5.5 Singularities of a Kirchhoff plate

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The handicap of a Kirchhoff plate is its lesser inner flexibility. Unlike a Reissner–

Mindlin plate, in which cross-sectional planes can rotate independently of

the position of the mid-surface, in a Kirchhoff plate the rotations are wedded

to the rotations of the mid-surface.

A Reissner–Mindlin plate can lie flat on the ground, giving no notice that

the cross-sectional planes in the interior tilt to the left; see Fig. 5.19, p. 434.

Or at a clamped edge the slab can perform a feat which is impossible for a

Kirchhoff plate: it can fold like sheet metal, and descend steeply.

This (relatively) inflexible behavior of a Kirchhoff plate can lead to problems

at corner points (see Table 5.1), as for example at angular points of a

hinged slab (see Fig. 5.12), because the gradient ∇w = [w,x , w,y ]T vanishes

at a hinged corner point. This is a consequence of the fact that the derivatives

in the direction of the two hinged edges (tangent vectors tR and tL) are zero:

Fig.

slab. At the corner

points the bending

moments become

430 5 Slabs

Table 5.1. Corner singularities of a Kirchhoff plate, [165]

Support conditions Bending moments Kirchhoff shear

clamped–clamped 180

◦

126

◦

clamped–hinged 129

◦

90

◦

clamped–free 95

◦

52

◦

hinged–hinged 90

◦

60

◦

hinged–free 90

◦

51

◦

free–free 180

◦

78

◦

Fig. 5.14. At the obtuse-angled corner points, the support reactions and bending

moments mxx become infinite

x

y

12.0

8.0

-88.0

17.2

x

y

12.0

8.0

402.1

402.1

5.6 Reissner–Mindlin plates 431

∇w • tR = 0 ∇w • tR = 0 ⇒ ∇w = 0 . (5.41)

At such points the plate is clamped:

w = w,x = w,y = 0. (5.42)

The singularity vanishes immediately if the rotations w,x and w,y are set free.

In a hinged plate with a rhombic shape, a strange singularity is observed

at the wide-angled corner points. The bending moment mxx tends to −∞

and the bending moment myy to +∞; see Fig. 5.13. Again by releasing the

rotations w,x and w,y the singularity disappears.

A skew bridge mainly carries the load from its lower wide-angled corner

to the upper wide-angled corner—this is the shortest path between the

two supports. Unfortunately the bending moments and the support reaction

(Kirchhoff shear) become singular precisely at these corner points; see Fig.

5.14.

If the lower edge of the bridge coincides with the x-axis (because w = 0)

the rotations w,x in the tangential direction are zero. In the terminology

of Reissner–Mindlin plates this would be a hard support, while it would be

considered a soft support if the rotations w,x were released. In a Kirchhoff

plate hinged supports are normally modeled as hard supports, w = w,x= 0,

but it eventually helps to release the rotations near critical points.

Whenever possible the flexibility of the supports should be taken into

account, because this helps to avoid stress peaks.

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