Thermal Stresses in Bars
In this chapter the concept of thermal stresses in bars is introduced for the simple case of a perfectly clamped bar subjected to arbitrary temperature change. The problems and solutions related to thermal stresses in bars are: a perfectly clamped bar, a
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Thermal Stresses in Bars
In this chapter the concept of thermal stresses in bars is introduced for the simple case of a perfectly clamped bar subjected to arbitrary temperature change. The problems and solutions related to thermal stresses in bars are: a perfectly clamped bar, a clamped bar with a small gap, a clamped circular frustum, a bar with variable cross-sectional area, two bars attached to each other, three bars fastened to each other, truss of three bars, and three bars hanging from a rigid plate.
13.1 Thermal Stresses in Bars When the temperature of a circular bar of length l changes from an initial temperature T0 to its final temperature T1 , the free thermal elongation λT of the bar is defined by λT = α(T1 − T0 )l = ατl
(13.1)
where α is the coefficient of linear thermal expansion which is measured in one per one degree of the temperature 1/K, and τ denotes the temperature change given by τ = T1 − T0
(13.2)
The free thermal strain is given by T =
λT = ατ l
(13.3)
When an internal force and the temperature change act simultaneously in the bar, the normal strain is given by = s + T (13.4)
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_13, © Springer Science+Business Media Dordrecht 2013
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13 Thermal Stresses in Bars
Fig. 13.1 A perfectly clampled bar
where s denotes the strain produced by the internal force. The strain s produced by the internal force is proportional to the normal stress σ σ E
s =
(13.5)
where E denotes Young’s modulus. Hooke’s law with the temperature change is =
σ + ατ E
(13.6)
When a perfectly clamped bar with length l and cross-sectional area A, shown in Fig. 13.1, is subjected to the uniform temperature change τ , the thermal stress is σ = −αEτ
(13.7)
If the temperature change τ (x) is a function of the position x, the free thermal elongation λT of the bar of length l is λT =
l
dλT =
0
The thermal strain T is
l
ατ (x) d x = α
τ (x) d x
(13.8)
0
α λT = T = l l
l
τ (x) d x
(13.9)
0
The thermal stress in the perfectly clamped bar is αE σ=− l
0
l
τ (x) d x
(13.10)
13.2 Problems and Solutions Related to Thermal Stresses in Bars
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13.2 Problems and Solutions Related to Thermal Stresses in Bars Problem 13.1. If the temperature in a mild steel rail with length 25 m is raised to 50 K, and the coefficient of linear thermal expansion for mild steel is 11.2×10−6 1/K, what elongation is produced in the rail? Solution. The elongation λT is from Eq. (13.1) λT = ατl = 11.2 × 10−6 × 50 × 25 = 14 × 10−3 m = 14 mm
(Answer)
Problem 13.2. The temperature of a bar of length 1 m of mild steel is kept at 300 K. If the temperature at one end of the bar is raised to 380 K and at the other end to 480 K, and the temperature distribution is linear along the bar, what elongation is produced in the bar? The coefficient of linear thermal expansion for mild steel is 11.2 × 10−6 1/K. Solution. The temperature rise τ (x) = T1 (x) − T0 is x τ (x)
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