work equivalent (consistent) normal loadswork-equivalent nodal forces corresponding to weight as a...
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![Page 1: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/1.jpg)
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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:
![Page 2: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/2.jpg)
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![Page 3: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/3.jpg)
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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.
![Page 4: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/4.jpg)
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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….
![Page 5: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/5.jpg)
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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.
![Page 6: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/6.jpg)
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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∑ ∫∫==
![Page 7: Work equivalent (consistent) normal loadsWork-equivalent nodal forces corresponding to weight as a body force (rectangular quadrilaterals for work-equivalence): LST has no vertex loads](https://reader033.vdocuments.net/reader033/viewer/2022052321/60a89f95fd193a081001e4eb/html5/thumbnails/7.jpg)
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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.