Structural Analysis II E


This course complements the fundamental training of civil engineers in the field of plates, shells and thin-walled bars. Relevant aspects concerning safety, stability and computational modeling are treated. The "engineering judgment" of students is enhanced by encouraging the interpretation of the modeling results.

General characterization





Responsible teacher

Rodrigo de Moura Gonçalves


Weekly - 5

Total - 70

Teaching language



Available soon


Subject collection (presentations, tables, exercises)
A Ugural, “Stresses in plates and shells”, McGraw-Hill, 1999.
L Castro, V Leitão, “Apontamentos sobre análise elástica linear de lajes”, IST, 2001.
L Castro, “Modelação de lajes com elementos de grelha”, IST, 2002.
JAC Martins, “Teoria Elástica Linear de Placas e Lajes”, IST, 1992.
S Timoshenko, S Woinowski-Krieger, “Theory of plates and shells”, McGraw-Hill, 1970.
A Reis e D Camotim, “Estabilidade Estrutural”, McGraw-Hill, 2001.
W Nash, “Resistência de Materiais” (Colecção Shaum’s outlines), McGraw-Hill, 2001.

Teaching method

Theoretical-practical classes. Critical thinking, reflection and interpretation of the results are stimulated. Several group exercises are executed.

Evaluation method

Available soon

Subject matter

Chapter 1. Introduction
Introduction. Plates, membranes and shells. Relevance of their study. Review of Continuum Mechanics.

Chapter 2. Influence lines
Moving loads. Influence functions. Direct and Indirect methods in statically determinate structures. Qualitative approach for statically indeterminate structures. 

Chapter 3. Thin plates subjected to transverse loads
Review of Kirchhoff plate theory. Fundamental hypotheses and relations. Boundary conditions. Symmetry and anti-symmetry simplifications. Statically and kinematically admissible solutions. Exact solutions. Grillage models. Properties of the grillage elements, loading and support conditions. Influence of the various parameters. Numerical methods: Finite Difference and Rayleigh-Ritz. Qualitative determination of influence surfaces. Aplications. 

Chapter 4. Thin plates subjected to in-plane loads
Linear analysis of plates. Fundamental hypotheses and relations. Navier equations. Boundary conditions. Symmetry simplifications. Airy stress function and solution through Finite Differences. The Rayleigh-Ritz method. Bifurcation of plates. Exact solutions. Numerical methods: Finite Difference and Rayleigh-Ritz. Reference to the post-buckling behavior of plates and to the effective width method. Linear analysis of thin-walled beams. Cross-section distortion. The Generalised Beam Theory (GBT) for unbranched open sections. Fundamental hypotheses and relations. Beam on elastic foundation analogy. Bifurcation of thin-walled beams. Local, distortional and flexural-torsional buckling. GBT equations for uniform compression. Reference to the determination of the distortional buckling resistance.

Chapter 5. Thin shells
Introduction. Fundamental concepts. Membrane theory. Fundamental hypotheses and relations. Axissymmetrically loaded shells of revolution. Assymetrically loaded conical and cylindrical shells. Hyperbolic and elliptic paraboloids. DMV theory for circular cylindrical shells subjected to axissymmetric transverse loads. Influence length. Comparison with membrane theory solutions. Stability and imperfection sensitivity. Gaussian curvature. Inextensional and extensional deformation. Form-finding. Case study.


Programs where the course is taught: