Nonlinear calculation of structural elements according to transforming design schemes

. An approach to the calculation of the processes of structural elements deformation and their environments in the case of a complex phased erection of building objects is proposed. On the basis of consistent equations of the geometrically nonlinear theory elasticity within the finite element approach to the geometry description, a technique for solving three-dimensional statics problems of concrete and reinforced concrete structural elements is implemented. On the basis of calculation by transforming calculation schemes new computational models of spatial elements deformation of building structures during their interaction with soil media have been developed. On the basis of the proposed methods the underground elements deformation problem of building structures was solved when carrying out technological measures related to the local strengthening of already erected underground facilities. The calculation results showed that the use of the proposed methodology makes it possible to implement the determination of the stress-strain state and displacement fields in the phased construction zone in a single design scheme.


Introduction
There is a wide enough range of problems for which the linear theory of elasticity gives acceptable results.However, it cannot be overlooked that the equations of the linear theory of elasticity are a very rough approximation and may be completely unacceptable when modeling the mechanical behavior of deformable solids in a realistic setting.Due to the development of modern and rapid development of computer technologies, it has become possible to solve the problems of nonlinear theory of elasticity taking into account the real operation of the structure and almost all physical and mechanical properties of the material.
The question of the correct formulation of nonlinear problems is of great importance.Usually, geometric and static equations are written in different coordinate systems, then for the correct recording of physical equations, static equations are usually reduced to the Cartesian coordinate system of body points before deformation.In this case, the so-called generalized stresses are introduced into consideration, which are not stresses in the exact sense of the word.However, the physical equations of the geometrically nonlinear theory of elasticity establish a connection precisely between the components of strain tensors and generalized stresses.A review of the literature of the last decades shows that the interest in the calculation of physically nonlinear elements of structures and continuous media taking into account geometrical nonlinearity is becoming more and more widespread, but there are lack of works that take into account the geometrically nonlinear theory of deformations, the geometrically nonlinear theory of stresses corresponding to it and the corresponding physical theory.Equations written in terms of generalized stresses are practically nonexistent.V.V. Novozhilov rigorously and in sufficient detail outlined the foundations of the nonlinear theory of elasticity.

Problem statement 2.1 Basic equations
For a deformable solid body 0 V bounded by a boundary surface 0 S in an orthogonal Cartesian coordinate system in a state of static equilibrium of the body, the variational equation can be written as where * i σ , * P * F and are the stress vector, the surface vector and the vector of body forces on the areas of the deformed elementary volume, referred to the units of area 0 dS and volume 0 dV respectively, u -the displacement vector.Let be ** , , u are the main basis vectors in the deformed state of the body, then the left side of equation ( 1) is transformed to the form: The true strains of elongation and shear are the quantities: and the true [7] stresses to V.V. Novozhilov are the components of the vectors  Taking into account relations (2), ( 3) and ( 4), the variational equation ( 1) can be written in terms of the components of the true stress and strain tensors.
For the model of elastoplastic deformation with isotropic hardening of the material, the plasticity condition [1] can be formulated as where only the scalar hardening parameter is chosen as the parameter of the internal state   , * i  is the intensity of true stresses, and * H is a monotonically increasing positive function approximating the true strain diagram of the material.
Then the increments of true stresses for beams of steel reinforcement can be represented as: In relations (1), the increments of the average true strain are denoted by 0 tr   , and the increments of the deviator of true strains are denoted by For "weak" soils, relations (6) take the form: and are also written for concrete and rocks in the form: The algorithm for solving the elastic-plastic deformation problem was implemented in the framework of finite-element discretization of the computational domain.

Calculation according to transforming calculation schemes
In the tasks of modeling mechanical processes occurring in complex building structures during the phased process of their erection, the concept of transforming structures (or mechanical systems) is introduced.Such structures undergo structural changes at certain stages of construction and move from one class to another.At the same time, new fields of stresses, deformations and displacements are formed in building structures at each stage.Moreover, the stress-strain state of the elements of such structures at the previous stage of construction can significantly affect the mechanical state of the structure at subsequent steps.In most cases, it is necessary to set the problems of mechanics taking into account geometric nonlinearity.So, the deformation process can be represented as a sequence of equilibrium states, and the transition from the current state to the next one is determined by the load increment, changes in the boundary conditions or in the computational domain.

Simulation of soil settlement in the area of the underground tunnel station in civil engineering
So, in Figure 1 a preliminary design solution for the construction of a complex of buildings of the Nagatinskaya office and shopping center in the area of the lifting facility of the metro is shown.The complex of buildings includes two high-rise buildings on reinforced concrete piles with bearings, which rest on clay.Between the two skyscrapers, it is planned to erect a seven-story building located on a reinforced concrete slab, which, in turn, rests on the extreme rows of reinforced concrete piles with thrust bearings.
Initially, bored piles are introduced into the ground, and then in the area of their bases, using high-pressure jet technology, a liquid concrete solution is injected, after which the socalled heel supports are formed after hardening.The process of building a complex of buildings with supports is divided into stages, the calculation itself is carried out according to transforming design schemes based on the methodology proposed in the work.The calculation area for a project with short bored piles and a protective casing is a piecewise homogeneous area of soil, to which the load from the building is transferred through the foundation slabs heel supports located on the sand-clay border (see Fig. 2).The piles are located both under the side buildings of the designed building and under the central one.It is assumed in the calculation that the boundary of the computational domain passes along the clay-limestone boundary.Figures 3 and 4 show the distribution of sediments in the ground in the area of the metro tunnel station for cases with and without a protective cover.

Conclusion
The application of the approach proposed in the study to solving problems of stage-by-stage erection of construction objects by transforming design schemes based on consistent equations of geometrically nonlinear elasticity theory significantly affects the final result of deformation of the entire structure.
e u the main basis vectors in the deformed state of the body, then for the vectors * i σ we can take the decomposition * * * i ij j   σR .Assuming the decomposition for vectors * i σ , where ** , then between the components * ij  and ij  there are dependencies[7], which can be transformed to the form:

Fig. 1 .Fig. 2 .
Fig. 1.Preliminary constructive solution for the construction of a complex of buildings in the metro station area.

Fig. 3 .Fig. 4 .
Fig. 3. Settlement of the calculation area for the case of calculation with a protective cover.