1. Compression Members

2. COLUMN STABILITY
A. Flexural Buckling
Elastic Buckling
Inelastic Buckling
Yielding
B. Local Buckling – Section E7 pp
and B4 pp
C. Lateral Torsional Buckling

3. AISC Requirements CHAPTER E pp 16.1-32 Nominal Compressive Strength
AISC Eqtn E3-1

4. AISC Requirements
LRFD

5. Design Strength

6. In Summary

7. Local Stability - Section B4 pp 16.1-14
If elements of cross section are thin LOCAL buckling occurs
The strength corresponding to any buckling mode cannot be developed

8. Local Stability - Section B4 pp 16.1-14
If elements of cross section are thin LOCAL buckling occurs
The strength corresponding to any buckling mode cannot be developed

9. Local Stability - Section B4 pp 16.1-14
Stiffened Elements of Cross-Section
Unstiffened Elements of Cross-Section

10. Local Stability - Section B4 pp 16.1-14
Compact
Section Develops its full plastic stress before buckling (failure is due to yielding only)
Noncompact
Yield stress is reached in some but not all of its compression elements before buckling takes place
(failure is due to partial buckling partial yielding)
Slender
Yield stress is never reached in any of the compression elements (failure is due to local buckling only)

11. Local Stability - Section B4 pp 16.1-14
If local buckling occurs cross section is not fully effective
Avoid whenever possible
Measure of susceptibility to local buckling
Width-Thickness ratio of each cross sectional element: l
If cross section has slender elements - l> lr
Reduce Axial Strength (E7 pp )

12. Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp

13. Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp

14. Slenderness Parameter - Limiting Values
AISC B5 Table B4.1 pp

15. Slender Cross Sectional Element: Strength Reduction E7 pp 16.1-39
Reduction Factor Q:
Q: B4.1 – B4.2 pp to

16. Slender Cross Sectional Element: Strength Reduction E7 pp 16.1-39
Reduction Factor Q:
Q=QsQa
Qs, Qa: B4.1 – B4.2 pp to

17. COLUMN STABILITY
A. Flexural Buckling
Elastic Buckling
Inelastic Buckling
Yielding
B. Local Buckling – Section E7 pp
and B4 pp
C. Torsional, Lateral/Torsional Buckling

18. Torsional & Flexural Torsional Buckling
When an axially loaded member becomes unstable overall
(no local buckling) it buckles one of the three ways
Flexural Buckling
Torsional Buckling
Flexural-Torsional
Buckling

19. Twisting about longitudinal axis of member
Torsional Buckling
Twisting about longitudinal axis of member
Only with doubly symmetrical cross sections with slender cross-sectional elements
Cruciform shape particularly vulnerable
Standard Hot-Rolled Shapes are NOT susceptible
Built-Up Members should be investigated

20. Flexural Torsional Buckling
Combination of Flexural and Torsional Buckling
Only with unsymmetrical cross sections
1 Axis of Symmetry: channels, structural tees, double-angle, equal length single angles
No Axis of Symmetry: unequal length single angles

21. Torsional Buckling Eq. E4-4 Cw = Warping Constant (in6)
Kz = Effective Length Factor for Torsional Buckling
(based on end restraints against twisting)
G = Shear Modulus (11,200 ksi for structural steel)
J = Torsional Constant

22. Lateral Torsional Buckling 1-Axis of Symmetry
AISC Eq. E4-5
Coordinates of shear center w.r.t centroid of section

23. Lateral Torsional Buckling No Axis of Symmetry
Fe is the lowest root of the
Cubic equation
AISC Eq. E4-6

24. In Summary - Definition of Fe
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
AISC Eqtn
E4-4
AISC Eqtn
E4-5
AISC Eqtn
E4-6

25. Column Strength

26. EXAMPLE Compute the compressive strength of a WT12x81 of A992 steel.
Assume (KxL) = 25.5 ft, (KyL) = 20 ft, and (Kz L) = 20 ft
FLEXURAL Buckling – X axis
WT 12X81 OK
Ag=23.9 in2
rx=3.50 in
Inelastic Buckling
ry=3.05 in

27. EXAMPLE FLEXURAL TORSIONAL Buckling – Y axis (axis of symmetry) OK
WT 12X81 OK
Ag=23.9 in2
rx=3.50 in
ry=3.05 in
y=2.70 in
Shear Center
tf=1.22 in
Ix=293 in4
Iy=221 in4
J=9.22 in4
Cw=43.8 in6

28. EXAMPLE FLEXURAL TORSIONAL Buckling – Y axis (axis of symmetry)
WT 12X81
Ag=23.9 in2
rx=3.50 in
ry=3.05 in
y=2.70 in
tf=1.22 in
Ix=293 in4
Iy=221 in4
J=9.22 in4
Cw=43.8 in6

29. EXAMPLE FLEXURAL TORSIONAL Buckling – Y axis (axis of symmetry)
WT 12X81
Ag=23.9 in2
rx=3.50 in
ry=3.05 in
y=2.70 in
tf=1.22 in
Ix=293 in4
Iy=221 in4
J=9.22 in4
Cw=43.8 in6

30. EXAMPLE FLEXURAL TORSIONAL Buckling – Y axis (axis of symmetry)
WT 12X81
Elastic or Inelastic LTB?
Ag=23.9 in2
rx=3.50 in
ry=3.05 in
y=2.70 in
tf=1.22 in
Ix=293 in4
Iy=221 in4
J=9.22 in4
Cw=43.8 in6

31. EXAMPLE FLEXURAL TORSIONAL Buckling – Y axis (axis of symmetry)
WT 12X81
Ag=23.9 in2
rx=3.50 in
ry=3.05 in
y=2.70 in
tf=1.22 in
Ix=293 in4
Iy=221 in4
Compare to FLEXURAL Buckling – X axis
J=9.22 in4
Cw=43.8 in6

32. Assumption : Strength Governed by Flexural Buckling
Column Design Tables
Assumption : Strength Governed by Flexural Buckling
Check Local Buckling
Column Design Tables
Design strength of selected shapes for effective length KL
Table 4-1 to 4-2, (pp 4-10 to 4-316)
Critical Stress for Slenderness KL/r
table 4.22 pp (4-318 to 4-322)

33. EXAMPLE
Compute the available compressive strength of a W14x74 A992 steel compression member. Assume pinned ends and L=20 ft. Use (a) Table 4-22 and (b) column load tables
(a) LRFD - Table 4-22 – pp 4-318
Fy=50 ksi
Table has integer values of (KL/r) Round up or interpolate

34. EXAMPLE
Compute the available compressive strength of a W14x74 A992 steel compression member. Assume pinned ends and L=20 ft. Use (a) Table 4-22 and (b) column load tables
(b) LRFD Column Load Tables
Tabular values based on minimum radius of gyration
Fy=50 ksi

35. Example II
A W12x58, 24 feet long in pinned at both ends and braced in the weak direction at the third points. A992 steel is used. Determine available compressive strength
Enter table 4.22 with KL/r=54.55 (LRFD)

36. Example II
A W12x58, 24 feet long in pinned at both ends and braced in the weak direction at the third points. A992 steel is used. Determine available compressive strength
Enter table 4.22 with KL/r=54.55 (ASD)

37. Example II
A W12x58, 24 feet long in pinned at both ends and braced in the weak direction at the third points. A992 steel is used. Determine available compressive strength
CAN I USE Column Load Tables?
Not Directly because they are based on min r (y axis buckling)
If x-axis buckling enter table with

38. Example II
A W12x58, 24 feet long in pinned at both ends and braced in the weak direction at the third points. A992 steel is used. Determine available compressive strength
X-axis buckling enter table with