Stress and Strain
Danial J. Neebel, Joseph R. Blandino, and David J. Lawrence,
College of Integrated Science and Technology
James Madison University
Instructor’s Portion
Summary
This lab illustrates the use of a strain gage. The gage is bonded near the
clamped end of a cantilever beam. Weights are applied to the free end of
the beam. The strain gage measures the axial elongation of the beam. The
students plot a force vs. strain curve using LabVIEW. They then analyze
the data and plot a stress vs. strain curve. From this curve, the students
determine the modulus of elasticity of the beam. The students must
determine the material used to fabricate the beam.
Uses
This exercise applies to mechanical engineering, physics, or a general
instrumentation course.
Equipment List
? Computer running Windows, Macintosh, Linux, Sun, or HP-UX
(visit http://www.ni.com/labview/lv_sysreq.htm for
requirements specific to your operating system)
? SCXI Bundle from National Instruments (part number 777448-37)
， LabVIEW Full Development System
， PCI-6024E Data Acquisition Board
， SCXI 1000 chassis
， SCXI 1349 cable
， SCXI 1180 feedthrough panel (not used in this experiment)
? SCXI Module from National Instruments
， SCXI 1121 – 4 Channel Isolation Amplifier
(part number 776572-21)
? SCXI Terminal Block from National Instruments
， SCXI 1321 – For use with the SCXI 1121
(part number 777687-21)
? Strain Gage
? Cantilever beam
? Metal plate
? Clamp
? Mass hanger and variety of masses
? Balance for accurately measuring the masses
? Websites
， Omega — www.omega.com
， National Instruments — www.ni.com
Setup
Computer-based measurement systems are used in a wide variety of applications. In laboratories, in field services and on manufacturing plant
floors, these systems act as general-purpose measurement tools well-suited for measuring voltage signals. However, many real-world sensors and transducers require signal conditioning before a computer-based measurement system can effectively and accurately acquire the signal. The front-end signal conditioning system can include functions such as signal amplification, attenuation, filtering, electrical isolation, simultaneous sampling, and multiplexing. In addition, many transducers require excitation currents or voltages, bridge completion, linearization, or high amplification for proper and accurate operation. Therefore, most computer-based measurement systems include some form of signal conditioning in addition to plug-in data acquisition (DAQ) devices. For more information on signal conditioning for computer-based data
acquisition systems you can find National Instruments Application Note 48 online at www.ni.com.
The LabVIEW VI is the Strain VI. Students input the beam dimensions
and the applied load. The students must be careful to use consistent units. The bridge circuit is balanced using the offset potentiometer on the VI
Stress and Strain 2
front panel. Using the offset potentiometer is preferable to having the students adjust the offset on the SCXI-1321 terminal block.
Figure 1. Basic Setup for a Beam Under a Stress Load Follow the steps listed to prepare the workstations for this experiment. The
instructions assume you are using the equipment list shown previously. Note: Most of the manuals that are referred to ship with National Instruments hardware and software. If you can’t find your hardcopy of the
manuals, you can get them online at http://www.ni.com/manuals. If
you encounter problems during setup, contact technical support at http://www.ni.com/support.
Before the Day of the Lab
1. Configure the cantilever beam and weight measurement device.
a. The beam is made of 6061 aluminum, but other materials can be
used. The beam dimensions are 12.5 ~ 1.0 ~ 0.125 inches.
b. The weight is applied 0.25 in. from the free end.
c. Attach the strain gage 10.188 in. from the free end.
d. The beam is clamped 11.0 in. from the free end.
Stress and Strain 3
e. The beam is clamped to the edge of a table.
f. A metal plate is used between the beam and the clamp. 2. Install LabVIEW (see the LabVIEW Release Notes for your version of
LabVIEW).
3. Install your PCI-6024E board (see the 6023E/6024E/6025E User
Manual).
4. Configure the SCXI 1121 (see the SCXI 1121 User Manual).
a. Enable shunt calibration for channel 0.
b. The SCXI 1121 has jumper settings that need to be made for this
experiment.
i. Set the gain to an appropriate value for your sensor. Think of
the gain as a scaling factor that makes the sensor voltage fit
well within the measurement system range. First, determine
the maximum voltage output of the sensor for the
temperature range it will experience. Then determine the
maximum voltage the system can measure. The gain is found
by dividing the system voltage by the sensor voltage. For
example, if your sensor outputs a maximum of 0.01 volts and
the measurement system has a range of 0 – 10 volts, you
would set the gain to 1000.
ii. Set the jumpers for a half-bridge completion network for
channel 0.
iii. Leave the rest of the jumpers in their factory settings.
c. Install the SCXI 1121 into slot 0 of the SCXI 1000 chassis.
d. Connect the strain gage to channel 0 of the SCXI 1321 terminal
block.
e. Connect the SCXI 1321 terminal block to the SCXI 1121 module. 5. Cable the PCI-6024E to the SCXI 1121 module with the SCXI 1349
(see the SCXI 1349 Shielded Installation Guide).
6. Configure the PCI-6024E board and the SCXI 1121 module (see the
NI-DAQ Release Notes for your version of NI-DAQ).
? When configuring the SCXI 1121 make sure the software settings
match the jumper settings on the physical module.
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7. Create the following directory in your LabVIEW folder:
\\LabVIEW\Experiments\Stress and Strain.
8. Copy Stress and Strain.llb into the Stress and Strain folder you
just created.
9. Conduct a run-through of the lab procedure the students will perform.
On the Day of the Lab
1. Power up the computer and SCXI chassis.
2. Make sure the students have enough masses at their workstations. References
? John A. Allocca, Transducers: Theory and Applications,
Reston Pub. Co.
? Wheeler, Anthony J. and Ganji, Ahmad R. (1996), Introduction
to Engineering Experimentation, Prentice-Hall Inc. Englewood
Cliffs, NJ.
Student’s Portion
Introduction
Strain gages are important to any device under stress or strain. Strain
gages provide solutions to many real world systems and testing. In this lab,
you will apply various loads to the cantilever beam and measure the strain
using a pre-written LabVIEW program.
Objective
? To use strain gages for measurement of strain.
? To determine the modulus of elasticity of a cantilever beam.
? To use Hooke’s Law to determine the modulus of elasticity of an
unknown material.
Theory
Design of Strain Gages
Strain gages are used as sensing elements in displacement or load
measurement systems. They can be used to measure extremely small Stress and Strain 5
deformations. Strain gages are made of fine wire with a certain cross-sectional area (A), initial length (L), and resistivity (：). The resistance of
the wire changes when the wire is displaced due to stress. The stress is an applied load acting over the cross-sectional area of the wire. The following equation defines the resistance (R) of the strain gage wire:
L：R ，A
It is common to use a gage factor (G) to describe the changes in resistance
((R) of the strain gage due to changes in length ((L) of the wire. The gage
factor is also used to compare various strain-gage materials:
(R/R(L；gage factor = G = where = is defined as strain (L/LL
So, strain is the change in length over the original length. The units are typically mm/mm or in/in. Sometimes the term ，strain (micro-strain) is -6used, meaning 10 in/in.
Stress and Modulus of Elasticity (Young’s Modulus)
By mounting strain gages to the surface (top and/or bottom) of a cantilever beam, you can measure the axial deformation of the beam (the deformation along the length of the beam) when a transverse load is applied to the end of the beam. The amount of deformation can be related to the strain using the previous gage factor equation. Figure 2 shows a strain gage bonded to the top surface of the cantilever beam.
Figure 2. Cantilever Beam with a Strain Gage Mounted on the Top When a load (P) is applied at the end of the beam, the tensile stress (；)
along the x-axis at the top surface is given as:
Mc;；;，;;；！？ I
where:
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? M = bending moment (units of N-m). M is the product of the effective
beam length and the force (P) applied at the end of the beam.
? c = distance from neutral axis of beam (m), Typically c = t/2, where t
is the beam thickness.
4? I = moment of inertia for a cross section of the beam (units are m).
3btThe moment of inertia is I = . 12
? P = load (N).
? L = effective beam length = distance between the point where the load
is applied and the center of the strain gage (m).
? b = beam width (m).
? t = beam thickness (m).
You can find the stress from the applied force and beam geometry, as
given by equation 1, and measure the strain with a strain gage. Conveniently, a relationship exists between the two. The modulus of elasticity (E) is an index of the stiffness of the material. When a load (that is, force) is applied to metal such as an aluminum alloy or steel, the strain of the material changes linearly as a function of the stress over a certain, usually small, range. Within this range, the beam exhibits elastic deformation (that is, elastic or reversible strain). When the deformation is not permanent, it is called elastic deformation. That is, when a force is applied, the beam will bend downward and elongate. When the force is removed, the beam will return to its original shape. The modulus of 2elasticity (E) is the ratio of stress to elastic strain and has units of N/m:
；( E = (2) (；
For this lab, strain will be measured by a strain gage bonded to the top surface of a beam. By applying a known load, the stress is calculated using Equation 1. After applying several different loads and measuring the strain directly using a strain gage, you can plot the stress-strain graph. The slope of this graph is the modulus of elasticity as predicted by Equation 2. Note: The location and position of the strain gage is critical. You must know if the position of the gage will result in axial, shear, torsional, or bending strains, or any combination.
Stress and Strain 7
Bridge Circuits
Usually, strain gages are connected in a Wheatstone bridge circuit as shown in Figure 3. The change in resistance of the strain gage due to an applied force can be measured as the output voltage of the Wheatstone
bridge circuit. The bridge circuit shown has a single strain gage and three fixed resistors and is called a quarter-bridge circuit.
Figure 3. Wheatstone Bridge with Strain Gage, R3
R3 is the strain gage resistance. Therefore,
?(RR34?)V = ，V0s?)R？RR？R2314??
When V = 0, the bridge is said to be balanced. The only way to balance 0
the bridge is to set:
RR34 - = 0 R？RR？R2314
or
RR = RR 1324
R is typically a variable resistor used to balance the bridge. In this 4
experiment, the offset potentiometer on the front panel of the LabVIEW VI is used to balance the bridge. Assuming the bridge is initially balanced,
R，R？(Rthen when the strain gage is strained,. The output of the 33
bridge circuit becomes:
(RR1V V0 = s；？；？R？R？(RR？R2314
Stress and Strain 8
Because (R is usually small compared to R, it can be neglected from the 3
denominator of the equation. Then, V becomes: 0
RR(1 V V0 = s；？；？R？RR？R2314
Note that Vis a linear function of (R. Now you can relate strain with the 0
output voltage based on the following derivation. Because the gage factor (G) is equal to ((R/R))/((L/L), and strain (；？ is
equal to (L/L=((R/R)/G, the conversion from strain to output voltage is:
；？；？R？RR？RV23240；， RRGV13s
You must know the gage factor (G) to get meaningful results.
Pre-Lab Preparation
? Read through the theory section from this experiment to understand
how strain gages are used to measure deformations. ? Find a reference and read about the modulus of elasticity. ? Find a table with the modulus of elasticity for common metals.
You will need this table to complete the lab.
? Bring a ruler and a formatted virus-free disk to the lab.
Workstation Details
Your workstation should have the following items: ? A computer with National Instruments LabVIEW software ? National Instruments DAQ board (inside computer). ? National Instruments SCXI chassis.
? Cantilever beam with mounted strain gage connected to SCXI input
channel 0.
? Masses for hanging on the cantilever beam.
? Balance for weighing masses.
Stress and Strain 9
Lab Procedure
1. Set up the software:
a. Launch LabVIEW.
b. Open the Stress and Strain.llb. The file is under
\\LabVIEW\Experiments\Stress and Strain on your
computer.
c. On the front panel of Strain VI, under Output File, enter the
filename a:\strain.dat.
d. Put a formatted, virus-free disk into the drive. e. On the front panel of the Strain VI, set:
– Scan rate = 512.
– Gage factor = The gage factor provided with the
manufacturer’s data for the gages used. Ask your instructor if
you do not know the proper gage factor.
– Gage resistance = the resistance provided with the
manufacturer’s specifications. This is not critical for this
experiment, but you should know this information.
– Applied weight = 0.
– Bridge circuit type = Quarter.
f. Measure the width, thickness, and effective beam length (distance
from the point where the mass is applied to the center of the strain
gage). Enter the values on your data sheet and on the Strain VI
front panel.
2. Balance the bridge:
a. Run the Strain VI and watch the strain output display beneath the
graph.
g. Use the offset potentiometer (use the dial as the coarse control and
the arrows next to the digital indicator as the fine control) to set
the strain output display to zero. Make sure the value is within
?0.000002.
h. Press the Keep Data button.
3. Take measurements with different loads or weights: a. Choose a mass and weigh the entire mass and hanger on the
balance to get an accurate measurement.
Stress and Strain 10