Course Details

# Structural Mechanics 1 (EVB)

BDA013 course is part of 1 study plan

BPC-EVB Winter Semester 1st year

The subject is focused on the interpretation of basic concepts of structural mechanics and elasticity of beam structures. Students will be able to solve reactions and internal forces of the plane statically determinate structures, of plane beams with straight and broken axis, to solve three-hinged broken beam with and without a bar, the planar composed beam systems and plane truss systems, to determine the position of centroid and the second order moments of cross-section. Basic principles, concepts and assumptions of the theory of elasticity and plasticity. Deflections. Strains. Stresses. Saint-Venant&apos;s principle of local action. Linear theory of the elasticity. Material laws, Stress-strain diagram. Analysis of a straight bar – basic assumptions. The interaction between the internal force components and the stress components, and between the internal force components and the external load. Basic cases of the loads of a bar. Simple tension and compression – the stress, the strain, the deflection. The influence of the temperature and the initial stresses. Simple shearing load. Simple bending load – calculation of the normal (axial) stresses. Design of the bent beams. The deflection of the bent beams. Differential equation of the deformation line. Method of initial parameters a Mohr’s method. Calculation of the tangent stresses – massive and thin-walled cross-sections. The consideration of the shear stress in the bent beam. The centre of the shear. Pure torsion and warping torsion. Free warping – massive circular and non-circular cross-section. Thin-walled closed and opened cross-section. Composed load cases of the bar. Spatial and biaxial bending. Tension (Compression) and bending in a plane. Eccentric torsion and compression. The core of the section. Design of the beams in the case of the composed (complex) load. The stability and the bucking strength of the compressed bars. Euler’s solution. The critical force and the critical stress. The strength approach to stability. A bar loaded by a bending and buckling load. The check of the buckling bars. The theory of the material strength and failure. The stress and strain state in a point of the body. The principal stress at the plane stress problem.

Course Guarantor

doc. Ing. Petr Frantík, Ph.D.

Institute

Institute of Structural Mechanics

Syllabus

1. Tasks of structural mechanics. Basic concepts, assumptions, principles and axioms of statics. Planar systems of forces, resultant, equivalence, equilibrium. Moment of force to point, couple.
2. Static models of planar structures, bonds and support in plane, load. Calculation of reactions. Components of the resultant internal forces of a straight beam. Differential relations of load, shear forces and bending moments. Straight planar statically indeterminate beams. Diagrams of internal forces.
3. The planar arched beam. The planar diagonal beam, the continuous load of the diagonal beam and its decomposition. Diagrams of internal forces.
4. Planar compound beam systems, indeterminate static and kinematic. Three-hinged angular beam without a tie bar and with tie bar, Gerber&apos;s beam. Reactions and diagrams of internal forces.
5. Plane truss structures, indeterminate static and kinematic. Calculation of axial forces in bars by general and simplified joint method, cross-sectional method and its Ritter adjustment.
6. Space systems of forces. Bounds and reactions of the solid body in space. Space-loaded straight beam. Diagrams of internal forces. Space-angular beam.
7. Surface, static moment, centre of gravity. Quadratic and deviating moments. Steiner&apos;s theorem. Principal axes of the cross section, principal quadratic moments. Mohr&apos;s circle. Radius of inertia, ellipse of inertia, polar quadratic moments.
8. Basic assumptions and concepts of the theory of elasticity, displacement, deformation, stress. Linear theory of elasticity, physical equation. Relation of components of internal forces and components of stresses. Simple tension and compression – stress and strain. Plastic response of materials, working diagram of elastic-plastic materials.
9. More general cases of tensile and compression. Statically indeterminate cases. Effect of initial stresses and temperature fields. Simple bend. Compound cases of beam loading. Angled and spatial bend. Eccentric tension and compression. Neutral axis position, core of cross section. Dimensioning beams stressed by compound loading.
10. Normal and shear stresses behind the bend. Solid and thin-walled cross-sections. Effect of shear on beam deformation. Simple shear and joints stressed to cut. The shear centre.
11. Free and bounded torsion. Bimoment. Normal and shear stresses. Free torsion of massive cross-sections, membrane analogy. Torsion of thin-walled open and closed cross-sections.
12. Deformation of bent beams. Differential equations of the bending function and its integration. Mohr&apos;s method.
13. Stability and buckling strength of beams in compression. Second order theory. Critical load. Euler&apos;s solution. Spatial stress and deformation at the point of the body. Principal stresses at plane stress problem. Extreme shear stresses. Mohr&apos;s circle of stresses.

Prerequisites

Knowledge of general physics, power, Newton&apos;s laws, trigonometric functions, vectors, differential and integral calculus.

Language of instruction

Czech

Credits

5 credits

Semester

winter

Forms and criteria of assessment

course-unit credit and examination

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

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

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/254404

Lecture

13 weeks, 2 hours/week, elective

Syllabus

1. Tasks of structural mechanics. Basic concepts, assumptions, principles and axioms of statics. Planar systems of forces, resultant, equivalence, equilibrium. Moment of force to point, couple.
2. Static models of planar structures, bonds and support in plane, load. Calculation of reactions. Components of the resultant internal forces of a straight beam. Differential relations of load, shear forces and bending moments. Straight planar statically indeterminate beams. Diagrams of internal forces.
3. The planar arched beam. The planar diagonal beam, the continuous load of the diagonal beam and its decomposition. Diagrams of internal forces.
4. Planar compound beam systems, indeterminate static and kinematic. Three-hinged angular beam without a tie bar and with tie bar, Gerber's beam. Reactions and diagrams of internal forces.
5. Plane truss structures, indeterminate static and kinematic. Calculation of axial forces in bars by general and simplified joint method, cross-sectional method and its Ritter adjustment.
6. Space systems of forces. Bounds and reactions of the solid body in space. Space-loaded straight beam. Diagrams of internal forces. Space-angular beam.
7. Surface, static moment, centre of gravity. Quadratic and deviating moments. Steiner's theorem. Principal axes of the cross section, principal quadratic moments. Mohr's circle. Radius of inertia, ellipse of inertia, polar quadratic moments.
8. Basic assumptions and concepts of the theory of elasticity, displacement, deformation, stress. Linear theory of elasticity, physical equation. Relation of components of internal forces and components of stresses. Simple tension and compression – stress and strain. Plastic response of materials, working diagram of elastic-plastic materials.
9. More general cases of tensile and compression. Statically indeterminate cases. Effect of initial stresses and temperature fields. Simple bend. Compound cases of beam loading. Angled and spatial bend. Eccentric tension and compression. Neutral axis position, core of cross section. Dimensioning beams stressed by compound loading.
10. Normal and shear stresses behind the bend. Solid and thin-walled cross-sections. Effect of shear on beam deformation. Simple shear and joints stressed to cut. The shear centre.
11. Free and bounded torsion. Bimoment. Normal and shear stresses. Free torsion of massive cross-sections, membrane analogy. Torsion of thin-walled open and closed cross-sections.
12. Deformation of bent beams. Differential equations of the bending function and its integration. Mohr's method.
13. Stability and buckling strength of beams in compression. Second order theory. Critical load. Euler's solution. Spatial stress and deformation at the point of the body. Principal stresses at plane stress problem. Extreme shear stresses. Mohr's circle of stresses.

Exercise

13 weeks, 3 hours/week, compulsory

Syllabus

1. Moment of force to a point, pair of forces. Concurrent system of forces in plane, general system of forces in plane.
2. Beam supports and types of loads. Calculation of support reactions. Internal forces diagrams of plane beams. Solution of basic types of beams: supported beams and cantilevers, straight beams with overhangs.
3. Supports reactions and internal forces diagrams of the beams with broken and curved axis. Decomposition of slant continuous loads. Support reactions and internal forces diagrams of the slant beam.
4. Three-hinged broken beam (with and without a bar) and plane arches. Beam with internal hinges - Gerber’s girder.
5. Planar trusses (hinged bar systems). Calculation of axial forces of trusses by method of sections, Ritter's solution.
6. Space systems of forces. General space system of forces. Constraints and reactions of rigid body in space. Straight bar with space loading, space cantilever beam with rectangular broken centre line, reactions and diagrams of internal forces and moments. Space beam with broken centre line, reactions and diagrams of internal forces and moments.
7. Centroid of planar cross-sections. Second order moments of planar cross-section, Steiner’s theorem. Mohr’s circle.
8. Simple tension – stress and strain state. More general cases of the tension (compression).
9. Statically indeterminate cases. The influence of the initial stress and the temperature field. Simple bending. Normal stress produced by bending. Design and check of bent girders. Spatial and biaxial bending.
10. Complex cases of the load of the beam. Eccentric tension and compression. The calculation of the position of the neutral axis, the core of the section. Design of the girders in a case of the complex load. Shearing stress in a bent beam. The centre of the shear. Shearing stress in the thin-walled girders.
11. Free warping of a massive and thin-walled (opened and closed) cross-section beams.
12. The differential equation of the deformation line. The integration of the thrust line diff. equation. The method of initial parameters, Mohr’s method.
13. Buckling strengths and the stability of the compressed bars. Euler’s solution. Critical force and critical stress. The strength approach to stability. A bar loaded by a bending and buckling load. The check of the buckling bars. The stress and strain state in a point of the body. The principal stress at the plane stress problem.