| Course Unit Code | 228-0235/01 |
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| Number of ECTS Credits Allocated | 5 ECTS credits |
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| Type of Course Unit * | Compulsory |
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| Level of Course Unit * | First Cycle |
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| Year of Study * | Third Year |
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| Semester when the Course Unit is delivered | Winter Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| KRE13 | prof. Ing. Martin Krejsa, Ph.D. |
| LEH061 | Ing. Petr Lehner, Ph.D. |
| Summary |
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| The subject of Elasticity and Plasticity II focuses on the basic and advanced tasks of mathematical theory of elasticity in the field of rod and surface supporting structures. Detailed information about basic variables and equations of elasticity theory, the most common types of simplified problems of planar problem solving in bearing walls, slabs and shells, and selected methods of solving them. Part of the subject is also an introduction to analytical and numerical solution of basic models of the subsoil, boundary plastic bearing capacity of simple statically determined and indetermined structures, numerical solution of stability tasks in slender columns under compression and introduction to energy variation methods. |
| Learning Outcomes of the Course Unit |
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| Understanding of basic quantities and equation of mathematical theory of elasticity. Ability to choose right calculation model for the problem. Ability to choose adequate method of solution. |
| Course Contents |
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Lectures:
1. Basic equations of elastic theory: Basic assumptions of the theory of elasticity, stress, deformation, stress analysis in the surroundings of the body point, components of tension tensor and their transformation, principal stress, differential equations of equilibrium, geometric equations, equations of compatibility, physical equations, basic system of equations of elasticity theory, types of boundary conditions.
2. Planar problem, wall equation: Plane state of strain, plane deformation state, solution of supporting walls, derivation of wall equation, solution of supporting walls by inverse method, planar problem in polar coordinates.
3. Solution of rectangular support walls by the Finite difference method: Differential method, important differential relations, meshing, wall equation with differential relations, boundary conditions, substitution frame analogy, calculation of Airy stress function and stress components in rectangular support wall.
4. Surface structures, support plates: Subdivision of support plates, assumptions for supporting plates, thin plate theory, stress components and specific internal forces, principal moments, plate equation, boundary conditions, methods of support plates solving, thick plates, Mindlin theory.
5. Solution of rectangular boards by the Finite difference method: Important differential relations, meshing, calculation of specific internal forces components, boundary conditions, description of the calculation method by the Finite difference method.
6. Circular and inter-circular plates (rotationally symmetrical plates) I.: Basic relations for radially symmetrical circular plates, geometric and physical conditions, specific internal forces, equilibrium conditions, plate equation, partial solution of plate equation, boundary conditions, examples of calculation.
7. Circular and inter-circular plates (rotationally symmetrical plates) II.: Basic relations for radially symmetrical inter-circular plates, internal support plates with varying thickness, boundary conditions, examples of calculation.
8. Shell structures: Membrane state of rotationally symmetric shells, equilibrium conditions, application of membrane state, examples of calculation of rotationally symmetrical shell structures, bending theory.
9. Subsoil models: Solid support (foundation foot) on flexible substrate, beam interaction with subsoil, Winkler and Pasternak subsoil model, half space, Boussinesq equations, analytical and numerical solution, calculation examples.
10. Non-linear behavior of materials, plasticity conditions, limit plastic resistance: Ideally elastic-plastic and rigid material, plasticity conditions for materials with the same and different tensile and compressive strengths, limit plastic resistance - static and kinematic solution.
11. Stability of rod structures: Using the principle of virtual work for solving rod stability.
12. Introduction to energy variation methods: Basic concepts and principles of variation in building mechanics. Detailed explanation of the deformation (Lagrange) variation principle, simple examples of use.
13. Introduction to Energy Variant Methods: Detailed explanation of the force (Castiglian) variation principle, simple examples of use.
14. Sample examples of selected tasks.
Tutorials:
1. Introduction to plane stress: stress tensor supporting walls and their transformation, principal stress.
2. Plane problem, wall equation: Solution of carrier walls using the Airy stress function and inverse method.
3. Solution of rectangular supporting walls by the Finite difference method I.
4. Solution of rectangular supporting walls by the Finite difference method II.
5. Solution of rectangular carrier plates by the Finite difference method I.
6. Solution of rectangular carrier plates by the Finite difference method II.
7. Circular and Inter-Circular Plates (rotationally symmetrical plates) I.: Analytical solution of rotationally symmetrical circular plates (with outer edge).
8. Circular and Inter-Circular Plates (rotationally symmetrical plates) II.: Solution of rotationally symmetrical inter-circular plates (with outer and inner edges).
9. Shell structures: Solutions of selected rotationally symmetrical shell structures, membrane state.
10. Subsoil models: Numerical solution of selected foundation structure on flexible substrate.
11. Nonlinear behavior of materials, plasticity conditions, limit plastic resistance: Limit plastic resistance of selected rod structures.
12. Stability of rod structures: Numerical solution of the selected case of a slender column under compression.
13. Introduction to Energy Variant Methods: Static solution of simple rod construction by Ritz method.
14. Presentation of semestral work. |
| Recommended or Required Reading |
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| Required Reading: |
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1. Russell C. Hibbeler, Mechanics of Materials (10th Edition), 896 pages, 2016, ISBN-13: 978-0134319650, ISBN-10: 0134319656.
2. James M. Gere, Stephen P. Timoshenko, Mechanics of Materials (4th Edition), 912 pages, 1996, ISBN-13: 978-0534934293, ISBN-10: 0534934293.
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1. Teplý, Šmířák: Pružnost a plasticita II. VUT Brno, ISBN 80-214-0498-1, 1993. dotisk CERN, 2000.
2. Brožovský, Materna: Základy matematické teorie pružnosti. VŠB-TU Ostrava, 2012.
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| Recommended Reading: |
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1. Russell C. Hibbeler, Engineering Mechanics: Statics (13th Edition), 672 pages, 2012, ISBN-13: 978-0132915540 , ISBN-10: 0132915545.
2. Boresi A. P., Schmidt, R. J.: Advanced Mechanics of Materials,John Wiley and Sons, Chichester, USA 2003 |
1. Bittnar, Šejnoha: Numerické metody mechaniky 1. ČVUT Praha, ISBN 80-01-00855-X, 1992.
2. Bittnar, Šejnoha: Numerické metody mechaniky 2. ČVUT Praha, ISBN 80-01-00901-7, 1992.
3. Dický, Mistríková, Sumec: Pružnosť a plasticita v stavebníctve 2. STU v Bratislavě, ISBN 80-227-2515-3, 2006.
4. Russell C. Hibbeler, Structural Analysis (10th Edition), 736 pages, 2017, ISBN-13: 978-0134610672, ISBN-10: 0134610679. |
| Planned learning activities and teaching methods |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 35 | 18 |
| Examination | Examination | 65 | 33 |