Cc Poe Stress Intensity Factor

8/6/2019 Cc Poe Stress Intensity Factor

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TECH LIBRARYKAFB, N M

IIllilllllllllllllll11111lllllI1I11. Report No.

NASA TR R-3583. Recipient's Catalog No.I. Government Accession No.I5. Report Date

May 1971

4. T itle and Subtitle

STRESS-INTENSITY FACTOR FOR A CRACKED SHEET WITHRIVETED AND UNIFORMLY SPACED STRINGERS

7. Author(s)

C. C. Poe, J r .

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, Va. 23365

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546

8. Performing Organization Report No.

L-682610. Work Unit No,1 126-14-15-0111. qntract or Grant No.

I13. Type of Report and Period Covered1 Technical Report14. Sponsoring Agency CodeI

15. Supplementary Notes

. .

16. Abstract

The stress-intensity fa ctor and forces in the most highly loaded rivet and strin gerwere calculated for a crac ked sheet with riveted and uniformly spaced stringers . Two sym-metr ical c as es of c rac k location were cons idered - the ca se of a cr ac k extending equally onboth sides of a str inge r and the case of a cr ac k extending equally on both sid es of a pointmidway between two stri nge rs. The complete res ults are prese nted on design graphs forsyste matic variatio ns of cr ack length, rive t spacing, st ring er spacing, and stringe r stiffness.

The results show that the stress-intensity factor for the stiffened sheet is significantly le ssthan that for an unstiffened she et, except for c rac k lengths much le ss than the s trin ger


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STRESS-INTENSITY FACTOR FOR A CRACKED SHEET WITH

RIVETED AND UNIFORMLY SPACED STRINGERSBy C. C. Poe, Jr.

Langley Researc h Center

SUMMARY

The stress-intensity factor and forc es in the most highly loaded rivet and str in ge rwere calculated for a cracked sheet with riveted and uniformly spaced stri nger s. Twosym met rica l ca se s of cr ack location we re considered - he c as e of a crack extendingequally on both sides of a s'tringe r and the c ase of a cra ck extending equally on both si de sof a point midway between two strin ger s. The complete resul ts a r e presented as designgraphs for syste matic variations of crac k length, rivet spacing, st ri ng er spacing, andst ri ng er stiffness. The re su lt s show that the stress-in tensity factor for the stiffened

sheet is significantly less than that for an unstiffened sheet, 'except for crack lengths muchle ss than the stringe r spacing. Also, the forc es in the most highly loaded st ri ng er andrive t asymptotically approach limiting values for increas ing crac k length. The st r es s -

intensity factor and the limiting values of the st ri ng er and rivet fo rce s are smal l e r fo rstiffer o r more closely spaced stringers o r for m or e closely spaced rivets.

INTRODUCTION

The design of a complex st ru ct ur e for maximum resid ual strength and maximum


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displacements at the r ive ts in the sheet and str ing ers to be equal. The stress-intensityfactor for the cracked sheet w a s determined by superimposing the stress-intensity fac-

to rs for the rive t forc es and for the applied uniaxial stress. Two ca se s of sy mme tri calcrack location were considered - he ca se of a cr ac k extending equally on both sides of ast rin ge r and the c as e of a cr ac k extending equally on both sides of a point midway betweentwo st ri ng er s. Although the points of attachment are referred to as being riveted, there su lt s apply equally well to spotwelded attachments.

The salien t effects of independently varying st ri ng er stiffness , st ri ng er spacing,

rivet spacing, and crack length are discussed in detail. The complete res ul ts for sys-tematic variations of s tr in ge r stiffness, s tr in ge r spacing, rivet spacing, and cra ck lengthare pre sen ted in the form of design graphs.

Aij

a

Bi

b

b0

SYMBOLS

displ aceme nt a t ith riv et becau se of fo rce of unity at jth rivet

half-crack length

disp lacem ent a t ith riv et because of applied uniaxial st r e s s of unity

str ing er spacing

specific value of s tr in ge r spacing


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P

Q

r , d

S

t

V

W

X,Y

X0,YO

Z

Z

(Y

rivet spacing

rivet force

plane polar coordinates

applied uniaxial stress

thickness

y-component of displacement

str ing er width

rectangular Cartesian coordinates

rivet coordinates

Westergaard stress function

complex variable

stringer-spacing reduction par am ete r


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-normal stress acting on crack surfa ce

shearing-stress componentsTxy TXZ y z

@ lyc$2

51

functions defined on page 14

function defined on page 16

Subscripts:

A pertaining to par t A of sketch (g)

pertaining to pa rt B of sketch (g)

i at ith rivet

j at jth rivet

lim limiting

S s t r inger

Superscript :


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-10 0 0 0 0 0 0 0

s 4 T i l -I 1

-1 0 0 0 0 0 0 0 ' . 0\ -

Sketch (a)

Sketch (b) shows the rivet fo rc es acting symmetri cally with respe ct to the cra ckwhen the crack extends equally on both sides of a str inger. The forces a r e also sym-me tri ca l for the other cra ck location. The row of riv ets col linear with the crac k ex ertno fo rce because of symm etry. As required by equilibrium, the rivet force s ac t in oppo-si te directions on the sheet and strin ge rs. All forces acting on the sheet and str in ge rswer e assumed to act at the midplane of the sheet.


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The equations nece ssa ry to determin e the unknown riv et fo rce s Qj were obtainedby equating the displacements at the rive t locations in the she et and stri nge rs. The dis-placements wer e writ ten in te rm s of the following influence coefficients: Aij and Bi,which represent the displacements in the cracked sheet at the ith riv et becau se of unitvalues of Qj and S, respe ctivel y; and Afj and Bf , which re pr es en t the displa ceme ntsin the s t r ing ers at the it h ri ve t becaus e of unit values of Qj and S%, respectively.Thus, the displacement a t the ith rivet in the sheet and stri ng er s can be written

E

and

(21'= AS&. + B? -S s1 1 3 1 1 Ej

respectively, where i and j refer to only those rivets in one quadrant of the she et.The negative sig n w a s introduced in equation (1) because the rivet for ces a ct in oppositedirect ions on the sheet and stri nge rs. Equating the displacements in equations (1)

and (2) and collecting terms gives

(i = 1 , 2, 3 , . . . ) (3 )

A truncated se t of th ese simu ltaneo us equations w as solved for the unknown rivet forces


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where K is the stress-inte nsity factor , and ter m s containing higher powers of r havebeen neglected.

B cau

-ketch (c)

f symmetry, the uniaxial s t r e s s S and the symm etrical r r a y of rivet

fo rc es produ ce only an opening mode of cra ck deformation in the stiffened sheet . Byusing the principle of sup erposition, the total-stress -inten sity factor can be written as

K = S G + (41j

where S f i is the component due to the uniaxial s t r e s s S, and EjQj is the compo-nent due to the sym met rica l se t of r ive t forces Qj. For the case of plane st re ss , thecoefficient is given (see ref. 17) for the sym met ric al se t of point forces shown i n


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where

Y

a1 =

and

(An e r r o r in ref. 17 has been co rrect ed inassumption is made that t he re are no rivet holes along the row of r iv et s through whichthe crack is assu med to advance.o r into a hole indicate that the effect of the holes on the stress-intensity factor is negli-gible except when the crack tip is within approximatel y a di amet er of the hole. Thus ,equations (4) and (5) will be accu rat e except when the crac k tip is close to a riv et hole.

of eq. (5).) In equations (4) and (5), the

The results in reference 18 fo r a crack extending fr om

Strin ger- Load- Concentration Fac tor

For convenience, a stringer-load-concentration factor L is defined as the ra t iof th i f i th t i t th t f li d t th t i


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Num eri ca1 Solution

Because the stresses in the sheet at large distances from the crack are unaffected

by the pre sen ce of the cr ack, the r ivet forces are negligibly s ma ll except in the vicinityof the crack. In the prese nt investigation only those riv ets within the rectangular regionshown in sketch (e) we re included in the solution of equations (3) . The height of th e region

L7

a -

Sketch (e)

w a s chosen to be equal to the cra ck length 2a o r to include 20 r ive ts per s t r inger,whichever was la rg er . The width of the region extended beyond the cra ck tips to includethe next stringers.ar ea were actually considered as unknowns in the solut ion of eqs . (3) .)calculations the computed value of the stress-intensity factor w a s found to be affected by

(Because of symm etry, only those ri vet f orc es in one quadrant of t h i sFrom preliminary


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Appendix B presents graphs of the stress-intensity factor and the forces in the mosthighly loaded rive t and str in ge r for systematic variations of the significant para meter s.

Thus, these graphs may be used as design graphs.

Stress-Intensity Factor

Figure 1 shows the effect of st rin ge r stiffness on the relationship between thestress-intensity factor and half-crack length for a cr ac k extending equally on both sidesof a str inger and a crack extending equally on both sides of a point midway between two

str ing ers . The str ess -in ten sit y facto r and the ha1.f-crack length and riv et spacing aremade nondimensional by dividing by the stre ss- int ens ity fac tor of an unstiffened shee t

S F a and the st ri ng er spacing b, respectively. The ra ti o of rivet to st rin ge r spacingis fixed a t p/b = 1/6. The stiff ness of the st ri ng er s is expressed as p , the rat io oftotal st ri ng er stiffness to total panel stiffness. Fo r str in ge rs of uniform s iz e and spacing

The curves show that the str ess -in ten sit y factor of the stiffened she et is essentiallyequal to that of an unstiffe ned sh ee t when the cra ck length is sma ll compared to str ing er

spacing. Moreover, fo r a crack located midway between two stringers, K = S p a forcr ac k lengths up to one-half the st ri ng er spacing.is sm al l compared to the str in ge r spacing, the st ri ng er s may give little improvement tothe strength of the cracked sheet.

Thus, in ca se s where the cr ack length

For longer cra cks, however, the str in ge rs can reduce


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s t r inge r s is fixed at p, = 0.5. The cur ves for various ratios of ri vet to stri nge r spacingshow that the stress-intensity factor is als o lower for mo re closely spaced rivets.

However, when the crack tip is near a stri nge r, the curves continue to de cr ea se withdecreasi ng rive t spacing, and very sm al l minimum values occur in the curves at eachstri nger . Thus, the effectiveness of the st ri ng er s in ar re st in g a crack may increasesignificantly with de creasing rive t spacing. It is important to note, however, that inte-gra l s t r ingers (a limiting cas e of riveted s tr in ge rs with sma ll rive t spacing) are not themost effective str inge rs for ar rest i ng o r impeding cracks because the cr ack fron t als otends to pro gre ss through the stri ngers. A s the r ivet spacing decr ease s, the str es s-intensity factor approaches the lower bound, which corresponds to the case of a continu-ously attached str inge r (ref. 9). When the cra ck tip is between st ri ng er s, this lowerbound is apparently reached with the s ma ll rivet spacings.

Figure 3 shows the effect of s tr in ge r spacing on the relationship between the s t r e s s -intensity factor and half-cra ck length for both c rac k locations.s t r inge r s is fixed at p = 0.5. In or de r to show dire ctly the effect of s tri ng er spacing,

the half-crack length and riv et spacin g a r e norma lized by dividing by a fixed value ofst ri ng er spacing bo where p/bo = 1/6. The curv es for the sm al le r values of s tr in ge rspacing, a! = 1/2 an d 1/4, show that the stress-intensity factor is also lower for mor eclosely spaced st rin ger s except at the minimum points. These exceptions occur at theminimum points because the mo re widely spaced st ri ng er s have a larger individual stiff-ne ss and, thus, a la rg er local effect.lower bound on the stress-intensity factor for decreasing stringer spacing.

The st iffness of the

The curve for a = 1/4 appears to represent a

It is important to note in figures 1 to 3 that the curves for the stress-intensity fac-


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that must be transferred from the sheet to the str i ng ers is btS pe r stringe r. By adding

this increment to the applied forc e the limiting force in a str i nge r becomes

By substituting equation (10) into equation (6) the limiting value of L becomes

btELlim = 1 +

-tSES

The right-hand side of equation (11) is the reciproca l of the stri nger stiffness ratio p.(See eq. (9).) Thus,

Values of Llim calcu lated by equation (12) are shown in fig ures 4 to 6 as horizontaldashed lines.most highly loaded st ri ng er can be much la rg er than the stress applied remotely to thestringers. Therefore, str i nge rs with sma ll st iffness a r e mo re l ikely to fai l than str i nge rs

with l arg e stiffness when the sh eet is cracked.

The curves in figure 4 show that for sm al l values of 1-1 the stress in the

Figures 5 and 6 show that for lar ge r rivet spacing and for la rg er stri ng er spacinglonger cracks are required for L to reach Llim.


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The limiting values of r iv et fo rce observed in figures 7 to 9 cannot be derived likethose for the stringer force given by equation (12). However, values of r ive t force for

very long crack s were estimated from graphs of ri vet force plotted against the inver seof crac k length. Th ese est ima ted values of limiting rive t fo rce are plotted in figure 10against p for the various values of CY and p/b. The curves show that the limitingvalue of ri ve t for ce is sma ller for st iffer or mo re closely spaced str in gers o r fo r moreclosely spaced rivets.

CONCLUDING REMARKS

The stress-intensi ty factor and forc es in the most highly loaded str in ge r and rivetw e r e calculated for a cracked sheet with riveted and uniformly spaced stri nge rs. Thecalculations we re made for two sym met rica l ca se s of crac k location - the case of a crackextending equally on both s id es of a st ri ng er and the case of a cr ack extending equally onboth s id es of a point midway between two stri nger s. The results a r e presented in the

form of design graph s fo r a system atic variation of crac k length, str in ge r stiffnes s,rivet spacing, and str in ge r spacing.

Fo r sm all crack s the stress-intensity factor for a stiffened sheet is essentiallyequal to that of an unstiffened sheet. Fo r longer crac ks the stress-in tensity factor isreduced significantly by the stri nge rs. The stress-in tensity factor is smal l e r fo r s t i f f e ror mo re closely spaced str ing ers or for mo re closely spaced rivets. However, thestress-intensity factor has a lower bound with res pec t to each of these par am et er s, andan optimum value of each pa ra me te r may exist when other design considerations a r e


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APPENDIX A

EQUATIONS FOR DETERMINING INFLUENCE COEFFICIENTS OF

CRACKED SHEET AND STRINGERS

Cracked Sheet

Applied uniaxial stress.- For the infinite sh eet containing a cra ck in ske tch (f), thedisplacement in the y-direction for a state of plane stress can be developed fro m refer-ence 19 as

where


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APPENDIX A - Continued

Rivet forces.- The displacem ents for a cracked sheet with symmetric ally appliedpoint forces, shown as p a r t C in sketch (g), a re obtained by the addition of the displace-

ments f or par ts A and B. Part A shows a sheet that is subjected to symmetrically appliedpoint for ces but does not contain a crack. The internal tensile st re ss u that existsalong y = 0 between 1x1 5 a is shown as u Part B shows a shee t that contains acra ck along y = 0, 1x1 6 a, with the compressive str es s -ayy applied to the crack s ur -face. The addition of parts A and B therefore re su lt s in uyy being ze ro along y = 0,1x1 5 a. The other boundary condition T~ = 0 along y = 0 is satisfied by symmetry.

- YY

YY' -

Yt

Par t A

Y

t

- x +

Part 0

Sketch (g)

Y


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APPENDIX A - Continuedr -

v = - 4 k 3ntE- u ) ln(x2 + y2) + (1 + v)

and(T = -

YY 4Trt + Y

Equation (14) cannot be used in its present form to calculate the displacement caused bythe force Q because ther e is a singularity at the origin. To circumvent this problem,the Doint forc e Q is assumed to be distributed uniformly over the rivet diameter d.-Replacing x by x - x in equation (14) and integrating with respect to x fromto -d , the displacement becomes

2

1

--d2

(1 6)

For d - 0, equation (16) reduces to equation (14). By translating the origin in equa-tion (16) s o that the point force is located at a point (xo,yo) and by adding the displace-ments for point forces located at @xo,*y0), the displacement field for p ar t A of ske tch (g)is written as


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and

2(x - xo)d

Yl=

2 ( x + x o )

2(Y - Yo)

2(Y4-

Yo)

da 2 =

3 =

4 =

d

d

By using the sam e procedure and equation (15), the st r e s s field for part A of sketch (g) is

written as

f

UYY= w p ) ( 1 4 4 + 2 + 44 ? + a 33 2 4 % )Y2+cY3 7

The stre ss cyy that must be removed in part A of sketch (g) is obtained from equa-tion (18) by setting y = 0 and x = 5. Thus,


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APPENDIX A - ContinuedThe displacement for par t B of sketch (g) is determined by using the displacements

for the point fo rces applied to the c rack s urfa ce in sketch (i) as a pseudo-Green's function.

Y

I

1Sketch ( i )

The Westergaard st r e s s , unction (see ref. 21) for this problem is given as

where z is the complex variable x + iy. For plane stress, the y-component of displace-ment is given by


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and


c2 = /By substituting tcYy5over the crack, the displacements for p a r t B of ske tch (g) become

for P in equation (22) and by integra ting with r esp ect to 5

VB = - a A d 5nE 0 YY

Now, by substituting equation (19) for FU, equation (23) can be written

(23)

By adding the displacem ents for p a rt s A and B (sketch (g)), which are given by equa-tions (17) and (24), the disp laceme nt fo r a cracked shee t with symm etrically applied pointforces becomes

r-


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X

Sketch (j )

Rivet forces.- The displacemen ts in the finite width str in ge r subjected to a pai r ofpoint forc es i n sketc h (k) can be approximated .by the displa cement s in the infinite she etsubjected to pa ir s of equal and uniformly spaced point forc es. The only boundary condi-tion for the finite width stringer that will not be satisfied is uxx = 0 along x = &w/2.


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APPENDIX A - Concluded

1 + ~ ) ( 3 v)Q8 ~ t s E s

v = (

where

and

for n = 1, 2, . . . . The influence coefficien ts AS. are determined from equation (27)by setting the rivet force Q equal to unity.

11


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APPENDIX B

DESIGN GRAPHS

Figures 11 to 16 show graphs of the stres s-in tens ity f actor, the stringer-loa d-concentration fa ctor for the mos t highly loaded str ing er, and the rive t force for the mos thighly loaded rivet plotted against the ratio a/b for various values of the rati os p/band p . The stress-int ensity factor and the rivet fo rce are made nondimensional bydividing by the stre ss-i nten sity fa ctor of an unstiffened shee t S F a and the force btS,

res pect ive ly . Values of d/p = 1/4, w/d = 5, andv

= 0 .3 were used to represent therive t diameter, st rin ge r width, and Poisson's ratio, respectively. Although the pointsof attachment are referred to as being riveted, the results apply equally well to spot-welded attachments. The rang e in values of p and the rat io p/b are sufficient toinclude most sti ffener-s heet configurations. The curves are shown for values of a /b 2 3and, thus, include crac k lengths up to si x time s the str ing er spacing. The graphs a r eshown for two symmetrical cases of crac k location - the case of a cra ck extendingequally on both sides of a stringer and the case of a cr ac k extending equally on bothsid es of a point midway between two str ing er s.

The curves fo r the sym met rica l ca ses of crac k location may be used to obtain an

upper bound on the stress-intensity factor for a nonsymmetrical c rack by entering a curvewith the lar ge r value of half-c rack length. The row of ri vet holes through which the cra ckis assume d to advance is not accounted for in the calculations for the stress-intensity fac-

tor. However, crack-propagation tim es calculated by using thes e values should be con-t i b f th ti i d f k t i iti t f h l Th t


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REFERENCES

1. Romualdi, J . P.; F r a s i e r, J . T.; and Irv in, G. R.: Crac k-Ext ensio n-For ce Nea r aRiveted Stiffener. NRL Rep. 4956, U.S. Navy, Oct. 11, 1957.

2. Sanders, J . Lyell, Jr.: Effec t of a String er on the S tr es s Concentration Due to a Cracki n a Thin Sheet. NASA TR R-13, 959. (Supe rsedes NACA TN 4207.)

3. Romualdi, Ja m es P.; and Sanders, Paul H.: Fr ac tur e Arr es t by Riveted Stiffeners.AFOSR TR 60-174, U.S. Air Force, Oct. 1960.

4. Isida, Makoto; and Itagaki, Yoshio: St re ss Concentration at the Tip of a CentralTransv erse Crack in a Stiffened Pla te Subjected to Tension.Fourth U.S. National Co ng res s of Applied Mechanics, Vol. Two, Amer. SOC. Mech.Eng., c.1962, pp. 955-969.

Proceedings of the

5. Leybold, Herbert A.: A Method fo r P redict ing th e Static Strength of a Stiffened SheetContaining a Sharp C entral Notch. NASA TN D-1943, 1963.

6. Greif, Robert; and Sanders, J. L., Jr.: The Effect of a Stringer on the Str es s in aCracked Sheet. Trans. ASME, Ser. E: J . Appl. Mech., vol. 32, no. 1, Mar. 1965,pp. 59-66.

7. Isida, M.: The Effect of a Stringer on the St re ss Intensity Fa cto rs fo r the Tension of aCra cke d Wide Plat e. Dept. Mech., Lehigh Univ., Sept. 1965.

8. Isida, M.; Itagaki, Y.; and Iida, S.: On the Crack Tip S tre ss Intensity Fa ctor f or theTension of a Cent rally Cra cked S tri p With Reinforced Edges. Dept. Mech., Lehigh


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14. Whaley, Richard E.; and Kurzhals , Peter R.: Fatig ue-C rack Propaga tion in Aluminum-Alloy Tens ion Pan els . NASA T N D-543, 1960.

15. Leybold, He rb ert A,: Resid ual Static Strength of Aluminum-Alloy Box BeamsContaining Fatigue Cr ac ks in the Tensi on Covers. NASA TN D-796, 1961.

16. Paris, Pa ul C.; and Sih, George C.: Stre ss Analysis of Cracks. Fra ctur e ToughnessTesting and Its Applications. Spec. Tech . Publ. No. 381, Am er. SOC. Testin g Mater.,c.1965, pp. 30-81..

17. Paris, Pa ul C.: Application of Muskehlishvili's Methods to the Ana lysis of Crack Tip

Stre ss Intensity Fa cto rs for Plane Problems. Pt. III. Inst. Research, Lehigh Univ.,June 1960.

18. Newman, J . C., Jr.: St res s Analysis of Simply and Multiply Connected RegionsContaining Cra ck s by the Method of Boundary Collocation. M.S. The sis , Vir gin ia

Polytech. Inst., May 1969.

19. Westergaard, H. M.: Bearing Pre ssu re s and Cracks. J . Appl. Mech., vol. 6, no. 2,

June 1939, pp. A-49 - A-53.20 . Love, A. E . H.: A Tre at ise on the Mathematical Theory of Elasticity. Fourth ed.

(F ir st Ame r. Print ing), Dover Publ., 1944, p. 209.

21. Irwin, G. R.: Analysis of Str ess es and Strains Near the End of a Crack Traversinga Plate. Trans. ASME, Ser. E: J. Appl. Mech., vol. 24, no. 3, Sept. 1957,pp. 361-364.


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tI I k -1 1 1 . 1 1 1 1 1

0 .5 10 15 2.0 2.5 3.0a/b

(a) Cr ack extending equally on both s ide s of s tri ng er.


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-.2 -

II a I 1 I I I t I I I I

0 ' .5 10 15 2b 2.5 3.0a/b

(a) Cr ack extending equally on both si de s of str in ger .


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t

*TS i

.2 ia

-1/4.2

I I I I 1 1 I I I I 1 I0 .5 Io 15 2.0 25 3.0

(a) Crack extending equally on both si de s of stringer.


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PO y- L b= p- 1L .3

5

I . 1 1 - j

15 2.0 2.5 3.0

a/b(a) Crack extending equally on both sides of str inger.


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b

P/b

1/12,h=p-l

- - -.0--

- -L

a/b

(a) Crack extending equally on both side s of str inger.


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a p 0

(a) Cr ack extending equally on both si de s of st rin ge r.


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*8tQ

btS

0 .5 1.0 1.5 2.0 2.5 3.0a/b

(a) Crack extending equally on both side s of s tr inger .


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.6 1

t

PAJI

0 .5 1.0 I 5 2.0 2.5 3 .O

(a) Crack extending equally on both sides of stringer.a /b


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QbtS-

c aI

t / /I 1 L . . . l - 1 - ! - - 1 I I I 1 1

0 .5 1.0 1.5 2 o 2.5 3.0a/b,

(a) Crack extending equally on both sides of stringer.


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l i m (Q/botS)Wb+W

0 I I 1 I I I A - I J

et a - 4


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WUI

a a

a - aa

-

t a - 4

.. . . . . . . . . . . . . . . . . . . . .I o

r -

.9 I

21 3

.aI13

* _ _ - _ _........ . . . . . . . ......... . . ..--*....l ...*.... ' , .- :,. . . . . . . . . . . . . . . . _ _ 1 _ _ . _ ...... ....,... . . . . . ... . . . .+ . . . . ........ - . . . . . . . . . _...< . . , ....0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

a / b

(a) p = 0.05.

Figure 11.- Stress-intensity factor fo r crack extending equally on both sides of string er.


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wQ, 1

I

.w

.9

.8

.7

.6

: .5

.4

.3

.2

. I

0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.(

213

113

I16

1/12

a /b

(b) 1-1 =0.10.

Figure 11. - Continued.


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o0 - 0

0 0 0

0P

...... _ _ _

.a

7P/b

I

21.3

.6

s 2 . 5I 3

I16

1/12

4

.3

.2

... . . . . . . . . . . . . . . .-

. . . . . . . . . . . . . . . . . . . . .

.I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3 .Oa/ b

(c) p = 0.30.

Figure 11.- Continued.

w


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w03

a/ b

(d) 1-1 =0.50.

Figure 11. - Continued.

-


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0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

wCD

a/ b

(e) 1-1 = 1.00.

Figure 11. - Concluded.

CP


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0

(a) 1.1 = 0.05.

Figure 12 . - Stringer-load-concentration factor in most highly loaded stringer for crack extendingequally on both si de s of str in ge r. (Scale of ordi nate is different for some par ts of thi s figure.)


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(b) 1.1 =0.10.



44/64

a/ b

(c) =0.30.



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a/ b

(d) p =0.50.

Figure 12.- Concluded.

AA


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Qbt S

_._-. ~~

_____-_- -~

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a/ b

(a) p = 0.05.

Figure 13 . - Forc e in most highly loaded rivet fo r crack extending equally on both sides of stringer.


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(b) p =0.10.

Figure 13 . - Continued.


48/64

I.o

.9

.8

.7

Qb tS

.6

.5

.4

.3

.2

.I

0

P/b

2/3I13I

I16

1/12

.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a /b

( c ) p =0.30.



49/64

0 .2 4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a /b

(d) 1.1 =0.50.



50/64

. . . , . . . . . . . . , . . .

. - . . -

. . . . ,

. . . . .. . . . . . .

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a /b

(e) p = 1.00.

Figure 13 . - Concluded.

-&&=


51/64

I.o

.9. . .

.8. . . . . . . - . . . . . . . .

.7

.6

P/ bI

213

1/12

... . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .. . . . . .

....

... . . . . . . . . . . . . . . . -___ ._____. . . . . . . . . . . . . . . . . . . . . . . . . - - - _ _ _ ~. . . . . . . . . . . . . . . . . ._L. . . . . . . . . . .

_ I

. . _ . _ , _ . _ . , * , . _ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,. ] :~ I:*I .- _ I ,

, , I , I . ,I . .. . . . . . . . .., ... ....,.. . . . . . . , . . , . . , . . . . _ .

0 .2 .4 .6 .8 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a / b

Figure 14.- Stress-intensity factor forcrack extending equally on both s id es of point midway between two str in ge rs .rpEo


52/64

.1

P0

0

0 0.f

_ _

P/bI

213

. .

I/3

I/6

I I?

- . . . . . . .

.. -. . . . . . . . . . . . . . . . . . . . . .. . . . .. . . .. . . . . . . . . . . . . . . . . ~ .. . .-

. . .. . . .

. . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . .

a / b

(b) p = 0.10.


e


53/64

P/bI

213

I/3

I16I/I2

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a / b

(C) /J = 0.30.

Figure 14. Continued.


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0

0

. . . .

. . . ._ _ _ . _ _ _ _ _ ~0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

a / b

(d) p =0.50.


0 0

P


55/64

0

c 0

P--.a

.9

.8

.7

.6

1

.3 213

113

.2 1161/12

. I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a/b

(e) p = 1.00.



56/64

I I

10

9

8

7

L 6

5

4

3

2

I

I 3

2/3

1

a/b

(a) p = 0.05.

Figure 15. - Stringer-load-concentration actor in most highly loaded stringer for crack extending equally on bothsi des of point midway between two str ing ers . (Scale of ordinate is different for some pa rt s of thi s figure.)


57/64

L

P i

1/12

1/6

2 /3

I

3

(b) p =0.10.

Figure 15 .- Continued.


58/64

a / b

( c ) 1-1 = 0.30.

Figure 15 .- Continued.


59/64

m4

a/ b

(d) p = 0 . 5 0 .



60/64

Figure 16.- Force in most highly loaded rivet for crack extending equallyon both sides of point midway between two stringers.

0P


61/64

0

I o

.9

.a

.7

.6

1/12

1/3213

I.3

.2

.I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a /b

(b) p = 0.10.


ua0


62/64

- e0

I o

.9

QbtS-

.0

.7

P/b

2/3t/3

I16

.6

.5 I

.4

1/12.3

- - __-

.2

.I

0 .2 .4 .6 .8 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a/b

(c) p =0.30.


0

0P


63/64

.31/12

.2

.I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 18 2.0 2.2 2.4 2.6 28 3.0a /b

(d) p =0.50.



64/64

r

0 . 2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0a/ b

(e) p = 1.00.

F i m e 16 . - Concluded.

m03Nm

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Cc Poe Stress Intensity Factor