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2E4: SOLIDS & STRUCTURES2E4: SOLIDS & STRUCTURES
Lecture 15Lecture 15
Dr. Bidisha Ghosh
Notes:
http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solid
s & Structures
Buckling of ColumnsBuckling of Columns
What is buckling?
Buckling is a large deformation produced under compressive load
in a
direction or plane normal to the direction of application of the
load.
Buckling is a form of instability, it occurs suddenly with large
changes
in deformation but little change in loading. For this reason it
is a
dangerous phenomenon that must be avoided in structural
design.
Buckling is not failure through yielding. Due to the shape of
a
structural element it can buckle under a load below the
ultimate
strength.
Whether a column will buckle or not depends on the
length of the column.
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Long ColumnsLong Columns
Long columns usually fail by elastic buckling.
The failure load is below ultimate strength of the
material.
The Euler formula is used to calculate failure strength in
long columns.
Short ColumnsShort Columns
Short columns generally dont fail by elastic buckling.
The failure stress is close to yield stress of the material.
The true short-columns do not have much practical
application.
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How do we know which is a short/long column?How do we know which
is a short/long column?
eLr
=
Ir
A=
Classification of ColumnsClassification of Columns
CD is Euler curve showing behaviour of long columns
Euler formula should not be used for slenderness ratio
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End ConditionsEnd Conditions
What was the pinned-pinned condition mentioned in connection
with Slenderness ratio?
Euler derived all formulae related to column buckling for
pinned-
pinned condition and later for other end support conditions,
those
formula were altered by using a constant, C
Instead of the actual length of the columns a new length termed
as the
effective length was used.
Effective length, 2
2e
LLC
=
Different EndDifferent End--ConditionsConditions
Check the load required to buckle!
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Failure StressFailure Stress
2
2cre
EIPLpi
=
2
2cr
cr
P EA
pi
= =
The radius of gyration provides a measure of the resistance
provided
by a cross-section to lateral buckling.
The radius of gyration is not a physical entity in itself. It is
a
relationship derived to make prediction of column behaviour
easy. The
radius of gyration is related to the size and shape of the
cross-section.
Columns will buckle in the direction of least
cross-sectional
stiffness (minimum value of I ).
A rectangular column will buckle in the direction of the
smaller
dimension in cross-section. A square column cross-section will
be
equally prone to buckling in both x and y directions. This is
because the
cross section will offer equal resistance to buckling in the
direction x
and y.
x
y
z
y
z
I y > I z
Cross-sectionP
P
Buckling Direction
A
b
h
CrossCross--sectional Shape & Bucklingsectional Shape &
Buckling
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Example Example
Determine the thickness of a round steel tubular strut,
375mm external diameter and 2.5m long, pin-jointed at
the ends, to withstand an axial load of 39000kN.
E=200GPa.
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Moment curvature equationMoment curvature equation
A quick touch on bending before learning buckling!
1.Moment at any section
2.Moment-curvature equation:
3.Buckling is an effect of combined compression &
bending
2
2d yM EIdx
=
Compression on A Slender ColumnCompression on A Slender
Column
From knowledge of bending,
Solve this equation..
1.
2. So,
2
2
2
2 0
d yM EI Pydx
d yEI Pydx
= =
+ =
22 2
2 0; where d y Pydx EI
+ = =
sin cos ;From boundary conditions, B=0 & sin 0y A x B x
A L
= +
=
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Compression on A Slender ColumnCompression on A Slender
Column
For any buckling to happen the second condition has to be true
and
that means,
2 2
2
0, ,2 ,3 .....
, ( 1,2,...)
L
Pand L n n
EIn EIP
L
pi pi pi
pi
pi
=
= = =
=
sin 0means, either 0 , sin 0A L
A or L
=
= =