Interaction of conical monolithic thin-walled reinforced concrete shells with the soil of the foundation

Conical monolithic thin-walled reinforced concrete blocks (shells) can be used as foundations for frame structures and buildings with load-bearing walls. As an analogue of columnar foundations, this type has the advantage of the speed of installation. It almost wholly excludes earthwork and formwork from the technological process, provided that the physical and mechanical properties of the soil base are improved. Shells are immersed in the supporting soil base with the help of dynamic or static loading. Additional soil base compaction occurs due to interaction with the conical outer surface of the shells. The work aims to develop the theoretical foundations for modelling, mathematical description and the possibility of optimizing the process of sinking monolithic thin-walled spatial reinforced concrete shells that model foundations for low-rise technological or residential facilities. Based on the analysis of the ”hammer-head-shell-soil” system, an elastic-viscous-plastic model with an attached soil mass was developed and presented, which is used as a basis for developing a system of nonlinear differential equations of the second order. The article provides recommendations on the impact of structural and technological factors on the failure rate and the contact stresses in the hammer-shell zone. The following factors have a significant influence on the failure of the shell: the height of the hammer lift, soil resistance, and the ratio of the masses of the hammer and the shell. The effect of shock-absorbing elements is insignificant. The greatest influence on the magnitude of the contact forces in the zone of the hammer-shell has: the height of the fall of the hammer and the stiffness of the resilient gasket.


Formulation of the problem
Monolithic and prefabricated reinforced concrete foundations traditionally used at present, along with known advantages, have great complexity of erection on the construction site, high material consumption and, accordingly, cost.The use of monolithic thin-walled spatial reinforced concrete shells (MTWSRCS) as foundations for low-rise technological or residential objects is economically feasible from the point of view of reducing the volume of earthworks by up to 90%, abandoning the use of formwork, shortening the time for the concrete to gain strength.In addition, the use of MTWSRCS, in comparison with classical types and methods of laying 1254 (2023) 012058 IOP Publishing doi:10.1088/1755-1315/1254/1/012058 2 foundations, allows you to significantly reduce the costs of concrete, reinforcement, and labor costs, which ultimately leads to a decrease in the estimated cost of zero-cycle works by up to 40% [1][2][3][4][5][6][7].
The issues of interaction processes of MTWSRCS with the soil of the foundation during dynamic or static immersion, as well as during the operation of the building, remain insufficiently researched and highlighted in the scientific literature.

Analysis of the latest sources of research and publications
There are some close, according to the physics of the process, the results of research on establishing the resistance of the foundation soils with the dynamic method of sinking classical reinforced concrete piles into it, conducted by: B.V. Bakholdin, L.Ya.Ginsburg, D.D. Barkan, etc. [8][9][10][11][12].Most researchers recommend determining the soil base's resistance by considering the so-called viscosity coefficient.
In the work of L.R. Stavnitzer established the existence of a critical rate of soil deformation, beyond which the magnitude of the reaction of the soil base practically does not depend on the rate of its deformation.B.V. Bakholdin, L.Ya.Ginzburg proved that the value of the critical rate of deformation of the soil relative to the piles depends on the static resistance of the soil.
Immersion in the soil of MTWSRCS, similar to the process of immersion of a classic reinforced concrete pile, the layers of the soil base are pushed to the sides and bent down.Due to the significantly greater conicity of MTWSRCS compared to the old one, soil deformations are brighter and visually noticeable.This leads to additional compaction of the soil under the conical part of the MTWSRCS, which is accompanied by the destruction of the natural structure of the upper layers of the soil, their mixing and partial bulging upwards.The result is an artificial soil base with improved physical and mechanical properties and correspondingly higher bearing capacity.The prospects of this type of foundation with MTWSRCS for soil foundations that are not prone to frost heave, are not over moistened and are not located in a climatic zone with a long period of negative average daily outdoor air temperatures should be noted.

The goal of the work
Based on practical experience using monolithic thin-walled spatial reinforced concrete shells as foundations for low-rise technological or residential facilities, develop theoretical foundations for modelling, mathematical description and the possibility of optimizing the immersion process.

Research results
Immersion of MTWSRCS is possible by impact and static load in the form of discrete or continuous gradual loading.Immersion by impact is a complex energy process, during which the potential energy of the hammer is transformed into the kinetic energy of the impact, which leads to the overcoming of the resistance forces of the soil base to the final and elastic movements under the surface of the MTWSRCS [2,3,[11][12][13].
At the same time, the impact energy developed by the falling load (hammer) is partially lost during the co-impact through the damper between the hammer and the shell, shaking the surrounding soil of the base, and only a part of it determines the final movements of the MTWSRCS.
During the impact of the shock load, the elastic deformation of the head C 1 , then the shell C 2 itself, and the soil of the base C 3 occurs, and only after that does the final vertical movement (immersion) of shell E occurs.After the shock pulse is exhausted, the elastic deformations are restored.Thus, the total elastic deformations are equal to: (1) The values of C 1 and C 2 are usually small and can be neglected in many cases, i.e. take C = C 3 .
In the idealized schemes of changes in C and E as the impact energy of the hammer ϵ increases, the elastic part of failure C increases at E = 0, until it reaches the limit value C (figure 1).With a further increase in ϵ , the value of C remains constant.The final failure of E begins to increase.Thus, the ultimate elastic deformation of soil C does not depend on the immersion parameters Q and H and is a soil characteristic.The dynamic nature of the load application, which qualitatively changes depending on the ratio of the three phases of the soil base, significantly influences the soil's deformation and resistance to the immersion of the shells.Due to the impossibility of considering all the features of the upper layers of the soil base of the construction site and obtaining accurate analytical expressions of dynamic resistance, it seems appropriate to use modelling methods for this purpose.At the same time, it is proposed to apply simple models that could reflect only the main properties of the system, and a large number of temporary features would be taken into account in a generalized way or through the values of calculated indicators.To simplify this model, the initial state of the soil and the features of its changes during deformation may not be taken into account.Just to reflect the final influence of the features of this soil on the development of resistance forces may be reflected.Such simplest soil models for the analysis of the "hammer-head-shell-soil" system are plastic and elastically plastic [9,[11][12][13][14][15] (figure 2).
The plastic model is built considering the following assumptions (figure 2, a).It is an absolutely solid body, and the soil surrounding it is motionless.Resistance on the side surface P b , etc. friction between the side surfaces of the shell and the soil, reduced to equivalent to all types of Coulomb dry friction (it is assumed that P b , does not depend on the speed of movement of the shell).
As the tests showed, reducing the dynamic side friction to the equivalent dry friction makes it possible to obtain fairly stable soil resistance values.
The frontal resistance P l is represented by a pinched weightless bottom.Overcoming the friction that develops on the side surface, the shell affects the bottom, which sinks if the force applied to it exceeds P l .
The relationship between the frontal resistance to the movement of the shell and its subsidence is given in a simplified form -in the form of a broken line OAB, and it is assumed that the force P l does not depend on the speed of movement of the shell.The area of the OABD diagram represents the work for one cycle of P e (figure 2, a).Despite its simplicity, this scheme can be quite useful, as it does not require setting many uncertain parameters of elasticity and viscosity.
The elastic-plastic model is distinguished by the presence of elastic ties that simulate the elasticity of the soil and intermediate elements (figure 2, b).At this research stage, it is assumed that the elasticity of the soil is manifested mainly at the point of contact of the end of the shell with the soil [16][17][18][19].Therefore, the assumption that the lateral resistance to immersion is characterized by dry friction remains valid for the scheme.
The mechanical model of frontal resistance is simplified as a bottom with a linear spring.When the casing is struck, the elastic deformation of the soil OA (compression of the spring) first occurs, and after the force in it reaches the value of the frontal resistance P l , irreversible compression of the bottom AB begins (figure 2, b).After the load is over, the BD is restored.This graph generally shows well the interaction of the shell with the soil.
The positive properties of this model are also that when using the energy approach to determine energy costs, i.e. the area of the force-displacement diagram, the contours of this diagram, as well as the condition that C = 0 does not affect the accuracy of the results.
The considered model does not take into account the influence of soil inertia, which is quite significant.Therefore, for further clarification, an elastic-plastic model with an attached soil mass is adopted (figure 3, c).In this model, the surrounding attached soil mass is represented by an equivalent elastic body resting on elastic supports.In this case, the body's weight is equal to the attached soil's weight.
When the shell moves after impact, the attached mass moves together with the shell until the elastic forces of the springs reach the values of the resistance on the side surface P b .After that, the shell begins to slide relative to these elements.Further interaction is similar to the interaction of the previous model.
The elastic-viscous-plastic model (figure 3, d) allows you to consider the soil's viscous resistance.Further, it is possible to clarify the nature of the change in the frontal resistance of the soil depending on the settlement during loading and unloading, which means applying instead of a simple Prandtl elastic-plastic model, an elastic-plastic model with compaction, etc.Thus, the number of possible variants of dynamic models can be pretty significant.These models will more and more fully reflect the influence of the main factors and their role in the immersion process.In all models analyzed above, the shell is considered as a completely rigid body.However, upon impact, the shell has an elastic deformation.In addition, the usual impact on the shell is performed through the elastic headband.Therefore, a spring can also represent the shell and the headband's elasticity.
Evaluating the possibility of using complex models for practical calculations, it should be pointed out that currently, there is not a sufficient amount of experimental data for a reasonable assignment of the numerical values of many indicators characterizing these models.In addition, the obtained mathematical expressions of the interaction of the "hammer-shell-soil" system will become more and more complex with the complexity of the models, and this complexity will not always contribute to an increase in accuracy, i.e. practical purposes.Taking this into account, for further research, it is most appropriate to use an elastic-viscous-plastic model considering the attached mass of the soil and the elasticity of the shell and the head (figure 3, d).This scheme allows you to take into account the change in the value of the attached mass of the soil and the change in the nature of the interaction in connection with this.
Thus, the dynamic model of the "hammer-head-shell-soil" system with elastic-viscous-plastic resistance of the soil is presented as follows.The elastic shell, which is in the soil, is struck with a hard undeformed hammer through the elastic head.
Under the influence of the impact, the shell acquires a reserve of kinetic energy, which is spent on overcoming soil resistance: with each impact, the shell sinks, initially elastically deforming the soil by the amount C, and then the plate moves in the soil by the amount of final failure E. Elastic deformations are restored after each impact.Each blow is considered as a single one, isolated from others, which means that before each subsequent impact, the shell, hammer and soil are at rest.The shock is absorbed by the reduced mass of the shell, taking into account part of the mass of the soil attached to it, and the subsequent movement is carried out only by the shell.The hammer impact effect is represented as the transfer of some part of the kinetic energy of the impact ϵ, which goes directly to the immersion of the shell [16][17][18]20].For the development of a mathematical model of the process of immersion in the ground base of conical-reinforced concrete shells, a calculation scheme is proposed (figure 4).
The mathematical model of this process is a system of nonlinear differential equations consisting of the equations of the "hammer-head" and "shell-soil" subsystems, which must be solved jointly.
The mathematical model "hammer-head-shell-soil" is a system of nonlinear differential equations of the second order: where m 1 , m 2 , m 3 , m 4 -are the masses of the hammer, head, and shell, respectively core and load; , P 23 -respectively, the forces acting on the contacts of the bodies, which collide; X 1 , X 2 , X 3 , X 4 -are, respectively, the coordinates of the hammer, head, shell and soil; F l , F b -frontal and lateral soil resistance, respectively; R b -is the ultimate resistance of the soil on the side surface.The solution of this system of equations gives the displacement and speed of all bodies, as well as the value of the forces acting on the contacts of colliding bodies at any moment in time.
Mathematical model (2) belongs to the class of simulation models, as it simulates in detail the submersion of shells by impact load.This nature of the model allows you to use it to study the process of computer immersion.In addition, implementing full-scale experiments on the immersion of shells is a rather expensive method that requires a lot of machine time.
Experience shows that the most interesting for practice is the study of the influence of the following factors on the immersion process: • ultimate resistance of the shell on the soil; • the ratio of the mass of the striking part to the mass of the submerged shell; • the thickness of the elastic gasket in the headrest; • lifting height of the impact part.
Wood or conveyor rubber can be used as cushioning material in the headrest.The soil's stiffness depends on the value of the lateral resistance so that the elastic failure does not exceed 1 cm.This value of elastic failure is most often encountered in practice.
Calculations on a computer were carried out for a hammer with a striking part weighing 500 kg.Mathematical planning of the experiment was used to organize calculations on a computer.The four-factor Hartley-Cohn plan was adopted [21,22].In order to obtain analytical dependencies between the parameters of the studied system, a correlation analysis of the obtained data was carried out.The regression equations of the type were used for the analysis: Co-impacts in the "hammer-head-shell-base" system are characterized by forces that appear at the contacts of the elements that are in direct contact during the impact.On (figure 4, a,  b, c) graphs of changes in time of these forces, calculated with the same initial data for some time, at a level of 0,03 s after the first contact of the hammer with the head, are given.The impulse of impact forces is characterized by several peaks and has a complex damping character.The contact of the shocking part with the head is irregular.Thus, for a period of 0,06 s, 11 co-impacts occurred at the "hammer-head" contact and ten at the "head-shell" contact.
This way, we can talk about the presence of an oscillatory process in the "hammer-headshell-shell-soil" system.The headrest oscillations are unstable.This follows from the fact that during slow, quasi-static immersion, oscillations do not occur.They occur only during impact and die out as the load decreases.
In real conditions, due to the inevitable eccentricities of load application, the headrest carries out not only translational movement along the vertical axis, but also angular oscillations.In the mathematical model, following the accepted assumptions, all movements occur only along the vertical axis, so the graph of the impact force on the shell is divided into a number of separate co-impacts.
The observed multiple co-impacts of the headpiece with the shell are due to a significantly shorter period of the headpiece's own oscillations compared to the duration of the force impact.Therefore, the headpiece can be represented as a mechanical oscillator, brought out of equilibrium by mechanical co-impact with the striking part and then oscillating between two massive bodies: the striking part and the shell.
The regularities noted above were reflected in the correlation dependences for the impact force.The impact force on the "hammer-headset" and "headset-shell" contacts can be determined by the following dependencies: where m m , m n , m o b, m c -are the masses of the hammer, head, shell, and core, respectively; a 1 , S 1 , and a 2 , S 2 -respectively, the thickness and area of the cushioning pads at the "hammer -headrest" and "headrest-shell" contacts; H 1 -hammer lift height; R -is soil resistance.The correlation ratio of these formulas is at least 0,95.This testifies to the correct selection of varied parameters.
The analysis of the given dependencies shows that the force of the impact is mainly determined by the height of the impact part and the stiffness of the elastic gasket.The ratio of the masses of the hammer and the shell has a smaller influence on the immersion process.The influence of the physical and mechanical properties of the soil base is insignificant.
Despite the periodic nature of the force impact, the shell sinks into the soil base quite smoothly.This is explained by the significant mass of the shell, as well as the inertial stabilization properties of the soil base.The shell receives a positive acceleration under the influence of the first co-impact, and then the shell acceleration changes its sign, and its motion slows down.
For the value of the final failure, the best result is given by the dependence of the form: Analysis of the influence of various factors on the size of the failure shows that the impact of the stiffness of the shock absorber can be neglected since the reduction in failure when the thickness of the lining increases from 0,05 m to 0,25 m does not exceed 10%.Therefore, for practical purposes, dependence (8) can be simplified:

Conclusions
• The theoretical foundations for modelling, mathematical description and the possibility of optimization of the process of immersion of monolithic thin-walled spatial reinforced concrete shells in the soil environment have been developed.• Evaluating the possibility of using complex models for practical calculations, it should be stated that, at the moment, there is not enough experimental data for a justified assignment of the numerical values of many indicators characterizing these models.• The following factors significantly influence the failure of the shell: hammer lifting height, soil resistance, and the ratio of hammer and shell masses.The effect of shock-absorbing elements is insignificant.• The most significant influence on the magnitude of the contact forces in the zone of the hammer-shell has: the height of the fall of the hammer and the stiffness of the elastic gasket.The influence of the ratio of the masses of the hammer and the shell, as well as the value of the soil resistance, is manifested to a much lesser extent.• It is promising to establish the stress-deformation state of the soil base depending on the method of immersion and during the structure's life.

3 Figure 1 .
Figure 1.Scheme of the dependence of C and E failures on the height of the fall of the hammer H.

Figure 2 .
Figure 2. Dependence between the frontal resistance to movement of the shell P and its settlement S with different models of interaction: a) -plastic model; b) -elastic-plastic model.

Figure 3 .
Figure 3. Dynamic models of the "hammer-head-shell-soil" system: a) -plastic model; b)elastic-plastic model; c) -elastic-plastic model with attached soil mass; d) -elastic-viscousplastic model with attached soil mass.

8 Figure 5 .
Figure 5. Forces acting on the contacts of colliding bodies: a) -hammer-head; b) -a headshell; c) -is the frontal resistance of the soil.