Stress concentration factors around a circular hole in a composite laminate are determined. The specific case investigated is a four layer (-45/45/45/-45 degs) graphite epoxy laminate. The factors are determined experimentally by means of electrical resistance strain gages, and analytically by using a hybrid finite element analysis

Ueng, C. E. S.

English

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STJBSS CONCENTRATION FACTORS AROUND A CIRCULAR 
HOLE IN LAMINATED COMPOSITES 
C. E. S. Ueng 
Georgia Institute of Technology 
SUMMARY 
This paper deals with the determination of stress concentration factors 
around a circular hole in a composite laminate. 
is a four layer (-45O/450/450/-45O) graphite epoxy laminate. The factors are 
determined experimentally by means of electrical resistance strain gages, and 
analytically by using a hybrid finite-element analysis. 
The specific case investigated 
I INTRODUCTION 
In thfs study, the laminar stress concentrations around a circular hole in 
an angle-ply composite laminate are determined for the axial tension loading 
case. Of particular interest is the largest value of 08 present at the peri- 
meter of the hole. This study proposes to determine these stresses experimen- 
tally and analytically. For the experimental analysis, electrical resistance 
strain gages are used. The analytic procedure uses the finite-element method 
of a two-dimensional hybrid model with an assumed stress field within the ele- 
ment and assumed displacements at the element interfaces, 
The stress concentration factors around a circular hole in an infinite, 
isotropic sheet have been determined analytically through various approaches 
and confirmed experimentally. For an infinite plate, the stress components 
around the hole are (ref. 1) 
Q ~ ( 1  . %)(I 2 + %) 2 sin 28 
1: r 2 
T&= 
rhere b is the radius of the hole and CT, is the applied load. The ratio of 
re/oo along the hole is plbtted as shown in figure 1. Obviously, the.maximum 
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o f  00 i s  t h r e e  t i m e s  
diameter perpendicular t o  t h e  d i r e c t i o n  of tension. 
o0, and occurs a t  0 = +, 90°, i .e . ,  a t  t h e  ends of t h e  
Due t o  t h e  increas ing  use of advanced laminated composites i n  f l i g h t  
s t r u c t u r e s  and o the r  p o t e n t i a l  appl ica t ions ,  t h e  stress concentration around a 
cutout i n  a f iber - re inforced  laminate has been t h e  subjec t  of research by 
seve ra l  i nves t iga to r s  i n  recent  years. Daniel and Rowland ( r e f .  2) used an 
experimental approach - t h e  Moiie technique, and determined t h e  s t r a i n  ( s t r e s s )  
concentration around a c i r c u l a r  hole i n  a tens ion  loaded an iso t ropic  p l a t e .  
Hyman e t  al .  ( r e f .  3) ca r r i ed  some exploratory tests on t h e  same problem. 
Franklin ( r e f .  4 )  a l s o  inves t iga ted  t h e  hole  stress concentrations i n  f i l a -  
mentary s t ruc tu res .  
help of finite-element method, Rybicki and Hooper ( r e f .  5) s tud ied  and obtained 
r e s u l t s  f o r  boron-epoxy lamina. In a reviewing a r t i c l e  ( r e f .  6 ) ,  G r i m e s  and 
Greimann gave an up-to-date o v e r a l l  p i c t u r e  about t h e  stress concentration 
around a c i r c u l a r  ho le  i n  a f iber - re inforced  composite. 
references are c i t e d  i n  t h i s  a r t i c l e .  
By using l i n e a r  e l a s t i c  plane stress conditions with t h e  
Several add i t iona l  
I n  an experimental study of or thot ropic  composite materials, Kulkarni, 
Rosen,and Zweben ( r e f .  7) have found t h a t  t h e  stress concentration f ac to r s  are 
a function of t he  hole diameter, up t o  a diameter of 2.54 cm (1 in . ) .  They 
observed t h a t  t h e  a c t u a l  number of fi laments severed by t h e  hole  determined 
t h e  s t r eng th  of t h e  specimen. 
The present problem of a general  angle-ply composite laminate with a 
c i r c u l a r  hole i s  f u r t h e r  complicated by t h e  i n t e r a c t i o n  of the  ind iv idua l  
layers .  
EXPERIMENTAL WORK 
Equipment 
The o r i e n t a t i o n  of t he  s t r a i n  gages around the  holes i s  shown f o r  each of 
t h e  four  specimens i n  f igu re  2. The gages w e r e  mounted adjacent t o  t h e  hole  
and w e r e  4 . 8  mm (3/16 in . )  wide, 120 ohm standard f o i l  gages. The specimens 
were mounted i n  clamp g r ips  and attached t o  a 90,000 N (20,000 l b . )  capac i ty  
load c e l l  through t h e  use of swivel bearings.  The gages were wired i n t o  t h e  
d i g i t a l  s t r a i n  ind ica to r  with the  ind ica to r  providing t h r e e  arms of t h e  Wheat- 
s tone  bridge required i n  t h e  electrical c i r c u i t .  The load ce l l  w a s  wired i n t o  
an e l e c t r i c a l  transducer and ca l ib ra t ed  t o  measure t h e  a x i a l  tension applied t o  
t h e  specimens. 
The s t r a i n  gages w e r e  applied t o  t h e  specimens using Eastman 910 adhesive, 
following standard prepara t ion  of t he  surfaces.  
T e s t  Specimen Data 
The four  t e s t i n g  specimens were provided by Lockheed-Geo’rgia A i r c r a f t  
Company. Their a s s i s t ance  i s  g r e a t l y  appreciated. 
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Material: graphi te  epoxy (Narmco 5209/T300) 
65% graph i t e  f i b e r ,  35% epoxy matrix 
Four l aye r  angle-ply ( -45°/450/450-450) 
Grip tabs  of f i b e r g l a s s  epoxy molded i n t e g r a l l y  wi th  specimens. 
For a un id i r ec t iona l  s i n g l e  l a y e r  t h e  macroscopic p rope r t i e s  are 
2 6 EoO = 137900 - 144795 MN/m (20 - 21(10) ps i )  
EgOO = 8274- 9653 MN/m (1 .2 -  1.4(10) p s i )  
G = 4.55 MN/m (0.66(10) ps i )  
2 6 
2 6 
The thickness of t h e  four specimens around t h e  hole  w a s  also c a r e f u l l y  
measured. Data w e r e  taken a t  e ight  s t a t i o n s ,  t h e  end po in t s  of  a hor izonta l  
diameter, a v e r t i c a l  diameter, and two more diameters which b i s e c t  t h e  hor i -  
zontal  and v e r t i c a l  d i r ec t ions .  The r e s u l t s  are shown i n  t a b l e  1. 
It can therefore  be concluded t h a t  t h e  assumed thickness 0.6350 nnn 
(0.0250 i n )  i s  q u i t e  reasonable. 
Testing Procedure 
F i r s t ,  t he  t e s t i n g  specimen was mounted i n  t h e  upper g r ips  of t h e  loading 
device. The loading i n d i c a t o r  and t h e  s t r a i n  ind ica to r  were zeroed and cal i -  
brated.  Then t h e  o the r  end of t h e  specimen was mounted i n  t h e  lower g r ips  of 
t h e  loading device. 
o r  500 l b ) ,  t h e  load w a s  then released. This w a s  repeated s i x  t i m e s  i n  order 
t o  eliminate t h e  s t r a i n  gage e r r o r  due t o  s t r a i n  hardening. 
20% of t h e  maximum load was  used each t i m e ,  and t h e  corresponding s t r a i n  read- 
ing  w a s  then taken. 
Af te r  t h e  specimen w a s  loaded up t o  t h e  100% load (2224 N 
An increment of 
The same s t eps  w e r e  followed f o r  t h e  o the r  t h ree  specimens 
Testing Results 
The data obtained from t h e  s t r a i n  gage t e s t i n g  was  t he  values f o r  €8 a t  
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four  d i f f e r e n t  l oca t ions  around t h e  hole.  
and displayed graphica l ly  i n  f igu re  3. 
These values are given i n  table 2,  
The s t r a i n  gage r e s u l t s  can be e a s i l y  repeated 
s t a b i l i t y  with repeated loadings. The values obtained a t  444.8 N (100 lb)  of 
load are not as r e l i a b l e  as t h e  incremental changes i n  s t r a i n  f o r  each incre- 
mental change i n  load. 
mens r e su l t ed  i n  an i n i t i a l  s t r a i n  of  some s igni f icance .  
and showed very good 
Normally,the t igh ten ing  of t h e  end clamps on t h e  speci-  
The a s s i s t ance  of M r .  W. H. Taylor i n  carrying out t h e  t e s t i n g  program i s  
acknowledged here.  
S t r e s ses  
Based upon t h e  ava i l ab le  mechanical p rope r t i e s  as previously mentioned, 
t he  stresses w e r e  ca lcu la ted  from the  s t r e s s - s t r a i n  r e l a t i o n  and t h e  t rans-  
formation r e l a t ions .  
corded s t r a i n s  are p lo t t ed  i n  f igu re  4 .  
The t angen t i a l  stress component 00 obtained from the  re- 
FINITE ELEMENT ANALYSIS 
The finite-element method used here i s  a two-dimensional hybrid approach. 
The v a r i a t i o n a l  p r i n c i p l e  used i s  t h a t  of minimum complementary energy with 
t h e  interelement stress con t inu i ty  enforced bymeans Of theLagrangemul t ip l i e r s .  
The elements used are shown i n  f igu re  5. 
The formulation of t he  problem a t  t h i s  s tage  follows a r a t h e r  standard 
fash ion  as t h i s  method i s  t y p i c a l l y  applied t o  many stress ana lys i s  problems. 
The stress function polynomial used 
3 2 $ = a x  + b x y +  
which r e s u l t s  i n  the  following stresses: 
It can be e a s i l y  v e r i f i e d  
t h e  equilibrium equations 
CT = 2 c x  xx 
CT = 6 a x  
YY 
i n  t h e  computer program i s  
2 3 
Cxy + d Y  
+ 6 dy 
+ 2 by 
T = -2 bx - 
XY 
t h a t  t hese  stress 
i n  the  absence of 
(3) 
components automatically s a t i s f y  
body forces .  
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Arranged i n  mat r ix  form, equations (3) become 
Col = [QI [a1 
where 
By Cauchy's r e l a t i o n  Ti = CT ,n one has 
i j  j '  
T 0 
T 
Y 
o r  
where 
(5) 
Following a somewhat standard fashion, t he  c i rcumferent ia l  stress around 
t h e  hole  i s  obtained and p l o t t e d  a l s o  i n  f igu re  4 f o r  t h e  comparison purpose. 
DISCUSSION OF RESULTS 
The r e s u l t s  presented i n  t h i s  paper represent an attempt t o  understand 
and p r e d i c t  t h e  stress concentration around a c i r c u l a r  ho le  i n  an angle-ply 
laminate. A s  shown i n  f i g u r e  4 ,  t h e  c i rcumferent ia l  stresses,based upon t h e  
finite-element method and t h e  one computed from t h e  recorded s t r a i n  da ta ,  are 
p lo t t ed  toge ther  f o r  comparison purpose. These two curves c ros s  each o the r  at 
a few p laces ,  but t h e  discrepancy at  some p laces  i s  up t o  35%. This degree of 
devia t ion  i s  not  hoped f o r ,  but it i s  to l e rab le .  Similar experience ind ica t e s  
t h a t  such a d i f f e rence  i s  by a l l  means possible.  
The stress concentration f a c t o r  a t  b / r  = 1 and 8 = t 900 i s  about 5.8 
7 15 
which is considerably higher than the classical factor 3 for an infinite, 
isotropic plate. 
stress concentration around such a circular hole. One possible reason for 
having such a high stress concentration factor is that a number of fibers were 
cut at the location of the hole. 
ments for transmitting the stresses. From an intuitive point of view, if 
the location of the hole is known in advance, then rerouting the fibers 
around the hole may cut down the high stress concentration factor. 
Therefore, special attention must be paid for the local 
This weakens the ability of the fiber ele- 
REFERENCES 
1. Timoshenko, S .  P. and Goodier, J. N.: Theory of Elasticity, Third Edition, 
McGraw-Hill, 1970, p. 91. 
2. Daniel, I. M. and Rowland, R. E.: Determination of Strain Concentration 
in Composites by MoirL Techniques, Journal of Composite Materials , vol. 
5, April 1971, pp. 250-254. 
3. Hyman, B. I., DeTurk, A., Diaz, R., and DiGiovanni, G.: Exploratory Tests 
on Fiber-Reinforced Plates with Circular Holes Under Tension, AIAA 
Journal, vol. 7, no. 9, September 1969, pp. 1820-1821. 
4. Franklin, H. G.: Hole Stress Concentrations in Filamentary Structures, 
Fibre Science and Technology, vol. 2, 1970. 
5. Rybicki, E. F., and Hooper, A. T.: Analytical Investigation of Stress 
Concentrations Due to Holes in Fiber Reinforced Plastic Laminated Plates, 
Two-Dimensional Models, AE’ML-TR-72-15, U.S. Air Force, 1972. 
6. Grimes, G. C. and Greimann, L. F.: Analysis of Discontinuities, Edge 
Effects, and Joints, Chapter 10, vol. 8, Structural Design and Analysis, 
Part 11, edited by C. C. Chamis, in Composite Materials, Academic Press, 
1974. 
7. Kulkarni, S. V., Rosen, B. W., and Zweben, C.: Load Concentration Factors 
for Circular Holes in Composite Laminates, Journal of Composite 
Materials, vol. 7, July 1973, pp. 387-393. 
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TABLE 1.- TEST SPECIMEN DATA 
TABLE 2.- TEST RESULTS 
7 17 
0 
-1 
Figure  1.- I s o t r o p i c  case, Oe/a0. 
x 
Specimen #l 
Specimen #3 
X 
Specimen 1 2  
Specimen #4 
Figure  2.- Locat ion of s t r a i n  gages. 
7 18 
e 
Figure 3 . -  Strain curve, (So = 6895 kN/m2 (1 ksi). 
0 From strain data 
@ Finite-element result 
Figure 4.- Stress concentration factors. 
7 19 
Figure 5.- Element assignment. 
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Stress concentration factors around a circular hole in laminated composites

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