4.6 Volume forces

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Volume forces are transformed into equivalent nodal forces by letting the load

act through the nodal unit displacements

4.7 Supports 347

Fig. 4.19. Equivalent nodal forces in load case g as a fraction of the total dead

weight G = γ Ae t for different elements. Note that the corner forces of the LSTelement

are zero, and those of the Q8 element even have a negative sign

fi =

_

Ω

p •ϕi dΩ . (4.53)

These equivalent nodal forces are in agreement with the elementary rules of

structural mechanics. Constant volume forces as g = [0, γ]T (gravity load) are

distributed in equal parts onto the nodes of a CST element (see Fig. 4.19),

while the corner nodes of a quadratic LST element are load-free, because the

integrals of the shape functions are zero in this load case; see Fig. 4.19. Note

that the corner forces of the Q8 element point upward, because the integrals of

the pertinent shape functions are negative. Of course the sum of all equivalent

nodal forces fi is equal to the total weight G · 1.

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