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Describe how FEA solves a problem.
The FE method converts a PDE into simultaneous equations, which are then solved easily by a computer to find a solution.
What does equilibrium mean?
Forces balance so there sum is zero, and there is no net force in one direction.
What is elasticity?
The linear relationship between force (or stress) and displacement
What is compatibility?
The displacement of things connected together must match up
What is Stress? What units is it measured in? What is the equation to link it to strain?
Force per unit area (Nm^-2). Stress = elastic modulus * strain
What is strain? What units is it measured in?
Displacement per unit distance, there are no units as it is a ratio
What is a modulus? What units is it measured in?
A material property (e.g. youngs modulus) that is a ratio of stress to strain in the elastic regime. units of N/(m^2)
What is Stiffness? What units is it measured in?
Ratio of force to displacement in the linear regime, same concept as modulus but property of structure not material. Measured in (NM^-1). Also used for angular stiffness, ratio of moment to angular Displacement.
What is Plane Stress?
An Assumption used in 2D analysis that there is no stress in other directions, e.g. thin plates
What is Plane strain?
An assumption used in 2D analysis that there is no strain in other direction, e.g. if analysis cross-section of something constrained
What does Linear Mean? Where would this occur?
Relationships between forces and displacements is linear. Requires both Materials in elastic regime and small displacements (e.g. sin(theta)=theta and cos(theta) = 1)
What does non-linear mean? Where would this occur?
Non Linear means when the relationship between forces and displacements is non linear. Could occur as an effect of material non-linearity, or large displacements causing material non-linearity.
What are Elements?
A piece of the structure that could be either a discrete physical component (e.g. a beam) or a region of a larger part. FE divides structures into elements.
What are Nodes?
Points that define the ends or corners of an element, FE "solves" to find the displacements of nodes.
What are Degrees of Freedom?
The possible ways in which something can move.
What is K?
The Stiffness matrix
What is M?
The Mass Matrix
What must a FE solution satisfy?
Equilibrium, Compatibility (displacements must match) and Elasticity (relationship between force, extension and stiffness)
What are the 6 generic steps for solving a FE problem?
1)Describe the geometry, the constraints and loading

2)Form the stiffness matrix for each element

3) Assemble the stiffness matrix for each element into a global stiffness matrix

4) Solve the resulting set of equations to obtain unknown displacements and forces at nodes

5) Calculate tensions in elements from nodal displacements

6) Display Results
What is the equation for an element stiffness matrix?
f = K u

force = (element stiffness matrix) * displacement
What conditions must be satisfied when assembling the global stiffness matrix?
Equilibrium - Total force at each node must balance

Compatibility - Displacement at every node must equal
When creating an element stiffness matrix, why is the matrix not just of a size 1x1?
As there are multiple nodes, each with a displacement and force constraint, hence a 22 or 33 etc matrix.
How do you generally solve the global matrix equations to find unknown forces and displacements?
Take individual matrix rows and re-arrange to get unknown displacements, then substitute to find unknown forces (or vice versa)
How do you generally calculate the tensions in each element, given you have solved for displacements and forces?
You only need displacements, sub these back into elasticity condition for each element to find tension in each bar.
Why are displacements more accurately estimated than stresses?
The primary solution gives nodal displacements, then another stage is required to find stresses. Hence, displacements are more accurately estimated than stresses.
What is the difference between a pin-jointed and non pin-jointed structure?
Pin-jointed, elements not rigidly connected at nodes, moments cannot be transferred. Non-pin jointed, vice versa. Pin jointing gives extra rotational DOF at a node.
Why would you use symmetry in FEA, how would you use it?
Used to reduce size of model, Find a line in the model where the geometry is mirrored, i.e. split in two, then using a displacement constraint boundary condition of 0 (i.e. for a horizontal mirror, vertical displacement uy = 0, and rotation in x and z is 0)
what will happen to a FE model as the element size is refined -> 0?
The results will converge to a stable solution
What is the purpose of model validation?
To check that the model has taken account of all relevant physical effects, that the model itself has been implemented correctly, and units are correct
What are some possible validation techniques?
Model a similar system with an analytical solution (if one exists), compare results to experiments (but this is expensive)
what is a rule of thumb for the accuracy of convergence?
element size e, result y, run again with element size e/2, result y2, then assume error for second model is of order |y2-y1|/|y2|
Give an example of when a quantity would not converge to a finite value as mesh is refined?
Stresses at a sharp internal corner, as mesh is refined value will keep increasing.
What aspect ratio is the best for elements? What value of side curvatures should be used as a maximum? Where should mid-side nodes be kept?
1, avoid curvatures of more than 30 degrees, Should be kept near middle of sides
Should corner nodes be joined to mid-side nodes?
No, join corner nodes to corner nodes, not mid-side nodes.
What specific information is needed to define a mesh?
Two tables of data, co-ordinates of all nodes in model, numbers of nodes that form each element in model
What information, besides meshing, is included in a FE input file?
Materials, Loading, Boundary conditions
What is the difference between continuum elements and bar and beam elements?
In continuum elements, element stiffness matrices are approx. (unlike exact for bar and beam), and the structure is discretised to elements controlled by the use, rather than elements representing physical components.
What are the steps to form a stiffness matrix for a continuum element?
1) Define shape functions that give displacement d(r) at point r, in terms of element nodal displacements u

2) Differentiatie d(r) to find expression for strain E(r) at point r

3) Use hookes law to find expression for stress o(r) at point r

4) Integrate strain energy density over whole element to get expression for total energy stored in element, U

5) Expression will have form U = 1/2u^t{Complex Matrix}u

6) Work done by nodal forces W = 1/2u^tKu

7) Equate W and U to find K
What are shape functions?
Shape functions interpolate nodal displacements to give the displacement at any point within element, can be linear or quadratic.
In 2d, how many components of strain are there?
3, Exx, Eyy, Exy

e = {Exx Eyy Exy} ^T
How do you transpose a matrix?
Write the rows of the matrix as the coloumns
What is the strain-displacement relationship for Exx, Eyy and Y(gamma) xy
Exx = d/dx * disp(x)
Eyy = d/dy * disp(y)
Yxy = 2Exy = 2 d/dydisp(x) + d/dx*disp(y)
What is the general matrix form of hooke's law, which relates stress to strain?
o = DE

Where D in 3D is a material property, but D in 2d also includes info about nature of structure in 3rd dimension, e.g. is it in plane stress or plane strain.

D is 'material stiffness matrix'
What is the equation for strain energy density at a point? How do you change this to total strain energy in an element?
Uo(r) = 1/2 o^T(r)E(r)

You integrate it 3 times, for 2d however, thickness = 1. If nodal displacement/strain relationship and material stiffness do not depend on position, then this is just multiplied by the volume, rather than integrated.
What is the equation for work done by a force? How do you calculate the total nodal work done?
The integral of the force wrt. distance moved.

W = int between 0 and u of f(u)*du

Calc total by summing work done for all nodes, one node is 1/2f1u1, so total sum

W = 1/2 f^T * u

and as f =Ku, f^T = u^T*K hence,

W = 1/2u^TKu
Summarise the 3 key steps to get an element stiffness matrix for a shape-function created element.
Write down N(r) - shape functions that give displacement r in terms of u
Compute expressions for terms in matrix B(r) = deltaN(r) (Differentiation of N(r))
Integrate B(r)^T D B(r) over volume of element to obtain K
What is the difference between 2D plane stress and plane strain material stiffness matrices?
Plain Strain - Ezz = Exz = Eyz = 0. 2D matrix is just relevant terms from 3D material stiffness matrix.

Plain Stress ozz = oxz = oyz = 0. Again, just relevant entries from 3D material compliance matrix
What are the advantages of higher order elements?
More rapid convergence to 'correct' answer

Better representation of curved parts, as elements can have curved edges.
What are the disadvantages of higher order elements?
Evaluating the element stiffness matrix is much more complex. (B is a function of position, hence have to integrate everything)
What is a parent element?
parent elements are simple generic elements in natural co-ordinates (e.g. triangle at 0,0 1,0 0,1)
What are shape functions used for w/ higher order elements?
To perform a co-ordinate transformation, i.e. map natural co-ordinates of parent element to global physical coordinates of actual elements.

Also used to interpolate nodal displacements to displacements within element.
In higher order elements that have been transformed from natural co-ordinates, is stiffness matrix integration performed in natural or global physical co-ordinates?
The variable is changed from physical to natural co-ordinates, hence the calculation is performed in natural co-ordinates.
What is the general property of a shape function? (Which expression has to be satisfied for shape functions to be found)
Ni(s,t) = 1 if i=j, 0 if i=/j.

x = N1(s,t)x1 + N2(s,t)x2 + N3(s,t)x3 = x1 at node 1, x2 at node 2 etc..

Sum of shape functions N(s,t) = 1.
What is an isoparametric element?
Elements for which the same shape function used for nodal position translation (from natural to global physical co-ordinates) can also be used to translate everything within the element, from these nodal values. This forms the stiffness matrix.
What is numerical integration by gauss quadrature? How do you apply this to solve an integral? Why is this an approximation?
Integral can be approximated by summing rectangles. By special choice of width x and weighting w, gauss found that the actual answer could be found very efficiently.

This is applied to solve an integral by evaluating the integral at a small number of points inside the element qi (called gauss points) and perform a weighted sum of these values. It is an approximation because although the shape functions are simple polynomials, the integral is not (if it was it could be easily integrated).
Describe some issues with guass quadrature integration? When can instability arise?
Generally FE over-estimates stiffness

More gaussian points means greater accuracy in integration.

But fewer gaussian points tends to under-estimate element stiffness so number is intentionally kept low to increase efficiency, and compensate for overall stiffness over-estimate.

Instability can arise if a possible deformation has 0 stress/strain at all gauss points.
What is the difference between static and dynamic models? What is the difference in the basic equations?
In dynamic models, f and u are functions of time t, and we also need to consider inertia and possible damping.
f(t) = Mddu(t) + Cdu(t) + Ku(t)
Where M is global mass matrix and C is global damping matrix.
Global mass matrix from F = M a (f = M ddu(t))
What is the difference between a lumped mass matrix and a consistent mass matrix?
Lumped mass matrix - mass shared equally between nodes, produces diagonal matrix

Consistent mass matrix - more mathematically rigorous way of obtaining element mass matrix by using general expression based on shape functions, like one used for element stiffness matrices.
How do you assemble a global mass matrix?
Add element mass matrices together, just like element stiffness matrices.
What is the equations governing the complete system at a harmonious excitation?
e^-i(omega)t = H (for clarity in this slide)

f(t) = Mddu(t) + C du(t) + Ku(t) (in time domain) becomes in freq domain

fH = -(omega)^2MuH - iomegaC*u*H + Kuu= -(omega)^2MuH - i*omega*C*u*H + K*u*H

removing common factor

f = (K - iomegaC - (omega^2)M)u

f = X(omega) * u

"stiffness matrix" is now complex, freq-dependent and includes inertia and damping
What is an important frequency-domain method? How is this solved?
The calculation of natural frequencies of a structure, to ensure they fall outside region of likely excitations. This undergoes resonance with no external forces (f = 0). Hence (K-omega^2M)u = 0.

This is an eigenvalue problem, eigenvalues are resonant freq that satisfy |K-omega^2*M| = 0. Eigenvectors describe mode shape.
What type of problems do frequency-domain methods solve? What type of problems do time-domain methods solve
Linear Steady-State problems where the result is simple harmonic motion. Time domain methods are used to solve transient problems where applied forces are complex functions of time (e.g. impacts, wave functions).
What is the difference between an implicit and an explicit method of solving Time-Domain methods?
Explicit - u(tn+1) = f(utn), requires time step to be smaller than some critical value

Implicit - Provides an expression f(u(tn+1),u(tn)) = 0 which must be solved iteratively (e.g. newton-raphson) to find u(tn+1), stable for much larger time-steps, widely used for non-linear analysis.
Why must the time step for an explicit method be smaller than some critical size? What is the equation for this critical size? What is the effect on solver time of reducing element size?
dt < Lmin/c. Where Lmin is size of smallest element in mode, and c is speed of sound in model (speed at which disturbances can move). If the time step is larger than this, the model will be unstable.
Reducing element size, and increasing accuracy, requires smaller time-steps, hence requiring more steps to be taken, and probably longer solving time.
What is the difference between linear and non-linear FE?
Linear FE - f = K *u, stiffness matrix K and loads f constant

Non-linear FE, K and/or f may be functions of displacement u, need to find displaced shape u that is the solution to f(u) = K(u)*u
Where and why does non-linearity occur?
material nonlinearity due to non-linear stress-strain relationship

Large displacements of structure so that nodal co-ordinates change significantly.

Non-linearities in BCs

Large strains to make strain-displacement ratio non-linear
Describe the incremental solution method?
Approximate a non-linear problem as multiple linear problems. Divide load into N smaller increments, find displacement at each increment, and sum resulting displacements.
What is the tangent stiffness matrix?
The tangent stiffness matrix relates incremental load to incremental displacement for the current shape of the structure. deltaf = Kt(un-1)* deltaun. Kt is the tangent stiffness matrix. assembled to global matrix in normal way.
What are the problems with an incremental method?
Displacement errors accumulated at each step

Not guaranteed to be stable, requirement for stability (number of increments N) hard to determine without trial and error

May require large N to reach stable solution with desired accuracy.
What is the iterative newton-raphson method?
Use repeat trial solutions that converge to the correct solution (i.e. residual error is acceptably small).

Can be seen from rearranging to get f(u) - K(u)u = 0

incorrect value of u will give residual error r rather than 0.
What is the equation for a newton-raphson iterative scheme?
x(n+1) = xn - g(xn) / g'(xn)
What is a problem with a newton-raphson iterative scheme? Why should this method be used in conjunction with an incremental technique?
Newton-Raphson will converge to a solution, but if there are multiple solutions it may not converge to a correct one. Therefore Newton-Raphson often used in conjunction with incremental technique so each step converges.
What determines the size of a stiffness matrix?
The number of nodes * number of Degrees of freedom. e.g. 2 nodes, 3 DOF (rotational as well as x,y translational), 6x6 matrix
What is the difference between a stiffness matrix for pin jointed beams, and a stiffness matrix for non-pin jointed beams?
Extra DOF, so matrix is now 6x6 rather than 4x4, and with an extra set of 6x6 terms with 12EI/L^2 as factor, seen in link http://imgur.com/pjnodJX
How do you find global co-ordinates, given a point in natural co-ordinates and shape functions?
From r(q) = N(q) * x in 2d, x(s,t) = the sum for all shape functions i to max of N(i)(s,t)x(i), where N(i) is the shape function for one corner of the element and x(i) is the related co-ordinate.

Do this for y as well, and you can also do this for dx/ds, dx/dt, dy/ds, dy/dt e.g. for dx/ds = sum of all dN/ds(i) * x(i).

E.g. for a quadratic element x = N(1)x(1) + N(2)x(2) + N(3)x(3) + N(4)x(4)
What is the equation for load that would cause yield in a bar?
Yield Load = x-sectional A * yield stress
Given a graph showing stress vs strain, how would you find the elastic/plastic modulus?
From the gradient of the graph - Delta Stress/delta Strain.

MPa over strain gives units of Pascals.
What is the relationship between Hz and angular frequency w?
w = 2pi Hz
What units are angular frequencys/natural frequencys in?
What is the equation that needs to be solved to find a natural frequency? How does this change when a periodic force is applied?
(K - w^2* M)u = 0, where w is natural frequency

= f rather than 0, and w is now period of force.
How do you convert mm^2 into m^2?
10^6 mm^2 = 1 m ^2
What is the difference between Quadrilateral and triangular elements?
Triangular elements - constant strain so strain increases in steps as you move across element

Quadrilateral elements - linear strain inside element
What is the difference between linear and quadratic elements?
Quadratic elements converge to a stable solution faster than linear elements, and provide a more accurate solution at large displacements.

Quadratic elements - 2nd order displacement (ax^2+bx+c), linear - 1st order displacement (ax+b)
When you know the compressive load in a beam due to a certain external force, how do you find the external force required to produce a different load?
You can scale the force up linearly, using the relationship

Force(required) = Load(required)/Load(current) * Force(current)

The same can be done for displacements