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TRANSCRIPT
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COMPRESSION
MEMBER/COLUMN:
Structural member
subjected to axial load
P
P
2Compression Module
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3Compression Module
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Compression Module 4
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Compression Module 5
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8/92Compression Module 8
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Strength design requirements:
PuP
n (P
aP
n/)ASD
Where = 0.9 for compression(= 1.67)ASD
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Axial Strength
Strength Limit States:
Squash Load
Global Buckling
Local Buckling
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Global
Buckling
Local
Flange
Buckling
Local
Web
Buckling
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Over-all buckling Flexural
Torsional Torsional-flexural
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Flexural Buckling
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Torsional buckling
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Flexural-Torsional buckling
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Flexural-Torsional buckling
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INDIVIDUAL COLUMN
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Squash Load
Fully Yielded Cross Section
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When a short, stocky column is loaded the strength is limited by
the yielding of the entire cross section.
Absence of residual stress, all fibers of cross-section yield
simultaneously at P/A=Fy.
P=FyA
yL0P
P
L0
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Results in a reduction in the effective
stiffness of the cross section, but the
ultimate squash load is unchanged.
Reduction in effective stiffness caninfluence onset of buckling.
24Compression Theory
RESIDUAL STRESSES
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RESIDUAL STRESSES
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P=FyA
yL0
No Residual Stress
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With residual stresses, flange tips yield first at
P/A + residual stress = FyGradually get yield of entire cross section.
Stiffness is reduced after 1styield.
RESIDUAL STRESSES
= Yielded
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With residual stresses, flange tips yield first at
P/A + residual stress = FyGradually get yield of entire cross section.
Stiffness is reduced after 1styield.
P=FyA
yL0
RESIDUAL STRESSES
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P=(Fy-Fres)A
1
No Residual Stress
= Yielded
Steel
1
= Yielded
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With residual stresses, flange tips yield first at
P/A + residual stress = FyGradually get yield of entire cross section.
Stiffness is reduced after 1styield.
P=FyA
yL0
RESIDUAL STRESSES
28Compression Theory
P=(Fy-Fres)A
1
Yielded
Steel
2
No Residual Stress
1
2
= Yielded
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With residual stresses, flange tips yield first at
P/A + residual stress = FyGradually get yield of entire cross section.
Stiffness is reduced after 1styield.
P=FyA
yL0
RESIDUAL STRESSES
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P=(Fy-Fres)A
1
Yielded
Steel
1
2
2
3
3
No Residual Stress
= Yielded
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With residual stresses, flange tips yield first at
P/A + residual stress = FyGradually get yield of entire cross section.
Stiffness is reduced after 1styield.
P=FyA
yL0
RESIDUAL STRESSES
P=(Fy-Fres)A
1
Yielded
Steel
1
2
2
3
3
Effects of Residual
Stress
4
304
No Residual Stress
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Euler Buckling
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Assumptions:
Column is pin-ended.
Column is initially perfectly straight.
Load is at centroid.
Material is linearly elastic (no yielding).
Member bends about principal axis (no twisting).
Plane sections remain Plane.
Small Deflection Theory.
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Euler Buckling
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PE =
divide byA, PE/A = , then with r2 =I/A,
PE/A = FE =
FE = Euler (elastic) buckling stress
L/r= slenderness ratio
2
2
L
EI
2
2
AL
EI
( )22
rLE
Re-write in terms of stress:
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Euler Buckling
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E
P
2
2
L
EIPE=
Stable Equilibrium
Bifurcation Point
Euler Buckling
P
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Dependant onIminandL2.
Independent of Fy.
L
PE 2
2
L
EIx
2
2
L
EIy
Minor axis buckling
For similar unbraced length in each direction,minor axis (Iyin a W-shape) will control strength.
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Major axis buckling
Euler Buckling
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Buckling controlled by largest value ofL/r.
Most slender section buckles first.
L/r
FE
( )22
rL
EFy
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Euler Buckling
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( )
2
2
e
EIP
KL
=
( )
2
2
e
EI
FKL
r
=
2
2
2
2
)2/1(
4
L
EI
L
EIPE
==
Similar to pin-pin,
withL =L/2.
Load Strength =
4 times as large.
EXAMPLE
KL
Set up equilibrium and solve
similarly to Euler buckling
derivation.
Determine a K-factor.
End Restraint(Fixed)
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Length of equivalent pin ended
column with similar elastic
buckling load,
Effective Length = KL
End Restraint(Fixed)
Distance between points ofinflection in the buckled shape.
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Compression Module 44
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Compression Theory 45
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47Compression Theory
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EULER ASSUMPTIONS
(ACTUAL BEHAVIOR)
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Fy
KL/r
2
2
=
r
KL
EFE
Experimental Data
Overall Column Strength
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0 = initial mid-span deflection of column
Initial Crookedness/Out of Straight
P
P
M = Po
o
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o
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P
2
2
L
EIPE =
o= 0
o
Elastic theory
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Initial Crookedness/Out of Straight
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P
2
2
L
EIPE =
o
= 0
o
Elastic theory
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Actual Behavior
Initial Crookedness/Out of Straight
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Buckling is not instantaneous.
ASTM limits of 0 = L/1000 or 0.25 in 20 feetTypical values are 0 = L/1500 or 0.15 in 20 feet
Additional stresses due to bending of the column,
P/AMc/I.
Assuming elastic material theory (never yields),
Papproaches PE.
Actually, some strength loss
small 0=> small loss in strengthslarge 0 => strength loss can be substantial
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Initial Crookedness/Out of Straight
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P
e
L
Load Eccentricity
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P
2
2
L
EIPE =
o= 0
Elastic theory
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If moment is significant section must be designed
as a member subjected to combined loads.
Buckling is not instantaneous.
Additional stresses due to bending of the column,
P/AMc/I.
Assuming elastic material theory (never yields),
Papproaches PE.
Actually, some strength loss
small e => small loss in strengthslarge e => strength loss can be substantial
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Load Eccentricity
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Fy
ET= Tangent Modulus
E
(Fy-Fres)
Test Results from an Axially Loaded Stub Column
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Inelastic Material Effects
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KL/r
2
2
=
r
KL
EFe
Inelastic Material Effects
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Elastic Behavior
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KL/r
2
2
=
r
KL
EFe
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Fy-Fres
Fy
2
2
=
r
KL
EF Tc
Inelastic
Elastic
Inelastic Material Effects
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KL/r
2
2
=
r
KL
EFe
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Fy-Fres
Fy
2
2
=
r
KL
EF Tc
Inelastic
Elastic
Inelastic Material Effects
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Elastic Buckling:ET =E
No yielding prior to bucklingFe Fy-Fres(max)Fe = predicts buckling (EULER BUCKLING)
Two classes of buckling:
Inelastic Buckling:Some yielding/loss of stiffness prior to buckling
Fe > Fy-Fres(max)
Fc - predicts buckling (INELASTIC BUCKLING)
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Inelastic Material Effects
O ll C l St th
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Fy
KL/r
2
2
=
r
KL
EFE
Experimental Data
Overall Column Strength
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O ll C l St th
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Major factors determining strength:
1) Slenderness (L/r).
2) End restraint (Kfactors).
3) Initial crookedness or load eccentricity.
4) Prior yielding or residual stresses.
Overall Column Strength
The latter 2 items are highly variable between specimens.
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LOCAL BUCKLING
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Local Buckling is related to Plate Buckling
Flange is restrained by the web at one edge.
Failure is localized at areas of high stress
(maximum moment) or imperfections.
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Local Buckling is related to Plate Buckling
Failure is localized at
areas of high stress
(maximum moment) or
imperfections.
Web is restrained by the flanges.
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Local Buckling is related to Plate Buckling
Failure is localized at
areas of high stress
(maximum moment) or
imperfections.
Web is restrained by the flanges.
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Local Buckling is related to Plate Buckling
Failure is localized at
areas of high stress
(maximum moment) or
imperfections.
Web is restrained by the flanges.
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Compression Module 74
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Compression Module 75
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Chapter E:
Compression Strength
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Compression Strength
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c= 0.90 (c= 1.67)
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Compression Strength
Compression Strength
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Specification considers the following conditions:
Flexural Buckling
Torsional BucklingFlexural-Torsional Buckling
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Compression Strength
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Compressive Strength
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Compression Strength
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The following slides assume:
Non-slender flange and web sections
Doubly symmetric members
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Compression Strength
Compression Strength
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Since members are non-slender and doubly symmetric,
flexural (global) buckling is the most likely potential failure
mode prior to reaching the squash load.
Buckling strength depends on the slenderness of the section,
defined as KL/r.
The strength is defined asPn= FcrAg Equation E3-1
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Compression Strength
EKLFy
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Fe = elastic (Euler) buckling stress, Equation E3-4
If , then Fcr = 0.877Fe. Equation E3-3
This defines the elastic buckling limitwith a reduction factor, 0.877, times the theoretical limit.
If , then . Equation E3-2
This defines the inelastic buckling limit.
yF
E.
r
KL714 y
F
cr F.F e
= 6580
yF
E.
r
KL714>
2
2
=
r
KL
E
Fe
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I l i M i l Eff
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KL/r
2
2
=
r
KL
EFe
Inelastic Material Effects
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Elastic Behavior
I l ti M t i l Eff t
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KL/r
2
2
=
r
KL
EFe
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Fy-Fres
Fy
2
2
=
r
KL
EF Tc
Inelastic
Elastic
Inelastic Material Effects
I l ti M t i l Eff t
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KL/r
2
2
=
r
KL
EFe
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Fy-Fres
Fy
2
2
=
r
KL
EF Tc
Inelastic
Elastic
Inelastic Material Effects
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Design Aids
Table 4-22
cFcras a function of KL/r
Tables 4-1 to 4-20
cPnas a function of KLy
Useful for all shapes.Larger KL/rvalue controls.
Can be applied to KLxby
dividing KLyby rx/ry.
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L l B kli C it i
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Local Buckling Criteria
Slenderness of the flange and web, , are used as criteria todetermine whether local buckling might control in the elastic or
inelastic range, otherwise the global buckling criteria controls.
Criteria r are based on plate buckling theory.
For W-Shapes
FLB, = bf
/2tf
rf
=
WLB, = h/tw rw =
yF
E.560
yF
E.491
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Local Buckling
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> r slender element
Failure by local buckling occurs.
Covered in Section E7
Many rolled W-shape sections are dimensioned such
that the full global criteria controls.
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Slenderness Criteria
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Per Section E.2
Recommended to provide
KL/rless than 200