1
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MECHANICAL AND MANUFACTURING
ENGINEERING
MMAN3400 Mechanics of Solids 2
Unsymmetrical Bending and
Shear Centre Experiments
Name: …………………………….. Student No. …………………...
2
SECTION 1: INTRODUCTION
1.1 Introduction
This lab guide describes how to set up and perform experiments related to unsymmetrical
bending and shear centre of beams. These two experiments clearly demonstrate the principles
involved and give practical support for your studies.
1.2 Description
Figure 1 and Figure 2 show the set up for the Unsymmetrical Bending and Shear Centre
experiment. It consists of a top plate and chuck, a bottom plate with two digital indicators, a
‘free end’ chuck and three specimens (‘U’, ‘L’ and rectangular). However, ‘L’ section will be
only used in this exercise. Also there is an attachment for finding the shear centre of each
beam.
Figure 1: Unsymmetrical bending and shear centre experiment
3
The top plate fastens to the top member of the Test Frame and the bottom plate on the bottom
member. The specimen beams secure into the top plate chuck, which indexes around in 22.5o
steps (giving 16 angular increments). The bottom plate has two digital indicators that can be
arranged at 90o to each other for the unsymmetrical bending experiment or parallel to each
other for the shear centre experiment.
The bottom chuck secures to the ‘free’ end of the specimen and contacts the two indicators.
This arrangement allows measurement of the end deflection of the specimen in two
directions. The force is applied to a peg on the bottom chuck. For the shear centre experiment
the indicators swing round and the shear beam attachment fixes to the bottom chuck.
1.3 How to set up the Equipment
The unsymmetrical bending and shear centre experiment fits into a Test Frame. Figure 1 and
Figure 2 show the Unsymmetrical Bending and Shear Centre experiment in the Frame.
Before setting up and using the equipment, always:
• Visually inspect all parts, including electrical leads, for dames or wear.
• Check electrical connections are correct and secure.
• Check all components are secure and fastenings are sufficiently tights.
• Position the Test Frame safely. Make sure it is on a solid, level surface, is steady, and
easily accessible.
Figure 2: Unsymmetrical bending and shear centre experiment in the structures frame
4
Never apply excessive loads to any part of the equipment.
These instructions may have been completed for you; if so go straight to the next section.
1. Place an assembled Test Frame (refer to the separate instructions supplied with the
Test Frame if necessary) on a workbench. Make sure the windows of the Test Frame
are easily accessible.
2. Referring to Figure 2, ensure there are two securing nuts in each of the top and bottom
members in the top slot. If none are present, move them from other positions on the
frame. When fitted them to approximately the position shown in Figure 2. Position the
top and bottom plates onto the frame as shown, and fix using the thumbscrews.
Figure 3: Setup for the experiments
SECTION 2: UNSYMMETRICAL BENDING
2.1 Theory
The beam is a cantilever beam with a load applied at the free end. The beam has two
principal axes, and , about which pure bending can take place. The principal axes and
pass through the centroid of the section but do not necessarily coincide with the arbitrary
geometrical axes of the section, axes and . Figure 4 shows the orientation of these axes.
Figure 4: Illustration of principal axes x and y and arbitrary axes u and v
5
If applying a moment about one of the principal axes then the beam will deflect in that
direction only and the simple bending formula can predict the deflection. However, if the
moment is at an angle to either of the axes then the beam will bend about both of the exes.
The free end deflection will have two components, one in the direction of pull () and one at
right angles to the pull (). If we were interested in predicting the magnitude of the
deflections, we would need to resolve the moment into components acting about the principal
axes. This leads to the following equations:
=
3
6
[(
1

+
1

) + cos 2 (
1

−
1

)]
=
3
6
sin 2 [
1

−
1

]
Where:
= Force,
= effective length of the specimen (m),
= Young’s modulus (GPa);
= Deflection in the direction of pull, direction (m),
= Deflection at right angles to the pull, direction (m),
= Angle of pull (o),
, = Principal second moments of area (m
4)
The Mohr’s Circle is an excellent graphical method to read off the cantilever deflections
(note: this refers to Mohr’s Circle for deflections, not the more commonly known Mohr’s
Circle used for stress and strain transformation). To construct a Mohr’s Circle of deflections
you would need to know at least the principal second moments of area ( and ). In this
experiment we will use the Mohr’s circle in reverse to establish values of the principal second
moments of area from deflections measured off the equipment in each direction ( and ) as
shown in Figure 5. We can then compare the principal second moments to the theoretical
values.
Figure 5: Mohr’s circle of deflections to find the principle second moments of area
6
The orientation and location of L section in the bottom chunk and the dimensions of the cross
section are shown in Figure 6 and Figure 7 respectively.
Figure 6: L section Specimen arrangement
Figure 7: Dimensions of L section cross section
2.2 Procedure
Preliminary steps are as follow:
(These steps may have been completed for you)
1- Set up the equipment as shown in Figure 1.
2- Loosen the two rearward facing thumbscrews on the indicator bosses, turn the
indicators inward to contact the inner two datum pegs and lock off the thumbscrews.
This set the 90 degree angle between the two indicators
3- Fit the L section specimen into the bottom chuck referring to Figure 6 for correct
position. Also fit the top of the specimen into the top chuck in the same relative
position, ensure that the specimen is set squarely and all of the screws are tight.
4- Fit the extension piece to the bottom chuck, hook the cord onto the groove and pass it
over the sliding pulley.
5- Undo the top chuck hand wheel and rotate the specimen so it is oriented as per Figure
4 when the angle of pull is zero. when you feel the chuck click into the correct
position tighten the hand wheel
7
6- Ensure that the indicators have about 10-11 mm forward and 2-3 mm backward travel
in this position. If not loosen the indicator top screw and slide the indicator to the
correct position, retighten the screw.
Main lab procedure is as follow:
1- Tap the frame sharply to reduce the effect of friction and zero the indicators each time
before measuring the deflections by holding the button on each indicator for about
two seconds.
2- Apply loads in 120 g increments, up to a maximum of 480 g on the end of the cord.
Ensure the cord remains parallel to the lines on the plate below.
3- Tap the frame sharply after adding each load. Record the resulting deflections (left
and right) in table below for each load.
4- Undo the top chunk hand wheel, rotate the specimen clockwise 22.5 degree (i.e. to the
next location) and tighten.
5- Zero the indicators, if required and then repeat the loading procedure, recording the
results in the table below.
: Head
Angle (o)
Reading measurements in mm
120 g 240 g 360 g 480 g
Left Right Left Right Left Right Left Right
0
22.5
45
67.5
90
112.5
135
157.5
180
6- Measure the length .
= … … … … … .
2.3 Results and questions
The students are required to answer the following questions:
1- U and V deflections can be obtained from the measurements as shown below. ‘Left’ is
the reading of the left indicator and ‘Right’ is the reading of the right indicator.
Resolve the left and right indicator readings into the and directions using the
following formulae: (Note: be careful with your signs as these values can be negative)
8
=
( + ℎ)
√2
=
( − ℎ)
√2
: Head
Angle (o)
Calculated values of deflections in mm
120 g 240 g 360 g 480 g

0
22.5
45
67.5
90
112.5
135
157.5
180
2- Plot graphs of and versus the pulling mass, , in grams for each head angle.
(There should be nine sets of graphs). Establish gradient of / and / on
each graph noting the results in mmg-1 in the table below and then convert these
values into the fundamental units of mN-1.
: Head
Angle (o)
/
(mmg-1)
/
(mmg-1)
/
(mN-1)
/
(mN-1)
1 0
2 22.5
3 45
4 67.5
5 90
6 112.5
7 135
8 157.5
9 180
3- Use this data to construct a Mohr’s circle by plotting the values of / versus /
for each head angle. The points should form a circle, if distorted draw a circle
that encompasses most of the points or draw two circles and average them.
4- Derive the following two equations using the equations on page 5 and the Mohr’s
circles in figure 5 and 8. After that calculate the experimental principal second
moments of area using the Mohr’s Circle and the following formulae:
9
=
3
=
3( – )
3
3( + )
= Measured length of the specimen (m),
= Young’s modulus (69 GPa for aluminium);
: Distance from origin to centre of Mohr’s Circle (mN-1),
: Radius of Mohr’s Circle (mN-1)
5- Calculate the theoretical values for the principal second moments of area using the
cross section in Figure 7?
6- Compare the theoretical values to the experimental values and comment on the
accuracy of your results and give possible reasons for any discrepancy between the
theoretical and experimental values either in terms of the analysis or in the equipment.
Is the graphical Mohr’s Circle method truly accurate? If not how could it be made
more so? You might refer to the graphs and formulas in this lab sheet.
Figure 8: Mohr’s Circle for second moments of area
SECTION 3: SHEAR CENTRE
3.1 Theory
Beam will always bend when loaded. Also the bending will be accompanied by a twisting
action unless the load is applied at a position known as the Shear Centre. Figure 9 shows a
‘U’ section loaded with a force F on its side. The load sets ups shearing stresses in the section
caused by the shear force. For equilibrium the vertical force must balance the applied load
and the two horizontal shearing forces must be equal and opposite. The two horizontal forces
form two moments, which combine to twist the section.
10
Figure 9: The twisting of a 'U' section under load
However, if the beam is loaded at its shear centre, S as shown in Figure 9 then the two
moments cancel out. The beam then bends but does not twist.
Figure 10: A 'U' section under load through its Shear Centre
In this experiment we will purposely load the ‘L’ section eccentrically, at is at positions each
side of the shear centre and measure the twisting action with the indicators each side of the
section. We can then ascertain the position of the shear centre, since it is the point of zero
twist (i.e. when the two indicator readings are identical)
3.2 Procedure
Preliminary steps are as follow:
(These steps may have been completed for you)
1- Loosen the two rearward facing thumbscrews on the indicator bosses, turn the
indicators outward to contact the outer two datum pegs and lock off the thumbscrews.
This sets the two indicators parallel.
2- Fit the ‘L’ Section into the bottom chuck for correct positions. Fit the top of the
specimen into the top chuck in the same relative position, ensuring that the specimen
is set squarely and all of the screws are tight.
11
3- Undo the top chuck hand wheel and rotate the specimen so it is orientated as per
Figure 10. When you feel and hear the chuck click into the correct position, tighten
the hand wheel.
4- Fit the shear centre beam to the bottom chuck as shown in Figure 10 and secure with
the extension piece.
Figure 11: Loading the beam eccentrically and measuring the twisting action
5- Ensure that the indicators have roughly equal travel forward and background on the
shear arm pegs. If not, loosen the indicator top screw, slide the indicator to the correct
position and tighten the screw.
Main lab procedure is as follow:
1- Tap the frame sharply to reduce the effect of friction and zero the indicators
2- Apply a load of 500 g to the left-hand notch (-25 mm). with the cord over the pulley,
ensure that the pulley and cord remain parallel to the lines on the plate below.
12
3- Record the resulting indicator reading in the table below.
4- Repeat with the same load at the other notch positions ensuring the cord remains
parallel at all times.
Eccentricity
of load (mm)
Left-hand
indicator
reading (mm)
Right-hand
indicator
reading (mm)
-25
-20
-15
-10
-5
0
5
10
15
20
25
3.3 Results and questions
The students are required to answer the following questions:
1- Plot a graph of the indicator readings in mm (y-axis) versus the eccentricity of the load
in mm (x-axis). Where the two lines intersect is the position of the shear centre
(it is helpful to visualise the position of the shear centre by sketching the section to the
x-axis scale on your graph).
2- Find the theoretical shear centre of the cross section, by referring to your textbook.
3- Compare your experiment result (Question 1) with your theoretical result (Question 2)
and comment on the accuracy of your results.

学霸联盟