This course is aimed at providing the students some fundamentals of solid mechanics needed to perform at least a preliminary operation of either design or verification of structural and mechanical systems undergoing some static loading conditions. Paradigm of this analysis is the beam, whose elementary approaches to compute the stress resultants and the occurring stress and strain are given together with a preliminary description of strength of materials in static behavior and related testing techniques. Course starts with the static equilibrium and shows how beamlike structures are constrained, loaded and to compute the external and internal reactions, together with the distribution of stress resultants along the beam line axis. This analysis is deeply performed at least in case of statically determined structure. Two and three dimensional examples will be proposed as well as a short description of rules for truss structures. A deep description of basic concepts of solid continuous mechanics is then proposed, by including definitions and computations of stress, strain and constitutive laws of materials under the assumption of linear elastic behaviour. Elastic and mechanical properties of material and static strength are then defined according to the standard tensile test. The elementary theory of beam is then described to allow the student computing both the stresses and the strains occurring in a one-dimensional structural element. De St Venant principle and related elaborations are then developed and applied to several examples to investigate the axial, flexural, torsional and shear behaviors. Some additional topics are proposed, concerning the computation of displacements and rotations in beam under a defined combined set of loading conditions. Elastic stability of slender beam under compression is evenly discussed and buckling phenomenon investigated. The last part of the course is aimed at showing the students how static structural analysis can be performed through the matrix form as an alternate approach to the energy theorems and virtual works principle. Discretization of static equilibrium equations and basic approaches to write them in matrix form are developed through the definition of stiffness matrices for all the typical loads and the solution of the static problem once the assembly operation is performed. Some examples of computation of reactions and stresses occurring into statically indeterminate structures are finally given to complete the scenario of static analysis used in mechanical design. Concept of static safety factor for design against yielding, rupture or buckling of beamlike structures is defined and applied to some examples of structures built in ductile and brittle materials as well as computation of equivalent (ideal) stress in multi-axial loading conditions.

At the end of this course it is required that student easily handle some typical tools of analytical methods for the static behavior prediction at least of beamlike structures.
Fundamental goals of the discipline are:
•a comprehensive knowledge, understanding and distinguishing of mechanical properties and strength of ductile and brittle materials; linear and nonlinear elastic behaviour conditions; concepts of stress, strain, displacement and rotation, described in both principal and non principal reference frames;
static failure criteria and safety factor; geometrical properties of plane figures, interpreted as cross section of beams; theory of beam and relations between stress and load for each static behavior foreseen by De St Venant. In addition student will get acquainted with the performing of static analysis through the matrix calculus, applied to system of assembled beam elements.
•providing some skills as the basic tools to:
1) simplify to a level of elementary scheme the layout of a beamlike mechanical component and perform a complete static analysis;
2) evaluate the degree of indeterminacy of the system;
3) calculate reaction forces of statically determinate structures (by analytical method) and indeterminate structures (at least by the matrix calculus);
4) calculate the internal stress resultant diagrams,
stresses, strains, displacements and rotations of each cross section of one-dimensional elements;
5) identify the critical points of the structure and compute the equivalent stress to be compared to the strength of material or even to buckling threshold;
6) handle the matrix form to analyse the structure, being able of writing the equations which describe the static equilibrium, by assembling the whole system and its matrices, then of solving for a straight computation of the safety factor and a prediction of the deformed shape of the whole structure.

Some typical mathematical tools (study of functions, computation of derivatives and integrals, matrix algebra, solution of eigenvalue/eigenvectors problems) and physics (basic concepts of kinematics and statics) and some basics of materials sciences (materials classes and properties).

Introduction

This course is aimed at providing the students some fundamentals of solid mechanics needed to perform at least a preliminary operation of either design or verification of structural and mechanical systems undergoing some static loading conditions. Paradigm of this analysis is the beam, whose elementary approaches to compute the stress resultants and the occurring stress and strain are given together with a preliminary description of strength of materials in static behavior and related testing techniques.

Course starts with the static equilibrium and shows how beamlike structures are constrained, loaded and to compute the external and internal reactions, together with the distribution of stress resultants along the beam line axis. This analysis is deeply performed at least in case of statically determined structure. Two and three dimensional examples will be proposed as well as a short description of rules for truss structures.

A deep description of basic concepts of solid continuous mechanics is then proposed, by including definitions and computations of stress, strain and constitutive laws of materials under the assumption of linear elastic behavior. Elastic and mechanical properties of material and static strength are then defined according to the standard tensile test.

The elementary theory of beam is then described to allow the student computing both the stresses and the strains occurring in a one-dimensional structural element. De St Venant principle and related elaborations are then developed and applied to several examples to investigate the axial, flexural, torsional and shear behaviors. Some additional topics are proposed, concerning the computation of displacements and rotations in beam under a defined combined set of loading conditions. Elastic stability of slender beam under compression is evenly discussed and buckling phenomenon investigated.

The last part of the course is aimed at showing the students how static structural analysis can be performed through the matrix form as an alternate approach to the energy theorems and virtual works principle. Discretization of static equilibrium equations and basic approaches to write them in matrix form are developed through the definition of stiffness matrices for all the typical loads and the solution of the static problem once the assembly operation is performed. Some examples of computation of reactions and stresses occurring into statically indeterminate structures are finally given to complete the scenario of static analysis used in mechanical design.

Concept of static safety factor for design against yielding, rupture or buckling of beamlike structures is defined and applied to some examples of structures built in ductile and brittle materials as well as computation of equivalent (ideal) stress in multi-axial loading conditions.

Goals, knowledge and skills

At the end of this course it is required that student easily handle some typical tools of analytical methods for the static behavior prediction at least of beamlike structures.

Fundamental goals of the discipline are:

•a comprehensive knowledge, understanding and distinguishing of mechanical properties and strength of ductile and brittle materials;
•linear and nonlinear elastic behavior conditions;
•concepts of stress, strain, displacement and rotation, described in both principal and non principal reference frames;
•static failure criteria and safety factor;
•geometrical properties of plane figures, interpreted as cross section of beams;
•theory of beam and relations between stress and load for each static behavior foreseen by De St Venant. In addition student will get acquainted with the performing of static analysis through the matrix calculus, applied to system of assembled beam elements.
•providing some skills as the basic tools to:
1) simplify to a level of elementary scheme the layout of a beamlike mechanical component and perform a complete static analysis;
2) evaluate the degree of indeterminacy of the system;
3) calculate reaction forces of statically determinate structures (by analytical method) and indeterminate structures (at least by the matrix calculus);
4) calculate the internal stress resultant diagrams, stresses, strains, displacements and rotations of each cross section of one-dimensional elements;
5) identify the critical points of the structure and compute the equivalent stress to be compared to the strength of material or even to buckling threshold;
6) handle the matrix form to analyse the structure, being able of writing the equations which describe the static equilibrium, by assembling the whole system and its matrices, then of solving for a straight computation of the safety factor and a prediction of the deformed shape of the whole structure.

Background

Some typical mathematical tools (study of functions, computation of derivatives and integrals, matrix algebra, solution of eigenvalue/eigenvectors problems) and physics (basic concepts of kinematics and statics) and some basics of materials sciences (materials classes and properties).

Contents

Topics dealt within this course are herein listed.

1. Statics : Basic concepts of static behavior of structures (force, moment, rigid and deformable bodies), loading conditions, constraints, static and kinematic determinacy, equilibrium conditions and equations. Computations of reactions, internal forces, diagrams. Beams, bars, trusses. Outlines of Virtual Work Principle and application to undetermined structures.
2. Stress : Stress vector, tensor, components. Principal stresses and direction, related computation. Mohr circles. Equivalent stress definition and computation.
3. Strain : Rigid body motion and strain definition in elastic body. Strain components, principal strain and direction. Stress-strain relations, Hooke’s law. Elastic properties of materials. Elastic energy storage.
4. Strength of materials : Tensile test, material behaviour and properties. Elastic coefficients. Yielding phenomenon, brittle and ductile materials. Safety factors in statics.
5. Beam theory : De Saint Venant principle, beam definition, loading conditions, axial, flexural, shear, torsional behaviors. Approximated solutions for torsion of rectangular cross sections, multiple rectangles and thin walled structures. Computation of stresses, strains, displacements and rotations. Shear centre. Coupled behavior. Buckling and elastic instability.
6. Matrix calculus applied to solid mechanics: Matrix form. Static equilibrium equations. Definition of stiffness matrix: computation in case of axial load, bending and shear, torsion, distributed load, thermal load. Reference frames: local and global. Assembly of the whole structure. Constraints application. Solution and computation of reactions, displacements, rotations, stresses and strains.

This course is organized in two parts. Lectures will give a straight presentation of relevant topics to be studied to perform a complete structural static analysis of some mechanical structures.

Practice hours will be offered to solve examples, numerical exercises and practical cases and even an exam simulation.

Textbooks: Some notes directly taken from the classes will be shared with students through the website.

Theoretical aspects presented during the lectures can be found on the following textbooks:
1.D.Gross, W.Hauger, J.Schroder, W.A.Wall, N. Rajapakse - "Engineering Mechanics 1: Statics", Springer.
2.V. Da Silva - "Mechanics and strength of materials", Springer.
3.J.D. Renton, "Applied elasticity : matrix and tensor analysis of elastic continua", Chichester, Horwood, New York: Wiley, 1987

or evenly, but textbooks are just partially dedicated to the above topics:
1.J. Beer, S.Johnston - "Solid mechanics", McGraw-Hill.
2.R.D. Cook – "Finite Element Modeling for Stress Analysis", John Wiley & Sons, 1995
3.K.J. Bathe – "Finite element procedures", Prentice Hall, 1995 e succ.

Exams: Final exam consists of two parts: written and oral, respectively.

•Written part: At the end of all the classes, student will be required to attend a written test, based on three exercises to be numerically solved. To attend the exam student will be asked of showing a valid identity document with a clear picture which could definitely certify the correspondence between candidate and document. In absence of this evidence the candidate will be rejected and the attendance to the test forbidden. All the material given by professor for the study of this discipline can be used during the written test.
Beware that students found with their portable phone switched on and communicating together with other persons inside or outside the classroom or evenly directly speaking with their colleagues during the exam will immediately consign their test and go outside the classroom.
Score of written test equal of higher than 18/30 will give the possibility to access to the oral exam.

•Oral exam will consists of at least few questions about the contents of the course, requiring a written demonstration, response, calculation or a graphical solution. Student is required to give an answer without consulting any material, note nor textbook.
Beware that if the student will be found out copying solutions from notes or from other candidates, exam will be evaluated as null and the candidate will repeat both the written and oral tests. In case the student will not attend the date of the oral exam, score of the written test will be considered accepted without any possible review and in case of null communication exam will be registered as rejected, being the oral part score set at zero.

•Written and oral tests have to be attended in the same exam session, strictly and only on the dates published through the website of Politecnico di Torino, for the two tests respectively.

If both the tests will be evaluated sufficient, final score will be computed as a weighted average between the marks of the written and oral exams respectively.
To complete the exam student has to pass both the tests. If the student even only fails the oral test the whole written and oral exam shall be completely repeated.