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Plate and Shell Analyses 189
M y = ∫ t/2 σ y zdz,( Nm m) (6.2)
⋅
/
t − /2
Twisting moment (per unit length):
⋅
M xy = ∫ t/2 τ xy zdz,( Nm m) (6.3)
/
t − /2
Shear Forces (per unit length):
Q x = ∫ t/2 τ xz dz,( N m) (6.4)
/
t − /2
Q y = ∫ t/2 τ yz dz,( N m) (6.5)
/
t − /2
Maximum bending stresses:
6 M 6 M
) max = ± x ) max =± y (6.6)
(σ x 2 ,(σ y 2
t t
Similar to the beam model, there is no bending stress at the mid-surface and the maxi-
mum/minimum stresses are always at z = ±t/2.
6.2.2 Thin Plate Theory (Kirchhoff Plate Theory)
The thin plate theory is based on assumptions that a straight line normal to the mid-
surface remains straight and normal to the deflected mid-surface after loading (Figure
6.4); that is
γ = γ = 0 (Negligible transverse shear deformations)
yz
xz
w
z x
w
x
FIGURE 6.4
Deflection and rotation after loading of a plate according to Kirchhoff plate theory.
