Interaction of Crippling and Tors Ional-flexural

7/28/2019 Interaction of Crippling and Tors Ional-flexural

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NASA CONTRACTORREPORT

Report No. 61287

INTERACTION OF CRIPPLING AND TORS IONAL-FLEXURALINSTABILITY FOR CENTRALLY LOADED COLUMNSBy J. C. WallsBrown Engineering, A Teledyne Company300 Sparkman DriveHuntsville, Alabama 35807June 1, 1969

Prepared forNASA-GEORGE C . M A RS H AL L S P A C E F L I G H T C E N T E RMarshall Space Flight Center, Alabama 35812


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TEC H N IC A L R EPOR T STAND AR D T IT L E P A1 1 . RE P ORT NO. 12. GOV E RNME NT A CCE S S ION NO. 13. R E C I P I E N T ' S C A T A L O G NO.NASA CR-6 1287 I4. TlTLE A N D S U B T I T L E 5. R E P O R T D A T EJune 1, 1969

6. P E R F O R M l NG O R G A N l Z A T l O N CUDEINTERACTION O F C RIPPLIN G AND TORSIONAL-FLEXURALINSTABILITY FOR CENTRALLY LOADED COLUMNSIJ. C. Walls

Brown Engineering, a Teledyne Company300 Sparkman Drive

George 6. Marsh all Space Flight CenterAstronautics Laboratory

Distributian of this r epo r t i s p r o v i d e d in the in te res t oinformation exchange. Responsibility for the contentsreside s in the author or organization that prepa red it.

Thi s report presents an empi r i cal t echni que to predi ct f ai l ure l oadsf or cent ral l y l oaded col umns w th t hi n- wal l ed open cross secti ons whi chmay f ai l by an i nteracti on of torsi onal - f l exural buckl i ng and cri ppl i ng.No attempt has been made to generate or correl ate test data to conf i rmthe accuracy of the techni que.

STAR ANNOUNCEMENT

~ F CForm 3292 M ~ Y969)


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FOREWORD

I t i s expected that the method presented i n thi spaper w l l prove val uabl e w thi n the l i mt at i onsset f orth.No attempt has been made to f ormul ate a t heoryto i ncl ude the coupl i ng of torsi on and f l exureand l ocal buckl i ng.A s an extensi on and modi f i cat i on of exi st i ngequati ons, however, i t provi des a ready tool f orsol ut i on of a part i cul ar cl ass of probl ems.

C E McCandl ess, Techni cal Edi tor


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TABLE OF CONTENTSSECTION PAGE

S U M M A R Y O . . . . . . . . . O . . . . . . . . . . . . O . . . 1I . INTRODUCTION. . . . . . e . . . . . . . . . . . . . . . . 1IS. cRIPPLINGSTRESS0 0 . 0 . . D . . . . . . . * . . 3111. TORSIONAL-FLEXURAL INSTABILITY. . e . . . . . . . 5

A. Two Axes of Symmetry. . . . . . . e . . . . . 55. One Axis of Symmetry. . . . . . . . . . . . . . .

C. General Cross S e c t i o n . . . . e . . . 6IVe INTERAGTIONANALYSISo o 0 o o e o 0 o o 0 e e o e o a o 0 o 7V. EXAMPLEPROBLEM. . o . . e D . e o . 11VI, C O N C L U S I O N S . o . . . . . D . . . . . . O . O D . . . . . . . 5

REFERENCES . . e e e e e e o . 16LIST OF ILLUSTRATIONS

FIGURE TITLE PAGE1. Dis to r t ion of Cross S e c t i o n an d S t r e s s D i s t r i b u t i o nf o r C r i p p l i n g . . . . . I . . . . . . . . . . . . 32, P l o t of P versus L' f o r Column, . . . . . . 83 , P l o t of J ohns on - E u l e r E qua t i on . . , . . . . e . 84 . P, versus L' for Example Problem. . . . . . . . 13

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LIST OF SYMBOLS

A10GJEL'P A

I'

p#PXPYPPCI X

4Cross-sec t iona l area ( in , * )= Moment of i n e r t ia of the se c t io n about its s h e a r c e n t e r ( i n , )= Shea r modulus of e l a s t i c i t y ( p s i )= T or s ion cons t an t ( i na 4)= Young's modulus of e l a s t i c i t y ( p s i )= Warping cons t an t o f t he se c t i on ( in O6 )= Ef fe ct iv e len gth of member ( in . )= C r i t i c a l l oa d i n p u re t o r s i o n a I b u ck li n g ( l b s )= C r i t i c a l l oa d i n t o r s i o n a l - f l e x u r a l b u ck li ng ( l b s )= C r i t i c a l E u l e r l o a d a bo ut x axis ( l b s )= C r i t i c a l E u l e r l oa d ab o ut y a x i s ( l b s )= Axia l l oad ( lb s )= C r i t i c a l l o ad i n combined c r i p p l i n g and t o r s i o n a l - f l e x u r a l b u ck l in g= Moment of i n e r t i a about x a x i s ( i n . 4 )(1bs)

ly = Moment of i n e r t i a about y axi s ( in O 4 )r o 1 - X, = Constant=xoyoP I, P2, P3 = Roots of cub ic equa t io n ( lb s)Ao , AI , A2 , A3Fce = Column failing s t ress fo r Johnson-Euler buckling ( p s i )F,, = Crippl ing s t re ss ( p s i )pPce = Column fa i l i n g load f o r Johnson-Euler buckl ing ( l b s)Pcs = Cr ipp l ing load ( lbs )Pe

= P o l a r r a d i u s of g y r a t i o n a bo ut s h e a r c e n t e r ( i n . )(4Dis tance f rom cen t r o id to shea r c en te r a long x ax i s ( i n . )= Dis tance f r om cen t r o id to shea r c en te r a long y ax i s ( i n . )

= Constants as d e f i n ed i n S e c t i on 1 1 1 - C

= Radius of g yr a t ion ( i n . )

= Minimum Eu ler buck ling load (l b s)

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SUMMARYAn e m p i r i c a l technique i s proposed f o r p r ed ic t in g f a i l u r e loads f o r

ce n t r a l l y loaded columns wi th th in-walled open c ro ss sec t io ns which mayf a i l by a combinat ion of t o r s i o n a l - f l e x u r a l b u ck l in g and c r i p p l i n g . Byknowing the to rs io na l- f le xu ra l buckling load and t he cr ippl - ing load f o ra co lumn. one can p r ed ic t t he i r i n t e r ac t ion by a modif ica t ion of t h eJohnson-E ule r equa t ion , o f t en used t o i n t e r a c t c r ip p l ing and E ule r -typebuck ling. Although no t e s t da ta wer e u t i l i z ed f o r compar ison , i t i sthought th a t the technique i s a n a c c u r a t e means o f p r e d i c t i n g f a i l u r eloads .

SECTION I a INTRQDUCTIONFor c en t r a l ly loaded columns wi th th in-wal led open c ro ss se c t io ns ,

which f a i l a t s t r e s s e s w i th i n t h e e l a s t i c ra ng e, t h e c r i t i c a l mode off a i l u r e i s o f t e n t o r s i o n a l b u ck li ng o r a combina t ion of to r s io na l andf l ex ur a l buckl ing. The c r i t i c a l mode depends pr ima r i ly on th e geometryof t h e c r o s s s e c t i o n and t h e l e n g t h of the column. Methods a r e av ai l-a b l e t o e v a l u a t e t h i s t o r s i o n a l o r t o r s i o n a l - f l e x u r a l b uc kl in g l o ad f o rmany va r i a t io ns of c r oss - sec t ion a l geomet ry and r e s t r a in t cond i t i ons .However, a l l of the present methods of e v al u at in g t o r s i o n a l o r t o r s i o n a l -f l e xu r a l buck ling load a re based on the assumpt ion tha t the c ross-sec t iona l shape does not change dur ing buckl ing; ices, t he t he or i e s con-s i de r pr imary f a i l u r e o f co lumns as opposed t o seconda r y f a i lu r e , cha r -a c t e r i z e d by d i s t o r t i o n o f t h e c r o s s s e c t i o n ,theory which would inc lud e coupl ing of to r s io n and f le xu re and lo ca lbuckling would be extremely complicated.

The formula t ion of a

For very s ho r t columns of th in-walled open c ro ss se c t i on s th ef a i l u r e stress i s de te rmined by t he c r ipp l in g s t r e s s method, which doesp ro v id e f o r l o c a l d i s t o r t i o n of e l em e nt s o f t h e c r o s s s e c t i o n .

Therefore , the coupl ing of these two fa i l u r e modes by an emp ir ica lmeans would provide a s i m p l e method to pr ed ic t f a i lu re load s of columnswhich may f a i l by a combinat ion of th e to rs io na l- f l ex ur al mode and t h ecr ip pl in g mode. This approach has previou sly been followed i n couplingcr ip pl in g and Euler buckling fo r c losed sec t io ns ( Johnson-Euler equa t ion) .The same approach w i l l be used i n t h i s r e p o r t t o c o u p le c r i p p l i n g andt o r s i o n a l - f l e x u r a l b u c k l i n g .


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SECTION 11. CRIPPLING STRESSWhen th e co rn er s of a t h in -wa l led sec t io n i n compr ession a r e r e -s t r a ine d aga i ns t any l a t e r a l movement, t he cor ne r ma te r i a l c an con t inue

t o be loaded even a f te r buckl ing has occured i n some elem ents of th es ec ti o n . When t h e s t r e s s i n t h e most s t ab le corners exceeds i t s c r i t i c a lv al ue F c r i t , t h e s e c t i o n l o s es i t s a b i l i t y t o suppor t any addi t iona lload , a n d f a i l s .

F igur e l ( a ) shows the c r oss - sec t io na l d i s t o r t i on occurr ing ove r onewavelength i n a t yp ica l t h in -wa l led sec t i on ,stress d i s t r i b u t i o n o ve r t h e c r o s s s e c t i o n j u s t b e f or e c r i p p li n g .

Figure l (b) shows the

C r i pp l i n g f a i l u r e o c c u rs a t extremely sh or t column leng ths. Thec r i p p l i n g l o a d of a member i s equa l t o t he p r oduct of t h e c r i p p l i n gs t ress and the ac tua l a r ea of t h e member.

Empirical methods of p r e d i c t i n g t h e c r i p p l i n g s t ress Fcs of extrudedand formed sheet metal elements are r e a d i ly a v a i l a b le i n t h e l i t e r a t u r e(Ref, 1 , 2 ) ,

F I G U R E 1. D I STO R TI O N OF CROSS SECTION A N D STRESS ~ I S T R I B ~ ~ I O NOR CRIPPLING

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SECTION 111, TORSIONAL-E'LEXURAL INSTABILITY

Ce nt ra l ly loaded columns can buckle i n one of thr ee po ss ibl e modes:(1) They can bend i n the plane of one of the p r in c i pa l axes ; o r (2 )they can t w i s t a bo ut t h e s h e a r c e n t e r a x i s ; o r (3) they can bend andt w i s t simultaneously. For any giv en member, dependin g on i t s l eng th andthe geometry of i t s c r o ss se c t i on , one of t he th r ee modes w i l l b e c r i t i -cal. Mode (1) i s th e well-known Eu ler typ e of bu ckl ing , Modes (2) and( 3 ) w i l l be discussed below,

A. Two Axes of SymmetryWhen the c ro ss s e c ti o n has two axes of symmetry, o r i s poin t

symmetric, th e shear cen te r and cen t ro id w i l l c oi n ci d e. I n t h i s c a s e ,the pur e ly to r s io na l buck ling load about t he shea r c en t e r ax i s i s givenby (Ref. 3)

(1)- - - - - - - - - Y - - - - - - - - - - - -- Appl - -O i-. (L ' I ' - ' 12where A = c r o s s - s e c t i o n a l areaIo = moment of i n e r t i a of t h e s e c t i o n a bo u t i t s s h e a r c e n t e rG = shea r modulus o f e l a s t i c i t yJ = t o r s i o n c o n s t a n tE = Young's modulus of e last ic i tyr = Warping consta nt of th e sec t i onL' = Ef fe ct iv e length of member

-- I, +Iy

Thus, for a c r o s s s e c t i o n w i th two axes of symmetry there are t h r eeva lues o f t he ax ia l l oad. They are t he f l ex ur a l buck ling loads abou tt h e p r i n c i p a l axes (Px and Py) and th e purely t or si on al buckling loadP+; then plc i s th e lowest value of th e th re e loads. The lowest valuew i l l depend on the shape of cr o ss se ct io n and len gth of member. Withtwo axes of symmetry, there i s no i n t e r a c t i o n and t h e column f a i l s i ne i th e r pure bending o r pur e tw i s t ing . Shapes i n t h i s c at egor y inc lude1 - sec t io ns , Z -sec t ions , and c r uc if o r m sec t ion s .

B. One Axis of SymmetryI f t h e c r o s s s e c t i o n ha s one axis of symmetry,say the x a x i s ,t h e e q u a t i o n t o o b t a i n t h e bu c kl in g l o ad s i s (Ref. 3)

(2)0 _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ - - _ _ _2 E I ro = 'Q

A9 p y = Y(L'1where Px =

(L' 2

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= A + :IO@nd

There are agai n three sol ut i ons to equati on (2), one of whi ch i s P1=Pyand represents purel y f l exural buckl i ng about the y axi s. The other t wo(P2, P3) are the roots of the quadrat i c termi nsi de the square bracket e-quated to zero, These two roots gi ve torsi onal - f l exural buckl i ng l oads.

------------ 3)2where K = 1 - (%)

Therefore, a si ngl y symmetr i cal secti on such as an angl e, channel , or hatcan buckl e i n ei ther of two modes:buckl i ng.and shape of the gi ven secti on. Theref ore, f or si ngl y symmetr i calsecti ons W: i s the l owest posi t i ve val ue of Pi , P2 and P3.

I

by bendi ng, or torsi onal - f l exuralWhi ch of these two actual l y occurs depends on the di mensi ons

6. General Cross Sect i onI n the general case of a col umn of thi n- wal l ed open cross sect i on,buckl i ng w l l occur by a combi nat i on of torsi on and bendi ng.f l exural or purel y t orsi onal buckl i ng cannot occuroto obtai n the buckl i ng l oads i s:

Purel yThe equat i on (Ref. 3)2 2 2 2 2ro (P-PY)(~px)(~~) - P yo (P-P) - P xo ( P- P~)= o - - - ( 4 )

Thi s equati on reduces to

where2 2A3 = - ( -Yo -xo 1 +1IO

A1 = Px P + P P +P@PxY Y @

Sol ut i on of thi s cubi c equat i on yi el ds three root s, Pi , P2, and P3.smal l est posi t i ve val ue w l l be the cr i t i cal l oad W. The

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SECTION IV, INTERACTION ANALYSIS

The tors ion a l - f lex ura l buckl ing load, Pt , obtained from Section 111,may, f o r sh or t column len gt hs , be g r e a t e r t han the c r ip p l in g load ob-ta ined f rom Sec t ion 11. However, the c r ip pl i ng load re pres ent s the upperl i m i t of th e load-carryin g cap aci ty of th e column (Fig. 2 ) . Thus, formembers whose lengths are such that cr ippling and t o r s i o n a l - f l e x u r a lb u c k l i n g i n t e r a c t , a means of interact ing the two loads i s d e s i r a b l e .

The well-known Jo hnso n-Eu ler eq ua ti on which prov ide s a means ofin t e r ac t ing c r ipp l ing and E ule r buck l ing i s u s u a l l y g i v e n i n t h e f ol lo w-

where Fce = column fa i l ing s t ressFcs = c r i p p l i n g s t ress f o r t h e g iv e n c r o s s s e c t i o nLP = r ad ius o f gyr a t ionE = Youngs modulus

= ef fe c t iv e length of member

T hi s equa t ion g ive s a pa r abo l i c cur ve s t a r t i n g f rom t h e c r i p p l i n g stressa t L/P= 0, and becomes tang ent t o th e Eul er curve a t a s t r e s s valueequa l t o one- hal f t he c r ipp l ing s t r e s s , Figure 3 shows a p l o t of t h i sequa t ion f o r a luminum a l lo y ma te r i a l f o r va r ious va lue s o f t h e c r i p p l i n gstress.

where Pce = column fa i l ing loadPcs = c r i p p l i n g l o a dPe = * 2E1 = minimum Euler buckling load(L)*

To p r o v i d e a r e l a t i v e l y s i m p l e means of coupl ing c r ippl ing andt o r s i o n a l - fl e x u r a l b u c k li ng , a l l t h a t i s r equ i r ed i s t o modify theJohnson-Euler equation (7) to the fol lowing equa t ion:

2( 8c 3 ------------- --------- --------- ---P, = P,, -

4w

By comparison of eq ua ti on s (7) and (8 ) i t can be seen tha t t heo n l y d i f f e r e n c e i s th a t Pe of eq ua t ion ( 7 ) i s replaced by Prc, where p?;

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P

.-LL0

\ r / - RIPPL ING STRESS-" I- - -I. - -

4L.L'

F IGURE 2. PLOT OF P VERSUS L' FOR COLUMN

1 J OH NS ON '!,1 C U R V E S 7 "1

F IGURE 3. P L O T OF JOHNSON-EULER EQUATION

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SECTION V. EXAMPLE PROBLEM

For t he s e ct io n shown below, p l o t Pc versus L ' using equa t ion (8).Also show Eu ler buc kling curve and Johnson-Euler curve .

c - c e n t r o i ds - s h e a r c e n t e rYGiven Pc, = 308,000 l b s .

A = 3.5 i n .I, = 22.5 i n ,

= 6.05 i n .X o = 2.74 i n ,G = 4.0 x 10 p s i

24

4

IY6

tiE = 10.5 x 10- p s i46r = 38.4 i n . J = .073 i n .

F o r c a l c u l a t i o n of P, aid P: r e f e r t o S e ct i on 111 B e96

22 2

n 10.5 x 10 G2.5) = 2.331 X 10n EI, =p x -(L ' l2 (L ' )

2Io = I,+ I + A xoIo = 22.5 +6.05 +3.5 (2.74)

4Io = 54.75 i n .

Y2

2

60

=18,660+254.39 x 10(L ' l2- 54.75

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r ' = -o = 54.75 = 15.650 A 3.5

p2,pi' 43 the col umn w l l f ai l i n pure torsi onal -of t ransi t i on f romt he cri ppl i ng l oad at: L ' = 0 to t orsi onal - f l exuralbuckl i ng at L ' = 43,f l exural buckl i ng. The J ohnson- Eul er curve and Eul er curve are shownf or compari son,

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as

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SECTION VI. CONCLUSIONSThe modified Johnson-Euler equation (Equation 8) is presented as amethod of predicting failure loads of centrally loaded columns withthinwalledopen cross sections which fail by an interaction of cripplingand torsional-flexural buckling.eccentric loads. Equation (8) should not be used with

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REFERENCES

1. Bruhn, E. F., Analysis and Desipn of Flipht Vehicle Structures,First Printing , Tri-State Offset Company Cincinnati Ohio, 1965.2, Astronautic Structures Manual, Marshall Space Flight Center,Huntsville, Alabama.3. Timoshenko, S. P,,and Gere, J. Me, heory of Elastic Stability,2nd ed., McGraw-Hill Book Company, Inc., New York, 1961.

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