Transcript 13. Buckling of Columns 13.2 IDEAL COLUMN WITH PIN SUPPORTS

13. Buckling of Columns
CHAPTER OBJECTIVES
• Discuss the behavior of
columns.
• Discuss the buckling of
columns.
• Determine the axial load
needed to buckle an ideal
column.
• Analyze the buckling with
bending of a column.
• Discuss inelastic buckling of a column.
• Discuss methods used to design concentric and
eccentric columns.
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13. Buckling of Columns
CHAPTER OUTLINE
1.
2.
3.
4.
5.
6.
7.
Critical Load
Ideal Column with Pin Supports
Columns Having Various Types of Supports
*The Secant Formula
*Inelastic Buckling
*Design of Columns for Concentric Loading
*Design of Columns for Eccentric Loading
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13. Buckling of Columns
13.1 CRITICAL LOAD
• Long slender members subjected to axial
compressive force are called columns.
• The lateral deflection that occurs is
called buckling.
• The maximum axial load a column can
support when it is on the verge of
buckling is called the critical load, Pcr.
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13. Buckling of Columns
13.1 CRITICAL LOAD
• Spring develops restoring force F = k, while
applied load P develops two horizontal
components, Px = P tan , which tends to push the
pin further out of equilibrium.
• Since  is small,
 = (L/2) and tan  ≈ .
• Thus, restoring spring
force becomes
F = kL/2, and
disturbing force is
2Px = 2P.
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13. Buckling of Columns
13.1 CRITICAL LOAD
• For kL/2 > 2P,
kL
P
stable equilibrium
4
• For kL/2 < 2P,
kL
P
4
unstable equilibrium
• For kL/2 = 2P,
kL
Pcr 
neutral equilibrium
4
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• An ideal column is perfectly straight before loading,
made of homogeneous material, and upon which
the load is applied through the centroid of the xsection.
• We also assume that the material behaves in a
linear-elastic manner and the column buckles or
bends in a single plane.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• In order to determine the critical load and buckled
shape of column, we apply Eqn 12-10,
d 2
13 - 1
EI 2  M
dx
• Recall that this eqn assume
the slope of the elastic
curve is small and
deflections occur only in
bending. We assume that
the material behaves in a
linear-elastic manner and
the column buckles or
bends in a single plane.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• Summing moments, M = P, Eqn 13-1
becomes
d 2  P 
13 - 2

  0
2 
dx
 EI 
• General solution is
 P 
 P 
  C1 sin 
x   C2 cos
x
 EI 
 EI 
13 - 3
• Since  = 0 at x = 0, then C2 = 0.
Since  = 0 at x = L, then
 P 
C1 sin 
L  0
 EI 
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• Disregarding trivial soln for C1 = 0, we get
 P 
sin 
L  0
 EI 
• Which is satisfied if
P
L  n
EI
• or
P
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n2 2 EI
2
L
n  1,2,3,...
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• Smallest value of P is obtained for n = 1, so critical
load for column is
 2 EI
Pcr  2
L
• This load is also referred to
as the Euler load. The
corresponding buckled
shape is defined by
x
  C1 sin
L
• C1 represents maximum
deflection, max, which occurs
at midpoint of the column.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• A column will buckle about the principal axis of the
x-section having the least moment of inertia
(weakest axis).
• For example, the meter stick shown will
buckle about the a-a axis and not
the b-b axis.
• Thus, circular tubes made excellent
columns, and square tube or those
shapes having Ix ≈ Iy are selected
for columns.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• Buckling eqn for a pin-supported long slender
column,
 2 EI
13 - 5
Pcr  2
L
Pcr = critical or maximum axial load on column just
before it begins to buckle. This load must not cause
the stress in column to exceed proportional limit.
E = modulus of elasticity of material
I = Least modulus of inertia for column’s x-sectional
area.
L = unsupported length of pinned-end columns.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• Expressing I = Ar2 where A is x-sectional area of
column and r is the radius of gyration of x-sectional
 2E
area.
13 - 6
 cr 
2
L r 
cr = critical stress, an average stress in column just
before the column buckles. This stress is an elastic
stress and therefore cr  Y
E = modulus of elasticity of material
L = unsupported length of pinned-end columns.
r = smallest radius of gyration of column, determined
from r = √(I/A), where I is least moment of inertia of
column’s x-sectional area A.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
• The geometric ratio L/r in Eqn 13-6 is known as the
slenderness ratio.
• It is a measure of the column’s flexibility and will be
used to classify columns as long, intermediate or
short.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
IMPORTANT
• Columns are long slender members that are
subjected to axial loads.
• Critical load is the maximum axial load that a
column can support when it is on the verge of
buckling.
• This loading represents a case of neutral
equilibrium.
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13. Buckling of Columns
13.2 IDEAL COLUMN WITH PIN SUPPORTS
IMPORTANT
• An ideal column is initially perfectly straight, made
of homogeneous material, and the load is applied
through the centroid of the x-section.
• A pin-connected column will buckle about the
principal axis of the x-section having the least
moment of intertia.
• The slenderness ratio L/r, where r is the smallest
radius of gyration of x-section. Buckling will occur
about the axis where this ratio gives the greatest
value.
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13. Buckling of Columns
EXAMPLE 13.1
A 7.2-m long A-36 steel tube
having the x-section shown is to
be used a pin-ended column.
Determine the maximum
allowable axial load the column
can support so that it does not
buckle.
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13. Buckling of Columns
EXAMPLE 13.1 (SOLN)
Use Eqn 13-5 to obtain critical load with
Est = 200 GPa.
Pcr 
 2 EI
L2
  

1
 200 10 kN/m   70 4 1 m / 1000 mm4
4

7.2 m 2
2
6
2
 228.2 kN
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13. Buckling of Columns
EXAMPLE 13.1 (SOLN)
This force creates an average compressive stress in
the column of
 cr
Pcr 228.2 kN 1000 N/kN 


A  752   70 2 mm2


 100.2 N/mm2  100 MPa
Since cr < Y = 250 MPa, application of Euler’s eqn
is appropriate.
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13. Buckling of Columns
EXAMPLE 13.2
The A-36 steel W20046 member shown is to be
used as a pin-connected column. Determine the
largest axial load it can support
before it either begins to buckle
or the steel yields.
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13. Buckling of Columns
EXAMPLE 13.2 (SOLN)
From table in Appendix B, column’s x-sectional area
and moments of inertia are A = 5890 mm2,
Ix = 45.5106 mm4,and Iy = 15.3106 mm4.
By inspection, buckling will occur about the y-y axis.
Applying Eqn 13-5, we have
Pcr 

 EI
2
L2
  
  

 2 200 106 kN/m 2 15.3 10 4 mm4 1 m / 1000 mm4
4 m 2
 1887 .6 kN
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13. Buckling of Columns
EXAMPLE 13.2 (SOLN)
When fully loaded, average compressive stress in
column is
Pcr 1887 .6 kN 1000 N/kN 
 cr 

A
5890 mm2
 320.5 N/mm
2
Since this stress exceeds yield stress (250 N/mm2),
the load P is determined from simple compression:
P
2
250 N/mm 
5890 mm2
P  1472 .5 kN
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
• From free-body diagram, M = P(  ).
• Differential eqn for the deflection curve is
d 2
P
P
13 - 7 
  
2
EI
EI
dx
• Solving by using boundary conditions
and integration, we get

 P 
   1  cos
x 
 EI 

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13 - 8
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
• Thus, smallest critical load occurs when n = 1, so
that
 2 EI
13 - 9
Pcr 
2
4L
• By comparing with Eqn 13-5, a column fixedsupported at its base will carry only one-fourth the
critical load applied to a pin-supported column.
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
Effective length
• If a column is not supported by pinned-ends, then
Euler’s formula can also be used to determine the
critical load.
• “L” must then represent the distance between the
zero-moment points.
• This distance is called the columns’ effective length,
Le.
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
Effective length
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
Effective length
• Many design codes provide column formulae that
use a dimensionless coefficient K, known as thee
effective-length factor.
13 - 10
Le  KL
• Thus, Euler’s formula can be expressed as
Pcr 
 cr 
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 2 EI
13 - 11
 KL 
2
 2E
 KL r 
2
13 - 12 
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13. Buckling of Columns
13.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS
Effective length
• Here (KL/r) is the column’s effective-slenderness
ratio.
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13. Buckling of Columns
EXAMPLE 13.3
A W15024 steel column is 8 m
long and is fixed at its ends as
shown. Its load-carrying capacity
is increased by bracing it about
the y-y axis using struts that are
assumed to be pin-connected
to its mid-height. Determine the
load it can support sp that the
column does not buckle nor
material exceed the yield stress.
Take Est = 200 GPa and Y = 410 MPa.
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13. Buckling of Columns
EXAMPLE 13.3 (SOLN)
Buckling behavior is
different about the x and y
axes due to bracing.
Buckled shape for each
case is shown.
The effective length for
buckling about the x-x axis
is (KL)x = 0.5(8 m) = 4 m.
For buckling about the y-y
axis, (KL)y = 0.7(8 m/2) = 2.8 m.
We get Ix = 13.4106 mm4 and Iy = 1.83106 mm4
from Appendix B.
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13. Buckling of Columns
EXAMPLE 13.3 (SOLN)
Applying Eqn 13-11,
 Pcr x 
 2 EI x
2
 KL x

 Pcr  y 
 KL 2y

  
4 m 
2
 Pcr x  1653 .2 kN
 2 EI y
  
 2 200 106 kN/m 2 13.4 10 6 m 4
  
  
 2 200 106 kN/m 2 1.83 10 6 m 4
2.8 m 2
 Pcr  y  460.8 kN
By comparison, buckling will occur about the y-y
axis.
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13. Buckling of Columns
EXAMPLE 13.3 (SOLN)
Area of x-section is 3060 mm2, so average
compressive stress in column will be
 cr
 
Pcr 460.8 103 N
2



150
.
6
N/mm
2
A
3060 m
Since cr < Y = 410 MPa, buckling will occur before
the material yields.
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13. Buckling of Columns
EXAMPLE 13.3 (SOLN)
NOTE: From Eqn 13-11, we see that buckling always
occur about the column axis having the largest
slenderness ratio. Thus using data for the radius of
gyration from table in Appendix B,
 KL   4 m1000 mm/m  60.4


66.2 mm
 r x
 KL   2.8 m1000 mm/m  114.3


24.5 mm
 r y
Hence, y-y axis buckling will occur, which is the same
conclusion reached by comparing Eqns 13-11 for
both axes.
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
• The actual criterion for load application on a column
is limited to either a specified deflection of the
column or by not allowing the maximum stress in
the column exceed an
allowable stress.
• We apply a load P to column
at a short eccentric distance
e from centroid of x-section.
• This is equivalent to applying
a load P and moment
M’ = Pe.
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
• From free-body diagram, internal moment in column
is
13 - 13
M   Pe   
• Thus, the general solution for the differential eqn of
the deflection curve is
P
P
13 - 14
  C1 sin
x  C2 cos
xe
EI
EI
• Applying boundary conditions to determine the
constants, deflection curve is written as
  P L  P 
 P  
13 - 15
  e tan
s   cos
x   1
 sin 
 EI  
  EI 2   EI 
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
Maximum deflection
• Due to symmetry of loading, both maximum
deflection and maximum stress occur at column’s
midpoint. Therefore, when x = L/2,  = max, so
max
  P L 
 e sec
  1
  EI 2  
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13 - 16 
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
Maximum deflection
• Therefore, to find Pcr, we require
 Pcr L 
sec

 EI 2 
Pcr L 

EI 2 2
Pcr 
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 2 EI
2
L
13 - 17 
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
The secant formula
• Maximum stress in column occur when
maximum moment occurs at the
column’s midpoint.
Using Eqns 13-13 and 13-16,
M  Pe  max 
 P L
13 - 18
M  Pe sec

 EI 2 
• Maximum stress is compressive and
P Mc
P Pec  P L 
 max  
;
 max  
sec

A I
A
I
 EI 2 
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
The secant formula
• Since radius of gyration r2 = I/A,
P  ec  L P 
13 - 19
 max  1  2 sec

A r
 2r EA 
• max = maximum elastic stress in column, at inner
concave side of midpoint (compressive).
• P = vertical load applied to the column. P < Pcr
unless e = 0, then P = Pcr (Eqn 13-5)
• e = eccentricity of load P, measured from the neutral
axis of column’s x-sectional area to line of action of
P.
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
The secant formula
• c = distance from neutral axis to outer fiber of
column where maximum compressive stress max
occurs.
• A = x-sectional area of column
• L = unsupported length of column in plane of
bending. For non pin-supported columns, Le should
be used.
• E = modulus of elasticity of material.
• r = radius of gyration, r = √(I/A), where I is computed
about the neutral or bending axis.
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
Design
• Once eccentricity ratio has been determined,
column data can be substituted into Eqn 13-19.
• For max = Y, corresponding load PY is determined
from a trial-and-error procedure, since eqn is
transcendental and cannot be solved explicitly for
PY.
• Note that PY will always be smaller than the critical
load Pcr, since Euler’s formula assumes
unrealistically that column is axially loaded without
eccentricity.
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13. Buckling of Columns
*13.4 THE SECANT FORMULA
IMPORTANT
• Due to imperfections in manufacturing or application
of the load, a column will never suddenly buckle,
instead it begins to bend.
• The load applied to a column is related to its
deflections in a nonlinear manner, so the principle of
superposition does not apply.
• As the slenderness ratio increases, eccentrically
loaded columns tend to fail at or near the Euler
buckling load.
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13. Buckling of Columns
EXAMPLE 13.6
The W20059 A-36 steel
column shown is fixed at its
base and braced at the top so
that it is fixed from
displacement, yet free to
rotate about the y-y axis.
Also, it can sway to the side in
the y-z plane. Determine the
maximum eccentric load the
column can support before it
either begins to buckle or the
steel yields.
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13. Buckling of Columns
EXAMPLE 13.6 (SOLN)
From support conditions, about the y-y
axis, the column behaves as if it was
pinned at the top, fixed at the base and
subjected to an axial load P.
About the x-x axis, the column is free at
the top and fixed at the base, and
subjected to both axial load P and
moment M = P(200 mm).
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13. Buckling of Columns
EXAMPLE 13.6 (SOLN)
y-y axis buckling:
Effective length factor is Ky = 0.7, so (KL)y = 0.7(4 m)
= 2.8 m = 2800 mm. Using table in Appendix B to
determine Iy for the section and applying Eqn 13-11,
Pcr  y 
 2 EI y
2
KL y

  

 
 2 200 103 N/mm2 20.4 106 mm4
2800 mm
2
 5136247 N  5136 kN
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13. Buckling of Columns
EXAMPLE 13.6 (SOLN)
x-x axis yielding:
Kx = 2, so (KL)x = 2(4 m) = 8 m = 8000 mm. From
table in Appendix B, A = 7580 mm2, c = 210 mm/2 =
105 mm, and rx = 89.9 mm, applying secant formula,
Px  ec   KL x Px 
 Y  1  2 sec

A  rx
EA 
 2rx

Px  200  105  8000
Px
250 
1
sec


2
7580 
89.9
 289.9  200103   7580 


1.895  10  Px 1  2.598 sec 1.143  10
6
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3
Px

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13. Buckling of Columns
EXAMPLE 13.6 (SOLN)
x-x axis yielding:
Solving for Px by trial and error, noting that argument
for secant is in radians, we get
Px  419368 N  419.4 kN
Since this value is less than (Pcr)y = 5136 kN, failure
will occur about the x-x axis.
Also,  = 419.4103 N / 7580 mm2
= 55.3 MPa < Y = 250 MPa.
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