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J. Cent. South Univ. (2014) 21: 1227−1241 DOI: 10.1007/s1177101420576
Temperature gradients in concrete box girder bridge under effect of cold wave GU Bin(顾斌) 1 , CHEN Zhijian(陈志坚) 2 , CHEN Xindi(陈欣迪) 3 1. College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China; 2. School of Earth Sciences and Engineering, Hohai University, Nanjing 210098, China; 3. College of Harbour, Costal and Offshore Engineering, Hohai University, Nanjing 210098, China © Central South University Press and SpringerVerlag Berlin Heidelberg 2014 Abstract: The temperature distributions of a prestressed concrete box girder bridge under the effect of cold wave processes were analyzed. The distributions were found different from those under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. A temperature predicting model that can accurately predict temperatures over the cross section of the concrete box girder was developed. On the basis of the analytical model, a twodimensional temperature gradient model was proposed and a parametric study that considered meteorological factors was performed. The results of sensitivity analysis show that the cold wave with shorter duration and more severe temperature drop may cause more unfavorable influences on the concrete box girder bridge. Finally, the unrestrained linear curvatures, selfequilibrating stresses and bending stresses when considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated against proposed model and several current specifications. The results show that the cold wave may cause more unfavorable effect on the concrete box girder bridge, especially on the large concrete box girder bridge. Therefore, it is necessary to consider the thermal effect caused by cold wave during the design stage. Key words: concrete box girder; temperature field; temperature gradient; cold wave
1 Introduction Concrete bridge structures are subjected to thermal effects due to the interaction with the environment. The interaction with daily climate changes will cause nonlinear temperature distributions in the structure while the interaction with seasonal variations in ambient air temperature will cause average temperature changes. Temperature variation is a major cause of movements in a concrete bridge. If these movements are restrained, large forces may develop and these forces sometimes cause damage. In the 1960s, ZUK [1] performed extensive investigations on the thermal behavior of highway bridges and concluded that the temperature distribution was affected by air temperature, wind speed, humidity and intensity of solar radiation. Afterwards, researchers and bridge engineers were aware of nonlinear temperature distribution developed across a bridge section and a lot of subsequent researches [2−5] were
made based on heat transfer theory and field observation data worldwide. Early studies of temperature effects on bridges generally focused on the temperature distributions or stresses for different forms of bridges by using field data or numerical simulation method. DILGER et al [2] studied temperature distribution and temperature stress of a composite bridge based on field test data, and subsequently, ELBADRY and GHAIL [3] developed a finite model to predict the temperature distribution over bridge cross sections from data related to the geometry, location, orientation, material and climate conditions. KENNEDY and SOLIMAN [4] proposed a simple but realistic vertical temperature gradient in concretesteel composite bridges based on a synthesis of several theoretical and experimental studies on prototype bridges. MIRAMBELL and AGUADO [5] carried out several parametric studies to analyze the influence of the crosssection geometry on the thermal response and stress distributions in concrete box girder bridges. In recent years, most of investigations mainly focus
Foundation item: Project(08Y60) supported by the Traffic Science’s Research Planning of Jiangsu Province, China Received date: 2012−08−20; Accepted date: 2013−04−19 Corresponding author: GU Bin, PhD Candidate; Tel: +86−15951986297; Email: [email protected]
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on the extreme values of temperature gradients in the bridge or temperature gradients across the non conventional sections. For example, SILVEIRA et al [6], TONG et al [7], LEI et al [8] and LARSSON and THELANDERSSON [9] all obtained the extreme temperature gradients through measured temperature data or meteorological history data. LI et al [10] proposed a vertical temperature gradient for deeply prestressed concrete boxgirder sections, LEE and KALKAN [11−12] developed both the vertical and horizontal temperature gradients for the BT63 concrete girder segment, and SONG et al [13] investigated the characteristics of temperature gradients in a high performance concrete box girder with unconventional cross section which has short overhanging slab and large bottom flanges. The above studies mainly concerned the temperature effects caused by solar radiation or nighttime radiation cooling, but few on thermal effects caused by cold wave. Different from the solar radiation or nighttime radiation cooling, the cold wave would cause lower temperature distribution in the external zone and higher temperature distribution in the internal zone. Therefore, the concrete box girder bridge would be affected by the effect of negative temperature difference in the deck slab, web and bottom slab, which could lead to significant bending stresses when considering the frame action of the cross section [5, 14]. However, most codes only provide the provisions of gradient effects through the depth of the cross section. In this work, the characteristics of the temperature gradients in the concrete box girder under the effect of cold wave are investigated through field monitoring and numerical simulation method. A twodimensional temperature gradient model was developed and a parametric study considering the meteorological factors was performed. The unrestrained linear curvatures, selfequilibrating stresses and bending stresses considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated with proposed model and several current specifications.
2 Measurements of temperatures 2.1 Description of bridge and monitoring system The Sutong Bridge with a main span of 1088 m across the Yangtze River links the city of Suzhou and Nantong (31.7°N and 121°E). The auxiliary shipping channel bridge of Sutong Bridge is the prestressed concrete boxgirder bridge with a continuous span of
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140+268+140=548 m (Fig. 1). The bridge spans in the north−south direction. There are two separate post tensioned bridges, each with a single cell, one carries the northbound traffic and the other carries the south bound traffic. The deck of the girders is 16.4 m wide, and the depth varies from 4.5 m at midspan to 15.0 m at the piers. There is a 110 mm asphalt concrete layer on the top of the boxgirder section.
Fig. 1 Auxiliary shipping channel bridge of Sutong Bridge (Unit: m)
In order to obtain crucial field data required for safety and serviceability evaluations of the bridge, and to achieve a better understanding on its behavior, a structural health monitoring system was initiated during construction of the bridge. The system consists of a large number of different types of sensors installed at different locations of the bridge, including temperature sensors, relative humidity/temperature probes, strain gauges, anemometers, accelerometers, displacement transducers and inclinometers. The interests of this work are the temperatures recorded on the girder sections along the bridge and the meteorological data recorded at the bridge site. The locations of girder sections and layout of temperature sensors are shown in Figs. 1 and 2. The relative humidity/temperature probes were installed both inside and outside the concrete box girder at midspan and the anemometers were installed above the top surface of the bridge at midspan. All data have been collected at 30 min intervals since February 2008. 2.2 Variations of temperature distributions during period of cold wave Cold wave is a significant disaster weather phenomenon which often occurs in spring and winter with characteristics of large temperature drop range and larger influence areas. Chinese cold wave grades code [15] suggests that a cold wave process should be defined when the drop range of daily minimum temperature is greater than 8 °C during 24 h, or greater than 10 °C during 48 h, or greater than 12 °C during 72 h, and the minimum temperature is lower than 4 °C. According to the regulations of code, variations of temperature distributions of the concrete box girder under the effect of several cold wave processes were captured during operation period of the Sutong Bridge.
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Fig. 2 Layouts of temperature sensors at section T1 (a) and section T2 (b) (Unit: cm)
Figure 3 shows the climatic data of one cold wave process from December 3 to December 6, 2008. As shown, the temperature drop duration was approximately 60 h, the temperature drop range of daily minimum temperature was about 13.4 °C, and the average wind speed was 8.8 m/s during the period of temperature drop. The variations of vertical temperature distributions of sections T1 and T2 are presented in Fig. 4. The vertical temperature distribution for section T1 is based on sensors T107, T106, T105, T102, T117, T118 and T119, and that for Section T2 is based on sensors T208, T207, T206, T205, T202 and T219. It is clear from the graph that the variations and patterns of vertical temperature distributions of sections T1 and T2 are different from each other due to the thickness of their web and bottom slab. The thicker the web and bottom
Fig. 3 Climatic data during cold wave process: (a) Air temperature and air temperature inside concrete box girder; (b) Air relative humidity; (c) Wind speed
slab get and the further away the outer surface, the lower the temperature drop amplitudes are expected. The variations of horizontal temperature distributions in deck slab are shown in Fig. 5. The horizontal temperature distribution for section T1 is based on sensors T104, T107, T108, T111 and T112, and that for section T2 is based on sensors T204, T208, T209, T213 and T214. As shown, the temperature of
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cantilever slabs decreases most significantly, while the temperature of deck slab over the box cell decreases slowly. The measured maximum horizontal temperature difference can reach 7.9 °C. Figure 6 shows the variations of horizontal temperature distributions along the web. The horizontal temperature distributions for section T1 is based on sensors T101, T102, T103, T113, T114 and T115,
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and that for section T2 is based on sensors T201, T202, T203, T215, T216 and T217. It can be seen that the temperature drop amplitudes in the web are affected by the thickness of web and distance from the outer surface. The measured maximum horizontal temperature gradient of the web is 4.5 °C, but the practical value would be greater than 4.5 °C, because the lowest temperature should appear on the outer surface of web.
Fig. 4 Variations of vertical temperature gradients in web plate at section T1 (a) and section T2 (b)
Fig. 5 Variations of horizontal temperature distribution in deck at section T1 (a) and section T2 (b)
Fig. 6 Variations of horizontal temperature distributions in web at section T1 (a) and section T2 (b)
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Figure 7 demonstrates the variations of horizontal temperature distributions in the bottom slab. The horizontal temperature distribution for section T1 is based on sensors T116, T118 and T120 and that for section T2 is based on sensors T217, T218 and T219. It can be found that the horizontal temperature difference in the bottom slab is very small and can be neglected. Based on the above analysis, it can be concluded that temperature distribution across the section of concrete box girder under the effect of cold wave is different from that under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. The horizontal temperature gradients in the web and cantilever slab should be considered when analyzing the temperature effect of concrete box girder bridge under the effect of cold wave. Moreover, the temperature is measured at limited points, and it is difficult and insufficient to investigate the bridge performance thoroughly by using the temperature data at limited points. Therefore, a temperature predicting model is needed to study the temperature gradients across the section of concrete box girder and its influencing factors under the effect of cold wave.
3 Prediction of temperature in bridge 3.1 Heat transfer For a bridge subjected to solar radiation, it can be assumed that thermal variation in the direction of the longitudinal axis is normally not significant [2], therefore, the temperature field of concrete box girder cross section under the cold wave at any time t can also be expressed by a twodimensional heat flow equation as
rc
æ ¶ 2T ¶ 2 T ö ¶T = k çç 2 + 2 ÷÷ ¶t ¶y ø è ¶x
(1)
where k is the isotropic thermal conductivity coefficient of concrete, W/(m∙°C); ρ is the density of concrete, kg/m 3 ; c is the specific heat of concrete, J/(kg∙°C).
The boundary conditions associated with Eq. (1) can be expressed by æ ¶T ¶T ö kç nx + n y ÷ + q = 0 ¶ x ¶y è ø
(2)
where nx and ny are direction cosines of unit outward vector normal to the boundary surfaces; q is the time rate of heat transferred between the surface and environment per unit area, W/m 2 . 3.2 Analysis of boundary conditions To calculate temperature field precisely in a finite element analysis, the natural boundary conditions must be carefully studied. Following factors (see Fig. 8) are considered: direct and diffuse solar radiation, cloud cover, ambient air temperature, wind speed, atmospheric thermal radiation, and thermal radiation of the bridge surfaces, reflection of the global solar radiation, thermal radiation of the ground surface and the reflection of the atmospheric thermal radiations. Thus, the heat balance in the bridge surface can be described as qS + qR + qB + qG = qCa + qRa
(3)
where qS is the time rate of heat absorbed by the bridge due to shortwave solar radiation, W/m 2 ; qR is the time rate of heat absorbed by the bridge due to reflection of global solar radiation, W/m 2 ; qB is the time rate of heat absorbed by the bridge due to atmospheric thermal radiation, W/m 2 ; qG is the time rate of heat absorbed by the bridge due to thermal radiation of the ground surface and reflection of the atmospheric thermal radiation, W/m 2 ; qCa is the time rate of heat transferred between air and the surface, W/m 2 , and qRa is the radiation emitted from the bridge, W/m 2 . Figure 9 shows the spatial position of the concrete box girder and the sun, where β is solar altitude, φ is the azimuth angle measured from the south and positive toward the east, η is the tilt angle between the horizontal and surface, θ is the solar incidence angle, and ψ is the
Fig. 7 Variations of horizontal temperature distributions in bottom slab at section T1 (a) and section T2 (b)
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Fig. 8 Natural boundary conditions of concrete box girder
Fig. 9 Spatial position of bridge and sun
surface azimuth angle between south and the normal to the surface and positive toward the east. Angles of β, φ and θ or the length of shadow Ls can be calculated through latitude and longitude of the bridge based on geometric theory. The rate of heat absorbed by the bridge due to short wave solar radiation is obtained by qS = a éë I B cos q + I D sin 2 (h / 2) ùû
(4)
where IB is hourly direct solar radiation normal to the beam, W/m 2 , ID is hourly diffuse solar radiation on horizontal surface, W/m 2 , and α is the solar radiation absorptivity of the surface material. Different from the clear day condition, the direct and diffuse solar radiation may also be influenced by
cloud cover fraction during the period of cold wave. Consequently, the temperature prediction model uses the product of cloud cover fraction and clear sky global radiation to calculate the direct and diffuse solar radiation. The ratio of global radiation for a given cloud cover fraction to clear sky global radiation has been shown to be independent of the solar incidence angle [16]: I G IGc = 1 - 0.75 N 3.4
(5)
where IG is hourly global irradiance on horizontal surface, W/m 2 ; IGc is hourly clear sky global irradiance with details of calculation shown in Ref. [16], W/m 2 ; N is cloud cover fraction. To estimate the total radiation on the surface of
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other orientations, the direct solar radiation and diffuse solar radiation must be calculated separately. de MIGUEL et al [17] correlated IG/ID with hourly clearness index kt, and the equations for the correlation can be expressed as
Tc is the surface temperature of bridge, °C, and the heat convection coefficient can be expressed as [22]
ì0.995 - 0.081k , k t £ 0.21 ï I D ï0.724 + 2.738k t - 8.32 kt2 + 4.967 k t 3 , =í I G ï 0.21 < k t £ 0.76 ï0.18, k > 0.76 t î
where v is the wind speed, m/s. The radiation emitted from the bridge can be calculated as
(6)
qR = az e cos 2 (h / 2)( I B sin b + I D )
(7)
where ζe is the reflection coefficient of the ground surface and the reflection coefficient of water can be fitted as [18]
x e = 3.5sin 4 b - 9.2 sin 3 b + 9.3sin 2 b - 4.6sin b + 1 (8) The rate of heat absorbed by the bridge due to atmospheric thermal radiation can be calculated as qB = e 0e a C0 (Ta + 273) 4 sin 2 (h 2)
(9)
where ε0 is the emissivity of the bridge; εa is the emissivity of the atmosphere; C0 is the Stefan Boltzmann constant 5.67×10 −8 , W/(m 2 ∙°C 4 ), and Ta is the temperature of ambient air, °C. The emissivity of the atmosphere can be expressed as [19]
(10) where ea is the atmospheric vapor pressure, mPa; Nm is the number of months. The details of calculation are shown in Refs. [20−21]. The rate of heat absorbed by the bridge due to thermal radiation of the ground surface and reflection of the atmospheric thermal radiation can be expressed as qG = e 0 [e e C0 (Te + 273) 4 + e a (1 - e e )C0 (Ta + 273) 4 ] × cos 2 (h / 2)
(11)
where Te is the temperature of the ground surface, °C; εe is the emissivity of the ground surface and the emissivity for water is 0.96 according to Ref. [18]. The rate of heat transferred between the air and the surface is expressed by Newton’s Law of cooling as qCa = hc (Tc - Ta )
(14)
3.3 Validation results The heat transfer model was validated using the field measurements taken from December 3rd to December 6th, 2008, when a strong cold wave occurred. The climatic information is shown in Fig. 4 and the mean cloud cover fractions from December 3 to December 6, 2008, are 0.6, 0.9, 1 and 0.5 respectively according to the meteorological record of local meteorological department. The physical properties of the materials are listed in Table 1. Table 1 Thermal and elastic material properties Material
Density, ρ/(kg∙m −3 )
Specific heat capacity, c/(J∙kg −1 ∙ o C −1 )
Concrete
2500
10.10
2100
8.86
Asphalt concrete
Thermal Absorptivity, Emissivity, conductivity, α ε0 −1 o −1 k/(W∙m ∙ C ) 1.17 0.90 0.95
Material Concrete Asphalt concrete
0.98
0.65
0.88
1 7
é ( N m + 2)π ù ü æ e a ö ú ýç T ÷ 6 ë û þ è a ø
e a =N + (1 - N ) í1.22 + 0.06 sin ê î
(13)
qRa = e 0 C0Tc 4
where an hourly clearness index, kt, can be defined as kt= IG/I0. The rate of heat absorbed by the bridge due to reflection of global solar radiation can be calculated as
ì
ìï 4v + 5.6, v £ 5 m/s h c = í 0.78 ïî 7.15 , v > 5 m/s
(12)
where hc is the heat convection coefficient, W/(m 2 ∙°C);
The commercial finite element package ANSYS was used to calculate temperature field of the concrete box girder. The heat transfer analysis was performed hourly for seven days. The initial temperature was assumed to be uniform with a value equal to the air temperature at the calculation starting time. The analysis of first three days was conducted to eliminate the effect of the assumed initial conditions. The concrete temperatures predicted by the model described were then compared with the measurements. Figures 10 and 11 show the comparisons between the predicted temperatures and the measurements at temperature sensors 1, 4, 7 and 19 of section T1, and at temperature sensors 1, 4, 8 and 19 of section T2. It can be seen that the results are in fairly good agreement. The average absolute error is calculated as [23] E AA =
å (| Tp - T 0 |) n
(15)
where EAA is the average absolute error, Tp is the predicted temperature, °C, T0 is the measured temperature,
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Fig. 10 Comparison of predicted and measured temperatures at section T1
Fig. 11 Comparison of predicted and measured temperatures at section T2
and n is the number of data used in the analysis. The average absolute errors for all measuring points range from 0.2 to 1.7 °C.
4 Analysis of temperature gradients model under effect of cold wave The temperature distributions across sections T1 and T2 at the lowest ambient air temperature are presented in Figs. 12 and 13. As expected, temperature distributions across sections T1 and T2 are both lower in the external zone and higher in the internal zone. Though both the temperature distributions are nearly symmetrical, the horizontal temperature differences in the cantilever slab and web are so large that it leads to significant bending stresses when considering the frame action of the cross section. Therefore, temperature distributions across section of concrete box girder under the effect of cold wave should be defined by twodimensional model.
Fig. 12 Temperature distribution across section T1
Fig. 13 Temperature distribution across section T2
4.1 Vertical temperature gradients Figure 14 presents the calculated vertical temperature gradients along the depth of sections T1 and
T2 when the ambient air temperature was the lowest. It can be found that the profiles of vertical gradients in the
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upper part of deck slab of sections T1 and T2 exhibit similar shapes under the same ambient conditions, but the temperatures in the webs and bottom slabs are different due to the thickness difference. For the vertical temperature gradients along the center line of web and bottom slab (A−A line), the temperatures increase dramatically in the deck slab, reaching the maximum at a depth of about 0.5 m, and then decrease gradually in the haunch and upper part of web. In the remainder of web, the temperatures keep nearly constant, while in the bottom slabs, the temperatures drop quickly. The curves of temperature gradients in the deck slab show a parabolic trend. The vertical temperature gradients along the depth of cantilever slab (B−B line located 1.5 m from the edge and C−C line is 4.1 m from the edge) of sections T1 and T2 are nearly the same, with the highest temperatures appearing near the middle of slabs.
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Fig. 15 Proposed vertical temperature gradients along depth: (a) Concrete box girder (not including cantilever slab); (b) Cantilever slab (Unit: m)
0.5 m, and can be expressed as Td ( y ) = Td [(0.5 - y ) / 0.5] a d
(16)
where y is the distance from the top surface, m; αd is the power of function. 2) Tw is the temperature in the lower part of web. The temperature along the haunch and upper part of web is defined as a linear variation from 0 at depth of 0.5 m to Tw at depth of 1.5 m. 3) The temperature along the bottom slab is defined as a linear variation from Tb at outer surface to Tbt at 0.6 m, and a constant Tbt to the inner surface of bottom slab. If the thickness of bottom slab db is less than 0.6 m, the temperature along the bottom slab is defined as a linear variation from Tb at outer surface to Tbt at inner surface of the bottom slab. 4) For the temperature gradients along the depth of cantilever slab, a piecewise linear temperature gradient distribution model is proposed, which is defined as a maximum temperature, Tcs(x), at middle depth and two equal minimum temperatures, Tcc(x), at the top surface and lower surface.
Fig. 14 Calculated vertical temperature gradients along A−A line (a) and B−B line and C−C line (b) of section T1 and section T2
Based on the discussions of the calculated vertical temperature gradients of different section size, a vertical temperature gradient model is proposed, as shown in Fig. 15. In the model, the notations are defined as follows: 1) Td(y) is the continuous temperature distribution increasing from Td at the top surface to 0 °C at depth of
4.2 Horizontal temperature gradients Based on the measurements, it can be assumed that horizontal temperature gradients in the bottom slab and deck slab over the box cell can be neglected. Figure 16(a) shows horizontal temperature gradients along the thickness of web (D−D line), when the ambient air temperature is the lowest. As shown, the horizontal temperature gradients along the web can reach −10 °C, and is affected by the thickness of web. As the web gets thicker, the temperature difference will be expected to be greater. The horizontal temperature gradients along top surface (E−E line) and the center line (F−F line) of the cantilever slab are presented in Fig. 16(b). It can be seen that the horizontal temperature gradients along the
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cantilever slab are rarely influenced by the thickness of web and bottom slab, and the temperature difference along E−E line of the cantilever slab is greater than that along F−F line.
gradients into account, a twodimensional temperature gradient model can be obtained, in which the concrete box girder crosssection is subdivided into four regions (Fig. 18).
Fig. 17 Proposed horizontal temperature gradients along thickness of web (a), top surface of cantilever slab (b) and center line of cantilever slab (c)
Fig. 18 Region division of concrete box girder crosssection
The temperature gradient of the four regions can be defined as follows: 1) The temperature gradient in region A can be expressed as Fig. 16 Calculated horizontal temperature gradients along D−D line (a) and E−E line and F−F line (b) of sections T1 and T2
Based on the discussions of the calculated horizontal temperature gradients in the web and cantilever slab, the horizontal temperature gradient models for the web and cantilever slab are proposed, as shown in Fig. 17. In the model, the notations are defined as follows: 1) Twd is the temperature at the outer surface of the web. Twd(xw) is the continuous temperature distribution increasing from Twd at the outer surface to 0 °C at the inner surface, and can be expressed as Twd ( xw ) = Twd [( d w - xw ) / d w ] a w
(17)
where dw is the thickness of web, m, αw is the power of function and xw is the distance from the outer surface, m. 2) Tcs and Tcc are the maximum negative horizontal temperature differences at the top surface and center line of cantilever slab, respectively. 4.3 Twodimensional temperature gradient model Taking both vertical and horizontal temperature
2 y æ d ( x ) ö TA ( x, y ) = TA ç x , c ÷ - 1 × 2 ø d c ( x) è é æ d c ( x ) ö ù - TA ( x, 0) ú êTA ç x, ÷ 2 ø ë è û
(18)
where TA(x, 0) and TA(x, 0.5dc(x)) are the temperatures at the top surface and middle depth of the cantilever slab, respectively, with distance x from the edge, °C; dc(x) is the thickness of the cantilever slab, m; y is the distance from the top surface, m. TA(x, 0) and TA(x, 0.5dc(x)) can be calculated as T TA ( x, 0) = Tcs + Td - cs x 4.45 T cc æ d ( x) ö TA ç x, c x ÷ = Tcc 2 4.45 è ø
(19) (20)
2) The temperature gradients in region B can be expressed as a ïìT [(0.5 - y ) / 0.5] d , y £ 0.5 m TB ( y ) = í d ïî Tw ( y - 0.5), 0.5 m < y £ 1 m
(21)
3) The temperature gradients in region C can be expressed as
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a ì é ( d - x ) ù w a w ïTw ( y - 0.5) + Tww ê w ú - T ww 0.5 , ï ë d w û ï y £ 1.5 m ï TC ( x, y ) = í a ï é ( d w - x ) ù w a w T + T ï w ú - T ww 0.5 , ww ê d w ë û ï ï y > 1.5 m î
(22) where x is the distance from the outer surface, m. 4) The temperature gradients in region D can be expressed as ìTbt , y > 0.6 m ï TD ( y ) = í (23) y ïTb + (Tb - Tbt ) min(0.6, d ) , y £ 0.6 m î b where y is the distance from the outer surface of bottom slab, m.
5 Influences of meteorological parameters Among the input meteorological parameters, temperature drop range, wind speed and cold wave duration can vary from different regions and different
Fig. 19 Sensitivity curves for Td (a) and Tw (b)
Fig. 20 Sensitivity curves for Twd (a) and Tb −Tbt (b)
time. As a result, there are always uncertainties in these parameters. Therefore, it is worthwhile to carry out sensitivity studies on the effects of these parameters on the temperature gradients. Taking section T2 for example, using the above meteorological data, for each analysis, only the parameter concerned varies with increase or decrease of 10% increment up to 50%, while the other parameters keep invariant. The sensitivity curves for the temperatures at the top surface of deck slab and in the lower part of web are shown in Fig. 19. It can be seen that they are hardly affected by other parameters but the temperature drop range. Moreover, an increase of 50% in temperature drop range will lead to the decrease of temperature at the top surface of deck slab or in the lower part of web by more than 35%. Figure 20 shows the sensitivity curves for the temperature differences in the web and bottom slab. As shown, the temperature differences in the web and bottom slab are rarely influenced by the wind speed, but greatly affected by the temperature drop range and cold wave duration. In the present model, the ambient temperature is applied to the outer surfaces of the web and bottom slab, and an increase in the temperature drop
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range of cold wave promotes heat loss from these surfaces to the surrounding and lowers the surface temperature most. Therefore, an increase in temperature drop range would lead to an increase in the temperature difference in the web and bottom slab. The drop of inner surface temperature lags behind that of the outer due to the poor conductivity of the concrete. As a result, the shorter the cold wave duration is, the smaller the drop of the inner surface temperature and the larger the temperature difference in the web and bottom slab are expected. Figure 21 shows the sensitivity curves for the maximum horizontal negative temperature difference, Tcc, at center line of cantilever slab. It can be seen that the maximum negative horizontal temperature difference at center line of cantilever slab is mostly affected by the temperature drop range. Generally, an increase of 50% in temperature drop range leads to the increase of the maximum negative horizontal temperature difference by about 40%.
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for highway bridges and culverts JTG D60—2004, the United States Highway Bridge Design Code AASHTO (2007) and British code BS 5400 are selected, as shown in Fig. 22. The parameter values of the proposed temperature gradient model are listed in Table 2.
Fig. 22 Vertical temperature gradient models (Unit: mm) Table 2 Parameters of proposed model Section Td/°C αd Tw/°C Tbt/°C Tb/°C Twd/°C αw Tcs/°C Tcc/°C
Fig. 21 Sensitivity curves for Tcc
From the results just presented, it can be concluded that the cold wave with shorter duration and more severe temperature drop effect may cause more unfavorable influences to the concrete box girder bridge.
T1
7
2
−1
T2
7
2 −4.6
To investigate the difference of temperature effect obtained from the proposed temperature gradient model and current code provisions, the unrestrained linear curvatures, selfequilibrating stresses and bending stresses considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated by proposed model and several current specifications. Three codes including the China general design specification
−9
−4 −7.7 −6.5 1.6 −4
−9
Curvatures calculated by the selected three current code provisions and proposed temperature gradient model are listed in Table 3. Comparison shows that the selected three current code provisions all underestimate the curvature at section T1 and this will lead to an underestimation of the temperature induced stresses. The curvature at section T2 calculated using the proposed temperature gradient model has opposite sign to all the others. This will increase the complexity of the temperature effect on the bridge. Table 3 Thermal curvatures calculated by proposed model and selected three current code provisions Model
6 Comparison of proposed temperature gradient model with different codes
1.1 −8.4 −10 3.1 −4
Thermal curvature/10 −6 m Section T1
Section T2
Proposed model
−1.51
1.04
JTG D60—2004
−0.83
−5.45
AASHTO
−0.48
−3.18
BS 5400
−0.69
−1.52
Selfequilibrating stresses sensitive to the shape of the temperature distribution were calculated against the selected three current code provisions and proposed temperature gradient model. The elastic modulus of concrete is set to be 36 GPa. Figure 23 illustrates the selfequilibrating stress distributions along A−A line. It can be seen that the proposed and the selected three
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current code provisions all present high tensile stresses over the top few decimeters. Table 4 lists the selfequilibrating tensile stress at the top surface of the deck slab. As shown, the tensile stress derived from the temperature gradient in BS 5400 is the largest, while the tensile stress derived from the proposed temperature gradient is the smallest. However, the compressive stress in the haunch and upper part of web obtained from the three selected current code provisions is all smaller than that from the proposed model. For the bottom slab, the selfequilibrating tensile stress at outer surface of bottom slab derived from the proposed model and temperature gradient in BS 5400 is much larger than that from the temperature gradient in JTG D60—2004 and AASHTO. This is because the JTG D60—2004 and AASHTO neglect the temperature differences in the bottom slab, but the proposed model and BS 5400 do not.
Figures 24 and 25 illustrate the selfequilibrating stress distributions along the thickness of web (D−D line) and along the depth of cantilever slab (B−B line). It can be seen that the tensile stresses at the out surface of web and lower surface of cantilever slab obtained from three current code provisions are far smaller than those from the proposed model. The tensile stresses at the out surface of web and cantilever slab of section T1 obtained from the proposed model, which can reach approximately 3 MPa, are greater than those of section T2.
Fig. 24 Selfequilibrating stress distributions along thickness of web (D−D line) (a) and along depth of cantilever slab (B−B line) (b) of section T1
Fig. 23 Selfequilibrating stress distributions along A−A line of section T1 (a) and section T2 (b) Table 4 Tensile stresses at top surface of deck slab (MPa) Model Section T1 Section T2 Proposed model
1.60
1.04
JTG D602004
2.18
1.83
AASHTO
1.62
1.40
BS 5400
2.63
2.23
The bending stresses derived from the proposed temperature gradient model and current code provisions are listed in Table 5. The maximum bending tensile stresses at the out surface of web and top and bottom slab obtained from the three current code provisions are also far smaller than those from the proposed model. The stresses at the out surface of web and bottom slab of section T1 are larger than those of section T2 as well. According to the comparisons, it is found that the cold wave may cause more unfavorable effect to the concrete box girder bridge, especially to the large concrete box girders bridge. The unrestrained self
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Fig. 25 Selfequilibrating stress distributions along thickness of web (D−D line) (a) and along depth of cantilever slab (B−B line) (b) of section T2 Table 5 Maximum bending tensile stresses at out surface of web and top and bottom slab Model
Maximum bending tensile stresses at Section T1/MPa
Maximum bending tensile stresses at Section T2/MPa
Deck slab
Web
Bottom slab
Deck slab
Web
Bottom slab
Proposed model
2.43
3.07
2.29
2.18
1.36
0.85
JTG D602004
1.82
0.28
−0.04
1.79
0.49
−0.02
AASHTO
1.41
0.26
−0.02
1.37
0.33
−0.02
BS 5400
2.02
0.6
1.6
2
0.88
1.02
equilibrating tensile stresses or bending tensile stresses at the out surface of web and bottom obtained from the proposed temperature gradient model are so large. When they are added to the stresses due to the other dead loads and live loads, they may give rise to cracking in a concrete box bridge. Moreover, the analyzed data in this work is not obtained under the worst cold wave. Table 6 lists the maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years in six typical cities of China. As shown, the maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years in the northern cities of China can reach 20.4, 23.2 and 26.5 °C, respectively, which is far greater than 13.4 °C in this work. As the temperature drop range is the main influencing factors, the concrete box girder would crack Table 6 Maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years of six typical cities City
Maximum temperature drop/°C 24 h
48 h
72 h
Beijing
−15.7
−21.8
−22.6
Chengdu
−9.5
−11.9
−12.1
Guangzhou
−14.4
−16.2
−17.4
Harbin
−20.4
−23.2
−26.5
Lanzhou
−12.9
−14.3
−15.3
Shanghai
−12.8
−14.4
−17.2
more easily under the worst cold wave. Thus, the design engineers should take temperature effects caused by cold wave into account during the design stage.
7 Conclusions 1) Temperature distribution across the section of concrete box girder under the effect of cold wave is different from that under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. The horizontal temperature gradients in the web and cantilever slab should be considered when analyzing the temperature effect of concrete box girder bridge under the effect of cold wave. 2) The developed temperature predicting model can accurately predict temperature distributions over the cross section of the concrete box girder under the effect of cold wave. The proposed twodimensional temperature gradient model can accurately predict the temperature gradient over the cross section of the concrete box girder under the effect of cold wave. 3) The cold wave with shorter duration and more severe temperature drop may cause more unfavorable influences on the concrete box girder bridge. The cold wave may cause more unfavorable effects to the concrete box girder bridge than the current code provisions, especially to the large concrete box girders bridge. The unrestrained selfequilibrating tensile stresses and
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bending tensile stresses at the outer surface of web and bottom under the effect of cold wave are so large that when the stresses are added due to the other dead loads and live loads, they may give rise to cracking in a concrete box bridge. 4) In the northern cities of China, the concrete box girder bridge would suffer more unfavorable effects from the cold wave and crack more easily.
References ZUK W. Thermal and shrinkage stresses in composite beams [J]. ACI Journal, 1961, 58(5): 327−339. [2] DILGER W H, BEAUCHAMP J C, CHEUNG M S, BEAUCHAMP J C. Field measurements of muskwa river bridge [J]. Journal of the Structural Division, 1981, 107(11): 2147−2161. [3] ELBADRY M M, GHAIL A. Temperature variations in concrete bridges [J]. Journal of the Structural Engineering, 1983, 109(10): 2355−2374. [4] KENNEDY J B, SOLIMAN M H. Temperature distribution in composite bridges [J]. Journal of Structural Engineering, 1987, 113(3): 475−482. [5] MIRAMBELL E, AGUADO A. Temperature and stress distributions in concrete box girder bridges [J]. Journal of Structural Engineering, 1990, 116(9): 2388−2409. [6] SILVEIRA A P, BRANCO F A, CASTANHETA M. Statistical analysis of thermal actions for concrete bridge design [J]. Journal of the International Association for Bridge and Structural Engineering, 2000, 10(1): 33−38. [7] TONG M, THAM L, AU F. Extreme thermal loading on steel bridges in tropical region [J]. Journal of Bridge Engineering, 2002, 7(6): 357−366. [8] LEI Xiao, YE Jianshu, WANG Yi. Representative value of solar thermal difference effect on PC boxgirder [J]. Journal of Southeast University: Natural Science Edition, 2008, 38(6): 1105−1109. (in chinese) [9] LARSSON O, THELANDERSSON S. Estimating extreme values of thermal gradients in concrete structures [J]. Materials and Structure, 2011, 44(8): 1491−1500. [10] LI D N, MAES M A, DILGER W H. Thermal design criteria for deep
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prestressed concrete girders based on data from confederation Bridge [J]. Canadian Journal of Civil Engineering, 2004, 31(5): 813−825. LEE J H, KALKAN I. Analysis of thermal environmental effects on precast, prestressed concrete bridge girders: Temperature differentials and thermal deformations [J]. Advances in Structural Engineering, 2012, 15(3): 447−459. LEE J H. Investigation of extreme environmental conditions and design thermal gradients during construction for prestressed concrete bridge girders [J]. Journal of Bridge Engineering, 2012, 17(3): 547−556. SONG Zhiwen, XIAO Jianzhuang, SHEN Luming. On temperature gradients in highperformance concrete box girder under solar radiation [J]. Advances in Structural Engineering, 2012, 15(3): 399−415. SVEINSON T N. Temperature effects in concrete box girder bridges [D]. Calgary: University of Calgary, 2004. GB/T 21987—2008, Grades of cold wave [S]. (in Chinese) MUBEER T, GUI M S. Evaluation of sunshine and cloud cover based models for generating solar radiation data [J]. Energy Conversion and Management, 2000, 41(5): 461−482. de MIGUEL A, BILBAO J, AGUIAR R, KAMBEZIDIS H, NEGRD E. Diffuse solar irradiation model evaluation in the north Mediterranean belt area [J]. Solar Energy, 2001, 70(2): 143−153. KEHLBECK F. The influence of solar radiation on the bridge structure [M]. LIU Xingfa, transl. Beijing: China Railway Publishing House, 1981: 9−29. (in Chinese) CRAWFORD T M, DUCHON C E. An improved parameterization for estimating effective atmospheric emissivity for use in calculating daytime downwelling longwave radiation [J]. Journal of Applied Meteorology and Climatology, 1999, 38(4): 474−480. ALDUCHOV O A, ESKRIDGE R E. Improved magnus form approximation of saturation vapor pressure [J]. Journal of Applied Meteorology, 1996, 35(4): 601−609. ZHAI Panmao, ESKRIDGE R E. Atmospheric water vapor over China [J]. Journal of Climate, 1997, 10(10): 2643−2652. SAETTA A, SCOTTA R, VITALIANI R. Stress analysis of concrete structures subjected to variable thermal loads [J]. Journal of Structural Engineering, 1995, 121(3): 446−457. CARINO N J, TANK R C. Maturity function for concretes made with various cements and admixtures [J]. ACI Material Journal, 1992, 89(2): 188−196. (Edited by FANG Jinghua)
Temperature gradients in concrete box girder bridge under effect of cold wave GU Bin(顾斌) 1 , CHEN Zhijian(陈志坚) 2 , CHEN Xindi(陈欣迪) 3 1. College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China; 2. School of Earth Sciences and Engineering, Hohai University, Nanjing 210098, China; 3. College of Harbour, Costal and Offshore Engineering, Hohai University, Nanjing 210098, China © Central South University Press and SpringerVerlag Berlin Heidelberg 2014 Abstract: The temperature distributions of a prestressed concrete box girder bridge under the effect of cold wave processes were analyzed. The distributions were found different from those under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. A temperature predicting model that can accurately predict temperatures over the cross section of the concrete box girder was developed. On the basis of the analytical model, a twodimensional temperature gradient model was proposed and a parametric study that considered meteorological factors was performed. The results of sensitivity analysis show that the cold wave with shorter duration and more severe temperature drop may cause more unfavorable influences on the concrete box girder bridge. Finally, the unrestrained linear curvatures, selfequilibrating stresses and bending stresses when considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated against proposed model and several current specifications. The results show that the cold wave may cause more unfavorable effect on the concrete box girder bridge, especially on the large concrete box girder bridge. Therefore, it is necessary to consider the thermal effect caused by cold wave during the design stage. Key words: concrete box girder; temperature field; temperature gradient; cold wave
1 Introduction Concrete bridge structures are subjected to thermal effects due to the interaction with the environment. The interaction with daily climate changes will cause nonlinear temperature distributions in the structure while the interaction with seasonal variations in ambient air temperature will cause average temperature changes. Temperature variation is a major cause of movements in a concrete bridge. If these movements are restrained, large forces may develop and these forces sometimes cause damage. In the 1960s, ZUK [1] performed extensive investigations on the thermal behavior of highway bridges and concluded that the temperature distribution was affected by air temperature, wind speed, humidity and intensity of solar radiation. Afterwards, researchers and bridge engineers were aware of nonlinear temperature distribution developed across a bridge section and a lot of subsequent researches [2−5] were
made based on heat transfer theory and field observation data worldwide. Early studies of temperature effects on bridges generally focused on the temperature distributions or stresses for different forms of bridges by using field data or numerical simulation method. DILGER et al [2] studied temperature distribution and temperature stress of a composite bridge based on field test data, and subsequently, ELBADRY and GHAIL [3] developed a finite model to predict the temperature distribution over bridge cross sections from data related to the geometry, location, orientation, material and climate conditions. KENNEDY and SOLIMAN [4] proposed a simple but realistic vertical temperature gradient in concretesteel composite bridges based on a synthesis of several theoretical and experimental studies on prototype bridges. MIRAMBELL and AGUADO [5] carried out several parametric studies to analyze the influence of the crosssection geometry on the thermal response and stress distributions in concrete box girder bridges. In recent years, most of investigations mainly focus
Foundation item: Project(08Y60) supported by the Traffic Science’s Research Planning of Jiangsu Province, China Received date: 2012−08−20; Accepted date: 2013−04−19 Corresponding author: GU Bin, PhD Candidate; Tel: +86−15951986297; Email: [email protected]
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on the extreme values of temperature gradients in the bridge or temperature gradients across the non conventional sections. For example, SILVEIRA et al [6], TONG et al [7], LEI et al [8] and LARSSON and THELANDERSSON [9] all obtained the extreme temperature gradients through measured temperature data or meteorological history data. LI et al [10] proposed a vertical temperature gradient for deeply prestressed concrete boxgirder sections, LEE and KALKAN [11−12] developed both the vertical and horizontal temperature gradients for the BT63 concrete girder segment, and SONG et al [13] investigated the characteristics of temperature gradients in a high performance concrete box girder with unconventional cross section which has short overhanging slab and large bottom flanges. The above studies mainly concerned the temperature effects caused by solar radiation or nighttime radiation cooling, but few on thermal effects caused by cold wave. Different from the solar radiation or nighttime radiation cooling, the cold wave would cause lower temperature distribution in the external zone and higher temperature distribution in the internal zone. Therefore, the concrete box girder bridge would be affected by the effect of negative temperature difference in the deck slab, web and bottom slab, which could lead to significant bending stresses when considering the frame action of the cross section [5, 14]. However, most codes only provide the provisions of gradient effects through the depth of the cross section. In this work, the characteristics of the temperature gradients in the concrete box girder under the effect of cold wave are investigated through field monitoring and numerical simulation method. A twodimensional temperature gradient model was developed and a parametric study considering the meteorological factors was performed. The unrestrained linear curvatures, selfequilibrating stresses and bending stresses considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated with proposed model and several current specifications.
2 Measurements of temperatures 2.1 Description of bridge and monitoring system The Sutong Bridge with a main span of 1088 m across the Yangtze River links the city of Suzhou and Nantong (31.7°N and 121°E). The auxiliary shipping channel bridge of Sutong Bridge is the prestressed concrete boxgirder bridge with a continuous span of
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140+268+140=548 m (Fig. 1). The bridge spans in the north−south direction. There are two separate post tensioned bridges, each with a single cell, one carries the northbound traffic and the other carries the south bound traffic. The deck of the girders is 16.4 m wide, and the depth varies from 4.5 m at midspan to 15.0 m at the piers. There is a 110 mm asphalt concrete layer on the top of the boxgirder section.
Fig. 1 Auxiliary shipping channel bridge of Sutong Bridge (Unit: m)
In order to obtain crucial field data required for safety and serviceability evaluations of the bridge, and to achieve a better understanding on its behavior, a structural health monitoring system was initiated during construction of the bridge. The system consists of a large number of different types of sensors installed at different locations of the bridge, including temperature sensors, relative humidity/temperature probes, strain gauges, anemometers, accelerometers, displacement transducers and inclinometers. The interests of this work are the temperatures recorded on the girder sections along the bridge and the meteorological data recorded at the bridge site. The locations of girder sections and layout of temperature sensors are shown in Figs. 1 and 2. The relative humidity/temperature probes were installed both inside and outside the concrete box girder at midspan and the anemometers were installed above the top surface of the bridge at midspan. All data have been collected at 30 min intervals since February 2008. 2.2 Variations of temperature distributions during period of cold wave Cold wave is a significant disaster weather phenomenon which often occurs in spring and winter with characteristics of large temperature drop range and larger influence areas. Chinese cold wave grades code [15] suggests that a cold wave process should be defined when the drop range of daily minimum temperature is greater than 8 °C during 24 h, or greater than 10 °C during 48 h, or greater than 12 °C during 72 h, and the minimum temperature is lower than 4 °C. According to the regulations of code, variations of temperature distributions of the concrete box girder under the effect of several cold wave processes were captured during operation period of the Sutong Bridge.
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Fig. 2 Layouts of temperature sensors at section T1 (a) and section T2 (b) (Unit: cm)
Figure 3 shows the climatic data of one cold wave process from December 3 to December 6, 2008. As shown, the temperature drop duration was approximately 60 h, the temperature drop range of daily minimum temperature was about 13.4 °C, and the average wind speed was 8.8 m/s during the period of temperature drop. The variations of vertical temperature distributions of sections T1 and T2 are presented in Fig. 4. The vertical temperature distribution for section T1 is based on sensors T107, T106, T105, T102, T117, T118 and T119, and that for Section T2 is based on sensors T208, T207, T206, T205, T202 and T219. It is clear from the graph that the variations and patterns of vertical temperature distributions of sections T1 and T2 are different from each other due to the thickness of their web and bottom slab. The thicker the web and bottom
Fig. 3 Climatic data during cold wave process: (a) Air temperature and air temperature inside concrete box girder; (b) Air relative humidity; (c) Wind speed
slab get and the further away the outer surface, the lower the temperature drop amplitudes are expected. The variations of horizontal temperature distributions in deck slab are shown in Fig. 5. The horizontal temperature distribution for section T1 is based on sensors T104, T107, T108, T111 and T112, and that for section T2 is based on sensors T204, T208, T209, T213 and T214. As shown, the temperature of
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cantilever slabs decreases most significantly, while the temperature of deck slab over the box cell decreases slowly. The measured maximum horizontal temperature difference can reach 7.9 °C. Figure 6 shows the variations of horizontal temperature distributions along the web. The horizontal temperature distributions for section T1 is based on sensors T101, T102, T103, T113, T114 and T115,
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and that for section T2 is based on sensors T201, T202, T203, T215, T216 and T217. It can be seen that the temperature drop amplitudes in the web are affected by the thickness of web and distance from the outer surface. The measured maximum horizontal temperature gradient of the web is 4.5 °C, but the practical value would be greater than 4.5 °C, because the lowest temperature should appear on the outer surface of web.
Fig. 4 Variations of vertical temperature gradients in web plate at section T1 (a) and section T2 (b)
Fig. 5 Variations of horizontal temperature distribution in deck at section T1 (a) and section T2 (b)
Fig. 6 Variations of horizontal temperature distributions in web at section T1 (a) and section T2 (b)
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Figure 7 demonstrates the variations of horizontal temperature distributions in the bottom slab. The horizontal temperature distribution for section T1 is based on sensors T116, T118 and T120 and that for section T2 is based on sensors T217, T218 and T219. It can be found that the horizontal temperature difference in the bottom slab is very small and can be neglected. Based on the above analysis, it can be concluded that temperature distribution across the section of concrete box girder under the effect of cold wave is different from that under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. The horizontal temperature gradients in the web and cantilever slab should be considered when analyzing the temperature effect of concrete box girder bridge under the effect of cold wave. Moreover, the temperature is measured at limited points, and it is difficult and insufficient to investigate the bridge performance thoroughly by using the temperature data at limited points. Therefore, a temperature predicting model is needed to study the temperature gradients across the section of concrete box girder and its influencing factors under the effect of cold wave.
3 Prediction of temperature in bridge 3.1 Heat transfer For a bridge subjected to solar radiation, it can be assumed that thermal variation in the direction of the longitudinal axis is normally not significant [2], therefore, the temperature field of concrete box girder cross section under the cold wave at any time t can also be expressed by a twodimensional heat flow equation as
rc
æ ¶ 2T ¶ 2 T ö ¶T = k çç 2 + 2 ÷÷ ¶t ¶y ø è ¶x
(1)
where k is the isotropic thermal conductivity coefficient of concrete, W/(m∙°C); ρ is the density of concrete, kg/m 3 ; c is the specific heat of concrete, J/(kg∙°C).
The boundary conditions associated with Eq. (1) can be expressed by æ ¶T ¶T ö kç nx + n y ÷ + q = 0 ¶ x ¶y è ø
(2)
where nx and ny are direction cosines of unit outward vector normal to the boundary surfaces; q is the time rate of heat transferred between the surface and environment per unit area, W/m 2 . 3.2 Analysis of boundary conditions To calculate temperature field precisely in a finite element analysis, the natural boundary conditions must be carefully studied. Following factors (see Fig. 8) are considered: direct and diffuse solar radiation, cloud cover, ambient air temperature, wind speed, atmospheric thermal radiation, and thermal radiation of the bridge surfaces, reflection of the global solar radiation, thermal radiation of the ground surface and the reflection of the atmospheric thermal radiations. Thus, the heat balance in the bridge surface can be described as qS + qR + qB + qG = qCa + qRa
(3)
where qS is the time rate of heat absorbed by the bridge due to shortwave solar radiation, W/m 2 ; qR is the time rate of heat absorbed by the bridge due to reflection of global solar radiation, W/m 2 ; qB is the time rate of heat absorbed by the bridge due to atmospheric thermal radiation, W/m 2 ; qG is the time rate of heat absorbed by the bridge due to thermal radiation of the ground surface and reflection of the atmospheric thermal radiation, W/m 2 ; qCa is the time rate of heat transferred between air and the surface, W/m 2 , and qRa is the radiation emitted from the bridge, W/m 2 . Figure 9 shows the spatial position of the concrete box girder and the sun, where β is solar altitude, φ is the azimuth angle measured from the south and positive toward the east, η is the tilt angle between the horizontal and surface, θ is the solar incidence angle, and ψ is the
Fig. 7 Variations of horizontal temperature distributions in bottom slab at section T1 (a) and section T2 (b)
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Fig. 8 Natural boundary conditions of concrete box girder
Fig. 9 Spatial position of bridge and sun
surface azimuth angle between south and the normal to the surface and positive toward the east. Angles of β, φ and θ or the length of shadow Ls can be calculated through latitude and longitude of the bridge based on geometric theory. The rate of heat absorbed by the bridge due to short wave solar radiation is obtained by qS = a éë I B cos q + I D sin 2 (h / 2) ùû
(4)
where IB is hourly direct solar radiation normal to the beam, W/m 2 , ID is hourly diffuse solar radiation on horizontal surface, W/m 2 , and α is the solar radiation absorptivity of the surface material. Different from the clear day condition, the direct and diffuse solar radiation may also be influenced by
cloud cover fraction during the period of cold wave. Consequently, the temperature prediction model uses the product of cloud cover fraction and clear sky global radiation to calculate the direct and diffuse solar radiation. The ratio of global radiation for a given cloud cover fraction to clear sky global radiation has been shown to be independent of the solar incidence angle [16]: I G IGc = 1 - 0.75 N 3.4
(5)
where IG is hourly global irradiance on horizontal surface, W/m 2 ; IGc is hourly clear sky global irradiance with details of calculation shown in Ref. [16], W/m 2 ; N is cloud cover fraction. To estimate the total radiation on the surface of
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other orientations, the direct solar radiation and diffuse solar radiation must be calculated separately. de MIGUEL et al [17] correlated IG/ID with hourly clearness index kt, and the equations for the correlation can be expressed as
Tc is the surface temperature of bridge, °C, and the heat convection coefficient can be expressed as [22]
ì0.995 - 0.081k , k t £ 0.21 ï I D ï0.724 + 2.738k t - 8.32 kt2 + 4.967 k t 3 , =í I G ï 0.21 < k t £ 0.76 ï0.18, k > 0.76 t î
where v is the wind speed, m/s. The radiation emitted from the bridge can be calculated as
(6)
qR = az e cos 2 (h / 2)( I B sin b + I D )
(7)
where ζe is the reflection coefficient of the ground surface and the reflection coefficient of water can be fitted as [18]
x e = 3.5sin 4 b - 9.2 sin 3 b + 9.3sin 2 b - 4.6sin b + 1 (8) The rate of heat absorbed by the bridge due to atmospheric thermal radiation can be calculated as qB = e 0e a C0 (Ta + 273) 4 sin 2 (h 2)
(9)
where ε0 is the emissivity of the bridge; εa is the emissivity of the atmosphere; C0 is the Stefan Boltzmann constant 5.67×10 −8 , W/(m 2 ∙°C 4 ), and Ta is the temperature of ambient air, °C. The emissivity of the atmosphere can be expressed as [19]
(10) where ea is the atmospheric vapor pressure, mPa; Nm is the number of months. The details of calculation are shown in Refs. [20−21]. The rate of heat absorbed by the bridge due to thermal radiation of the ground surface and reflection of the atmospheric thermal radiation can be expressed as qG = e 0 [e e C0 (Te + 273) 4 + e a (1 - e e )C0 (Ta + 273) 4 ] × cos 2 (h / 2)
(11)
where Te is the temperature of the ground surface, °C; εe is the emissivity of the ground surface and the emissivity for water is 0.96 according to Ref. [18]. The rate of heat transferred between the air and the surface is expressed by Newton’s Law of cooling as qCa = hc (Tc - Ta )
(14)
3.3 Validation results The heat transfer model was validated using the field measurements taken from December 3rd to December 6th, 2008, when a strong cold wave occurred. The climatic information is shown in Fig. 4 and the mean cloud cover fractions from December 3 to December 6, 2008, are 0.6, 0.9, 1 and 0.5 respectively according to the meteorological record of local meteorological department. The physical properties of the materials are listed in Table 1. Table 1 Thermal and elastic material properties Material
Density, ρ/(kg∙m −3 )
Specific heat capacity, c/(J∙kg −1 ∙ o C −1 )
Concrete
2500
10.10
2100
8.86
Asphalt concrete
Thermal Absorptivity, Emissivity, conductivity, α ε0 −1 o −1 k/(W∙m ∙ C ) 1.17 0.90 0.95
Material Concrete Asphalt concrete
0.98
0.65
0.88
1 7
é ( N m + 2)π ù ü æ e a ö ú ýç T ÷ 6 ë û þ è a ø
e a =N + (1 - N ) í1.22 + 0.06 sin ê î
(13)
qRa = e 0 C0Tc 4
where an hourly clearness index, kt, can be defined as kt= IG/I0. The rate of heat absorbed by the bridge due to reflection of global solar radiation can be calculated as
ì
ìï 4v + 5.6, v £ 5 m/s h c = í 0.78 ïî 7.15 , v > 5 m/s
(12)
where hc is the heat convection coefficient, W/(m 2 ∙°C);
The commercial finite element package ANSYS was used to calculate temperature field of the concrete box girder. The heat transfer analysis was performed hourly for seven days. The initial temperature was assumed to be uniform with a value equal to the air temperature at the calculation starting time. The analysis of first three days was conducted to eliminate the effect of the assumed initial conditions. The concrete temperatures predicted by the model described were then compared with the measurements. Figures 10 and 11 show the comparisons between the predicted temperatures and the measurements at temperature sensors 1, 4, 7 and 19 of section T1, and at temperature sensors 1, 4, 8 and 19 of section T2. It can be seen that the results are in fairly good agreement. The average absolute error is calculated as [23] E AA =
å (| Tp - T 0 |) n
(15)
where EAA is the average absolute error, Tp is the predicted temperature, °C, T0 is the measured temperature,
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Fig. 10 Comparison of predicted and measured temperatures at section T1
Fig. 11 Comparison of predicted and measured temperatures at section T2
and n is the number of data used in the analysis. The average absolute errors for all measuring points range from 0.2 to 1.7 °C.
4 Analysis of temperature gradients model under effect of cold wave The temperature distributions across sections T1 and T2 at the lowest ambient air temperature are presented in Figs. 12 and 13. As expected, temperature distributions across sections T1 and T2 are both lower in the external zone and higher in the internal zone. Though both the temperature distributions are nearly symmetrical, the horizontal temperature differences in the cantilever slab and web are so large that it leads to significant bending stresses when considering the frame action of the cross section. Therefore, temperature distributions across section of concrete box girder under the effect of cold wave should be defined by twodimensional model.
Fig. 12 Temperature distribution across section T1
Fig. 13 Temperature distribution across section T2
4.1 Vertical temperature gradients Figure 14 presents the calculated vertical temperature gradients along the depth of sections T1 and
T2 when the ambient air temperature was the lowest. It can be found that the profiles of vertical gradients in the
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upper part of deck slab of sections T1 and T2 exhibit similar shapes under the same ambient conditions, but the temperatures in the webs and bottom slabs are different due to the thickness difference. For the vertical temperature gradients along the center line of web and bottom slab (A−A line), the temperatures increase dramatically in the deck slab, reaching the maximum at a depth of about 0.5 m, and then decrease gradually in the haunch and upper part of web. In the remainder of web, the temperatures keep nearly constant, while in the bottom slabs, the temperatures drop quickly. The curves of temperature gradients in the deck slab show a parabolic trend. The vertical temperature gradients along the depth of cantilever slab (B−B line located 1.5 m from the edge and C−C line is 4.1 m from the edge) of sections T1 and T2 are nearly the same, with the highest temperatures appearing near the middle of slabs.
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Fig. 15 Proposed vertical temperature gradients along depth: (a) Concrete box girder (not including cantilever slab); (b) Cantilever slab (Unit: m)
0.5 m, and can be expressed as Td ( y ) = Td [(0.5 - y ) / 0.5] a d
(16)
where y is the distance from the top surface, m; αd is the power of function. 2) Tw is the temperature in the lower part of web. The temperature along the haunch and upper part of web is defined as a linear variation from 0 at depth of 0.5 m to Tw at depth of 1.5 m. 3) The temperature along the bottom slab is defined as a linear variation from Tb at outer surface to Tbt at 0.6 m, and a constant Tbt to the inner surface of bottom slab. If the thickness of bottom slab db is less than 0.6 m, the temperature along the bottom slab is defined as a linear variation from Tb at outer surface to Tbt at inner surface of the bottom slab. 4) For the temperature gradients along the depth of cantilever slab, a piecewise linear temperature gradient distribution model is proposed, which is defined as a maximum temperature, Tcs(x), at middle depth and two equal minimum temperatures, Tcc(x), at the top surface and lower surface.
Fig. 14 Calculated vertical temperature gradients along A−A line (a) and B−B line and C−C line (b) of section T1 and section T2
Based on the discussions of the calculated vertical temperature gradients of different section size, a vertical temperature gradient model is proposed, as shown in Fig. 15. In the model, the notations are defined as follows: 1) Td(y) is the continuous temperature distribution increasing from Td at the top surface to 0 °C at depth of
4.2 Horizontal temperature gradients Based on the measurements, it can be assumed that horizontal temperature gradients in the bottom slab and deck slab over the box cell can be neglected. Figure 16(a) shows horizontal temperature gradients along the thickness of web (D−D line), when the ambient air temperature is the lowest. As shown, the horizontal temperature gradients along the web can reach −10 °C, and is affected by the thickness of web. As the web gets thicker, the temperature difference will be expected to be greater. The horizontal temperature gradients along top surface (E−E line) and the center line (F−F line) of the cantilever slab are presented in Fig. 16(b). It can be seen that the horizontal temperature gradients along the
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cantilever slab are rarely influenced by the thickness of web and bottom slab, and the temperature difference along E−E line of the cantilever slab is greater than that along F−F line.
gradients into account, a twodimensional temperature gradient model can be obtained, in which the concrete box girder crosssection is subdivided into four regions (Fig. 18).
Fig. 17 Proposed horizontal temperature gradients along thickness of web (a), top surface of cantilever slab (b) and center line of cantilever slab (c)
Fig. 18 Region division of concrete box girder crosssection
The temperature gradient of the four regions can be defined as follows: 1) The temperature gradient in region A can be expressed as Fig. 16 Calculated horizontal temperature gradients along D−D line (a) and E−E line and F−F line (b) of sections T1 and T2
Based on the discussions of the calculated horizontal temperature gradients in the web and cantilever slab, the horizontal temperature gradient models for the web and cantilever slab are proposed, as shown in Fig. 17. In the model, the notations are defined as follows: 1) Twd is the temperature at the outer surface of the web. Twd(xw) is the continuous temperature distribution increasing from Twd at the outer surface to 0 °C at the inner surface, and can be expressed as Twd ( xw ) = Twd [( d w - xw ) / d w ] a w
(17)
where dw is the thickness of web, m, αw is the power of function and xw is the distance from the outer surface, m. 2) Tcs and Tcc are the maximum negative horizontal temperature differences at the top surface and center line of cantilever slab, respectively. 4.3 Twodimensional temperature gradient model Taking both vertical and horizontal temperature
2 y æ d ( x ) ö TA ( x, y ) = TA ç x , c ÷ - 1 × 2 ø d c ( x) è é æ d c ( x ) ö ù - TA ( x, 0) ú êTA ç x, ÷ 2 ø ë è û
(18)
where TA(x, 0) and TA(x, 0.5dc(x)) are the temperatures at the top surface and middle depth of the cantilever slab, respectively, with distance x from the edge, °C; dc(x) is the thickness of the cantilever slab, m; y is the distance from the top surface, m. TA(x, 0) and TA(x, 0.5dc(x)) can be calculated as T TA ( x, 0) = Tcs + Td - cs x 4.45 T cc æ d ( x) ö TA ç x, c x ÷ = Tcc 2 4.45 è ø
(19) (20)
2) The temperature gradients in region B can be expressed as a ïìT [(0.5 - y ) / 0.5] d , y £ 0.5 m TB ( y ) = í d ïî Tw ( y - 0.5), 0.5 m < y £ 1 m
(21)
3) The temperature gradients in region C can be expressed as
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a ì é ( d - x ) ù w a w ïTw ( y - 0.5) + Tww ê w ú - T ww 0.5 , ï ë d w û ï y £ 1.5 m ï TC ( x, y ) = í a ï é ( d w - x ) ù w a w T + T ï w ú - T ww 0.5 , ww ê d w ë û ï ï y > 1.5 m î
(22) where x is the distance from the outer surface, m. 4) The temperature gradients in region D can be expressed as ìTbt , y > 0.6 m ï TD ( y ) = í (23) y ïTb + (Tb - Tbt ) min(0.6, d ) , y £ 0.6 m î b where y is the distance from the outer surface of bottom slab, m.
5 Influences of meteorological parameters Among the input meteorological parameters, temperature drop range, wind speed and cold wave duration can vary from different regions and different
Fig. 19 Sensitivity curves for Td (a) and Tw (b)
Fig. 20 Sensitivity curves for Twd (a) and Tb −Tbt (b)
time. As a result, there are always uncertainties in these parameters. Therefore, it is worthwhile to carry out sensitivity studies on the effects of these parameters on the temperature gradients. Taking section T2 for example, using the above meteorological data, for each analysis, only the parameter concerned varies with increase or decrease of 10% increment up to 50%, while the other parameters keep invariant. The sensitivity curves for the temperatures at the top surface of deck slab and in the lower part of web are shown in Fig. 19. It can be seen that they are hardly affected by other parameters but the temperature drop range. Moreover, an increase of 50% in temperature drop range will lead to the decrease of temperature at the top surface of deck slab or in the lower part of web by more than 35%. Figure 20 shows the sensitivity curves for the temperature differences in the web and bottom slab. As shown, the temperature differences in the web and bottom slab are rarely influenced by the wind speed, but greatly affected by the temperature drop range and cold wave duration. In the present model, the ambient temperature is applied to the outer surfaces of the web and bottom slab, and an increase in the temperature drop
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range of cold wave promotes heat loss from these surfaces to the surrounding and lowers the surface temperature most. Therefore, an increase in temperature drop range would lead to an increase in the temperature difference in the web and bottom slab. The drop of inner surface temperature lags behind that of the outer due to the poor conductivity of the concrete. As a result, the shorter the cold wave duration is, the smaller the drop of the inner surface temperature and the larger the temperature difference in the web and bottom slab are expected. Figure 21 shows the sensitivity curves for the maximum horizontal negative temperature difference, Tcc, at center line of cantilever slab. It can be seen that the maximum negative horizontal temperature difference at center line of cantilever slab is mostly affected by the temperature drop range. Generally, an increase of 50% in temperature drop range leads to the increase of the maximum negative horizontal temperature difference by about 40%.
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for highway bridges and culverts JTG D60—2004, the United States Highway Bridge Design Code AASHTO (2007) and British code BS 5400 are selected, as shown in Fig. 22. The parameter values of the proposed temperature gradient model are listed in Table 2.
Fig. 22 Vertical temperature gradient models (Unit: mm) Table 2 Parameters of proposed model Section Td/°C αd Tw/°C Tbt/°C Tb/°C Twd/°C αw Tcs/°C Tcc/°C
Fig. 21 Sensitivity curves for Tcc
From the results just presented, it can be concluded that the cold wave with shorter duration and more severe temperature drop effect may cause more unfavorable influences to the concrete box girder bridge.
T1
7
2
−1
T2
7
2 −4.6
To investigate the difference of temperature effect obtained from the proposed temperature gradient model and current code provisions, the unrestrained linear curvatures, selfequilibrating stresses and bending stresses considering the frame action of the cross section, were derived from the proposed temperature gradient model and current code provisions, respectively. Then, a comparison was made between the value calculated by proposed model and several current specifications. Three codes including the China general design specification
−9
−4 −7.7 −6.5 1.6 −4
−9
Curvatures calculated by the selected three current code provisions and proposed temperature gradient model are listed in Table 3. Comparison shows that the selected three current code provisions all underestimate the curvature at section T1 and this will lead to an underestimation of the temperature induced stresses. The curvature at section T2 calculated using the proposed temperature gradient model has opposite sign to all the others. This will increase the complexity of the temperature effect on the bridge. Table 3 Thermal curvatures calculated by proposed model and selected three current code provisions Model
6 Comparison of proposed temperature gradient model with different codes
1.1 −8.4 −10 3.1 −4
Thermal curvature/10 −6 m Section T1
Section T2
Proposed model
−1.51
1.04
JTG D60—2004
−0.83
−5.45
AASHTO
−0.48
−3.18
BS 5400
−0.69
−1.52
Selfequilibrating stresses sensitive to the shape of the temperature distribution were calculated against the selected three current code provisions and proposed temperature gradient model. The elastic modulus of concrete is set to be 36 GPa. Figure 23 illustrates the selfequilibrating stress distributions along A−A line. It can be seen that the proposed and the selected three
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current code provisions all present high tensile stresses over the top few decimeters. Table 4 lists the selfequilibrating tensile stress at the top surface of the deck slab. As shown, the tensile stress derived from the temperature gradient in BS 5400 is the largest, while the tensile stress derived from the proposed temperature gradient is the smallest. However, the compressive stress in the haunch and upper part of web obtained from the three selected current code provisions is all smaller than that from the proposed model. For the bottom slab, the selfequilibrating tensile stress at outer surface of bottom slab derived from the proposed model and temperature gradient in BS 5400 is much larger than that from the temperature gradient in JTG D60—2004 and AASHTO. This is because the JTG D60—2004 and AASHTO neglect the temperature differences in the bottom slab, but the proposed model and BS 5400 do not.
Figures 24 and 25 illustrate the selfequilibrating stress distributions along the thickness of web (D−D line) and along the depth of cantilever slab (B−B line). It can be seen that the tensile stresses at the out surface of web and lower surface of cantilever slab obtained from three current code provisions are far smaller than those from the proposed model. The tensile stresses at the out surface of web and cantilever slab of section T1 obtained from the proposed model, which can reach approximately 3 MPa, are greater than those of section T2.
Fig. 24 Selfequilibrating stress distributions along thickness of web (D−D line) (a) and along depth of cantilever slab (B−B line) (b) of section T1
Fig. 23 Selfequilibrating stress distributions along A−A line of section T1 (a) and section T2 (b) Table 4 Tensile stresses at top surface of deck slab (MPa) Model Section T1 Section T2 Proposed model
1.60
1.04
JTG D602004
2.18
1.83
AASHTO
1.62
1.40
BS 5400
2.63
2.23
The bending stresses derived from the proposed temperature gradient model and current code provisions are listed in Table 5. The maximum bending tensile stresses at the out surface of web and top and bottom slab obtained from the three current code provisions are also far smaller than those from the proposed model. The stresses at the out surface of web and bottom slab of section T1 are larger than those of section T2 as well. According to the comparisons, it is found that the cold wave may cause more unfavorable effect to the concrete box girder bridge, especially to the large concrete box girders bridge. The unrestrained self
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Fig. 25 Selfequilibrating stress distributions along thickness of web (D−D line) (a) and along depth of cantilever slab (B−B line) (b) of section T2 Table 5 Maximum bending tensile stresses at out surface of web and top and bottom slab Model
Maximum bending tensile stresses at Section T1/MPa
Maximum bending tensile stresses at Section T2/MPa
Deck slab
Web
Bottom slab
Deck slab
Web
Bottom slab
Proposed model
2.43
3.07
2.29
2.18
1.36
0.85
JTG D602004
1.82
0.28
−0.04
1.79
0.49
−0.02
AASHTO
1.41
0.26
−0.02
1.37
0.33
−0.02
BS 5400
2.02
0.6
1.6
2
0.88
1.02
equilibrating tensile stresses or bending tensile stresses at the out surface of web and bottom obtained from the proposed temperature gradient model are so large. When they are added to the stresses due to the other dead loads and live loads, they may give rise to cracking in a concrete box bridge. Moreover, the analyzed data in this work is not obtained under the worst cold wave. Table 6 lists the maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years in six typical cities of China. As shown, the maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years in the northern cities of China can reach 20.4, 23.2 and 26.5 °C, respectively, which is far greater than 13.4 °C in this work. As the temperature drop range is the main influencing factors, the concrete box girder would crack Table 6 Maximum drop of daily minimum temperature during 24, 48 and 72 h in 50 years of six typical cities City
Maximum temperature drop/°C 24 h
48 h
72 h
Beijing
−15.7
−21.8
−22.6
Chengdu
−9.5
−11.9
−12.1
Guangzhou
−14.4
−16.2
−17.4
Harbin
−20.4
−23.2
−26.5
Lanzhou
−12.9
−14.3
−15.3
Shanghai
−12.8
−14.4
−17.2
more easily under the worst cold wave. Thus, the design engineers should take temperature effects caused by cold wave into account during the design stage.
7 Conclusions 1) Temperature distribution across the section of concrete box girder under the effect of cold wave is different from that under the effect of solar radiation or nighttime radiation cooling and should not be simplified as one dimensional. The horizontal temperature gradients in the web and cantilever slab should be considered when analyzing the temperature effect of concrete box girder bridge under the effect of cold wave. 2) The developed temperature predicting model can accurately predict temperature distributions over the cross section of the concrete box girder under the effect of cold wave. The proposed twodimensional temperature gradient model can accurately predict the temperature gradient over the cross section of the concrete box girder under the effect of cold wave. 3) The cold wave with shorter duration and more severe temperature drop may cause more unfavorable influences on the concrete box girder bridge. The cold wave may cause more unfavorable effects to the concrete box girder bridge than the current code provisions, especially to the large concrete box girders bridge. The unrestrained selfequilibrating tensile stresses and
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bending tensile stresses at the outer surface of web and bottom under the effect of cold wave are so large that when the stresses are added due to the other dead loads and live loads, they may give rise to cracking in a concrete box bridge. 4) In the northern cities of China, the concrete box girder bridge would suffer more unfavorable effects from the cold wave and crack more easily.
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