1.24 Why finite element support reactions are relatively accurate

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When the same problem is solved with various FE programs, it is often found

that agreement in the support reactions is quite good, and it is soon recognized

that they change little when the mesh is refined. The reason is that influence

functions for support reactions have a particularly simple shape. They are the

deflection curves or surfaces if the support settles by one unit of deflection;

see Fig. 1.70. These simple shapes are easy to approximate, even on a coarse

mesh.

Things are different if the wall is not isolated, but is instead in contact

with other walls. Now if the wall settles, the slab cracks in the transition zone

between the rigid supports, w = 0, and the sagging wall, w = 1; see Fig. 1.71.

In practice the neighboring walls will not be perfectly rigid, but will move too,

so the discontinuity in the transition zone will not be so sharp. But accuracy

will certainly suffer. Short intermediate supports will be affected more than

longer walls.

100 1 What are finite elements?

Fig. 1.70. Slab: a) plan view; b) influence function for the sum of the support

reactions in wall W1; c) in wall W2

1.24 Why finite element support reactions are relatively accurate 101

X

Y

Z

Fig. 1.71. Floor plate: a) plan view; b) influence function for the support reaction

in the wall extending vertically; c) 3-D view of the FE approximation; d) contour

lines of the influence function

Example

Consider the slab in Fig. 1.71. The exact influence function G0 for the support

reaction of the wall (the one extending vertically) is displayed in Fig. 1.71 b,

and the FE approximation is shown in Fig. 1.71 d. The latter is the shape of

the slab if the nodes of the FE mesh that lie on the wall are pushed down by

w = 1 cm. This produces the deflection w = 1.85 cm at the distant node yk;

see Fig. 1.71 d. This number is exactly the sum of the equivalent nodal forces

of the wall if a force P = 1 kN is placed at this node

_

i

fi =

_

Ω

Gh0(y) p(y) dΩ = Gh0

(yk) · 1kN = 1.85 kN . (1.299)

The exact value (on a very fine mesh) for the support reaction is 2.2 kN, so

the FE solution underestimates the support reaction on this mesh by about

16%. When a uniform load is applied, the error in the support reaction is

about 5%.

102 1 What are finite elements?

-15.4

-14.0

-17.6

-17.3

6.6

1.6

6.5

6.36.3

199.4

254.5

254.5

-53.3

Fig. 1.72. Floor plate on rigid supports: a) plan view and FE mesh; b) support

reactions under gravity load; c) influence function for the nodal force A, and d) for

the nodal force B

Peaks

Support reactions tend to oscillate and end in high peaks—in particular near

the ends of free-standing walls and at corner points; see Fig. 1.72. This is

easily understood by looking at the influence functions for the nodal force

directly at the end of the wall, node A, and at a node further back, node B.

The node up front has a much larger influence area than the node behind it,

and the force at node B is most often negative, simply because a movement

w = 1 of this node lifts the part in front upwards.

Point supports

A Kirchhoff plate is about the only 2-D structure that can safely be placed on a

point support. A Reissner–Mindlin plate or a plate (shear wall) simply ignores

point supports, because a single point can travel freely in any direction; see

Fig. 1.73. A Kirchhoff plate will finally succumb to even the smallest moment

1.24 Why finite element support reactions are relatively accurate 103

Fig. 1.73. a) The exact influence function for the support reaction B is zero. One

single point of the plate (= the support) can move downward by one unit length

without disturbing the plate b) The FE approximation to the influence function

closely follows the beam solution

M, and will not try to prevent the point of attack from rotating. Rather it

will let it loose so that it can rotate freely. Polynomial shape functions cannot

accomplish such remarkable feats. If one point moves, the whole neighborhood

follows suit, and therefore such influence functions always turn out wrong.

But the situation can be saved if the mathematical idea of a point support

is abandoned, and instead the supports are allowed to have finite extent. Then

the support reaction acts over a small surface area, and it may be assumed

that the influence function for such a patch of forces comes close to the shape

if the node is moved by one unit of displacement.

Cantilever plate

According to the theory of elasticity, the support reaction of the cantilever

plate in Fig. 1.74 should be zero. The reason that it is not zero is that an FE

program cannot generate the exact influence function; see Fig. 1.73 a. Instead

it produces the shape in Fig. 1.73 b which closely follows the deflection curve

of a beam. This is also why the FE support reaction B is identical to the beam

solution. At least for bilinear elements (Q4), this tendency prevailed even when

the mesh was refined (8 → 32 → 128 → 512 elements) as illustrated by the

results in Table 1.5. While the support reaction hardly changed, the stresses

near the point support increased with each refinement step.

Hence, point supports can be freely used, and the results are also reasonable,

in the sense of beam analysis, only stresses in the neighborhood of such

supports are meaningless.

104 1 What are finite elements?

Fig. 1.74. Cantilever plate with a point support and equivalent beam model

Table 1.5. Results for the plate (beam) in Fig. 1.74

Elements Support reaction B (kN) Min σ (kN/m2) near the support

8 47.72 −480

32 47.54 −963

128 47.49 −1960

512 47.46 −3854

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