“Computational Civil Engineering - "Intersections" International Journal

16 A.G. Popa, H.L. CucuThe finite element formulation is based on the 4-node isoparametric rectangularelement with 2 degrees of freedom (dof) per node (Figure 2,a). For a higheraccuracy in bending, shear is assumed to be constant over the finite element, equalto the value in the center.For 3-D structures, an additional dof – representing the translation normal to theelement’s plane – is considered in each node, in order to describe the spatial stateof displacements (Figure 2,b).Figure 2. The finite element PLAN-12The shape functions are assumed to be the bi-linear Lagrange polynomialNi1= i 14( 1+ξξ ) ⋅( + ηη ,i) i = 1, 2, 3, 4, (1)furnishing linear displacements on the sides of the element (Figure 3,c).The analysis was performed for three meshes:- meshwork U 1 : 24 elements (1. 0 × 1.0 m)- meshwork U 2 : 96 elements ( 0 . 5× 0.5 m)- meshwork U 3 : 384 elements ( 0. 25 × 0.25m).It is assumed that the numbering of the nodes and elements starts at the right cornerof the section and it is left handed (Figure 1).The three meshes were analyzed for the following loading cases (Figure 3):- case (a): symmetrical uniform distributed forces;- case (b): horizontal symmetrical concentrated forces;- case (c): horizontal force acting on the direction of the web.