Learning objectives
To present basic concepts and tools for structural design in the linear regime, with reference to the theory of one-dimensional structures (cables and beams) and two-dimensional structures (plates and shells).
To present basic concepts of the non-linear behaviour of structures, with reference to material and geometric non-linearities. The aim is to supply the tools for evaluating the load-carrying capacity of a structure, related to the attainment of material strength (plastic limit analysis) and the loss of structural stiffness (stability of equilibrium).
Prerequisites
Scienza delle Costruzioni (Structural Mechanics).
Course unit content
Static equilibriumo of cables
The influence lines for statically determinate elastic frames
- Fundamentals
- Use of the influence lines
Plane stress and plane strain states
Plane stress state
- Formulation: Stress function or Airy function
- Simple cases related to rectangular plates
Plane strain state
- Formulation
Plane stress and stain state for solids of revolution
- Formulation
- Thick cylindrical vessel
- Concentrated force acting on the edge of an elastic half-plane
Bending of plates
- Fundamentals: the kinematic hypotheses
- Components of displacements, strains, stresses; internal reaction characteristics
- Differential equation of the elastic surface (or differential equation of Germain-Lagrange)
- Boundary conditions
- Moments with reference to a point
- Finite difference method: approximate solutions for rectangular plates
Simply supported rectangular plates
- Navier solution
- Lévy solution
Shells of revolution
- Introduction
- Membrane regime
- Bending regime
Plastic limit analysis
- Constitutive laws of materials. Generalised Hooke's law for isotropic, orthotropic and transversally isotropic materials.
- Material non-linear behaviour: plasticity (yielding function, isotropic and kinematic hardening, associative and non-associative flow rule). Yielding criteria for structural materials.
- Perfectly plastic behaviour and plastic collapse. Plastic collapse of beams under bending: plastic hinge, limit moment for symmetric and non-symmetric cross-sections, combined actions and limit curves.
- Incremental analysis of elastic-plastic frames. Collapse mechanisms. Theorems of limit analysis (static and kimenatic theorems).
- Frames under proportional point loads (method of the combination of mechanisms) and under distributed loads. Frames under non-proportional loads.
- Limit loads of plates (yield line theory and strip method).
Stability of equilibrium.
- Discrete elastic systems: stationarity and minimum of total potential energy, theory of the second order, critical Euler load (static and energetic criteria).
- Flexural stability of axially compressed beams: fundamental cases, frames, beams with curved axis.
- Torsional stability of beams under axial compressive load or bending.
- Stability of plates.
- Determination of the critical load: Rayleigh-Ritz method, finite element method. Post-critical behaviour.
- Beams under axial compressive load and bending: effects of material non-linearity and imperfections on load-carrying capacity, stability curves.
- Snap-through instability of shallow arches.
Bibliography
O.Belluzzi “Scienza delle Costruzioni” Vol.I e Vol.III, Zanichelli, Bologna.
A.Carpinteri “Scienza delle Costruzioni” Vol.II, Pitagora Editrice, Bologna.
L. Corradi dell’Acqua “Meccanica delle strutture” Vol.I e Vol.II, McGraw-Hill, Milano.
E. Giangreco. Teoria e Tecnica delle Costruzioni. Vol.III, Liguori Editore, Napoli.
P. Pozzati. Teoria e Tecnica delle Strutture. Vol.II, UTET, Torino.
E.Viola“ Esercitazioni di Scienza delle Costruzioni/1. Pitagora Editrice, Bologna.
M. JIRASEK - Z.P. BAZANT: “Inelastic analysis of structures”, J.Wiley & Sons, New York, 2001.
D.O. BRUSH – B.O. ALMROTH: “Buckling of bars, plates and shells”, McGraw-Hill, New York, 1975.
A. CARPINTERI: "Analisi non-lineare delle strutture”, Ed. Pitagora, Bologna, 1998.
L. CORRADI DELL’ACQUA: “Instabilità delle strutture”, CLUP, Milano, 1978.
L. CORRADI DELL’ACQUA: “Meccanica delle strutture”, Voll. 1,3, McGraw-Hill, Milano, 1995.
S.P. TIMOSHENKO – J.M. GERE: “Theory of elastic stability”, McGraw-Hill, New York, 1961.
A. CARPINTERI: “Meccanica dei Materiali e della Frattura”, Ed. Pitagora, Bologna, 1992.
D. BROEK: “Elementary engineering fracture mechanics”, Martinus Nijhoff Publishers, 1982.
Teaching methods
Theory supported by exercises.
