MDME: MANUFACTURING, DESIGN, MECHANICAL ENGINEERING 

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Compare FEA&Formulas (Simple)


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 applyboth 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 (Ivonaff 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.

 

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)
 


SIMPLY SUPPORTED BEAM

simple beam

This steel beam is loaded with weights and the dimensions measured.
Note that this is a truss beam turned sideways - in the weak direction.
The material is mild steel (hot dip galvanised).
You will need to model this in INVENTOR, so take all the necessary dimensions to build the frame. Take care when measuring the flat bar on each side since these are doing all the work (use a micrometer to get accurate measurement).

Calculating a maximum safe load

Flat depth: h = 19.3, 19.35, 19.67, 19.25. 19.25, 19.25  Average = 19.345

Flat breadth: b = 6.78, 6.875, 6.5, 6.875, 6.83  Average = 6.772

Beam length: L = 2167

Beam width: BW = 140

Second moment of Area: Ixx = bh3/12 = 6.772*19.345^3/12 = 4085.5 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 = 19.345/2 =  9.6725 mm
Ix - the second moment of area about the neutral axis x = 4085.5 mm4

This is for one flat, but we have two flats, so total  = 4085.5 * 2 = 8171 mm4

Re-arrange to find max allowable bending moment M

M = σ Ix / y = 100*8171/9.6725 =  84476 Nmm 

From Beam Bending Equations

Max Bending Moment: M = WL/4

so W = 4M/L = 4*84476/2167 = 156 N (about 16 kg)

Maximum load applied at the end of the beam is 16 kg.

Weight of beam is about 7kg, so we can add about 10kg.

You must use the actual dimensions, not these!



SIMPLE BEAM


Experimental Results
kgDeflection (mm)
054.5
252
449.5
646.5
844
1041.5
1140.5
1238.5

Cantilever Beam. 

Using the steel beam, compare the deflection of various loadings. 

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

Simple FEA setup.
  • Add a pad of material under each end - 30mm x full width x 0.1mm deep. This allows us to select a smaller section of material for our constraints.
  • Now apply a Fixed constraint to the outside EDGE of each pad.
  • Make sure gravity is turned on.
  • Apply a force of 12kg *9.81 on the top surface.
Deflection

The deflection can be found using the beam bending formulas.


Where:
  • W = 12*9.81 N   (12 kg convertd to Newtons)
  • L = 2167 mm  (Distance between support points)
  • E = 200000 MPa  (Is this accurate?)
  • I = 8171 mm4 

The Report

Follow the following headings in your report

  1. Model the rig using solid modelling software (Inventor). Take measurements, sketch it on paper, then model it. Ignore the triangulated webbing, and add three cross-members (one on each end and one in the middle).
    Using only the MAXIMUM load from the experiment, determine the max stress and deflection using FEA. Ensure your mesh size and constraints are correctly applied.
    Apply the load on the centre cross-member. See Younis: Chapter 1: The Stress Analysis Environment (pdf 5.3 MB) 


    General CAD arrangement
    Pad under the end supports
    Step-down on centre cross-member
       
  2. Using the same maximum load as the FEA, calculate (by hand) the max stress and max deflection. (Calculating only with the lengthwise beams, ignoring the cross members)
  3. Explain any differences between the three methods (Formulas, FEA, Experiment). Is there any adjustment that could be made to get the results to match (e.g. adjust Modulus, Gravity)
  4. Give examples and estimate an approximate % error in each of the following;
    • measurement tolerances
    • calculation rounding
    • theoretical approximations of formulas
    • variability of material properties like modulus, density, gravity
    • approximations in the FEA method
    • inaccurate model assumptions (think about mounting, point of loading, etc)
Lab instructions
  • Add weights to the centre of the beam. (Take care not to overload the beam)
  • Measure the deflection of the beam somehow. You need to measure the maximum deflection, which is at the centre of the beam.

Web Links
www.learneasy.info

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