This document discusses two-dimensional photoelasticity techniques for stress analysis. It describes various methods for separating principal stresses at interior points of a photoelastic model, including using a lateral extensometer, properties at free boundaries, Laplace's equation, shear-difference method, and oblique incidence method. It also covers scaling stresses between models and prototypes for various applications, including static, thermal, and dynamic cases. As an example, it discusses using photoelasticity to optimize the design of sheet pile sections for cofferdams.

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

2. 2
Module-3 Part-B
TWO DIMENSIONAL PHOTOELASTICITY
1. STRESS SEPERATION:
In the analysis of isochromatic patterns, it was shown that the principal-stress
difference − could be determined directly and that the maximum shear stress
could be determined provided the two principal stresses are of opposite sign. Also, at
free boundaries, the principal stress normal to the boundary is zero; therefore, the
isochromatic data yield directly the value of the other principal stress. At interior
regions of the model, individual values for the two principal stresses cannot be
obtained directly from the isochromatic and isoclinic patterns without using
supplementary data or employing numerical methods.
STRESS SEPERATION TECHNIQUES
Many techniques have been proposed to determine the individual values of
1 and 2. Such techniques which are commonly used are described here.
(a) Use of Lateral Extensometer
From Hooke’s law, for the plane state of stresses,
x = 1/E (x - y)
y = 1/E (y - x)
z = - /E (x + y)
If h is the original thickness of the model before being stressed and h + h is the
thickness after loading, then
z = h /h
If the change in thickness can be determined, then a knowledge of  and E of the
model material will enable one to determine (x + y) from the above equations.
x + y = - E/ z = - E/ h /h
The instrument that is used to determine the change in thickness, i.e. the change in the
lateral dimension of the model, is called the lateral extensometer. Since (x + y) and
(x - y) are known, the individual values of x and y can be calculated.

3. 3
Alternatively, as (1 + 2) = (x + y) and (1 - 2) are known, 1 and 2 can be
determined.
(b) At the Free Boundary
When a boundary of the model is not loaded directly, it is called a free
boundary. The normal and shear stresses on a plane tangential to a free boundary are
therefore zero. The principal stress axes are normal and tangential to the boundary.
One of the principal stresses, say 2, is zero. Hence the isochromatics near a free
boundary gives the values of the nonvanishing principal stress 1.
Fig (14) Stresses of a free boundary
(c) Use of Laplace Equation (analytical separation method):
We know that if the body forces x and y are constant so that 2
 = 0, then
the first invariant of stress I1 = x + y is a harmonic function, i.e. it satisfies
Laplace’s equation. Thus
2
(x + y) = 2
(1 + 2) = 0
Or
2
/ x2
(1 + 2) + 2
/ y2
(1 + 2) = 0
From the theory of potential functions, we have the following result: if a
function V is harmonic and continuously differentiable in a closed regular region R
and vanishes at all points of the boundary of R, then it vanishes at all points of R.

4. 4
i
j
i
j
The above result has an important consequence. Let V1 and V2 be two
harmonic functions in a closed regular region R and let them take on the same
boundary values. The function (V1 -V2) is therefore harmonic in R and reduces to 0 on
the boundary. Hence, from the above result, (V1 -V2) vanishes throughout R. The
resulting statement is therefore: A function, harmonic and continuously differentiable
in a closed regular region R is uniquely determined by its values on the boundary of
R.
Since (1 + 2) is harmonic, the above statement can be applied. The values of
(1 + 2) on the boundary of the photo-elastic model can be established from the
photo-elastic fringe pattern, since on a free boundary 1 + 2 = 1 - 2 = 1 or 2. For
these values of (1 + 2) on the boundary, their values inside the region are unique.
Numerical methods are available to determine the values of (1 + 2) from the
known boundary values. Standard computer programmes are also available.
Alternatively, one can adopt the electrical analogy method. It is known that the
voltage distribution in a uniformly conducting medium satisfies Laplace equation, i.e.
2
V / x2
+ 2
V / y2
= 0
Where V (x,y) is the voltage distribution. Therefore, if voltages proportional to
the sum of the principal stresses are applied to the boundary of a uniformly
conducting medium having the shape of the model, the resulting voltage at any
internal point will be proportional to the value of (1 + 2). In practice, “Teledeltos”
paper which consists of a uniform layer of graphite over a thin paper is used for
making the model.
(d) Shear – difference Method
This is a step-by-step integration process along a selected line starting from a
point where one of the normal stresses is known. Generally, the initial point lies on
the boundary where the individual values of 1 and 2 are known. The method makes
use of one of the differential equations of equilibrium
x / x + τxy / y = 0
Or
y / y + τxy / x = 0
In the absence of body of forces. The line of integration, called the x- or y- axis, joins
the points of interest where the individual values of x and y (or 1 and 2) are

5. 5
i
j
known. Assuming that this line is called the x-axis, the first equation above can be
integrated from the initial point i up to the desired point j. Thus
∂ σx / σx = - ∂ τxy / ∂y dx
(σx) j = (σx) i - ∂ τxy / ∂y dx
Using the finite difference form, the above equation can be written as
j
(σx) j = (σx) i - ∑ Δ τxy / Δy Δx
i
The values of xy / y are calculated from the values of xy determined along
Fig (15) Elements for Shear – difference method
two lines CD and EF Fig (15) which are y/2 apart from the line of integration, i.e.
the x – axis. In practice, the intervals x and y are usually equal to 0.2mm.
In general, the boundary at the initial point i will not be normal to the x-axis.
Let  be the angle between the tangent to the boundary and the x-axis Fig (16). At the

6. 6
boundary, one of the principal stresses (say 2) is zero and the other stress (i.e. 1)
tangential to the boundary can be evaluated from the isochromatics.
2
=
0
1
1

i
Fig (16) Rectangular stress components at a free boundary
(e) Oblique Incidence Method
Secondary principal Stresses
So far, the incident ray has been assumed to be normal to the face of the
model. If the incidence is not normal, then it is said to be an oblique incidence. Let the
incident ray be in the xz-plane and let Ф be the angle between the z-axis and the
incident ray Fig (17). For normal incidence along the z-axis, the stresses which cause
photo-elastic effect are x, y and xy or equivalently, the principal stresses 1 and 2
corresponding to these stresses. However, experiments reveal that for oblique
incidence, the stress components which cause photo-elastic effect are not the primary
stresses (as represented by x, y and xy), but the secondary stresses. The secondary
stresses corresponding to a given direction are

7. Fig (17) Model under oblique incidence
those rectangular stress components whose vector lies completely in a plane
perpendicular to the given direction. For example, for the normal incidence
stress components whose vectors lie completely in
in the xy – plane, are 
lie completely in a plane perpendicular to
incidence along the y-axis, the appropriate stress component is
are zero for a plane stress case. For incidence along the x
component is y. These are called secondary rectangular stress components
corresponding to the give
corresponding to the secondary rectangular stress components are called secondary
principal stresses. These are denoted usually by
The secondary principal stresses for the oblique incidence
1,
From fig.17 and transformation of stress
x'
Hence,
1', 2' = (x cos2
Ф + y) / 2
Similarly, for incidence along the
7
Fig (17) Model under oblique incidence
those rectangular stress components whose vector lies completely in a plane
perpendicular to the given direction. For example, for the normal incidence
stress components whose vectors lie completely in the plane perpendicular to
x, y and xy. For the direction Z
'
O, the stress vectors which
lie completely in a plane perpendicular to Z
'
O are  'x,  'y = y and x
axis, the appropriate stress component is x only since
are zero for a plane stress case. For incidence along the x-axis, the appropriate stress
. These are called secondary rectangular stress components
corresponding to the given direction of incidence. The principal stresses
corresponding to the secondary rectangular stress components are called secondary
principal stresses. These are denoted usually by 
'
1 and 
'
2.
The secondary principal stresses for the oblique incidence Z
'
O are
, 2 = (x'+y') / 2 + √[((x' - y') / 2))2
+ 2
x'y']
From fig.17 and transformation of stress
' = x cos2
Ф; y' = y'; x'y' =xy cos Ф
) / 2 + √[((x cos2
Ф - y) / 2))2
+ 2
xy cos2
Ф]
Similarly, for incidence along the y – axis,
1' = x and 2' = 0
those rectangular stress components whose vector lies completely in a plane
perpendicular to the given direction. For example, for the normal incidence ZO, the
the plane perpendicular to ZO, i.e.
, the stress vectors which
x'y' fig (17). For
only since z and xz
axis, the appropriate stress
. These are called secondary rectangular stress components
n direction of incidence. The principal stresses
corresponding to the secondary rectangular stress components are called secondary
]

8. 8
SCALING MODEL-TO-PROTOTYPE STRESSES
In the application of the photo-elastic method of stress analysis, it is tacitly
assumed that the prototype structure behaves as a linearly elastic structure within the
range of design loads. The photo-elastic models reproduce elastic behaviour both at
the room temperature as well as at the stress freezing temperature used for three-
dimensional analysis.
A close examination of the governing equations of the mathematical theory of
elasticity shows that for a singly connected body, these are entirely independent of the
elastic constants, Young's modulus and Poisson’s ratio of the material in the absence
of body forces or when the body force field is uniform as in the case of the
gravitational field. In other case Poisson's ratio does have a small effect. Even in the
case of multiple connected bodies, if the forces around individual openings are such
that the resultant force in each case is zero, the stress distribution remain independent
of the elastic constants. It is thus possible in most practical cases to apply the photo-
elastic model results to the prototype.
(a) Scaling for Stress and Deflection
For two-dimensional structures, the most significant dimensionless ratio is the ratio
(σhl /P) and for deflection, δEh /P, where σ is the stress at any point, δ the
displacement at the point, P the applied load, h the thickness and l a typical length
dimension. If subscripts p and m refer to the prototype and model respectively we
may write the scale relations as follows:
σp = [(Pp / Pm) . (hm / hp). (lm / lp) σm]
δp = [(Pp / Pm) . (Em / Ep). (hm / hp) δm]
In general, if the pressure σ0 on the body is defined rather than the load, we obtain the
simple relations
σp = (σ0p / σ0m) σm
σp = [(σ0p / σ0m) (Em / Ep) (lp / lm)] δm
(b) Scaling for Temperature
The thermal stress problem is generally considered in two parts. If the temperature
change ΔT is known at all the points and the coefficient of thermal expansion α is
given, in a linear structure we have for plane stress

9. 9
σp = [(Ep / Em) (σp / σm) (ΔTp / ΔTm) σm
For plane strain the right-hand expression is to be multiplied by (1- νm) / (1- νp), where
ν denotes Poisson’s ratio.
The other part is the determination of temperature at any time t. If a = k / pc denotes
the diffusivity of the material, where k is the thermal conductivity, ρ the density and c
the specific heat, the time scale is given by
tp = (l2
p / l2
m) (am / ap) tm
(c) Scaling for Dynamic Cases
In an elastic system the frequency n of any specified natural mode depends on the
length l that designates the size of the system, the mass density ρ of the material and
the Young's modulus E (ignoring the effect of Poisson's ratio ν). The most general
form of a dimensionally homogeneous equation among these variables is
n α [(1 / l) (E/ ρ) ½
]
This leads to the model law for frequency,
np = [(lm / lp) (Ep / Em) ½
(ρm / ρp) ½
] nm
When the same material is used for the model and prototype,
np = (lm / lp) nm
TWO-DIMENSIONAL APPLICATIONS
The following examples of practical problems solved by two-dimensional photo-
elastic techniques illustrate the wide application and utility of photo-elasticity. The
simultaneous use of other types of models, including mathematical models, is
necessary in taking many a design decision.
(a) Design of Sections for Flat Web Sheet Piles
Flat web steel sheet piles are extensively used in the construction of cellular coffer
dams by interlocking a number of sheet piles in the form of a circular shell which is
then filled with sand. Such coffer dams are required during the construction of
barrages, jetties and breakwaters.
One of the major design problems is the optimization of the radii of curvatures and
other dimensions to achieve the maximum interlock strength for any specified yield
strength of steel. Such a problem arose during the design of flat web sheet piles to be
fabricated at Bhilai Steel Plant (in the state of Madhya Pradesh, India) and ultimately

10. proposed to be used for the construction of the Farakka barrage in West Bengal, India.
Two dimensional photo-
fundamental mechanism of structural action of the interlock and there by in arriving at
a procedure for the development of an optimum section satisfying the functional and
fabrication requirements.
Photoelastic models of nine different designs were tested reproducing half the
cross-section of each pile. The models were fabricated from 6 mm thick CR
The lateral forces on the interlock elements due to the filler material are very small
and hence the interlock was tested under a direct pull. A typical isochromatic pattern
for the interlock can be obtain. From isochromatics, the boundary stresses are
computed. The typical distribution of the boundary stresses for one of the nine cases
is shown in Fig (6).
Fig (6) Tangential boundary stress distribution (stress concentration factor at
X is 2.6 and at
Photo-elastic analysis enabled the determination of: (i) the most highly stressed zones
and the corresponding stress concentration factors, (ii) the points of contact, and (iii)
the direction of the resultant force at the point of contact.
(b) Reinforcement Design Around Opening in Mass Concrete Structures
A large number of openings, such as
constructed in all modern high dams. These introduce local disturbances in the
10
proposed to be used for the construction of the Farakka barrage in West Bengal, India.
-elastic tests have proved very useful in elucidating the
fundamental mechanism of structural action of the interlock and there by in arriving at
a procedure for the development of an optimum section satisfying the functional and
f nine different designs were tested reproducing half the
section of each pile. The models were fabricated from 6 mm thick CR
The lateral forces on the interlock elements due to the filler material are very small
the interlock was tested under a direct pull. A typical isochromatic pattern
for the interlock can be obtain. From isochromatics, the boundary stresses are
computed. The typical distribution of the boundary stresses for one of the nine cases
Fig (6) Tangential boundary stress distribution (stress concentration factor at
is 2.6 and at Y 2.5 based on average tension in web)
elastic analysis enabled the determination of: (i) the most highly stressed zones
d the corresponding stress concentration factors, (ii) the points of contact, and (iii)
the direction of the resultant force at the point of contact.
(b) Reinforcement Design Around Opening in Mass Concrete Structures
A large number of openings, such as galleries, sluiceways of shafts are required to be
constructed in all modern high dams. These introduce local disturbances in the
proposed to be used for the construction of the Farakka barrage in West Bengal, India.
roved very useful in elucidating the
fundamental mechanism of structural action of the interlock and there by in arriving at
a procedure for the development of an optimum section satisfying the functional and
f nine different designs were tested reproducing half the
section of each pile. The models were fabricated from 6 mm thick CR-39sheets.
The lateral forces on the interlock elements due to the filler material are very small
the interlock was tested under a direct pull. A typical isochromatic pattern
for the interlock can be obtain. From isochromatics, the boundary stresses are
computed. The typical distribution of the boundary stresses for one of the nine cases
Fig (6) Tangential boundary stress distribution (stress concentration factor at
2.5 based on average tension in web)
elastic analysis enabled the determination of: (i) the most highly stressed zones
d the corresponding stress concentration factors, (ii) the points of contact, and (iii)
(b) Reinforcement Design Around Opening in Mass Concrete Structures
galleries, sluiceways of shafts are required to be
constructed in all modern high dams. These introduce local disturbances in the

11. 11
general stress field resulting in high stress concentrations at some locations and
reversal of stress, from compressive to tensile, at others. A typical isochromatic
pattern around a gallery is shown can be obtain. The stress distribution and stress
concentration can be easily determined from these isochromatics. For the design of
openings in mass concrete however, as the concrete is not designed to take tension, it
is the total tension T across a critical section which is important for design. The
tension T across any section is defined by
T = σn ds
Where σn is the tensile stress normal to the section and the integral is taken from the
boundary (where generally the maximum stress occurs) to the point where the tensile
stress vanishes. For the determination of σn along the critical section, it is necessary to
determine the sum of the principal stresses by one of the standard methods and then
determine the principal stresses. The photo-elastic interferometer has also been used
for this purpose.
The United States Bureau of Reclamation arrived at valuable design data for a
standard circular roof gallery 5' X 8' (0.152 m x 0.244 m) by photo-elastic analysis
using the photo-elastic interferometer. The data enable the evaluation of the total
tension T at various sections for any given average stress in the dam in the absence of
such a gallery. The designers, therefore, tended to adopt this shape even if the
functional requirements and ease in construction indicated other shapes. Thus for the
226m high Bhakra dam in the Punjab, India, for increasing the head room for grouting
and drilling equipment as well as for introducing simplicity in framework rectangular
galleries were preferred. These were designed on the basis of photo-elastic analysis.
Similar shapes were adopted for Koyna dam Hirakud dam and Nagarjunasagar dam,
all in India.
A series of photo-elastic tests were conducted using Columbia resin as the model
material. Opening of maximum dimensions 30mmwere made on sheets of size
350 mm x 210mm and 6 mm thick. The load was applied by a special lever-and-link
arrangement, giving uniform tension in the central zone. Isochromatic fringe orders
were measured by Tardy's method and the sum of the principal stresses were
measured by a lateral extensometer enabling the separation of the principal stresses.
The general results obtained for the uni-axial stress field are given in Fig (7). The
theoretical results obtained for various elliptical openings are combined with the

12. experimental data and the resulting chart enables the designer to select the most
suitable shape at a glance. Similar charts for the bi
prepared. If
rectangular gallery with known
thickness is given by A
steel, p the uniform vertical compressive principal stress, w the width and h the height
of opening.
Fig (7) Total tensile force across critical sections for galleries of various shapes
12
experimental data and the resulting chart enables the designer to select the most
uitable shape at a glance. Similar charts for the bi-axial stress have also been
prepared. If k denotes the value of T/pW read from these charts for any
rectangular gallery with known h/w and d/w ratio, the area of tensile steel per
thickness is given by As = k (pw/σs) where σs is the allowable stress in
steel, p the uniform vertical compressive principal stress, w the width and h the height
Fig (7) Total tensile force across critical sections for galleries of various shapes
experimental data and the resulting chart enables the designer to select the most
axial stress have also been
from these charts for any
ratio, the area of tensile steel per unit
is the allowable stress in
steel, p the uniform vertical compressive principal stress, w the width and h the height
Fig (7) Total tensile force across critical sections for galleries of various shapes

13. 13
PROPERTIES OF AN IDEAL PHOTOELASTIC MATERIAL
The development of photoelasticity for the analysis of problems has been due to the
identification of a suitable material for models from among scores of plastics
available in the market. Therefore, selection of a suitable material for preparation of
photo-elastic models forms an important step in photo-elastic analysis. The plastics
available exhibit various properties and selection depends upon the specific
requirements of a given situation. It is difficult to find a material that satisfies all the
requirements. However, some of the important properties of an ideal photo-elastic
material are classified as follows:
(a) Optical properties: Transparency
Sensitivity
Linearity
(b) Structural properties: Isotropy
Homogeneity
(c) Mechanical properties: Modulus of elasticity
Poisson's ratio
Ultimate strength
Yield limit
Creep
(d) Thermal properties: Temperature sensitivity
Exothermic reaction
Thermal conductivity
(e) Production properties: Castability
Machinability
Shrinkage stresses
(f) Environmental properties: Curing period
Time-edge effect
A photo-elastic model material has to be necessarily transparent if not crystal
clear. Since most of the plastics are transparent, difficulty does not arise in adherence
to this basic requirement. The photo-elastic material must have high sensitivity in
order that a large number of fringes are observed for low levels of loads on the model.
Some photo-elastic materials with high sensitivity like urethane rubber and gelatin are

14. 14
used for the study of stresses developed due to self-weight and for problems of stress-
wave propagation. However, when photo elastic analysis is required to be carried out
for cases where body force is not of consequence and rigidity is desirable, a photo-
elastic material with a relatively low sensitivity may be selected. Since the results of
photo-elastic analysis are to be transferred to the prototype, a linearity should exist
between the stress levels and the order of fringes in photo-elastic models. Hence,
while using polymers, the loading on the model must be such that number of fringes
should not be too many, this anyway may not necessarily come in the way of
sensitivity of photo-elastic data, since fringe multiplication and sharpening techniques
or compensation techniques can be used along with the other techniques to improve
the sensitivity.
To conform with the basic assumption in mechanics of solids, the photo-
elastic material must be isotropic and homogeneous. This property is normally met
with as the molecular chains of polymer are randomly oriented. A photo-elastic
material should not distort too much under load so as to change the geometry of the
model. This makes the photo-elastic results inaccurate. Especially in the analysis of
stresses in the neighborhood of cracks, the distortion should not alter the geometry at
the crack tip. Therefore, a material with a high modulus of elasticity should preferably
be used. The loading on the model should not be too large for the stress level to cross
the yield limit since in this region- elastic relations are no longer applicable. Even if
the material is used for photo-elasticity, the stress level should not reach the ultimate
strength resulting in breaking of the photo-elastic model. Use of material with high
yield and ultimate levels is recommended if a material with lower sensitivity is to be
used for analysis. To eliminate the uncertain influence of Poisson's ratio on the stress
distribution, it is suggested that the Poisson's ratio of the model and prototype be as
close as possible. Most of the materials used for photo-elastic analysis are visco-
elastic and therefore exhibit mechanical and optical creep. The best way to overcome
this effect would be to carry out the analysis immediately after loading. Alternately, a
curve of material fringe value with time has to be obtained. This curve must be used
during photo-elastic analysis, when the analysis exceeds a limited time. In most static
tests, investigations are generally carried out after sufficient time has elapsed beyond
the creep process.
The material should exhibit a favourably constant material fringe value over a
wide range of temperature variation in the vicinity of room temperature. However, at

15. 15
the critical temperature of the material, the material fringe value drops down
considerably, but it should remain constant above this temperature. Further the
exothermic reaction during mixing and setting processes should low enough to obtain
stress-free casting.
The material must just have the required flowability in order that castings of
complicated shapes can be made. The castings must be capable of being machined
and should therefore be capable of being filed, turned, milled, ground and if required,
bored, in order that a model of complex nature can be prepared. The material should
be free of shrinkage stresses, which might creep in during casting and machining
operations. These stresses distort the true fringe pattern, sometimes to such an extent
as to yield a wrong solution. If the material selected is susceptible to formation of
these stresses, the fringe pattern corresponding to no-load condition must be recorded
for analysis.
When a photo elastic model is exposed to atmosphere, water vapour from the
environment diffuses into the model near the boundaries. This results in the formation
of fringes that are parallel to the boundaries. The stresses that are responsible for the
formation of these fringes are called time-edge stresses and the effect is called the
time-edge effect. The material selected should exhibit the time-edge effect of a very
low order. Two-dimensional photo-elastic models or slices from a three-dimensional
model should be analysed immediately to avoid formation of fringes due to the time-
edge effect. Alternately, it is suggested that the slices be either stored in a desiccator
or an incubator at a temperature slightly above the room temperature.
The cost of photo-elastic model material should be as low as possible,
Properties of some of the photo-elastic materials are listed above. While no definite
rules exist as to the selection of a photo-elastic model material, a material that best
suits the loading condition, complexity of the model shape and nature of analysis,
should be chosen.

16. 16
Problems
1. A fringe order of 3.0 was observed at a point in a stresses model with a light
having wave length of 598nm. Assuming that the velocity of propagation
remains the same. What would be the order of the fringe at the same point
with a light having a wave length of 548nm?
2. A fringe of order 2.5 was observed at a point in a 2D photo-elastic model with
a light having a wave length of 598nm. Assuming that the stress optic
coefficient ‘c’ remains a constant. What would be the order of the fringe at the
same point if a light of wave length 542nm is to be used?
3. The fringe order observed at a point in a stressed model is 3.45 with mercury
light (=548.1 nm). The material fringe constant in tension is 20kn/m. if the
model has a thickness of 0.6 cm, calculate the maximum shear stress at the
point.
4. The material fringe constant in tension for a certain photo-elastic model is 18
kN/m. when calibrated with sodium light (=548.1 nm). The model under
investigation has a thickness of 6mm. If the model is observed with mercury
with light (=548.1 nm) and the stress 1-2 at a point is 18KPa. What fringe
order will be observed? Assume that ‘c’ is independent of .