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Work equivalent (consistent) normal loads

■ Mechanical loads: concentrated loads, surface traction, body forces.

◆ Normal surface traction on a side of a plane element whose sides remain straight (q is force/length):

✦ Work-equivalent nodal forces:

2

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Distributed Shear Traction■ Shear traction on a side of a plane element whose sides remain straight (q is force/length):

■ In (b), a Q4 element and two CSTs share the top midnode so that the nodal loads from Q4 and the right CST are combined.

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Quadratic Normal Surface Traction■ Quadratic normal surface traction on a side of a plane element whose sides may deform quadratically:

■ Work equivalent nodal loads lead to greater accuracy than lumped loads. But….

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Uniform Body Force■ Work-equivalent nodal forces corresponding to weight as a

body force (rectangular quadrilaterals for work-equivalence):

■ LST has no vertex loads and vertex loads of Q8 are upwards!■ The resultant in all cases is W, the weight of the element.

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Equivalence of nodal forces and weight

■ To show that the nodal forces are work-equivalent to the element weight for a Q4 element of unit thickness:

total work of the nodal forces=(v1+v2+v3+v4)W/4

total work of the body force:

■ By performing the indicated integration, the two work expressions can be shown to be equal.

dxdyvyxNdxdyAWyxv i

iiA

W ∫∫ ∑∫∫ =

=

4

1),(),(

( ) ii

iAW vdxdyyxN ),(

4

1∑ ∫∫==

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Connecting beam and plane elements■ Since all of the previous plane elements have translational dofonly, no moment can be applied to their nodes. ■ Therefore the connection (a) of a beam and a plane elements cannot transmit a moment and the beam element can freely rotate.(Singular K!)

■ A solution is in (b) where beam is extended. Rotational dof at A, B and C are associated with the beam elements only. A plane element with drilling dof would also work but is not recommended.