5.7 Singularities of a Reissner–Mindlin plate

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Slab models are usually developed by starting with a 3-D elastic continuum

and then simplifying the kinematics. Depending on the assumptions made in

this process, the result is either a Kirchhoff plate or a Reissner–Mindlin plate.

The loss of accuracy in the transition from a 3-D model to a 2-D model

is often felt the most near the boundary where the models differ most. With

regard to the Reissner–Mindlin plate we speak of a boundary layer effect [9],

[238]. To some extent this effect seems to be due to the fact that we cannot put

an elastic solid on a line support such as a hinged support (see Sect. 1.16, p.

55), and the Reissner–Mindlin plate—not surprisingly—seems to suffer more

from such adverse effects.

In a Reissner–Mindlin plate we have a choice of two support conditions

for a hinged edge: the hard support, w = w,t = 0, which corresponds to the

hinged support of a Kirchhoff plate, and the soft support, w = 0, where the

slope w,t = w,x tx +w,y ty in the tangential direction is released, w,t     = 0; see

Fig. 5.20.

As in a Kirchhoff plate, the boundary conditions and the angle of the

boundary points determine when and where singularities will develop; see

Table 5.2 [205].

Table 5.2. Corner singularities of a Reissner–Mindlin plate

Boundary conditions Bending moment Shear force

clamped–clamped 180

◦

180

◦

sliding edge–sliding edge 90

◦

180

◦

hard support–hard support 90

◦

180

◦

soft support–soft support 180

◦

180

◦

free-free 180

◦

180

◦

clamped–sliding edge 90

◦

180

◦

clamped–hard support 90

◦

180

◦

clamped–soft support ≈ 61.70

◦

(ν = 0.29) 180

◦

clamped–free ≈ 61.70

◦

(ν = 0.29) 90

◦

sliding edge–hard support 45

◦

90

◦

sliding edge–soft support 90

◦

180

◦

sliding edge–free 90

◦

90

◦

hard support–soft support ≈ 128.73

◦

180

◦

hard support–free ≈ 128.73

◦

90

◦

soft support–free 180

◦

90

◦

5.7 Singularities of a Reissner–Mindlin plate 437

Shear locking

The main advantages of a Reissner–Mindlin plate are essentially the relaxed

continuity requirements for the shape functions, and its inner “richness” of

kinematic variables. On the other hand shear locking can become a problem.

The transition from a Reissner–Mindlin plate model—relatively thick slabs

(foundation slabs)—to a Kirchhoff plate model, which is the standard model

for thin slabs, can cause problems.

Clearly the Reissner–Mindlin plate theory subsumes the Kirchhoff slab

theory, because the transition from the former to the latter model is simply

achieved by setting the shearing strains to zero:

γx = w,x +θx = 0 γy = w,y +θy = 0. (5.53)

Because in normal slabs shear deformations are negligible, a Reissner–Mindlin

plate should behave like a Kirchhoff plate. But this does not happen. The

more the slab thickness h shrinks, the more a Reissner–Mindlin plate tends

to stiffen, until the slab ultimately seems to freeze.

This is primarily a problem of the finite elements. If the equations could be

solved exactly, then if h tends to zero the Reissner–Mindlin results should tend

(in the sense of the strain energy [238], p. 263) to the results of a Kirchhoff

plate.

Shear locking is best explained by studying the example of a Timoshenko

beam, u = [w, θ]T , see Fig. 5.21. The strain energy product is

a(u,ˆu) =

_ l

0

[EI θ

_ ˆθ

_ + GAs (w

_ + θ) ( ˆ w

_ + ˆθ)] dx , (5.54)

so that with appropriate unit displacements (2 at each node)

ϕ1(x) =

_

w10

_

ϕ2(x) =

_

0

θ2

_

_ _ _

node 1

ϕ3(x) =

_

w30 _

ϕ4(x) =

_

0

θ4

_

_ _ _

node 2

. . .

(5.55)

for example linear functions

wi(x) = l − x

l

θj(x) = x

l

(5.56)

a result like

(KB +KS)u = f (5.57)

is obtained, where the entries in the matrix KB come from the bending terms,

and the entries in KS from the shear terms:

438 5 Slabs

Fig. 5.21. Cantilever beam

kB

ij =

_ l

0

EI θ

_

i θ

_

j dx kS

ij =

_ l

0

GAs (w

_

i + θi) (w

_

j + θj) dx . (5.58)

If the cantilever beam in Fig. 5.21 is modeled with just one element, the

end deflection is [70],

w(l) =

12(h/l)2 + 20

12(h/l)2 + 5

· 1.2 Pl

GAs

(5.59)

where As is the equivalent shear cross-sectional area. For a short beam, l   1,

the first fraction is approximately 1 and the end deflection is identical to the

shear deformation

w(l) = 1.2 Pl

GAs

. (5.60)

If l ! h, i.e., if the length l is much greater than the width h of the beam,

the first fraction is about 20/5 and the end deflection will be much too small:

w(l) = 4 · 1.2 Pl

GAs

= 9.6 Pl

E bh

(G = 0.5 E, ν = 0) (5.61)

compared with the exact value (let l/h = 8)

w(l) = Pl3

3 EI

= 4 Pl

E bh

_

l

h

_2

= 256 Pl

E bh

. (5.62)

This is shearlocking.

The reason for this stiffening effect is the different sensitivity of the bending

stiffness EI and the shear stiffness GAs with respect to the width h of the

beam:

EI = bh3

12 GAs _ Gbh (rectangular cross section) . (5.63)

If the width h—and thus the shear deformations γ = w_ + θ—tend to zero,

the bending stiffness decreases much faster than the shear stiffness. As in an

equation such as

5.8 Reissner–Mindlin elements 439

Fig. 5.22. Plate elements: a) Bathe–Dvorkin element b) DKT element. At the nodes

of a DKT element, the shearing strain is zero, θxi = −∂w/∂xi, θyi = −∂w/∂yi

(1 + 105) u = 10 (solution u = 9.9999 · 10−5 ≈ 0) , (5.64)

where the large second term enforces a nearly zero solution, the matrix KS—

because of the rapidly increasing influence of GAs—tends to dominate (5.57).

If (5.57) could be solved exactly, the increasing influence of GAs would be

canceled by the opposing tendency of w_ + θ = γ _→ 0.

All this holds for slabs as well: the transition from a Reissner–Mindlin

plate model (average to large slab thickness) to a Kirchhoff plate model (small

thickness) does not succeed numerically.

A whole catalog of measures has been proposed to circumvent shearlocking.

Reduced integration is the best-known remedy. Although a better and

simpler approach is to raise the order of the polynomials. This holds in similar

situations as well, where internal constraints force an element to sacrifice

all the degrees of freedom to satisfy the constraints, so that nothing is left to

describe the movements of the element [70].

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