1.0 TEMPERATURE STRESSES AND STRAINS
1.1 Introduction
Temperature stresses are the stresses that are induced in a body due to changes in temperature.
Temperature stresses are set up in a body when the temperature of the body is raised or lowered and the
body is not allowed to expand or contract freely. But, if the body is allowed to expand or contract freely,
no stresses will be set up in the body. Consider a body which is heated to a certain temperature, Let:
𝐿 = Original length of the body
𝑇 = Rise or fall in temperature
𝐸 = Young’s Modulus of Elasticity
𝛼 = Coefficient of linear thermal expansion
𝛿𝑙 = Extension of the rod due to temperature rise
If the rod is free to expand, then extension of the rod will be given by the formula below:
𝛿𝑙 = 𝛼𝑇𝐿
Figure 3.1
This change in length is shown in Figure 3.1 (a) in which AB represents the original length and BB’
represents the increase in length due to a rise in temperature. Now, suppose an external compressive
load P, is applied at B’ so that the rod is decreased in its length from 𝐿 + 𝛼𝑇𝐿 to 𝐿 as shown in Figure 3.1
(b) and (c), then:
𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ
𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝛼𝑇𝐿
=
𝐿 + 𝛼𝑇𝐿
, 𝛼𝑇𝐿
≈ , 𝑠𝑖𝑛𝑐𝑒 𝛼𝑇𝐿 𝑖𝑠 𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙 𝑖𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛 𝑡𝑜 𝐿
𝐿
∴ 𝜀 = 𝛼𝑇 … (𝑖)
𝜎
𝐵𝑢𝑡 𝐸 =
𝜀
𝑆𝑜 𝜎 = 𝜀 × 𝐸
∴ 𝜎 = 𝛼𝑇𝐸 … (𝑖𝑖)
𝐴𝑛𝑑 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑜𝑟 𝑡ℎ𝑟𝑢𝑠𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑜𝑑, 𝑃 = 𝑆𝑡𝑟𝑒𝑠𝑠 × 𝐴𝑟𝑒𝑎
𝑃 = 𝛼𝑇𝐸 × 𝐴
𝑃 = 𝛼𝑇𝐸𝐴 … (𝑖𝑖𝑖)
If the ends of the body are fixed to rigid supports, so that its expansion is prevented, then compressive
stress and strain will be set up in the body. These stresses and strains are known as temperature stresses
and temperature strains.
𝐸𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 (𝑜𝑟 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝑝𝑟𝑒𝑣𝑒𝑛𝑡𝑒𝑑
∴ 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑛, 𝜀𝑇 =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
𝛿𝑙
=
𝐿
𝛼𝑇𝐿
=
𝐿
𝜀𝑇 = 𝛼𝑇 … (𝑖𝑣)
𝐴𝑛𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑒𝑠𝑠, 𝜎𝑇 = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 × 𝐸
= 𝜀𝑇 × 𝐸
𝜎𝑇 = 𝛼𝑇𝐸 … (𝑣)
1.2 Temperature Stress and Strain when the supports yield
If the supports yield by an amount equal to δ, then the actual expansion will be given by:
𝐸𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑖𝑠𝑒 𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 − 𝛿
= 𝛼𝑇𝐿 − 𝛿
𝐴𝑐𝑡𝑢𝑎𝑙 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝛼𝑇𝐿 − 𝛿
∴ 𝐴𝑐𝑡𝑢𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝐿
1.1 Introduction
Temperature stresses are the stresses that are induced in a body due to changes in temperature.
Temperature stresses are set up in a body when the temperature of the body is raised or lowered and the
body is not allowed to expand or contract freely. But, if the body is allowed to expand or contract freely,
no stresses will be set up in the body. Consider a body which is heated to a certain temperature, Let:
𝐿 = Original length of the body
𝑇 = Rise or fall in temperature
𝐸 = Young’s Modulus of Elasticity
𝛼 = Coefficient of linear thermal expansion
𝛿𝑙 = Extension of the rod due to temperature rise
If the rod is free to expand, then extension of the rod will be given by the formula below:
𝛿𝑙 = 𝛼𝑇𝐿
Figure 3.1
This change in length is shown in Figure 3.1 (a) in which AB represents the original length and BB’
represents the increase in length due to a rise in temperature. Now, suppose an external compressive
load P, is applied at B’ so that the rod is decreased in its length from 𝐿 + 𝛼𝑇𝐿 to 𝐿 as shown in Figure 3.1
(b) and (c), then:
𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ
𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝛼𝑇𝐿
=
𝐿 + 𝛼𝑇𝐿
, 𝛼𝑇𝐿
≈ , 𝑠𝑖𝑛𝑐𝑒 𝛼𝑇𝐿 𝑖𝑠 𝑣𝑒𝑟𝑦 𝑠𝑚𝑎𝑙𝑙 𝑖𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛 𝑡𝑜 𝐿
𝐿
∴ 𝜀 = 𝛼𝑇 … (𝑖)
𝜎
𝐵𝑢𝑡 𝐸 =
𝜀
𝑆𝑜 𝜎 = 𝜀 × 𝐸
∴ 𝜎 = 𝛼𝑇𝐸 … (𝑖𝑖)
𝐴𝑛𝑑 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑜𝑟 𝑡ℎ𝑟𝑢𝑠𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑜𝑑, 𝑃 = 𝑆𝑡𝑟𝑒𝑠𝑠 × 𝐴𝑟𝑒𝑎
𝑃 = 𝛼𝑇𝐸 × 𝐴
𝑃 = 𝛼𝑇𝐸𝐴 … (𝑖𝑖𝑖)
If the ends of the body are fixed to rigid supports, so that its expansion is prevented, then compressive
stress and strain will be set up in the body. These stresses and strains are known as temperature stresses
and temperature strains.
𝐸𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 (𝑜𝑟 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 𝑝𝑟𝑒𝑣𝑒𝑛𝑡𝑒𝑑
∴ 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑛, 𝜀𝑇 =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
𝛿𝑙
=
𝐿
𝛼𝑇𝐿
=
𝐿
𝜀𝑇 = 𝛼𝑇 … (𝑖𝑣)
𝐴𝑛𝑑 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑒𝑠𝑠, 𝜎𝑇 = 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑛 × 𝐸
= 𝜀𝑇 × 𝐸
𝜎𝑇 = 𝛼𝑇𝐸 … (𝑣)
1.2 Temperature Stress and Strain when the supports yield
If the supports yield by an amount equal to δ, then the actual expansion will be given by:
𝐸𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑖𝑠𝑒 𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 − 𝛿
= 𝛼𝑇𝐿 − 𝛿
𝐴𝑐𝑡𝑢𝑎𝑙 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝛼𝑇𝐿 − 𝛿
∴ 𝐴𝑐𝑡𝑢𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝐿
