The course aims to provide a general methodology for evaluating the static and kinematic behavior of structural systems with particular reference to elastic beams (one-dimensional solid according to the De St. Venant theory). Some hints on the resistance of materials and on the stability of the elastic equilibrium are illustrated. The tools and language are those of mathematical physics, continuum mechanics and related calculation methods.
Beyond the teaching material of the theoretical and application classes, provided in the form of PDF files downloadable from the Moodle platform, the following books are relevant to the study of the subject:
a) Fundamental books:
1. C. Borri, M. Betti, E. Marino. Lectures on Solid Mechanics, Firenze, FUP, 2008.
2. L. Galano, P.M. Mariano. Eserciziario di Meccanica delle strutture. Roma, CompoMat, 2011.
3. R. Baldacci. Scienza delle costruzioni, Torino, Vol. I (UTET, 1970), Vol. II (UTET, 1976).
4. Taliercio, A., & Perego, U. (2023). Fundamentals of Structural Mechanics. Società Editrice Esculapio.
b) Other texts and further readings:
1. P.M. Mariano, L. Galano, Fondamenti di Meccanica dei Solidi, Ed. Bollati Boringhieri, Torino, ISBN: 978-88-339-5893-4.
2. O. Belluzzi. Scienza delle costruzioni, Bologna, Zanichelli, 1970.
3. E. Benvenuto. La Scienza delle Costruzioni ed il suo sviluppo storico. Sansoni, 1981.
4. E. Viola. Scienza delle costruzioni, Bologna, Pitagora, 1992.
5. R. Camiciotti, A. Cecchi. Esercizi di Scienze delle Costruzioni, Firenze, Morelli, 1992.
6. L.C. Dell'Acqua. Meccanica delle strutture, Milano, McGraw-Hill libri Italia, 1992.
7. L. Gambarotta, L. Nunziante, A. Tralli. Scienza delle costruzioni, Milano, McGraw-Hill, 2003.
8. F. Angotti, A. Borri. Lezioni di scienza delle costruzioni, Roma, DEI, 2005.
Learning Objectives
The student will be able to model and solve statically determined structures (beams and beam systems), and carry out safety and functionality checks.
The learning outcomes consist in understanding the basic concepts and principles of Structural Mechanics (Mechanics of structures, strength of materials, elasticity, equilibrium stability, etc.) in order to acquire the necessary tools to deal with structural analysis both of one-dimensional and continuous elements.
With reference to the Dublin 1 descriptor (knowledge and understanding), the student will acquire theoretical and modeling knowledge (verified during the exam) to the study of the mechanical behavior of solids, beams and statically determined beams in the linear elastic range (see the program to this effect).
With reference to the Dublin 2 descriptor (applying knowledge and understanding), the student will acquire skills in applying theoretical knowledge to practical cases concerning, for example, the solution of plane elastic beams, the calculation of elastic displacements, the analysis of the state of stress and strain for beams subjected to the following internal actions: normal force, bending, eccentric normal force, torsion, and shear. The student will also acquire the basic knowledge related to the stability problems of the elastic equilibrium, with practical applications.
With reference to the Dublin 3 descriptor (autonomy of judgment - making judgements), the student will acquire autonomy of judgment in the choice of approaches for modeling and structural analysis with particular reference to: 1) verification of resistance through the determination of the internal actions in statically determined plane elastic beams, 2) calculation of the state of stress in the beam sections using the De Saint Venant model, 3) determination of the equation of the elastic line of flat beams with straight axis, 4) evaluation of the critical buckling load of compressed members and in the safety checks of flat elastic beams.
Prerequisites
According to the plan of studies. Important requisites for profitably following the lessons are a good knowledge of the contents of the courses: Mathematical Analysis I and II, Geometry, Physics I, and Continuum Mechanics.
Teaching Methods
Classroom lessons. Classroom exercises on all topics.
Further information
See the Moodle platform (e-learning) activated for the course and possibly the same platform as last academic year's course.
Course program
A. CONTINUUM MECHANICS OF ELASTIC MATERIALS
1.Constitutive laws (elastic solid) and energy theorems.
The elastic solid (R. Hooke's law): constitutive equations; elastic and linear elastic material; homogeneity and isotropy. Lamé constants (Young's modulus and Poisson's ratio). Theoretical limits of Poisson's ratio. Coincidence between the principal directions of stress and strain. Formulation of the elastic-static problem in terms of displacements (Navier) and stresses (Beltrami-Michell). Deformation work. Elastic potential. Clapeyron's theorem. Betti's theorem (first principle of reciprocity). Castigliano's theorem. Kirchhoff's uniqueness theorem. Principle of Superposition of Effects.
2.Principle of Virtual Works
Equation of virtual work for deformable continua: equilibrium, compatibility and equation of virtual work. The principle of virtual works in the direct form and in the inverse form.
3.De Saint Venant's problem
Characterization of the De Saint Venant solid. Hypotheses on geometry, loads, constraints, stress state. Internal actions. De Saint Venant's postulates. Loading cases:
• Simple normal force: State of stress and state of deformation. Volumetric expansion coefficient, volume change, displacement components, normal stress stiffness. Elastic potential and deformation work. Design and verification problems.
• Pure bending: loading plane and loading axis. Stress state, neutral axis, relationship between stress axis and neutral axis. Monomial expressions of stress, internal moment couple. State of deformation: deformation components, coefficient of volume expansion and volume variation, displacement components. Flexural plane and axis; relationship between bending axis and stress axis; deviated bending and simple bending. Elastic potential and deformation work. Verification and design problems. resistance modules. Maximum resistant moment; choice of the most convenient stress axis. Simple bending: deformation of the beam’s center line, deformation of the longitudinal fibers, rotation and deformation of the section, flexural stiffness.
• Not centered Normal force: load center and axis. Stress state: neutral axis, monomial and binomial expressions of normal stress. Relationship between stress axis and neutral axis, relationship between center of stress and neutral axis. Properties of the central core of inertia. Strain state: strain components, displacement components, volumetric expansion coefficient and volume change. Elastic potential and deformation work. No-tension strength solids. Verification and design problems.
• Simple torsion (and Bredt theory): Thin-walled tubular beams: Bredt theory (1st and 2nd formula).
• Shear (and Jourawski theory): Shear stress: Jourawski theory. State of stress: expressions of the tangential components of stress. state of deformation. Elastic potential and deformation work. Shear factor. Shear center. Thin-walled beams (open and closed profiles). Verification and design problems. Influence of shear on the deformation of beams under bending (basic concepts).
B. MECHANICS OF THE ELASTIC BEAM
1.Statics of beams and of statically determined beam systems
Real structure and calculation scheme. Structural elements (classification). External actions: loads and distortions (classifications). Response to external actions: state of internal stress.
External and internal constraints: kinematic and static characteristics, multiplicity of constraints. Analytical determination of the support reactions. Internal actions characteristics diagrams. Indefinite equilibrium equations. Calculation of displacements of points and rotations of sections. Design and verification of sections. The deflected beam: differential equation of the elastic line and its integration [identification of the boundary conditions (static and kinematic)].
2.Isostatic lattice truss structures
Conditions for statically determined systems. Calculation of stresses in members: principle of virtual work, equilibrium equations of nodes and Ritter's method. Calculation of the displacements of the nodes with the principle of virtual work.
3.Geometry of areas.
Centroid of an area, first order (or static) moments of flat areas; second order moments of inertia; transposition theorem (or Huygens-Steiner theorem); moments of inertia for rotations of the reference axes). Inertia polarity. Central ellipse of inertia of a plane area. Definition of conjugate axes. Graphical method for determining the relative center. Central core of inertia of a flat area.
C. STRENGHT CRITERIA
Fundamental theories of strength. Conditions of failure, resistance and safety level. Criterion of allowable stresses. Von Mises criterion. Special cases of stress states: slab, de Saint Venant beam, pure shear. Fundamental theories of strength. Conditions of failure, resistance and security. Maximum (Galileo) and minimum (Navier) stress criterion. Maximum (De Saint Venant) and minimum (Grashoff) expansion criterion. Maximum tangential stress criterion (Tresca). Beltrami criterion. Special cases of stress states for the Tresca criterion: slab, beam of D.S. Venant, pure cut.
D. STABILITY OF THE ELASTIC EQUILIBRIUM
Definition of the critical buckling load; structures with concentrated deformability and tip loaded rods. Euler's formula. Hints on other formulas in the elastic-plastic case. Axially loaded beam: safety assessment. Slenderness of a beam.
Sustainable Development Goals 2030
This course contributes to the realization of the UN objectives of the 2030 Agenda for Sustainable Development.