Building dynamics

Learning objectives

The main objective of the course is to provide the student with the foundations for the knowledge of the procedures and general tools of construction dynamics, providing the theoretical principles and practical applications of the study of bodies with one degree of freedom, discrete systems with multi degrees of freedom and continuous systems . Concepts of random dynamics, soil dynamics and evaluation of vibrations on buildings will also be introduced. The teaching offer is completed by applied laboratory lessons where students will have to develop assigned design activities.

Course content

PART 1
1 DOF systems

• Load types: periodic, non-periodic, deterministic, random.
• Free vibrations of elementary oscillators (one degree of freedom systems). Undamped, Unforced Oscillator: Pulse Period and Natural Frequency, Maximum Amplitude, Phase Angle, Energy Approach and Application: Portal to 1 DOF.
• Damped, unforced oscillator: viscous damping, critical, subcritical, supercritical, energy approach and application: Portal to 1 DOF.
• Damped, forced oscillator: homogeneous and particular solution. Transfer function. Phases: transient, regime and decay. Pseudostatic response, dynamic amplification coefficient, delay time, resonance. Case of the Takoma Narrows Bridge. Inelastic collisions, perfectly elastic collisions, generic forcing: integral formulation and incremental formulation.
• Seismic analysis of elementary oscillators: response spectrum, design response spectrum (NTC 2008).
• Rotating machine on the ground, structure-base interaction.
• Numerical resolutions: method of central differences, Newmark's method, stability.

PART 2
Multi DOF discrete systems

• Discrete undamped and unforced systems. Matrix notation: matrix of masses. Eigenvalue and eigenvector problem and their physical meaning. Modal shapes.
• Discrete damped and unforced systems. Damping matrix: classic or non-diagonal.
• Forced discrete systems. Integral formulation. Multi DOF seismic analysis. Drag matrix, synchronous motion, non-synchronous motion.
• Modal analysis with response spectrum, combination rules: ABS, SRSS, CQC.
• Periodic motion, Fourier transform and properties. Transition from the time domain to the frequency domain. Duhamel integral.

PART 3
Continuous systems

• Equilibrium equations. Longitudinal bar, wave speed, method of separation of variables. Free end, bound end, mixed end and bar subject to beating.
• Waves in the continuum: equations of equilibrium, congruence and bond. Plane waves: P waves, S waves and Reyleigh waves. Comparison and physical meaning, geometric attenuation.
• Shear layer. Free oscillations, forced oscillations, damped oscillations. Kelvin model. Harmonic motion and layer amplification function. Undamped homogeneous ground, damped homogeneous ground on rigid rock and on elastic rock.
• Beam vibrations: supported and cantilevered.

PART 4
Applications

• Random dynamics (outline): random functions and processes, probability distribution function and probability density. Normal or Gaussian random variable. Weakly and strongly stationary processes. Power spectral density. Single DOF systems, multi DOF systems. Monte Carlo simulation.
• Dynamic soil-structure interaction (outline): kinematic interaction and inertial interaction, radiation damping. Direct approach, substructure approach. Impedance function. Period lengthening, radiant damping.
• Insulation at the base of structures (notes): linear theory and efficiency of insulation systems: increase in flexibility and energy dissipation. Types: elastomeric bearings and sliding bearings. Mathematical and mechanical models.
• Dynamics of vibrations on buildings.

PART 5
GENERAL INFO

• 1 DOF systems
• Multi-DoF systems
• Modal response spectrum analysis of an AC building
• Sizing of an insulation system 

Method of teaching
Lectures, design exercises, seminars and individual work.

Prerequisites

Building science 1 and 2, Rational mechanics, Building technology 1 and 2.

Bibliography

• EC8 CEN (2004): Eurocode 8 Design of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings. EN 1998-1, Brussels.
• NTC 2008 (2008): DM of the Ministry of Infrastructures and Transport of 14/01/2008. New Technical Standards for Buildings. OJ no. 29 of 04.02.2008, SO no. 30 (in Italian).

Assessment methods and criteria

The exam includes an oral test and verification of the exercises carried out during the course.