10: Buckling of Plates and Sections
- Page ID
- 21538
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Most of steel or aluminum structures are made of tubes or welded plates. Airplanes, ships and cars are assembled from metal plates pined by welling riveting or spot welding. Plated structures may fail by yielding fracture or buckling. This chapter deals with a brief introduction to the analysis of plate buckling. A more complete treatment of this subject is presented in the 2.081 course of Plates and Shells, which is available on the Open Course. For additional reading, the following monographs are recommended:
- Stephen P. Timoshenko and James M. Gere, Theory of Elastic Stability.
- Don. O. Brush and Bo. O. Almroth, Buckling of Bars, Plates and Shells.
- 10.1: Governing Equations and Boundary Conditions
- This page covers column and plate buckling, analyzing a rectangular plate under compressive loading with simply supported boundary conditions. It establishes stress boundary conditions and equilibrium equations connecting membrane forces and strain. Displacement derivations show that compression causes lateral expansion due to the Poisson ratio. It emphasizes the need for practical experimental setups or finite element models to maintain the plate's freedom in the in-plane direction.
- 10.2: Buckling of a Simply Supported Plate
- This page covers plate buckling governed by specific loading conditions, focusing on a solution using harmonic functions that meet boundary conditions. It details bending moments, buckling coefficients, and their dependency on aspect ratios and wave numbers. Key findings on minimum buckling loads are presented, along with the impact of limiting lateral expansion on buckling behavior.
- 10.3: Effect of Boundary Conditions
- This page covers boundary conditions in the buckling analysis of rectangular plates, exploring configurations like simply supported, clamped, and free edges. It presents ten combinations of these conditions and associated buckling coefficients influenced by plate aspect ratios. The page cites Timoshenko and Gere's analytical solutions and includes examples with angle elements and columns under simply supported conditions.
- 10.4: Buckling of Sections
- This page covers the behavior of cold-formed or welded profiles in engineering, focusing on buckling characteristics and common cross-sectional shapes. It highlights how some profiles buckle under loads while others need complex analysis involving bending moments. The relationship between buckling coefficients and prismatic column dimensions is detailed, along with the transition to combined bending/compression states, paving the way for subsequent discussions on post-buckling behavior.
- 10.5: Post-buckling Response of Plates (Advanced)
- This page explores the mechanics of plates under axial compression, detailing the transition from pre-buckling to post-buckling states. It highlights the effects of uniaxial compression and introduces displacement equations that account for edge conditions following buckling. Total potential energy is analyzed, leading to equilibrium equations for in-plane and out-of-plane displacements.
- 10.6: Ultimate Strength of Plates
- This page examines post-buckling plate behavior, emphasizing in-plane compressive stress distribution and von Karman's effective width concept. It details how variations in compressive stresses affect load distribution, particularly at plate edges. The effective width is defined to optimize load capacity, and the page concludes by presenting empirical adjustments to enhance the predictive accuracy of von Karman's theory for engineering thin-walled structures.
- 10.7: Effect of Initial Imperfection
- This page discusses geometric imperfections in plates caused by manufacturing, welding distortion, or mishandling during transport. It explains how these imperfections can be modeled using a Fourier series, particularly the first mode of deformation, and modifies key definitions for curvatures and strains. The adjusted expressions account for initial imperfections and revert to standard forms when they are absent.