S3-1
S3: Strain Gage Rosettes
(Revised August 4, 2009)
LABORATORY SAFETY ALERT
1. All general laboratory Safety Practices must be followed.
OBJECTIVE
To experimentally determine the normal stresses in a cylindrical pressure vessel and
compare the values to those from thin-walled pressure vessel theory.
INTRODUCTION
In this experiment, strain gage rosettes are used to determine the principal values and
principal directions of stress and strain in a thin-walled cylinder. Each student should bring a
compass and a ruler to the laboratory.
THEORY
A single strain gage can be applied to a specimen to measure the normal strain in a
particular direction. By itself, it is sufficient for situations, such as the standard tension test,
where only the strain in a single direction is required. In many cases, however, stress/strain
analyses involve situations in which there are complicated components and/or loading
conditions, and little is known about the nature of the strain field to be investigated.
In such cases, a configuration having three or more gages is needed. Such an
arrangement is known as a strain gage rosette. The gage readings can be used to find the
magnitude and direction of the maximum and minimum normal strains.
The common types of rosette gages are the three-element rectangular rosette (which
will be used in this lab) and the delta rosette. There are also four element variations of the
rosette. The fourth strain gage provides an extra reading which can be used to verify the
accuracy of the rosette analysis.
Principal Strain Equations
The three-element rectangular rosette employs gages placed at the 0, 45 and 90 degree
positions as shown in Figure 1. Strain gages A, B, and C directly measure normal strains ε A ,
εB , and εC ; that is, they measure the normal strains along the A, B, and C axes, respectively.
In two-dimensional problems, the in-plane principal strains ε max and ε min may be
determined from these measured values by
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(1)
using a transformation of coordinates to calculate rectangular components ε x , ε y , and
γ xy , and then
(2)
performing a second transformation to obtain ε max and ε min .
As shown in Figure 1, the x-y coordinate system is selected so that the x-axis coincides with
the axis of gage A, and the y-axis coincides with the axis of gage C.
y
C
B
θ
C
θ
B
A
x
θ A = 0°
θ B = 45°
θC = 90°
Figure 1: Gage positions in a three-element rectangular rosette
Consequently,
εx = εA
(1)
and
ε y = εC .
(2)
To obtain an expression for γ xy , we may use a coordinate transformation between the x-y
system and a system consisting of the B-axis and an axis perpendicular to B. Under this
tensor transformation, we have that
ε BB = ε xx cos(B, x ) cos(B, x )
+ ε xy cos(B, x ) cos(B, y )
+ ε yx cos(B, y ) cos(B, x )
(3)
+ ε yy cos(B, y ) cos(B, y )
where ε BB = ε B , ε xx = ε x , ε yy = ε y , ε xy = γ xy 2 , and
cos(B , x ) = the cosine of the angle between the B-axis and the x-axis, and
cos(B , y ) = the cosine of the angle between the B-axis and the y-axis.
For the given rosette, cos(B , x ) = cos(B , y ) = cos 45° , so substituting equations (1) and (2)
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into equation (3) yields
or
1
1⎛γ ⎞ 1⎛γ ⎞ 1
ε B = ε A + ⎜ xy ⎟ + ⎜ xy ⎟ + ε C
2
2⎝ 2 ⎠ 2⎝ 2 ⎠ 2
(4)
γ xy = 2ε B − ε A − ε C .
(5)
By using equations (1), (2), and (5), we can obtain the x-y components of the strain tensor
from measured values ε A , ε B , and ε C . To determine the principal strains, we may use
Mohr's circle, as follows.
For a two-dimensional state at a point within a loaded body, each direction corresponds
to a normal strain ε and a shearing strain γ /2. A graphical/mathematical representation of
the state of strain at the point may be obtained by sketching the curve defined by all ordered
pairs (ε , γ 2 ) . The resulting curve is a circle centred on the ε -axis with each graph point on
the circle corresponding to a different physical direction within the body. We may sketch the
circle by plotting (ε x , − γ xy 2) and (ε y , γ xy 2) as shown in Figure 2.
Figure 2: Mohr's circle for strain
Then, any strain component, including the principal strains εmax and εmin, may be determined
by simple trigonometry. Referring to Figure 2, the centre of the circle is given by
ε0 =
εx + εy
(6)
2
while, for example, the radius is given by
R0 =
(ε y − ε 0 )2 + (γ xy 2)2
.
(7)
Once the centre and the radius of the circle have been determined, the principal strains are
simply given by
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and
ε max = ε 0 + R0
(8)
ε min = ε 0 − R0 .
(9)
To determine the principal directions, we note that a physical rotation of θ (within the
body) corresponds to an angle of 2θ along the circle, and we recall that each graph point on
the circle corresponds to a direction or axis within the body. For the case given in Figure 2,
the principal axis corresponding to εmax is physically at an angle of θ clockwise from the yaxis, where
tan 2θ =
γ xy 2
ε y − εo
.
(10)
Similarly, the principal axis corresponding to εmin is at an angle of (180° − 2θ ) 2
counterclockwise from the y-axis, or equivalently, at an angle of θ clockwise from the xaxis.
Principal Stresses
Once the principal strains have been determined by the preceding method, they can be
used to determine the principal stresses σmax and σmin. If the physical point under
consideration is in a state of plane stress, generalised Hooke’s law with respect to the
principal directions gives
σ max =
E
(ε max + νε min )
2
1 −ν
(11)
and
σ min
E
(ε min + νε max )
=
2
1 −ν
where E is Young’s modulus and ν is Poisson’s ratio for the material.
Thin-Walled Cylinder Theory
Figure 3(b) shows a free body diagram of a finite length of a thin-walled cylinder.
From an equilibrium of forces in the circumferential (or hoop) direction,
Q = 2 Ltσ c
(12)
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where
Q = 2 RLp .
(13)
Figure 3: Free body diagrams of a pressurised thin-walled cylinder
and
p is the internal pressure,
Q is the force due to pressure p ,
σ c is the circumferential stress in the cylinder,
L is the length of the cylinder element,
t is the wall thickness, and
R is the cylinder inner radius.
Hence,
σc =
pR
.
t
(14)
Figure 3(c) shows the free body diagram that is used to calculate the longitudinal stress
in a closed thin-walled cylinder. Axial equilibrium gives
where
Q = 2πRtσ long
(15)
2
Q = pπR .
(16)
Hence, the longitudinal stress, σ long , in the walls of a closed cylinder is given by
σ long =
pR
.
2t
(17)
In an open cylinder, σ long is theoretically zero. Also, for both open and closed cylinders, the
shearing stress associated with the circumferential and longitudinal directions is theoretically
zero, and therefore σc and σlong are the principal stresses (σmax and σmin, respectively).
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APPARATUS
Thin-Walled Cylinder:
Outer Diameter = 3.060 + 0.002"
Inner Diameter = 2.960 + 0.006"
Modulus of Elasticity = 30 ×106 psi
Poisson's Ratio = 0.3
Strain Indicator:
VISHAY, P-3500
Portable Digital Strain Indicator
Strain and Balance Unit:
VISHAY, SB-10
Portable Switch & Balance Unit
Strain Rosettes:
I) Shinkoh, 3 arm, 45 degree Rosette
Gage Factor F = 2.14
Resistance 120 Ω
II) Kyowa, 3 arm, 45 degree Rosette
Gage Factor F = 2.115
Resistance 120 Ω
The cylinder used in this experiment was designed to allow two possible configurations
to be set up as in Figure 4. When the piston is moved off its stop, it is said to be in an open
position. In this case, pressure exerted axially on the cylinder is transmitted to the frame,
thus eliminating longitudinal stresses due to end pressure. The closed position exists when
the piston is allowed to rest on its stop. The cylinder now becomes a closed tank with
longitudinal and circumferential (hoop) stresses due to the internal pressure.
The VISHAY P-3500 Portable Digital Strain Indicator is designed to amplify, measure,
and display the millivolt per volt output of strain gages in microinches per inch. The
instrument generates a regulated 4.8 Volt bridge excitation voltage and will accommodate a
1/4, 1/2, or full bridge gage configuration. Gages with resistances of 120 or 350 ohms and
with gage factors ranging from 0.5 to 9.500 can be used with the instrument.
The VISHAY SB-10 Switch & Balance Unit allows the outputs from a group of up to
ten quarter, half, or full-bridge strain gage configurations to be successively monitored on a
single VISHAY P-3500 Digital Strain Indicator.
The two strain rosettes to be used in this experiment are located approximately six and
ten inches away, respectively, from the bleeder valve (see Figure 7). The first strain rosette
is aligned in a hoop orientation but the second strain rosette has been rotated 29 degrees
clockwise from this orientation.
S3-7
Figure 4: Thin-walled cylinder configurations
PROCEDURE
Each group is to obtain the principal stresses developed in the cylinder under open and
closed end conditions using strain gage rosettes, and is to compare these experimental values
to the values predicted using classical thin cylinder theory.
1)
Identify the two strain Rosettes and corresponding wires.
2)
Connect the two strain Rosettes to the SB-10 Switch & Balance Unit (see Figure 5).
a) Connect the red wire of each active gage to the P+ terminal of the SB-10 Switch &
Balance Unit.
b) Connect the white wire of each active gage to the S- and D terminal of the SB-10
Switch & Balance Unit.
c) Note which channel corresponds to which strain gage.
3)
Connect the SB-10 Switch & Balance Unit to the P-3500 Strain Indicator as shown in
Figure 6 by using the supplied color coded connecting cable.
4)
Push the Gage Factor button down on the P-3500 and set the Gage Factor to the correct
value for each strain gage by unlocking the rotary switch and rotating it to the desired
position.
5)
Make sure the Bridge button is set to ¼ - ½ bridge. When the bridge button is set, select
1X on the MULT button.
S3-8
SB-10
red wire
white wire
1
2
3
4
5
6
7
9
9
10
P+ P- S- S+ D
Figure 5: Wiring from rosettes to SB-10 Switch & Balance Unit
6)
Set CHANNEL SELECTOR switch on the SB-10 to first channel in use.
7)
Adjust each channel to zero using the rotary switch for the channel selected.
8)
Bleed air out of the cylinder.
a) Tighten PRESSURISING Knob.
b) Loosen BLEEDER valve.
c) Slowly pump oil into cylinder until air bubbles no longer appear in the bleeder
line.
d) Close BLEEDER valve.
9)
Set cylinder to either Open or Closed position by turning END SCREW at the end of
the cylinder clockwise for the Open position and counterclockwise for the Closed
position. Cylinder pressure should be at 0 psi before attempting to change the screw
position.
10)
Tighten PRESSURISING Knob at base of pump.
11) Pressurise cylinder to approximately 400 psi by operating pump. Close the pressurising
valve. Record strain gage values by turning CHANNEL switch on the SB-10 Unit to
appropriate channel and reading value from the P-3500 Unit DISPLAY. Hold pressure
at approximately 400 psi until all strain gage values have been recorded.
12) Lower pressure to approximately 100 psi by loosening PRESSURISING Screw.
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13) Perform test three times in the closed position and three times in the open position.
P-3500
SB-10
P+
P+
P-
P-
D
EXT
S-
S-
S+
S+
GND
D120
D350
GND
Figure 6: Indicator/Switching Unit Wiring Interconnection
200
Pressure Gauge
Bleader Valve
300
400
100
0
500
PSI
Oil Line
Thin Walled Cylinder
End Screw
Oil Line
Oil Line
Oil Line
Oil Reservoir
Pump
Pressurizing Knob
Pressurizing Valve
Figure 7: Schematic diagram of thin-walled cylinder assembly
S3-10
EXPERIMENTAL DATA
Measured Strain (μ in in )
Test
Run
Gage
Open
1
2
Closed
3
avg
1
2
A
B
C
D
E
F
Pres.
Rosette I
Rosette II
Gage Position Record
3
avg
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EXPERIMENTAL ANALYSIS
In the analysis, consider 4 cases:
(i)
(ii)
(iii)
(iv)
Rosette I, open cylinder position,
Rosette II, open cylinder position,
Rosette I, closed cylinder position, and
Rosette II, closed cylinder position.
For all four cases,
• Use the experimental strain data and Mohr's circle to find principal strains ε max and ε min .
• Use Hooke's law to find principal stresses σ max and σ min .
• Compare σ max and σ min to the values predicted by thin-walled pressure vessel theory
(i.e., to the theoretical values of σc and σlong).
For each of Rosettes I and II in the open position, use the experimentally obtained values of
ε max and ε min and generalised Hooke’s law to calculate Poisson's ratio assuming σlong = 0.
Compare your calculated values to the expected value of ν = 0.3 .
For each of the open and closed positions, use your strain data and Mohr’s circle to
determine the angle between the middle gage of Rosette II and the hoop direction of the
cylinder. Show this angle on a sketch of the rosette.
For all the above calculations, use the average strain values (from the 3 runs) and neglect any
transverse effects in the strain gages.
LOG BOOK
Record all experimental data in your logbook.
Summarise, with an example, the procedure for reducing strain gage data to principal stress
data for a biaxially stressed member such as a cylinder. Each student should record
sufficient information to be able to use this equipment to analyze the stresses on a typical
machine component at a later date.
Write in your logbook the equations giving σc and σlong according to thin-walled cylinder
theory. Substitute the values of p, R, and t, into these equations to determine the theoretical
values of σmax and σmin.
Record any additional experimental observations and related discussion and conclusions.
S3-12
REPORT REQUIREMENTS
1.
The report should follow the basic format outlined at the beginning of the ME 318 Lab
Manual.
2.
Provide your table of experimental data in the body of the report.
3.
In an appendix, give sample calculations of σmax and σmin from experimentally measured
strains. Also, in the appendix, give sample calculations of Poisson’s ratio and the
angular orientation of Rosette II.
4.
In the body of the report, provide the theoretical solutions (including equations (14) and
(17), with proper referencing).
5.
In the body of the report, provide table(s) summarising the experimental analysis
comparing all theoretical and experimental values of stress, Poisson’s ratio, and
Rosette.II orientation. Discuss all comparisons. Discuss the implications of rosette
orientation with respect to determination of σmax and σmin.
Marks are given largely on the basis of:
- quality of writing;
- results and discussion;
- correctness of statements; and
- completeness (all report requirements must be included and the Logbook Requirements
must be incorporated, in an appropriate way, into your report).
BIBLIOGRAPHY
Beer, F.P., Johnston Jr., E.R., and DeWolf, J.T. (2006) Mechanics of Materials (4th ed.)
New York: McGraw-Hill.
Budynas, R.G. (1999) Advanced Strength and Applied Stress Analysis (2nd ed.) New York:
McGraw-Hill.
Perry, C.C., and Lissner, H.R. (1962) The Strain Gage Primer (2nd ed.) New York:
McGraw-Hill.
Shigley, J.E., and Mischke, C.R. (1989) Mechanical Engineering Design (5th ed.) New
York: McGraw-Hill.
Ugural, A.C., and Fenster, S.K. (1987) Advanced Strength and Applied Elasticity, (2nd SI
ed.) New York: Elsevier.