Table of contents  
What is Thermal Stress?  
Thermal Stress and Strain  
Thermal Stress in Varying CrossSection  
Thermal Stress in Composite Bars  
Solved Numericals 
Causes of Thermal Stress
Thermal stress occurs under the following conditions:
Suppose a bar that is free to expand. Consider the below figure; assume the crosssectional area A, length of bar L, and coefficient of thermal expansion is ‘α’.
‘t’ is the change in temperature
Δ is the free thermal expansionFree thermal expansion for the above case is given by
Δ = Lαt
If the temperature is reducing then there will be a contraction given by
Δ = Lαt
In this particular condition, no thermal stress is developed as there is no restriction for expansion or contraction. Stress is developed only when there is restriction or resistance to free motion.
Deformation due to force P_{T }= P_{T}L/AE
Deformation due to force P_{T }= ∆
P_{T}L/AE = Lαt
σ_{T }= αtE
where σ_{T} = Thermal Stress
σ_{T} is compressive stress when there is an increase in temperature because when there is a temperature rise, the body tries to expand but boundaries will try to keep it in its original position, so the thermal forces will be compressive.
σ_{T} is tensile when there is a decrease in temperature because when there is a temperature drop, the body tries to contract, but the boundaries will try to keep it in its original position, so the thermal forces will be tensile.
Case 2: Thermal Stress when Support Yield
Free Expansion = Lαt
Net Deformation = Δ – δ (where δ is the support yield or gap)
Net Deformation = Lαt – δ
Deformation due to force P_{T}= P_{T}L/AE
Deformation due to force P_{T} = Net Deformation
Lαt – δ= P_{T}L/AE
Lαt – δ= σ_{T}L/E
The force in both bars is the same.
(Force)_{1} = (Force)_{2}
σ_{1}A_{1}= σ_{2}A_{2}
σ_{1}= σ_{2}A_{2}A_{1}
Free Expansion = Δ = Δ_{1} + Δ_{2}
= L_{1}α_{1}t + L_{2}α_{2}t
Deformation due to force P = PL_{1}/A_{1}E_{1 }+ PL_{2}/A_{2}E_{2}
Deformation in bars due to P = Free Expansion
PL_{1}/A_{1}E_{1 }+ PL_{2}/A_{2}E_{2 }= L_{1}α_{1}t + L_{2}α_{2}t
σ_{1}L_{1}/E_{1 }+ σ_{2}L_{2}/E_{2 }= (L_{1}α_{1 }+ L_{2}α_{2})t
Composite bars are the bars in which there will be bars of different materials whose coefficient of thermal expansion is different and are clubbed together so that they act as a single unit and there will be the same explanation in both bars. As both bars act as a single unit, the deformation in both bars will be the same. Thermal stress in composite bars can be given by the following formula:
Q1. A steel rod with a crosssectional area of 0.25 in^{2} is stretched between two fixed points. The tensile load at 70°F is 1200 lb. What will be the stress at 0°F? At what temperature will the stress be zero? Assume α = 6.5 × 10^{6 }in / (in·°F) and E = 29 × 10^{6} psi.
Solution: For the stress at 0°C:
For the temperature that causes zero stress:
Q2. A steel rod is stretched between two rigid walls and carries a tensile load of 5000 N at 20°C. If the allowable stress is not to exceed 130 MPa at 20°C, what is the minimum diameter of the rod? Assume α = 11.7 µm/(m·°C) and E = 200 GPa.
Solution:
Q3. Steel railroad reels 10 m long are laid with a clearance of 3 mm at a temperature of 5°C. At what temperature will the rails just touch? What stress would be induced in the rails at that temperature if there were no initial clearance? Assume α = 11.7 µm/(m·°C) and E = 200 GPa.
Solution:
Required stress:
37 videos39 docs45 tests

1. What is thermal stress and how does it differ from mechanical stress? 
2. How does thermal stress affect materials with varying crosssections? 
3. What are the factors that influence thermal stress in composite bars? 
4. How can thermal stresses be minimized or controlled in structures? 
5. Why is it important to consider thermal stress in engineering design? 

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