BC

1
Members subjected to combined axial loads and bending moments are called
“Beam

Columns”. Examples of this, are floor or roof beams resisting later wind
loads. Top chord truss elements supporting roof loading causing bending, etc.
Interaction Formula:
1
BC
0.9)
(
1.0
M
M
P
P
or
1.0
resistance
effects
load
Σ
b
c
n
b
u
n
c
u
where:
P
u
= factored axial compression.
M
u
= factored bending moment.
P
n
= nominal axial strength.
M
n
= nominal bending strength.
w w w w w w w
Top chord with bending
BC

2
Equation
(BC

1
)
is
the
basic
of
AISC
design
criteria
as
stated
in
(chapter
H)
of
AISC
–
LRFD
specs
:

1a)
H1
Equation
(AISC
1.0
M
M
M
M
P
2
Pu
0.2,
P
Pu
For
1a)
H1
Equation
(AISC
1.0
M
M
M
M
9
8
P
Pu
0.2,
P
Pu
For
ny
b
uy
nx
b
ux
n
c
n
c
ny
b
uy
nx
b
ux
n
c
n
c
BC

3
Example BC

1
The
beam

column
shown
in
Figure
below
is
pinned
at
both
ends
and
is
subjected
to
the
factored
loads
shown
.
Bending
is
about
the
strong
axis
.
Determine
weather
this
member
satisfy
the
appropriate
AISC
Specification
interaction
equation
.
Solution
From the column load tables (Table 4.1) the
axial
compressive design strength of W8x58 with
F
y
=50 ksi and K
y
L
y
=17 ft
c
P
n
= 286 kips
BC

4
From the beam design charts (Table 3

10 page 3
–
125) for
un braced length of L
b
=17, and C
b
=1.0
b
M
n
= 202 k.ft.
For this condition and this loading : C
b
=1.32 (table 3.1)
b
M
n
= 1.32 x 202 = 267 k.ft.
b
M
p
= 224 k.ft. (Table 3.2 page 3

18).
b
M
n
= 224 k.ft.
b
M
p
OK
1
0.889
224
93.5
9
8
386
200
M
M
9
8
P
P
1.1a
H
AISC
Use
0.2
0.518
386
200
P
P
ft.
k
93.5
4
17
22
M
:
moment
maximum
factored
The
nx
b
ux
n
c
u
n
c
u
u
This member satisfies the AISC specifications.
BC

5
Moments caused by eccentricity of axial load cannot be ignored for beam

columns.
P.
δ
8
wL
Moment
Maximum
2
The
value
of
(P∙
)
is
called
“Moment
magnification”
due
to
initial
beam
column
initial
crookedness
or
from
bending
due
to
transverse
load
(
)
.
It can be proven that a beam column with initial
crookedness (e) and initial moment (M
o
= P
u
∙e),
that the total moment becomes:
M = P
u
( e + y
max
)
e
u
o
P
P
1
1
M
M
BC

6
where:

M = Magnified moment.
M
o
= Initial moment (due to initial crookedness or more often due to transverse loads).
1.15
1567
200
1
1
P
P
1
1
Factor
ion
Amplificat
kips.
1567
17.1
x
(55.9)
x29000
π
P
55.9
3.65
12
x
17
x
1.0
r
L
K
Bending
of
Axis
r
L
K
r
KL
re
whe
.Ag
r
KL
E
π
P
e
u
2
2
e
x
x
x
x
x
x
2
2
e
So M = 1.15 M
o
= 1.15
93.5 = 107.5 k∙ft.
load.
buckling
Euler
L
EI
π
Pe
2
Example BC

2
Compute the amplification factor for example (BC

1)
BC

7
Moment amplification is covered in chapter C of the AISC code.
Two amplification factors are used in LRFD:

A)
* One to account for amplification due to deflection.
B)
* One to account for amplification due to frame
sideway to lateral forces in unbraced frames.
LRFD account for both effects:
M
u
= M
r
= B
1
M
nt
+ B
2
M
lt
AISC C2

1a
Where:
M
r
= M
u
= factored load combination as affected by amplification.
M
nt
= Maximum moment assuming no sidesway (no translation)
M
lt
= Maximum moment caused by sidesway (lateral translation).
(M
lt
= 0 for braced frames)
B
1
= amplification factor for braced frames.
B
2
= amplification factor for unbraced frames.
BC

8
The
maximum
moment
in
a
beam

column
depend
on
the
end
bending
moments
in
a
braced
frame,
the
various
cases
are
accounted
for
by
a
factor
(Cm)
as
follows
:
1.0
P
P
α
1
Cm
B
e1
r
1
(AISC C2

2)
BC

9
Where:
Cm = Coefficient whose value taken as follows:
1: If there are no transverse loads acting on the member,
M
1
/
M
2
is
a
ratio
of
the
bending
moments
at
the
ends
of
the
member
.
M
1
is
the
end
moment
that
is
smaller
in
absolute
value,
M
2
is
the
larger,
and
the
ratio
is
positive
for
moment
bent
in
reverse
curvature
and
negative
for
single

curvature
bending
.
Reverse
curvature
(a
positive
ratio)
occurs
when
M
1
and
M
2
are
both
clockwise
or
both
counterclockwise
.
2
1
m
M
M
0.4
0.6
C
(AISC Equation C2
–
4)
BC

10
2
.
For
transversely
loaded
members,
C
m
can
be
taken
as
0
.
85
if
the
ends
are
restrained
against
rotation
and
1
.
0
if
the
ends
are
unrestrained
against
rotation
(pinned)
.
End
restraint
will
usually
result
from
the
stiffness
of
members
connected
to
the
beam

column
.
The
pinned
end
condition
is
the
one
used
in
the
derivation
of
the
amplification
factor
;
hence
there
is
no
reduction
for
this
case,
which
corresponds
to
C
m
=
1
.
0
.
Although
the
actual
end
condition
may
lie
between
full
fixity
and
a
frictionless
pin,
use
of
one
of
the
two
values
given
here
will
give
satisfactory
results
.
Evaluation of C
m
Factor
2
2
e1
u
r
KL
EI
π
P
P
P
1.0
α
(AISC C2
–
5)
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