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COPPOTELLI GIULIANO
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programma)
Avvisi: http://www.ingaero.uniroma1.it/index.php?option=com_content&view=article&id=2166&Itemid=2471&lang=it
Materiale didattico: http://www.ingaero.uniroma1.it/index.php?option=com_content&view=article&id=2164&Itemid=2469&lang=it
COURSE SYLLABUS
1) Introduction to the main properties of aero-space structural components and their use. (2 hrs)
2) Aeronautical and Space mechanical environment characterizing the operating life of structures. (4 hrs)
3) Introduction to the mechanical modeling of structures and materials:
a. elastic and failure models(2 hrs)
b. membrane and flexural modeling: elementary equations from continuum mechanics and "De Saint-Venant problem (4 hrs)
4) Semi monocoque model for aerospace stressed shell structures (20 hrs - practices included)
5) Introduction to the Finite element Method (linear structural behavior) (30 hrs)
a. Weak formulation for continuum solid and structures. Virtual work principle and derivation of the equations of equilibrium
b. Finite domain, shape functions, discretization process and elemental matrices
c. 1D, 2D, and 3D applications: rod, beam
d. Discrete loading vector, Iso-parametric finite elements, backgrounds on Gauss integration technique, finite element assembling techniques
e. Solution technique, close form convergence analysis
f. Some real case applications concerning aero-space structures static behavior using commercial code
6) Introduction to aero-space technologies (8 hrs)
a. Metallic materials technologies. Composite material technologies: lamination, filament winding.
b. Curing process for thermo-setting resins and thermo-plastics materials.
c. Some laboratory hands-on practices in manufacturing composite structural components
7) Introduction to structural dynamics (16hrs)
a. Background on linear harmonic oscillator dynamics.
b. System properties and dynamic response: resonance conditions and convolution integral.
c. System dynamics for multi-degree of freedom (MDOF) system in space discretized domain.
d. Eigenvectors and eigenvalues, response in the modal domain for MDOF systems.
e. Background on damping modeling and frequency response function.
f. Some laboratory hands-on practices in structural dynamics
8) Background in isotropic shell structure models (6hrs)
DETAILED PROGRAM
Course Introduction. Textbooks, preliminary required background.
Short overview on most important topics relevant to aerospace structures.
Typical mechanical enviroment affecting the operating life of aerospace structures.
Definition of load factor and most used load-factor charts.
Derivation of the V-n diagrams and its characteristics points. Some recals on FAA/EASA regulations.
Qualitative characterization of the stress field in wing structures subjected to different loading corresponding to flight envelope typical points.
Modification of the flught envelop due to gust encountering.
Examples in deriving the flight envelope of a given aircraft.
Introduction to the beam elastic model. Linear elastic model and 2D eqilibrium equations.
Solution to the differential equations for tension, shear, and bending elastic reaction to a given loading condition.
Evaluation of the shear and bending moment distribution for a cantilever beam.
Some examples in evaluating the shear and bending distributions along a beam structure: cantilever beam with uniformly distributed load; simply supported beam with uniformly distributed load, and point looad.
Cantilever beam with a prabolic shape loading.
Description on the effects of external loading on a elastic body. Definition of a stress in a point and direction. Name designation of the stresses in a 3d field.
Stress transformation between reference frames.: 2D case. formulas for the evaluation of the principal stress (shear) and their corresponding principal planes.
Mohr's circle for stresses.
3D stress field at a point. Tenorial nature of the stress.
Principal stresses and directions form the eigen-vlue problem associated to the stress tensor. Stress invariants I1, I2, and I3.
Example problem on the evaluaiton of the principal stress field characteristics. Deformation and strain field at a point. 1D and 2D equations; generalization to the 3D field case.
Saint-Vanant compatibility equations.
Strain at a point. Derivation of the strain in a different reference of frame. Principal value and direction for strain. Mohr's circle for strain.
Constitutive equations. Isotropic and homoeneous materials. E, n, G and 3D strain/stress relations.
Stress-Strain characterization of ductile and fragile materials. Some details on ductile behavior, Baushinger's effect.
Elastic-plastic models: purely plastic, linear plastic, and exponential plasticity.
Introduction to the failure criteria: 1D and 2D stress/strain field.
Failure criteria for both ductile and fragile materials. Safety margin. Rankine, De Saint-Venant, Belttrami, and Von Mises failure criteria; example on their use.
Some Examples in solving complex beam configrations: evaluation of shear and bending. Assignemnt of a homework dealing with a generic beam truss.
Introduction to mechanical environment of space structures. Definition of worst conditions and limiting criteria for structural design.
Saint-Venant principle. Introduction to the torsional problem of uniform bars and definition of "Prandtl" stress function and "Saint-Venant" warping function.
Kinematics of a general shape section under torque. Compatibility equations for torsion and traction-free boundary condition.
Use of the Prandtl function to solve for torsional rigidity.
Additional details on spectral properties of a stress/strain field. Examples on how to solve the eigenvalue problem.
Torsion in bars with circular cross sections. Traction-free boundary condition, compatibility equations, resulting torque, radial and tangential shear stresses, warping.
Torsion of bars with narrow rectangular cross-section. Traction-free boundary condition, compatibility equations, resulting torque, torsional rigidity, and warping. Composition of a number of thin-walled members.
Torsion in closed single-cell thin-walled sections. Shear stress in the "s" curvilinear abscissa.
Torsion in closed single-cell thin-walled sections. Shear stress in the "n" direction.
Definition of the shear flow and its properties. Resulting Torque laong the entire single-cell thin-walled section.
Resulting force (magnitude, direction, and position) of a partial countour. Evaluation of the twist angle. Torsional constant.
Comparison of open/closed thin walled section torsional constants. Numerical example concerning the torsional problem associated to a three-stringer thin-walled beam structure subjected to torque.
Multicell thin-walled sections: Torque equilibrium, comaptibility condition. Numerical examples concerning 2-cell cross-sections with/without diagonal web.
Warping in ope thin-walled sections: Primary and secondary warping. Derivation of the warp function. Numerical example on a C-like cross section.
Warping in closed thin-walled sections: derivation of the warp funcion. Numerical example on a 4-stringers box structure.
Bending and flexurar shear in beam structures. Assumptions for the displacement field for a one-way transversal loading. Derivation of the equation of motion in a linear formulation for a symmetric cross-section: Bernoulli-Euler Beam Equation.
Analysis of the bi-directiona bending. Definition of the rotation function about the intrinsic y- and x-axis. Simplifying assumptions concerning the shear strain. Derivation of the general bending equations for a generic transverse cross-section under bi-directional loading.
Definition of the neutral axis and plane and its identification from the cross-section geometrical properties and loading conditions.
Application example on a single-cell box beam with four stringers and on a thin-walled Z-section subjected to a one-way bending moment. Determination of the centroid of the "effective" section; moment of inertia; neutral axis/plane; stress distribution.
Transverse shear due to transverse force in symmetric sections: analysis of narrow rectangular section.
Timoshenko beam model to take into account transverse shear deformation. Deformaton/strain relations; relationship between shear force/bending moment and the displacement field. Timoshenko's beam equations. Shear force effect in typical boundary conditions.
Application example on a cantilever web/two-stringer beam subjected to constant shear flow at the free-end.
Compensation of the shear rigidity for constant transverse shear strain hypothesis. Discussion on how to determine the "effective" shear-carrying area in curved and inclined thin-walled sections.
Flexural shear flow in open thin-walled sections: analysis of symmetric and antisymmetric sections.
Application example concerning a C-section with bending carrying webs. Stringer-web C-section.
Application example concerning a S-shaped two-stringer section.
Multiple shear flow Junctions: analysis of section without and with concentrated areas. Equilibrium at the junction node including the jump due to the variation of the normal stress
Shear center in open sections. General method for the determination of the location of the shear center: example concerning a curved web and a 4-stringer C-shaped section.
Closed thin-walled sections and combined flexural & torsion shear flow. Approach based on a fictitius longitudinal cut to solve for the shear flows: shear flow due to the shear flow produced by the shear force in open section and unknown constant shear flow.
Use of the twist rate equation as a compatibility equation to dtermine the costant shear flow in terms of the shear center location.
Application example concerning a 4-stringer closed thin-walled section.
Determination of the shear center location in an example 4-stringer thin-walled section case.
Method to solve for the flexural shera flows based on the statically determinate shear flow. Example concerning a three-stringer single thin-walled section with a vertical shear loading.
Analysis of thin-walled sections subjected to a system of forces and solution of the shear flow problem using the statically-determined shear flow-based approach. Example problem concerning a three-stringer thin-walled section beam.
Analysis of closed multi-cell sections. Method for solving the shear flow distribution based on fictitius cut of the section; introdiction of the compatibility equations and the bending moment equilibrium equation for solving the costantt shear flow for each cell.
Guidance rules on how to cut multi-cell thin closed sections.
Some example problems to familiarize with thin-walled sections beams generally used in the aerospace structure design: find the shear flow due to a vertical shear loading and the shear center location of a two-cell box beam with 6 stringers ;
find the shear flow due to a vertical shear load of a 8 stringer closed thin-walled section having horizontal symmetry.
Review problems in beam modeling and design. Analysis of a a hyperstaticEuler-Bernoulli beam structure with uniform distributed shear loading: shear/bending and displacement/rotation distribution along the beam span.
Analysis and design of an open thin walled section subjected to a vertical shear: Determination of the normal stresses and the shear flow. Identification of the shear center location.
Review problems concerning the aerospace structures having closed thin-walled section: evaluation of the shear and bending spanwise distribution; evaluation of the shear flow, compression/tension stresses, shear center for two distinct cross sections.
Introduction to plate structures and their categorization in thin with small displacement, thin with large displacement, and thick plates. Kirchoff's plate hypothesis.
Kirchoff's plate model: derivation of the kinematic and stress relations, equilibrium equation written in term of the transverse displacement.
Remarks on the boundary conditions: trasformation of the torque distribution into point force at the boundaries. Clamped end, simply supported end. Free edge and sliding edge.
Navier's solution method for solving the bending problem in plates. Soution in terms of series of complete and orthogonoal functions satisfying the boundary conditions and projection of the external load onto the previous functions.
Example problem of a square simply supported plate with an uniform loading pressure. Some remarks on the convergency of the solution.
Introduction to Shell structures: gategorization and main hypothesis on kinematics. Motivational example based on a pressurized dome structure.
Description of the main geometrical and structural parameters of a shell in the form of a surface of revolution and introduction to its analysis when subjected to an axis-symmetric load.
Derivation of the equilibrium equations for the shell in the form of a surface of revolution subjected to an axis-symmetric load. Alternate approach for the equation of the equilibrium in the meridian direction.
Particular cases of shells in the form of surface of revolution: spherical shell, conical shell, cylindrical shell.
Shell in the form of a surface of revolution loaded with a non axis-symmetrical load. Derivation of the equilibrium equation.
Recals on problems in structural dynamics. Derivation of the equation of motion for a simple harmonic oscillator with voscous damping (SDOF system).
Definition of the Transfer Function (TF) and and Frequency Transfer Function (FRF). System poles and modal parameters.
Analysis of the system response to the variation of the value of the damping: damped, underdamped and critical damped dynamic response. Derivation of the Residuals.
Sketching rules to follow when plotting the Amplitude/Phase or Real/Imaginary of the Frequency Response Function.
Sensitivity of the FRFs to variation of the stiffness, mass, and damping; compliance lines, mass lines, and shift of the natural frequency due to change of the damping factor.
Analysis of MDOF system. Derivation of the TF and FRF matrices and their relation with the space model and modal model.
Example solution of a 2-DOF lumped parameter system subjected to a dynamic loading: sokution of the characteristic equation for the eigenvalues and determination of the corresponding eigenvectors.
Some recalls on damping and their common use: viscous and hysteretical models.
Dynamics of continuos structures. Derivation of the wave equation and its solution using the separation of variable approach.
Example problem in determining the eigenvalues and eigenvectors of a string fixed at its ends, cantilever rod, supported shell, and a simply supported plate.
Dynamic of systems described with discrete parameters. Techniques for the soluiton of the free response using the system eigen-properties. Orthogonality properties of the eigen-characteristics.
Solution techniques for the prediciton of the dynamic response.
Evaluation of the rigid body motion using the system stiffness matrix.
Introduction to the static and dynamic condensation: main motivations and brief description of the currently used methods.
Static structural condensation using Guyan Technique
Dynamic condensation/expansion using Component Mode Synthesis (CMS) using Craig-Bampton component modes
Dynamic condensation using System Equivalent Expansion/Reduction Process (SEREP)
Assessment of the accuracy of the prediciton of the normal modes from condensed model using MAC, NCO, and CO.
Introduction to the effective modal mass: physical meaning, main applications in space enginnering.
Derivation of the effective modal mass for a SDOF system. Relation between the response funcion, effective modal mass, amplification factor, and dinamic reaction at the base.
Derivation of the effective modal mass for a MDOF system. Characterization of the Rigid Body Mode Matrix, Matrix of the Modal Participation Factors, Matrix of Generalized Masses. Properties of the effective modal mass matrix.
S.P. Timoshenko and J.N. Goodier, "Theory of Elasticity," International Student Edition.
S.P. Timoshenko and S. Woinowsky-Krieger, "Theory of Plates and Shells," McGraw-Hill International Editions.
P.C. Chou and N.J. Pagano, "Elasticity - Tensor, Dyadic and Engineering Approaches," Dover Publications, Inc.
T.H. Megson, "Aircraft Structures for engineering Students," Elsevier Aerospace Engineering.
C.T. Sun, "Mechanics of Aircraft Structures," Wiley
P. Santini, "Costruzioni Aeronautiche (in Italian), Vol. I and Vol. II, ESA Ed.
J.S. Przemieniecki, "Theory of Matrix Structural Analysis," Dover Publications, Inc.
M.C.Y Niu, "Airframe Structural Design; Practical Design Information," Conmilit.
O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, "the Finite Element Method," Butterworth-Heinemann.
K.J. Bathe, "Numerical Methods in Finite Element Analysis," Prenctice-Hall.
L. Meirovitch, "Computational Methods in Structural Dynamics," Sijthoff & Noordhoff Ed.
Course notes provided by the professor(s).
(Date degli appelli d'esame)
