MOHR’S CIRCLE AND STRAIN GAUGE ROSETTE
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Summary
Contents List
1 Introduction
The Mohr’s circle is a graphical method to determine the effect of a coordinate
rotation on a tensor quantity. Within engineering it finds application in the effect of
coordinate rotation on stress, strain, second moment of area and moment of inertia.
In this experiment,
2 Objectives
3 Apparatus
An aluminium alloy beam clamped at one end within a rig containing a cam whose
full-range rotation leads to a repeatable tip deflection of   1 2  12.7 mm , as
depicted in Figure 1. The beam has dimensions breadth b  25.4 mm , depth
d  6.35 mm and length (to the cam) L  254 mm . Three strain gauges are mounted
on the upper surface 94 mm from the clamped end; these gauges are mounted at 15 ,
45 and 75 to the beam’s longitudinal x-axis, as shown in Figure 2. Each of these
gauges may be selected, in turn through a switch, to be one arm of a Wheatstone
bridge arrangement [1 research and give reference]. A dummy strain gauge provides
temperature compensation.
94 mm
x
L=254 mm
y, v
1 beam
Figure 1. Cantilever
4 Procedure
1.
The strain gauge amplifier should be warmed up, and set to the correct gauge
factor (G.F.) of 2.1.
2.
Select gauge 1 and balance (zero) the bridge with the beam unloaded. Rotate
the lever to bend the beam, and take the strain gauge reading.
3.
Repeat step 2. for gauges 2 and 3; note that it will be necessary to re-balance
the bridge for each gauge whilst unloaded – this is necessary because the cables from
each gauge to the switch will have slightly different length and hence slightly
different electrical resistance.
The measurements should be recorded as
Beam condition
Gauge 1
Gauge 2
Gauge 3
Unloaded
0
0
0
Loaded
where the strain gauge readings are in microstrain.
94 mm
1
2
x
3
z
Figure 2. Strain gauges 1, 2 and 3 mounted at 15°, 45 °, and
75 °, respectively, to the x-axis
2
5 Analysis of results
On graph paper, construct the Mohr’s circle for strain and determine the principal
strains 1 and  2 . For this educational example, the maximum strain 1 should be in
the direction of the longitudinal x-axis, that is 1   x . Inevitably there will be
experimental errors, so discuss possible reasons why the maximum strain obtained
may not be predicted to occur in the x-direction. Determine a value of the Poisson’s
ratio,  , for the material. Compare your experimental values with those from the
theory presented below.
6 Theoretical predictions
For the rectangular beam cross-section, the second moment of area is
I
bd 3 25.4  6.353

 542 mm4  542 1012 m 4 .
12
12
The tip deflection of a cantilever beam subject to a force W at the free-end is  
WL3
3EI
[give reference]. Assuming a Young’s modulus for aluminium of E  70 109 Nm2 ,
then the required force to produce a deflection of 12.7 mm is
W
3EI  3  70 109  542 1012 12.7 103

 88.21 N .
L3
0.2543
W
M
94 mm
x
Q
L=160 mm
y, v
Figure 3. Bending moment at strain gauge location.
3
At the strain gauge location, Figure 3, the bending moment M is
M  W  0.16  14.11 Nm . Now employ the expression  x 
reference], and on the upper beam surface y  
x 
14.11  3.175 103 
542 10
12
My
[give
I
6.35
mm =  3.175 103 m , to give
2
 82.66 106 Nm 2 .
There is zero stress in the y- and z-directions, so the Hooke’s law reduces to
x 
Thus  x 
x
E
,  y   z   x .
82.66 106
 1181106  1181  ,  y   z  0.3 1181  354  .
9
70 10
7 Discussion
8 Conclusions
References
1. PP Benham, RJ Crawford and CG Armstrong, Mechanics of Engineering
Materials, 1996, second edition, Harlow, Addison Wesley Longman.
2.
Bibliography
Nomenclature
In alphabetical order, Greek symbols at end
b
breadth
d
depth
E
Young’s modulus
I
second moment of area
L
length
M
bending moment
W
load
4
x, y, z Cartesian coordinates

direct strain

direct stress

deflection
Appendices
If required.
Notes
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