Download - Compression Notes14th
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Developed by Scott CivjanUniversity of Massachusetts, Amherst
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COMPRESSION MEMBER/COLUMN: Structural member subjected to axial load
P
P
2Compression Module
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Compression – AISC Manual 14th Ed
Compression Members: Chapter E: Compression Strength Chapter I: Composite Member
Strength Part 4: Design Charts and Tables Chapter C: Analysis Issues
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Strength design requirements:
Pu Pn (Pa Pn/Ω)ASD
Where = 0.9 for compression(Ω = 1.67)ASD
4Compression Module
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Axial Strength
Strength Limit States:• Squash Load• Global Buckling• Local Buckling
5Compression Module
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Global Buckling
Local Flange Buckling
Local Web Buckling
6Compression Module
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INDIVIDUAL COLUMN
7Compression Theory
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Squash LoadFully Yielded Cross Section
8Compression Theory
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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
yL0
P
PL0
9Compression Theory
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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 can influence onset of buckling.
10Compression Theory
RESIDUAL STRESSES
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P=FyA
yL0
No Residual Stress
11Compression Theory
With residual stresses, flange tips yield first at P/A + residual stress = Fy
Gradually get yield of entire cross section.
Stiffness is reduced after 1st yield.
RESIDUAL STRESSES
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With residual stresses, flange tips yield first at P/A + residual stress = Fy
Gradually get yield of entire cross section.
Stiffness is reduced after 1st yield.
P=FyA
yL0
RESIDUAL STRESSES
12Compression Theory
P=(Fy-Fres)A 1
No Residual Stress
= YieldedSteel
1
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With residual stresses, flange tips yield first at P/A + residual stress = Fy
Gradually get yield of entire cross section.
Stiffness is reduced after 1st yield.
P=FyA
yL0
RESIDUAL STRESSES
13Compression Theory
P=(Fy-Fres)A 1
= YieldedSteel
2
No Residual Stress
1
2
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With residual stresses, flange tips yield first at P/A + residual stress = Fy
Gradually get yield of entire cross section.
Stiffness is reduced after 1st yield.
P=FyA
yL0
RESIDUAL STRESSES
14Compression Theory
P=(Fy-Fres)A 1
= YieldedSteel
1
2
2
3
3
No Residual Stress
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With residual stresses, flange tips yield first at P/A + residual stress = Fy
Gradually get yield of entire cross section.
Stiffness is reduced after 1st yield.
P=FyA
yL0
RESIDUAL STRESSES
Compression Theory
P=(Fy-Fres)A 1
= YieldedSteel
1
2
2
3
3
Effects of Residual Stress
4
154
No Residual Stress
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Euler Buckling
16Compression Theory
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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.
17Compression Theory
Euler Buckling
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E
P
2
2π
L
EIPE
Stable Equilibrium
Bifurcation Point
Euler Buckling
P
18Compression Theory
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Dependant on Imin and L2.Independent of Fy.
L
PE 2
2π
L
EI x
2
2π
L
EI y
Minor axis buckling
For similar unbraced length in each direction, “minor axis” (Iy in a W-shape) will control strength.
19Compression Theory
Major axis buckling
Euler Buckling
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PE =
divide by A, PE/A = , then with r2 = I/A,
PE/A = FE = FE = Euler (elastic) buckling stressL/r= slenderness ratio
2
2π
L
EI
2
2π
AL
EI
22π
rL
E
Re-write in terms of stress:
20Compression Theory
Euler Buckling
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Buckling controlled by largest value of L/r. Most slender section buckles first.
L/r
FE
22π
rL
EFy
21Compression Theory
Euler Buckling
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EULER ASSUMPTIONS(ACTUAL BEHAVIOR)
22Compression Theory
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0 = initial mid-span deflection of column
Initial Crookedness/Out of Straight
P
P
M = Po
o
23Compression Theory
o
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P
2
2π
L
EIPE
o= 0
o
24Compression Theory
Initial Crookedness/Out of Straight
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P
2
2π
L
EIPE
o= 0
o
Elastic theory
25Compression Theory
Initial Crookedness/Out of Straight
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P
2
2π
L
EIPE
o= 0
o
Elastic theory
26Compression Theory
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/A Mc/I.
Assuming elastic material theory (never yields), P approaches PE.
Actually, some strength losssmall 0 => small loss in strengthslarge 0 => strength loss can be substantial
27Compression Theory
Initial Crookedness/Out of Straight
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Pe
L
Load Eccentricity
28Compression Theory
P
2
2π
L
EIPE
o= 0
Elastic theory
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Pe
L
Load Eccentricity
29Compression Theory
P
2
2π
L
EIPE
o= 0
Elastic theory
Actual Behavior
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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/A Mc/I.
Assuming elastic material theory (never yields), P approaches PE.
Actually, some strength losssmall e => small loss in strengthslarge e => strength loss can be substantial
30Compression Theory
Load Eccentricity
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2
2
πe
EIP
KL
2
2
πe
EIF
KLr
2
2
2
2
)2/1(
ππ4
L
EI
L
EIPE
Similar to pin-pin, with L’ = 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)
31Compression Theory
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Length of equivalent pin ended column with similar elastic buckling load,
Effective Length = KL
End Restraint (Fixed)
Distance between points of inflection in the buckled shape.
32Compression Theory
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Handout on K-factorsEquivalentLength.pdf
33Compression Theory
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Fy
ET= Tangent Modulus
E
(Fy-Fres)
Test Results from an Axially Loaded Stub Column34Compression Theory
Inelastic Material Effects
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KL/r
2
2π
rKL
EFe
Inelastic Material Effects
35Compression Theory
Elastic Behavior
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KL/r
2
2π
rKL
EFe
36Compression Theory
Fy-Fres
Fy
2
2π
rKL
EF T
c
Inelastic
Elastic
Inelastic Material Effects
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KL/r
2
2π
rKL
EFe
37Compression Theory
Fy-Fres
Fy
2
2π
rKL
EF T
c
Inelastic
Elastic
Inelastic Material Effects
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Elastic Buckling: ET = E No yielding prior to buckling Fe Fy-Fres(max)
Fe = predicts buckling (EULER BUCKLING)
Two classes of buckling:
Inelastic Buckling:Some yielding/loss of stiffness prior to bucklingFe > Fy-Fres(max)
Fc - predicts buckling (INELASTIC BUCKLING)
38Compression Theory
Inelastic Material Effects
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Fy
KL/r
2
2π
rKL
EFE
Experimental Data
Overall Column Strength
39Compression Theory
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Fy
KL/r
2
2π
rKL
EFE
Experimental Data
Inelastic Material effects Including Residual Stresses
Out of Straightness
Overall Column Strength
40Compression Theory
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Major factors determining strength:1) Slenderness (L/r).2) End restraint (K factors).3) Initial crookedness or load eccentricity.4) Prior yielding or residual stresses.
Overall Column Strength
The latter 2 items are highly variable between specimens.
41Compression Theory
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Chapter E: Compression Strength
42Compression – AISC Manual 14th Ed
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c= 0.90 (c= 1.67)
43
Compression Strength
Compression – AISC Manual 14th Ed
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Specification considers the following conditions:
Flexural BucklingTorsional BucklingFlexural-Torsional Buckling
44
Compression Strength
Compression – AISC Manual 14th Ed
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Compressive Strength
45Compression – AISC Manual 14th Ed
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The following slides assume: Non-slender flange and web sections Doubly symmetric members
46
Compression Strength
Compression – AISC Manual 14th Ed
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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
47
Compression Strength
Compression – AISC Manual 14th Ed
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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
F
cr F.F e
y
6580
yF
E.
r
KL714
2
2π
rKL
EFe
48Compression – AISC Manual 14th Ed
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KL/r
2
2π
rKL
EFe
Inelastic Material Effects
49
Elastic Behavior
Compression – AISC Manual 14th Ed
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KL/r
2
2π
rKL
EFe
50
Fy-Fres
Fy
2
2π
rKL
EF T
c
Inelastic
Elastic
Inelastic Material Effects
Compression – AISC Manual 14th Ed
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KL/r
2
2π
rKL
EFe
51
Fy-Fres
Fy
2
2π
rKL
EF T
c
Inelastic
Elastic
Inelastic Material Effects
Compression – AISC Manual 14th Ed
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KL/r
2
2π
rKL
EFe
52
Fy
Inelastic
Elastic
Inelastic Material Effects
yF
F
cr F.F e
y
6580
ecr F.F 8770
yF
E.714
0.44Fy
Compression – AISC Manual 14th Ed
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Design Aids
Table 4-22cFcr as a function of KL/r
Tables 4-1 to 4-20cPn as a function of KLy
Useful for all shapes.Larger KL/r value controls.
Can be applied to KLx by dividing KLy by rx/ry.
53Compression – AISC Manual 14th Ed
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Slenderness Criteria
54Compression – AISC Manual 14th Ed
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Per Section E.2
Recommended to provide KL/r less than 200
55Compression – AISC Manual 14th Ed
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LOCAL BUCKLING
56Compression Theory
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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.
57Compression Theory
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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.
58Compression Theory
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Local Buckling is related to Plate Buckling
Flange is restrained by the web at one edge.
59Compression Theory
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.
60Compression Theory
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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.
61Compression Theory
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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.
62Compression Theory
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Local Buckling: Criteria in Table B4.1 Strength in Chapter E: Members with Slender
Elements
63Compression – AISC Manual 14th Ed
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Local Buckling CriteriaSlenderness of the flange and web, , are used as criteria to determine 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
64Compression – AISC Manual 14th Ed
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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.
65
Local Buckling
Compression – AISC Manual 14th Ed
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Section E7: Compression StrengthMembers with Slender Elements
66Compression – AISC Manual 14th Ed
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THE FOLLOWING SLIDES CONSIDER SLENDER FLANGES AND SLENDER WEBSNOT COMMON FOR W-SHAPES!!
67Compression – AISC Manual 14th Ed
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Fe = elastic (Euler) buckling stress For a doubly symmetric section, Equation E3-4
If , then Fcr = 0.877Fe. Equation E7-3
This defines “elastic” buckling limit similar to non-slender elements. Q has no impact in this region.
If , then Equation E7-2
This defines “inelastic” buckling limit.
yQF
E.
r
KL714
yF
QF
cr F.QF e
y
6580
yQF
E.
r
KL714
2
2π
rKL
EFe
68Compression – AISC Manual 14th Ed
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Q = Reduction Factor for local buckling effects.Equations E7-4 to E7-16
Q = 1 when section is non-slender. No reduction from Section E3.
Q = QsQa for slender sections.
Qs = Reduction Factor for slender unstiffened element
Qa = Reduction Factor for slender stiffened element
69
Compression Strength – Slender Sections
Compression – AISC Manual 14th Ed
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For projections from rolled shapes (except for single angles)Base on slenderness b/t. (ratio is bf/2tf for a W-shape)
For b/t ≤ Qs = 1.0 Equation E7-4 yF
E.560
For < b/t <
Equation E7-5
E
F
t
b.-.Q y
s
7404151
yF
E.031
yF
E.560
For b/t ≥ Equation E7-6 2690
tbF
E.Q
y
s yF
E.031
70
Qs for Unstiffened Elements
Compression Strength – Slender Sections
Compression – AISC Manual 14th Ed
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For projections from built-up shapes (except for single angles)Base on slenderness b/t.
For b/t ≤ Qs = 1.0 Equation E7-7 y
c
F
Ek.640
For < b/t <
Equation E7-8
c
ys Ek
F
t
b.-.Q
6504151
y
c
F
Ek.171
y
c
F
Ek.640
For b/t ≥ Equation E7-9 2900
tbF
Ek.Q
y
cs
y
c
F
Ek.171
71
Qs for Unstiffened Elements
Compression Strength – Slender Sections
Compression – AISC Manual 14th Ed
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w
c
th
k4
kc shall not be taken less than 0.35 nor greater than 0.76 for calculation purposes.
72
Qs for Unstiffened Elements
Compression Strength – Slender Sections
Compression – AISC Manual 14th Ed
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For single angles and stems of T sectionssee sections E7.1c and E7.1d respectively.
73
Qs for Unstiffened Elements
Compression Strength – Slender Sections
Compression – AISC Manual 14th Ed
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Qa = Ae/Ag
Ag = gross cross sectional area of the member
Ae = effective area of the cross section based on the reduced effective width be
74
Compression Strength – Slender Sections
Qa for Stiffened Elements
Compression – AISC Manual 14th Ed
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Base on slenderness b/t. (ratio is h/tw for a W-shape)
f = Fcr as calculated assuming Q = 1.0or, conservatively, can use f = Fy.
For b/t ≥
Equation E7-17
0 34
1 92 1e
E . Eb . t b
bf ft
f
E.491
75
Compression Strength – Slender Sections
Qa for Stiffened Elements
Compression – AISC Manual 14th Ed
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FULL STRUCTURE BEHAVIOR
76Compression Theory
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ALIGNMENT CHARTORDIRECT ANALYSIS METHODS
77Compression Theory
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Does not redistribute restraining moments into girders/beams.
ALIGNMENT CHART
“Traditional Method”
Determine effective length, KL, for each column.
Basis for design similar to individual columns.
78Compression Theory
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DIRECT ANALYSIS METHOD
Analysis of entire structure interaction.
Include lateral “Notional” loads.
All members must be evaluated under combined axial and flexural load.
No K values required.
Reduce stiffness of structure.
79Compression Theory
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ALIGNMENT CHART METHODIS USED FOR THE FOLLOWING SLIDES
80Compression Theory
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ALIGNMENT CHART
“Traditional Method”
Determine effective length, KL, for each column.
Basis for design similar to individual columns.
Does not redistribute restraining moments into girders/beams.
81Compression Theory
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K-FACTORS FOR END CONSTRAINTS
No Joint Translation Allowed – Sidesway Inhibited0.5 K 1.0
Joint Translation Allowed – Sidesway Uninhibited1.0 K
82Compression Theory
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K-FACTORS FOR END CONSTRAINTS
Behavior of individual column unchanged (Frame merely provides end conditions).
Two categories, Braced Frames, 0.5 K 1.0Sway Frames, K ≥ 1.0
83Compression Theory
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Floors do not translate relative to one another in-plane.
Typically, members are pin connected to save cost.
84Compression Theory
Sidesway Prevented
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Assume girder/beam infinitely rigid or flexible compared to columns to bound results.
K=0.7K=0.5
K=1K=0.7
Sidesway Prevented
85Compression Theory
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Shear Wall
Idealized Equivalent
86Compression Theory
Sidesway Prevented
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Shear Wall
Idealized Equivalent
87Compression Theory
Sidesway Prevented
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Shear Wall
Idealized Equivalent
88Compression Theory
Sidesway Prevented
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Typically, members are pin-connected to save cost (K = 1).
If members include fixity at connections, Alignment Chart Method to account for rotational restraint (K < 1).
Typical design will assume K = 1 as a conservative upper bound (actual K ≈ 0.8 not much difference from K = 1 in design).
89Compression Theory
Sidesway Prevented
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Floors can translate relative to one another in-plane.
Enough members are fixed to provide stability.
Number of moment frames chosen to provide reasonable force distribution and redundancy.
90Compression Theory
Sway Frame
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Assume girder/beam infinitely rigid or flexible compared to columns to bound results.
K=2K=1
K = ∞K=2
Sway Frame
91Compression Theory
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Moment Frame
92Compression Theory
Sway Frame
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93Compression Theory
Moment Frame
Sway Frame
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94Compression Theory
Moment Frame
Sway Frame
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Calculate “G” at the top and bottom of the column (GA and GB).
G is inversely proportional to the degree of rotational restraint at column ends.
I = moment of inertia of the membersL = length of the member between joints
girders
columns
LEILEI
G
95Compression Theory
Alignment Charts
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Alignment Charts
Separate Charts for Sidesway Inhibited and Uninhibited
Sidesway Inhibited(Braced Frame)
Sidesway UnInhibited(Sway Frame)
96Compression Theory
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Alignment Charts
Separate Charts for Sidesway Inhibited and Uninhibited
Sidesway Inhibited(Braced Frame)
Sidesway UnInhibited(Sway Frame)
97Compression Theory
GtopX
GbottomX
GtopX
Gbottom
X
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Alignment Charts
Separate Charts for Sidesway Inhibited and Uninhibited
Sidesway Inhibited(Braced Frame)
Sidesway UnInhibited(Sway Frame)
98Compression Theory
GtopX
Gbottom
X
K
K
GtopX
GbottomX
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Use the IN-PLANE stiffness Ix if in major axis direction, Iy if in minor axis. Girders/Beams are typically bending about Ix when column restraint is considered.
Only include members RIGIDLY ATTACHED (pin ended members are not included in G calculations).
If column base is “pinned” – theoretical G = ∞. AISC recommends use of 10.If column base is “fixed” – theoretical G = 0.
AISC recommends use of 1.99Compression Theory
Alignment Charts
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
100Compression Theory
Alignment Charts
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Let’s evaluate the assumptions.
101Compression Theory
Alignment Charts
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
102Compression Theory
Alignment Charts
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If the column behavior is inelastic,
Yielding decreases stiffness of the column.
Relative joint restraint of the girders increases.
G therefore decreases, as does K.
Decrease is typically small.
Conservative to ignore effects.
Can account for effects by using a stiffness reduction factor, , times G.
(SRF Table 4-21)103Compression Theory
Alignment Charts
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
104Compression Theory
Alignment Charts
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These conditions can be directly accounted for, but are generally avoided in design.
Partial restraint of connections and non-uniform members effectively change the rotational stiffness at the connections.
Only include members RIGIDLY ATTACHED (pin ended members are not included in G calculations).
105Compression Theory
Alignment Charts
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
106Compression Theory
Alignment Charts
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Calculation of G accounts for rotational stiffness restraint at each joint based on assumed bending.
girders
columns
LEI
m
LEI
G
For other conditions include a correction factor “m” to account for actual rotational stiffness of the girder at the joint.
107Compression Theory
Alignment Charts
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Far end pinned
Bending Stiffness =
Bending Stiffness =
Bending Stiffness =
Sidesway Inhibited (Braced)Assumption: single curvature
bending of girder.
Far end fixed
108Compression Theory
Alignment Charts
2EI
L
3EI
Lm = (3EI/L)/(2EI/L) = 1.5
m = (4EI/L)/(2EI/L) = 2
4EI
L
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Far end pinned
Sidesway Uninhibited (Sway)Assumption: reverse curvature
bending of girder.
Far end fixed
Bending Stiffness =
Bending Stiffness =
Bending Stiffness =
109Compression Theory
Alignment Charts
6EI
L
3EI
Lm = (3EI/L)/(6EI/L) = 1/2
4EI
Lm = (4EI/L)/(6EI/L) = 2/3
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
110Compression Theory
Alignment Charts
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Design typically checks each story independently, based on these assumptions.
In general, columns are chosen to be a similar size for more than one story. For each column section this results in sections with extra strength in upper floors, and close to their strength in lower floors.
Actual conditions can be directly accounted for, but are generally ignored in design.
111Compression Theory
Alignment Charts
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ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
112Compression Theory
Alignment Charts
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This case will be addressed first, with the concept valid for general conditions as well.
In a story not all columns will be loaded to their full strength.Some are ready to buckle, while others have additional strength.
An extreme case of this is a “leaner” column.
113Compression Theory
Alignment Charts
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“LEANER” COLUMNS
114Compression Theory
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Leaner Columns
For this structure note that the right columns are pinned at each connection, and provide no bending restraint.Theoretically G at top and bottom is infinite.
115Compression Theory
Moment Frame Leaner Columns
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Theoretically the column has an infinite KL.Therefore, the strength should be zero.
For Leaner Columns:G top= InfinityG bottom= InfinityTherefore K= Infinity
KL= Infinite
So the column has no strength according to the alignment chart
116Compression Theory
Leaner Columns
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MomentFrame
Leaner Columns
Consider only applying a small load to the right columns
117Compression Theory
Leaner Columns
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MomentFrame
Surely a small load could be applied without causing instability! (Due to connection to the rest of the structure)
Leaner Columns
Consider only applying a small load to the right columns
118Compression Theory
Leaner Columns
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PA
K = infinity
Pn= zero
PA
K < infinity
Pn> zero
Actual ConditionChart
Provided that the moment frame is not loaded to its full strength, it can provide some lateral restraint to the leaner columns. This is indicated by the spring in the figure above.
119Compression Theory
Leaner Columns
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P
Note that the result of a vertical force trying to translate through displacement, is a lateral load of value P/H applied to the system.
P/H
H
P/H
P
120Compression Theory
Leaner Columns
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leaner
1 2 3 4
P1P2
P3 P4
ΣP = ΣPe
ΣP = P1+P2+P3+P4
ΣPe = P1e+P2e+P3e+P4e=P1e+P4e
In the elastic range, the “Sum of Forces” concept states that the total column capacities can be re-distributed
121Compression Theory
Leaner Columns
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leaner
1 2 3 4
P1P2
P3 P4
If P2 = P2e
Reach failure even if
ΣP < ΣPe
However, the total load on a leaner column still must not exceed the non-sway strength.
122Compression Theory
Leaner Columns
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A system of columns for each story should be considered.
Actual design considers inelastic behavior of the sections, but the basic concept is the same.
The strength of the story is the load which would cause all columns to sway.
The strength of an individual column is the load which would cause it to buckle in the non-sway mode (K=1).
123Compression Theory
Leaner Columns
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EXAMPLE DEMONSTRATION –SEE YURA VIDEOS
124Compression Theory
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Once the limit against lateral buckling and lateral restraint is reached, the entire story will exhibit sidesway buckling.
In general, each story is a system of columns which are loaded to varying degrees of their limiting strength.
Those with additional strength can provide lateral support to those which are at their sidesway buckling strength.
125Compression Theory
Alignment Chart
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Alignment Chart
ALIGNMENT CHART ASSUMPTIONS:1) Behavior is purely elastic.2) All members have constant cross section.3) All joints are rigid.4) Sidesway Inhibited (Braced) – single curvature bending of
girders.5) Sidesway Uninhibited (Sway) – reverse curvature bending of
girders.6) Stiffness parameter of all columns is equal.7) Joint restraint is distributed to columns above and below the joint
in proportion to EI/L of the columns.8) All columns buckle simultaneously.9) No significant axial compression force exists in the girders.
126Compression Theory
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Axial load reduces bending stiffness of a section.
In girders, account for this with reduction factor on EI/L.
127Compression Theory
Alignment Chart
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If bending load dominates, consider the member a “girder” with reduced rotational stiffness at the joint (axial load reduction).
If axial load dominates, consider member a “column” with extra strength to prevent the story from buckling (sum of forces approach).
It is helpful to think in terms of members controlled by axial force or bending, rather than “girders” and “columns.”
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Alignment Chart
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Alignment Chart Issues
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To account for inelastic column effects,stiffness reduction factors, a,used to reduce EI of the columns.
Stiffness Reduction FactorsTable 4-21
Alignment Chart
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If beams have significant axial load, they provide less rotational restraint.
1-Q/Qcr
Q = axial loadQcr = axial in-plane buckling strength with K=1
Reduce rotational stiffness component (EI/L) of beams with modification,
This is also valid for “columns” at a joint (multiple stories), which carry minimal axial load compared to their strengths.
Alignment Chart
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To account for story buckling concept, all columns must reach their capacity to allow for story failure.Revise K to account for story effects.
Alignment Chart
2
22
2
2
2
2 8
5
π
π
n
n
r
r
K
LK
EIΣ
ΣP
PL
EIK
Kn2 = K factor directly from the alignment chartPr = Load on the column (factored for LRFD)
K2 from Equation C-A-7-8
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DIRECT ANALYSIS METHODIS USED FOR THE FOLLOWING SLIDES
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DIRECT ANALYSIS METHOD
Analysis of entire structure interaction.
Include lateral “Notional” loads.
No K values required.
Reduce stiffness of structure.
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DIRECT ANALYSIS METHOD
Further evaluation of this method is included in the module on “Combined Forces.”
135Compression Theory