Thermal loading

There are two thermal effects which can induce stresses in bridges. The first is a uniform temperature change which results in an axial expansion or contraction. If restrained, such as in an arch or a frame bridge, this can generate significant axial force, bending moment and shear. The second effect is that due to differential changes in temperature. If the top of a beam heats up relative to the bottom, it tends to bend; if it is restrained from doing so, bending moment and shear force are generated.

Uniform changes in temperature result from periods of hot or cold weather in which the entire depth of the deck undergoes an increase or decrease in temperature. Both the draft Eurocode and the British standard specify contour plots of maximum and minimum ambient temperature which can be used to determine the range of temperature for a particular bridge site. The difference between ambient temperature and the effective temperature within a bridge depends on the thickness of surfacing and on the form of construction (whether solid slab, beam and slab, etc.). The American approach is much simpler. In ‘moderate’ climates, metal bridges must be designed for temperatures in the range −18 °C to 49 °C and concrete bridges for temperatures in the range −12 °C to 27 °C. Different figures are specified for ‘cold’ climates.

It is important in bridge construction to establish a baseline for the calculation of uniform temperature effects, i.e. the temperature of the bridge at the time of construction. It is possible to control this baseline by specifying the permissible range of temperature in the structure at the time of completion of the structural form. Completion of the structural form could be the process of setting the bearings or the making of a frame bridge integral. In concrete bridges, high early temperatures can result from the hydration of cement, particularly for concrete with high cement contents. Resulting stresses in the period after construction will tend to be

relieved by creep although little reliable guidance is available on how this might be allowed for in design. Unlike in-situ concrete bridges, those made from precast concrete or steel will have temperatures closer to ambient during construction. The AASHTO code specifies a baseline temperature equal to the mean ambient in the day preceding completion of the bridge. The British Standard and the draft Eurocode specify no baseline.

As is discussed inChapter 4, integral bridges undergo repeated expansions and contractions due to daily or seasonal temperature fluctuations. After some time, this causes the backfill behind the abutments to compact to an equilibrium density. In such cases, the baseline temperature is clearly a mean temperature which relates to the density of the adjacent soil.

In addition to uniform changes in temperature, bridges are subjected to differential temperature changes on a daily basis, such as in the morning when the sun shines on the top of the bridge heating it up faster than the interior. The reverse effect tends to take place in the evening when the deck is warm in the middle but is cooling down at the top and bottom surfaces. Two distributions of differential temperature are specified in some codes, one corresponding to the heating-up period and one corresponding to the cooling-down period. These distributions can be resolved into axial, bending and residual effects as will be illustrated in the following examples. As for uniform changes in temperature, the baseline temperature distribution is important, i.e. that distribution which exists when the structural material first sets. However, no such distribution is typically specified in codes, the

implication being that the distributions specified represent the differences between the baseline and the expected extremes. Transverse temperature differences can occur when one face of a superstructure is subjected to direct sun while the opposite side is in the shade. This effect can be particularly significant when the depth of the superstructure is great.

Cracking of reinforced concrete members reduces the effective cross-sectional area and second moment of area. If cracking is ignored, the magnitude of the resulting thermal stresses can be significantly overestimated.

The effects of both uniform and differential temperature changes can be determined using the method of ‘equivalent loads’. A distribution of stress is calculated corresponding to the specified change in temperature. This is resolved into axial, bending and residual distributions as will be illustrated in the following examples. The corresponding forces and moments are then readily calculated. Methods of analysing to determine the effects of the equivalent loads are described inChapter 3.

Example 2.1: Differential temperature I

The bridge beam illustrated inFig. 2.3is subjected to the differential increase in temperature shown. It is required to determine the effects of the temperature change if it is simply

supported on one fixed and one sliding bearing. The coefficient of thermal expansion is 12×10−6and the modulus of elasticity is 35000 N/mm2.

The applied temperature distribution is converted into the equivalent stress distribution of

Fig. 2.4(a)by multiplying by the coefficient of thermal expansion and the modulus of elasticity. There is an ‘equivalent’ axial force and bending moment associated with any distribution of temperature. The equivalent axial force can readily be calculated as the sum of products of stress and area:

Fig. 2.3 Beam subject to differential temperature change

Fig. 2.4 Components of imposed stress distribution: (a) total distribution; (b) axial component; (c) bending component; (d) residual stress distribution

This corresponds to a uniform axial stress of 579600/(600× 1200)=0.81 N/mm2 as illustrated in Fig. 2.4(b). However, this beam is supported on a sliding bearing at one end and is

therefore free to expand. Thus, there is in fact no axial stress but a strain of magnitude 0.81/35000=23×10−6.

The equivalent bending moment is found by taking moments about the centroid (positive sag):

as illustrated in Fig. 2.4(c). As the beam is simply supported, it is free to rotate and there is in fact no such stress. Instead, a strain distribution is generated which varies linearly in the range ±1.11/35 000=±32×10−6. The difference between the applied stress distribution and that which results in axial and bending strains is trapped in the section and is known as the residual stress distribution, illustrated inFig. 2.4(d). It is found simply by subtracting Figs.2.4(b)and(c)

from 2.4(a).

Example 2.2: Differential temperature II

For the beam and slab bridge illustrated inFig. 2.5(a), the equivalent axial force, bending moment and residual stresses are required due to the differential temperature increases shown in Fig. 2.5(b). The coefficient of thermal expansion is αand the modulus of elasticity is E.

Fig. 2.5 Beam and slab bridge subject to differential temperature: (a) cross-section; (b) imposed distribution of temperature

Table 2.2 Calculation of force

Block Details Force

a 3αE (2.4×0.15)= 1.080αE

b 1.890αE

c 0.150αE

d 0.100αE

Fig. 2.6 Division of section into blocks: (a) cross-section; (b) corresponding imposed stress distribution

By summing moments of area, the centroid of the bridge is found to be, below the top fibre. The bridge is split into two halves, each of area, 0.70 m2and second moment of area, 0.064 86 m4. The temperature distribution is converted into a stress distribution inFig. 2.6and divided into rectangular and triangular blocks. The total tensile force per half is then found by summing the products of stress and area for each block as shown inTable 2.2.

The total force of 3.22αE corresponds to an axial tension of 3.22αE/0.70= 4.60αE. Similarly moment is calculated as the sum of products of stress, area and distance from the centroid as outlined inTable 2.3 (positive sag). The total moment of −0.718αE corresponds to stresses (positive tension) of:

Table 2.3 Calculation of moment

Block Details Moment

a −0.262αE

b −0.506αE

c −0.012αE

d 0.062αE

Fig. 2.7 Resolution of stress distribution into axial, bending and residual components: (a) total distribution; (b) axial component; (c) bending component; (d) residual stress distribution

Hence the applied stress distribution can be resolved as illustrated inFig. 2.7. The residual distribution is found by subtracting the distributions of Figs.2.7(b)and (c)from the applied distribution ofFig. 2.7(a).