Structural analysis of frames and trusses by virtual works principle.
Introduction to F.E. method for statics and dynamics
Dynamics of SDOF and MDOF systems
Introduction to seismic analysis of SDOF and MDOF systems
R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Concepts and Applications of Finite Element Analysis, fourth ed., Wiley, 2002
K.-J. Bathe, Finite Elements Procedures, Upper Saddle River, NJ: Prentice-Hall, 1996
O.C. Zienkiewicz, R.L. Taylor, The finite element method for solid and structural mechanics, Oxford: Elsevier Butterworth Heinemann, 2005
L. Facchini, Elementi di dinamica delle strutture, soc. ed. Esculapio (BO)
R. W. Clough, J. Penzien, Dynamics of Structures, 3rd Edition - Computers & Structures, Inc, 2003.
A. K. Chopra, Dynamics of Structures, Earthquake Engineering Research Institute, 1980.
A. K. Chopra, Dynamics of Structures - Theory and Applications to Earthquake Engineering, Prentice-Hall, 1995.
C. Gavarini, Dinamica delle strutture, ESA, 1978.
Provide knowledge to properly analyze a strucural system and specifically:
- build a numerical model of the structure
- analyze the model under both static and dynamic actions an particularly under earthquake
- critically analyze the obtained results
Calculus: derivatives, integrals over multi-dimensional domains
Linear algebra, matrix and vector calculus, spectral theorem
Material point and rigid body dynamics
Strength of materials, energetic theorems, De St. Venant theory
Classroom lectures and exercises.
Autonomous and individual computer exercises
Type of Assessment
Final test can be given only after the completion of the analysis of a structural system carried out by means of a FEM code (SAP2000, ANSYS, Code Aster, Straus or other, to be agreed with the profesor).
Final test will consist of:
1 - a written exercise
2 - oral question
3 - discussion of the analysis of the structural system
1 - Introduction
Principle of virtual work for continua with virtual forces or virtual displacements
Computation of work done by concentrated and/or distributed loads
Principle of virtual work for a plane beam and for a spatial beam.
Usage of PVW to find equilibrium conditions for a Timoshenko beam and plane frames.
2 - Introduction to F.E.M.
Truss and beam elements. Degrees of freedom, shape functions, stiffness matrix, nodal loads. Topological matrix, assembly of stiffness matrix, restraint conditions, determination of reactions. Stress recovery. Intermediate degrees of freedom and stiffness matrix. Springs. Thermal deformations. Distributed loads. LAgrange polynomials.
Releases. Thermal loads.
3 - Structural dynamics
Equations of dynamic equilibrium. Lumped and consistent mass matrices.
3a - SDOF systems
Free and forced oscillations, with or without damping. Circular frequency, frequenct, natural period. Static deflection and dynamic amplification. Phase angle. Damped frequency.
Multi-harmonic forces, perodic forces, Fuorier analysis. Proprtires, Riemann-Lebesgue lemma, dynamic admittance function, Dirac's delta function, unitary impulse response. Effect of ground motion. Accelerometers and vibrometers.
3.b MDOF systems
Euation of motion, modal shapes, modalmatrix, Rayleigh ratio. Static condensation. Method of Stodola-Vianello. Modelling of structural damping, Caughey-O'Kelly condition. Caughey, Rayleigh and modal damping models.
Forced oscillations, modal loads, modal reduction.
Dynamic testing: vibrodine, band widthmethod, logarithmic decrement method.
Fourier analysis of MDOF systems, transfer matrix.
4 - fundamentals of seismic analysis
Modelling of earthquakes over limited extensions; equation of motion of a multi-storey structure. modal participation factors, participating masses. Principal directions of a building.
Elastic and inelastic design response spectra.
Static and dynamic linear analysis. CQC and SRSS combination rules.
Introduction to nonlinear systems. Duffing and EPP oscillators. Newmatk-Hall method and equivalent linearization.
Ritz and Lanczos vectors.
Tuned mass dampers.