Method of full discretization in joint calculation in time of the system “construction - foundation - soil”

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Introduction1 Prospective transition of CIS to the European design standards in the construction industry requires development of new and modification of known engineering methods for calculation and design of construction [1-3]. Creation and development of such methods should be based on fundamental research that can become the basis for the development of principally new, innovative technologies [4]. One of such directions is the further development and perfection of methods of analysis of structures and grounds, more precise estimate of evolution of the deformation stress state (DSS) and the yield of the building foundation in time. It is important to consider inelastic and nonlinear properties of building materials and structures, spatial and temporal in homogeneity, i.e. construction technology and other factors influencing their deformability, strength, durability and stability [5]. The basis for calculating any projected building object is its closed mathematical model, usually represented by systems of differential or integro-differential equations with boundary and initial conditions. However, the analytical solution of such systems of equations can be obtained only in special cases for objects with a simple geometric shape with simple boundary and initial conditions. The way out was found by introducing various types of discretization of a continuous problem in which an infinite set of numbers representing the unknown function is replaced by a finite number of unknown parameters. As a result, the system of differential equations for real-world problems tends to turn into large systems of algebraic equations that can be solved only with the help of modern computer technology [6-9]. Method of full discretization (FDM) - special modification of the finite element method (FEM) for the solving of various problems of the elastic creep, offered by N. Ter-Emmanuilyan [10]. FDM - universal, comparatively simple and obvious engineering method being “not step-by-step” in time. It gives an opportunity to determine discrete values of displacements, deformations and stresses in a calculated interval of time. The method is developed both in variant of displacements, and in variant of forces. It can be combined with other engineering numerical methods, such as a method of boundary elements, a method of finite differences and others. [11]. The FDM allows to take into account: a heterogeneous creep and ageing of materials of any constructions and foundation soils; physical and geometrical nonlinearity; plasticity; anisotropy; different modular elastic creep; influence of temperature; presence stressed enforcement and normal armatures in ferro-concrete; discrete diagrams of erection of constructions (increase or reduction of volumes, change of operational loadings, etc.) [12]. The method is applied at the decision of a wide class of engineering problems of a linear and nonlinear elastic creep. For example: plane problems; axisymmetric; three-dimensional; single-layered and multilayered plates and envelopes; bar and thin-walled systems; stability of plates and bars; contact problems; rock”; reinforced concrete pipe - backfilling; a heterogeneous thick-walled shell with steel facing at loading and unloading; research of evolution of stressdeformation state of the vertical supported shaft at drivage with the preset speed; calculation in time reinforced concrete wall panels with holes; calculation of multilayered plates in view of a creep of some layers; buckling of flexible plates; buckling of a rod and cylindrical bend of a plate with initial camber; research on model Shenly at conservative and following loadings; calculation of prestressed ferroconcrete rods; combined calculations of growing buildings and constructions and their bases [13]: § an evolution of stress-deformation state of a foundation ferroconcrete plate on a soil base; § a problem about of influence of non-simultaneity of erection of buildings on evolution of the stressdeformation state in constructions and basis at the constrained building of city territories; § calculations of the box-shaped substructure and the basis of high-altitude television tower on KokTyube mountain near Almaty city in three variants of statement of a task: plane, quasi-spatial and spatial; § calculation of a road embankment and its basis; § calculation in time four-level a ferro-concrete construction and its basis, etc. 6. Bases of the method Stress-deformation state (SDS) the elastic creep, homogeneous and isotropic body loaded in the age of τ = τ1 at small deformations in static problems completely is determined, if all are known 15 components of a vector: ë û f (xi ,t, τ) = éuT (xi ,t,τ) εT (xi ,t,τ)σT (xi ,t,τ)ù, (i =1, 2,3), as functions of coordinates and time, satisfying in each point to system of the matrix-vector equations: thermoelastic creep problems; a short-time high-tem- æ -nT J 0 ö æ u ö æ 0 ö perature creep of metals, etc. ç ÷ ç ÷ ç ÷ The mathematical justification of a FDM as a ver- ç 0 J -Lt ÷ ×ç ε ÷ + ç 0 ÷ = 0. (1) sion of a method of weighted residuals and also ap- ç 0 0 n ÷ ç σ ÷ ç ρ ÷ proximation and discretization error in numerical solutions is considered. The appropriate algorithms of the solutions of linear and nonlinear problems of elasticity, elastic creep and plasticity are constructed. è ø è ø è ø And to boundary conditions in movings on S1 and in superficial forces on S2. In system (1) n (3×6) - a matrix of linear differential operators on coordi- The package of application software for engineers and T researchers is developed. nates; ρ = [ XYZ ] - a vector of volumetric forces; The wide class of modelling and applied engineering problems are solved: calculation of evolution of ns - a matrix directing cosines an external normal to a surface, but with replacement of operators of diffestress-deformation state in the system “tunnel lining - rentiation ¶ ¶xi ,... on cosines cos (n , xi ),... ; J - a unit matrix; Lt 6´6 § a matrix of integro-differentual t 1 k δ = % ik δi (τ) dτ, (6) operators of an elastic creep with 12 nonzero elements Lij from which Lt11 has the following kind: or, is simplified, δ% k -1 tk - tk -1 t ò δ + δ = ik i,k -1 . (7) E (τ1 ) τ ë E(τ) û 1 t é 1 ù dK Lt11 = + C (t, τ1 ) + ò ê + C (t,τ)ú dτ = 1 t d K Thus ik 2 τ = δ(t, τ1 ) + ò δ(t, τ) dτ . 1 (2) δ(t, τ) = 1 E (τ) + C (t, τ). (8) Matrix Lt in usual designations is constructed on the basis of the equations of a condition for linear three-dimensional elastocreep the bodies, received by N.H. Arutjunjan [14]. For the decision of system of the equations (1) the numerical method of the decision based on full, existential digitization (FDM) was offered [10]. Generally triaxial the SDS, from six scalar integrated equations of a condition making the second group of the matrix equations (1) it is received, after sampling, the system of the algebraic equations having the form of generalized Hooke’s law: Digitization of objects in FDM on geometry is ε = E-1σ § μ E-1 (σ + σ )ü carried out as well as in FEM at the decision elastic and elastoplastic problems. x x y z ï K K K K K K ï , (9) The limited time piece (day, years) digitize р time points. For uniaxial the discrete form of the equation of a condition looks like the intense condition: xy xy ï γ = 2E-1 (J + μ) τ ý K K K K K K ï þ where µ=n Π. i ε = σ δ + t j dσ(τ) dτ δ (τ) dτ, An auxiliary matrix: i i i1 å ò i j =2 t æ 1 0 0 0 K 0 ö ç -1 1 0 0 K 0 ÷ (i = 1, 2, 3,K, p), ( j = 2, 3,K, p). (3) ç 0 -1 1 0 K 0 ÷ 0 0 -1 1 K 0 ÷ j-1 In (3) integral it is broken for the sum of inte- ç ÷ P= ç ÷ , ç çK K K K K K÷ (10) grals, the derivative is replaced differential. The formula (3), after introduction of matrix restrictions, gets a kind conterminous under the form with çç è 0 0 0 0 ø -1 1 ÷÷ Hooke’s law: ε = E-1σ, (4) n - the bottom triangular matrix generated from sizes nij: where ε and σ - vectors of discrete values. ìïν (ti , τ j ), νij = í (i £ j ) , ν (t, τ) = ε% (t, τ) . (11) The return square-bottom triangular matrix of matrix “module” Е describing elasticity, hereditary creep îï 0, (i < j ) ε0 (τ) and ageing of a material: At construction of matrixes Е and µ it is possible to use theories of creep or the data of base experiments. As a result of generalization of physical pari- æ δ11 0 K 0 ö ç ÷ ties(ratio) of the linear theory of creep and matrix δ K 0 -1 ç δ21 - δ% 22 % 22 ÷ “Hooke’s law” (10), the matrix form of the law of E p´ p = ç K K K K ÷, ç ÷ (5) a linear elastic creep is received in a general view: δ - δ δ - δ δ K ç % p1 p 2 % % % ÷ p 2 p3 pp σt = Dt εt , (12) è ø where Dt - the generalized matrix of a linear elastic r ik in which sizes δ% are calculated under the formula creep having in scalars 6р the order, σt and εt - 6рdimensional (3р-dimensional in flat problems) on an opportunity to change in a settlement interval of components tensors and to time points of vectors of stresses and deformations. time are taken into account; Rt - a vector of a varia- At a conclusion of the formula calculation of the matrix of rigidity of a final element generalized in time for quasistatic problems of an elastic creep the principle of possible movings Lagrange is used. In result the formula is received: ble or a constant in time of central loading. 7. Results of numerical modelling Let us consider joint in time calculation of twotier reinforced concrete building and its foundation k t = ò B1 Dt B1dv, (13) in the spatial formulation in the example of modular r T V where В1 - a matrix of communication of components of movings and deformations in time in FE: r r station tanks of biological sewage treatment with capacity of 6400 m3/day designed for filtration fields in the village Tasboget, Akmola Oblast, Kazakhstan (Figure 1). εt = B1qt . (14) Looks like a rectangular matrix generally about 6p×3np (n - number of units in FE). Matrix В1 turns out from a usual matrix in way of expansion of each scalar member Вkℓ in diagonal blocks - matrixes of the order р with constant diagonal elements Вkℓ. k t The matrix r of rigidity generalized in time elastocreep a final element has the order in р time the greater, than the order kr - matrixes of rigidity of an elastic element due to replacement of scalar elastic constants Е and n bottom triangular matrixes Е and µ the order р. The generalized matrix of rigidity Кt of system elastocreep elements which is square, block, the order, generally, 3Np (N - the general (common) number of units of elements of system) further is resulted. For uniformity on properties of an elastic creep of a body, this matrix can be received very simply - Figure 1. General view of the biological treatment of sewage The tanks of the modular station are made of concrete B25 with the following mechanical properties: initial modulus is 30 000 MPa, the calculated resistance to axial compression is 14.5 MPa, the calk t as well as r · by expansion of scalars Е and n in culated resistance to axial tension is 1.0 MPa. There was used the reinforcement armature of Class A-III matrixes Е and µ р the order. At calculation of the designs consisting from non-uniform on properties or age of materials, blocks of matrix Кt of system are calculated only by summation on the elements containing units i and j of global numbering of the appropriate members of generalized matrixes of rigidity of final elements: with mechanical properties: modulus of elasticity is 200 000 MPa, the design resistance is 367.7 MPa. In building practice erection of any constructions is an example ph growing in time. Depending on concrete conditions, process of escalating of viscoelastic bodies can occur both discretely, and is continuous. The account of a time history of develop- K tij = å r 'ij k . r tij (15) ment and loading bodies frequently results in qualitative changes in their mechanical behaviour. At designing large ground and underground constructions, Further allowing linear matrix algebraic equait is especial in conditions of city building, perfortion concerning a required vector qt of components mance of stage-by-stage geotechnical calculation, of movings in time of all units of system is submitted: since process of construction and finishing an operation phase is expedient. Thus results of calculations can differ from usual on the order, and sometimes Kt × qt = Rt , (16) and with change of a mark [15]. The model of concrete with “smeared” reinforwhere Kt - generalized stiffness matrix of system (SMS) in which the kinematic boundary conditions having cement were used in the calculation of the building to account for the rheological properties of the material of the tanks. In this case, the behaviour of elastic creeping material (concrete or reinforced concrete with “smeared” reinforcement) can be described by equations of the hereditary theory of aging with a measure of creep by S.V. Aleksandrovskiy [16]. δ(t, τ) = 1 (1 - e-βτ ) + ψ(τ) - E γτ L (t - τ) = D(t - τ)α-1 , where Δ, α are the creep parameters. The influence of the weight of a building on the deformation stress state of the soil mass is considered with assumption of smallness of the deformation with respect to the size of the deformed region. The same assumption remains valid when the yield of a building -ψ(τ) e - A2 + D (τ)é1 - e-α(1-t ) ù , is considered near walled pits, underground structu- where D(τ) 2 eγτ - A = C1 - C3 + A1 - A3 ë û A3 , ψ = C3 + , res, natural slopes, etc. The hypothesis about the smallness of the deformation is not applicable only when deformations comparable to the size of the deformable region (i.e. the process of sliding slope, buckling, etc.) are considered [19]. τ τ the values for parameters of creep are taken in accordance with recommendations of N.H. Aratyunyan [14]: α = d -1, β = 0.206 d -1, γ = 0.03 d -1, А1 = 4.62·10 -5 d/(kgf/cm2), А2 = 1, А3 = = 3.416·10 -5 d/(kgf/cm2), С1 = 0.975·10 -5 (kgf/cm2) -1, С3 = 0.756·10 -5 (kgf/cm2) -1 are the parameters of creep, Eo =30 000 MPa. Then we use the measure and kernel of Zh.S. Erzhanov [17] to describe the behaviour of elastic creeping ground: é ù Since the modern theory of creep contains a number of errors [20-23], we correct these errors by selecting coefficients; we perform the adapting of creep parameters for specific structures using experimental data [24]. The analytical model of building is shown in Figure 2, a. Due to the presence of symmetry plane, only half of the tank is considered in the calculation. This design scheme is split into 6212 octanodal prismatic volume elements connected in 7714 nodes. Boundary conditions in the form of rod connections are shown in Figure 2, b. δ(t, τ) = 1 1 + Δ (t - τ)α Figure 3 shows the projections of the tank with E ê α ú ë û obtained from the difference kernel building axes. а b Figure 2. Design scheme of the tank Загружение 1 1 3 4 5 E E 11.2 D D 6.2 C C B B A A 0 Y X 1 3 4 5 Figure 3. Frontal and upper projections of the tank Building of the tank is carried out in two stages. The first stage is the concreting of the base plate. The duration of this phase is 30 days (5 time points) (Figure 4, a). The own weight of the base plate is taken into account at this stage. The second stage is the concreting of the walls of the tank. The own weight of the base plate and the tank walls is taken into account at this stage. The duration of this stage is 90 days (5 time points) (Figure 4, b). In the third stage the tank is filled with water. The own weight of the base plate and the tank walls is taken into account at this stage. Duration of this stage is 480 days (10 time points) (Figure 4, b). a b Figure 4. Stages of construction of building Figure 5. Deformed diagram computational region in 600 days The timeline for calculations is taken uneven, with increasing intervals to the end of the scale. This makes it possible to estimate the evolution of the deformation stress state of the tank, both during periods of construction and initial operation. Information about the components of nodal displacements in 7714 nodes for the 20 time points is obtained as a result of the calculations. The components of vectors of deformations and stresses for each time point, at all 6212 finite elements are calculated. Thus, the complete picture of evolution of the displacement vectors, deformations and stresses in space and time is obtained. Let us consider some typical results of calculations of the change of the deformation stress state of the reservoir in time. Software module in AutoCAD was developed during the study. This module was used for graphical interpretation of the results obtained. The input data for this module are parameters of FEM of the construction model (nodes, elements, supports, loads, stiffness) and the results of solution of the problem MFD for given construction (vectors of displacements, deformations and stresses in each of the time points of the interval). The figure shows the deformed scheme of computational region (scale deformation increased 30-fold) for the last time point - in 600 days after the construction of buildings. Figure 6, a, b, c, d shows the contours of vertical displacements for t4 = 20, t9 = 70, t14 = 200, t20 = 600 days, which illustrate the evolution of vertical displacements. Isolines of the horizontal components of displacements for time points t9 = 70, t20 = 600 days are shown in Figure 7 (a, b). Figure 8 shows diagrams of yield of tank elements located along the axis 4, and Figure 9 shows diagrams of yield of tank elements along C-axis for four time points corresponding to the four stages of erection: t4, t9, t14 and t20 (respectively curves 1, 2, 3, 4). Figure 9 shows the displacements (yield) of more typical node that is located at the intersection of the axes C and 5 depending on time for the entire range of the study. Analysis of the solution reveals the significant incremental growth of displacement in time (Figure 10) and deformation (on the example of the vector components of ground deformations in the element number 2451 - Figure 11). Evolution of stress tensor components (on the example of the components of the stress vector in the ground element number 2451 - Figure 12) within each time step virtually are unchanged. a b c d Figure 6. Contours of vertical displacements for times t4 (a), t9 (b), t14 (c), t20 (d) a b c d Figure 7. Contours of horizontal components of displacements for times t9 (b), t20 (d) 11..2 6.2 0 Vertical displasements m10-3 0,00E+00 4887 4937 4987 5037 5087 5137 5187 5237 5287 5337 5387 5437 5487 5537 5587 -1,00E-03 4 -2,00E-03 9 -3,00E-03 14 -4,00E-03 20 -5,00E-03 Node numbers Figure 8. Diagrams of yield of nodes (m) along the axis 4. Curves correspond to the times: 1 - t4; 2 - t6; 3 - t14; 4 - t20 11..2 6.2 0 0,00E+00 Vertical displasements m10-3 -1,00E-03 -2,00E-03 -3,00E-03 -4,00E-03 -5,00E-03 4513 4514 4515 4516 4517 4518 4519 4520 4521 4522 4523 4524 4525 4526 4527 4528 4529 4530 4531 4532 4533 4534 4535 4536 4 9 14 20 Node numbers Figure 9. Diagrams of yield (m) of nodes along the axis C. Curves correspond to the times: 1 - t4; 2 - t6; 3 - t14; 4 - t20 Vertical displasements m10-3 0,00E+00 -1,00E-03 -2,00E-03 -3,00E-03 -4,00E-03 -5,00E-03 0 50 100 150 200 250 300 350 400 450 500 550 600 Time point(day) Figure 10. Diagrams of yield (m) of element at the intersection of the axes of C-5. Curves correspond to times: 1 - t4; 2 - t6; 3 - t14; 4 - t20 1 2 0 50 100 150 200 250 300 350 400 450 500 550 600 3 Time point(day) 4 5 6 4,00E-04 Deformations 2,00E-04 0,00E+00 -2,00E-04 -4,00E-04 -6,00E-04 -8,00E-04 Stresses MPa Figure 11. Dependence of the components of vector in the element of deformation of the soil mass number 2451 on time Time point(day) Figure 12. Dependence of the components of the stress vector in the element of the soil mass number 2451 on time Conclusions The developed method can be used in calculations and design of large-scale structures in the form of buildings, dams, and other objects taking into account their joint performance with the soil and bedrock, the temporal and spatial inhomogeneity due to creep materials. In addition, the developed software packages are convenient because of their user-friendly interface, advanced automation data input and processing of the results. Methodology developed in practice and the results obtained allow to predict the change of DSS in time considering construction projects with high accuracy over a long period of operation, even with the possibility of reconstruction. Such consideration can lead to significant changes of DSS (by an order of magnitude and higher) for the entire period of operation.

About the authors

Tatyana N. Ter-Emmanuilyan

Russian University of Transport

Author for correspondence.
Email: tanya_ter@mail.ru

Grand Ph.D., Professor, Department of Theoretical Mechanics

9 Obraztsova St., bldg. 9, Moscow, 127994, Russian Federation

Igor O. Polumordvinov

Kazakh-British Technical University

Email: tanya_ter@mail.ru

Assistant Professor, Faculty of Oil and Gas Industry

59 Tole bi St., Almaty, A05H1T2, Republic of Kazakhstan

References