Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 4, pp. 1003-1012, Warsaw 2013

STRESS CONCENTRATION ON THE CONTOUR OF A PLATE OPENING:

ANALYTICAL, NUMERICAL AND EXPERIMENTAL APPROACH

Nikola Momcilovic, Milorad Motok, Tasko Maneski

University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Serbia

e-mail: nmomcilovic@mas.bg.ac.rs; mmotok@mas.bg.ac.rs; tmaneski@mas.bg.ac.rs

The objective of the paper is to analyze the stress concentration factor (SCF) in the corner
of a rectangular plate opening with small radii of curvature and various methods for its
derivation. Besides the finite element method, as the most used approach today, there are
some analytical and experimental procedures that can obtain stress concentration results in
such spots. An analytical approach can deliver prediction of the SCF around the corner but
cannot illustrate the stress field opposite to the finite element method. On the other hand,
finite element analysis needs much computation time to deal with very sensitive and fine
mesh generation around concentration zones. Experimental devices, such as strain gauges
are not able to performmeasurements on areas where high gradient of stress occurs due to
their lack of sensitiveness and larger dimensions compared to the measured part of a struc-
ture. The paper presents Digital Image Correlation (DIC) technique that obtains not only
stress concentration,where other devices fail, but also provides full displacement, strain and
stress field’s even where a high gradient of stress exists. These threemethods are discussed,
compared and illustrated on the model of a plate with rectangular opening subjected to
tension.

Keywords: stress concentration factor (SCF), digital image correlation(DIC), plate opening,
small radius of curvature

1. Introduction

Plate openingswith some kind of rounded corners are inevitable in structures, whethermade by
amanufacturingprocess or for the reasonof decreasing stress concentration. Stress concentration
in zoneswitha small radiusof curvatureonacontour of theplate openingseemsavery significant
issue in strength problems (Kopecki, 2011; Laughalam et al., 2011; Rezaeepazhand and Jafari,
2010;Motok, 1997), especially in ship structures suchasdeckopenings, scallops, aircraft fuselages
or tail planes. The presence of an openingmakes even a simple structure model complicated to
analyze regardless of the method being used. In order to deal with this problem, three possible
approaches appear: analytical, numerical and experimental. Every one of them has advantages
and issues, although their ongoing techniques are constantly being updated andmodernized.

Stress concentration in a plate under tension and weakened by a hole was recognized long
time ago, and basic analytical solutions were presented by Muskhelishvilli (1953) and further
utilized by Savin (1961). More recently, some other solutions were derived (Laughalam et al.,
2011; Motok, 1997). The main problem of the analytical approach seems to be the difficulty
to analytically express the contour of a rectangular opening with a small and variable corner
radius.

As far as numerical methods are concerned, a very fine mesh is needed in order to obtain
realistic stress results around the critical zone. Although nowadays modern finite element ba-
sed software has possibility to analyze models with large degrees of freedom that significantly
increases the computation time.



1004 N.Momcilovic et al.

Experimentalmethods remain themost confidentmethods tomeasure real behavior of struc-
tures with openings (Maneski andMilosevic-Mitic, 2010; Guo et al., 2009; Toubal et al., 2005).
Nevertheless, it is very hard (almost impossible) to “catch” stress concentration by means of
classical measuring devices (strain gauges, inductive sensors) where a large gradient exists. For-
tunately, some “state of the art” experimental nondestructive methods, like Digital Image Cor-
relationMethod –DIC (Sutton, 2008; Hild andRoux, 2006; Lecompte et al., 2006), holographic
interferometry, speckle interferometry,Moiré interferometry are available today.Theholographic
and speckle interferometry are used for displacements and shapemeasurement of optically rough
surfaces at sensitivity of the order of the wavelength of light. In the holographic interferometry
(Kreis, 2005; Jones and Wykes, 1989), three-dimensional information about the surface of an
object is recorded by a hologram. Changed geometry of the surface, upon being subjected to
load, is compared to the formerly recorded state. Surface changes are observed as a series of
interference fringes superimposed on the image of the object in order to obtain displacements.
The speckle interferometry (Jones andWykes, 1989) is based on a random interference pattern
observed when coherent laser light is scattered from a rough surface. The information about
the object, in Moiré interferometry (Post, 1987), is acquired as interference fringe patterns (or
contour maps) of the displacement fields with high sensitivity. Furthermore, trustworthy strain
distributionsare extracted fromthesepatterns.Thismethod, as oppose to classical inteferometry
that is most effective for out of plane displacements, is widely recognized for obtaining in-plane
displacements. Digital Image Correlation is discussed in Section 4 regarding the experimental
approach being used in the presented measurement.
In this paper, analytical, numerical and experimental techniques that may obtain stress

concentration results for the discussed problem properly are presented. The idea is to illustrate
and compare those methods on the example of a plate weakened by a rectangular opening with
a small corner radius of curvature and subjected to tension.
The analyzed plate (Fig. 1) with overall dimensions of 1000mm×1000mm×2mm is made

of structural steel S235JRG2with themodulus of elasticity E =200000N/mm2, Poisson’s ratio
ν =0.3 and the yield stress σy =240N/mm

2. The plate is weakened by a rectangular opening
(300mm×300mm) with corner radii of 0.5, 1, 2 and 3mm. That allowed exploration pf one
experimental model to provide results for four different corner radii. The plate is subjected to
in-plain tension.

Fig. 1. Analyzed plate



Stress concentration on the contour of a plate opening ... 1005

2. Analytical approach

The formula defined byMotok (1997) is chosen as an analytical procedure to obtain SCF results
for the analyzed model. The algorithm is derived from Muskhesvili’s solution for the problem
of an infinite plate subjected to tension and weakened by a hole the contour of which can be
defined bymeans of conformal mapping (Fig. 2). These solutions were practically applicable for

Fig. 2. Mapping of a unit circle into a square

a narrow spectrum of the corner radius of curvature only. The functions of a complex variable
ω(ξ) for mapping a unit circle into a square, rectangular or hexagonal contours can be derived
from the Schwarz-Christoffel integral. If complete series (i.e. with an infinite number of terms)
resulting from the integral are used, the mapped contours achieve shapes of regular polygons
with straight sides and no curvatures in the corners. By decreasing the number of terms used in
themapping, the curvilinear contours canbeobtained.The radius of curvature on their sides and
in the corners is greater as the number of used terms is smaller. Therefore, using an appropriate
number of terms, the contours of an arbitrary small corner can be analyzed. However, since
the conformal mapping presumes making use of functions of the complex variable, evaluation
of stresses demands using of basic equations of theory of elasticity in a complex form. Derived
final formula (2.1) for SCF in the corner of a square opening of an arbitrary small radius of
curvature still appears to be easy to use (Ci coefficients in Table 1 are improved versions of the
original ones (Motok, 1997)). SCF is taken as the ratio of maximal principal stress in the corner
on the contour and the nominal tension load applied to the plate; a/r is the ratio of the overall
dimension of the opening (a=300mm) and the corner radius

SCF=
10
∑

i=0

Ci
(a

r

)i

(2.1)

Table 1.Ci coefficients

i Ci i Ci

0 3.1002 6 −1.753E-14

1 0.087697 7 1.4196E-17

2 −5.322E-04 8 −7.158E-21

3 2.4609E-06 9 2.0415E-24

4 −7.304E-09 10 −2.514E-28

5 1.4011E-11

Expression (2.1) is simple, ready for instant use and applicable for the presented problem.
The formula is limited to local strength analyses only, but due to its straightforwardness, it can
be especially practicable in quick solving of practical problems.



1006 N.Momcilovic et al.

3. Finite element approach

As a standardmethod incorporated inmodern software engineering tools, the finite elementme-
thod (FEM)presents a powerful numerical device for obtaining the stress field in a concentration
zone of a structure, especially regarding small geometrical discontinuities, such as openings, cut-
outs, various imperfections (Laughalam et al., 2011; Madani et al., 2010; Kopecki, 2010; Guo et
al., 2009; Ukadgaonker and Kakhandki, 2005; Wu and Mu, 2003; Chen, 1993). Still, the need
for an extremely finemesh around corners increases computation time drastically.ModernFEM
based software such as one used for the purpose of this analysis –Komips (Maneski, 2005) works
with an extremely high number of degrees of freedom and is able to generate a large density
mesh around areas of high stress gradient. Yet, the radius of 0.5 or 1mm, such as in this case, is
still a very small dimension (compared to the dimensions of the opening and plate themselves),
and it is not easy to produce a mesh that would illustrate the high stress gradient properly.
Based on several models of the same plate with different mesh density (Fig. 3), it is concluded
that the minimum of eight divisions is needed to describe the radius geometrically and obtain
convergence in the stress concentration field. It means that what seems to be a simple model
of the plate has to bemeshed with a significantly increased number of elements just because of
the presence of these four corner areas. The final number of shell elements in themodel is 8787.

Fig. 3. Finite element model

4. Experimental approach

In this paper, theDigital ImageCorrelation (DIC) is usedas an experimentalmethod todetermi-
ne the stress field and stress concentration factors on each radius. DIC represents a non-contact
“state of the art” technique to measure and acquire full displacement, strain and stress fields
of the structure, which is not possible using classical devices. Strain gauges cannotmeasure the
full field and, however small, they cannot be placed on such a small corner radius (r=0.5mm).
TheDigital ImageCorrelation compares the reference and deformed image by tracking their

same physical points. The reference image is divided into “subsets” that represent a square
matrix of pixels (Hild andRoux, 2006; Stoilov et al., 2012). The digital images are recorded and
processed using an image correlation algorithmwhich is based on the tracking of the grey value
pattern in local subsets (Sutton, 2008). In order to determine a correspondence of each reference
subset to the respective calculated subset in the deformed image, the correlation coefficient is
defined (Lecompte et al., 2006). The software calculates then the average gray scale intensity
over the subset in the reference and deformed images and compares them.
The Aramis system, based on the DIC technique and used in this research, could be one

of the first modern experimental methods that are able to “catch” stress results and full stress
and strain fields in these small areas. The system consists of 3D cameras and a computer unit



Stress concentration on the contour of a plate opening ... 1007

with the Aramis software. Two 3D cameras record the measurement and the computer unit
(incorporated with the Aramis software) processes the results by recording full displacement
fields and calculating strains and stresses (Ghafoori andMotovalli, 2011; Yu andDehmer, 2010;
Eriksen et al., 2010).

Before even starting the measurement, calibration (Fig. 4) of the DIC installation needs to
be performed. The calibration means taking photos of a series of images on an already defined
calibration panel. This procedure defines 3D measuring volume and is extremely important for
themeasurement to be successful. If the parameters such as the camera angle, distance between
cameras and measured object are in the prescribed range, the calibration is supposed to be
performed correctly. Since the whole system is very sensitive, every movement of the cameras,
sensors, lenses or modification of illumination in the laboratory can disrupt the measurement
completely (Fazzini et al., 2010). The measured surface has to be sprayed. The spray creates
visible dots that make contrast between them and the surface. The system is monitoring the
displacement of these dots. The level of contrast is directly linked to the accuracy of the process.
A potential error can also be caused due to limited system resolution, statistical errors induced
by calibration, subpixel effects resulting from limited camera resolutions, intrinsic uncertainties
of the correlation algorithm, etc. (Schreier et al., 2006). During the experiment performed, all
error parameters (such as calibration deviation, defined angles and distance, illumination, etc.)
are kept within the prescribed ranges to acquire more accurate results. Therefore, the obtained
values are to be takenwith the strain inaccuracy up to 0.01%and stress inacurracyup to 20MPa
for the usedmaterial (GOMmbH, 2009).

Fig. 4. Camera calibration

The Aramis system records dots, partitions dots into pixels and the pixels into smaller
units called facets. Smaller facets capture localized effects better and therefore they are more
convenient for a particular experiment of this research. By tracking the facet distance, position
its their relative movement while changing the loads, the Aramis acquires the displacement,
strain and stress fields. After performing the experiment, the DIC system is able to illustrate
the results in form of von Mises stress, principal stresses or in each direction.

The experimental installation (Fig. 5) beside the DIC system, includes an acquiring system
and an additional computer unit to obtain and process the information from the load cell and
additional strain gauges. The strain gauges are placed in the middle of each side of the plate,
symmetrically, with the purpose to prove that the load is subjected correctly (if not, the gauges
would measure different strains). To ensure an equal load distribution along the edge of the
plate, two stiff bars are welded on each side of the plate to transfer the force without deforming



1008 N.Momcilovic et al.

the plate itself. The load is subjected via a hydraulic compressor and cylinder, and increased
gradually. Themaximal subjected load has been kept around 20kN in order to stay in the elastic
area of the material behavior concerning the expected high SCF around the corners.

Fig. 5. Experimental installation

5. Results

For the analytical formula of the SCF (Motok, 1997) applied to the model investigated in this
paper, the obtained results are presented in Table 2. The SCF is taken as the ratio of maximal
principal stress in the point and the nominal tension load; a/r represents the ratio of the overall
dimension of the opening and corner radius.

Table 2. SCF obtained analytically

a [mm] r [mm] a/r SCF

300 0.5 600 16.7

300 1.0 300 12.7

300 2.0 150 9.8

300 3.0 100 8.3

In the numerical analysis, the load subjected to themodel in FEM calculation is 20kN, and
the principal stress field for each radius is presented in the Fig. 6.

For the experimental procedure, the measurement had to be performed each time for each
radius since the camera could record only one spot at onemoment.The results of the experiment
are illustrated in Fig. 7.

The analytical formula, finite element analysis and digital image correlation method offered
fine matching of the results (Table 3, Fig. 8). Generally, there are some differences for smaller
radii of corner curvatures and a very good correspondence for larger ones. The tendency of
the results in all methods is the same. The finite element analysis provided the full stress field
of the concerning zone, despite difficulties in mesh generation due to large differences in the
dimension between the critical zone and the rest of themodel. The DIC technique gave the full
field results and even the change in the gradient, which is a step forward having in mind the
available classical measuring devices. Rarely seen (but expected), a high SCF (especially for 0.5
and 1.0mm of the radius) is well caught by all three methods.



Stress concentration on the contour of a plate opening ... 1009

Fig. 6. Principal stress filed obtained by FEM, load applied – 20kN

Fig. 7. Principal stress field obtained by the Aramis system

Fig. 8. SCF – comparison of the methods



1010 N.Momcilovic et al.

Table 3. SCF – comparison of the methods

a [mm] r [mm] a/r Analytical FEM Experiment

300 0.5 600 16.7 20.9 16.57

300 1.0 300 12.7 15.2 11.33

300 2.0 150 9.8 9.27 9.87

300 3.0 100 8.3 7.98 7.03

6. Conclusion

Corners of a rectangular plate openings are high stress concentration zones in daily structure
application, especially ifmadewith a significantly small radius of curvature. The paper presents
and compares three possible methods for finding stress concentration factors (SCF) in such
models. The chosen analytical method appears very easy to use, due to a simple formula, but
cannotprovide the stress gradientfieldas thefinite element analysis can.Ananalytical definition
of the opening contour by means of conformal mapping remains a problem. As expected, the
finite element analysis is able to provide the full stress field in the contour of the radius and
to illustrate such a stress gradient. However, a very fine mesh density and adjusting is needed
to obtain valid results, which increases computation time vastly. Until recently, there was no
experimental method that could have been able to catch stress concentration of such small
spots of radii of 0.5 or 1mm. In such zones, the stress gradient changes within a millimeter
distance. Strain gauges, inductive sensors and other classical devices are too large to be evenly
placed on the spot. The Aramis system, used in this research, with the incorporated Digital
Image Correlation technique, was capable to validate the numerical and analytical procedures,
obtain the SCF and illustrate the stress and strain fields. All three methods produced very
similar results, especially where larger corner radii of openings are placed. For those cases, both
analytical and FEMapproaches differ around 6% from the experimental results. For spots with
higher SCFs, the numerical method gave a slightly greater divergence but still illustrated the
stress gradient change in the zone very well.
The analytical formula showed almost no difference from theDIC results even in these areas.

It can be stated that theDICvalidated chosen analytical andnumerical approaches as legitimate
solutions for the problem. TheDIC is still developing and is under research. Although used over
the past few years, there are rare available data for the application of this experimental method
to catch significantly high gradients. The technique is extremely sensitive and depends on the
illumination, camera angle, contrast made by spraying, which all can influence the accuracy of
the results (up to 20MPa for this experiment, thus smaller values of the SCF for 2 or 3mm
radius becomemore sensitive) but definitely presents up-to-date solution to record and calculate
the displacement, strain and stress filed of any measuring volume.

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Manuscript received September 19, 2012; accepted for print April 23, 2013