Thermal actions or actions due to temperature are indirect actions due to movement of structures in response to temperature changes within specific time interval. Structures react to temperature changes by expanding when temperature rises and shrinking when temperature falls. As for bridges, stresses induced by thermal movement are important design criteria, most especially for serviceability limit state. Consequently, bridges are verified to ensure that thermal movement will not cause overstressing of the structure, either by the provision of movement joints in simple span bridges or by including the effects in the design for integral or continuous bridges.
Categorization of Bridge Decks for Thermal Actions
For the purpose of thermal effects, bridge decks are grouped into
- Steel decks: These types of deck are dubbed deck type 1. Examples are: steel box girder, steel truss or plate girder
- Composite deck: These types of deck are dubbed deck type 2. They are decks that are made from combination of structural steel and concrete. Examples are: concrete deck on steel box, truss or plate girder
- Concrete deck: These types of deck are dubbed deck type 3. Examples are: concrete slab deck, concrete beam, concrete box girder
Constituent Components of Temperature Profiles
The distribution of thermal stress in any structural element can be complex. For simplicity and ease of evaluation, these temperature components can be split into four essential constituent components and treated in isolation. These constituent components of a temperature profile are listed below.
- A uniform temperature component, ΔTu
- A linearly varying temperature difference component about the z-z axis, ΔTmy
- A linearly varying temperature difference component about the y-y axis, ΔTmz
- A non-linear temperature difference component, ΔTE
- . This results in a system of self-equilibrated stresses which produce no net load effect on the element.
Temperature Components on Bridge Deck
For bridge decks, thermal actions are typically represented by two broad components:
- Uniform temperature component
- Temperature difference component
The temperature difference component encompasses three distinct temperature profiles, as illustrated in the diagram below. These profiles are further categorized into vertical and horizontal temperature differences.
The forthcoming sections provide further details on uniform temperature component and temperature difference component on bridge decks.
Uniform Temperature Component
The uniform temperature component refers to a consistent temperature change across the entire cross-section of a bridge deck, affecting its full length uniformly. This uniform change causes the deck to expand or contract. Such movements are either accommodated through the use of expansion joints and bearing supports, which allow free movement, or the resulting forces from restrained movement are accounted for in the structural design.
The extent of expansion or contraction experienced by a bridge is primarily influenced by the maximum (Tₘₐₓ) and minimum (Tₘᵢₙ) shade air temperatures. These values should be derived from isotherms provided in the relevant National Annexes. The characteristic temperatures must represent conditions at mean sea level in open country, with an annual exceedance probability of 0.02, corresponding to a 50-year return period. For other return periods, the isotherm values should be adjusted according to Annex A of EN 1991-1-5.
Once the characteristic shade air temperatures are determined, they are converted into the maximum (Tₑ,ₘₐₓ) and minimum (Tₑ,ₘᵢₙ) effective bridge temperatures using Figure 6.1 of EN 1991-1-5.
Range of Uniform Bridge Temperature Component
Bridge expands up to the maximum bridge temperature and also contracts down to the minimum temperature components from a particular temperature at the stage of restraint. If there is no enough data to confirm the temperature at the stage of restraints, then the average temperature during construction should be taken as the average temperature during construction. Otherwise, the initial temperature should be taken as 10oc as recommended by Annex A of EN 1991-1-5.
Thus, the characteristic value of the maximum contraction range of the uniform bridge temperature component is:
ΔTN,con = To – Tₑ,ₘᵢₙ
and the characteristic value of the maximum expansion range of the uniform bridge temperature component:
ΔTN,exp = Tₑ,ₘₐₓ – To
The overall range of the uniform bridge temperature component is:
ΔTN = Tₑ,ₘₐₓ – Tₑ,ₘᵢₙ
For the design of expansion joints, the standard allows member countries to specify maximum range of uniform bridge temperature component for both expansion and contraction. If no specifications are provided the standard recommends: (ΔTN,exp + 20)ºC and (ΔTN,conc + 20)ºC, for expansion and contraction respectively. However, if initial temperature at which the bearings are set is specified then the recommended values are (ΔTN,exp + 10)ºC and (ΔTN,conc + 10)ºC, respectively.
Estimating Uniform Thermal Movement and stress in Bridges
The thermal movement in bridges can be estimated using the general formular of linear thermal expansion.
ΔL = α⋅L⋅ΔT
While the stress due to restrained thermal movement can be estimated using the below expression:
σ = α x Ec x ΔT
where:
α is the coefficient of thermal expansion
Ec is the elastic modulus
ΔT change in temperature
L is the length of the bridge deck
Temperature Difference Component
The exposure of bridge deck to direct sunlight over a prescribed period of time always result in a maximum heating (top surface warmer) and a maximum cooling (bottom surface warmer) temperature variation. When the top surface is warmer than the bottom it is called positive thermal gradient. Conversely, when the top surface is cooler than the bottom it is called negative thermal gradient.
The temperature difference component can be divided into:
- Vertical temperature difference component
- Horizontal temperature difference component
Vertical Temperature Difference Component
Vertical temperature difference occurs due to non-uniform heating and cooling through the depth of bridge decks. During the day, the top surface (deck slab) is more exposed to sunlight and atmospheric conditions, while the underside (soffit) remains shaded. However, at night or in cold conditions the top surface cool faster than the underside. This creates a temperature gradient between the top and bottom surfaces. These gradients cause differential expansion and contraction within the deck cross-section, leading to internal stresses.
The vertical temperature difference can be classified into two approaches based on whether the non-linear temperature profile is considered:
- Approach 1 (EN 1991-1-5): Assumes a simplified, equivalent linear temperature profile by ignoring non-linearity.
- Approach 2 (EN 1991-1-5): Incorporates a more refined, non-linear temperature profile for greater accuracy.
Vertical Linear Component (Approach 1)
In this approach, the variation of temperature is assumed to be linear over the entire depth of the bridge deck. The effect of vertical temperature differences is considered by using an equivalent linear temperature difference component with ΔTm,heat (for heating) and ΔTm,cool(for cooling) . These values should be applied between the top and the bottom of the bridge deck and are based on a depth of surfacing of 50mm for both road and railway bridges. For depths of surfacing other than 50mm, the values of the temperature component should be multiplied by a modification factor ksur. Recommended values for ksur are given in Table 2.7 of the standard which is reproduced below.
Vertical temperature component with non-linear effects.
In this approach, the temperature variation is represented as a non-linear gradient between the top and bottom surfaces. This refined method better reflects thermal stresses in bridge decks. The heated top tends to expand, but the cooler bottom restrains it, generating non-linear strain. This restraint produces reactive forces, and if these acts eccentrically relative to the deck centroid, they also generate reactive moments. The resulting thermal stresses (forces and moments) must be checked against allowable stress limits.
Diagrams of non-uniform temperature for the three types of bridge decks with recommended values of vertical temperature differences are given in Figures 6.2a – 6.2c of EN 1991-1-5 and are reproduced below. These values are valid for 40 mm surfacing depths for deck type 1 and 100 mm for deck types 2 and 3. For other depths of surfacing Annex B of EN 1991-1-5 should be consulted.
Steps in Estimating Stress in bridge decks Using the Non-linear Approach
Below are the steps in estimating thermal stress in bridges due to vertical temperature difference:
- Determine the vertical temperature difference (across various depth by using figure 6.2a – 6.2c of EN 1991-1-5
- Calculate the cross-sectional properties of the girder (e.g: Area, Second Moment Area, centroidal distance from the top and soffit etc.)
- Determine the restraining stresses at various level in the cross-section using: σ = α x Ec x ΔT
- Calculate the forces F in different segments of the cross section using: F = Average stress x Area
- Calculate the moment due to the restraining forces about the centroidal axis of the beam.
- Calculate the final stress in the cross section as the sum of the initial restraining stress plus the stress due to axial force and moment in order to produce a self-equilibrating stress system.
Horizontal temperature difference component
Horizontal temperature difference component needs not to be taken into account for the design of bridge decks. When seldom necessary, a value of 5ºCis recommended as characteristic value of the linear difference of temperature between the outer edges of the deck.
