Semester 5 & 6

GM 13-5 Structural Mechanics 1: Elastic Solids and Theory of Beams

Module total : 92 hours (42 hrs Course/TD (*) + 24 hrs TP + 4 hrs evaluation + 22 hrs personal work) - 3 credits

Objectives

• Learn to describe and understand a constraint condition - deformation in a continuous environment. Be able to use deformation measurements by extensometry on the structures.
• Learn how to model a beam type structure and learn how to perform traditional sizing calculations in terms of resistance and deformation (engineer approach).
• Discuss the concepts of elastic energy from deformation of a structure (energy approach of structures) Address the concepts of instability of structures (buckling).

Pre-requisites and links to other modules
Mathematics :

- Matrix calculus: own values, own vectors, diagonalisation, change of basis.

- Integral calculation: differential equations, integrals.

Resistance of Materials :

- Geometric definition of a beam.

- Assumptions: on the material, on the efforts applied, on the deformation, on the small displacements, Saint-Venant.

- Concept of constraint.

- Phenomenon of concentration of constraints.

- Description of the traction test.

- Properties of the straight sections: centre of gravity, static moments, quadratic moments, main axes.

- Torsor of cohesion and layout of diagrams, equation of local balance of a beam.

- Simple stresses : traction-compression, pure shear, bending (calculation of the distorted elastic), twisting of beams with circular cross-section.

- Compound stresses (overlay principle).

- Order 1 hyperstatism.

Course / TD (42 hrs (*) + OS assessment (2 x 2 hrs = 4 hrs)

 (*) : 12 hrs of TO per 1/2 promotion (or even 1/4 of promotion) depending on the sites.

Linear elasticity (indicative duration : 14 hrs)

- Reminder of mathematics: matrix calculus and operators.

- Deformation study (tensor of deformations, Mohr's circles, flat deformation condition).

- Study of constraints (vector constraint, tensor, Mohr's circles, flat constraint condition, equation of local balance, Cauchy's reciprocity).

- Elastic linear behaviour (traction test, Hooke's law, potential elastic energy, generalised elastic area, permissible constraint, equivalent constraint, extensometry by gauges).

- A special elastic solid : beam.

Theory of beams (indicative duration : 28 hrs)

- Basic assumptions of the theory.

- Reminder: geometry of sections (centre of gravity, quadratic moments, etc.) and beam statics (Basic Principle of Static, principle of cutting, diagrams etc.).

- Tensor of constraints associated with the straight section of a beam.

- Relationship between the constraints and cohesion efforts.

- Study of simple stresses (tensors of constraints, deformations, fields of displacement and potential elastic energy).

- Compound stresses and criterion/criteria for sizing (concentration of constraints phenomenon, safety coefficients).

- Energy from structures.

- Study of structures composed of beams (gantries).

- Resolution of hyperstatic systems.

- Buckling.

- Specific cases of sizing (exercises applied) : Assemblies (fastenings) ; Shocks (taking account of the dynamic effects: for example lift cable) ; Thermal expansion (thermo-elastic 1 D).

6 sessions of TP (24 hrs)

- TP1 Characterisation of the mechanical characteristics of a metallic material : Young's modulus, Poisson ratio, constraint limits, etc. (traction and/or compression test, deformation of a beam by bending).

- TP2 Bending - twisting of a tube: RdS approach and elasticity (Mohr's circle, extensometry), overlay principle.

- TP3 Study of a structure in flat elasticity : RdS approach and elasticity (in cylindrical coordinates), use of the photoelasticimetry.

- TP4 Isostatic beam by bending : diagrams, measurement of the deformed, study of sections, deflected bending, Maxwell Betti's theorem of reciprocity.

- TP5 Energy of structures approach : isostatic and hyperstatic gantries and/or trellis structures.

- TP6 Curve beams or Trellis type structures.

GM 13-6 Structural Mechanics 2 : Methods for solving Statics and Dynamics problems

Module total : 92 hours (42 hrs Course/TD (*) + 24 hrs TP + 4 hrs evaluation + 22 hrs WP) - 3 credits

Taxonomic level : ③ (application and analysis)

Objectives

• Learn how to put a problem of linear elasticity into equations and learn how to implement a resolution method. Learn the basic principles of the method of finite elements applied to the matrix calculus of structures, the formulation and properties of the elements used.
• Be able to use computer calculation tools in Mechanical Engineering.
• Understand and learn to describe the concepts of frequencies and own modes, and depreciation.

Pre-requisites and links to other modules

- Dynamics of the non-deformable solid: matrix of inertia, Fundamental Principle of Dynamics.

- System with a degree of freedom: free and forced oscillations, various cases of depreciation (0<ξ<1, ξ=1, ξ>1).

- Module GM 13-5.

Course / TO (42 hrs (*)) + evaluation OS (2 x 2 = 4 hrs)

(*) 12 hrs of TD per 1/2 promotion (or even 1/4 of promotion) depending on the sites.

Calculation methods in elasticity (indicative duration : 12 hrs)

- Method of displacement (Navier's equations).

- Methods of constraints (Beltrami's equations).

- Special cases : flat deformations, flat constraints, Airy's function, concentration of constraints.

- Thermo-elasticity : Hooke's law equations – Duhamel.

Method of Finite Elements applied to Structural Mechanics (indicative duration : 14 hrs)

- General presentation.

- Continuous environments and discrete structures<: the concept of finite element, elementary and global, Principle of Virtual Works.

- Matrix writing of structural mechanics : equilibrium equations, generalised Hooke's law, relations between deformation and displacement, energy from deformation.

- Matrix of rigidity and matrix equilibrium equations of nodes F=Ku: elementary level, assembling of matrices, setting up on a system of springs.

- 1D elements (bar, beam).

- 2D elements (membranes, plates and shells).

- 3D elements.

- Modelling and mesh (practical advice) : General recommendations ; Fineness of the mesh ; Continuity ; Rigid body movements.

Introduction to Structure Dynamics (indicative duration : 16 hrs)

- System with a degree of freedom: reminders, bandwidth, transfer function, resolution of the differential equations of the movement, viscous and structural depreciations.

- Lagrange's equations.

- System with two degrees of freedom.

- System with n degrees of freedom : Putting into equations of movement (Basic Principle of Dynamics, Lagrange's equations) ; Putting into equations by the Method of Finite Elements ; Reminder : matrix of straightness.

Download semester 5 & 6 course program

Semester 9 & 10 (Master in Mechanics)

• Non linear analysis for metallic structures (course : 20h)
• Large displacements
• Plasticity
• Contact analysis
• Mechanical behaviour of polymers (course : 20h)
• Visco-elasticity, visco-plasticity, creep, relaxation (time-dependent deformations)