Chapter 6 Introduction to Finite Element Analysis

Question 1

FEM software tools

Answer 1

1. CAD, standard parts, material database
2. CAE: preprocessor, finite element solver, postprocessor
3. CAD, CAM

Question 2

3 types of error

Answer 2

1. solution error (usually negligible)
2. discretization error (how many domains do I need for a valid answer?)
3. idealization error (is the model correct in terms of physics?)

Question 3

classification of finite elements

Answer 3

1. 1D, 2D, 3D
2. linear, quadratic, cubic

Question 4

1D elements

Answer 4

provide (usually) axial stiffness (e.g. stringers)

CROD, CBAR

Question 5

2D elements

Answer 5

covering surfaces (e.g. skin)

CTRIA, CQUAD4

Question 6

3D elements

Answer 6

e.g. interfaces of VTP and HTP, detailed finite element models

Question 7

essential boundary condition

Answer 7

Dirichlet BC

directly imposed to a DOF (supports)

Question 8

natural boundary condition

Answer 8

Neumann BC

loading

Question 9

Direct Stiffness Method (DSM)

Answer 9

breakdown:
1. disconnection
2. localization
3. member (elements) formation

assembly and solution:
4. globalization
5. merge
6. application of BCs
7. solution
8. recovery of derived quantities

Question 10

rules for assembly

Answer 10

compatibility: joint displacement of all members meeting at a joint must be the same

equilibrium: sum of forces by all members that meet at a joint must balance the external force applied to that joint

Question 11

Tonti diagram of governing equations

Answer 11

given (problem data) (external):
1. prescribed end displacements
2. distributed axial load q(x)
3. prescribed end loads

unknowns (internal):
1. axial displacements u(x)
2. axial strains e(x)
3. axial dorce F(x)

Question 12

minimum potential energy (MPE) principle

Answer 12

nature wants to stay at equilibrium

for conservative structural systems, of all the kinematically admissible deformations, those corresponding to the equilibrium state extremize (i.e., minimize or maximize) the total potential energy

if the extremum is a minimum, the equilibrium state is stable.

Question 13

consistent nodal force vector

Answer 13

external loads are sometimes applied to a whole element (e.g. pressure), but in FEM the forces must be applied only on nodes, hence we “lump” the loads to the nodes

Question 14

each node must be connected to […] stiffness values

Answer 14

each node must be connected to 6 stiffness values (each for one DOF), otherwise a singularity will occur

Question 15

discretization error goes to zero when …

Answer 15

… when the number of finite elements goes to infinity

Question 16

stress gradient and refinment

Answer 16

the higher the stress gradient, the more finite elements are required e.g. at the following positions: holes, cross-sectional changes, inner radii, in-plane bending