## Learning objectives

Knowledge and understanding

The Course presents basic concepts related to equilibrium of forces and to structural mechanics. Such concepts are needed to understand the main aspects of the structural design and, after examining the constitutive laws of the mechanical behaviour of materials, aims at describing in depth the concepts of equilibrium and deformation for elastic solids.

Applying knowledge and understanding

At the end of the Course, each student should be able to model simple structural systems (elastic frames), to determine their equilibrium conditions, to describe the mechanical behaviour of statically determinate elastic frames and the mechanical behaviour of statically indeterminate elastic frames, and to identify, to formulate and to solve the structural problems of the architectural design.

Communication skills

At the end of the Course, each student should know all the technical words related to the topics treated.

## Prerequisites

- - -

## Course unit content

The topics treated in the Course are the following ones:

(1) Systems of forces

(2) Geometry of areas

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

(4) Statically determinate framed structures

(5) Internal beam reactions

(6) Particular problems

(7) Analysis of stresses (for three-dimensional solids)

(8) Analysis of strains (for three-dimensional solids)

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

(10) Theory of elasticity (for deformable three-dimensional solids)

(11) Strength criteria. Criteria by Rankine, Grashof, Tresca, von Mises

(12) The problem of De Saint-Venant

(13) Computation of displacements for framed structures

(14) Statically indeterminate framed structures

## Full programme

(1) Systems of forces

- Introduction

- Decomposition of forces

- Definition of forces and couples, both distribuited and concentrated

- Funicular curve

Exercises

(2) Geometry of areas

- Introduction

- Static moment and centroid

- Moments of inertia

- Laws of transformation

- Principal axes and moments of inertia

- Mohr’s circle

Exercises

(3) 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

Exercises

(4) Statically determinate framed structures.

Cardinal equations of statics; kinematic procedure; auxiliary equations

Exercises

(5) Internal beam reactions.

- Direct method; differential method (indefinite equations of equilibrium for plane beams)

- Diagrams of characteristics of internal beam reactions

Exercises

(6) Particular problems

- Closed-frame structures

- Plane trusses

Exercises

(7) 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

Exercises

(8) Analysis of strains (for three-dimensional solids)

- Rigid displacements, strain tensor

- Strain components: dilatations and shearing strains

- Principal strain directions

Exercises

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

Exercises

(10) 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)

Exercises

(11) Strength criteria

- Criteria by Rankine

- Criteria by Grashof

- Criteria by Tresca

- Criteria by von Mises

Exercises

(12) 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

Exercises

(13) Computation of displacements for framed structures

- Differential equation of the elastic line

- Theorem of virtual work for deformable beams

- Constraints (like thermal distortions and constraint settlements)

Exercises

(14) Statically indeterminate framed structures

- Theorem of virtual work: structures subjected to loads and constraints (like thermal distortions and constraint settlements)

Exercises

## Bibliography

Recommended books:

- A. Carpinteri, “Scienza delle Costruzioni” Vol.1, Pitagora Ed., Bologna

- A. Carpinteri, “Scienza delle Costruzioni” Vol.2, Pitagora Ed., Bologna

- M. Capurso, "Lezioni di Scienza delle Costruzioni", Ed. Pitagora, Bologna

- E. Viola, “Esercitazioni di Scienza delle Costruzioni – Vol.1: Strutture isostatiche e geometria delle masse”, Pitagora Ed., Bologna

- E. Viola, “Esercitazioni di Scienza delle Costruzioni – Vol.2: Strutture iperstatiche e verifiche di resistenza”

- L. Boscotrecase, A. Di Tommaso, “La statica applicata alle costruzioni”, Patron, Bologna

All books are available by the library (Biblioteca Politecnica di Ingegneria e Architettura).

Additional educational material:

- Documentation provided by the teacher (Centro Documentazione - Ingegneria – Sede Didattica)

## Teaching methods

The Course consists of theoretical lectures and practical tutorials. For each topic treated in the Course, exercises are solved so that each student can determine the solutions of the theoretical problems explained just before such practical tutorials.

The theoretical lectures are delivered by employing transparencies, which the students can get at the Documentation Office.

For each theoretical topic treated, practical tutorials are planned according to two modes:

- at first, by employing transparencies (which the students can get at the Documentation Office) to explain the solution methods;

- then, students solve some exercises in the lecture hall, and a common discussion on the difficulties to solve them follows.

## Assessment methods and criteria

The final test of the Course of Laboratory of Structures (Structural Mechanics) consists of a written test which is weighted as follows:

- 70% application of theoretical concepts to practical cases, i.e. exercises (applying knowledge and understanding)

- 20% questions on theoretical concepts (knowledge and understanding)

- 10% ability to present scientific topics with the right technical words (communication skill)

## Other information

Students must compulsorily attend all lectures and tutorials.