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Steel Designers' Manual - 6th Edition (2003)
Analysis of skeletal structures 303
the out-of-plane displacement is not zero, and this condition is referred to as plane
stress idealization.
For an isotropic plate, the general equation relating the displacement, w,
perpendicular to the plane of the plate element is given by
∂ ∂
+ + (9.10)
∂ ∂ ∂
where q is the normal applied load per unit area in the z direction which will, in
general, vary with x and y. The term D is the flexural rigidity of the plate, given by
3 Et
D =
(9.11)
2
12( 1 - � )
The main difficulty in using this approach lies in the choice of a suitable
displacement function, w, which satisfies the boundary conditions. For loading
conditions other than the simplest, an exact solution of this differential equation is
virtually impossible. Hence approximate methods (e.g. multiple Fourier series) are
utilized. Once a satisfactory displacement function, w, is obtained, the moments per
unit width of the plate may be derived from
∂
4
4
4
w w w q
2
=
4
2 2 4
x x y ∂y
D
M D w w
x y
M D w
∂
x =- +
∂
w
y
y x
w
Mxy Myx D
x y
∂ Ê
ˆ
Á
˜
Ë ∂ ¯
∂
=- +
∂
∂ Ê
ˆ
Á
˜
Ë ∂ ¯
=- = ( - ) ∂
2
2
� 2
2
2
2
� 2
2
2
1 �
∂ ∂
(9.12)
For orthotropic plates, the stiffness in x and y directions is different and the
equations are suitably modified as given below:
D
(9.13)
where Dx and Dy are the flexural rigidities in the two directions.
In view of the difficulty of using classical methods for the solution of plate
problems, finite element methods have been developed in recent years to provide
satisfactory answers.
w
4
4
4
∂
∂ w ∂ w
x + 2Dxy
+ Dy = q
4
2 2
4
∂x
∂x∂y ∂y
9.5 Analysis of skeletal structures
The evaluation of the stress resultants in members of skeletal frames involves the
solution of a number of simultaneous equations. When a structure is in equilibrium,
every element or constituent part of it is also in equilibrium. This property is made
use of in developing the concept of the free body diagram for elements of a structure.