Structural Stability

Module aims

• To give students a rigorous grounding in the behaviour of structural components and systems that suffer from failure due to geometric, rather than material, nonlinearity; the principal features being that failure primarily occurs in the elastic range and due to buckling. It is a course based on fundamental mechanics that is designed to give the theoretical background to the more practical design-based modules.

Learning outcomes

On successfully completing this course unit, students will have:

• A rigorous grounding in the theory of structural stability and nonlinear structural behaviour.
• An appreciation of the potential failure modes that can occur due to geometric nonlinearity.
• The techniques to classify post-buckling phenomena.
• The techniques to analyse geometrically perfect and imperfect systems for structural stability.
• An appreciation of the differences between linear and nonlinear buckling analysis.
• An understanding of how basic structural components behave when they are subject to instability.
• The techniques to analyse basic structural components that are susceptible to instability.
• An appreciation of the fundamental basis of design rules concerned with structural instability.

Module syllabus

• Introduction to potential energy methods for single degree-of-freedom elastic systems. Axioms connecting potential energy to equilibrium and stability. General Theory approach. Determination of bifurcation points and classification of stability of equilibrium for post-buckling responses for geometrically perfect systems. Imperfect systems: determination of imperfection-sensitivity.
• Instabilities in struts and columns: direct equilibrium and energy formulations; Euler load and the elastica; effective length concept. Approximate methods of analysis: Rayleigh and Timoshenko methods. Ultimate strength of real columns using the Perry-Robertson formulation and the description of the method for designing steel columns in Eurocode 3.
• Multiple degree-of-freedom elastic systems: diagonalised systems; elimination of passive coordinates; non-trivial fundamental paths; introduction to mode interaction.
• Instabilities in beams: direct equilibrium and energy formulations, critical moment for lateral-torsional buckling, general loading cases and effective lengths and the description of the method for designing steel beams in Eurocode 3.
• Instabilities in rigid framed structures: analysis using stability functions and limitations.
• Instabilities in plates: critical and post-buckling of plated structures under compression and shear.
 No. Topic Staff 01 Introduction to stability concepts. Physical demonstrations of instabilities. Perturbation of static systems. Principles of minimum total potential energy: equilibrium and stability. Rolling ball analogy MAW 02 Energy formulation for elastic systems. Evaluation of strain energy from direct and bending stresses and strains. MAW 03 Single degree-of-freedom (SDOF) systems. Stability of analysis of systems with exact formulations. General Theory: Perturbation methods for SDOF problems. Calculation of critical buckling loads and classification of post-buckling behaviour. MAW 04 Analysis of systems with imperfections. Imperfection sensitivities for different distinct post-buckling responses. MAW 05 Buckling of struts and columns: Perfect column and effective lengths for different boundary conditions. Approximate analytical methods due to Rayleigh and Timoshenko. MAW 06 Real columns: Perry–Robertson approach to calculate ultimate strength of inelastic columns. Implementation in structural design codes. MAW 07 Multiple degree-of-freedom (MDOF) systems I: Linear analysis. Diagonalisation. MAW 08 Lateral-torsional buckling of beams. Evaluation of elastic critical moment. Lateral restraints and buckling lengths. Implementation of structural design procedures. MAW 09 MDOF systems II: Nonlinear analysis. Elimination of passive coordinates. Non-trivial fundamental paths. MAW 10 Buckling of plates: Critical buckling under axial and shear stresses. Post-buckling analysis.  Ultimate behaviour: von Kármán “Effective Width” concept and Winter's design curve. MAW 11 MDOF systems III: Introduction to mode interaction – Augusti's model. Secondary and compound bifurcations. Discussion of wider consequences on structural response and imperfection sensitivity. MAW

Teaching methods

Each session is 3 hours long. The course has lectures and supporting tutorials. Staff and GTAs will be available to answer specific questions.

Assessments

Assessment is by written examination only.

Supplementary

• Elastic instability phenomena

Thompson, J. M. T.

Wiley

• A general theory of elastic stability

Thompson, J. M. T.

London : Wiley-Interscience

• Background to buckling

Allen, Howard G.

McGraw Hill

• Stability of structures elastic, inelastic, fracture and damage theories

Bazant, Z. P.

World Scientific Publishing

• Theory of elastic stability

Timoshenko, Stephen

2nd, McGraw Hill

• Theory of Elastic Stability [electronic resource].

Timoshenko, Stephen P.

Dover Publications

• The behaviour and design of steel structures to EC3

4th ed., London : Taylor & Francis

• The behaviour and design of steel structures to ec3

4th ed., London ; New York : Taylor & Francis