Course detail

# Elasticity and Plasticity

Basic equations of theory of elasticity, stress and strain analysis in point, two-dimensional problems – plane stress and plane strain, axisymmetric problems, duality solution of the problem, energy theorems, variational methods, theory of thick and thin plates, theory of shells, static solution of foundation structures, models of soil, basics of elastic-plastic analysis, physical equations for elastic-plastic material with hardening, analysis of elastic-plastic state structures, the plastic limit of load carrying capacity.

Department

Institute of Structural Mechanics (STM)

Learning outcomes of the course unit

After completing the course, students will know the basic equation describing the linear elasticity behavior of the body. They know the solution of problems of elasticity. They will be able to use the principle of virtual work for solving simple problems. Understand the use of Ritz and Galerkin methods. Students can model the structure as a 2-D elasticity problem (plane stress, deformation) and knows the theory of plates. Marginally is informed about the theory of shells. Students will learn the basics elastic-plastic analysis and is able to determine the plastic limit load carrying capacity of beam and slab structures.

Prerequisites

Diagrams of internal forces on a beam, the meaning of the quantities: stress, strain and displacement, Hook’s law, equilibrium conditions for a beam, physical and geometrical equations for a beam. Stress states of beam and combinations thereof. Statically indeterminate beams systems and force and displacement methods of solution. The matrix notation of solutions.

Co-requisites

Not applicable.

Recommended optional programme components

Extended theoretical knowledge of nonlinear mechanics including its applications in analysis of structures should be gained on a voluntary specialized seminar.

Not applicable.

Planned learning activities and teaching methods

During lectures, standard model of theory explanation using the projector and blackboard is used. In the training course, students themselves solve tasks on a paper.

Assesment methods and criteria linked to learning outcomes

Conditions to get credit are active presence in training course (two absences are allowed) and successful evaluated of two tests. Tests are consists of both theoretical questions and practical tasks.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. A brief historical reference of the theory of elasticity. Fields in the theory of the continua and the definition of state variables.
2. Basic equations of elasticity. The derivation of geometric equations and physics equations. The properties of the strain and stress tensors. The equilibrium conditions and compatibility conditions.
3. Analysis of stress and strain in point. Plane stress and plane strain. Levy condition. Airy‘s stress function. Procedure for solving plane stress.
4. Axisymmetric problems - basic equations of plane problem in polar coordinates. Rheological models of material.
5. The deformation of non-force effects. Display of stress (Becker-Westergard, Mohr).
6. The potential energy of deformation and strain of work. Energy principles. The principle of virtual work and variational methods in continuum mechanics.
7. Theory of plates. Types of plates, boundary conditions. Special types of plates.
8. Analytical solution of plates in a rectangular coordinate system. Approximate solution of plates.
9. Introduction to the theory of shells. Membrane and bending state of stress. Internal forces with shells.
10. Cylindrical shells - basic equations of the bending theory of cylindrical shells. Flat shell.
11. Static solution of foundations. Models of soil.
12. Basics elastic-plastic analysis. The physical equations for elastic-plastic material with hardening.
13. Analysis of elastic-plastic state. The limit state plastic bearing capacity of beam structures.

Aims

During the course the student will obtain knowledge about basic quantities and relations of theory of elasticity for solid, beam, plane and plate structures. He will be skilled in the basic laws of mechanics - the principle of the virtual work and the principle of minimum of potential energy - and variational methods - Ritz method and finite elements method. After finishing the course he will be able to apply these methods on mentioned types of structures, to derive finite elements and to use computational programs based on finite elements methods in practise.

Specification of controlled education, way of implementation and compensation for absences

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Classification of course in study plans

• Programme N-P-C-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory
branch S , 1. year of study, winter semester, 5 credits, compulsory
branch S , 1. year of study, winter semester, 5 credits, compulsory

• Programme N-K-C-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory

• Programme N-P-E-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory

• Programme N-K-C-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory

• Programme N-P-E-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory
branch S , 1. year of study, winter semester, 5 credits, compulsory

• Programme N-K-C-SI (N) Master's

branch S , 1. year of study, winter semester, 5 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. A brief historical reference of the theory of elasticity. Fields in the theory of the continua and the definition of state variables.
2. Basic equations of elasticity. The derivation of geometric equations and physics equations. The properties of the strain and stress tensors. The equilibrium conditions and compatibility conditions.
3. Analysis of stress and strain in point. Plane stress and plane strain. Levy condition. Airy‘s stress function. Procedure for solving plane stress.
4. Axisymmetric problems - basic equations of plane problem in polar coordinates. Rheological models of material.
5. The deformation of non-force effects. Display of stress (Becker-Westergard, Mohr).
6. The potential energy of deformation and strain of work. Energy principles. The principle of virtual work and variational methods in continuum mechanics.
7. Theory of plates. Types of plates, boundary conditions. Special types of plates.
8. Analytical solution of plates in a rectangular coordinate system. Approximate solution of plates.
9. Introduction to the theory of shells. Membrane and bending state of stress. Internal forces with shells.
10. Cylindrical shells - basic equations of the bending theory of cylindrical shells. Flat shell.
11. Static solution of foundations. Models of soil.
12. Basics elastic-plastic analysis. The physical equations for elastic-plastic material with hardening.
13. Analysis of elastic-plastic state. The limit state plastic bearing capacity of beam structures.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. The calculation of stress and strain using equations of elasticity - the relationship between stress and strain.
2. The principal stresses (stress invariants), the calculation for different cases of stress.
3. Strength and plasticity criteria - calculation of equivalent stress by various theories.
4. The graphical representation of stress. The Mohr’s method.
5. Determining the work of external forces. Application of Lagrange and Castigliano's theorem. Calculation the strain energy.
6. Analytical solutions of wall – Airy stress function.
7. Principle of virtual work. Practical use of Castigliano‘s method.
8. Approximations of the line deflection of the beam by Ritz's method.
9. Application of the Galerkin method for solving simple problems of elasticity
10. Classical solutions of plates - method of an infinite series.
11. Calculation graph of internal forces of a cylindrical shell.
12. Determination a limit plastic resistance of the beam and plate.
13. Analysis of the formation of plastic hinges on a simple frame structure.

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