Technical Advice Note ASBD- Method of Calculating the Effect of Differential Temperature

Overview In Autodesk Structural Bridge Design it is possible to define any temperature profile and apply it to a simple or compound section and obtain a set of self-balancing stresses and a relaxing moment and force. What is the theory behind the derivation of these values and what are they used for?

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David Geeves David Geeves Ltd 09/12/2020

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Technical Advice Note

ASBD- Method of Calculating the Effect

of Differential Temperature

Introduction

An arbitrary section with cross sectional area A with a material having an elastic modulus E and a

coefficient of thermal expansion has a non-linear temperature profile applied to it.

From this a stress profile can easily be calculated as z = Etz assuming that the section is fully

restrained.

If the section is part of a simply supported beam then the beam will deflect and the restraining

moments and force will tend to zero leaving just the self-equilibrating stresses. These stresses are

termed the primary differential temperature effects.

For statically indeterminate structures then either restraining moments and forces or equivalent

temperature loads, tu and tg, need to be applied to a structural model to represent the re-

distribution of the restraining forces and moments. These are termed the secondary effects

Basic Theory The fully restrained stress profile can be split into three components

1. A uniform stress distribution – giving rise to a restraining axial force

2. A linear flexural stress distribution – giving rise to a restraining moment

3. A non-linear self-equilibrating stress profile.

The calculation of the restraining force and moment are shown below where fully restrained stresses

f0 are integrated over the area of the section.

The calculation of tu and tg are also illustrated

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Technical Advice Note

ASBD- Method of Calculating the Effect

of Differential Temperature

f = f0 – f1- f2

tu is the equivalent uniform temp

tg is the equivalent temp gradient

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Technical Advice Note

ASBD- Method of Calculating the Effect

of Differential Temperature

Example A simple Composite steel composite girder section using Eurocode temperature profiles

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Technical Advice Note

ASBD- Method of Calculating the Effect

of Differential Temperature

Summary The calculations above are quite straightforward for simple sections with one material property

but are a little more complicated with compound sections, such as composite beams, as the

varying material propertied need to be taken into the integral.

With regards to how these values are used in the beam design when considering a refined

analysis (not a line girder analysis):

The stress profile f is used as the primary differential temperature stresses in the beam.

The secondary stresses are determined by first applying relaxing forces and moments

(the negative of the restraining forces and moments, F and M from above) to a

structural analysis model of the beam(s). This can be done in one of three ways:

1. For grillages, apply the relaxing moments as “Temperature Primary Loads” to

each longitudinal beam. This is suitable for spans that have no redundancy in

the axial direction of the beams as you can only apply moments

2. For Grillages, apply the relaxing moments and forces as “Beam Element Loads”

which can be added as point loads, in the local axes direction, at either end of

the beam element with an opposite signs in value. Both moments and axial

forces can be applied here but the sign convention is not very clear because the

loads are applied with respect to the local axis direction and the local axes may

not be consistent. Even with beams all in the x-y plane, where the local x axis is

consistent, the “J” and “K” ends of the beam may be different. However, if care

is taken then this method is very effective for grillage type structures.

3. Convert the relaxing moments and forces to equivalent uniform and gradient

temperature loads, as shown above. For grillages, these values van be applied

as “beam member temperature loads” and the sign convention is a lot clearer

(+ve axial temperatures for heating and +ve gradient temperature where the

top of the element is hotter than the bottom).

For other forms of structural model, such as offset beams, FE Webs, and full

finite elements then this method is also suitable, as temperature loads can be

applied to FE element as well as beam members. In general, the design beam

will be represented by a virtual beam member in these forms of analysis. So this

means there is a little work to do to determine the temperature loads for the

individual virtual member components but this will be based upon the uniform

and gradient temperature for the whole virtual beam section as described in the

body of this report.

4. Initial strain loads can also be used instead of temperature loads but they do

exactly the same thing as temperature loads and the strains can be obtained

directly from the temperatures and thermal coefficients.

Both Primary and secondary stresses should then be included and added in any limit

state stress validation in combinations which include temperature load effects.