8: Stability of Elastic Structures
- Page ID
- 21524
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 8.1: Prelude to Stability of Elastic Structures
- This page examines static equilibrium and potential energy, highlighting that equilibrium is achieved when the first variation of potential energy is zero. It discusses stability conditions around equilibrium using a rigid body example and defines stable, unstable, and neutral equilibria based on the second variation's sign. The analysis is further applied to elastic bodies, enhancing the understanding of equilibrium stability across various systems.
- 8.2: Trefftz Condition for Stability
- This page examines Erich Trefftz's energy criterion for stability in elastic structures through a one-degree-of-freedom column example. It details the behavior of a hinged rigid column under compressive load, highlighting critical load \(P_c\) as the point where stability shifts. The page outlines equilibrium paths, noting that the primary path becomes unstable beyond \(P_c\), while the secondary path remains stable.
- 8.3: Stability of Elastic Column Using the Energy Method
- This page covers the extension of the Trefftz stability condition for elastic columns subjected to bending and compression, detailing the calculations for potential energy and critical buckling load. It introduces various shape functions, notably a sinusoidal solution, which is shown to be the exact solution for buckling, with its coefficient verified against prior polynomial solutions.
- 8.4: Effect of Structural Imperfections
- This page explores how rotated, imperfect columns react to vertical loads, emphasizing the impact of structural imperfections on equilibrium paths. It outlines the connection between bending moments and presents a magnification factor that reflects the amplification of imperfections under load. Furthermore, it describes the modeling of imperfections in a pin-pin elastic column using differential equations, revealing a sinusoidal solution.
- 8.5: Stability in Tension
- This page explores instability in tension for specific materials, focusing on local neck formation in incompressible round bars under tensile stress. It details changes in volume and cross-sectional area while maintaining constant volume, relating axial strain to length and area changes. True stress is defined, and variations in potential energy are analyzed to clarify stability conditions.
- 8.6: Plastic Buckling of Columns
- This page examines the critical buckling load and stress in pin-pin columns, emphasizing the relationship between buckling behavior and slenderness ratio. It notes that slender columns experience lower buckling stress, while shorter columns encounter higher stress. The critical slenderness ratio for yield stress is derived, with examples for mild steel.
- 8.7: Mode Transition (Advanced)
- This page examines the moment equilibrium equation for a pin-pin supported column, focusing on how imperfections affect deformation, modeled by Fourier series. It presents conditions for deformation solutions and introduces the potential energy method to analyze stability and critical buckling loads.