# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2018/2019

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# Undergraduate Course: Real Structural Behaviour and its Analysis 5 (CIVE11002)

 School School of Engineering College College of Science and Engineering Credit level (Normal year taken) SCQF Level 11 (Year 5 Undergraduate) Availability Available to all students SCQF Credits 10 ECTS Credits 5 Summary This course develops the student's comprehension of the nonlinear behaviour of structures. The concepts of geometrical and material nonlinearity are introduced and followed by numerical methods employed for modelling nonlinearities through the medium of finite element analysis. These advanced topics give the student the ability to analyse realistic systems with confidence. The student will develop and understand of many aspects of structural behaviour and its modelling. The course prepares the student well for a career in computational modelling in civil or structural engineering. Course description L1 Introduction Structure and aims of the course. Subject in the context of theoretical and applied mechanics and structural engineering practice. The limitations of linear analysis and associated assumptions of small displacement and unchanged geometry. The need for going beyond linear analysis. The concept of equilibrium path and critical points along the path with appropriate examples. L2 Sources of nonlinearity and types of problems How do nonlinearities arise and what types of problems in structural engineering they produce. How can these problems be dealt with mathematically. L3 Analysis of nonlinear problems I Nonlinear analysis using the stiffness method and the finite element method. Formulation of a non-linear truss element with a geometric stiffness. Applicaton to examples of linear bifurcation analysis (LBA) to solve elastic critical load problems. L4 Analysis of nonlinear problems II Geometrically nonlinear analysis (GNA) of simple problems using the truss element with load increments and Newton iterations. L5 Analysis of nonlinear problems III Beam-column elements with combined bending and axial force, geometric stiffness matrix. Solution of simple LBA and GNA type problems. L6-9 Fundamentals of continuum mechanics Eulerian and Lagrangian frames of reference, Green and Almansi strain measures and corresponding (Piola-Kirchoff) stress measures, deformation gradient, total Lagrangian, updated Lagrangian and corotational approaches to GNA. L10 Introduction to material nonlinearity; linear elasticity; nonlinear elasticity; viscoelasticity; elastoplasticity; elasto-viscoplasticity. L11 1D elastoplasticity 1 Concepts of hardening, softening and perfect plasticity; load and displacement control; uniaxial behaviour of different materials _\$ú steel, aluminium, concrete, Gray cast iron, rubber. L12 1D elastoplasticity 2 Solution nonlinear problems; issues associated with satisfying equilibrium and constitutive law; example problems; nonlinear solution in the context of FE analysis. L13 Numerical solution approaches Concept of tangent stiffness; incremental methods; incremental-iterative methods; Newton Raphson method; modified Newton Raphson method; convergence criterial. L14 Multiaxial stress Nonlinear models for multiaxial states; principal sresses and stress invariants; convenient form of invariants for plasticity; recap of linear rlstic stress-starin relations. L15 Yield criteria Concept of yielding in a multiaxial stress state; Rankine, von Mises, Tresca, Mohr Coulomb and Drucker Prager yield criteria; representation in principal stress space; hydrostatic axis and deviatoric plane; deviatoric plane and plane stress representations; expressing criteria in principal stress and stress invariant forms. L16 Multiaxial plasticity 1 Hardening, softening and perfect plasticity; Bauschinger effect; decomposition of strain; incremental stress-strain relations; flow rule; consistency condition; tangential modulus matrix. L17 Multiaxial plasticity 2 Elastic predictor _\$ú plastic corrector concept; numerical evaluation of the flow vector; evaluation of flow vector terms for Rankine, von Mises,Tresca, Mohr Coulomb and Drucker Prager yield criteria; issues associated with singular regions; evaluation of hardening parameters. L18 Revision
 Pre-requisites Students MUST have passed: Computer Methods in Structural Engineering 3 (CIVE09027) OR Finite Element Methods for Solids and Structures 4 (CIVE10022) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites None High Demand Course? Yes
 Not being delivered
 On completion of this course, the student will be able to: describe material and geometric nonlinearities through terms such as equilibrium path, limit load, collapse, bifurcation, and snap-through buckling etc.show an understanding of large displacement behaviour including the need for more precise measures of stress and strain and associated analysis methodsuse nonlinear finite element analysis to manually solve simple problems with geometrically nonlinear behaviour including stability and bifurcationdistinguish between the roles of eigenvalue and non-linear analysis of geometrically nonlinear structural systemssolve simple 1D plasticity problems through hand calculations
 McGuire, Gallagher, Ziemian (2000) "Matrix Structural Analysis, 2nd Edition". Wiley, London, UK.
 Graduate Attributes and Skills Not entered Keywords Nonlinear structural analysis,geometric nonlinearity,material nonlinearity,large displacement ana
 Course organiser Dr David Rush Tel: (0131 6)50 6023 Email: D.Rush@ed.ac.uk Course secretary Mr Craig Hovell Tel: (0131 6)51 7080 Email: c.hovell@ed.ac.uk
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