Structural Mechanics AB ( 9 CFU )

**Prof. Andrea Carpinteri **
Phone: 0521.905918 - Fax: 0521.905924 E-mail.

andrea.carpinteri@unipr.it

Objectives

To present basic concepts and tools for structural design, with reference to statically determinate and indeterminate elastic frames (beam systems).

Program

**Geometry of areas**. Introduction. Static moment and centroid. Moments of inertia. Laws of transformation. Principal axes and moments of inertia. Mohr’s circle.

**Simple (beams) and complex (frames) structural systems**. Plane beams and frames. Problem of structural system equilibrium: kinematic definition of plane constraints; static definition of plane constraints (constraint reactions) and cardinal equations of statics. Framed structures: statically determinate (or isostatic); hypostatic; statically indeterminate (or hyperstatic). Principle of superposition.

**Statically determinate framed structures**. Three methods: cardinal equations of statics; auxiliary equations; the principle of virtual work.

**Internal beam reactions**. Three methods: direct method; differential method (indefinite equations of equilibrium for plane beams); the principle of virtual work. Diagrams of characteristics of internal beam reactions.

**Particular problems**. Closed-frame structures. Plane trusses. Symmetric frames.

**Analysis of stresses** (for three-dimensional solids). Stress tensor, equations of Cauchy, law of reciprocity. Principal stress directions, Mohr’s circles. Plane stress condition and Mohr’s circle. Boundary conditions of equivalence and indefinite equations of equilibrium.

**Analysis of strains** (for three-dimensional solids). Rigid displacements, strain tensor. Strain components: dilatations and shearing strains. Principal strain directions. Equations of compatibility.

**The theorem of virtual work** (for three-dimensional solids).

**Theory of elasticity** (for deformable three-dimensional solids). Real work of deformation, elastic material, linear elasticity, homogeneity and isotropy, linear elastic constitutive equations. Real work of deformation: Clapeyron’s theorem; Betti’s theorem. The problem of a linear elastic body: solution uniqueness theorem (or Kirckhoff’s theorem).

**Strength criteria**. Criteria by Rankine, Grashof, Tresca, von Mises.

**The problem of De Saint-Venant**. Fundamental hypotheses, indefinite equations of equilibrium, elasticity equations and boundary conditions. Centred axial force, flexure (bending moment), biaxial flexure, eccentric axial force, torsion, bending and shearing force.

**Computation of displacements for framed structures**. Differential equation of the elastic line; theorem of virtual work for deformable beams; thermal distortions and constraint settlements.

**Statically indeterminate framed structures**. Theorem of virtual work: structures subjected to loads, thermal distortions and constraint settlements.

**Instability of elastic equilibrium**. Euler’s critical load and free length of deflection; omega method.

Laboratory activities

Theory supported by exercises.

Examination methods

Written and oral examination.

Prerequisites

Analisi A-B, Analisi C, Geometria, Meccanica Razionale.

Suggested textbooks

Documentation provided by the teacher.

A. CARPINTERI: "Scienza delle Costruzioni", Vol. 1 e 2, Ed. Pitagora, Bologna.

A. CARPINTERI, "Structural Mechanics", E&FN Spon, London.

E. VIOLA: "Esercitazioni di Scienza delle Costruzioni", Ed. Pitagora, Bologna.

M. CAPURSO: "Lezioni di Scienza delle Costruzioni", Ed. Pitagora, Bologna.

V. FRANCIOSI: "Fondamenti di Scienza delle Costruzioni ", Ed. Liguori, Napoli.

A. MACERI: "Scienza delle Costruzioni", Accademica, Roma.

A. CASTIGLIONI, V. PETRINI, C. URBANO: "Esercizi di Scienza delle Costruzioni", Ed. Masson Italia, Milano.