SEC. 18—4. SOLUTION TECHNIQUES; STABILITY ANALYSIS 599

Expanding g in a Taylor series about= g(k) + — +

9i,i 91,2 91n

== 92,1 92,2 92,n

L0xrJ'

and retaining only the first two terms results in the convergence measure

(x — = a (18—39)

where lies between xk and For convergence, the norm of 'g. must beless than unity.

The generalized Newton-Raphson method consists in first expandingabout

= + + = 0

where

= = [T'—j —

= (18—40)

=

Neglecting the second differential leads to the recurrence relation

= 18 41= + ( — )

The corresponding convergence measure is

— = (18—42)

Let us now apply these solution techniques to the structural problem. Thegoverning equations are the nodal equations referred to theqlobcil system frame,

1?e — = 0 (1843)

where contains the external nodal forces and — is the summation ofthe member end forces incident on node i. One first has to rotate the memberend forces, (18—26), from the member frame to the global frame using

=

k° =

In our formulation, the member frame is fixed, i.e. is constant. We introducethe displacement restraints and write the final equations as -

e m 1844Pm P1 + + KU

600 ANALYSIS OF GEOMETRICALLY NONLINEAR SYSTEMS CHAP. 18

Note that K and depend on the axial forces while Pr depends on both theaxial force and the member rigid body chord rotation, if the axial forces aresmall in comparison to the member buckling loads, we can replace K withK1, the linear stiffness matrix.

Applying successive substitution, we write

KU = —

and iterate on U, holding K constant during the iteration:

= P1 — (18—45)

We employ (18—45) together with an incremental loading scheme since K isactually a variable. The steps are outlined here:

1. Apply the first load increment, and solve for U(I), using K K1.

2. Update K using the axial forces corresponding to Pe(l)• Then applyPe(2) anditerate on

K (ni_p _p(n—1)(1) e(1) e(2) r

3. Continue for successive load increments.

A convenient convergence criterion is the relative change in the Euclideannorm, N, of the nodal displacements.

N ='\ . (18—46)

— 1 e (a specified value)Jabs va]uc

This scheme is particularly efficient when the member axial forces are smallwith respect to the Euler loads since, in this case, we can take K = K1 duringthe entire solution phase.

In the Newton-Raphson procedure, we operate on Vi according to (18—41):

= — 4,(fl)

Now, Pe is prescribed so that

= due to+ dPr + K LW + (18—47)

= —

where denotes the tangent stiffness matrix. The iteration cycle is

AU(n) = —

18 48= + ( — )

We iterate on (18—48) for successive load increments. This scheme is moreexpensive since has to be updated for each cycle. However, its convergencerate is more rapid than direct substitution. If we assume the stability functions

SEC. 18—4. SOLUTION TECHNIQUES; STABILITY ANALYSIS 601

are constant in forming due to AU, the tangent stiffness matrix reduces to

dI( 0 dP, 0(18—49)

K + Kr

where K. is generated with (18—28). We include the incremental member loadsin at the start of the iteration cycle. Rather than update at each cycle,one can hold fixed for a limited number of cycles. This is called mod (fledNewton-Raphson. The convergence rate is lower than for regular Newton-Raphson but higher than successive substitution.

We consider next the question of stability. According to the classical stabilitycriterion,t an equilibrium position is classified as:

stable — > 0neutral d2W,,, — d2We 0 (18—50)

unstable d2 W,, d2 < 0

where d2 is the second-order work done by the external forces during adisplacement increment AU, and is the second-order work done by themember end forces acting on the members. With our notation,

d2w, Pe)TAU

= (dAU

(18—51)

= AUTK, AU

and the criteria transform to

/ / \T < 0 stable(AU)TK, AU

—Pe) AU = 0 neutral (18—52)

> 0 unstable

The most frequent case is Pe prescribed, and for a constant loading, the tangentstiffness matrix must be posil.ive definite,

To detect instability, we keep track of the sign of the determinant of thetangent stiffness matrix during the iteration. The sign is obtained at no cost(i.e., no additional computation) if Gauss elimination or the factor methodare used to solve the correction equation, (18—48). When the determinantchanges sign, we have passed through a stability transition. Another indicationof the existence of a bifurcation point (K1 singular) is the degeneration of theconvergence rate for Newton-Raphson. The correction tends to diverge andoscillate in sign and one has to employ a higher iterative scheme.

Finally, we consider the special case where the loading does not producesignificant chord rotation. A typical example is shown in Fig. 18—4. Both the

t See Sees. 7—6 and 10—6.

602 ANALYSIS OF GEOMETRICALLY NONLINEAR SYSTEMS CHAP. 18

frame and loading are symmetrical and the displacement is due only to short-ening of the columns. To investigate the stability of this structure, we deletetthe rotation terms in K, and write

K

K is due to a unit value of the load parameter The member axialforces are determined from a linear analysis. Then, the bifurcation problemreduces to determining the value of 2 for which a nontrivial solution of

(K + 2K;)AU 0 (18—54)

exists. This is a nonlinear eigenvalue problem, since K = K(2).

______________________

12X

I I

Fig. 18—4. Example of structure and loading for which linearized stability analysisis applicable.

In linearized stability analysis, K is assumed to be K1 and one solves

K, AU = —2K AU (18-55)

Both K, and K; are symmetrical. Also, K1 is positive definite, Usually, onlythe lowest critical load is of interest, and this can be obtained by applyinginverse iterations to

(—K;)Au1 (18—56)

REFERENCES

1. TIMOSHENKO, S. P., and J. M. GERE: Theory of Elastic Stability, 2d ed., McGraw-Hill,New York, 1961.

2. KOLLBRUNNSR, C. F., and M. MEIsTER: Knicken, Biegedriilknicken, Kippen. 2d ed.,Springer-Verlag, Berlin, 1961.

3. BLEICH, F.: Buckling Strength of Metal Structures, McGraw-Hill, New York, 1952.

t Set Pi P2 = = 0 in (18—28).See Refs. 11 and 12 of Chapter 2.

REFERENCES 603

4: BLYRGERMEISTEa, G.. and F!. STEUP: Stabilidhsrheorie, Part 1, Akademie-Verlag.Berlin, 1957,

5. CFJtLVER, A. H., ed.: Thin- Walled Structures, Chatto & Windus, London, 1967.6. VLASOV, V. Z.: Thin Walled Elastic Beams, Israel for Scientific Transla-

tions, Office of Technical Services: U.S. Dept. of Commerce, Washington, D.C., 1961.7. LIVESLItY, R. K.: Matrix Methods of Structural Anal vsis, Pergamon Press, London,

1964.

8. AROYRIS, J. H.: Recent Advances in Matrix Methods of Structural Analysis, PergamonPress, London, 1964.

9. HILDEBRAND, F. B.: Introduction to Numerical Analysis, McGraw-Hill, New York,1956.

10. GALAMBOS, T. V.: Structural Members and Frames, Prentice Hall, 1968.11. BRUSU, D. and B. ALMROTH: Buckling of Bars, Plates, and Shells, McGraw-Hill,

New York, 1975.

http://www.cvisiontech.com/pdf_compressor_31.html

index

Associative multiplication, 8Augmented branch-node incidence ma-

trix, 124, 222Augmented matrix, 33Axial deformation, influence on bending

of planar member, 472

Bar stiffness matrix, 180Bifurcation; Neutral equilibriumBimoment, MqS, 373Branch-node incidence table, 121, 145

Cç1, Cr, C,,r—coefficients appearing incomplementary energy expressionfor restrained torsion, 387, 388, 416

Canonical form, 58Cartesian formulation, principle of vir-

tual forces for a planar member, 465Castigliano's principles, 176Cayley—Hamilton Theorem, 63Center of twist, 383, 389Characteristic values of a matrix, 46Chord rotation. p. 586Circular helix, definition equation, 84, 86Circular segment

out-of-plane loading, 504restrained warping solution, 509

Classical stability criterioncontinuum, 256member system, 603truss, 170

Closed ring, out-of-plane loading, 503Cofactor, 19Column matrix, 4Column vector, 4Complementary energy

continuum, 261member system, 572planar curved member, 434restrained torsion, 385; 387, 388unrestrained torsion-flexure, 301

Conformable matrices, 8, 35Connectivity matrix, member system, 563Connectivity table for a truss, 121, 143.Consistency, of a set of linear algebraic

equations, 31, 44

Constraint conditions treated. with La-grange multipliers, 76, 80

Curved memberdefinition of thin and thick, 434thin, 487slightly twisted, 487

Defect, of a system of linear algebraicequations, 31

Deformationfor out-of-plane loading of a circular

member, transverse shear, twist, andbending, 498

for planar member, stretching andtransverse shear vs. bending, 454

Deformation constraintsforce method, 573displacement method, 576variational approach, 583

Deformed geometry, vector orientation,239

Degree of statical indeterminacymember, 555, 567truss, 210

Determinant, 16, 37, 39Diagonal matrix, 10Differential notation for a function, 70, 72,

79Direction cosine matrix for a bar, 119Discriminant, 40, 59Distributive multiplication, 8

Echelon matrix, 29Effective shear area, cross-sectional prop-

erties, 302Elastic behavior, 125, 248End shortening due to geometrically non-

linear behavior, 589Engineering theory of a member, basic

assumptions, 330, 485Equivalence, of matrices, 27Equivalent rigid body displacements, 334,

414, 430Euler equations for a function, 73Eulerian strain, 234

605

606 INDEX

First law of thermodynamics, 248Fixed end forces

prismatic member, 523thin planar circular member, 528

Flexibility matrixarbitrary curved member, 515circular helix, 534planar member, 462prismatic member, 345, 521thin planar circular member, 526

Flexural warping functions, 296, 300/nFrenet equations, 91

Gauss's integration by parts formula. 254Geometric compatibility equation

arbitrary member, 499continuum, 259, 264member system, 569planar member, 463, 466prismatic member, 355truss, 160, 212, 216, 223unrestrained torsion, 279, 315

Geometric stiffness matrix for a bar, 200Geometrically nonlinear restrained torsion

solution, 595Green's strain tensor, 234

Hookean material, 126, 249Hyperelastic material, 248

Incremental system stiffness matrixmember system, 601truss, 193

Inelastic behavior, 125Initial stability

member system, 562truss, 137

Invariants of a matrix, 59, 62Isotropic material, 252

Kappus equations, 592Kronecker delta notation, 11

Lagrange multipliers, 76, 80, 583Lagrangian strain, 234Lamé constants, 253Laplace expansion for a determinant, 20,

38Linear connected graph, 218Linear geometry, 120, 143, 237Linearized stability analysis, 602Local member reference frame, 92

Marguerre equations, 449, 456Material compliance matrix, 249Material rigidity matrix, 249Matrix iteration, computational method,

201

Maxwell's law of reciprocal defiections.356

Member, definition, 271Member buckling, 588Member force displacement relations. 537,

546, 556Member on an elastic foundation, 384, 369Mesh, network, 220Minor, of a square array, 19Modal matrix, 52Modified Neuton-Raphson iteration, 601Moment, MR,Mushtari's equations, 444

Natural member reference frame, 92Negative definite, 58Negligible transverse shear deformation,

planar member, 443, 454, 498Network, topological, 220Neutral equilibrium, 170, 256, 601Newton-Raphson iteration, member sys-

tem, 598Normalization of a vector, 49Null matrix, 4

One-dimensional deformation measures,335, 338, 432

arbitrary member, 491Orthogonal matrices and trnasformations,

50, 53Orthotropic material, 250, 251

Permutation matrix, 42, 135Permutation of a set of integers, 16, 37Piecewise linear material, 126, 146Plane curve, 98, 425Poisson's ratio, 252Positive definite matrix, 58, 63Positive semi-definite matrix, 58Postmultiplication, matrix, SPotential energy function, member system,

571Premuftiplication, matrix, 8Primary structure

member system, 568planar member, 463prismatic member, 354truss, 211

Principle minors, 55Principle of virtual displacements

member system, 570planar member, 442

Principle of virtual forcesarbitrary member, 490, 492, 512member systens, 571planar member, 435, 458prismatic member, 338, 351

Quadratic forms, 57

INDEX 607

Quasi-diagonal matrix, 15, 38Quasi-triangular matrix, 39

Radius of gyration, 434Rank of a matrix, 27, 42, 43Rayleigh's quotient, 75, 79Reissner's principle

continuum, 270member, 383, 414member system, 573

Relative minimum or maximum value of afunction, relative extrema, 66

Restrained torsion solution, prismaticmember

linear geometry, 391nonlinear geometry, 595

Restrained torsion stress distribution andcross-sectional parameters

channel section, 401multicell section, 411symmetrical I section, 398thin rectangular cell, 407

Rigid body displacement transformation,109

Rotation transformation matrix, 101, 232Row matrix, 4

Self-equilibrating force systems, 160, 211,258

member systems, 568Shallow member, assumptions, 448Shear center, 297, 300, 309, 378, 389Shear flow, 287Shear flow distribution for unrestrained

torsiOn, 308Similarity transformation, 53, 62Simpson's rule, 475Singular matrix, 22Skew symmetrical matrix, 11Small strain, 120, 235Small-finite rotation approximation, 238Square matrix, 4Stability of an equilibrium position, 171,

195Stability functions (4), prismatic member,

589Statically equivalent force system, 103, 106Statically permissible force system, 159,

216, 257Stationary values of a function, 67, 79Stiffness matrix

arbitrary curved member, 516, 520

modification for partial end restraint,535

prismatic member, geometrically non-linear behavior, 588, 595

prismatic member, linear geometry, 522Strain and complementary energy for pure

torsion, 280Strain energy density. 248Stress and strain component trnasforma-

tions. 249Stress components

Eulerian, 242Kirchhoff, 246

Stress function, torsion, 276Stress resultants and stress couples, 272Stress vector, 240Stress vector transformation, 242Submatrices (matrix partitioning), 12, 36Successive substitution, iterative method

member system, 597truss, 193

Summary of system equations, force equi-ibrium and force displacement, 561

Symmetrical matrix, Il, 35System stiffness matrix

member system, 548, 550, 565truss, 179, 180, 188, 206

Tangent stiffness matrixfor a bar, 193prismatic member, 590, 596

Tensor invariants, 232Torsion solution, rectangle, 281Torsional constant, J, 276, 278, 323Torsional warping function, 274, 377Transverse orthotropic material, 252Transverse shear deformation

planar member, 454, 498prismatic member, 355

irapezoidal rule, 474Tree, network, 220Triangular matrix, 12Two-hinged arch solutions, 467, 470

Unit matrix, 10

Variable warping parameter, f, for re-strained torsion, 372

Vector, definition (mechanics), 4/n

Work done by a force, definition, 153, 156