COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
A Study on Temperature Distribution in a Cross Section of Concrere
Box Girder Bridges
Y. P. Tan *, D. J. Han
Department of Civil Engineering, South China University of Technology, Guangzhou, 510640 China
Email: coca1234@163.com
Abstract It is known that nonlinear temperature variation over a cross section of long-span continuous box-girder
beam bridges or continuous rigid-frame box girder bridges can cause longitudinal stresses and that in some cases such
kind of stresses may reach or even exceed those induced by the live loads and that temperature cracking may occur in
the structure component. However, the mechanism of such phenomenon is not very clear yet. Therefore, to predict the
stresses caused by temperature distribution is important for a correct design of the bridge structures. The purpose of the
paper is to present a practical computation method to simulate the temperature distribution over cross sections of a
concrete box girder bridge. The relative factors considered include the bridge geometry, the geographical location, the
bridge orientation, the material properties and ambient climatic conditions.
In this paper heat flow equations over a concrete bridge cross section and heat boundary conditions of outside and
inside the box-girder are first derived. And then numerical solution for daily time variation of air temperature is
presented. The relationships between inside concrete temperature and ambient climatic conditions that include solar
radiation, relative humidity, and wind speed etc. are established. A prediction for variation of concrete temperature is
obtained. The results are compared to the observed values measured from The Guangzhou Guanyinsha Bridge that is a
single-cell box-girder bridge. Finally, numerical method and computer programming are given to predict the
temperature distributed in concrete bridge cross sections. This can be used to generate the thermal loads for finite
element analysis.
Key words: temperature variation, climatic conditions, box-girder bridge, numerical simulation
INTRODUCTION
There have been some research works imported in the literature. Three-dimensional temperature variation of
post-prestressed concrete box-girder bridges are studied by McClure [4]. It is found that longitudinal temperature
variation and transversal temperature variation are not remarkable based on by the thermocouples the measuring data
and attenuation analysis, Thus three-dimensional heat conduction can be simplified as a problem of vertical variation
along the cross section. Based on studying the relation of the meteorological data and temperature distribution of a
bridge, Zuk [5] points out that parameters to effect the temperature distribution include air temperature, wind, solar
radiation and material. And the approximation equations of maximum temperature difference between the top surface
and the bottom surface of the beam is presented. Based on the measuring data of Madway Bridge, Maker assumes that
the temperature of continuous box-girder bridges is linear distributed, and temperature of the web and the bottom slab
is constantly distributed. However, later with the development of further studies, it is acquainted that the temperature
distribution along the cross section is nonlinear. Therefore, D. A. Stephenson presents that the temperature distribution
through the thickness can be expressed as an exponential function according to the amplitude of surface temperature.
Based on field measurement , Shiu [6] and Nam Shiu K. [7] put forward patterns of the thermal gradients and
longitudinal strain of box-girder section every day in each season. Branco [8] studies the thermal gradients and its
calculation method of a concrete double box girder. According to experimental data, American Potgieter [9], Rao [10]
and Carlin L. [2] give maximum positive and negative temperature gradients of a single box girder. Parameters
considered include different area in America, different top surface and the climate condition including solar radiation,
wind speed, environment temperature and so on.
⎯ 960 ⎯
On temperature stress aspect, firstly, Fritz Leonbardt gives examples of serious crack which takes place in several
prestressed box-girder bridge in Germany, and estimates transversal temperature stress. Furthermore, he discusses
quantitatively the temperature stress of thick-wall box-girder and conclusion that the temperature stress is the main
cause of gives cracking occurred in prestressed box-girders. In the paper “Influence of Bridge Structure Caused by
Solar Radiation”, F.Kehlbeck (1974) has studied systematically how the meteorological factors to influence on each
part of the surface temperature of a concrete bridge. A matrix-form expression of temperature function is proposed.
According one-dimensional nonstationary thermal conduction theory, restrained stresses and self restraint stresses of
thick slabs are found. Because this complicated method deals with many factors, it is not easy to apply to engineering
practice. Mamdouh [1] has established finite element formulation, and gives a summary for the parameters which
influence the temperature distribution and stresses in concrete including section shape, season and pavement. Anna
Saetta [12] gives heat conduction equation, and boundary conditions of thermal analysis, also gives finite element
equation for temperature load. Barr [13] points out that the differences between the thermal expansion coefficients of
the concrete and the prestressing cables and the differences of temperature inside the concrete result in deviations in
prestresses and concrete stresses as well as the deflection.
A numerical method and corresponding computer programming are given to predict the temperature distributed in
concrete bridge cross sections in this paper. Therefor, some fitted equations such as the air temperature and the relative
humidity at any time are presented based on test results, and the time difference and the solar declination are fitted
according to Chinese Astronomical Ephemeris. The surface temperature of the box girder are solved by the
above-mentioned parameters.
EQUATIONS OF HEAT CONDICTION
1. Equations of heat flow Because longitudinal temperature variation is neglected, the variation of temperature T over
a bridge cross section at any time t is governed by the well-known heat flow equation (1).
⎛ ∂ 2T ∂ 2T ⎞
∂T
k⎜ 2 + 2 ⎟ +Q = ρ ⋅c
∂t
∂y ⎠
⎝ ∂x
(1)
in which k is the isotropic thermal conductivity coefficient of units W/m⋅°C; Q is the rate of heat per unit volume
3
generated within the body, W/m3; ρ is the density, kg/m ; and c is the heat, J/(kg⋅°C).
If the energy is transferred by the surrounding media to or from the boundary surface, the boundary conditions
associated with Eq.(1) can be expressed by
⎛ ∂T
∂T ⎞
k ⎜⎜
nx +
ny ⎟ + q = 0
∂y ⎟⎠
⎝ ∂x
(2)
in which n x and n y are the direction cosines of the unit outward normal to the boundary surface respectively; q is the
heat flux input or loss per unit area, W/m2. The heat flux at the surface of a bridge structure includes solar radiation,
convection and irradiation and so on.
2. Heat flux If the bridge structure arrives at the state of thermal equilibrium subjected to the periodic heat
exchange for a period of time, the equilibrium equations of the heat flux on the surface at different positions
of the box girder could be established based on the balance of the heat flux flowing forward and backward to
the box girder.
On the top surface of the deck:
q B + q K = q J + q H + qGa
⎛ qσm⋅TL + q am
⎛ TV ⎞
ε BL ⋅ C s ⋅ ⎜
⎟ + α K ⋅ (TV − TA ) = ABK ⋅ J 0 ⋅ sinh⋅ ⎜⎜
2
⎝ 100 ⎠
⎝
4
On the bottom surface of the bottom slab:
q B + q K = q R + qUR
⎯ 961 ⎯
⎞
T
⎟ + γ ⋅ C s ⋅ ⎛⎜ A ⎞⎟
⎟
⎝ 100 ⎠
⎠
4
(3)
⎛ qσm⋅TL + q am
⎛ TV ⎞
⎜
(
)
sinh
T
T
r
A
J
α
+
⋅
−
=
⋅
⋅
⋅
⋅
⎟
K
V
A
uk
BK
0
⎜
2
⎝ 100 ⎠
⎝
4
ε BL ⋅ C s ⋅ ⎜
⎞
⎟ + ε BL ⋅ C s
⎟
⎠
⎛T ⎞
⋅⎜ A ⎟
⎝ 100 ⎠
4
(4)
On the surface of the web under the shadow:
q B + q K = q J + q H + q R + qGa + qUR
4
ε BL
⎛T ⎞
4
⋅ Cs ⋅ ⎜ V ⎟ + α K ⋅ (TV − TA ) = 0.25 ABK ⋅ J 0 ⋅ sinh{qam (ruK + 1) + qσm⋅TL (ruK − 1)
⎝ 100 ⎠
β
β⎞ ⎛T ⎞
⎛
+ cos β [ qam ( ruK − 1) + qσm⋅TL ( ruK + 1)]} + ε BL ⋅ cs ⋅ ⎜ ε a sin 2 + cos 2 ⎟ ⋅ ⎜ A ⎟
2
2 ⎠ ⎝ 100 ⎠
⎝
4
(5)
On the surface of the web under the sunlight:
q B + q K = q J + q H + q R + qGa + qUR
4
⎛ TV ⎞
⎟ + α K ⋅ (TV − TA )
⎝ 100 ⎠
ε BL ⋅ C s ⋅ ⎜
⎞
⎛π
= ABK ⋅ J 0 ⋅ qσm⋅TL cos⎜ + h − β ⎟ ⋅ cos(a s − a w ) + 0.25 ABK ⋅ J 0 ⋅ sinh{q am ( ruK + 1) + qσm⋅TL (ruK − 1)
⎝2
⎠
+ cos β [q (ruK − 1) + qσ
m
a
m ⋅TL
β
β⎞ ⎛T ⎞
⎛
(ruK + 1)]} + ε BL ⋅ cs ⋅ ⎜ ε a sin 2 + cos 2 ⎟ ⋅ ⎜ A ⎟
2
2 ⎠ ⎝ 100 ⎠
⎝
4
(6)
in which ε BL is the radiation factor of the structure; ABK is the short wave absorption coefficient; qσm⋅T is the total
transmission coefficient; q am is the absorb transmission coefficient; ruK is the ground short wave reflection factor,
%; β is the angle between horizon and the surface; a s is the azimuth angle of the sun; a w is the surface azimuth angle;
L
TV is the absolutely radiation temperature on the surface of structure, K.
q B , q K , q H , q R , qGa , qUR refers to the heat
flux aroused from radiation of the structure, convection exchange, the sky radiation, the reflection of the solar radiation
and the sky radiation, the atmospheric counter radiation, the reflection of the radiation of the earth surface
surroundings and the atmospheric counter radiation respectively.
Eqs. (3-6) indicates that the equation can only be solved when the solar radiation, the temperature, cloud amount and
the solar azimuth at any time in every day are all known. The azimuth angle of the sun is relevant to solar declination,
hour angle, solar altitude and all these can be calculated by the correlation formulas. The heat flux depends on the
bridge orientation, material properties of structure and the bridge geometry equally.
Figure 1: Geometry defining incidence angle of solar radiation
⎯ 962 ⎯
PARAMETER OF AMBIENT CLIMATIC CONDITIONS
1. The air temperature outside box girder The air temperature is a physical parameter in meteorology that
denotes hot and cold degree of the air. And it’s variation with time effects the temperature of concrete.
According to measurement data at the site of Guangzhou Panyu Guanyinsha Bridge and the weather data provided by
the local meteorological bureau, it is known the maximum temperature occurs periodically at 13:00~15:00 the
minimum at 6:00~7:00 daily (Fig. 2). As a result of the solar radiation in the daytime, the air temperature ascends
apparently, and it decreases laxly at night time. It can be seen that the air temperature change basically following the
path of sine curve or cosine curve with respect to the time. But a piecewise continuous function may fit better with the
actual temperature curve. Furthermore, the air temperature variation is different at the daytime and at night time. The
average daily temperature is higher than the mean value of daily maximum and minimum air temperature. The
measurement time should be increases in order to obtain the average daily temperature accurately. The formula of the
air temperature can be fitted as follows:
Measured
Computed
Measured
Computed
Figure 2: Actual temperature compared
with calculated value
Figure 3: Daily maximum and minimum
temperature in the year of 2005
For 0 ≤ t ≤ t 0 − 8 and t 0 ≤ t < 24
⎛ t ⎞
⎛ t ⎞
⎛
⎞
⎛
⎞
t
t
⎟⎟ + b2 ⋅ ΔTA sin⎜⎜
⎟⎟ + b3 ⋅ ΔTA cos⎜⎜
⎟⎟ + b4 ⋅ ΔTA sin⎜⎜
⎟⎟
TA = TA + b1 ⋅ ΔTA cos⎜⎜
−
−
−
−
30
t
30
t
60
2
t
)
60
2
t
0 ⎠
0 ⎠
0 ⎠
0 ⎠
⎝
⎝
⎝
⎝
(7)
For t 0 − 8 < t < t 0
⎛ t ⎞
⎛ t ⎞
⎛ t
⎞
⎛ t
⎞
⎟⎟ + b2 ⋅ ΔTA sin⎜⎜
⎟⎟ + b3 ⋅ ΔTA cos⎜⎜
⎟⎟ + b4 ⋅ ΔTA sin⎜⎜
⎟⎟
TA = TA + b1 ⋅ ΔTA cos⎜⎜
−
6
6
2
12
2
12
−
t
−
t
−
t
t
⎝ 0
⎠
⎝ 0
⎠
⎝ 0
⎠
⎝ 0
⎠
(8)
in which t is the time, in hour; t 0 is the time corresponding to the maximum air temperature, also in hour; ΔT A is the
intraday temperature difference, °C; T A is the intraday mean temperature. If the measuring data is lacking, it can be
given as follows:
T A = α (T A max − T A min )
(9)
in which α is a coefficient, always taken a value 1.01-1.05; TA max is the intraday maximum temperature, °C; TA min is
the intraday minimum temperature, °C.
Eqs. (7-9) can only be applied to the situation without such unusual situations as the great temperature change for the
cases of freeze, storm and typhoon and so on.
Fig. 2 shows the maximum and minimum air temperature variation curve of Guangzhou city in the year of 2005.
According to the results, the air temperature dispersed greatly, especially in the spring season. Therefore, the general
⎯ 963 ⎯
formula could not be fitted and the feasible alternative is to collect the air temperature datum provided by the local
meteorological bureau or to measure it on site. Anyway, the rule of the air temperature variation is quite similar. The
annual temperature change can be calculated according to measuring result in one year.
Table 1: Coefficients in Eqs.(7) and (9)
time
b1
b2
b3
b4
0 ≤ t ≤ t0−8
−0.94
1.85
0.71
−4.3
t0−8 < t < t0
−0.64
0.64
−2.14
−0.88
t0 ≤ t < 24
−0.23
0.18
0.23
0.23
2. The air temperature inside box girder Based on measurement data at the site (Fig. 4), the temperature varied
slightly inside the box girder due to the weak thermal conductivity and the prevention from the direct solar radiation.
The daytime temperature difference inside the box girder is basically within 2°C when the maximum temperature is
less than 26°C and merely 3°C when it arrives at 30°C. Anyway, the air temperature equals to the average daily
temperature approximatively.
Figure 4: Air temperature inside the box girder
3. Wind speed Wind is aroused by air motion relative to the ground, and refers to horizontal direction in meteorology.
It is denoted by wind direction and wind speed. The magnitude of wind speed has an influence on the heat exchange
directly. The convection exchange coefficient may be approximately expressed with the following equation:
α K ≈ 2.6[ 4 ΔT + 1.54 w]
(10)
Wind speed
3
2.5
2
1.5
1
0.5
0
0:00
4:48
9:36
14:24
19:12
0:00
Time
Figure 5: Wind speed curve on March 6, 2006
⎯ 964 ⎯
Figure 6: Wind speed curve on March 9-13, 2006
in which α K is the convection exchange coefficient, in W/m2⋅K; ΔT is the difference between the surface
temperature and the surrounding air temperature, K; w is the wind speed, m/s.
It can be seen that the regularity of the wind speed showed in Figure 5 is not very obvious according to the
meteorological data of Panyu meteorological bureau announced online on March 6, 2006. The maximum wind speed
appears at 2:50; the minimum value at 14:50. They are 2.9m/s and 0.3m/s respectively. The maximum value is quite
different from the minimum. And the wind speed change appears in a short time. Figure 6 shows the meteorological
data of foshan meteorological bureau announced online on March 9-13, 2006. The wind speed reaches maximum value
2m/s on March, 9~10 at afternoon and also on March, 11 at morning and night. It reaches 4m/s on March, 12 at
afternoon. It is quite difficult to determine the wind speed at any time every day for its great dispersion and
non-periodic variation. It cannot be fitted with formula either. Yet it is more complicated for wind speed at any time in
the whole year. In the current literatures about the weather, only such datum as the whole year instantaneous maximum
wind speed, the annual mean wind speed, the monthly mean wind speed, the accumulated maximum wind speed in ten
minutes and so on are given as reference. There is no computational formula suitable for a region being proposed.
4. Relative humidity The relative humidity denotes the percentage of the water-vapour pressure of the air sample to
the saturated water-vapour pressure. The relative humidity variation can change the water content of the concrete, and
affects the thermal gradient of the concrete box-girder. This parameter is useful for obtaining the pattern of the thermal
gradient according to the weather datum. Generally, it decends as the air temperature ascends. The equation of the
relative humidity is analogous to the air temperature that can be fitted as follows:
For 0 ≤ t ≤ t 0 − 8 and t 0 ≤ t < 24
⎞
⎛ t
⎞
⎛ t
⎞
⎛ t
⎞
⎛ t
+ π ⎟⎟
RH = R H + b1 ⋅ ΔRH cos⎜⎜
+ π ⎟⎟ + b4 ⋅ ΔRH sin⎜⎜
+ π ⎟⎟ + b3 ⋅ ΔRH cos⎜⎜
+ π ⎟⎟ + b2 ⋅ ΔRH sin⎜⎜
30
30
60
60
t
t
t
−
t
−
−
−
0
0
0
0
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎝
(11)
For t 0 − 8 < t < t 0
⎞
⎛ t
⎞
⎛ t
⎞
⎛ t
⎞
⎛ t
+ π ⎟⎟
RH = R H + b1 ⋅ ΔRH cos⎜⎜
+ π ⎟⎟ + b4 ⋅ ΔRH sin⎜⎜
+ π ⎟⎟ + b3 ⋅ ΔRH cos⎜⎜
+ π ⎟⎟ + b2 ⋅ ΔRH sin⎜⎜
⎠
⎝ 2t0 − 12
⎠
⎝ 2t0 − 12
⎠
⎝ t0 − 6
⎠
⎝ t0 − 6
(12)
in which ΔRH is the intraday relative humidity difference, %; R H is the intraday mean relative humidity, %.
Figure 7: Relative humidity curve
Figure 7 shows relative humidity-time curve when the weather is fine or cloudy. The relative humidity appears
minimum value as the air temperature arrives topmost. But the rain will bring large deviation.
Guanyinsha Bridge is a concrete box-girder bridge with 4% single transverse gradient. According to the measuring
data, the temperature of the right and left web surface is different and the relative humidity has different influence on
the thermal gradient of the web. Basd on agreat amount of measuring data, curve fitting is done for the relative
temperature ΔT, which is difference between the temperature at a point of the web and the minimum temperature at a
point of the web and the minimum temperature at the web. The fitted formula is given as
⎯ 965 ⎯
ΔT = a0 ⋅ H ⋅ RH ⋅ T 2 + a1 ⋅ H ⋅ RH ⋅ T + a2 ⋅ H ⋅ RH + a3 ⋅ H ⋅ T 2 + a4 ⋅ H ⋅ T + a5 ⋅ H + a6 ⋅ RH ⋅ T 2
+ a7 ⋅ RH ⋅ T + a8 ⋅ RH + a9 ⋅ T 2 + a10 ⋅ T + a11
(13)
in which RH is the mean of the relative humidiy provided by the local meteorogical bureau at that day; T is the ambient
temperature; H is the depth from the top surface; the values of parameters a0 − a11 are shown in Table 2.
From the Eq.(13), it is seen that ΔT increases linearly in the direction of the height of the web, which is similar to the
positive temperature gradient in national code. ΔT reduces as the relative humidity increases. And ΔT increases as the
air temperature goes up. But the increasing rate is getting slower as the air temperature goes up.
In conclusion, a similar function between the relative humidity and the thermal gradient can be built based on site
measurement for the bridges.
Table 2: Coefficient a0 − a11 in Eq. (13)
a0
a1
a2
a3
a4
a5
0.1<H<0.55m
0.046032
−2.177
25.638
−2.6032
12148
−1413.5
0<H<0.1m
−0.25
12.368
−150.73
14.625
−735.67
8970.3
0.1<H<0.55m
0.097619
−4.6595
55.357
−6.0476
287.24
−3397.7
0<H<0.1m
−0.2043
9.4776
−107.14
11.873
−557.96
6305.9
a6
a7
a8
a9
a 10
a11
0.1<H<0.55m
−0.024857
1.1756
−13.828
1.40577
−65.601
762.84
0<H<0.1m
0.0047222
−0.27778
3.7961
−0.31556
20.041
−274.68
0.1<H<0.55m
−0.027714
1.3328
−15.96
1.6324
−77.992
929.43
0<H<0.1m
0.0024774
−0.080911
0.28983
−0.15963
6.5283
−40.927
position
right web
left web
position
right web
left web
5. Cloud amount The cloud is visual cluster which constituted by lots of drips and ice crystals floating in the air. Data
of the cloud types, the height of the cloud and the cloud amount are collected in routine meteorological observation.
The humid air move upward and adiabatically expanse with falling temperature and they reach a saturation state and
form the cloud. So ascending airs current with enough vapors are necessary for the formation of cloud. Three processes
should be considered in simulating the numerical model of cloud: dynamics, thermodynamics and cloud physics.
Therefore, there exist two issues: one is that only the approximate numerical solution is attained for a group of
differential equations that express these processes; the other is the insufficient understanding to the dynamics process
that relates to air mixtures inside and outside of the cloud. So the determination of the instantaneous cloud amount is
very difficult and can only be based on the actual measurement and the datum provided by the local meteorological
bureau.
Relationship between the atmospheric counter radiation and the ambient air temperature is as follows:
⎛T ⎞
Ga = γ a ⋅ Cs ⋅ ⎜ A ⎟
⎝ 100 ⎠
4
(14)
in which Ga is atmospheric counter radiation, W/m2; T A is ambient air temperature, K; C s is the blackbody radiation
factor, takes 5.775×10−4 W/m2⋅K4; γ a is the coefficient, takes 0.82 in cloudless day (cloud amount is 0), 0.94 when the
sky is fully covered by cloud and mist (cloud amount is 1) and linear interpolating between them when the cloud
amount is 0-1.
It is obvious that the coefficient relative to the cloud amount γ a is the key factor of the calculation of atmospheric
counter radiation. Uncertainty of cloud amount determines that it is difficult to calculate atmospheric counter radiation
accurately.
⎯ 966 ⎯
SOLAR ALTITUDE AND AZIMUTH ANGLE OF THE SUN
The time difference and the solar declination change basically following the path of sine curve or cosine curve along
with the time, and reflect a periodic variation. They can be fitted at any time each year according to 1978~1988
Chinese Astronomical Ephemeris.
The formula of the time difference each 4 years since 1980 as follows:
⎛πD⎞
⎛πD⎞
⎛πD⎞
⎛πD⎞
TD = c1 ⋅ d ⋅ cos ⎜
⎟ + c2 ⋅ d ⋅ sin ⎜
⎟ + c3 ⋅ d ⋅ cos ⎜
⎟ + c4 ⋅ d ⋅ sin ⎜
⎟+ f
⎝ k ⎠
⎝ k ⎠
⎝ 2k ⎠
⎝ 2k ⎠
(15)
in which the value of c1 − c4 , d , k , f are shown in Table 3; D is the day of the year.
The formula of the solar declination each 4 years since 1980 as follows:
⎛ πD ⎞
⎛ πD ⎞
⎛ πD ⎞
⎛ πD ⎞
⎟⎟ + m2 ⋅ n ⋅ sin⎜⎜
⎟⎟ + m3 ⋅ n ⋅ cos⎜⎜
⎟⎟ + m4 ⋅ n ⋅ sin ⎜⎜
⎟⎟
⎝ p ⎠
⎝ p ⎠
⎝ 2p ⎠
⎝ 2p ⎠
δ = m1 ⋅ n ⋅ cos⎜⎜
(16)
in which the value of m1 − m4 , n, p are shown in Table 4.
The formula of other year is similar to Eq. (15) and Eq. (16), but the coefficients are different.
Table 3: Coefficient in Eq. (15)
c1
c2
c3
c4
d
k
f
0 < D ≤ 43
−0.38687
−0.79065
0.65969
−0.9289
9
75
−5.273
43 < D ≤ 135
−0.15709
−0.93193
−0.08145
0.00257
9
92
−5.273
135 < D ≤ 208
0.88687
−0.46625
−0.01196
0.02105
5
73
−1.73
208 < D ≤ 308
−0.93421
−0.27583
0.032574
−0.0268
11.4
100
4.97
D > 308
−1.3581
−0.50699
−0.35023
0.39714
11.4
100
4.97
Table 4: Coefficient in formula (16)
m1
m2
m3
m4
n
p
0 ≤ D < 173
−0.85534
0.16487
−0.13118
0.14006
23.44
181
173 ≤ D < 357
−0.90709
0.15625
0.087908
0.07173
23.44
184
D ≥ 357
−1.1397
0.12142
−0.14581
0.01014
23.44
181
Based on celestial bodies circulating (earth, sun), the solar altitude may be expressed with the following equation:
sinh = sin Φ ⋅ sin δ + cos Φ ⋅ cos δ ⋅ cos ω
(17)
It is an equation of the solar altitude at any time. It shows that the solar altitude is determined by latitude of the location
Φ (south is positive), solar declination δ and hour angle ω .
The azimuth angle of the sun can be computed as follows:
sin a s = cos δ ⋅ sin ω / cosh
(18)
COMPUTER PROGRAMMING
Computer programming is given to calculate the simulating temperature field. The flow chart is shown in Fig. 8. The
input data includes year, the daily maximum and minimum air temperature, the time corresponding to the daily
maximum air temperature, the daily relative humidity, surface azimuth angle, angle between horizon and the surface,
the absolutely radiation temperature on the surface of structure, the short wave absorption coefficient, the ground short
⎯ 967 ⎯
wave reflection factor etc. The annual air temperature and the annual relative humidity can refer to the datum in the
past years.
Input
calculate TD , , ,
calculate h and a s
calculate RH and T A each hour
calculate q B ,q K ,q H , q R ,
qGa ,q UR using Equ.(14)~(17)
solve T using
formula(8)
solve T V
close
Figure 8: Flow chart
The solar radiation can be either got from the meteorological data announced by the local meteorological bureau, or
calculated according to the total transmission coefficient. Point relative humidity as a function of time is calculated
using Eq. (11) and (12), and point temperatures using Eqs.(7-9). The heat flux flowing forward and backward to the
box girder is built according to Eqs. (3-6), and solves the temperature of the box-girder surface. The results can be used
to generate the thermal loads for finite element analysis of solid element. The positive temperature gradients of beam
element are arrived by Eq. (13).
CONCLUSION
The data required in the analysis of the temperature distribution of the prestressed concrete bridge are the latitude and
altitude of the structure and its orientation, the cross section geometry, the material properties and deck surface cover
and parameters pertaining to the ambient climatic conditions.
The instantaneous cloud amount and the instantaneous wind speed are uncertain parameters of the ambient climatic
conditions, and their fitted formula can’t be presented. The regularity of the ambient air temperature and the relative
humidity variations continuous is not very obvious every day, and obvious in a year. Some parameters such as the time
difference, the solar declination, the instantaneous air temperature and the instantaneous relative humidity can be
calculated as a result of their change basically following the path of sine curve or cosine curve along with the time.
A method is presented for computer programing which is predicted by the temperature distribution in concrete bridge
cross sections.
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