Prof. P. A. Minaev.

On the Question of the Distribution of Stresses in Granular Bodies

In the Proceedings of the Tomsk Technological Institute (vol. XXXIV, 1914) my work “On the Distribution of Stresses in Granular Bodies” was printed, in which I propose a new theory of earth pressure, based on considering granular bodies as elastic bodies.

The proposed theory finds confirmation in the experiments of Strohschneider. These experiments served, first, to determine the law of the distribution of stresses in sand under the action of a single load; secondly, to determine the limiting loads for various diameters of a circular base placed on the surface of the sand; and thirdly, to determine the limiting loads for a circular base placed at various depths below the surface of the sand. The last two series of experiments, as is easy to see, essentially constitute the determination of the resistance of the soil and of the depth of foundation embedment.

Strohschneider’s experiments*) were carried out with very sensitive, accurate instruments, but the thickness of the sand layer in the experiments was very small: it did not exceed 5 cm.

Fig. 1.

) Elastische Druckverteilung und Drucküberschreitung in Schüttungen.*—Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Wien, 1912.

To determine the law of distribution of stresses in sand, the apparatus shown in Fig. 1 was used. An opening was made in the bottom of the experimental box, into which a small glass capsule \(K\), closed by a thin rubber membrane \(M\), was inserted. The capsule was filled with water and connected, by means of a tube, with an intermediate device \(Z\), filled with petroleum jelly. The petroleum jelly separated the water of the capsule from the alcohol with which the tube \(S\) was filled for measuring pressure. The pressure of the alcohol column was transmitted through the capillary tube \(C\), where in the alcohol there was a small air bubble, whose position during the experiments was observed with a microscope. The apparatus was loaded with sand with the cock \(H\) closed; in the tube \(S\) there was first established, corresponding to the expected pressure of the sand, the height of the alcohol column, and then the cock \(H\) was carefully opened; so that the air bubble in the capillary \(C\) would not change its position, the pressure could be regulated by means of the micrometer screw \(B\). With this method of carrying out the experiments it could be expected that the membrane did not bulge and, consequently, that no motion of the sand particles occurred. In order to determine the stresses at various horizontal distances of the load from the membrane, the load was moved along a special beam with distances from the axis of the membrane marked on it. The determination of the stresses was made each time with a new filling of the box with sand.

The results of the experiments are presented in Fig. 2. The experimental curves,

Fig. 2.

depicting the law of distribution of stresses in different horizontal planes, have the same character as the curves obtained from Boussinesq’s formula.

\[ \widehat{zz}=-\frac{3P}{2\pi}\frac{z^{3}}{(r^{2}+z^{2})^{5/2}}\;*), \]

derived for an infinite solid mass bounded by a horizontal plane, but the magnitudes of the stresses obtained from the experiments are greater than those given by Boussinesq’s formula. This discrepancy between experiment and theory is explained by the fact that, in a thin layer of sand, owing to the insufficient loading by its own weight, the particles of sand are not pressed closely enough against one another, and therefore the propagation of pressure cannot extend infinitely far, as is assumed in the theory, but is limited to a finite magnitude.

From his experiments Strohschneider concludes that the difference between the experimental and theoretical magnitudes of the stresses decreases with an increase in the thickness of the sand layer, and he assumes that, practically, it may be accepted that at a depth of 1 meter the stresses in the sand will be the same as in a solid mass.

Fig. 3.

*) In this formula \(\widehat{zz}\) is the stress in the direction of the vertical axis \(z\), \(r\) is the horizontal distance from this axis; the origin of coordinates is taken at the point of application of the force. A derivation of the formula may be found, for example, in the course on the theory of elasticity by S. P. Timoshenko, p. 216.—St. Petersburg, 1914.

To determine the resistance of sand to indentation, i.e., to determine the resistance of the soil and the depth of embedment of the foundation, the instrument shown in Fig. 3 was used. A round base \(B\), rigidly connected by means of a thin rod to the vessel \(F\), was placed in the experiments both on the surface of the sand and at various depths beneath its surface. The weight of the vessel and the base was balanced by the load \(G\); loading of the base was effected by mercury, which was introduced in a thin stream into the vessel \(F\).

Fig. 4 presents the results of experiments to determine the resistance of the soil. In these experiments the base was placed on the surface of the sand. The experiments were carried out with bases having areas of \(1\ \text{cm}^2\), \(2\ \text{cm}^2\), \(3\ \text{cm}^2\), \(5\ \text{cm}^2\), \(7\ \text{cm}^2\), and \(10\ \text{cm}^2\). The curves in Fig. 4 indicate the relation between the load on the base and the amount of indentation of the base into the sand. At first the amount of indentation is proportional to the load; then, approximately at a load equal to one half

Fig. 4.

of the limiting value, the indentation begins to increase more rapidly than the load, and finally, when the load reaches the limiting value, the base continues to sink into the sand without an increase of load.

The relation obtained from these experiments between the magnitude of the limiting loads and the diameter of the base is well expressed by the formula

\[ P = 0.18 \frac{\pi}{4} D^3, \]

or

\[ p = \frac{P}{\frac{\pi}{4}D^2} = 0.18D, \]

where \(p\) is the limiting pressure per sq. cm.

Indeed we have:

If as the limiting pressure we take the pressure corresponding to the moment when the indentation of the base into the sand begins to increase faster than the load, we obtain:

\[ p = 0.09D \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (1) \]

Below we shall see that, for bases in the form of a circle, the relationship obtained according to our theory between the limiting pressure on the soil and the diameter of the base is exactly the same as that given by Strohsсhneider’s experiments (formula 1), i.e. that the limiting pressure is proportional to the diameter of the base.

In Fig. 5 are presented the results of experiments for determining

Fig. 5.

the depth of embedment of the foundation. In these experiments a base with an area of 1 cm² was placed at various depths below the surface of the sand. The curves in Fig. 5 indicate the relationship between the load on the base and the magnitude of the indentation of the base into the sand under the action of this load. Here, as in the preceding experiments, at first the magnitude of the indentation is proportional to the load; then, approximately at a load equal to half the limiting load, the indentation begins to increase

of rapid loading, and finally, when the load reaches its limiting value, the base continues to sink into the sand without any increase in load.

The relation between the depth of embedment in the sand of the base, the diameter of the base, and the limiting load may, for these experiments, be expressed as:

\[ h=-0.08D+2.6p, \]

where \(D\) is the diameter of the base (in the experiments \(D=1.13\) cm), and \(p\) is the limiting pressure per sq. cm.

Indeed, we have:

If, as the limiting pressure, one takes the pressure corresponding to the moment when the indentation of the base into the sand begins to increase under rapid loading, then the above relation is expressed as follows:

\[ h=-0.16D+5.2p \ . . . . . . . . \tag{2} \]

As will be seen below, a formula of the same type also expresses theoretically the relation between the depth of embedment of a foundation, the diameter of the base, and the pressure.

Let us now determine \(p\) (formula 1) and \(h\) (formula 2) theoretically. In the present case the problem reduces to determining the depth of embedment of the foundation and the magnitude of the soil resistance under a foundation in the form of a circle. This case belongs to the spatial problems of the theory of elasticity.

We shall find its solution under the assumption (see our above-cited work) that the pressure over the base is distributed uniformly and that the dangerous point lies on the axis of the structure.

To find the stresses at the dangerous point due to the action of the weight of the entire structure, we shall use Boussinesq’s solution, given for the case of a solid infinite mass bounded by a horizontal plane and subjected to the action of a single concentrated force. Taking the origin of coordinates at the point of application of the force and directing the \(z\)-axis vertically downward, we obtain this solution in the form**):

) See S. P. Timoshenko, Course of the Theory of Elasticity, p. 218; 1914.

\[ \left. \begin{aligned} \widehat{rr} &=\frac{P}{2\pi}\left[ -\frac{3r^{2}z}{(r^{2}+z^{2})^{5/2}} -(1-2\sigma)\left(\frac{z}{r^{2}(r^{2}+z^{2})^{1/2}}-\frac{1}{r^{2}}\right) \right],\\[6pt] \widehat{zz} &=-\frac{3P}{2\pi}\frac{z^{3}}{(r^{2}+z^{2})^{5/2}},\\[6pt] \widehat{\theta\theta} &=\frac{P}{2\pi}(1-2\sigma)\left[ \frac{z}{(r^{2}+z^{2})^{3/2}} +\frac{z}{r^{2}(r^{2}+z^{2})^{1/2}} -\frac{1}{r^{2}} \right],\\[6pt] \widehat{rz} &=-\frac{3P}{2\pi r^{2}}\frac{rz^{2}}{(r^{2}+z^{2})^{5/2}} . \end{aligned} \right\} \qquad\cdots\ (3) \]

where \(z\) and \(r\) are the cylindrical coordinates of the point under consideration; \(\widehat{rr}\) and \(\widehat{\theta\theta}\) are the normal stresses acting on vertical areas respectively perpendicular and parallel to the radius vector \(r\); \(\widehat{zz}\) is the normal stress acting on a horizontal area perpendicular to the \(z\)-axis; \(\widehat{rz}\) is the shearing stress acting on vertical and horizontal areas in the directions \(r\) and \(z\); \(\sigma\) is Poisson’s ratio.

Let us now divide our load by a system of radii and concentric circles into infinitely small elements \(p\,ds\) (Fig. 6), where \(p\) is the intensity of the load, \(ds=r\,dr\,d\alpha\) is the area element.

Fig. 6.

For the point \(a\) (Fig. 6), taken on the axis of the structure, the stresses from each such elementary load will be found from formulas (3),

where, instead of \(P\), one need only insert \(pds=prdrd\alpha\) and the corresponding \(z\) and \(r\). In order, for the point \(a\), to determine the stresses on some area from the action of all elementary loads, it is necessary to sum the stresses from the action of the individual loads. We note that, in the present case, owing to symmetry, for every point lying on the axis of the structure, the normal stresses for all vertical areas will be identical, and the tangential stresses equal to zero; the tangential stresses on horizontal areas will likewise be equal to zero, so that the state of stress at the point \(a\) will be determined by the total normal stress acting on some vertical area, for example by the stress \(X_x\), acting on an area whose normal is the \(x\)-axis (Fig. 6), and by the total normal stress \(Z_z\), acting on a horizontal area (normal: the \(z\)-axis).

The stresses \(X_x\) and \(Z_z\) from the action of one elementary load will be:

\[ X_x=\widehat{rr}\cos^2\alpha+\widehat{\theta\theta}\sin^2\alpha, \]

\[ Z_z=\widehat{zz}. \]

The total stresses \(X_x\) and \(Z_z\) from the action of all elementary loads will be found by integrating these expressions:

\[ X_x=\int_0^{2\pi}\int_0^R\left(\widehat{rr}\cos^2+\widehat{\theta\theta}\sin^2\alpha\right), \]

\[ Z_z=\int_0^{2\pi}\int_0^R\widehat{zz}, \]

or, substituting here for \(\widehat{rr}\) and \(\widehat{\theta\theta}\) their values from (3), where instead of \(P\) one must substitute \(pr\,dr\,d\alpha\):

\[ X_x=\int_0^{2\pi}\int_0^R \frac{p}{2\pi}\left[ -\frac{3r^2z}{(r^2+z^2)^{5/2}} -(1-2\sigma)\left( \frac{z}{r^2(r^2+z^2)^{1/2}}-\frac{1}{r^2} \right) \right]\cos^2\alpha\cdot r\,dr\,d\alpha+ \]

\[ +\int_0^{2\pi}\int_0^R \frac{p}{2\pi}(1-2\sigma)\left[ \frac{z}{(r^2+z^2)^{3/2}} +\frac{z}{r^2(r^2+z^2)^{1/2}} -\frac{1}{r^2} \right]\sin^2\alpha\cdot r\,dr\,d\alpha= \]

\[ = \frac{p}{2}\int\limits_0^R\left[-\frac{3r^2z}{(r^2+z^2)^{5/2}}+(1-2\sigma)\frac{z}{(r^2+z^2)^{3/2}}\right]r\,dr = \]

\[ = p\left[-1+\frac{3}{2}\frac{z}{\sqrt{R^2+z^2}}-\frac{1}{2}\frac{z^3}{\sqrt{(R^2+z^2)^3}}+\left(\frac{1-2\sigma}{2}\right)\left(1-\frac{z}{\sqrt{R^2+z^2}}\right)\right] \]

\[ Z=\int\limits_0^{2\pi}\int\limits_0^R\left(-\frac{3p}{2\pi}\cdot\frac{z^3}{(r^2+z^2)^{5/2}}\right)r\,dr\,d\alpha = -3p\int\limits_0^R\frac{z^3}{(r^2+z^2)^{5/2}}r\,dr = \]

\[ = p\left[-1+\frac{z^3}{\sqrt{(R^2+z^2)^3}}\right]. \]

The expression for the stress \(X_x\) contains the unknown value of Poisson’s ratio \((\sigma)\); taking it for a loose body equal to \(0.5\), then:

\[ \left. \begin{aligned} X_x &= p\left[1-1.5\frac{z}{\sqrt{R^2+z^2}}+0.5\frac{z^3}{\sqrt{(R^2+z^2)^3}}\right] \\ Z_z &= p\left[1-\frac{z^3}{\sqrt{(R^2+z^2)^3}}\right] \end{aligned} \right\} \quad \ldots \quad (4), \]

where compressive stresses are taken as positive.

If the point \(a\) under consideration (Fig. 6) is a dangerous point, then for it the following condition must be satisfied:

\[ \frac{ p\left[1-1.5\frac{z}{\sqrt{R^2+z^2}}+0.5\frac{z^3}{\sqrt{(R^2+z^2)^3}}\right]+n_1\Delta(z+h) }{ p\left[1-\frac{z^3}{\sqrt{(R^2+z^2)^3}}\right]+\Delta(z+h) } = n \quad . \quad (5), \]

where

\[ n=\tg^2\left(\frac{\pi}{4}-\frac{\varphi}{2}\right),\qquad n_1=\tg^2\left(\frac{\pi}{4}+\frac{\varphi}{2}\right). \]

Whence

\[ h=-z+\frac{p}{\Delta(n_1-n)} \left[ -1+n+1.5\frac{z}{\sqrt{R^2+z^2}} -(0.5+n)\frac{z^3}{\sqrt{(R^2+z^2)^3}} \right]. \quad (6) \]

To find the maximum value of \(h\), we take the derivative with respect to \(z\) and set it equal to zero:

\[ \frac{dh}{dz} = -1+\frac{p}{\Delta(n_1-n)} \left[ 1.5\frac{R^2}{\sqrt{(R^2+z^2)^3}} -3(0.5+n)\frac{R^2z^2}{\sqrt{(R^2+z^2)^5}} \right] =0 \]

or

\[ -\sqrt{(R^2+z^2)^5} +\frac{1.5p}{\Delta(n_1-n)} \left[ R^2(R^2+z^2)-(1+2n)R^2z^2 \right] =0 \quad . \quad (7) \]

Let us find the roots of this equation that make \(h\) a maximum, for \(\Delta=0{,}0015\ \mathrm{kg/cm.^3}\)*) and \(\varphi=32^\circ\). Then:

\[ n_1=tg^2\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)=3{,}255;\quad n=tg^2\left(\frac{\pi}{4}-\frac{\varphi}{2}\right)=0{,}307. \]

Put

\[ z=mR \quad \text{and} \quad 1+m^2=u^2. \]

Equations (6 and 7) can now be represented in the form:

\[ h=-\frac{m}{2}D+\frac{p}{0{,}00437}\left[-0{,}693+1{,}5\,\frac{m}{\sqrt{1+m^2}}-0{,}807\,\frac{m^3}{\sqrt{(1+m^2)^3}}\right]\tag{8}, \]

\[ u^5+422\,\frac{p}{D}u^2-1108\,\frac{p}{D}=0 \ . \ . \ . \ . \ . \tag{56}, \]

where \(D\) is the diameter of the base.

The roots of equation (56) that make \(h\) a maximum are most simply found graphically, by putting:

\[ y_1=u^5 \]

\[ y_2=-\left(422\,\frac{p}{D}u^2-1108\,\frac{p}{D}\right). \]

The determinations made for the quantities \(u\) show that these quantities, and together with them the quantities \(m\), vary very little with changes in \(D\) and \(p\), so that for practical applications expression (58) may be replaced by the simpler one:

\[ h=-0{,}64D+21{,}3p \ . \ . \ . \ . \ . \ . \ . \ . \ . \tag{10} \]

The results of computations of \(h\) by formulas (8) and (9), and also by formula (10), for \(D=1{,}13\ \mathrm{cm.}\) (which corresponds to the area of the base in Strohschnеider’s experiments of \(1\ \mathrm{cm^2}\)) are as follows:

We have seen that the results of Strohschnеider’s experiments on determining the depth of embedment of a foundation can be expressed by a formula of the same form as formula (10), but only with other coefficients (see form. 2). The depth of embedment according to formula (57) comes out approximately 4 times greater than is given by Strohschnеider’s experiments (form. 2). Here, evidently, the same phenomenon is repeated as with respect to Kurdyumov’s experiments.

*) In Strohschnеider’s experiments the values of \(\Delta\) and \(\varphi\) are not indicated.

The large experimental resistance of the foundation bed to heaving, as compared with the theory, may be explained, first, by the fact that we determine the depth of embedment of the foundation at the moment when limiting equilibrium occurs at only one dangerous point; and, second, by the fact that the magnitude of internal friction (in the opinion of many authors) is always greater than the angle \((\varphi)\) of the natural slope.

The magnitude of the soil resistance at the surface is determined from formulas (10), in which, instead of \(h\), one must substitute zero; we obtain:

\[ p=\frac{0{,}64}{21{,}3}D=0{,}03D \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (11) \]

Consequently, in this case as well, the theory gives the same dependence between \(p\) and \(D\) as do Strohschneider’s experiments, i.e., the resistance of the soil is proportional to the diameter of the foundation; but the theoretical values of the soil resistance are three times smaller than the experimental ones.

P. Miliev.