Plate Bending Element, W.E. Haisler
Plate Bending Element
1. Introduction
In this section, we consider the development of a rectangular plate finite element that undergoes
bending only. It is assumed that the student is familiar with the general finite element
methodology discussed in Chapter 7 of Allen and Haisler (such as the treatment of plane
stress/strain problems using a 3-node triangular element), or handout A04 on FEM theory.
As in Chapter 7 of A&H and in handout A04, we will follow the following general
procedure for establishing the stiffness matrix for an element (linearly elastic, small strains):
* Identify the appropriate nodal degrees of freedom {q} ,
* Assume the appropriate solution for the displacement field over the element w( x, y ) ,
* Obtain the corresponding shape functions {N ( x, y )} for {q} so that
w( x, y)  {N ( x, y)}T {q} ,
* Write the strains in terms of the nodal dof {q} , i.e., { }  [ B]{q} , where [B] contains
derivatives of the shape functions,
* The element stiffness matrix takes the usual form of [k ]   [ B]T [ D][ B]dV where [D]
V
is the matrix of elastic properties in the stress-strain relation { }  [ D]{ } .
2. Plate Degrees of Freedom and Displacement Function
For the rectangular plate bending element, we choose degrees of freedom at the ends of each
edge which are analogous to that of beam theory, i.e., a transverse displacement and normal
rotation about each axis. Figure 1 shows a rectangular plate with the following degrees of
freedom at each of the four comer nodes: wi , xi , yi where wi ,  xi  wi / y and  yi  wi / x
are the transverse displacement, and rotations about the x and y axis, respectively, at node i.
w2
w3
y
y
 y2
a
a
 x2
2
1
z
3
4
b
2
z
3
x
 x1
b
1
 y1
Fig. 1 Rectangular Plate Geometry and Nodal Degrees of Freedom
1
4
 y4
 x3
x
w4
w1
 y3
 x4
Plate Bending Element, W.E. Haisler
Note that  yi is a vector in the negative y direction. Note that node "1" is located at the lower
left comer of the plate (x=-a, y=-b) and that the nodes are numbered 1,2,3,4 in a CW direction
around the plate. We assume that the plate dimensions are given by 2a and 2b as shown in Figure
1 and that the x-y coordinate system is located at the center of plate. The 12 degrees of freedom
are arranged in the {q} matrix as:
{q}T  {w1  x1  y1 w2  x 2  y 2 w3  x3  y 3 w4  x 4  y 4 }T
(1.1)
We assume that w(x,y) is some function over the plate geometry. A general quartic polynomial
expression for w(x,y) can be written as 15 terms
w( x, y )  a1  a2 x  a3 y  a4 x 2  a5 xy  a6 y 2
 a7 x3  a8 x 2 y  a9 xy 2  a10 y 3  a11x 4  a12 x3 y  a13 x 2 y 2  a14 xy 3  a15 y 4
The expression for w(x,y) is referred to as a complete fourth-order polynomial since it contains
all fourth-order products of x and y. However, consistent with the assumption of 12 degrees of
freedom for the rectangular plate element, it is necessary to choose only the most appropriate 12
terms for w(x,y). Consequently, we will choose the complete set of cubic terms (10 terms) and
the two quartic terms x3 y and xy 3 :
w( x, y )  a1  a2 x  a3 y  a4 x 2  a5 xy  a6 y 2
 a7 x3  a8 x 2 y  a9 xy 2  a10 y 3  a11x3 y  a12 xy 3
(1.2)
Note: the displacement function above is one of the earliest plane bending element formulations
investigated. From the 5 quartic terms, we chose the two terms which are symmetric ( x3 y and
xy 3 ). However, those working in finite element theory have shown that the term x 2 y 2 may be
an important term; however, we cannot include it since we can only choose 12 terms in the
displacement function. FEM developers have utilized many other displacement functions and
some turn out to be much better than the above. Some involve sub-dividing the element into
three triangular regions and patching a displacement field over each triangular element. Most of
this work is an attempt to satisfy stress continuity conditions along the edges and improve the
convergence characteristics of the element for stresses.
In defining the geometry, as well as the displacement functions, it is expedient to define
normalized coordinates  and  such that   x / a and   y / b . In the normalized ( , )
coordinate system, the plate coordinates are given by 1    1 and 1    1 .
In order to obtain the shape functions in the x,y coordinate system, we follow the
standard procedure and write one equation for each of the 12 degrees of freedom as follows"
2
Plate Bending Element, W.E. Haisler
w1  q1  w( x  a, y  b)  a1  a2 (a)  a3 (b)  a4 (a) 2  a5 (a)(b)  a6 (b) 2
 a7 (a)3  a8 (a) 2 (b)  a9 ( a)( b) 2  a10 ( b)3  a11( a)3 ( b)  a12 ( a)( b)3
 x1  q2 
w
 a3  a5 (a)  2a6 (b)  a8 (a) 2  2a9 (a)(b)  3a10 (b) 2
y x  a, y b
(1.3)
 a11 (a)3  3a12 (a)(b) 2
and so forth through the 12th equation:
 y 4  q12 
w
 a2  2a4 (a)  a5 (b)  3a7 (a) 2  2a8 (a)(b)  a9 (b) 2
x xa, y b
 3a11 (a)2 (b)  a12 (b)3
(1.4)
These 12 equations may be solved for the 12 unknown coefficients a1, a2 ,..., a12 (which will be
in terms of the 12 nodal degrees of freedom qi ). Substituting the solution for the coefficients
into w(x,y), and rearranging into the form of
w( x, y)  {N ( x, y)}T {q}
(1.5)
gives the following for some of the shape functions:

N1( x, y)  18 (1  x / a)(1  y / b) 2  x / a  y / b  ( x / a)2  ( y / b) 2

(1.6)
or, in terms of non-dimensional coordinates ( , ) using the transformation   x / a and
  y/b:

N1( ,)  18 (1   )(1  ) 2       2   2

(1.7)
Similarly,
N 2 ( , )  (b / 8)(1   )(1   ) 2 (1   )
N3 ( , )  (a / 8)(1   ) 2 (1   ) 2 (1   )
and so forth through N12 ( ,) . Note that {N} is a (12x1) column matrix (since we have 12
degrees of freedom).
3. Stress and Strain Relations
The stress-strain relations (for plane stress) and strain-displacement relations can be
written as (see the Bending and Extension of Plates discussion):
3
(1.8)
Plate Bending Element, W.E. Haisler
{ }  [ D]  { T }
(1.9)
where
0
1 



[ D] 
 1
0
2 

1 
 0 0 (1  ) / 2 
1 
ET  
and { }  thermal strain 
1 
1   
0 
E
T
(1.10)
and the strain-displacement relations are given by (only have the bending part)
    z  2 w / x 2 
 xx  

{ }   yy     z  2 w / y 2 
  

2
 xy  2 z  w / xy 
(1.11)
The assumed displacement function for w(x,y) may be substituted into the strain expressions to
obtain
{ }  [ B]{q}
(1.12)
where
  z  2 N1 / x 2
 z  2 N 2 / x 2 ...  z  2 N12 / x 2 


(1.13)
[ B]    z  2 N1 / y 2
 z  2 N 2 / y 2 ...  z  2 N12 / y 2 


2
2
2
 2 z  N1 / xy 2 z  N1 / xy ... 2 z  N1 / xy 
In the above, it should be noted that Ni  Ni ( x, y) and that [B] is a (3x12) matrix whose
elements are functions of (x, y and z). For example,
B11  
3
4a 2
z ( x / a)(1  y / b)
B12  0
B21  
(1.14)
3
4b 2
z ( y / b)(1  x / a)
It should be clear that we need to transform these to non-dimensional coordinates ( , ) . For
example, we obtain
3
B11   2 z (1   )
4a
B12  0
(1.15)
B21  
4
3
4b 2
z (1   )
Plate Bending Element, W.E. Haisler
4. Element Stiffness Matrix
We are now ready to construct the stiffness matrix using the general relation
[k ]   [ B]T [ D][ B]dV
V
(1.16)
In this case, the differential volume element dV  tdxdy and the integral is over the range x=-a
to +a and y=-b to +b. However, since [B] is written in terms of the non-dimensional coordinates,
we need to transform the volume integral from (x,y) to ( , ) coordinates. Hence, in nondimensional coordinates, dV  abt d d and integration is over the range   1 to  1 and
  1 to  1 . The stiffness matrix then becomes
[k ]  abt 
1 1

[ B]T [ D][ B]d d
1 1
(1.17)
Note that the element stiffness matrix, [k], is a (12x12) matrix (recall that [B] is (3x12) and [D]
is (3x3)).
It should be noted that the purpose of transforming from dimensional (x,y) to nondimensional ( , ) Cartesian coordinates was two-fold: 1) the form of the displacement function
w(x,y) and resulting [N] matrix is simpler in form, and 2) the integration is simpler because of
non-dimensional limits. In addition, because the limits are -1 to +1, Gaussian quadrature may be
easily used to do the integration numerically.
5. Element Force Matrix
The element force matrix {F} is obtained for the plate element in the same manner as is done for
elements like the CST (see chapter 7, A&H). In this case, the applied load is a distributed
pressure acting normal to the surface (in +z direction). Recall the external potential expression,
V    pz wdA where the integral is over the area, A, which the normal load pz acts (normally
A
2a x 2b). Substituting the displacement function (1.5) into V gives
V    pz wdA   
b  a

b  a
A
p z {N }T {q}dxdy
(1.18)
Transforming to ( , ) and noting that the element dof {q} may be taken outside the integral
gives
1 1
(1.19)
V    ab  pz {N}T d d {q}  {F}T {q}
1 1


The quantity in the parenthesis can be identified as the generalized nodal force matrix {F}.
Hence, we have
{F }  ab 
1 1

1 1
5
p z {N }d d
(1.20)
Plate Bending Element, W.E. Haisler
Note that {F} is a (12x1) column matrix (recall that {N} was also a 12x1 matrix - see equations
(1.5)-(1.8). Note also that {F} contains both forces and moments, i.e., F1 is a force in the z
direction at node 1, F2 is a moment about the x-axis at node 1, F3 is a moment about the y-axis
at node 1 and so on.
If thermal effects are being considered, this will add another term to the element force matrix.
See any finite element text for including this.
6. Assembly of Elements and Global Equations of Equilibrium
The assembly of element stiffness and force matrices to form global equations of equilibrium is
exactly the same as for any other element type.
The boundary conditions for a structure consisting of plate elements is similar to beam elements;
i.e., we may specify that transverse displacement wi , rotation about the x-axis  xi and rotation
about the y-axis  yi are known at global nodes. As with any finite element analysis, care must
be exercised that sufficient boundary conditions are given to prevent rigid body motion.
Physically, this means that when the load is applied, the plate is prevented from translating the x,
y and directions; and from rotating about the x and y-axis as a rigid body. If rigid body motion
is not suppressed, the stiffness matrix will be singular and solution is not possible. Rigid
body modes are permissible only when a dynamic analysis is conducted (mass matrices are
included).
7. Strain and Stress for each Element
Once the global equations of equilibrium have been solved for the global nodal displacements
and rotations, the strain and stress for each element may be obtained. We simply use for each
element { }  [ B]{q} and { }  [ D]  { T } where {q} are displacements (and rotations) for the
element in question and are obtained for the global displacement vector.
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