## Learning objectives

Knowledge and understanding:

At the end of the course, students will have acquired the basic knowledge of the mechanics of deformable elastic bodies, with particular reference to the technical theory of beams.

Skills:

The student will be able to perform the design and verification of some simple structural elements, with particular reference to the organs of machines.

Making judgments:

The student will have the tools to critically evaluate the output of a structural software.

Communication skills:

The student must possess the ability to clearly present a technical report on the design of simple structural elements.

## Prerequisites

It is essential to have a basic knowledge of calculus, linear algebra and physics.

## Course unit content

The course aims to provide a unified treatment of the main aspects of the mechanical behavior of structures, referring mainly to the linear elastic response but also with nods to the behavior beyond the elastic limit.

The first part of the course is devoted to the evaluation of the static schemes and strain in elastic solids under assigned external actions. Structural theories for beams and the evaluation of the load bearing capacity of structures will subsequently be addressed.

The topics covered are more specifically as follows.

Basic principles of mechanics. Fundamental equations of statics.

Constraints, Free Body Diagrams, calculation of constraint reactions in variously articulated bodies.

Diagrams of stress characteristics in elongated bodies (slender beams).

Stress tensor. Strain tensor. Mohr’s graphical representation. Strength criteria for ductile and brittle materials.

Constitutive relations. Linear elastic constitutive relation. Thermal variations.

De Saint Venant problem. Normal stress. Twist. Simple bending. Shear flow.

Technical theory of beams. Calculation of the elastic curve under the most varied load conditions. Solution of statically undetermined structures.

Notes on buckling. Euler beam.

The application aspect is taken care of through the introduction of a large number of solved examples and exercises.

## Full programme

1. Fundamental principles of mechanics

Concept of force. Moment of a force. Condition for equilibrium. Engineering applications.

2. Introduction to the mechanics of deformable bodies

Analysis of deformable bodies. Uniaxial loading and deformation. Statically determinate situations. Statically indeterminate situations.

3. Forces and moments transmitted by slender members.

Distributed loads. Resultant of distributed loads. Differential equilibrium relationships. Singularity functions. Three-dimensional problems.

4. Stress and strain

Stress. Plane stress. Equilibrium of a differential element in plane stress. Stress components associated with arbitrarily oriented faces in plane stress. Mohr’s circle representation of plane stress. Mohr’s circle representation of a general state of stress. Analysis of displacement. Definition of strain components. Relation between strain and displacement in plane strain. Strain components associated with arbitrary sets of axes. Mohr’s circle representation of plane strain. Mohr’s circle representation of a general state of strain. Measurement of strain.

5. Stress-strain-temperature relations

The tensile test. Idealization of stress-strain curves. Elastic stress-strain relations. The effect of temperature on strain. Complete equations of elasticity. Criteria for initial yielding. Behavior beyond yielding in the tensile test. Ductile fracture. Brittle fracture.

6. Torsion.

Geometry of deformation of a twisted circular shaft. Stress obtained from stress-strain relations. Equilibrium requirements. Stress and deformation in a twisted circular shaft. Torsion of elastic hollow circular shafts. Stress analysis in torsion; combined stress. The onset of yielding in torsion. Torsion of rectangular shafts. Torsion of hollow, thin-walled shafts.

7. Stress due to bending.

Geometry of deformation of symmetrical beam subjected to pure bending. Stress obtained from stress-strain relations. Equilibrium requirements. Stress and deformation in symmetrical elastic beams subjected to pure bending. Stress in symmetrical elastic beams transmitting both shear forces and bending moments. Bending of unsymmetrical beams. Shear flow in thin-walled open sections; shear center. Stress analysis in bending; combined stresses. The onset of yielding in bending.

8. Deflection due to bending.

The moment-curvature relation. Integration of the moment-curvature relation. Superposition. The load-deflection differential equation.

9. Stability of equilibrium. Buckling.

Elastic stability. Examples of instability. Elastic stability of flexible columns. Instability as a mode of failure.

## Bibliography

Recommended text:

S. Crandall, N. Dahl, T. Lardner, An introduction to the mechanics of solids, McGraw-Hill, 1978 ISBN-13 :978-0-07-013441-6

Further reading:

O. Belluzzi, Building Science, Vol I, Freeman, 1973.

F.P. Beer, É.R. Johnston, Jr., J.T. Dewolf: Solid Mechanics. Elements of building science (second edition). McGraw-Hill, Milano, 2002.

L. Corradi Dell'Acqua, Structural Mechanics, Vol I, Mac-Graw Hill, 2010.

Additional material presented during the lectures.

## Teaching methods

We provide lectures and exercises on the blackboard. As a rule, lectures will follow as much as possible the recommended text, so that the student can critically review what has been developed in the classroom. From time to time, homework exercises will be assigned; clarification will be given during office hours.

## Assessment methods and criteria

The exam is based on a written test and an oral test. In the written test, the student will be asked to solve a few exercises of the same type as those carried out in the classroom during recitation. The oral test will be aimed at verifying the learning of the basic theoretical knowledge.

## Other information

It is strongly recommended to attend the course.