1.36 Support conditions

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An FE solution satisfies geometric boundary conditions such as

plate: u = 0, v= 0 slab: w = 0 w,n = 0 (1.509)

pointwise, while static boundary conditions

plate: Sn =￣t slab: mn = ￣m vn = ￣v (1.510)

10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

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10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00

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2.65 2.54 2.21 1.65 -2.87 -3.25 1.42 1.76 2.04 2.44

2.56 5.02 4.68 2.29 -5.55 -5.31 2.13 3.45 3.56 1.91

2.25 4.81 4.19 0.63 -4.75 -3.18 1.32 2.54 2.80 1.29

1.66 2.45 1.48 -3.51 -3.05 2.75 -1.11 1.15 2.30 1.06

-2.97 -5.86 -5.28 1.10 -2.14 -8.15 2.81 4.32 3.36 1.25

-3.26 -5.38 -2.89 8.73 -11.26-17.27 10.44 6.71 3.86 1.31

1.50 2.43 1.67 -9.06 6.53 12.33 -6.66 1.35 2.28 0.97

1.81 3.50 2.06 -0.26 5.62 7.20 0.44 0.41 1.48 0.85

2.04 3.47 2.51 2.09 3.78 4.04 1.92 1.29 2.00 1.39

2.43 1.87 1.21 1.04 1.35 1.36 0.89 0.77 1.37 2.21

184 1 What are finite elements?

Fig. 1.132. The resultant forces are the same, R = Rh

are only satisfied in the weak sense, i.e., along a free edge of a slab it is only

guaranteed that the support reaction vn together with the surface load ph and

interelement jump terms vΔ

n in the neighborhood of the edge (see Sect. 1.34,

p. 177) contribute no work through any of the nodal unit displacements of

the edge nodes:

_

Ω

ph ϕi dΩ +

_

k

_

Γk

vΔ

n ϕi ds +

_

Γ

vn ϕi ds = 0. (1.511)

The same holds for the bending moment mh

n, which is nonzero along a free or

hinged edge. The distribution of mh

n is skillfully balanced in such a way that

mh

n annihilates the work done by the other terms:

fi =

_

Ω

ph ϕi dΩ +

_

k

_

Γk

mΔn

∂ϕi

∂n

ds +

_

Γ

mh

n

∂ϕi

∂n

ds = 0. (1.512)

The same logic applies of course to edge loads. The substitute FE edge loads

are only weakly equivalent to the true load.

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