# NUMERICAL METHODS FOR STRUCTURAL ANALYSIS (UNIT 1) cod. 1004340

1° year of course - Second semester
Professor
Scienza delle costruzioni (ICAR/08)
Field
Ingegneria civile
Type of training activity
Characterising
48 hours
of face-to-face activities
6 credits
hub: -
course unit
in - - -

Integrated course unit module: NUMERICAL METHODS FOR STRUCTURAL ANALYSIS

## Learning objectives

The course is aimed to present concepts and tools for computational mechanics applied to generic solid structures

## Prerequisites

Students are recommended to follow previously the courses: Mathematical analysis, Geometry, Theoretical mechanics, Structural mechanics

## Course unit content

Basic concepts in computational mechanics.
Variational methods.
Residual methods.
Basic concepts of the finite element method
Isoparametric formulation
Structural discretisation with finite elements.
Use of finite elements in non linear problems
Some more aspects about the finite element method.

## Full programme

Basic concepts in computational mechanics.Introduction to the finite element method: displacement method for plane beam structures.Variational methods.Weak and strong form of a physical problem. Natural and essential boundary conditions. Variational principles. Virtual work theorem. Approximate polynomial solution. Bubnov-Galerkin method. General formulation of a problem by using finite elements: differential and integral forms.Minimum potential energy principle. Displacement field approximation. Rayleigh-Ritz method.Residual methods.Weighted residual method: subdomain method, collocation method, least square method, Galerkin method. The finite element method as a particular case of the Weighted residual method.Basic concepts of the finite element methodAlgebraic static and dynamic equilibrium equations of a structure discretized by finite elements. Stiffness matrix and nodal force vector . Stiffness matrix assembling. Treatment of boundary conditions and their classification: linear and non linear, single freedom constraints, multi freedoms constraints. Master-slave method, penalty method, Lagrange's multipliers method.Structural discretisation with finite elements.Choice of the finite element and of the shape functions. Shape functions in the local reference system and their derivatives. Examples of linear shape functions.
Some more aspects about the finite element methodFlow-chart of a simple program for finite element analysis.
Substructuring. Post-processing of the results. Accuracy of the solutions, reduced integration, hourglass modes, incompressible materials.Practical activities During the course, practical and theoretical exercises will be held with the aid of programs running on PCs to get the students confident with numerical techniques applied to the analysis of structures. Convergence tests and critical results assessment.

## Bibliography

Stuff provided by the teachers.
Cook, R.D., Malkus D.S., Plesha, M.E.: “Concept and application of finite element analysis”, 4th edition, John Wiley & Sons, 2002.
Zienkiewicz, O.C.: “The finite element method”, Mc Graw-Hill, 2000.
Corradi dell’Acqua, L.: "Meccanica delle strutture", Vol. 1,2 e 3, Mc Graw-Hill, 1995.
Hughes, T.J.R.: “The finite element method. linear static and dynamic finite element analysis”, Prentice Hall, 1987.
Owen, D.R.J., Hinton, E.: “Finite elements in plasticity”, Pineridge Press, 1980.
Bathe, K.J., “Finite element procedures”, Prentice Hall, 1996.

## Teaching methods

Theoretical readings and practical activities. Development of a project consisting in a FE program written by the students.

## Assessment methods and criteria

Oral examination concerning the subjects treated during the course and the developed project.

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