MOM12 Column Buckling
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Transcript of MOM12 Column Buckling
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Column Buckling
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Stability and Buckling
Examples of Columns Conventional Design of Columns
Eulers Formula for Pin-ended Columns
Buckling Modes
Eulers Formula for Cantilevered Columns
Extension of Eulers Formula
Buckling in Orthogonal Planes
Applicability of Eulers Formula Failure Diagram
Critical Stress for Intermediate Columns
Design of Columns
Ways to Improve Column Stability2
Contents
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Stability is characterized as the ability of a structure to maintain
its (stable) equilibrium under working conditions.
Stability and Buckling
Buckling is the behavior of a structure losing its equilibriumunder working conditions. This is another type of failure criterion,
in addition to strength (fracture/yielding), stiffness (deformation)
and fatigue criteria.
Buckling occurs suddenly and results in catastrophic accident. 3
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Examples of Columns
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In the design of columns, cross-
sectional area is selected such that
- allowable stress is not exceeded
P
A
- deformation falls within
specifications
L P
L AE
After these design calculations, many
discover that the column is unstable
under loading and that it suddenly
becomes sharply curved or buckles.
Conventional Design of Columns
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Consider an axially loaded beam.
After a small perturbation, thesystem reaches a neutral
equilibrium configuration such that
2
2
2
2
2 2
2 2
2
0
" 0,
sin cos
0 0
0;
cr
cr
cr
cr
Pd w Mw
dx EI EI Pd w
wdx EI
Pw k w k
EI
w A kx B kx
w w L
B
EInkL n P
L
Eulers Formula for Pin-ended Columns
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2 2 2
2 2 2
4 9; ;cr cr cr
EI EI EI
P P PL L L
Buckling Modes
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A column with one fixed and one
free end, will behave as the upper-half of a pin-connected column.
The critical loading is calculated
from Eulers formula,
lengthequivalent2
4
4
2
2
2
2
2
2
2
2
LL
iLE
iLE
L
EI
L
EIP
e
rre
cr
e
cr
Eulers Formula for Cantilevered Columns
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tcoefficienlengthlength;equivalent
;2
2
2
2
2
2
2
2
e
rre
cr
e
cr
L
iL
E
iL
E
L
EI
L
EIP
Extension of Eulers Formula
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Assuming the same material and cross-sectional dimension, which
of the following four columns is most susceptible to buckling?
2l
5l 7l
9l
2
2 Euler Formulacr
EIP
l
(a) (b) (c) (d)
Sample Problem
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x
yz y
Buckling inx-z:
y = 12
2
y
cr y
EIP
L
Buckling inx-y:
z= 0.5
2
20.5
zcr z
EIP
L
Buckling in Orthogonal Planes
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For the column shown, find the relationship between the critical
loads corresponding to buckling inx-y andx-zplane respectively.
l2b
b
z
y
Pcr
2
2
2
2
3
3
2 124
2 12
zcr z
z
y
cr y
y
cr z z
cr y y
EIP
l
EIP
l
b bP I
P I bb
Solution:
What if the cross-section is increased to 2b2b?
Sample Problem
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l
2r r
r
r
(a) (b) (c)
Pcr
For the column shown, find the relationship between the critical
loads corresponding to cross-sections described by (a), (b) and (c).
42 2 2 4
2 2 2
2 2 4
2 2
4
2 2 2 2 4
2 2 2
2
64 4: :
1 1: :
64 4 64 12
11: :16 3
12 12
acr a
cr a cr b cr c
bcr b
c
cr c
rEI E E rP
l l l P P P
EI E rP
l l
rEI E E rP
l l l
Solution:
Sample Problem
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Which of the following best describes the relationship between
criticalP1 andP2? (a)P1 =P2; (b)P1P2; (d) not sure.
a
a
P1
A B
C D
a
a
P2A B
C D1P
1P
0
012P 0
0
2P
2P
22P
Solution:
2 2
1 12 2
2 2
2 22 2
12
2 22N
N
:
:
AD
AB
E I E I a F P P
aa
E I E I b F P P
a a
Sample Problem
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Critical buckling stress
P: proportion limit;
p: critical slenderness ratio.
2
2 2 2 22
2 2 2
slenderness ratio
rcr
crcr
r
r
EIP
E AiP E EL
A L A L iP
A
L i
Applicability of Eulers formula
2
2cr P PP
E E
206 GPaQ235: 100
200 MPaP
P
E
Applicability of Eulers Formula
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Previous analyses assumed
stresses below the proportionallimit and initially straight,
homogeneous columns
Experimental data demonstrate
- for largeLe/ir, crfollowsEulers formula and
depends uponEbut not Y.
- for intermediateLe/ir, cr
depends on both YandE.
- for smallLe/ir,cr is
determined by the yield
strength Yand notE.
2 2
cr E Y
rL i
Failure Diagram
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O
Strength
analysis
Y
P
Y bacr
2 2
cr E
Y
P
Stability
analysis
Y cr Y 3. Short columns:
1. Long columns:2 2
2 2P cr p
P
E E
2. Intermediate columns:
YY P p cr Y a
a bb
Critical Stress for Intermediate Columns
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For the truss shown,F, , andLACare given. Find 0 < < /2, under
which columnAB andBCreach critical stability simultaneously.
2 2
2 2 2cos
cosAB
AB
EI EIF F
l l
2
2 22
2
2 2
coscos
cot tan
sin sin
EIF
l
EI
F l
A C
B
F
Solution:
1. Eulers equation for columnAB:
2 2
2 2 2sin
sinBC
CB
EI EIF F
l l
2. Eulers equation for columnBC:
3. If columnAB &BCreach critical stability simultaneously:
Sample Problem
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1. Method of safety factor
cr
st
st
cr
st
st
FF F
n
F
A n
2. Method of discount factor
st
F
A
: discount/stability factor
nst: safety factor
[Fst]: allowable load
[st]: allowable stress
Stability analysis of columns
- Stability check
- Cross-section design
- Allowable load/stress
Design of Columns
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2 2
22
2 2
2 2
cr
e
cr
r
EI EIP
L L
E E
L i
Increase moment of inertia for a given
cross-sectional area.
Ways to Improve Column Stability
Selection of materials.
Decrease effective column length (L).
For a group of columns, make each one
equally stable (avoid the last straw).
z r z y r yi i
Coordination of end conditions and
moment of inertia in orthogonal planes:
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A Q275 structural steel column is cylindrically pinned at both ends,
as shown in the following figure. IfFmax = 60 kN and nst= 3.5,check the column stability.
Solution:
lz=800m
m
h =45mm
x
y
ly=77m
m
b=20mm
x
z
F
lz
F
ly
- Inx-yplane, both ends are pinned.
1 80012.99 , 61.9
12.992 3
z z zz z
z
I lhi mm
A i
Sample Problem
Determine the principal buckling plane.
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- Inx-zplane, both ends are constrained.
0.5 7705.77 , 66.7
5.772 3
y y y
y y
y
I lbi mm
A i
y > z, hence column buckling first happens inx-zplane.
For Q-275 structural steel p = 96, and therefore y = 66.7 < p = 96
2
6 6
280 0.00872 241 MPa
(241 10 )(20 45 10 ) 217 kN
2173.62
60
cr y
cr cr
crst
F A
Fn n
F
The column is stable.
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For the composite beam and column structure shown,EAB =EBD =E,
dAB = dBD = d,L:d= 30, P.BD = 100. FindP, under whichBD reachesthe critical condition.
Solution:P
A C B
D
EI
EA L
LL: 120
4 P
L LBD
i d
2
2cr BD
EIF
L
Deformation compatibility:
33 85
6 3
BD BDF L F LPL
EI EI EA
3 2
18000
EdP
Sample Problem
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Stability and Buckling
Examples of Columns Conventional Design of Columns
Eulers Formula for Pin-ended Columns
Buckling Modes
Eulers Formula for Cantilevered Columns
Extension of Eulers Formula
Buckling in Orthogonal Planes
Applicability of Eulers Formula Failure Diagram
Critical Stress for Intermediate Columns
Design of Columns
Ways to Improve Column Stability 25
Contents