Beam Columns - AISC Interaction Equations

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15 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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