fundamentstroy) on loess-like loam. Curve 1 in Fig. (a) is plotted by formula (3); curve 2—by Schleicher’s formula.

Fig. 4. Smolikov’s experiments (“Fundamentstroy”) on sandy soil.

Fig. 5. Press’s experiments on sand; \(\gamma = 1.52\ \mathrm{t}/\mathrm{m}^3\).

Fig. 6. Press’s experiments on clay; \(\gamma = 2.3\ \mathrm{t}/\mathrm{m}^3\).

of medium-grained sands \((\gamma = 1.52\ \text{t}/\text{m}^3)\) and heavy brown clays \((\gamma = 2.3\ \text{t}/\text{m}^3)\).

From examination of the graphs (Figs. 1–6), which illustrate our analysis of the experimental data, it is evident that, with a proper choice of the elastic constants \(k\) and \(m\), the experimental results agree satisfactorily with the conclusions of the theory. At the same time, attention should be paid to the fact that satisfactory agreement of the results is obtained both for clayey and for sandy soils. It is indicative that curves analogous to those given above, constructed for comparison by Schleicher’s formula (curves 2 in Figs. 2 and 3a), deviate substantially from the data obtained from experiments.

Fig. 7. Comparison of settlements of plates \((F_1 = 0.09\ \text{m}^2;\ F_2 = 1.0\ \text{m}^2)\) with settlements of an elevator \((F_{\mathrm{red}} = 1310\ \text{m}^2)\), erected on a thick layer of medium-grained sands.

Satisfactory fitting of the theoretically constructed curves, based on the proposed model, to the experimental data was also obtained with a very large difference in the sizes of the plates. Thus, we compared settlements of foundations of actual structures with settlements of plates placed in foundation pits or in their immediate vicinity. As an illustration, we confine ourselves here to presenting the results of processing one such experiment, the data for which were taken by us from the book by Engineer V. I. Rybakov [1].

Figure 7 presents graphs constructed from the settlements of plates with areas of 0.09 and \(1.0\ \text{m}^2\), and of the foundation of an elevator with a reduced area of \(1310\ \text{m}^2\). They confirm the conclusion made above and give a clear picture of the extrapolation of settlements for structures with a large bearing area, obtained with the aid of the model of an elastic foundation proposed by us.

Results of experimental verification of the properties of the approximate model, analogous to those presented above, were also obtained by us in a number of other cases.

5. Conclusions

The analysis of experimental data carried out by us made it possible to formulate the following conclusions.

1. The settlements of geometrically similar plates always increase more slowly than their perimeters increase (or, what is essentially the same thing, \(\sqrt{F}\)).

1. Within known limits, soil may be regarded as a linearly deformable medium, but this latter should be ascribed properties not entirely identical with the properties of an elastic half-space.

2. By disposing of the values of the constants \(k\) and \(m\), one can obtain a satisfactory agreement of the data of calculations by formula (2) with experimental data.

3. As a rule, the coefficients \(k\) and \(m\), especially the coefficient \(m\), prove to be rather stable quantities for a given category of soils and given ranges of loads, and depend little on the shape and dimensions of the bases of the stamps.

The results of the analysis of experimental data carried out by us confirm the possibility of applying the indicated model of an elastic foundation. This model, like the model of Prof. M. M. Filonenko-Borodich, although it suffers from a certain schematic character and does not make it possible to analyze the stress state of the foundation, opens possibilities for developing a method of calculating foundation beams and plates that is more accurate, but no more complicated, than calculation by the method of the subgrade-reaction coefficient. It is free from the principal shortcomings, noted above, of the Fuss–Winkler model, since it makes it possible to take into account not only local, but also general deformations of the surface of an elastic foundation. With its aid one can solve all the problems that are solved in the theory of calculating foundation beams and plates by the method of the subgrade-reaction coefficient, and, in addition, problems which cannot be solved at all by that method. These include, for example, problems on the influence of the dimensions and shape of foundations on the magnitude of their settlement, on the influence of loading, and on the mutual influence of closely spaced foundations. A special article will be devoted to the solution of these latter problems. Here we shall confine ourselves only to considering problems of the calculation of foundation beams and plates resting on an elastic foundation of the proposed approximate model. The remaining sections of the present work are devoted to this.

6. Differential equation of the deflected axis of a beam on an elastic foundation

The derivation of the differential equation of the deflected axis of a beam resting on an elastic foundation of the adopted model is essentially no different from the analogous derivation for a beam resting on a Fuss–Winkler elastic foundation. Here too one may start from the general equation of bending of a bar of constant cross section:

\[ EJw^{\mathrm{IV}} = q(x) - r(x), \tag{4} \]

in which \(q(x)\) denotes the intensity of the distributed load applied to the beam, and \(r(x)\) the line reaction of the elastic foundation. The difference will consist in the fact that, in the case under consideration, the reaction of the foundation is determined not by the hypothesis of the subgrade-reaction coefficient, but by a more complicated formula, into which, in addition to the reaction of the foundation located directly under the beam, the edge reactive forces also enter. With the notation adopted in Fig. 8, this formula will have the form:

\[ r(x) = 2bkw + 2mw. \tag{5} \]

Substituting into equation (4), in place of \(r(x)\), its value (5), we obtain:

\[ EJw^{\mathrm{IV}} = q(x) - 2(kb + m)w. \]

Now, transferring to the left-hand side of the equality all terms except \(q(x)\), and carrying out simple transformations, we shall have:

\[ w^{\mathrm{IV}} + \frac{2}{EJ}(kb + m)w = \frac{q(x)}{EJ}. \tag{6} \]

Denoting, for simplicity of notation,

\[ \sqrt[4]{\frac{kb+m}{2EJ}}=\lambda \ [\mathrm{m}^{-1}], \tag{7} \]

we reduce equation (6) to the form

\[ w^{\mathrm{IV}}+4\lambda^{4}w=\frac{q(x)}{EJ}, \tag{8} \]

which is more convenient for the subsequent exposition.

Thus, the solution of the problem of bending a beam of constant cross-section resting on an elastic foundation of the adopted model is reduced to the integration of a linear differential equation, completely analogous to the equation following from the Fuss–Winkler hypothesis.

Fig. 8

In it, \(\lambda\) is a generalized elastic characteristic depending not only on the dimensions and stiffness of the beam, but also on two elastic constants characterizing the elastic foundation of the proposed model, instead of one according to the bed-coefficient method.

From consideration of formula (7) it may be concluded that inaccuracy in determining the values of the elastic constants \(k\) and \(m\) leads to a small error in the calculation of all design forces, except for settlement forces, since these constants stand in formula (7) under a fourth-root radical. The magnitudes of settlements depend linearly on both elastic constants.

Let us compare equation (8) with the differential equation for the bending of a beam resting on the elastic foundation of Prof. M. M. Filonenko-Borodich. The latter has the form

\[ w^{\mathrm{IV}}-2a^{2}w''+4\lambda^{4}w=\frac{q(x)}{EJ}, \tag{9} \]

where

\[ a=\sqrt{\frac{bH}{EJ}}\ [\mathrm{m}^{-1}], \]

\(H\) is the membrane tension, and \(b\) is the half-width of the beam.

The three-term equation (9) differs from equation (8) by the presence of the middle term on the left-hand side, which takes into account the supporting action of the portion of the membrane located beneath the beam due to the curvature of its middle surface. Calculations show that, for the ratios of beam parameters and values of elastic constants characteristic of real soils encountered in practice, the influence of the middle term in equation (9) proves to be very small in comparison with its other terms. This means that the integrals of equations (8) and (9), obtained in solving the same problems, differ very little from one another. Such a difference, which we established by comparing the results of solutions of a number of particular problems, did not exceed 1.5 percent, which is significantly lower than possib-

of an error in the initial data—the elastic constants of the foundation \(k\) and \(m\). Consequently, the refinement of solutions obtained by integrating differential equation (9) instead of equation (8) has no practical value, although it is associated with more complicated calculations.

Thus, the solution of various problems concerning the calculation of beams resting on an elastic foundation of the proposed model is reduced to the integration of the binomial differential equation (8), whose solution is well developed and widely known in computational practice. Consequently, replacement of the Fuss–Winkler model by a more accurate model of an elastic foundation does not lead to complication of the mathematical methods of solution and makes it possible to use in full the rather rich and perfected computational apparatus of the theory based on the adoption of the hypothesis of the bed coefficient. This is one of the essential merits of the method being developed. At the same time, the solutions obtained by the method we propose will differ substantially from the solutions obtained by the method of the bed coefficient, since in our case the generalized elastic characteristic \(\lambda\) of the differential equation of beam bending is related by a somewhat different dependence to the two elastic constants of the foundation than is the characteristic of the Winkler equation with the bed coefficient.

7. Boundary Conditions

Integration of differential equation (8) is carried out under boundary conditions determined in each case by the conditions of the problem. The boundary conditions for an infinitely long beam, as well as for fixed and hinged ends of beams lying on an elastic foundation, do not differ in any way from the analogous conditions occurring in the solution by the method of the bed coefficient. Nor does one of the boundary conditions at the free end of a beam differ from them, namely the condition that the bending moment be equal to zero, and consequently also the second derivative of the deflections,

\[ \text{a) } \quad M(0)=-EJw''(0)=0;\quad w''(0)=0. \]

The second condition has a certain peculiarity. It consists in the fact that, owing to the settlement of the elastic foundation beyond the limits of the beam, the transverse force at the free end is not equal to zero, as it would be in the solution according to the Fuss–Winkler hypothesis, but proves to be proportional to the magnitude of the deflection of the end of the beam (Fig. 9). We shall write this condition separately for the left and right ends of the beam.

Fig. 9.

At the left end of the beam, for \(x=0\), we shall have:

\[ \text{b) } \quad [Q]_{x=0}=Q_0=2bm'[w]_{x=0}=\zeta W_0=-EJ[w''']_{x=0}, \]

or

\[ \left. \begin{aligned} EJw'''(0)+\zeta w(0)&=0; \quad \text{at the right end of the beam for}\\ x=l \qquad EJw'''(l)-\zeta w(l)&=0,\quad \text{where } \zeta=2bm. \end{aligned} \right\} \tag{10} \]

This boundary condition, unusual for solutions by the subgrade-modulus method, somewhat complicates the solution for beams of finite length whose ends are free.

8. Calculation of Beams of Finite Length Resting on an Elastic Foundation

Since the calculation of infinitely long beams by the proposed method is formally no different from their calculation by the subgrade-modulus method, we shall not dwell here on the solution of this problem, but shall proceed directly to the calculation of beams of finite length.

Taking as the basis for solving problems on the bending of beams lying on an elastic foundation the adopted model, the differential equation (8), entirely analogous to the equation of the subgrade-modulus method, we have obtained the possibility, in constructing solutions, of using the mathematical apparatus of this method. Thus, for calculating beams of finite length we shall use the method of initial parameters, developed by Soviet scholars, which is the simplest and most accessible from both the theoretical and the applied standpoint. In comparison with other methods it has the advantage that with its aid any problems can be solved without integrating the differential equation of a beam on an elastic foundation, since ready formulas are available for determining all calculated quantities by this method:

\[ \left. \begin{aligned} w &= W_0 Y_1+\frac{\varphi_0}{\lambda}Y_2-\frac{M_0}{EJ\lambda^2}Y_3-\frac{Q_0}{EJ\lambda^3}Y_4+\Phi(x),\\ \varphi=w'&=-4\lambda W_0Y_4+\varphi_0Y_1-\frac{M_0}{EJ\lambda}Y_2-\frac{Q_0}{EJ\lambda^2}Y_3+\Phi'(x),\\ M=-EJw''&=4EJ\lambda^2W_0Y_3+4EJ\lambda\varphi_0Y_4+M_0Y_1+\frac{Q_0}{\lambda}Y_2-EJ\Phi''(x),\\ Q=-EJw'''&=4EJ\lambda^3W_0Y_2+4EJ\lambda^2\varphi_0Y_3-4\lambda M_0Y_4+Q_0Y_1-EJ\Phi'''(x). \end{aligned} \right\} \tag{11} \]

Here \(Y_i\) \((i=1,2,3,4)\) are linearly independent solutions of equation (8)—the functions of N. P. Puzyrevsky–A. N. Krylov, determined by the formulas:

\[ \left. \begin{aligned} Y_1&=\operatorname{Ch}\lambda x\cdot\cos\lambda x,\\ Y_2&=\frac{1}{2}\bigl(\operatorname{Ch}\lambda x\sin\lambda x+\operatorname{Sh}\lambda x\cdot\cos\lambda x\bigr),\\ Y_3&=\frac{1}{2}\operatorname{Sh}\lambda x\cdot\sin\lambda x,\\ Y_4&=\frac{1}{4}\bigl(\operatorname{Ch}\lambda x\cdot\sin\lambda x-\operatorname{Sh}\lambda x\cdot\cos\lambda x\bigr), \end{aligned} \right\} \tag{12} \]

\(W_0,\ \varphi_0,\ M_0\) and \(Q_0\) are the initial parameters—the values of the calculated quantities at the origin of coordinates, determined from the boundary conditions; and \(\Phi(x)\) is a particular solution, depending on the type of external load and determined in exactly the same way as in the solution by the subgrade-modulus method.

Since the boundary conditions for beams with supported and fixed ends do not differ in any way from the conditions occurring in the solution of such beams by the subgrade-modulus method, the determination for them of the initial parameters when solving the problem we have posed presents nothing new. Therefore we shall not dwell on it here, but shall take up the solution of a beam with free ends, since

it differs somewhat from the analogous solution by the method of the bed coefficient.

Suppose it is required to calculate a beam resting on an elastic foundation, with free ends, loaded by an arbitrary load.

Taking the direction of the coordinate axes as shown in Fig. 10, and recalling what was said in Section 7 about the boundary conditions for the case under consideration, we shall have:

\[ \begin{aligned} &\text{for } x=0 \qquad [M_0]_{x=0}=0; \qquad [Q_0]_{x=0}=\zeta[W_0]_{x=0}, \tag{13a}\\ &\text{for } x=l \qquad [M_l]_{x=l}=0; \qquad [Q_l]_{x=l}=-\zeta[W_l]_{x=l}. \tag{13b} \end{aligned} \]

Now, to solve the problem posed, we shall use the formulas of the method of initial parameters. Substituting into (11) the initial values of the bending moments and transverse forces, determined by the boundary conditions at the left end of the beam (13a), we obtain formulas for finding the computed quantities:

Fig. 10.

\[ \left. \begin{aligned} \text{a)}\quad w&=\left(Y_1-\frac{\zeta Y_4}{EJ\lambda^3}\right)W_0+\frac{\varphi_0}{\lambda}Y_2+\Phi(x),\\[4pt] \text{b)}\quad \varphi&=-\left(4\lambda Y_4+\frac{\zeta Y_3}{EJ\lambda^2}\right)W_0+\varphi_0Y_1+\Phi'(x),\\[4pt] \text{c)}\quad M&=\left(4EJ\lambda^2Y_3+\frac{\zeta Y_2}{\lambda}\right)W_0+4EJ\lambda\varphi_0Y_4-EJ\Phi''(x),\\[4pt] \text{d)}\quad Q&=(4EJ\lambda^3Y_2+\zeta Y_1)W_0+4EJ\lambda^2\varphi_0Y_3-EJ\Phi'''(x). \end{aligned} \right\} \tag{14} \]

Using the boundary conditions at the right end of the beam (13b), with the aid of formulas (a), (c), and (d), we find the magnitude of the deflection \(W_0\) and the angle of rotation \(\varphi_0\) of the beam section at the origin of coordinates. Substituting the initial values \(W_0\) and \(\varphi_0\) thus found into formulas (14), we convert the latter into expressions for computing the ordinates of the diagrams of the computed quantities.

When solving symmetric beams carrying a symmetric load, it is convenient to place the origin of coordinates at the middle of the beam, since this gives some simplification of the solution, especially in the case when \(Q_0=0\).

9. Examples of calculation of beams resting on an elastic foundation of the proposed model

1. A beam of rectangular section of the dimensions indicated in Fig. 11 rests on an elastic foundation characterized by the constants: \(k=500\ \text{t}/\text{m}^3\); \(m=1500\ \text{t}/\text{m}^2\). The modulus of elasticity of the beam is \(E=3.5\cdot10^5\ \text{t}/\text{m}^2\). It is required to calculate it for the action of the three concentrated forces shown in Fig. 11.

We find the generalized elastic characteristic \(\lambda\).

\[ \lambda=\sqrt[4]{\frac{500\cdot 1+1500}{2\cdot3.5\cdot10^5\cdot0.1667}}=0.3618\ \text{m}^{-1}. \]

Taking the origin of coordinates at the middle of the beam, we shall have the following boundary conditions:

\[ \begin{aligned} &\text{for } x=0 \quad [\varphi_0]_{x=0}=0, \qquad [Q_0]_{x=0}=-50\ \text{t};\\ &\text{for } x=l=12\ \text{m}; \quad [M]_{x=l}=0, \qquad [Q]_{x=l}=-\zeta Wl,\\ &\text{where } \zeta=2bm=2.0\cdot1500=3000\ \text{t/m}. \end{aligned} \]

Taking the boundary conditions into account and using formulas (10), we compose the equations for determining the initial parameters:

\[ 4EJ\lambda^{2} W_0Y_3(\lambda l)+M_0Y_1(\lambda l)+\frac{Q_0}{\lambda}Y_2(\lambda l)-\frac{P_2}{\lambda}Y_2[\lambda(l-\xi_1)]=0 \]

\[ \begin{aligned} 4EJ\lambda^{3} W_0Y_2(\lambda l)-4\lambda M_0Y_4(\lambda l)+Q_0Y_1(\lambda l)-P_2Y_1[\lambda(l-\xi_1)] ={}&-\zeta\left\{W_0Y_1(\lambda l)-\frac{M_0}{EJ\lambda^2}Y_3(\lambda l)\right.\\ &\left.-\frac{Q_0}{EJ\lambda^3}Y_4(\lambda l)+\frac{P_2}{EJ\lambda^3}Y_4[\lambda(l-\xi_1)]\right\}. \end{aligned} \]

For brevity of notation we denote \(4EJW_0=\overline{W}_0\) and substitute into the written equations, in place of the letters, their numerical values¹

Fig. 11. Diagrams of bending moments constructed as a result of calculating the beam by different methods. Diagram 1—by the proposed method; 2—by the foundation-coefficient method; 3—by the elastic half-space method.

\[ \begin{gathered} Y_1(\lambda l)=Y_1(2.17)=-2.502;\qquad Y_2(\lambda l)=0.6129;\qquad Y_3(\lambda l)=1.785;\qquad Y_4(\lambda l)=1.525;\\ Y_1[\lambda(l-\xi)]=Y_1(0.43)=0.9943;\qquad Y_2(0.43)=0.4295;\qquad Q_0=-50\ \text{t};\qquad P_2=60\ \text{t};\\ \lambda=0.3618\ \text{m}^{-1}. \end{gathered} \]

Doing this, we obtain the equations:

\[ 0.23366\,\overline{W}_0-2.502M_0-155.93=0, \]

\[ -0.00314\,\overline{W}_0-2.908M_0+149.08=0, \]

solving which, we find the following values of the initial parameters:

\[ \overline{W}_0=1202.3\ \text{tm}^3, \]

\[ M_0=49.96\ \text{tm}. \]

Substituting these values into formulas (11), we obtain the equations of the calculated quantities. In particular, for the bending moments the equation will be:

\[ M=157.38Y_3(\lambda x)+49.96Y_1(\lambda x)-1.382Y_2(\lambda x)-165.84Y_2[\lambda(x-\xi_1)]. \]

The diagram of bending moments constructed according to this equation is presented in Fig. 11. For comparison, the diagrams of bending

moments obtained by calculating this same beam by different methods¹.

A comparison of the diagrams shows that the magnitudes of the bending moments obtained by our method of calculation prove to be intermediate between the magnitudes obtained by the method of the bed coefficient and by the theory of an elastic half-space.

1. A railway sleeper of the dimensions indicated in Fig. 12 rests on a foundation characterized by the elastic constants:

\[ k=240\ \text{t}/\text{m}^3,\qquad m=470\ \text{t}/\text{m}^2. \]

The modulus of elasticity is

\[ E=10^6\ \text{t}/\text{m}^2,\qquad J=\frac{bh^3}{6}=8.533\cdot 10^{-5}\ \text{m}^4. \]

The generalized elastic characteristic of the differential equation of bending will be

\[ \lambda=\sqrt[4]{\frac{240\cdot 0.125+470}{2\cdot 10^6\cdot 8.533\cdot 10^{-5}}} =1.308\ \text{m}^{-1}. \]

Taking the origin of coordinates at the middle of the beam, we shall have the following boundary conditions:

\[ \begin{array}{lll} \text{for} & x=0 & [\varphi_0]_{x=0}=0, \qquad [Q_0]_{x=0}=0,\\[4pt] \text{for} & x=l=1.35\ \text{m} & [M_l]_{x=l}=0, \qquad [Q_l]_{x=l}=-\zeta W_l, \end{array} \]

where

\[ \zeta=2bm=2\cdot 0.125\cdot 470=117.5\ \text{t}/\text{m}. \]

In view of these boundary conditions, using formulas (11), we compose equations for determining the initial parameters \(M_0\) and \(W_0\):

\[ 4EJ\lambda^2 W_0Y_3(\lambda l)+M_0Y_1(\lambda l) -\frac{P_1}{\lambda}Y_2[\lambda(l-\xi_1)]=0, \]

\[ 4EJ\lambda^3 W_0Y_2(\lambda l)-4\lambda M_0Y_4(\lambda l) -P_1Y_1[\lambda(l-\xi_1)] \]

\[ =-\zeta\left\{ W_0Y_1(\lambda l) -\frac{M_0Y_3(\lambda l)}{EJ\lambda^2} +\frac{P_1}{EJ\lambda^3}Y_4[\lambda(l-\xi_1)] \right\}. \]

Denoting \(4EJW_0=\overline{W}_0\) and substituting in the written equations, instead of the letters, their numerical values:

\[ Y_1(\lambda l)=Y_1(1.77)=-0.5977;\quad Y_2(\lambda l)=1.1984,\quad Y_3(\lambda l)=1.3970; \]

\[ Y_2[\lambda(l-\xi_1)]=Y_2(0.77)=0.7610,\quad Y_4(0.77)=0.0760, \]

we obtain the equations

\[ 2.3902\,\overline{W}_0-0.5977M_0-0.5818P=0, \]

\[ 2.4762\,\overline{W}_0-5.7347M_0-0.8948P=0, \]

solving which, we find the following values of the initial parameters:

\[ \overline{W}_0=0.2291P, \]

\[ M_0=-0.05710P. \]

Substituting them into formulas (11), we obtain the equations of the calculated quantities. In particular, for the bending moments the equation will be:

\[ M=\{0.3920Y_3(\lambda x)-0.05710Y_1(\lambda x)-0.7645Y_3[\lambda(x-\xi_1)]\}P. \]

The diagram of bending moments constructed from this equation is shown in Fig. 12. For comparison, the same figure also shows the diagrams of bending moments obtained by calculating a railway sleeper according to the theory of an elastic half-space¹ and by the method of the bed coefficient for \(k_z = 4000\ \mathrm{t}/\mathrm{m}^3\).

Fig. 12. Diagrams of bending moments constructed as a result of calculating a railway sleeper by various methods. Diagram 1—by the proposed method; 2—by the method of the bed coefficient; 3—by the method of the elastic half-space.

1. In conclusion, we present the results of calculating a beam, in the first case loaded along its entire length by a uniformly distributed load (Fig. 13), and in the second by a load varying according to a linear law.

The elastic constants of the foundation are:

\[ k = 500\ \mathrm{t}/\mathrm{m}^3,\qquad m = 1500\ \mathrm{t}/\mathrm{m}^2. \]

The modulus of elasticity of the beam is \(E = 2 \cdot 10^6\ \mathrm{t}/\mathrm{m}^2\), the moment of inertia is \(J = 0.03815\ \mathrm{m}^4\), and the width of the beam footing is \(2b = 1.5\ \mathrm{m}\).

When such a beam is calculated by the method of the bed coefficient, it turns out that no bending moments arise in it, and its elastic line is a straight line whose equation is

\[ W(x)=\frac{q(x)}{2b\cdot k_z}. \]

In other words, a beam loaded throughout by a load distributed according to a linear law, resting on a Fuss–Winkler foundation, settles without undergoing bending.

Fig. 13. Beam under a uniformly distributed load \(q = 10\ \mathrm{t}/\mathrm{m}\). Diagram \(M\) (\(\mathrm{t\,m}\)); deflection line \(W\cdot 10^4\).

Fig. 14. Beam under a linearly varying load from \(q_0 = 10\ \mathrm{t}/\mathrm{m}\) to \(q_\ell = 30\ \mathrm{t}/\mathrm{m}\), with \(\ell = 12\ \mathrm{m}\). Diagram \(M\) (\(\mathrm{t\,m}\)); diagram \(W\cdot 10^4\ \mathrm{m}\).

If the beam under consideration rests on an elastic half-space, then its bending will occur. The beam also bends in the case where it rests on the elastic foundation adopted by us, as is seen from the diagrams presented in Figs. 13 and 14.

10. On the calculation of plates resting on an elastic foundation

Considering the equilibrium of an infinitely small element of an elastic thin plate cut out far from its contour, we obtain the differential equation of the middle surface of this plate, completely analogous to the equation following from the hypothesis of the subgrade coefficient:

\[ \nabla^{2}\nabla^{2} W(x_{1}y)+\lambda^{4}W=\frac{q(x_{1}y)}{D}, \tag{15} \]

where \(D\) is the cylindrical rigidity of the plate, \(q(x_{1}y)\) is the load applied to the plate, and \(\lambda\) is the generalized elastic characteristic of the plate, determined by the formula

\[ \lambda=\sqrt[4]{\frac{K}{D}}\,[m^{-1}]. \]

The solution of various problems on the bending of plates resting on an elastic foundation of the type adopted by us is thus reduced to the integration of equation (15). Here, as in the solution of beams resting on an elastic foundation, the difference in comparison with solutions by the method of the subgrade coefficient will consist only in the boundary conditions.

With the exception of one of the conditions on the free edge of the plate, the integral of equation (15) must satisfy all boundary conditions that occur in the theory of thin plates. The special boundary condition on the free edge consists in the fact that the transverse force, as in the calculation of beams on an elastic foundation, is not equal to zero, but proves to be proportional to the deflection of the free edge of the plate,

\[ [N_{nz}]_{s}=-mW_{s}. \tag{16} \]

If \(s\) is the contour of the plate and \(n\) is the normal to it directed outward, then, taking into account the value of the transverse force on the contour (16), instead of the known combined boundary condition on the free edge of thin plates

\[ N_{nz}+\frac{\partial H}{\partial S}=0, \]

in the case under consideration we shall write the condition

\[ N_{nz}+\frac{\partial H}{\partial S}+mW_{s}=0, \]

which, in expanded form, is rewritten as follows:

\[ \frac{\partial^{3}W_{s}}{\partial n^{3}}+(2-\mu)\frac{\partial^{3}W_{s}}{\partial n\partial S^{2}}+mW_{s}=0. \tag{17} \]

Satisfying this condition constitutes the only difference between the solutions by the method proposed by us and the solutions by the method of the subgrade coefficient. Of course, the solution of plates that have no free edges, as well as the solution of infinitely large plates, is completely the same by both methods and leads to identical results.

We do not have the opportunity here to dwell in greater detail on the calculation of plates resting on an elastic foundation; therefore we shall confine ourselves only to the following indication, which distinguishes our calculation from calculation by the method of the bed coefficient.

11. General conclusions

1. The calculation of beams on an elastic foundation by the proposed method requires almost no changes to the widely known and fairly perfected apparatus of the theory based on the Fuss–Winkler hypothesis.

2. However, in our case the generalized elastic characteristic \(\lambda\) of the differential equation for beam bending and the boundary conditions at the free end differ from the corresponding characteristic and boundary conditions of the bed-coefficient method. As a consequence, the results of calculations by the two methods may differ substantially from one another.

3. This difference may lead both to overestimation and to underestimation of the values of the calculated moments. As a rule, calculations by the proposed method give intermediate values of the extreme bending moments in comparison with the moments obtained in calculations by the bed-coefficient method and by the elastic half-space method.

In general, calculation by our method gives results that qualitatively coincide with the analogous results given by solutions of the theory of the elastic half-space.

1. We believe that, for the calculation of foundation beams and plates in whose bases a moderate and significant development of regions of plastic deformation is expected, it is more rational to use the method we propose than the bed-coefficient method. In the calculation of structures in whose bases the regions of plastic deformation are negligibly small, application of the elastic half-space method gives results more closely corresponding to reality.

2. The method we propose still requires careful verification in experiments set up specifically for this purpose. From the results of the experiments, statistical material must be accumulated that will make it possible to establish the values of the elastic constants \(k\) and \(m\) characteristic of soil foundations frequently encountered in practice. We intend to undertake this work in the future. Even now, however, it is obvious that our method makes it possible, much better than the bed-coefficient method, to reflect the actual stress state of foundation beams and plates and to solve a substantially broader class of problems encountered in engineering practice.

References

1. Rybakov V. I. Settlements of the foundations of structures, 1937.

2. Savinov O. A. Pressure of a rigid rectangular punch on an elastic foundation. Proceedings of the Leningrad Institute of the Trust for Deep Works, issue 2, 1941.

3. Filonenko-Borodich M. M. Some approximate theories of an elastic foundation. Scientific Notes of Moscow State University, issue 46, 1940.

4. Khakimov Kh. R. Experimental studies of foundation deformations. Hydrotechnical Construction, No. 9, 1939.

5. Die Bautechnik H. 42, 1930.

1. Borrowed from the book by V. I. Kuznetsov, Problems of the Static Calculation of the Track Superstructure, 1940. ↩↩↩