AIMS AND CONTENT

AIMS AND LEARNING OUTCOMES

The learning objectives are:

• to understand the differences between the force method and the displacement method for the analysis of statically indeterminate structures;

• to apply the displacement method and introduce its extension to more general matrix formulations, including an introduction to finite elements;

• to analyze the dynamic response of single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems subjected to generic loading and seismic actions, both in the time and frequency domains;

• to understand the basic principles of equilibrium instability, the concept of critical load, and the modeling of systems subjected to second-order effects;

• to introduce the basic concepts of limit analysis and plastic design for beam systems.

At the end of the course, the student will be able to:

• analyze and solve simple statically indeterminate beam systems using the displacement method;

• interpret the dynamic response of structures subjected to external loads and seismic actions;

• understand and assess the stability conditions of structures;

• apply the basic concepts of plastic analysis to evaluate the load-bearing capacity of simple structural systems;

• critically interpret the results obtained and evaluate their implications in terms of structural safety and performance.

TEACHING METHODS

Lectures will be delivered at the board with the aid of projections and in-class exercises, aimed at providing practical examples and qualitative interpretation of results.

Working students and students with certified learning disabilities, disabilities, or other special educational needs should contact the instructor at the beginning of the course to agree on teaching and examination methods that, while respecting the course objectives, take into account individual learning needs.

SYLLABUS/CONTENT

Introduction, Course Program, and Examination Methods

Displacement Method, Discrete Formulation, and Introduction to Discretization Methods
Comparison between the force method and the displacement method for the analysis of statically indeterminate beam systems, including basic cases with imposed end displacements (rotations) for fixed-fixed and fixed-supported beams. Application examples and qualitative considerations on stiffness. Equilibrium equations and their matrix form. Structures with nodal displacements only: shear-type frames and Vierendeel beams. Application examples with qualitative interpretation of results. Generalized equilibrium equations, structures with nodal displacements and rotations. General analytical procedure (by induction). Discrete formulation of the displacement method: procedural steps, generic beam element (shear-inflexible), force-displacement relationships, local stiffness matrix for a straight, shear-inflexible beam element, local-to-global displacement relations, global equilibrium at nodes, stiffness matrix assembly, direct assembly method; boundary conditions (perfect and inelastic constraints); final step for obtaining the solution; calculation of equivalent nodal forces for distributed loads on beam elements; application example. Introduction to discretization methods: Ritz method, finite element method as discretization of a continuous domain with local approximation, operational advantages and analogies/differences with the discrete displacement method formulation.

Dynamic Analysis of Beam Systems
Derivation of equations of motion for planar beam systems (generalization of the discrete formulation) under active and seismic forces (example: generic planar beam system). Equations of motion for a generic forced planar shear-type frame. Free undamped vibrations of N degree-of-freedom (DOF) systems. Basic N-DOF example: two-DOF planar reinforced concrete shear-type frame. Modal vibration shapes of prismatic cantilevers, examples of modal analysis of real structures. Forced undamped vibrations of N-DOF systems. Structural damping ratio. Forced damped vibrations of N-DOF systems. Equation of motion for a forced damped single degree-of-freedom (SDOF) system. SDOF systems: free undamped vibrations; free damped vibrations, analytical solution for lightly damped systems, notes on logarithmic decrement; forced vibrations, seismic loading cases; notes on real examples modeled as SDOF systems; undisturbed forced vibrations, direct analytical approach (impulse force, impulse response function, convolution integral). Frequency domain analysis of SDOF systems: Fourier transform, frequency response function (FRF), dynamic amplification factor, structural filtering role; qualitative examples of response to harmonic loads, wind excitation, and seismic excitation. Seismic response spectra: definition, pseudo-spectra, equivalent static force, example. Seismic analysis of N-DOF systems: modal participation factor, participating mass, use of response spectra, notes on empirical modal combination rules, application example. Equivalent static forces due to seismic action on N-DOF systems.

Introduction to Stability Problems
Linear and linearized theory, planar mathematical pendulum example, pre-stress state and concept of geometric stiffness. Discrete examples with concentrated elastic stiffness, critical load and equilibrium bifurcation, limitations of linearized theory. Euler’s column, Euler critical load and corresponding buckling mode, effective buckling length under different boundary conditions, slenderness and Euler’s hyperbola, brief notes on practical stability curves, omega method, exercise. Imperfect elastic column, solution and discussion. Beam deflection equation with second-order effects, example. Subcritical compression of beams subjected to transverse loads (sinusoidal and generic transverse loading), amplification factor and bifurcation diagram; notes on local and global buckling of truss beams. Prestressed continuous systems, total potential energy, 1D continuous model (extensible beam), discussion on total potential energy, minimum potential energy theorem; specialization of total potential energy (second-order) for the extensible beam model. Finite element buckling analysis: polynomial finite element (planar beam), rewriting of potential energy in matrix form, definition and meaning of the geometric stiffness matrix, second-order force-displacement relationship, eigenvalue problem for load multipliers.

Introduction to Plastic Analysis
Concepts of load-bearing capacity, ductility, plastic collapse, assumptions in plastic analysis. Main behavioral characteristics of ductile materials, ideal elastic-perfectly plastic model. Example of plastic collapse (rods of different lengths connected by a rigid body), collapse mechanism. Elasto-plastic beams: pure bending, rectangular cross-section (double symmetry), moment-curvature relationship in the elasto-plastic phase, plastic moment capacity of rectangular sections. Sections with single-axis symmetry, example of T-section, moment-curvature diagram and capacity of common section shapes. Incremental plastic analysis of beam systems: concept of plastic hinge, examples of statically determinate systems, fixed-fixed beam. Notes on plastic limit analysis theorems, beam systems subjected to proportionally increasing concentrated loads, mechanism combination method, application example.

RECOMMENDED READING/BIBLIOGRAPHY

Course handouts available on Aulaweb

Steen Krenkr, Jan Høgsberg (2013). Statics and Mechanics of Structures. Springer

Angelo Luongo, Manuel Ferretti, Simona Di Nino (2022). Stabilità e biforcazione delle strutture - Sistemi statici e dinamici. Esculapio

Alberto Carpinteri (2023). Scienza delle Costruzioni 2. Esculapio

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/en/corsi/11969/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral test, which can be taken after completing a simple written exercise on the displacement method

ASSESSMENT METHODS

The exam consists of a simple written exercise on the displacement method, carried out in class, followed by an oral exam on the other topics covered in the course.
Passing the written exercise is a prerequisite for taking the oral exam.

FURTHER INFORMATION

Students are invited to contact the professor for any information not included in the teaching schedule.