Introduction to FEA

Question 1

Why/what FE

Answer 1

-solves value boundary problems
-breaks down the infinite to the finite
*For displacement problems, it finds >= 1 disp. at each node
*Shape function relates the nodal displacements to the domain, also limits the accuracy and convergence of the solution
*element are only affected by associated nodes

Question 2

Boundary conditions

Answer 2

-Essential: nodal disp @ supporting boundaries
*Primary unknown is nodal displacements
*Secondary are derived from the primary, strains and stresses and the displacement field

-Natural: equivalent nodal forces away from supporting boundaries

Question 3

Assembly

Answer 3

1.Discrete system
- Dividing the system into finite number of elements that have known characteristics
-Assembled to form the whole system

2.Get {u}:
-equilibrium between applied nodal F {P} and nodal resistive F

{R} = [K]{u} - {Pp} for linear response
*{Pp} is the equivalent nodal F from distributed loads, lack of fit or change in temp

-Global matrices constructed from local ones using Incidence vectors {phi}

3.Apply BC
-Natural and essential are mutually exclusive with the no. of natural BC = no. unknowns
- Essential BC are applied by setting {P}i +{Pp}i to the fixed displacement and [K]ii = 1 with all others in the row equal to zero

Question 4

Formulation of finite element equations

Answer 4

Apply virtual work to get [K]^e + {Pp}^e for any element
- external work by the loads is equal to the internal work by the stresses over the strains

{sigma} = D
-for linear materials
-{epsilon0} = alpha.delta T for change in temp, generally the strain at no stress

delta {u}^e.{R}^e is the work done by nodal resistance
- external since are considering elements only
- rearrange for {R}^e which is applicable for linear and non-linear materials

Question 5

Stiffer response of linear FEA

Answer 5

(W_FE = ({P} + {Pp}) transposed {u}) <= W_exact
- so for a single point load the displacement for FE is smaller than the exact
- convergence from below by increasing the number of elements
- Approximation of disp is better than that of its derivative so stress and strains are not as accurate as {u}