(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

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

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