# Stability Of Structures: Basic Concepts

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1 23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1

2 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for testing stability (3) classifies models and analysis methods (4) concludes by contrasting exact versus linearized determination of critical loads ASEN 3112 Lecture 23 Slide 2

3 Terminology Related to Mechanical Systems: Table 23.1 of Lecture 23 Term Definition System A functionally related set of components regarded as a physical entity. Configuration The relative disposition or arrangement of system components. State The condition of the system as regards its form, structure or constitution. Degrees of Freedom A set of state variables that uniquely characterizes the state. Abbrv.: DOF Kinematic DOF A DOF that is directly linked to the system geometry; e.g., a displacement. Input The set of all actions that can influence the state of a system, or component. Output The set of all quantities that characterize the state of a system, or component. Model A mathematical idealization of a physical system. Discrete Model A model with a finite # of DOF. Often expressed as vector equations. Continuous Model A model with an infinite # of DOF. often expressed as a ODE or PDE. Event A change in the state produced by an agent. Behavior A pattern of events. Reference State A state of the system adopted as base or origin to measure relative changes. Often the same as the undeformed state (cf. Table 23.2). Motion The change in system geometry, as measured from a reference state. Kinematics The study of system motion independently of force agents. Kinetics The study of forces as action agents, and their effect on the system state. Kinematic Constraint Any condition that restricts the system motion, usually expressed in terms of kinematic DOF. Also simply called constraint. Environment A set of entities that do not belong to the system, but can influence its behavior. Open System A system that is influenced by entities outside the system (its environment). Closed System A system that is not affected by entities outside the system. Interaction The mutual effect of a system component, or group of such components, on other components. Forces The action agents through which effects are transmitted between system components, or between environment entities and system components. Internal Forces Forces that act between system components. External Forces Forces that act between environment entities and system components. Constraint Force A force manifested by removing a constraint while keeping it enforced. Reaction Force A constraint force that is an external force. Applied Load An external force specified as data. Also simply called load. ASEN 3112 Lecture 23 Slide 3

5 What is Structural Stability? ASEN Structures Broadly speaking: The power to recover equilibrium It is an essential requirement for all structures ASEN 3112 Lecture 23 Slide 5

6 Two Stability Scenarios ASEN Structures Static Stability Stability of static equilibrium configurations of a mechanical system (e.g., structures) Dynamic Stability Stability of the motion of a dynamic system (e.g., a vehicle trajectory) In this course we consider only static stability ASEN 3112 Lecture 23 Slide 6

7 Testing Static Stability: Varying Applied Loads Scale applied loads by a parameter λ - this is called the load factor or the loading parameter If λ = 0, structure is under zero load, and takes up a reference undeformed configuration C We assume that the structure is stable at λ = 0 Now apply loads by varying λ monotonically away from zero. Structure assumes equilibrium configurations C(λ) 0 ASEN 3112 Lecture 23 Slide 7

8 Testing Static Stability: Perturb and Release To test stability of a specific configuration C(λ d), freeze λ = λ d. Apply a small perturbation (for example, a tiny deflection) and let it go! The structure is set in motion. Three possible outcomes are pictured in the next slide. ASEN 3112 Lecture 23 Slide 8

9 Testing Static Stability: Perturbation Outcomes Equilibrium configuration Apply an allowed perturbation Motion oscillates about equilibrium configuration or decays toward it Subsequent motion outcome Transition between stable and unstable S: Stable* N: Neutrally stable Motion is either unbounded, or oscillates about, or decays toward, another equilibrium configuration U: Unstable * Strictly speaking, S requires stability for all possible admissible perturbations ASEN 3112 Lecture 23 Slide 9

10 Ex. - Identify Stability Outcomes for 5 Heavy Rollers Shown Below - All 5 Are in Static Equilibrium Configurations Under Gravity Gravity ;; ;; ;;; (A) (B) (C) (D) (E) ASEN 3112 Lecture 23 Slide 10

11 Testing Static Stability: Critical Load ASEN Structures For sufficiently small λ = λ d assume that the structure is stable (S). It becomes neutrally stable (N) at a value λ = λ cr This value is called a critical load parameter or critical load factor. The associated load is called the critical load (or loads, if more than one). The determination of the critical load(s) is a key goal of the stability analysis. ASEN 3112 Lecture 23 Slide 11

12 Testing Static Stability: Clarifications As described, the stability test is dynamic since there is a before event: apply perturbation there is an after event: what happens upon release ASEN Structures Thus time is involved. However, under certain conditions (next slide) time may be factored out. If so, a dynamic response analysis is no longer needed, which greatly simplifies test procedures. Those are called static stability criteria. Another question: how small is a "small perturbation"? In examples later, they are supposed to be infinitesimal. ASEN 3112 Lecture 23 Slide 12

13 Assumptions That Allow Use of Static Stability Criteria Linearly elastic material displacements and rotations, however, are not necessarily small Loads are conservative (= derivable from potential) gravity and hydrostatic loads are conservative aerodynamic and propulsion loads are generally not ASEN 3112 Lecture 23 Slide 13

14 Setting Up Stability Equations for Static Stability Criteria Equilibrium Method use FBD in perturbed equilibrium configuration, look for nontrivial solutions as function of λ Energy Method ASEN Structures set up total potential energy of system, analyze whether its Hessian is positive definite (S), nonnegative definite (N), or indefinite (U), as function of λ Only the Equilibrium Method will be used in this course ASEN 3112 Lecture 23 Slide 14

15 Models To Predict Critical Loads Of Structures Discrete Models Finite # of DOF Lead to discrete set of equations and algebraic eigenproblems for critical loads Continuous Models Infinite # of DOF Lead to ordinary or partial differential equations in space and trascendental eigenproblems for critical loads ASEN 3112 Lecture 23 Slide 15

16 Stability Model Subclassification ASEN Structures Discrete Models Lumped parameter Finite element discretizations Continuous Models Ordinary Differential Equations (ODE) for 1D models Partial Differential Equations (PDE) for 2D and 3D models Only those in red are considered in these Lectures ASEN 3112 Lecture 23 Slide 16

17 Load-Deflection Response Plot ASEN Structures Load or load factor Primary equilibrium path Initial linear response R Reference state Representative deflection ASEN 3112 Lecture 23 Slide 17

18 Critical Points Displayed in Response Plot ASEN Structures Load or load factor Limit point (snapping) Primary equilibrium path R B L Bifurcation point (buckling) Reference state Terms in red are those in common use by structural engineers Equilibrium paths Representative deflection ASEN 3112 Lecture 23 Slide 18

19 Importance of Critical Points in Structural Design ASEN Structures General property in static stability: Transition from stability to instability always occurs at a critical point ASEN 3112 Lecture 23 Slide 19

20 First Critical Point Found Along Primary Equilibrium Path is That of Interest in Design Load or load factor B L Critical load, or load factor, of interest R Primary equilibrium path Representative deflection Reference state ASEN 3112 Lecture 23 Slide 20

21 First Critical Point Found Along Primary Equilibrium Path is That of Interest in Design - 2 Load or load factor L Critical load, or load factor, of interest for design R Primary equilibrium path Reference state B Representative deflection ASEN 3112 Lecture 23 Slide 21

22 Geometrically Exact Versus Linearized Prebuckling (LPB) Stability Analysis Geometrically Exact Analysis Exact geometry of deflected structure taken into account Can handle both bifurcation and limit points Loading must be conservative (to stay within statics) Inelastic behavior and imperfections may be accommodated Linearized Prebuckling (LPB) Analysis Deformations prior to buckling are neglected Perturbations of the reference configuration involve only infinitesimal displacements and rotations Only linear elastic behavior and conservative loading No geometric or loading imperfections The critical state must be a bifurcation point ASEN 3112 Lecture 23 Slide 22

23 Geometrically Exact Analysis Example 1: The Hinged Rigid Cantilever (HRC) Column P = λ P ref A L rigid k B Load P remains vertical, even if the column tilts This is necessary so the applied loads are conservative ASEN 3112 Lecture 23 Slide 23

24 Geometrically Exact Stability Analysis of HRC Column: FBD ASEN Structures Perturb column by a tilt angle θ, which is arbitrary (not necessarily small). Taking moments with respect to hinge B yields k θ = P v = λ P A ref L sin θ k θ λ P L sin θ = 0 ref This equilibrium equation has two solutions: v = L sin θ A A M = k θ θ P = λ P ref A' L θ = 0 for any λ λ = P k ref L θ sin θ B P These define the equilibrium paths for the untilted (vertical) and tilted column, respectively ASEN 3112 Lecture 23 Slide 24

25 Geometrically Exact StabilityAnalysis of HRC Column: Bifurcation Point The two equilibrium paths, plotted at right, intersect at the bifurcation point θ = 0, λ = λ, in which cr λ Equilibrium path of tilted column λ = cr k P L ref whence the critical load is P = λ P cr cr ref = k L Bifurcation point Primary equilibrium path: vertical (untilted) column Stable Unstable λ cr = k /(P ref L ) Note that the column keeps taking additional load after buckling, which suggests a safe configuration from the standpoint of postbuckling reserve strength θ ASEN 3112 Lecture 23 Slide 25

26 Linearized Prebuckling (LPB) StabilityAnalysis of HRC Column Assume tilt angle is very small: θ << 1 so that sin θ ~ θ and cos θ ~ 1. Replacing into the exact equilibrium expression yields the LPB stability equation λ ASEN Structures (k λ P L) θ = 0 ref One solution is θ = 0 (untilted column). For a nonzero θ the expression in parentheses must vanish, whence λ = cr k P L ref P cr = λ cr P ref = k L Bifurcation point λ cr = k /(P ref L ) θ This is the same result given by the geometrically exact analysis for the critical load. But note that the LPB analysis does not provide any information on postbuckling behavior. See diagram above ASEN 3112 Lecture 23 Slide 26

27 Geometrically Exact Analysis Example 2: The Propped Rigid Cantilever (PRC) Column P = λ P ref v = L sin θ A C A P = λ P ref A ; u = L (1 cos θ) k ; C' A θ A' rigid L L B B P Load P remains vertical, even if the column tilts As the column tilts, spring remains horizontal - see right Figure ASEN 3112 Lecture 23 Slide 27

28 Geometrically Exact Stability Analysis of PRC Column: FBD ASEN Structures Perturb column by a tilt angle θ, which is arbitrary (not necessarily small). Taking moments with respect to hinge B yields k v A L cos θ = P v A 2 k L sin θ cos θ λ P ref L sin θ = 0 Important: do not cancel out sin θ yet! As in the previous example, this equilibrium equation has two solutions: θ = 0 for any λ λ = k L P ref cos θ v = L sin θ A A F = k v A A B θ P = λ P ref A' L cos θ These define the equilibrium paths for the untilted (vertical) and tilted column, respectively ASEN 3112 Lecture 23 Slide 28

29 Geometrically Exact Stability Analysis\ of HRC Column: Bifurcation Point The two equilibrium paths, plotted at right, intersect at the bifurcation point θ = 0, λ = λ, in which λ = cr cr k L P ref whence the critical load is Bifurcation point Primary equilibrium path: vertical (untilted) column λ Equilibrium path of tilted column λ cr = k L/P ref P = λ P cr cr ref = k L 90 Stable Unstable +90 θ Note that the load capacity decreases after buckling, which suggests an unsafe configuration from the standpoint of postbuckling reserve strength ASEN 3112 Lecture 23 Slide 29

30 Linearized Prebuckling (LPB) Stability Analysis of PRC Column Assume tilt angle is very small: θ << 1 so that sin θ ~ θ and cos θ ~ 1. Replacing into the exact equilibrium expression k L sin θ cos θ λ P ref sin θ = 0 yields the LPB stability equation (k L λ P ) θ = 0 ref One solution is θ = 0 (untilted column) as before. For a nonzero θ the expression in parentheses must vanish, whence Bifurcation point λ λ cr = k L/P ref θ λ cr = k L P ref P = λ P = k L cr cr ref This is the same result found using geometrically exact analysis for the critical load. But the LPB analysis doesn't provide any information on postbuckling behavior. See diagram above ASEN 3112 Lecture 23 Slide 30

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