1. 1 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity PLASTICITY (inelastic behaviour of materials)

2. 2 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity   RHRH Elastic materials when unloaded return to initial shape (strains caused by loading are reversible) Plastic strains are irreversible Plastic strains occurs when loads are high enough   RHRH RmRm ReRe Linear elastic material Elastoplastic material Brittle material Permanent plastic strain arctanE

3. 3 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity  RHRH  Linear elasticity  ReRe  Elasticity with ideal plasticity Different idealisations of tensile diagram for elasto-plastic materials ReRe   Elasto-plastic material with plastic hardening ReRe   Stiff material with plastic hardening ReRe   Stiff material with ideal plasticity  RHRH  Typical real material

4. 4 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Elasto-plastic bending for   M x z Elastic range y z A Side view Neutral axis Beam cross-section z max Neutral axis Centre of gravity

5. 5 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity y z A z max Elastic neutral axis Centre of gravity Elastic limit moment Plastic limit moment z’ y’ z’ x’ Elasto-plastic bending Plastic neutral axis

6. 6 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Elasto-plastic bending Plastic limit moment z’ y’ z’ x’ A A1A1 A2A2 z1’z1’ z2’z2’ CoG of A 1 N1N1 CoG of A 2 N2N2 z’ y’

7. 7 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Limit elastic moment y z A Limit plastic moment z’ y’ z1’z1’ z2’z2’ A/2 k  1 – shape coefficient Elasto-plastic bending

8. 8 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity y c = y o z A1A1 A2A2 d  WSAb hh bh plyc  22 244 1 2 W bh spr  2 6     WSASAb hh b hh bh plyo               12 2 24244 y c = y o z b h A1A1 A2A2 MR bh e  2 6 MR bh e  2 4 k M M W W pl spr  15. W d   3 32   WSA dd d plyc  22 1 24 42 36 1 2 3   k W W pl spr  32 6 17 .

9. 9 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity 6 2 222 k = 1.76 55 5 5 5 20 k = 1.42 343 2 7 3 k = 1.52 1 1 6 4 9 10 k = 2.38 1 8 2 5 9 1 k = 1.45 12 15 k = 2.34 k=1,5  k=? MC riddle: k=k(  )=? Loading plane

10. 10 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Limit elastic capacity Limit plastic capacity Ratio of plastic to elastic capacities k Tension Bending Plastic limit of a cross-section   Elasticity with ideal plasticity Statically undetermined Statically determined No plastic gain Plastic gain Homogeneous distribution Non- homogeneous distribution

11. 11 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Limit analysis of structures Length and cross-section area of both bars: l, A Elastic solution   11 P From equilirium:  Plastic solution Statically determined structures Stress in bars: In limit elastic state: Limit elastic capacity: Limit plastic capacity:

12. 12 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Length and cross-section area of both bars: l, A Elastic solution Displacement compatibility: Equilibriuim :  Elastic limit capacity – plastic limit in bar #2 Plastic limit capacity – plastic limit in bars #1 and #2   1 2 1 P  Statically undetermined structures Limit analysis of structures

13. 13 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity 020503040607080 90 1,00 1,20 1,365 1,40 1,10 10    67,5 o 1,30 Capacity of the 3-bar structure due to plastic properties

14. 14 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Limit analysis of beams Concept of plastic hinge z’ x’ Trace of the cross-section plane according to the Bernoulli hypothesis Beam axis Plastic hinge:

15. 15 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Moment – curvature interdependence In elastic range:In plastic range: 1 1 k Limit analysis of beams

16. 16 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Statically determined structures Bending moment Curvature Plastic zone spreading Plastic hinge Limit analysis of beams

17. 17 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Statically indetermined structures Limit elastic moment Limit plastic moment Unstable mechanism! Shear forces diagram

18. 18 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity Limit analysis by virtual work principle In limit plastic state the moment distribution due to given mechanism is known. Example: l/2 On this basis limit plastic capacity can be easily found, however, the ratio of plastic to elastic capacity is unavailable. In a more complex case one has to consider all possible mechanisms. The right one is that which yields the smallest value of limit plastic capacity.

19. 19 /18 M.Chrzanowski: Strength of Materials SM2-011: Plasticity  stop