# STRUCTURAL MECHANICS

## Learning outcomes of the course unit

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

## Prerequisites

Analisi Matematica 1, Analisi Matematica 2, Geometria, Meccanica Razionale.

## Course contents summary

Geometry of areas.

Simple (beams) and complex (frames) structural systems.

Statically determinate framed structures.

Internal beam reactions.

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

Analysis of stresses (for three-dimensional solids).

Analysis of strains (for three-dimensional solids).

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

Theory of elasticity (for deformable three-dimensional solids). 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: 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.

## Course contents

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.

## Recommended readings

Documentation provided by the teacher.

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

Alberto 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.

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

## Teaching methods

Theory supported by exercises.

## Assessment methods and criteria

Written and oral examination.