Beam

Columns
Members Under Combined Forces
Most beams and columns are subjected to some degree of both bending and axial load
e.g. Statically Indeterminate Structures
P
1
P
2
C
E
A
D
F
B
Interaction Formulas for Combined Forces
0
.
1
Resistance
Effects
Load
e.g. LRFD
If more than one resistance is involved consider interaction
0
.
1
2
1
LS
n
i
i
LS
n
i
i
R
Q
R
Q
Basis for Interaction Formulas
Tension/Compression & Single Axis Bending
0
.
1
n
b
u
n
c
u
M
M
P
P
0
.
1
ny
b
uy
nx
b
ux
n
c
u
M
M
M
M
P
P
Tension/Compression & Biaxial Bending
Quite conservative when compared to actual ultimate strengths
especially for wide flange shapes with bending about minor axis
AISC Interaction Formula
–
CHAPTER H
AISC Curve
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
r = required strength
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
c = available strength
REQUIRED CAPACITY
P
r
P
c
M
rx
M
cx
Mry
Mcy
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
Axial Capacity P
c
877
.
0
44
.
0
or
71
.
4
658
.
0
otherwise
F
QF
F
QF
E
r
KL
if
QF
F
e
y
e
y
y
F
QF
cr
e
y
g
cr
n
A
F
P
Axial Capacity P
c
Elastic Buckling Stress corresponding to the controlling mode of
failure (flexural, torsional or flexural torsional)
F
e
:
Theory of Elastic Stability (Timoshenko & Gere 1961)
Flexural Buckling
Torsional Buckling
2

axis of symmetry
Flexural Torsional
Buckling
1 axis of symmetry
Flexural Torsional
Buckling
No axis of symmetry
2
2
/
r
KL
E
F
e
AISC Eqtn
E4

4
AISC Eqtn
E4

5
AISC Eqtn
E4

6
Effective Length Factor
2
2
2
r
L
EA
P
cr
2
2
5
.
0
r
L
EA
P
cr
2
2
7
.
0
r
L
EA
P
cr
Fixed on bottom
Free to rotate and translate
Fixed on bottom
Fixed on top
Fixed on bottom
Free to rotate
Effective Length of Columns
A
B
I
g
L
g
I
g
L
g
I
c
L
c
I
c
L
c
Assumptions
•
All columns under
consideration reach buckling
Simultaneously
•
All joints are rigid
•
Consider members lying in the
plane of buckling
•
All members have constant A
Define:
g
g
c
c
A
L
I
L
I
G
g
g
c
c
B
L
I
L
I
G
Effective Length of Columns
Use alignment charts
(Structural Stability Research Council SSRC)
LRFD Commentary Figure C

C2.2 p 16.1

241,242
Connections to foundations
(a) Hinge
G is infinite

Use G=10
(b) Fixed
G=0

Use G=1.0
Axial Capacity P
c
LRFD
n
c
c
P
P
strength
e
compressiv
design
n
c
P
0.90
n
compressio
for
factor
resistance
c
Axial Capacity P
c
ASD
c
n
c
P
P
strength
e
compressiv
allowable
c
n
P
1.67
n
compressio
for
factor
safety
c
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
Moment Capacity M
cx
or M
cy
2
2
2
078
.
0
1
ts
b
o
x
ts
b
b
cr
r
L
h
S
Jc
r
L
E
C
F
r
b
p
b
r
p
x
cr
p
p
r
p
b
r
p
p
b
p
b
p
n
L
L
L
L
L
M
S
F
M
L
L
L
L
M
M
M
C
L
L
M
M
for
for
for
x
y
r
S
F
M
7
.
0
REMEMBER TO CHECK FOR NON

COMPACT SHAPES
Moment Capacity M
cx
or M
cy
r
p
r
p
x
cr
p
p
r
p
r
p
p
p
p
n
M
S
F
M
M
M
M
M
M
for
for
for
REMEMBER TO ACCOUNT FOR LOCAL
BUCKLING IF APPROPRIATE
Moment Capacity M
cx
or M
cy
n
b
c
M
M
b
n
c
M
M
LRFD
ASD
90
.
0
b
67
.
1
b
Demand
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
Axial Demand P
r
u
r
P
P
LRFD
ASD
a
r
P
P
factored
service
Demand
2
.
0
0
.
1
9
8
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
2
.
0
0
.
1
2
c
r
cy
ry
cx
rx
c
r
P
P
for
M
M
M
M
P
P
Second Order Effects & Moment Amplification
W
P
P
M
y
y
max
@ x=L/2 =
d
M
max
@ x=L/2 =
M
o
P
d
wL
2
/8 + P
d
additional moment causes additional
deflection
Second Order Effects & Moment Amplification
Consider
M
max
=
M
o
P
D
additional moment causes additional
deflection
D
P
H
H
P
Second Order Effects & Moment Amplification
•
Total Deflection cannot be Found Directly
•
Additional Moment Because of Deformed Shape
•
First Order Analysis
•
Undeformed Shape

No secondary moments
•
Second Order Analysis (P

d
and P

D
)
•
Calculates Total deflections and secondary moments
•
Iterative numerical techniques
•
Not practical for manual calculations
•
Implemented with computer programs
Design Codes
AISC Permits
Second Order Analysis
or
Moment Amplification Method
Compute moments from 1
st
order analysis
Multiply by amplification factor
Derivation of Moment Amplification
L
x
e
y
o
sin
Derivation of Moment Amplification
Moment Curvature
EI
M
dx
y
d
2
2
M
P
y
y
P
M
o
L
x
e
y
o
sin
y
L
x
e
EI
P
dx
y
d
sin
2
2
L
x
EI
Pe
y
EI
P
dx
y
d
sin
2
2
2
nd
order nonhomogeneous DE
Derivation of Moment Amplification
L
x
EI
Pe
y
EI
P
dx
y
d
sin
2
2
Boundary Conditions
0
0
@
y
x
0
@
y
L
x
L
x
B
y
sin
Solution
Derivation of Moment Amplification
L
x
EI
Pe
L
x
B
EI
P
L
x
B
L
sin
sin
sin
2
2
1
1
2
2
2
2
2
2
PL
EI
e
PL
EI
e
L
EI
P
EI
Pe
B
Solve for B
Substitute in DE
L
x
B
y
sin
1
P
P
e
e
Derivation of Moment Amplification
o
e
e
e
y
P
P
L
x
e
P
P
L
x
P
P
e
L
x
B
y
1
1
sin
1
1
sin
1
sin
Deflected Shape
Derivation of Moment Amplification
L
x
P
P
e
L
x
e
P
e
sin
1
sin
Moment
y
y
P
M
o
e
P
P
L
x
e
P
1
1
sin
M
o
(
x
)
Amplification
Factor
Braced vs. Unbraced Frames
lt
nt
r
M
B
M
B
M
2
1
ASD
for
for LRFD
strength
moment
required
a
u
r
M
M
M
Eq. C2

1a
Braced vs. Unbraced Frames
lt
nt
r
M
B
M
B
M
2
1
Eq. C2

1a
M
nt
= Maximum 1
st
order moment assuming no
sidesway occurs
M
lt
= Maximum 1
st
order moment caused by sidesway
B
1
= Amplification factor for moments in member
with no sidesway
B
2
= Amplification factor for moments in member
resulting from sidesway
Braced Frames
y
y
P
M
o
P
P
M
e
o
1
1
Braced Frames
Braced Frames
2

C2
Equation
AISC
1
1
1
1
e
r
m
P
aP
C
B
P
r
= required axial compressive strength
= P
u
for LRFD
= P
a
for ASD
P
r
has a contribution from the P
D
effect and is given by
lt
nt
r
P
B
P
P
2
Braced Frames
2

C2
Equation
AISC
1
1
1
1
e
r
m
P
aP
C
B
a = 1 for LRFD
= 1.6 for ASD
2
1
2
1
L
K
EI
P
e
Braced Frames
C
m
coefficient accounts for the shape of the moment
diagram
Braced Frames
C
m
For Braced & NO TRANSVERSE LOADS
4

C2
AISC
4
.
0
6
.
0
2
1
M
M
C
m
M
1
: Absolute smallest End Moment
M
2
: Absolute largest End Moment
Braced Frames
C
m
For Braced & NO TRANSVERSE LOADS
2

C2
Commentary
AISC
1
1
e
r
m
P
aP
C
C2.1

C
Table
Commentary
AISC
1

2
2
L
M
EI
o
o
d
COSERVATIVELY C
m
= 1
Unbraced Frames
lt
nt
r
M
B
M
B
M
2
1
Eq. C2

1a
M
nt
= Maximum 1
st
order moment assuming no
sidesway occurs
M
lt
= Maximum 1
st
order moment caused by sidesway
B
1
= Amplification factor for moments in member
with no sidesway
B
2
= Amplification factor for moments in member
resulting from sidesway
Unbraced Frames
Unbraced Frames
Unbraced Frames
1
1
1
2
2
e
nt
P
P
a
B
a
= 1.00 for LRFD
= 1.60 for ASD
nt
P
= sum of required load capacities for all columns in
the story under consideration
2
e
P
= sum of the Euler loads for all columns in the
story under consideration
Unbraced Frames
2
2
2
2
L
K
EI
P
e
H
m
e
HL
R
P
D
2
Used when shape is known
e.g. check of adequacy
Used when shape is NOT known
e.g. design of members
Unbraced Frames
2
2
2
2
L
K
EI
P
e
I = Moment of inertia about axis of bending
H
m
e
HL
R
P
D
2
K
2
= Unbraced length factor corresponding to the
unbraced condition
L = Story Height
R
m
= 0.85 for unbraced frames
D
H
= drift of story under consideration
S
H = sum of all horizontal forces causing
D
H
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