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Experiment # 4:
Beams Strain Measurements
Using Strain Rosettes in Aluminum
Beam
1) To study the strain measurements of a simply supported aluminum beam in a general case of plane stress by means of the Mohr 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.
Objectives :
Let’s assume that the State of Stress at a point like Q is known:
The most general state of stress at a point
may be represented by 6 components:
σx , σy , σz : Normal Stresses
τxy , τxz , τzy : Shear Stresses
( Note: τyx= τxy , τxz = τzx , τzy = τyz )
The element at point Q is in plane stress if : σz= τxz = τzy
0
0
0
zy
zx
z
Q Q
Plane Stress :
Stresses in xy coordinates
Wedge-shaped stress element
Forces acting on the element
Transformed stresses in x′y′
Thus, if we know the state of stress at a point, all the stresses acting on different planes at that point can be calculated. And the value of transformed stresses are:
Why the stress transformation is important?
Stress Transformation :
Moreover, for the normal transformed strains we have :
Now, if by using some strain gages the normal strains are measured in 3 directions, then we would be able to figure out the state of strain at that point :
өA өB
өC
C x
y
xA
xB xC
Strain Measurement :
When the state of strain at a point is known the corresponding stresses
can be calculated from the Constitutive Law which is the generalization of Hooke’s law :
Where :
E : modulus of elasticity
G : shear modulus of elasticity
ν : Poisson's ration
Constitutive Law :
For measuring the strain in one direction uni-axial
electrical strain gages are used.
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For measuring the strain in three different
directions strain rosettes are used. Strain rosettes are three strain gages positioned in a rosette-like layout.
Therefore by measuring three linearly
independent strain in three direction, the components of the plain stress state at the point where strain rosette is located can be calculated.
How does an electrical strain gage work?
Rectangular Strain-Rosette
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Strain Gages :
Two types of Strain Rosettes are used in this experiment:
1) Delta Strain Rosette (60° Rosette) :
In this case we have :
өA = 0° , өB = 60° , өC = 120°
- Therefore by plugging this values and also the measured
εA , εB and εC in the strain transformation formulas we will
have :
)(3
2
)22(3
1
BCxy
ACBy
Ax
εεγ
εεεε
εε
xA
xB xC
Strain Rosettes :
2) Rectangular Strain Rosette (45° Rosette) :
In this case we have :
өA = 0° , өB = 45° , өC = - 45°
and for this strain rosette :
xA
xB
xC
BCxy
ACBy
Ax
Strain Rosettes :
For this experiment 6 strain rosettes have been cemented to an
aluminum beam as indicated; Gages 2, 4 and 6 are 45° rosettes and 1, 3 and 5 are 60° rosettes.
P=10,000 lbs
x
y y
z
P/2 P/2
1.5”
1.5”
1.5”
1.5”
9.0” 9.0”
L= 36.0”
6.0”
2.0”
V
x
P/2
-P/2
+ _
+
PL/4
M
x
1
2
3
4
5
6
Aluminum Beam
E = 10.3 106 psi
G = 4.1 106 psi
υ = ?
Application of Strain Rosettes in This Test:
Moreover the state of stress at any point on the beam can be calculated
from the appropriate formulas.
For example to calculate the stresses at a point like i on
the beam we have:
P=10,000 lbs
x
y
P/2 P/2
1.5”
1.5”
1.5”
1.5”
9.0” 9.0”
L= 36.0”
1
2
3
4
5
6
i
yi
a
y
z
b = 2.0”
6.0” yi
a
bI
QV
I
yM
z
i
xy
y
z
i
ix
.
.
0
.
)2
( a
ybaQ ii
i
M : Bending moment at the section
V : Shear force at the section
Qi : First moment of area about the neutral axis
Iz : Moment of inertia
b : Section width
yi : Location of point i
Calculating the State of Stress:
P =15000 lbs
x
y
P/2 P/2 L= 36.0”
1
2
3
4
5
6
Using strain rosettes the state of strain and stress is measured at points 1 to 6
The theoretical state of stress is calculated at
point 1 to 6
Experimental Mohr Circle Theoretical Mohr Circle compare
τ
theoretical experimental
σ
Comparison of Mohr Circles at point x
Experiment Scope:
Mohr Circle:
(σy, τxy)
(σx, τxy)
http://www.engin.umich.edu/students/ELRC/me211/flash3/coach_stress_03.swf
An Interesting Link:
http://www.engin.umich.edu/students/ELRC/me211/mohr.html
Thank you !
Questions?
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