Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 4, pp. 847-858, Warsaw 2015
DOI: 10.15632/jtam-pl.53.4.847

ANALYZING SQUARE PLATE IN DIAGONAL COMPRESSION USING
BELTRAMI-MICHELL METHODOLOGY

Djelloul Rezini
LCGE Laboratory, USTO – Mohamed Boudiaf University of Oran, Algeria

e-mail: redjellou@yahoo.fr

Abdelkrim Khaldi
LRTTFC Laboratory, USTO – Mohamed Boudiaf University of Oran, Algeria

Kouider Bouzid
Metallurgy Department, USTO – Mohamed Boudiaf University of Oran, Algeria

Yahia Rahmani
LRTTFC Laboratory, USTO – Mohamed Boudiaf University of Oran, Algeria

This paper is concerned with the “Study of photoelasticity and a photoelastic theoretical
investigation of the stress distribution in Square Blocks subjected to concentrated diagonal
loads”, a thesis topic by M.M. Frocht who developed the well-known semi empirical Shear
Stress Difference method. Indeed, the use of the Beltrami-Michell methodology remains
quick, when complemented by photoelasticity to acquireDirichlet’s conditions. The synergy
of bothmethods is enhancedwith the use of the finite differencemethod. In addition, a finite
element analysis has provided results that will be a supplementary reference for validation.
The results obtained have been of lower cost than those obtained by Frocht.

Keywords: photoelasticity, hybrid method, Beltrami-Michell, square-plate, diagonal com-
pression

1. Introduction

Theobjective of thispaper is topresentanalternative solution to the stressdistribution in square
blocks to that presented in Frocht’s thesis entitled “Study of photoelasticity and a photoelastic
theoretical investigation of the stress distribution in Square Blocks subjected to concentrated
diagonal loads” (Frocht, 1931). Furthermore, some features of the stress distribution are exami-
ned by both techniques. We will show, moreover, maps of the whole field of each of the stress
components individually, which the method by Frocht cannot provide. In addition, the finite
element analysis has provided results that will be a supplementary reference for the validation.
Stressanalysis is a classical subject in thefieldof elasticity theory.Exact solutions topractical

problems are not easily determined because of the exact incorporation of the associated physical
conditions. The usually used displacement approach degrades the order of the approximation,
especially when the state of stress is of principal interest. Therefore, in the alternative direct
formulation in terms of stress, the tensor stress field is used as the principal unknown variable
(Boresi et al., 2011). This technique is associated with the elliptic equation of Beltrami-Michell
forming a well-posed Dirichlet’s problem which can be partitioned into two parts; the first
approach provides a harmonic field function subject to certain boundary conditions, while the
second one investigates the behavior of this function in the neighbourhood of the boundary.
Actually, a correct treatment of boundary conditions is one of themajor obstacles to the reliable
solution to practical problems.



848 D. Rezini et al.

Although, photoelasticity provides an incomplete solution within the field, nevertheless it
always gives a complete one on the boundaries (Frocht, 1941). Isoclinic information will not
be needed here, since the isochromatics on the boundaries is the sufficient information (Rezini,
1984) and will be thoroughly used for the proposed stress analysis. However, enhanced with
the use of Finite Difference Method (FDM), the aim of this paper is to propose a photoelastic-
-numerical hybrid method for stress analysis within any irregularly shaped domain in plane
stress conditions. Thus, the Square Block will be an appropriate application for the validation
of the present model.

2. Elastic plane stress boundary value problem

The plane stress problem considered is referred to as an elastic planar layer subjected to sym-
metric in-plane loading in the case of the diagonal compressed Square Block. Given the plane
stress assumptions, this problem is firstly governed by the necessary equilibrium conditions. As
a result of the forces applied at the boundary, the stress tensor σ, a symmetric rank two tensor
field, satisfies the following equilibrium condition either in the absence or for constant body
forces for each i

2∑

j=1

∂σij
∂xj
=0 (2.1)

Moreover, the plane stress problem is defined as the elastic planar layer Ω on the stress as-
sumption that (σ33 = σ13 = σ23 = 0); (x1,x2,x3) ∈ Ω. Furthermore, the stress field in the
middle layer is reasonably described by the three field components σ11(x1,x2), σ22(x1,x2), and
σ12(x1,x2) which form the symmetrical stress tensor, of which the trace is expressed as follows

trace(σ)=Σσ(x1,x2)= σ11+σ22 (2.2)

In termsof stress, the invariantΣσ(x1,x2) has the fundamentalproperties of aharmonic function
that satisfies the Laplace-potential equation. In fact, if the Laplace operator is applied to the
trace of the stress tensor given in (2.2), thewell-known Laplace potential equation for the stress
sum is obtained

2∑

j=1

∂2Σσ(x1,x2)
∂x2j

=0 (2.3)

Equation (2.3) as stated is the compatibility condition in termsof stress for planar version.When
equilibrium equations (2.1) are derived in a suitable form and subtracted from each other, the
following equality in terms of the second order derivatives will result

∂2σ11
∂x21

=
∂2σ22
∂x22

(2.4)

In a similarway, by addingboth equations, the determiningPoisson equation for the shear stress
is obtained, and is given as follows

∆σ12 =−
∂2(σ11+σ22)

∂x1∂x2
(2.5)

where ∆ is referred to as the two-dimensional Laplace operator. Laplace’s potential equation
(2.3) may be rewritten in the extended form shown below

∆σ11 =−
∂2σ22
∂x21

−
∂2σ22
∂x22

(2.6)



Analyzing square plate in diagonal compression... 849

The incommoding member in equation (2.6) will be replaced by its equivalent one given in
equation (2.4), so that an equation of the Poisson type for the normal stress component is
obtained

∆σ11 =−
∂2(σ11+σ22)

∂x21
(2.7)

In a cyclic permutation, the equation for the normal stress in the other direction is established
likewise

∆σ22 =−
∂2(σ11+σ22)

∂x22
(2.8)

Equations (2.5), (2.7) and (2.8) are thewell-knownBeltrami-Michell equations;with the require-
ment that the potential equation solution of the sum of stresses given in (2.3) at the stress field
must be available. Other formulations shown in the literature may lead to the same equations
(Hetnarski and Ignaczak, 2011). Furthermore, it is essential, however, to notice the fact that
theBeltrami-Michell equations are independent of elastic constants for the two-dimensional case
(Muskhelishvili, 1975).

3. Photoelasticity treatment of the problem

Thephotoelastic experiment (Ramesh, 2008; Frocht, 1941, 1948) provides only two experimental
parameters for each point: the isochromatic fringe order N which is proportional to (σ1 −σ2)
and θ, the isoclinic parameter.However, three characteristics are needed to fully define the stress
state. Photoelasticity provides therefore an incomplete solution as is symbolically sketched below

(σ11,σ22,σ12)⇐⇒
{
N isochromatic order

θ isoclinic parameter

Both the two-dimensional basic equation of photoelasticity and the procedure arewell documen-
ted in the literature (Sharafutdinov, 2012). Photoelasticity and the SSD-method form a single
semi empirical hybrid method which calculates stress components within a grid starting from
initial values at the boundary points. The SSD-method is based on the step-by-step integration
of one of the differential equations of equilibrium. In aCartesian (x,y) plane, for any particular
y-ordinate, equation (2.1) can be integrated in the x-direction

σx(x1)= σx(x0)−
x1∫

x0

∂τxy
∂y

dx (3.1)

where the partial derivative ∂τxy/∂y is a function of the x-coordinate. In practice, τxy is deter-
mined by photoelasticity from the isochromatic fringe order N and the isoclinic parameter θ
viewed in the (x,y)-plane (Fig. 1) using the following photoelasticity law

τxy =
Nf

2t
sin2θ (3.2)

where f and t are the material constant and thickness of the model, respectively. When the
loaded Square Block model is viewed in the polariscope, a fringe pattern called isochromatic is
apparent.
The Square Block photoelastic data are shown in Fig. 1 where the stress patterns are on the

right-hand side, and the corresponding isoclinics are on the left-hand side. The stress separation



850 D. Rezini et al.

Fig. 1. Isoclinics and isochromatic fringes data of the diagonally compressed Square Block

will be carried out along the straight line, indicated by X. Along the horizontal line y = const,
for the SSD-method, the central-difference between 2 auxiliary lines parallel to y at a distance
of ∆y, the partial differential quotient of the shear stress is approximated as follows

∂τxy
∂y
∼=
∆τxy
∆y

(3.3)

Substituting expression (3.3) into equation (3.1) andusing elementary summation along the grid
nodes in the photoelasticity law, equation (3.2) leads to the following separated stresses both in
the x and y-directions

σy
∣∣
i
= σy

∣∣
0
+

i∑

k=1

(∆τ
∆x

∆y
)

k
σx
∣∣
i
= σx

∣∣
0
+

i∑

k=1

(∆τ
∆y

∆x
)

k
(3.4)

It is shown that even if the error introduced by themeasurement of the isochromatic parameter
may not be important, the error due to the isoclinic (stress direction)measurement can be very
significant (Kuske, 1959, 1971). Since the SSD-method involves step-by-step integration, the
error introduced will be cumulative.
Concerning the boundary values, the state of stress is most evident on the free boundaries.

At each point of the free boundary either σ1 or σ2 vanishes; the stress fringe pattern gives
immediately the numerical value of the remaining principal stress. In addition, the sign of the
boundary stress can be determined by considering the external loading or by “nail” pressure on
the boundary (Kuske and Robertson, 1974). The required boundary potentials arise automati-
cally from photoelastic study of the specimen in which the principal stress sum can be obtained
readily because they are identical to the principal stress difference on the free boundary (Mangal
and Ramesh, 1999; Petrucci, 1997; Pinit and Umezaki, 2007), from which the equality of their
absolute values at the boundary is deduced

σB =
∣∣∣(σ1+σ2)∂Ω

∣∣∣ =
∣∣∣(σ1−σ2)∂Ω

∣∣∣ (3.5)

Besides, the harmonic state of the stress invariant satisfies Laplace equation (2.3); and it is
usually used to determine the interior potential of the stress sumwith the necessary knowledge
of boundary conditions (Fernàndez et al., 2010). The accuracy of the boundary values of the
stress sum is absolutely decisive for the results. This is true because the stress sum occurs each
time as a source function of the set of the Beltrami-Michell equations. According to the law of
photoelasticity, the stress sum is proportional to the fringe order on the free boundaries

(σx +σy)
∣∣
∂Ω
= κN(x,y)

∣∣
∂Ω

(3.6)



Analyzing square plate in diagonal compression... 851

Knowing the inclination angle θ (or the slope) at a given point on the model edge and the
tangential stress σB = κN(x,y) at the same point, the Cartesian stresses components are easily
computed using separately the equilibrium conditions as follows

σx = σB cos
2θ σy = σB sin

2θ τxy =−σB sinθcosθ (3.7)

Commonly, only one black-and-white photograph of the boundary-isochromatic fringe patterns
of the model for the input data is needed. The simplest way of fringe data recording is to use a
digitizing tablet and to “click” (pick up) on the centre of each fringe at a density that adequately
represents the fringe centreline (such as Gauss distribution) on the breadth (Rezini, 1984).
This simple acquisition technique has the significant advantage of allowing the user maximum
control of the number and location data. Per boundary, the coordinates of the boundary points
are recorded along with their isochromatic values and tabulated in an array (xi,yi,Ni), where
i =1, . . . ,n labelling each recorded point in this array. It is also all of the entire work necessary
for the data inputprocedure. It is to be pointed out that thematrices of each systemof the finite
difference algebraic equations will be automatically occupied, taking into account the recorded
boundaryvalues, asdescribedabove. In thenext step, aprocessing subprogram(solver) is started
in order to perform the numerical analysis. An output program shall be provided.

4. Finite difference implementation on the Square Block

Due to its definition, the FDM is the most important and certainly still dominant numerical
method in the partial differential equations modelling. In order to approximate the solution of
a partial differential equation by its nodal values at a set of grid points, the FDMoften requires
themesh points to be uniformly placed, so that the derivatives of the unknown function at grid
points can be accurately approximated. Difficulties principally occur in the FDMdiscretization
of the homogeneous and mixed derivatives at irregular shaped boundaries. To overcome this
restriction and in order to solve partial differential equations that are defined on domains with
irregular geometry, numerous variants are proposed in several works (Collatz, 1960; McKenney
et al., 1996). Many procedured exist on this subject; e.g. McKenney et al. (1996) propose a fast
Poisson solver for complex geometries. In this way, Zhang (1998) developed amultigrid solution
to Poisson’s equation. In the present implementation, a flexible 5-point stencil is systematically
used for the discretization of the homogeneous and the mixed partial derivatives occurring in
both Laplace equation (2.3) and in the right-hand side of Poisson equations (2.5), (2.7) and
(2.8).

Fig. 2. Unequal armed difference star: (a) normal grid, (b) skewed grid

Consider the general case of a group of five points, of which spacing is non-uniform, arranged
in an unequal-armed star.We represent each distance by αih, where αi represent the respective



852 D. Rezini et al.

fractions of the standard spacing h from the discretization central point as shown in Fig. 2a.
Based on Taylor’s theorem, the derivative of a function at a point can be approximated by the
linear combination of function values at nearby points including the function itself (Dahlquist
andBjorck, 1974; Smith, 1985). Thedefinition of difference quotients and of the discrete Laplace
operator also has to bemodified.This leads to the Shortley-Weller approximation (Shortley and
Weller, 1938)

∆φ
∣∣
h
= C+0 Φ0+C

+
1 Φ1+C

+
2 Φ2+C

+
3 Φ3+C

+
6 4Φ4 =

{
0 Laplace

f
∣∣
k=0

Poisson
(4.1)

In this symbolic equation, (4.1), the coefficients of the finite difference equations are defined
depending on the spacing to the boundaries. The Shortley-Weller coefficients for the discrete
Laplace operator ∆|h will be distinctly expressed as follows

C+1 =
1

h2α1(α1+α3)
C+2 =

1
h2α2(α2+α4)

C+3 =
1

h2α3(α3+α1)
C+4 =

1
h2α4(α4+α2)

C+0 =−
4∑

k=1

C+k

(4.2)

The motivation of the following approach is to produce an accurate method for treating
mixed derivatives. Near the boundary, the approximation of the second order mixed derivatives
requires a 9-point stencil. This task is not easy to handle with respect to the aimed “automati-
zation” of themethod.A transformation by rotation 45◦ of the grid (stencil) to obtain a 5-point
formula is also suggested. Notice that the 2-dimensional Taylor series approach, according to
the approximation of mixed derivatives, leads to a similar form of Shortley-Weller’s coefficients
in a 45◦ skewed stencil as shown in Fig. 2b

∂2φ

∂x∂y

∣∣∣
h
= C×0 Φ0+C

×
1 Φ1+C

×
2 Φ2+C

×
3 Φ3+C

×
4 Φ4 (4.3)

The finite difference coefficients for mixed derivates (4.3) can be expressed in individual terms
as follows

C×1 =
1

2h2β1(β1+β3)
C×2 =

−1
2h2β2(β2+β4)

C×3 =
1

2h2β3(β3+β1)
C×4 =

−1
2h2β4(β4+β2)

C×0 =−
4∑

k=1

C×k

(4.4)

The above symbols (+,×) are used for normal and skewed network stencil, respectively. By
definition, the irregular 5-point-difference star is present when at least one of its star boughs
(arms) becomes smaller than the mesh size h. The irregular difference star is the most general
case and for the regular ones, the following condition is required αk = βk =1, k ∈{1,2,3,4}.
Furthermore, in this study, the direct method which produces an exact solution in a finite

number of operations is used (Duff et al., 1986). When solving successively the Laplace and the
Poisson systems of equations, the errors in the solutions arise only from the rounding off and
truncation errors in the computational calculation (Smith, 1985; Collatz, 1960). This method is
tested on several application examples and is found to produce very good results (Rezini, 1984).



Analyzing square plate in diagonal compression... 853

5. Results of methods and discussion

Before the PCs had appeared, the SSD-methodwas done bymanual calculations. One imagines
the excess of labour to investigate the SquareBlock problem.However, becausewe already have
the results of Frocht (1948, pp. 265-269), the feasibility of this method will be proved on this
application example. Isochromatics (in orders) have been superimposed for convenience over the
isoclinic parameters in the field around the straight line X, as depicted in Fig. 1. The stress-
integration can be started from the initial values, and the stress-separationwill be accomplished
according to numerical integrations (3.4), as mentioned previously. The initial stress values are
obtained from the boundary values at a point from conditions (3.7). For the stress-separation
along the straight line, instead of using relation (3.4), Frocht proceeds as follows

σx
∣∣
i
= σx

∣∣
0
−

i∑

0

∆τxy
(∆x
∆y

)

σy = σx ±
√
(σ1−σ2)2−4τ2xy

(5.1)

The plus orminus signs in front of the radical in (5.1)1 correspond to the two possible positions
of the centre of Mohr’s circle. Accordingly, σ1 lies furthermost on the right and σ2, being the
furthermost, on the left of the circle.
The initial value of σx|0 for the step-by-step integration is evaluated from the boundary

value of q0 (fringe order negative) and θ0, which are 0.80fringe and 45◦, respectively. The initial
stresses in both the x direction and the y direction get the following respective values

σx
∣∣
0
= q0cos

245◦ =−0.8 ·0.5=−0.40fringe (5.2)

and

σy
∣∣
0
= q0 sin

245◦ =−0.8 ·0.5=−0.40fringe (5.3)

The initial shear stress value becomes

τxy
∣∣
0
=−q0 sin45◦ cos45◦ =−

q0
2
=+0.8 ·0.5=+0.40fringe (5.4)

In this case, it is important to point out the difference between both procedures: in Frocht’s
SSD-process, the stress component σy is determined bymeans ofMohr’s circle statement (5.1)2;
while the isoclinic parameter θ is used for the integration.The cumulative error in the integration
process has also an effect on the stress-separation. It is noted that the determination of θ
isoclinic parameter in its physical range remains, generally, adifficultproblem(Fernàndez, 2011).
Several works have been published concerning this topic; see for example (Mangal andRamesh,
1999; Siegmann et al., 2011; Ajovalasit and Zuccarello, 2000). Besides, in the Beltrami-Michell
BoundaryValue Problem (B-MBVP), each stress component being subject to its values on the
boundary formaBVP individually. In addition, it is noted that the SquareBlock edges direction
(slope) is a principal stress direction itself in the case of a free boundary. Otherwise, this shows
the long process in achieving the results by means of the SSD-method (see diagram; Fig. 3).
Mainly, the fact it is noticed that the separation of stresses is established along a straight line
only.
The need for full-field separation of stresses ismore advantageous, and this is whyB-MBVP

ismore appropriate. In order tomake full comparisons, the same representation as that of Frocht
(Fig. 3) is adapted.
In order to re-examine the stress distribution for further validation, a numerical analysis has

been carried out using the finite element analysis for the same Square Block loading situation.



854 D. Rezini et al.

Fig. 3. Cartesian stress components distributions on section X (Frocht, 1948, pp. 265-269)

Fig. 4. Cartesian stress component distributions on section X (referring to this work)

The FEA has been performed using ANSYS software. PLANE82 elements have been used to
model the Square Block subjected to diagonal compression. This type of element provides more
accurate results formixed (quadrilateral-triangular) automaticmeshes and can tolerate irregular
shapes without as much loss of accuracy. Such 8-node elements have compatible displacement
shapes and are well suited to model curved boundaries (Madenci andGuven, 2005).
After preliminary tests, the meshing configuration shown in Fig. 5 has been opted for; a

number of 3491 PLANE82-elements (10720 Nodes) have been used.
The static balance across section X is then stated when the sum of vertical forces vanishes.

This condition of equilibrium in the loading direction provides the necessary (check) verifica-
tion of each method used, permitting the comparison of the numerical results. All geometrical
dimensions and the load necessary for the result verification are shown in Fig. 5. The units are
also kept Anglo-American. Furthermore, the stress distributions along the straight line X will
be comparedwith those of the SSD-method and of FEA.Accordingly, the equilibriumacross the
horizontal section may be re-examined. The normal stress σy acting within each cross section
must balance the load Py each time. The area under the σy-curve (Fig. 3) has been measured
bymeans of a planimeter and found to be 11.2 in2. The length of the half section 0.6h has been
found to be 4.065in (Frocht, 1941, pp. 270). Hence

σy =
11.2
4.065

=2.75in=2.2fringes



Analyzing square plate in diagonal compression... 855

Fig. 5. Meshing configuration of the Square Block and associated boundary conditions

Fig. 6. Cartesian stress component distributions on section X using FEA

Multiplying this value by themodel compressive fringe value of 284psi, we have: σy =2.2·284=
625psi. The force Py acting on the section of length 0.6h is then given by: Py = 625 · 1.855 ·
0.255·0.6=177.5lb. This result differs from the applied load of 180lb approximately by−1.5%.
Concerning our results, the integral of theσy-curve (Fig. 6) is calculated. By taking into account
our scale, the following result is obtained: σy =2.18 ·284=619psi. Hence, the calculated force
resulting from our calculation method is: Py =619 ·1.855 ·0.255 ·0.6=175.7lb.
Therefore, the error induced is approximately−2.4%. This discrepancy is not a crucial one.

The reasons leading to this errormagnitudewill be discussed later. In contrast, the FEA results
are the most accurate (see Table 1) in this analysis.

Table 1.Comparison of present results with Frocht’s and FEA

Result of Py [lb] Error [%]

Frocht 177.50 −1.5
FEA 179.75 −0.1
this work 175.70 −2.4

Referring to the available experimental results, those achieved by the photoelastic-numerical
hybrid method are in good agreement with a relative difference of about −1%. Furthermore,
it is worth emphasizing that the occurring small deviations in the results obtained are due
to the procedure of the graphical representation rather than to inaccuracy in the calculation.



856 D. Rezini et al.

Moreover, the component stress distribution is automatically interpolated at the grid points
in the neighbourhood of the straight line X, which explains once more the deviation of the
results. On the one hand, the designated lines along which the results will be compared cannot
be identical in the absolute. Moreover, there is a significant difference in terms of the process
used to expose stress distributionmaps in the reference work (Fig. 4) likewise for FEA (Fig. 6).
Separating the result data in individual components usually provides stresses at only discrete
locations. For this purpose, the field results are presented in Figs. 7 and 8.

Fig. 7. Isopachic stress patterns; LHS calculated and RHSmeasured

Fig. 8. Full-field distributions of each stress

6. Conclusion

In the present study, validation of the stress state results obtained by two comparativemethods
has been carried out using supplementary FEAnumericalmodels. A very good qualitative agre-
ement between the three methods has been found. As treated by Frocht, the true whole field
isoclinic angle and phase difference are the key data to obtain the stress distribution in a bire-
fringent model. The feasibility of this method has been proved by the Square Block experiment
under in-plane compression, wherein only boundary isochromatic fringe data are needed. Fur-
thermore, the developed SSD-method by Frocht still provides accurate results. The aim of this
paper is to showthe efficiency of anadditionalmethod for stress-separationwithoutneedingfield
information as in the case of the SSD-method; and to quickly obtain results with an acceptable
accuracy. Based on the numerical solving of theB-MBVPofDirichlet’s type, thismethod redu-
ces the photoelastic data to the minimum, in order to separate the prevailing stresses. Further,
the method facilitates the automatic process for generating stress component maps (Ajovalasit
and Zuccarello, 1998) and reconstructing the full-field stress tensor using little equipment. In



Analyzing square plate in diagonal compression... 857

regard to large utilization in the experimental-numerical photoelasticity, the next step will be
to develop an automatic system that allows the computer to control both the image acquisition
of boundary isochromatic fringes and the shape recognition simultaneously. In fact, the present
paper has shown that the successful use of a hybrid method can thereby reduce the amount of
measurements required, increase the accuracy of numerical results and, at the same time, allow
quick determination of the complete stress distribution within the studied part.

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Manuscript received October 7, 2013; accepted for print April 17, 2015