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Thermally induced uniform strains and curvatures calculated using equivalent linear temperature distributions
Nuclear Engineering and Design 250 (2012) 42–52
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Thermally induced uniform strains and curvatures calculated using equivalent linear temperature distributions Sungjin Bae ∗ Bechtel Power Corporation, 5275 Westview Drive, Frederick, MD 21703, USA
h i g h l i g h t s
Changes of temperature distributions were investigated using the FE heat transfer analyses. Validity of thermal curvature calculated using linear thermal gradient was examined. Thermal gradient in ACI 349 can be used for concrete sections with reinforcing steel. Thermal gradient in ACI 349 underestimates curvature of a section with liner. The proposed thermal gradient takes into account the effect of liner on thermal curvatures.
a r t i c l e
i n f o
Article history: Received 21 January 2012 Received in revised form 15 May 2012 Accepted 24 May 2012
a b s t r a c t Thermal effects due to varying temperature are considered in the design of reinforced concrete structures for nuclear power plants. The commentary to Appendix E of ACI 349 provides a method of converting a nonlinear temperature distribution to an equivalent linear temperature distribution. The linear temperature distribution consists of two components: (1) a uniform temperature change and (2) a thermal gradient. Each component of the linear temperature distribution is used for calculating free thermal uniform strain and curvature. Further, thermal-induced loads can be estimated by taking into account the degree of restraints. The ACI 349 method for calculating free thermal uniform strain and curvature using a linear temperature distribution is developed based on a pure concrete section. As a result, the calculated free thermal uniform strain and curvature do not account for the effect of steel liner plate and reinforcing bars. The objective of this paper is to examine free thermal uniform strains and curvatures calculated using equivalent linear temperature distributions. Concrete sections with different thickness and the existence of steel liner plate and reinforcing bars were considered. Finite element analyses were performed and free thermal uniform strain and curvature were obtained. These results were compared with those calculated by the ACI 349 method. It was found that the equivalent linear temperature distribution method in ACI 349 can be used for predicting free thermal uniform strains and curvatures for pure concrete sections and concrete sections with steel reinforcement if rebars are equally placed on both faces of a concrete section or the steel area ratio is less than 1%. However, when a steel liner is attached to concrete, free thermal curvature can be significantly underestimated. Modified expression for the equivalent linear temperature distribution was proposed to account for concrete sections with liner. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Reinforced concrete structures for nuclear power plants are designed for thermal effects (ACI 349, 2006; ASME, 2011). When a structure is exposed to temperature change, concrete and steel
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expand or contract. If the structure is restrained for the thermal expansion or contraction, thermal stresses and loads are generated. When a structure is subjected to transient temperature rise, temperature distribution in a member becomes nonlinear. Commentary of ACI 349 (2006) provides a method of converting a nonlinear temperature distribution to an equivalent linear temperature distribution, which facilitates the calculation of free thermal uniform strain and curvature. Then, restrained thermal uniform strain and curvature can be obtained by taking into account the degree of restraint. Finally, thermal-induced loads can be estimated
(outside) Equivalent linear temperature distribuon, TL Tm ΔT
Actual nonlinear temperature distribuon, TNL
Tb +y
t Fig. 1. Equivalent linear temperature distribution.
using restrained thermal uniform strain and curvature (ACI 349.1R, 2007; Elgaaly, 1988; Gou et al., 1997; Gurfinkel, 1971; Kohli and Gürbüz, 1975; Nakamura and Maruyama, 1997; Vecchio, 1987) It is important to note that the ACI 349 expressions for calculating the equivalent linear temperature distribution are based on a pure concrete section. The influence of steel liner and reinforcing bars on thermal uniform strains and curvatures is not addressed. Therefore, the validity of the obtained free thermal uniform strain and curvature calculated from the linear temperature distribution can be questionable when a concrete section contain steel liner or reinforcing bars. The objective of the paper is to investigate the validity of free thermal uniform strain and curvature calculated using the linear temperature distribution for various concrete section details. Concrete sections studied were varied by changing section depth, steel liner and rebars. The calculated free thermal uniform strain and curvature were compared with the results from finite element analyses. Finally, modified expressions were proposed for calculating the equivalent linear temperature distribution for a concrete section with steel liner or rebars. 2. Equivalent linear temperature distribution in ACI 349 (2006)
Step (i). Obtain the actual nonlinear temperature distribution TNL through the concrete thickness. Step (ii). Determine mean temperature Tm as 1 t
t/2
TNL dy −(t/2)
(1)
175
Case 1
150
300
Case 2
250
125 100
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50 100
1 hour
5 days
25
50
1 day
100 days
0 2 3 6 0.1 104 105 10 107 0.3 1 3 10 30 1030010300030000300000 3000000 30000000
Time (sec) Fig. 3. Accident air temperatures.
where t is the depth of a concrete section and y is the distance from the centroid of a concrete section (see Fig. 1). The expression given in ACI 349 for calculating the mean temperature is incorrect. The mean temperature in Fig. RE.1 of ACI 349 is defined in terms of temperature change TNL − Tb , where Tb is the base (stress-free) temperature. This contradicts the uniform temperature change Tm − Tb shown in ACI 349 (2006) and ACI 349.1R (2007). Hence, the correct expression for the mean temperature is Eq. (1). Step (iii). Determine the thermal gradient T, which produces the same uncracked moment about the centerline of the section as does the nonlinear temperature distribution. For rectangular sections, the thermal gradient T can be calculated as T =
The actual nonlinear temperature distributions can be converted to the equivalent linear temperature distributions per ACI 349 code (2006):
Tm =
350
43
(°C)
(inside)
Ambient Temperature at Inner Face (°F)
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
12 t2
t/2
TNL · y dy
(2)
−(t/2)
where y is the distance from the centroid of the concrete section (see Fig. 1). The integral is the first moment of the area under TNL about the centerline of the section. The equivalent linear temperature distribution TL can be presented as the sum of a mean temperature of Tm and a thermal gradient T. Step (iv). Free thermal uniform strain εth,free due to uniform temperature change Tm − Tb is calculated as ˛(Tm − Tb ) for beams/columns and as ˛(Tm − Tb )/(1 − ) for plate members, where ˛ is concrete coefficient of thermal expansion (in./in./◦ F or mm/mm/◦ C) and is Poisson’s ratio of concrete. Similarly, free thermal curvature change th,free due to thermal gradient T is
Fig. 2. Concrete sections used for heat transfer analysis.
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Fig. 4. Finite element models for heat transfer and structural analyses.
Fig. 5. Results of heat transfer and structural analyses.
calculated as ˛T/t for beams/columns and as ˛T/[t(1 − )] for plate members. The base temperature Tb in Fig. 1 is the temperature at which a concrete member is free of thermal stresses is the temperature during concrete curing. It is related to the temperature variation during concrete curing. A temperature of 70 ◦ F (21 ◦ C) is typically used (ACI 349, 2006). Note that the base temperature only affects the uniform temperature change but not the thermal gradient. The equivalent linear temperature distribution converts the nonlinear temperature change across a section to uniform temperature change Tm − Tb and thermal gradient T. Free thermal uniform strain εth,free and free thermal curvature change th,free can be calculated from the equivalent linear temperature distribution as shown in Step (iv) above. It is important to note that the thermal forces or moments are proportional to the degree of restraint. If a structure is not restrained and thermal uniform strain and curvature can develop freely, the temperature change will not induce any thermal forces or moments. For example, axisymmetric structures like nuclear concrete containments are unrestrained for uniform temperature
change except boundary regions but fully restrained for thermal gradient. As such conditions, thermal forces and moments resulting from thermal uniform strain are insignificant when compared with those that result from thermal curvature.
3. Heat transfer mechanism When air temperature or pipe surface temperature arises, heat is transferred to the surrounding concrete structure by convection and/or radiation. Heat that is absorbed by concrete is transferred away by means of conduction. Finally the structure loses heat to ambient air or contacting media by convection/or and radiation. When a liner is attached to concrete, heat is transferred to the liner by convection and/or radiation. The heat transfer from the liner to the concrete depends on the interface conditions (BN-TOP3, 1983): i. Bonded: The liner and the concrete establish a perfect bond due to the curing of the concrete. The interface offers no real
Table 1 Properties of concrete and steel (Drysdale, 1999).
Compressive strength (or yield strength) Modulus of elasticity Poisson ratio Coefficient of thermal expansion Density Thermal conductivity Specific heat
Concrete
Steel
5000 psi (34.5 MPa) 4030 ksi (27,786 MPa) 0.17 5.5 × 10−6 /◦ F (5.5 × 10−6 /◦ C) 140 lb/ft3 (2300 kg/m3 ) 0.8 BTU/(h ft ◦ F) (1.4 W/m ◦ C) 0.21 BTU/lb ◦ F (880 J/kg ◦ C)
60 ksi (414 MPa) 29,000 ksi (200,000 MPa) 0.17 5.5 × 10−6 /◦ F (5.5 × 10−6 /◦ C) 490 lb/ft3 (7850 kg/m3 ) 26.5 BTU/(h ft ◦ F) (45.8 W/m ◦ C) 0.11 BTU/lb ◦ F (460 J/kg ◦ C)
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6 hrs, 12 hrs, 18 hrs & 1 day
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0 0
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(°C)
0 sec, 30 min & 1 hr
Temperature (°F)
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(°C)
Temperature (°F)
(a) Case 1 accident temperature
0
12
24
0 36
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
48
(b) Case 2 accident temperature Fig. 6. Temperature distributions of 48-in. wall.
resistance to heat transfer. Thus, the heat transfer is limited only by the thermal properties of the concrete. ii. Contact with no bond: No bond exists between the liner and the concrete but substantial contact is maintained due to pressure loads on the liner. Small voids may exist at the interface under this condition and, therefore, the interface acts as an additional resistance to heat transfer. The value of the joint conductance is a strong function of the surface properties and the pressure applied at the interface. iii. No contact: The liner and the concrete have physical separation. This condition severely limits the heat transfer at the interface. Here heat transfers by conduction through the air in the gap, convection across the gap, and radiation between the interface surfaces. The heat transfer is limited to a small amount. The bonded condition provides the most efficient heat transfer, producing higher temperature in concrete and larger free thermal curvature. The bonded condition is used in this study for analyzing temperature distributions. 4. Finite element heat transfer and structural analyses Finite element (FE) analyses were performed in two steps to have temperature distributions, free thermal uniform strains and curvatures: (1) heat transfer analyses were conducted to get temperature distributions and (2) structural analyses were performed by applying the temperature distributions and free thermal uniform strains and curvatures were obtained. ANSYS program (2011) was used for this purpose. The temperature distributions from FE
analyses were converted to linear temperature distribution and free thermal uniform strains and curvatures were calculated. These free thermal uniform strains and curvatures were compared with those from the FE analyses. In this way, the effect of section thickness, steel liner and reinforcing bars on the estimated thermal uniform strains and curvatures was examined. Fig. 2 illustrates concrete sections used in the study. Concrete walls having a thickness of 48 in. (1220 mm) and 24 in. (610 mm) were studied. A concrete wall with a section thickness of 48-in. (1220 mm) and a ¼-in. (6.4 mm) steel liner was also included to investigate the effect of liner on heat transfer and structural behavior. A bonded condition was used at the interface between the liner and the concrete. Heat transfer analyses were performed by applying air temperatures to concrete surfaces. Two accident temperature profiles with different durations of the maximum temperature were used for the analyses, as shown in Fig. 3. The air temperature is 100 ◦ F (38 ◦ C) initially and it rises to 300 ◦ F (149 ◦ C) at 10 s linearly, maintains the constant temperature for 1 h (or one day) and decreases to 100 ◦ F (38 ◦ C) at 5 days (or 100 days) in a log-linear manner. The accident air temperature was applied to the inner surface of the concrete wall or the liner when the liner was attached. The ambient temperature at the outer face was maintained to 30 ◦ F (−1 ◦ C) during analyses. Only convection and conduction were considered in the analyses as radiation can be ignored for temperatures below 650 ◦ F (345 ◦ C) (Edward Pope, 1997). Fig. 4 illustrates a schematic view of FE analysis procedures, including models, input loads and boundary conditions used in the analyses. A concrete section was modeled using a thermal plane
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
350
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6 hrs, 12 hrs & 18 hrs
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0 sec, 30 min & 1 hr
(°C)
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0 12
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0
48
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Distance from Inner Surface of Concrete
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Cooling stage
(a) Case 1 accident temperature
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0 sec, 30 min & 1 hr 250
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6 hrs, 12 hrs, 18 hrs & 1 day
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(°C)
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Temperature (°F)
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25 50
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0
12
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48
0
12
24
36
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
48
(b) Case 2 accident temperature Fig. 7. Temperature distributions of 24-in. wall.
element (PLANE55) for heat transfer analyses (see Fig. 4(a)). Temperature distributions of the section were analyzed in the time domain by applying the accident air temperature at the inner face and the ambient air temperature at the outer face. A temperature distribution in Fig. 5(a) is a typical result from heat transfer analyses. Then, thermal plane elements (PLANE55) were converted to structural plane stress elements (PLANE182) for structural analyses (see Fig. 4(b)). The elastic modulus of concrete was assigned to the concrete section. When steel rebars were modeled, the elastic modulus of steel and areas of reinforcing bars were specified. Pin-roller supports were used at the bottom of the wall and a rotationally rigid condition was set at the top to model a continuous member. Such boundary conditions permit free thermal deformation but enforce that the strain compatibility that a plane section remains plane. Deformed shapes resulting from structural analyses were used to calculate free uniform strains εansys and curvatures ansys (see Fig. 5(b)). Bi-axial effects were not included in the analyses since plane stress elements were used. Alternatively, free uniform strains εlinear and curvatures linear can be conveniently calculated using equivalent linear temperature distributions. The accuracy of free uniform strains and curvatures estimated using equivalent linear temperature distributions was examined by comparing with those resulting from FE analyses. Table 1 summarizes mechanical and thermal properties of concrete and steel used in this study. The convective coefficient of air at inner and outer faces of the wall was 4.4 BTU/h ft2 ◦ F (25 W/m2 ◦ C) in the analyses.
5. Discussion of temperature distributions Temperature distributions were generated in time sequence from heat transfer analyses, as shown in Figs. 6–8. The following observations are made from the figures: i. The initial temperature distribution was linear with temperature close to 100 ◦ F (38 ◦ C) at the inside face and 30 ◦ F (−1 ◦ C) at the outside face as a results of a steady-state condition. The gap between temperatures at the concrete (or liner) surface and the air is due to the convective heat transfer. ii. The temperature distribution becomes nonlinear with the increase of the accident air temperatures. Comparison of 48in. (1220 mm) and 24-in. (610 mm) thick concrete walls in Figs. 6 and 7 suggests that the temperature rises at the inner surface are similar regardless of the section thickness. However, temperature rises slowly across a concrete section for a thick concrete wall. iii. Temperature changes at the outside face are negligible for wall thicknesses and accident temperatures considered in this study. iv. In the cooling stage, the nonlinearity of temperature distributions reduces and becomes linear at the end. v. Comparison of temperature distributions for case 1 and case 2 accident temperatures shows that the contacting concrete temperature increases close to the maximum air temperature as the maximum air temperature sustained longer. vi. Comparison of Figs. 6 and 8 shows that the liner has negligible effect on the concrete temperature distributions. This is because
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
47
Fig. 8. Temperature distributions of 48-in. wall with liner.
(a) the liner is continuously bonded to the concrete, (b) the liner has high thermal conductivity (see Table 1) and (c) the liner is thin. As a result, temperatures of the liner and the contacting concrete surface are essentially the same (see Fig. 8). A linear steady-state temperature distribution existed initially in the concrete wall. The concrete temperatures at inside and outside surfaces are closely related to the air temperatures. As the accident air temperature rises, thermal energy is provided by the air, the concrete temperature rises and the temperature distribution becomes nonlinear. If the accident air temperature sustains at a maximum for sufficiently long duration, the concrete temperature distribution will reach a linear steady-state condition with the contacting concrete temperature close to the maximum air temperature. This linear steady-state temperature distribution is the concrete temperature distribution which produces the highest thermal gradient. Fig. 9 shows comparisons of the initial and maximum steady-state temperature distributions with the concrete temperature distributions obtained from heat transfer analyses. The steady-state temperature distributions were plotted as straight lines connecting the inside and outside air temperatures linearly. The convective heat loss between ambient air and concrete was not accounted for in these steady-state conditions. Fig. 9 suggests that all concrete temperature distributions are essentially bounded by the initial and final steady-state temperature distributions. The maximum steady-state temperature
distribution can be considered as upper-bound concrete temperature distribution. It is important to note that the maximum steady-state temperature distribution is only achievable when the maximum air temperature is sustained for a sufficiently long duration. 6. Discussion of free thermal uniform strains and curvatures calculated using equivalent linear temperature distributions Free thermal uniform strains and curvatures were calculated using equivalent linear temperature distributions as ˛(Tm − Tb ) and ˛T/t, respectively. Biaxial effect was not included in the calculations to be consistent with the ANSYS FE analyses, where the plane stress element was used. The results for pure concrete sections are shown in Fig. 10. The equivalent linear temperature distributions provided accurate estimates of free thermal uniform strains and curvatures. Fig. 11 shows the effect of steel liner on free thermal uniform strains and curvatures. The free uniform strains calculated using linear temperature distributions showed good agreement with the FE analysis results. However, free thermal curvatures were consistently underestimated by up to 16.2%. Therefore, it is concluded that the steel liner needs to be considered for accurate estimation of thermal curvatures. The results for concrete sections with steel rebars are shown in Fig. 12. Rebars with the steel area ratio of 1% were provided on
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Fig. 9. Comparisons of steady-state and transient-state temperature distributions.
inside or outside face. Similar to the liner, rebars had negligible influence on the predicted free thermal uniform strains. However, free thermal curvatures could be underestimated by up to 5.1% for rebars on the inside face and overestimated by up to 3.5% for rebars on the outside face, respectively. The measured accuracy will depend on the concrete cover and the amount of rebars. Errors associated with the predicted thermal curvature will be smaller if rebars are placed on both faces as the error related to each rebar will be cancelled out. If rebars are equally placed on both faces of sections and/or the steel area ratio is less than 1%, the use of equivalent linear temperature distribution will provide a reasonable prediction of thermal curvature with the expected error less than 5.1%.
7. Proposed equivalent linear temperature distribution for concrete sections with liner The use of equivalent linear temperature distribution in ACI 349 (2006) can underestimate free thermal curvature significantly if the liner is not taken into account. For the investigated concrete section, where the wall thickness of 48 in. and the liner thickness of ¼ in. were used, the thermal curvatures was underestimated by close to 16%.
Several approaches can be used to account for the liner in estimating thermal curvatures: i. FE analysis with explicit modeling of steel liner. FE structural analyses can be performed. The liner should be included in the FE model. The concrete temperature can be applied as nonlinear or linear distribution. However, the applied liner temperature should be the actual temperature. ii. ACI 349 method with liner temperature as external load. The ACI 349 method (2006) may be used if the thermal force due to steel liner is considered separately. A constraint condition is required to estimate the thermal force of the liner. If a fully constrained condition is used, the thermal force of the liner will be ˛(TNL,liner − Tm )·Aliner ES . The thermal curvature can be obtained from the force equilibrium of the internal stresses due to the thermal gradient and the liner thermal force. iii. Modified linear thermal gradient. The expression for calculating linear thermal gradient in ACI 349 can be modified to include the effect of a steel liner on thermal curvature, as shown in Eq. (A3). Detailed descriptions of deriving the proposed expression are provided in Appendix A. The proposed expression can be used to calculate the thermal curvature of a concrete section with steel liner.
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
500
Max. steady state
Case 2 temperature
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-6
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ANSYS Linear temp. method
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30 Case 1 temperature
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Inial steady state
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(b) 24-in. walls Fig. 10. Accuracy of equivalent linear temperature method for rectangular concrete section only.
In order to vilify the proposed expression, free thermal curvatures were calculated using Eq. (A3) and compared with the FE analysis results. The comparisons shown in Fig. 13 demonstrate that the proposed thermal gradient provides accurate estimates of free thermal curvatures for concrete sections with liner. Note that
there is no need to separately consider the liner as the effect of steel liner on thermal curvature is accounted by the use of the proposed thermal gradient in Eq. (A3). In other words, the thermal gradient from Eq. (2) should be used if the liner is separately included in the calculation or modeled in the FE analysis.
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12345678910111213
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Fig. 11. Accuracy of equivalent linear temperature method for concrete section with liner.
(× 10 /mm)
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(b) 48-in. walls with rebars on outside face ( ρs,out = 1%) Fig. 12. Accuracy of equivalent linear temperature method for concrete sections with rebars.
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Case 2 temperature
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the results from the FE analyses. The following conclusions can be reached based on the research reported herein:
2.5
30 1.0
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10
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0 1
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7
8
9 10 11 12 13
Time Fig. 13. Accuracy of proposed thermal gradient for concrete sections with liner.
8. Conclusions Changes of temperature distributions were investigated using the FE heat transfer analyses. The validity of the use of equivalent linear temperature distribution for calculating free thermal uniform strain and curvature was examined by comparing with
i. Temperature changes at the outside face are negligible for the given wall thickness and accident temperature. ii. The impact of the liner on concrete temperature distributions is negligible when (a) the liner is continuously bonded to the concrete, (b) the liner has high thermal conductivity and (c) the liner is thin. As a result, temperatures of the liner and the contacting concrete surface are essentially the same (see Fig. 8). iii. All concrete temperature distributions are essentially bounded by the initial and maximum steady-state temperature distributions. The maximum steady-state temperature distribution can be considered as upper-bound concrete temperature distribution, which produces the highest thermal gradient. It is important to note that the maximum steady-state temperature distribution is only achievable when the maximum air temperature sustains for sufficiently long duration. iv. The equivalent linear temperature distribution can be used to calculate free thermal uniform strains and curvatures for concrete sections without and with steel reinforcement when rebars are placed equally on both faces or the amount of rebars is less than 1.0% of the gross concrete section on each face. v. For concrete sections with steel liner, the equivalent linear temperature distribution in ACI 349 may underestimate thermal curvatures significantly. On the other hand, the estimated thermal uniform strains are acceptable. The modified expression
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
for calculating the thermal gradient is proposed to account for the liner, as shown below. As the effect of the liner on the structural response is taken into account in the thermal gradient, the proposed expression provides an accurate estimate of thermal curvature without separately considering the liner.
t/2 T =
Appendix A. Derivations of equivalent linear temperatures for rectangular sections A.1. Rectangular concrete section The uniform temperature change Tm is related to the area under the temperature profiles. The area under TNL is
QNL = Qconc + Qliner =
t/2
TNL dy
Tm is defined as the mean temperature and, therefore, the area under the mean temperature Tm is
=
T · t 2 T + Tm + 12 2
Therefore, the mean temperature Tm can be presented as −(t/2)
Tm =
TNL dy
(A1)
t
As such, the uniform temperature change can be expressed as Tm − Tb , where Tb is the base temperature. It is important to note that the formula calculating the mean temperature in Fig. RE.1 of ACI 349-06 is corrected to Eq. (A1) in order to be consistent with the definition of the uniform temperature change, which is Tm − Tb . The equivalent liner temperature gradient T is derived based on the moment of area under the temperature profiles. The moment of area under TNL is
t/2
QNL =
t 2
Aliner b
(n · tliner )
= T
t/2
TNL · y dy + TNL,liner ·
T · t 2 A + TL,liner · yliner × n · liner 12 b
AL = Tm · t
−(t/2)
TNL · y dy + TNL,liner · yliner × n ·
where TNL is the nonlinearly distributed temperature; TNL,liner is the temperature of a liner plate. The liner temperature is equal to the temperature at the contacting concrete surface for a thin steel liner; y is the distance from centroid of concrete section; n is the ratio of elastic moduli = Es /Ec in which Ec is the elastic modulus of concrete and Es is the elastic modulus of steel; Alinear /b is the area of liner per unit width of a concrete member, which is equal to the thickness of liner plate, tliner ; and yliner is the distance from centroid of liner plate to centroid of concrete section. For thin liners, the liner thickness can be ignored, resulting in t/2 for rectangular sections. The moment of area under TL with liner plate is QL =
−(t/2)
t/2
t/2
−(t/2)
(t 2 /12) + (t/4)(n · tliner )
ANL =
With the liner plate, the moment of area under TNL is
QNL =
T · y dy + (TNL,liner − Tm )(t/2)(n · tliner ) −(t/2) NL
51
t
2
(n · tliner )
t t2 + (n · Aliner ) + Tm 12 4
t 2
(n · tliner )
Therefore, the thermal gradient for a rectangular concrete section with steel liner attached is
t/2
T =
−(t/2)
TNL · y dy + (TNL,liner − TTm )(t/2)(n · tliner ) (t 2 /12) + (t/4)(n · tliner )
(A3)
In a similar way, the following expression can be used to calculate the thermal gradient for a rectangular concrete section that contains rebars:
t/2
T =
−(t/2)
TNL · y dy + (t 2 /12) +
ns
i=1
(TNLs,i − Tm )ys,i (n − 1)(As,i /b)
i=1
2 /t)(n − 1)(A /b) (ys,i s,i
ns
(A4)
where ns is the number of rebar layers; TNLs,i is the temperature at rebar layer, i; ys,i is the distance from centroid of concrete section to centroid of rebar layer, i; and As,i is the area of rebar layer, i.
TNL · y dy −(t/2)
Similarly, the moment of area under TL can be calculated QL =
1 2
T t 2 2
2
×
3
·t
=
References
T · t 2 12
By equating above two expressions, the thermal gradient T is T =
12 t2
t/2
TNL · y dy
(A2)
−(t/2)
A.2. Rectangular concrete section with steel liner plate When a steel liner plate is placed on the face of concrete, which is common for nuclear concrete containments in the US, the liner needs to be considered for calculating thermal gradient. Since the liner has negligible impact on the calculated mean temperature (see Fig. 11), the effect of the liner on mean temperature is not considered here. The overall temperature distribution across a concrete section remains essentially the same when the interface between the liner and the concrete is assumed to a bond condition due to high thermal conductivity of steel and thin thickness of the liner plate (see Figs. 6 and 8).
ACI 349, 2006. Code Requirements for Nuclear Safety-Related Concrete Structures (ACI 349-06) and Commentary, ACI 349-06. American Concrete Institute. ACI 349.1R, 2007. Reinforced Concrete Design for Thermal Effects on Nuclear Power Plant Structures, ACI 349.1R-07. American Concrete Institute. ANSYS Inc., 2011. ANSYS Release 13.0. ASME, 2011. Code for Concrete Containments, 2010 ASME Boiler and Pressure Vessel Code, Section III, Division 2. The American Society of Mechanical Engineers, New York, NY. BN-TOP-3, 1983. Performance and Sizing of Dry Pressure Containments, Revision 4. Bechtel Power Corporation. Drysdale, D., 1999. An Introduction to Fire Dynamics, 2nd ed. John Wiley & Sons Ltd. Edward Pope, J., 1997. Rules of Thumb for Mechanical Engineers – A Manual of Quick, Accurate Solutions to Everyday Mechanical Engineering Problems. Elsevier. Elgaaly, M., 1988. Thermal gradients in beams, walls, and slabs. ACI Struct. J. 85, 76–81. Gou, P.F., Nakamura, M., Marayama, K., Ehlert, G., Ogawa, S., 1997. Analysis of reinforced concrete shell elements subjected to membrane forces, bending moments and transverse shear forces encountered in RCCV. In: Proceedings of 14th Conference on Structural Mechanics in Reactor Technology (SMiRT), H06/4, pp. 227–234. Gurfinkel, G., 1971. Thermal effects in walls of nuclear containments – elastic and inelastic behavior. In: Proceedings of 1st Conference on Structural Mechanics in Reactor Technology (SMiRT), J 3/7, pp. 277–296.
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S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
Kohli, T.D., Gürbüz, O., 1975. Optimum design of reinforced concrete for nuclear containments, including thermal effects. In: Proceedings of the Second ASCE Specialty Conference on Structural Design of Nuclear Plant Facilities, New Orleans, LA, pp. 1292–1319. Nakamura, M., Maruyama, K., 1997. Stress evaluation of RC shell elements based on three-dimensional equilibrium. In: Proceedings of 14th Conference on Structural Mechanics in Reactor Technology (SMiRT), H06/3, pp. 219–226. Vecchio, F.J., 1987. Nonlinear analysis of reinforced concrete frames subjected to thermal and mechanical loads. ACI Struct. J. 84, 492–501.
Sungjin Bae is a Senior Structural Engineer at Bechtel Corporation, Frederick, MD. He received his BS and MS from Hanyang University, Seoul, Korea, and his PhD from the University of Texas at Austin, Austin, TX. He is a member of ACI Committee 209, Creep and Shrinkage of Concrete; 349, Concrete Nuclear Structures; Working Group on Design (SC-3C) of Joint ACI-ASME Committee 359, Nuclear Concrete Containments; and Joint ACI-ASCE Committee 441, Reinforced Concrete Columns. His research interests include the behavior and performance-based design of concrete columns and design of nuclear structures, foundations for dynamic equipment, and chimneys.
Contents lists available at SciVerse ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Thermally induced uniform strains and curvatures calculated using equivalent linear temperature distributions Sungjin Bae ∗ Bechtel Power Corporation, 5275 Westview Drive, Frederick, MD 21703, USA
h i g h l i g h t s
Changes of temperature distributions were investigated using the FE heat transfer analyses. Validity of thermal curvature calculated using linear thermal gradient was examined. Thermal gradient in ACI 349 can be used for concrete sections with reinforcing steel. Thermal gradient in ACI 349 underestimates curvature of a section with liner. The proposed thermal gradient takes into account the effect of liner on thermal curvatures.
a r t i c l e
i n f o
Article history: Received 21 January 2012 Received in revised form 15 May 2012 Accepted 24 May 2012
a b s t r a c t Thermal effects due to varying temperature are considered in the design of reinforced concrete structures for nuclear power plants. The commentary to Appendix E of ACI 349 provides a method of converting a nonlinear temperature distribution to an equivalent linear temperature distribution. The linear temperature distribution consists of two components: (1) a uniform temperature change and (2) a thermal gradient. Each component of the linear temperature distribution is used for calculating free thermal uniform strain and curvature. Further, thermal-induced loads can be estimated by taking into account the degree of restraints. The ACI 349 method for calculating free thermal uniform strain and curvature using a linear temperature distribution is developed based on a pure concrete section. As a result, the calculated free thermal uniform strain and curvature do not account for the effect of steel liner plate and reinforcing bars. The objective of this paper is to examine free thermal uniform strains and curvatures calculated using equivalent linear temperature distributions. Concrete sections with different thickness and the existence of steel liner plate and reinforcing bars were considered. Finite element analyses were performed and free thermal uniform strain and curvature were obtained. These results were compared with those calculated by the ACI 349 method. It was found that the equivalent linear temperature distribution method in ACI 349 can be used for predicting free thermal uniform strains and curvatures for pure concrete sections and concrete sections with steel reinforcement if rebars are equally placed on both faces of a concrete section or the steel area ratio is less than 1%. However, when a steel liner is attached to concrete, free thermal curvature can be significantly underestimated. Modified expression for the equivalent linear temperature distribution was proposed to account for concrete sections with liner. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Reinforced concrete structures for nuclear power plants are designed for thermal effects (ACI 349, 2006; ASME, 2011). When a structure is exposed to temperature change, concrete and steel
∗ Tel.: +1 301 228 8821. E-mail addresses: [email protected], [email protected] 0029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nucengdes.2012.05.032
expand or contract. If the structure is restrained for the thermal expansion or contraction, thermal stresses and loads are generated. When a structure is subjected to transient temperature rise, temperature distribution in a member becomes nonlinear. Commentary of ACI 349 (2006) provides a method of converting a nonlinear temperature distribution to an equivalent linear temperature distribution, which facilitates the calculation of free thermal uniform strain and curvature. Then, restrained thermal uniform strain and curvature can be obtained by taking into account the degree of restraint. Finally, thermal-induced loads can be estimated
(outside) Equivalent linear temperature distribuon, TL Tm ΔT
Actual nonlinear temperature distribuon, TNL
Tb +y
t Fig. 1. Equivalent linear temperature distribution.
using restrained thermal uniform strain and curvature (ACI 349.1R, 2007; Elgaaly, 1988; Gou et al., 1997; Gurfinkel, 1971; Kohli and Gürbüz, 1975; Nakamura and Maruyama, 1997; Vecchio, 1987) It is important to note that the ACI 349 expressions for calculating the equivalent linear temperature distribution are based on a pure concrete section. The influence of steel liner and reinforcing bars on thermal uniform strains and curvatures is not addressed. Therefore, the validity of the obtained free thermal uniform strain and curvature calculated from the linear temperature distribution can be questionable when a concrete section contain steel liner or reinforcing bars. The objective of the paper is to investigate the validity of free thermal uniform strain and curvature calculated using the linear temperature distribution for various concrete section details. Concrete sections studied were varied by changing section depth, steel liner and rebars. The calculated free thermal uniform strain and curvature were compared with the results from finite element analyses. Finally, modified expressions were proposed for calculating the equivalent linear temperature distribution for a concrete section with steel liner or rebars. 2. Equivalent linear temperature distribution in ACI 349 (2006)
Step (i). Obtain the actual nonlinear temperature distribution TNL through the concrete thickness. Step (ii). Determine mean temperature Tm as 1 t
t/2
TNL dy −(t/2)
(1)
175
Case 1
150
300
Case 2
250
125 100
200
75 150
50 100
1 hour
5 days
25
50
1 day
100 days
0 2 3 6 0.1 104 105 10 107 0.3 1 3 10 30 1030010300030000300000 3000000 30000000
Time (sec) Fig. 3. Accident air temperatures.
where t is the depth of a concrete section and y is the distance from the centroid of a concrete section (see Fig. 1). The expression given in ACI 349 for calculating the mean temperature is incorrect. The mean temperature in Fig. RE.1 of ACI 349 is defined in terms of temperature change TNL − Tb , where Tb is the base (stress-free) temperature. This contradicts the uniform temperature change Tm − Tb shown in ACI 349 (2006) and ACI 349.1R (2007). Hence, the correct expression for the mean temperature is Eq. (1). Step (iii). Determine the thermal gradient T, which produces the same uncracked moment about the centerline of the section as does the nonlinear temperature distribution. For rectangular sections, the thermal gradient T can be calculated as T =
The actual nonlinear temperature distributions can be converted to the equivalent linear temperature distributions per ACI 349 code (2006):
Tm =
350
43
(°C)
(inside)
Ambient Temperature at Inner Face (°F)
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
12 t2
t/2
TNL · y dy
(2)
−(t/2)
where y is the distance from the centroid of the concrete section (see Fig. 1). The integral is the first moment of the area under TNL about the centerline of the section. The equivalent linear temperature distribution TL can be presented as the sum of a mean temperature of Tm and a thermal gradient T. Step (iv). Free thermal uniform strain εth,free due to uniform temperature change Tm − Tb is calculated as ˛(Tm − Tb ) for beams/columns and as ˛(Tm − Tb )/(1 − ) for plate members, where ˛ is concrete coefficient of thermal expansion (in./in./◦ F or mm/mm/◦ C) and is Poisson’s ratio of concrete. Similarly, free thermal curvature change th,free due to thermal gradient T is
Fig. 2. Concrete sections used for heat transfer analysis.
44
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
Fig. 4. Finite element models for heat transfer and structural analyses.
Fig. 5. Results of heat transfer and structural analyses.
calculated as ˛T/t for beams/columns and as ˛T/[t(1 − )] for plate members. The base temperature Tb in Fig. 1 is the temperature at which a concrete member is free of thermal stresses is the temperature during concrete curing. It is related to the temperature variation during concrete curing. A temperature of 70 ◦ F (21 ◦ C) is typically used (ACI 349, 2006). Note that the base temperature only affects the uniform temperature change but not the thermal gradient. The equivalent linear temperature distribution converts the nonlinear temperature change across a section to uniform temperature change Tm − Tb and thermal gradient T. Free thermal uniform strain εth,free and free thermal curvature change th,free can be calculated from the equivalent linear temperature distribution as shown in Step (iv) above. It is important to note that the thermal forces or moments are proportional to the degree of restraint. If a structure is not restrained and thermal uniform strain and curvature can develop freely, the temperature change will not induce any thermal forces or moments. For example, axisymmetric structures like nuclear concrete containments are unrestrained for uniform temperature
change except boundary regions but fully restrained for thermal gradient. As such conditions, thermal forces and moments resulting from thermal uniform strain are insignificant when compared with those that result from thermal curvature.
3. Heat transfer mechanism When air temperature or pipe surface temperature arises, heat is transferred to the surrounding concrete structure by convection and/or radiation. Heat that is absorbed by concrete is transferred away by means of conduction. Finally the structure loses heat to ambient air or contacting media by convection/or and radiation. When a liner is attached to concrete, heat is transferred to the liner by convection and/or radiation. The heat transfer from the liner to the concrete depends on the interface conditions (BN-TOP3, 1983): i. Bonded: The liner and the concrete establish a perfect bond due to the curing of the concrete. The interface offers no real
Table 1 Properties of concrete and steel (Drysdale, 1999).
Compressive strength (or yield strength) Modulus of elasticity Poisson ratio Coefficient of thermal expansion Density Thermal conductivity Specific heat
Concrete
Steel
5000 psi (34.5 MPa) 4030 ksi (27,786 MPa) 0.17 5.5 × 10−6 /◦ F (5.5 × 10−6 /◦ C) 140 lb/ft3 (2300 kg/m3 ) 0.8 BTU/(h ft ◦ F) (1.4 W/m ◦ C) 0.21 BTU/lb ◦ F (880 J/kg ◦ C)
60 ksi (414 MPa) 29,000 ksi (200,000 MPa) 0.17 5.5 × 10−6 /◦ F (5.5 × 10−6 /◦ C) 490 lb/ft3 (7850 kg/m3 ) 26.5 BTU/(h ft ◦ F) (45.8 W/m ◦ C) 0.11 BTU/lb ◦ F (460 J/kg ◦ C)
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
250
150
300
150
125
250
125
100
6 hrs, 12 hrs & 18 hrs
200
175
75
150
50 100
100
200
75 150
50 100
25 50
(°C)
0 sec, 30 min & 1 hr
350
(°C)
Temperature (°F)
300
175
Temperature (°F)
350
45
25 50
0
18 hours, 1 day & 5 days
0
0
0 0
12
24
36
0
48
12
24
36
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
48
175
350
175
300
150
300
150
125
250
125
250
6 hrs, 12 hrs, 18 hrs & 1 day
200
100 75
150
50 100
200
100
1 day
75 150
50 100
25 50
0
25 50
5, 10, 50 days & days
0
0 0
12
24
36
48
(°C)
0 sec, 30 min & 1 hr
Temperature (°F)
350
(°C)
Temperature (°F)
(a) Case 1 accident temperature
0
12
24
0 36
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
48
(b) Case 2 accident temperature Fig. 6. Temperature distributions of 48-in. wall.
resistance to heat transfer. Thus, the heat transfer is limited only by the thermal properties of the concrete. ii. Contact with no bond: No bond exists between the liner and the concrete but substantial contact is maintained due to pressure loads on the liner. Small voids may exist at the interface under this condition and, therefore, the interface acts as an additional resistance to heat transfer. The value of the joint conductance is a strong function of the surface properties and the pressure applied at the interface. iii. No contact: The liner and the concrete have physical separation. This condition severely limits the heat transfer at the interface. Here heat transfers by conduction through the air in the gap, convection across the gap, and radiation between the interface surfaces. The heat transfer is limited to a small amount. The bonded condition provides the most efficient heat transfer, producing higher temperature in concrete and larger free thermal curvature. The bonded condition is used in this study for analyzing temperature distributions. 4. Finite element heat transfer and structural analyses Finite element (FE) analyses were performed in two steps to have temperature distributions, free thermal uniform strains and curvatures: (1) heat transfer analyses were conducted to get temperature distributions and (2) structural analyses were performed by applying the temperature distributions and free thermal uniform strains and curvatures were obtained. ANSYS program (2011) was used for this purpose. The temperature distributions from FE
analyses were converted to linear temperature distribution and free thermal uniform strains and curvatures were calculated. These free thermal uniform strains and curvatures were compared with those from the FE analyses. In this way, the effect of section thickness, steel liner and reinforcing bars on the estimated thermal uniform strains and curvatures was examined. Fig. 2 illustrates concrete sections used in the study. Concrete walls having a thickness of 48 in. (1220 mm) and 24 in. (610 mm) were studied. A concrete wall with a section thickness of 48-in. (1220 mm) and a ¼-in. (6.4 mm) steel liner was also included to investigate the effect of liner on heat transfer and structural behavior. A bonded condition was used at the interface between the liner and the concrete. Heat transfer analyses were performed by applying air temperatures to concrete surfaces. Two accident temperature profiles with different durations of the maximum temperature were used for the analyses, as shown in Fig. 3. The air temperature is 100 ◦ F (38 ◦ C) initially and it rises to 300 ◦ F (149 ◦ C) at 10 s linearly, maintains the constant temperature for 1 h (or one day) and decreases to 100 ◦ F (38 ◦ C) at 5 days (or 100 days) in a log-linear manner. The accident air temperature was applied to the inner surface of the concrete wall or the liner when the liner was attached. The ambient temperature at the outer face was maintained to 30 ◦ F (−1 ◦ C) during analyses. Only convection and conduction were considered in the analyses as radiation can be ignored for temperatures below 650 ◦ F (345 ◦ C) (Edward Pope, 1997). Fig. 4 illustrates a schematic view of FE analysis procedures, including models, input loads and boundary conditions used in the analyses. A concrete section was modeled using a thermal plane
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
350
350
175
150
300
150
125
250
125
250 200
100
6 hrs, 12 hrs & 18 hrs
150
75 50
100
Temperature (°F)
0 sec, 30 min & 1 hr
(°C)
100
200
75
150
50 100
25 50
25 50
0 12
24
36
0
48
12
24
36
48
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
(a) Case 1 accident temperature
300
0 sec, 30 min & 1 hr 250
350
175
150
300
150
125
250
125
100
6 hrs, 12 hrs, 18 hrs & 1 day
200
175
75
150
50 100
Temperature (°F)
350
(°C)
0
Temperature (°F)
0
18 hours, 1 day & 5 days 0
0
100
200
75
150
50 100
25 50
0
0
(°C)
Temperature (°F)
300
175
(°C)
46
25 50
0
1, 5, 10, 50 days & 100 days 0
0
12
24
36
48
0
12
24
36
Distance from Inner Surface of Concrete
Distance from Inner Surface of Concrete
Heang stage
Cooling stage
48
(b) Case 2 accident temperature Fig. 7. Temperature distributions of 24-in. wall.
element (PLANE55) for heat transfer analyses (see Fig. 4(a)). Temperature distributions of the section were analyzed in the time domain by applying the accident air temperature at the inner face and the ambient air temperature at the outer face. A temperature distribution in Fig. 5(a) is a typical result from heat transfer analyses. Then, thermal plane elements (PLANE55) were converted to structural plane stress elements (PLANE182) for structural analyses (see Fig. 4(b)). The elastic modulus of concrete was assigned to the concrete section. When steel rebars were modeled, the elastic modulus of steel and areas of reinforcing bars were specified. Pin-roller supports were used at the bottom of the wall and a rotationally rigid condition was set at the top to model a continuous member. Such boundary conditions permit free thermal deformation but enforce that the strain compatibility that a plane section remains plane. Deformed shapes resulting from structural analyses were used to calculate free uniform strains εansys and curvatures ansys (see Fig. 5(b)). Bi-axial effects were not included in the analyses since plane stress elements were used. Alternatively, free uniform strains εlinear and curvatures linear can be conveniently calculated using equivalent linear temperature distributions. The accuracy of free uniform strains and curvatures estimated using equivalent linear temperature distributions was examined by comparing with those resulting from FE analyses. Table 1 summarizes mechanical and thermal properties of concrete and steel used in this study. The convective coefficient of air at inner and outer faces of the wall was 4.4 BTU/h ft2 ◦ F (25 W/m2 ◦ C) in the analyses.
5. Discussion of temperature distributions Temperature distributions were generated in time sequence from heat transfer analyses, as shown in Figs. 6–8. The following observations are made from the figures: i. The initial temperature distribution was linear with temperature close to 100 ◦ F (38 ◦ C) at the inside face and 30 ◦ F (−1 ◦ C) at the outside face as a results of a steady-state condition. The gap between temperatures at the concrete (or liner) surface and the air is due to the convective heat transfer. ii. The temperature distribution becomes nonlinear with the increase of the accident air temperatures. Comparison of 48in. (1220 mm) and 24-in. (610 mm) thick concrete walls in Figs. 6 and 7 suggests that the temperature rises at the inner surface are similar regardless of the section thickness. However, temperature rises slowly across a concrete section for a thick concrete wall. iii. Temperature changes at the outside face are negligible for wall thicknesses and accident temperatures considered in this study. iv. In the cooling stage, the nonlinearity of temperature distributions reduces and becomes linear at the end. v. Comparison of temperature distributions for case 1 and case 2 accident temperatures shows that the contacting concrete temperature increases close to the maximum air temperature as the maximum air temperature sustained longer. vi. Comparison of Figs. 6 and 8 shows that the liner has negligible effect on the concrete temperature distributions. This is because
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
47
Fig. 8. Temperature distributions of 48-in. wall with liner.
(a) the liner is continuously bonded to the concrete, (b) the liner has high thermal conductivity (see Table 1) and (c) the liner is thin. As a result, temperatures of the liner and the contacting concrete surface are essentially the same (see Fig. 8). A linear steady-state temperature distribution existed initially in the concrete wall. The concrete temperatures at inside and outside surfaces are closely related to the air temperatures. As the accident air temperature rises, thermal energy is provided by the air, the concrete temperature rises and the temperature distribution becomes nonlinear. If the accident air temperature sustains at a maximum for sufficiently long duration, the concrete temperature distribution will reach a linear steady-state condition with the contacting concrete temperature close to the maximum air temperature. This linear steady-state temperature distribution is the concrete temperature distribution which produces the highest thermal gradient. Fig. 9 shows comparisons of the initial and maximum steady-state temperature distributions with the concrete temperature distributions obtained from heat transfer analyses. The steady-state temperature distributions were plotted as straight lines connecting the inside and outside air temperatures linearly. The convective heat loss between ambient air and concrete was not accounted for in these steady-state conditions. Fig. 9 suggests that all concrete temperature distributions are essentially bounded by the initial and final steady-state temperature distributions. The maximum steady-state temperature
distribution can be considered as upper-bound concrete temperature distribution. It is important to note that the maximum steady-state temperature distribution is only achievable when the maximum air temperature is sustained for a sufficiently long duration. 6. Discussion of free thermal uniform strains and curvatures calculated using equivalent linear temperature distributions Free thermal uniform strains and curvatures were calculated using equivalent linear temperature distributions as ˛(Tm − Tb ) and ˛T/t, respectively. Biaxial effect was not included in the calculations to be consistent with the ANSYS FE analyses, where the plane stress element was used. The results for pure concrete sections are shown in Fig. 10. The equivalent linear temperature distributions provided accurate estimates of free thermal uniform strains and curvatures. Fig. 11 shows the effect of steel liner on free thermal uniform strains and curvatures. The free uniform strains calculated using linear temperature distributions showed good agreement with the FE analysis results. However, free thermal curvatures were consistently underestimated by up to 16.2%. Therefore, it is concluded that the steel liner needs to be considered for accurate estimation of thermal curvatures. The results for concrete sections with steel rebars are shown in Fig. 12. Rebars with the steel area ratio of 1% were provided on
48
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
Fig. 9. Comparisons of steady-state and transient-state temperature distributions.
inside or outside face. Similar to the liner, rebars had negligible influence on the predicted free thermal uniform strains. However, free thermal curvatures could be underestimated by up to 5.1% for rebars on the inside face and overestimated by up to 3.5% for rebars on the outside face, respectively. The measured accuracy will depend on the concrete cover and the amount of rebars. Errors associated with the predicted thermal curvature will be smaller if rebars are placed on both faces as the error related to each rebar will be cancelled out. If rebars are equally placed on both faces of sections and/or the steel area ratio is less than 1%, the use of equivalent linear temperature distribution will provide a reasonable prediction of thermal curvature with the expected error less than 5.1%.
7. Proposed equivalent linear temperature distribution for concrete sections with liner The use of equivalent linear temperature distribution in ACI 349 (2006) can underestimate free thermal curvature significantly if the liner is not taken into account. For the investigated concrete section, where the wall thickness of 48 in. and the liner thickness of ¼ in. were used, the thermal curvatures was underestimated by close to 16%.
Several approaches can be used to account for the liner in estimating thermal curvatures: i. FE analysis with explicit modeling of steel liner. FE structural analyses can be performed. The liner should be included in the FE model. The concrete temperature can be applied as nonlinear or linear distribution. However, the applied liner temperature should be the actual temperature. ii. ACI 349 method with liner temperature as external load. The ACI 349 method (2006) may be used if the thermal force due to steel liner is considered separately. A constraint condition is required to estimate the thermal force of the liner. If a fully constrained condition is used, the thermal force of the liner will be ˛(TNL,liner − Tm )·Aliner ES . The thermal curvature can be obtained from the force equilibrium of the internal stresses due to the thermal gradient and the liner thermal force. iii. Modified linear thermal gradient. The expression for calculating linear thermal gradient in ACI 349 can be modified to include the effect of a steel liner on thermal curvature, as shown in Eq. (A3). Detailed descriptions of deriving the proposed expression are provided in Appendix A. The proposed expression can be used to calculate the thermal curvature of a concrete section with steel liner.
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
500
Max. steady state
Case 2 temperature
2.5 2.0
50
-6
400
ANSYS Linear temp. method
60
Curvature (× 10 /in.)
Case 1 temperature
300 200 100 0
1.5
Case 2 temperature
Max. steady state
30 Case 1 temperature
1.0
20 0.5
10
Inial steady state
-100
40
-6
-6
600
Uniform strain (× 10 )
70
ANSYS Linear temp. method
(× 10 /mm)
700
49
Inial steady state
-200
0
0 1
2
3
4
5
6
7
8
9 10 11 12 13
1
2
3
4
5
6
Time
7
8
9 10 11 12 13
Time
(a) 48-in. walls
Max. steady state
500
Max. steady state
Case 2 temperature
Case 2 temperature Case 1 temperature
50
2.0
-6
400
2.5
60
Curvature (× 10 /in.)
Case 1 temperature
300 200 100 0
1.5
30 1.0
20 10
Inial steady state
-100
40
-6
-6
Uniform strain (× 10 )
70
ANSYS Linear temp. method
600
-200
ANSYS Inial steady state Linear temp. method
2
3
4
5
6
7
8
9 10 11 12 13
0.5 0
0 1
(× 10 /mm)
700
1
2
3
4
5
6
Time
7
8
9 10 11 12 13
Time
(b) 24-in. walls Fig. 10. Accuracy of equivalent linear temperature method for rectangular concrete section only.
In order to vilify the proposed expression, free thermal curvatures were calculated using Eq. (A3) and compared with the FE analysis results. The comparisons shown in Fig. 13 demonstrate that the proposed thermal gradient provides accurate estimates of free thermal curvatures for concrete sections with liner. Note that
there is no need to separately consider the liner as the effect of steel liner on thermal curvature is accounted by the use of the proposed thermal gradient in Eq. (A3). In other words, the thermal gradient from Eq. (2) should be used if the liner is separately included in the calculation or modeled in the FE analysis.
700
300 200
Case 1 temperature
100 0
2.0
50 Case 2 temperature
40
1.5 Case 1 temperature
30 1.0
20 0.5
10
-100
0
0
-200 12345678910111213
12345678910111213
Time
2.5
-6
Case 2 temperature
ANSYS Linear temp. method
Time
Fig. 11. Accuracy of equivalent linear temperature method for concrete section with liner.
(× 10 /mm)
400
-6
500
60
Curvature (× 10 /in.)
-6
Uniform strain (× 10 )
600
70 ANSYS Linear temp. method
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
500 Case 2 temperature
300
Case 1 temperature
200 100 0
2.5 2.0
50
-6
400
ANSYS Linear temp. method
60
Curvature (× 10 /in.)
-6
600
Uniform strain (× 10 )
70
ANSYS Linear temp. method
Case 2 temperature
40
1.5
Case 1 temperature
-6
700
30 1.0
20 0.5
10
-100
0
0
-200 1
2
3
4
5
6
7
8
(× 10 /mm)
50
1
9 10 11 12 13
2
3
4
5
6
Time
7
8
9 10 11 12 13
Time
(a) 48-in. walls with rebars on inside face ( ρs,in = 1%)
300
2.0
50
-6
Case 2 temperature Case 1 temperature
200 100 0
2.5
Case 2 temperature
40
1.5
Case 1 temperature
-6
500 400
ANSYS Linear temp. method
60
Curvature (× 10 /in.)
-6
Uniform strain (× 10 )
70
ANSYS Linear temp. method
600
30 1.0
20 0.5
10
-100 -200
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13
(× 10 /mm)
700
1
2
3
4
5
6
Time
7
8
9 10 11 12 13
Time
(b) 48-in. walls with rebars on outside face ( ρs,out = 1%) Fig. 12. Accuracy of equivalent linear temperature method for concrete sections with rebars.
70
ANSYS Linear temp. method
Case 2 temperature
40
1.5
-6
Case 1 temperature
(× 10 /mm)
2.0
50
-6
Curvature (× 10 /in.)
60
the results from the FE analyses. The following conclusions can be reached based on the research reported herein:
2.5
30 1.0
20 0.5
10
0
0 1
2
3
4
5
6
7
8
9 10 11 12 13
Time Fig. 13. Accuracy of proposed thermal gradient for concrete sections with liner.
8. Conclusions Changes of temperature distributions were investigated using the FE heat transfer analyses. The validity of the use of equivalent linear temperature distribution for calculating free thermal uniform strain and curvature was examined by comparing with
i. Temperature changes at the outside face are negligible for the given wall thickness and accident temperature. ii. The impact of the liner on concrete temperature distributions is negligible when (a) the liner is continuously bonded to the concrete, (b) the liner has high thermal conductivity and (c) the liner is thin. As a result, temperatures of the liner and the contacting concrete surface are essentially the same (see Fig. 8). iii. All concrete temperature distributions are essentially bounded by the initial and maximum steady-state temperature distributions. The maximum steady-state temperature distribution can be considered as upper-bound concrete temperature distribution, which produces the highest thermal gradient. It is important to note that the maximum steady-state temperature distribution is only achievable when the maximum air temperature sustains for sufficiently long duration. iv. The equivalent linear temperature distribution can be used to calculate free thermal uniform strains and curvatures for concrete sections without and with steel reinforcement when rebars are placed equally on both faces or the amount of rebars is less than 1.0% of the gross concrete section on each face. v. For concrete sections with steel liner, the equivalent linear temperature distribution in ACI 349 may underestimate thermal curvatures significantly. On the other hand, the estimated thermal uniform strains are acceptable. The modified expression
S. Bae / Nuclear Engineering and Design 250 (2012) 42–52
for calculating the thermal gradient is proposed to account for the liner, as shown below. As the effect of the liner on the structural response is taken into account in the thermal gradient, the proposed expression provides an accurate estimate of thermal curvature without separately considering the liner.
t/2 T =
Appendix A. Derivations of equivalent linear temperatures for rectangular sections A.1. Rectangular concrete section The uniform temperature change Tm is related to the area under the temperature profiles. The area under TNL is
QNL = Qconc + Qliner =
t/2
TNL dy
Tm is defined as the mean temperature and, therefore, the area under the mean temperature Tm is
=
T · t 2 T + Tm + 12 2
Therefore, the mean temperature Tm can be presented as −(t/2)
Tm =
TNL dy
(A1)
t
As such, the uniform temperature change can be expressed as Tm − Tb , where Tb is the base temperature. It is important to note that the formula calculating the mean temperature in Fig. RE.1 of ACI 349-06 is corrected to Eq. (A1) in order to be consistent with the definition of the uniform temperature change, which is Tm − Tb . The equivalent liner temperature gradient T is derived based on the moment of area under the temperature profiles. The moment of area under TNL is
t/2
QNL =
t 2
Aliner b
(n · tliner )
= T
t/2
TNL · y dy + TNL,liner ·
T · t 2 A + TL,liner · yliner × n · liner 12 b
AL = Tm · t
−(t/2)
TNL · y dy + TNL,liner · yliner × n ·
where TNL is the nonlinearly distributed temperature; TNL,liner is the temperature of a liner plate. The liner temperature is equal to the temperature at the contacting concrete surface for a thin steel liner; y is the distance from centroid of concrete section; n is the ratio of elastic moduli = Es /Ec in which Ec is the elastic modulus of concrete and Es is the elastic modulus of steel; Alinear /b is the area of liner per unit width of a concrete member, which is equal to the thickness of liner plate, tliner ; and yliner is the distance from centroid of liner plate to centroid of concrete section. For thin liners, the liner thickness can be ignored, resulting in t/2 for rectangular sections. The moment of area under TL with liner plate is QL =
−(t/2)
t/2
t/2
−(t/2)
(t 2 /12) + (t/4)(n · tliner )
ANL =
With the liner plate, the moment of area under TNL is
QNL =
T · y dy + (TNL,liner − Tm )(t/2)(n · tliner ) −(t/2) NL
51
t
2
(n · tliner )
t t2 + (n · Aliner ) + Tm 12 4
t 2
(n · tliner )
Therefore, the thermal gradient for a rectangular concrete section with steel liner attached is
t/2
T =
−(t/2)
TNL · y dy + (TNL,liner − TTm )(t/2)(n · tliner ) (t 2 /12) + (t/4)(n · tliner )
(A3)
In a similar way, the following expression can be used to calculate the thermal gradient for a rectangular concrete section that contains rebars:
t/2
T =
−(t/2)
TNL · y dy + (t 2 /12) +
ns
i=1
(TNLs,i − Tm )ys,i (n − 1)(As,i /b)
i=1
2 /t)(n − 1)(A /b) (ys,i s,i
ns
(A4)
where ns is the number of rebar layers; TNLs,i is the temperature at rebar layer, i; ys,i is the distance from centroid of concrete section to centroid of rebar layer, i; and As,i is the area of rebar layer, i.
TNL · y dy −(t/2)
Similarly, the moment of area under TL can be calculated QL =
1 2
T t 2 2
2
×
3
·t
=
References
T · t 2 12
By equating above two expressions, the thermal gradient T is T =
12 t2
t/2
TNL · y dy
(A2)
−(t/2)
A.2. Rectangular concrete section with steel liner plate When a steel liner plate is placed on the face of concrete, which is common for nuclear concrete containments in the US, the liner needs to be considered for calculating thermal gradient. Since the liner has negligible impact on the calculated mean temperature (see Fig. 11), the effect of the liner on mean temperature is not considered here. The overall temperature distribution across a concrete section remains essentially the same when the interface between the liner and the concrete is assumed to a bond condition due to high thermal conductivity of steel and thin thickness of the liner plate (see Figs. 6 and 8).
ACI 349, 2006. Code Requirements for Nuclear Safety-Related Concrete Structures (ACI 349-06) and Commentary, ACI 349-06. American Concrete Institute. ACI 349.1R, 2007. Reinforced Concrete Design for Thermal Effects on Nuclear Power Plant Structures, ACI 349.1R-07. American Concrete Institute. ANSYS Inc., 2011. ANSYS Release 13.0. ASME, 2011. Code for Concrete Containments, 2010 ASME Boiler and Pressure Vessel Code, Section III, Division 2. The American Society of Mechanical Engineers, New York, NY. BN-TOP-3, 1983. Performance and Sizing of Dry Pressure Containments, Revision 4. Bechtel Power Corporation. Drysdale, D., 1999. An Introduction to Fire Dynamics, 2nd ed. John Wiley & Sons Ltd. Edward Pope, J., 1997. Rules of Thumb for Mechanical Engineers – A Manual of Quick, Accurate Solutions to Everyday Mechanical Engineering Problems. Elsevier. Elgaaly, M., 1988. Thermal gradients in beams, walls, and slabs. ACI Struct. J. 85, 76–81. Gou, P.F., Nakamura, M., Marayama, K., Ehlert, G., Ogawa, S., 1997. Analysis of reinforced concrete shell elements subjected to membrane forces, bending moments and transverse shear forces encountered in RCCV. In: Proceedings of 14th Conference on Structural Mechanics in Reactor Technology (SMiRT), H06/4, pp. 227–234. Gurfinkel, G., 1971. Thermal effects in walls of nuclear containments – elastic and inelastic behavior. In: Proceedings of 1st Conference on Structural Mechanics in Reactor Technology (SMiRT), J 3/7, pp. 277–296.
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Kohli, T.D., Gürbüz, O., 1975. Optimum design of reinforced concrete for nuclear containments, including thermal effects. In: Proceedings of the Second ASCE Specialty Conference on Structural Design of Nuclear Plant Facilities, New Orleans, LA, pp. 1292–1319. Nakamura, M., Maruyama, K., 1997. Stress evaluation of RC shell elements based on three-dimensional equilibrium. In: Proceedings of 14th Conference on Structural Mechanics in Reactor Technology (SMiRT), H06/3, pp. 219–226. Vecchio, F.J., 1987. Nonlinear analysis of reinforced concrete frames subjected to thermal and mechanical loads. ACI Struct. J. 84, 492–501.
Sungjin Bae is a Senior Structural Engineer at Bechtel Corporation, Frederick, MD. He received his BS and MS from Hanyang University, Seoul, Korea, and his PhD from the University of Texas at Austin, Austin, TX. He is a member of ACI Committee 209, Creep and Shrinkage of Concrete; 349, Concrete Nuclear Structures; Working Group on Design (SC-3C) of Joint ACI-ASME Committee 359, Nuclear Concrete Containments; and Joint ACI-ASCE Committee 441, Reinforced Concrete Columns. His research interests include the behavior and performance-based design of concrete columns and design of nuclear structures, foundations for dynamic equipment, and chimneys.