5.8 Reissner–Mindlin elements

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A multitude of plate elements are based on the Reissner–Mindlin plate model.

Only three such elements, the Bathe–Dvorkin element, the DKT element and

the DST element, will be discussed here because they are probably the most

popular.

Bathe–Dvorkin element

It seems that this element (see Fig. 5.22 a) was first developed by Hughes and

Tezduyar [122], and later extended by Bathe and Dvorkin [27] to shells.

The element is an isoparametric four-node element with bilinear functions

for the deflection w and the rotations θx and θy. According to the equations

γx = w,x +θx γy = w,y +θy , (5.65)

440 5 Slabs

the polynomial shape function for w should have a degree one order higher

than the rotations. To begin, one chooses for w a ninth-order Lagrange

polynomial—4 corner nodes + 4 nodes at the mid-sides + 1 node at the

center of the element—and a matching bilinear polynomial for the rotations.

The idea is to calculate the shearing strains γx and γy independently of the deflection

at the center and deflections at the mid-side nodes. Hence the stiffness

matrix only depends on the deflections w at the four corner points, therefore

bilinear functions for w suffice. This simplification is based on the observation

that the shearing strains parallel to the sides of the element at the mid-side

nodes are independent of the deflections at the mid-side nodes and at the

center of the element.

The main advantage of this element is the easy transition from thick slabs

to thin slabs, so that the element is universally applicable. The bending moment

mxx in the principal direction—which is assumed here to be the x-axis—

is constant, and the bending moment myy is linear. As in the Wilson element,

quadratic terms can be added, so that the bending moments also vary linearly

in the x-direction. The shear forces qx and qy are of course constant.

DKT element

The DKT element (see Fig. 5.22 b) can be considered a modified Kirchhoff

plate element or a modified Reissner–Mindlin plate element [236].

The derivations starts with a Reissner–Mindlin plate element and the assumption

is that the shearing strains in the element are zero [29]. Hence the

strain energy product is simply the scalar product of the bending moments

and the curvature terms:

a(u,u) =

_

Ω

[mxx θx,x + mxy θx,y + myx θy,x + myy θy,y ] dΩ . (5.66)

But the element shape functions satisfy the condition γx = γy = 0 only at

discrete points, namely the corner points of the triangular element and the

mid-side nodes of the element, which is why this element is called a discrete

Kirchhoff triangle.

For the rotations, linear functions are chosen:

θx =

_3

i=1

θxi ϕi(x) θy =

_3

i=1

θyi ϕi(x) . (5.67)

The deflection w is instead only defined along the edge, and interpolated

by Hermite polynomials. Next one assumes: a) linear rotations θn (= slope)

on each side, b) zero shearing strains at the corner points and at the midside

nodes. In particular the latter assumption, θα = w,α, makes it possible to

couple the rotations to the deflection w, and it is thereby possible to reduce the

model to the 3×3 degrees of freedom wi, w,xi , w,yi at the three corner points,

so that the result is a triangular plate element with the nine “natural” degrees

5.9 Supports 441

of freedom. Because there are no shape functions for the deflection w inside

the element, a calculation of consistent equivalent nodal forces is not possible.

In other words, it is up to the user how to distribute the load over the nodes.

To calculate shear forces additional assumptions must be introduced.

The DKT element is very popular, because with minimal effort (C0-

functions suffice) a triangular element with the nodal degrees of freedom

w, w,x , w,y is obtained. But of course this element also is nonconforming [51].

DST element

The DST element is closely related to the DKT element [31]. Unlike the DKT

element, the shearing strains γi are not zero at the nodes. The starting point

is a weak formulation of the Reissner–Mindlin equations using a Hellinger–

Reissner functional with w,ϕx, ϕy, γx, γy, qx, qy as independent variables. By

a corresponding weak coupling of the terms (L2 orthogonality), a triangular

element can be derived in which each of the three nodes has degrees of freedom

w,ϕx, ϕy. The name discrete is justified: as in a DKT element the shearing

strains γx and γy are coupled at only three points (the mid-side nodes) to the

other degrees of freedom w,ϕx, ϕy.

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