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Simulation analysis of thermal stress of RCC dams using 3-D finite element relocating mesh method
Advances in Engineering Software 32 (2001) 677±682
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Simulation analysis of thermal stress of RCC dams using 3-D ®nite element relocating mesh method Yaolong Chen*, Changjiang Wang, Shouyi Li, Ruijun Wang, Jing He Xi'an University of Technology, Southern Golden Flower Road, Xi'an 710048, Shaanxi Province, People's Republic of China Received 9 January 2001; revised 5 April 2001; accepted 21 May 2001
Abstract The 3-D ®nite element relocating mesh method is developed for simulation analysis of temperature and thermal stress distribution in a roller compacted concrete dam during the construction period. According to the relation between speci®c properties and age of concrete, some meshes are merged into a larger mesh or a few larger meshes when the age of the concrete is appropriate. Using this method, the total number of elements and nodes were remarkably reduced when the dam height was increased. When the change in elastic modulus, creeps and hydration heat is within the limits permitted by design criteria, the relocating of mesh will start. Using this method, a 3 D simulation analysis of thermal stress in a roller compacted concrete (RCC) high dam can be realized by microcomputer and appeared at the construction site. On the basis of real factors during the construction period, an engineer can predict the distribution of temperature and thermal stress in the RCC dam. Therefore, engineers can take appropriate measures to control the concrete temperature to reduce the thermal stress and avoid crack development within the dam. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Roller compacted concrete gravity dam; Simulation analysis; 3-D FEM; Thermal stress; Relocating mesh method
1. Introduction During the construction period of a roller compacted concrete (RCC) dam, the thickness of each layer is usually 0.3±0.5 m. Hence, an RCC dam with a height of 200±300 m may consists of several hundreds up to thousand layers. The heat will be exchanged between the top surface of a new placement layer and the circumference, and between the bottom surface and the old layer or base rock. Since the gradient of temperature and the stress in an RCC dam is great in the vertical direction, when calculating thermal stress in the dam during construction period, the mesh sizes of the region has to be 0.3±0.5 m in order to reduce the calculation error. The size of the mesh is usually same as the thickness of the layer. After generating the mesh, it is very dif®cult to carry out a 3 D ®nite element simulation analysis by a microcomputer due to the great number of elements and nodes [1]. When the thermal stresses in the Three Gorges Project (TGP) concrete gravity dam were studied in 1989, a kind of element having two layers or several layers was developed [2]. On the basis of this principle, the relocating mesh method is developed in this paper. Using the relocating * Corresponding author. Tel.: 186-29-328-1434; fax: 186-29-323-5545. E-mail address: [email protected] (Y. Chen).
mesh method, the 3 D simulation analysis of the thermal stress in a high RCC dam can be completed by microcomputer and performed at the construction site. 2. Time for starting relocation of the mesh According to the relationship between properties and age of concrete, when the age of concrete is appropriate, some meshes are merged into a larger mesh or a few larger meshes. This method is called relocating mesh method. Using this method, the total number of elements and nodes were remarkably reduced when the dam height was increased. The calculation time and the storage space of the computer were greatly reduced. The time for starting relocation of the mesh must be established. It should assure that the error of relocating mesh was controlled within the limits permitted by the design criteria. The properties of concrete, such as elastic modulus, creep and hydration heat depend on the age of concrete. The in¯uence of these factors should be taken into account. In the upper concrete layers of RCC dam, the elastic modulus, creeps and hydration heat change with respect to time; each thin concrete layer is meshed as one layer of element. In the lower concrete layers of RCC dam, the
0965-9978/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0965-997 8(01)00025-4
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Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Fig. 2. Finite element mesh for the relocating model.
The layers whose age is greater than t p1 are merged into a larger mesh by relocating. Fig. 1. Finite element mesh for the calculating model and the discretized model for RCC dam.
difference of elastic modulus, creeps and hydration heat between each thin layer is very small; these thin layers can be merged into a larger mesh. The calculating model and discretized model of an RCC dam are shown in Fig. 1. The relocating mesh model is shown in Fig. 2. 2.1. The elastic modulus effect The elastic modulus of RCC at time t can be written as E t E0 1 2 e
2lt
;
1
where E0 is the ®nal elastic modulus, l is a coef®cient of placement temperature [3]. If t i, t j are the age of layers i and j of RCC, we obtain E ti 2 E tj # 11 ; E tj
2
where 1 1 is the acceptable error; E ti ; E tj are the elastic modulus of layers i and j of RCC, respectively. If the layers from layer i to j were merged into a larger layer, the average elastic modulus of these layers is used as elastic modulus of the larger layer. The error of elastic modulus of the relocating meshes is smaller than 1 1/2. Substituting Eq. (1) into Eq. (2) yields Eq. (3). 1 ti 2 ln 1 1 11 e2ltj 2 11 : l
3
After ageing of RCC, the increment of elastic modulus of RCC is very small. When the age of RCC is t p1, the difference of elastic modulus between layers is smaller than 1 1, and the layers whose ages are greater than t p1 are merged into a larger mesh. The average elastic modulus of the layers is used as the elastic modulus of the larger layer [8]. From Eq. (2) and let t ! 1; yields 1 1 11 e2ltj 2 11 ! 0; and for tj tp1 ; we get Eq. (4) 1 11 tp1 2 ln : 4 l 1 1 11
2.2. Thermal insulation temperature rise effect The thermal insulation temperature rise u (t ) of RCC can be written as
u t u0 1 2 e2mt ;
5
where u 0 is the ®nal thermal insulation temperature rise, m is the hydration heat coef®cient of cement. If t i, t j are the ages of layers i and j of RCC, respectively, we can obtain
u ti 2 u tj # 12 ; u t j
6
where 1 2 is the acceptable error; u (t i), u (t j) are the ®nal thermal insulation temperature rise of layers i and j of RCC, respectively. Again, we obtain 1 12 tp2 2 ln : 7 m 1 1 12 It means that the layers whose age is greater than t p2 are merged into a larger mesh by relocating, and the error is smaller than 1 2. 2.3. The creep effect The creep of RCC can be written as C t; t A 1 Bt2s 1 2 e2g t2t ;
8
where t is the time, t is the age of concrete; A, B, s, g are the experimental coef®cients in concrete creep. In Eq. (8), let t ! 1; then the creeps whose age is greater than t are C 1; t A 1 Bt2s :
9
If t i, t j are the ages of layers i and j of RCC, respectively, we can obtain C ti 2 C tj # 13 ; C tJ where 1 3 is the acceptable error.
10
Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Substituting Eq. (9) into Eq. (10) we get Eq. (11)
tp3
B 1 2 13 A1 3
1=S
:
3. Finite element model 11
This implies that the layers whose age is greater than t p3 are merged into a larger mesh by relocating and the error is smaller than 1 3 [7,8]. For example, assume a RCC gravity dam, with height 156 m and maximum width at base 120 m, which is shown in Fig. 1. Also, assume that the monthly concrete placement is 6 m in height and the thickness of each thin layer is 0.3 m, the total number of placement layers is 20. Each layer is placed within 6 h and there is a long placement interval of 5 days after continuously placing ®ve layers. When the next 21st thin layer is placed, the age of concrete in the bottom ten layers is 30±22 days. The ten thin layers are merged into a larger layer (3 m) whose average age is 26 days. The relation between elastic modulus and the age of concrete is E t 29:96 1 2 e20:094t : The following can be obtained: E 26 2 E 22 3%; E 22
679
E 30 2 E 26 2:1%: E 26
The above error for relocating is only 3%. The relation between the thermal insulation temperature rise of hydration heat and the age of concrete is
u t 16:87 1 2 e20:24t : The following can be obtained
u 26 2 u 22 0:3%: u 22 The error for relocating is only 0.3%. The relationship between the creep and the age of concrete is C t; t 0:945 1 159:539t20:733 1 2 e20:2 t2t £ 1026 ; while t 3 days, we have C 26; 3 2 C 22; 3 1:2%; C 22; 3 while t 7 days, we have C 26; 7 2 C 22; 7 2:9%: C 22; 7 It shows that the maximum error for relocating is only 3%, and it is within the limits permitted by design criteria.
3.1. Foundation region The thermal stress is caused by the change of temperature within a constrained body. After the concrete of dam is built, the hydration heat raises the temperature of dam [4]. Part of the hydration heat in the dam will be transmitted to the foundation. Usually, the distance to which the hydration heat is transmitted is less than 30 m. In this model, for calculating the thermal stress, the depth of the foundation is 60 m, the length from heel to upstream boundary is 60 m and the length from toe to downstream boundary is 60 m as well. This model is shown in Fig. 1. 3.2. Timetable of construction According to the construction plan of the RCC dam, the total RCC construction time is planed to be 33 months from March of the ®rst year to November of the third year. The dam was constructed to 156 m in height. The numerical model used allows simulation of the actual evolutionary construction process of the dam. After continuously placing ®ve thin layers of RCC, usually, there is a long placement interval, each layer takes 6 h, and the thickness of the layer is 0.3 m. If the increase of the dam is 3 m in a month, the placement interval will be 13.75 days. From September of the ®rst year to May of the second year, the average monthly increase of the dam is 6 m. The placement interval is 6.25 days. From June to August of the second year, the average monthly increase of the dam is 3 m. The placement interval is 13.75 days. From September of the second year to May of the third year, the average monthly increase of dam is 6 m. The placement interval is 6.25 days. From June to November of the third year, the average monthly increase of the dam is 3 m. The placement interval is 13.75 days. 3.3. Meshes and time step The numerical model used for the 3-D thermal analysis consists of a ®nite element discretization of the central cross-section of the dam. The eight-node hexahedral isoparametric elements and six-node pentahedral isoparametric elements are used in the numerical model. The mesh represents the body of the dam (from elevation 0 to 156 m), plus the foundation (from elevation 260 to 0 m). The dam is formed by 520 RCC layers 0.30 m thick. In the 3-D model, every layer of the dam is discretized into 10 £ 120 1200 elements (X-direction, 10 elements; Y-direction, 120 elements). The size of maximum element is 2.2 m £ 1:0 m £ 0:3 m at elevation 0 m. In other words, the length of the element in X-direction is 2.2 m, and the length of the element in Y-direction is 1.0 m, and the height of the element in Z-direction is 0.3 m. As the height of the dam increases, the size of the elements become smaller. Thus, the total number of elements of the dam body is
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Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Fig. 3. Temperature curve at the center for RCC dam in normal placement temperature.
Fig. 4. Temperature curve at the center for RCC dam when placement temperature is 158C.
520 £ 10 £ 120 624 000: The directions of the foundation of the dam is 2.2 m £ 2:0 m £ 2:0 m. In other words, the length of the element in X-direction is 2.2 m, and the length of the element in Y-direction is 2.0 m, and the height of the element in Z-direction is 2.0 m. The total number of elements of the foundation of the dam is 10 £ 120 £ 30 36 000: The total number of elements of the numerical model is 624 000(dam body) 1 36 000(foundation of the dam) 660 000. The RCC dam has transverse the joints every 22 m along axial direction (X-direction) of the dam. The boundary condition in the X-direction is that the two end joints of monolith (two transverse joints) are as thermal insulation boundary. In order to reduce the calculation time and storage space of computer, the proposed relocating mesh method is employed. After placing 20 layers of RCC, the bottom ten layers are merged into a greater layer and the size of element in the Z-direction is increased from 0.3 to 3 m. Thus, the maximum size of the mesh for the fresh concrete layers is 2.2 m £ 1:0 m £ 0:3 m, while the maximum size for the old concrete layers is 2.2 m £ 1:0 m £ 0:3 m. By using the relocating mesh method for the simulation of the construction process of RCC dam, when the dam lifts to an elevation of 156 m, except the upper most 10 layers of 0.3 m The thickness of the other 51 layers is 3 m. The maximum total number of elements of the dam body is 10 £ 1200 1 51 £ 1200 73 200: The maximum total number of elements of the numerical model is 73 200(dam body) 1 36 000(foundation of the dam) 109 200. It means that the total number of elements of the numerical model is reduced by about 83%. The ®rst of March of the ®rst year is the starting time of simulation, and the time step used is 0.25 day. The relocating mesh method and the construction procedures are
implemented in fortran by authors. The temperature and the stress distributions in every layer are obtained using this fortran code. 4. Temperature distribution during construction 4.1. Placement temperature equal to the average air temperature During the construction of the RCC dam, the temperature controlling measures are not used and the RCC placement temperature is simulated as the average air temperature. The initial temperature of the activated elements is automatically set equal to the corresponding placing temperature. For the simulation case, the placing temperature of each layer is being concluded as equal to the ambient temperature. The initial temperature of the foundation rock is assumed to vary linearly from the ambient temperature at the surface to 188C at the bottom of the model. During the construction of the dam, the temperature at the boundaries in contact with the air is automatically set equal to the ambient temperature at the corresponding data. To this end, the ambient temperature is automatically adjusted to follow the average cyclic seasonal thermal variation in the area. In the construction period of 33 months from March of the ®rst year to November of the third year, the variations of temperature distribution at the center of dam are shown in Fig. 3, and the highest temperature of the dam is 36.08C. 4.2. Placement temperature controlled at 158C From Fig. 3, it can be seen that there are three high temperature regions in the dam, and they occur in summer.
Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Fig. 5. Maximum tensile stress curve of RCC dam in normal placement temperature.
In order to reduce the highest temperature within the dam, the RCC placement temperature must be controlled and reduced. When the RCC placement temperature is controlled at 158C, the temperature distribution at the center of the dam is shown in Fig. 4. It can be seen from Fig. 4 that there are three high temperature regions and two low temperature regions. The patterns of the temperature distribution in the dam shown in Fig. 3 are similar to those in Fig. 4. However, the highest temperature is 29.28C in Fig. 4, which is 6.88C lower than the highest 36.08C in Fig. 3.
5. Thermal stress distribution during construction 5.1. Thermal stresses at placement temperature equal to the averaged air temperature 5.1.1. Maximum stress during construction During the construction period, stresses in the dam chane with time to time. Engineers are particularly interested in the maximum tensile stress during construction period [5,6]. The maximum tensile stress in the axial direction of the dam s Xmax varies from 0.60 to 1.30 MPa. The maximum tensile stress occurs around the upstream surface in January of the second year, meanwhile, the tensile stress around the downstream surface s X is 1.20 MPa, and the tensile stress at the center of dam is the smallest. Around the height of 18 m of the dam, the tensile stresses around the upstream surface and downstream surface are about 1.00±1.30 MPa. The tensile stresses along the upstream to downstream direction s Ymax varies from 0.30 to 0.69 MPa. The maximum tensile stress (0.69 MPa) occurs in the upstream surface at a height of 16 m of the dam in January of the second year. The tensile stresses at the elevation points of the upstream surface are about 0.30±0.69 MPa. The tensile stresses around downstream surface at the height of 12± 27 m are 0.50±0.69 MPa. The tensile stresses inside the dam are 0.45±0.50 MPa. The tensile stress in the vertical direction s Zmax varies from 0.80 to 1.92 MPa. The maximum stress occurs around the upstream surface in January of the second year. At the
681
Fig. 6. Stress variation curve along with time at up face of 1.5 m height of RCC dam in normal placement temperature.
height of 24 m, the maximum tensile stresses around the upstream surface are about 1.50±1.90 MPa. The tensile stresses around the downstream surface are about 0.80± 1.32 MPa. The tensile stresses inside the dam are the smallest. When the height of the dam is over 36 m, the tensile stresses are lower. Fig. 5 shows the distribution of the maximum tensile stresses along the height of the dam. 5.1.2. Variation of stresses during construction During the construction period the stresses of each point varies, the variations of the stresses in some region are very large. The variations of stress along with time for three points at a height of 1.5 m of the dam are shown in Fig. 6. When the construction started in March, the maximum compressive stress around the surface at a height of 1.5 m dam occurs in August, but the maximum tensile stress occurs in January. The variation of the compressive stresses is similar to the variation of air temperature. The stresses around the center of the dam vary slightly with the change in air temperature. The stresses around the center of the dam are varies with the age of concrete as well. At the age of 90 days, the variation of stresses in the RCC dam is great. After the age of 100 days, the variation of stresses in the RCC dam becomes smaller and smaller. 5.2. Thermal stresses at placement temperature controlled at 158C 5.2.1. Maximum stresses during construction The maximum tensile stress in the axial direction of the dam s Xmax changes from 0.40 to 1.09 MPa along with the height of dam. The maximum tensile stress occurs around the upstream surface in January of the second year. At the same level and time, the stress around the downstream surface s X is 0.70 MPa. The tensile stress at the center of dam is smallest. At a height of 15 m of the dam, the tensile stresses around the upstream and downstream surfaces are about 0.80±1.09 MPa. The tensile stresses in the stream direction s Ymax change from 0.08 to 0.46 MPa along the height of dam. The maximum stress occurs in January of the second year. The tensile
682
Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Fig. 7. Maximum tensile stress curve of RCC dam when placement temperature is 158C.
stresses around the upstream surface are smaller than 0.30 MPa. The tensile stresses around the downstream surface at a height of 24 m of the dam are 0.30±0.46 MPa. The tensile stresses inside the dam are 0.20±0.35 MPa. The tensile stress in the vertical direction s Zmax changes from 1.04 to 1.76 MPa along the height of the dam. The maximum tensile stress occurs in January of the second year. At a height of 24 m of the dam, the maximum tensile stresses at the upstream surface are 1.40±1.76 MPa. The tensile stresses at downstream surface are 0.80±1.32 MPa. The tensile stresses inside the dam are lower. When the height of the dam is above 36 m, the tensile stresses of every point are lower. Fig. 7 shows the distribution of tensile stresses along the height of the dam. 6. Conclusions The 3 D analysis of the temperature and the thermal stress in RCC high dam can be carried out using microcomputer according to the thin layer placement construction plan. If the time for starting relocation of the mesh is suitable, the error caused by relocating the mesh for temperature and stress distribution is controlled within the limits permitted by design criteria. This method, for the 3 D FEM simulation analysis of multi-layer concrete structure at a high RCC dam using microcomputer, is appropriate and can be applicable in engineering design. The simulation analysis of the temperature ®eld in the dam during the construction period suggests that when the RCC placement temperature is the average air temperature,
the highest temperature of the dam is 36.08C; when the RCC placement temperature is controlled at 158C, the highest temperature of the dam is 29.28C. It is concluded that reducing the RCC placement temperature is a very useful method for controlling the highest temperature in the dam. By changing the starting date and construction plan, the temperature ®eld in the dam is changed, then simulating the different construction programs, a better RCC construction plan can be determined. The simulation analysis of the thermal stress, considering the creep of concrete, shows that the stresses around the surfaces of the dam are greater than in other areas, and then change with the air temperature as well. When the air temperature increases, the compressive stresses will occur. When the air temperature decreases, the tensile stresses will occur. The maximum tensile stress is 1.92 MPa in this case. The tensile stresses inside the dam are lower than 0.7 MPa whether RCC placement temperature is the average air temperature or RCC placement temperature is controlled at 158C.
References [1] Xi'an University of Technology, The thermal stresses analysis of RCC gravity dam in construction period. Xi'an, China, Report No: 26, 1995. [2] Xi'an University of Technology, The thermal stresses analysis of SAN GORGE gravity dam in construction period by using viscoelastic FEM. Xi'an, China, Report No: 75, 1989. [3] Barrett PR, Foadian. Thermal structural analysis methods for RCC dam, The Third ASCE specialty conference on RCC. Rcc 3. New York: ASCE, 1992 p. 407±422. [4] Ditchey EJ, Schrader. Mocksville dam temperature studies. ICOLD Sixteenth Congress, vol. III, Q62, June 1988; San Francisco. p. 379± 96. [5] Hirose T, Nagayama, Takemura, Soto H. A study of control of temperature cracks in large roller compacted dams. ICOLD Sixteenth Congress, vol. III, Q62, 1988; San Fransico. p. 119±35. [6] Jackson HE, Thermal cracking in roller compacted concrete at Galesville Dam. COLD Sixteenth Congress, vol. V, 1988; San Francisco. p. 462±5. [7] Bofang Zhu, Ping Xu. Thermal stresses and temperature control of concrete gravity dams without longitudinal joint. Proceedings of International Conference held at University of Dundee, September 1999; Scotland, UK. p. 127±34. [8] Bofang Zhu, Ping Xu. New methods for thermal stresses analysis simulating construction process of concrete dam. Proceedings of Tenth International Conference for Numerical Methods in Thermal Problems, July 1997; Swansea, UK. p. 743±52.
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Simulation analysis of thermal stress of RCC dams using 3-D ®nite element relocating mesh method Yaolong Chen*, Changjiang Wang, Shouyi Li, Ruijun Wang, Jing He Xi'an University of Technology, Southern Golden Flower Road, Xi'an 710048, Shaanxi Province, People's Republic of China Received 9 January 2001; revised 5 April 2001; accepted 21 May 2001
Abstract The 3-D ®nite element relocating mesh method is developed for simulation analysis of temperature and thermal stress distribution in a roller compacted concrete dam during the construction period. According to the relation between speci®c properties and age of concrete, some meshes are merged into a larger mesh or a few larger meshes when the age of the concrete is appropriate. Using this method, the total number of elements and nodes were remarkably reduced when the dam height was increased. When the change in elastic modulus, creeps and hydration heat is within the limits permitted by design criteria, the relocating of mesh will start. Using this method, a 3 D simulation analysis of thermal stress in a roller compacted concrete (RCC) high dam can be realized by microcomputer and appeared at the construction site. On the basis of real factors during the construction period, an engineer can predict the distribution of temperature and thermal stress in the RCC dam. Therefore, engineers can take appropriate measures to control the concrete temperature to reduce the thermal stress and avoid crack development within the dam. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Roller compacted concrete gravity dam; Simulation analysis; 3-D FEM; Thermal stress; Relocating mesh method
1. Introduction During the construction period of a roller compacted concrete (RCC) dam, the thickness of each layer is usually 0.3±0.5 m. Hence, an RCC dam with a height of 200±300 m may consists of several hundreds up to thousand layers. The heat will be exchanged between the top surface of a new placement layer and the circumference, and between the bottom surface and the old layer or base rock. Since the gradient of temperature and the stress in an RCC dam is great in the vertical direction, when calculating thermal stress in the dam during construction period, the mesh sizes of the region has to be 0.3±0.5 m in order to reduce the calculation error. The size of the mesh is usually same as the thickness of the layer. After generating the mesh, it is very dif®cult to carry out a 3 D ®nite element simulation analysis by a microcomputer due to the great number of elements and nodes [1]. When the thermal stresses in the Three Gorges Project (TGP) concrete gravity dam were studied in 1989, a kind of element having two layers or several layers was developed [2]. On the basis of this principle, the relocating mesh method is developed in this paper. Using the relocating * Corresponding author. Tel.: 186-29-328-1434; fax: 186-29-323-5545. E-mail address: [email protected] (Y. Chen).
mesh method, the 3 D simulation analysis of the thermal stress in a high RCC dam can be completed by microcomputer and performed at the construction site. 2. Time for starting relocation of the mesh According to the relationship between properties and age of concrete, when the age of concrete is appropriate, some meshes are merged into a larger mesh or a few larger meshes. This method is called relocating mesh method. Using this method, the total number of elements and nodes were remarkably reduced when the dam height was increased. The calculation time and the storage space of the computer were greatly reduced. The time for starting relocation of the mesh must be established. It should assure that the error of relocating mesh was controlled within the limits permitted by the design criteria. The properties of concrete, such as elastic modulus, creep and hydration heat depend on the age of concrete. The in¯uence of these factors should be taken into account. In the upper concrete layers of RCC dam, the elastic modulus, creeps and hydration heat change with respect to time; each thin concrete layer is meshed as one layer of element. In the lower concrete layers of RCC dam, the
0965-9978/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0965-997 8(01)00025-4
678
Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Fig. 2. Finite element mesh for the relocating model.
The layers whose age is greater than t p1 are merged into a larger mesh by relocating. Fig. 1. Finite element mesh for the calculating model and the discretized model for RCC dam.
difference of elastic modulus, creeps and hydration heat between each thin layer is very small; these thin layers can be merged into a larger mesh. The calculating model and discretized model of an RCC dam are shown in Fig. 1. The relocating mesh model is shown in Fig. 2. 2.1. The elastic modulus effect The elastic modulus of RCC at time t can be written as E t E0 1 2 e
2lt
;
1
where E0 is the ®nal elastic modulus, l is a coef®cient of placement temperature [3]. If t i, t j are the age of layers i and j of RCC, we obtain E ti 2 E tj # 11 ; E tj
2
where 1 1 is the acceptable error; E ti ; E tj are the elastic modulus of layers i and j of RCC, respectively. If the layers from layer i to j were merged into a larger layer, the average elastic modulus of these layers is used as elastic modulus of the larger layer. The error of elastic modulus of the relocating meshes is smaller than 1 1/2. Substituting Eq. (1) into Eq. (2) yields Eq. (3). 1 ti 2 ln 1 1 11 e2ltj 2 11 : l
3
After ageing of RCC, the increment of elastic modulus of RCC is very small. When the age of RCC is t p1, the difference of elastic modulus between layers is smaller than 1 1, and the layers whose ages are greater than t p1 are merged into a larger mesh. The average elastic modulus of the layers is used as the elastic modulus of the larger layer [8]. From Eq. (2) and let t ! 1; yields 1 1 11 e2ltj 2 11 ! 0; and for tj tp1 ; we get Eq. (4) 1 11 tp1 2 ln : 4 l 1 1 11
2.2. Thermal insulation temperature rise effect The thermal insulation temperature rise u (t ) of RCC can be written as
u t u0 1 2 e2mt ;
5
where u 0 is the ®nal thermal insulation temperature rise, m is the hydration heat coef®cient of cement. If t i, t j are the ages of layers i and j of RCC, respectively, we can obtain
u ti 2 u tj # 12 ; u t j
6
where 1 2 is the acceptable error; u (t i), u (t j) are the ®nal thermal insulation temperature rise of layers i and j of RCC, respectively. Again, we obtain 1 12 tp2 2 ln : 7 m 1 1 12 It means that the layers whose age is greater than t p2 are merged into a larger mesh by relocating, and the error is smaller than 1 2. 2.3. The creep effect The creep of RCC can be written as C t; t A 1 Bt2s 1 2 e2g t2t ;
8
where t is the time, t is the age of concrete; A, B, s, g are the experimental coef®cients in concrete creep. In Eq. (8), let t ! 1; then the creeps whose age is greater than t are C 1; t A 1 Bt2s :
9
If t i, t j are the ages of layers i and j of RCC, respectively, we can obtain C ti 2 C tj # 13 ; C tJ where 1 3 is the acceptable error.
10
Y. Chen et al. / Advances in Engineering Software 32 (2001) 677±682
Substituting Eq. (9) into Eq. (10) we get Eq. (11)
tp3
B 1 2 13 A1 3
1=S
:
3. Finite element model 11
This implies that the layers whose age is greater than t p3 are merged into a larger mesh by relocating and the error is smaller than 1 3 [7,8]. For example, assume a RCC gravity dam, with height 156 m and maximum width at base 120 m, which is shown in Fig. 1. Also, assume that the monthly concrete placement is 6 m in height and the thickness of each thin layer is 0.3 m, the total number of placement layers is 20. Each layer is placed within 6 h and there is a long placement interval of 5 days after continuously placing ®ve layers. When the next 21st thin layer is placed, the age of concrete in the bottom ten layers is 30±22 days. The ten thin layers are merged into a larger layer (3 m) whose average age is 26 days. The relation between elastic modulus and the age of concrete is E t 29:96 1 2 e20:094t : The following can be obtained: E 26 2 E 22 3%; E 22
679
E 30 2 E 26 2:1%: E 26
The above error for relocating is only 3%. The relation between the thermal insulation temperature rise of hydration heat and the age of concrete is
u t 16:87 1 2 e20:24t : The following can be obtained
u 26 2 u 22 0:3%: u 22 The error for relocating is only 0.3%. The relationship between the creep and the age of concrete is C t; t 0:945 1 159:539t20:733 1 2 e20:2 t2t £ 1026 ; while t 3 days, we have C 26; 3 2 C 22; 3 1:2%; C 22; 3 while t 7 days, we have C 26; 7 2 C 22; 7 2:9%: C 22; 7 It shows that the maximum error for relocating is only 3%, and it is within the limits permitted by design criteria.
3.1. Foundation region The thermal stress is caused by the change of temperature within a constrained body. After the concrete of dam is built, the hydration heat raises the temperature of dam [4]. Part of the hydration heat in the dam will be transmitted to the foundation. Usually, the distance to which the hydration heat is transmitted is less than 30 m. In this model, for calculating the thermal stress, the depth of the foundation is 60 m, the length from heel to upstream boundary is 60 m and the length from toe to downstream boundary is 60 m as well. This model is shown in Fig. 1. 3.2. Timetable of construction According to the construction plan of the RCC dam, the total RCC construction time is planed to be 33 months from March of the ®rst year to November of the third year. The dam was constructed to 156 m in height. The numerical model used allows simulation of the actual evolutionary construction process of the dam. After continuously placing ®ve thin layers of RCC, usually, there is a long placement interval, each layer takes 6 h, and the thickness of the layer is 0.3 m. If the increase of the dam is 3 m in a month, the placement interval will be 13.75 days. From September of the ®rst year to May of the second year, the average monthly increase of the dam is 6 m. The placement interval is 6.25 days. From June to August of the second year, the average monthly increase of the dam is 3 m. The placement interval is 13.75 days. From September of the second year to May of the third year, the average monthly increase of dam is 6 m. The placement interval is 6.25 days. From June to November of the third year, the average monthly increase of the dam is 3 m. The placement interval is 13.75 days. 3.3. Meshes and time step The numerical model used for the 3-D thermal analysis consists of a ®nite element discretization of the central cross-section of the dam. The eight-node hexahedral isoparametric elements and six-node pentahedral isoparametric elements are used in the numerical model. The mesh represents the body of the dam (from elevation 0 to 156 m), plus the foundation (from elevation 260 to 0 m). The dam is formed by 520 RCC layers 0.30 m thick. In the 3-D model, every layer of the dam is discretized into 10 £ 120 1200 elements (X-direction, 10 elements; Y-direction, 120 elements). The size of maximum element is 2.2 m £ 1:0 m £ 0:3 m at elevation 0 m. In other words, the length of the element in X-direction is 2.2 m, and the length of the element in Y-direction is 1.0 m, and the height of the element in Z-direction is 0.3 m. As the height of the dam increases, the size of the elements become smaller. Thus, the total number of elements of the dam body is
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Fig. 3. Temperature curve at the center for RCC dam in normal placement temperature.
Fig. 4. Temperature curve at the center for RCC dam when placement temperature is 158C.
520 £ 10 £ 120 624 000: The directions of the foundation of the dam is 2.2 m £ 2:0 m £ 2:0 m. In other words, the length of the element in X-direction is 2.2 m, and the length of the element in Y-direction is 2.0 m, and the height of the element in Z-direction is 2.0 m. The total number of elements of the foundation of the dam is 10 £ 120 £ 30 36 000: The total number of elements of the numerical model is 624 000(dam body) 1 36 000(foundation of the dam) 660 000. The RCC dam has transverse the joints every 22 m along axial direction (X-direction) of the dam. The boundary condition in the X-direction is that the two end joints of monolith (two transverse joints) are as thermal insulation boundary. In order to reduce the calculation time and storage space of computer, the proposed relocating mesh method is employed. After placing 20 layers of RCC, the bottom ten layers are merged into a greater layer and the size of element in the Z-direction is increased from 0.3 to 3 m. Thus, the maximum size of the mesh for the fresh concrete layers is 2.2 m £ 1:0 m £ 0:3 m, while the maximum size for the old concrete layers is 2.2 m £ 1:0 m £ 0:3 m. By using the relocating mesh method for the simulation of the construction process of RCC dam, when the dam lifts to an elevation of 156 m, except the upper most 10 layers of 0.3 m The thickness of the other 51 layers is 3 m. The maximum total number of elements of the dam body is 10 £ 1200 1 51 £ 1200 73 200: The maximum total number of elements of the numerical model is 73 200(dam body) 1 36 000(foundation of the dam) 109 200. It means that the total number of elements of the numerical model is reduced by about 83%. The ®rst of March of the ®rst year is the starting time of simulation, and the time step used is 0.25 day. The relocating mesh method and the construction procedures are
implemented in fortran by authors. The temperature and the stress distributions in every layer are obtained using this fortran code. 4. Temperature distribution during construction 4.1. Placement temperature equal to the average air temperature During the construction of the RCC dam, the temperature controlling measures are not used and the RCC placement temperature is simulated as the average air temperature. The initial temperature of the activated elements is automatically set equal to the corresponding placing temperature. For the simulation case, the placing temperature of each layer is being concluded as equal to the ambient temperature. The initial temperature of the foundation rock is assumed to vary linearly from the ambient temperature at the surface to 188C at the bottom of the model. During the construction of the dam, the temperature at the boundaries in contact with the air is automatically set equal to the ambient temperature at the corresponding data. To this end, the ambient temperature is automatically adjusted to follow the average cyclic seasonal thermal variation in the area. In the construction period of 33 months from March of the ®rst year to November of the third year, the variations of temperature distribution at the center of dam are shown in Fig. 3, and the highest temperature of the dam is 36.08C. 4.2. Placement temperature controlled at 158C From Fig. 3, it can be seen that there are three high temperature regions in the dam, and they occur in summer.
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Fig. 5. Maximum tensile stress curve of RCC dam in normal placement temperature.
In order to reduce the highest temperature within the dam, the RCC placement temperature must be controlled and reduced. When the RCC placement temperature is controlled at 158C, the temperature distribution at the center of the dam is shown in Fig. 4. It can be seen from Fig. 4 that there are three high temperature regions and two low temperature regions. The patterns of the temperature distribution in the dam shown in Fig. 3 are similar to those in Fig. 4. However, the highest temperature is 29.28C in Fig. 4, which is 6.88C lower than the highest 36.08C in Fig. 3.
5. Thermal stress distribution during construction 5.1. Thermal stresses at placement temperature equal to the averaged air temperature 5.1.1. Maximum stress during construction During the construction period, stresses in the dam chane with time to time. Engineers are particularly interested in the maximum tensile stress during construction period [5,6]. The maximum tensile stress in the axial direction of the dam s Xmax varies from 0.60 to 1.30 MPa. The maximum tensile stress occurs around the upstream surface in January of the second year, meanwhile, the tensile stress around the downstream surface s X is 1.20 MPa, and the tensile stress at the center of dam is the smallest. Around the height of 18 m of the dam, the tensile stresses around the upstream surface and downstream surface are about 1.00±1.30 MPa. The tensile stresses along the upstream to downstream direction s Ymax varies from 0.30 to 0.69 MPa. The maximum tensile stress (0.69 MPa) occurs in the upstream surface at a height of 16 m of the dam in January of the second year. The tensile stresses at the elevation points of the upstream surface are about 0.30±0.69 MPa. The tensile stresses around downstream surface at the height of 12± 27 m are 0.50±0.69 MPa. The tensile stresses inside the dam are 0.45±0.50 MPa. The tensile stress in the vertical direction s Zmax varies from 0.80 to 1.92 MPa. The maximum stress occurs around the upstream surface in January of the second year. At the
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Fig. 6. Stress variation curve along with time at up face of 1.5 m height of RCC dam in normal placement temperature.
height of 24 m, the maximum tensile stresses around the upstream surface are about 1.50±1.90 MPa. The tensile stresses around the downstream surface are about 0.80± 1.32 MPa. The tensile stresses inside the dam are the smallest. When the height of the dam is over 36 m, the tensile stresses are lower. Fig. 5 shows the distribution of the maximum tensile stresses along the height of the dam. 5.1.2. Variation of stresses during construction During the construction period the stresses of each point varies, the variations of the stresses in some region are very large. The variations of stress along with time for three points at a height of 1.5 m of the dam are shown in Fig. 6. When the construction started in March, the maximum compressive stress around the surface at a height of 1.5 m dam occurs in August, but the maximum tensile stress occurs in January. The variation of the compressive stresses is similar to the variation of air temperature. The stresses around the center of the dam vary slightly with the change in air temperature. The stresses around the center of the dam are varies with the age of concrete as well. At the age of 90 days, the variation of stresses in the RCC dam is great. After the age of 100 days, the variation of stresses in the RCC dam becomes smaller and smaller. 5.2. Thermal stresses at placement temperature controlled at 158C 5.2.1. Maximum stresses during construction The maximum tensile stress in the axial direction of the dam s Xmax changes from 0.40 to 1.09 MPa along with the height of dam. The maximum tensile stress occurs around the upstream surface in January of the second year. At the same level and time, the stress around the downstream surface s X is 0.70 MPa. The tensile stress at the center of dam is smallest. At a height of 15 m of the dam, the tensile stresses around the upstream and downstream surfaces are about 0.80±1.09 MPa. The tensile stresses in the stream direction s Ymax change from 0.08 to 0.46 MPa along the height of dam. The maximum stress occurs in January of the second year. The tensile
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Fig. 7. Maximum tensile stress curve of RCC dam when placement temperature is 158C.
stresses around the upstream surface are smaller than 0.30 MPa. The tensile stresses around the downstream surface at a height of 24 m of the dam are 0.30±0.46 MPa. The tensile stresses inside the dam are 0.20±0.35 MPa. The tensile stress in the vertical direction s Zmax changes from 1.04 to 1.76 MPa along the height of the dam. The maximum tensile stress occurs in January of the second year. At a height of 24 m of the dam, the maximum tensile stresses at the upstream surface are 1.40±1.76 MPa. The tensile stresses at downstream surface are 0.80±1.32 MPa. The tensile stresses inside the dam are lower. When the height of the dam is above 36 m, the tensile stresses of every point are lower. Fig. 7 shows the distribution of tensile stresses along the height of the dam. 6. Conclusions The 3 D analysis of the temperature and the thermal stress in RCC high dam can be carried out using microcomputer according to the thin layer placement construction plan. If the time for starting relocation of the mesh is suitable, the error caused by relocating the mesh for temperature and stress distribution is controlled within the limits permitted by design criteria. This method, for the 3 D FEM simulation analysis of multi-layer concrete structure at a high RCC dam using microcomputer, is appropriate and can be applicable in engineering design. The simulation analysis of the temperature ®eld in the dam during the construction period suggests that when the RCC placement temperature is the average air temperature,
the highest temperature of the dam is 36.08C; when the RCC placement temperature is controlled at 158C, the highest temperature of the dam is 29.28C. It is concluded that reducing the RCC placement temperature is a very useful method for controlling the highest temperature in the dam. By changing the starting date and construction plan, the temperature ®eld in the dam is changed, then simulating the different construction programs, a better RCC construction plan can be determined. The simulation analysis of the thermal stress, considering the creep of concrete, shows that the stresses around the surfaces of the dam are greater than in other areas, and then change with the air temperature as well. When the air temperature increases, the compressive stresses will occur. When the air temperature decreases, the tensile stresses will occur. The maximum tensile stress is 1.92 MPa in this case. The tensile stresses inside the dam are lower than 0.7 MPa whether RCC placement temperature is the average air temperature or RCC placement temperature is controlled at 158C.
References [1] Xi'an University of Technology, The thermal stresses analysis of RCC gravity dam in construction period. Xi'an, China, Report No: 26, 1995. [2] Xi'an University of Technology, The thermal stresses analysis of SAN GORGE gravity dam in construction period by using viscoelastic FEM. Xi'an, China, Report No: 75, 1989. [3] Barrett PR, Foadian. Thermal structural analysis methods for RCC dam, The Third ASCE specialty conference on RCC. Rcc 3. New York: ASCE, 1992 p. 407±422. [4] Ditchey EJ, Schrader. Mocksville dam temperature studies. ICOLD Sixteenth Congress, vol. III, Q62, June 1988; San Francisco. p. 379± 96. [5] Hirose T, Nagayama, Takemura, Soto H. A study of control of temperature cracks in large roller compacted dams. ICOLD Sixteenth Congress, vol. III, Q62, 1988; San Fransico. p. 119±35. [6] Jackson HE, Thermal cracking in roller compacted concrete at Galesville Dam. COLD Sixteenth Congress, vol. V, 1988; San Francisco. p. 462±5. [7] Bofang Zhu, Ping Xu. Thermal stresses and temperature control of concrete gravity dams without longitudinal joint. Proceedings of International Conference held at University of Dundee, September 1999; Scotland, UK. p. 127±34. [8] Bofang Zhu, Ping Xu. New methods for thermal stresses analysis simulating construction process of concrete dam. Proceedings of Tenth International Conference for Numerical Methods in Thermal Problems, July 1997; Swansea, UK. p. 743±52.