Thermal Stresses in Beams
In this chapter, based on the Bernoulli-Euler hypothesis, thermal stresses in beams subjected to thermal and mechanical loads are recalled. Thermal stresses in composite and curved beams, and thermal deflections in beams subjected to a symmetrical thermal
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Thermal Stresses in Beams
In this chapter, based on the Bernoulli-Euler hypothesis, thermal stresses in beams subjected to thermal and mechanical loads are recalled. Thermal stresses in composite and curved beams, and thermal deflections in beams subjected to a symmetrical thermal load are treated. Furthermore, solutions for stresses in curved beams are included. Problems and solutions for beams subjected to various temperature field or various boundary conditions are presented. [see also Chap. 23.]
14.1 Thermal Stresses in Beams 14.1.1 Thermal Stresses in Beams We consider the thermal stresses in beams under the Bernoulli-Euler hypothesis.1 The neutral axis passes through the centroid of the cross section of the beam which is defined by y dA
0
(14.1)
A
where dA denotes a small element area of the cross section at a distance y from the neutral plane. The thermal stress is given by σx
αEτ E0 E ρy
(14.2)
where ρ denotes the radius of curvature at the neutral plane and 0 denotes the axial strain at the neutral plane. When the beam is subjected to an axial force N and a 1
The plane which is perpendicular to the neutral axis before deformation remains plane and perpendicular to the neutral axis after deformation. 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_14, © Springer Science+Business Media Dordrecht 2013
317
318
14 Thermal Stresses in Beams
mechanical bending moment MM , the axial strain 0 and the curvature 1/ρ at the neutral plane y 0 are 1 N EA
1 MM EI
0 1 ρ
αEτ (y) dA
(14.3)
A
αEτ (y)y dA
(14.4)
A
where I denotes the moment of inertia of the cross section which is defined by y2 dA
I
(14.5)
A
The thermal stress is
αEτ (y) A1
σx (y)
y
I
MM
N
αEτ (y) dA A
αEτ (y)y dA
(14.6)
A
The thermal stress in the beam with free boundary conditions under only thermal loads is σx (y)
αEτ (y) A1
A
αEτ (y) dA
y I
αEτ (y)y dA
(14.7)
A
The thermal stress in the beam with rectangular cross section with width b and height h is σx (y)
αEτ (y) h1 12y h3
h/2
h/2
αEτ (y) dy
h/2
αEτ (y)y dy
(14.8)
h/2
Next, we consider the thermal stress in the beam subjected to an arbitrary temperature change τ (x, y, z). The thermal stress σx is σx
αEτ (x, y, z) E0 E ρy E ρz y
(14.9)
z
where 0 and ρy , ρz denote the axial strain and the radii of curvature in y and z directions at the centroid of the cross section. When the external force and moments act
14.1 Thermal Stresses in Beams
319
on the beam, the conditions of the equilibrium of both the force and the moments are σx dA
σx ydA
N,
A
MMz ,
A
σx zdA
MMy
(14.10)
A
where N denotes the axial force, and MMy and MMz mean the mechanical bending moments with respect to y and z axes, respectively. z 0 are from The axial strain 0 and the curvatures 1/ρy and 1/ρz at y Eq. (14.10) P (14.11) 0 EA 1 ρy
Iy Mz Iyz My 2) E(Iy Iz Iyz
(14.12)
1 ρz
Iz My Iyz Mz 2) E(Iy Iz Iyz
(
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