1. Design of Combined Bending and Compression Members in Steel

2. Combined stresses Bi-axial bending Bending and compression

3. Multi-story steel rigid frame

4. 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)

5. Vierendeel steel truss cycle bridge Beaufort Reach, Swansea

6. Typical crane columns

7. neutral axis f max = f a + f bx + f by < f des ( P f / A ) + ( M fx / S x ) + ( M fy / S y ) < f des (P f / A f des ) + (M fx / S x f des ) + (M fy / S y f des ) < 1.0 (P f / P r ) + (M fx / M r ) + ( M fy / M r ) < 1.0 x x f bx = M fx / S x M fx y y f by = M fy / S y M fy f a = P f / A PfPf

8. Cross-sectional strength P f /P r M f /M r 1.0 Class 1 steel sections (P f / P r ) + 0.85(M fx / M r ) + 0.6( M fy / M r ) < 1.0 other steel sections (P f / P r ) + (M fx / M r ) + ( M fy / M r ) < 1.0

9. 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

10. Beam-column in a heavy industrial setting BMD

11. Moment amplification δ oδ o δ max P P P E = Euler load

12. Interaction equation Axial load Bending about y-axis Bending about x-axis ω 1 = moment gradient factor (see next slide)

13. Moment gradient factor for steel columns with end moments M1M1 M2M2 ω 1 = 0.6 – 0.4(M 1 /M 2 ) ≥ 0.4 i.e. when moments are equal and cause a single curvature, then ω 1 = 1.0 and when they are equal and cause an s-shape, then ω 1 = 0.4

14. Steel frame to resist earthquake forces Warehouse building, Los Angeles

15. Moment gradient factor for other cases ω 1 = 1.0 ω 1 = 0.85ω 1 = 1.0ω 1 = 0.6 ω 1 = 0.4 v

16. 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 (C rx ) 3.Laterally unsupported Buckling about y-axis (C ry ) Lateral torsional buckling (M rx ) Use moment amplification factors Usually the most critical condition Note: M ry never includes lateral-torsional buckling

17. Example of different support conditions These two columns supported in y-direction by side wall This column unsupported y direction x direction