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Thermal Stresses in Plates

In this chapter the thermal stresses and the deflection in thin rectangular plates subjected to the temperature change in the thickness direction only are recalled. The basic equations are developed with the Kirchhoff-Love hypothesis. Next, the basic equations for the thermal bending of circular plates with various boundary conditions are summarized. A number of problems for rectangular and circular plates are presented.

20.1 Basic Equations for a Rectangular Plate We consider a thermal stress in a plate shown in Fig. 20.1, due to uniform temperature in flat surface as a simple case of thermal stresses in plates. When the temperature change τ varies in the thickness direction only, and the plate is subjected to the same bending along both x and y axes, the strain components x and  y are x =  y = 0 +

z ρ

(20.1)

where 0 and ρ denote the in-plane strain and the radius of curvature at the neutral plane of z = 0, respectively. When the plate is subjected to the in-plane force P per unit length and the bending moment M M per unit length in the x and y directions, the thermal stress component σx (= σ y ) is given by  12M M 1 h/2 P αE  + −τ (z) + z+ τ (z) dz σx (= σ y ) = h h3 1−ν h −h/2   12z h/2 + 3 τ (z)z dz (20.2) h −h/2

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_20, © Springer Science+Business Media Dordrecht 2013

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20 Thermal Stresses in Plates

Fig. 20.1 A plate

Fig. 20.2 Displacement

For the pure thermal stress problems without external forces, Eq. (20.2) reduces to    1 h/2 αE  12z h/2 −τ (z) + τ (z) dz + 3 τ (z)z dz (20.3) σx (= σ y ) = 1−ν h −h/2 h −h/2 It is seen that the thermal stress given by Eq. (20.3) for the plate are 1/(1 − ν) times the values for the beam given by Eq. (14.8). Next, we discuss the general treatment of the thermal bending problems of an isotropic thin plate with thickness h under Kirchhoff-Love hypothesis that the plane initially perpendicular to the neutral plane of the plate remains a plane after deformation and is perpendicular to the deformed neutral plane. Referring to Fig. 20.2, the displacement components u  and v  in the in-plane direction x and y at the arbitrary point z of the plate are u = u − z

∂w , ∂x

v = v − z

∂w ∂y

(20.4)

where u, v, and w are displacement components in the x, y, and z direction at the neutral plane (z = 0). The strain components in the in-plane direction are ∂u  = ∂x ∂v  = = ∂y 1  ∂u  = 2 ∂y

x x =  yy x y

∂u ∂2w −z 2 ∂x ∂x ∂v ∂2w −z 2 ∂y ∂y   1  ∂u ∂2w ∂v ∂v  = −z + + ∂x 2 ∂y ∂x ∂x∂ y

(20.5)

20.1 Basic Equations for a Rectangular Plate

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Hooke’s law is  1  σx x − νσ yy + ατ E  1  σ yy − νσx x + ατ = E 1 1+ν = σx y = σx y 2G E

x x =  yy x y

(20.6)

The stress components are  ∂2w  ∂v E  ∂u ∂2w  + ν − z − (1 + ν)ατ + ν 1 − ν 2 ∂x ∂y ∂x 2 ∂ y2  ∂2w  ∂u E  ∂v ∂2w  + ν − z − (1 + ν)ατ = + ν 1 − ν2 ∂ y ∂x ∂ y2 ∂x 2  ∂u E ∂v ∂2w  = + − 2z 2(1 + ν) ∂ y ∂x ∂x∂ y

σx