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Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring, and Model Test of Thermal Stresses in Massive Concrete Structures
16 Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring, and Model Test of Thermal Stresses in Massive Concrete Structures
16.1
Full Course Simulation Analysis of Concrete Dams
In the construction process, the concrete dam is divided into many blocks which are further divided into many horizontal layers with a thickness of 1.53.0 m. Generally, there are several years from the beginning to the completion of construction of a dam. Due to the variation of air temperature, the hydration heat, and pipe cooling, the variation of temperature field is very complex. It is difficult to compute the thermal stresses in the construction period by the traditional methods of structural mechanics. As the number of the monitoring instruments is small, it is also difficult to give the actual stress field of a concrete dam in the construction process by the instrumental monitoring. Only the finite element method (FEM) can be used to give the simulation computation of concrete dams considering the process of construction and all the factors which influence the temperature and stress fields of the dam. If necessary, it can give the factor of safety by overload computation. The peculiarities of simulation computation are as follows: (1) adopt the incremental FEM; (2) simulate the whole course of construction, the dam is divided into blocks and layers; (3) consider the following factors—the variation of the ambient and the interior temperatures, the variation of the mechanical and thermal properties with age of concrete, the artificial cooling, etc.; (4) the influence of the opening and grouting of joints; (5) give the factor of safety by overload computation, if necessary. The scope of simulation computation depends on the purpose of analysis and the type of structure. Generally, only one dam block is computed for a gravity dam. In the analysis of the thermal stresses in the course of construction of arch dams, only one dam block or three adjacent dam blocks are computed. In the analysis of the stress state of an arch dam after impounding of water, it is necessary to conduct simulation computation of the whole dam [26, 28, 35, 37, 52, 57, 78]. The envelopes of the first principal stresses of the 3D whole course simulation computation of the Jinghong gravity dam are shown in Figure 16.1. Thermal Stresses and Temperature Control of Mass Concrete. DOI: http://dx.doi.org/10.1016/B978-0-12-407723-2.00016-6 © 2014 Tsinghua University Press. Published by Elsevier Inc. All rights reserved.
334
Thermal Stresses and Temperature Control of Mass Concrete
16.2
Dynamic Temperature Control and Decision Support System of Concrete Dam
In the past, the allowable temperature differences and maximum temperature of concrete are given in the design report of concrete dams. Technical measures are taken in the stage of construction to satisfy the needs of design. Experience shows that this is not sufficient for the temperature control of high concrete dams. 610.00
600.00
590.00
580.00
Elevation (m)
570.00
560.00
550.00
540.00
530.00
520.00
510.00
500.00 –20 480.00
–10
0
10
20
30 X (m)
40
50
60
70
Figure 16.1 The envelopes of the first principal stresses of the 3D simulation computation of Jinghong gravity dam (stress unit: MPa).
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
335
In the construction process, there may be some changes of the conditions assumed in the design but generally there is no analysis and monitoring of these changes. This may be the cause of cracking of concrete. A decision-support system of dynamic temperature control of high concrete dams has been developed by Zhu Bofang, Zhang Guoxin, and Xu Ping. The functions of the system are as follows: 1. Whole course simulation computation of the temperature and stress fields of the dam by 3D FEM. The output of the system includes the actual geometrical figure of the shape of the dam blocks, the temperature and stress fields at any time. 2. Back analysis of the thermal properties of concrete and the effect of superficial thermal insulation are conducted in the process of construction to give the practical thermal properties of concrete and the actual effect of the superficial thermal insulation. 3. Forecast of the temperature and stress fields of the dam. In the process of construction, based on the actual temperature and stress fields of the concrete blocks which have been constructed, according to the predetermined schedules of progress and technical measures of temperature control, simulation computation is conducted to predict the temperature and stress fields in the future and check the effect of the technical measures and the schedule of progress. 4. Decision support of the control of temperatures and thermal stresses of the dam. The experiences of the experts in the world about the design, construction and temperature control, the specifications, and examples of design and construction of concrete dams are collected and systematized. Based on the above data and the results of simulation computation of the temperature and stress fields, information will be given to the engineers to help them to modify the measures of temperature control and construction schedule to prevent the cracking of a dam. 5. Operation platform and database for comprehensive management of the system.
This system had been successfully applied in the Zhougongzhai concrete arch dam with a height of 126.5 m. The dam was constructed from December 2003 to April 2006. The first principal stresses on January 15, 2005 are shown in Figure 16.2.
16.3
Numerical Monitoring of Concrete Dams
At present concrete dams are monitored by instruments in the period of construction and operation. Instrumental monitoring is important and can be used to judge
y
Si – Stress 300 261.11 222.22 183.33 144.44 105.55 66.664 27.775 –11.114 –50
z x
Figure 16.2 The first principal stress of Zhougongzhai concrete arch dam on January15, 2005 (unit: 1022 MPa).
336
Thermal Stresses and Temperature Control of Mass Concrete
whether the dam is working normally, but there are some drawbacks, so it is suggested to add numerical monitoring to instrumental monitoring of concrete dams.
16.3.1 The Drawbacks of Instrumental Monitoring 1. There are few cross sections of monitoring. Due to the restraint of many practical factors, generally there are two to four cross sections embedded with instruments. Practically, there is no instrument in most dam blocks. 2. It is difficult to give the whole picture of temperature and stress fields even in the cross section embedded with instruments. A concrete dam is constructed layer by layer. The variation of temperature and stress fields in each layer is very complex. In order to monitor the temperature and stress variations, at least three rows of instruments must be embedded in each layer of concrete. For a dam block with 100 layers, there are 300 rows of instruments to be embedded. Practically, there are only 34 rows of instruments in the observed cross section, so it is impossible to give the whole picture of temperature and stress fields and factor of safety even in the monitored section.
16.3.2 Numerical Monitoring It is suggested to add numerical monitoring in the construction of concrete dams. The content of numerical monitoring includes the following: 1. Whole dam and whole course simulation computation of the temperature field and stress field by 3D FEM. 2. Loading, including temperature, self-weight, water pressure, seepage flow, and initial stress. 3. Properties of materials, including the elastic, inelastic, and creep deformations and the influences of joints. 4. In addition to the room test, the actual properties of materials are determined by back analysis of observed results of the prototype. 5. Forecast the temperature and stress fields of the dam.
16.3.3 The Important Functions of Numerical Monitoring 1. The function of numerical monitoring in the period of construction By numerical monitoring, we can understand the temperature and stress fields of the dam in the period of construction and technical measures may be adopted to prevent possible cracks, if there are any problems. 2. The function of numerical monitoring in the period of operation It is possible to give appraisals of a dam’s work frequently in the period of operation from the results of numerical monitoring.
A concrete gravity-arch dam with maximum height 76.3 m and crest length 419 m was constructed in 19681972. There was no efficient temperature control in the construction period of the dam so some cracks appeared in the dam. We gave a whole dam and whole course inelastic finite element simulation analysis to the dam. All the
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
337
Displacement (mm)
Figure 16.3 The transverse joints and cracks in the computing model of a concrete gravityarch dam. 5.0
0.0
–5.0 1994
1995
1996
1997 Calculated value
1998
1999
2000
2001
2002
Observed value
2003 Year
Figure 16.4 Comparison of the observed displacements with the displacements given by simulation computation of FEM of a concrete gravity-arch dam.
faults in the foundation, the joints and principal cracks in the dam, and the construction process are simulated in the computation. The joints and cracks in the computing model are shown in Figure 16.3. The computed displacements and the observed displacements of the dam are shown in Figure 16.4; they are close to each other. Based on the results of simulation computation, an overload analysis is given by FEM: (1) With consideration of stress history and influence of joints, the safety factor is k 5 1:11 (in winter) and 1.14 (in summer). (2) Without consideration of stress history and influence of joints, k 5 1:67. (3) If efficient temperature control was conducted in the period of construction as required by specifications of concrete dam, the safety factor is k 5 3:26. The failure of the downstream face of the dam under loading is shown in (Figure 16.5).
16.4
Model Test of Temperature and Stress Fields of Massive Concrete Structures
If the adiabatic temperature rise of concrete is θðtÞ 5 θ0 ð1 2 e2st Þ, the differential equation of heat conduction and the initial and boundary conditions of the prototype are as follows:
338
Thermal Stresses and Temperature Control of Mass Concrete
Figure 16.5 Failure of the downstream face of a concrete gravity-arch dam after overloading. Equation of heat conduction: 2 @T @ T @2 T @2 T θ0 @ð1 2 e2st Þ 5a 1 1 1 2 2 2 @t @x @y @z @t
ð16:1Þ
Initial condition: Tð0; x; y; zÞ 5 T0 ðx; y; zÞ
ð16:2Þ
Boundary condition: λ
@T 1 βðT 2 Tc Þ 5 0 @n
ð16:3Þ
If the adiabatic temperature rise of the material of the model is θm ðtm Þ 5 θ0m ð1 2 e2sm tm Þ, the differential equation and the conditions of the model are: Equation of heat conduction: 2 @Tm @ Tm @2 Tm @2 Tm θ0m @ð1 2 e2sm tm Þ 1 5 am 1 1 2 2 2 @tm @tm @xm @ym @zm
ð16:1aÞ
Initial condition: Tm ð0; xm ; ym ; zm Þ 5 T0 ðxm ; ym ; zm Þ
ð16:2aÞ
Boundary condition: λm
@Tm 1 β m ðTm 2 Tcm Þ 5 0 @nm
ð16:3aÞ
Let Ct 5 t/tm ; CL 5 x/xm ; Ca 5 a/am CT 5 T=Tm ; Cθ 5 θ0 =θ0m ; Cλ 5 λ/λm ; Cβ 5 β/β m ; Cs 5 s/sm
ð16:4Þ
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
339
where the subscript “m” represents the model. Substitution of Eq. (17.51) into Eqs. (16.1)(16.3) yields 2 CT @Tm Ca CT @ Tm @2 T m @2 T m Cθ θ0m @ð1 2 e2CS Ct sm tm Þ 5 a 1 1 1 m 2 2 2 @tm Ct @tm @xm @ym @zm Ct CL ð16:1bÞ CT CL Tm ð0; xm ; ym ; zm Þ 5 CT CL T0 ðxm ; ym ; zm Þ
ð16:2bÞ
Cλ CT @Tm λm 1 Cβ CT β m ðTm 2 Tcm Þ 5 0 CL @nm
ð16:3bÞ
Comparing Eqs. (16.1b)(16.3b) with Eqs. (16.1a)(16.3a), it is clear that, in order to make the temperature field of the model similar to that of the prototype, the following conditions of similarity must be satisfied: CT Ca CT Cθ 5 5 ; 2 Ct Ct CL Ct 5 CL2 =Ca ;
Cλ CT 5 Cβ CT ; CL
Cβ 5 Cλ =CL ;
Cθ 5 CT ;
Cs Ct 5 1
ð16:5Þ
Cs 5 1=Ct
ð16:6Þ
There are eight parameters and four equations in Eq. (16.5), thus four parameters may be given freely and the other four parameters must be computed by Eqs. (16.5) and (16.6). For example, let CT 5 1; Ca 5 1:1; Cλ 5 1:1; CL 5 20, from Eq. (16.6), we have: Ct 5 202 /1:1 5 364;
Cβ 5 1:1/20 5 0:055;
Cθ 5 1;
Cs 5 1/364
In order to satisfy the similarity conditions, for the period of annual variation (P 5 1 year) in the prototype, the period of variation in the model is Pm 5 1=364 year 5 1d; for the duration of cold wave Q 5 2 2 4d in the prototype, in the model Qm 5 0:13 0:26 h; in the prototype β 5 80 kJ/(m2 h C), in the model, β 5 4.40 kJ/(m2 h C). These conditions are possible to be fulfilled in the model, but it is difficult to make the rate of hydration heat in the model sm 5 s=364. Thus, it is possible to make a model test of the temperature field for the period of operation and it is difficult to conduct a model test of the temperature field for the period of construction. In the construction period, the thermal stresses are related to the modulus of elasticity EðτÞ and unit creep Cðt; τÞ which are functions of time t and age τ. It is difficult to make EðτÞ and Cðt; τÞ to satisfy the similarity conditions. So it is difficult to do a model test for thermal stress in the period of construction. The modulus of elasticity of concrete is nearly constant in the period of operation, so it is possible to do a model test for elastic thermal stress of mass concrete
340
Thermal Stresses and Temperature Control of Mass Concrete
in the period of operation. Because the creep of concrete is difficult to satisfy the similarity condition, it is difficult to do a model test for viscoelastic thermal stress even in the period of operation. In 2010, one engineer had made a 1:7 model test for the thermal stresses in the period of construction, and the material for model is also concrete, so Ct 5 1:0; Ca 5 1:0; Cs 5 1:0, from Eq. (16.5), CL 5 1:0. Thus, the model must have the same dimensions as the prototype, and the model with CL 5 7 is not similar to the prototype.
16.1
Full Course Simulation Analysis of Concrete Dams
In the construction process, the concrete dam is divided into many blocks which are further divided into many horizontal layers with a thickness of 1.53.0 m. Generally, there are several years from the beginning to the completion of construction of a dam. Due to the variation of air temperature, the hydration heat, and pipe cooling, the variation of temperature field is very complex. It is difficult to compute the thermal stresses in the construction period by the traditional methods of structural mechanics. As the number of the monitoring instruments is small, it is also difficult to give the actual stress field of a concrete dam in the construction process by the instrumental monitoring. Only the finite element method (FEM) can be used to give the simulation computation of concrete dams considering the process of construction and all the factors which influence the temperature and stress fields of the dam. If necessary, it can give the factor of safety by overload computation. The peculiarities of simulation computation are as follows: (1) adopt the incremental FEM; (2) simulate the whole course of construction, the dam is divided into blocks and layers; (3) consider the following factors—the variation of the ambient and the interior temperatures, the variation of the mechanical and thermal properties with age of concrete, the artificial cooling, etc.; (4) the influence of the opening and grouting of joints; (5) give the factor of safety by overload computation, if necessary. The scope of simulation computation depends on the purpose of analysis and the type of structure. Generally, only one dam block is computed for a gravity dam. In the analysis of the thermal stresses in the course of construction of arch dams, only one dam block or three adjacent dam blocks are computed. In the analysis of the stress state of an arch dam after impounding of water, it is necessary to conduct simulation computation of the whole dam [26, 28, 35, 37, 52, 57, 78]. The envelopes of the first principal stresses of the 3D whole course simulation computation of the Jinghong gravity dam are shown in Figure 16.1. Thermal Stresses and Temperature Control of Mass Concrete. DOI: http://dx.doi.org/10.1016/B978-0-12-407723-2.00016-6 © 2014 Tsinghua University Press. Published by Elsevier Inc. All rights reserved.
334
Thermal Stresses and Temperature Control of Mass Concrete
16.2
Dynamic Temperature Control and Decision Support System of Concrete Dam
In the past, the allowable temperature differences and maximum temperature of concrete are given in the design report of concrete dams. Technical measures are taken in the stage of construction to satisfy the needs of design. Experience shows that this is not sufficient for the temperature control of high concrete dams. 610.00
600.00
590.00
580.00
Elevation (m)
570.00
560.00
550.00
540.00
530.00
520.00
510.00
500.00 –20 480.00
–10
0
10
20
30 X (m)
40
50
60
70
Figure 16.1 The envelopes of the first principal stresses of the 3D simulation computation of Jinghong gravity dam (stress unit: MPa).
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
335
In the construction process, there may be some changes of the conditions assumed in the design but generally there is no analysis and monitoring of these changes. This may be the cause of cracking of concrete. A decision-support system of dynamic temperature control of high concrete dams has been developed by Zhu Bofang, Zhang Guoxin, and Xu Ping. The functions of the system are as follows: 1. Whole course simulation computation of the temperature and stress fields of the dam by 3D FEM. The output of the system includes the actual geometrical figure of the shape of the dam blocks, the temperature and stress fields at any time. 2. Back analysis of the thermal properties of concrete and the effect of superficial thermal insulation are conducted in the process of construction to give the practical thermal properties of concrete and the actual effect of the superficial thermal insulation. 3. Forecast of the temperature and stress fields of the dam. In the process of construction, based on the actual temperature and stress fields of the concrete blocks which have been constructed, according to the predetermined schedules of progress and technical measures of temperature control, simulation computation is conducted to predict the temperature and stress fields in the future and check the effect of the technical measures and the schedule of progress. 4. Decision support of the control of temperatures and thermal stresses of the dam. The experiences of the experts in the world about the design, construction and temperature control, the specifications, and examples of design and construction of concrete dams are collected and systematized. Based on the above data and the results of simulation computation of the temperature and stress fields, information will be given to the engineers to help them to modify the measures of temperature control and construction schedule to prevent the cracking of a dam. 5. Operation platform and database for comprehensive management of the system.
This system had been successfully applied in the Zhougongzhai concrete arch dam with a height of 126.5 m. The dam was constructed from December 2003 to April 2006. The first principal stresses on January 15, 2005 are shown in Figure 16.2.
16.3
Numerical Monitoring of Concrete Dams
At present concrete dams are monitored by instruments in the period of construction and operation. Instrumental monitoring is important and can be used to judge
y
Si – Stress 300 261.11 222.22 183.33 144.44 105.55 66.664 27.775 –11.114 –50
z x
Figure 16.2 The first principal stress of Zhougongzhai concrete arch dam on January15, 2005 (unit: 1022 MPa).
336
Thermal Stresses and Temperature Control of Mass Concrete
whether the dam is working normally, but there are some drawbacks, so it is suggested to add numerical monitoring to instrumental monitoring of concrete dams.
16.3.1 The Drawbacks of Instrumental Monitoring 1. There are few cross sections of monitoring. Due to the restraint of many practical factors, generally there are two to four cross sections embedded with instruments. Practically, there is no instrument in most dam blocks. 2. It is difficult to give the whole picture of temperature and stress fields even in the cross section embedded with instruments. A concrete dam is constructed layer by layer. The variation of temperature and stress fields in each layer is very complex. In order to monitor the temperature and stress variations, at least three rows of instruments must be embedded in each layer of concrete. For a dam block with 100 layers, there are 300 rows of instruments to be embedded. Practically, there are only 34 rows of instruments in the observed cross section, so it is impossible to give the whole picture of temperature and stress fields and factor of safety even in the monitored section.
16.3.2 Numerical Monitoring It is suggested to add numerical monitoring in the construction of concrete dams. The content of numerical monitoring includes the following: 1. Whole dam and whole course simulation computation of the temperature field and stress field by 3D FEM. 2. Loading, including temperature, self-weight, water pressure, seepage flow, and initial stress. 3. Properties of materials, including the elastic, inelastic, and creep deformations and the influences of joints. 4. In addition to the room test, the actual properties of materials are determined by back analysis of observed results of the prototype. 5. Forecast the temperature and stress fields of the dam.
16.3.3 The Important Functions of Numerical Monitoring 1. The function of numerical monitoring in the period of construction By numerical monitoring, we can understand the temperature and stress fields of the dam in the period of construction and technical measures may be adopted to prevent possible cracks, if there are any problems. 2. The function of numerical monitoring in the period of operation It is possible to give appraisals of a dam’s work frequently in the period of operation from the results of numerical monitoring.
A concrete gravity-arch dam with maximum height 76.3 m and crest length 419 m was constructed in 19681972. There was no efficient temperature control in the construction period of the dam so some cracks appeared in the dam. We gave a whole dam and whole course inelastic finite element simulation analysis to the dam. All the
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
337
Displacement (mm)
Figure 16.3 The transverse joints and cracks in the computing model of a concrete gravityarch dam. 5.0
0.0
–5.0 1994
1995
1996
1997 Calculated value
1998
1999
2000
2001
2002
Observed value
2003 Year
Figure 16.4 Comparison of the observed displacements with the displacements given by simulation computation of FEM of a concrete gravity-arch dam.
faults in the foundation, the joints and principal cracks in the dam, and the construction process are simulated in the computation. The joints and cracks in the computing model are shown in Figure 16.3. The computed displacements and the observed displacements of the dam are shown in Figure 16.4; they are close to each other. Based on the results of simulation computation, an overload analysis is given by FEM: (1) With consideration of stress history and influence of joints, the safety factor is k 5 1:11 (in winter) and 1.14 (in summer). (2) Without consideration of stress history and influence of joints, k 5 1:67. (3) If efficient temperature control was conducted in the period of construction as required by specifications of concrete dam, the safety factor is k 5 3:26. The failure of the downstream face of the dam under loading is shown in (Figure 16.5).
16.4
Model Test of Temperature and Stress Fields of Massive Concrete Structures
If the adiabatic temperature rise of concrete is θðtÞ 5 θ0 ð1 2 e2st Þ, the differential equation of heat conduction and the initial and boundary conditions of the prototype are as follows:
338
Thermal Stresses and Temperature Control of Mass Concrete
Figure 16.5 Failure of the downstream face of a concrete gravity-arch dam after overloading. Equation of heat conduction: 2 @T @ T @2 T @2 T θ0 @ð1 2 e2st Þ 5a 1 1 1 2 2 2 @t @x @y @z @t
ð16:1Þ
Initial condition: Tð0; x; y; zÞ 5 T0 ðx; y; zÞ
ð16:2Þ
Boundary condition: λ
@T 1 βðT 2 Tc Þ 5 0 @n
ð16:3Þ
If the adiabatic temperature rise of the material of the model is θm ðtm Þ 5 θ0m ð1 2 e2sm tm Þ, the differential equation and the conditions of the model are: Equation of heat conduction: 2 @Tm @ Tm @2 Tm @2 Tm θ0m @ð1 2 e2sm tm Þ 1 5 am 1 1 2 2 2 @tm @tm @xm @ym @zm
ð16:1aÞ
Initial condition: Tm ð0; xm ; ym ; zm Þ 5 T0 ðxm ; ym ; zm Þ
ð16:2aÞ
Boundary condition: λm
@Tm 1 β m ðTm 2 Tcm Þ 5 0 @nm
ð16:3aÞ
Let Ct 5 t/tm ; CL 5 x/xm ; Ca 5 a/am CT 5 T=Tm ; Cθ 5 θ0 =θ0m ; Cλ 5 λ/λm ; Cβ 5 β/β m ; Cs 5 s/sm
ð16:4Þ
Simulation Analysis, Dynamic Temperature Control, Numerical Monitoring
339
where the subscript “m” represents the model. Substitution of Eq. (17.51) into Eqs. (16.1)(16.3) yields 2 CT @Tm Ca CT @ Tm @2 T m @2 T m Cθ θ0m @ð1 2 e2CS Ct sm tm Þ 5 a 1 1 1 m 2 2 2 @tm Ct @tm @xm @ym @zm Ct CL ð16:1bÞ CT CL Tm ð0; xm ; ym ; zm Þ 5 CT CL T0 ðxm ; ym ; zm Þ
ð16:2bÞ
Cλ CT @Tm λm 1 Cβ CT β m ðTm 2 Tcm Þ 5 0 CL @nm
ð16:3bÞ
Comparing Eqs. (16.1b)(16.3b) with Eqs. (16.1a)(16.3a), it is clear that, in order to make the temperature field of the model similar to that of the prototype, the following conditions of similarity must be satisfied: CT Ca CT Cθ 5 5 ; 2 Ct Ct CL Ct 5 CL2 =Ca ;
Cλ CT 5 Cβ CT ; CL
Cβ 5 Cλ =CL ;
Cθ 5 CT ;
Cs Ct 5 1
ð16:5Þ
Cs 5 1=Ct
ð16:6Þ
There are eight parameters and four equations in Eq. (16.5), thus four parameters may be given freely and the other four parameters must be computed by Eqs. (16.5) and (16.6). For example, let CT 5 1; Ca 5 1:1; Cλ 5 1:1; CL 5 20, from Eq. (16.6), we have: Ct 5 202 /1:1 5 364;
Cβ 5 1:1/20 5 0:055;
Cθ 5 1;
Cs 5 1/364
In order to satisfy the similarity conditions, for the period of annual variation (P 5 1 year) in the prototype, the period of variation in the model is Pm 5 1=364 year 5 1d; for the duration of cold wave Q 5 2 2 4d in the prototype, in the model Qm 5 0:13 0:26 h; in the prototype β 5 80 kJ/(m2 h C), in the model, β 5 4.40 kJ/(m2 h C). These conditions are possible to be fulfilled in the model, but it is difficult to make the rate of hydration heat in the model sm 5 s=364. Thus, it is possible to make a model test of the temperature field for the period of operation and it is difficult to conduct a model test of the temperature field for the period of construction. In the construction period, the thermal stresses are related to the modulus of elasticity EðτÞ and unit creep Cðt; τÞ which are functions of time t and age τ. It is difficult to make EðτÞ and Cðt; τÞ to satisfy the similarity conditions. So it is difficult to do a model test for thermal stress in the period of construction. The modulus of elasticity of concrete is nearly constant in the period of operation, so it is possible to do a model test for elastic thermal stress of mass concrete
340
Thermal Stresses and Temperature Control of Mass Concrete
in the period of operation. Because the creep of concrete is difficult to satisfy the similarity condition, it is difficult to do a model test for viscoelastic thermal stress even in the period of operation. In 2010, one engineer had made a 1:7 model test for the thermal stresses in the period of construction, and the material for model is also concrete, so Ct 5 1:0; Ca 5 1:0; Cs 5 1:0, from Eq. (16.5), CL 5 1:0. Thus, the model must have the same dimensions as the prototype, and the model with CL 5 7 is not similar to the prototype.