CC BY
8
Salyamova Klara Djabbarovna, D. Sc., professor,
Rumi Dinara Fuadovna, Ph.D., senior researcher
Turdikulov KhusanboyXudoynazarovich, junior researcher,
Institute of Mechanics and Seismic Stability of Structures of the Academy of Sciences of the Republic of Uzbekistan
E-mail: klara_51@mail.ru
ANALYSIS OF SEISMIC STABILITY OF RETAINING EARTH STRUCTURES WITH ACCOUNT OF DISSIPATIVE PROPERTIES OF SOIL
Abstract: Formulation and solution of the problem of natural and forced vibrations and strain state of retaining earth structures (earth dams) under dynamic (seismic) influences are given in the paper with account of dissipative properties of soil, its moisture content and hydrostatic pressure on the upstream side.
Keywords: earth structures, stresses, strains, natural and forced oscillations, dynamic characteristics.
Introduction. Design, construction and operation of important hydro-technical structures in seismic regions, as is the territory of Uzbekistan, require continuous improvement of computational methods for assessing their strength and stability under various loads. Currently used regulatory methods for calculating hydro-technical structures for seismic effects are limited to a one-dimensional elastic scheme of structure [1; 2]. In the previously published works [3; 4] the authors have solved the problems of the dynamics to define stress-strain state of earth structures under basic static loads in a plane elastic statement.
In the proposed paper, dynamic problems concerning determination of oscillations (natural and forced ones) and strain state under dynamic effect are considered taking into account dissipative properties of soil, its moisture content and hydrostatics. The applied design schemes and methods for solving these problems are consistent with modern require-
ments for earthquake-resistant engineering of earth hydro-technical structures; this is reflected in the proceedings of the XIV World Conference on Earthquake Engineering (October 12-17, 2008, Beijing, China).
Intense and long seismic effects can significantly affect dynamic behavior and entire pattern of stress distribution in earth hydro-technical structures located in the zone of high seismic activity. The reliability and strength of such structures, the damage of which can cause great loss to the population of the underlying territories, are determined by the reliability of the forecast of stress-strain state of structures subjected to intense dynamic loads, especially in resonant mode.
The aim of the study is to assess the effect of structural features of the object, its real geometry, soil properties that make up the dam body, hydrostatic pressure on the upstream face and seismic effect causing the structure's resonant oscillations on its stress-strain state and dynamic behavior.
Figure 1. Design scheme of a retaining earth structure
Solution of the problem of unsteady forced oscillations of plane model of retaining earth structure is given in the paper on the example of earth dam of the Rezaksay reservoir [5], operated in the region with 7 point seismicity. The problem is solved numerically by the finite element method. The effect is represented by harmonic two-component acceleration - the record of real accelerogram of an earthquake. For a
8 A = SAa + 8Aq + 8AP + 8Au = -ja{j 8s{jdV +
given kinematic effect, the displacements and stresses in the elements of the structure are determined at each moment of time, i. e. dynamic behavior and stress state are defined as functions of time at any point of the structure.
Statement ofthe problem. To study dynamic behavior of earth structure on a rigid basis under seismic effect, the principle of virtual displacements is used with the d'Alembert principle jpgSvdV + jyh8vdS -jpiiSudV = 0 (1)
The integrals in (l) represent the work of elastic forces, mass forces (pg), hydrostatic pressure on the surface of upstream face (jh) and the work of inertial forces.
In the absence of hydrostatics, the surface of the side slopes and the dam crest are load-free, static boundary conditions on these surfaces are represented as
C. n = 0, (2)
where n is the vector of the normal to the surface.
An effect of hydrostatics on the surface of the upstream face of the dam, located in a homogeneous incompressible fluid of the reservoir, is reduced to setting a pressure linearly increasing with depth on the surface of the slope
P = YZ (3)
where z is the depth measured from the free surface of water; Y is the specific weight of water.
Boundary conditions at the lower boundary of the base are rigid, which is expressed at the absence of horizontal and vertical displacements:
y = 0: Su\r=0,= 0; Sv\=, = 0. (4)
Discretization of the model is represented by elements of triangular and rectangular shape; their matrices of stiffness and masses are given in monographs describing the FEM, for example, in [5].
As a result of the finite element discretization, the problem of unsteady forced oscillations of plane model of a dam is reduced to solving a matrix system of second-order inhomoge-neous differential equations with a time-dependent right side [M ]{q} + [C ]{q} + [K ]{q} = {P (t)} (5)
where [M] and [K] are the matrices of mass and stiffness of the model; {P(t)} is the load to be taken into account, which in general case depends on time (kinematic effect [M]{u0(t)}), on coordinates (hydrostatic pressure on the upstream face), body weight of the dam (distributed over the dam); {q} is the sought for displacements of the nodes of the finite element mesh, determined by the iterative Newmark method [6].
Attenuation matrix [C] describes the internal friction caused by viscosity of the medium. To describe the absorbing, dissipative properties of soil and to obtain the resolving system of equations, the Kelvin-Voigt dynamic model of vis-coelastic medium is used.
atj = xes^ + 2Geij + X'QS^ + 2G '¿tj (6)
where ct,,, ¿^ , ¿.. are the components of stress, strain and strain rate tensors;
X,G are the Lame constants; X',G' are the corresponding
coefficients of viscosity of the medium; Q = — +s2 + s3);
1 3
0 = — (g +s2 + s3) are the average strain and strain rate;
is th^Kronecker symbol.
The use of such model in structure calculation (built
from earth materials) for seismic effects allows one to take
into account energy absorption in soil due to material viscosity, friction between solid particles, water-soil skeleton interaction under irreversible plastic strains, etc. In addition, this model makes it possible to evaluate absorptive capacity of earth structures, depending on the frequency spectrum of the structure.
Description of viscoelastic behavior is achieved by representing the components of strains and average deformations in a complex form £{j = £0{j exp(ia>t), 0 = 00 exp(ia>t), as a result the strain rates and the rate of volume change are equal to e. = irneoij exp(mt), 9 = ia>00 exp(iat).
The use of complex modulus l(m) = A-imX'; G(ioi) = G - iaG' allows to obtain a complex expression of the Hooke's law
atJ =A(m)9StJ + 2G(ia)sij,
(7)
an explicit expression for the dissipation matrix [ C], entering equation (5) before the derivatives of nodal displacements, is obtained as
[C]=n[K], (8)
where n= À.' + 2G' is the positive constant.
Thus, the resolving system of differential equations takes the form
[M ]{q} +V[K ]{q} + [K ]{q} = {P(t )} (9)
where q is the viscosity coefficient.
The matrix of damping coefficients in (9) is proportional to the matrix of quasi-elastic coefficients, this case is called internal friction and is associated with the manifestation of viscous properties of material (in our case - of soil).
As a result of transformation of equation (9), a system of separate equations is obtained
{q} +ndiag (œ. 2){q} + diag (œ. 2){q} = [M ]-1{P (t )} (10) To fit the value of n the formula is used that links the coefficient f with the coefficients of friction q and frequencies (w.)
2nrlM _ V
¥
-, hence ni
CO. 2ncoi
Taking into account the range of variation Y (0,2 < Y - 0,35) [6] and the spectrum of fundamental frequencies of the dam (3^5.4 Hz), the following limits for the coefficient n for the dam soils 0,006 < ^ < 0,0175 are obtained, from which the average value q = 0.01 is selected, later used in calculations of dynamic behavior of earth structures under kinematic harmonic effect.
Considering the problem of non-stationary forced oscillations of the Rezaksay dam, the horizontal kinematic action is represented by a harmonic function with the frequency of natural oscillations of the structure. Initial conditions are assumed to be homogeneous (zero): at t=0: {q0}=0, { q0 }=0. Duration of the effect is 2 seconds. The entire time interval taken in calculation is 3-4 seconds:
\Asin(2n pt) 0 < t < 2c
(11)
[ 0 t > 2c
After the cessation of the effect a free oscillation mode is set in the structure. The structure in the resonant mode makes 6-7 oscillations in 2 seconds; analyzing this it is possible to draw necessary conclusions about its dynamic behavior in seismic process.
Results. This kinematic effect with a frequency equal to the fundamental frequency of natural oscillations of the structure (p = w = 3.1 Hz) is used to demonstrate the dynamic behavior of the structure in a dangerous resonant mode. In addition, it should be noted that such a harmonic effect with a period of T= 0.05 ^ 0.3 sec can be classified as
u,M
* A n ft ft
\ \ l\
i 1 I \l V v
}L\L\L\ V
;!/ x, V \y V7 \V W / v/
:
0.0 1 .0 cek .0 J
a) without hydrostatics and moistening u,M
z A- L
a A A / \AAA /
z
nek
c) with hydrostatics and soil moistening
Figure 2. Horizontal (—*—*—) and vertical (-
seismic one, since its frequency range coincides with the frequency range of seismic effects. For example, the predominant period of the 1976 Gazli earthquake was about T = 0.1 sec. Thus, an artificially chosen effect may be a substitute for a real accelerogram. The amplitude of acceleration A is taken equal to 0.1g, which corresponds to the intensity of the 7-point earthquake zone.
The studies have been carried out on three options of calculation of the Rezaksay dam under given kinematic effect: 1 - with account of its own weight and soil viscosity; 2 - with account of weight, soil viscosity and hydrostatic pressure on the upstream face; 3 - with account of moisture content of soil below the depression curve.
u,M
nelc
b) with hydrostatics
d) contour change of the upstream face at options b - (---) and c - (----)
-) displacements of the dam crest with dissipation in soil
under kinematic effect
Displacements of structure points and the forms of strain of the structure are obtained on the basis of results of solutions are. Figure 2 a-c shows horizontal and vertical oscillations of a point on the dam crest for each of the options of oscillations, and figure 2 d shows a comparison of strains of dam contours at the end of each process (a-c). At the beginning of the effect, as seen from the figures, an increase in the amplitudes of horizontal oscillations, collinear to the direction of the effect, is observed.
An account of internal friction in soil contributes to the stabilization of oscillatory process with time and gradual damping of oscillations after the cessation of the effect. In the absence of hydrostatics, horizontal oscillations occur relative to the neutral position, and vertical ones relative to the position of static equilibrium, determined by structure displacement under its own weight (Fig. 2 a). After the cessation of the effect (t > 2 sec), the oscillations quickly attenuate and strain
state of the dam at the end of the process is characterized by low residual strains horizontally and by vertical settling at the level of static equilibrium position (Fig. 2 d).
Studying the effect of hydrostatic pressure of water on the upstream face on the dam dynamics, it is assumed that the level of reservoir filling corresponds to the design one - 77 m. In calculation it is taken into account that the hydrostatic pressure vector P (h) increases linearly with depth h and is directed perpendicular to the upstream face, i.e. it has components in both horizontal and vertical directions
P (h) = {yh sin(a),yh cos(a)} (12)
Here y is the specific weight of water; a is the angle of inclination of the upstream face; h is the depth of the point on the upstream face of the dam, measured from the level of the free surface of water.
Thus, in calculation presented in (Fig. 2 b), two loads are involved: horizontal one - kinematic according to harmonic law (ll) with the natural frequency of the structure and surface one - static (12) on the part of the upstream face that contacts water. Internal friction in soil with a viscosity coefficient of n = 0.01 is also taken into account. Horizontal and vertical displacements of the dam crest obtained with account of hydrostatic pressure (12) on the upstream face are shown in (Fig. 2 b). A comparative analysis of dam displacements obtained under horizontal kinematic effect without (Fig. 2 a) and with hydrostatics (Fig. 2 b) has shown that in a filled reservoir, dam displacements in horizontal and vertical directions almost doubled. Strain state of the dam at the end of the process versus initial unstrained one is shown in (Fig. 2 d) by long dash lines.
This solution corresponds to the case of filled reservoir with an upstream face reliably protected from moisture. At insufficient protection of the face from moisture in the dam, water is filtered and the parameters of soil below the depression curve change (Fig. 1). The problem of oscillations of the inhomogeneous Rezaksay dam under kinematic effect (11)
is solved in this statement, taking into account own weight, hydrostatic pressure of water and soil moistening.
To select physico-mechanical parameters of moisture-content of gravel-pebble soils, data on soil mechanics [7] were used, according to which the density of soil mass with 10% moisture-content increases and equals to p = 2.2 f/m3. Taking into account velocity of propagation of longitudinal waves in such soil (v = 1 km/s), the value E = 2200 MPa is obtained for the Young's modulus. Soils of the prisms (gravel) and loamy soils of the kernel are assumed to be un-moistened, and its physico-mechanical parameters (density, Young's modulus and Poisson's ratio) remain unchanged.
Crest displacements of the Rezaksay dam obtained under such assumptions, accounting its own weight, hydrostatic pressure and moisture-content under kinematic effect with a frequency equal to the fundamental frequency of natural oscillations of the structure, are shown in (Fig. 2 c), and strain state at the end of the process is shown by dash line in (Fig. 2 d) (short dash lines). In the same figure, the line of long dash lines shows the strain state of the dam, obtained without considering soil moisture-content.
The graphs in (Fig. 2 a-c) show a twofold increase in the horizontal residual strains in the crest as compared to vertical ones with account of hydrostatics and soil moistening. A comparison of the contours of the structure in (Fig. 2 d) indicates an increase in the curvature of the upstream face in near-crest zone of the moistened dam, which can cause its collapse.
Conclusions. Solution of the problem of forced unsteady oscillations of earth retaining structures using as example the earth dam of the Rezaksay reservoir has shown that:
- an account of internal friction of soil material over time stabilizes the amplitudes of horizontal and vertical oscillations;
- water pressure almost doubles the displacements of the dam in both horizontal and vertical directions, and soil moisture-content of the upstream fill increases the curvature of near-crest zone of the upstream face, increasing the chances of its collapse.
References:
1. Guidelines to Account Seismic Effects in Design of Hydro-technical Structures (SNiP II-A.12-75).- Leningrad: Vedeneev VNIIG, 1977.- 164 p.
2. ShNK 2.06.11-04 Construction in Seismic Areas. Hydro-technical Structures. Gosarkhitekstroy.- Tashkent. 2004.- 54 p.
3. Salyamova K. D., Rumi D. F. Numerical Analysis of Stress-strain State of the "Earth Structure-Foundation" System // J. Siberian Federal University. SFU. Tekhnika. Tehnologies. 2016.- 9 (4).- P. 516-535.
4. Salyamova K. D., Rumi D. F. Dynamics of Earth Dams // Publishing House: LAP LAMBERT Academic Publishing Germany. 2015.- 146 p.
5. Zenkevich O. K. The Finite Element Method in Engineering.- Moscow: Mir. 1975.- 542 p.
6. Obraztsov I. F., Saveliev L. M., Khazanov Kh. S. The Finite Element Method in Problems of Structural Mechanics of Aircraft.- Moscow: Higher School, 1985.- 392 p.
7. Dalmatov B. I. Soil Mechanics, Bases and Foundations.- Moscow: Stroiizdat. 1981.- 319 p.
