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Experience shows that temperature induced cracking is still one of the main concerns in the design and construction of RCC dams, especially in the case of large RCC dams. The special features associated with the construction of RCC dams, thin layer, large exposure area, highascending speed as well as high content of fly ash in the cementitious materials make the temperature and thermal stress analysis for RCC dams a bigger challenge than for the conventional concrete gravity dams. Based on comprehensive studies several new methods have been proposed and adopted in an effort to develop a more realistic and computational more efficient model for the temperature and thermal stress analysis of RCC dams. Theeffects of material properties, climatic conditions and construction process on the distributionand development of temperature and stresses in the dam could be studied using the model.
In comparison with conventional concrete dams, RCC dams have many advantages withrespect to thermal cracking control. The lower potential of cracking of RCC is a result of thelower cement (on the average only 75-85 kg/m3 [Dunstan, M.R.H.-1999]) and water content,generally lower elastic modulus and higher creep rates. Even so the thermal analysis for RCCdams is just as important as for conventional concrete dams because thermal cracks have been observed in some completed RCC dams. This is due to the following reasons:
The rapid method of construction associated with RCC dams creates an almost adiabaticbehaviour of the material in the centre of the dam, as there is no time to dissipate the heatgenerated before placing the next layer. Because of the much bigger ratio of surface tovolume of RCC dams compared with conventional concrete dams (factor 5), it is possible inRCC dams that the solar radiation contributes more heat to the structure than that generatedby the cement hydration. On the other side the potential of cooling, for example due to agentle breeze, which blows over the warm and large surface of a thin concrete lift, is alsobigger. Therefore the thermal analysis for RCC dams is much more challenging than forconventional concrete dams.
The concrete lifts, which are placed at different time, have different thermal and mechanicalproperties as well as different boundary conditions. In order to appropriately calculate thetemperature and thermal stress fields, the spatially and temporally changing materialproperties, boundary conditions and geometry should be considered as realistically as possibleand reasonable. For this purpose numerical simulation methods based on the finite elementanalysis are widely used.
In traditional finite element methods the domain concerned is discretized into very smallelements layer by layer, which brings about extreme high costs in computing and makes arealistic simulation for large RCC dams (H>200 m) almost unfeasible.
The main objective of this contribution is to show the main feature of the ongoing develop-ment of a computationally effective method to analyse and calculate the transient temperaturefield and the related deformations and stresses in large RCC dams. With the developedmethod the concrete properties, construction process and climate conditions can be simulatedat reasonable computing cost and more realistically.
The spatial and temporal changing temperature field in a RCC dam can be evaluated startingfrom the Fourier’s differential equation
(1) 
and in combination with the boundary and initial conditions.
In (1), T [Co] represents the temperature, c [J / (kg · K)] the specific heat, p [kg/ m3] the density; d is the dimension of the domain W, ki [ W/ m · K] the thermal conductivity; q [W / m3] the rate of heat generation. Boundaries with specified temperature and heat flux havebeen treated with the usual methods. Special attention, see below, has been given to theboundaries exposed to the open air in order to consider influence of the solar radiation andevaporation on the temperature development in RCC dams because of the large ratio ofsurface to volume.
The heat exchange between the surface of a concrete structure and its environment takes placein the form of convection, radiation, conduction and possibly in the form of latent heat (evaporation and condensation). The energy balance on the surface of the structure in theopen air can be approximated with
(2) qn = qH + qL - Rn
in which qn is the heat flux normal to the surface of the structure, qH the sensible heat flux through convection and conduction, qL the latent heat flux through evaporation andcondensation and Rnnet radiation. All terms are expressed in W / m2.
Sensible heat transfer, the heat exchange between structure surface and the fluid (air or water)in its environment, can be calculated with the following formula:
(3)
qH = ac ·(T0 - Ta)
where ac [W /m2 · K] is the convective heat transfer coefficient, Ta the temperature of the fluid, and T0the temperature of the structure surface.
Net radiation on the surface of open-air structure results from the net incoming shortwaveradiation qG and the net outgoing long-wave radiation qE:
(4)Rn = qG - qE
Net incoming shortwave radiation can be calculated with
(5)qG = (1.0 - aG) · G
where aG is the albedo (reflection coefficient) of the structure surface with respect to globalradiation. It varies with the texture, roughness and moisture content of the surface as well as the angle of incidence of the sun. G is the global radiation incident upon the structure surface. It can be measured on the field or estimated using solar radiation models.
The heat exchange between the structure surface and the atmosphere through long-wave(wave length > 3µm) radiation can be estimated with the Stefan-Boltzmann equation
(6)qE = ar · (T0 - Ta)
(7)ar = e · s · (T02 + T2a)(T0+Ta)
where ar [W/(m2·K)] is the radiative heat transfer coefficient, s = 5.67035x10-8 [W/(m2·K4)] the Stefan-Boltzmann constant, T0 and Ta the temperatures of the surface and the atmosphere in Kelvin, e[-] the radiation exchange coefficient.
After RCC has been placed and compacted, the lift surface must be maintained in a dampcondition. This is usually done through water spraying.
Because of RCC’s large exposed lift surface, the evaporation is responsible for significantquantities of heat exchange between the lift surface and the atmosphere. This phenomenonalways causes a heat loss from the concrete lift and partly balances energy from shortwavesolar flux and therefore has a significant influence on the temperature field calculation.
In practice, the evaporation on newly placed concrete has been usually estimated withMenzel’s formula or the ACI monograph which was derived from the Menzel formula.
Because of their origin Menzel’s formula and the ACI monograph are only suitable forestimating the total evaporation or the average evaporation rate over a relative long period oftime. For shorter duration the actual evaporation will be notably underestimated in the hotsunny days or daytime while in cloudy days or at night notably overestimated [Hasanain, G.S.;Khallaf, T.A.; Mahamood, K., -1989]. For this reason the Penman-Brutsaert Model[Brutsaert, W. -1982], which is well approved in practice, has been used for the short timeprediction of diurnal latent heat flux on wet surfaces:
From (2), (3), (4) and (6) results the heat transfer boundary condition:
(8)
qn = (ac + ar)
(T0 - Ta) + qL - qG = a(T0 - Ta) + qL - qG
where ar is calculated with (7) and ac is determined in the process of calculating qL · qG can be taken from the field measurements or calculated with solar radiation models. The heattransfer boundary condition (8) must be solved together with (1) step by step by means of aniterative method.
Hydration of concrete is a highly exothermic and thermally activated reaction. An adequate numerical simulation of the associated thermal problem requires the evaluation of the rate of hydration heat q liberated at every instant during the process. In practice, the adiabatic temperature rise Tad can be measured through experiments. Assuming that the specific heatc and the density of concrete p keep constant during the hydration process, the rate of hydration heat q can be calculated with
(9)
where Tad is the adiabatic temperature rise measured in the adiabatic test, and  the final value of Tad.
is the hydration degree. Under a temperature regime different from that under which Tad is measured x be calculated with
(10)
where b and n are material constants, te the maturity or equivalent age. The following formulation, which is equivalent to the generally accepted Arrhenius-type maturity function, is used [Carino, N.J., Tank, R.C. -1992]:
(11)
where B is a material constant called temperature sensitivity factor; Tref the reference temperature for which maturity equals the real time. With (10) and (11) the evolution of the hydration degree x under a variable temperature regime can be calculated according to:
(12)
RCC dams are constructed in layers of 30 to 50cm. The properties of the newly placed concrete layers change rapidly and are strongly influenced by the construction process and the environmental conditions such as air temperature, solar radiation and wind. By usingtraditional finite element methods, the computation domain is discretized into small elements layer by layer and small time steps are used throughout the whole construction process. Thusthe construction process and the actual temperature development in a RCC dam can be well simulated. For small dams this method is still workable. But for large RCC dams the demandon hardware equipments as well as computing, hence cost, is even in 2D cases very high. Inorder to improve the computational efficiency, among others, the adaptive compound layermethod and the adaptive time step method have been used in the ongoing work.
The adaptive compound layer method is a modified compound layer method [Zhu, B. -1994].
In the compound layer method the domain considered is divided into two regions. As the damrises, in the newly placed layers the temperature is computed layer by layer while in the lower region, when the hydration process has almost finished and thermal and mechanical properties of the concrete of adjacent layers are not too much different from each other, these adjacentlayers are combined into one layer and a coarser mesh is used. In the adaptive compoundlayer method the requirements for combining the layers are somewhat relaxed. For some adjacent layers, as long as the hydration degree of each layer reaches above a certain level andthe thermal and mechanical properties of different layers can be linearly interpolated, theselayers can be combined to form a thicker layer. The mesh size within each layer is controlled by the a posteriori estimation of the discretization error in space.
While the adaptive compound layer method controls the discretization in space, the adaptive time step method [Bornemann, F.A. -1991] controls the discretization in time. Short timesteps are used in the region of sharp temperature variation (in the region of newly placedlayers), and long time steps are used in the lower region of moderate temperature variation. In each region the time steps are calculated in such a way that they are adapted to the timegradient of the solution of the temperature field. This is realised through a posteriori estimation of the discretization error in time.
As a result of temperature gradients and restraint conditions, thermal stress occurs and mayinduce cracks in concrete. Therefore, the prediction of thermal stress is very important indesign and construction in order to control thermal cracking in the dam. Much work withrespect to conventional mass concrete dams has been done [Giesecke, J. -1968, Marx, W. -1987].
As hydration proceeds, the mechanical properties of the concrete, elastic modulus, strengthand creep, change with time. Higher temperature accelerates the hydration and the agingprocess of the concrete. In the past, hydration degree or maturity has been widely used in analysing the effects of the temperature on the aging process. Experimental evidence shows that the mechanical properties of the concrete depend not only on the degree of hydration but also on the kinetics of the hydration reaction. Higher curing temperature not only acceleratesthe development of concrete strengths but also affects the ultimate values of them. And the same temperature has different influence on the development of concrete properties at different ages. For these reasons Cervera et al [Cervera, M., Oliver, J., and Prato, T. -1999] introduced the concept of aging degree k , defined as a normalized strength of concrete:
(13)fc(k) = k · fc,¥
where fcis the compressive strength and fc,¥ its final value. The evolution of k depends on the evolution of the hydration degree x. and the kinetics of the hydration reaction:
(14)
where Af and Bf are material constants, Tref is the reference temperature for the determination of fc, TT the maximum temperature at which hardening of concrete may occur; nT is a material property controlling the sensibility of the compressive strength development to thecuring temperature. Af, Bf and nTcan be experimentally determined through adiabatic and compressive strength tests. x. can be calculated with (12).
According to most codes of practice, tensile strengths and the elastic modulus can be relatedto the compressive strength, and therefore to the aging degree, in the following form:
(15)
where , ft,¥ and E are tensile strengths and the elastic modulus of concrete; ft,¥ and E¥ are their final values.
The total strain e in the concrete can be decomposed into a stress related part es and a stress unrelated part es. The stress related strain includes elastic strain
ee and creep strain ec, while the stress unrelated part includes the free shrinkage es and thermal strain eT. In the form of strain rate one has:
(16)
where a dot above each strain component denotes its first derivative with respect to time.
Shrinkage strain es includes temperature and moisture movement induced and autogenousshrinkage. For the thermal stress analysis of RCC dams, usually only the latter one isconsidered. The thermal strain eT can be defined in terms of the temperature and the thermal expansion coefficient aT in the form:
(17)
eT = aT(T-Tref)
with the reference temperature Tref taken as equal to the temperature reached at the end of thesetting phase, when the hydration degree e = eset.
The stress related strain can be modelled with a rheologic model (Fig. 1), which consists of a spring, a Kelvin chain and a single dashpot. The material properties in the model are the stiffnesses E(t), Ei(t), i = 1,2···,n and viscosities hs(t), hi(t), i = 1,2···,n
. The single dashpot h(S) is included to consider the transitional thermal creep and the long-term aging.
Fig. 1 Stress Related Strain Components
The influence from thermal effects on the creep properties is twofold. The first part is due tothe increase in the rate of hydration with temperature that reduces the creep at highertemperatures. The second part is due to the increase in water mobility with increasedtemperature that reduces the viscosity of the concrete. The first part is usually considered byrelating the compliance function to the hydration degree or aging degree. The second part is afunction of a tensile microprestress [Bazant, Z. P., Hauggaard, A. B., Baweja, S., and Ulm, F.-J. -1997] carried by the bonds and bridges crossing micropores in the hardened cement gel:
(18)
where eTTC is the transitional thermal creep produced by the microprestress S, s the stress, c1 a material constant. If humidity effects are not considered, the microprestresses due to thetransitional thermal effect can be estimated by solving the following equation [Hauggaard, etla.-1999]:
(19)
where c2 and c3are material parameters, T· denotes the change of temperature in time. c2, c3 and the initial value of the microprestresses can be experimentally determined.
For the i-th Kelvin cell (a parallelly connected strain and dashpot pair) the incremental equilibrium equation is:
(20)
The creep strain rate ec is the sum of all the strain rates in the Kelvin chain and thetransitional thermal creep rate. The stress rate can then be calculated with:
(21)
where E is the elastic modulus.
An incremental procedure is adopted for the stress analysis. The time interval [t0, t] is subdivided into N time steps. With the assumption that the strain rate and material propertiesare constant within each time step (tr-1<=t<=tr), one has a closed form solution of (20). Therate form of stress-strain relation (21) can then be transformed into the following incrementalform:
The above described methods regarding the temperature development have been implementedin the finite element program TESAR (Temperature and Stress Analysis for RCC Dams), which is partly based on Kaskade [Beck, R.; Erdmann, B.; Roitzsch, R. -1995]. The program modules for the thermal stress calculation are still under development. With TESAR theconstruction process and the temperature field development of Longtan RCC dam in Chinahas been simulated and will be briefly shown here as a practical example. The required datawere taken from an internal report [Yue, Y; Hu, P.; Huang, S. -1994]
The Longtan RCC dam is located on theupper reach of Hongshui River. Themaximum dam height will be 192m inthe first stage and 216.5m at the secondstage. Fig.2 shows the non-overflowsection of the Longtan RCC Dam. The upstream and downstream facing consist of 50 cm conventional mass concrete (CMC). The cushion layer on thebottom is 5m CMC.
The development of hydration heat iscalculated with (9), the materials constants , Tadd¥ for RCC and CMC are determined thought the evaluation of the measured adiabatic temperature rise and shown in Table 1.
Table 1 Thermal Parameters for RCC and CMC in Longtan RCC Dam
| Concrete type | Tadd¥ | B | N |
| CMC | 26.8150 | 8.9412 | 0.925781 |
| RCC | 16.882 | 14.9621 | 0.871875 |
The required thermal parameters for the calculation of the temperature field of Longtan RCC Dam are listed in Table 2.
Table 2 Thermal Parameters Required for the Calculation of the Temperature Field
| Parameter | Unit | RCC | CMC | Foundation |
| Thermal Conductivity K | kJ/(m·h·K) | 9.270 | 9.270 | 8.374 |
| Specific Heat c | kJ/(kg·K) | 0.9673 | 0.9672 | 0.9672 |
| Density p | kg/ m3 | 2400.0 | 2450.0 | 2400.0 |
As shown in Fig 3, boundaries N1 will beregarded as adiabatic boundaries. At boundaries C1, C2, C3, C 4 only theseasonal variation of the air temperature is considered. At boundary C5, in addition to the seasonal variation, the diurnal variationof the air temperature is also considered.
Because solar radiation measurements at thedam site are not available, the effects of solarradiation and evaporation are not considered here. Their influence on the development ofthe temperature field in a RCC dam will beshown below in a parametric study.
The Coefficient of surface heat transfer is 2 11.5 W/(m2 · K) for an average wind speed of 1m/s at the dam site.
For the placement temperature ofconcrete it will be assumed that it doesnot exceed a temperature limit T0. If the average daily temperature is lower than T0, the average daily temperature will be used in the simulation,otherwise T0
In this simulation a constructionschedule (Table 3) suggested by the designer of the Longtan RCC Dam willbe used.
Table 3 Construction Schedule
| Time | Elevation(m) |
| Feb. 1 – Apr.30 Year 5 | 222-227 |
| Jun. 1. – Aug.31 Year 5 | 227-240 |
| Sept. 15 – Dec.15 Year 5 | 240-253 |
| Dec.16 Year 5 – Feb.15 Year 6 | 253-260 |
| Feb.16 – Apr.30 Year 6 | 260-270 |
| May 1 – Nov.15 Year 6 | 270-312 |
| Nov 16 – Dec. 15 Year 6 | 312-318 |
| Dec. 16 Year 7 – Mar. 15 Year 7 | 318-357 |
| Mar. 16 – Mar. 31 Year 7 | 357-362 |
| Apr.1 – May 15 Year 7 | 362-382 |
For comparison, the temperature fields in the Longtan RCC Dam during construction havebeen calculated with program TESAR under the same conditions with the following two methods:
- Conventional method with fixed time step = 2 hours;
- Adaptive compound layer and adaptive time step method
The temperature fields at the time of the completion of concrete placement are shown in Fig.4.
It can be seen that the difference of the results is negligibly small. But with the new developed method the computation time (less than 50 minutes) has been much less than that with the traditional method (about 9.5 hours).
In order to estimate the effects of solar radiation, wind velocity, evaporation and diurnalchange of the air temperature on the temperature development in a RCC dam, a virtual RCCstructure is analysed with a 1-D model.
The structure consists of 10 RCC lifts. The construction began on October 5, 1998 and wascompleted on October 14. A RCC lift of 50 cm thickness was placed at 7:30 each day.
As in 4.3 N1 is the adiabatic boundary. For boundary C5 the solar radiation and evaporation are also considered in addition to the seasonal and diurnal air temperature fluctuation. The necessary meteorological data (air temperature, relative air humidity, sun shine duration, airpressure, wind velocity and global solar radiation) for the calculation of evaporation were taken from the Web-Service of NTUA (Department of Water Resources, Hydraulic & Maritime Engineering, National Technical University of Athen). For the period of consideration the measured global solar radiation is not complete. The missing data are supplemented through the solar radiation model. The evaporation on the concrete surface is calculated with the Penman-Brutsaert Model, provided that the RCC lift surface is kept wet all the time, i.e., that sufficient water on the lift surface is available for evaporation. The convective heat transfer between air and concrete surface is calculated iteratively during thecalculation of evaporation.
Under conditions described above, the temperature development in the structure from the construction beginning until one year after the completion of the construction is calculated.
The hydration heat and other thermal parameters are the same as in section 4.2.
The temperature development at different locations in the structure is displayed in Fig. 5. It can be observed from the figure that diurnal fluctuations of the air temperature, solar radiation and evaporation only affect the temperature near the surface of the concrete, while the cold and heat waves (lasting several days to several weeks) can influence the temperature in the structure until 2 m under the surface. Below that depth only the seasonal variations of the air temperature play a main role.
In order to show the effects of solar radiation and evaporation, three situations are considered in the calculation of the temperature development in the structure.
- Both solar radiation and evaporation are considered in the calculation;
- Only solar radiation is considered;
- Neither Solar radiation nor evaporation is considered in the calculation.
Fig.6 and Fig.7 show the temperature development under these situations. The effects of solarradiation and evaporation on the temperature evolution in RCC structures are not to overlook.
Fig. 4 Isothermal Lines of Longtan RCC Dam at the Completion of ConcretePlacement, Calculated with TESAR using the Conventional Method(left) and the New Method (right)
Fig. 5 Temperature Change at Different Locations During the First Year afterthe Completion of the Concrete Placement
Fig. 6 Comparison of Temperature Development Calculated under DifferentConditions
Fig. 7 Comparison of Temperature Development Calculated under DifferentConditions
This paper presents some main features of the ongoing development of a computationally effective method to analyse and calculate the transient temperature field and thermal stresses in large RCC dams. Numerical procedures based on the research results have been developedand implemented in the program TESAR. With this program the construction process of a RCC Dam in China is simulated and a series of parametric studies have been conducted. The results obtained suggest the following conclusions:
The developed procedures are able to simulate the construction process and temperature development in large RCC dams efficiently and realistically. For the simulation of temperature development in RCC dams, all major influencing factors such as construction schedule, temperature coupled hydration, solar radiation, evaporation, wind speed and airtemperature are considered.
With the adaptive compound layer method and the adaptive time step method, the required computing time is only a small fraction of the time needed for the computation with theconventional method.
Because of the specific characteristics of RCC construction, the influence of the solar radiation and evaporation on temperature development is significant.
Diurnal changes of the air temperature, solar radiation and evaporation have a noticeable influence on the temperature evolution of the surface layer. Both the magnitude and distribution of the wind speed have significant effects on the temperature evolution of early age concrete.
Temperature effects on concrete creep can be considered through the concepts ofmaturity/aging degree and microprestresses. They are easy to implement in the numerical FE-procedure.
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