Mechanics of Materials Laboratory
Cantilever Flexure Test
David Clark
Group C:
9/15/2006
Abstract
A cantilever beam, a beam supported at one point, has been used many times in
countless designs and structures. It is important to understand the behavior of this setup
to avoid any type of failure that might occur if this design is improperly used or executed.
The following experiment utilizes a cantilever test fixture, strain gages, and basic
principles of Statics to determine both theoretical and actual stress along a cantilever
beam. For the 2024-T6 aluminum beam tested, the measured strain was found to be 903,
601, and 293 microstrain along a 1, 4, and 7 inch spacing respectively. The calculated
strain was 1205, 751, and 297 microstrain along the same interval. This deviance in
measured and calculated values demonstrates the need to test all conditions and better
understand the limitations of calculations.
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Table of Contents
1. Introduction & Background.............................................................................3
1.1. General background.................................................................................3
1.2. Calculating stress using Statics................................................................3
1.3. Load Estimate by Strain Relations and Hooke's Law..............................4
1.4. Load Estimation by Deflection................................................................5
2. Equipment and Procedure................................................................................6
2.1. Equipment and Setup...............................................................................6
2.2. Test procedure for measuring the difference of two strain gages............7
2.3. Test procedure for measuring individual strain gages.............................8
3. Data, Analysis & Calculations.........................................................................9
3.1. Known information..................................................................................9
3.2. Results......................................................................................................9
3.3. Load calculations...................................................................................10
3.4. Stress calculations..................................................................................10
4. Results............................................................................................................12
4.1. Graphical Results...................................................................................12
4.2. Comparison of Results...........................................................................12
5. Conclusions....................................................................................................13
6. References......................................................................................................14
7. Raw Notes......................................................................................................15
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1. Introduction & Background
1.1. General background
A cantilever beam refers to any beam that is supported at only one point. This
type of design has been used many times in countless designs and structures. It is
important to understand the behavior of this setup to avoid any type of failure that might
occur if this design is improperly used or executed.
1.2. Calculating stress using Statics
Analysis utilizing basic principles of Statics establishes a cantilever beam
experiences a vertical, horizontal, and moment reaction at the point being supported and a
point force at a length, L.
The stress is found using the elastic flexure formula where terms M, y, and I are
explained below.
Equation 1
M is the moment at the point of loading. For a steady cantilever beam, it is
expessed as:
Equation 2
where P is the applied load, L is the length between the supporting and loading point, and
x is the distance between the clamp and the strain gage.
y is the distance measured from the neutral axis to the point under consideration.
For a simple cantilever setup, this is expressed as:
Equation 3
3
where t is the thickness of the beam.
Finally, I, is the centroidal moment of inertia for the beam. This is expressed as:
Equation 4
where b is the length of the base and t is the thickness.
Combining Equations 1 through 4, the first relation utilizing measurable parameters
can be expressed.
Equation 5
Equation 5 returns units of pounds per square inch (psi) when P is in pounds and
L, x, b, and t are in units of inches.
1.3. Load Estimate by Strain Relations and Hooke's Law
Stress is directly proportional to strain by a constant, E, known as the elastic
modulus. This is expressed mathematically as:
Equation 6
Combining Equations 5 and 6, the strain can be expressed using Equation 7.
Equation 7
An important behavior studied in the following experiment is the correlation
between strain, ε, and the distance along with length, L. The first derivative of Equation 7
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with respect to distance, x, can be expressed as in Equation 8. Simple algebraic
manipulation can be used to solve for the applied force, P, as in Equation 9.
Equation 8
Equation 9
This difference of two strain gages can then be used to find the force, P.
Equation 10
1.4. Load Estimation by Deflection
The load can also be calculated in terms of deflection. This is derived from the
expression,
Equation 11
Equation 2 and 4 are utilized to express M and I respectively. E, the modulus of
elasticity, is known for 2024-T6 aluminum to be 10.4 × 106.
Integrating Equation 11 twice with the known conditions of dy/dx = 0 at x = 0, and
y=0 at x=0, y can be expressed as,
Equation 12
The deflection in the following experiment is easily known. Therefore,
substituting δend for the difference in deflection and solving for P,
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Equation 13
2. Equipment and Procedure
2.1. Equipment and Setup
1. Cantilever flexure frame: A simple apparatus to hold a rectangular beam
at one end while allowing flexing of the specimen upon the addition of a
downward force.
2. Metal beam: In this experiment, 2024-T6 aluminum was tested. The beam
should be fairly rectangular, thin, and long. Specific dimensions are
dependant to the size of the cantilever flexure frame and available weights.
3. P-3500 strain indicator: Any equivalent device that accurately translates
to the output of strain gages into units of strain.
4. Three strain gages:
5. Micrometers and calipers:
The specimen should be secured in the flexure frame such that an applied force
can be placed opposite of the securing end of the fixture. Three strain gages should be
mounted such that the long metal traces run parallel to the length of the beam. The center
of the three gages should be mounted one inch, four inches, and seven inches from the
end of the clamp in the fixture.
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Figure 1
Whenever taking a reading from a strain gage, consult the strain gage
measurement device for optimal setup. The instructions below explain the setup used in
using the P-3500 strain indicator.
2.2. Test procedure for measuring the difference of two strain
gages
The first setup creates a half-bridge setup to find the difference between two
strain gages.
Figure 2
As shown in the diagram, the direction of stress is opposite of the other to read
positive strain. This returns the difference of the two strains.
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With a known initial deflection, the strain indicator was balanced to read zero
strain. Without adjusting the balance, the difference in strain for gage 2 to 3 was
measured and recorded.
To generate a change in strain, a 300 με increase was added by applying a point
load on the bar. The deflection was recorded and the difference in strain for 1 to 2, as
well as 2 to 3, was measured and recorded.
2.3. Test procedure for measuring individual strain gages
To demonstrate how this differential method of calculating strain is equivalent to
the individual strain measurements, the net strain reading for each single strain gage was
measured and recorded using a quarter-bridge configuration.
Figure 3
D, the "dummy" resistance, is needed to balance the bridge. Any uncertainty
within the accuracy of this resistance greatly influences the accuracy of the strain
indicator and should have the same impedance as R3.
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3. Data, Analysis & Calculations
3.1. Known information
The gage factor for the strain gage used was 2.085 and the transverse sensitivity
was 1.0. Both these factors are dependant upon the strain gage used and are generally
given by the manufacturer.
Table 1
3.2. Results
Table 2
Table 3
The difference in deflection between the initial and final position was 0.291
inches.
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3.3. Load calculations
The first load estimate is calculated using Equation 10 and the differences in
strain gages, as recorded in table 1. An example calculation is shown below.
Equation 14
The strain gradient is determined by finding the slope of the strain versus position
graph. For these results, see section 4.
A second load estimate was calculated using Equation 9 and the slope of the strain
versus position graph.
Equation 15
The third load estimate was calculated using Equation 13.
Equation 16
The comparison of the three load estimates is listed in the results section.
3.4. Stress calculations
Three estimates for stress can be determined. The first stress estimate is found
using the calculated force from the differential strain gage measurement setup. This is
expressed using Equation 5. Equation 17 is a sample calculation for the determination of
the stress at gage 1.
Equation 17
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A second stress estimate can be determined using the calculated point load found
in Equation 16.
Equation 18
The third stress estimate uses Hooke's Law to correlate strain to stress. Using
Equation 6, the stress can be found as follows:
Equation 19
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4. Results
4.1. Graphical Results
Strain vs Position
y = -101.67x + 1005.7
0
100
200
300
400
500
600
700
800
900
1000
0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000
Position
Str
ain
(μ
ε)
4.2. Comparison of Results
Figure 4
The table below lists the three load estimates using all three methods.
Table 4
Table 5 contains the first and second stress estimate using both calculated load points.
1 7.96875 8426 125362 4.96875 5254 78163 1.96875 2082 3097
σ x (psi) (P = 11.014)
σ x (psi) (P = 16.387)(L - x)Station
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Table 5
Table 6 catalogs the results from using Hooke's Law.
Station x (in) σ (psi)1 1 9391.22 4 6250.43 7 3047.2
Table 6
Table 7 summarizes and compares the different stress values generated
throughout the experiment.
σ1 (psi) σ2 (psi) σ3 (psi)P = 11.014 8426 5254 2082P = 16.387 12536 7816 3097
Hooke's Law 9391 6250 3047
Stress Summary and Comparison
Table 7
Table 8 shows the error between measured and calculated strain.
Table 8
5. Conclusions
Due to the large margin of error from the measured and calculated results, the
experimental results are not acceptable for practical application. Any design utilizing a
cantilever setup that experiences stresses close to the yield point of the material need to
be more rigorously tested. At maximum deflection, strain gage 1 exhibited a 25% error
from the calculated value. One cause for this error occurs because the equations used are
accurate in small deflections and loads easily handled by the material tested. Also,
Hooke's law is only valid for a portion of the elastic range for some materials, including
aluminum (Wikipedia). Although the net deflection in this experiment was small, the
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stress put upon the material in testing was theoretically 12,536 psi, nearly 84% of the
yield stress. Strain gage three experienced approximately 2% error, whereas the stress
that that point was theoretically 3,097 psi, or 21% of the yield stress. Therefore,
whenever a cantilever setup is used in high stress or deflection applications, thorough
testing and a suitable safety factor must be considered.
6. References
Gilbert, J. A and C. L. Carmen. "Chapter 8 – Cantilever Flexure Test." MAE/CE 370 –
Mechanics of Materials Laboratory Manual. June 2000.
Kuphaldt, Tony R. (2003). "Chapter 9 – Electrical Instrumentation Signals."
AllAboutCircuits.com. Retrieved September 19, 2006, from Internet:
http://www.allaboutcircuits.com/vol_1/chpt_9/7.html
"Hooke's Law." Wikipedia. Retrieved September 22, 2006, from Internet:
http://en.wikipedia.org/wiki/Hooke%27s_law
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7. Raw Notes
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16
17
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