E 410
MECHANICAL ENGINEERING SYSTEMS LABORATORY
EXPERIMENT 5
STRESS ANALYSIS BY USING STRAIN GAGES
OBJECTIVE
The objective of this experiment is to become familiar with the electric
resistance strain gage techniques and utilize such gages for the
determination of unknown quantities (such as strain and stress) at
prescribed conditions of a cantilever beam and a thin walled pressure vessel.
INTRODUCTION
There are various types of experimental methods to analyze strains and
stresses at a point. Strain gage methods use either electrical or mechanical
means to measure strains. In these types of strain gages, electrical
resistance strain gages are the most accurate and widely used ones.
This experiment consists of three parts, all utilizing electric resistance strain
gages. You will perform three experiments.
In these experiments, gages will be used to determine
1. the flexural rigidity of a cantilever beam,
2. internal pressure in a pressure vessel along with principal stresses at
a given point on the vessel, the Poisson’s ratio of the vessel material
and the gage orientation with respect to principal directions,
3. structural properties (damped and undamped natural frequency and
damping ratio) of a singe degree of freedom vibratory system,
PART I
CANTILEVER BEAM TEST
CONCEPT
In this experiment, a cantilever beam is used as a force transducer to
determine the applied force. Three axial strain gages are used in two gage
locations as shown in Fig. 1. At gage location 1, the gage B on the lower
surface is located precisely under the gage A which is located on the top
surface. Gages A and B measure bending strains that are of equal
magnitudes but of opposite signs. Any resistance change in the active gage
resulting from strains of the like sign (e.g. produced by axial loads) will be
canceled since the active gages are in adjacent arms of the Wheatstone
bridge. The gage C on the upper surface is located 300 mm from the free end
of the beam. This gage also measures bending strains.
Page 1 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
PROCEDURE
1. The strain gauges are wired as Ch.1 quarter bridge and Ch.2 half bridge.
2. Press the BAL button on strain indicator twice to balance the bridges
when unloaded, wait for the device to perform autobalancing then press
RECORD to save the values.
3. The device is ready to read the strain in
.
4. Set the dial gage at the free end of the beam and adjust to zero.
5. Apply the given known load to the free end of the beam.
6. Measure strains at locations 1 (Ch.2) and 2 (Ch.1) and measure the
deflection ( D) at the free end of the beam.
7. Remove the load from the beam.
8. Apply the given unknown load P at point E.
9. Measure the strains at gage locations 1 (Ch.2) and 2 (Ch.1).
GIVEN QUANTITIES
Fig. 1. Schematic representation of experimental set up in cantilever beam
test
Gauge factor of the strain gauges 2.075
REQUIRED QUANTITIES
1. Calculate the flexural rigidity, EI, of the beam.
2. Calculate the height, h, of the beam
3. Determine the distance, L (distance between the free end of the beam and
the gage location 1)
4. Determine the distance, x (distance between the applied load and free
end of the beam)
5. Calculate the applied unknown load P
Page 2 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
PART II
THIN WALLED PRESSURE VESSEL TEST
CONCEPT
Cylindrical pressure vessels, hydraulic cylinders, and pipes carrying fluid at
high pressures develop both radial and tangential stresses with values
which are dependent upon the radius of the element under consideration.
When the wall thickness of the cylindrical pressure vessel is about onetwentieth or smaller than its radius, the radial stress which results from
pressurizing the vessel is quite small compared to the tangential stress.
Under these conditions the tangential stress can be assumed to be
uniformly distributed across the wall thickness. When this assumption is
made, the vessel is called thin walled pressure vessel.
Consider a cylindrical vessel of inner radius r and wall thickness t,
containing a fluid under pressure. Because of the axi-symmetry of the vessel
and its contents, it is clear that no shearing stresses are created on the
element.
Fig. 2. Schematic representation of thin walled pressure vessel
The normal stresses 1 and 2, shown in Fig. 2 are therefore principal
stresses. The stress 1 is known as the hoop (circumferential) stress and the
stress 2 is called the longitudinal stress. Principal stresses then can be
calculated as
1
pr
,
t
where
2
pr
2t
p = internal pressure
r = inner radius of the cylinder
t = wall thickness of the cylinder
In this experiment, a three element rectangular rosette forming an unknown
angle with the axis of the vessel is used to determine the gage pressure in
the cylindrical vessel (Fig. 3).
If you consider gage A to be x direction, and gage C in y direction as shown
in Fig.3, corresponding strains become:
Page 3 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
Fig. 3 Configuration of strain gages on pressure vessel
A
xx
,
C
yy
,
B
1
(
2
xx
yy
Principal strains become:
xy
)
xx
1,2
1
(
2
yy
2
the corresponding angle is: tan 2
xx
yy
)2
2
xy
xy
xx
yy
Thus, the principal stresses can be calculated as:
E
1
1
2
1
2
and
E
2
1
2
2
1
PROCEDURE
Connect the strain gages to the strain indicator (use Quarter Bridge
configuration)
1. Set the gage factor setting to 2.10
2. Balance the indicator
3. Load the pressure vessel
4. Read the strain values from the indicator.
GIVEN QUANTITIES
Gage factor Sg = 2.10
Outer diameter do = 112.5 mm
Inner diameter di = 107.9 mm
Modulus of Elasticity E = 200 GPa
REQUIRED QUANTITIES
1.
2.
3.
4.
5.
Determine the Poisson’s ratio of the cylinder material.
Determine the unknown gage angle, .
Calculate principal strains and their directions.
Calculate principal stresses.
Determine the inner pressure of the vessel.
Page 4 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
PART III
DAMPED FREE VIBRATION TEST
CONCEPT
In this experiment, an inverted L beam is used as to measure bending
moment. Two axial strain gages are used in as shown in Fig. 4. Gages A and
B measure bending strains that are of equal magnitudes but of opposite
signs. Any resistance change in the active gage resulting from strains of the
like sign (e.g. produced by thermal and axial loads) will be canceled since
the active gages are in adjacent arms of the Wheatstone bridge.
A
B
Figure 4 Damped Free Vibration Test Setup
Two of the important properties of single-degree of freedom systems are the
undamped natural frequency, n, and the damping ratio (approximating the
actual energy dissipation by a viscous damper), . These quantities may be
determined experimentally as follows:
Logarithmic decrement
ln
X t
X t Td
X t
1
ln
n
X t nTd
where Td is the damped period, X is the amplitude and n is the number of
cycles.
Damping ratio
Page 5 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
2
4
2
2
(for small
compared to 2 )
Damped natural frequency
d
1
2
n
The ratio of mechanical energy dissipated in one cycle to the energy at the
beginning of the cycle is
4
En
En
1 e
1
2
1 e
2
PROCEDURE
Connect the strain gages to the strain data logger channel 1 (half
bridge),
Turn on the data logger,
Double click the code VreeVibration.m (available on the desktop)
which will run Matlab® as well
Null the bridge output by pressing the red button on the back face of
the data logger (leftmost one is for channel 1)
Run the code to start data collection and immediately drop the weight
to start vibrations
REQUIRED QUANTITIES
Determine the damped period, Td with three measurements, a single
cycle close to the beginning of data acquisition, a single cycle towards
the end of data acquisition and all cycles including the above two.
Determine the logarithmic decrement, , for the beginning one cycle
and many cycles in the course of vibrations to determine the damping
ratio.
Estimate the damping ratio using two logarithmic decrements
obtained.
Estimate the undamped natural frequency, n, using the two damped
periods and two damping ratios obtained.
Discuss very briefly the reasons that caused the two values obtained
are not the same.
Discuss the possible sources of electrical noise before the vibrations
start.
Page 6 / 7
ME 410 – Experiment 5: Stress Analysis by using Strain Gages
ASSIGNMENT
(Only for the long reports)
Make a detailed research about the _______________________________ method.
(The method will be assigned during the experiment.)
REFERENCES
1.
Daily, J. W. and W. F. Riley., “Experimental Stress Analysis”,
McGraw-Hill, 1965.
2.
Timoshenko, S.P. and Goodier, J. N., “Theory of Elasticity”, McGrawHill, 1982.
3.
Peterson, R.E. “Stress Concentration Design Factors”, John Wiley and
Sons, 1953.
4.
Tse, F. S., Morse, I. E., Hinkle, R. T., Mechanical Vibrations, Theory
and Applications, Second Edition, Allyn and Bacon, 1978.
Page 7 / 7