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During the experimental study of the process of vibro-driven piles (O. A. Savinov, A. Ya. Luskin, M. G. Tseitlin, S. V. Plekhanova, 1954; N. A. Preobrazhenskaya, 1958; D. D. Barkan, 1959) and analysis of the results of studies of the lateral and frontal soil resistance (I. I. Artobolevsky, A. P. Bessonov, N. P. Raevsky, 1954; A. S. Golovachev, 1960; M. N. Goldstein, V. M. Goldstein, 1963), the following experimental facts were established:

- at amplitudes and frequencies of vibrations smaller than certain values, depending on the type of soil and the size of the surface of the submerged element, the pile does not move relative to the soil, but oscillates with it;
- when the amplitude at a certain frequency reaches a critical value, the pile breaks down relative to the adjacent soil (ground slippage); with increasing frequency, the critical amplitude of the breakdown decreases; in clay soils, due to the presence of cohesive forces, the value of critical amplitudes is 1.5–2 times higher than in sand;
- after a breakdown has occurred, the lateral resistance becomes significantly less than the static one (2-4 times in clay soils, 4.5-6 times in water-saturated sands);
- with sufficient intensity of vibrations in the conditions of failure, the influence of elasticity and inertia of the soil along the side surfaces of the pile is small;
- when vibrating into low-moisture sands and dense clays, no decrease in resistance is observed.

On the basis of these experimental facts, computational models of the interaction of the submerged (retrievable) element with the soil are constructed.

The general equations of immersion (extraction) into the soil of an element considered as a solid body can be written in the following form:

(1)

Determination of the most effective vibration modes depending on the type of functions and and the study of the patterns of these processes, taking into account changes in soil resistance and are problems of the theory of vibration penetration and vibration recovery.

In theoretical studies of the process of vibrational immersion of elements into the soil, various calculation models of the interaction between the immersed element and the soil are used.

One of the most widely used at present, the elastic-plastic model of soil resistance in the study of pile driving, was first used by N. M. Gersevanov. A special experimental study with direct measurement of the soil resistance force and pile settlement showed the legitimacy of using an elastic-plastic model of soil resistance during vibrational driving of piles.

It should be noted that for the construction of models of vibration penetration under elastic-plastic resistance of the soil, clearly formulated by O. A. Savinov and A. Ya. Luskin, conditions for effective pile driving by vibration include factors such as the need to break the pile relative to the adjacent soil, the minimum value of the amplitudes, the values of the driving pressure, etc.

In the elastoplastic model, the relationship between dynamic soil resistance and pile settlement can be represented as a Prandtl diagram: such a model is characterized by the presence of an ideal (without dissipative losses) spring (the elastic component of resistance) located between the pile and a weightless plug. The movement of the plug is possible if the force applied to it exceeds the resistance force of the soil, considered as the force of dry friction at the stage of plastic deformation of the soil.

Dynamic soil resistance to pile driving is a non-linear function of displacement and has a viscoelastic-plastic character (A. S. Golovachev, 1960; Yu. R. Perkov. V. M. Shaevich, 1974; V. V. Bakholdin, L. Ya. Ginzburg, 1975) . The viscous component of soil resistance manifests itself when the pile slides relative to the ground. Its dependence on the oscillation velocity is essentially non-linear with a soft characteristic, while at low oscillation velocities (5–10 cm/s), values are achieved that change little with increasing velocity. This gives grounds to consider the viscous component of soil resistance as a constant force in the calculation models of vibration penetration.

The second (after elastic-plastic) of the most commonly used soil resistance models is purely plastic, in which it is assumed that dry friction forces act between the side surfaces of the pile and the soil. In a purely plastic model, dynamic drag is represented as a weightless plug, the movement of which is possible if the force applied to it exceeds the soil resistance force, considered as a dry friction force. Thus, in a purely plastic model, to ensure the pile sinking, the sum of the forces applied to it must exceed the soil resistance force constant for one plunge cycle along the lateral and frontal surfaces.

In addition to these two basic models, some studies also use more complex models of the soil resistance mechanism: elastoplastic with hardening, elasto-viscoplastic with regard to the added mass of soil, etc.

Experimental data indicate that, depending on the soil conditions, the relationship between the lateral and frontal resistances of the soil, the type of vibration effect, etc., one or another component of the dynamic soil resistance may be dominant. These features should be taken into account in the accepted calculation models.

So, for example, when immersing and extracting tubular elements in the mode of longitudinal-rotational vibrations, the resistance of the soil is determined depending on the ratio between the longitudinal and rotational components of the vibration velocity, and the soil surrounding the pile can be considered motionless, since the vibration amplitude of the tubular element is two to three orders of magnitude higher amplitude of ground vibrations. When driving elements with a developed frontal surface (piles or pile shells) in which a soil plug has formed, it is necessary to consider an elastic-plastic model of soil frontal resistance. When setting casing pipes under the action of longitudinal shocks and static forces, during horizontal penetrations with soil extraction, etc., a purely plastic resistance model can be considered.

For engineering calculations, it is necessary to have reasonable data on the values of soil resistance forces used in a particular calculation model.

The most reasonable estimate of the average soil resistance forces is based on the energy expended during vibrational immersion (with measured displacement parameters.) It should be noted that this approach is justified when estimating the values of dynamic lateral resistance and from a physical point of view. As the speed of oscillations of the submerged body increases, the lateral resistance of the soil decreases, tending to a certain value, correspondingly reduced in comparison with the static value of the resistance. This experimental fact equally applies not only to resistances that have the nature of dry friction, but also to viscous resistance forces, which also reach a limiting value and can be interpreted as energetically equivalent forces of dry friction. It is this approach that underlies the method for determining the forces of lateral soil resistance, which can be used in calculations.

So, for example, according to A. S. Golovachev, the forces of dynamic lateral resistance during vibration immersion (extraction) are less than the corresponding static ones in wet sands by 3.5-5 times, in water-saturated sands by 4.5-6 times, in loams by 2.5-4 times.

The question of quantifying drag forces in various soils has been less studied. In particular, the issue of changing the resistance of soil saturated with water under the pile end has not been practically studied. This resistance can significantly decrease due to the effect of weighing and decompacting the soil under the pile tip during vibrational immersion.

A. S. Golovachev’s research showed that the dynamic frontal resistance during vibration penetration in hard and semi-hard clay soils remains approximately equal to the static one and decreases by 20-30% in plastic soils due to the formation of a softened layer during vibration under the pile end. In this case, the dynamic frontal resistances during vibration and shock immersion are practically equal to each other.

In the practice of engineering calculations of the parameters of vibrational driving of piles, the values of dynamic frontal resistances, taking into account the possibility of sinking in various soil conditions, are taken equal to their static values.

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