Mechanics of Solids.
Laboratory 3: Statically indeterminate torsional stresses.
Submission requirements
Carefully read through the instructions below, watch the video of this practical and use the prerecorded
data for your lab report. Submit this in a pdf format on Learnline. You do not have to record the data yourself this year.
General.
When a shaft transmits a torque, torsional stresses are induced and the shaft twists between the
point where the torque is applied and the torque reaction point.
Theory.
A tube supported at both ends with a centrally applied torque will be statically indeterminate as the only equation of equilibrium will be
TA + TB – TC = 0
Thus, we need another compatibility equation to solve the unknowns.
This comes from the fact that we know the angle of twist of each segment AC & BC must be the same and in opposite directions. In other words, there is no angle of twist of point A relative to point B.
This can be written ØAC + ØCB = 0
Also,
JG
= TL
Thus
+ = 0
JG
T L
JG
T L AC AC BC BC
where AC BC T andT are the internal torques in section AC and section CB respectively and AC L and BC L are the lengths of section AC and section BC.
Therefore
AC AC BC BC T L +T L = 0
Mechanics of Solids.
So, as the location of the applied torque shifts, the proportion of the torque that each segment carries will vary. When the applied torque is in the middle of the tube, the torque is the same magnitude in both segments.
For this prac, you are required to compare the theory and your measurements from three different
aspects.
1. Determining the torque applied by the lever and the torsional stress induced in the tube as calculated by the angle of twist of the lever.
2. Determining the torque applied to and the torsional stress induced in the tube by the lever as calculated by the force measured in your load cell.
3. Determining the torque applied to and the torsional stress induced in the tube by the lever as calculated by the strain measurement from the tube itself. For this last section, you will need to convert torsional strain () as measured by the strain gauge (microstrain) into shear strain () (radians). Later in the semester (strain transformations) you will learn that XY = 2 XY . This can then be used in Hooke’s Law ( = G) to calculate the shear stress.
For your calculations, you may use the following data:
Tube Dimensions
Outer Diameter (mm) 19.1 mm
Inner Diameter (mm) 16.3 mm
Length 938 mm
Position of lever Left of lever: 497 mm Right of lever: 441 mm
Gsteel 75 GPa
Load Cell Dimensions
Cross sectional Area (mm2) 20.0 mm x 3.0 mm = 60 mm2
Ealuminium 70 GPa
Length from Load Cell to tube (mm) 141.93 mm
Get your hands dirty! Not this year though. Read through the instructions instead and keep your hands clean.
1. Connect the rosette strain gauges to the preamp in an order that makes sense to you so
that you can interpret the data later on. It is advisable to draw a diagram of the apparatus
and record which gauge is connected to which channel.
2. Fit your load cell (from the first prac) into the apparatus.
3. Set up the NI DAQ data acquisition equipment to record the output from all 7 channels.
Make sure you remember to connect to the analogue side of the NI DAQ and set the channels to a voltage range of 5 volts. You can trim the ‘zero’ point of each channel by using the adjusting knobs on the preamp.
Mechanics of Solids.
4. Start logging and record a shunt calibration trace for each of the active channels before
starting the actual test.
5. For each turn of the nut, the lever is pulled in by 2.5 mm. It is recommended that you do a
full turn and wait a moment before doing another full turn. This will make it easier to follow
the data during your analysis (you will be able to see when you were turning the nut and
how many turns were executed).
6. Complete a maximum of 5 turns. Doing more than this will interfere with your load cell.
7. Unwind the nut until it is loose and repeat until you have reliable data.
8. Test your own load cell (i.e. each member of the group must complete their own test).
Report
Your prac report should be semi formal, only briefly outlining what was done and why (i.e. aim,
apparatus & method), with the main focus being on the discussion and analysis of the results.
Ensure your report answers the following points (show all working where calculations are
required).
1. Draw a free body diagram of the system.
2. Calculate the torque applied by the lever and the torsional stress induced in the steel tube
using the angle of rotation of the lever as the source data.
3. Calculate the torque applied by the lever and the torsional stress induced in the steel tube
using the force measured in your load cell as the source data.
4. Calculate the torque applied by the lever and the torsional stress induced in the steel tube
from the torsional strain measured in the steel tube from the rosette strain gauges as the
source data.
Mechanics of Solids.
5. The strains as measured by the rosette gauges along the X and Y axes were very close to
zero with maximum strain occurring on the middle strain gauge (which is at 45° to the X & Y
axes). Please explain why this is the expected case and why there was a small output from
the X & Y gauges. (X here is defined as along the tube and Y across it).
6. Please explain which of your results (from questions 2 – 4) correlated with each other (say
within 15%) and which did not correlate, stating possible reasons for the differences with
justifications for these reasons.
7. Did your load cell appear to work successfully? What could the possible causes of error be
with your load cell?
8. If the middle bearing (near the torque arm) was removed, would the torque applied to the
tube be higher or lower with the same number of turns of the nut?
