nasa beam+column stability

7/27/2019 Nasa Beam+Column Stability

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T E C H N IC A L N O T E N A S A TN_ D-5782e 1

THE STRUCTURAL INSTABILITYF BEAM C OLU MN S, FRAMES, A N D ARCHES

JeweZZ M . Thomm iC. MarshaZZ Space FZight Center 1Ah . 35812

A E R O N A U T I C S A N D S PA C E A D M I N I S T R A T I O N W A S H I N G T O N , D. C M A Y 1970


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TECH LIBRARY KAFB, NMllllllllllllllllllllllrllllllllll~llllll~~1. REPORT NO. GOVERNMENT ACCESSION NO. 0132513NASA TN D-5782 I I4. TIT LE A ND S UBTITLE 15. REPORT DATEA Finite Element Approach to th e Struc tura l Instability of 16: PzO:tz: ORGANIZAT ION CODEBeam Columns, F ra me s, and Arch es

7. AUTHOR S)Je r re l l M. Thomas9. PERFORMING ORGANIZATION NAME AN0 ADDRESSGeorge C. Marsh all Space Flight Cen ter

Mars hall Space Flight Cen ter, Alabama 3581212. SPONSORING AGENCY NAME AN0 ADDRESS

Nati ona l Aeronautics and Space AdministrationWashington, D.C. 20546~~15. SUPPLEMENTARY NOTES

Pre pa re d by Astronautics LaboratoryScience and Engineering Dire ctor ate

16. ABSTRACT

i8 PERFORMING ORGANIZATION REPOR rM-35110. WORK UN IT NO.981-10-10-0000-50-00-08

11 1. CONTRACT OR GRANT NO.

13. T Y P E OF REPOR-; & PERIOD COVERE

I Technical NoteSPONSORING AGENCY CODEI-.I.I

A nonlinear st if he ss m atrix f or a beam column element subjected to nodal forc es andto a uniformly distr ibute d load is developed fro m the principle of virtu al displacements andthe bifurcation theory of el as tic stability. Thr ee ca ses of applied load behavior a r e conside reThe buckling load of a uniform circular arch is calculated as an example.

17. KEY W O R D S 18. OlS TRlB UT lON S TA TE ME NTUnclass i f i ed - Unlimited

19. SECURITY CLASSIF. (of this report) 20. SECL IF. (of this pege) 21. NO. OF PAGES 22. PRICE*iUnclassified I Unclassified 58 3.00


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T A B L E OF CONTENTS

INTRODUCTION.... ................................... 1G E N E R A L T H E O R Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Basic Assumptions and Limitations . . .. . ... . . .. .. .. .. .. . 3Method of Analysis .. . .. . ... .. .. .... . . . . . . .. .. . . . .. 4Development of Element Stiffness M atrix . . . . . . . . .. .. . .... 5

Descri ption of Element . . . . .. . .. .. . . .. .. . . . .. . . 5Displacement Functions . . . . .. . .. . . .. . .. . . .. ... 5Development of Equilibrium Equations . . . . . .. . . . . . . . 7Bifurcation Theory of Instability ....... . . . .. . . . . . . 16onventional Stiffness M a t r i x , K .... . . . . . . . . . . . . . 20Geometric Stiffness Matrix, ,G . . . . . . . . . . . . . . . . . . . 21Load Behavior Matrix, L. . . . . . . . . . . . .. . . . . . . .. . 28

Coordinate Transformation ... . . . . . . . . . . . . . . . . . . ... . . 32Maste r Stiffness M a t r i x . . . . . . . . . . . . .. . . . . . . . . . . .. . . . 35Stability Cri terion . .. . . . . . . . . . .. . . . . . . . . . . . .. . .... 37Pri mar y Equilibrium State. . .. .. . . . . .. . .. . . . . . . .. . ... 38

APPLICATIONOF THE THEORY ........................... 39EXAMPLE P R O B L E M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41C O N C L U S I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44REFERENCES ........................................ 46

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l l l l I l l I I I I

L I S T OF I L L U S T R A T I O N SF i g k e T it le Page

i. Beam Column Element ............................. 52. Loads Remain Normal to Element. ..................... 103. Loads Remain Parallel to Original Direction . . . . . . . . . . . . . . 114. Loads Remain Direct ed Toward a Fixed Point . . . . . . . . . . . . . 125. Coordinate Transformation .......................... 326. General Structure ................................ 357. Uniform Circular Arch. ............................ 42

L I S T O F T A B L E STable Title Page

I. Comparison of Arch Buckling Loads .................... 44

A C K N O W L E D G M E N TThis analysis was done at the suggestion of Dr. C. V. Smith of Georgia

Institute of Technology. Mr . John Key ass ist ed in checking certain portionsof the analysis. M r . Dan Roberts of Computer Sciences Corporation did all ofthe programming for the example problem. The contributions of the se peoplea r e grate fully acknowledged.

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L I S T O F SYMBOLSSymbol Meaning

A Cross sectional area of elementE Modulus of elasticity of materialF Force applied to element node-F Force applied to element node re fere nce d to undeformed

element coordinate systemF Proportionality constants between forces, F, and lateral

load, pG = [GI Geometric stiffness matrix defined by equation (46)

I Moment of ine rtia of eleme nt cr o s s sectionK = [K] Conventional stif fness matri x defined by equation (45) K f = [ Kf]* Total element stiffness matrix (see equation (44))K= K] Element stiffness matrix in global coordinate system

KO = KO1 Nonlinear stiffness matrix defined by equation (91)

L Length of elem entL = [L ] Load behavior stiffnes s mat rixkF = LF1 Nodal force behavior stiffness ma trixL ILJ Lateral load behavior stiffness matrixP1 Distance from elem ent node to point through which the FXnodal forces are directed

V


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L I S T OF SYMBO L S (Continued)Symbol Meaning

M Bending moment applied to elemen t nodeM = M Bending moment applied to element node referenced to

undeformed element coordinate systemm, n Unit vectors in element coordinate systemiii, n Unit vector s in global coordinate syst emP Lat eral load applied to elemen tQ Nodal force o r momentQ = { Q } Column vector of nodal forces and momentsg = {a} Column vector of nodal forces and moments referenced toglobal coordinate system

Nodal displacement o r rotationColumn vector of nodal displacementsColumn vector of nodal displacements in global coordinatesystem

R Distance fro m ele ment ce nterlin e to point through which late ralforces are directed; also radius of circu lar a rch

T = [TI Coordinate transformation matrix defined by equation (81)

U Strain energy of the elementU Displacement in x direction-U Displacement in Z direction

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SymbolV

W-Wx, zz, zY

P

6 U , 6V

kExx9

K

UXdlw

L I S T OF SYMBOLS (Continued)Meaning

Volume of the elementDisplacement in z directionDisplacement in E directionElement coordinatesGlobal coordina tesConstant coefficientAngle between element and global coordinate s yst em sDisp lacem ent functions defined by equation ( 6 )Change in str ai n energy, virtual work of externalforcesStrain of element in x direction at any pointStrain of element midsurface in x directionRotation of element nodeCurvature of beam element centerlineStress in x direction at any point in the elementDisplacement functions defined by equation ( 5 )Unit rotation ve ctor

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Symbol

Subscript1, 2

9 x; r m c

Superscript0I-1T

L I S T OF SYMBOLS C o n c l u d e d )Meaning

Mat rice s defined by equations (55) through (61)

MeaningDenotes left and right nodes respecti vely of the elements;also denotes a particular displacement function, yDenotes a particular @ displacement functionDenotes a particul ar elemen t of a general structureDenotes a part icul ar node of the eleme nt or one of thedisplacement functions, y or @Indicates no summation on double subs crip t, iDirection of the coordinates x and zFi rs t and second derivatives with resp ect to x

MeaningDenotes displacements of the str uct ure j ust pri or to bucklingDenotes displacements of the stru ctu re just after bucklingInv ers e of a matrixTranspose of a matrix

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A F I N IT E 'E L E M E N T A P P R O A C H T O T HE S T R U C T U R A LI N S T A B L l T Y O F B E A M C OLU MN S,F R A M E S, A N D A R C H E SS U M M A R Y

Fr om the principle of vi rtual displa ceme nts and the bifurcation theory ofelastic stability a stiffness matri x is developed for a beam column elemen t withsh ea r, moment, and axial load applied to the ends ( nodes) of the e lemen t and auniformly distri buted load applied along th e span of the element. The stiffnessmat rix developed re lat es the forc es applied at the nodes t o the displa ceme nts ofthe nodes and is a function of the magnitude of the applied load. Th re e types ofbehav ior of the applied loads are considered as separate cases. These are: theloads remain normal to the deformed element; the loads remain parallel to t heiroriginal direction; and, the loads rema in directed toward a fixed point.

A method is given for using the element stiffness matri x to predi ct thebuckling load for a st ruc tu re which may be repre sented by beam column elements.A s an example, the buckling load of an ar ch fo r each of th e thr ee load-behaviorcases is calculated and compared to known solutions.

I N T R O D U C T I O NIn recen t year s int eres t in the solution of st ruc tura l problems by the

finite element method has greatl y increased. This inter est can be attributed tothe fact that the digita l computer h as made pos sible the solution of a gre at numbe r of complex, y et pra cti cal , proble ms by this method.

Those acti ve in publishing results in the field include a group at theUniversity of Washington and The Boeing Company [ 1-91, J. H. Argyris at theUniversity of London [ I O , 111, and R. H. Gallagher and his associa tes at BellAerosy stems Company [ 12- 141.

Basically the method cons ists of r epresen ting an actual st ruc tu re by anidealized st ru ct ur e made up of el eme nts for which relationshi ps between thefo rce s applied to points (no des) on the boundary of the elem ent and the displacements of the nodes is known. Thi s rela tio nsh ip is conveniently rep rese nte d by


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an element stif fnes s matrix. If all of the element stiffness matrices are transformed to a common (global) coordinate sy ste m, the sum of.forces applied tothe elements meeting at each node is equated to the externall y applied forc e atthat node, and the disp lace men ts of el eme nts which meet at each node a r e equated,a s e t of equations relating the externally applied for ces t o the nodal displace mentsresults. When this set is expressed in matrix form, the matrix relating the externally applied forc es to the nodal displacements is called the ma ste r stiffnessma trix. Then fo r any given combination of known applied extern al fo rc es andnodal displac ements the above re lationsh ips may be algebrai cally manipulated todetermine internal loads and displacements throughout the structure.

Once a sufficient vari ety and quality of el ement stiffness ma tr ic es a r eavailable to idealize a particular structure, the remainder of the above procedureto solve a par ti cul ar problem becomes tedious but quite mechanical. Hence theadvantage of the computer in solving such problems is obvious.

Although the use of such techniques in solving complex st ru ct ur al problems is now an everyday occur ren ce, a gr ea t dea l of developmental activity i nthe field still persi sts fo r several reasons. First, there is no such thing as acorre ct stiffness matri x for a part icul ar eleme nt which excludes other stiffnessmatrices for that same element as incorrect. A number of co rr ec t stif fnes smatrices may be developed for the same element i f different assumptions ar emade for the derivation. Each of these ma tr ic es may have its own advantagesfor some particular application and all of them may converge to identical answersi f the st ruc tu re to be analyzed is broken into sm al l enough elements. It is thesea rch fo r the mo st advantageous element for parti cula r applications or for all-inclusive eleme nts which gives r i s e to much of the cu rre nt rese arc h.

Second, many deriva tions have been based on geom etrical considerat ionsonly and have som eti mes led to inc orrec t stiTfness mat ric es o r mat rice s withpoor convergence charact eris tics . The rece nt trend has been to base derivati onson basic elasticity principles and to use mo re c ar e in assuring that adjacentelemen t deformations conform to each other not only a t the nodes but along thecomplete boundary 15, 15-19].

Third, extension of the finite element techniques to include a larger c lassof problem s has re ceiv ed a grea t deal of re cen t attention. In particular, thesolution of nonlinear problem s including la rg e deformations and stru ct ur alinstability ha s been of i nt ere st [ 1-3, 5, 8, 9, 13-15, 17, 201.

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This rep ort was m otivated to s om e extent by all three of the above considerations. The objective was to develop a beam column element, beginningwith fundamental principles, which would account for the applied load behaviorand could be used for structural stability analysis.

It is assumed that the reader is fam ilia r with the conventional mat rix andindex notation used throughout the report. Since the re is no stringent spac elimitation, the work is prese nted in an unusual amount of detai l with the presumption that the development could se rv e as a guide fo r the development ofcurved or three-dimensional elements.

GENERAL THEORYBasic Assumption s and Lim itations

The usual Euler-Bernoulli beam assumptions a r e made for developingthe beam column element stiffness matrix. These are basic engineering assumptions and may be listed as follows:

I. The material of the element is homogeneous and isotropic.2. Plane sections rem ain plane a fter bending.3 . The stress-s train curve is identical in tension and compression.4. Hooke's law holds.5. The effect of tra ns ve rse sh ea r is negligible.6 . The deflections are sm al l compared to the cro ss sectionai dimensions.7. The loads act in a single plane passi ng through a principal axis of

inertia of the cr os s section.8. N o initial curvature of the element exists.9. No local type of instability will oc cur within the element.

I O . Loads are applied quasi-statically.

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The above ass umptions l imi t the app lication of results to s t ruc tures thatlie in a plane and are loaded in the sa m e plane. Out-of-plane and tor sionalbuckling are not treated.

Method of A n a l y s i sThe work done during a virtua l displac emen t of the elemen t is equated

to zero, as presented by Hoff in Reference 21 and derived m or e rigorously byLanghaar in Reference 22, to obtain the equilibrium equations fo r the element.The bifurcation concept of e last ic stab ility as pre sen ted by Novozhilov i n Reference 23, Chapter V , is used to postulate that two possib le sets of,displacementswhich satisf y the equilibrium equations may exist under the sa me magnitude ofextern al load i f the magnitude is such that a s t ructure is unstable. Each of thesesets of displacements is substituted in tur n into the equilibrium equations. Theresulting sets of equations are combined to obtain a relationship between thenodal fo rc es and the nodal displa ceme nts during buckling. When placed in ma tri xfor m this relationship becomes

where Qi is the column ma tri x of nodal fo rce s , is the column mat ri x of nodaldisplacgments, K is the usual beam element stiffness matrix , G is the geometrics t if fness ma t r ixrand L is the load behavior stiffness matrix. G and L bothcontain the magnit udezf external load as a factor. Thus an e le ke nt syiffnessma trix can be derived which is a function of the applied load including the appliedload behavior.

A number of suc h element s may be combined to rep res en t a particularstructure, s o that equation I ) applies to the entir e st ru ctu re and K + G + L isthe ma st er stiffne ss matrix . Boundary conditions may be applied To r&ucg thesi ze of the ma st er stiffness matrix. Fo r instability to exist the determinant ofthe ma ste r stiffness mat rix mu st vanish. Hence, an eigenvalue problem isformulated where the eigenvalues are the magnitudes of the applied load at whichthe structure is unstable.

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Development of Element' Stiffness Ma tr ixDes crip tion of Element. Consider a beam column element as shown in

Figure 1 which is subjected to nodal forc es and moments and a uniformly distributed load, p, applied along its length. It is desired to determine a stiffnessma tri x for the element which may be used to calculate the stability of s tru ctu resmad e up of such elemen ts. The stiffness matrix is to account fo r the fact thatthe components of the nodal for ce s or distributed load, o r both, may be a functionof the element displacements. Three case s ar e to be considered: the loadsremain normal to the deformed elemeht; the loads remain para llel to theiroriginal direction; and, the loads rema in directed toward a fixed point.

FIGURE I. BEAM COLUMN ELEMENTDisplacement Functions. If it is assumed that the late ral displacement

of the beam element of Figure i may be represen ted by

and the longitudinal disp lacem ent is given by

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then the si x unknown co nst ant s, a i may be determined in terms of the nodaldisplacements from the elem ent boundary conditions,

w = w i at x = Ow, X = e l at x = Ow = w 2 a t x = Lw , X = -e2 at x = Lu = u i a t x = Ou = u 2 a t x = L .

The resulting displacement functions a r e

w = w i(1 - 3 - + 2 9 + w 2 ( 3 2L3+ e , ( - x + 2 -2L2 + e2( 5)= w . @ . ( X I1 1

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The se displacement functions have been used by o thers fo r the beamcolumn element (see Reference 5, for example). However, it should be mentioned that a n inconsistency ar is e s when the cubic function is used for the lateraldispla cemen t of the pre se nt elem ent which has a distribu ted load. This is illustrat ed by taking the third der iva tive of w, which should be the equation for thesh ea r load on the element, and observing that a constant results. But the element has a distributed load, and the sh ea r should obviously be a linear function,not a constant. Since the change in she ar ov er the length of an element becomesnegligible in the limit as the element becomes small er and small er, this inconsistency is not unacceptable and it will be s een that adequate res ults are obtained.The co rre ct fourth ord er function could not be used fo r w because there is noother boundary condition available fo r evaluating another constant in equation ( 2 ) .This will be discus sed fur ther in the conclusions.

Development of Equilibrium Equations. The element equilibrium equations will now be developed fro m the principle of v irtual displac emen ts,

6 U - 6 V = O 7 )where 6 U is the change in str ain en ergy and 6V is the work done by the exte rnalforces during a virtual displacement.

For the uniaxial state of stress assumed to exist in a beam the changein strain energy is given by

6U = J ax6 exdvVHere, X is the st ra in of the beam at any point of the c ro ss section and is givenby

where is the st ra in of the beam mid -surfac e, z is the distance of the point inxxthe beam from the mid-surface, and

K = Wxx xx

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is the beam curvature. The nonlinear strain-displacement relationship1Exx = u , x + 2 w x2 + - u , x22

is used for the mid-surface stra in. The u , ~X term in this expression is notknown to have been used in previous derivations of nonlinear beam elementstiffness mat rice s. Substituting equations 9 ) , ( I O ) , and (11) into equation8 ) ,

6 U = s [,,cxx6t xx + E I Kxx 6Kxx1dxLwhered = 6 u , + d ( i w , ; ) + d (& ;) = 6 u , x + w , x 6w, x + u , x 6% xxx X

+ z I u, 2 t u , + w , dw, + u , ~ u , ~ ) + - E I w , ~ ~ ~ w, ~I w,: + z dxL X X x x X

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+ higher orde r terms. (14)The virtual work of the e xternal for ces acting on the element is given by:

where the fo rc e components depend upon the load behavior. Thr ee load behaviorcases w i l l be considered.

Case I. Loads Remain Normal to the Deformed Element (Fig. 2)P z = P

-pX - - P K x-F = F . + F exi xi zi -F = F - F ezi zi xi -Z. = M.1 1

Then

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= p c o s 9 H p 4pz II

FIGURE 2. LOADS REMAIN NORMAL TO ELEMENTCase 11. Loads Remain Parallel to The ir Original Direction (Fig. 3)

P, = Pp = oX- -Fxi - Fxi-FZi = FZiM. = M.1 1

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FIGURE 3. LOADS RENIAIN PARALLEL TO ORIGINAL DIRECTION

6V = Fx16u.1 + Fz1dwi + M. 6 8 . i - s p6wdx 19)l LCase 111. Loads Remain Direc ted Toward a Fixed Point (Fig. 4)Pz = P

PUP, =- UiFxi = Fx1+ F Z i ~- WiFZi = FZi - Fxik = M.1 1


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Fxi + F Z i % 6 ~ .+(FZi - Fxi?)6w i

FIGURE 4. LOADS REMAIN DIRECTED TOWARD A FIXED POINT

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-u

Substituting equations ( 14), (17) , ( 19), and ( 21) into equation (7), weobtain the following resu lts :

Case I.IEA(USx + z w , 2 + 3 2)6u,x + EA(u,xw,x)6W,xx 2 'x

+ EIw ,xx 6w,-1 dx = (Fxi ;FZi Bi-)6ui +(FZi - Fxi ,)6wiCase 11.

dx = F . 6u . + F . 6 ~ .Mise i+ p6wdxxx LCase 111.

LFr om the displacement functions, equations ( 5) and ( 6

u, X = u.1 yl , xw, X =wi cPi x (25)w,xx = wi cPi - .

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-

And the above variations can be written,6u = y . 6Ui1

-6u, X Yi,x 6ui6w = Gi6Wi6WYx = @i,xdWi

- 6w .6WYn - @i,xx iUpon substituting equations (25 ) and (26 ) into equations (22) through(24) and recalling that Oi = wi +2, the following equations are obtained:

Case I.

) 6 u i - ( F z i - F xi wi + 2)6w i-

Case 11.

- Fx16 u . - F z16w. -M.6w i + 2 - [email protected]=O1 1 1 1 1L14


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/

1L

+( Ju.y.J , X wk q k,x)'i,x

+ FZi )hi-iPSwi +-u. y y 6ui

Case 111.

+ [email protected] 1 ,xx Pj , x x(FZi - Fxi ') 6wi - Mi 6wi+

R J J 1 (29)

For independent v irtu al displacements , 6u. and 6wi the equilibrium1equations are obtained fro m equations ( 2 7 ) through ( 2 9 ) .Case I.

I 3j , xy i ,x + - w w $j k ,x@ .J , Xy i , x + - u u y2 k 2 k j k,xy j , xy i ,x

EAuj yj , x wk Pk , x P i , x +EIw.@. I , = Pj y = - P ( @ i ) ] hL ,

Case 11.i2 k wj @k,x @. 3[..('jyj,xYi,x +-w J , X yi , ~ + 2 % ~ j ~ k , x ~ j , x ' i , xL

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_.. . - . .. .._._ .II I 111 I ,._..

Case 111.I 3bA ' ,x i, x ZWk"j 'k, x 'j ,x 'i, x ;% "j ' x ' ,x 'i, x

W i-- Qzi + Fxi -1 = o (35)Equations ( 3 0 ) through ( 3 5 ) a r e the final equilibrium equations for the

beam column element.Bifurcation Theory of Instability. Let a solution of the equilibrium

equations be uo w oi' According to the bifurcation concept of instabi lity, at thei'instability load magnitude th er e is another set of displacements, arbitrarilyclose to the first set, which. al so sat is fie s the equations of equilibrium . Denotethis second set of nodal displacements by uo1 u1 w? + w1i .i ' i

Upon substituting this new solution into the equilibrium equations, th er eresults fo r Case I

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---- -. ........_ .. . .. . ..- -. . .

k yj , x @k , x @i , x

(37)

Similar results are obtained for Cases I1 and 111,If these equations are expanded, if the q? state terms (which themselves1

satisfy the equations) are canceled, and if only linea r te rm s in the ar bit rari lysm al l q state are retained, the following sets of equations result.1

Case I.oF ($'j,xyi,x w ji W k g k , x+.J , X y.1 , x + 3Y(:ulyk, x ,x 'i, x

+pw @. y. d x - Fzi 0'i = O ( 3 8 )J JYX 11 i , x @k , x yj , x + w ij \ '+ .i , x @.J , X yk , x + EIW +J i , x x +j , x xdx+ Fx1e-= 0 ( 3 9 )

Case 11.

Y. +W W0 k @k , x @.J , X y.1 , ~ J, ~ + 3 q u i yk,xy j, x 'i, x dx = 017


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r

~ [ E A ( u ~ w ~ ' i y x 1 ,x+. Jk , x j , x +w iu kO9. J , X k , x + E I W ~ i , = +j , = JdxCase 111.

i , x @k , x yj , x +w ij \ @ .1,x @ .J , X k , x

These equations may be expressed i n ma tri x notation sa- three casesas follows:

o r[Ki l (q'} = 0 )

where[K1] = [ K 1 + [GI+ [Lp] + [LFl

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--------- - - - - - - - -1s 3EA yk,xyj,x'i,x d x s EAwL'k,x'j,xYi,x

L

- IE A w 0 i , x k ,x yj , x dx Is EA i , x @.J , X yk,x ' L[GI (q'} =for all cases.

I

The L matrices are different for each case.Case I .

[Lpl {si} =

[L,l{ql} =

[ L 1 = [L,I=[OlPCase 11.

Case 111.

(47 1

48)

(49)


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Conventional Stiffhess Matrix 5, Equation (45) rep res ents the conventional beam column e l e m e n t a t r i x , which is suitable for use whenthere is no in ter es t in nonlinear effects o r instability. The elements of thismat rix may be derived fro m the general expressio n, equation (45) . Forexample,

fro m equations ( 5) and 6 ) . Then,

Therefore,

L L 2 I2EId x = EI(-j $+%) dx =- L30 0

Other elemen ts of K a r e obtained sim ilarly to yield:

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- -

-- -- ----

U I uz-AE LEL L FxiAE AE- L L 0 0 0 0 Fx2

12EI 12EI 6EI 6EI0 L3 L3 L2 L2 F Z i[Kl = 0 0 12EI 12EI 6EI 6EI (52)L3 L3 L2 L2 F Z 2

0 0 MI0 0 M2-

Geometric Stiffness Matrix G. Equation (46) is the general form of the~geo met ric stiffn ess mat rix which accounts for the effect of lo ads existing in theelemen t on the stif fnes s of the elem ent. For example, it is well known that theaxial load in a beam column has an appreciable effect on the l ate ral stiffness.The geometric stiffness matrix is an adjustment to the conventional stiffnessmat rix to account for such effects. Such ma tric es have als o been refe rred to inthe literature as stability coefficient mat rice s and incremental stiffness matrices.Elemehts of the G matrix will now be derived.

LEvalua tion of

0 @k,x@j , x Y i , x d x .F o r i = 1 ,

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when k = I,= I ,

For i = 2- 9d

Other values of j and k are evaluated sim ilar ly to yield

22

~ - - - . . - - 1 1 ~ . ~ . 1 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 . 1 111.111111111111.11.11111.1111111111111111111 1111111 II 1111 I I I 1111 1111111 11111 I I I 111


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-- - -

------

L IOL'k.x , x 2, x d x = I - I (54)0 iOL iOL 30 30

4-30-L

Evaluation of0 'i,x'j,x 'k,x dx .

For i = I

j = I,k = 1L

0when j = I k = 2 .

6 65L2 5L2J ' l , x C p k , x Y j , xdx =0 i0L iIOL 5 5 )1 i__IOL i0L

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- - L - -OL IOL

L.f '3,x'k,x'j,x dx =

-- - L - -OL IOL

Ldx =0 '4 ,x 'k,xyjyx

LEvaluation of

0 yk, Yj , Yi , dx .

Then,

24

~ . .. .... ,.


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- -- - - - --- - - -

Evalllation of 1 @ . @. y dx .0 1YX JYX k , x

The above matr ices a re now multiplied by the cyo state displacements asindicated in equation (46) , k

- -6 -5L2 5L2 IOL IOL6 6 1 I-5L2 5L2 IOL IOL- I 4 I-iOL IOL 30 30

25


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--

--------

L

Similarly,L r 0 0 0w o s 'I,x'k,x'j,x dx = [wlw2 el0 e2Jk O

26

- - I -IOL IOL 30 30-

-6 6-5L2 u6 6-5L2 5L2I I-IOL i 0L (64)I I-IOL IOL-


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The above matric es a r e combined to form the G matrix (Equation ( 46) ) :

27


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[GI =

Load Behavior Matrix. The effect of applied load behavior on the elementstiffness is obtained by adjus ting the 5 matr ix wi th the km at r i x , where

[ L l = [ LP '1 + [LFl . 75)The L ma tr ix will now be derived fro m equations (47) throug,, (51) for each ofthe t&ee load behavior cases.

Case I.

2 8

I


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--- -

Then, -- I L L-0 0 2 12 12I - A0 0 2 12 12

0 0 0 . o 0( 7 6 10 0 0 0 0

0. 0 0 0 0

0 0 0 0- 0-L-F may be written directly from equation (48) in view of equations ( 3 8 )

and (39 ) ;

0 0 0 - FZ20 0 0[ L F l = P F X l 77)0 Q 0 "2 0 0 00 0 0 0

where F s are proportionality cons tants between the indicated fo rc es and p.The magnitudes of the applied force s' are assumed to rem ain in a constant ratioto each other and to the lateral load, p, during loading of the element. Th is

29


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--- -

-assumption is made he re and in the Stability Criterio n section for illu strativ epurposes. The assumption is easily alt ered to study buckling under p re ss ur ewith specified nodal forces or buckling under nodal for ces with specifiedpressure. Case 11.

[ L 1 = [LFl = i o 1PCase 111.

dx

L="[(x-r+3 X2 x3 ) uj +( - ) 11R 0

L 0 0 06RL 0 0 03R0

[ LP I ' P

0

030

I I


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--

L may be written direct ly fro m equation (51) ;-F

O 0 0

F Z 2R 0 0F X l0 1 0

[LFl = pFx20 0 1

0 0 0

0I - 0 0 -00000031


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Coor di nate TransformationThe developments thus far have been in a coordinate sys tem which wasoriented so that the x-axis coincides with the element centerli ne. To combine

sev era l elements fo r solution of a particular problem, it is necessary to obtainthe stiffness matrices of the elements in a common or global coordinate system.This system w i l l be denoted by x z as shown in Figure 5.

FIGURE 5. COORDINATE TRANSFORMATIONThe figure shows that vectors in the two coordinate syst ems transform

according to the relationshipsm = i cos p + ii s in pn = ii cos ,8 - GI s in p-o = w .

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Writing in ma tri x notation, we see that

Thus the displacements and for ces of th e elemen t trans form according to

I -T I FZ i

I0 I T F Z 2 I

II G2

33


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or

If the nodal for ces and displacements are relate d by the stiffness K i(equation (44), then

where K i has been rearra nge d to conform to the Q and q matrices. Upon sub-stit uti ni equations 82) and (83) into equation (8z) ,

But in this c ase ,

34


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where if is the stiffness matrix of the elem ent in the global coordinate system.

Master Stiffness MatrixThe stiffness ma trix fo r complete s t ruc ture is obtained by combining the

element stiffness ma tric es in the global coordinate system. This sectiondes cri bes the method of accomplishing this combination.

Consider pa rt of a n over all stru ctu re schematically repres ented byFig ure 6. The Is in Figure 6 are intended to r epr ese nt the total load vector1

FIGURE 6. GENERAL STRUCTURE3 5


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I

at the point of application,

Load-displacement relationships for each of the th re e elements in theglobal coordinate system are expressed fr om equation ( 87) as follows:

By combining the above relationships with the fact that gi is the same forboth elements b and c , and noting that

and that sim ila r relationships hold for the other nodes, the ma ste r stiffnessmatr ix is obtained:

36


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- _ - - K a i - i . i - l +Eb i - 1 , i - 1 K b i - i , j 0- - 0 % i , i - 1 K bii +Ec i i Kci , + 1

-0 0 K . txc i + l , i K c i + l , i + l d i + l , i t0 0 - _ -

Note that thi s is a relatio nship between the externally applied loadsand the nodal displacements of th eas sem ble d stru ctu re i n the global coordinatesystem.

S t a b iIity C r i t e r i o nElements with stiffness ma tric es of the type given by equation (44) maybe assembled into a mas ter s t if fness matr ix to represent a structu re subjectedto the c rit ical (buckling) magnitude of applied loads. Thus,

[[Kl + P [Eol]{G1)= (0)where,

p[liol = [GI + [Ep] + [E] 9 ( 9 1 )and the null matr ix of applied extern al loads indicates that the st ruct ur al stiffnesshas vanished under the critical load magnitude ( a physical interp retatio n ofinstability). Boundary conditions may be applied to re duc e the s iz e of the s e t ofequations (90) and the new reduced set again denoted by the equations (90 ) .

The E, G, and ma tric es above are obtained by assembly of el ement"Pmatr ices as descr ibed in the preceding sections, The E matrix is m or e Fconveniently obtained by di re ct application of equations (48) and (51) o eachnode of the assembled str uc tu re in the global coordinate syste m.

A nontrivial solution of equations ( 90) will exist only when the determinantof the matr ix + pE, vanishes;

This is an eigenvalue problem w here the magnitudes of applied load, p, at whichinstability will occ ur are the eigenvalues.37


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111111111 1 1 1 1 1 I I I I I I , I

Primary Equi l ibr ium Sta te

It should be noted at this point that the foregoing solution is contingentupon a knowledge of the prima ry equilibrium state. That is, the G matrix cannot be formulated until the qodisplacements are known in te rm s the appliedloads. Given a s e t of applied loads, the nonlinear equilibrium equations (30)through ( 3 5 ) may be solved fo r the qo displacements. An iterative procedurefo r inco rporating this solution into the stability problem is given in Reference 14.

Another approach which is consis tent with many cl ass ical stability problem solutions is to lin earize equations ( 3 0 ) through (35 ) for the purpose ofobtaining the pr im ar y state solution. If this is done these equations become forall three cases

( 9 3 )

A l l te rm s i n the above equations have previously been evaluated andexpressed i n matr ix form except the [email protected] rm which is as follows:1

0 -0 \

38


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or00L-2

{ P @ > = P L2L2-12L212

Equations ( 9 3 , 94) may then be expressed in the form

Fro m this expression for each element, the ma st er stiffness matri x forthe entire s tructur e is assembled as before. Then for the complete stru ctu re,

A furt her simplification fo r obtaining 9' which has al so been usedextensively in clas sic al problems is the use of a mem brane solution for zo. Thestructure is assumed to take no loads in bending before buckling. Fo r manypractical problems such a solution can readil y be obtained by inspection, elementary equilibrium considerations, o r from the literature. For simplicity thisapproach is used in the example problem of th is re port.

A P P L I C A T I O N OF TH E T H EOR YThis section is intended to give a ste p by ste p approach for applying the

theory to the solution of p ract ica l problems. For problems with sev eral elements

39


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either a gene ral computer program would need to be developed o r a relativelysimple computer program for each application could be developed as w a s donefo r the example to follow later.

One begins by dividing the s tru ct ur e to be analyzed into dis cr et e elements,the number depending upon the des ire d accuracy . Fro m the known prop ertiesand loading on ea ch el emen t the K matrix can be calc ulate d fro m equation (52)

and the L and L ma tric es determined from the appropriate equations (76), (77), P -F( 78 ) , ( 79 ), o r ( 8 0 ) . Note that the criti ca l magnitude, p, of the applied loadingremains as the unknown to be de termin ed.

The qo primary state is now determ ined from a known me mbran e solutiono r from equztion (9 7). The K matrix in equation (97) is assembled by usingequations (81) and ( 8 6 ) to tr&sform elements to the global coordinate sys temand equation (89) fo r combining the elemen ts. Boundary conditions a r e appliedand equation (97) is solved fo r qo. Once the qo state is determined the Gmatrix can be found from equatl'uon 74) , using the definitions (53) throcgh ( 6 1 ) .

The K ma tr ix defined by equation (44 ) is now known fo r each element.Thes e a r e agsembled into a ma ste r stiffness matrix by equations (86) and (89) ,and boundary conditions are applied to the resulting set of equat ions to obta inequations (90) . Eigenvalues of the chara cteristi c determinant, equations (92),which is obtained from equation ( 9 0 ) , a r e the magnitudes of the buckling load.The mode s hape f or each buckling load c.an be deter mine d by substituting eacheigenvalue in tu rn into equations (90) and solving fo r the r elativ e amplitudes ofdisplacements, 6i'

EXAMPLE PROBLEMCircular Arch With Uniform Pressure

The above application proc edur e w a s applied to the uniform circu lar arc hshown in Fig ur e 7. The known me mbran e solution fo r a circular arc h underuniform pressure is

4 0


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r

for all elements.The G matrix is formulated as indicated by equation (74) :

m

EA

41


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- -

- -- -

- --

I Illlllll ll l l lll I l l I I

+zw

MOMENT OF INERT IA = I = 0.314159 i n 4 (13.1 .m44YOUNGS MODULUS = E = lo p s i 6.9 x 1010 N/m )

FIGURE 7. UNIFORM CIRCULAR A R C HOther elements of the ,G matrix a r e calculated similarly to yield for all

elements

3 -3 0 0 0 0L L Fxl3 -3 0 0 0 0L L Fx2

- - -0 0 5L 5L 10 10 F Z l[GI = P R - - - 6 _ - I _ - 99)0 O 5L 5L i o 10 Fz2

I i 4L L- - - _ - 0 O i o 10 30 30 MII i L 4L.- 0 O 10 10 30 30- M2-where the four te rm s in the upper left cor ner can be traced directly to the

u term in equation ( ii ) .,x42


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- -

Now fo r al l eleme nts the K matrix is given by equation (52) , the L"Pmatric es fo r the three load cases a r e given by equations (76), (78), and (79),respectively, and all L matric es a re null.

-FThe K , G, and L matrices are now all rearranged to the order ,-F

II FXlI 4 2 FZ III - - - - Fx2II Ki2 F Z 2I

M2

A l l element ma tric es a r e now rotated to the global coordinate s yst em ,x, y by the operation,

as indicated by equation ( 8 6 ) . The T m atrix is of cou rse calculated separatelyfor each element a s indicated by equation (81) whe re p is the angle measuredclockwise from the element x axis to the i axis.

The element matr ice s a r e now combined to form the m as te r stiffnessmatr ix as indicated by equation ( 89 ) . The size of this matrix is reduced byapplying the boundary conditions,

uI= w i = ~ i = y = w l = ~= o (102)where Y, wl' 81 are dis placem ent s of the righ t end of the last element. From

43


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this reduced mat rix the characteristi c determinant ( 92) is formed and eigenvaluesare calculated. The above operations we re perform ed on a computer for idealizations of 2, 3, 6, 9, and 12 elements. A hand calculation was also made fo r thetwo-element arch. Results of the analysis are shown in Table I. A s can be seenfrom th e table, comp arison with known re su lt s is excellent and there is rapidconvergence to the exact solutions. The exact solutions w e r e obtained from theanalys is by Wempner and Kesti; [241. The 12-element solution for Case I1 hasalso been previously obtained by the finite eleme nt method in Refere nce 14.

TABLE I. COMPARISON OF ARCH BUCKLING LOADS

NumberofElements

2369

12Exact solution

~~_ _ _ _

Buckling Pr ess ure ( lb/in)~- ~-Case I Case 11 Case I1194. I 94. I 94. I57.1 64.7 67.457.4 62. 6 64.456.6 61. 9 63.656.4 61. 6 63.356.87 60. 95 63.46

CONCLUS IONS

The results of the example problem above show that the theorydeveloped he re may be quite useful in solving practical engineering stabilityproblems .

44


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The impli cit assum ption of uniform s he ar on the el eme nt mentionedin Development of E leme nt Stiffness Matrix sect ion does not appea r to adver selyaffect the res ult s. However, i f desir ed, this assumption could possibly beeliminated by the technique introduced by Pian [ 191. Also, the u t e rm of, xequation (11) apparently has a negligible effect on the res ul ts fo r the parti cula rexample solved, sinc e the exact solutions used for co mparison do not containeffects of this term.

A ve ry im portant item to note is that the pres ent theory applies tounconservative systems (repreqented by Case I ) as well as conservativesys tem s sin ce the principle of virtual work was used. It is known, however,that certain unconservative syste ms a r e stable in the static sen se (consideredh e re ), but unstable in the dynamic sense. (See Reference 25, page 152through 156. ) Thus the present analysis could not be expected to adequatelytreat this type of problem, and great care should be used in applications tounconservative systems.

Fru itfu l extensions of t he pre sen t theory would likely be found in thearea of cu rved and three-dimensional elements and in a general automation ofthe theory application.

George C. Marshall Space Flight CenterNational Aeronautics and Space Administration

Mars hall Space Flight Center , Alabama 35812, May I,1968981- I O - I O - 0000-5 0- 00 - 008

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I l l I l l I I

REFERENCE S1. Green , B. C. : Stiffness M atrix f o r Bending of A Rectangular Elementwi th In i ti a l Membrane S t re s ses . S t ruc tura l Ana lys i s Res ea rc h

Memorandum No. 45, Th e Boeing Company, Seattle, Wa sh ., Aug. 1962.2. Kapur , K. K. and Har tz , B. J. : Stability of Plates Using the Fin i te

Element Method. J. Eng. Mech. Div. ASCE, vol. 92, 1966, pp. 177195.

3. Lunder, C. A. : Derivation of a Sti f fness Matr ix fo r a Right Tr iangula rPlate in Bending and Subjected to Init ial Str es se s. M. S. The s i s , Dept.of Aero naut ics and Astro naut ics, Univ. of Washington, Seattle, Wash. ,Jan. 1962.

4. Martin, H. C. : Introduction to M atrix Methods of St ructu ral Analysis .McGraw-H ill Book Co., Inc. , 1966.

5. Martin, H. C. : On the Derivation of Stiffness Matr ices fo r the Analysisof La rg e Def lec t ion and Stabil i ty Prob lems. Conference on Matr ixMethods in Stru c tura l Mechanics, Wright-P atterson AFB, Dayton, 1965.

6. Melosh, R. J. and Mer r i t t , R. G. : Evalua t ion of Sp ar Matr ic es fo rSt i f fness Analyses . J. Aerospace Sci. , vol. 25, 1958 , pp. 537-543.

7. Turne r , M. J . ; Clough, R . W.; M artin , H. C .; and Topp, L. J . :Stiffness and Deflection Analysis of Complex Struct ures . J. Aeron. Sci. ,vol. 23, no. 9, Sept. 1956 , pp. 805-824.

8. Turner , M. J. ;Dill, E. H. ;Ma rt in , H. C. ; and Melosh, R. J. : L a r g eDeflections of Str uct ure s Subjected to Heating and Exter nal Loads. J.Ae ro sp ac e Sci . , vol. 27, 1959, pp. 97-106.

9. Turner , M. J. ; Mar tin , H. C. ; and Weik el, R. C. : Further Developmentand Applicat ions of the Stiffness Method. Pr es en te d at AGARD Structuresand Mate r ia l s Pane l ( Paris, F r a n c e ) , July 1962. Published in AGARDograph 72 , Pe rgamon Press, 1964 , pp. 203-266.

IO. A r g y r i s , J. H. : Matr ix Analys is of Three-dimensional Elas t ic Media ,Smal l and Large Displacements. AIAA J . , vol. 1, 196 5, pp. 45-51.

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REFERENCES (Co ntinued)11. A r g y r i s , J. H. : Recent Advances in Matrix Methods of Struc tura l

Analysis . Pr og re ss in Aeronaut ical Sc iences, Pe r ga m on Press , 1964.12. Gallagher, R. H. : A Cor relat ion Study of Methods of Matr ix Struc tura l

Analysis . AGARDograph 69, Jul. 1962.13. Gallagher, R. H. and Padlog, J. : Dis c re t e E lement Approach to

St ruc tura l Instability Analysis. AIAA J. , vol. I no. 6, Jun. 1963,pp. 1437-1439.

14. Gallagher, R. H. ; Padlog, J . ; Galletly, R. ; Mallett, R. : A Discre teElem ent Pro ced ure f or Thin-Shell Instabil i ty Analysis . AIAA J. ,vol. 5, no. I , Ja n. 1967, pp. 138-145.

15. A r c h e r , J . S. : Consis tent Matr ix Formu la t ions fo r Struc tura l Analys isUsing Influen ce-coeffic ient Tech niq ues. AIAA J. , vol. 3 , 1965, pp.1910- 1918.

16. Melosh, R . J . : Bas is f o r Deriva t ion of Matr ices fo r the Dire c t St i ffnessMethod. AIAA J. , vol. 1, no. 7 , Jul . 196 3, pp. 1631-1637.

17. Oden, J . T. : Calculation of Geometr ic St i f fness Matr ices fo r ComplexStructures. AIAA J. , vol. 4, no. 8, Aug. 1966 , pp. 1480-1482.

18. Pian, T. H. H . : Deri vati on of Elem ent Stif fnes s Mat ric es. AIAA J . ,vol.2, no. 3 Mar . 196 4, pp. 576-577.

19. P ian , T. H. H. : Derivation of Element Stiffness Matr ice s by AssumedSt re ss Distr ibu tions. AIAA J. , vol. 2, no. 7, Jul. 1964, pp. 1333-1336.

20. Nav ara tn a, D. R. : Elastic Stability of Shells of Revolution by theVaria t ional Approach Using Di scr e t e Elements . ASRL TR 139-1 ,BSD TR 66-261, Jun. 1966.

21. Hoff, N. J . : The Analys is of Stru ctur es. John Wiley and Sons, 1956.22. Langhaar, H. L. : Ener gy Methods in Applied Mechanic s. Joh n Wiley

and Sons, 1962.

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REFERENCES (Concluded)23. Novozhilov, V. V. : Foundations of the Nonlinear Theory of Elasticity.

Graylock Press (Rochester , N. Y. , 1953.24. Wempner, G. A. and Kes ti, N. E. : On the Buckling of Ci rcular Arc he s

and Rings. Proce eding s of the Fou rth U. S. National Congress ofApplied Mechanics (Am. SOC. of Mech. Eng. , N. Y. , 1962), vol. 2.

25. Timoshenko, S. P. and Gere, J. M. : Theory of Ela stic Stability.McGraw-Hill Book Co. , 1961.

nasa beam+column stability

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