Experimental Strain Analysis by Strain Gage Rosettes
Strain gage rosettes are simply a
Y
gage 2
cluster of three strain gages, arranged Y’
around a point, whose combined readings
X’
45
allow us to experimentally measure the
gage 1
X
complete in-plane strain state at a point.
From these data, we may, with a
45
knowledge of the way in which the strain
gage 3
is related to the stress (e.g., Hooke’s Law),
estimate the components of stress at this point.
A sketch of a typical strain gage rosette is shown at right. This rosette consists of two
gages arranged at 90 angles to each other, with a third gage at 45 angles to the first two
(another common arrangement is to have three gages separated by 120). By virtue of their
construction, strain gages respond only to a lengthening or shortening of the gage along the gage
axis. Hence they can measure only the normal strain along the gage axis. If we take the x- axis
to be parallel to gage 1 as shown, then, obviously,
o
o
 x  1
It can be shown that components of strain transform in almost exactly the same manner
under a rotation of coordinate axes as do components of stress. Hence the equations discussed in
Mech 12 can be used here if we simply replace stress by strain components (see section 6.10,
Beer & Johnson, Mechanics of Materials). In particular, with reference to the sketch above, we
have
 xy
1
1
sin 90 o
2
2
2

1
1
 y'   3  ( x   y )  ( x   y ) cos 90 o  xy sin 90 o
2
2
2
 x'   2  ( x   y )  ( x   y ) cos 90 o 
or, setting  x   1 ,
 xy
1
2
2

1
 3  ( 1   y )  xy
2
2
 2  ( 1   y ) 
If we now solve these for y and xy, we have
 y   2   3  1
 xy   2   3
Thus the three strain gage readings  1 ,  2 ,  3 give us the three in-plane strain components
 x ,  y ,  xy .
If we know something about how the strains relate to the stresses, then we can use the
measured strains to estimate the stresses at the point where the rosette is applied. Suppose, for
example, that Hooke’s Law holds (linear elasticity). If the normal to the surface of the specimen
to which we apply the strain gages is taken to be the z-axis, and if the surface is unloaded (the
usual case), then all the z-components of stress  z   xz   yz  0 . In terms of the strain
components, Hooke’s Law then reads
E
 x   y
12
E
y 
 y   x
12
 xy  G xy
x 
c
c
h
h
Here E and  are Young’s modulus and Poisson’s ratio, respectively. The shear modulus G is
related to these two by
G
E
2 1
a f
y
The Experiment
For this experiment, we will utilize a length of
0.25 in.
fiberglass beam, formed in the shape of a wide-flange section.
z
The cross-sectional dimensions are as shown in the sketch at
4 in.
right.
rosettes
Strain gage rosettes are mounted halfway along the
beam length at two locations: on one side of the web at the
neutral axis and on the bottom surface of the lower flange. The
beam is supported by angle-iron supports at either end (simple
4 in.
support), and loaded with a point load applied
through the Instron machine (see sketch at y
P
right).
rosettes
a
The purpose of the experiment is to
measure the normal and shearing strains at
both locations, and compare them to the
predictions from strength of materials (Mech
12). Let us first consider the location on the
L/2
L/2
bottom of the beam. Here the gages are
arranged as indicated on the sketch on the next
page, with the gages labeled as they actually are on the beam. The only thing that has changed
here from our previous analysis is that we have relabeled the axes. Hence we have
x
x  2
 z  1   3   2
 xz   1   3
Z (side of beam)
gage 1
Z’
45o
X’
gage 2
Hence, from the measured values 1, 2, and 3,
we can determine the strains x, z, and xz.
A similar situation holds with the rosette
mounted on the side of the beam along the
beam centerline (neutral axis). The situation is
as shown on the sketch at right, again with the
gages labeled as they actually are on the beam.
The analysis is identical to that preceding, and
gives
X
(beam axis)
45o
Bottom rosette
gage 3
gage 7
gage 8
Y (top of beam)
X (beam axis)
Side rosette
 x  8
 y  7  9  8
gage 9
 xy   7   9
We will acquire the six strain values 1 - 6 using LabVIEW, and from these we can
calculate the strain components along the beam axes. We will take data at eleven different load
levels: P  0, 200, 400, 600, 800, 1000, 800, 600, 400, 200, and 0 lbs. At each load level, enter
the load value on the LabVIEW panel, and click on the button to start data acquisition. After all
the data are acquired, LabVIEW will fit each set of strain readings with a least-squares linear fit,
and will display the results on a plot. Make a copy of each plot.
We now want to compare the results with the predictions of the simple beam models we
have been studying in Mech 12. To do this, it will be more convenient to look at the slope of the
least-squares fit through the strain data, d/dP. This will ameliorate some of the effects of the
scatter in the data.
Let us first of all examine the situation at the rosette mounted on the bottom beam flange,
along the beam centerline. Here beam theory would predict the following stresses:
x 
Mc
;  z   xz  0
Iz
where M is the bending moment at the point in the beam where the rosette is applied, c is the
distance from the neutral axis (beam centerline) to the bottom surface (2 inches in this case), and
I is, as usual, the area moment of inertia about the z- axis. In terms of strains, this would be
x 
Mc
;  z   x ;  xz  0
EI z
Likewise, for the rosette mounted on the side of the beam web, on the neutral axis, we
would expect from beam theory that
 x   y  0 ;  xy 
VQ
It
where V is the shearing force at the point along the beam where the rosette is applied and t is the
thickness of the web (0.25 inches, in this case). In terms of strain components, then,
 x   y  0 ;  xy 
VQ
GIt
We will be comparing these predictions to the slope of the linear fits to the experimental
strain vs. load data generated by LabVIEW. Hence we will want to calculate at the bottom
rosette the quantity
and
d x
c dM

dP EI dP
d z
d
  x
dP
dP
dM/dP may be calculated from simple statics. For fiberglass, take E = 2.8 x 106 lbs/in2 and  =
0.35. Likewise, for the side-mounted rosette, we will need
d xy
dP

Q dV
GIt dP
and, once again, dV/dP may be calculated from statics. For fiberglass, take G = 1.0 x 106 lbs/in2.
The quantities dx/dP and dxy/dP calculated from these equations may now be compared to the
experimentally-measured values.