SM 2 011 Plasticity PLASTICITY inelastic behaviour of

SM 2 -011: Plasticity PLASTICITY (inelastic behaviour of materials) M. Chrzanowski: Strength of Materials /18

SM 2 -011: Plasticity Rm RH Elastic materials when unloaded return to initial shape (strains caused by loading are reversible) Brittle material Linear elastic material Plastic strains occurs when loads are high enough Plastic strains are irreversible M. Chrzanowski: Strength of Materials Re RH Elastoplastic material arctan. E Permanent plastic strain 2/18

SM 2 -011: Plasticity Different idealisations of tensile diagram for elasto-plastic materials RH Re Re Linear elasticity Stiff material with ideal plasticity Elasticity with ideal plasticity RH Re Re Typical real material M. Chrzanowski: Strength of Materials Elasto-plastic material with plastic hardening Stiff material with plastic hardening 3/18

SM 2 -011: Plasticity Elasto-plastic bending for for Elastic range z Centre of gravity Neutral axis z zmax y x Neutral axis M A Beam cross-section Side view M. Chrzanowski: Strength of Materials 4/18

SM 2 -011: Plasticity Elasto-plastic bending Elastic limit moment Plastic limit moment Elastic neutral axis z z’ Centre of gravity A z’ zmax y x’ y’ Plastic neutral axis M. Chrzanowski: Strength of Materials 5/18

SM 2 -011: Plasticity Elasto-plastic bending Plastic limit moment z’ z’ z’ A 1 N 1 A y’ N 2 x’ A 2 Co. G of A 1 z 1’ z 2’ y’ Co. G of A 2 M. Chrzanowski: Strength of Materials 6/18

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

SM 2 -011: Plasticity z Wspr = A 1 b h 2 Wpl = 2 S yc (A 1 ) = 2 b h h = 2 4 4 b h 2 6 é æ ö ù b h 2 Wpl = S yo (A 1 ) - S yo (A 2 ) = b h h - êb h çè - h ÷ø ú = 2 4 ë 2 4 û 4 h yc= yo A 2 M = Re b h 2 6 b M = Re k = Wspr = A 1 A 2 = M z d M yc= yo M. Chrzanowski: Strength of Materials W pl b h 2 4 = 1. 5 W spr p d 2 4 d 2 d 3 = Wpl = 2 S yc (A 1 ) = 2 1 2 4 3 p 6 p d 3 32 k = W pl W spr = 32 6 p = 1. 7 8/18

SM 2 -011: Plasticity 5 2 2 3 20 6 k = 1. 42 k = 1. 76 k = 1. 52 7 5 2 2 5 5 5 2 1 1 3 4 3 5 1 9 10 8 k = 2. 38 1 4 6 k = 1. 45 15 k = 2. 34 2 12 9 MC riddle: a Loading plane k=1, 5 M. Chrzanowski: Strength of Materials k=k(a)=? k=? 9/18

SM 2 -011: Plasticity Plastic limit of a cross-section Elasticity with ideal plasticity Nonhomogeneous distribution Homogeneous distribution Statically determined Limit elastic capacity No plastic gain Limit plastic capacity Ratio of plastic to elastic capacities k Tension Bending Plastic gain Statically undetermined M. Chrzanowski: Strength of Materials 10/18

SM 2 -011: Plasticity Limit analysis of structures Statically determined structures Length and cross-section area of both bars: l, A Elastic solution 1 a a 1 Plastic solution From equilirium: P Stress in bars: In limit elastic state: Limit elastic capacity: Limit plastic capacity: M. Chrzanowski: Strength of Materials 11/18

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

SM 2 -011: Plasticity 1, 40 1, 365 1, 30 67, 5 o 1, 20 1, 10 1, 00 a [ o] 0 10 20 30 40 50 60 70 80 Capacity of the 3 -bar structure due to plastic properties M. Chrzanowski: Strength of Materials 90 13/18

SM 2 -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: M. Chrzanowski: Strength of Materials 14/18

SM 2 -011: Plasticity Limit analysis of beams Moment – curvature interdependence In elastic range: In plastic range: k 1 1 M. Chrzanowski: Strength of Materials 15/18

SM 2 -011: Plasticity Limit analysis of beams Statically determined structures Plastic hinge Bending moment Curvature Plastic zone spreading M. Chrzanowski: Strength of Materials 16/18

SM 2 -011: Plasticity Unstable mechanism! Statically indetermined structures Limit elastic moment Limit plastic moment Shear forces diagram M. Chrzanowski: Strength of Materials 17/18

SM 2 -011: Plasticity Limit analysis by virtual work principle In limit plastic state the moment distribution due to given mechanism is known. Example: On this basis limit plastic capacity can be easily found, however, the ratio of plastic to elastic capacity is unavailable. l/2 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. M. Chrzanowski: Strength of Materials 18/18

SM 2 -011: Plasticity stop M. Chrzanowski: Strength of Materials 19/18