SCIENZA DELLE COSTRUZIONI

The learning objectives of Structural Analysis and Strenght of Materials consist in reaching the ability to identify structural models of real structures, to classify different structural schemes composed by deformable beams with homogeneous material, to evaluate the internal state of stress, to identify of the deformed configuration and, finally, to execute the necessary checks for the safety assessment according to the admissible stress criterion. Furthermore, skills toward the execution of the beam cross section design and identification, based on predetermined typological choices, and also the determination of carrying capacity of beam and frame structures are developed. The above mentioned learning objectives represent also a fundamental requirement to develop structural analysis of composite material structures such reinforced concrete.

The teaching is mainly developed according to frontal theoretical lectures given by the professor, by making use of a traditional board, by recalling basic theoretical background and subsequently by introducing new concepts, theories and model propes of the structural engineering to be widely exposed and discussed. For each topic, as decribed in detail in the section "Detailed Course Content", a wide variety of exercises will be solved in class which are fundamental for the comprehension of the application and the deepening of the theoretical concepts. Some exercises will be briefly introduced such that the student may take the opportuning to face a challenge independently by presenting in the sequent lecture the difficulties and doubts arisen during the solution procedure.

• Statics and kinematics of the unconstrained rigid body

Kinematics of the point, kinematics of the rigid body, principle of virtual work, static equilibrium equations.

• Statics and kinematics of the constrained rigid body

Static and kinematic definition of the 2D constraints, kinematic efficacy of the constraints, disconnection methodology, general methodology for the static and kinematic analysys, grafic determination of the constraint reactions.

• The sectional internal forces in the plane beams

Equilibrium equations, graphical and analytical methods for the internal forces distribution diagrams

• Statics, kinematics and sectional internal forces of beam like and frame structures

Static-kinematic description of the plane internal constraints, absolute and relative rotation centres, geometry of the kinematic mechanisms, kinematic efficacy of the internal constraints, disconnection methodology, general and auxiliary equation methods for the evaluation of the constraint reactions, equilibrium equations, graphical and analytical methods for the evaluation of the internal forces diagrams, frame structures composed of straight beams, closed and multi-connected systems, symmetrical and emisymmetrical systems.

• Dimensional and typological classification of structures

Mono-dimensional structures and principle of conservation of the plane cross-sections after the deformation, bi-dimensional structures and principle of conservation of the segment normal to the middle plane after the deformation.

• The elastic straight beam

Deformation parameters of the straight beam, compatibililty conditions of the straight beam, Bernoulli-Navier beam model and Timoshenko beam model, inelastic distributed and concentrated deformations, inealstic deformation due to termic variations, constraint inelastic displacements.

• Statics and kinematics of plane trusses

Statically impossible, determinate and undeterminate trusses, general moethd for static and kinematic analysis, internal axial forces evaluation, equilibrium equations and compatibility equations in explicit form and in compact matricial form, equilibrium and compatibility matrices, discussion of the equilibrium matrix based on the degree of indeterminacy of the structure, alternative methods for the evaluation of internal axial forces.

• Stress analysis of the continuum Cauchy body

Stress vector on a plane, Cauchy hypothesis, stress matrix, Cauchy formula, reciprocity principle of the tangential stress, equilibrium equations and boundary conditions, normal stress vector and resultant tangential stress, stress invariants, characteristic equation, stress state classification, principal stress reference system, Mohr circles and properties, pole of the normal directions and the plane direction, maximum tangential stress, plane stress state, purely tangential stress state, uniaxial stress state, hydrostatic stress state, octaedric stresses.

• Strain analysis of the continuum Cauchy body

Regular displacement field and strain field, infinitesimal displacements, dilatation and shear defromation components, strain matrix, decomposition of the configuration change of the elementary volume into a rigid and pure deformation, cubic dilatation, Cauchy formula for the strain components, principal strain components and principal strain directions, Mohr circel for strain components, compatibility equations and boundary conditions.

• Costitutive laws and energy theorems

Linear elasticity, tension test and torsion test, σ −ε diagrams for ductile materials and fragile materials, limits of linear elasticity, yielding and collapse, Hooke's law and elastic constants, total strain as sum of elastic and inelastic components, equilibrium and compatibililty equations in compact matricial form, the linear elastic problem of the Chauchy continuum body, discussion on governing equations, statically admissible solutions and kinematically admissible solutions, principle of virtual work and possible interpretations, deformation work and independence of the loading history, external work and Clapeyron theorem, Betti's and Maxwell's reciprocity theorem, theorem of unit force, principle of minimum potential energy and principle of minimum complementary energy, method for structural analysis: displacement method and force method with the use of the relevant equations, application of the displacement method to trusses.

• Safety criteria

General concepts on crisis criteria for materials and cosequent safety criteria, admissible stress criterion for uni-axial and multi-axial stress states based on collapse criteria of maximum principal stress (Galileo-Rankine) and maximum principal strain (de Saint Venant-Grashof) as well as yielding criteria of Tresca and Mises.

• De Saint Venant problem

De Saint Venant principle and applications to engineering problems, uniform axial force, bending, biaxial bending, bending moment and axial force, no-tension materials, simplified theory of torsion: hydrodynamic analogy and membrane analogy, centre of torsion, polar-symmetric cross sections, results for the rectangular cross section, thin rectangular cross sections, thin walled closed cross section, thin walled open cross section, simplified shear theory, Jouraski formula, symmetric cross section, thin walled open and closed cross sections, determination of the shear centre, rectangular I and C shaped cross sections, comparison of bending and shear deformations for a cantilever beam, safety assessment for beams.

• Elastic analysis of beams and frame like structures

The unit force theorem for beam and frame like structures, differential equation of the beam axis deformed configuration. Mohr's analogy: conjugate beam, fictitious load diagrams, evaluation of strain and displacement components as internal forces of the conjugate beam.

• Force method for beam like structures

Static indeterminacy and disconnection method, principal structure, unknown forces, compatibililty equations, application of the principle of virtual work for the evaluation of the compatibility equation coefficients and load dependent terms.

• Instability of the elastic equilibrium configuration

Axially loaded elastic beam and Euler limit load for different boundary conditions, generalized Euler formula and limit slenderness, omega method for the stability assessment.

[1] G. Muscolino, G.Falsone : Introduzione alla Scienza delle Costruzioni, Pitagora Editrice Bologna.

[2] E. Viola, Esercitazioni di Scienza delle Costruzioni, Vol. I, Vol. II, Pitagora Editrice Bologna.

[3] L. Corradi Dell’Acqua, Meccanica delle Strutture, Vol. I, McGraw-Hill.