Indian Journal of Engineering & Materials Sciences

Vol. 15, December 2008, pp. 452-458

Effect of fibre orientation on stress concentration factor in a laminate with central

circular hole under transverse static loading

N D Mittal* & N K Jain

Department of Applied Mechanics, Maulana Azad National Institute of Technology, Bhopal 462 007, India

Received 12 April 2007; revised received 17 June 2008

The effect of fibre orientation (θ) on stress concentration factor (SCF) in a rectangular composite laminate with central

circular hole under transverse static loading has been studied by using finite element method. The percent variations in

deflection with fibre orientation are also compared with deflection in laminate without hole. Studies are carried out for three

D/A ratios (where D is hole diameter and A is plate width). The results are obtained for four different boundary conditions.

Three different types of materials are used for whole analysis to find the sensitivity of stress concentration with elastic

constants. A finite element study is made for whole analysis of laminate with a central hole under transverse static loading.

Keywords: Finite element method, Stress concentration factor, Composite, Laminate, material properties, Fibre orientation,

Transverse loading

A laminated composite plate with central circular hole

have found widespread applications in various fields

of engineering such as aerospace, marine, automobile

and mechanical. Stress concentration arises from any

abrupt change in geometry of plate under loading. As

a result, stress distribution is not uniform throughout

the cross-section. Failures such as fatigue cracking

and plastic deformation frequently occur at points of

stress concentration. Hence, for the design of a

laminated composite plate with central circular hole,

stress concentration factor plays an important role and

accurate knowledge of stresses and stress

concentration factor at the edges of hole under in

plane or transverse loading are required. Analytical

solutions are available in the literature for prediction

of SCF in different types of abrupt changes in shape.

Shastry and Raj1 have analysed the effect of fibre

orientation for a unidirectional composite laminate

with finite element method by assuming a plane stress

problem under in plane static loading. Paul and Rao2,3

presented a theory for evaluation of stress

concentration factor of thick and FRP laminated plate

with the help of Lo-Christensen-Wu higher order

bending theory under transverse loading. Xiwu

et al.4,5

evaluated stress concentration of finite

composite laminates with elliptical hole and multiple

elliptical holes based on classical laminated plate

theory. Iwaki6 worked on stress concentrations in a

plate with two unequal circular holes. Ukadgaonker

and Rao7 proposed a general solution for stresses

around holes in symmetric laminates by introducing a

general form of mapping function and an arbitrary

biaxial loading condition into the boundary

conditions. Ting et al.8 presented a theory for stress

analysis by using rhombic array of alternating method

for multiple circular holes. Chaudhuri9 worked on

stress concentration around a part through hole

weakening a laminated plate by finite element

method. Mahiou and Bekaou10

studied for local stress

concentration and for the prediction of tensile failure

in unidirectional composites. Toubal et al.11

studied

experimentally for stress concentration in a circular

hole in composite plate. Younis12

investigated by

reflected photoelasticity method that the assembly

stress are the result of contact and bearing stresses

between the bolts and member, contributes to

reducing stresses around the circular holes in a plate

under uniaxial tension. Peterson18

has developed good

theory and charts on the basis of mathematical

analysis and presented excellent mythology in

graphical form for evaluation of stress concentration

factors in isotropic plates with different types of

abrupt change, but no results are presented for

orthotropic and laminated plate.

In this paper, a study of rectangular laminated

composite plate with central circular hole for the

effect of fibre orientation on stress concentration

factor under transverse static loading is made. The

analytical treatment for such type of problem is very

difficult and hence the finite element method is __________

*For correspondence (E-mail: ndmittal@yahoo.com)

MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE

453

adopted for whole analysis. The purpose of this

research work is to investigate the effect of fibre

orientations on SCF for normal stress in X, Y

directions (σx, σy), shear stress in XY plane (τxy) and

von mises (equivalent) stress (σeqv) in a single layer

laminate plate with central circular hole. Three types

of different composite materials of different material

properties are used for analysis to find out the

sensitivity of SCF with respect to elastic constants.

The work also illustrates the variation of SCF versus

D/A ratio in a lamina at different fibre orientations.

The deflections in transverse direction (Uz) for

different cases are also calculated.

Description of Problem

To study the influence of fibre orientation upon

deflection and SCF for different stresses, a laminated

composite plate of dimension 200 mm × 100 mm × 1

mm with a central circular hole of diameter D

subjected to a total transverse static load of P Newton

(which is uniformly distributed on whole plate) for all

cases is analysed by finite element method. The

analysis is carried out for three different D/A ratios.

Figure 1a shows the basic model of the problem.

Finite Element Analysis An eight nodded linear layered structural 3-D shell

element with six degrees of freedom at each node

(specified as Shell99 in ANSYS package) was

selected based on convergence test and used through

out the study. Each node has six degrees of freedom,

making a total 48 degrees of freedom per element. In

order to construct the graphical image of the

geometries of the three different models for different

D/A ratios, a laminated plate examined using the

ANSYS (Advanced Engineering Simulation). It was

necessary to input the basic geometric elements such

as points, lines and arcs. Mapped meshing are used

for all models so that more elements are employed

near the hole boundary. Due to the un-symmetric

nature of different models investigated, it was

necessary to discretize the full laminated plate for

finite element analysis. Main task in finite element

analysis is selection of suitable element type.

Numbers of checks and convergence test are made for

selection of suitable element type from different

available elements and to decide the element length.

Results were then displayed by using post processor

of ANSYS programme. For some simple problems of

plates, the finite elements results are also assessed

with available theoretical and experimental results in

literature and it in concluded that the finite elements

results are acceptable. Figure 1b provides the example

of the discretized models for D/A =0.2, used in study.

Results and Discussion

Numerical results are presented for three different

D/A ratio as 0.1, 0.2 and 0.5. Three different

orthotropic composite materials are used for analysis.

The material properties are given in Table 1.

Where; E, G and µ represent modulus of elasticity,

modulus of rigidity and poisson’s ratio respectively.

Four types of plates (a)-(d) are analysed. In plate (a)

all edges are simply supported, in plate (b) one edge is

Fig. 1a — Details of model analysed in study (A laminated plate

with central hole under uniformly distributed static loading of P

Newton in transverse direction)

Table 1—The material properties

Materials

Properties

Boron/

aluminium

Silicon carbide/

ceramic

Woven glass/

epoxy

Ex

Ey

Ez

Gxy

Gyz

Gzx

µxy

µyz

µzx

235 GPa

137 GPa

137 GPa

47 GPa

47 GPa

47 GPa

0.3

0.3

0.3

121 GPa

112 GPa

112 GPa

44 GPa

44 GPa

44 GPa

0.2

0.2

0.2

29.7 GPa

29.7 GPa

29.7 GPa

5.3 GPa

5.3 GPa

5.3 GPa

0.17

0.17

0.17

Fig. 1b — Typical example of finite element mesh for D/A=0.2

INDIAN J. ENG. MATER. SCI., DECEMBER 2008

454

fixed, in plate (c) two edges are simply supported and

two edges are fixed, in plate (d) all edges are fixed.

Figure 2 provides the boundary conditions at all edges

of plates (a), (b), (c) and (d).

The variation of SCF for different stresses and

percent variation in Uz with different fibre

orientations are presented in Figs 3-11. It has been

noted that these are the maximum values in the plates.

In case of plates (a) and (c), the maximum stress

concentration for all stresses is always occurred on

boundary of hole, i.e., values of SCF for different

stresses are plotted for boundary of hole, where, in

case of plates (b) and (d), the maximum stress

concentration is occurred on supports, i.e., values of

SCF for different stresses are plotted for supports.

Maximum Uz is always occurred at boundary of hole,

hence, the percent variation in Uz is plotted for

boundary of hole in all the cases.

Variations of SCF for σx, σy, τxy for different D/A

ratios with respect to fibre orientations in plates (a),

(b), (c), and (d) made of different composite materials

are shown in Figs 3-5. Following observation can be

Fig. 2 — Boundary conditions at all edges of plates (a), (b), (c)

and (d)

Fig. 3 — Variation of SCF (for σx, σy, τxy) versus fibre

orientations in plates (a), (b), (c) and (d) of boron/aluminum

material

Fig. 4 — Variation of SCF (for σx, σy, τxy) versus fibre

orientations in plates (a), (b), (c) and (d) of silicon

carbide/ceramic material

Fig. 5 — Variation of SCF (for σx, σy, τxy) versus fibre

orientations in plates (a), (b), (c) and (d) of woven glass/epoxy

material

MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE

455

made from Figs 3-5. In case of plate (a); for D/A=0.1

and 0.2, maximum SCF is obtained for σx for almost

all the values of θ and attaining maximum at θ=90°,

but for D/A=0.5 maximum SCF is obtained for τxy for

almost all the values of θ and attaining maximum at

θ=90° for all materials. Figures illustrate that at any

fibre orientation, SCF for σx, σy, τxy decrease with

increase of D/A ratio for all materials. It is also

clear from figures that SCF for σx, σy, τxy obtained

maximum when θ=90° for all D/A ratios and

materials. For all D/A ratios and materials, it has been

seen that SCF for σy is always lesser then SCF for σx

at almost all the values of θ. Maximum value of SCF

is coming as 3.5 in case of woven glass/epoxy

composite material at θ=90° for D/A=0.1 for σx. In

case of plate (b); maximum SCF is obtained for τxy for

almost all the values of θ and attaining maximum

value at θ=90° for all D/A ratios and materials. For all

Fig. 6 — Variation of SCF (for σeqv) versus fibre orientations in

plates (a), (b), (c) and (d) of boron/aluminum material

Fig. 7 — Variation of SCF (for σeqv) versus fibre orientations in

plates (a), (b), (c) and (d) of silicon carbide/ceramic material

Fig. 8 — Variation of SCF (for σeqv) versus fibre orientations in

plates (a), (b), (c) and (d) of woven glass/epoxy material

Fig. 9 — Percent variation in Uz versus fibre orientations in plates

(a), (b), (c) and (d) of boron/aluminum material

INDIAN J. ENG. MATER. SCI., DECEMBER 2008

456

materials, it has been seen that SCF for σx is not

varied with D/A ratio for all values of θ, SCF for σy is

not varied with D/A ratio when θ changes from 0° to

75° and 105° to 180° but SCF for σy is slightly varied

with D/A ratio when θ changes from 75° to 105° and

SCF for τxy increases with increase of D/A ratio for all

values of θ. Figures illustrate that the variation of SCF

for σx is much small with respect to θ and fluctuated

near about 1.0 for all materials, in case of woven

glass/epoxy material, it become exactly 1.0 for all

values of θ and D/A ratios. Maximum SCF is coming

as 4.7 in woven glass/epoxy composite material at

θ=90° for D/A=0.5 for τxy. In case of plate (c); for

boron/aluminium and woven glass/epoxy materials,

maximum SCF is obtained for τxy for almost all the

values of θ and attaining maximum at θ=90° for all

D/A ratios. For silicon carbide/ceramic material,

maximum SCF is obtained for σy for all values of θ

but variation is much small with respect to θ for all

D/A ratios. For all materials, SCF for σx increases with

increase of D/A ratio and SCF for σy decreases with

increase of D/A ratio for almost all the values of θ.

Maximum SCF is coming as 3.8 in woven

glass/epoxy composite material at θ=90° for D/A=0.5

for τxy. In case of plate (d); maximum SCF is obtained

for τxy for all values of θ and attaining maximum at

θ=90° for all D/A ratios and materials. It has been

seen that SCF for τxy and σy decrease with increase of

D/A ratio and SCF for σx increases with increase of

D/A ratio for all values of θ and materials. Figures 3-5

show that variation of SCF for σy is much small with

respect to θ for all materials and in case of woven

glass/epoxy, variation is negligible. Maximum SCF is

coming as 3.9 in woven glass/epoxy composite

material for D/A=0.2 for τxy. Figures 3-5 illustrate that

in case of plate (a), stress concentration for σx played

an important role where stress concentration for σy

played a significant role for all materials at any fibre

orientations. For plates (b), (c) and (d), stress

concentration for σx and σy played significant role and

SCF for these stresses varied from 1.0 to 2.0 for all

D/A ratios, materials and all values of θ. It has been

seen also, that sometimes SCF obtained less then 1.0.

In case of plate (d), the effect of stress concentration

for σx and σy is almost negligible. For plates (b), (c)

and (d), stress concentration for τxy played an

important role and has significant value for all cases

and sometimes it obtained more then 4.0. It is

observed that the SCF follows a symmetric trend with

respect to 90° in all cases. For woven glass/epoxy

laminate, SCF follows a symmetric trend with respect

to 45° when orientation changes from 0°

to 90°

and

to135° when orientation changes from 90° to 180°

due to same value of Ex and Ey.

Variations of SCF for σeqv for different D/A ratios

with respect to θ in plates (a), (b), (c), and (d) made of

different composite materials are shown in Figs 6-8. It

has been observed that, for all plates of different

materials, SCF decreases with increase of D/A ratio.

In case of plate (a); following observation can be

Fig. 10 — Percent variation in Uz versus fibre orientations in

plates (a), (b), (c) and (d) of silicon carbide/ceramic material

Fig. 11 — Percent variation in Uz versus fibre orientations in

plates (a), (b), (c) and (d) of woven glass/epoxy material

MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE

457

made from figures. In case of boron/aluminium

material, SCF increases continuously when θ changes

from 0° to 15°, decreases when θ changes from 15° to

30°, again increases when θ changes from 30° to 75°,

attaining a maximum value when orientation is at 75°

and then decreases when θ changes from 75° to 90°

for all D/A ratios. In case of silicon carbide/ceramic

material, SCF decreases continuously when θ changes

from 0° to 45°, attaining a minimum value when

orientation is at 45° and then again increases when θ

changes from 45° to 90°, attaining a maximum value

when orientation is at 90° for all D/A ratios. In case of

woven glass/epoxy, SCF increases continuously when

θ changes from 0° to 15°, decreases when θ changes

from 15° to 45°, again increases when θ changes from

45° to 75°, attaining a maximum value when

orientation is at 75° and then decreases when θ

changes from 75° to 90° for all D/A ratios. Maximum

SCF are coming as 2.2 at θ=75°, 1.9 at θ=90° and 2.1

at θ=75° for D/A=0.1 in boron/aluminium, silicon

carbide/ceramic and woven glass/epoxy composite

materials respectively. It is observed that the SCF for

σeqv also follows a symmetric trend with respect to

90° in all cases. For woven glass/epoxy laminate, SCF

follows a symmetric trend with respect to 45° when

orientation changes from 0°

to 90° and to135°

when

orientation changes from 90° to 180°. It is clear from

figures that, for all materials and D/A ratios,

maximum stress concentration occurred in case of

plate (a) for all values of θ and for plate (a) SCF

varied from 1.3 to 2.3 for different cases. It is also

observed that, in case of plate (c), some significant

stress concentration occurred. But in case of plates (b)

and (c), the effect of stress concentration is much

small, and in case of plate (d), it is almost negligible

for all cases. For plate (d), the variation of SCF with

respect to θ is also negligible for all D/A ratios and

materials; SCF is fluctuated near about 1 for all cases.

In case of plate (b); it has been seen that the effect of

D/A ratio on SCF is negligible for all values of θ and

materials. In case of all plates, the trend of variation

of SCF with respect to θ is different for different

material, i.e., variation of SCF depends up on elastic

constants. In case of plate (a); SCF obtained always

greater then 1.0 for all values of θ, D/A ratios and

materials but in case of plates (b), (c), and (d), SCF

obtained less then 1.0 in some cases.

The variation of percent variation in Uz for

different D/A ratios with respect to θ in plates (a), (b),

(c), and (d) made of different composite materials are

shown in Figs 9-11. The percent variation in UZ has

been calculated with respect to laminate without hole

for same case. Following observation can be made

from Figs 9-11. In case of plates (a), (b) and (c), Uz

increases with increase in D/A ratio, but in case of

plate (d) Uz increases when D/A ratio increase from

0.1 to 0.2 and then decreases when D/A ratio increase

from 0.2 to 0.5 for all values of θ and materials. For

boron/aluminium and silicon carbide/ceramic plates

(a), (c) and (d), the maximum and minimum

deflection occurred at θ=90° and θ=0° respectively,

but in case of plate (b) maximum deflection occurred

when θ=0° and minimum occurred when θ=90°. In

case of woven glass/epoxy material; percent variation

in Uz is almost constant with respect to θ for all D/A

ratios and plates (maximum variation is obtained up

to 5%). It has been observed that maximum percent

variation occurred for plate (a) and minimum

occurred for plate (d). It has been also seen that, per

cent variation in Uz is obtained less then 0% at some

values of θ for all D/A ratios, plates and materials.

Conclusions

In general; for plates (a) and (c), the maximum

stress concentration is always occurred on hole

boundary and in case of plates (b) and (d), the

maximum stress concentration is occurred on

supports. The SCF for σx, σy, σeqv play an important

role in plate (a), a significant role in plate (c) and

negligible role in plates (b) and (d). The SCF for τxy

plays, an important role in plates (b), (c), (d) and a

significant role in plate (a). It has been observed that

SCF for all stresses decrease with increase in D/A

ratio, where deflection increases with increase in D/A

ratio for almost all values of θ, materials and plates.

For plates (a), (c) and (d), maximum Uz always

occurred at θ=90° and for plate (b), maximum Uz

always occurred at θ=0° for all D/A ratios. Maximum

SCF for τxy always occurred at θ=90° for all cases. It

is also observed that SCF for all stresses and

deflection follow a symmetric trend with respect to

90° fibre orientation. In case of composite materials

those have same modulus of elasticity in X and Y

directions SCF for all stresses and deflection follow a

symmetric trend with respect to 45° when orientation

changes from 0

or 90° and to135° when orientation

changes from 90° or 180°. In case of all plates, the

trend of variation of SCF with respect to θ is different

for different material, i.e., variation of SCF depends

up on elastic constants. It has been also seen that the

INDIAN J. ENG. MATER. SCI., DECEMBER 2008

458

SCF is most sensitive to material properties and

directly depend on the ratio of Ex/Ey and Ex/Gxy. The

results obtained, show that for higher values of these

ratios, SCF for all stresses may also be higher.

References

1 Shastry BP & Raj GV, Fibre Sci Technol, 10 (1977) 151.

2 Paul T K & Rao K M, Comput Struct, 33 (1989) 929.

3 Paul T K & Rao K M, Comput Struct, 48 (1993) 311.

4 Xiwu X, Liangxin S & Xuqi F, Comput Struct, 57 (1995) 29.

5 Xiwu X, Liangxin S & Xuqi F, Int J Solids Struct, 32 (1995)

3001.

6 Iwaki T, Int J Eng Sci, 18 (1980) 1077.

7 Ukadgaonker V G & Rao D K N, Compos Struct, 49 (2000)

339.

8 Ting K, Chen K T & Yang W S, Int J Pressure Vessels

Piping, 76 (1999) 503.

9 Chaudhuri R A, Comput Struct, 27 (1987) 601.

10 Mahiou H & Bekaou A, Compos Sci Technol, 57 (1997)

1661.

11 Toubal L, Karama M & Lorrain B, Compos Struct, 68 (2005)

31.

12 Younis N T, Mech Res Commun, 33 (2006) 837.

13 Sinclair G B, Int J Mech Sci, 22 (1980) 731.

14 Meguid A, Eng Fract Mech, 25 (1986) 403.

15 Giare G S & Shabahang R, Eng Fract Mech, 32 (1989) 757.

16 Troyani N, Gomes C & Sterlacci G, J Mech Design (ASME),

124 (2002) 126.

17 Fillipini M, Int J Fat, 22 (2000) 397.

18 Peterson R E, Stress concentration design factors (John

Wiley and Sons, New York), 1966.

19 Daniel I M & Ishai O, Engineering mechanics of composite

materials (Oxford University Press, New York), 1994.

20 Ross C T F, Advance finite element methods (Horwood

Publishing Limited, Chichester), 1998.