ANALYTICAL METHODS FOR THE STRESS CONCENTRATION ANALYSIS OF MULTILAYERED ANISOTROPIC COMPOSITES WITH

PIN-LOADED HOLES

Werner Hufenbach, Bernd Grber, Martin Lepper, Bingquan Zhou

Institut fr Leichtbau und Kunststofftechnik (ILK), Technische Universitt Dresden, 01062 Dresden, Germany

KEYWORDS: stress concentrations, pin-loaded holes, multilayered anisotropic composites, ana-lytical methods.

INTRODUCTION

The use of fibre- and textile-reinforced multilayered composites in particular for load-carrying and safety-relevant applications has been expanded considerably during recent years. Hereby tex-tile semi-finished reinforcing products in form of multi-axial knitted, woven or braided performs have gained increasing importance compared to uni- or bidirectional fibre-reinforced layers.

The analysis of the stress concentration behaviour of pin-loaded holes is for many structures of great importance since holes in composites structures can often be considered as design drivers and since related issues occur in form of subproblems in the evaluation of rivets, screws, etc. Whilst in literature analytical solutions for the concentration problem of fibre-reinforced multilay-ered composite plates with cut-outs and elastic inclusions can be found [1, 2] ( mostly based on the fundamental works of LEKHNITSKII [3]), such publications are not found very often for generally multilayered anisotropic composites with pin-loaded holes [4, 5].

ANALYTICAL CALCULATION METHOD

Since the composite structures dealt with here are considered to be thin-walled, they can be glob-ally described by means of the classical laminate theory (CLT). Starting with the structural law as given by the CLT and expanding the classical plate theory in such a way, that the equilibrium equations for the plate are supplemented by the membrane forces a generalised plate equation for composites is obtained that can be represented as a system of coupled partial differential equations (PDE).

I the case of the problem of a pin-loaded hole, the mechanical equivalent of an infinite plate with a centric cut-out loaded with a sinusoidal normal distribution of edge forces on a continuous part of the notch edge (Eq. 1) (partial edge load, see Fig. 1(a)).

[ [

=

=

SB

SBBB

BB

B

n

s

sspsN\0

,sin)(~ 1001

00

fr

fr (1)

For obtaining solutions for the PDE-system, the method of complex-valued displacement functions is called upon. Hereby it is obvious, that the static equilibrium conditions on the overall plate are no longer fulfilled, because of the asymmetric partial edge load. So a virtual resultant R = (Rx, Ry)T is introduced as an auxiliary variable inside the cut-out (see for example Fig. 1(a) and [5]). By adding some additional boundary conditions caused by the introduction of the virtual resultant, a complex LAURENT series expansion and a comparison of coefficients on the notch edge, a linear system of equations is created to determine the unknowns.

NUMERICAL VERIFICATION

To verifyof the developed calculation methods for multilayered composites with pin-loaded holes, extensive numerical finite-element (FE) investigations have been carried out. The acquired numerical results established by the finite-element method show a very good correlation with the pre-calculated analytical solutions and thus endorse the developed theory in an impressive way (see for example Fig. 1(b)).

Based on this strong support, the developed calculation methods offer a very good basis for the development of new fast dimensioning-tools for the end-user dealing with multilayered unidirec-tional or textile-reinforced composites with pin-loaded holes.

CONCLUSION

For the problem of multilayered composites with pin-loaded holes examined here, analytical solu-tion methods based on complex valued displacement functions have been developed. Here, we should emphasize that not only stresses, distortions and displacements directly at the edge of the notch can be calculated, but also their distribution throughout the entire plate. A vast number of FE calculations were performed for the verification of the developed calculation methods. A comparison of these numerical values and the results obtained by means of the developed analytical solution methods showed a high level of concurrence and therefore support the developed calculation method in a strong way.

REFERENCES

1. Hufenbach W., Grber B., Langkamp A., Lepper M. and M., Kroll, L. Advanced calculation methods for notched hybrid composites with textile-reinforced polymers. Journal of Plastics Technology, Vol. 3, No. 2, 2007.

2. Engels H. and Becker W. Closed-form analysis of external patch repairs on laminates. Composite Structures, Vol. 56, No. 3, pp 259-268, 2002.

3. Lekhnitskii S. G. Anisotropic Plates. Transl. 2nd russ. edition: S. W. Tsai and T. Cheroin, New York et al.: Gordon and Breach, 1968.

4. Echavarra C., Haller P. and Salenikovich A. Analytical study of a pin-loaded hole in elastic orthotropic plates. Composite Structures, Vol. 79, pp. 107-112, 2007.

5. Grber B. Beitrag zur Strukturanalyse von anisotropen Schichtverbunden mit elastischen Einschlssen und Bolzen. PhD-Thesis, Technische Universitt Dresden, 2004.

y

x

Ry

Rx

OB1

0

0

s

SB

0

Nn~

00

p0

2232

normal force distributionon notch bondary

Nn~

s0

1

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

0.0 45.0 90.0 135.0 180.0 225.0 270.0 315.0 360.0

notch edge coordinate []

stre

ss [M

Pa]

+45-layer

+45-layer

+45-layer

-45-layer

-45-layer

-45-layer

sr

t rj

s j

FEM: -45-layerFEM: +45-layer

analytical

y,90

CFK[+45/-45]S

x,0DJ

(a) Cut-out with partial edge load (b) Comparison of analytical and numerical results

Fig.1. Mechanically equivalent model and analytical and numerical results