Stability Of Structures: Basic Concepts

17

.

Stability Of Structures: Basic Concepts

17–1

17–2

Lecture 17: STABILITY OF STRUCTURES: BASIC CONCEPTS

TABLE OF CONTENTS Page

§17.1. Introduction §17.2. Testing Stability §17.2.1. Stability of Static Equilibrium . . . . . . . . . . . §17.2.2. Stability of Dynamic Equilibrium . . . . . . . . . §17.3. Static Stability Loss §17.3.1. Buckling Or Snapping? . . . . . . . . . . . . . §17.3.2. Response Diagrams . . . . . . . . . . . . . . §17.3.3. Stability Models . . . . . . . . . . . . . . . §17.3.4. Stability Equations Derivation . . . . . . . . . . §17.4. Exact Versus Linearized Stability Analysis §17.4.1. Example 1: The HCR Column: Geometrically Exact Analysis §17.4.2. Example 1: The HCR Column: LPB Analysis . . . . . §17.4.3. Example 2: The PCR Column: Geometrically Exact Analysis §17.4.4. Example 2: The PCR Column - LPB Analysis . . . . . §17.5. Discrete Stability Analysis As Eigenproblem §17.5.1. Example 3: A Cantilevered Two-Strut Column . . . . . §17.5.2. Example 4: A Pinned-Pinned Three-Strut Column . . . .

17–2

. . . . . . . . . . . .

17–3 17–3 17–3 17–4 17–5 17–5 17–5 17–6 17–7 17–7 17–8 17–9 17–10 17–12 17–13 17–13 17–15

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§17.2

TESTING STABILITY

§17.1. Introduction The term stability has both informal and formal meanings. As regards the former, the American Heritage Dictionary lists the following three. 1. Resistance to sudden change, dislodgment, or overthrow. 2a. Constancy of character or purpose: tenacity; steadfastness. 2b. Reliability; dependability. Related verb: to stabilize. Related adjective: stable. Opposite terms: stability loss, instability, to destabilize, unstable. The formal meaning is found in engineering and sciences, concerning stability of systems.* Broadly speaking, structural stability can be defined as the power to recover equilibrium. It is an essential requirement for all structures. Jennings† provides the following historical sketch: “Masonry structures generally become more stable with increasing dead weight. However when iron and steel became available in quantity, elastic buckling due to loss of stability of slender members appeared as a particular hazard.” §17.2. Testing Stability The stability of a mechanical system, and of structures in particular, can be tested (experimentally or analytically) by observing how it reacts when external disturbances are applied. Here we have to distinguish between statics and dynamics. §17.2.1. Stability of Static Equilibrium For simplicity we will assume that the structure under study is elastic, since memory effects such as plasticity or creep introduce additional complications such as historical dependence, which are beyond the scope of the course. The applied forces are characterized by a loading parameter λ, also called a load factor. Setting λ = 0 means that the structure is unloaded, at which it takes up an equilibrium configuration C0 = (0) called the undeformed state. Furthermore, assume that this state is stable in the sense defined below. As λ is varied from 0 the structure deforms and assumes equilibrium configurations C(λ). These are assumed to be (I) continuously dependend on λ and (II) stable for sufficienttly smaller values of λ. How is stability tested? Freeze λ at a specific value, say λd where d connotes “deformed.” The associated equilibrium configuration is Cd = C(λd ). Apply a perturbation to Cd , and remove it. What sort of perturbation? An action that may disturb the state, for example a tiny load or a small imposed deflection. More precise restrictions on such perturbations are described later. For now we will generically call admissible perturbations those that are allowed in the application of the test. The perturbation triggers subsequent motion of the system. Three possible outcomes are sketched in Figure 17.1. S

For all admissible perturbations, the structure either returns to the tested configuration Cd or executes bounded oscillations about it. The equilibrium at λd is called stable.

* A system is a functionally related group of components forming or regarded as a collective entity. This definition uses “component” as a generic term that embodies “element” or “part,” which connote simplicity, as well as “subsystem,” which connotes complexity. In this course we shall be concerned about mechanical systems governed by Newtonian mechanics, with focus on structures. † A. Jennings, Structures: From Theory to Practice, Taylor and Francis, London, 2004, Chapter 7.

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Equilibrium configuration

Apply an admissible perturbation

Motion oscillates about equilibrium configuration or decays toward it Subsequent motion

Transition between stable and unstable

Motion is either unbounded, or oscillates about, or decays toward, another equilibrium configuration

S: Stable*

N: Neutrally stable

U: Unstable

* Strictly speaking, S requires stability for all possible admissible perturbations Figure 17.1. Stability test outcomes.

U

If for at least one admissible perturbation the structure moves to (decays to, or oscillates about) another configuration, or “takes off” in an unbounded motion, the equilibrium is unstable.

N

The transition from S to N occurs at a value λ = λcr called the critical load factor. The configuration Ccr = C(λcr ) at the critical load factor is said to be in neutral equilibrium. The quantitative determination of this transition is a key objective of stability analysis.

The foregoing classification leaves several gaps and details unanswered. First, speaking about “moving” or “returning” introduces time into the picture. Indeed the concept of stability is necessarily dynamic in nature† There is a before: prior to applying the perturbation, and an after: what happens upon removing it. Many practical methods to assess critical loads, however, factor out the time dimension as long as certain conditions‡ are verified. Those are know as static criteria. Second, the concept of “perturbation” as “small imposed change” is imprecise. How small is a “tiny load” or a “slight deflection”? The idea is made more mathematically precise later when we introduce linearized stability, also called “stability in the small.” This is a natural consequence of assuming infinitesimal configuration changes. §17.2.2. Stability of Dynamic Equilibrium Stability of motion is a more general topic that includes the static case as a particular one. (As previously noted, the concept of stability is essentially dynamic in nature.) Suppose that a mechanical system is moving in a predictable manner. For example, a bridge oscillates under wind, an airplane is flying a predefined trajectory under automatic pilot, a satellite orbits the Earth, the Earth orbits the Sun. What is the sensitivity of such a motion to changes of parameters such as initial conditions? If the system includes stochastic or chaotic elements, like turbulence, the analysis will require probabilistic methods. † For example, Bazant and Cedolin in Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories,Dover, 2003, comment on p. 144: “Failure of structures is a dynamical process, and so it is obviouly more realistic to approach buckling and instability from a dynamical point of view.” ‡ E.g., conservative loading: applied loads derive from a potential.

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§17.3

STATIC STABILITY LOSS

To make such problems mathematically tractable it is common to restrict the kind of motions in such a way that a bounded reference motion can be readily defined. For example a bounded periodic motion of a oscillating structure. Departures under parametric changes are studied. Transition to unbounded or unpredictable motion is taken as a sign of instability. An important application of this concept are vibrations of structures that interact with an external or internal fluid flow: bridges, buildings, airplanes, fluid pipes. The steady speed of the flow may be taken as parameter. At a certain airflow or liquid-flow speed, increasing oscillations may be triggered: this is called flutter. Or a non-oscillatory unbounded motion happens: this is called divergence. A famous example of flutter in a civil structure was the collapse of the newly opened Tacoma-Narrows suspension bridge near Seattle in 1940 under a moderate wind speed of about 40 mph. Modeling and analysis of dynamic instability is covered in other courses, primarily at the graduate level because it requires fancier math tools and heavier use of complex calculus. In this course we consider only static stability. Moreover, the class of problems, methods and examples will be severely constrained so as to fit into three lectures. §17.3. Static Stability Loss As just noted, we restrict attention to stability of static equilibrium. Two assumptions are introduced: Linear Elasticity. The structural material is, and remains, linearly elastic. Displacements and rotations, however, are not necessarily small. Conservative Loading. The applied loads are conservative, that is, derivable from a potential. For example, gravity and hydrostatic loads are conservative. On the other hand, aerodynamic and propulsion loads (wind gusts on a bridge, rocket thrust, etc) are often nonconservative. The main reason for the second restriction is that loss of stability under nonconservative loads is inherently dynamic in nature, and thus lies beyond the scope of the course. §17.3.1. Buckling Or Snapping? Under the foregoing restrictions, two types of instabilities may occur: Bifurcation. Structural engineers use the more familiar name buckling for this one. The structure reaches a bifurcation point, at which two or more equilibrium paths intersect. What happens after the bifurcation point is traversed is called post-buckling behavior. Snapping. Structural engineers use the term snap-through or snap buckling for this one. The structure reaches a limit point at which the load reaches a maximum. What happens after the limit point is traversed is called post-snapping behavior. Bifurcation points and limit points are instances of critical points. The importance of critical points in static stability analysis stems from the following property: Transition from stability to instability can only occur at critical points Reaching a critical point may lead to immediate destruction (collapse) of the structure. This depends on its post-buckling or post-snapping behavior, and nature of the material. For some scenarios the knowledge of such behavior is important since immediate collapse may lead to loss of life. On the other hand, there are some configurations where the structure keeps resisting significant — or even increasing — loads after traversing a critical point. Such “load-sustaining” designs are obviously preferable from a safety standpoint. 17–5

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Load or load parameter

(a)

Load or load parameter

Equilibrium path Terms in red are those in common use by structural engineers

L

Equilibrium paths

B

Initial linear response

Bifurcation point (buckling)

Representative deflection

R

(b) Limit point (snap)

R

Reference state

Representative deflection

Reference state

Figure 17.2. Graphical representation of static equilibrium paths and their critical points: (a) a response path with no critical points; (b) multiple response paths showing occurrence of two critical points types: bifurcation point (B) and limit point (L).

§17.3.2. Response Diagrams To illustrate the occurrence of static instability as well as critical points we will often display loaddeflection response diagrams.‡ This is a plot of equilibrium configurations taken by a structure as a load, or load parameter, is gradually and continuously varied. The load (or load parameter) is plotted along the vertical axis while a judiciously chosen representative deflection is plotted along the horizontal axes. A common convention is to take zero deflection at zero load. This defines the reference state, labeled as point R in such plots. A continuous set of equilibrium configurations forms an equilibrium path. Such paths are illustrated in Figure 17.2. The plot in Figure 17.2(a) shows a response path with no critical points. On the other hand, the plot in Figure 17.2(b) depicts the occurrence of two critical points: one bifurcation and one limit point. Those points are labeled as B and L, respectively. §17.3.3. Stability Models Stability models of actual structures fall into two categories: Continuous. Such models have an infinite number of degrees of freedom (DOF). They lead to ordinary or partial differential equations (ODEs or PDEs) in space, from which stability equations may be derived by perturbation techniques. Obtaining nontrivial solutions of such equations generally leads to trascendental eigenproblems, even if the underlying model is linear. Discrete. These models have a finite number of DOF in space. Where do these come from? Often they emerge as discrete approximations to the underlying continuum models. Two common discretization techniques are: (1) Lumped parameter models, in which the flexibility of the structure is localized at a finite number of places. One common model of this type for columns: joint-hinged rigid struts supported by extensional or torsional springs at the joints. (2) Finite element models that include the so-called geometric stiffness effects. ‡ Students should be familiar with this visualization technique if they have done tension or torsion mechanical tests.

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§17.4

EXACT VERSUS LINEARIZED STABILITY ANALYSIS

Stability equations for discrete models may be constructed using various devices. For lumped parameter models one may resort to either perturbed equilibrium equations built via FBDs, or to energy methods. For FEM models only energy methods are practical. All techniques eventually lead to matrix stability equations that take the form of an algebraic eigenproblem. Both continuous and lumped-parameter discrete models for column problems are presented later in this Lecture, and in the following two Lectures. §17.3.4. Stability Equations Derivation The equations that determine critical points are called characteristic equations in the applied mathematics literature.* In structural engineering the names stability equations and buckling equations are common. For structural stability analysis, two methods are favored for deriving those equations. Equilibrium Method. The equilibrium equations of the structure are established in a perturbed equilibrium configuration. This is usually done with Free Body Diagrams (FBD). The resulting equations are examined for the occurrence of nontrivial perturbed equilibrium configurations. Such perturbed configurations are obtained by disturbing parametrized equilibrium positions by admissible buckling modes. If those equilibrium equations are linearized for small perturbations, one obtains an algebraic eigenproblem. The eigenvalues give values of critical loads while eigenvectors yield the buckling mode shapes. Energy Method. The total potential energy of the system is established in terms of the degrees of freedom. The Jacobian matrix of the potential energy function, taken with respect to those degrees of freedom, is established and tested for positive definiteness as the load parameter (or set of parameters) is varied. Loss of such property occurs at critical points. These may be in turn categorized into bifurcation and limit points according to a subsequent eigenvector analysis. The energy method is more general for structures subject to conservative loading. It has two major practical advantages: (1) merges naturally with the FEM formulation and so it can be efficiently implemented in general-purpose codes, and (2) requires no a priori assumptions as regards admissible perturbations. But since energy methods are not covered at the undergraduate level, only equilibrium methods will be presented here. Such techniques are necessarily restricted to simple 1D problems amenable to FBDs, but that approach is sufficient for an introduction to stability analysis. §17.4. Exact Versus Linearized Stability Analysis As noted above, we will use only the equilibrium method to set up stability equations. Its key feature is that FBDs must take the perturbed configuration into account. For certain simple problems it is possible to establish the equations using the exact geometry of the deflected structure. Such analyses will be called geometrically exact. A couple of examples follow to illustrate exact versus linearized results.

* Often this term is restricted to the determinantal form of the stability eigenproblem. This is the equation whose roots give the eigenvalues, which can be interpreted physically as critical loads, or as critical load parameters.

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(a)

(c)

(b)

P = λ Pref

vA = L sin θ

A

A θ

P = λ Pref A'

L rigid k

MB = k θ

;;; B

λ Equilibrium path of tilted column

Bifurcation point

L

λ cr

Equilibrium path of straight (untilted) column

B P

θ Stable Unstable

Figure 17.3. Geometrically exact analysis of the hinged cantilevered rigid (HRC) column: (a) untilted column, (b) tilted column, (c) equilibrium paths intersecting at bifurcation point.

§17.4.1. Example 1: The HCR Column: Geometrically Exact Analysis Consider the configuration depicted in Figure 17.3(a). A rigid strut of length L stabilized by a torsional spring of stiffness k > 0 is axially loaded by a vertical dead load P = λPr e f , in which Pr e f is a reference load and λ a dimensionless load parameter. The load remains vertical as the column tilts. (Observe that k has the physical dimension of force × length, i.e. of a moment.) This configuration will be called a hinged cantilevered rigid column, or HCR column for brevity. The definition P = λk/L renders λ dimensionless, which is convenient for result presentation. As state parameter we pick the tilt angle θ as most appropriate for hand analysis. For sufficiently small P the column remains vertical as in Figure 17.3(a), with θ = 0. The only possible buckled shape is the tilted column shown in Figure 17.3(b). This Figure depicts the FBD required to analyze equilibrium of the tilted column. Notice that θ is not assumed small. Taking moments with respect to the hinge B as sketched we obtain the following equilibrium equation in terms of λ and θ: k θ = Pr e f d = λ Pr e f L sin θ

⇒

kθ −λ

Pr e f sin θ = 0. L

(17.1)

The equation on the right hand side has the two solutions θ = 0 for any λ,

λ=

k Pr e f

θ . L sin θ

(17.2)

These pertain to the untilted (θ = 0) and tilted (θ = 0) equilibrium paths, respectively. Since lim(θ/ sin θ ) → 1 as θ → 0, the two paths intersect when λcr =

k Pr e f L

,

or

Pcr =

k . L

(17.3)

The two paths are plotted in Figure 17.3(c). The intersection (17.3) characterizes a bifurcation point B. The diagram shows four branches emanating from B. Three are stable (full lines) and one is unstable 17–8

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§17.4


(dashed line). Note that the applied load may rise beyond the critical Pcr = λcr Pr e f = k/L by moving to a tilted configuration. It is not difficult to show that the maximum load occurs if θ → 180◦ , for which P → ∞; this is a consequence of the assumption that the column is rigid (and that it may fully rotate by that amount about the hinge). §17.4.2. Example 1: The HCR Column: LPB Analysis The geometrically exact analysis that leads to (17.2) has the advantage of providing a complete solution. In particular, it shows what happens after the bifurcation point B is traversed. For this configuration the structure maintains load-bearing capabilities while tilted, which is the hallmark of a safe design. But for more complicated cases this approach becomes impractical as it involves solving systems of nonlinear algebraic or differential equations. Even for the Euler column presented in Lecture 18, a geometrically exact analysis leads to elliptic functions. Often the engineer is interested only in the critical load. This is especially true in preliminary design scenarios, when the main objective is to assess safety factors against buckling. If so, it is more practical to work with a linearized version of the problem. Technically the full technical name is linearized prebuckling (LPB) analysis. This approach relies on the following assumptions: •

Deformations prior to buckling are neglected. This permits the analysis to be carried out in the reference configuration geometry.

•

Perturbations of the equilibrium configuration produce only infinitesimal displacements and rotations.

•

The structure remains linearly elastic up to buckling.

•

Both structure and loading do not exhibit any imperfections.

•

The critical state is a bifurcation point.

We apply these rules to the HCR column of Figure 17.3(a). The equilibrium equation (17.1) is linearized by assuming an infinitesimal tilt angle θ << 1, whence sin θ ≈ θ and that expression becomes Pr e f θ = 0, (17.4) k−λ L This is the stability equation. Since the product of two numbers is zero, at least one must be zero: θ = 0 or k − λ Pr e f /L = 0, or both. The solution θ = 0 reproduces the untilted configuration. For buckling to occur, we must have θ = 0. If so, the expression in parenthesis must vanish, which requires λ = λcr = k/(Pr e f L). This reproduces the critical load given in (17.3). Note that this analysis yields only the critical load. It does not provides any information on postbuckling behavior. If this is necessary, a more comprehensive analysis, such as done in the previous subsection, is required. It should be noted that if the loss of stability is by snap buckling, it cannot be obtained by LPB. The main reason is that deformations prior to buckling are essential in the determination of limit points, whereas the first LPB assumption listed above explicitly neglects those.

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17–10


; ;;

(a)

P = λ Pref

C

A k L rigid

; ;;

(b) vA = L sin θ

A

C'

P = λ Pref A'

uA = L (1− cos θ)

θ

L

;; B

B P

Figure 17.4. Geometrically exact analysis of a propped cantilevered rigid (PRC) column with extensional spring remaining horizontal: (a) untilted column, (b) tilted column.

(a)

vA = L sin θ

A FA = k vA

λ

(b) P = λ Pref A'

θ L cos θ

Bifurcation point Equilibrium path of straight (untilted) column

Equilibrium path of tilted column

λ cr = k L/Pref B

−90

P

Stable Unstable

+90

θ

Figure 17.5. Geometrically exact analysis of a propped cantilevered rigid (PRC) column with extensional spring remaining horizontal: (a) FBD for tilted equilibrium, (b) equilibrium paths intersecting at bifurcation point.

§17.4.3. Example 2: The PCR Column: Geometrically Exact Analysis Consider next the configuration pictured in Figure 17.4(a); this differs from the HRC column in the type of stabilizing spring. A rigid strut of length L is hinged at B and supports a vertical load P = λPr e f at end A. The load remains vertical as the column tilts. The column is propped by an extensional spring of stiffness k attached to A. This configuration will be called a propped cantilevered rigid column; or PCR column for short. The only DOF is the tilt angle θ. For the geometrically exact analysis is it important to know what happens to the spring as the column tilts. One possible assumption is that it remains horizontal, as depicted in Figure 17.4(b). If so the FBD in the tilted configuration will be as shown in Figure 17.5(a). Taking moments about B yields λPr e f sin θ = k L sin θ cos θ. 17–10

(17.5)

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§17.4

; ;

ηL

(a)


P = λ Pref A

C

; ;

(b) vA = L sin θ

A

P = λ Pref A'

C

k


θ

L L

rigid

;; B

B P

Figure 17.6. Geometrically exact analysis of a propped cantilevered rigid (PRC) column with wall-attached extensional spring: (a) untilted column, (b) tilted column. ϕ = angle A'CA positive CW

vA = L sin θ

A

C

P = λ Pref A'

FA = k ds ds : tilting spring elongation


θ

B P Figure 17.7. Geometrically exact analysis of a propped cantilevered rigid (PRC) column with wall-attached extensional spring: (a) FBD for tilted equilibrium, (b) equilibrium paths intersecting at bifurcation point. 4

λ

3

−60

−40

2

η = 1/4

2

1

1

0.5

−20

20

1 0.75

−60

40

−40

−20

0.8

η=1

0.5

0.6

0.25 −60

−40

−20 −0.25 −0.5

20 −1 1

λ

20

40

60

η = 1/2

−0.5

angle θ (deg)

−1

λ

1.5

80

λ η=5

0.2 −60

−1

60

0.4

80

angle θ (deg)

−0.75

40

angle θ (deg)

−40 −20

angle θ (deg) 20

40

60

80

Figure 17.8. Geometrically exact analysis of a propped cantilevered rigid (PRC) column with wall-attached extensional spring: λ vs./ θ response diagrams for four values of η (defined in Figure 17.6(a)), with Pr e f = k = L = 1.

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17–12

This has two solutions θ = 0 for any λ ,

λ=

kL cos θ. Pr e f

(17.6)

These yield the vertical and tilted equilibrium paths, respectively, plotted in Figure 17.5(b) on the λ versus θ plane. The two paths intersect at λ = k L/Pr e f , which is a bifurcation point. Consequently the critical load parameter is λcr = k L/Pr e f and the critical load is Pcr = λcr Pr e f = k L. Of the 4 branches that emanate from the bifurcation point B, only one (the θ = 0 path for λ < λcr is stable. Once B is reached the tilted column supports only a decreasing load P, which vanishes at θ = ±90◦ . Consequently this configuration is poor from the standpoint of post-buckling safety. three are unstable. Another reasonable assumption is that the spring attachment point to the wall, called C in Figure 17.6(a), stays fixed. The distance AC is parametrized with respect to the column length as ηL, in which η is dimensionless. If the column tilts, the spring also tilts as pictured in Figure 17.6(b). The geometrically exact FBD for this case is shown in Figure 17.6(b). As can be observed, it is considerably more involved than for the stay-horizontal case. We quote only the final result for the equilibrium path equations: k L η cos θ + sin θ . θ = 0 for any λ, λ = (17.7) Pr e f η + sin θ Figure 17.8 shows response plots on the λ versus θ plane for the four cases η = 14 , η = 12 η = 1 and η = 5, drawn with Pr e f = 1, k = 1, and L = 1. Although the bifurcation points are in the same location, the post-buckling response is no longer symmetric for ±θ. That deviation from symmetry is most conspicuous when η < 1, since if so the spring tilting has a noticeable effect if θ < 0. The sharp drop of λ towards −∞ occurs when the spring and the tilted column are nearly aligned. As η → ∞ one recovers the solution (17.2), as may be expected. §17.4.4. Example 2: The PCR Column - LPB Analysis To linearize this problem, again assume θ is so small that sin θ ≈ θ and cos θ ≈ 1. Then the equilibrium equations of the foregoing cases collapse to (λ Pr e f L − k L) θ = 0.

(17.8)

This has the equilibrium paths θ = 0 and λ = k L/Pr e f . Consequently λcr =

kL , Pr e f

or

Pcr = λcr Pr e f = k L .

(17.9)

This result is independent of assumptions on how the spring wall-attachment point behaves after the column buckles. This is to be expected since linearization filters out that information.

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§17.5 (a)

k

DISCRETE STABILITY ANALYSIS AS EIGENPROBLEM

A

3k/2

L

v deflections + to the right

vB B

B'

rigid

B

L/2

P A'

A

rigid

L/2

vA

(b)

P

C

C

Figure 17.9. Cantilevered two-strut column. (a): structure; (b): deflected shape defined by two DOF v A and v B ;

§17.5. Discrete Stability Analysis As Eigenproblem When the discrete model has n ≥ 2 DOFs, the determination of critical loads by the LPB approach leads to a matrix algebraic eigenproblem of order n. The n eigenvalues provide the critical loads, and the corresponding eigenvectors give the associated buckling shapes. This is illustrated through several examples worked out below. §17.5.1. Example 3: A Cantilevered Two-Strut Column The problem investigated here in detail is the buckling of the two-hinged-cantilever column depicted in Figure 17.9(a).† Rigid links of equal length L/2 are connected at the joints A, B and at the bottom C by frictionless hinges. The column is propped by two extensional springs attached at A and B with the stiffnesses shown in the figure. Data: Length L and spring stiffness k. Required: Set up the stability equations as a matrix eigenproblem, find critical loads in terms of the data and display buckling shapes as separate diagrams. Draw an arbitrary, but kinematically admissible, buckling shape as in Figure 17.9(b). This shape is completely defined by the two lateral deflections v A and v B of points A and B, respectively, positive to the right. Note: The deflections are highly exaggerated in the figure for visibility; they are actually infinitesimally small. Figure 17.10(c1,c2,c3) shows free body diagrams (FBD) of links AB, BC and of the whole column ABC, respectively. To facilitate visualization applied forces are pictured in black, spring forces in blue, whereas internal reactions are shown in red. The two lateral deflections v A and v B are taken as independent degrees of freedom (DOF). Consequently, two equilibrium equations are required obtained from FBDs are required. Three combinations of two FBDs are possible: AB and BC, AB and whole column, or BC and whole column. In all FBDs translational force equilibrium holds identically by construction. The remaining moment equilibrium conditions become part of the matrix stability equation. Since the three FBDs are linearly dependent † Adapted from E. P. Popov, Engineering Mechanics of Solids

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17–14


P

vA

A

A

A' FA

P

vA

A' FA

(c1) FBD of link AB

vB

FA +FB B B'

FB

B

P P (c2) vB

FA +FB

C

P

B'

FB

FBD of whole column

FBD of link BC

FA +FB

vB

FA +FB

B'

B

(c3)

Applied forces in black, spring forces in blue, reactions in red

Restoring spring forces + to the left

C

P

FA = k vA , FB = 3kvB /2

Figure 17.10. Cantilevered two-strut column. (c1,c2,c3): FBDs of link AB, link BC and whole column, respectively. FBD color convention: applied forces in black, spring foces in blue, reactions in red.

[check the figure: merging (c1) and (c2) gives (c3)], the results of the analysis should be exactly the same, although the stability matrix equations will be different. For the ensuing derivation we pick (c1) and (c3). For link AB, taking moments with respect to the displaced hinge point B’, positive CW, gives (17.10) M B = P(v A − v B ) − FA ( 12 L) = P(v A − v B ) − 12 Lkv A = 0 For the whole column ABC, taking moments with respect to the hinge C, positive CW, gives (17.11) MC = Pv A − FA L − FB ( 12 L) = Pv A − k Lv A − 32 k( 12 L)v B = 0 Characteristic Equation. Combining (17.10) and (17.11) in a matrix equation gives the stability equation (also known as the characteristic equation): −P P − 12 k L 0 vA = , or Av = 0. (17.12) 3 0 vB P − kL −4kL Matrix A is called the stability matrix or characteristic matrix. System (17.12) has two kinds of solutions. The trivial solution v = 0, or v A = v B = 0, corresponds to the undeflected (vertical) column, which holds for any P. To find the critical loads we must have v = 0, that is, at least one of the {v A , v B } must be nonzero. Consequently A must be singular or, equivalenet, have zero determinant. Since the entries of A depend on P, this is an algebraic eigenproblem in P. Critical Loads. The eigenvalues are the solution of the characteristic equation det(A) = 0,

or

P 2 − 74 k L P + 38 k 2 L 2 = 0

(17.13)

Solving this quadratic for P gives the two critical loads: Pcr 1 = 14 k L = 0.25 k L ,

Pcr 2 = 32 k L = 1.50 k L .

17–14

(17.14)

17–15

§17.5

DISCRETE STABILITY ANALYSIS AS EIGENPROBLEM

Pcr = Pcr1 = 0.25 kL 1

B'

Pcr2 = 1.50 kL 1

A'

A'

2/3

−1

B'

Figure 17.11. Buckling mode shapes for the cantilevered two-strut column.

The smallest one is the critical load: Pcr = Pcr 1 = 14 k L = 0.25 k L .

(17.15)

Buckling Mode Shapes. Replacing Pcr 1 into (17.14) gives the following equation to determine the eigenvector v1 : k L −3 −3 v A1 0 −1 v A1 = ⇒ v1 = = c1 (17.16) 0 1 v B1 v B1 4 −1 −1 where c1 is an arbitrary nonzero scaling factor. Replacing Pcr 2 into A gives the following equation to determine the eigenvector v2 : k L 2 −3 v A2 v A2 0 3/2 = ⇒ v2 = = c2 0 1 v B2 v B2 4 4 −6

(17.17)

where c2 is an arbitrary nonzero scaling factor. One common nomalization condition for eigenvectors is to make their largest component equal to +1. This is done for (17.16) and (17.17) by taking c1 = 1 or c1 = −1 (either one works), and c2 = 2/3, respectively. The two eigenvectors normalized as per that criterion are

1 v1 = −1

for Pcr 1 ,

1 v2 = 2/3

for Pcr 2 .

(17.18)

The buckling shapes defined by (17.18) are plotted in Figure 17.11. §17.5.2. Example 4: A Pinned-Pinned Three-Strut Column This is a slight varint of one treated in Jennings (op. cit.). Figure 17.12(a) shows a column built with three rigid struts of equal length, pinned at the joints, and with two extensional spring of stiffness k as lateral supports at the internal joints B and C. Determine the critical loads of this configuration by linearization. Figure 17.12(b) depeicts how the structure might displace after buckling. Since the structs are rigid and cannot change length, joints A, B and C will necesasrily moved down while the springs tilt; 17–15

17–16


(a)

P

(b)

(c)

P>Pcr

A

A' (1/3) (2FB +FC ) = (1/3) k (2vB +vC )

rigid

L/3

k B

FB = k vB

B'

v deflections + to the right

rigid

L/3

k C

B' vC

FC = k vC

C'

rigid

L/3

P A vB

C'

(1/3) (FB +2FC ) = (1/3) k (vB +2vC )

D

D

D

Figure 17.12. Stability analysis of three-strut column propped by extensional springs: (a) reference configuration; (b) admissible buckled shape showing realistic geometry change (struts do not change length); (c) linearized configuration: B and C displace by small horizontal movements whereas A stays fixed.

such geometry is correct for geometrically exact analysis. The linearized version is shown in Figure 17.9(c); here B and C move horizontally by v B and v D , respectively, and A does not move vertically. The horizontal reactions R A and R D shown in Figure 17.9(c) are obtained in terms of FB and FD by statics on taking moments with respect to D and A, respectively. The two FBDs used to form the stability equations are shown in Figure 17.13. We take moments with respect to C’ in the left FBD and with respect to B’ in the right FBD:

MC = −P vC − FB

L 2L + RA = 0, 3 3

M B = −P v B − FC

vB

P RA = (1/3) (2FB +FC ) = (1/3) k (2vB +vC )

P FB = k vB

A vB

FB = k vB

B' vC

FC = k vC

B' vC

FC = k vC

L 2L + RD = 0. 3 3

C'

RD = (1/3) (FB +2FC ) = (1/3) k (vB +2vC )

D

C' P

P

Figure 17.13. Free Body Diagrams for the 3-strut column of Figure 17.12.

17–16

(17.19)

17–17

§17.5

DISCRETE STABILITY ANALYSIS AS EIGENPROBLEM Pcr1 = kL/9

Pcr2 = kL/3

A

A

Buckling mode #1 (antisymmetric)

Buckling mode #2 (symmetric)

B 1

B 1

−1 C

C 1

D

D

Figure 17.14. Critical loads and buckling mode shapes for the 3-strut column of Figure 17.12.

Substituting R A = (2FB + FC )/3, R D = (FB + 2FC )/3, FB = k v B and FC = k vC , and casting (17.19) in matrix form we obtain the stability equations kL 2 1 vB vB =P , (17.20) vC vc 9 1 2 This is a 2 × 2 matrix eigenproblem with P as the eigenvariable. The characteristic equation is k2 L 2 4k P L kL 2 1 1 0 −P = (17.21) − + P 2. C(P) = det 0 1 9 1 2 27 9 This is a quadratic polynomial in P. Setting C(P) = 0 gives the two critical loads: Pcr 1 = k L/9 and Pcr 2 = k L/3, as roots. The corresponding eigenvectors, normalized to 1 as largest entry, are −1 1 , v2 = . (17.22) v1 = 1 1 These are plotted in Figure 17.14. Note that the mode shape corresponding to the lowest critical load, Pcr 1 = k L/9, is antisymmetric. This is in sharp contrast to the Euler column buckling treated in Lecture 18. An alternative eigenvalue problem is obtained on premultiplying by the inverse of the LHS matrix in (17.20). This gives 3P 2 −1 vB vB = , (17.23) −1 2 v vC kL c an eigenproblem which has the same solutions. This is the form derived by Jennings (except for scale factors).

17–17

Stability Of Structures: Basic Concepts

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