Sources of Nonlinearity

We live in a complex, nonlinear world, but what are the sources of nonlinearity in finite element analysis (FEA)?

1. Contact Nonlinearity - is significant when the functioning of an assembly involves the interaction of parts.
2. Material Nonlinearity - such as time-dependent (visco) behaviours, plasticity and damage effects amongst many other characteristics commonly seen in nonlinear materials.
3. Geometric Nonlinearity - occurs when the structure experience large strain/large deformations (This is typically when strains are larger than 10%) which causes stress stiffening of structures.

Plane Stress vs. Plane Strain vs. Axisymmetric Elements

A summary of the differences between plane stress, plane strain and axisymmetric elements.

Element	Plane stress	Plane strain	Axisymmetric
Required input geometric property	Thickness.	Thickness.	None. User should find out the angle of segment assumed by the FE software.
Assumptions	• σz=0 • εz=-v/E(σx+ σy)	• εz=0 • σz=v(σx+ σy)	Axisymmetric loading and structure.
Degree of freedom per node	Translations Ux, Uy	Translations Ux, Uy	Translations Ur, Uz
Stress output	σx, σy, σxy	σx, σy, σz, σxy	σr, σz, σrz, σθ
Strain output	εx, εy, εz, εxy	εx, εy, εxy	εr, εz, εrz, εθ

Also see
Types of elements in the element library

Steps in FEA: An Overview

In the FEA vs. FEM article, we distinguished the two by introducing FEM as the numerical foundation of FEA. It is the underlying method that makes FEA works. Let us now look at the following steps involved in running a general FEA and their conceptual difference should become even clearer when compared with the Steps in FEM.

     The common steps for carrying out a general purpose FEA is summarised below:

1. Model Idealisation
2. Spatial Domain Identification
3. Element Selection
4. Mesh Discretisation
5. Material and Geometric Definition
6. Boundary Conditions
7. Pre-analysis Checks
8. Job Submission
9. Results Verification

1. Idealise Geometric Model

     The first step involves defeaturing small details of the model. Defeaturing is necessary but must always be done carefully as it is often the major source of error of an analysis. Assumptions made in this step will affect the results strongly and must be coherent throughout the analysis. For example, fillets are often omitted as they are difficult to mesh. However, defeaturing fillets in the region of interest for stress can be a hazardous decision due to stress singularity, which is common in structures with sharp re-entrant corners.

Thin Shell vs. Thick Shell

Thick shells are capable of modelling transverse shear deformation whilst thin shells do not. Thick shells are governed by the Mindlin–Reissner (thick shell) theory. As the shell thickness decreases, the problem tends to favour Kirchhoff (thin shell) theory which neglect the inclusion of transverse shear deformation. This is pretty much similar to the thin (Euler-Bernoulli) vs. thick (Timoshenko) beams comparison.

     Typical thickness for thin shell is <5% whilst thick shell theory applies within the 5-10% range. Anything significantly >10% should not be modelled using plate theories.

A comparison of the differences between thin and thick shell theories.

Theory	Thin shell Kirchoff-Love	Thick shell Mindlin-Reissner
Thickness vs. percentage of in-plane dimensions	Thickness < 5%	5% < thickness < 10%
Key assumptions	• Plane remains plane • Normal remains normal • Thickness is not affected by deformation	• Plane remains plane
Degree of freedom per node	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz
Transverse shear deformation	No	Yes

See also
Types of elements in the element library

Membrane, Plate and Shell Elements

A comparison of the differences between membrane, plate and shell elements.

Element	Membrane	Plate	Shell
Engineering components	Pressure vessels, oil tanks, ship hull, wing skin
Analogous to	Truss	Beam with no axial stiffness	Beam
Number of nodes	Usually three: One each at the top, middle and bottom
Displacement degree of freedom	• Translations Ux, Uy	• Translations Uz • Rotations Rx, Ry	• Translations Ux, Uy, Uz • Rotations Rx, Ry, Rz
Stress output	• In-plane stresses	• Bending stresses • Additional transverse shear stresses for thick plates	• In-plane stress • Bending stress • Additional transverse shear stresses for thick shells

Also read
Types of elements in the element library

Thin Beam vs. Thick Beam

Standard beam theory (Euler-Bernoulli bending theory) assumes no deformation by shear. This can be safely applied to thin beams that are long and slender. For short and deep beams however, displacement due to shear becomes hugely important which is included in the formulation of thick beam element. Below is an exaggerated example of shear deformation:

Displacement due to shear that standard beam theory does not take into account

     This effect is the key that distinguish between the Euler-Bernoulli and Timoshenko (thick beam theory) bending theories. See below for a direct comparison between thin and thick beams.

Table illustrating the differences between thin (Euler-Bernoulli) beams and thick (Timoshenk) beams. Diagram in "look" section is adapted from Wikipedia.

Also read
Types of elements in the element library