Retrofit of Steel Moment Frame Connections: Current Testing Program

1
Teaching Modules for Steel Instruction
Compression Member Design
Developed by Scott Civjan University of
Massachusetts, Amherst
2
P
COMPRESSION MEMBER/COLUMN Structural member
subjected to axial load
P
3

• Compression Members
• Chapter E Compression Strength
• Chapter I Composite Member Strength
• Part 4 Design Charts and Tables
• Chapter C Analysis Issues

4
Strength design requirements Pu ? ?Pn (Pa ?
Pn/O)ASD Where ? 0.9 for compression (O
1.67)ASD
5
Axial Strength

• Strength Limit States
• Squash Load
• Global Buckling
• Local Buckling

6
Local Flange Buckling
Global Buckling
Local Web Buckling
7
INDIVIDUAL COLUMN
8
Squash Load Fully Yielded Cross Section
9
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/AFy.
P
PFyA
D
L0
P
D
eyL0
10
RESIDUAL STRESSES
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.
11
RESIDUAL STRESSES
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.
No Residual Stress
PFyA
D
eyL0
11
Compression Theory
12
Yielded Steel
RESIDUAL STRESSES
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.
1
No Residual Stress
PFyA
P(Fy-Fres)A
1
D
eyL0
12
Compression Theory
13
Yielded Steel
RESIDUAL STRESSES
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.
1
2
No Residual Stress
PFyA
2
P(Fy-Fres)A
1
D
eyL0
13
Compression Theory
14
Yielded Steel
RESIDUAL STRESSES
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.
1
2
No Residual Stress
PFyA
3
2
P(Fy-Fres)A
1
3
D
eyL0
14
Compression Theory
15
Yielded Steel
RESIDUAL STRESSES
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.
1
2
No Residual Stress
PFyA
3
2
Effects of Residual Stress
P(Fy-Fres)A
1
3
D
eyL0
Compression Theory
4
15
16
Euler Buckling
17
Euler Buckling

• 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.

18
Euler Buckling
P
P
Bifurcation Point
Stable Equilibrium
D
D
s
E
e
19
Euler Buckling
Dependant on Imin and L2. Independent of Fy.
For similar unbraced length in each direction,
minor axis (Iy in a W-shape) will control
strength.
PE
Major axis buckling
Minor axis buckling
L
19
Compression Theory
20
Euler Buckling
Re-write in terms of stress
PE divide by A, PE/A , then with r2
I/A,
PE/A FE   FE Euler (elastic) buckling
stress L/r slenderness ratio
21
Euler Buckling
Buckling controlled by largest value of L/r.
Most slender section buckles first.
Fy
FE
L/r
22
EULER ASSUMPTIONS (ACTUAL BEHAVIOR)
23
Initial Crookedness/Out of Straight
?0 initial mid-span deflection of column
P
Do
Do
M PDo
P
24
Initial Crookedness/Out of Straight
P
Do 0
D
Do
25
Initial Crookedness/Out of Straight
P
Do 0
Elastic theory
D
Do
25
Compression Theory
26
Initial Crookedness/Out of Straight
P
Do 0
Elastic theory
Actual Behavior
D
Do
26
Compression Theory
27
Initial Crookedness/Out of Straight
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 loss small ?0 gt small
loss in strengths large ?0 gt strength loss can
be substantial
ASTM limits of ?0 L/1000 or 0.25 in 20
feet Typical values are ?0 L/1500 or 0.15 in
20 feet
28
Load Eccentricity
e

P
P
Do 0
Elastic theory
D
L
D
D
28
Compression Theory
29
Load Eccentricity
e

P
P
Do 0
Elastic theory
D
L
Actual Behavior
D
D
29
Compression Theory
30
Load Eccentricity
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 loss small e gt small
loss in strengths large e gt strength loss can
be substantial
If moment is significant section must be
designed as a member subjected to combined loads.
31
End Restraint (Fixed)

Set up equilibrium and solve similarly to Euler
buckling derivation. Determine a K-factor.
EXAMPLE
KL
Similar to pin-pin, with L L/2. Load Strength
4 times as large.
32
End Restraint (Fixed)
Effective Length KL
Length of equivalent pin ended column with
similar elastic buckling load,
Distance between points of inflection in the
buckled shape.
33
Handout on K-factors EquivalentLength.pdf
34
Inelastic Material Effects
Fy
s
ET Tangent Modulus
(Fy-Fres)
E
e
Test Results from an Axially Loaded Stub Column
35
Inelastic Material Effects
Elastic Behavior
s
KL/r
35
Compression Theory
36
Inelastic Material Effects
Fy
Inelastic
Fy-Fres
Elastic
s
KL/r
36
Compression Theory
37
Inelastic Material Effects
Fy
Inelastic
Fy-Fres
Elastic
s
KL/r
37
Compression Theory
38
Inelastic Material Effects
Two classes of buckling
Elastic Buckling ET E No yielding prior to
buckling Fe ? Fy-Fres(max) Fe predicts
buckling (EULER BUCKLING)
 Inelastic Buckling Some yielding/loss of
stiffness prior to buckling Fe gt
Fy-Fres(max) Fc - predicts buckling (INELASTIC
BUCKLING)
39
Overall Column Strength
Fy
Experimental Data
KL/r
40
Overall Column Strength
Inelastic Material effects Including Residual
Stresses
Fy
Experimental Data
KL/r
Out of Straightness
40
Compression Theory
41
Overall Column Strength
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.
The latter 2 items are highly variable between
specimens.
42

• Chapter E
• Compression Strength

43
Compression Strength
fc 0.90 (Wc 1.67)
44
Compression Strength
Specification considers the following conditions
Flexural Buckling Torsional Buckling Flexural-Tors
ional Buckling
45

• Compressive Strength

46
Compression Strength

• The following slides assume
• Non-slender flange and web sections
• Doubly symmetric members

47
Compression Strength
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 as Pn FcrAg Equation
E3-1
48
If , then . Equation E3-2 This defines
the inelastic buckling limit.
If , then Fcr 0.877Fe Equation E3-3 This
defines the elastic buckling limit with a
reduction factor, 0.877, times the theoretical
limit.
Fe elastic (Euler) buckling stress,
Equation E3-4
49
Inelastic Material Effects
Elastic Behavior
s
KL/r
49
50
Inelastic Material Effects
Fy
Inelastic
Fy-Fres
Elastic
s
KL/r
50
51
Inelastic Material Effects
Fy
Inelastic
Fy-Fres
Elastic
s
KL/r
51
52
Inelastic Material Effects
Fy
Inelastic
0.44Fy
Elastic
s
KL/r
52
53
Design Aids
Table 4-22 fcFcr as a function of KL/r
Useful for all shapes. Larger KL/r value controls.
Tables 4-1 to 4-20 fcPn as a function of KLy
Can be applied to KLx by dividing KLy by rx/ry.
54

• Slenderness Criteria

55
Per Section E.2 Recommended to provide KL/r less
than 200
56
LOCAL BUCKLING
57
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.
58
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.
58
Compression Theory
59
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.
59
Compression Theory
60
Local Buckling is related to Plate Buckling
Web is restrained by the flanges.
Failure is localized at areas of high stress
(maximum moment) or imperfections.
61
Local Buckling is related to Plate Buckling
Web is restrained by the flanges.
Failure is localized at areas of high stress
(maximum moment) or imperfections.
61
Compression Theory
62
Local Buckling is related to Plate Buckling
Web is restrained by the flanges.
Failure is localized at areas of high stress
(maximum moment) or imperfections.
62
Compression Theory
63

• Local Buckling
• Criteria in Table B4.1
• Strength in Chapter E Members with Slender
  Elements

64
Local Buckling Criteria Slenderness of the flange
and web, l, 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
65
Local Buckling
? gt ?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.
66

• Section E7
• Compression Strength
• Members with Slender Elements

67
THE FOLLOWING SLIDES CONSIDER SLENDER FLANGES AND
SLENDER WEBS NOT COMMON FOR W-SHAPES!!
68
If , then Equation E7-2 This defines
inelastic buckling limit.
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.
Fe elastic (Euler) buckling stress For a
doubly symmetric section, Equation E3-4
69
Compression Strength Slender Sections
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
70
Compression Strength Slender Sections
Qs for Unstiffened Elements
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
For lt b/t lt Equation E7-5
For b/t Equation E7-6
71
Compression Strength Slender Sections
Qs for Unstiffened Elements
For projections from built-up shapes (except for
single angles) Base on slenderness b/t.
For b/t Qs 1.0 Equation E7-7
For lt b/t lt Equation E7-8
For b/t Equation E7-9
72
Compression Strength Slender Sections
Qs for Unstiffened Elements
kc shall not be taken less than 0.35 nor greater
than 0.76 for calculation purposes.
73
Compression Strength Slender Sections
Qs for Unstiffened Elements
For single angles and stems of T sections see
sections E7.1c and E7.1d respectively.
74
Compression Strength Slender Sections
Qa for Stiffened Elements
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
75
Compression Strength Slender Sections
Qa for Stiffened Elements
Base on slenderness b/t. (ratio is h/tw
for a W-shape)
For b/t Equation E7-17
f Fcr as calculated assuming Q 1.0 or,
conservatively, can use f Fy.
76
FULL STRUCTURE BEHAVIOR
77
ALIGNMENT CHART OR DIRECT ANALYSIS METHODS
78
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.
79
DIRECT ANALYSIS METHOD
Analysis of entire structure interaction.
Include lateral Notional loads.
Reduce stiffness of structure.
All members must be evaluated under combined
axial and flexural load.
No K values required.
80
ALIGNMENT CHART METHOD IS USED FOR THE FOLLOWING
SLIDES
81
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.
82
K-FACTORS FOR END CONSTRAINTS
No Joint Translation Allowed Sidesway
Inhibited 0.5 ? K ? 1.0
Joint Translation Allowed Sidesway
Uninhibited 1.0 ? K ? ?
83
K-FACTORS FOR END CONSTRAINTS
Two categories, Braced Frames, 0.5 ? K ?
1.0 Sway Frames, K 1.0
Behavior of individual column unchanged (Frame
merely provides end conditions).
84
Sidesway Prevented
Floors do not translate relative to one another
in-plane.
Typically, members are pin connected to save cost.
85
Sidesway Prevented
Assume girder/beam infinitely rigid or flexible
compared to columns to bound results.
K0.5
K0.7
K1
K0.7
86
Sidesway Prevented
Equivalent
Idealized
Shear Wall
87
Sidesway Prevented
Equivalent
Idealized
Shear Wall
87
Compression Theory
88
Sidesway Prevented
Equivalent
Idealized
Shear Wall
88
Compression Theory
89
Sidesway Prevented
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 lt 1).
Typical design will assume K 1 as a
conservative upper bound (actual K 0.8 not much
difference from K 1 in design).
90
Sway Frame
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.
91
Sway Frame
Assume girder/beam infinitely rigid or flexible
compared to columns to bound results.
K2
K1
K 8
K2
92
Sway Frame
Moment Frame
92
Compression Theory
93
Sway Frame
Moment Frame
93
Compression Theory
94
Sway Frame
Moment Frame
94
Compression Theory
95
Alignment Charts
Calculate G at the top and bottom of the column
(GA and GB).
I moment of inertia of the members L length
of the member between joints
G is inversely proportional to the degree of
rotational restraint at column ends.
96
Alignment Charts
Sidesway Inhibited (Braced Frame)
Sidesway UnInhibited (Sway Frame)
Separate Charts for Sidesway Inhibited and
Uninhibited
96
Compression Theory
97
Alignment Charts
Sidesway Inhibited (Braced Frame)
Sidesway UnInhibited (Sway Frame)
Separate Charts for Sidesway Inhibited and
Uninhibited
97
Compression Theory
98
Alignment Charts
K
K
Sidesway Inhibited (Braced Frame)
Sidesway UnInhibited (Sway Frame)
Separate Charts for Sidesway Inhibited and
Uninhibited
98
Compression Theory
99
Alignment Charts
Only include members RIGIDLY ATTACHED (pin ended
members are not included in G calculations).
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.
If column base is pinned theoretical G 8.
AISC recommends use of 10. If column base is
fixed theoretical G 0. AISC recommends use
of 1.
100
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

101
Alignment Charts
Lets evaluate the assumptions.
102
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

102
Compression Theory
103
Alignment Charts
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, t, times G.
(SRF Table 4-21)
104
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

104
Compression Theory
105
Alignment Charts
Only include members RIGIDLY ATTACHED (pin ended
members are not included in G calculations).
Partial restraint of connections and non-uniform
members effectively change the rotational
stiffness at the connections.
These conditions can be directly accounted for,
but are generally avoided in design.
106
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

106
Compression Theory
107
Alignment Charts
Calculation of G accounts for rotational
stiffness restraint at each joint based on
assumed bending.
For other conditions include a correction factor
m to account for actual rotational stiffness of
the girder at the joint.
108
Alignment Charts
Sidesway Inhibited (Braced) Assumption single
curvature bending of girder.
Bending Stiffness
Far end pinned
Bending Stiffness
m (3EI/L)/(2EI/L) 1.5
Far end fixed
Bending Stiffness
m (4EI/L)/(2EI/L) 2
109
Alignment Charts
Sidesway Uninhibited (Sway) Assumption reverse
curvature bending of girder.
Bending Stiffness
Far end pinned
Bending Stiffness
m (3EI/L)/(6EI/L) 1/2
Far end fixed
Bending Stiffness
m (4EI/L)/(6EI/L) 2/3
110
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

110
Compression Theory
111
Alignment Charts
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.
Design typically checks each story independently,
based on these assumptions.
Actual conditions can be directly accounted for,
but are generally ignored in design.
112
Alignment Charts

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

112
Compression Theory
113
Alignment Charts
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.
This case will be addressed first, with the
concept valid for general conditions as well.
114
LEANER COLUMNS
115
Leaner Columns
Leaner Columns
Moment Frame
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.
115
Compression Theory
116
Leaner Columns
For Leaner Columns G top Infinity G bottom
Infinity Therefore K Infinity KL Infinite So
the column has no strength according to the
alignment chart
Theoretically the column has an infinite
KL. Therefore, the strength should be zero.
117
Leaner Columns
Consider only applying a small load to the right
columns
Moment Frame
Leaner Columns
117
Compression Theory
118
Leaner Columns
Consider only applying a small load to the right
columns
Moment Frame
Leaner Columns
Surely a small load could be applied without
causing instability! (Due to connection to the
rest of the structure)
119
Leaner Columns
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.
120
Leaner Columns
P
D
PD/H
H
PD/H
P
Note that the result of a vertical force trying
to translate through displacement, D, is a
lateral load of value PD/H applied to the system.
121
Leaner Columns
P3
P2
P4
P1
2
3
1
4
leaner
SP SPe SP P1P2P3P4 SPe P1eP2eP3eP4eP
1eP4e
In the elastic range, the Sum of Forces concept
states that the total column capacities can be
re-distributed
122
Leaner Columns
P3
P2
P4
P1
2
3
1
4
leaner
If P2 P2e Reach failure even if SP lt SPe
However, the total load on a leaner column still
must not exceed the non-sway strength.
123
Leaner Columns
Actual design considers inelastic behavior of the
sections, but the basic concept is the same.
A system of columns for each story should be
considered.
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 (K1).
124
EXAMPLE DEMONSTRATION SEE YURA VIDEOS
125
Alignment Chart
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.
Once the limit against lateral buckling and
lateral restraint is reached, the entire story
will exhibit sidesway buckling.
126
Alignment Chart

• ALIGNMENT CHART ASSUMPTIONS
• Behavior is purely elastic.
• All members have constant cross section.
• All joints are rigid.
• Sidesway Inhibited (Braced) single curvature
  bending of girders.
• Sidesway Uninhibited (Sway) reverse curvature
  bending of girders.
• Stiffness parameter of all columns is equal.
• Joint restraint is distributed to columns above
  and below the joint in proportion to EI/L of the
  columns.
• All columns buckle simultaneously.
• No significant axial compression force exists in
  the girders.

126
Compression Theory
127
Alignment Chart
Axial load reduces bending stiffness of a section.
In girders, account for this with reduction
factor on EI/L.

128
Alignment Chart
It is helpful to think in terms of members
controlled by axial force or bending, rather than
girders and columns.
If axial load dominates, consider member a
column with extra strength to prevent the story
from buckling (sum of forces approach).
If bending load dominates, consider the member a
girder with reduced rotational stiffness at the
joint (axial load reduction).
129

• Alignment Chart Issues

130
Alignment Chart
To account for inelastic column
effects, stiffness reduction factors, ta, used to
reduce EI of the columns.
Stiffness Reduction Factors Table 4-21
131
Alignment Chart
If beams have significant axial load, they
provide less rotational restraint.
Reduce rotational stiffness component (EI/L) of
beams with modification,
1-Q/Qcr
Q axial load Qcr axial in-plane buckling
strength with K1
This is also valid for columns at a joint
(multiple stories), which carry minimal axial
load compared to their strengths.
132
Alignment Chart
To account for story buckling concept, all
columns must reach their capacity to allow for
story failure. Revise K to account for story
effects.
K2 from Equation C-A-7-8
Kn2 K factor directly from the alignment
chart Pr Load on the column (factored for LRFD)
133
DIRECT ANALYSIS METHOD IS USED FOR THE FOLLOWING
SLIDES
134
DIRECT ANALYSIS METHOD
Analysis of entire structure interaction.
Include lateral Notional loads.
Reduce stiffness of structure.
No K values required.
135
DIRECT ANALYSIS METHOD
Further evaluation of this method is included in
the module on Combined Forces.