“Computational Civil Engineering - "Intersections" International Journal

26 H. L. Cucu, A. G. Popa[ N](3x24)⎡N⎢=⎢⋅⎢⎣ ⋅1⋅N⋅1⋅⋅N1− A ⋅l− A ⋅ m11− A ⋅ n1222A ⋅lA ⋅ m11A ⋅ n1111A ⋅lA ⋅ m11A ⋅ n1333⋅⋅⋅ N4⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅N⋅4⋅⋅N4− A ⋅l− A ⋅ m44− A42⋅ n22A ⋅lA ⋅ m44A ⋅ n4111A4⋅l3⎤⎥A4⋅ m3⎥A ⋅ ⎥4n3⎦(13)with:A1ζζζζ= N1⋅ t ⋅ ; A2= N2⋅ t ⋅ ; A3= N3⋅ t ⋅ ; A4= N4⋅ t ⋅2222ηNode 4 Node 3(14)t/2t/2x,z,a)ONode 1y,vNode 4ζ ηNode 3OuξNode 2z';Θ z’wx';Θ x’vy';Θ yNode 11.00O1.00Node 21.001.00ξ1.0Node 81.0b)1.001.001.00Node 5Node 1z,Node 4ζηξNode 61.00wuNode 7vNode 3x,uOy,Node 2Figure 1. The two new hybrid-strain finite element proposed(a – the element “HYBFLAT”; b – the element “HYBBRICK”)· step 3) the interpolation of the strains on the finite element, through the straininterpolation matrix [ P ]:[ P](6 x18 )⎡1⎢⎢⋅⎢⋅= ⎢⎢⋅⎢⋅⎢⎣⋅η ζ⋅ ⋅⋅ ⋅⋅ ⋅⋅ ⋅⋅ ⋅η ⋅ζ⋅⋅⋅⋅⋅⋅1⋅⋅⋅⋅⋅ξ⋅⋅⋅⋅⋅ζ⋅⋅⋅⋅⋅ξ ⋅ζ⋅⋅⋅⋅⋅ ⋅⋅ ⋅1 ξ⋅ ⋅⋅ ⋅⋅ ⋅(6 x18 )⋅⋅η⋅⋅⋅⋅⋅ξ ⋅ζ⋅⋅⋅⋅⋅⋅1⋅⋅⋅⋅⋅ζ⋅⋅⋅⋅⋅⋅1⋅⋅⋅⋅⋅ξ⋅⋅⋅⋅⋅⋅1⋅ ⎤⋅⎥⎥⋅ ⎥⎥⋅ ⎥⋅ ⎥⎥η⎦(15)