# NON-LINEAR ANALYSIS OF MATERIALS AND STRUCTURES cod. 1002221

2° year of course - Second semester
Professor
Scienza delle costruzioni (ICAR/08)
Field
A scelta dello studente
Type of training activity
Student's choice
48 hours
of face-to-face activities
6 credits
hub:
course unit
in - - -

## Learning objectives

To present basic concepts of the non-linear behaviour of structures, with reference to material and geometric non-linearities. The collapse of structures due to fatigue and fracture is also presented. The aim of the course is to supply the tools for evaluating the load-carrying capacity of a structure, related to the attainment of material strength (plastic limit analysis), the loss of structural stiffness (stability of equilibrium) and the fatigue and fracture collapse.

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## Course unit content

Plasticity
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.

Plastic limit analysis.
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).

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.

Fracture mechanics and fatigue.
Collapse due to fracture. The problem of cracked bodies. Linear elastic fracture mechanics. Westergaard's solution. Stress intensity factor (SIF) concept. Energetic aspects and fracture energy (Griffith approach). Cohesive cracks. Cyclic (fatigue) loading of constant amplitude. Experimental approach (Whöler curves) and analytical approach (Paris-Erdogan law).

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## Bibliography

• 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.

## Assessment methods and criteria

Oral examination possibly integrated complemented by a written examination

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