Beams

BEAMS

A structural member loaded in the transverse direction to the longitudinal axis.

Internal Forces: Bending Moments and Shear

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Stability

Structural Steel - Characteristics Buckling:

Instability due to slenderness

Stability

Elastic Buckling

Load Deflection

500 450 400 350 300 250 200 150 100 50 0 0 4 8 12

Deflection (in)

16 20 FEM Test 24

Limit States

Limit States

Limit States

Limit States

Classification of Shapes

Compact Section Non-Compact Section Web Local Buckling Flange Local Buckling

Bending Strength of Compact Shapes

Lateral Torsional Buckling

Bending Strength of Compact Shapes

L p

 1 .

76

r y E F y

Bending Strength of Compact Shapes Laterally Supported Compact Beams

M n



M p



F y Z x L b



L p



1 .

76

r y E F y

Bending Strength of Compact Shapes

Bending Strength of Compact Shapes Elastic Buckling

L b



L r

 1 .

95

r ts E

0 .

7

F y M n



F cr S x



M p Jc S x h o

1  1  6 .

76   0 .

7

F y S x h o EJc

  2

F cr



C b

 

L b

2

r ts E

 2

1



Jc

0 .

078

S x h o

 

L b r ts

  2

M r

 0 .

7

F y S x

Elastic Buckling

C b

= factor to account for non-uniform bending within the unbraced length

C b

 2 .

5

M

max  12 .

5 3

M A M

 max 4

M B

 3

M C R m

 3 .

0 See AISC table 3-1 p 3.10

M max A B C L/4 L/4 L/4 L/4

Elastic Buckling

Elastic Buckling

Elastic Buckling

C b

= factor to account for non-uniform bending within the unbraced length

C b

 2 .

5

M

max  12 .

5 3

M A M

 max 4

M B

 3

M C R m

 3 .

0 R m = 1 for doubly symmetric cross sections and singly symmetric subject to single curvature See textbook p 190 for other cases

Elastic Buckling

C b

= factor to account for non-uniform bending within the unbraced length

F cr



C b

 

L b

2

r ts E

 2 1  0 .

078

Jc S x h o

 

L b r ts

  2

Elastic Buckling

C b

= factor to account for non-uniform bending within the unbraced length

r

2

ts



I y C w S x c

  1  

h

for 2

o I C y w

doubly for symmetric channels I shapes

h o

= distance between flange centroids =

d-t f

Bending Strength of Compact Shapes

Bending Strength of Compact Shapes Inelastic Buckling

L p



L b



L r M n



C b

  

M p

 

M p



M r



L b L r

 

L p L p

   

M p

Linear variation between M p and M r

M r

 0 .

7

F y S x

Nominal Flexural Strength – Compact Shapes

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 F cr



C b

 

L b

2

r ts E

 2 1  0 .

078

Jc S x h o

 

L b r ts

  2

Nominal Flexural Strength – NON-Compact Shapes Most W- M- S- and C- shapes are compact A few are NON-compact NONE is slender Webs of ALL hot rolled shapes in the manual are compact

FLB and LTB

Built-Up welded shapes can have non-compact or slender webs

FLB, WLB, LTB (AISC F4 and F5)

Nominal Flexural Strength – NON-Compact Shapes WLB

M n

  

M p

for

 

M N p

/A

 

M

for



p

  

M

rolled

p r

  

r

 

shapes

 

p p

in



M p

Manual for



for



r p

   

b

 

r λ



b

2

t f f λ p



0 .

38

E F y λ r



1 .

0

E F y