Thermal Stresses in Beams
Beam are one of the basic elements of structural design problems. Thermal stresses in beams are discussed in this chapter.
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Thermal Stresses in Beams
Beam are one of the basic elements of structural design problems. Thermal stresses in beams are discussed in this chapter. The elementary beam theory is employed and the equations for thermal stress and thermal deflection are presented.
23.1 Thermal Stresses in Beams In accordance with Euler-Bernoulli hypothesis, a beam deflects in such a way that its plane sections remain plane after deformation and perpendicular to the beam neutral axis. Now, consider a beam under axial and lateral loads in x-y plane, as shown in Fig. 23.1 Consider two line elements of the beam, EF and GH, which are straight and along the axial direction with equal lengths before the load is applied. Element EF lies on the neutral axis, while the element GH is at a distance y from the neutral axis. The beam is assumed to be under the bending and axial loads so that it deflects in lateral direction. Considering Euler-Bernoulli hypothesis, the elongation of EF and GH elements may be written as E F = (1 + 0 )EF H = (1 + )GH G H ry + y G = ry EF
(23.1)
where and 0 are strains of GH and EF elements, respectively, and ry is the radius of curvature of the beam axis at y = 0 in the xy-plane. Dividing the second of Eq. (23.1) by the first equation and using the last of Eq. (23.1) gives 1+
(1 + ) y = ry (1 + 0 )
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_23, © Springer Science+Business Media Dordrecht 2013
(23.2)
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23 Thermal Stresses in Beams
Fig. 23.1 Deflection of an element of beam
Then, using the small deformation theory, yields = 0 +
y y y + 0 ∼ = 0 + ry ry ry
(23.3)
Now, consider a beam with thermal gradients along the y and z-directions. In accordance with Euler-Bernoulli assumption, the axial displacement is a linear function of the coordinates y and z in the plane of cross section of the beam. Thus u = C1 (x) + C2 (x)y + C3 (x)z
(23.4)
where C1 , C2 , and C3 are coefficients, which are functions of x, the beam axis. Assuming thermal loading only, these coefficients may be obtained using the boundary conditions. Since the beam is in static equilibrium, the axial force and bending moments in y and z directions must vanish. These conditions in terms of the axial stress in the beam yield the following relations:
σxx dA = 0, A
σxx ydA = 0, A
σxx zdA = 0
(23.5)
A
where dA = dydz. Equations (23.5) are sometimes called the equilibrium equations of the beam. In order to find C1 , C2 , and C3 , the axial strain is written from Eq. (23.4) as xx =
dC1 dC2 dC3 du y z = + y+ z = 0 + + dx dx dx dx ry rz
(23.6)
where ry and rz are the radii of curvatures of the beam axis in xy and xz planes, respectively, and 0 is the axial strain of the beam on the x-axis. The stress, according to Hooke’s law is (23.7) σxx = E(xx − αθ)
23.1 Thermal Stresses in Beams
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where θ = T − T0 . Thus σxx = E[0 +
y z + − αθ] ry rz
(23.8)
Substituting Eq. (23.8) in Eq. (23.5) and no
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