A general framework for modelling long term behaviour of earth and concrete dams

B.A. Schrefler1, F. Pesavento1, L. Sanavia1, S. Secchi2 and L. Simoni1

1 Department of Structural and Transportation Engineering, Via F. Marzolo 9, 35131 Padova, Italy 2 CNR ISIB, Padova, Italy

E-mail: bas@dic.unipd.it

Abstract

Many problems are linked with the long term behaviour of dams, both earth-dams and concrete dams. They range from hydraulic fracturing to alkali-silica reaction and to repair work in concrete dams, from seismic behaviour to secondary consolidation in earth-dams. A common framework for simulation of such systems is shown, based on the mechanics of multiphase porous media. The general model is particularized to specific situations and several examples are shown.

Introduction

Dams are very challenging structures, depending on their geometrical complexity and the variety of stress/strain histories they can undergo. Other problems are related to the material they are composed of: different and complex phenomena can take place ranging from creep, shrinkage, thermal and hygral effects, chemical reactions in concrete dams to dynamical behaviour, stability and liquefaction of earth dams. Traditional structural models account for some of them in purely phenomenological way, i.e. by assuming simple constitutive relationships based on experience, as, for instance, in the case of shrinkage. In other cases these simple relationships are not available and quantitative solution of the problem at hand is not possible, for instance in the case of ASR effects. In the present discussion the material of the dam body (soil or concrete) is assumed to be a multiphase porous medium composed of a deformable solid skeleton and one or several fluids saturating the interstitial voids. This is a more general frame, which allows for a deeper analysis of the physics of the phenomena and the solution of a wider set of mechanical problems.

Governing and Constitutive equations

The general mathematical model for thermo-hydro-mechanical problems taking place in a medium composed of a deformable solid skeleton and one or several fluids saturating the interstitial voids encompasses the equations for

the conservation of the usual thermodynamic properties of every component. Each one of these equations contains, in its general form, terms representing the interaction effects of all the present fields (coupling terms). The mathematical model is represented by [1],[2] – one mass balance equation for each constituent (solid, liquid and gaseous phase). For the sake of simplicity, mass balance equation for solid skeleton is usually summed with the equation of each fluid phase. The ensuing equation comprises a rate capacity term related to changes of that constituent contained in a representative volume element dependent on deformation of all components, flux terms, dependent on transport law of the fluid phase (usually of Darcy’s type and/or Fick’s type), source/sink terms and exchange terms of the quantity with other phases; – the linear momentum balance (equilibrium) equation for the whole multiphase system. This is based on Terzaghi’s effective stress generalized to partially saturated conditions. When dealing with soil dynamic problems, the acceleration frequencies are supposed to be low as in the case of earthquake motion, which allows the relative acceleration of the pore fluid to be disregarded [3]. Hence, only an inertia term has to be added to the equilibrium equation; – the energy balance equation for the mixture solid plus fluid(s), assuming that all phases of the material are locally in thermodynamic equilibrium (hence their temperatures are the same). When mechanical terms are neglected, internal energy depends on temperature only; in particular it is related to rate of heat capacity of the mixture at constant volume. Volume heat sources are frequently retained, owing that they can represent very different phenomena, e.g. heat production due to hydration of concrete or to plastic work. Hence they represent coupling effects between chemo/mechanical and thermal fields. Source terms may also arise along the boundary and represent frictional effects. Energy terms related to phase changes, convective and flux contributions complete the balance equation. The last ones are generally expressed by means of Fourier’s law. More simple models can be easily obtained from the previous one by properly omitting some terms or equations. For instance, when isothermal problems are analysed, the energy balance can be disregarded or when gaseous phase is always at constant pressure (for instance the atmospheric one, [3])

K - 5 61 Long Term Behaviour of Dams

the corresponding mass balance can be dropped. A proper choice of state variables for hygro-thermo-chemo-mechanical modelling of concrete, in particular at early ages for repair work, as well as for soil is of great importance. From a practical point of view, the physical quantities used, should be easily measurable during experimental tests, and from a theoretical point of view, they must uniquely describe the thermodynamic state of the medium. They should assure a good numerical performance of the computer code, based on the resulting mathematical model, as well. The number of the state variables may be reduced if existence of local thermodynamic equilibrium at each point of the medium is assumed. In such a case all constituents of the porous medium have the same temperature and several additional equations valid only at thermodynamic equilibrium may be used to reduce the number of variables. Use of solid skeleton displacement vector, gas pressure and temperature (the same for all constituents of the porous medium) is rather obvious. But a choice of variables describing hygrometric state or advancement of cement hydration is not so obvious. Capillary pressure is the moisture state variable used by the authors for analysis of concrete and it is clearly related to stresses through the generalized Bishop stress tensor. For soil, water and gas pressures are used and in case of static air phase assumption the gas balance equation can be dropped. In real applications there is an obvious necessity to reduce the model complexity by introducing simplifying assumptions, and partial models are often used in practice. Of course the admissibility and the correctness of such simplifications have to be carefully assessed to obtain meaningful numerical models and reliable forecasts. The mathematical model is completed by the constitutive equations. In the simplest formulation, linear relationships are generally used; however several nonlinearities are also incorporated for the solution of specific problems. For the mechanical behaviour of the overall system traditional elasto-plastic models are introduced; cohesive models are used in fracture mechanics applications and damage models for diffused degradation of the material strength. Fluid permeability may depend on porosity or capillary pressure; other nonlinearities of material properties due to temperature, gas pressure, moisture content and material degradation are also taken into account. In advanced formulations, special constitutive models can accommodate, even though in phenomenological model, very complicated problems as the alkali silica reactions, thermo-chemical strains, creep and shrinkage and other variation of material parameters due to aging. For the spatial discretisation of governing equations we have chosen the finite element method, following in particular the continuous Galerkin approach [4]. Special techniques are sometimes necessary to handle situation in which geometric singularities play a fundamental role, as, for instance, in the presence of discrete fracture problems. In similar cases continuous changes in the spatial mesh are automatically made (a posteriori mesh refinement) based on the evolution

of the crack path and information on the solution are transferred from the old to the new mesh. Once the spatial approximation is obtained, the equations are discretised in the time domain. Usually this step is accomplished by means of the finite difference method (generalised trapezoidal and Newmark method) [4], but finite element approximation in time (discontinuous Galerkin method) has also been used. In the latter technique computational difficulties depending in the increment of the unknowns (usually doubled) are balanced by the availability of time discretisation error measures, which allows for time adaptivity and optimal distribution of time steps increments with an overall gain of the solution process [5].

Applications

Three different applications are presented in the following to show the versatility and the efficiency of the multiphase approach previously summarized. Hydraulic fracture propagation The propagation of a discrete fracture caused by water pressure in a cohesive medium (concrete or rock) is firstly analysed. Water pressure could be caused artificially (as in extraction engineering) or could be caused by the filled reservoir water. In this application, failure conditions as a consequence of overtopping wave acting on a concrete gravity dam are investigated. Fracture propagates in mode II conditions, which are accounted for as in [6], usually enucleating from a geometric singularity. The problem is similar to ICOLD benchmark [7], but the dam foundation is also considered, which has been assumed homogeneous with the dam body (Figure 1). In such a situation, the crack path is unknown. Applied loads are the dam self-weight and the hydrostatic pressure due to water in the reservoir growing from zero to the overtopping level h (which is higher than the dam). The geometry of the dam is shown in together with an intermediate discretisation. For more details on material parameters and boundary/initial conditions, we refer to [8]. A continuous updating of the geometry and discretisation are made according to crack movements. This also allows for the proper representation of the process zone and the cohesive forces. This is shown in Figure 2, which represents the process zone when the fracture length is 3.5 m and corresponds to an intermediate step of the analysis when the water level is 78 m. Finally, the formation of the fluid lag is studied. This is due to the different velocities of propagation of the crack tip and the one of seepage inside the fracture. Depending on the characteristics of the fluid lag, different water pressure distributions inside the process zone take place resulting in different stress field in the tip area. Figure 3 depicts cohesive stresses and water pressure distribution, putting in evidence the fluid lag zone. In similar cases, crack path cannot be forecast; hence the traditional use of special/interface elements to simulate

Long Term Behaviour of Dams 62 K - 5

fracture propagation in large structures is prevented. The alternative to the successive re-meshing is the use of cumbersome discretisations of the areas interested by fracture, but also this strategy is not viable in analyzing cohesive behaviour of dams. Further, the used technique for the analysis of the nucleation of the fracture does not require the presence of an initial notch and requires a very limited amount of information to be initially defined.

h

80 m

70

80 m

10

p=0

5

5

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80 m

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80 m

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5

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5

Figure 1: Problem geometry for ICOLD benchmark and

intermediate discretisation

Figure 2: Zoom near the fracture for maximum principal

stress contour Dynamic behaviour of an earth dam Seismic behaviour of an earth dam is shown as the second example. In this case a partial saturated zone also appears which needs an extension of the above model to partial saturation [1],[2]. In the following the simulation was performed with the passive air phase assumption, i.e. with air pressure in the unsaturated region equal to the atmospheric pressure [3]. This dam is located near Perugia in Italy and

was stressed by the Umbria-Marche earthquake of September 1997 while the water level in the lake was only 5 m [9].

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Coh

esiv

e tra

ctio

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Pa]

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d pr

essu

re [M

Pa]

PressureCohesive tractions

Figure 3: Pressure and cohesive forces distribution within the

crack: formation of the fluid lag Some damage was observed in the dam and displacements were measured after the earthquake. Numerical simulations of the seismic behaviour of the dam are performed by using the Swandyne code [3] and a comparison with the in situ damage is carried out. The dam and the surrounding soil are discretised with triangular isoparametric finite elements as can be seen from Figure 6. Small strains, small displacements and plane strain conditions are assumed. The most important material parameters used in the computation are obtained from the data of the in situ geotechnical analysis. The others are estimated by direct comparison between the given materials and similar well known soils. In particular, the solid skeleton is considered to be elasto-plastic and the Mohr-Coulomb and Pastor-Zienkiewicz [3] laws are used for cohesive and granular materials, respectively. The partial saturation conditions are described using Safai-Pinder’s law. First, an initial elastic static analysis is performed assuming the gravity and external water pressure to be applied without dynamic effects. The phreatic line is hence obtained, revealing the existence of a zone above it where capillary pressures develop. The cohesion resulting from these capillary pressures is taken into account in the model. Then, a full non-linear dynamic computation is performed, using vertical and horizontal accelerations of the Nocera Umbra registration as seismic input excitation, see Figure 4. As far as the results are concerned, the calculated displacements of the dam are close to those measured, Figure 5. The numerical analysis reveals, also, the damage accumulated by the concrete barrier in the core of the foundation of the dam as indicated in Figure 6 where the equivalent plastic strain contour after 200 s is shown. As a consequence, the barrier should become more pervious. Water pressures distribution after 200s is shown in Figure 7. From Figures 5b-c it appears that the displacements are increasing for a considerable

K - 5 63 Long Term Behaviour of Dams

period after the end of the earthquake. This was also observed by Zienkiewicz et al. [3] and is aided by the redistribution of pore pressures. The good agreement obtained between in situ and simulated damage reveals the validity of the proposed approach.

Horizontal acceleration (Nocera Umbra)

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Figure 4: Acceleration histories of Nocera Umbra earthquake

Measured horizontal displacement

Measured vertical displacement

Datum point Disp. (m) Datum point Disp. (m) 3 0.079 3 -0.247 4 0.114 4 -0.250 5 0.071 5 -0.150

A

horizontal displacement (node 168)

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disp

lace

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t (m

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C Figure 5: Measured (A) and computed (B,C) displacements

on the top of the dam

Figure 6: Equivalent plastic strain contour at 200s

Figure 7: Pore water pressures contour at 200s. Pore water

tractions are positive.

Mechanical behaviour of concrete at early stages In this section the general model presented briefly in the first part of this work will be particularized to the case of cementitious materials at early ages like for example concrete during the hydration process and to the long term behaviour assessment of concrete structures. This version of the model can be fruitfully applied for the cases of repairing works of damaged concrete structures exposed to aggressive environments, e.g. dams subject to silica-alkali reactions. In addition to what described in the previous sections it is necessary to formulate an evolution equation accounting for

Long Term Behaviour of Dams 64 K - 5

the chemical reaction the hydration/hardening process is based on. To meet this requirement a non-dimensional measure related to the chemical reaction extent, known as hydration degree, can be defined as follows:

Γhydr = χ / χ∞ = mhydr / mhydr∞ (1)

where mhydr means mass of hydrated water (chemically combined), χ is the hydration extent and χ∞, mhydr∞ are the final values of hydration extent and mass of hydrated water, respectively. From the macroscopic point of view, hydration of cement is a complex interactive system of competing chemical reactions of various kinetics and amplitudes. For this reason, a thermodynamically based approach is used herein, see [4],[5]. In this approach the hydration extent χ is the advancement of the hydration reaction and its rate is related to the affinity of the chemical reaction through an Arrhenius-type relationship, as usual for thermally activated chemical reactions, [10],[11]. This formulation can be written in terms of hydration degree, defined as in Equation (1), and relative humidity by means of a function βφ(φ), (φ is the relative humidity), [10],[11]:

( ) ( )exphydr ahydr

d EA

dt RTϕβ ϕΓ

Γ ⎛= Γ −⎜⎝ ⎠

⎞⎟ (2)

where AΓ is the normalized affinity (it accounts both for chemical non-equilibrium and for the nonlinear diffusion process), Ea is the hydration activation energy, and R is the universal gas constant. An analytical formula for the description of the normalized affinity can be found in [12]. Taking into account that the material properties change with time, i.e. they are a function of hydration degree, the constitutive relationship describing the stress-strain behaviour of concrete can be written in the following form:

( ) ( )s stot c ch t tot c ch tη = − − − + − − −D Dε ε ε ε ε ε ε ετ (3)

where D is the tangent matrix of the material, εch is the chemical strain, εc is the creep strain, εt is the thermal strain (infinitesimal deformations) and, finally, s sη τ is the so-called “effective stress tensor”. The chemical strains account for thermo-chemical processes that take place in concrete at early ages, for further details see [10],[11]. Creep is modelled here by means of the solidification theory, [13], for the description of the basic creep, and microprestress theory, [14], for the description of the long-term creep and the stress induced creep (part of the so called drying creep). In the following, a brief description of the necessary constitutive relationships will be presented. According to the solidification theory proposed by [13] aging is considered as the result of a solidification process involving basic constituents, which do not have aging properties. In view of this theory the total creep strain increment, dεc, is decomposed in two components, the visco-elastic, and the viscous flow strain increments, dεv and dεf:

dεc = dεv + dε f =

dγΓhydr

+1η

Gηsτ sdt (4)

where γ is the visco-elastic microstrain and η is the apparent microscopic viscosity. The matrix G is defined in such a way that , where t and ε are stress and strain vectors, respectively, [8]. In the current model concrete is treated as a multiphase porous medium, so the stresses in Equation (4) should be interpreted as the effective ones,

ε = E−1Gt

s sη τ , [10] and not the total ones, tTotal, as in the original theory by Bazant [13]. In such a way it is possible to couple the free shrinkage with the creep, obtaining creep strains even if the concrete structure is externally unloaded. The capillary forces represent, in fact, an “internal” load for the microstructure of the material skeleton. It is now possible to define the creep compliance function [15]

J (t, ′t ) = ′q1 + Φ t,t '( )+1/η (5)

The apparent macroscopic viscosity, η, is not constant in time and is defined by means of microprestress theory, while the micro-compliance function Φ(t,t’) can be expanded in Dirichlet series. For further details see [10]. By applying this version of the general model, the assessment of the hydration process and autogenous shrinkage evolution for a high performance cement paste is presented and discussed. The results of the numerical computations are compared with the ones of the experimental tests performed by Lura et al. [16]. The autogenous shrinkage is a phenomenon the physical nature of which is still not well understood. In recent years various approaches, taking into account possible factors influencing autogenous shrinkage such as surface tension in the cement gel, disjoining pressure and tension in capillary water, have been proposed to explain this phenomenon. The main parameters (for the fully matured material) used in our simulations are listed in [10]. Initially the HCP was at temperature To= 293.15K and relative humidity φo= 99.9%RH, while hydration degree was equal to Γhydr0= 0.3 (i.e. material had already some rigidity). The boundary conditions corresponded to adiabatic heat exchange and sealed conditions for mass exchange. Figure 8a shows the comparison of the hydration degree calculated from the experimental data for heat of hydration [16] and the evolution resulting from FE calculation. A good agreement of the results can be noticed. The relative humidity development in time clearly shows the self-desiccation process of the material, Figure 8b. The RH values measured during the test have been scaled to 100%RH because we refer to the RH due to menisci formation, neglecting the effects of salts dissolved in the pore fluid. Figure 9 shows the measured deformation and the shrinkage strains calculated in [16] in comparison with the present simulations. The agreement is good especially as far as the shrinkage strains are concerned. Some significant differences between the values of strains measured

K - 5 65 Long Term Behaviour of Dams

during the test and those calculated using the capillary tension model can be ascribed to the creep of the cement paste which becomes more evident at lower RH, e.g. for longer time span. This effect is considered due to the coupling between the creep and the shrinkage via the effective stress tensor, [15].

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]

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Figure 8: Comparison of the simulation results with the experimental data [16] concerning the evolution of autogenous phenomena during hydration of a high-

performance cement paste: A) hydration degree, B) relative humidity.

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RELATIVE HUMIDITY [%]

STR

AIN

[-]

total strain - experimenttotal strain - numericalshrinkage - Lura et al.shrinkage - numerical

Figure 9: Comparison of the simulation results with the

experimental data [16] concerning the evolution of autogenous deformations in a high-performance cement paste

during hydration.

Conclusion

A multiphase approach for the dam body material has been presented as a general framework which allows the analysis of a wide variety of problems.

Acknowledgements

The authors would like to thank the University of Padua, Italy, research grant: STPD08JA32_004 for the financial support.

References

[1] Lewis R.W., Schrefler B.A. (1998). The finite element method in the static and dynamic deformation and consolidation of porous media, Wiley, Chichester.

[2] Schrefler, B.A. (2002). Mechanics and Thermodynamics of Saturated-Unsaturated Porous Materials and Quantitative Solutions. Applied Mechanics Review. Vol. 55(4), pp. 351-388.

[3] Zienkiewicz O.C., Chan A., Pastor M., Schrefler B.A., Shiomi T. (1999). Computational Soil Dynamics with Special Reference to Earthquake Engineering, J Wiley, Chichester.

[4] Zienkiewicz O.C., Taylor R.L. (2000). The Finite Element Method, Fifth edition, Vol. 2: The Basis, Butterworth-Heinemann, Oxford.

[5] Simoni L., Secchi S., Schrefler B.A. (2008). Numerical Difficulties and Computational Procedures for Thermo-Hydro-Mechanical Coupled Problems of Saturated Porous Media. Computational Mechanics, Vol. 43, pp. 179-189.

[6] Camacho G.T., Ortiz M. (1996). Computational modelling of impact damage in brittle materials. Int. Journal of Solids and Structures Vol. 33, pp. 2899-2938.

[7] ICOLD (1999). Fifth International Benchmark Workshop on Numerical Analysis of dams, Theme A2. Denver, Colorado.

[8] Schrefler B.A., Secchi S., Simoni L. (2006). On adaptive refinement techniques in multifield problems including cohesive fracture. Computer Methods in Applied Mechanics and Engineering Vol. 195, pp. 444-461.

[9] Briseghella L., Sanavia L., Schrefler B.A. (1999). Seismic analysis of earth dams using a multiphase model. Proc. of the IXth Italian National Congress “L’Ingegneria Sismica in Italia”, Turin (Italy).

[10] Gawin D., Pesavento F., Schrefler B.A. (2006). Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: Hydration and hygro-thermal phenomena. Int. J. Numer. Meth. Engng Vol. 67(3), pp. 299-331.

[11] Gawin D., Pesavento F., Schrefler B.A. (2006). Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete. Int. J. Num. Meth. Engng Vol. 67(3), pp. 332-363.

[12] Cervera M, Olivier J, Prato T. (1999). A thermo-chemo-mechanical model for concrete. II: damage and creep. J. Engrg. Mech. (ASCE) Vol. 125(9), pp. 1028-1039.

[13] Bazant Z.P., Prasannan S. (1989). Solidification theory for concrete creep. I: for-mulation. J. Engrg. Mech. (ASCE) Vol. 115, pp. 1691-1703.

[14] Bazant Z.P., Hauggaard A.B., Baweja S, Ulm F.-J. (1997). Microprestress-solidification theory for concrete creep. I: aging and drying effects. J. Engrg. Mech. (ASCE) Vol. 123(11), pp. 1188-1194

[15] Gawin D., Pesavento F., Schrefler B.A. (2007). Modelling creep and shrinkage of concrete by means of effective stress. Mat. and Struct. Vol. 40, pp. 579-591.

[16] Lura P., Jensen O.M., van Breugel K. (2003). Autogenous shrinkage in high-performance cement paste: an evaluation of basic mechanisms. Modelling of autogenous relative humidity change and autogenous deformation. Cement and Concrete Research Vol. 33, pp. 223-232.

Long Term Behaviour of Dams 66 K - 5