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

A B P 1 P 2 C D E F

Interaction Formula

P r P c

 8 9

M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

P r

2

P c



M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

REQUIRED CAPACITY

P r M rx Mry P c M cx Mcy

P r P c

 8 9

M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

P r

2

P c

    

M rx M cx



M ry M cy

     1 .

0

for P r P c

 0 .

2

Axial Capacity P c

P n



F cr A g F cr

           0 .

658 0

QF y F e

.

877

F e QF y if KL r

or

otherwise F



e

4  .

71 0 .

44

E QF QF y y

Axial Capacity P c

F e

:

Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional) 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

F e

  

KL

2 /

E r

 2 AISC Eqtn E4-4 AISC Eqtn E4-5 AISC Eqtn E4-6

Axial Capacity P c

LRFD

P c

 

c P n



c

 resistance factor for compressio n  0.90



c P n

 design compressiv e strength

Axial Capacity P c

ASD

P c



P n



c



c

 safety factor for compressio n  1.67

P n



c

 allowable compressiv e strength

Moment Capacities

P r P c

 8 9

M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

P r

2

P c

    

M rx M cx



M ry M cy

     1 .

0

P for P c r

 0 .

2

Moment Capacity M cx or M cy

M n

   

M



C b

 

F cr p

  

S

for

M x



p



L M b



M p



p L p

for 

M L r r

 

L b L r L b

 

L p L p

   

M p

for

L p



L b



L r M r

 0 .

7

F y S x F cr



C b

 

L b

2

r ts E

 2 1  0 .

078

Jc S x h o

 

L b r ts

  2

REMEMBER TO CHECK FOR NON COMPACT SHAPES

Moment Capacity M cx or M cy REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

M n

     

M

  

M F cr p S p

for

x

   

M M p

  

p M r p

for 

r

  

r

   

p



p

    

M p

for 

p

   

r

Moment Capacity M cx or M cy

LRFD

M c

 

b M n



b

 0 .

90 ASD

M c



b



M



b n

 1 .

67

Axial Demand

P r P c

 8 9

M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

P r

2

P c

    

M rx M cx



M ry M cy

     1 .

0

P for P c r

 0 .

2

LRFD

P r



P u

factored

Axial Demand P r

ASD

P r



P a

service

Demand

P r P c

 8 9

M rx M cx



M ry M cy

 1 .

0

for P r P c

 0 .

2

P r

2

P c

    

M rx M cx



M ry M cy

     1 .

0

P for P c r

 0 .

2

Second Order Effects & Moment Amplification

P P W 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 P H H D P M max = M o  P D additional moment causes additional deflection

Design Codes AISC Permits Second Order Analysis or Moment Amplification Method

Compute moments from 1 st order analysis Multiply by amplification factor

Braced vs. Unbraced Frames

M r



B

1

M nt



B

2

M lt

Eq. C2-1a

M r

  required moment strength

M u

for LRFD 

M a

for ASD

Braced vs. Unbraced Frames

M r



B

1

M nt



B

2

M lt

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

B

1 

1



C m



aP r P e

1  

1 AISC Equation C2 2

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

P r



P nt



B

2

P lt

Braced Frames

B

1 

1



C m



aP r P e

1  

1 AISC Equation C2 2

a = 1 for LRFD = 1.6 for ASD

P e

1   

K

1 2

EI L

 2

Braced Frames

C m coefficient accounts for the shape of the moment diagram

Braced Frames

C m For Braced & NO TRANSVERSE LOADS

C m

 0 .

6  0 .

4  

M M

2 1   AISC C2 4 M 1 : Absolute smallest End Moment M 2 : Absolute largest End Moment

Braced Frames

C m For Braced & NO TRANSVERSE LOADS

C m

 1    

aP r P e

1   AISC Commentary C2 2    2 d

o EI M o L

2 1 AISC Commentary Table C C2.1

COSERVATIVELY C m = 1

Unbraced Frames

M r



B

1

M nt



B

2

M lt

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

B

2  1 

a

1  

P e P nt

2  1 a = 1.00 for LRFD = 1.60 for ASD 

P nt

= sum of required load capacities for all columns in the story under consideration 

P e

2 = sum of the Euler loads for all columns in the story under consideration



P e

2    

K

2 2

EI L

 2

Unbraced Frames

Used when shape is known e.g. check of adequacy 

P e

2 

R m

 D

HL H

Used when shape is NOT known e.g. design of members

Unbraced Frames



P e

2    

K

2 2

EI L

 2 

P e

2 I = Moment of inertia about axis of bending 

R m

 D

HL H

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

• 6.2-1 • 6.2-2 • 6.5-2 • 6.5-6 • 6.6-1

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