1. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education STATICS OF PLANE BAR STRUCTURES GEOMETRICAL RIGIDITY

2. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education General assumptions of statics  static equilibrium (RÓWNOWAGA STATYCZNA)  stability (RÓWNOWAGA STATECZNA)  principle of rigidity (ZASADA ZESZTYWNIENIA)  geometric rigidity (GEOMETRYCZNA NIEZMIENNOŚĆ)  principle of superposition (ZASADA SUPERPOZYCJI)  continuity axiom (KONTINUUM MATERIALNE)

3. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Static equilibrium object in a state of rest; balance of forces  3D – 6 equations of equilibrium  2D – 3 equations of equilibrium (in 3 different forms) 2 particular cases (only 2 eq. independent): concurrent forces parallel forces

4. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education w Stability Lyapunov criterion of stability: response of structure to small excitation or disturbance 3 possible states of equilibrium:  stable (RÓWNOWAGA TRWAŁA)  neutral (OBOJĘTNA)  unstable (NIETRWAŁA)

5. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Principle of rigidity structures are composed of elements so rigid that their deformations don’t influence the statics considering the statics we can neglect the displacements of the structures and treat all elements as perfectly rigid it follows that the stability of the structures is determined by its pure geometry

6. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Geometric rigidity  structure is either geometrically rigid or a mechanism  for the mechanisms the methods of dynamics (not statics) should be applied  there are no mechanisms in civil engineering (with rare exceptions, eg. anti-seismic systems)  so, all structures should be (geometrically) rigid  there is no static equilibrium of not rigid structures

7. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Preliminaries  a rigid body has only one instantaneous centre of rotation (the plane motion)  two pins are enough to immobilize a rigid body  the connection by two not parallel bars is equivalent to the connection by the hinge

8. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Geometric rigidity – methods easy to use but sometimes fail:  2 elements theorem  3 elements theorem always effective but sometimes „not user friendly”:  virtual velocities scheme  numerical computations: stiffness matrix is not singular

9. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education 2 elements theorem rigid connection of 2 elements is the connection by 3 bars which directions don’t intersect at one point (also at the ideal point at infinity where the parallel lines intersect)

10. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education 3 elements theorem rigid connection of 3 elements is the connection of the elements one-by-one with a pair bars which directions don’t intersect at the collinear points A B C

11. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education 3 elements theorem - contd connection „in triangle” is geometrically rigid triangular shape, but 4 elements = mechanism not triangular shape, but 3 elements = rigid body

12. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Virtual velocities scheme the structure is geometrically rigid if and only if there is no consistent scheme of the virtual velocities  usually the stability is proved by contradiction: we start with the assumption that some move is possible and show the results are contradictory  contradiction at some points doesn’t prove the theorem for the whole structure  geometrical rigidity is proved if and only if all elements are immobilized

13. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Simple example von Mises truss free body stability 2 el. th. 2 1 hinge = 2 bars 3 el. th. hinge = 2 bars 1 2 3 v. v. s. 1,3 2,3

14. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Simple example - contd rigidity with the constraints 2 el. th. 3 el. th. v. v. s. 1 2 3rd bar 2 3 one point, 2 vel.: not possible, so it’s pinned each element pinned at 2 points 1

15. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Example 2 12 3 1,2 1,3 2,3 not collinear

16. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Example 2 - contd two elements connected by 3 bars = rigid body

17. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Example 3 12 3 only one bar here but there are two bars here no conclusion 3 all elements used – not rigid 1,2 1,3

18. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Example 3 - contd each element immobilized = rigid body

19. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Example 4 1 2 3 1,22,3 1,3 1 2 3 1,2 1,32,3 bad choice: - points are collinear - some bars haven’t been used 1 2 3 1,2 1,3 2,3 no conclusion

20. Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education Recapitulation  (in civil engineering) each structure must be stable  if the static equilibrium cannot be achieved then the structure must be unstable  the analysis of the free body rigidity is useful for subsequent static solution  the presented methods have to be practiced and mastered