# THEORY OF COSTRUCTION

## Learning outcomes of the course unit

Knowledge and understanding

The Course presents basic concepts to complete all concepts needed to understand the main aspects of the structural design and, after presenting the constitutive laws of the mechanical behaviour of materials, aims at describing in depth the equilibrium concept and deformation of elastic solids.

Applying knowledge and understanding

At the end of the Course, each student should be able to describe the mechanical behaviour of statically indeterminate elastic frames and to identify, formulate and 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.

## Course contents summary

The topics treated in the Course are the following ones:

A1) Systems of forces

A2) Geometry of areas

A3) Simple (beams) and complex (frames) structural systems

A4) Statically determinate framed structures

A5) Internal beam reactions

B1) Analysis of stresses (for three-dimensional solids)

B2) Analysis of strains (for three-dimensional solids)

B3) The theorem of virtual work (for three-dimensional solids).

B4) Theory of elasticity (for deformable three-dimensional solids)

B5) Strength criteria. Criteria by Rankine, Grashof, Tresca, von Mises

B6) The problem of De Saint-Venant

B7) Computation of displacements for framed structures

B8) Statically indeterminate framed structures

## Course contents

A1) Systems of forces

- Introduction

- Decomposition of forces

- Definition of forces and couples, both distribuited and concentrated

- Funicular curve

Exercises

A2) Geometry of areas

- Introduction

- Static moment and centroid

- Moments of inertia

- Laws of transformation

- Principal axes and moments of inertia

- Mohr’s circle

Exercises

A3) 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

A4) Statically determinate framed structures.

Cardinal equations of statics; kinematic procedure; auxiliary equations

- Closed-frame structures

- Plane trusses

Exercises

A5) Internal beam reactions.

- Direct method; differential method (indefinite equations of equilibrium

for plane beams)

- Diagrams of characteristics of internal beam reactions

Exercises

B1) 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

B2) Analysis of strains (for three-dimensional solids)

- Rigid displacements, strain tensor

- Strain components: dilatations and shearing strains

- Principal strain directions

Exercises

B3) The theorem of virtual work (for three-dimensional solids)

Exercises

B4) 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

B5) Strength criteria

- Criteria by Rankine

- Criteria by Grashof

- Criteria by Tresca

- Criteria by von Mises

Exercises

B6) 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

B7) 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

B8) Statically indeterminate framed structures

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

Exercises

## Recommended readings

Boscotrecase L., Di Tommaso A., Statica applicata alle costruzioni, Patron.

Beer, F. P., E. R. Johnston & J. T. DeWolf, Meccanica dei Solidi – Elementi di Scienza delle Costruzioni, III edizione, McGraw-Hill, Milano (2006).

Comi C., Corradi Dell'Acqua L., Introduzione alla Meccanica Strutturale (II Ed.), McGraw-Hill, 2007;Cesari F., Del Re V., Esercizi di Meccanica delle Strutture, McGraw-Hill, 2010

## 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 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 informations

Students must not attend all lectures and tutorials.