1、NASATechnicalPaper3192June 1992Stress Concentrationsfor Straight-Shankand Countersunk Holesin Plates Subjectedto Tension, Bending,and Pin LoadingK. N. Shivakumarand J. C. Newman, Jr.,fr, AI!.t_r-qt4.Ai_F A_Q C4.UNTERSUNK HOLESPLATL_S qU:,JFCTE_rl TO TENSIQN_ qENDING,PI_ L _ArING (NASA) “_6 pF,q_ N_2
2、-25997INANDUncl asHI/39 009288_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASATechnicalPaper31921992National Aeronautics andSpace AdministrationOff
3、ice of ManagementScientific and TechnicalInformation ProgramStress Concentrationsfor Straight-Shankand Countersunk Holesin Plates Subjectedto Tension, Bending,and Pin LoadingK. N. ShivakumarAnalytical Services _I, and wedge load-ing P. App(mdix A explains how tire stressconcentrations for a pin-load
4、ed hole were approxi-mated flom the remote tension and wedge loadingsolutions. The wedge loading was imposed on thehole boundary a.s a normal pressure loading that hasa cosine distribution and is assumed to be constantthrough the plate thickness. The surface pressure isdefined as (2l/rcrt)cos O (ref
5、s. 18 and 19) and wasapplied over the angle = +90 . The angle O isnleasure(t fronl the y-axis. (See fig. 2(a).) The bend-ing nlOlllelll ell wa.s applied as an equivalent reinot, estress thal varies linearly through the plate thick-ness. For c()un/ersullk holes, two loading types, re-mote tensiolt an
6、d remote bending, were considered.(See fig. 2(t).) Because of the lack of understandingof 3-I) load transfer between the rivet arm tile coun-tersunk h_le, rivet (pin) loading was not eonsidere(tin the current study.Provided by IHSNot for ResaleNo reproduction or networking permitted without license
7、from IHS-,-,-Definition of Stress-ConcentrationFactorAlthoughthedefnitionofthestress-concentrationfactor is givenin many classical books on theoryof elasticity and in stress-concentration handbooks,many of these solutions are associated with 2-D con-figurations. For 3-D configurations, however, thes
8、tress concentration varies along the structural dis-continuity, such zus along the bore of the hole. Herein,tile stress-eoncentration factor is defined as the stressat any location along the bore of the hole normalizedby a characteristic stress (related to applied loading).For configurations and loa
9、ding conditions consideredin this study, tile highest stresses occurred along thebore of the hole at. the intersection of the hole surfaceand the y = 0 plane. Even for the case of pin loading,the peak stresses occurred at = 90 because tile pincontact angle was assumed to be 90 . (See appen-dix A for
10、 details.) Tire stress-concentration factorsfor the three loading conditions are defined as follows.Remote tension. The stress-concentration fac-tor fl)r tension Kt is the hoop stress Gyy at = 90 along the bore of tile hole normalized by the appliedremote tension stress S and is given byKt(z) - Gyy,
11、z,( _ (1)SRemote bending. The stress-concentration fac-tor for bending K b is the hoop stress Oyy at 0 = 90along the bore of tile hole normalized by the remoteouter-fiber bending stress 65I/t 2 and is given by- (2)6M/t 2Wedge loading. The stress-concentration factorfor wedge loading Kw is the hoop s
12、tress Gyy at = 90 along the bore of the hole normalized bythe average bearing stress P/2rt and is given by(z) - yy(z) (3)P/2rtPin loading. The stress-concentration factor forpin loading Kp is obtained from a superposition ofremote tension and wedge loading. (See appendix A.)Tire factor Kp is defined
13、 as the hoop stress Gyy at= 90 along the bore of the hole normalized by theaverage bearing stress P/2rt and is given byK/z) - P/2, (4)Finite-Element ModelingA three-dimensional finite-element code FRAC3Ddeveloped at NASA Langley Research Center for an-alyzing cracked isotropic and anisotropic solids
14、 wasused in this study. The code is based on the 20-nodeisoparametric eleinent formulation. The stiffness ma-trix and the consistent load vectors were generatedwith the 2 by 2 by 2 Gaussian quadrature fornmla.Tile program uses a vector skyline Choleski decom-position algorithm (ref. 20) for solving
15、matrix equa-tions of equilit)rium. The plates with the straight-shank hole and remote tension and wedge loadingwere symmetric about the :r - 0, y = 0, and z - 0planes. The reinote t)ending was symmetric aboutthe x = 0 and y = 0 planes and antisynmmtric aboutthe z = 0 plane. Because of these conditio
16、ns, onlyone-eighth of the straight-shank hole plate was nlo(t-eled. The FRAC3D code has an option to imposesymmetry and antisymnwtry t)oundary conditions.The plate with the countersunk hole was symmetricabout the x = 0 and y = 0 planes; hence, one-fourthof the plate was modeled. Tile F-E model inclu
17、desthe full thickness of the plate.Because many configurations were to be analyzed,a simple 3-D modeling procedure was developed togenerate the finite-element meshes. In this proce-(lure, a 2-D F-E mesh in the :r-y plane was gener-ated with refined elements near tile hole boundary.Then tile 2-D mesh
18、 was translated in the z-direction(with appropriate x-y transformation to account forthe countersunk hole). Typical 3-D F-E meshes forone-eighth of a straight-shank hole in a plate andfor one-quarter of a countersunk hole in a plate areshown in figure 3.For all straight-shank hole models, tile half-
19、thickness of tile plate was divided into six layers ofunequal thickness. The layer thicknesses (startingfrom the z = 0 midplane) were 15, 13, 10, 6, 4,and 2 percent of the total plate thickness. The smallthickness layers were used in the high-stress-gradientregions (near the free surface). The F-E m
20、odel had936 elements and 4725 nodes (14 175 degrees of free-dom). For different values of r/t, tile hole radiuswas kept constant and the plate thickness was scaledby t/r. Tile F-E mesh for r/t = 1.0 is shown infigure 3(a).In the countersunk hole, there are three regionswhere tile stress gradient is
21、high: near the two freesurfaces of the plate and at the countersink edge.Therefore, different through-the-thickness idealiza-tions were used for different countersink edge loca-tions b/t. Table 1 gives tire details of the F-E ide-alizations used for countersunk holes with b/t = O,Provided by IHSNot
22、for ResaleNo reproduction or networking permitted without license from IHS-,-,-0.25,0.50,and0.75.Figure3(b)showsatypicalF-Emodel(r/t = 0.25andb/t = 0.50) for one-quarter ofa plate with a countersunk hole.Comparison With Other SolutionsThe present 3-D stress-c(mcentratioI_ factors(SCFs) for the strai
23、ght-shank hole are comparedwith Folias and _Tangs solution (ref. 6) for re-mote tension and with F/cissners solution (ref. 8)for remote bending. Three-dimensional stress-concentration solutions for wedge loading or simu-lated pin loading have not been reported in tile litera-ture. For countersunk ho
24、les, the present solutions arecompared with Chengs photoelastic measurements(ref. 17) for thick plates subjected to tension andbending.Straight-Shank HoleRemote tension. The distribution of the stress-concentration factor Kt along the bore of the hole forremote tension is shown in figure 4 for vario
25、us valuesof r/t. The stress concentrations are symmetricabout the midplane (z/t = 0). Note the expandedscale on the ordinate axis.) In all these cases, theplate width and height, were selected large enough(w/r = h/r = 5) so that ht values are not greatlyaffected by the finite plate. The SCFs for r/t
26、 _ 0.5, the maxinmin Kt occurs atz/t = 0 (midplaim). For thicker plates (r/t 0.5),but the maximum SCF is slightly interior from thefree surface (Iz/tl 1.5.The difference between results for Reissners solu-tion and the present results for r/t less than unityis about 4 to 8 percent. Neut)ers thin-plat
27、e the-ory, KI, = (5 _- u)/(3 + r,), is inadequate even fortit = 2.5 and produces values about 6 percent lowerthan those ti)r Reissners solution and the presentresults.Countersunk HoleCheng (ref. 17) measured 3-D stress-concentrationfactors for countersunk holes in thick plates usingit phot.()(qastic
28、 slice technique for both tension andbending. (ih_ngs photoelastie models for tension(model 7) and for bending (model 8) were analyzedthrough I he generation of separate F-E meshes. Thegeometric parameters of models 7 and 8 are given intable 2. For both models, b/t = 0.6 and 0(. = 90 .The SCFs f()r
29、the two configurations at tile criticallocations arc t)resented in table 2. The F-E resultsshow thai the maximum SCF for remote tension oc-curs slightly away from the countersink edge and inthe straight-shank portion of the hole (at z/t = 0.08,whereas the countersink edge is at z/t = 0.1). Themaximu
30、m SCI: calculated from the F-E analysis iswithin 3 percent of Chengs measured value. (Seetable 2; note that percent error is defined as thedifference between solutions divided by the largeststress-concentration value.) For bending, three lo-cations on lhc hole (z/t = -0.5, 0.1, and 0.5) wereconsider
31、ed f()r comparison. The difference betweenChengs measurements and the present solution isabout 2.5 t)_r(:(mt at z/t = -0.5, but the. (tifferenceProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-isabout8percentat thecountersinkedge(z/t = 0.1)and alongth
32、e countersinkflank (z/t = 0.5). Aspreviouslyobservedfor straight-shankholesin thickplates,themaximumSCFis not at tile freesurface(z/t = -0.5) butisslightlyinteriortothefreesurface(z/t = -0.48). Thedropin SCFat thefreesurfaceis attributedto tile well-knownfree-boundary-layereffect(refs.21and22).Effec
33、t of Countersink Parameterson SCFThetwo parametersthat caninfluencethe SCFfor countersunkrivetholesarctile countersinkangle0c and the countersink depth t - b. (See fig. l(b).)The effects of these two parameters on SCF at coun-tersunk holes in plates subjected to tension and bend-ing were analyzed.Co
34、untersink AngleThe effect of small variations in the countersinkangle 0,: on tension and bending SCF was analyzedwith Chengs model 7 configuration (ref. 17). Fig-ure 6 shows the distribution of Kt and K b along thebore of the hole for 0c = 80 , 90 , and 100 . A changein 0c of +10 from the reference
35、angle of 90 changesmaximum Kt by about 3.5 percent at the countersinkedge. However, the variation in Kt is Inuch smallerat all other locations on the hole boundary. For the+10 variation in Oc, Kb varies less than 1 percent.(See fig. 6(b).) These results are for a thick plate,where r/t = 0.24. For th
36、in plates, used in aircraftapplications (r/t of about 2), the effect of Oc varia-tion on SCF is of the same order as that shown fortile thick plates.Countersink DepthAs previously mentioned, the countersink-depth-to-plate-thickness ratio is defined as 1 - (b/t), wherebit represents the ratio of the
37、straight-shank depthto the plate thickness. For convenience, b/t is used asa depth parameter. Figure 7 shows tile distributionof Kt and K b along the bore of the hole (-0.5 _ 1.5 and t)ecomes nonlinearfor thick plates (r/t _ 1.0. For r/t = 0.5and 0.25, the nmximum SCF is not at the free surface(z/t
38、= 0.5); it is h)cated in the interior of the plate(Iz/tl 0.s).Now that the SCF equations t)r remote tensionand wedge loading have been establishe(t, the SCFequation for simulate(t t)in loading is written asK, + (,-/,)Kt (7)Kv= 2Equation (7) is restricted to r/w = 0.2 because Ktand K_, are generated
39、for a plate with r/w - 0.2. The(levohq)menl of equation (7) is given in apl)endix A.The results from equation (7) are shown in figure 11.Of course, lhese results strew the saine trends a.sthos( show_ in figures 8 and 9 for tension and wedgeloading, respectively.Countersunk HolesThe configurations of
40、 the countersunk hole dic-t.ate thai tw,) separate SCF equations be fit.: oneequation for the straight-shank part (-0.5 _ z/t (b/t 0.5) and the other equation for the counter-sunk porti()n (b/t- 0.5) _ z/t _ 0.5). Furthermore,separate (!qmflions were developed for each value ofb/t. A g(meral i)olyno
41、nfial series equation in terlns ofr/t and 2/! was fit to the F-E results with the least-squares t)r()ce_lure previously discussed. The SCF(_(uations aI_! giveIl t)y3 ,1t,-,= Z Z u(“ttY( /tY (s)i=0 j=0for -0.5 z/t (b/t - 0.5) and: 4=Z Z - b + - t,)J_:() ):t)(9)for (b/t) 0.5 _ z/t _ 0.5. Equations (8)
42、 and (9)apply over 0.25 _ r/t _ 2.5. Coefficients aij and 2ijfor varicms values of b/t are given in tables 6 and 7for remole tension and remote bending, respectively.Figures 12 and 13 show comparisons t)etween resultsfor e(tuati()ns 8) and (9) with the F-E results h)rrenl()t(_ lcnsion and remote ben
43、ding, respectively.Thc (_(tual.i()n results and F-E results agree well,except near the free surface for thick plates. Even forthick plates, the maximmn SCF is within 2 percentof the F-E results for all b/t values. Note that t.hcbending SCF at z/t = 0.5 for the straight-shank hoh_(b/t := .00. s(!e fi
44、g. 10) is slightly less than that atz/t = -0.5 i()r the countersuilk hole with b/t = 0.50(see fig, 13(c).In appendix B, a FORTRAN program is giwmto evalual( lhe SCFs for straight-shank and coun-tersunk hoh,s sut)jected t.o remote tension, remotebending, pin hm(ting, and wedge loading. This pro-gram
45、is bas(d (m equations (5) to (9) with the coefli-cient,s ln(sclded in tables 3 to 7. This program maybe used to generate three-dimensional SCFs for anyvalue of b/l and l/t and at any location along thet)or(_ of the hole. To generate the SCFs for values ofb/t olh(,r ih;m those used in this study, an
46、interpola-tion s(hen_(, b(,tween the available sohltions has beenilnt)lement(d in the program.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Concluding RemarksA comprehensivethree-dimensionalstress-concentrationanalysisof straight-shankand coun-ters
47、unk (rivet) holesin a large plate subjectedto variousloadingconditionsencounteredin ser-vice wasconducted.The plate materialwasas-sulnedto beisotropic,with a Poissonsratioof 0.3.Three-dimensionalfinite-elementanalyseswereper-formedwith 20-nodeisoparametricelements.Stress-concentrationfactorsfor wide
48、rangesof hole-radius-to-plate thicknessand countersink-depth-to-platethicknessratioweregenerated.Thecountersinkan-glewasvariedfrom80 to l/0 insometypicalcases,It)litthe anglewasheldconstantat 100 for mostcases.Forstraight-shankholes,threetypesof load-ing,remotetension,remotebending,andwedgeload-ing,
49、 wereconsidered;for the countersunkholeonlyremotetensionandremotebendingwereconsidered.Series-typeequationswerefit tothefinite-elementre-sults.Theseequationsgenerallyagreedwithin1per-centof thefinite-elementresults.Tensionstress-concentrationfactor (SCF) fora countersunkholewasabout 37 percenthigherthanthe classical(2-D) solutionfor a
