ABSTRACT
SubstitutionofEqs.(12.1.21)through(12.1.26)intoEq.(12.1.27)resultsinthe followingshapefunctions.
where 1
TheseshapefunctionssatisfytheconditionsgiveninEqs.(12.1.19)and(12.1.20). Singulareight-nodeshapefunctionscanbedevelopedfromthefollowinginter-
polation:
1 u(~,TJ)=2(Hr(0ut48+H(~)u236]+(Ht(TJ)Ut25+H2(ry)u437]
whereUijkisdefinedinEqs.{12.1.21)through(12.1.26).Theshapefunctionsare shownbelow:
Previously developed derivative singular shape functions are used to interpolate the variables like the displacements while standard shape functions are used for interpolation of the geometric coordinate variables like x and y for the mapping from the natural coordinate system to the physical coordinate system. Previous derivations can be applied to higher order singular elements in 2-D or 3-D with either point singularity or line singularity.
