Structural Mechanics
2.080 Recitation 9
Semester Yr
Recitation 9
Buckling of Sections
Consider the box column shown:
Find the critical buckling load.
Column can buckle
globally (Euler) or
locally (plate)
Global (Euler) Buckling
Pc =
⇡ 2 EI
L2
Ix = 2
b 1 h3
+ b1 h
12
✓
b2
2
◆2 !
+2
hb32
12
b1 h3 b1 b22 h b32 h
+
+
2
6
6
✓ ◆2 !
3
b2 h
hb3
b1
Iy = 2
+ b2 h
+2 1
12
2
12
=
=
b2 h3 b21 b2 h b31 h
+
+
2
6
6
Ix < Iy ! Will buckle about x-axis
⇡2E
So Pc = 2
L
✓
b1 h3 b1 b22 h b32 h
+
+
6
2
6
◆
Local (Plate) Buckling
Treat each side as an individual plate.
9-1
Structural Mechanics
2.080 Recitation 9
Semester Yr
* Assume uniform compression ! stress in each plate is the same.
1 i1 =
Plate Q:
P1
hb1
2 i2 =
Plate Q:
P2
hb2
For a plate simply supported on the loaded edges:
Pc =
✓
D=
kc ⇡ 2 D
b
Eh3
12(1 _ ⌫ 2 )
(where kc depends on BC on other sides and dimensions)
◆
So
icr,1 =
Pc1
kc1 ⇡ 2 D
=
hb1
hb21
and
icr,2 =
kc2 ⇡ 2 D
Pc2
=
hb2
hb22
All plates buckle at the same time, so
icr,1 = icr,2
⇡2D
kc1
hb12
kc2 ⇡ 2 D
=
! kc2 = kc1
hb22
Total load = 2P1 + 2P2
! Pc,tot
✓
✓
b2
b1
◆2
kc1 ⇡ 2 D kc2 ⇡ 2 D
= 2Pc1 + 2Pc2 = 2
+
b2
b1
"
#
✓ ◆2
1
1
b2
= 2⇡ 2 Dkc1
+
·
b1
b2
b1
✓
◆
2⇡ 2 Dkc1
b2
=
1+
b1
b1
9-2
◆
Structural Mechanics
2.080 Recitation 9
Semester Yr
But what is kc1 ??
- In general, the adjacent plates on the unloaded edges will cause a bending moment (some­
where between simply supported and fully clamped)
b2
(Plot on page 16 gives kc1 as function of )
b1
(assumes L/b1 > 5)
Example
h = 2 mm
b1 = 100 mm
b2 = 50 mm
Find the length, L, that marks transition between global and local buckling.
Pc,global =
=
Pc,local =
=
=
✓
◆
⇡ 2 E b1 h3 b1 b22 h b32 h
+
+
6
2
6
L2
✓
◆
2
3
⇡ E 100(2)
100(50)2 (2) 503 (2)
⇡2E
+
=
+
(291, 800 mm4 )
6
2
6
L2
L2
✓
◆
2⇡ 2 Dkc1
b2
1+
(From plot: kc1 ' 5.2)
b1
b1
✓
◆
2⇡ 2 Eh3 kc1
b2
1
+
b1
b1 (12)(1 _ ⌫ 2 )
2
3
2⇡ E(2) (5.2)
(1 + 0.5) = ⇡ 2 E(0.114 mm2 )
100(12)(1 _ 0.32 )
Let Pc,global = Pc,local and solve for L:
⇡2E
(291, 800) = ⇡ 2 E(0.114)
L2
L ' 1600 mm = 1.6 m
9-3
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2.080J / 1.573J Structural Mechanics
Fall 2013
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