ABSTRACT: Nonlinear temperature variation over the cross section of a simply-supported bridge produces longitudinal stresses (eigenstresses). In a continuous bridge, additional continuity stresses develop. Parametric studies are conducted in order to find the effects of various parameters on the eigen and continuity stresses. A method is presented and reference is made to a computer program to predict the temperature distribution over bridge cross sections from data related to the geometry, location, orientation, material and climatological conditions.
INTRODUCTION
Bridge structures are subject to complex thermal stresses which vary-continuously with time. The magnitude of these stresses depends upon the temperature variation within the structure and this depends upon the geographic location and the orientation of the bridge, climatological conditions, cross section geometry and thermal properties of the material and the exposed surfaces.
Although many bridge designers recognize that the temperature variations can produce high stresses (12,16), little guidance is given in bridge design codes on how these stresses can be accurately calculated.
The distribution of temperature throughout the cross section of a bridge structure must be known if the resulting stresses, reactions and deformations are to be calculated. Theoretical analysis of temperature distribution throughout the cross section of a typical bridge structure is complex because temperature varies with time and may also vary from section to section. In a concrete bridge with constant cross-sectional properties over a long length, it can be assumed that the temperature is constant over the bridge length but varies through the depth and within the width of the cross section. Thus, the temperature field to be determined at any time t is essentially two-dimensional; T = T(x,y,t). Therefore, in this paper, a method of analysis based on two-dimensional finite elements is described to determine the time-dependent temperature variation within the cross section of a concrete bridge of arbitrary geometry and orientation for a given geographic location and environmental conditions. The finite element formulation for the analysis of transient heat flow in a two-dimensional body is treated by several authors (e.g., Refs. 15 and 17) and thus is only briefly reviewed here. The variation in temperature with time can be treated using Crank-Nicholson's time-stepping finite difference scheme (2). A different numerical technique is used here employing Galerkin's weighted residuals; this results in better accuracy,
'Grad. Student, Dept. of Civ. Engrg., The University of Calgary, Calgary, Alberta, Canada T2N 1N4.
2Prof. of Civ. Engrg., The University of Calgary, Calgary, Alberta, T2N 1N4, Canada.
avoids numerical oscillations, and allows use of larger time increments. To apply a finite element analysis successfully for a bridge structure,
the boundary conditions must be accurately modeled. Of particular importance are the amount of heat gain or loss by solar radiation and surface convection and irradiation from or to the atmosphere. In this paper, the solar radiation is considered dependent on the angle of altitude of the sun which varies with the seasons and the time of day. The convection and irradiation depend on the time-varying temperatures of the surface and the air. The latter is considered different inside and outside a box-girder bridge. The equations chosen to idealize the boundary conditions in this case are discussed in the following sections.
A series of transient finite element analyses are performed to study the influences of various parameters including bridge axis orientation, ambient temperature extremes, wind speed, surface cover and section shape on the thermal response of concrete bridges of various cross section types.
EQUATIONS OF HEAT FLOW AND BOUNDARY CONDITIONS
The variation of temperature T over a bridge cross section at any time t is governed by the well-known heat flow equation (1)
/d2T d2T\ dT
Kdx2 dy2} dt
in which k = isotropic thermal conductivity coefficient of units Btu/(h ft °F) or W/m °C; Q = rate of heat per unit volume generated within the body (e.g., by hydration of cement), Btu/(h cu ft) or W/m3 ; p = density, lb/cu ft or kg/m3 ; and c = specific heat, Btu/(lb °F) or 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
/dT BT \ k[TX
n* + J-yn>)+^° (2)
in which nx and ny = direction cosines of the unit outward normal to the boundary surface; and q = boundary heat input or loss per unit area, Btu/(h ft2) or W/m2.
The rate of energy transfer q at the surfaces of a bridge structure is the sum of solar radiation, convection and irradiation
q = qs + qc + qr (3)
in which qs(s,t) is the solar radiation, qc(s,t) is the convection and qr(s,t) is the irradiation from the surface to the surrounding air. Each of these quantities varies with the position s of the point considered on the surface and the time.
The heat transfer components of Eq. 3 are treated by the following simple expressions. The heat gain due to sun rays, i.e., shortwave radiation, received by the structure can be expressed by
qs = al (4)
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in which I(s,t) is the total solar radiation on surface s at time t. This depends upon the parameter s defining the position of the point considered and the angle of incidence of sun rays; in shaded areas, 1 = 0 . The dimensionless coefficient a is the fraction of I absorbed by the surface of the structure. A method of calculating the time dependent value of I will be discussed later in this paper.
The heat lost to or gained from the surrounding air by convection as a result of temperature differences between the bridge surface and the air is given by Newton's law of cooling as
qc = hc(T -T.) (5)
in which hc = convection heat transfer coefficient, Btu/(h ft2 °F). or W/ (m2 °C); T(s,t) = temperature of the surface, in degrees Fahrenheit or Celsius; and Ta(t) = air temperature at time t. As previously mentioned, two values of Ta are considered for the ambient air outside and inside a box-girder bridge. The convection heat transfer coefficient hc, is a function of many variables such as wind speed, surface roughness, and geometric configuration of the exposed structure. Its value is usually determined experimentally or calculated by empirical formulae (11,14).
The heat transfer between the bridge surface and the surrounding atmosphere due to long wave radiation, i.e. thermal irradiation, produces a nonlinear boundary condition which can be modeled by Stefan-Boltz-man radiation law as
qr = Cse[(T + T*)4 - (Ta + T*f] (6)
in which Cs = Stefan-Boltzman constant = 18.891 X 1(T8 Btu/(h sq ft °R4) or 5.677 x 1CT8 W/(m2 °K4); e = emissivity coefficient relating the radiation of the bridge surface (a gray body) to that of an ideal black body (0 s e < 1); and T* = a constant (459.67) used to convert temperature in degrees Fahrenheit, °F, to degrees Rankine, °R, or = 273.15 to convert temperature in degrees Celsius, °C, to degrees Kelvin, °K.
It is convenient to rewrite Eq. 6 in quasi-linear form
qr(8,t) = hr(T - T.) (7)
in which hr(s,t) is a radiation heat transfer coefficient defined as (11)
hr = Cse[(T + T*)2 + (T„ + T*)2](T +Ta + 2T*) (8)
This coefficient cannot be calculated unless T(s,t) is known. However, in a time incremental solution, an approximate value of hr at any instant can be calculated by using earlier values of T, Once the radiation coefficient is calculated, it can be treated similarly to the convection coefficient hc, and the effects of heat flow by convection and radiation can thus be combined in an overall heat transfer coefficient
h = hc + hr (9)
FINITE ELEMENT FORMULATION
The problem thus defined is easily solved by finite element method. The cross section of the bridge is divided into a finite number of discrete elements. Two types of elements are used (Fig. 1): bilinear quadrilateral
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MNTERIOR QUADRILATERAL ELEMENT
FIG. 1.-—Finite Elements for Temperature Analysis in a Two-Dimensional Body
interior elements and linear one-dimensional fictitious elements to represent the boundaries. The temperature within a finite element e can be approximated by
T(x,y,ty = [N]{TY (10)
in which [N] = linear shape functions and {T}e = column vector of four or two nodal temperatures for an interior or boundary elements, respectively. By applying the variational principle of finite element method and assembling the element contributions over the cross section, Eqs. 1 and 2 reduce to (17)
[C] ^ - } + [K]{T} - {F} = 0 dt
(11)
in which {T(t)} is time-dependent vector of temperatures at the nodes; {F(t)} = thermal "load" vector for the system (further discussed below); [C] and [K] = heat capacity and conduction matrices of the system generated by assemblage of individual element matrices. The heat capacity matrix for an element is given by
[C] -I Pc[N]T[N]dV (12)
in which V indicates integration over the volume of the element. The system conduction matrix, [K], is generated by assemblage of two
types of element matrices: conductivity matrix [K]^, for quadrilateral interior elements and the convection-radiation matrix, [K]e
cr, for boundary elements which are given by
[K]l = k [B]T[B]dV (13) Jv
in which [B] = a matrix consisting of the first derivatives of the element shape functions with respect to x and y
[B] =
dJN]
dx
d[N] (14)
The convection-radiation matrix, [K]lr, of a boundary element e at time
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t can be expressed by
[Kjl = J h[N]T[N]dS (15) S8
in which Se - surface area of the element e; and h(s,t) = overall heat transfer coefficient (Eq. 9). It should be noted that [K]cr is a function of time and hence [K].
The system thermal load vector, {F(t)} is generated by assemblage of element thermal load vector {F(t)}e of individual elements. For an interior quadrilateral element, the vector {FY is
Jv {F}e= Q[N]TdV (16)
Jv
For a boundary element, the vector {F}e = sum of contributions of the heat gain due to solar radiation, {Qs(t)}
e, and convection and irradiation losses, {Qa(t)Y- Thus
{FY = {QsY + {QcrY (17)
in which {QSY = qs[N]TdS, (18) Js<
{QcrY = J Tah[N]TdS (19) Js'
In the analysis, the integrals for the quadrilateral elements (Eqs. 12, 13 and 16) are calculated numerically over each element using Gauss quadrature technique.
NUMERICAL SOLUTION FOR TIME VARIATION OF TEMPERATURE
Eq. 11 is a set of linear differential equations that can be solved for nodal point temperatures as a function of time. Assume that the time span is divided into a number of finite increments each At and that temperatures and thermal loads are varying linearly within each increment (Fig. 2(a)). Over each time increment Af, temperature and thermal loads can be approximated, respectively, by (Fig. 2(b))
{T} = AMI},- + N^i (20)
TIME VARIATION OF TEMPERATURE FOR A TYPICAL NODE •
N l - 7 vN i
a) TIME SPAN DIVIDED INTO FINITE INCREMENTS b> LINEAR SHAPE FUNCTIONS FOR A FINITE TIME ELEMENT At
FIG. 2.-—Variation of Temperature with Time
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{F} = N^i + N,{F}, (21)
[K]{T} = N,([K]{T})i + NydKHTfy (22)
in which N, = 1 (23)
N< = — (24) 'At K ' Application of Galerkin's weighted residual method and choosing N ;
as the weighting function give
Nj{L}dt = {0} '. (25) o
in which {L} represents the residual or error in the solution of Eq. 11 and is given by
{L} = [C] ^ + [K]{T} - {F} (26) at
Substituting Eqs. 20-24 and Eq. 26 into Eq. 25 and performing the integration yield the following recurrence equation
( A l t C ] + i [ K ] - ) { r } - = fetC]-3[X]"){r}" + ; {F}n+i + I {F} (27)
which can be solved to find {T}n+1 from {T}„, provided the initial value {T}i at t = 0 is known. It has been shown that the results obtained from Eq. 27 are more accurate and oscillate less than those obtained from the well-known Crank-Nicholson technique, even when large time increments are used (6).
When solving Eq. 27 for the nodal point temperatures {T}„+1, an approximate value must be adopted for frr,„+i (involved in generation of [K]„+1 and of {F}„+i). It has been shown that hr defined by Eq. 8 is only slightly temperature dependent (13) and consequently iteration can be avoided as /zr,„+i can be extrapolated from the values of hr at time steps n and n — 1 giving
hr,n+1 = 2hr,n - ftr#„_! (28)
COMPUTER IMPLEMENTATION
A computer program FETAB (7) based on the method of analysis just described is developed to calculate the time-dependent temperature distribution within the cross section of a concrete bridge of arbitrary geometry and orientation when subjected to climatological conditions. Time-varying environmental conditions of air temperature, solar radiation intensity, and thermal irradiation are modeled mathematically by equations with variable coefficients which are input into the computer program.
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{
Diurnal Air Temperature Variation.—The diurnal variation of ambient air temperature is assumed to follow a sinusoidal cycle (4,8) between the minimum air temperature, min T„ and the maximum air temperature, max T„. Thus
2 iT( f -9 Ta(t) = A s i n — ^ - 2 d + B ( 2 9 )
in which t = hour of the day; A = amplitude of the sine wave, or one-half the daily range of air temperature, i.e. A = 1/2 (max T„ - min T„); B = average daily temperature, i.e., B = 1/2 (max T„ + min Ta); and £ is a lag factor equal to 9.0, assuming the minimum air temperature to occur at 3:00 a.m. and the maximum air temperature to occur at 3:00 p.m.
In a concrete box section, the temperature of the still air inside the box depends on the time variation of the temperature field of the inside surface of the box. Considering the energy balance between the interior surface and the air enclosed between two cross sections of the box a unit distance apart, the following equation is obtained (similar approach is used in Ref. 4)
h[Ts - Tb]ds = P«c« V>—* (30) o "t
in which Ts(s,t) = inside surface temperature at time t; Tb(t) = air temperature inside the box; s = length of the inner perimeter of the box; p„ = density of air (0.0767 lb/cu ft or 1.228 kg/m3); c„ = its specific heat (0.171 Btu/(lb °F) or 716 J/(kg °Q); and Vb = air volume inside the box. As the interior surface of the box is divided into finite boundary elements, Eq. 30 can be rewritten as
in which nb = number of boundary elements inside the box; As, = length of boundary element i; and hi(t) and Tsi(t) represent, respectively, the average of the two nodal values of the overall heat transfer coefficient and surface temperature. Putting
C=PacaVb (32)
H=J£hiAsi (33) i= l
«b
F = ^hJsiASi (34) i=i
Eq. 31 becomes
dTb C—^ + HTb = F (35)
at
which is of the same form as Eq. 11. Thus, Galerkin's method (Eq. 25) can now be applied to give the following recurrence equation
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1 2 At 3 ' ' Af 3 Tb,n + ^ f B+1
1 (36)
which is analogous to Eq. 27 and from which Th,n+i c a n be obtained knowing Tbi„.
Solar Radiation.—The equations used to calculate the solar radiation intensity are briefly mentioned below; for more detailed presentation see Ref. 4.
The rate of solar energy incident upon a surface normal to the sun rays
h = IKKT (37)
in which Ix = rate at a point on the outer edge of the earth's atmosphere; this rate is called the solar constant, and KT = a transmission coefficient accounting for the attenuation of solar radiation by the atmosphere
KT=0.9 katu/sin (6, + 5°) (38)
in which ka = ratio of atmospheric pressure to pressure at sea level; tu = a turbidity factor accounting for the effect of clouds and air pollution; and 0„ = solar altitude.
When the sun rays make an angle 9 with the normal to the surface, the rate of solar radiation becomes
I = I„ cos i (39)
The angle 0 can be described in terms of several angles defining the position of the sun relative to an observer on the earth and the orientation of the surface relative to the surface of the earth as follows (5) (see Fig. 3)
cos 6 = sin 8 sin <|> cos p - sin 8 cos <|> sin p cos 7 + cos 8 cos <j> cos p cos T
+ cos 8 sin <j> sin p cos 7 cos T + cos 8 sin p sin 7 sin T (40)
in which $ = latitude of the location (north positive); 8 = solar declination, i.e., the angular position of the sun at solar noon with respect to the plane of the equator (north positive); p = angle between the horizontal and the surface; 7 = surface azimuth angle, i.e., the angle between the normal to the surface and the local meridian, the zero point
N0R1Z0NAL OVERHANG PROVIDING SHADE
FIG. 3.—Geometry Defining Incidence Angle of Solar Radiation
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being due south, east positive and west negative; and T = hour angle, solar noon being zero, and each hour equaling 15° of longitude with mornings positive and afternoons negative. The declination 8 can be found by the approximation
/ 284 + D\ 8 = 23.45 sin I 360 I (41)
in which D = day of the year. The solar altitude 9„ is equal to 90 - 9 with 9 calculated by setting p = 0 in Eq. 40, representing a horizontal surface.
Eq. 39 is, of course, applicable only between tsr and t ss, which are the hours of sunrise and sunset, and can be calculated by
1 tsr =12 cos x (-tan 8 tan 4>) (42)
1 and tss = 12 + — cos x (-tan 8 tan <f>) (43)
15 Also, Eq. 39 does not apply on a shaded surface. The height of the shade of the overhanging slab on the web of a box-girder bridge (Fig. 3) is given by
tan9a lsh = lc (44)
sin (90 + 7 - 7') sin p - cos P tan 9„
in which lc = length of the overhanging slab; p = angle between the web and the horizontal; and 7' = azimuth angle of the sun (11)
, /cos 8 sin T \ 7' = sin"1 — (45)
V cos9„ / '
STRESSES DUE TO TEMPERATURE
Temperature stresses can be high in both the longitudinal and transverse directions. In this paper, only the longitudinal stresses are briefly discussed. In a simply-supported bridge the temperature stresses in the longitudinal direction may be calculated assuming the bridge to act as a beam, by (see Fig. 4)
(N Mox Moy \
CENTROSD •/ / / / / f// / SJ-J.,
T ( x . y )
(TENSION) "
M0y
FIG. 4.—Sign Convention for Symbols Employed in Eq. 46
TEMPERATURE RISE
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ZERO REACTIONS AND MOMENTS
- CURVATURE = ^ = ~— Jf Ty dx dy
MULTIPLIERS =
REACTIONS- i ^ ™ i
f 24/19 f 30/19 16/19 f 6/19 ^ 3 0 / 1 9 ^4/19 REACTIONS
^ lO/lO KJ/ 18/19 B/19
FIG. 5.—Reactions and Bending Moments In Continuous Beams Due to Temperature Rise (see Fig. 4)
This is referred to as eigenstress; its value is zero when the temperature distribution over the cross section is planer. Eq. 46 may be proved by the general displacement method of structural analysis (9). Terms N0, Mox and M^ are resultants of stress <J0 = —EaT that would be required if the strain due to temperature is artificially restrained.
N„ = J / v0dx dy (47)
Mox = J ! (T„y dx dy (48)
M-oy = / / a0x dx dy (49)
The thermal curvatures tyx and »|/y induced in the vertical and horizontal directions, respectively, are
Mox
**—wx (50)
and *»=-^ f (51)
In continuous bridges, the curvature \\ix is restrained and statically indeterminate reactions and internal forces will develop producing continuity stresses which must be added to the eigenstresses. Fig. 5 shows the reactions and continuity moments produced by a temperature rise in continuous beams of different number of spans. It can be seen that, for a given bridge, continuity stresses are proportional to the induced curvature fyx.
In the following section the eigenstresses and the curvatures are con-
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c) BOX-GIRDER
FIG. 6.—Bridge Cross Sections Analyzed to Study the Effects of Section Shape and Depth
sidered as indicators of the order of magnitude of stresses in continuous bridges. The effect of various parameters on the temperature distribution and the corresponding eigenstresses and curvatures will be considered.
PARAMETRIC STUDY
Consider the effects of varying the orientation of the bridge axis, air temperature extremes, wind speed, surface cover and shape and dimensions of the cross section on the eigenstresses and the curvatures. The bridge with cross section shown in Fig. 6(c) is assumed to be located at latitude cj> = 51.03° N and altitude 3,445 ft (1,050 m) above sea level (conditions at Calgary, Canada). The daily extremes of ambient air temperature on March 21 are: max Ta = 41° F (5° C) and min Ta = 5° F (-15° C), and the wind speed = 3.28 ft/s (1.0 m/s). The top surface is
TABLE 1 .—Thermal and Elastic Material Properties
Material property (1)
Thermal conductivity, k, Btu/(h ft °F) (W/m °C) .
Specific heat, c, Btu/(lb °F) (J/(kg °Q)
Solar absorptivity, a Emissivity, e Coefficient of thermal
expansion, a, o F - l ( 0 C - 1 )
Density, p, lb/cu ft (kg/m3)
Modulus of elasticity, E, ksi (MPa)
Concrete (2)
0.87 (1.5)
0.23 (960) 0.5 0.88
4.4 X 10~6
(8 X 10"6)
150 (2,400)
3,970 (27,386)
Asphalt (3)
0.54 (0.93)
0.22 (920) 0.9 0.92
11.1 x 10"6
(20 x HT6)
130 (2,100)
14.5 (100)
Deck covered with a thin
layer of snow (4)
—
— 0.15 0.3 —
—
•—
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BOUNDARY ELEMENT
INTERIOR ELEMENT
FIG. 7.—Finite Element Idealization of Half-Cross Section of Fig. 6(c)
concrete and the azimuth angle 7 = 90°. The physical properties of the material(s) are listed in Table 1. The bridge with these conditions is here considered as a reference case for which the results are compared with other cases varying one or the other of the parameters.
The finite element idealization of half-cross section is depicted in Fig. 7. A time increment At of 1.0 hr is used in the analysis. Initial temperatures are assumed uniform at a value equal to the minimum air temperature (at 3:00 a.m.). A 72-hr analysis period is adopted, with the same climatological cycle repeated for the three days. Choice of other initial temperatures, or continuation of the analysis for a number of days more than three, results in little change in the temperature cycles. Thus, the results of the third day presented below are considered representative of the actual conditions on the day considered.
Effect of Season and Orientation of Bridge Axis.—The bridge shown in Fig. 6(c) is analyzed for chosen days of Spring, Winter and Summer. Additional data required for the analysis are given in Table 2. For preparation of this table Eqs. 41-43 are used and the ambient air temperature extremes are assumed using weather records of Calgary, Canada (3). For
TABLE 2.—-Climatological Conditions for Different Seasons
Variable
0) Date and number of the
day in the year, D Solar declination, 8°,
(Eq. 41) Solar constant, Isc,
Btu/(h ft2) (W/m2)(5)
Time at sunrise, tsr, hour (Eq. 42)
Time at sunset, ts,, hour (Eq. 43)
Relative atmospheric pressure, fcj10)
Turbidity factor, t£0)
Spring (2)
March 21 = 81
0.0
429 (1,353)
6:00
18:00
0.885 1.8
Winter (3)
December 21 = 356
-23.45
443 (1,398)
8:15
15:45
0.885 1.8
Summer (4)
June 21 = 173
23.45
414.6 (1,308)
3:45
20:15
0.885 3.5
(a) Ambient Air Temperature Extremes (3)
max T„ min T.
°F CQ °F (°C)
41(5) 5 (-15)
14 (--22 (-
10) 30)
86 (30) 50 (10)
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X
45
E 4 0
35
""30
25
20
^ i d P 15
h — — — Y = 90° _ _ _ _ _ y = 22.5°
I - 6|
4
2
0
'T V SUMMER^ // V
V
•y = o° / / fl y \ \ \
\ . \ \ v w
' / ^SPRING \ \
WINTER ̂
0 2 4 6 MIDNIGHT
J_ I I I 8 10 12 2 4 6
NOON DAY TIME t (HOURS)
_L__ 8 10 12
MIDNIGHT
FIG. 8.—Hourly Variation of Thermal Curvature for Different Orientations and Different Seasons
each of the three seasons, the azimuth angle 7 which defines the orientation of the bridge (Fig. 3) is decreased in steps of 22.5° from the North-South position (7 = 90°) to the East-West orientation (7 = 0°).
Fig. 8 shows the hourly variation of thermal curvature tyx for different orientations of the bridge axis and for different seasons. As can be seen, the effect of bridge orientation on thermal curvature is not very pro-
TEMPERATURE T (°F) EIGENSTRESS a ( Pli)
WINTER CONDITIONS ( ) " 90° )
. -16 0.9 C 1 • —»«i_
j COMPRESSION
o.a i f i
1
/ TENSION j
EIGENSTRESS <r SPRING CONDITIONS ( y » 0 * )
m__. | IP ?
EIGENSTRESS <r {pal}
SUMMER CONDITiONS {Y ' _ _ . - ' )
FIG. 9.—Temperature and Elg@nstress Distributions Along the Web Center Line Corresponding to Maximum Curvatures for Different Seasons
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1,-1 f t - l 40
35
30
ID"
o & X
20
CU
RVA
TU
5
5
" 12
-
8
-" 6 \ \
_ 4
2
s.v-. \ ^ - v* . X,
I 1 1 j , 1 °0 2 4 6 MIDNIGHT
//^"^\\ •*y~\\"--
/ V\\ / \ \ " \ / \ v / \ v
y /TEMPERATURE RANGE= 36°F(20°C)
/ TEMPERATURE RANGE'54°F (30°C)
TEMPERATURE RANGE = 72°F (40°C)
1 , 1 i 1 • I . 1 • • 8 0 I 2 2 4 6 8 K ) I 2
NOON MIDNK3H1 DAY TIME t (HOURS)
FIG. 10.—Hourly Variation of Thermal Curvature for Different Ambient Temperature Extremes (Spring Conditions, 7 = 0°)
nounced during Spring or Summer, while in Winter, a change of 44% can occur in the maximum curvature when the bridge orientation is changed from 7 = 90° to 7 = 0°. Thermal curvatures are in general greater in Summer than they are in Spring or Winter. This is to be expected because in Summer the intensity of solar radiation received on the deck surface is higher and the shadow provided by the overhanging cantilever slab on the web is longer.
Temperature and eigenstress distributions along the web center line corresponding to maximum curvatures for different seasons are shown in Fig. 9. Compressive stresses occur in the upper and lower layers of the cross section, whereas tensile stresses are present in the web. Tensile stresses may be induced in the upper and lower surfaces at the times of early morning and late evening when these surfaces become cooler
-I6 -12 -8 - 4 0 4 8
— ~ - ^ z ^ ^ — ' "rem w^ 1 1 1 i 1 I
olTEMPERATURE
>
Z 16
1 1
-2.4 -16 -OB 0 0.8 i f i ( M
COMPRESSION
-300 -200 -100
u^J, ' ' ' '
^ " " " 1
/ ' '
/// y/.-•'' TENSION
D 100 200 300 bJEIGENSTRESS (psi)
100 £00 300 C1C0NT1NUITY STRESS (pai)
FIG. 11.—Temperature and Stress Distributions Along the Web Center Line Corresponding to Maximum Curvature (at 4:00 p.m.) for Different Ambient Tempera ture Extremes (Spring Conditions, y = 0°)
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30
25
It!
I 1 0
3 5
0 -5
WIND SPEED = 3.28 ft/s(l.0m/s),
WIND SPEED* 13.12ft/s(4.0m/U
2 MIDNIGHT
I . "r-v-r-:" I . I I 10 12 2
NOON DAY TIME t (HOURS)
8 10 12 MIDNIGHT
FIG. 12.—Hourly Variation of Thermal Curvature for Different Wind Speeds (Spring Conditions, 7 = 0°)
than the interior of the cross section. Eigenstresses are in general larger in Summer than they are in Spring or Winter.
Effect of Daily Air Temperature Extremes.—Increasing the daily range of the ambient air temperature increases both thermal curvatures and eigenstresses (Figs. 10 and 11(b)), but the increase is greater for the latter. An increase in curvature means an increase in continuity stresses. Figs. 11(b) and (c) show the relative magnitudes of the eigenstresses and the continuity stresses for the bridge shown in cross section in Fig. 6(c) assumed as continuous beam over two spans, each 130 ft (40 m). The eigen and continuity stresses can be both tensile over the middle portion of the web and can be significant in a case where the shear stresses due
10 15 20 25 30 35 40 45 50 TEMPERATURE T («F)
200 -100 0 100 200 EIGENSTRESS a ( psi)
WIND SPEED = 3.28 f l /s ( l .0m/s) WIND SPEED = I3. l2ft /s(4.0m/s) WIND SPEED = 41.0 ft/s (12.5 m/s)
AIR TEMPERATURE T„ =40.4 °F (4.66°C)
FIG. 13.—Temperature and Eigenstress Distributions Along the Web Center Line Corresponding to Maximum Curvature (at 4:00 p.m.) for Different Wind Speeds (Spring Conditions, 7 = 0°)
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TABLE 3.—-Convection Heat Transfer Coefficient at Different Surfaces of Box-Girder Bridge for Different Wind Speeds (Adopted from Ref. 10)
Bridge surface (1)
Top surface of concrete deck Asphalt cover Bottom surface of cantilever slab Inside surfaces of box" Outer surfaces of the webs Outer surface of bottom slab
to gravity loads are high. The continuity stresses are shown for a section over the central support of the bridge continuous over two equal spans; these stresses are independent of span length.
Effect of Wind Speed.—The wind speed affects the convection coefficient as shown in Table 3 which is adopted here from Ref. 10. Using these coefficients for the bridge in Fig. 6(c) gives the results depicted in Figs. 12 and 13. It can be seen that higher wind speeds tend to bring the temperature of the bridge surfaces closer to that of the ambient air. Higher wind speeds result in smaller curvatures and consequently reduced continuity stresses. The wind speed has smaller effect on the eigenstresses.
Effect of Surface Cover.—Asphalt has higher absorptivity and emis-sivity coefficients compared to gray concrete surfaces (see Table 1). Also, the presence of a thin layer of fresh snow drastically reduces these coef-
40
35
30
b 5 20
i io
I ,
•i , r i I2F
ASPHALT COVER
SNOW COVER
0 2 MIDNIGHT
10 12 2 4 NOON
DAY TIME I (HOURS)
8 10 12 MIDNIGHT
FIG. 14.-™Hourly Variation of Therma! Curvature for Different Surface Covers on Concrete Deck (Spring Conditions, 7 = 0°)
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mft
0.25- _,
• NO COVER
• 2-IN(50mm) ASPHALT COVER
SNOW COVER
10 15 20 25 30 35 40 45 50 55 TEMPERATURE T (°F)
-2.0
NO COVER
2-IN (50rrm) ASPHALT CC^ER
SNOW COVER
• • w y - f * --200 -100 0 100
EIGENSTRESS <r (psi I
FIG. 15.—Temperature and Eigenstress Distributions Along the Web Center Line Corresponding to Maximum Curvature for Different Surface Covers on Concrete Deck (Spring Conditions, 7 = 0°)
ficients (18). The bridge in Fig. 6(c) (with d = 7.4 ft (2.25 m)) is analyzed adopting the absorptivity and emissivity coefficients in Table 1. The results in Figs. 14 and 15 indicate that the presence of a 2 in. (50 mm) asphalt overlay on the concrete deck results in an increase in the top surface temperature and in the thermal curvature and eigenstresses induced in the cross section. On the contrary, snow reduces each of these variables. A surface cover of dense concrete is therefore more favorable with respect to surface temperature and stresses compared to asphalt cover.
FIG. 16.—Variation of Temperature, Eigenstresses and Curvature in Cross Sections of Fig. 6 for the Same Depth (Summer Conditions, 7 = 22.5°)
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Critical Temperature Fields.—From the results just presented, it can be concluded that the temperature fields that induce the largest curvature and eigenstresses in a concrete box-girder cross section of a given geometry may develop under the following:
1. During Summer when the solar energy received on the deck surface is a maximum and the webs are protected by the shade of the overhangs.
2. When the daily range of ambient air temperature is large. 3. When the wind speed is a minimum. 4. When the deck is covered with an asphalt wearing surface.
Effect of Cross Section Shape Depth.—In this section, solid slab, cellular slab and box-girder cross sections with variable depth are analyzed (Fig. 6). All analyses presented in this section are based on Calgary Summer climatological conditions and bridge axis azimuth angle 7 = 22.5° (conditions which were shown in Figs. 8 and 9 to be unfavorable).
I 10 15 20 25 3035___40 .CO .-2.4 i 6 -Qfi r> OB 16 , (MP°>
W_j i ' ' i 1 1 —L -200-100 0 00 200 50 60 70 80 90 00
TEMPERATURE T (8F) EIGENSTRESS tr (psi)
a) SOLID SLAB. DEPTH = 0.25 m
50 60 TO 60 90 100 TEMPERATURE T CF)
-400 -200 0 200"1
EIGENSTRESS <r (psi)
c) BOX - GIRDER
FIG. 17.™Variation of Temperature and Eigenstresses in Cross Sections of Fig 6 with Different Depths (Summer Conditions, 7 = 22,5°)
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Fig. 16 shows that bridges of the same depth but with the three cross section shapes considered have almost the same temperature distribution, eigenstresses and curvatures. On the other hand, the temperature distribution varies considerably with the section depth. Fig. 17 which gives the results of the analyses for bridges varying in depth d between 0.82 ft (0.25 m) and 7.5 ft (2.25 m) indicates that the eigenstresses are higher when the depth is larger.
The above conclusions may be helpful in the development of code provisions or simplified rules for bridge design.
SUMMARY AND CONCLUSIONS
A method is presented and a reference is made to a computer program which predicts the temperature distribution in concrete bridge cross sections. The data required in the analysis are the latitude and altitude of the structure and its orientation, the cross section geometry, thermal properties of concrete and deck surface cover (if any) and parameters pertaining to the climatological conditions.
The temperature distribution over the cross section of a bridge is generally non-planar and results in longitudinal stresses even in a simply-supported bridge; these stresses are referred to as eigenstresses. Additional continuity stresses also develop due to temperature in a continuous, or statically indeterminate bridge.
The temperature distribution that produces largest eigen and continuity stresses in a concrete bridge of a given cross section develops in Summer when solar radiation is maximum and the length of shade of the overhanging part over the webs is large. Also unfavorable conditions exist when the daily range of ambient temperature is large and the wind speed is minimum. An asphalt deck cover accentuates the stresses due to temperature.
Analyses of cross sections of various depths indicate that bridges with larger depth have larger eigenstresses. The temperature distribution and the corresponding eigenstresses and curvatures are not much different in the three types of bridge sections considered, namely solid or cellular slabs and box-girders.
ACKNOWLEDGMENT
The research work reported in this paper was financially supported by the Natural Sciences and Engineering Research Council of Canada which is greatly appreciated.
APPENDIX.—REFERENCES
1. Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, England, 1959.
2. Crank, J., The Mathematics of Diffusion, Clarendon Press, Oxford, England, 1975.
3. "Daily Data Summaries—No. 24," Atmospheric Environment Service, Canada Ministry of the Environment, Information Canada, Ontario, Canada, June, 1972.
4. Dilger, W. H., Ghali, A., Chan, M., Cheung, M. S., and Maes, M. A., "Tem-
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perature-Induced Stresses in Composite Box-Girder Bridges," Journal of Structural Engineering, ASCE, Vol. 109, No. 6, June, 1983, pp. 1460-1478.
5. Duffle, J. A., and Beckman, W. A., Solar Energy Thermal Processes, John Wiley and Sons, Inc., New York, N.Y., 1974.
6. Elbadry, M. M., "Thermal Response of Concrete Box-Girder Bridges," thesis, presented to the University of Calgary, at Calgary, Alberta, Canada, in 1982, in partial fulfillment of the requirements for the degree of Master of Science in Engineering.
7. Elbadry, M. M., and Ghali, A., "User Manual for Computer Program FETAB: Finite Element Thermal Analysis of Bridges," Research Report No. CE82-10, Dept. of Civ. Engrg., The University of Calgary, Calgary, Canada, Oct., 1982.
8. Emanuel, J. H., and Hulsey, J. L., "Temperature Distributions in Composite Bridges," Journal of the Structural Division, ASCE, Vol. 104, No. ST1, Proc. Paper 13474, Jan., 1978, pp. 65-78.
9. Ghali, A., and Neville, A. M., Structural Analysis—A Unified Classical and Matrix Approach, 2nd ed., Chapman and Hall, London, England, 1978.
10. Kehlbeck, F., "Einfluss der Sonnenstrahlung bei Brilckenbauwerken," Werner-Verlag, Diisseldorf, Germany, 1975.
11. Kreith, F., Principles of Heat Transfer, 3rd ed., Intext Educational Publishers, New York, N.Y., 1973.
12. Leonhardt, F., and Lippoth, W., "Folgerungen aus Schaden an Spannbeton-briicken," Beton-und Stahlbetonbau, Vol. 65, Heft, No. 10, Germany, Oct., 1970, pp. 231-244.
13. Maes, M. A., "Effects of Environmental and Material Characteristics on the Behaviour of Concrete Structures," thesis, presented to the University of Calgary, at Calgary, Alberta, Canada, in 1980, in partial fulfillment of the requirements for the degree of Master of Science in Engineering.
14. McAdams, W. H., Heat Transmission, 3rd ed., McGraw-Hill Book Co., Inc., New York, N.Y., 1954.
15. Myers, G. E., Analytical Method in Conduction Heat Transfer, McGraw-Hill Book Co., Inc., New York, N.Y., 1971.
16. Priestley, M. J. N., "Effects of Transverse Temperature Gradients on Bridges," Central Laboratories, New Zealand Ministry of Works, Report No. 394, Oct., 1971.
17. Segerlind, L. J., Applied Finite Element Analysis, John Wiley and Sons, Inc., 1976.
18. Zarem, A. M., and Erway, D. D., (Eds.), "Introduction to the Utilization of Solar Energy," University of California Engineering and Sciences Extension Series, McGraw-Hill Book Co., Inc., New York, N.Y., 1963.
