Università Roma Tre

The Analysis of Aeronautical Structures course is part of the activities of the Construction and aerospace structures (ING-IND/04 SSD).

The teaching program is structured to provide students with knowledge and skills in the structural design of aeronautical components, using methods widely used in the aircraft detailed design phase.

The teaching program is divided into 36 lectures (equal to 9 CFU) divided into the following eight main sections:

1) Tensor Calculus Fundamentals: N-th order tensors. Tensor operations. Curvilinear coordinates. Covariant and contravariant base vectors. Vectors and tensors in curvilinear coordinates. Differential operators in curvilinear coordinates.

2) Kinematics of Deformable Continua: Lagrangian and Eulerian descriptions of motion. Finite strain theory. Deformation gradient tensor. Polar decomposition theorem. Lagrangian and Eulerian finite strain tensors (Cauchy-Green and Eulero-Almansi). Rates of deformation tensors. Linearization of the finite strain theory (infinitesimal strain theory). Material and spatial descriptions of the continuity equation.

3) Dynamics of Deformable Continua: Material and spatial forms of linear momentum balance. Cauchy and Piola-Kirchhoff stress tensors. Material and spatial forms of the angular momentum balance. Material and spatial forms of the mechanical energy balance.

4) Thermodynamics of Deformable Continua: Material and spatial forms of the energy balance. Stokes' heat flux theorem. Material and spatial forms of the thermodynamic energy balance. The second law of thermodynamics.

5) Constitutive relations theory: Noll's axioms. Limitations on the constitutive relations due to the second law of thermodynamics. Constitutive relations for thermoelastic materials: definition of isothermal elastic tensor, thermal stress tensor, and thermal conductivity tensor. Constitutive equation for linear isotropic thermoelastic materials.

6) Thermoelastic problems in aeronautical structures: Gibbs and entropy evolution equations. Uncoupled thermoelastic formulation. Initial boundary value problem of the heat conduction equation. Thermal stress analysis for elastic bodies subjected to external and thermal loads: Euler-Bernoulli beam and Kirchhoff plate. Eigenfunction method for the solution of the thin plate bending problem.

7) Finite Element method: Strong and weak forms of the uncoupled thermoelastic problem. The relation between strong and weak forms and boundary conditions. Virtual work principle. Discretization and definition of shape functions. Shape function choice criteria. Evaluation of element mass, stiffness, and damping matrices. Evaluation of equivalent nodal loads vector. Assembly procedure. The imposition of displacement constraints. Conformal elements. Non-conformal elements – patch test. Standard methods for shape functions construction. Applications in aeronautical problems: truss, beam, plate, and shell.

8) Introduction to the code Simulation Mechanical: Geometric preprocessor; material properties definition; constraints and external loads imposition; solution methods and post-processing. Structural analysis of a wing and/or a fuselage.

- M.E., Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981 (for contents 1, 2, 3, and 5 of the syllabus)

- Boley, B.A, Weiner. J.H., Theory of Thermal Stresses, John Wiley & Sons, New York, 1960 (for contents 4, 5, and 6 of the syllabus)

- Thomas J.R., Hughes, ‘The Finite Element Method – Linear Static and Dynamic Finite Element Analysis,’ Dover, 2000 (for content 7 of the syllabus)

- T.H.G., Megson, Aircraft Structures for Engineering Students, Arnold, London, 1999 (for content 7 of the syllabus)

- Lecture notes by the teacher (for all the contents of the syllabus)

The educational material used by the teacher from time to time is indicated during lectures. The lecture notes are available on the Moodle platform to facilitate their use for attending and non-attending students. On the same platform, are also made available the specifications of the project the students have to perform during the year, as well as a collection of written tests of previous exams, to provide students with a valid and realistic test bench for the final exam.

The Analysis of Aeronautical Structures course is part of the activities of the Construction and aerospace structures (ING-IND/04 SSD).

The teaching program is structured to provide students with knowledge and skills in the structural design of aeronautical components, using methods widely used in the aircraft detailed design phase.

The teaching program is divided into 36 lectures (equal to 9 CFU) divided into the following eight main sections:

1) Tensor Calculus Fundamentals: N-th order tensors. Tensor operations. Curvilinear coordinates. Covariant and contravariant base vectors. Vectors and tensors in curvilinear coordinates. Differential operators in curvilinear coordinates.

2) Kinematics of Deformable Continua: Lagrangian and Eulerian descriptions of motion. Finite strain theory. Deformation gradient tensor. Polar decomposition theorem. Lagrangian and Eulerian finite strain tensors (Cauchy-Green and Eulero-Almansi). Rates of deformation tensors. Linearization of the finite strain theory (infinitesimal strain theory). Material and spatial descriptions of the continuity equation.

3) Dynamics of Deformable Continua: Material and spatial forms of linear momentum balance. Cauchy and Piola-Kirchhoff stress tensors. Material and spatial forms of the angular momentum balance. Material and spatial forms of the mechanical energy balance.

4) Thermodynamics of Deformable Continua: Material and spatial forms of the energy balance. Stokes' heat flux theorem. Material and spatial forms of the thermodynamic energy balance. The second law of thermodynamics.

5) Constitutive relations theory: Noll's axioms. Limitations on the constitutive relations due to the second law of thermodynamics. Constitutive relations for thermoelastic materials: definition of isothermal elastic tensor, thermal stress tensor, and thermal conductivity tensor. Constitutive equation for linear isotropic thermoelastic materials.

6) Thermoelastic problems in aeronautical structures: Gibbs and entropy evolution equations. Uncoupled thermoelastic formulation. Initial boundary value problem of the heat conduction equation. Thermal stress analysis for elastic bodies subjected to external and thermal loads: Euler-Bernoulli beam and Kirchhoff plate. Eigenfunction method for the solution of the thin plate bending problem.

7) Finite Element method: Strong and weak forms of the uncoupled thermoelastic problem. The relation between strong and weak forms and boundary conditions. Virtual work principle. Discretization and definition of shape functions. Shape function choice criteria. Evaluation of element mass, stiffness, and damping matrices. Evaluation of equivalent nodal loads vector. Assembly procedure. The imposition of displacement constraints. Conformal elements. Non-conformal elements – patch test. Standard methods for shape functions construction. Applications in aeronautical problems: truss, beam, plate, and shell.

8) Introduction to the code Simulation Mechanical: Geometric preprocessor; material properties definition; constraints and external loads imposition; solution methods and post-processing. Structural analysis of a wing and/or a fuselage.

- M.E., Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981 (for contents 1, 2, 3, and 5 of the syllabus)

- Boley, B.A, Weiner. J.H., Theory of Thermal Stresses, John Wiley & Sons, New York, 1960 (for contents 4, 5, and 6 of the syllabus)

- Thomas J.R., Hughes, ‘The Finite Element Method – Linear Static and Dynamic Finite Element Analysis,’ Dover, 2000 (for content 7 of the syllabus)

- T.H.G., Megson, Aircraft Structures for Engineering Students, Arnold, London, 1999 (for content 7 of the syllabus)

- Lectures notes by the teacher (for all the contents of the syllabus)

The educational material used by the teacher from time to time is indicated during lectures. The lecture notes are available on the Moodle platform to facilitate their use for attending and non-attending students. On the same platform, are also made available the specifications of the project the students have to perform during the year, as well as a collection of written tests of previous exams, to provide students with a valid and realistic test bench for the final exam.