NFEM.Ch28.pdf

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28Bifurcation:

LinearizedPrebuckling II

28–1

Chapter 28: BIFURCATION: LINEARIZED PREBUCKLING II 28–2

TABLE OF CONTENTS

Page§28.1. Introduction 28–3

§28.2. State Decomposition at Bifurcation Point 28–3

§28.3. LPB Assumptions 28–3

§28.4. LPB Limitations 28–5§28.4.1. When LPB Works . . . . . . . . . . . . . . . . 28–7§28.4.2. And When It Does Not . . . . . . . . . . . . . . 28–7§28.4.3. Extending LPB the Applicability . . . . . . . . . . . 28–8

§28. Exercises . . . . . . . . . . . . . . . . . . . . . . 28–9

28–2

28–3 §28.3 LPB ASSUMPTIONS

§28.1. Introduction

This Chapter continues with the subject of linearized prebuckling (LPB) bifurcation analysis. Itgoes deeper than Chapter 27 in that it probes the assumptions (so far stated without proof) behindLPB, and the practical modeling implications that emanate from these assumptions.

To present some of the derivations in mathematical terms it is necessary to introduce the conceptof state decomposition at the bifurcation point and to define homogeneous and particular solutions.This is done briefly in §28.2, primarily to introduce notation for §28.3 and following. The detailedmathematical analysis of this decomposition, beyond LPB, is relegated to Chapters 29ff.

§28.2. State Decomposition at Bifurcation Point

Recall from previous Chapters that an isolated bifurcation point at λcr is characterized by a singulartangent stiffness at the equilibrium configuration,

K(ucr , λcr ) z = 0, (28.1)

and by the normalized null eigenvector (buckling mode) z �= 0, ‖z‖ = 1, being orthogonal to theincremental load vector:

qT z = zT q = 0. (28.2)

Assume that we have located a bifurcation point B and computed the buckling mode z. Our nexttask is to examine the structural behavior in the neighborhood of B. We shall be content with lookingat the so-called branching direction information. This information characterizes the tangents to theequilibrium branches that cross at B.

To carry out this task we borrow from algebraic ODE theory. Consider the variation in the statevector u measured from its value uB at buckling:

�u = u − uB (28.3)

Divide this increment by �t , t being the pesudotime parameter introduced in Chapter 3, and passto the limit:

u = limt→0

�u�t

. (28.4)

This variation rate u from the bifurcation point can be decomposed into a homogeneous solutioncomponent σz in the buckling mode direction, and a particular solution component y. The vectorKyis orthogonal to z:

u = (y + σ z)λ, yT z = zT K y = 0, (28.5)

The particular solution solves the system

K y = q, yT K z = 0, (28.6)

which is simply the first-order incremental equation Ku = qλ augmented by an orthogonalityconstraint. Imposing this constraint removes the singularity (rank deficiency) of K. The geometricinterpretation of this decomposition on the control space of a system with two DOF is shown inFigure ?, which is patterned after Figure 5.1. The geometric interpretation of this decompositionon the y, z plane is shown in Figure ?.

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λ

B2

1B

u1u2

Particular solution "follows loading"

Homogeneous solution "follows buckling"

y

zFundamental

(primary) path

Secondarypath

Figure 28.1. State decomposition at bifurcation point B: view in 3D control-state space. Vectors y and zare mutually orthogonal (proved in text). Together they define a plane in control-state space. All important

physics happens in that plane, regardless of how many DOF the model has: 2 or a million.

§28.3. LPB Assumptions

With the notation introduced in §28.2 we may now state precisely several key assumptions invokedin linearized prebuckling (LPB). (Those collected in item (II) have already been formally statedand used in previous Chapters)

(I) The external loading is conservative and proportional:

f = q0 + λq. (28.7)

and the structure is linearly elastic. In other words, the total residual equations are derivablefrom a potential energy function. As a consequence, K is symmetric: K = KT .

(II) The displacements and displacement gradients prior to the critical state are negligible in thesense that (a) the material stiffness matrix can be evaluated at the reference configuration, and(b) the geometric stiffness is proportional to the control parameter λ:

KM ≡ K0, KG ≡ λK1, (28.8)

in which K0 is the material matrix evaluated at the reference configuration, also called thelinear stiffness, and K1 is the reference geometric stiffness. As discussed in the previousChapter, the singular stiffness criterion det K = 0 leads to the eigenproblem

(K0 + λK1) z = 0. (28.9)

(III) The particular solution y defined in (28.5) and (28.6) is obtained by solving

(K0 + λcr K1) y = q (28.10)

under the constraint yT K z = 0. Observe that from assumption (I) q is constant.

28–4

28–5 §28.4 LPB LIMITATIONS

y

z

.T

||z||2= 1

B

Particular

Homogeneous

σ z λ

.y λ

Actual motion of the system at B

Orthonormalityconditions

z K y = 0

Figure 28.2. State decomposition at bifurcation point B, projected on the y-z plane.

We now prove that if these assumptions hold, all critical points determined from the LPB eigenprob-lem are bifurcation points, that is, zT q vanishes. To show that, premultiply both sides of (28.10)by zT :

zT q = zT (K0 + λ K1)y = yT (K0 + λK1)z = yT (Kz) = 0. (28.11)

Note that the transformation zT Ky = yT Kz holds because K0 and K1 are symmetric on account ofthe conservativeness assumption (I).

Remark 28.1. Bifurcation points are classified in later sections into various types: unsymmetric, stable-symmetric, stable-unsymmetric, and so on. It will be shown later that, under most common assumptions, LPBbifurcation points are generally of symmetric type. The LPB model does not provide, however, informationas to the post-bifurcation stability, so we cannot say whether the bifurcation point is stable-symmetric orunstable-symmetric.

§28.4. LPB Limitations

Linearized prebuckling (LPB) is used extensively in engineering design. Many standard books instructural stability1 only treat this technique. In its finite element version LPB is a feature availablein many finite element programs. Exercising this feature has the advantages of avoiding a fullnonlinear analysis, which can be expensive and time-consuming. Given its practical importance,structure designers (and most especially aerospace designers) should be familiar with the range ofapplicability of LPB. The limitations are discussed next.

1. Conservative loading. LPB is a restricted form of the static criterion, also known as Euler’stest; cf. §27.2.1. If the loads are not conservative, the dynamic criterion should be used, atleast to check out whether a flutter condition may occur. If the dynamic criterion shows thatstability is lost by divergence, one may regress to the singular-stiffness test criterion.

1 For example, Timoshenko’s 1936 textbook [619].

28–5


�� Figure 28.3. Structures that are adequately modeled by LPB assumptions.

negligible deformationprior to buckling

λ

v

B

R

Figure 28.4. Type of response expected under LPB assumptions. (Branch intersectionat B not shown for clarity)

2. Loss of stability must be by symmetric bifurcation. If the first critical point is a limit point orasymmetric bifurcation,2 LPB is not strictly applicable although in some cases it may providea sufficiently good approximation. Lacking experimental confirmation or a priori knowledge,the only practical way to check whether the first critical point is symmetric bifurcation is togo through a full nonlinear analysis.

3. Prebuckling deformations must be small. This assumption fits well many engineering struc-tures because of the nature of construction materials. The structures that best fit these assump-tions are straight columns, frameworks and flat plates, as illustrated in Figure 28.3 Care mustbe exercised for arches, shells, very thin members, and for imperfection-sensitive structuresin general.

4. Elastic material behavior. If the material is inelastic the structure is not internally conservative.Then the tangent stiffness depends on the prior deformation history, and the LPB eigenproblemloses meaning. The topic of inelastic buckling (in particular creep and plastic buckling) is anenormous subject that falls outside the scope of this course.

2 Symmetric bifurcation occurs when bucking in the z and −z directions is equally likely. Asymmetric bifurcation occurswhen one of the directions is physically more likely; for example axially compressed cylinders buckle inwards. Thisclassification of critical points is covered in more detail in Chapter 11 and following.

28–6

28–7 §28.4 LPB LIMITATIONS

λ

v

L

R

BLPB Actual

(a)

λ

v

L

R

B

LPB

Actual

(b)

Figure 28.5. Structures that are poorly modeled by LPB assumptions.

5. Applied loads should not depend nonlinearly on the displacements. Such a dependence usuallyintroduces nonconservative effects, thus voiding the conservative-loading assumptions. Evenis the loads remain conservative, the reference geometric stiffness would depend on the loadlevel, thus leading to a nonlinear eigenproblem.

6. The effect of imperfections is negligible. Some structures are highly imperfection sensitive inthat the first critical load is strongly affected by the presence of imperfections. In such casesobviously LPB is of limited value or outright irrelevant.

§28.4.1. When LPB Works

The systems that best fit the LPB model are symmetrically loaded structures such as straight columnsand in-plane-loaded plates (laminas) which are not excessively thin. See Figure 28.3. The lateralbuckling of such structures occurs following very small deformations, as typified by the responsesketch in Figure 28.4,

§28.4.2. And When It Does Not

Two examples of structures that are not properly treated by the LPB model are shown in Figure 28.5.The LPB predictions are way off in both cases, but for different reasons.

28–7


Case (a) is an axially compressed cylindrical shell made up of almost flat panels joined by curvedpanels, forming like a “curved triangle” cross section seen in some combat helicopters and theSpace Shuttle fuselage. There is a substantial redistribution of stresses due to changes on geometry.The structure eventually collapses at a limit point substantially over the predicted LPB load. Thelatter is therefore overly safe.

On the other hand, the axially compressed circular cylinder of case (b) is highly imperfection-sensitive structure that fails at a substantially lower load than that predicted by LPB. Consequentlythe LPB prediction is highly unsafe.

§28.4.3. Extending LPB the Applicability

One way to broaden the application of the LPB model is to update the reference configuration3 sothat the prebuckling deformations are reduced. If this is done the control parameter λ is of coursemeasured from the latest reference configuration and consequently becomes a true stage controlparameter. Limitations on the conservativeness of applied loads and types of critical point, however,cannot be readily circumvented by this “staging” technique.

3 As naturally done in the CR description, in which the deformational displacements are measured from a continuouslyvarying configuration, and also in the Updated Lagrangian description.

28–8

28–9 Exercises

Homework Exercises for Chapter 28

Bifurcation: Linearized Prebuckling II

EXERCISE 28.1 [A:15] Find the particular solution y at the lowest bifurcation load of the two-bar exampleof Chapter 27.

EXERCISE 28.2 [A:15] Find the particular solution y at the symmetric and antisymmetric bifurcation loadsof the one-element Euler column example of Exercise 24.4.

EXERCISE 28.3 [A:25] The first order residual rate equations is r = 0, where r is given by

r = Ku − qλ = 0, (E28.1)

(E28.1) holds at a bifurcation point where K and q are the tangent stiffness matrix and incremental load vector,respectively, at bifurcation. Decompose u = (y + σz)λ, where y is the particular solution and z �= 0 thebuckling mode normalized to length one. Show that the first-order differential equation system (E28.1) cannotgive information on the “buckling mode amplitude” σ because one gets σ = 0/0. (Hint: premultiply thatequation by an appropriate vector.)

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