fem past papers

What is convergence?

When you refine the finite element mesh by decreasing the element size, the displacement predicted should approach the exact solution

How do you ensure convergence of a finite element method?

You must satisfy the completeness and compatibility requirements

What are the 2 requirements of completeness

The finite element approximation of the displacement vector must be able to represent an arbitrary constant rigid body motion

The finite element approximation of the displacement vector must be able to represent an arbitrary constant strain state

What is the compatibility condition

Aka the conforming condition.

The finite element approximation of the displacement vector must vary in a continuous manner over element boundaries

What are the 2 approaches to test for convergence?

H adaptivity and p adaptivity

What is h adaptivity?

When mesh refinement is borough about by subdividing elements in problem areas and a subsequent analysis will contain more elements where they are found to be required (h is element size)

What is p-adaptivity?

The mesh refinements if brought about by keeping the same mesh, and therefore the same total number of elements and increasing the order of the elements (p is the polynomial order)

What are the changes in the set up when you assume plane stress for FEM?

Plane stress means you use 2d elements and only consider in plane displacements

A different De is used (in formula sheet, De is the think you multiply ε by to get σ)

Integration is done over a 2D area not volume

What are the impacts on FEM if you assume plane stress?

Very efficient computation

Not suitable for thick structures as they underestimate constraint effects

Overestimates fracture toughness

Common in problems where thickness is negligible compared to other dimensions

Thickness can be used as a multiplier to convert 2D forces to 3D force effects

What changes in FEM set up if you assume plane strain?

Use 2d elements

A different De is used than for a plane stress

Assume that εz = 0, but σz not 0

Must specify a unit thickness for volume, as it is modelled as infinite

How to find inverse of a 3×3 matrix

1) find the determinant of A, A^-1=1/detA * [C]

2) find C, C= -1^(i+j) * M_(i,j)

3)Find M_(i,j), where you cross out row I and column j and find determinant of remaining matrix and put it at position ij in M.

4) find each point in C, then multiply by the 1/detA

How to find plane stress from big matrix equation

Write in terms of ε= Kσ

Cancel out the zero σ terms ( you only know out of plan stress is zero since it’s plane stress)

Find the inverse of the remaining matrix to get σ = Αε

What does it mean if model is axisymmetric?

The geometry, loading and boundary conditions are symmetric about an axis, typically the z axis

What are the changes to FEM if model is axisymmetric

Use 2d elements in the r-z plane, includes circumferential component in formulations

Track normal strain in the out of plane, circumferential direction

Specialised axisymmetric stiffness matrix is formed, incorporating effects due to rotation around axis, hoop stress

Integration includes a 2πr weighting factor to account for rotation around the axis

Only works if the symmetry is perfect

Application of plane stress

Thin plates

Sheet metal

Anything where the through thickness stresses are very much smaller than the in plane stresses

Applications of plane strain

Long shafts, underground cavities, strip footing (out of plane stress not zero)

Applications of axisymmetric assumptions

Pressure vessels, nozzles, pipes under symmetric loads, anything with rotational symmetry

How does corrosion accelerate fatigue growth?

the surface pits created by corrosion act as crack nucleation sites

How does fatigue accelerate corrosion damage?

the protective oxide layer is repeatedly broken

what are the temperature effects on corrosion?

fracture toughness reduces at lower temperatures

creep behaviour with non zero mean stress

more rapid macro crack growth at higher temperatures

what happens to number of cycled to failure for immersed corrosion?

there is no longer a fatigue limiit, even at very low stress amplitude there is still a finite number of cycles to failure. This is because during each cycle the corrosion, like rust, gets deeper and deeper into the material.

what variables change the poisiton of modified S-N curves?

frequency of loading

material type

material composition

level of humidity

temperature

flow rate

when is corrosion not as important?

for low cycle fatigue, (when strain life method is used) then there is less time for the corrion to take effect as it is in the corrosive environment for less time before failure. Plastic strain is much more damaging.

what effect do corrosive environments have on paris law coefficients

higher paris coefficients for higher crack growth rates

how much of an effect does corrosion have at a high delta K?

Less of an effect as there are higher growth rates so less time for corrosion to have an effect

measures to improve resistance to corrosion fatigue

induce residual stresses - if you induce compressive stresses before, then when there are surface cracks it reduces the stresses first. (limiting fatigue)

surface protection (limiting corrosion) : paint, cathodic protection, ceramic polymer coatings

Where should you put fine mendium and coarse mesh?

Put fine mesh where there is a displacement constant and shaft geometric feature s

Dur the medium mesh where there are sharp geometric feature

Put coarse mesh everywhere else

What are the types of boundary condition and when do you use them?

Homogenous Dirichlet - for fixed pointa

Mixed homo dirichlet/neuman - for rollers to rep symmetry

Non zero in homo Neuman - for fractions applied

What order polynomial will the B matrix hold for a 3D problem?

A 3rd order polynomial

How does the number of gauss quadrature points change for irregular elements?

if the elements is not regular the jacobins transformation between local and global coordinates will vary through the element. So should use a higher order GQ to capture the variation

How to use gauss quadrature

Write out the integral

Do sum of (number of gauss points) the function of the inside of the integral * wi

Then use the gauss points for the variable we are integrating around, get the function of these gauss points, multiply by the weight

Add these values together to get the integral

Write out a table of

j, xj , wj , f(xj), f(xj)*wj

How to find the strain from given displacements

ε= B* d

To find the b matrix we need to find the jacobian

To find the jacobian use x= sum of Ni*xi, where xi is the global coordinate of the point, this gives (x,y) in terms of ξ η

Differentiate your x and y in terms of the local coordinates

Find the inverse of this, which is the inverse jacobian

Then get the b matrix values from J^-1 * δN/δξ = δN/δx

Reorder the displacements of the nodes into local order

Multiply the displacements by the B matrix to get the strains

How to get nodal forces from traction

Write out shape functions

Sub in the local coordinate (if know ) of the traction surface to reduce it

The equation fi=integral(Ni*t)

Write out the nodal forces in local coordinates

How to use the strong for statement of equilibrium

Rearrange so that you get fb in terms of the differentials of the stresses

Find the cuchy stress derivatives by finding differential of displacement given

Rewrite into the strain vector using differential of displacement is strain

Use the equation to get stress from the strain

Differentiate the stresses that you find