introductio to column buckling
Post on 15-Jan-2016
276 Views
Preview:
DESCRIPTION
TRANSCRIPT
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 1
INTRODUCTION TO COLUMN BUCKLING
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 2
INTRODUCTION TO COLUMN BUCKLING
•Introduction•Elastic buckling of an ideal column•Strength curve for an ideal column•Strength of practical column•Concepts of effective lengths•Torsional and torsional-flexural buckling•Conclusions
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 3
INTRODUCTION
• Compression members: short or long
• Squashing of short column
• Buckling of long column
• Steel members more susceptible to
buckling compared to RC and PSC
members
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 4
INTRODUCTION
A “long” column failsby predominant buckling
A “short” column fails by compression yield
Buckled shape
Fig 1: “short” vs “long” columns
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 5
ELASTIC BUCKLING OF EULER COLUMN
Assumptions:
• Material of strut - homogenous and linearly elastic
• No imperfections (perfectly straight)
• No eccentricity of loading
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 6
The governing differential equation is
02
2
yEI
P
dx
yd cr .
x
y
Pcr
ELASTIC BUCKLING OF EULER COLUMN
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 7
2
2
EI
Pcr
Lowest value of the critical load
ELASTIC BUCKLING OF EULER COLUMN
Buckling load Vs Lateral deflection Relationship
9
4
1
Unstable buckling modes
crPP /
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 8
Conclusions of Euler buckling analysis:
• Strut can remain straight for all values of P
• When P = Pcr the strut buckles in the shape
of a half sine wave
• At higher values of loads, other sinusoidal buckled shapes are possible. It is seen that for for higher values of Pcr , the column is in
unstable equilibrium.
ELASTIC BUCKLING OF EULER COLUMN
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 9
Mean compressive buckling stress, cr is given by
ELASTIC BUCKLING OF EULER COLUMN
2
2
2
2
2
22
2
2
)/(
E
r
ErE
A
IE
A
P
cr
cr
cr
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 10
Elastic buckling stress(cr) defined by (2E/2 )
= /r
cr
Fig. 4 Euler buckling relation between cr and
ELASTIC BUCKLING OF EULER COLUMN
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 11
STRENGTH CURVE FOR AN IDEAL STRUT
fy
y
Yield plateau
Fig. 5 Idealized elastic-plastic relationship for steel
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 12
Strength curve for an axially loaded initially straight pin-ended column
B1f
f y A
c = /r
Plastic yield definedby ff = y
Elastic buckling (cr )defined by 2 E/ 2
AC
B
STRENGTH CURVE FOR AN IDEAL STRUT
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 13
• Column fails when the compressive stress is greater than or equal to the values defined by ACB.
• AC Failure by yielding (Low slenderness ratios)
• CB Failure by bucking ( c )
STRENGTH CURVE FOR AN IDEAL STRUT
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 14
f /fy
1.0
= (fy/cr)1/2
1.0
Elastic buckling
Plastic yield
Strength curve in a non-dimensional form
STRENGTH CURVE FOR AN IDEAL STRUT
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 15
FACTORS AFFECTING STRENGTH OF A COLUMN IN PRACTICE:
• Effect of initial out of straightness
• Effect of eccentricity of applied
loading
• Effect of residual stress
• Effect of a strain hardening and the
absence of clearly defined yield
point
• Effect of all features taken together
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 16
Effect of initial out of straightness
F
y x
y0
a0
Pin-ended strut with initial imperfection
P
x
ay
sin00
P enhances the deflection by the factor
)(1
1
crPP
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 17
O O1
P
Pcr
PpPy
Pf C
D
Actual elastic-plastic response
Curve A
Curve B
Initial imperfection (a0)
Load deflection responseof a strut with initial imperfection
Ideal bifurcation type
buckling
Effects of imperfection(elastic behaviour)
Strength(plastic unloading curve)
Stress distribution at D
M
M
Stress distribution at C
fy
Stress distributions at C and D
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 18
Strength curves for strut with initial imperfection
Lower bound curve
Data from collapse tests (marked x)
Elastic buckling curve
f
fy
= /r
X X
X X
X
X X X
X
X X
XX
X X X
X X X
Strut
P
P
Low slenderness ratios effect of initial imperfections is negligible
Intermediate slenderness ratios lower bound curve is below fy and cr curves
High slenderness ratios lower bound curve is close to the cr curve
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 19
Effect of eccentricity of applied loading
X
X X X X
X X
X
X
X X
X
X X
Strength curve for eccentrically loaded columns
Deflectedshape afterloading
P
P
Axis ofthe column
e
Data from collapsetests
Elastic bucklingcurve
Lower bound curve
f
fy
Behaviour is similar to that of initial out of straightness
Difference is noticed in the reduction of load carrying capacity for stocky members even for low values of
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 20
Effect of residual stress b b
a a
b b
a a b b
aa
Various stages of rolling a steel girder
(a) (b) (d)(c)
• Residual stress differential heating and cooling during rolling and forming
• Self equilibrating system of stresses
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 21
Residual stresses in flanges
Residual stressesin web
Residual stresses distribution (no applied load)
Residual stresses in anelastic section subjectedto a mean stress a
(net stress = a +r)
The influence of residual stresses
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 22
a
fy
p
r
Stub column yieldswhen a = fy
av
Mean axial stress vs mean axial strainin a stub column test
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 23
f
fy
fy - r
Elastic critical buckling
Columns with residual stresses
(E/fy)1/2 = /r
Buckling of an initially straight columnhaving residual stresses
The difference between buckling and plastic squash load is most pronounced when
21
yf
Er
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 24
Effect of strain hardening and absence of clearly defined yield point
Strain hardening athigh strains
fy
Stress-strain relationship for Steels exhibiting strain hardening
• Ignoring the effect of strain-hardening provides a margin of safety
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 25
fy
p
a
fy
p
a 0.2% proof stress
0.2%
Lack of clearly defined yield Lack of clearly defined yield with strain hardening
• Above >p, the material exhibits non-linear
behaviour• When the yield point is not defined, the yield
stress is generally taken as 0.2% proof stress
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 26
Effect of all features taken together
a
fy
/r
Data from collapse tests
Theoretical elastic buckling
Lower bound curve
(E/fy)1/2
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 27
CONCEPT OF EFFECTIVE LENGTHS
Po in t o f in flection
l
B u ckled m o d e fo r d ifferen t en d co nd itio ns
2 l
l/2 2/l
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 28
Effective lengths in different planes (No sway and sway columns)
Columns with partial rotational restraint
P
e
P
P
e
P
P
e
P
e
(a) (b) (c) (d)
No swaye always
Swaye always
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 29
ACCURACY IN USING EFFECTIVE LENGTHS
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 30
Torsional and Torsional-Flexural Buckling of columns
Flexural buckling Torsional buckling
Folded plate twists under axial loadPlate with unsupported edges
Twisted position
Original position
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 31
+C1
X1
O
Y1
+ CX
O
Y
Y0
X0
u
v
Torsional -flexural buckling deformations.
C’
Shear centre
Rayleigh-Ritz energy method is used to obtain the-Ritz energy method is used to obtain thecritical loadcritical load
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 32
BUCKLING MODES
In general there are 3 buckling loads, i.e. Euler
buckling about x and y axes and flexural
torsional buckling loads
Doubly symmetric section
• Buckling about x and y axes (One of
these is lowest)
• Flexural torsional buckling load (we
disregard this)
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 33
BUCKLING MODES
Singly symmetric sections
• Euler buckling load about weak axis
• Flexural Torsional buckling load
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 34
CONCLUSIONS
• Elastic buckling load dependent on the
slenderness ratio
• Factors affecting column strength ( viz. residual
stresses etc.) considered in design practice
• ‘Effective length’ concept of columns
• Elastic torsional and torsional-flexural buckling
©Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG 35
THANK YOU
top related