1. Beam-Columns

2. Members Under Combined Forces
Most beams and columns are subjected to some degree of both bending and axial load
e.g. Statically Indeterminate Structures P1 P2 C E A D F B

3. Interaction Formulas for Combined Forces
e.g. LRFD
If more than one resistance is involved consider interaction

4. Basis for Interaction Formulas
Tension/Compression & Single Axis Bending
Tension/Compression & Biaxial Bending
Quite conservative when compared to actual ultimate strengths
especially for wide flange shapes with bending about minor axis

5. AISC Interaction Formula – CHAPTER H
AISC Curve
r = required strength
c = available strength

6. REQUIRED CAPACITY
Pr Pc
Mrx Mcx
Mry Mcy

8. Axial Capacity Pc

9. Axial Capacity Pc
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

10. Effective Length Factor
Free to rotate and translate
Fixed on top
Free to rotate
Fixed on bottom
Fixed on bottom
Fixed on bottom

11. Effective Length of ColumnsA B
Ig Lg
Ic Lc
Assumptions
All columns under consideration reach buckling Simultaneously
All joints are rigid
Consider members lying in the plane of buckling
All members have constant A
Define:

12. Effective Length of Columns
Use alignment charts (Structural Stability Research Council SSRC)
LRFD Commentary Figure C-C2.2 p ,242
Connections to foundations
(a) Hinge
G is infinite - Use G=10
(b) Fixed
G=0 - Use G=1.0

13. Axial Capacity Pc
LRFD

14. Axial Capacity Pc
ASD

16. Moment Capacity Mcx or Mcy
REMEMBER TO CHECK FOR NON-COMPACT SHAPES

17. Moment Capacity Mcx or Mcy
REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

18. Moment Capacity Mcx or Mcy
LRFD
ASD

19. Demand

20. Axial Demand Pr
LRFD
ASD
factored
service

21. Demand

22. Second Order Effects & Moment Amplificationy P W
x=L/2 = d
x=L/2 = Mo + Pd = wL2/8 + Pd
additional moment causes additional deflection

23. Second Order Effects & Moment Amplification
Consider
Mmax = Mo + PD
additional moment causes additional deflection

24. Second Order Effects & Moment Amplification
Total Deflection cannot be Found Directly
Additional Moment Because of Deformed Shape
First Order Analysis
Undeformed Shape - No secondary moments
Second Order Analysis (P-d and P-D)
Calculates Total deflections and secondary moments
Iterative numerical techniques
Not practical for manual calculations
Implemented with computer programs

25. Moment Amplification Method
Design Codes
AISC Permits
Second Order Analysis or
Moment Amplification Method
Compute moments from 1st order analysis
Multiply by amplification factor

26. Derivation of Moment Amplification

27. Derivation of Moment Amplification
Moment Curvature P M
2nd order nonhomogeneous DE

28. Derivation of Moment Amplification
Boundary Conditions
Solution

29. Derivation of Moment Amplification
Solve for B

30. Derivation of Moment Amplification
Deflected Shape

31. Derivation of Moment Amplification
Mo(x)
Amplification Factor

32. Braced vs. Unbraced Frames
Eq. C2-1a

33. Braced vs. Unbraced Frames
Eq. C2-1a
Mnt = Maximum 1st order moment assuming no sidesway occurs
Mlt = Maximum 1st order moment caused by sidesway
B1 = Amplification factor for moments in member with no sidesway
B2 = Amplification factor for moments in member resulting from sidesway

34. Braced Frames

35. Braced Frames

36. Braced Frames
Pr = required axial compressive strength
= Pu for LRFD
= Pa for ASD
Pr has a contribution from the PD effect and is given by

37. Braced Frames
a = 1 for LRFD
= 1.6 for ASD

38. Braced Frames
Cm coefficient accounts for the shape of the moment diagram

39. Braced Frames Cm For Braced & NO TRANSVERSE LOADS
M1: Absolute smallest End Moment
M2: Absolute largest End Moment

40. Braced Frames
Cm For Braced & NO TRANSVERSE LOADS
COSERVATIVELY Cm= 1

41. Unbraced Frames
Eq. C2-1a
Mnt = Maximum 1st order moment assuming no sidesway occurs
Mlt = Maximum 1st order moment caused by sidesway
B1 = Amplification factor for moments in member with no sidesway
B2 = Amplification factor for moments in member resulting from sidesway

42. Unbraced Frames

43. Unbraced Frames

44. Unbraced Frames
a = 1.00 for LRFD
= 1.60 for ASD
= sum of required load capacities for all columns in the story under consideration
= sum of the Euler loads for all columns in the story under consideration

45. Used when shape is known e.g. check of adequacy
Unbraced Frames
Used when shape is known
e.g. check of adequacy
Used when shape is NOT known
e.g. design of members

46. Unbraced Frames
I = Moment of inertia about axis of bending
K2 = Unbraced length factor corresponding to the unbraced condition
L = Story Height
Rm = 0.85 for unbraced frames
DH = drift of story under consideration
SH = sum of all horizontal forces causing DH