# loads_etc - ■ To show that the nodal forces are...

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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:

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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.

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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.

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6 Equivalence of nodal forces and weight
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Unformatted text preview: ■ 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=(v 1 +v 2 +v 3 +v 4 )W/4 total work of the body force: ■ By performing the indicated integration, the two work expressions can be shown to be equal. dxdy v y x N dxdy A W y x v i i i A W ∫∫ ∑ ∫∫ = = 4 1 ) , ( ) , ( ( ) i i i A W v dxdy y x N ) , ( 4 1 ∑ ∫∫ = = 7 Connecting beam and plane elements ■ Since all of the previous plane elements have translational dof only, 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....
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loads_etc - ■ To show that the nodal forces are...

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