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Beam-Columns. Members Under Combined Forces Most beams and columns are subjected to some degree of both bending and axial load e.g. Statically Indeterminate.

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Presentation on theme: "Beam-Columns. Members Under Combined Forces Most beams and columns are subjected to some degree of both bending and axial load e.g. Statically Indeterminate."— Presentation transcript:

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 P1P1 P2P2 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 P r P c M rx M cx Mry Mcy

7

8 Axial Capacity P c

9 Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional) Fe:Fe: Theory of Elastic Stability (Timoshenko & Gere 1961) Flexural BucklingTorsional 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 Fixed on bottom Free to rotate and translate Fixed on bottom Fixed on top Fixed on bottom Free to rotate

11 Effective Length of Columns A B I g L g I c L c 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 P c LRFD

14 Axial Capacity P c ASD

15

16 Moment Capacity M cx or M cy REMEMBER TO CHECK FOR NON- COMPACT SHAPES

17 Moment Capacity M cx or M cy REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

18 Moment Capacity M cx or M cy LRFDASD

19 Demand

20 Axial Demand P r LRFDASD factoredservice

21 Demand

22 Second Order Effects & Moment Amplification W P P M y y x=L/2 =  M x=L/2 =    P  wL 2 /8 + P  additional moment causes additional deflection

23 Second Order Effects & Moment Amplification Consider M max =    P  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-  and P-  ) Calculates Total deflections and secondary moments Iterative numerical techniques Not practical for manual calculations Implemented with computer programs

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

26 Derivation of Moment Amplification

27 Moment Curvature M P 2 nd 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 Moment Mo(x)Mo(x) Amplification Factor

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

33 Braced vs. Unbraced Frames Eq. C2-1a M nt = Maximum 1 st order moment assuming no sidesway occurs M lt = Maximum 1 st order moment caused by sidesway B 1 = Amplification factor for moments in member with no sidesway B 2 = Amplification factor for moments in member resulting from sidesway

34 Braced Frames

35

36 P r = required axial compressive strength = P u for LRFD = P a for ASD P r has a contribution from the P  effect and is given by

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

38 Braced Frames C m coefficient accounts for the shape of the moment diagram

39 Braced Frames C m For Braced & NO TRANSVERSE LOADS M 1 : Absolute smallest End Moment M 2 : Absolute largest End Moment

40 Braced Frames C m For Braced & NO TRANSVERSE LOADS COSERVATIVELY C m = 1

41 Unbraced Frames Eq. C2-1a M nt = Maximum 1 st order moment assuming no sidesway occurs M lt = Maximum 1 st order moment caused by sidesway B 1 = Amplification factor for moments in member with no sidesway B 2 = Amplification factor for moments in member resulting from sidesway

42 Unbraced Frames

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44 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 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 K 2 = Unbraced length factor corresponding to the unbraced condition L = Story Height R m = 0.85 for unbraced frames  H = drift of story under consideration  H = sum of all horizontal forces causing  H


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