Page 91 - Finite Element Analysis with ANSYS Workbench

82 Chapter 5 Plate Bending analysis

an application example at the end of the chapter to demonstrate the
advantages of the finite element method.

5.1 Basic Equations

5.1.1 Differential Equation
Derivation of the differential equation for plate bending
is similar to that for the beam bending as explained in Chapter 3. A
thin plate with its thickness of h that lies in the x-y plane is shown
in the figure. The plate is subjected to the pressure of (, )p x y on
the upper surface causing the deflection of w in the z-direction and
the in-plane displacement of u and v in the x- and y-direction,
respectively.

xy
p (, )

u
x

v w

y h
z

The basic assumption of plane section remains plane
z w x
before and after deflection leads to the relations of u   
and v  z w y . Together with the additional assumption of the


deflection w that varies with x and y only, w  w (, ) , the
x
y
governing differential equation representing equilibrium condition
of plate bending can be derived in the form,
M x  2   2 2 M xy   2 M y  
p
x  2  xy  y  2

where the bending moment components are,