The Reduction of Stress Concentration Around the Hole in an Isotropic Plate Using Composite...

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Engineering Fracture Mechanics Vol. 32, No. 5, pp. 757-166, 1989 0013 7944/893.00 + 0.00Printed in Great Britain. 0 1989PergamonPresspk.

THE REDUCTION OF STRESS CONCENTRATIONAROUND THE HOLE IN AN ISOTROPIC PLATE

USING COMPOSITE MATERIALSG. S. GIARE and R. SHABAHANG

Aeronautical Engineering Department, Embry-Riddle Aeronautical University, Prescott,AZ 86301, U.S.A.

Abstract-The stress distribution around a hole in a finite isotropic plate reinforced by compositematerial has been calculated using a finite element analysis. The results agreed very closely withexperimental investigations. With the present reinforcement technique, the stress concentrationfactor of 2.72 for a finite plate near a hole reduces to the factor of one. The theoretical andexperimental results have shown that the present reinforced around hoop-wound carbon fiber discprovides an efficient reinforcement around circular holes. This technique could be very useful inreducing the stress concentration factor of orthotopic plates of composite materials. The techniquemay be found to be particularly useful in composite laminate plates in which they could beincorporated during manufacturing process.

NOMENCLATUREirlEECEAIGKPPCs112f3x9 Yc80r12Pfl

Radius of holeOuter radius of reinforcing discRadius greater than holeCross-sectional areaYoungs modulusModulus of graphite/epoxyModulus of aluminumShear modulusStress concentration factorDirect loadLoad in compositeUniform applied tensile stressThickness of aluminum plateThickness of reinforcement ringThickness of filament winding of graphite/epoxyLongitudinal and transverse plate coordinate directionsDirect strainAngular coordinatesPoissons ratioDirect stressShear stressAluminum alloy plateComposite materialPressure lb/in2Load transfer efficiency factor

1. INTRODUCTIONTHE REDUCTIONof stress concentration around cut-outs in a flat plate has been studied for a longtime. An early paper by Gurney[ l] dealt with circular holes in plane stress, and analytical solutionswere given for holes reinforced with thin uniform annuli of rectangular cross sections subjectedto uniform stress in plane direct and shear loads. Later on, Reissner and Morduchow[2] consideredthe same problem, but with the reinforcement in the form of a bead which was treated as a curvedbeam. The idea of designing a bead such that the stresses in the plate remained unchanged by thecut-out was introduced {neutral holes), and a theoretical analysis by MansfieldpI gave the shapea neutral hole and the variation of the cross-sectional area of the reinforcement for any particularloading system. Since this early work, many papers have been published on the subject, and designinformation is now available in ref.[4], which deals with the reinforcement of circular, elliptical andrectangular holes in isotropic plates.

757


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1% G. S. GIARE and R. SHABAHANG

D. 0. McKenzie and J. P. H. Webber studied analytically and experimentally hole reinforce-ment in isotropic and orthotopic plates using filament-wound discs[5]. The present work is a stepfurther, based on what was done in[5].In the present studies, the isotropic plate is reinforced using special winding techniques withthe use of graphite fiber/epoxy systems. The unidirectional fiber layer maintains continuity of stressaround the hole of the isotropic plate. In this paper, finite element analysis in the linear elastic

region has been made, using the Coade program. The results are verified with experimentalinvestigations. The stress concentration factor for a finite isotropic plate in tension is 2.72. Whenthe plate is reinforced with graphite fiber, according to the technique presented in this paper, thestress concentration factor is only 1.25.

2. TH ORYThe aluminum plate with a hole, reinforced by a CFRP disc and further strengthened bygraphite fiber winding, is shown in Fig. l(a). The construction of the specimen is described inSection 5.

3.6--

---

S = 1684 PSI-

IO. ol ~.

Fig. 1 (b). Reinforcement of a circular hole by a hoop-wound CFRP disc on each side of the plate.

:

1.s:-

------

--:I684 PSI

Fig. 1 a). Reinforcement of Alum. plate with a CFRP disc and further reinforcement with graphite/epoxy.


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Stress concentration around a hole 759

-\1020 lb = = IO20 lb

RADIAL PRESSUREP= 483 PSI

B

c

1020Ib = \zl020fbA L

Fig. 2. Tensile fibers inducing compressive force.

When the specimen is loaded in tension, the graphite winding AA and aluminum alloy platebehave as a composite structure. The longitudinal graphite fiber is in tension, as the graphite fiberpassed through disc 2, fibers induces compressive force on the ring, as shown in Fig. 2. Thiscompressive force causes the reduction of stress concentration factors around the hole. For theinitial load of 200 lb, the load carried by the graphite fiber/epoxy composite and aluminum alloyplate may be calculated as follows:

L = Q = CA,p = P, + PA,

a/E = q/E, = o EAP = cTc.A, + a,,*A *,P =200lbPC l/4,90 P = 40.8 lb

EA, = 10 x 1061b/in2E, = 10 x lo6 lb/in2

P JP, = 4.9v, = 0.25

vAL= 0.30.A, = 0.0341 in*

AAl = 0.1271 in*.The load carried by P, produces compressive forces around the ring of the hole, as shown in Fig. 2.This reduces the stress concentration. The magnitude of tensile force in composite winding isequivalent to l/4.9 of the applied load. This ratio of loading can be controlled by varying themodulus of elasticity of fiber winding.

3. FINITE ELEMENTThe finite element analysis was done using Coade software on an IBM AT computer. Infinite element analysis, an aluminum alloy plate 1 x 20 x 40 with a hole of 2 diameter wasanalysed. The a/w ratio was 0.1. The same ratio was maintained in experimental investigation.

EFM2,1--P


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760 G. S. GIARE and R. SHABAHANGIn the Coade program, the stress concentration around the hole in the isotropic material

was modified as follows:MSH 2D was changed to accommodate two materials, one marked 2 consisting of a graphitedisc around the hole and the rest of the isotrophic material marked I. Figure 3(a) shows the finiteMSH 2D of the aluminum alloy plate with CFRP disc, and 3(b) shows the magnified view of meshnear hole of the plate.

Theoretically, the I/4.9 P load transfer by filament graphite fiber/epoxy composite wasmodeled in finite element analysis as load acting inward to hoie, as shown in Fig. 4.The hoop load carried by the graphite fiber produces a radial pressure P on the diameter of

disc 2, as follows:P, 2t. I = 0,

P = 483 lb/ino;2r = P/4.712 = 1020 lb

d = diameter of disc = 3.Xp = radial pressure in lb,/&q = load transfer efficiency factor

= 0.90 (ASSUMPTION).

s=500 PSI

L 10.0 _I(a)

I = ALUMINUM PLATE2 = COMPOSITE MATERIAL

(b)Fig. 3. (a) Finite MSH Z&-of Alum. plate with CFRP disc. (b) magnified view near the hole showingtwo different materials.


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Fig. 4. Finite MSH 2LAlum. plate with CFRP disc loaded at different node.

It produces a radial pressure of p = -483 lb/in* acting radially towards the center. Pressure at eachnode is resolved in a vertical component opposing the applied load using eq. (1) and applied-459 lb/in at node 25-26, - 392 lb/in2 at node 26-27, -295 lb/in2 at node 27-28, - 187 lb/in2 atnode 28-29, - 90 lb/in2 at node 29-30. Figure 5 shows the radial pressure resolved in x-x direction,

P P cos26. (1)

pg= = 1020 lb

GRAPHITE WINDING

GRAPHITE RING

ALUMINUM PLATE

= 1020 lb

25 26 459 PSI26 27 39227 26 29526 29 I6729 30 90

Fig. 5. Radial load resolved in x-x-direction.

GRAPHITE WINDING


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762 G S GIARE and R SHABAHANG

4. ELASTIC SOLUTION OF INFINITE PLATE WITH HOLEWhen an infinite width plate is loaded in tension with the hole the maximum stress, cfl, is given

as follows:fsH= S/2[1 + al/r*] - S/2( 1 + 3a4/r4 cos 28. (.2)

At the hole when r = a, r_~@ 3, hence K = 3. For finite width in this study, 2ajw = 0.1.K = 2.72 has been obtained using Coade finite element analysis.

5. EXPERIMENTSUniaxial tension tests were carried out on aluminum alloy plates, with holes reinforced by

filament-wound carbon fibers discs are shown in Fig. l(b). The present aluminum alloy plate witha hole reinforced by CFRP disc and further strengthened by graphite fiber winding is shown inFig. l(a). The aluminum plate is l/32 x 3.8 x 10 is cut from aluminum alloy sheet. A hole of0.38 in diameter is bored on a drilling machine. The hoop-wound CFRP reinforcing discs werecut from thick walled tube manufactured in the lab. on a lathe at 8 = fgg using 6K mediummodulus roving graphite fiber/epoxy system. Two CFRP discs l/32 thick and 0.69 in diameterare bonded to the aluminum plate on either side surrounding the hole. Two tabs of glass-fiberlaminate l/S x l/2 x l/2 are bonded symmetrically along the central line at station A and Aof the aluminum plate (as can be seen in Fig. l(a). The above said tabs of the same material anddimensions are bonded on the other side of the plate. After the glass fiber tabs are cured, a slotl/32 thick and l/16 deep parallel to the aluminum plate and touching it is cut into each tab. Thisgroove keeps the graphite fiber/epoxy winding in position. Two l/4 thick, l , diameter perspexwashers are clamped with l/4 diameter bolts and nuts on either side of the CFRP disc. Thesewashers give a space of l/32 between aluminum plate and washer, to keep the graphite fiberimpregnated in the proper position. The graphite fiber is impregnated with epoxy and woundaround the slot of glass fiber epoxy laminate at station A and disc 2. The other layer is then woundaround A and disc 2. This alternate winding process is continued until six layers of 6K tow fibersare completed. The same process is carried out on the other side of the aluminum plate. Care istaken that the plate remains straight, while the winding is done. The winding is pressed againstthe aluminum plate and allowed to cure. After curing, the perspex washers are removed. With handlay out, the graphite composite of 33% I/; is obtained.

In total 12 aluminum alloy plates, l/32 x 3.8 x lo, were cut. A set of 4 aluminum alloyplates were tested with 0.380 diameter holes. A set of 4 aluminum alloy plates reinforced withCFRP disc were made. A set of 4 aluminum alloy plates were bonded with CFRP discs and furtherreinforced with graphite fiber/epoxy as shown in Fig. l(a) were made also. Figure 6 shows thephotographs of all the three specimens. All holes were made 0.380 in diameter.In these specimens, the reinforcing discs were placed symmetrically on each side of the hole.Each specimen was instrumented with 120 ohms electrical resistance strain gauges. Strain gaugeswere fixed at x = s/g, 314 and 1. Strain gauges were used, made by Measurements Group, withgauge length of 4 mm. Single gauges were bonded to the surfaces of the plates and reinforcing discsto obtain the distribution of strain in the vicinity of the hole. The hole was small; the strain gaugecould not be fitted on the inner surface. Using eq. (2), the stress concentration factor was calculatedat the inner edge of the hole.All the specimens up to 500 lb tension load only were tested on the tensile testing machinewith a 2000 lb load cell, using strain indicator, P - 3500, manufactured by Instruments Division,Measurements Group, Raleigh, North Carolina. From the strains obtained by the gauges, thestresses are calculated, and hence stress concentration factors were obtained.

6. RESULTS AND DISCUSSIONStress concentration factors were calculated, using Coade finite element program, for

aluminum plate, for aluminum plate with disc, and for aluminum plate reinforced with disc withfurther reinforcement with graphite fibers. Figure 7 shows the stress concentration factor vs r/a


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Fig. 6. Photograph of the three specimens.


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Stress concentration around a hole

4l

I

A ALUMINUM PLATE3 \

A ALUMINUM DISC0 ALUMlNUM DISC WINDING

o/..........I 2 3 4 5 6 7 8 9 IO IIR/A

765

Fig, 7. Stress across minimum section alum. plate, CFRP-reinforced alum. and Present specimenb a = 1.8125, 1,/t, = 3.0 specimen.

for all the three specimens. For aluminum plate, the stress con~ntrat~on is 2.72 at the edge of holeand reduces to one at r/a equal to 10. When the aluminum plate reinforced with orthotropic discof t,ltl = 3.0 and diameter of disc d = 0.69 (b/a = 1.8125) the stress concentration is 1.92 nearthe hole and approaches to one at r a equal to 10. This result agrees closely with the result presentedby D. 0. McKenzie and J. P. H. Webbert for aluminum plate reinforced with a disc of CFRP.When the aluminum plate is reinforced with disc of CFRP and with the present reinforcement,the stress concentration is only 0.92, near the hole and approaches to one at the edge of the plate.Experimental results for all the same specimens are shown in Fig. 8. Stress concentrations are veryclose to the theoretical predictions.

CONCLUSIONThe stress dist~bution around a hole in a finite isotropic plate reinforced by compositematerial, has been calculated, using finite element analysis hence stress concentration factor of 0.9,

4

A ALUMINUM PLATE3 D ALUMINUM DISC

40 ALUMINUM DISC WINDING

0123456?89 10 IIR/A

Fig. 8. Experimental stress distribution of stress across-Minimum Section of Alum. plate, CFRPDISC-reinforced. Alum SCPresent Specimen. b,h = 1.8125, f,/t, = 3.0.


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766 G. S. GIARE and R. SHABAHANGis evaluated. The stress concentration factor using an experimental technique is 1.25. Results agreevery closely with expe~mental investigations.

The theoretical and experimental results have shown that the present reinforced aroundhoop-wound carbon fiber disc provides an efficient reinforcement around circular holes. Thistechnique could be very useful in reducing the stress concentration factor of orthotropic plates ofcomposite materials.

The present reinforcement technique may be found to be particularly useful in compositelaminate plates, where they could be incorporated during manufacturing process.Acknowledgemenrs-The authors are thankful to Professor Blame Butler. Chairman, Aeronautical Engineering Departmentof Embry-Riddle Aeronautical University, for his encouragement and support. The authors are also thankful to MissSantwana Giare for helping in testing and discussion. It is a pleasure to acknowledge the partial support of N.S.F. grantfor the development of the composite laboratory at Embry-Riddle Aeronautical University.

REFERENCES[I] C. Gurney, An analysis of the stresses in a flat plate with a reinforced circular hole under edge forces. ARC R & M1834 (1938).[2] H. Reissner and M. Morduchow. Reinforced circular cutouts in plane sheets. NACA TN 1852 (1949).[3] E. H. Mansfield, Neutral holes in plane sheet-reinforced holes which are elastically equivalent to the uncut sheet, Q,Jl Mech. appl. Math. VI, 370 (1953).[4] Elastic stress concentration factors single reinforced and unreinforced holes in infinite plates of isotropic materials.Engineering Sciences Data Unit, ESDU Item No 70005 (1970).[5] D. 0. McKenzie, and J. P. H. Webber, Aeronaut Q V 26 Pt. 4, p. 254-274 (November 1975).[6] S. G. Lekhnitskii, Anisoiropic Plates, 2nd edn, Chapter 6. Gordon and Breach, New York (1968).[7] G. S. Giare, Fracture toughness of unidirectional fibre reinforced composites n mode II. Engng Fracture ML 20,I l-21 (1984).

(Received 29 February 1988)

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