1. Compression Members

2. Compression Members Compression members are susceptible to BUCKLING
BUCKLING – Loss of stability
Axial loads cause lateral deformations (bending-like deformations)
P is applied slowly
P increases
Member becomes unstable - buckles

3. Column Theory Pcr depends on Length of member Material Properties
Axial force that causes Buckling is called Critical Load and is associated to the column strength
Pcr depends on
Length of member
Material Properties
Section Properties

4. Column Theory - Euler Buckling

5. Column Theory - Euler Buckling

6. Column is perfectly straight The load is axial, with no eccentricity
Assumptions
Column is perfectly straight
The load is axial, with no eccentricity
The column is pinned at both ends
No Moments
Need to account for other boundary conditions

7. Other Boundary Conditions
Free to rotate and translate
Fixed on top
Free to rotate
Fixed on bottom
Fixed on bottom
Fixed on bottom

8. Other Boundary Conditions
In general
K: Effective Length Factor
LRFD Commentary Table C-C2.2 p

9. Effective Length Factor

10. Column Theory - Column Strength Curve

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

12. AISC Requirements
LRFD

13. AISC Requirements
ASD

14. AISC Requirements
ASD – Allowable Stress

15. Design Strength

16. Alternatively
Inelastic Buckling

17. In Summary

18. LOCAL BUCKLING
A. Flexural Buckling
Elastic Buckling
Inelastic Buckling
Yielding
B. Local Buckling – Section E7 pp
and B4 pp
C. Lateral Torsional Buckling

19. 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

20. 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

21. 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

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

23. 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)

24. 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 )

25. Slenderness Parameter - Section B5 pp 16.1-12
Cross Sectional Element
Slenderness l
Stiffened
Unstiffened b t h tw
l=h/tw
l=b/t=bf/2tw

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

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

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

29. Slenderness Parameter - Limiting Values

30. Slenderness Parameter - Limiting Values

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

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

33. Flange is not slender, OK
Example I
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
FLANGES - Unstiffened Elements
Flange is not slender, OK

34. Example I Web is not slender, OK
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
WEB - Stiffened Element
Web is not slender, OK

35. Example I
Investigate a W14x74, grade 50 in compression for local stability
W14x74: bf-10.1 in, tf=0.785 in
PART 1 – Properties: Slender Shapes are marked with “c”

36. Example II
Determine the axial compressive strength of an HSS 8x4x1/8 with an effective length of 15 ft with respect to each principal axis. Use Fy=46 ksi.
HSS 8x4x1/8
7.652 in
8 in
1.5 t =
Ag=2.70 in2
h/t=66.0
rx=2.92 in2
b/t=31.5
ry=1.71 in2

37. Example II Maximum OK Inelastic Buckling Nominal Strength HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
Maximum OK
Inelastic Buckling
Nominal Strength

38. Example II Local Buckling SLENDER HSS 8x4x1/8 Ag=2.70 in2 rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
Local Buckling
SLENDER

39. Example II Local Buckling
HSS 8x4x1/8
Ag=2.70 in2
rx=2.92 in2
ry=1.71 in2
h/t=66.0
b/t=31.5
Local Buckling
Stiffened Cross-Section – Rectangular w/ constant t
Qs=1.0
AISC E7.2
Case (b) applies provided that
Code allows f=Fy to avoid iterations
Aeff: Summation of Effective Areas of Cross section based on reduced effective width be

40. Example II
Aeff: be

41. Example II
Aeff: be
7.652 in
8 in
1.5 t =
Loss of Area

42. Example II
Loss of Area
Reduction Factor

43. Example II Local Buckling Strength Inelastic Buckling Same as before
Nominal Strength

44. Example II Local Buckling Strength Nominal Strength CONTROLS
Lateral Flexural Buckling Strength
LRFD
ASD

45. 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)