Cantilever (FEA/Formulas)
The whole point of using
formulas is to be able to predict the behaviour of an object. We could
use simple formulas for simple shapes or we can turn to Finite Element
Analysis (FEA) for complex shapes. Here, we apply both methods to a
simple cantilever to see how they compare to the real thing.
Lecture Notes: empty.pdf
empty.one
Lecture Video: Empty
Beam
Bending Standard formulas have been derived for
common arrangments of loaded beams. See Beam
Bending tables here.
An example is shown at right. For a simple
cantilever beam, where a weight W is applied at the end of the
cantilever of length L; Maximum Bending
Moment = WL Therefore, from the bending
stress theory; where;
σ is the bending stress
(Ivanoff uses fb) M -
the moment about
the neutral axis y -
the distance to the neutral axis Ix -
the second
moment of area about the neutral axis x
Maximum Deflection
= WL3/(3EI) which occurs at the end of the beam. |
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Predicting Stress and Deflection
In this exercise we will use two methods to predict the
behaviour of a cantilever beam made of aluminium.
The
two methods are:
1. Empirical formulas. (As listed
above). These can be reviewed in MEM30006A
Stresses.
2. Finite Element Analysis. A
computer calculated approximation.
Both the maximum stress and the maximum deflection will be
compared.
In class..
Calculations:
How to calculate I
- by
formula
- by AutoCad
- by
Inventor
Setting up formulas in Excel.
Adding Forces in Excel (Mathematical addition of
forces)
Calculating
a maximum safe load
Beam depth: h
= 6.35 mm (1/4 inch)
Beam breadth: b
= 38 mm (1.5 inches)
Second moment of Area: Ixx
= bh3/12 = 38*6.35^3/12 = 810.8 mm4
Maximum allowable stress: 100 MPa
Maximum
allowable bending moment from bending stress equation;
where;
σ is the bending stress = 100
MPa
M - the moment about
the neutral axis = ?
y -
the distance to the neutral axis = 6.35/2 = 3.18 mm
Ix - the second
moment of area about the neutral axis x =
810.8 mm4
Re-arrange to find
max allowable bending moment M
M = σ Ix / y =
100*810.8/3.18 = 25497 Nmm
From Beam Bending Equations
Max Bending Moment: M = WL
so W = M/L =
25497/500 = 51 N (about 5 kg)
Maximum
load applied at the end of the beam is 5kg.
You
must use the actual dimensions, not these!
Cantilever Beam
This report must be submitted as a single
Word document containing no attachments (all images etc embedded).
Submitted to WSI Online assignment by the due date.
Failed
reports will be returned for a second attempt. No third attempt is
permitted without documented special circumstances.
Cantilever
Beam. Using the
aluminium beam jig, compare the deflection of various loadings on a
cantilever. Assume the weight of the beam
is the starting datum for measurement. Weights are added and the
deflection measured. You must record all relevant
information (beam material, cross-section, length, mass etc), then
calculate the deflection by; 1. Analytical methods
(see bending
table) 2. FEA analysis. 3.
Laboratory measurement | |
Cantilever
Arrangement.
Build
CAD model as appropriate
Here is a rough sketch of
the beam rig, including the mounting frame;
A full assembly is optional (we will be doing FEA on assemblies in a later assignment)
Larger image
Different ways of constraining the beam for realistic FEA results
What do we really need to model here?
It is always good to begin analysis with something very simple first. Then you can gradually refine it - if necessary. Keep it simple!
The cantilever does not need to be modelled as a complete assembly.
Even a simple beam with a fixed constraint on the end face could
suffice.
However, this fixed constraint completely eliminates any
movement of this face, so it does not get a uniform stress along the
mounting face. There is higher stress in the middle (41 MPa) and lower
on the outside (only 25MPa). This is not correct, the rigid constraint
is causing stress errors. | |
Now the rigid face constraint is replaced with a rigid edge constraint to reduce how rigidly the face is locked up.
We now add a frictionless face constraint to prevent rotation.
The
stresses look more uniform when close to the mounting, but now we have
big stress concentrations on each end of the rigid line. These are at
77 MPa, so the whole colour bar has been re-scaled (hence the beam is
green rather than red, even though the stresses are the same as before)
| |
Another way to reduce stress concentration issues at the mounting
end is to apply a simple fixed face constraint (which is normally
dangerous), but isolate it by thickening the beam to lower stresses.
These stresses look a lot better, but they are not perfect.
Do you see any issues with this design in terms of being a good representation of the real object? | |
Here is another way to accurately measure maximum stress in the cantilever without having issues with stress concentrations at the constraint.
Build the cantilever longer and then measure stress at the correct 722mm from the free end using probe. (With exactly adjusted location, In this case the cantilever is 822mm long, so we want stress at 100mm from the constraint)
Now we have results that are closely matching the equations.
There are issues though - like we can't get deflection from this model. |
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ABOUT GRAVITY
- Deflection in the experiment already includes gravity force
(weight) of the beam itself, so we are not measuring any gravity but only the effect of the added mass.
So we should not include gravity in the FEA analysis to determine deflection.
- Stress in the experiment combines both the weight of the beam and the added mass. However - we did not measure this stress directly (this would require an electronic strain gauge). So to correctly determine the total stress we need to add stress due to weight (distributed load) plus stress due to force (concentrated load).
Total Stress = Stress due to weight + Stress due to force
We can do this by formula if we add both results together (the 2nd and 3rd rows of the beam bending table).
We can also do this in FEA by simply turning on gravity. Make sure you use the same value for g!
- Note: If we do this, it will just add gravity to the hand calc and to FEA, so nothing would really change other than a slight increase in stress.
The Report
Follow the following headings in your report
- EXPERIMENT: Using a unique load (different to everyone else in the
class), measure the maximum deflection at the end of the cantilever.
EXPERIMENTAL RESULTS
(a) Load = 1.68389kg, Max Deflection = 28.89mm
(b) Load = 3.5kg + 0.5 lb, Max Deflection = 63mm
(c) Load = 2kg, Max Deflection = 32.5mm
Other Load/Deflection combinations:
kg |
Deflection |
1 |
16.5 |
1.5 |
24.35 |
2 |
32.5 |
2 |
33.3 |
2.5 |
41.7 |
3 |
49.45 |
Lab
instructions
- Use a unique load. (Take care not to overload the beam)
- Hang weights off the end, but try to estimate the most
precise line of action of the weight force.
- Measure
the deflection at the end of the beam somehow. You need
to measure the maximum deflection, which is on the very end of
the centilever.
- CALCULATION: Using the same load as you used in the experiment in part (3), calculate of max stress and max deflection by formula. We do not know exactly what type of aluminium this is, but we will assume 6061 as definied in Inventor. Look up the modulus that Inventor uses for 6061 and use that value in your calculations.
- FEA: For
the same load again, determine the max stress and deflection using FEA.
Ensure your mesh size and constraints are correctly applied. See Younis:
Chapter 1: The Stress Analysis Environment (pdf 5.3 MB). Model the rig using solid modelling
software. Take measurements, sketch it on paper, then model it. Trial
different methods of mounting the rigid end of the beam. Determine if
there are any stress concentration problems and how big they are (how
many MPa away from the "correct" value we obtained in step 2). Of the two methods (formulas and FEA), which one
is the most accurate? Explain.
- DISPLACEMENT PROBLEM: Experimental displacement values are higher than hand calcs or FEA. Comment on any adjustments that could be made to bring
the FEA values closer to the experimental values. Consider any other sources of flexibility in the experiment. E.g.
- Modulus E (Try adjusting E in the formula
deflection to equal to the experiment. Would this be a reasonable value
for modulus? Check this value against the Inventor value, or published data, or on MATWEB. How much variation is there in the modulus of various aluminums?)
- Inaccurate measurements? Comment on which dimensions are most critical.
- Flexible mounting frame? Comment on the rigidity of the mount
- Poor FEA analysis? Comment on convergence of mesh size.
- CONCLUSION: The whole point of this report is to see if FEA can match the experiment and theory.
- Explain what you did that gave the best stress match between FEA and formula.
- Explain what you did that gave the best deflection match between FEA and experiment - or give some reasons why the experiment has more displacement then calculations/FEA.
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