ABSTRACT
In this chapter, analytical solutions for deflections and stresses of simply supported rectangular plates are developed using the Navier method, the Le´vy method with the state-space approach, and the Ritz method. The Navier solutions can be developed for rectangular plates when all four edges are simply supported. The Le´vy solutions can be developed for rectangular plates with two opposite edges simply supported and the remaining two edges having any possible combination of free, simply supported, and clamped/fixed boundary conditions. The Ritz method can be used to determine approximate solutions for more general boundary conditions, provided that suitable approximation functions that satisfy the geometric boundary conditions can be constructed. The governing equations of rectangular plates are best described using Cartesian
rectangular coordinates, as discussed in Chapter 3. For the linear analysis of plates studied here, we consider plates subjected to transverse distributed load q and thermal loads due to temperature change ∆T . When inplane forces (Nˆxx, Nˆyy, Nˆxy) and the nonlinear terms are (N ) omitted, the equation governing the bending, Eq. (3.8.5), becomes uncoupled from Eqs. (3.8.3) and (3.8.4). For an orthotropic plate, we have
+ 2Dˆ12 ∂4w0 ∂x2∂y2
+ kw0
= q − Ã ∂2MTxx ∂x2
+ 2 ∂2MTxy ∂x∂y
! (6.1.1)
where Dˆ12 = (D12 + 2D66) (6.1.2)
and the thermal moments MTxx, M T xy, andM
T yy are defined by [see Eqs. (3.6.10) and
(3.6.15)] ⎧ ⎨ ⎩
⎫ ⎬ ⎭ =
⎧ ⎨ ⎩ Q11α1 +Q12α2 Q12α1 +Q22α2
⎫ ⎬ ⎭
∆T (x, y, z) z dz (6.1.3)
of Elastic Plates and
ij i coefficients of respectively, of an orthotropic plate, and ∆T (x, y, z) is the temperature increment above a reference temperature (at which the plate is stress free). Note that MTxy is identically zero for isotropic or orthotropic plates. For isotropic plates (D11 = D22 = Dˆ12 = D, α1 = α2 = α), Eq. (6.1.1) simplifies
to
D
µ ∂4w0 ∂x4
+ 2 ∂4w0 ∂x2∂y2
+ ∂4w0 ∂y4
¶ + kw0 = q −
1− ν
à ∂2MT ∂x2
+ ∂2MT ∂y2
! (6.1.4)
where
MT = Eα Z h
∆T (x, y, z) z dz (6.1.5)
Equation (6.1.4) can be expressed in terms of the Laplace operator ∇2 as
D∇2∇2w0 + kw0 = q − 1
(1− ν)∇ 2MT (6.1.6)
Simply supported boundary conditions on all four edges of a rectangular plate can be expressed as
w0(0, y) =0, w0(a, y) = 0, w0(x, 0) = 0, w0(x, b) = 0 (6.1.7)
Mxx(0, y) = 0, Mxx(a, y) = 0, Myy(x, 0) = 0, Myy(x, b) = 0 (6.1.8)
where the bending moments are related to the transverse deflection by
Mxx = − Ã D11
Myy = − Ã D12
! −MTyy (6.1.9)
Mxy = −2D66 ∂2w0 ∂x∂y
where
D11 = E1h
12(1− ν12ν21) , D22 =
12(1− ν12ν21)
D12 = ν12E2h3
12(1− ν12ν21) , D66 =
(6.1.10)
and a and b denote the inplane dimensions along the x and y coordinate directions of a rectangular plate. The origin of the coordinate system is taken at the upper left corner of the mid-plane (Figure 6.1.1).
