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 be able to: 

  • understand the theory of structural stability and nonlinear structural behaviour, 
  • appreciate the differences between linear and nonlinear buckling analysis, 
  • appreciate of the potential failure modes that can occur due to geometric nonlinearity, 
  • classify post-buckling behaviour and infer imperfection sensitivity, 
  • analyse geometrically perfect and imperfect systems for structural stability, 
  • understand of how basic structural components behave when they are subject to instability, 
  • analyse basic structural components that are susceptible to instability, 
  • understand and apply the 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.   





Introduction to stability concepts. Physical demonstrations of instabilities. Perturbation of static systems. Principles of minimum total potential energy: equilibrium and stability. Rolling ball analogy



Energy formulation for elastic systems. Evaluation of strain energy from direct and bending stresses and strains.



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.



Analysis of systems with imperfections. Imperfection sensitivities for different distinct post-buckling responses.



Buckling of struts and columns: Perfect column and effective lengths for different boundary conditions. Approximate analytical methods due to Rayleigh and Timoshenko.



Real columns: Perry–Robertson approach to calculate ultimate strength of inelastic columns. Implementation in structural design codes.



Multiple degree-of-freedom (MDOF) systems I: Linear analysis. Diagonalisation.



Lateral-torsional buckling of beams. Evaluation of elastic critical moment. Lateral restraints and buckling lengths. Implementation of structural design procedures.



MDOF systems II: Nonlinear analysis. Elimination of passive coordinates. Non-trivial fundamental paths.



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.



MDOF systems III: Introduction to mode interaction – Augusti's model. Secondary and compound bifurcations. Discussion of wider consequences on structural response and imperfection sensitivity.


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.


Assessment is by written examination only. 

Reading list


Module leaders

Professor Ahmer Wadee