(1) Beam Bending -Review --Plane sections remain plane. y M ρ M x ρ : radius of curvature; k = 1/ρ :curvature; M: moment Compatibility : εxx= - y/ρ Independent of material behavior compression y neutral axis εxx tension Equilibrium: M = -σxx y dA Independent of material behavior A M y σxx (2) Elastic beam bending σ E elastic material behavior : σxx =E εxx 1 ε compression y neutral σ ε ρ) axis xx =E xx = -E (y/ σxx=-M y/I σxx tension linear stress distribution Μ 2 M = -σxx y dA = E /ρ y dA M=EI / ρ A A EI I : second moment of area 1 1/ρ linear moment-curvature relationship B 2 for rectangular beams 3 I = y dA H I = BH /12 A for I-beams B 3 I = (BH -bh3 )/12 H h b/2 b/2 (3) Elastic -plastic beam bending σ elastic-perfectly plastic material behavior σy E 1 ε nonlinear Moment-curvature relationship σ y y M p M ->Mp :fully plastic regime: |σ|=σy at all fibers of beam σxx σ y y Me M>Me : elastic-plastic regime: |σ|=σy at outer fibers of beam σxx EI y : elastic regime: |σ| <σy at all points M<Me σ 1 xx 1/ρ In the elastic regime σxx=-M y /I -> | σmax | =M ymax /I Yielding initiates at the outer fibers of the beam when | σmax |= σy. This corresponds to a bending moment Me : Me = (σy I )/ymax 2 for rectangular beams : Me = (σy BH )/6 σ Elastic -plastic beam bending (4) σy elastic-perfectly plastic material behavior ε εy = σy/E For M>Me the beam is in the elastic-plastic regime: the core of the beam (between y=- c and y=c) is in the elastic regime, while the outer fibers are in the plastic regime (σ=σy). ε σ y for -c<y<c = /E y c c εy σy −ε −σ y εxx = -y/ρ y σxx -c -c For |y| = c , ε=-σy/E --> -y/ρ = −c/ρ=-σy/E--> the extension of the elastic region is given by: c= ρ (σy/E) The moment-curvature relationship is then given by: For a rectangular beam : c -c ymax 2 M = -σxx y dA = E /ρ y dA + -σy y dA + -σy y dA A -c -ymax c 2 2 M = σy (3H -4c ) B/12 σy For very large curvatures c-->0 and the moment approaches the y limit moment Mp, where the entire section is in the plastic regime σxx M p = -σy y dA 2 A for rectangular beams : Mp = (σy BH ) / 4 = 1.5 Me Elastic -plastic beam bending . Unloading /springback (5) σ material behavior: elastic unloading E E ∆σ = ∆ε ∆σ E 1 1 ε ∆ε The beam unloads elastically σ y y σxx M ∆(1/ρ)=∆M / EI =-M loaded /EI M loaded ∆M (1/ρ)unloaded =(1/ρ)loaded-M loaded /EI EI EI 1 1 (1/ρ) (1/ρ) 1/ρ unloaded loaded ∆(1/ρ) Stress distribution upon unloading σ y y y y σ xx σxx σxx stress distribution in the Change in stress Residual stress loaded configuration upon unloading distribution σ loaded ∆σ=(Mloaded y)/I σresidual M =0.