Thermally Induced Instability
In this chapter the thermoelastic buckling of beam-columns subjected to both in-plane and lateral loads is recalled for a built-in edge, a simply supported edge and a free edge. Furthermore, the thermoelastic buckling of rectangular and circular plates is
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Thermally Induced Instability
In this chapter the thermoelastic buckling of beam-columns subjected to both in-plane and lateral loads is recalled for a built-in edge, a simply supported edge and a free edge. Furthermore, the thermoelastic buckling of rectangular and circular plates is also recalled. The problems and solutions for the buckling behavior of beam-columns with various boundary conditions are given. The stress-displacement relations, the relations between the resultant forces and the stress function are treated in illustrative problems.
21.1 Instability of Beam-Column When the deflection w of the beam-column is produced in the z axial direction, the fundamental equation for thermal bending problems is My d2w =− dx 2 EIy
(21.1)
My = MMy + MTy
(21.2)
where MTy =
αEτ (x, z)z dA
(21.3)
A
Iy =
z2 dA
(21.4)
A
in which My is the total bending moment about y axis, MMy is the bending moment about y axis due to the force, MTy is the thermally induced bending moment about y axis, and Iy is the moment of inertia about the y axis. The boundary conditions are expressed as follows: 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_21, © Springer Science+Business Media Dordrecht 2013
535
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21 Thermally Induced Instability
[1] A built-in edge w = 0,
dw =0 dx
(21.5)
[2] A simply supported edge w = 0, EIy
d2w + MTy = 0 dx 2
(21.6)
[3] A free edge EIy
d2w d3w dw dMTy + MTy = 0, EIy 3 + P + =0 2 dx dx dx dx
(21.7)
where P means the axial compressive force. The bukling load Pcr for each boundary condition is
Pcr
⎧ ⎪ π 2 EIy ⎪ ⎪ ⎪ ⎪ l2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ π EIy ⎨ 4l 2 = ⎪ ⎪ 2.046π 2 EIy ⎪ ⎪ ⎪ ⎪ l2 ⎪ ⎪ ⎪ ⎪ ⎪ 4π 2 EIy ⎪ ⎩ l2
for simply supported edges for a cantilever beam-column (21.8) for simply supported edge and built-in edge for built-in edges
21.2 Instability of Plate The governing equation of the thermal stress function F, which is available to the in-plane deformation is (21.9) ∇ 2 ∇ 2 F = −∇ 2 NT where NT means the thermally induced resultant force defined by Eq. (20.9), and ∇2 =
∂2 ∂2 + ∂x 2 ∂y2
(21.10)
Here, the thermal stress function F is defined by Nx =
∂2F ∂2F ∂2F , Ny = , Nxy = − 2 2 ∂y ∂x ∂x∂y
(21.11)
21.2 Instability of Plate
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and Nx , Ny , Nxy denote the resultant forces per unit length of the plate defined by Eq. (20.8). The boundary conditions for the in-plane deformation are [1] A built-in edge un = 0, us = 0
(21.12)
Nn = 0, Nns = 0
(21.13)
[2] A free edge where ui (i = n, s) mean in-plane displacement components, n and s are the normal and tangential directions of the boundary surface of the plate. On the other hand, the governing equation of deflection w is ∇2∇2w =
1 ∂2w ∂2w ∂2w 1 p− ∇ 2 MT + Nx 2 + Ny 2 + 2Nxy D 1−ν ∂x ∂y ∂x∂y
(21.14)
where p is the lateral load, D means the bending rigidity defined by Eq. (20.12) and MT denotes thermally induced resultant moment defined by Eq. (20.9). The boundary conditions for the out-plane deformation are [1] A built-in edge
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