Thermal Stress and Strain

Topics

• Introduction to thermal stress

• Coefficient of thermal expansion

• Mathematical approach of thermal stress

• Impact of thermal stress at real world (bridge concrete and steel)

Introduction to thermal stress

         When a material is exposed to change in temperature it experiences a change in dimension. In the absence of constraints, this change in dimension is called as thermal expansion or contraction, when this expansion or contraction is restricted, a stress is induced in the material called as ‘thermal stress’.

         Thermal expansion of a material is directly proportional to the change in temperature,

ɛ α ΔT

         To remove the proportionality, a constant known as ‘coefficient of thermal expansion’ is introduced.

ɛ = α.ΔT

         Thermal stress = Thermal strain x Youngs Modulus = α.ΔT.E

Coefficient of thermal expansion

         Coefficient of thermal expansion is the measure of change in dimension of a material per unit change in temperature.

         Coefficient of thermal expansion is categorized in to three categories

• Coefficient of linear thermal expansion

• Coefficient of areal thermal expansion

• Coefficient of volumetric thermal expansion

Unit of coefficient of thermal expansion is K-1

Mathematical Approach of thermal stress

         Thermal stress and expansion have a great impact on real world scenario, making it crucial to mathematically analyse the values of thermal stress and thermal expansion to prevent failure.

         Here are some real-world cases where thermal expansion is considered,

• Gaps in railway track to prevent misalignment of track due to thermal expansion. 

• Expansion joints in bridges to prevent formation of cracks due to thermal stress

• Application of tooth filler in to layers to prevent thermal contraction etc.,

In this article we will consider three cases of thermal expansion

• Thermal stress on bars with linearly varying cross section

• Thermal stress on sections with partial restriction

• Thermal expansion sections in series 

• Thermal expansion of compound sections

Thermal stress on sections with partial restriction

Consider the bar of length L, with a space of value ‘a’ provided for thermal expansion when the value of thermal expansion exceeds the value ‘a’, a stress is created on it to determine the value of thermal stress the following method is used.

Value of thermal expansion is given by the equation Δl = L.α.ΔT                ……….eq (1)

Strain (ɛ) = Δl / L

Since the bar is permitted to expand for a length of ‘a’ stress is developed for the change in length which exceeds ‘a’ thus strain = (L.α.ΔT – a) / L                                    ……….eq (2)

Stress = strain x youngs modulus                                                                  ……….eq (3)

Substituting eq (2) and eq (1) on eq (3) we get,

                   σ = (L.α.ΔT – a).E / L

Thermal stress on bars with varying cross section

Consider a bar of length L with diameter d at one end and D at the other end with a linear increment, change in length for a value of load P on the bar is given by,

         Δl= 4PL / E.D.d.π                                      

Can also be written as,

         P = Δl.E.D.d.π / 4.L                                    ………eq(4)

Δl / L can be written as L.α.t for thermal strain

substituting the value of Δl / L on eq(4) we get,

P = α.t.E.D.d.π / 4

σ = P/A

σ = α.t.E.D.d.π / 4   x  4 / πd’ ^2

for σmax      ,d’ = d

σmax = α.t.E.D / d

Thermal expansion sections in series 

When bars of various cross sections are connected in series thermal stresses and strain experienced by each bar varies, following method is adapted to find out required parameter from the give values for thermal stress on bars at series.

Points to be considered

• Total strain is equal to the sum of stain of individual bars.

• Yielding due to thermal stress must be subtracted to the total strain since they don’t contribute to the development of stress on the bar.    

By substituting equation for strain on the relation various equations can be derived based on the requirement.

Thermal expansion of compound sections

Compound sections are materials that consist of layers of materials with various coefficient of thermal expansion.

To solve the problems on this topic the following concept is adapted

Consider a bar with cross section containing two different material one at the inner and the other at the outer.

         Strain on both the materials remain constant.

         Thermal strain on material 1 + Tensile strain on material 1 = Thermal strain on material 2 - Compressive strain on material 2.

         By substituting the equations for the strain required relations can be obtained.

Note: In the above case coefficient of expansion on material 2 is considered to be greater than that of material 1.