# MECHANICS OF STRUCTURES cod. 1006595

1° year of course - First semester
Professor
- Andrea SPAGNOLI
Scienza delle costruzioni (ICAR/08)
Field
Ingegneria civile
Type of training activity
Characterising
72 hours
of face-to-face activities
9 credits
hub: PARMA
course unit
in ITALIAN

## Learning objectives

Knowledge and understanding:
At the end of the course, the student will acquire the theoretical basis and the application tools of advanced structural analysis in both the linear and non-linear regimes, with particular reference to structural theories of one-dimensional elements (cables, trusses and beams) and two-dimensional (plates and shells). With regard to the non-linear regime, the basic concepts for the calculation of both plastic design and buckling design of one- and two-dimensional civil engineering structures will be provided.

Applying knowledge and understanding:
At the end of the course, the student will acquire the ability to understand the mechanical behavior of the linear elastic two-dimensional structures. He/she will also acquire the methodologies for calculating the load bearing capacity of a structure, related to the attainment of the material resistance (plastic collapse) or the loss of structure stiffness (buckling collapse).

Communication skills:
At the end of the course, the student should have acquired a command of the technical language to allow a proper and effective presentation of the results.

## Prerequisites

It is useful to have familiarity with the basic features of Microsoft Excel e Matlab.

## Course unit content

The course is organized in four modules.

Module 1: Plates
Statement of the problem
Kinematic hypotheses of Kirchhoff
Components of displacements, strains, stresses; internal reaction characteristics
Differential equation of the elastic surface (or differential equation of Germain-Lagrange)
Boundary conditions
Principal moments
Approximate solutions through finite difference method
Simply supported rectangular plates: Navier solution and Lévy solution

Module 2: Shells
Characteristics of surfaces
Shells of devolution
Membrane regime for axisymmetric shells (reservoir, pressure vessels, domes, etc.): internal reactions and displacements
Bending regime in cylindrical shells

Module 3: Plastic limit analysis
Material non-linear behaviour: plasticity (yielding function, isotropic and kinematic hardening, associative and non-associative flow rule). Yielding criteria for structural materials.
Perfectly plastic behaviour and plastic collapse. Plastic collapse of beams under bending: plastic hinge, limit moment for symmetric and non-symmetric cross-sections, combined actions and limit curves.
Incremental analysis of elastic-plastic frames. Collapse mechanisms. Theorems of limit analysis (static and kimenatic theorems).
Lattice structures and frames under proportional point loads (method of the combination of mechanisms) and under distributed loads. Frames under non-proportional loads.
Limit loads of plates (yield line theory and strip method).

Module 4: Stability of equilibrium
Discrete elastic systems: stationarity and minimum of total potential energy, theory of the second order, critical Euler load (static and energetic criteria).
Flexural stability of axially compressed beams: fundamental cases, frames, beams with curved axis.
Equilibrium of cables as a particolar case of the differential equations of beam-columns.
Torsional stability of beams under axial compressive load or bending.
Stability of plates.
Determination of the critical load: Rayleigh-Ritz method, finite element method. Post-critical behaviour.
Beams under axial compressive load and bending: effects of material non-linearity and imperfections on load-carrying capacity, stability curves.
Snap-through instability of shallow arches.

## Bibliography

Recommended textbooks:
O.Belluzzi “Scienza delle Costruzioni” Vol.I e Vol.III, Zanichelli, Bologna.
A. Carpinteri "Analisi non-lineare delle strutture”, Ed. Pitagora, Bologna, 1998.
L. Corradi dell’Acqua “Meccanica delle strutture” Vol.II e Vol.III, McGraw-Hill, Milano.

L. Corradi dell’Acqua “Instabilità delle strutture”, CLUP, Milano, 1978.
M. Jirasek, Z.P. Bazant “Inelastic analysis of structures”, J.Wiley & Sons, New York, 2001.
S.P. Timoshenko, K.S. Woinowsky “Theory of plates and shells”, McGraw-Hill, New York, 1987.
S.P. Timoshenko, J.M. Gere “Theory of elastic stability”, McGraw-Hill, New York, 1985.

Teaching material available via Elly platform

## Teaching methods

The course consists of theoretical lessons and practical exercises. For each topic, exercises are planned so that the student can deal with the resolution of the problems previously formulated in theoretical form.
The theoretical lessons and practical exercises are carried out on an electronic whiteboard together with slides, which are available on Elly platform.
Lessons will be delivered face-to-face in classrooms, but they will also be made accessible remotely in interactive mode (synchronous), via MS Teams platform.
Notes on electronic whiteboard produced by the teacher during lessons will be made available on Elly platform.

## Assessment methods and criteria

The assessment of student learning is formulated on the basis of a final written exam followed by a brief oral discussion.
The final exam is weighted as follows:
- 70% application of the theory to exercises (applying knowledge and understanding);
- 20% theoretical questions (knowledge and understanding);
- 10% clarity of presentation (communication skills).
Exam modalities could change, due to the SARS-CoV-2 emergency. Variations will be communicated in due time.

## Other information

It is strongly recommended to attend lessons.