44939 - Structural mechanics

Learning outcomes

The course aims at increasing the hypotheses and methodologies acquired during the course of Scienza delle Costruzioni (Mechanics of Solids and Structures), while introducing the bases for the analysis of the dynamic and nonlinear structural behavior.

Course contents

MATRIX STRUCTURAL ANALYSIS
1)System with three springs and two free nodes: equations and primal scheme of physical theories. 2)Proof of the virtual work principle, equilibrium and congruence equations, relationship between operators. 3)Principles of virtual forces and virtual displacements, alternative formulations of congruence and equilibrium. 4)System with three springs and two free nodes: equations and dual scheme of physical theories. 5)Deformation work, elastic and complementary energy, alternative formulation of constitutive equations, equations in the scheme of physical theories. 6)Principle of stationary and minimum total potential energy and complementary potential energy. 7)Formulation of fundamental equations through the stationary principles, equations and scheme of physical theories. 8)Stiffness matrix for axial load: scheme of equations and variables, unitary displacement method and direct method. 9)Stiffness matrix for axial load: scheme of equations and variables, Castigliano's theorem method and stationary total potential energy method. 10)Stiffness matrix of a system of rods in series: direct method, Castigliano's theorem method, collocation matrices. 11)System matrix of a system of rods in series: assembling of stiffness matrices, imposing of boundary conditions and general formulation, graphical illustration of the assembling procedure. 12)Stiffness matrix of the rod in the global reference system: direct method, alternative procedure. 13)Plane truss: vector of nodal loads and nodal displacements, relationships in expanded form, tolopogical matrix, sum of the stiffness matrices, formal solution of the problem. 14)Bending stiffness matrix: equations in the scheme of physical theories, shape functions, evaluation of the stiffness matrix through the direct method. Bending stiffness matrix: equations in the scheme of physical theories, shape functions, evaluation of the stiffness matrix through Castigliano's theorem method. 16)Construction of the 8x8 stiffness matrix of a clamped-simply supported beam with concentrated loads. Formal solution of the linear system. 17)Construction of the 8x8 stiffness matrix of a clamped-simply supported beam with concentrated loads. Numbering of nodes in order to group known and unknown terms. 18)Finite elements: introduction, structure discretization and displacement model. 19)Triangular element stiffness matrix. 20)Equivalent nodal loads for the triangular finite element. Rectangular element. 21)Scheme of variables and equations of a finite element. Deduction of the balance equation. 22)Three-dimensional element. 23)Higher order finite elements. 24)Equations of motion of a discreet system. Consistent mass matrix for the two degrees of freedom beam. 25)Description of the Strait of Messina bridge. 26)Torsion stiffness matrix.

ELASTIC WAVES

1)Equations and operators of equilibrium, congruence and elastic constitutive relations. 2)Field equilibrium equations in terms of displacement: extended matrix and operator notation in the static and dynamic case. 3)Scheme of physical theories. 4)Three dimensional longitudinal waves.

FORMULATION OF THE CABLE PROBLEM
1)Equilibrium, congruence, constitutive equations and fundamental equation. 2)Classical or strong formulation of the problem. Weak or generalized formulation. 3)Linear operators, functionals, variational formulation. 4)String with concentrated and distributed load (E. Viola, Esercitazioni S.d. Costruzioni 1).

FORMULATION OF THE PLANE BEAM AND OF THE PLATE PROBLEM
1)Geometry of deformations, generalized strains and displacement components, congruence equations in different notations. 2)Field equilibrium equations and elastic constitutive equations. 3)Principle of virtual work, fundamental equation, scheme of physical theories. 4)Elastic strain energy in terms of strains, of internal stresses and in mixed form. 5)Total potential energy, stationarity and minimum principles. 6)Equation of bending for the Timoshenko beam with concentrated force and couple. 7)Flexural vibration of thin plates. Kinematic hypothesis and strain components. 8)Stresses and internal force resultants. 9)Field equilibrium equations. 10)Elastic constitutive equations and fundamental equation. 11)Scheme of physical theories in the static and dynamic case. 12)Free vibration of plates.

GENERALIZED DIFFERENTIAL QUADRATURE METHOD
1)Definition of differential quadrature, calculation of first order derivatives and n-order derivatives. 2)Lagrange polynomials, evaluation of the weighting coefficients for the first and n order derivatives. 3)Kinds of domain discretization and their characteristics. 4)Application of the G.D.Q. Method to the power and square root functions. Comparison of numerical results obtained for the two functions. 5)“Delta points” technique and its application to the Euler-Bernoulli beam. 6)Euler-Bernoulli beam with varying cross-section: deduction of the fundamental equation. 7)Application of the G.D.Q. Method to beams with constant and varying cross sections, fundamental system and boundary conditions, evaluation of the kinematic parameters and internal force resultants. 8)Numerical examples: cantilevered beam with constant cross-section, simply supported beam with varying cross-section (linear variation of heigth); clamped beam with quadratic variation of heigth. 9)Matlab: matrix operations (inverse, transpose, determinant, eigenvalues and matrix composition, right and left division, Gauss method). 10)Difference between script and function files and their syntax. 11)Control instructions: syntax and logical meaning of return, for, while, if, else, elseif and end. 12)Logical and relational operators: syntax and differences. 13)Meaning and syntax of: clc, clear, format , input, menu, subplot, plot, set, gca , gcf, text, line, linspace, interp1, spline. 14)Three dimensional matrices in Matlab: syntax and application of the G.D.Q. Method.

FUNDAMENTALS OF STRUCTURAL DYNAMICS
1)Introduction to dynamic structural analysis: mathematical modelling of the dynamic problem, geometric or structural model, external loads model, mechanical or rheological material model. 2)Review and unified formulation of the problems: complex numbers, harmonic motions, Newton's second law. 3)Free motion of the single degree of freedom oscillator: simple non damped oscillator, equation of motion, frequency and period, amplitude and phase of motion, structural identification, equivalent oscillator, example 3.1. 4)Simple damped oscillator, motion equation, critically damped system, overdamped system, underdamped system. 5)Harmonic excitation of single degree of freedom systems: simple undamped oscillator, solution of the forced oscillator, amplitude and phase of steady state response, frequency response. 6)Resonance condition (basics). 7)Harmonic excitation in presence of damping, motion equation, dynamic amplification coefficient. 8)Frequency response, diagrams of dynamic amplification coefficient and phase, quasi static zone, resonance and seismographic zone. 9)Input base motion, solution in terms of relative displacement, amplitude of response and of input base motion, scheme of functioning of shaker (exercise 4.2). 10)Damping in single degree of freedom systems: kinds of damping, viscous damping, dissipated energy in the viscous damper, hysteretic damping (structural), friction damping (Coulomb), hysteresis cycle for Coulomb friction, equivalent viscous damping (example 5.1). 11)Methods for the evaluation of damping, logarithmic decrease for viscous damping, curve of response to resonance, bandwidth method (example 5.2). 12)Basics on energy methods: principle of energy conservation, Rayleigh method (example 6.1). 13)Lagrange equation for the single degree of freedom system. 14)Hamilton's principle, alternative form of Lagrange equation. 15)Periodic excitation and harmonic analysis, periodic functions and Fourier's series. 16)Dirichlet's theorem, example 8.1. 17)Generical applied forces and impulse loads: impulse and momentum, impulse excitation, response function to impulse, impulse applied at the initial instant, example 9.1. 18)Arbitrary excitation, particular load cases: constant force, rectangular impulse, example 9.2. 19)Two degrees of freedom systems: writing and solution of the equation of motion: method of dynamic equilibrium. 20)N degrees of freedom systems: equations of motion, free vibration: eigenvalue problem, frequency spectrum and modal shapes. Example 11.1. 21)Orthogonality of vibration modes, generalized mass and stiffness. 22)Orthonormal condition. 23)Expansion theorem and modal analysis. 24)Non homogeneous initial conditions: normal equations, determination of constants. 25)Damped N degrees of freedom system: matrix formulation, weakly damped structures, proportional damping. 26)Canonical form reduction. 27)Imposed constraint motion: motion differential equations, natural frequencies and modal shapes, generalized or modal stiffness and mass matrices, uncoupled motion equations, modal space response, modal response. 28)Rayleigh quotient. 29)Generalized multi degrees of freedom systems: compliance and stiffness matrix. Example of calculation of stiffness coefficients through direct method. 30)Vibration of thin plates. Kinematic hypothesis and strain components. 31)Stress and internal force resultants. 32)Field equilibrium equations and elastic constitutive equations. 33)Fundamental equation, scheme of physical theories, energy variable. 34)Free vibration of plates, rectangular plates.

Readings/Bibliography

VIOLA  E., “Introduzione all'analisi matriciale delle strutture”, Pitagora Editrice, Bologna, 1996.

VIOLA  E., “Fondamenti di Dinamica e Vibrazione delle Strutture”, Pitagora Editrice, Bologna, 2001, Vol.1.

Teaching methods

The course syllabus is covered entirely during classes. Exercises will be solved in class.

Assessment methods

Written test and oral test.

Teaching tools

In the classroom, transparencies and projected slides will be used.

Office hours

See the website of Erasmo Viola