1
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
3
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.
4
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….
5
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.
6
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∑ ∫∫==
7
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.