Aeronautical engineering

A) BASIC EQUATIONS linearized mechanics of solids and space-discretized structures - Approach weak and Finite Element Method (FEM) A.1 Basic concepts of continuum mechanics solid Preliminary concepts of continuum mechanics solid. Eulerian and Lagrangian point of view. Local bases. Derivation and integration of Eulerian domain (time-varying) and Lagrangian. Reynolds transport theorem in local form. Principle of conservation of mass and its locations. According to the Reynolds transport theorem. Axioms of continuum mechanics. Local forms of the principle of conservation of momentum, theorems of Cauchy. Localization of the moment of momentum: symmetry of the stress tensor and the theorem of conservation of mechanical power. Existence of the carrier density of heat. Local form of conservation of Energy. Concept of deformation for a solid. Strain rate. (Second) Principle of thermodynamics (Clausius postulate) and its local form. Local form of energy storage in the form of entropy for solids. Reversible and irreversible processes for solids. Elastic solids. Linear elastic solids. Characterization of linear elastic solids. Linear isotropic elastic solids (Hookean solids). Elastic constants and their constraints. A.2 Formulation of the problem in weak form: FEM global equations Formulation of the problem in the weak form of the method of travel. Law of virtual work (weak formulation) and Lagrange equations for elastic solids with linear hypotheses in the field of movement of the Finite Element method (function-based "curtain"). Lagrange equations for a linear elastic solid. Weak formulation approach (as "Galerkin" approach). Structure and properties of the mass and stiffness matrices (for isotropic elastic solid and elastic). A.3 1D, 2D and 3D Finite Elements. Local equations and Assembly. Practical approach the MEF: the local view "element" with the displacement field, depending on form, link kinematic and constitutive element. Writing equilibrium in weak form: mass and stiffness matrices of the element. The element 2D triangular three-node plane (membrane in plane stress): effect of the nodes. Element 1D to two nodes of rod and torsion bar (stiffness matrix of element); element 1D to two nodes of the Euler beam (stiffness matrix of element); general case of element 1D to two nodes in the space (stiffness matrix total of element). 2D plane quadrilateral element with four nodes (membrane in plane stress): effects of the increase of the nodes. Element 2D quadrilateral plate plane bending (notes on 'generic quadrilateral element flexed-membrane). 3D tetrahedral element for the elastic problem general hints on increasing the number of nodes and the item box. General assembly of the EF based on the weak formulation: sum of partitioned matrices. Assembly of the element matrices and load element. Imposition of the boundary conditions. Notes on the properties and characteristics of the finite element completeness, compatibility, convergence and geometric invariance. B) METHODS OF SOLUTION for linearized mechanics of solids and space-discretized structures B.1 Free and forced response Problem formulation of the second order ODE by the Lagrange equations linearized. Associated eigenvalue problem. Intrinsic properties and three theorems on the problem of eigenvalues associated. Free response. Closed form solution of the homogeneous problem: identification of the modes and frequencies of vibration with eigenvectors and eigenvalues. Eigenvalue problem of the relationship between two matrices and the problem of eigenvalues standard. Lagrangian coordinates FEM v.s. modal coordinates. Convergence of the solution. Forced response of linear ODE systems of the second order with constant coefficients for the structural dynamics. Calls on Laplace and Fourier transforms and inverse-transform, transforms elementary fictions. Transfer matrices (Laplace domain) and frequency response functions (Fourier domain) based on spatial and spectral. Matrix of transfer functions and matrix functions Frequency Response for a dynamic vibration: Fourier series; integral and Fourier transform. Matrix of Frequency Response Functions for a dynamic vibration: Fourier transform as the limit of Fourier series. Meaning of the functions of frequency response to an input simple harmonic. Meaning of the matrix of the functions of frequency response to multiple inputs of harmonic synchronized. Resonance conditions for systems with N dofs, modal appropriation and the orthogonality conditions of the load. B.2 Modeling and analysis of damped structures General thermodynamic considerations on the damping in the continuous solid. Linear models for viscous solids. Spatial discretization (Eq. Lagrange) in the linear viscous solids. General properties of the matrix of viscous damping. Linear models for solid weak damping: viscous damping matrix, estimation of eigenvalues and eigenvectors for weakly damped systems theory (Rayleigh-Basile), modal damping, proportional damping. Work on non-viscous damping weak. C) BASIC EQUATIONS AND SOLUTIONS on linearized mechanics of space-continuous solids Dynamics of space-continuous systems. Structural linear operators (examples, 1D, 2D, and 3D; domestic product, a series of orthogonal functions. Added and self-adjoint operators. Betti-Castigliano's theorem. Spectral Three theorems. Example: the eigenfunctions of the auction. The method of eigenfunctions for the solution of problems elastic linear space-continuous: a) dynamic response problems free (the eigenfunctions as natural modes of vibration); b) Problems of static response (example of a response of a plate resting in the four sides, solution to the Navier). D) two-dimensional THIN STRUCTURES D.1 Isotropic and composite plate structures The model of purely isotropic plate in bending (with mid-plane membrane deformable) rectangular small displacements: kinematic field, strain and stress; equilibrium. Equilibrium equations and the end plate in bending purely homogeneous isotropic. Equations of multi-layer plate. Boundary conditions. Solution of the load plate with diaphragm and imperfection (displacement field) initial prescribed: solution eigenfunctions. Thermal problems for purely bending plates: reduction of the problem to the second order and resolution with the Navier eigenfunctions. Equations of isotropic plate with mid-plane deformable membrane: field and kinematic equations, constitutive equations and equilibrium equations written in the deformed configuration. Problem in the plane and out of plane (one-way coupling). Example guide of the rectangular plate resting in the four sides with load a membrane in the two opposite sides: i) static solution of the problem in the plane; ii) solution of the problem flexural dynamic with the method of eigenfunctions and discussion of stability: determination of the limit load buckling and deformed critical. Elastic properties, strength and thermo-elasticity of structural materials; physical characteristics of composites, composites with long fibers. Characterization of a linear elastic lamina of unidirectional composite long fiber as a continuous 2D orthotropic. Solution method of eigenfunctions for orthotropic plates. D.2 Shell structures General aspects of shell structures: membrane and bending models. The case of spherical shell for the membrane and bending behavior of the shells. The basic equations of the membrane shell behavior of shells of revolution with axisymmetric load. The analytical solution in terms of stress state for a generic geometry axisymmetric shell with axisymmetric load. Some analytical solutions: i) spherical shell pressure closed ii) closed dome subjected to its own weight (change sign circumferential stresses), iii) open dome with top load carrying its weight. iv) the closures of spherical or elliptical cylindrical pressure, v) conical shell vi) cylindrical shell vii) shell ring (optional argument). The basic equations of the membrane shells of revolution with NOT-axisymmetric load. Some analytical solutions for axisymmetric shells to axisymmetric load: i) closed cylindrical tank containing heavy fluid ii) spherical shell subjected to wind (only basic equations). Shells flexural behavior: the basic hypothesis in the theory of shells bending, the presence of curvature in the characterizations of effort, constitutive local and global. Shells with flexural behavior: equations with the displacement method for the circular cylinder with axisymmetric load. Solution of the static problem free with only the boundary conditions (homogeneous solution). Some solutions for cylinder with axisymmetric load: i) static response to a load ring, ii) the static forces and moments of extremity and thermal stress in a cylindrical shell iii) Evidence closures of spherical and ellipsoidal pressurized cylindrical shells ( optional argument). iv) critical load for a cylinder with compressed simply-supported end. E) Optimal Design and single-and multi-objective optimization for aeronautical structures Introduction on the use of optimization techniques in structural engineering. General formulation of the problem of structural optimization: cost function, design variables and constraints. Analytical solution and the Kuhn-Tucker conditions. Outline of numerical methods for analytical problems (stepest descendent and Newton method). Examples of mono-objective structural optimization. An outline of the principles and applications of multi-objective optimization: Pareto optimality, algorithms based on gradient and genetic algorithms. F) NUMERICAL EXERCISES - Static response of a wing-box type structure with the Finite Element method. Comparisons with analytical solutions of elementary beam structure having bending and torsion behavior. - Dynamic response of a structure type wing box with the Finite Element method. - Static response of a rectangular plate with a static load distributed and concentrated by the method of eigenfunctions (Navier) and modal analysis: comparison between numerical and analytical solutions. - Use of pre-and post-processors for geometry generation of shell structures. Generation of thin structures by rotation and extrusion. Finite element discretization and generation of an input file. Example of a cylindrical shell. Comparison with analytical solution. Example of a truncated cylindrical dome subject to its own weight and to a load annular. Comparison with analytical solutions. - Finite Element Analysis for the stability of the linear elastic field. Determination of the critical load limits and deformed critical. - Determination of the critical load of a structure plate and of a cylindrical shell structure with membrane axial load.

Prerequisites of the course are knowledge of solid continuum mechanics, simplified one-dimensional structures (Euler-Bernoulli beam model), simplified wing-box theory, buckling stability of elementary structures, dynamics of discrete space systems.

The notes of the course, edited by the teacher, made available to the students, fully cover the program; to learn more we indicate: - Stresses in Shells, W. Flugge, Springer 1973. - Theory of Plates and Shells (2nd Edition) - Timoshenko & S. Woinowsky, Mc-Graw-Hill, 1976. - J. Inman: Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey, 1996. - K. G. McConnell: Vibration Testing: Theory and Practice, John Wiley, 1995. - L. Meirovitch: Elements of Vibration Analysis, McGraw Hill, 2nd ed., 1986. - Géradin, M., Rixen, D., Mechanical Vibration - Theory and Application to Structural Dynamics, John Wileys & Sons, 1997.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.

Attendance of the face-to-face course is strongly recommended.

The assessment of the acquired knowledge is carried out with a final written test on theoretical questions and on an oral test in which both theoretical and design aspects are requested on the numerical exercises carried out during the delivery of the course.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.

A) BASIC EQUATIONS linearized mechanics of solids and space-discretized structures - Approach weak and Finite Element Method (FEM) A.1 Basic concepts of continuum mechanics solid Preliminary concepts of continuum mechanics solid. Eulerian and Lagrangian point of view. Local bases. Derivation and integration of Eulerian domain (time-varying) and Lagrangian. Reynolds transport theorem in local form. Principle of conservation of mass and its locations. According to the Reynolds transport theorem. Axioms of continuum mechanics. Local forms of the principle of conservation of momentum, theorems of Cauchy. Localization of the moment of momentum: symmetry of the stress tensor and the theorem of conservation of mechanical power. Existence of the carrier density of heat. Local form of conservation of Energy. Concept of deformation for a solid. Strain rate. (Second) Principle of thermodynamics (Clausius postulate) and its local form. Local form of energy storage in the form of entropy for solids. Reversible and irreversible processes for solids. Elastic solids. Linear elastic solids. Characterization of linear elastic solids. Linear isotropic elastic solids (Hookean solids). Elastic constants and their constraints. A.2 Formulation of the problem in weak form: FEM global equations Formulation of the problem in the weak form of the method of travel. Law of virtual work (weak formulation) and Lagrange equations for elastic solids with linear hypotheses in the field of movement of the Finite Element method (function-based "curtain"). Lagrange equations for a linear elastic solid. Weak formulation approach (as "Galerkin" approach). Structure and properties of the mass and stiffness matrices (for isotropic elastic solid and elastic). A.3 1D, 2D and 3D Finite Elements. Local equations and Assembly. Practical approach the MEF: the local view "element" with the displacement field, depending on form, link kinematic and constitutive element. Writing equilibrium in weak form: mass and stiffness matrices of the element. The element 2D triangular three-node plane (membrane in plane stress): effect of the nodes. Element 1D to two nodes of rod and torsion bar (stiffness matrix of element); element 1D to two nodes of the Euler beam (stiffness matrix of element); general case of element 1D to two nodes in the space (stiffness matrix total of element). 2D plane quadrilateral element with four nodes (membrane in plane stress): effects of the increase of the nodes. Element 2D quadrilateral plate plane bending (notes on 'generic quadrilateral element flexed-membrane). 3D tetrahedral element for the elastic problem general hints on increasing the number of nodes and the item box. General assembly of the EF based on the weak formulation: sum of partitioned matrices. Assembly of the element matrices and load element. Imposition of the boundary conditions. Notes on the properties and characteristics of the finite element completeness, compatibility, convergence and geometric invariance. B) METHODS OF SOLUTION for linearized mechanics of solids and space-discretized structures B.1 Free and forced response Problem formulation of the second order ODE by the Lagrange equations linearized. Associated eigenvalue problem. Intrinsic properties and three theorems on the problem of eigenvalues associated. Free response. Closed form solution of the homogeneous problem: identification of the modes and frequencies of vibration with eigenvectors and eigenvalues. Eigenvalue problem of the relationship between two matrices and the problem of eigenvalues standard. Lagrangian coordinates FEM v.s. modal coordinates. Convergence of the solution. Forced response of linear ODE systems of the second order with constant coefficients for the structural dynamics. Calls on Laplace and Fourier transforms and inverse-transform, transforms elementary fictions. Transfer matrices (Laplace domain) and frequency response functions (Fourier domain) based on spatial and spectral. Matrix of transfer functions and matrix functions Frequency Response for a dynamic vibration: Fourier series; integral and Fourier transform. Matrix of Frequency Response Functions for a dynamic vibration: Fourier transform as the limit of Fourier series. Meaning of the functions of frequency response to an input simple harmonic. Meaning of the matrix of the functions of frequency response to multiple inputs of harmonic synchronized. Resonance conditions for systems with N dofs, modal appropriation and the orthogonality conditions of the load. B.2 Modeling and analysis of damped structures General thermodynamic considerations on the damping in the continuous solid. Linear models for viscous solids. Spatial discretization (Eq. Lagrange) in the linear viscous solids. General properties of the matrix of viscous damping. Linear models for solid weak damping: viscous damping matrix, estimation of eigenvalues and eigenvectors for weakly damped systems theory (Rayleigh-Basile), modal damping, proportional damping. Work on non-viscous damping weak. C) BASIC EQUATIONS AND SOLUTIONS on linearized mechanics of space-continuous solids Dynamics of space-continuous systems. Structural linear operators (examples, 1D, 2D, and 3D; domestic product, a series of orthogonal functions. Added and self-adjoint operators. Betti-Castigliano's theorem. Spectral Three theorems. Example: the eigenfunctions of the auction. The method of eigenfunctions for the solution of problems elastic linear space-continuous: a) dynamic response problems free (the eigenfunctions as natural modes of vibration); b) Problems of static response (example of a response of a plate resting in the four sides, solution to the Navier). D) two-dimensional THIN STRUCTURES D.1 Isotropic and composite plate structures The model of purely isotropic plate in bending (with mid-plane membrane deformable) rectangular small displacements: kinematic field, strain and stress; equilibrium. Equilibrium equations and the end plate in bending purely homogeneous isotropic. Equations of multi-layer plate. Boundary conditions. Solution of the load plate with diaphragm and imperfection (displacement field) initial prescribed: solution eigenfunctions. Thermal problems for purely bending plates: reduction of the problem to the second order and resolution with the Navier eigenfunctions. Equations of isotropic plate with mid-plane deformable membrane: field and kinematic equations, constitutive equations and equilibrium equations written in the deformed configuration. Problem in the plane and out of plane (one-way coupling). Example guide of the rectangular plate resting in the four sides with load a membrane in the two opposite sides: i) static solution of the problem in the plane; ii) solution of the problem flexural dynamic with the method of eigenfunctions and discussion of stability: determination of the limit load buckling and deformed critical. Elastic properties, strength and thermo-elasticity of structural materials; physical characteristics of composites, composites with long fibers. Characterization of a linear elastic lamina of unidirectional composite long fiber as a continuous 2D orthotropic. Solution method of eigenfunctions for orthotropic plates. D.2 Shell structures General aspects of shell structures: membrane and bending models. The case of spherical shell for the membrane and bending behavior of the shells. The basic equations of the membrane shell behavior of shells of revolution with axisymmetric load. The analytical solution in terms of stress state for a generic geometry axisymmetric shell with axisymmetric load. Some analytical solutions: i) spherical shell pressure closed ii) closed dome subjected to its own weight (change sign circumferential stresses), iii) open dome with top load carrying its weight. iv) the closures of spherical or elliptical cylindrical pressure, v) conical shell vi) cylindrical shell vii) shell ring (optional argument). The basic equations of the membrane shells of revolution with NOT-axisymmetric load. Some analytical solutions for axisymmetric shells to axisymmetric load: i) closed cylindrical tank containing heavy fluid ii) spherical shell subjected to wind (only basic equations). Shells flexural behavior: the basic hypothesis in the theory of shells bending, the presence of curvature in the characterizations of effort, constitutive local and global. Shells with flexural behavior: equations with the displacement method for the circular cylinder with axisymmetric load. Solution of the static problem free with only the boundary conditions (homogeneous solution). Some solutions for cylinder with axisymmetric load: i) static response to a load ring, ii) the static forces and moments of extremity and thermal stress in a cylindrical shell iii) Evidence closures of spherical and ellipsoidal pressurized cylindrical shells ( optional argument). iv) critical load for a cylinder with compressed simply-supported end. E) Optimal Design and single-and multi-objective optimization for aeronautical structures Introduction on the use of optimization techniques in structural engineering. General formulation of the problem of structural optimization: cost function, design variables and constraints. Analytical solution and the Kuhn-Tucker conditions. Outline of numerical methods for analytical problems (stepest descendent and Newton method). Examples of mono-objective structural optimization. An outline of the principles and applications of multi-objective optimization: Pareto optimality, algorithms based on gradient and genetic algorithms. F) NUMERICAL EXERCISES - Static response of a wing-box type structure with the Finite Element method. Comparisons with analytical solutions of elementary beam structure having bending and torsion behavior. - Dynamic response of a structure type wing box with the Finite Element method. - Static response of a rectangular plate with a static load distributed and concentrated by the method of eigenfunctions (Navier) and modal analysis: comparison between numerical and analytical solutions. - Use of pre-and post-processors for geometry generation of shell structures. Generation of thin structures by rotation and extrusion. Finite element discretization and generation of an input file. Example of a cylindrical shell. Comparison with analytical solution. Example of a truncated cylindrical dome subject to its own weight and to a load annular. Comparison with analytical solutions. - Finite Element Analysis for the stability of the linear elastic field. Determination of the critical load limits and deformed critical. - Determination of the critical load of a structure plate and of a cylindrical shell structure with membrane axial load.

Prerequisites of the course are knowledge of solid continuum mechanics, simplified one-dimensional structures (Euler-Bernoulli beam model), simplified wing-box theory, buckling stability of elementary structures, dynamics of discrete space systems.

The notes of the course, edited by the teacher, made available to the students, fully cover the program; to learn more we indicate: - Stresses in Shells, W. Flugge, Springer 1973. - Theory of Plates and Shells (2nd Edition) - Timoshenko & S. Woinowsky, Mc-Graw-Hill, 1976. - J. Inman: Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey, 1996. - K. G. McConnell: Vibration Testing: Theory and Practice, John Wiley, 1995. - L. Meirovitch: Elements of Vibration Analysis, McGraw Hill, 2nd ed., 1986. - Géradin, M., Rixen, D., Mechanical Vibration - Theory and Application to Structural Dynamics, John Wileys & Sons, 1997.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.

Attendance of the face-to-face course is strongly recommended.

The assessment of the acquired knowledge is carried out with a final written test on theoretical questions and on an oral test in which both theoretical and design aspects are requested on the numerical exercises carried out during the delivery of the course.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.

A) BASIC EQUATIONS linearized mechanics of solids and space-discretized structures - Approach weak and Finite Element Method (FEM) A.1 Basic concepts of continuum mechanics solid Preliminary concepts of continuum mechanics solid. Eulerian and Lagrangian point of view. Local bases. Derivation and integration of Eulerian domain (time-varying) and Lagrangian. Reynolds transport theorem in local form. Principle of conservation of mass and its locations. According to the Reynolds transport theorem. Axioms of continuum mechanics. Local forms of the principle of conservation of momentum, theorems of Cauchy. Localization of the moment of momentum: symmetry of the stress tensor and the theorem of conservation of mechanical power. Existence of the carrier density of heat. Local form of conservation of Energy. Concept of deformation for a solid. Strain rate. (Second) Principle of thermodynamics (Clausius postulate) and its local form. Local form of energy storage in the form of entropy for solids. Reversible and irreversible processes for solids. Elastic solids. Linear elastic solids. Characterization of linear elastic solids. Linear isotropic elastic solids (Hookean solids). Elastic constants and their constraints. A.2 Formulation of the problem in weak form: FEM global equations Formulation of the problem in the weak form of the method of travel. Law of virtual work (weak formulation) and Lagrange equations for elastic solids with linear hypotheses in the field of movement of the Finite Element method (function-based "curtain"). Lagrange equations for a linear elastic solid. Weak formulation approach (as "Galerkin" approach). Structure and properties of the mass and stiffness matrices (for isotropic elastic solid and elastic). A.3 1D, 2D and 3D Finite Elements. Local equations and Assembly. Practical approach the MEF: the local view "element" with the displacement field, depending on form, link kinematic and constitutive element. Writing equilibrium in weak form: mass and stiffness matrices of the element. The element 2D triangular three-node plane (membrane in plane stress): effect of the nodes. Element 1D to two nodes of rod and torsion bar (stiffness matrix of element); element 1D to two nodes of the Euler beam (stiffness matrix of element); general case of element 1D to two nodes in the space (stiffness matrix total of element). 2D plane quadrilateral element with four nodes (membrane in plane stress): effects of the increase of the nodes. Element 2D quadrilateral plate plane bending (notes on 'generic quadrilateral element flexed-membrane). 3D tetrahedral element for the elastic problem general hints on increasing the number of nodes and the item box. General assembly of the EF based on the weak formulation: sum of partitioned matrices. Assembly of the element matrices and load element. Imposition of the boundary conditions. Notes on the properties and characteristics of the finite element completeness, compatibility, convergence and geometric invariance. B) METHODS OF SOLUTION for linearized mechanics of solids and space-discretized structures B.1 Free and forced response Problem formulation of the second order ODE by the Lagrange equations linearized. Associated eigenvalue problem. Intrinsic properties and three theorems on the problem of eigenvalues associated. Free response. Closed form solution of the homogeneous problem: identification of the modes and frequencies of vibration with eigenvectors and eigenvalues. Eigenvalue problem of the relationship between two matrices and the problem of eigenvalues standard. Lagrangian coordinates FEM v.s. modal coordinates. Convergence of the solution. Forced response of linear ODE systems of the second order with constant coefficients for the structural dynamics. Calls on Laplace and Fourier transforms and inverse-transform, transforms elementary fictions. Transfer matrices (Laplace domain) and frequency response functions (Fourier domain) based on spatial and spectral. Matrix of transfer functions and matrix functions Frequency Response for a dynamic vibration: Fourier series; integral and Fourier transform. Matrix of Frequency Response Functions for a dynamic vibration: Fourier transform as the limit of Fourier series. Meaning of the functions of frequency response to an input simple harmonic. Meaning of the matrix of the functions of frequency response to multiple inputs of harmonic synchronized. Resonance conditions for systems with N dofs, modal appropriation and the orthogonality conditions of the load. B.2 Modeling and analysis of damped structures General thermodynamic considerations on the damping in the continuous solid. Linear models for viscous solids. Spatial discretization (Eq. Lagrange) in the linear viscous solids. General properties of the matrix of viscous damping. Linear models for solid weak damping: viscous damping matrix, estimation of eigenvalues and eigenvectors for weakly damped systems theory (Rayleigh-Basile), modal damping, proportional damping. Work on non-viscous damping weak. C) BASIC EQUATIONS AND SOLUTIONS on linearized mechanics of space-continuous solids Dynamics of space-continuous systems. Structural linear operators (examples, 1D, 2D, and 3D; domestic product, a series of orthogonal functions. Added and self-adjoint operators. Betti-Castigliano's theorem. Spectral Three theorems. Example: the eigenfunctions of the auction. The method of eigenfunctions for the solution of problems elastic linear space-continuous: a) dynamic response problems free (the eigenfunctions as natural modes of vibration); b) Problems of static response (example of a response of a plate resting in the four sides, solution to the Navier). D) two-dimensional THIN STRUCTURES D.1 Isotropic and composite plate structures The model of purely isotropic plate in bending (with mid-plane membrane deformable) rectangular small displacements: kinematic field, strain and stress; equilibrium. Equilibrium equations and the end plate in bending purely homogeneous isotropic. Equations of multi-layer plate. Boundary conditions. Solution of the load plate with diaphragm and imperfection (displacement field) initial prescribed: solution eigenfunctions. Thermal problems for purely bending plates: reduction of the problem to the second order and resolution with the Navier eigenfunctions. Equations of isotropic plate with mid-plane deformable membrane: field and kinematic equations, constitutive equations and equilibrium equations written in the deformed configuration. Problem in the plane and out of plane (one-way coupling). Example guide of the rectangular plate resting in the four sides with load a membrane in the two opposite sides: i) static solution of the problem in the plane; ii) solution of the problem flexural dynamic with the method of eigenfunctions and discussion of stability: determination of the limit load buckling and deformed critical. Elastic properties, strength and thermo-elasticity of structural materials; physical characteristics of composites, composites with long fibers. Characterization of a linear elastic lamina of unidirectional composite long fiber as a continuous 2D orthotropic. Solution method of eigenfunctions for orthotropic plates. D.2 Shell structures General aspects of shell structures: membrane and bending models. The case of spherical shell for the membrane and bending behavior of the shells. The basic equations of the membrane shell behavior of shells of revolution with axisymmetric load. The analytical solution in terms of stress state for a generic geometry axisymmetric shell with axisymmetric load. Some analytical solutions: i) spherical shell pressure closed ii) closed dome subjected to its own weight (change sign circumferential stresses), iii) open dome with top load carrying its weight. iv) the closures of spherical or elliptical cylindrical pressure, v) conical shell vi) cylindrical shell vii) shell ring (optional argument). The basic equations of the membrane shells of revolution with NOT-axisymmetric load. Some analytical solutions for axisymmetric shells to axisymmetric load: i) closed cylindrical tank containing heavy fluid ii) spherical shell subjected to wind (only basic equations). Shells flexural behavior: the basic hypothesis in the theory of shells bending, the presence of curvature in the characterizations of effort, constitutive local and global. Shells with flexural behavior: equations with the displacement method for the circular cylinder with axisymmetric load. Solution of the static problem free with only the boundary conditions (homogeneous solution). Some solutions for cylinder with axisymmetric load: i) static response to a load ring, ii) the static forces and moments of extremity and thermal stress in a cylindrical shell iii) Evidence closures of spherical and ellipsoidal pressurized cylindrical shells ( optional argument). iv) critical load for a cylinder with compressed simply-supported end. E) Optimal Design and single-and multi-objective optimization for aeronautical structures Introduction on the use of optimization techniques in structural engineering. General formulation of the problem of structural optimization: cost function, design variables and constraints. Analytical solution and the Kuhn-Tucker conditions. Outline of numerical methods for analytical problems (stepest descendent and Newton method). Examples of mono-objective structural optimization. An outline of the principles and applications of multi-objective optimization: Pareto optimality, algorithms based on gradient and genetic algorithms. F) NUMERICAL EXERCISES - Static response of a wing-box type structure with the Finite Element method. Comparisons with analytical solutions of elementary beam structure having bending and torsion behavior. - Dynamic response of a structure type wing box with the Finite Element method. - Static response of a rectangular plate with a static load distributed and concentrated by the method of eigenfunctions (Navier) and modal analysis: comparison between numerical and analytical solutions. - Use of pre-and post-processors for geometry generation of shell structures. Generation of thin structures by rotation and extrusion. Finite element discretization and generation of an input file. Example of a cylindrical shell. Comparison with analytical solution. Example of a truncated cylindrical dome subject to its own weight and to a load annular. Comparison with analytical solutions. - Finite Element Analysis for the stability of the linear elastic field. Determination of the critical load limits and deformed critical. - Determination of the critical load of a structure plate and of a cylindrical shell structure with membrane axial load.

Prerequisites of the course are knowledge of solid continuum mechanics, simplified one-dimensional structures (Euler-Bernoulli beam model), simplified wing-box theory, buckling stability of elementary structures, dynamics of discrete space systems.

The notes of the course, edited by the teacher, made available to the students, fully cover the program; to learn more we indicate: - Stresses in Shells, W. Flugge, Springer 1973. - Theory of Plates and Shells (2nd Edition) - Timoshenko & S. Woinowsky, Mc-Graw-Hill, 1976. - J. Inman: Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey, 1996. - K. G. McConnell: Vibration Testing: Theory and Practice, John Wiley, 1995. - L. Meirovitch: Elements of Vibration Analysis, McGraw Hill, 2nd ed., 1986. - Géradin, M., Rixen, D., Mechanical Vibration - Theory and Application to Structural Dynamics, John Wileys & Sons, 1997.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.

Attendance of the face-to-face course is strongly recommended.

The assessment of the acquired knowledge is carried out with a final written test on theoretical questions and on an oral test in which both theoretical and design aspects are requested on the numerical exercises carried out during the delivery of the course.

The course is held with traditional frontal lessons (with further parallel remote connection in a pandemic emergency) both for the more theoretical aspects and in carrying out practical and design exercises.