,

I

Calculation of Geometric Stiffness

Matrices for Complex Structures

J. T. ODENUniversitr of Alabama, Hunts~'ille, Ala.

Reprintedfrom AIAA JournalVolume 4, Number 8, Pages 1480-1482, August 1966

Copyright. 1966, by the American Institute of Aeronautics and Astronautics.and reprinted by permission of the copyright owner

(2)

(3)

(4)

(5)

(6)

(9)

(8)

iii = ![(OW'/OXi) + (OWi/OXi»)

v

where

and

placement DUi. If tt; nrc the components of displacement inthe new configuration,

where Wi arc arbitrary kinematically admisi'ible functions anda is a small positive constant.

According to NOVOllhilov,llclassical elastic instability of astructure is characterized by a change from an equilibriumconfiguration in which rotations aro small to a configurationin which rotations substantially exceed the strain components.Ilence, if configuration 0 is one of unstable equilibrium, weassume that the strains in the new l'onfiguration C are givenby the formula

The quantities Wki represent rotations of line elements in con-fi~lration C. Indices in Eq. (3) and in equations to followobey the summation convention.

Similarly, the stress components in the new equilibriumconfiguration nre

where (Til arc stresses in the initial configuration and iTi; andaii are stresses due to ii; and the rotations, respectively.

Following a development similar to that of Ref. 11, wenow evaluate the total strain energy in the new equilibriumconfib'llration:

v = vo + aU + a:(j + higher order terms (7)

where UO is the total potential energ)' in the initial configura..tion,

(1)and Y is the volume of the element.

If the displacement components Wi in the new configurationnow are given small variations, the variation of VO is zero be-cause it is not a function of the Wi, and the variation in Ulends to equilibrium conditions in C but gives no informationconcerning the stability of the system. The problem ofelastic stability thus is concerned with variations in the term

Reprinted from AIAA JOURNALCopyright, 1966, by the AmericlIll Iustitute of Aeronautics lind Astrollllutics, and reprinted by permission of the copyright oWller

Introduction

Culculation of Geometric StiffnessMutrices for Complex Structures

J. T. ODEN*

Univel'ftity of Alabama, Huntsville, Ala.

..l pplications of the direct stiffness method to problems in.IX classical elastic stability have been concerned with thederivation of so-called geometric stiffness matrices that ac-count for the second-order effects of displacements on theequilibrium equations. Using these matrices, an eigenvalueproblem is established from which critical loads can be evalu-ated. Geometric stiffness matrices for simple bar elementsand beam-columns have been derived from purely geometricconsiderations by several authurs.I-4 Similar proceduresalso have been presented for stability analyses of triangularplate elements in plane stress,6 rectangular plate elements inbending,' and tetrahedral elements of three-dimensionalbodies.7 Martin8 presentcd a technique for deriving geo-metric stiffness matrices fOl"bar elements, beam-columns,and triangular plate clements in both plane stress and bend-ing; and Kapur and Hartz9 derived a geometric stiffnessmatrix for rectangular plates. Martin's paper contains abrief survey of the literature on this subject ..

This note presents a general formula. for evalua.ting geo-metric stiffness matrices for the stability analysis of generaldiscrete stntctural systems. The formula provides a meansfor the direct calculation of geometric stiffness matrices thatare consistent with any kinematically admissible displace-ment field as!mmed for the element. The development fallswithin the framework of the classical theory of elastic stabil-itylO and is based upon Noyozhilov'sll interpretation of theinstability of elastic systems.

Stubilit), Theor)'

Consider an elastic body in equilibrium at a certain con-figuration 0, and letuiOdenote the components of displacementparallel to a fh:ed cartesian coordinate system x; in this con-figuration. According to the linear small-deflection theory,the components of strain developed in the body are gi\'en bythe formula

Now consider another equilibrium configuration C whichdiffers from the initial configurati.on by a small virtual dis-

Received :March 9, 1966; revision received May 9, 1966.• Associate Professor of Engineering l\lecbanics, Research In-

stitute. Member AIAA.

AUGUST 1966 TECHNICAL NOTES 1481

from the definitions of 00and ifI, the elements of K. are func-tions of certllin loads parameters. Critical loads for thestructure are determined from the condition

where "det" indicates the detenninant of the argumentwithin. It is easily shuwn that the general furmula for K. inEq. (19) leads to consistent geometric stiffness matrices forVllriOUStypes of finite clements.

U defined in Eq. (9). Furthermore, on comparing Eqs, (1)and (4), it is seen that €i/ and iij differ only in the displace-ment field used in their evaluation; both are lineal' in the dis-placement gradients. Because, in finite element formula-tions, the displacements are 8.."5umedto be known in tenns ofthe genemlized node displacements, the first term on the rightside of J~q. (9) leads to the stiffness matrix Ko of a stable ele-ment in C. Thus, the remaining tenn in Eq. (9) can he usedto calculate the geometric stiffness matrix, K •.

det(Ko + K.) = 0 (20)

Geumetric Sti ffncss ]\'Iatrix

The del"ivation of stiffness matrices for finite elements oftenis based on 1111 approximate displllccment field of the form

where u is the vectur of displaccment components, cll is 11:3 Xn matrix whose elements arc functions of the coordinates,and v is an n X 1 column matrix of generalized displacements.The discrete model thus has 11 degrees of freedom, The sixindependent components of strain and stress then are elllcu-lated by the formulas

Examplc,

As a simple example, consider a stmight prismatic bar ofIcnj!;th L which is subjected to a compressive force P in itsinitial configurat.ion. In this case, Eqs. (14) aud (15) he-come

u = cl'v (10)

and

P [1 0 0]00=--000A 0 0 0

(21)

where rand dare 6 X 1 column matrices containing the strainand stress components, respectively; D is a matrix of differ-ential operators; E is a matrix of material constants defin-ing the stress-strain relationship, and C = Dcll.

In the case of stability !tnl~lyses, we introduce the addi-tional matrices

where A is the cross-sectional area of the bar.If a lineal' valiation in the longitudinal component of dis-

placement and a cubic variation in the transverse componentof displacement are assumed, thc matrices of Eq. (10) acquirethe following forms:

£ = Du = Cv

d = Er = ECv

(11)

(12)

u = lUi, It,, Usl

(22)

(23)

(24)

(25)

x/Loo

- x2/L t X3/L2J

oo

1 - 4x 3x2

---y- + £2

o 0 0 0]1/10 0 -6/5£ 1/102L/15 0 -1/10 - £/30 (27)o 0 0 0-1/10 0 6/5L -1/10- L/30 0 -1/10 2L/15

o1 - 3(x/L)2 + 2(x/L)3

o

o6/5L1/10

o-6/5L1/10

oo

-6x 6X2

L2 + l,~

o3(x/L)2 - 2(x/L)3

o

o 0 0 ]000~ ~2 _~ k2 ~~

o L2 - £3 L + L2

Finally, substituting Eqs. (21), (22), and (26) into Eq, (19)and integrating throughout the volume of the bar gives thegeometric stiffness matrix

Here Va and VIS arc the longitudinal displacements and V21and V22 are the transverse displacements of ends 1 and 2 ofthe bar, 81 and 82 are the end slopes, and x is a coordinatemeasured along the bar's geometric axis with oribrinat end 1.

The matrix \I~ is now obtained by introducing Eq. (25)into Eq. (lG):

(17)

(18)

(15)

(19)

(13)

(14)

Ko = Iv CTEC dV

D = tVT(Ko + K.)v

00 0

(}X3 OX2

W = t I 00

o . (16)OX3

- -Itllv = \IvOXI

0 0 0i)x2 OXI

where

and'

and the scalar function

Here w represents the rotation vector. It is related to thegeneralized displacements as follows:

where \}t is a matrix whose elements arc known functions ofthe coordinates. The quantities Ui/ in EllS. (14) and (15)are the stress components in the initial configuration of thestructure. These are known functions of the external loadsand arc determined by analyzing the structure using small-deflection theory

Substituting Eqs. (11-16) into Eq. (9) gives

Here Ko is the usual stiffness matrix obtained by the lineartheory, and K. is the geometric stiffness matrix. Note that,

This result agrees with that given by Gallagher and Padlog3

and Martin.8

1482 AIAA JOURNAL VOL. 4, NO.8

References

I Turner, 1\1. J., Dill, E. H., Martin, II. C., aud Melosh, R. J.,"Large deflections of structures subjected to heating and externalloads," J. Aerospace Sci. 27,97-106 (lH59).

I Archer, J. S., "Consistent matrL't formulations for struc-tural analysis using influence-coefficient techniques," AIAA Pre-print 64-488; also AIAA J. 3,1910-1918 (1965).

I Gallagher, R. II. and PadIog, J., "Discrete element approachto structural instability analysis," AIAA J. 6, 1437-14:iU (1903).

• Argyris, J. H., "Recent advance-; in Inlltrix methods ofstructural analysis," Progre$s in Aeronautical &ienca (PergamonPrC.'lS,Oxford, 19(4).

• Turner, M. J., Martin, H. C., and Weikel, R. C., "Furtherdevelopment and applicationll of the stilTnCS8method," AGARD-ograph 72 (Pergamon Pre.'lS, Paris, 19(4), pp. 203-266.

I Greene, B. C., "Stiffness matrix for bending of a rectangularplate element with initial membrane stre.'lSes," The BoeingCo., Structural Analysis Research Memo. 45 (1962).

1 Argyris, J. H., "Matrix analysis of three-dimensional elasticmedia, small and large displacements." AIAA J. 1,45-51 (1965).

I l\lartin, H. C., "On the derivation of stiffness matrices forthe analysis of large deflection and stability problems," Con-ference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base (1965).

I Kapur, K. K. and Hartz, B. J., "Stabilit.y of plates using thefinite element method," J. Eng. Mech. Div. Am. Soc. CivilEngrs. 92, 177-195 (1900).

10 Timoshenko, S. P. aud Gere, J. M., Theory of Elastic Staliility(McGraw-lIill Book Co. Inc., New York, 19(1), 2nd ed.

11 Novozhilov, V. V., Foundations of the Nonlinear Theory ofElasticity (GTllylock Prei.~, H.ochc.~ter, 1053), 153-176.