Experiment 4
Transcript of Experiment 4
STRENGTH OF MATERIALS LABORATORY
EXPERIMENT 4
STRAIN MEASUREMENTS USING STRAIN ROSETTES IN
ALUMINUM BEAMS
Submitted by Antonio Sampedro and Sebastian Vieira
Date Performed April 2, 2007
Date Submitted April 23, 2007
Instructor Sun Punurai
SPECIMEN
Material Aluminum Beam
Section 6 in x 2 in
Clear Span 36 in
Abstract
This experiment used six strain gages Rosettes placed at different locations to
experimentally measure the strains on a simply supported aluminum beam when a 25,000
lb load was applied. Three of the Rosettes were oriented at 45 degrees and the other three
were oriented at 60 degree. Two of the Rosettes were on the neutral axis of the test
specimen. The other four strain gages were half the distance from the neutral axis to the
top and bottom of the specimen. The dimensions of the beam used in the experiment were
6in high by 2in wide. A computer on the testing machine recorded the measured strains in
each Rosette under the loaded condition and also calculated an experimental value for the
strain at that point under the load.
Introduction
1 – To study the strain measurement of a simply supported aluminum beam in a general
case of plane stress by means of the Mohr’s Circle analysis.
2 – To verify theoretical computations of the combined stresses at several point on a
beam with the experimental results.
3 – To experimentally determine the combined stresses (the actual state of stress) at
several points on a beam using the Strain Rosettes.
Theory
1 – Plane Stresses
Consider an element in plane stress as shown below; this element is infinitesimal in size
and can be sketched as a parallelepiped. σx and σy are designated as normal stresses
acting on the x-axis and t-face of the element, respectively. The shear stress τxy acts on
the x-face in the direction of the y-axis, and τyx on the y-face in the direction of the x-
axis. They are equal. The positive sign conventions of these plane stresses are depicted in
the figures below
(a) – Two-dimensional view in x-y axis
(b) – Two-dimensional view in x-y and x’-y’ axis
The transformation of stresses with respect to the {x,y,z} coordinates to the stresses with
respect to {x',y',z'} is performed via the equations,
Where θ is the rotation angle between the two coordinate sets (positive in the
counterclockwise direction).
– Strain gage Rosettes
Since each strain gage measures the normal strain in only one direction, at least three
strain gages are needed to determine the strains in a plane stress elelment, as indicated as
A,B, C in figure below
Three strain gages, A,B, and C arranged in an element
Applying this equation to each of the three strain gages results in the following
system of equations,
These equations are then used to solve for the three unknowns, x, y, and γxy
Special Cases of Strain Rosette Layouts
Case 1: 45º strain rosette aligned with the x-y axes, i.e., θa = 0º, θb = θc = 45º.
Case 2: 60º strain rosette, the middle of which is aligned with the y-axis, i.e., θa = 30º, θb
= θc = 60º.
In the case of biaxial stress, Hooke’s law of plane stress-strain relation or the constitutive
law for a linearly elastic material is given by:
Where E = Young’s modulus or modulus of elasticity
v = Poisson’s ratio
G = Shear modulus of elasticity
When the Hooke’s law holds, or the beam behaves in a linearly elastic manner, the
following normal and shear stresses from the flexural and shear formulas as seen in most
standard Strength of Materials text books, can be used:
Flexure formula:
And Shear formula:
Where M = bending moment about the z-axis
I = moment of inertia about the z-axis
y = distance from the z-axis
V = Shear force in y-axis
b = width of the cross-section
Q = first moment of the cross-sectional area outside of the point in the cross
section where the stress is being found.
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Experiment Procedure
This test is conducted using a Vishay Micro Measurement System 5000 data acquisition
system. This equipment is capable of conditioning and reading signals from strain gage,
thermocouples, L VDTs, high range signals like those coming from DCDTs and
tiltmeters or any device providing scaled voltage signals. In this experiment the System
5000 is used to measure and reduce the data from 5 rosettes, comprising 15 strain gages,
and the applied load.
1 – Start the computer and load the Strain Smart program
2 – Open, arm, and start the Lab4 program
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3 – Apply a 25,000 lb load the beam
4 – Click on the Read button.
5 – Release the load and turn off the testing machine.
6 – Identify the data file, and using the Strain Smart program reduce the data into Excel
comma delimited format.
7 – Copy the data.
Analysis of Data
Rosette #1
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 9 = 112,500 lbs-in
y = 1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = My / I = 112,500 * 1.5 / 36 = 4687.5 = 4.69 ksi
σy = 0
Q = Ay = 2*1.5*2.25 = 6.75 in3 b = 2in
τxy = VQ/bI = 12,500*6.75 / (2*36) = 1171.875 = 1.17 ksi
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Rosette #2
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 9 = 112,500 lbs-in
y = 1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = 0
σy = 0
Q = Ay = 2*3*1.5 = 9 in3 b = 2in
τxy = VQ/bI = 12,500*9 / (2*36) = 1562.5 = 1.56 ksi
Rosette #3
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 9 = 112,500 lbs-in
y = -1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = My / I = 112,500 * (-1.5) / 36 = - 4687.5 = - 4.69 ksi
σy = 0
Q = Ay = 2*4.5*0.75 = 6.75 in3 b = 2in
τxy = VQ/bI = 12,500*6.75 / (2*36) = 1171.875 = 1.17 ksi
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Rosette #4
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 18 = 225,000 lbs-in
y = 1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = My / I = 225,000 * 1.5 / 36 = 9375 = 9.375 ksi
σy = 0
Q = Ay = 2*1.5*2.25 = 6.75 in3 b = 2in
τxy = VQ/bI = 12,500*6.75 / (2*36) = 1171.875 = 1.17 ksi
Rosette #5
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 18 = 225,000 lbs-in
y = 1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = 0
σy = 0
Q = Ay = 2*3*1.5 = 9 in3 b = 2in
τxy = VQ/bI = 12,500*9 / (2*36) = 1562.5 = 1.56 ksi
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Rosette #6
P = 25,000 lbs = 12,500 lbs
M = 12,500 * 18 = 225,500 lbs-in
y = - 1.5 in
I = 1/12 * bh3 = 1/12 * 2 *63 = 36 in4
σx = My / I = 225,000 * (- 1.5) / 36 = -9.375 ksi
σy = 0
Q = Ay = 2*1.5*2.25 = 6.75 in3 b = 2in
τxy = VQ/bI = 12,500*6.75 / (2*36) = 1171.875 = 1.17 ksi
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