ÇANKAYA UNIVERSITY
©2013
The descriptive text for the experiments contained in this Manual is
copyrighted material by TecQuipment Ltd (2008). It is intended solely
for use by students of Çankaya University who are registered for this
course. Any other use falls outside the scope of the permission that has
been granted to Çankaya University by the Vendor, and is not allowed.
TABLE OF CONTENTS
MM1: Structures Test Frame and Digital .................................................................... 5
MM2: Bending Moment in a Beam ............................................................................ 11
Experiment 3.1: Bending Moment Variation at the Point of Loading ......................... 14
Experiment 3.2: Bending Moment Variation away from the Point of Loading ............ 15
MM3: Shear Force in a Beam ..................................................................................... 17
Experiment 3.1: Shear Force Variation with Increasing Load ................................... 20
Experiment 3.2: Shear Force Variation for Various Load Conditions ....................... 21
MM4: Deflections of Beams and Cantilevers ............................................................ 23
Experiment 4.1: Deflection of a Cantilever ................................................................ 26
Experiment 4.2: Shear Deflection of a Simply Supported Beam ................................. 28
Experiment 4.3: The Shape of a Deflected Beam ........................................................ 30
Experiment 4.4: Circular Bending.............................................................................. 31
MM5: Bending Stress in a Beam ................................................................................ 35
Experiment 5.1: Bending Stress in a Beam ................................................................ 39
MM6: Torsion of Circular Sections ........................................................................... 45
Experiment 6.1: Torsional Deflection of a Solid Rod ................................................ 48
Experiment 6.2: The Effect of Rod Length on the Torsional Deflection ...................... 50
Experiment 6.3: Comparison of Solid Rod and Tube .................................................. 51
MM12: Buckling of Struts .......................................................................................... 53
Experiment 12.1: Buckling Load of as Pinned-End Strut ............................................ 57
Experiment 12.2: The Effect of End Conditions on the Buckling Load ....................... 59
MM1: Structures Test Frame and Digital
Force Display
- Description
- Operating Conditions
- Setting up the Equipment
- Assembling the Test Frame
- Connecting the Digital Force Display
- Care and Maintanence
5
Description
Operating Conditions
The Structures Test Frame (STR1) is a sturdy
aluminium frame designed to stand on a workbench.
The experiments in the TecQuipment Structures Range
mount in or on this frame. It has a capacity of 5 kN and
a window opening of 700 × 450 mm. Figure 1 shows
the Structures Test Frame.
Table 1 shows the working limits of both the Test
Frame and the Digital Force Display. Do not exceed any
of these limits.
Self-positioning
nuts
Slots for selfpositioning nuts
Storage temperature
range:
–25°C - +55°C (when packaged for
transport)
Operating temperature
range:
+5°C - +40°C
Operating relative
humidity range:
80% at temperatures ≤ 31°C
decreasing linearly to 50% at 40°C
Operating environment:
Indoor (laboratory)
Altitude up to 2000 m
Overvoltage category 2 (as
specified in EN61010-1).
Pollution degree 2 (as specified in
EN61010-1).
STR1A Power
Supply Voltage
'Window'
Input 90 VAC to 264 VAC
50 Hz to 60 Hz at 1A
Output 12 VDC at 5 A
Centre Positive
Table 1 Operating limits
Adjustable
levelling feet
Figure 1 Structures test frame
The Digital Force Display (STR1A) displays force in
two ranges (0 - 20 N and 0 - 500 N). It is for use with
the piezoelectric force sensors and proving ring load
cells of experiments in the TecQuipment Structures
Range. An automatic data acquisition (ADA) output
allows connection to a separate Automatic Data
Acquisition Unit (STR2000).
Before setting up and using the equipment, always:
• Visually inspect all parts (including electrical leads)
for damage or wear. Replace as necessary.
• Check electrical connections are correct and secure.
Electrical maintenance must only be carried out by a
competent person.
• Check all components are secured correctly and
fastenings are sufficiently tight.
• Position the Test Frame safely. Make sure it is
mounted on a solid, level surface, is steady, and
easily accessible.
When the equipment is in use, always:
• Make sure students are adequately supervised.
• Comply with any statutory requirements that are in
force about the installation, operation and
maintenance of this apparatus in the country where
it is to be used.
• Ensure excessive loads are never applied to any part
of the equipment.
Setting up the Equipment
The ‘Packing Contents List’ supplied is a list of items
provided with the apparatus to enable normal use of it
during the warranty period. If any item is missing or
damaged, contact TecQuipment or the importer.
A protective coating may have been applied to parts
of this apparatus to prevent corrosion during transport.
If so, remove the coating with paraffin or white spirit,
and a cloth or brush.
WARNING – If you do not use the equipment as
described in these instructions, its protective parts may
not work correctly.
6
MM1: Structures Test Frame and Digital Force Display
Assembling the Test Frame
Lie the frame ‘window’ on a workbench with the
uncapped ends pointing outwards.
Stand the frame up on its feet, and adjust so it sits
squarely and evenly on all four feet. Lock the feet
position with the locking nut.
Figure 4 Adjusting the self-positioning nuts
Figure 2 Fitting the legs
Referring to Figure 2, determine which way the legs
face (ensure that the threaded hole for the feet face
outwards). Using the special cap head screws secure
both legs to the frame.
Threaded
holes
for feet
Test frame
mounting holes
Test frame
leg
While setting up some experiments in the structures
range, you may have to adjust the self-positioning nuts
in the test frame. To do this, referring to Figure 4;
1. Push the bottom of the nut with a thin steel rule. The
nut should rotate 90° in the rail exposing the spring.
2. Slide the rule between the spring and the rail and
ease the nut out with a finger.
To reposition the nut simply place it into the rail in the
rotated position and push back and upwards into the
rail.
Connecting the Digital Force Display
Digital
Force Display
Washer
1
2
3
4
Power Supply
From Mains
Electrical Supply
ADA
OUTPUT
FORCE INPUTS
N
POWER ON
DISPLAY INPUT:
1
INPUT
12V
2
3
4
Adjustable
locking nut
Levelling foot
Anti-slip
rubber inserts
Figure 3 Fitting the self-levelling feet
Fit the anti-slip rubber inserts into each of the four
levelling feet. Screw a locking nut onto the feet, and
screw the feet into the threaded holes in the frame legs.
STR1A
DIGITAL FORCE DISPLAY
TecQuipment supply a mains-to-dc converter (power
supply) with the Digital Force Display.
There are no user serviceable parts on the Power
Supply. If it fails, renew it.
Use the mains cable supplied with the Power Supply to
connect it to your electrical supply using these colours:
Brown – Live
Blue – Neutral
Green/Yellow – Earth
WARNING - Connect the cable to the supply through
a plug and socket. The apparatus must be connected to
earth.
WARNING - The mains supply connector at the
Power Supply is its mains disconnect device. Make
sure it is always easily accessible.
7
MM1: Structures Test Frame and Digital Force Display
Care and Maintenance
NOTE: Renew or replace faulty or damaged parts or
detachable cables with an equivalent item of the same
type or rating.
To clean the Test Frame and Force Display, use a damp
cloth.
Figure 5 The digital force display
Figure 5 shows the Digital Force Display. It fits on the
Test Frame by means of two thumbscrews through the
top of the casing.
The Digital Force Display connects to experiments
using the lead with a four-way mini DIN plug at each
end. Connect one end of the lead to any of the four
sockets marked “Force Input” on the Digital Force
Display. Connect the other end of the lead to the socket
on the experiment marked “Force Output”. All four
channels can be connected at the same time.
To display a force reading, turn the control on the
front panel to the number of the force input socket you
want to read. The display will automatically set itself to
a 0 - 20 N range or a 0 - 500 N range, dependent on the
type of transducer (piezoelectric or load ring) the
experiment uses.
To use the Automatic Data Acquisition Unit with
the Digital Force Display, use the six-way mini-DIN
lead provided with the Automatic Data Acquisition
Unit. Connect the socket marked “ADA Output” on the
Digital Force Display to the socket marked “Force
Input” on the Automatic Data Acquisition Unit.
8
If the Force Display fails, first check its Power Supply
still works by trying it with another Force Display or
testing its low voltage output with a multimeter.
If the Power Supply works but the Force Display does
not, then contact TecQuipment or your local agent.
If the Power supply does not work, check its electrical
supply is connected. Do not try to open the Power
Supply or the Force Display. There are no userserviceable parts inside.
MM1: Structures Test Frame and Digital Force Display
NOTES:
9
10
MM2: Bending Moment in a Beam
- Introduction and Description
- Experiments
1- Bending Moment Variation at the Point of Loading
2- Bending Moment Variation away from the Point of
Loading
11
SECTION 1 INTRODUCTION AND DESCRIPTION
Figure 1 Bending moment in a beam experiment
Introduction
How to Set up the Equipment
This guide describes how to set up and perform Bending
Moment in a Beam experiments. It clearly demonstrates
the principles involved and gives practical support to
your studies.
The Bending Moment in a Beam experiment fits into a
Test Frame. Figure 2 shows the Bending Moment of a
Beam experiment assembled in the Frame.
Before setting up and using the equipment, always:
Description
Figure 1 shows the Bending Moment in a Beam
experiment. It consists of a beam, which is ‘cut’ by a
pivot. To stop the beam collapsing a moment arm
bridges the cut onto a load cell thus reacting (and
measuring) the bending moment force. A digital display
shows the force from the load cell.
A diagram on the left-hand support of the beam
shows the beam geometry and hanger positions. Hanger
supports are 20 mm apart, and have a centre slot, which
positions the hangers. The moment arm is 125 mm long.
12
 Visually inspect all parts, including electrical leads,
for damage or wear.
 Check electrical connections are correct and secure.
 Check all components are secured correctly and
fastenings are sufficiently tight.
 Position the Test Frame safely. Make sure it is
mounted on a solid, level surface, is steady, and
easily accessible.
Never apply excessive loads to any part of the
equipment.
MM2: Bending Moment in a Beam
Figure 2 Bending moment of a beam experiment in the structures frame
Steps 1 to 4 of the following instructions may already
have been completed for you.
1. Place an assembled Test Frame (refer to the separate
instructions supplied with the Test Frame if
necessary) on a workbench. Make sure the ‘window’
of the Test Frame is easily accessible.
2. There are four securing nuts in the top member of
the frame. Slide them to approximately the positions
shown in Figure 3.
3. With the right-hand end of the experiment resting on
the bottom member of the Test Frame, fit the lefthand support to the top member of the frame. Push
the support on to the frame to ensure that the internal
bars are sitting on the frame squarely. Tighten the
support in position by screwing two of the
thumbscrews provided into the securing nuts (on the
front of the support only).
4. Lift the right-hand support into position and locate
the two remaining thumbscrews into the securing
nuts. Push the support on to the frame to ensure the
internal bars are sitting on the frame squarely.
Position the support horizontally so the rolling pivot
is in the middle of its travel. Tighten the
thumbscrews.
5. Make sure the Digital Force Display is ‘on’. Connect
the mini DIN lead from ‘Force Input 1’ on the
Digital Force Display to the socket marked ‘Force
Output’ on the left-hand support of the experiment.
Ensure the lead does not touch the beam.
6. Carefully zero the force meter using the dial on the
left-hand beam of the experiment. Gently apply a
small load with a finger to the centre of the beam and
release. Zero the meter again if necessary. Repeat to
ensure the meter returns to zero.
Note: If the meter is only ±0.1 N, lightly tap the frame
(there may be a little stiction and this should overcome
it).
13
SECTION 2 EXPERIMENTS
Experiment 1: Bending Moment Variation at the Point of Loading
This experiment examines how bending moment varies
at the point of loading. Figure 3 shows the force diagram
for the beam.
Check the Digital Force Display meter reads zero with
no load.
Place a hanger with a 100 g mass at the ‘cut’. Record
the Digital Force Display reading in a table as in Table
1. Repeat using masses of 200 g, 300 g, 400 g and
500 g.
Convert the mass into a load (in N) and the force
reading into a bending moment (Nm). Remember;
Bending moment at
= Displayed force  0.125
the cut (in Nm)
Calculate the theoretical bending moment at the cut and
complete Table 2.
Figure 3 Force diagram
The equation we will use in this experiment is:
BM (at cut) = Wa
Mass
(g)
l  a 
Experimental
Theoretical
Load Force
bending moment bending moment
(N)
(N)
(Nm)
(Nm)
0
l
You may find the following table useful in converting
the masses used in the experiments to loads.
100
200
300
400
Mass (Grams)
Load (Newtons)
100
0.98
200
1.96
300
2.94
400
3.92
500
4.90
500
Table 2 Results for Experiment 1
Plot a graph which compares your experimental results
to those you calculated using the theory.
Comment on the shape of the graph. What does it tell
us about how bending moment varies at the point of
loading? Does the equation we used accurately predict
the behaviour of the beam?
Table 1 Grams to Newtons conversion table
14
MM2: Bending Moment in a Beam: Student Guide
Experiment 2: Bending Moment Variation away from the Point of Loading
This experiment examines how bending moment varies
at the cut position of the beam for various loading
conditions. Figure 4, Figure 5 and Figure 6 show the
force diagrams.
Figure 6 Force diagram
We will use the statement:
“The Bending Moment at the ‘cut’ is equal to the
algebraic sum of the moments caused by the forces
acting to the left or right of the cut.”
Figure 4 Force diagram
Check the Digital Force Display meter reads zero with
no load.
Carefully load the beam with the hangers in the
positions shown in Figure 4, using the loads indicated in
Table 3. Record the Digital Force Display reading in a
table as in Table 2.
Convert the force reading into a bending moment (in
Nm). Remember;
Bending moment at
= Displayed force  0.125
the cut (in Nm)
Calculate the support reactions (RA and RB) and
calculate the theoretical bending moment at the cut.
Repeat the procedure with the beam loaded as in
Figure 5 and Figure 6.
Comment on how the results of the experiments
compare with those calculated using the theory.
Figure 5 Force diagram
W2
(N)
Figure
W1
(N)
4
3.92
5
1.96
3.92
6
4.91
3.92
Force
(N)
Experimental
bending moment
(Nm)
Table 3 Results for Experiment 2
15
RA
(N)
RB
(N)
Theoretical
bending moment
(Nm)
MM2: Bending Moment in a Beam
NOTES:
16
MM3: Shear Force in a Beam
- Introduction and Description
- Experiments
1- Shear Force Variation with Increasing Load
2- Shear Force Variation for Various Load
Conditions
17
SECTION 1 INTRODUCTION AND DESCRIPTION
Figure 1 Shear forces in a beam experiment
Introduction
How to Set Up the Equipment
This guide describes how to set up and perform Shear
Force in a Beam experiments. It clearly demonstrates
the principles involved and gives practical support to
your studies.
The Shear Force in a Beam experiment fits into a Test
Frame. Figure 2 shows the Shear Force of a Beam
experiment assembled in the Frame.
Before setting up and using the equipment, always:
Description
Figure 1 shows the Shear Force in a Beam experiment.
It consists of a beam which is ‘cut’. To stop the beam
collapsing a mechanism, (which allows movement in
the shear direction only) bridges the cut on to a load cell
thus reacting (and measuring) the shear force. A digital
display shows the force from the load cell.
A diagram on the left-hand support of the beam
shows the beam geometry and hanger positions. Hanger
supports are 20 mm apart, and have a central groove
which positions the hangers.
18
• Visually inspect all parts, including electrical leads,
for damage or wear.
• Check electrical connections are correct and secure.
• Check all components are secured correctly and
fastenings are sufficiently tight.
• Position the Test Frame safely. Make sure it is
mounted on a solid, level surface, is steady, and
easily accessible.
Never apply excessive loads to any part of the
equipment.
MM3: Shear Force in a Beam
Figure 2 Shear force of a beam experiment in the structures frame
Steps 1 to 4 of the following instructions may already
have been completed for you.
1. Place an assembled Test Frame (refer to the separate
instructions supplied with the Test Frame if
necessary) on a workbench. Make sure the ‘window’
of the Test Frame is easily accessible.
2. There are four securing nuts in the top member of
the frame. Slide them to approximately the positions
shown in Figure 3.
3. With the right-hand end of the experiment resting on
the bottom member of the Test Frame, fit the lefthand support to the top member of the frame. Push
the support on to the frame to ensure that the
internal bars are sitting on the frame squarely.
Tighten the support in position by screwing two of
the thumbscrews provided into the securing nuts (on
the front of the support only).
4. Lift the right-hand support into position and locate
the two remaining thumbscrews into the securing
nuts. Push the support on to the frame to ensure the
internal bars are sitting on the frame squarely.
Position the support horizontally so the rolling pivot
is in the middle of its travel. Tighten the
thumbscrews.
5. Make sure the Digital Force Display is ‘on’.
Connect the mini DIN lead from ‘Force Input 1’ on
the Digital Force Display to the socket marked
‘Force Output’ on the left-hand support of the
experiment. Ensure the lead does not touch the
beam.
6. Carefully zero the force meter using the dial on the
left-hand beam of the experiment. Gently apply a
small load with a finger to the centre of the beam
and release. Zero the meter again if necessary.
Repeat to ensure the meter returns to zero.
Note: If the meter is only ±0.1 N, lightly tap the frame
(there may be a little ‘stiction’ and this should
overcome it).
19
SECTION 2 EXPERIMENTS
Experiment 1: Shear Force Variation with an Increasing Point Load
This experiment examines how shear force varies with
an increasing point load. Figure 3 shows the force
diagram for the beam.
W
a
Remember,
40 mm
RA
RB
'Cut'
Check the Digital Force Display meter reads zero with
no load.
Place a hanger with a 100 g mass to the left of the
‘cut’ (40 mm away). Record the Digital Force Display
reading in a table as in Table 2. Repeat using masses of
200 g, 300 g, 400 g and 500 g. Convert the mass into a
load (in N).
l
Shear force at the cut = Displayed force
Calculate the theoretical shear force at the cut and
complete the table.
Figure 3 Force diagram
The equation we will use in this experiment is:
W .a
Shear force at cut, S c =
l
Mass
Load
Experimental shear
Theoretical shear
(g)
(N)
force (N)
force (N)
0
100
200
Where a is the distance to the load (not the cut)
Distance a = 260 mm
300
400
You may find the following table useful in converting
the masses used in the experiments to loads.
Mass (Grams)
Load (Newtons)
100
0.98
200
1.96
300
2.94
400
3.92
500
4.90
500
Table 2 Results for Experiment 1
Plot a graph which compares your experimental results
to those you calculated using the theory.
Comment on the shape of the graph. What does it
tell us about how shear force varies due to an increased
load? Does the equation we used accurately predict the
behaviour of the beam?
Table 1 Grams to Newtons conversion table
20
MM3: Shear Force in a Beam
Experiment 2: Shear Force Variation for Various Loading Conditions
This experiment examines how shear force varies at the
cut position of the beam for various loading conditions.
Figure 4, Figure 5 and Figure 6 show the force
diagrams.
Figure 6 Force diagram
We will use the statement:
“The Shear force at the ‘cut’ is equal to the
algebraic sum of the forces acting to the left or right
of the cut.”
Figure 4 Force diagram
Check the Digital Force Display meter reads zero with
no load.
Carefully load the beam with the hangers in the
positions shown in Figure 4, using the loads indicated in
Table 2.
Record the Digital Force Display reading as in
Table 3. Remember,
Shear force at the cut (N) = Displayed force
Calculate the support reactions (RA and RB) and
calculate the theoretical shear force at the cut.
Repeat the procedure with the beam loaded as in
Figure 5 and Figure 6.
Comment on how the results of the experiments
compare with those calculated using the theory.
Figure 5 Force diagram
Figure
W1
W2
Force
Experimental shear
RA
RB
Theoretical shear
(N)
(N)
(N)
force (N)
(N)
(N)
force (Nm)
4
3.92
5
1.96
3.92
6
4.91
3.92
Table 2 Results for Experiment 2
21
MM3: Shear Force in a Beam: Student Guide
NOTES:
22
MM4: Deflections of Beams and Cantilevers
- Introduction and Description
- Experiments
1- Deflection of a Cantilever
2- Shear Deflection of a Simply Supported Beam
3- The Shape of a Deflected Beam
4- Circular Bending
23
SECTION 1 INTRODUCTION AND DESCRIPTION
Figure 1 Deflection of Beams and Cantilevers experiment
Introduction
This guide describes how to set up and perform
experiments on the deflection behaviour of beams and
cantilevers. The equipment clearly demonstrates the
principles involved and gives practical support to your
studies.
Description
Figure 1 shows the Deflections of Beams and
Cantilevers experiment. It consists of a backboard with
a digital dial test indicator. The digital dial test indicator
is on a sliding bracket which allows it to traverse
accurately to any position along the test beam. Two
rigid clamps mount on the backboard and can hold the
beam in any position. Two knife-edge supports also
fasten anywhere along the beam. Scales printed on the
backboard allow quick and accurate positioning of the
digital dial test indicator, knife-edges and loads.
Look at the reference information on the backboard.
It is useful and you may need it to complete the
experiments in this guide.
How to Set up the Equipment
The Deflections of Beams and Cantilevers experiment
fits into a Test Frame. Figure 2 shows the Deflections of
Beams and Cantilevers experiment in the Frame.
Before setting up and using the equipment, always:
• Visually inspect all parts, including electrical leads,
for damage or wear.
• Check electrical connections are correct and secure.
• Check all components are secure and fastenings are
sufficiently tight.
• Position the Test Frame safely. Make sure it is on a
solid, level surface, is steady, and easily accessible.
Never apply excessive loads to any part of the
equipment.
24
MM4: Deflections of Beams and Cantilevers
Figure 2 Deflections of Beams and Cantilevers experiment in the structures frame
The following instructions may already have been
completed for you.
them to roughly the positions of the thumbscrews
shown in Figure 2.
1. Place an assembled Test Frame (refer to the separate
instructions supplied with the Test Frame if
necessary) on a workbench. Make sure the
‘window’ of the Test Frame is easily accessible.
3. Lift the backboard into position and have an
assistant secure it by threading the thumbscrews into
the securing nuts. If necessary, level the backboard
by loosening the thumbscrews on one side,
repositioning the backboard, and tightening the
thumbscrews.
2. There are two securing nuts in each of the side
members of the frame (on the inner track). Slide
25
SECTION 2: EXPERIMENTS
Experiment 1: Deflection of a Cantilever
In this experiment, we will examine the deflection of a
cantilever subjected to an increasing point load. We will
repeat this for three different materials to see if their
deflection properties vary.
0
10
20
30
40
50
60
70
80
90
100
110
1 20 130
140
150
1 60 170
180
190 200
210
Z ER O/ AB S
P RE S ET
0
10
20
30
40
50
60
70
80
90
100
110
12 0 130
140 1 50
160
170
180 19 0 200
220
2 30 240
250
260 27 0
280
290
260
280 29 0
3 00 310
320
33
300
3 20 33
O N /O F F
TO L .
210 22 0
230
240 2 50
270
310
Remove any clamps and knife edges from the
backboard. Set up one of the cantilevers as shown in
Figure 3.
Slide the digital dial test indicator to the position on
the beam shown in Figure 3, and lock it using the
thumbnut at the rear. Slide a knife-edge hanger to the
position shown.
Tap the frame lightly and zero the digital dial test
indicator using the ‘origin’ button.
Apply masses to the knife-edge hanger in the
increments shown in Table 1. Tap the frame lightly each
time you add the masses. Record the digital dial test
indicator reading for each increment of mass.
Repeat the procedure for the other two materials and
fill in a new table.
200 mm
Material
E value: ___________ Nm
–2
4
I: _________________ m
Width b: ____________ mm
Depth d: ____________ mm
W
Mass
Actual deflection
Theoretical deflection
Figure 3 Cantilever set-up and schematic
(g)
(mm)
(mm)
You may find the following table useful in converting
the masses used in the experiments to loads.
100
Mass (Grams)
Load (Newtons)
100
0.98
200
1.96
300
2.94
400
3.92
500
4.90
200
300
400
500
Table 1 Results for Experiment 1 (beam 1)
Table 1 Grams to Newtons conversion table
Material
As well as the information given on the backboard you
will need the following formula:
Deflection =
0
WL3
3EI
where:
E value: ___________ Nm
4
I: _________________ m
–2
Width b: ____________ mm
Depth d: ____________ mm
Mass
Actual deflection
Theoretical deflection
(g)
(mm)
(mm)
0
W = Load (N)
L = Distance from support to position of loading
(m);
–2
E = Young’s modulus for cantilever material (Nm );
4
I = Second moment of area of the cantilever (m ).
Using a vernier gauge, measure the width and depth of
the aluminium, brass and steel test beams. Record the
values next to the results tables for each material and
use them to calculate the second moment of area, I.
26
100
200
300
400
500
Table 2 Results for Experiment 1 (beam 2)
MM4: Deflections of Beams and Cantilevers
Material
E value: ___________ Nm
4
I: _________________ m
–2
Width b: ____________ mm
Depth d: ____________ mm
Mass
Actual deflection
Theoretical deflection
(g)
(mm)
(mm)
0
100
200
300
400
On the same axis, plot a graph of Deflection versus
Mass for all three beams. Comment on the relationship
between the mass and the beam deflection. Is there a
relationship between the gradient of the line for each
graph and the modulus of the material?
Calculate the theoretical deflection for each beam
and add the results to your table and the graph. Does the
equation accurately predict the behaviour of the beam?
Why is it a good idea to tap the frame each time we
take a reading from the digital dial test indicator?
Name at least three practical applications of a
cantilever structure.
500
Table 3 Results for Experiment 1 (beam 3)
27
MM4: Deflections of Beams and Cantilevers
Experiment 2: Deflection of a Simply Supported Beam
In this experiment, we will examine the deflection of a
simply supported beam subjected to an increasing point
load. We will also vary the beam length by changing the
distance between the supports. This means we can find
out the relationship between the deflection and the
length of the beam.
As well as the information given on the backboard
you will need the following formula:
3
Part 1
Using a vernier gauge, measure the width and depth of
the aluminium test beam. Record the values next to the
results table and use them to calculate the second
moment of area, I.
Remove any clamps from the backboard. Setting
length between supports l to 400 mm, set up the beam
as shown in Figure 4.
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
Actual deflection
Theoretical deflection
(g)
(mm)
(mm)
0
100
400
500
Table 4 Results for Experiment 2 (fixed beam
length variable load)
Part 2
Set up the beam with the length l at 200 mm. Ensure the
digital dial test indicator and load hanger are still central
to the beam, as shown in Figure 5.
0
PRESET
10
20
30
40
50
60
70
800
90
100 110
12
120
130
140
150
160
170
180
190
200
210
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230
240
250
260
270
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290
330
340
350
360
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380
390
400
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440
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460
470
480
490
330
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440
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460
470
80 490
480
500
510
520
530
540
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560
570
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590
500 510
5
520
530
540
550
560
570
580
590
10
20
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40
50
60
70
80
90
100 110
120 130
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
320 330 340
10
20
30
40
50
60
70
80
90
100 110
120 130
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
370 380 390
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
590
350 360
370 380 3900
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
590
ON/OFF
ZERO/ABS
PRESET
350 360
TOL.
320 330 340
ON/OFF
ZERO/ABS
0
320
Depth d: ____________ mm
Mass
0
10
Width b: ____________ mm
I: _________________ m
300
where:
W = Load (N);
L = Distance from support to support (m);
–2
E = Young’s modulus for cantilever material (Nm );
4
I = Second moment of area of the cantilever (m ).
0
–2
4
200
WL
48EI
Maximum deflection =
E value: ___________ Nm
300
TOL.
310
320
l =200 mm
l = 400 mm
200 mm
200 mm
W
Figure 5 Simply supported beam set-up and
schematic (fixed beam load with variable length)
W
Figure 4 Simply supported beam set-up and
schematic (fixed beam with variable load)
Slide the digital dial test indicator into position on the
beam and lock it using the thumbnut at the rear. Slide a
knife-edge hanger to the position shown.
Tap the frame lightly and zero the digital dial test
indicator using the ‘origin’ button.
Apply masses to the knife-edge hanger in the
increments shown in the results table. Tap the frame
lightly each time, and record the digital dial test
indicator reading for each increment of mass.
28
Lightly tap the frame and zero the digital dial test
indicator using the ‘origin’ button. Apply a 500 g mass
and record the deflection in Table 5. Repeat the
procedure for each increment of beam length.
From Table 4 plot a graph of Deflection versus
Applied Mass for a simply supported beam. Comment
on the your graph. Inspect the ruling equation of the
beam. What is the relationship between the deflection
and the beam length? Test your assumption by filling in
the empty column of Table 5 with the correct variable.
Plot a graph.
MM4: Deflections of Beams and Cantilevers
Length (mm)
Deflection (mm)
Name at least one example where this type of bending is
desirable and one where it is undesirable.
200
260
320
380
440
500
560
Table 5 Results for Experiment 2 (fixed beam load
variable length)
29
MM4: Deflections of Beams and Cantilevers
Experiment 3: The Shape of a Deflected Beam
This experiment shows how the deflection of a loaded
beam varies with span.
Traverse the loaded beam with the digital dial test
indicator recording the deflections.
Position from
0
10
20
30
40
50
60
70
80
90
100 110
10
120 130
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
320 330 340
350 360
370 380 390
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
left (mm)
590
PRESET
Loaded
Deflection
(mm)
0
ON/OFF
ZERO/ABS
Datum
reading (mm) reading (mm)
TOL.
20
0
10
20
30
40
50
60
70
80
90
90
100 110
110
100
120 130
120
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
320 330 340
350 360
370 380 390
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
590
40
60
80
100
600 mm
150
x
200
250
200 mm
200 mm
300
350
W
400
Figure 6 Simply supported beam set-up and
schematic
450
500
Remove any clamps from the backboard and set up the
beam as shown in Figure 6.
Slide the digital dial test indicator to the zero
position on the beam and, using the ‘±’ button, set it so
a downward movement reads negative. Do not lock the
digital dial test indicator. Slide a knife-edge hanger to
the correct position on the beam.
Tap the frame lightly. Roughly zero the digital dial
test indicator using the ‘origin’ button. Record the
actual ‘datum’ value in Table 6.
Carefully slide the digital dial test indicator to the
positions shown in Table 6 (note the change in the
increments after 100 mm). Remember to tap the frame
each time you take a reading. Record the ‘datum’ value
at each position.
Apply a 500 g mass to the knife-edge hanger and
return the digital dial test indicator to the zero position.
Make sure the digital dial test indicator stylus passes
through the gap in the knife-edge hanger.
30
550
600
Table 6 Results for Experiment 3
Work out the true deflection from the datum and loaded
values. Why is it important to take datum values in this
experiment?
Plot a graph of deflection versus position along the
beam. What shape does the beam adopt outside the
bounds of the knife-edge supports? Why is that?
Using a suitable method calculate the true deflection
of the beam (within the bounds of the knife-edge
supports) and add the data to the graph. Does the
method you have used accurately predict the shape of
the deflected beam?
MM4: Deflections of Beams and Cantilevers
Experiment 4: Circular Bending
In this experiment, we apply loads to a simply
supported beam at its end to induce a moment and thus
produce circular bending. As well helping to establish
an important relationship, this test is an accurate method
for measuring Young’s modulus.
h
C
R = Radius of curvature (m);
C = Chord (m);
h = Height of chord (m).
R
0
10
20
30
40
50
60
70
80
90
100 110
120 130
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
320 330 340
0
10
20
30
40
50
60
70
80
90
90
100 110
110
100
120 130
120
140 150
160 170
180 190 200
210 220
230 240 250
260 270
280 290
300 310
370 380 390
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
590
350 360
370 380 390
400 410 420
430 440
450 460 470
480 490
500 510
520 530 540
550 560
570 580
590
ON/OFF
ZERO/ABS
PRESET
350 360
TOL.
320 330 340
Figure 8 Radius of curvature
100 mm
400 mm
W
100 mm
W
Figure 7 Circular bending set-up and schematic
In this experiment we will be using the following
formula:
M
E
=
R
I
where:
M = Applied moment (Nm);
R = Radius of curvature (m);
–2
E = Young’s modulus for cantilever material (Nm );
4
I = Second moment of area of the cantilever (m ).
Using a vernier, measure the width and depth of the
aluminium, brass and steel test beams. For each
material, record the values next to the results tables and
use them to calculate the second moment of area, I.
Remove any clamps from the backboard and set up
the beam as shown in Figure 7.
Slide the digital dial test indicator into position on
the beam and lock it using the thumbnut at the rear.
Slide a knife-edge hanger on to each end of the beam as
shown.
Tap the frame lightly and zero the digital dial test
indicator using the ‘origin’ button.
Tapping the frame lightly each time, apply masses to
the knife-edge hangers in increments as shown in
Table 7. Record the digital dial test indicator reading for
each increment of mass.
Repeat the procedure for the other two specimen
materials filling in a new table.
You will also need to use the following mathematical
relationship:
R =
C 2 + 4h 2
8h
Material: _______________________
–2
E value: _____ Nm
Width, b: ____ mm
Mass at each end
Deflection
Applied moment
Radius of
(g)
(mm)
(Nm)
curvature (m)
0
100
200
300
400
500
Table 7 Results for Experiment 4 (beam 1)
31
Depth, d: ____ mm
I: ___________ m
1/R
M/I (× 10 )
9
4
MM4: Deflections of Beams and Cantilevers
Material: _______________________
–2
E value: _____ Nm
Width, b: ____ mm
Mass at each end
Deflection
Applied moment
Radius of
(g)
(mm)
(Nm)
curvature (m)
Depth, d: ____ mm
I: ___________ m
1/R
M/I (× 10 )
4
9
0
100
200
300
400
500
Table 8 Results for Experiment 4 (beam 2)
Material: _______________________
–2
E value: _____ Nm
Width, b: ____ mm
Mass at each end
Deflection
Applied moment
Radius of
(g)
(mm)
(Nm)
curvature (m)
Depth, d: ____ mm
1/R
I: ___________ m
4
9
M/I (× 10 )
0
100
200
300
400
500
Table 9 Results for Experiment 4 (beam 3)
From the load values calculate the applied moment in
Nm. From the deflection calculate values for the radius
of curvature in m. Then complete the table by
calculating 1/R and M/I.
32
Plot a graph of M/I versus 1/R. Is this a linear
relationship? If so, what is the value of the gradient.
MM4: Deflections of Beams and Cantilevers
NOTES:
33
34
MM5: Bending Stress in a Beam
- Introduction and Description
- Experiments
1- Bending Stress in a Beam
35
SECTION 1.0 INTRODUCTION AND DESCRIPTION
Figure 1 Bending stress in a beam experiment
Introduction
This guide describes how to set up and perform Bending Stress in a Beam experiments. The equipment clearly
demonstrates the principles involved and gives practical support to your studies.
Description
Figure 1 shows the Bending Stress in a Beam experiment. It consists of an inverted aluminium T- beam, with strain
gauges fixed on the section (the front panel shows the exact positions).
The panel assembly and Load Cell apply load to the top of the beam at two positions each side of the strain gauges.
Loading the beam in this way (rather than loading the beam at just one point) has two main advantages:
•
It allows a gauge to be placed on the top of the beam.
•
The constant bending moment area it creates gives better strain gauge performance and avoids stress concentration
close to the gauge positions.
Strain gauges are sensors that experience a change in electrical resistance when stretched or compressed.
Strain gauges are made from a metal foil formed in a zigzag pattern. They are only a few microns thick so they are
mounted on a backing sheet. The backing sheet electrically insulates the zigzag element and supports it so it does not
collapse when handled.
The T-beam has strain gauges bonded to it. These stretch and compress the same amount as the beam, so measure strain
in the beam. If you look carefully at the equipment you will notice there is another set of strain gauges. These are called
36
MM5: Bending Stress in a Beam
dummy gauges. The dummy gauges, and how the way they are connected in the electrical circuit, help reduce inaccurate
readings caused by temperature changes and thermal expansion.
The Digital Strain Display converts the change in electrical resistance of the strain gauges to show it as displacement
(strain). It shows all the strains sensed by the strain gauges, reading in microstrain.
Look at the reference information on the unit. It is useful and you may need it to complete the experiments in this guide.
How to Set up the Equipment
The Bending Stress in a Beam experiment fits into a Test Frame. Figure 2 shows the Bending Stress in a Beam
experiment in the Frame.
Before setting up and using the equipment, always:
•
Visually inspect all parts, including electrical leads, for damage or wear.
•
Check electrical connections are correct and secure.
•
Check all components are secure and fastenings are sufficiently tight.
•
Position the Test Frame safely. Make sure it is on a solid, level surface, is steady, and easily accessible.
Never apply excessive loads to any part of the equipment.
The following instructions may already have been completed for you.
1. Place an assembled Test Frame (refer to the separate instructions supplied with the Test Frame if necessary) on a
workbench. Make sure the ‘window’ of the Test Frame is easily accessible.
2. There are two securing nuts in each of the side members of the frame (on the inner track). Move one securing nut
from each side to the outer track (see STR1 instruction sheet). Slide them to about the positions shown in Figure 2.
Fix the two supports on to the frame in the same position.
3. Slide two nuts into position to hold the load cell. Fix the load cell leaving the screws slightly loose.
4. Lift the beam into position and level the ends of the beam with the frame.
5. Position the load cell so the hole in the fork reaches the hole of the loading position, and it is vertical. Tighten the
load cell using the 6 mm A/F hexagonal key. Secure the fork using a pin.
6. Make sure the Digital Force Display is ‘on’. Connect the mini DIN lead from ‘Force Input 1’ on the Digital Force
Display to the socket marked ‘Force Output’ on the left-hand side of the load cell.
7. With no load on the load cell (the pin should turn), use the control on the front of the load cell to set the reading to
around zero.
8. Make sure the Digital Strain Display is ‘on’ and set to gauge configuration 1. Matching the number on the lead to
the number on the socket, connect the strain gauges to the strain display. Leave the gauges for five minutes to warm
up and reach a steady state.
37
MM5: Bending Stress in a Beam
Figure 2 Bending stress in a beam experiment in the structures frame
38
MM5: Bending Stress in a Beam
39
SECTION 2.0 EXPERIMENTS
Experiment 1: Bending Stress in a Beam
Figure 3 Beam set-up and schematic
As well as the information given on the unit you will need the following formulae:
E = σ
--ε
Where:
σ = Stress (Nm-2)
ε = Strain
E = Young’s modulus for the beam material (Nm–2)
(Typically 69 x 109 Nm-2 or 69 GPa)
and
M
----- = σ
--I
y
(The bending equation)
where:
M = Bending moment (Nm)
I = Second moment of area of the section (m4)
σ = Stress (Nm-2)
y = Distance from the neutral axis (m)
Ensure the beam and Load Cell are properly aligned. Turn the thumbwheel on the Load Cell to apply a positive (downward) preload to the beam of about 100 N. Zero the Load Cell using the control.
Take the nine zero strain readings by choosing the number with the selector switch. Fill in Table 1 with the zero force
values.
Increase the load to 100 N and note all nine of the strain readings. Repeat the procedure in 100 N increments to 500 N.
Finally; gradually release the load and preload.
Correct the strain reading values for zero (be careful with your signs!) and convert the load to a bending moment then
fill in Table 2.
From your results, plot a graph of strain against bending moment for all nine gauges (on the same graph).
•
What is the relationship between the bending moment and the strain at the various positions?
40
MM5: Bending Stress in a Beam
•
What do you notice about the strain gauge readings on opposite sides of the section? Should they be identical?
•
If the readings are not identical, give two reasons why.
Gauge
number
Load (N)
0
100
200
300
400
500
70
87.5
1
2
3
4
5
6
7
8
9
Table 1 Results for Experiment 1 (uncorrected)
Gauge
Bending moment (Nm)
Number
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
17.5
35
52.5
Table 2 Results for Experiment 1 (corrected)
41
MM5: Bending Stress in a Beam
Gauge
Nominal
Actual
Number
Vertical
position
(mm)
Vertical
Bending moment (Nm)
position
(mm)
0
1
0
2,3
8
4,5
23
6,7
31.7
8,9
38.1
Table 3 Averaged strain readings for Experiment 1
Calculate the average strains from the pairs of gauges and enter your results in Table 3 (disregard the zero values).
Carefully measure the actual strain gauge positions and enter the values into Table 3. Plot the strain against the relative
vertical position of the strain gauge pairs on the same graph for each value of bending moment. Take the top of the beam
as the datum.
Calculate the second moment of area and position of the neutral axis for the section (use a vernier to measure the exact
size of the section) and add the position of the neutral axis to the plot.
•
What is the value of strain at the neutral axis?
•
Calculate the maximum stress in the section by turning the strains into stress values (at the maximum load).
Compare this to the theoretical value.
•
Does the bending equation accurately predict the stress in the beam?
42
MM5: Bending Stress in a Beam
NOTES:
43
44
MM6: Torsion of Circular Sections
- Introduction and Description
- Experiments
1- Torsional Deflection of a Solid Rod
2- The Effect of Rod Length on the Torsional
Deflection
3- Comparison of Solid Rod and Tube
45
SECTION 1 INTRODUCTION AND DESCRIPTION
Figure 1 Torsion of circular sections experiment
Introduction
How to Set up the Equipment
This guide describes how to set up and perform
experiments on the torsion of circular sections. It clearly
demonstrates the principles involved and gives practical
support to your studies.
The Torsion of Circular Sections experiment fits into a
Test Frame. Figure 2 shows the Torsion of circular
sections experiment assembled in the Frame.
Before setting up and using the equipment, always:
Description
Figure 1 shows the Torsion of Circular Sections
experiment. It consists of a backboard with chucks for
gripping the test specimen at each end. The right-hand
chuck connects to a load cell using an arm to measure
torque. A protractor scale on the left-hand chuck
measures rotation. A thumbwheel on the protractor
scale twists specimens. Sliding the chuck along the
backboard alters the test specimen length.
The backboard has some formulae and data printed
on it. Note this information – it will be useful later.
46
• Visually inspect all parts, including electrical leads,
for damage or wear.
• Check electrical connections are correct and secure.
• Check all components are secured correctly and
fastenings are sufficiently tight.
• Position the Test Frame safely. Make sure it is on a
solid level surface, is steady and easily accessible.
Never apply excessive loads to any part of the
equipment.
MM6: Torsion of Circular Sections
Figure 2 Torsion of circular sections in the structures frame
Steps 1 to 3 of the following instructions may already
have been completed for you.
1.
2.
3.
Place an assembled Test Frame (refer to the
separate instructions supplied with the Test Frame
if necessary) on a workbench. Make sure the
‘window’ of the Test Frame is easily accessible.
There are two securing nuts in each of the side
members of the frame (on the inner track). Move
one to the outer track (see STR1 instruction sheet)
then slide them to approximately the positions
shown by the thumbscrews in Figure 2.
Lift the backboard into position and have an
assistant secure the backboard with thumbscrews
into the securing nuts. If necessary, level the
backboard by loosening the thumbscrews on one
side and tightening when ready.
4.
5.
47
Make sure the Digital Force Display is ‘on’.
Connect the mini DIN lead from ‘Force Input 1’ on
the Digital Force Display to the socket marked
‘Force Output’ on to the right underside of the
backboard.
Carefully zero the force meter using the dial.
Gently apply a small torque to the left-hand chuck
and release. If necessary, zero the meter again.
SECTION 2 EXPERIMENTS
Experiment 1: Torsional Deflection of a Solid Rod
This experiment examines the relationship between
torque and angular deflection of a solid circular section.
Further work will show how the properties of the
material affect this relationship.
With a pencil and a rule, mark the steel and brass rods
with these distances from the left-hand end (note that
the rubber tip is on the right-hand end):
• 15 mm,
• 315 mm,
• 365 mm,
• 415 mm,
• 465 mm,
• 515 mm.
Force
Torque, T
Angular deflection
(N)
(Nm)
(°)
0
0
0
1
2
3
4
5
Table 3 Results for a brass rod
Wind the thumbwheel down to its stop. Position the
steel rod from the right-hand side with the rubber tipped
end sticking out. Line up the first mark with the lefthand chuck (note the jaws of the chuck move outward
as they close!). Tighten it fully using the chuck key in
the three holes.
Undo the four thumbnuts which stop the chuck from
sliding. Slide the chuck until the last mark (515 mm)
lines up with the right-hand chuck. This procedure sets
the rod length at 500 mm. Fully tighten the right-hand
chuck using the chuck key in each of the three holes.
Wind the thumbwheel until the force meter reads
0.3 N to 0.5 N. Zero the force meter and the angle scale
using the moveable pointer arm. Wind the thumbwheel
so the force meter reads 5 N and then back to zero. If
the angle reading is not zero check the tightness of the
chucks and start again.
Take readings of the angle every 1 N of force: you
should take the reading just as the reading changes.
Take readings to a maximum of 5 N of force. Enter all
the readings into Table 2. To convert the load cell
readings to torque multiply by the torque arm length
(0.05 m).
Repeat the set up and procedure for the brass rod
and enter your results in Table 3.
Force
Torque, T
Angular deflection
(N)
(Nm)
(°)
0
0
0
1
2
3
4
5
Table 2 Results for steel rod
48
From your results, on the same graph plot torque versus
angle for both rods
Comment on the shape of the graph. What does it
tell us about how angle of deflection varies because of
an increased torque? Name at least three applications or
situations where torsional deflection would undesirable
and one application where it could be desirable or of
use.
Take a look at the formulas on the backboard that
predicts the behaviour of the rods. What would happen
to the relative stiffness of the rod if the diameter were
increased from 3 mm to 4 mm?
MM6: Torsion of Circular Sections
Further Work
Measure the diameter of both the rods with the vernier
as accurately as you can (remember the affect of a small
error in the diameter!). Calculate J values for each rod
using the formulae on the backboard of the equipment.
Fill in Tables 4 and 5 from your experimental results
to establish values of TL and Jθ. Remember you must
convert your angle measurements from degrees to
radians (2π radians = 360°).
Diameter of brass section, d
_________ mm
Polar moment of inertia, J
_________ × 10
Length L
0.5 m
Torque
(Nm)
Angular deflection,
θ (rad)
TL
−12
m
Jθ × 10−
4
13
0
0.05
0.10
Diameter of steel section, d
_________ mm
Polar moment of inertia, J
_________ × 10− m
Length L
0.5 m
Torque
(Nm)
Angular deflection,
θ (rad)
0.15
12
TL
4
0.20
0.25
Jθ × 10−
13
Table 5 Calculated values for a brass rod
0
0.05
0.10
0.15
0.20
Plot a graph of TL against Jθ. Examine the torsion
formula and say what the value of the gradient
represents. Does the value compare favourably with
typical ones?
0.25
Table 4 Calculated values for a steel rod
49
MM6: Torsion of Circular Sections
Experiment 2: The Effect of Rod Length on Torsional Deflection
This experiment examines the relationship between
torsional deflection and rod length at a constant torque.
If you have completed Experiment 1 you will have
already completed some of the following steps. In
which case you can leave the brass rod in place at
500 mm long.
With a pencil and a rule, mark the steel and brass
rods these distances from the left-hand end (note that
the rubber tip is on the right-hand end):
the angle reading is not zero check the tightness of the
chucks and start again.
Wind the thumbwheel so the torque is 0.15 Nm (a
reading of 3 N) and note down the angle in Table 6.
Reduce the length of the rod to the next mark (450 mm)
and reset. Take a reading of angle at the same torque
and record. Repeat this procedure for lengths down to
300 mm.
Dia. of brass rod
• 15 mm,
• 315 mm,
• 365 mm,
• 415 mm,
• 465 mm,
• 515 mm.
Length (m)
_____ mm
Torque, T
0.15 Nm
Angular deflection (°)
0.30
0.35
0.40
Wind the thumbwheel down to its stop. Position the
steel rod from the right-hand side with the rubber tipped
end sticking out. Line up the first mark with the lefthand chuck (note the jaws of the chuck move outward
as they close!). Tighten it fully using the chuck key in
each of the three holes.
Undo the four thumbnuts which stop the chuck from
sliding. Slide the chuck until the last mark (515 mm)
lines up with the right-hand chuck. This procedure sets
the rod length at 500 mm. Fully tighten the right-hand
chuck using the chuck key in each of the three holes.
Wind the thumbwheel until the force meter reads
0.3 N to 0.5 N. Zero the force meter and the angle scale
using the moveable pointer arm. Wind the thumbwheel
so the force meter reads 5 N and then back to zero. If
50
0.45
0.50
Table 6 Results for a brass rod
Plot a graph of angular deflection against rod length.
Comment on the shape of the plot.
On most front-wheel drive vehicles have unequal
length drive shafts (from side-to-side). This is because
of the gearbox position being at one end of the engine.
This mismatch in length causes an undesirable effect on
the steering as the car accelerates (that is, as torque
from the engine increases). Why is that? What could
eliminate the effect?
MM6: Torsion of Circular Sections
Experiment 3: Comparison of Solid Rod and Tube
This experiment compares the torsional deflection of a
solid rod and a tube with a similar diameters.
With a pencil and a rule mark the brass tube and
brass rods at 15 mm and 515 mm from the left-hand end
(the end without the rubber tip).
Wind the angle thumbwheel down to its stop.
Position the brass tube in from the right-hand side with
the rubber tip end sticking out. Line up the first mark
with the left-hand chuck (note the jaws of the chuck
move outward as they close!). Tighten it fully using the
chuck key in each of the three holes.
Undo the four thumbnuts that stop the chuck from
sliding. Slide the chuck until the last mark (515 mm)
lines up with the right-hand chuck. This sets the rod
length at 500 mm. Fully tighten the right-hand chuck
using the chuck key in each of the three holes.
Wind the thumbwheel until the force meter reads
0.3 N to 0.5 N. Zero the force meter and the angle scale
with the moveable pointer arm. Wind the thumbwheel
so the force meter reads 5 N and then back to zero. If
the angle reading is not zero check the tightness of the
chucks and start again.
Take readings of the angle every 1 N of force: you
should take the reading just as the reading changes.
Take readings to a maximum of 5 N of force. Enter all
the readings into Table 7. To convert the load cell
readings to torque multiply by the torque arm length
(0.05 m).
If you have completed Experiment 1, enter your
results for the solid brass rod in Table 7. If not, repeat
the set up and procedure for the solid brass rod.
Force
(N)
Torque
(Nm)
Rod angular
deflection (°)
Tube angular
deflection (°)
0
1
2
3
4
5
Table 7 Results for brass rod and tube
Calculate the J values for the solid rod and tube. To
calculate J for a tube, find J for a solid of the same
diameter then subtract J for the missing material in the
centre. Examine your results and the J values you have
calculated and comment on the effect of the missing
material.
Assuming a density of 8450 kgm−3 for brass, work
out the nominal mass per unit length of both the tube
and the solid rod. Comment on the efficiency of
designing torsional members out of tube instead of solid
material.
51
MM6: Torsion of Circular Sections
NOTES:
52
M12: Buckling of Struts
- Introduction and Description
- Experiments
1- Buckling Load of as Pinned-End Strut
2- The Effect of End Conditions on the Buckling
Load
53
SECTION 1 INTRODUCTION AND DESCRIPTION
Figure 1 Buckling of struts experiment
54
MM12: Buckling of Struts
Introduction
This guide describes how to set up and perform
experiments related to the Buckling of Struts. The
equipment clearly demonstrates the principles involved
and gives practical support to your studies.
Description
Figure 1 shows the Buckling of Struts experiment. It
consists of a back plate with a load cell at one end and a
device to load the struts at the top. There are five
aluminium alloy struts included in a holder on the back
plate Printed on the equipment are a number of
equations and pieces of information that you will find
useful while using the equipment
How to Set Up the Equipment
The Buckling of Struts experiment fits into a test frame.
Figure 2 shows the Buckling of Struts experiment in the
Structures Test Frame. Before setting up and using the
equipment, always:
• Visually inspect all parts (including electrical leads)
for damage or wear. Replace as necessary.
• Check electrical connections are correct and secure.
Only a competent person must carry out electrical
maintenance.
• Check all components are secured correctly and
fastenings are sufficiently tight.
• Position the Test Frame safely. Make sure it is on a
solid, level surface, is steady, and easily accessible.
Never apply excessive loads to any part of the
equipment.
The following instructions may have already been
completed for you. If so, go straight to Section 2.
1. Place an assembled Test Frame (refer to the separate
instructions supplied with the Test Frame if
necessary) on a workbench. Make sure the ‘window’
of the Test Frame is easily accessible.
Figure 2 Buckling of struts experiment in the structures frame
55
MM12: Buckling of Struts
2.
3.
4.
On the Test Frame there are securing nuts in the
bottom groove of the top member and the top grove
of the bottom member. In each member slide two of
these to approximately the positions shown in
Figure 2.
Lift up the STR12 unit onto the frame and have an
assistant secure the unit to the frame using the
thumbscrews and washers provided.
Make sure the Digital Force Display is ‘on’.
Connect the mini DIN lead from ‘Force Input 1’ on
the Digital Force Display to the socket marked
‘Force Output’ on the right-hand side of the unit.
56
5.
Carefully zero the force meter using the dial on the
front panel of the experiment. Gently apply a small
load with a finger to the top of the load cell
mechanism and release. Zero the meter again if
necessary. Repeat to ensure the meter returns to
zero.
Note: If the meter is only ±1 N, lightly tap the frame
(there may be a little ‘stiction’ and this should overcome
it).
SECTION 2 EXPERIMENTS
Experiment 1: Buckling Load of a Pinned-End Strut
Compressive members can be seen in many structures.
They can form part of a framework for instance in a
roof truss, or they can stand-alone; a water tower
support is an example of this.
Unlike a tension member which will generally only
fail if the ultimate tensile stress is exceeded, a
compressive member can fail in two ways. The first is
via rupture due to the direct stress, and the second is by
an elastic mode of failure called Buckling. Generally,
short wide compressive members that tend to fail by the
material crushing are called columns. Long thin
compressive members that tend to fail by buckling are
called struts.
When buckling occurs the strut will no longer carry
any more load it will simply continue to displace i.e. its
stiffness then becomes zero and it is useless as a
structural member.
work. The struts provided have an l/k ratio of between
520 and 870 to show clearly the buckling load and the
deflected shape of the struts. In practice struts with an
l/k ratio of more than 200 are of little use in real
structures.
We will use the Euler buckling formula for a pinned
strut:
Pe = π2EI/L2
where:
Pe
E
I
L
=
=
=
=
Euler buckling load (N);
Young’s modulus (Nm−1);
Second moment of area (m4);
Length of strut (m).
Referring to Figure 3, fit the bottom chuck to the
machine and remove the top chuck (to give 2 pinned
ends). Select the shortest strut, number 1, and measure
the cross section using the vernier provided and
calculate the second moment of area, I, for the strut.
Adjust the position of the sliding crosshead to accept
the strut using the thumbnuts to lock off the slider.
Ensure that there is the maximum amount of travel
available on the handwheel thread to compress the strut.
Finally tighten the locking screws.
Carefully back off the handwheel so that the strut is
resting in the notch but not transmitting any load; rezero
the forcemeter using the front panel control.
Carefully start to load the strut. If the strut begins to
buckle to the left, “flick” the strut to the right and vice
versa (this reduces any errors associated with the
straightness of the strut). Turn the handwheel until there
is no further increase in load (the load may peak and
then drop as it settles into the notches).
Record the final load in Table 1 under ‘buckling
load’. Repeat with strut numbers 2, 3, 4 and 5 adjusting
the crosshead as required to fit the strut. Take more care
with the shorter struts, as the difference between the
buckling load and the load needed to obtain plastic
deformation is quite small. Try loading each strut
several times until a consistent result for each strut is
achieved.
Strut
Figure 3 Experimental layout (pinned ends)
In this experiment we will load struts until they buckle
investigating the effect of the length of the strut. To
predict the buckling load we will use the Euler buckling
formulae. Critical to the use of the Euler formulae is the
slenderness ratio, which is the ratio of the length of the
strut to its radius of gyration (l/k). The Euler formulae
become inaccurate for struts with a l/k ratio of less than
125 and this should be taken into account in any design
57
number
Length
(mm)
1
320
2
370
3
420
4
470
5
520
Buckling load
(N)
Table 1 Results for Experiment 1
Examine the Euler buckling equation and select an
appropriate parameter to establish a linear relationship
MM12: Buckling of Struts
between the buckling load and the length of the strut
(Hint: remember π, E and I are all constants).
Calculate the values and enter them into Table 1
with an appropriate title. Plot a graph to prove the
relationship is linear. Compare your experimental value
to those calculated from the Euler formula by entering a
theoretical line onto the graph. Does the Euler formula
predict the buckling load?
It would be useful at this stage to calculate the
gradient of the experimental results for use in
Experiment 2.
58
MM12: Buckling of Struts
Experiment 2: The Effect of End Conditions on the Buckling Load
Follow the same basic procedure as Experiment 1, but
this time remove the bottom chuck and clamp the
specimen using the cap head screw and plate to make a
pinned-fixed end condition. Record your results in
Table 2 and calculate the values of 1/L2 for the struts.
Note that the test length of the struts is shorter than in
Experiment 1 due to the allowance made for clamping
the specimen.
Strut
number
Length
(mm)
1
300
2
350
3
400
4
450
5
500
Buckling load
(N)
2
1/L
-2
(m )
Table 2 Results for Experiment 2 (pinned-fixed)
Now fit the top chuck with the two cap head screws and
clamp both ends of the specimen, again this will reduce
the experimental length of the specimen and you will
have to calculate new values for 1/L2. Take care when
loading the shorter struts near to the buckling load.
NOTE
Do not continue to load the struts after the
buckling load has been reached otherwise the
struts will become permanently deformed!
Figure 4 Experimental layout for pinned-fixed
conditions
Enter your results into Table 3.
Strut
number
Length
(mm)
1
280
2
330
3
380
4
430
5
480
Buckling load
(N)
2
1/L
-2
(m )
Table 3 Results for experiment 2 (fixed-fixed)
Plot separate graphs of buckling load versus 1/L2 and
calculate the gradient of each line. Establish ratios
between each end condition (taking the pinned-pinned
condition as 1).
Examine the Euler buckling formulae for each end
condition and confirm that the experimental and
theoretical ratios are similar.
Figure 5 Experimental layout for fixed-fixed
conditions
59
MM12: Buckling of Struts
NOTES:
60