1.35 The output

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To assess the accuracy of FE results correctly, it must be understood how an

FE program processes the raw output and how it displays it on the screen.

The load case ph

In general the equivalent load case ph is not displayed on the screen, because

a user not well-acquainted with FE techniques would be irritated.

Support reactions

One would assume that an FE program outputs the support reactions of the

FE load case ph. These forces plus the forces that have been reduced to the

supports at the start would be the true support reactions. But instead what

is displayed on the screen are the equivalent support reactions, the equivalent

nodal forces spread along the supports to simulate a continuous support.

182 1 What are finite elements?

1.130. Onnodal

forces fi

Formally, what happens is that the program converts the element volume

loads ph and interelement line loads lh into equivalent nodal support reactions

by letting these loads act through the nodal unit displacements of the

supports:

fi =

_

Ω

ph ϕi dΩ +

_

Γ

lh ϕi ds

ϕi = unit displacement of a support node .

Because in the neighborhood of supports there are probably more loads ph

and lh pointing upward (having a negative sign) the net result will be a series

of equivalent nodal forces that point upward, i.e., which support the slab.

Basically all this was already done when the global stiffness matrix was

assembled. Hence the stiffness matrix K must only be multiplied by the nodal

unit displacements:

Ku = f ← list of equivalent nodal forces . (1.507)

These equivalent nodal forces fi (kNm) are then transformed into equivalent

line forces (kN/m). Assuming a linear distribution between two nodes, this

would result in a distribution such as

l(x) =

1

2

×

_

fi

le

(1 − x

le

) + fi+1

le

x

le

_

0 < x < le . (1.508)

In Fig. 1.131 and Fig. 1.130 the two versions can be seen side by side. The

first figure shows the distribution of the support reactions as they appear

tions are the evenly

The support reacscreen

appearance.

spread equivalent

Fig.

1.36 Support conditions 183

Fig. 1.131. Slab a) system and loading, b) principal moments, c) element surface

loads, d) vertical forces along the interelement boundaries

on-screen—these are the transformed equivalent nodal forces fi—while the

second figure displays the “true” support reactions, where it is seen that the

slab is not only supported by the walls but by negative element surface loads

as well. Note also that the support reactions do not end abruptly at the ends

of the walls, but continue beyond these points.

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