Experimental stress analysis module 5 Part-B (ME832) notes by Dr. Mohammed Imran

1. 1
GHOUSIA COLLEGE OF ENGINEERING
RAMANAGARAM-562159
EXPERIMENTAL STRESS ANALYSIS
[15ME832]
Dr. MOHAMMED IMRAN
ASST PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING

2. 2
Module-5 Part-B
MOIRE FRINGES TECHNIQUE
INTRODUCTION
Moire is a French word and its meaning is watered silk, which is a common optical
phenomenon being observed daily if a fold of silk or any other finely woven fabric is
allowed to slip on another fold and viewed against a light background simmering zig-
zag patterns, localized between two folds of fabric. Such patterns are also seen in
mismatched window blinds or superimposed fine wire meshes. In other words,
whenever a geometric figure repeating itself regularly such as a set of straight lines,
dots, meshes, etc. is superimposed on another similar but not of identical
pattern, moire fringes are visible.
Initially moire fringe effect was a problem and needed to be eliminated because these
fringes were due to errors in spacing, parallelism, and straightness of lines in rulings
used for diffraction gratings. But now the technique has found innumerable
applications wherever a precise measurement of relative movement is required such
as in machine tool control, metrology, strain analysis, and study of dislocations in
crystals. The simple obstruction theory of light in which an opacity in one grid
obstructs the light from the transparency in the other grid explains the formations of
fringes.
1. STRAIN ANALYSIS THROUGH MOIRE FRINGES
In this technique, a grid is fixed on the surface of the model to be analysed for strains.
When the load is applied on the model, the grid on the model called the model grid is
deformed. If another grid, similar to the undeformed model grid, and called the master
grid, is placed in front of the deformed (or strained) model grid and light is allowed to
pass through, fringes are visible in the field of view. These fringes are called ‘moire
fringes’ and can be used to compute the strains at various points of the model. It is
usual to use grids consisting of equally spaced straight parallel non-diffracting lines
though grids from circular, radial, or other non-parallel arrays can also be used for the
formation of Moire fringes.
The moire technique for strain measurement is ideally suited for the measurement of
strains greater than or equal to 200 microstrain.

3. For the analysis of moire fringe pattern to cal
approaches can be used, i.e.
(i) Geometrical approach and
(ii) Displacement approach.
In the geometrical approach
of the rulings of the model grid and master grid. C
fringe pattern, it is possible to deduce expressions for the distance between the model
grid lines and the change in their direction caused by strain in terms of the pitch of
lines on master grid and the fringe characteristic
characteristics such as their separation and inclination of the master grid
lines enables one to compute the
In displacement approach
displacements. Consider, for example, a simple tension test specimen with a line grid
fixed on it and let a simple uniform tension be applied to the test specimen in a
direction normal to the grid lines. The applied
model grid. If another grid with the same pitch as the unstretched model grid be
superimposed on it, a series of bright and dark fringes will be seen in the field of
view.
Figure.1 shows the formation of dark and bright fringes. We have bright fringes at
C, and E and dark fringes at
Figure.1 Light and dark moire fri
The distance between the centres of the two consecutive bright or dark fringes is equal
to the master pitch divided by strain. The fringes can thus be considered as the loci of
the points of equal displacement in a direction norm
3
For the analysis of moire fringe pattern to calculate strains in model, two distinct
approaches can be used, i.e.
Geometrical approach and
Displacement approach.
geometrical approach, fringe pattern is regarded as the result of intersections
of the rulings of the model grid and master grid. Considering the geometry of the
fringe pattern, it is possible to deduce expressions for the distance between the model
grid lines and the change in their direction caused by strain in terms of the pitch of
lines on master grid and the fringe characteristics. The measurement of
characteristics such as their separation and inclination of the master grid
enables one to compute the normal and shear strains at the point of observation.
displacement approach, moire fringe pattern is used for determining
Consider, for example, a simple tension test specimen with a line grid
fixed on it and let a simple uniform tension be applied to the test specimen in a
direction normal to the grid lines. The applied tension will change the pitch of the
model grid. If another grid with the same pitch as the unstretched model grid be
superimposed on it, a series of bright and dark fringes will be seen in the field of
shows the formation of dark and bright fringes. We have bright fringes at
and dark fringes at B and D.
Light and dark moire fringe tension or compression
The distance between the centres of the two consecutive bright or dark fringes is equal
to the master pitch divided by strain. The fringes can thus be considered as the loci of
the points of equal displacement in a direction normal to the rulings. No doubt a
culate strains in model, two distinct
fringe pattern is regarded as the result of intersections
onsidering the geometry of the
fringe pattern, it is possible to deduce expressions for the distance between the model
grid lines and the change in their direction caused by strain in terms of the pitch of
s. The measurement of fringe
characteristics such as their separation and inclination of the master grid
at the point of observation.
fringe pattern is used for determining
Consider, for example, a simple tension test specimen with a line grid
fixed on it and let a simple uniform tension be applied to the test specimen in a
tension will change the pitch of the
model grid. If another grid with the same pitch as the unstretched model grid be
superimposed on it, a series of bright and dark fringes will be seen in the field of
shows the formation of dark and bright fringes. We have bright fringes at A,
nge tension or compression
The distance between the centres of the two consecutive bright or dark fringes is equal
to the master pitch divided by strain. The fringes can thus be considered as the loci of
al to the rulings. No doubt a

4. 4
simple uniaxial tension test piece is considered but the argument can be extended to a
general strain field and the moire fringes can be considered as the loci of points of
constant displacement. Measurement on these fringes gives us the displacement
in x and y directions and strains εxx, εyy, γxy are calculated from these displacements in
a usual manner.
Now the model grid is fixed to the structure or a component under investigation. The
master grid is used as a reference grid,
(i) For the purpose of producing and analyzing moire pattern and
(ii) Establishing directions of the co-ordinate axes.
The distance between points of any two consecutive ruling on the master grid is called
master pitch, p. The direction perpendicular to the ruling and lying in the plane of
master grid is known as principal direction and is denoted by r. The direction parallel
to the ruling is often called the secondary direction and is denoted by s.
The model grid undergoes the same deformation as the model itself because it is fixed
to the model. Therefore, after the model is loaded, the pitch of the model grid changes
from point to point. Say the model grid pitch in any deformed state is p′. The centre-
to-centre separation between any two consecutive dark or bright fringes is called
the inter-fringe. Say the interfringe between two closely spaced fringes measured
along the normal to the fringe is denoted by δ (See Figure.1). Sometimes it is more
convenient to measure the interfringe along r and s directions. In such case, distances
are denoted by δr, and δs, respectively
.
2. GEOMETRICAL APPROACH
Pure normal strains perpendicular to grid lines: Consider the ideal case of a test
specimen subjected to uniaxial tension along the principal direction r. The master grid
is superimposed over model grid without any rotation. On account of normal strain,
the pitch of the model grid is changed and the ruling on the model and master grids
are no longer in alignment, consequently bright and dark fringes are formed and are
visible in the field of view.
Say n = number of lines on the reference grid between two successive bright or dark
fringes.
Then n – 1 = number of lines on the model grid between two successive bright or dark
fringes (if test piece is in tension)

5. or n + 1 = number of lines on the model grid between two successive bright or dark
fringes, if test piece is in compress
Then
Moreover
where δ is interfringe spacing.
From these equations
In the elastic region
Similarly in the case of compression
or
From Eq. (3) we can determine strain
measuring the interfringe spacing
Pure rotation: To obtain an expression for the relative orientation in terms of the
fringe spacing and the grid pitch, let us consider
p, making a small angle as shown in
rhombus is formed by two pairs of grid lines from the geometry of the figure, we can
write
5
1 = number of lines on the model grid between two successive bright or dark
fringes, if test piece is in compression.
δ = np = (n − 1)p′
δ = (n − 1)p (1 + εxx),
is interfringe spacing.
, therefore (on the case of tension)
Similarly in the case of compression
(3)
we can determine strain εxx knowing the value of master pitch
measuring the interfringe spacing δ.
To obtain an expression for the relative orientation in terms of the
fringe spacing and the grid pitch, let us consider two line grids of identical pitch
making a small angle as shown in Figure 2. It is obvious that the diagonal
rhombus is formed by two pairs of grid lines from the geometry of the figure, we can
1 = number of lines on the model grid between two successive bright or dark
(1)
(2)
knowing the value of master pitch p and
To obtain an expression for the relative orientation in terms of the
two line grids of identical pitch
. It is obvious that the diagonal AB of the
rhombus is formed by two pairs of grid lines from the geometry of the figure, we can

6. or
For small value of θ < 10°,
measured values of δ, interfringe spacing:
Combined shear and normal strains
the acute angle measured from the fixed master grid lines to the model grid lines is
denoted by θ. The angle measured from the fixed master grid lines to a fringe at a
point in the direction of θ
B be the point of observation. The master grid lines with pitch
model grid lines with pitch
by thick lines. The angles
angle.
ϕ = ∠ EBD angle made by the model grid line with the master grid line.
ϕ = ∠ EBA = angle measured from the fixed master grid line to a moire fringe at the
point B in the direction of
ϕ = ∠ EBA = angle measured
point in the direction of θ.
δ = BF = interfringe spacing.
Now
or
6
Figure.2 Pure rotation
(4)
< 10°, . Thus Eq. (4) enables us to calculate
, interfringe spacing:
Combined shear and normal strains (or rotation and stretching/contraction):
the acute angle measured from the fixed master grid lines to the model grid lines is
measured from the fixed master grid lines to a fringe at a
is designated by ϕ. This angle ϕ can be acute or obtuse. Let
B be the point of observation. The master grid lines with pitch p are shown. The
model grid lines with pitch p′ are also shown. The resulting moire fringes are shown
θ and ϕ are shown in Figure 11.3. In this case
angle made by the model grid line with the master grid line.
= angle measured from the fixed master grid line to a moire fringe at the
point B in the direction of θ.
= angle measured from the fixed master grid line to a moiré fringe at the
= interfringe spacing.
to calculate θ from the
rotation and stretching/contraction): Say
the acute angle measured from the fixed master grid lines to the model grid lines is
measured from the fixed master grid lines to a fringe at a
can be acute or obtuse. Let
are shown. The
are also shown. The resulting moire fringes are shown
. In this case ϕ is an obtuse
angle made by the model grid line with the master grid line.
= angle measured from the fixed master grid line to a moire fringe at the
from the fixed master grid line to a moiré fringe at the

7. or
Since p is known and δ and
(5).
If gratings with a very fine pitch are used to measure small angles of rotation,
then . Eq. (5)reduces to
or
Now DE = p, pitch of master grid
DK = p′, pitch of deformed
It is also observed that
or
Thus
But from Eq. (5)
Substituting the value of sin(
7
Figure.3
or
(5)
and ϕ can be measured, rotation θ can be calculated from
If gratings with a very fine pitch are used to measure small angles of rotation,
reduces to
(5)
, pitch of master grid
, pitch of deformed model grid
or
Thus
Substituting the value of sin(ϕ − θ) in Eq. (7), we get
can be calculated from Eq.
If gratings with a very fine pitch are used to measure small angles of rotation,
(7)

8. Moreover,
Therefore
Now substituting the value of
Langrangian strain in the direction
Eulerian strain the direction
3. DISPLACEMENT APPR
A moire fringe is a locus of points having the same magnitude of displacements in the
principal direction of master grating. Such a locus is called an
moire fringe, an isothetic
the height of a point on the surface above a reference plane represents
the displacement of the point
isothetic patterns are obtained using line gr
respectively,on the surface of a specimen under investigation. From these moire
gratings u and v displacements are determined by noting down the order of
fringes Nx and Ny.
Then
u = Nx p
v = Ny p
8
(9)
substituting the value of in Eq. (9) we can write
(10)
Langrangian strain in the direction r
n the direction r will be
3. DISPLACEMENT APPROACH
A moire fringe is a locus of points having the same magnitude of displacements in the
principal direction of master grating. Such a locus is called an isothetic.
moire fringe, an isothetic pattern, can be visualized as a displacement surface where
the height of a point on the surface above a reference plane represents
displacement of the point in the principal direction of master grating. Two
isothetic patterns are obtained using line gratings perpendicular to x-axis and
respectively,on the surface of a specimen under investigation. From these moire
displacements are determined by noting down the order of
p
p (11)
(8)
A moire fringe is a locus of points having the same magnitude of displacements in the
isothetic. Therefore, a
pattern, can be visualized as a displacement surface where
the height of a point on the surface above a reference plane represents
in the principal direction of master grating. Two
axis and y-axis,
respectively,on the surface of a specimen under investigation. From these moire
displacements are determined by noting down the order of

9. The Cartesian components of strain can be computed from the derivatives of
displacements as follows:
The slopes of displacements
displacement curves of u and
Figure .4(a) shows the moire fringes when the model grating is perpendicular to
axis. Order of the fringes N
Figure .4 The moire fringes when the model grating is perpendicular to x
9
The Cartesian components of strain can be computed from the derivatives of
(12)
slopes of displacements as above are obtained by drawing tangents to the
and v fields along x and y axes.
shows the moire fringes when the model grating is perpendicular to
Nx, are marked (–3 to + 4) as shown.
The moire fringes when the model grating is perpendicular to x
(out of plane displacement)
The Cartesian components of strain can be computed from the derivatives of
as above are obtained by drawing tangents to the
shows the moire fringes when the model grating is perpendicular to x-
The moire fringes when the model grating is perpendicular to x-axis

10. Lines along x
displacement u along AB and
pitch of master grating.
Now Figure .5(a) shows the moire fringes when the model grating is
to y-axis. Order of the fringes
Lines AB and CD along x
displacement v along AB and
pitch of the master grating. From the plots of
v versus x, and v versus y,
relationships given by Eqs (12)
10
x and y axes say AB and CD are drawn. The
and CD are plotted by noting that u = Nx., p,
shows the moire fringes when the model grating is
axis. Order of the fringes Ny are marked (–2 to + 6) as shown in
and y axes are drawn. The
and CD is plotted by noting that v = Ny. p
pitch of the master grating. From the plots of u versus
strains at any point are determined by u
Eqs (12).
Figure.5 Out of plane slopes.
are drawn. The
, p, where p is the
shows the moire fringes when the model grating is perpendicular
2 to + 6) as shown in Figure.5(a).
axes are drawn. The
p where p is the
x, u versus y,
strains at any point are determined by using the

11. Problem .1 Two gratings of density 25 line per mm are used for strain analysis in a
uniform strain field of uniaxial tensile strain. If the distance between the fringes is
uniform all over the field and equal to 3.6 mm, determine the Eulerian and
Langrangian strains. What would be the strains if it were a compressive strain field?
Solution: Master grating pitch,
Interfringe spacing, δ = 3.6 mm
n = number of lines on the reference grid between successive bright or dark fringes.
Uniaxial tensile strain
Number of lines on the model grid between two successive bright or dark fringes
= n − 1 = 90 − 1 = 89
Pitch on model grid, p′ = 3600/89 = 40.45 micron
Langrangian strain,
Eulerian strain
Uniaxial compressive strain
n = 90
n + 1 = 90 + 1 = 91, number o
or dark moire fringes
Model grid pitch,
Langrangian strain,
11
Two gratings of density 25 line per mm are used for strain analysis in a
uniform strain field of uniaxial tensile strain. If the distance between the fringes is
uniform all over the field and equal to 3.6 mm, determine the Eulerian and
. What would be the strains if it were a compressive strain field?
Master grating pitch,
= 3.6 mm
= number of lines on the reference grid between successive bright or dark fringes.
Number of lines on the model grid between two successive bright or dark fringes
′ = 3600/89 = 40.45 micron
Uniaxial compressive strain
+ 1 = 90 + 1 = 91, number of lines of the model grid between two successive bright
Two gratings of density 25 line per mm are used for strain analysis in a
uniform strain field of uniaxial tensile strain. If the distance between the fringes is
uniform all over the field and equal to 3.6 mm, determine the Eulerian and
. What would be the strains if it were a compressive strain field?
= number of lines on the reference grid between successive bright or dark fringes.
Number of lines on the model grid between two successive bright or dark fringes
f lines of the model grid between two successive bright

12. Eulerian strain,
Problem .2 When a grating of pitch 40 lines per mm is given a slight rotation
respect to a second grating of the same pitch, moire fringes are formed making an
angle ϕ with respect to second grating. Determine the angle
spacing δ, if the angle ϕ is equal to (
Solution: Master grating pitc
Model grating pitch,
So,
or
θ = 2ϕ − π = 2 × 60 − 180 = −60°
Angle of rotation,
θ = −60°
Moreover,
(ii) ϕ = 110°
sinϕ = sin(ϕ
sinϕ = sin(π
ϕ = π − ϕ +
2 ϕ − π = θ
2 × 110° − 180° =
12
When a grating of pitch 40 lines per mm is given a slight rotation
respect to a second grating of the same pitch, moire fringes are formed making an
with respect to second grating. Determine the angle θ and interfringe
is equal to (i) 60° and (ii) 110°.
Master grating pitch,
ϕ = 60°
ϕ = π − ϕ + θ
− 180 = −60°
= 110°
ϕ − θ), because p = p′
π − ϕ + θ)also
+ θ
− 180° = θ
When a grating of pitch 40 lines per mm is given a slight rotation θ with
respect to a second grating of the same pitch, moire fringes are formed making an
and interfringe

13. θ = 40°, angle of rotation.
Now,
4. Out of plane displacement can be measured using Moir
There are various techniques to solve plane stress case and obtain u
displacement fields. Similarly there are techniques to determine the lateral
displacement uz in plane stress problems (due to Poisson’s effect). In the case of plate
bending, the lateral displacement becomes much larger compared to u
displacements.
a) Moire techniques for lateral displacements due to poison’s ratio effect
Moire Technique
i. Collimated illumination and Recording Moire
ii. Point illumination and Point Recording Moire
b) Lighten berg's photo reflective Moire.
c) Salet-lkeda Moire Technique.
d) Reflected-Image Moire method.
Reflected Image Moire method:
In this technique the image of the master grating MGl illuminated by a light source
gets reflected by the polished surface of the specimen and interferes with the second
master grating MG2 whose pitch is the same as that of the first master grating. In
practice, the second master grating is placed on the ground glass plate of the camera
and the camera lens is adjusted to give unit magnification of the image of the first
master grating on the specimen. The moire pattern yields the partial slope contours of
the surface of the specimen along the primary direction of the line gratings along the
other perpendicular axis yields partial slope contours along the new direction. The
position of the camera lens corresponding to which a minimum number of fringes or
no moire fringes at all are observed for no load on the specimen corresponds to the
zero-fringe position. To improve the sensitivity of moire data, the camera lens can be
moved away from this position to obtain initial equispaced parallel fringes with no
13
= 40°, angle of rotation.
4. Out of plane displacement can be measured using Moire.
There are various techniques to solve plane stress case and obtain u
displacement fields. Similarly there are techniques to determine the lateral
in plane stress problems (due to Poisson’s effect). In the case of plate
bending, the lateral displacement becomes much larger compared to u
Moire techniques for lateral displacements due to poison’s ratio effect
Collimated illumination and Recording Moire
Point illumination and Point Recording Moire
Lighten berg's photo reflective Moire.
lkeda Moire Technique.
Image Moire method.
Reflected Image Moire method:
In this technique the image of the master grating MGl illuminated by a light source
gets reflected by the polished surface of the specimen and interferes with the second
master grating MG2 whose pitch is the same as that of the first master grating. In
ctice, the second master grating is placed on the ground glass plate of the camera
and the camera lens is adjusted to give unit magnification of the image of the first
master grating on the specimen. The moire pattern yields the partial slope contours of
he surface of the specimen along the primary direction of the line gratings along the
other perpendicular axis yields partial slope contours along the new direction. The
position of the camera lens corresponding to which a minimum number of fringes or
oire fringes at all are observed for no load on the specimen corresponds to the
fringe position. To improve the sensitivity of moire data, the camera lens can be
moved away from this position to obtain initial equispaced parallel fringes with no
There are various techniques to solve plane stress case and obtain ux and uy
displacement fields. Similarly there are techniques to determine the lateral
in plane stress problems (due to Poisson’s effect). In the case of plate
bending, the lateral displacement becomes much larger compared to ux and uy
Moire techniques for lateral displacements due to poison’s ratio effect-shadow
In this technique the image of the master grating MGl illuminated by a light source
gets reflected by the polished surface of the specimen and interferes with the second
master grating MG2 whose pitch is the same as that of the first master grating. In
ctice, the second master grating is placed on the ground glass plate of the camera
and the camera lens is adjusted to give unit magnification of the image of the first
master grating on the specimen. The moire pattern yields the partial slope contours of
he surface of the specimen along the primary direction of the line gratings along the
other perpendicular axis yields partial slope contours along the new direction. The
position of the camera lens corresponding to which a minimum number of fringes or
oire fringes at all are observed for no load on the specimen corresponds to the
fringe position. To improve the sensitivity of moire data, the camera lens can be
moved away from this position to obtain initial equispaced parallel fringes with no

14. 14
load on the specimen. This amounts to a differential image moire and facilitates easier
determination of slope contours.
Fig: Out of plane displacement can be measured using Moire.
5. Out of plane slope can be measured using Moire (Salet-Ikeda Moire Technique).
In this technique a multiple light source illuminates the reflective surface of the
specimen. The light reflected from the reflective surface of the specimen is viewed
through either n slit or a pinhole as shown in Fig. shown below. The multiple light
source consists of a coarse grating illuminated by a white light diffuser. The coarse
grating can have either parallel-line gratings or equispaced circular and radial-line
gratings. This can also be a crossed grating. A collimating lens L1 is placed at the
focal length from the coarse grating M. The collimated beam of light passes through a
second collimating lens L2 which is kept at focal length from either the slit or pinhole
after getting reflected from the reflective surface of the specimen. As in the previous
case, a moire pattern is produced by doubly exposing the same film to the images of
the multiple source grating on the specimen before and after loading the specimen.
Analysis techniques are exactly as in the previous case. The size of the slit or pinhole
solely depends upon the pitch of the master grating and not on the curvature of the
specimen. It has been observed that clearly defined slope contours are observed for a
slot width or pinhole diameter equal to half the pitch of the coarse master grating.

15. 15
Fig: Out of plane slope can be measured using Moire
6. Applications.
 Strain distribution in 2D bodies in the elastic and elastoplastic states can be
determined conveniently, particularly around sharp corners.
 In plane and out of plane displacements and slopes can be determined easily.
 Moire’s methods can be used to study dynamic problems.
 Strains in curved surfaces can be conveniently analysed.
 Strain in metal structure due to welding and machining can be determined.
 Moire’s fringes can be used to determine isopachics and thus can act as an
extensometer.
7. Advantages.
 Strain distributions can be determined from purely geometric relationships
derived from the interference of the specimen reference of ratings.
 Strain distribution in actual components can determined and there is no
necessity of making models.
 Individual components of strain are determined directly, which is not possible
in other methods of stress analysis.
 Strains at very high temperatures can be determined conveniently.
 Moire’s fringe method is not restricted to elastic analysis only as is the case
other methods.
 The gratings do not spoil the specimen as is the case in the brittle lacquer and
strain-gauge methods.

16. 16