Design of Combined Bending and Compression Members in Steel

Combined stresses

Bi-axial bending

Bending and compression

Multi-story steel rigid frame

• Rigid frames, utilizing moment connections, are well suited for specific types of buildings where diagonal bracing is not feasible or does not fit the architectural design

• Rigid frames generally cost more than braced frames (AISC 2002)

Vierendeel steel truss cycle bridge Beaufort Reach, Swansea

Typical crane columns

neutral axis

fmax = fa + fbx + fby < fdes

( Pf / A ) + ( Mfx / Sx ) + ( Mfy / Sy ) < fdes

(Pf / Afdes) + (Mfx / Sxfdes) + (Mfy / Syfdes) < 1.0

(Pf / Pr) + (Mfx / Mr) + ( Mfy / Mr) < 1.0

x

x

fbx = Mfx / Sx

Mfx

y

yfby = Mfy / Sy Mfy

fa = Pf / A

Pf

Cross-sectional strengthPf/Pr

Mf/Mr

1.0

1.0

Class 1 steel sections

(Pf / Pr) + 0.85(Mfx / Mr) + 0.6( Mfy / Mr) < 1.0

other steel sections

(Pf / Pr) + (Mfx / Mr) + ( Mfy / Mr) < 1.0

Slender beam-columns

• What if column buckling can occur ?

• What if lateral-torsional buckling under bending can occur ?

Use the appropriate axial resistance and moment resistance values in the interaction equation

Beam-column in a heavy

industrial setting

BMD

Moment amplification

δ o δmax

P

P

0max

0max

1

1

1

1

MPP

M

PP

E

E

PE = Euler load

Interaction equation

0.111

11

ry

fy

Ey

y

rx

fx

Ex

x

r

f

M

M

PPM

M

PPP

P

Axial load

Bending about y-axis

Bending about x-axis

ω1 = moment gradient factor (see next slide)

Moment gradient factor for steel columns with

end moments

M1

M2

ω1 = 0.6 – 0.4(M1/M2) ≥ 0.4

i.e. when moments are equal and cause a single curvature, then ω1 = 1.0

and when they are equal and cause an s-shape, then ω1 = 0.4

Steel frame to resist earthquake forces

Warehouse building, Los Angeles

Moment gradient factor for other cases

ω1 = 1.0 ω1 = 0.85ω1 = 1.0 ω1 = 0.6 ω1 = 0.4

v

Design of steel beam-columns

1. Laterally supported• Cross-sectional strength

2. Supported in the y-direction• Overall member strength• Use moment amplification factor• Use buckling strength about x-axis (Crx)

3. Laterally unsupported• Buckling about y-axis (Cry)• Lateral torsional buckling (Mrx)• Use moment amplification factors• Usually the most critical condition

Note: Mry never includes lateral-torsional buckling

Example of different support conditions

These two columns supported in y-directionby side wall

This column unsupported

y direction

x direction