MIT - 16.
20
Fall, 2002
16.20 HANDOUT #5
Fall, 2002
Stability and Buckling
Bifurcation Buckling and Snap-Through Buckling
bifurcation point
BIFURCATION BUCKLING
Z
Perfect Column: X
EI = constant
d 4w d 2
w
• Governing Equation: EI 4 + P 2
= 0
dx dx
P P
• Solution: w = Asin x + B cos x + C + D x
EI EI
Paul A. Lagace © 2002 Handout 5-1
MIT - 16.20 Fall, 2002
n 2π 2 EI nπ x
• Simply supported: Pcr = mode shape: w = Asin
l2 l
π 2 EI
Euler buckling load: Pcr =
l2
• General Case:
c π 2 EI
Pcr = c = coefficient of edge fixity
l2
• Various Boundary Conditions
– Simply-supported (pinned)
w=0
d 2w
M = EI 2 = 0
dx
– Fixed end (clamped)
w=0
dw
= 0
dx
– Free end
d 2w
M = EI 2 = 0
dx
d d 2w
S = EI 2 = 0
dx dx
– Sliding
d d 2w
S = EI 2 = 0
dx dx
dw
= 0
dx
Paul A. Lagace © 2002 Handout 5-2
MIT - 16.20 Fall, 2002
– Free end with axial load
d 2w
M = EI 2 = 0
dx
d d 2w dw
S = E I 2 = − P0
dx dx dx
– Vertical spring
d 2w
M = EI 2 = 0
dx
d d 2w
S = EI 2 = kf w
dx dx
– Torsional spring
w=0
d 2w dw
M = EI 2 = − kT
dx dx
• Various Configurations
torsional
spring kT
c=1 c= 4 c = 0.25 1<c<4
Paul A. Lagace © 2002 Handout 5-3
MIT - 16.20 Fall, 2002
• Important Definitions
12
- radius of gyration = ρ = (I A)
- slenderness ratio = L ρ
- effective length = L′ =
L
c
• Effects of Initial Imperfections
d 4w d 2w
Governing equation still: EI 4 + P 2 = 0
dx dx
--> Boundary Conditions Change: Primary Moment = -eP
P
1 − cos EI l
P P
w = e sin x + cos x − 1
P EI EI
sin l
EI
π 2 EI
Pcr =
l2
increasing eccentricity (e/ll)
P
1 − cos l
d 2w EI P P
M = E I 2 = − eP sin x + cos x
dx P EI EI
sin l
EI
Paul A. Lagace © 2002 Handout 5-4
MIT - 16.20 Fall, 2002
BEAM-COLUMN
• Resultant Relations
dF d d w
= − px − S ≈ -px
dx dx dx
dS d dw
= pz + F
dx dx dx
dM
= S
dx
• Governing Equation:
d2 d 2w d d w
EI − F = pz
dx 2 dx 2 dx d x
• Buckling of Beam-Column:
d 2w
EI 2 + P w = M primary
dx
Paul A. Lagace © 2002 Handout 5-5
MIT - 16.20 Fall, 2002
OTHER ISSUES
• Fracture/Failure via “squashing”
Euler curve
compressive yield
actual behavior
P
σ = = σcu for “squashing”
A
compressive ultimate
• Progressive Yielding
use tangent modulus
Paul A. Lagace © 2002 Handout 5-6
MIT - 16.20 Fall, 2002
• Nonuniform Beams
d2 d 2w d 2w
EI 2 + P 2 = 0
dx 2 dx dx
• Plates
π 2 EI
Pcr = 2
Pin-sliding
l (1 − ν 2 )
simply-supported
isotropic plate
Free mπ x nπ y
w = wmn sin sin
a b
• Cylinders
local “crippling”
σcr(linear) = 0.606 E h (isotropic)
R
due to
imperfections: σ cr( actual ) ≈ (0.15 to 0.9) σ cr( linear )
• Reinforced Plates
Consider buckling/crippling of elements of stiffness as well
as of panels
Paul A. Lagace © 2002 Handout 5-7
MIT - 16.20 Fall, 2002
• Postbuckling
actual behavior w/post-
buckling capability
perfect linear behavior
behavior with imperfection
dθ 1 d 2w
large deformations curvature = = 2 2
ds dw ds
1−
ds
Basic Equation:
2
1 dw d 2w P
1 + + H.O.T. + w = 0
2 ds ds 2
E I
Use Galerkin Method (minimize residuals)
Paul A. Lagace © 2002 Handout 5-8
