What do you think the eigenvectors of the element stiff- ness matrix represent? 1. a basis in which the stiffness matrix would be diago- nal (if rotated to that basis) 2. a set of nodal displacements for the element corre-

Numerical integration Consider He 4-D integral =+(s 1 Seek n-point apploxiwisfions G~2M =1 are the weights and

How many zero eigenvalues do you think any element stiffness matrix (regardless of the type of finite element nterpolation should have in 2D and 3D, respectively?

Finite element wodel of a beam(Euler-Bernollia- governing eQuations

Why is the element stiffness matrix singular in a finite element formulation? 1. So that it can accomodate rigid element dis- placements without introducing spurious nodal 2 Because we made a mistake in the formula- tion the stiffness matrix should not be sin- g 3. Because we havent enforced any displace ment boundary conditions(it's a variational approach after all) Statement(1)

The correct global local node mapping for the quadraticelement mesh in the figure is

By looking at the potential energy of an element, what can you conclude about the properties required for the basis functions of an euler -bernoulli beam element? They should be differentiable twice

The reaction on the left end is not exact because 1. The order of interpolation is too low, a higher order of interpolation would give the right reaction 2. The distributed load attributed to node one does not ke it into the solution

The finite element method In FEMi we derale finite element equations fro PVD swe- SWe and obtained: K0=R:=4…n waere n:number of element nodal p Ue: elenent nodal displace ents

The finite element melod I In FEM I We derived basis functions of arbitrary order for Hhe rod Model