Contact stresses in bolted joints of composite laminates

Composite Structures 6 (1986) 57-75

Contact Stresses in Bolted Joints of Composite Laminates L. IngvarEriksson* The Royal Institute of Technology, Department of Aeronautical Structures and Materials, S-10044, Stockholm, Sweden

ABSTRACT The designing of bolted joints of composite laminates requires, like all structural design, relevant failure criteria and methods for determining the stress distribution. This paper deals with stress analysis of bolted joints using the finite element method. There are several possible ways to treat the bolt~hole contact problem in the finite element model. One is to assume a certain contact pressure distribution acting on the boundary of loaded holes, for example a cosine distribution; another is to assume radial displacements equal to zero on the hole boundary, which yields a distributed contact reaction to the applied load. Performing a complete contact analysis is a third possibility, which obviously is the most accurate. In this paper, contact stresses and stresses in the vicinity of the hole boundary are calculated with account taken of the contact problem. Effects of laminate elastic properties, clearance, friction, load magnitude and bolt stiffness are studied. A comparison of the results with experimentally obtained data shows that most of these parameters affect the stress distribution significantly.

1 INTRODUCTION C o m p o s i t e materials are frequently used in m o d e m aircraft. Just as in any structure made of conventional material, loads must be transmitted from one m e m b e r to another through joints. Very often bolted joints require the thickest laminate in the structure, leading to a concentration of weight in * Present address: Saab-Scania AB, Saab Aircraft Division, TKHKA, S-58188LinkOping, Sweden. 57 Composite Structures 0263-8223/86/$03-50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain

58

L. Ingvar Eriksson

joint regions and a tapered laminate. Overdesigned joints can therefore easily eliminate some of the weight savings obtainable through the use of composite materials. Hence, an adequate design theory must be developed. The fundamentals of such a procedure are stress analysis and failure criteria. Knowledge of the stress distributions in the vicinity of the boundary of the hole is a basic requisite for prediction of failure. Most analyses usually ignore the bolt/hole interaction and assume rather than evaluate a certain stress distribution on the boundary of loaded holes. Such a procedure may be severely erroneous. A cosine distribution was assumed in Refs 1 and 2. Collings 3 also assumed a cosine distribution but added Coulumb frictional forces to act in the tangential direction of the hole. Wilson et al. 4 fixed displacement in the load direction and in the perpendicular direction of the load, respectively. Comparison with experimentally obtained data gave poor agreement. Matthews et al. 5 had the hole periphery loaded via an arrangement of pinjointed bars connected between the hole center and finite element nodes on the loaded side of the hole. Agarwal 6 and Conti 7 assumed that the radial displacements on half of the hole circumference are zero. Of the studies dealing with the bolt/hole interaction problems in terms of analytical complex variable theory, de Jong 8 studied the case of a rigid, frictionless pin with a perfect fit clearance in an infinitely large plate. So did Oplinger and Gandhi, 9but they included friction and clearance. Mangalgiri and Dattaguru ~° included clearance only. Zhang and Ueng tt included friction only. Hyer and Klang 12 found in an excellent piece of work that laminate elastic properties, friction and clearance are parameters that significantly affect the stress distribution on the hole boundary. Rowlands et al. 13used the finite element method and evaluated the effect of friction and clearance in joints of wood and glass and boron composites. They assumed a rigid bolt. The effect of friction in Ref. 13, however, is in disagreement with the results presented in this paper and in Ref. 12. It would be of even greater interest to investigate how the contact problem affects the stress distribution a small distance inside the laminate from the hole boundary. Most failure criteria proposed for the bolted joint application use stresses a small distance inside the laminate from the boundary, since the prediction of failure would otherwise be too conservative. Solving a contact problem which includes friction is a difficult problem.

Contact stresses in bolted joints of composite laminates

59

Due to the irreversible character of frictional forces the solution has to be traced incrementally. ~4The use of Coulumb's law as a constitutive relation leads to both mathematical and numerical difficulties. It has been proved ~5that in certain cases the solution is not unique.

2 PROCEDURE The finite element method (FEM) was chosen as a tool to solve the problem, mainly because joints in aircraft structures are generally subjected to arbitrary loads, which would be too complicated to attack with an analytical method.

Shear-out plane ), Bearing plane

1

2

~f

3

Fig. 1. Load cases studied in this paper: (1) single row joint loaded in tension; (2) single row joint loaded in compression; (3) single row joint loaded by a tension by-pass load.

In this paper stresses are calculated for three different load cases (see Fig. 1). For the first joint, numerical results were obtained in order to assess the effects of the various parameters of laminate elastic properties, clearance, friction, load magnitude and bolt stiffness on the stress distributions on the hole boundary and a small distance away from the hole boundary. Results are compared with results obtained in Ref. 12 as well as with those obtained with an assumed cosine distribution applied on the hole boundary. For the second joint, experimentally obtained axial strains along the bearing plane are compared with results obtained from a contact analysis and with results obtained from an FE analysis where radial displacements on half of the hole circumference were assumed to be zero. Lastly, for the third joint, experimentally obtained axial strains along the net-section plane are compared with results obtained from the filled hole contact analysis and also with results obtained from an open hole FE analysis.

60

L. lngvar Eriksson

3 A P P R O A C H TO SOLVING T H E P R O B L E M The general purpose FE computer program A S K A (a finite element system from S A A B - S C A N I A AB and IKOSS GmbH) was used to solve the problem. The contact subroutine in A S K A is developed by Torstenfelt. ~4 In the frictionless case, the minimization of the potential energy is performed under consideration of the inequality constraints arising in the contact zone. A linear transformation of the unknown displacements in the contact area is performed to obtain a global

I

L.

z

x

QUAM 8

Fig. 2. FE model.

TRIM 6

Contact stresses in bolted joints of composite laminates

61

minimum. The total load is applied and the contact area is found by iteration. In a general contact problem, including friction, the solution is obtained by a combined incremental and iterative algorithm because the frictional forces are dependent on the load history. A typical distorted FE model is shown in Fig. 2. The composite plate is mainly modeled with eight-node quadratilateral anisotropic plane membrane elements, QUAM8, i.e. through-the-thickness effects are ignored. The bolt is modeled with six-node triangular plane membrane elements, TRIM6, with the same thickness as the laminate.

4 N U M E R I C A L RESULTS

4.1 Effects of laminate elastic properties To evaluate the effects of laminate elastic properties, stresses on the hole boundary and in the vicinity of the hole boundary were computed for three different laminates, defined in Fig. 3 and Table 1. Here, the frictionless contact problem was solved with a perfect fit between the bolt and the hole. The bolt used in the analyses is representative of a titanium bolt with Young's modulus E = 110 GPa and Poisson's ratio v = 0.29. Also a cosine distribution was applied to laminate C. e

I S .r-o°t d

t

ne

J

P

.,,.

wx

Wx

~.

(__ =.~..

=4)

d d t : l a m i n a t e thickness

Fig. 3. Laminate studied in this paper.

a~o

r

62

L. Ingvar Eriksson

TABLE 1 Laminate Stiffnesses of T300/914C Graphite/Epoxy System Percentage of plies in directions 01901+-45 A: 25/25/50 B: 69/6/25 C: 6/69/25

Ex (GPa)

Ey (GPa)

G xy (GPa)

51.4 102 24.2

51.4 24.2 102

19.3 111 112

/)xy

0.33 0.44 0.10

Since a membrane analysis is performed, the stacking sequence is unimportant. The stresses in Figs 4 to 6 were normalized by the average bearing stress S = P/dt

Figure 4(a) and (b) shows the radial stress, as a function of ~p, on the hole boundary and a distance a0 inside the laminate (a0 = 1-26 ram, d = 6 m m ) , respectively. The figures show that the radial stress strongly depends on the laminate properties. The peak stress seems to move off the center line for laminates with high stiffness in the y-direction. Of the contact stress distributions which were calculated, the one obtained for laminate B shows good agreement with the cosine distribution. The others differ considerably. The results presented in Fig. 4(a) are in general agreement with those presented in Ref. 12. Figure 5 shows the tangential stress at a distance a0 from the hole boundary as a function of ~. It is clear that the tangential stress also strongly depends on the laminate properties. Laminate C, being stiffest in the y-direction, has the largest stress for ¢ = 0. It is interesting to observe that differences are small at ,¢ = 90 °, indicating that for this simple load case axial stress along the net-section plane is not sensitive to the laminate stiffness. Figure 6 shows the shear stress at a0 as a function of ~. Laminate properties also influence the shear stress. Laminate B, being stiffest in the x-direction, has the highest shear stress at location ~ = 50 °. Figure 7 shows the shear stress along the shear-out plane. The shear stress was normalized by the average shear stress -c = P/2et

The laminate properties also affect the shear stress along the shear-out

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(Degrees)

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Fig. 8. Axial stress along bearing plane.

66

L. Ingvar Eriksson

plane. The peak stress occurs at approximately x / e = 0-15 from the hole boundary. Finally, Figure 8 shows the axial stress along the bearing plane. The axial stress was normalized by the far field stress o = P/wyt

Again, the laminate properties strongly affect the stress distributions. Even at a distance x / d = 0.5 away from the hole boundary the stress levels are quite different for the various laminates analyzed.

4.2 Effects of clearance To evaluate the effects of clearance, laminate C in Fig. 3 was analyzed. The frictionless contact problem was again solved with the same titanium bolt in the hole as previously. Three different clearances were used (see Fig. 9). (Negative clearance, h, means press fit.) 1) k =

6/r=O.O0

2) X =

5/r=0.01

3) k =

5/r=-0.0066

Fig. 9. Definition of clearance.

Figures 10(a) and (b) show the radial stress as a function o f ¢ at the hole boundary and at a distance a0 inside the laminate, respectively. Figure 10(a) shows that the contact area increases the smaller the clearance is. The peak stress, however, is about the same in magnitude but differs in location. At a distance a0 away from the hole boundary, differences are still significant. Here the peak stresses also differ in magnitude. The results presented in Fig. 10(a) are in general agreement with those obtained in Ref. 12.

4.3 Effects of friction To evaluate the effects of friction laminate A in Fig. 3 was analyzed for three different coefficients of friction: /.~ = 0-0, 0.2 and 0.5. The

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~P ( Degrees

80 )

Fig. 10. Radial stress versus ~.

100

67

68

L. lngvar Eriksson

clearance was set equal to zero and the same titanium bolt as previously was used. Friction affects the contact stress distribution in an interesting way. Figure 11 shows the radial contact stress as a function of ~. With frictional

~

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.o

o .i=

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40

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Fig. 11. Radial contact stress versus ~.

Fig. 12. Frictional forces acting on the hole boundary.

forces present, the radial stress decreases. This is physically correct, since frictional forces will transfer axial load (see Fig. 12), and due to equilibrium the radial stress must decrease. These results are in general a g r e e m e n t with those obtained in Ref. 12. Figure 13 shows the tangential stress at a0 as a function of ¢. Friction decreases the tangential stress at ¢ = 0 °, but increases the maximum tangential stress (at ~ = 80 °) by about 15%. Figure 14 shows the shear stress at a0 as function of ~. The presence of friction decreases the shear stress.

4.4 Effects of load magnitude To evaluate the effects of load magnitude, laminate C in Fig. 3 was analyzed for two different load levels. The frictionless contact problem was solved with a clearance h = 0.01 and with the same titanium bolt as previously in the hole. Figure 15 shows the radial contact stress as a function of ~.

Contact stresses in bolted joints of composite laminates -t ~lO

-1 ~:10

:1..5

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0 e-

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( Degrees

)

Fig. 13. T a n g e n t i a l stress at ao versus ~o.

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-

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60

-

= 0.0

i

40

!

60

~P ( Degrees ) Fig. 15. Radial stress versus ~o.

80

L. Ingvar Eriksson

70

Even the frictionless contact problem is non-linear, since the contact area varies with load. 4.5 Effects of bolt stiffness To evaluate the effect of bolt stiffness, laminate C in Fig. 3 was analyzed with different bolts in the hole. The frictionless contact problem was solved with a perfect fit between the bolt and the hole. The mechanical properties of the bolts are shown in Table 2. 4

TABLE 2 Mechanical Properties of the Bolts

Bolt material

Young' s modulus ( G Pa )

P oisson's ratio

206 110

0.31 0.29

Steel Titanium

U)

i

~.5

0

2 0

~.0 ....,.....,." '

g

0.5 C 0 m

0.0 0

20

40

80

80

( Degrees ) Fig. 16. Radial stress versus ~.

)0

Contact stresses in bolted joints of composite laminates

71

Figure 16 shows the radial contact stress around the hole as a function of ~p for both the joints. The bolt stiffness has a very small effect on the radial contact stress distribution. This is in general agreement with Ref. 12. 5 TEST RESULTS Two types of tests were performed for comparison with the numerical results which were obtained (see Figs 17 and 18). In specimen A, strain gages were placed along the bearing plane and in specimen B along the net-section plane. Table 3 gives mechanical data used for both specimens. In both tests a titanium bolt with Young's modulus E = 110 000 MPa and Poisson's ratio v = 0-29 was used. Specimen A was loaded in a double shear test fixture to a load level of 3 kN. Axial strain, ex, along the bearing plane was measured. The experi-

e

i 90 o

Bearing plane

x

w x = 171 mm Wy

oo-

Wx

Wy = 36 mm e =

36 mm

d =

6ram

Fig. 17. Test specimenA.

y Net-section|

Wy

plane

~

~90o

d

~

0o

WX

Fig. 18. Test specimenB.

w x = 190 mm

Wy =

48 rnm

d =

8ram

72

L. Ingvar Eriksson TABLE 3 Laminate Properties of T300/914C Graphite/Epoxy Laminate

Percentage o f plies in directions 0]90] +-45 A: 20/60/20 B: 25/25/50

Ex (GPa)

Ey (GPa)

G xy (GPa)

v xy

40.6 51-4

91.4 51-4

9-60 19.3

0-10 0.33

mentally obtained strains were compared with results obtained from two different FE models: (1) A n FE model which takes into account the contact problem with friction included: the coefficient of friction was set equal to 0.3, which should not be so far from reality for titanium-carbon/epoxy joints. The clearance was set equal to zero, which is close to the actual clearance in the test specimen. (2) An FE model with radial displacement fixed on half of the hole circumference: as already mentioned this is a commonly used m e t h o d in the literature. This way of treating the bolt/hole interaction is equal to assuming a frictionless rigid bolt with a perfect fit, if the actual contact area is known in advance. The outcome of the comparison is presented in Fig. 19, where the strain, Ex, has been normalized by the far field strain, ~x. Except for test point No. 3, which most likely is unreliable, the strains calculated with the contact model are in good agreement with measured strains. For small values of x/d there is poor agreement with model 2. Test specimen B was loaded in tension to a load level of 40 kN. Axial strain, Ex, along the net-section plane was measured. The experimentally obtained strains were comparedwith numerical results obtained from two different FE models. (1) An FE model which takes into account the contact problem with friction included: again the coefficient of friction was assumed to be 0.3 and the clearance was set equal to zero. (2) As a comparison an open hole model was analyzed. The outcome of the comparison is presented in Fig. 20, where the strain, Ex, has been normalized by the far field strain, L. Near the hole boundary,

Contact stresses in bolted joints of composite laminates

73

5.50 5.00

Contact model

i,

.....

@)

l l

4.50

l Displacement model

x

H

ItO 4 . 0 0 bJ

3.50

"\

3.00

~

2.50

\

o

~

\

\ \

x

\

2.00

\

Test points

% %

.E *d

t .50

4""- ..,....~ .~......:.

x

5 t .00

I

I

I

I

0

I

2 Normalized distance,

x/d

Fig. 19. Axial strain along bearing plane.

where friction plays an important role, the two FE models differ. As far as the test results are concerned, it is difficult to make any relevant conclusions, since the interesting part close to the hole boundary is not measured. Anyway, the results presented may be helpful when planning new investigations.

6 CONCLUSION It has been shown that laminate elastic properties, friction and clearance significantly affect the stress distributions on the hole boundary as well as at a small distance inside the laminate, whereas the bolt stiffness has a very small effect. Further, it has been shown that the frictionless contact problem is non-linear and that the bolt stiffness has a small effect on the

L. Ingvar Eriksson

74

3.50 Contact model

®

3.00

Open hole model

X

lw X

2.50 ~I

©

OJ

\ •

~ Test points

I 1.50

\\\

0

.__.

I

.00

t'@

0.50 0.0

I

0.5

!

t.0

.5

Normalized distance, y / d

Fig. 20. Axial strain along net-section plane.

radial contact stress distribution. This is important information to be aware of when developing failure criteria. If wrong stresses are used to predict failure, no one can tell how much of a disagreement depends on the failure criterion itself. On the other hand, good agreement can be a lucky coincidence. The failure criterion may work on a particular laminate and load case, but will fail for others.

ACKNOWLEDGEMENT

The work presented here was supported by The National Swedish Board for Technical Development, STU, and Saab-Scania Aircraft Division, Linkrping, Sweden.

Contactstressesin boltedjoints of composite laminates

75

REFERENCES 1. Waszczak, J. P. and Cruse, T. A., Failure mode and strength predictions of anisotropic bolt bearing specimens, Journal of Composite Materials, 5 (1971) 421. 2. Chang, Fu-Kuo, Scott, Richard A. and Springer, George S., Failure of composite laminates containing pin-loaded holes--method of solution, Journal of Composite Materials, 18 (1984) 255. 3. Collings, T. A., On the bearing strengths of CFRP laminates, Composites, July (1982) 241. 4. Wilson, D. W., Gillespie, J. W., York, J. L. and Pipes, R. B., Failure analyses of composite bolted joints, Center of Composite Materials, University of Delaware, NASA-CR-163732, July 1980. 5. Matthews, F. L., Wong, C. M. and Chryssafitis, S., Stress distribution around a single bolt in fibre-reinforced plastics, Composites, July (1982) 316. 6. Agarwal, B. L., Static strength prediction of bolted joint in composite materials, AIAA Journal, 18(11) (1980) 1371-5. 7. Conti, Paolo, Influence of geometric parameters on the stress distribution around a pin-loaded hole in a composite laminate, Composites Science and Technology, 25 (1986) 83-101. 8. de Jong, Theo, Stresses around pin-loaded holes in elasticity orthotropic or isotropic plates, Journal of Composite Materials, 11 (1977) 313. 9. Oplinger, D. W. and Gandhi, K. R., Analytical studies of structural performance in mechanically fastened composite plates, AMMRC MS 74-8, 1974, 221-40. 10. Mangalgiri, P. D. and Dattaguru, B., Unbonded smooth rigid circular pin in an orthotropic plate, Res Mechanica, 12 (1984) 143-50. 11. Zhang, Kai-da and Ueng, Charles E. S., Stresses around a pin-loaded hole in orthotropic plates with arbitrary loading direction, Composite Structures, 3 (1985) 119-43. 12. Hyer, M. W. and Klang, E. C., Contact stresses in pin-loaded orthotropic plates, Virginia Polytechnical Institute and State University, USA, NASACR-173475, April 1984. 13. Rowlands, R. E., Rahman, M. U., Wilkinson, T. L. and Chiang, Y. I., Single- and multiple-bolted joints in orthotropic materials, Composites, July (1982) 273. 14. Torstenfelt, B., Finite elements in contact and friction applications, PhD Thesis, Division of Solid Mechanics and Strength of Materials, Department of Mechanical Engineering, Link6ping University, Sweden, August 1983. 15. Klarbring, A., Contact problems in linear elasticity-friction laws and mathematical programming applications, PhD Thesis, Division of Solid Mechanics and Strength of Materials, Department of Mechanical Engineering, Link6ping University, Sweden, August 1985.