Rectangular element with corner nodes (12 degrees of freedom)

4.3.1 Shape functions

Consider a rectangular element of a plate ijkl coinciding with the xy plane as shown in Fig. 4.7. At each node, n, displacements a_nare introduced. These have three compo-nents: the ®rst a displacement in the z direction, w_n, the second a rotation about the x axis, …^_x†_n and the third a rotation about the y axis …^_y†_n.^

The nodal displacement vectors are de®ned below as a_i. The element displacement will, as usual, be given by a listing of the nodal displacements, now totalling twelve:

a^eˆ

A polynomial expression is conveniently used to de®ne the shape functions in terms of the 12 parameters. Certain terms must be omitted from a complete fourth-order

Note that we have changed here the convention from that of Fig. 4.3 in this chapter. This allows transfor-mations needed for shells to be carried out in an easier manner. However, when manipulating the equations of Chapter 5 we shall return to the orginal de®nitions of Fig. 4.3. Similar diculties are discussed by Hughes,⁵⁰and a simple transformation is as follows:

^h ˆ Th where T ˆ 0 1 ÿ1 0

 

124 Plate bending approximation

polynomial. Writing

w ˆ ₁‡ ₂x ‡ ₃y ‡ ₄x²‡ ₅xy ‡ ₆y²‡ ₇x³‡ ₈x²y

‡ ₉xy²‡ ₁₀y³‡ ₁₁x³y ‡ ₁₂xy³

 Pa …4:39†

has certain advantages. In particular, along any x constant or y constant line, the displacement w will vary as a cubic. The element boundaries or interfaces are composed of such lines. As a cubic is uniquely de®ned by four constants, the two end values of slopes and the two displacements at the ends will therefore de®ne the displacements along the boundaries uniquely. As such end values are common to adjacent elements continuity of w will be imposed along any interface.

It will be observed that the gradient of w normal to any of the boundaries also varies along it in a cubic way. (Consider, for instance, values of the normal @w=@y along a line on which x is constant.) As on such lines only two values of the normal slope are de®ned, the cubic is not speci®ed uniquely and, in general, a discon-tinuity of normal slope will occur. The function is thus `non-conforming'.

The constants ₁ to ₁₂ can be evaluated by writing down the 12 simultaneous equations linking the values of w and its slopes at the nodes when the coordinates take their appropriate values. For instance,

w_iˆ ₁‡ ₂x_i‡ ₃y_i‡

@w

@y

 

iˆ ^_xiˆ ₃‡ ₅x_i‡


ÿ @w

@x

 

iˆ ^_yiˆ ÿ₂ÿ ₅y_iÿ


Listing all 12 equations, we can write, in matrix form,

a^eˆ Ca …4:40†

l k

2a

2b

i j ξ

η

z

y

x w (f_w) θˆy (f_θy)

θˆx (f_θx) Forces and corresponding displacements

Fig. 4.7 A rectangular plate element.

Rectangular element with corner nodes (12 degrees of freedom) 125

where C is a 12  12 matrix depending on nodal coordinates, and a is a vector of the 12 unknown constants. Inverting we have

a ˆ C^ÿ1a^e …4:41†

This inversion can be carried out by the computer or, if an explicit expression for the sti€ness, etc., is desired, it can be performed algebraically. This was in fact done by Zienkiewicz and Cheung.²⁶

It is now possible to write the expression for the displacement within the element in a standard form as

u  w ˆ Na^eˆ PC^ÿ1a^e …4:42†

where

P ˆ …1; x; y; x²; xy; y²; x³; x²y; xy²; y³; x³y; xy³†

The form of the B is obtained directly from Eqs (4.28) and (4.33). We thus have

Lrw ˆ We can write the above as

Lrw ˆ Qa ˆ QC^ÿ1a^eˆ Ba^e and thus B ˆ QC^ÿ1 …4:43†

It is of interest to remark now that the displacement function chosen does in fact permit a state of constant strain (curvature) to exist and therefore satis®es one of the criteria of convergence stated in Volume 1.^

An explicit form of the shape function N was derived by Melosh³⁶ and can be written simply in terms of normalized coordinates. Thus, we can write for any node

N^T_i ˆ¹₈…1 ‡ ₀†…1 ‡ ₀†

with normalized coordinates de®ned as:

 ˆx ÿ x_c

a where ₀ˆ _i

 ˆy ÿ y_c

b where ₀ˆ _i

If 7to 12are zero, then the `strain' de®ned by second derivatives is constant. By Eq. (4.40), the corre-sponding a^ecan be found. As there is a unique correspondence between a^eand a such a state is therefore unique. All this presumes that C^ÿ1does in fact exist. The algebraic inversion shows that the matrix C is never singular.

126 Plate bending approximation

This form avoids the explicit inversion of C; however, for simplicity we pursue the direct use of polynomials to deduce the sti€ness and load matrices.

4.3.2 Stiffness and load matrices

Standard procedures can now be followed, and it is almost super¯uous to recount the details. The sti€ness matrix relating the nodal forces (given by lateral force and two moments at each node) to the corresponding nodal displacement is

K^eˆ …

^eB^TDB dx dy …4:46†

or, substituting Eq. (4.43) into this expression, K^eˆ C^ÿT

The terms not containing x and y have now been moved from the operation of integrating. The term within the integration sign can be multiplied out and integrated explicitly without diculty if D is constant.

The external forces at nodes arising from distributed loading can be assigned `by inspection', allocating speci®c areas as contributing to any node. However, it is more logical and accurate to use once again the standard expression (4.31) for such an allocation.

The contribution of these forces to each of the nodes is

f_iˆ

The integral is again evaluated simply. It will now be noted that, in general, all three components of external force at any node will have non-zero values. This is a result that the simple allocation of external loads would have missed. The nodal load vector for uniform loading q is given by

f₁ ˆ₁₂¹ qab

The vector of nodal plate forces due to initial strains and initial stresses can be found in a similar way. It is necessary to remark in this connection that initial strains, such as may be due to a temperature rise, is seldom con®ned in its e€ects on curvatures.

Usually, direct (in-plane) strains in the plate are introduced additionally, and the complete problem can be solved only by consideration of the plane stress problem as well as that of bending.

Rectangular element with corner nodes (12 degrees of freedom) 127