1. 1Combined Forces Theory Developed by Scott Civjan University of Massachusetts, Amherst

2. DIRECT ANALYSIS 2Combined Forces Theory

3. DIRECT ANALYSIS METHOD Analysis of Entire Structure Interaction Include Lateral “Notional” Loads All Members Must be Evaluated Under Combined Axial and Flexural Load No K values required Reduce Stiffness of Structure 3Combined Forces Theory

4. Moment M Axial Force P DIRECT ANALYSIS METHOD Initially consider a “traditional” analysis P nKL PyPy Axial Strength is defined as P nKL which includes K factors (P y indicates crushing) MpMp Bending Strength is defined as M n, assumed here to be M p for a laterally braced member 4Combined Forces Theory

5. Moment M P nKL Axial Force P PuPu MuMu MpMp Elastic 2 nd Order (Nominal Loads) Actual Response PyPy DIRECT ANALYSIS METHOD Typical design accounts for interaction by calibrating the member design to column curves Actual response produces a higher internal moment in the member. This is accounted for in calibrating the member check, but does not get transferred into adjacent members and connections 5Combined Forces Theory

6. Moment M P nKL Axial Force P PuPu MuMu MpMp Elastic 2 nd Order (Nominal Loads) Actual Response PyPy DIRECT ANALYSIS METHOD 6Combined Forces Theory

7. Moment M P nL Axial Force P PyPy MpMp DIRECT ANALYSIS METHOD Bending Strength is defined as M n, assumed here to be M p for a laterally braced member Axial Strength is defined as P nL which assumes K=1 for all cases Now consider the “Direct” analysis Design Curve is therefore shifted upwards from previous assumptions P nKL 7Combined Forces Theory

8. Moment M P nL Axial Force P PyPy PuPu MuMu MpMp DIRECT ANALYSIS METHOD Elastic 2 nd Order (Direct Analysis includes Notional Loads and Reduced Stiffness) Direct Analysis accounts for interaction by including lateral “notional” loads which increase moment, reducing stiffness and calibrating the member design to K=1 analysis Actual Response Actual response should then match the internal moment, transferring this moment into adjacent members and connections during analysis 8Combined Forces Theory

9. Moment M P nL Elastic 2 nd Order (Direct Analysis includes Notional Loads and Reduced Stiffness) Actual Response Axial Force P PyPy PuPu MuMu MpMp DIRECT ANALYSIS METHOD 9Combined Forces Theory

10. “Notional” Loads Notional loads are a function of the gravity load being applied Notional loads are applied as a lateral load at each floor level in the direction that adds to the destabilizing effects of the load combination being considered Notional loads can account for geometric imperfections, inelasticity of members, and other non-ideal conditions DIRECT ANALYSIS METHOD 10Combined Forces Theory

11. H+P/LH+P/L  L Recall that a vertical load acting through a displacement  is similar to the application of a horizontal load P  /L Therefore, a notional load can be considered the equivalent of an assumed geometric imperfection  “Notional” Loads H  P L DIRECT ANALYSIS METHOD 11Combined Forces Theory

12. DIRECT ANALYSIS METHOD Analysis and Calibration With proper calibration design strength approaches the actual response Calibration consists of a combination of notional load values and reduction in member stiffness Analysis is referenced to K=1 member capacities 12Combined Forces Theory

13. Appendix 7: Direct Analysis Method 13Combined Forces Spec 13th Ed

14. K=1 for all analysis Rigorous Second Order Analysis Required (P-  and P-  ) (Such as verified computer analysis or amplified first order analysis) Direct Analysis Method REQUIRED if  2nd Order /  1st Order >1.5 (B 2 >1.5) (Section C2.2) Analysis 14Combined Forces Spec 13th Ed

15. Rigorous Second Order Analysis Typically computer analysis performed Direct Analysis Method Many programs neglect P-  analysis Often not a significant effect, but this should be checked (low B 1 factor from AISC Section C2 indicates it can be neglected) 15Combined Forces Spec 13th Ed

16. If  P r <0.15P eL analysis can neglect P-  Direct Analysis Method Where:  = 1.0 (LRFD), 1.6 (ASD) P r = Required Axial Compressive Strength P eL = Euler Buckling Strength in the Plane of Bending (K=1) Equation A-7-1 16Combined Forces Spec 13th Ed

17. Apply Notional Loads Reduce Flexural Stiffness EI* Reduce Axial Stiffness EA* Direct Analysis Method Steps 17Combined Forces Spec 13th Ed

18. Apply Notional Loads N i =0.002Y i N i = Notional Lateral Load Applied at Level i Y i = Gravity Load at Level i from Load Combinations Direct Analysis Method 18Combined Forces Spec 13th Ed Notional loads are applied to ALL load combinations unless second order to first order drift ratio is ≤ 1.5. Then apply as minimum lateral load per Appendix 7.3.

19. Reduce Flexural Stiffness EI* EI*=0.8  b EI Required for all members who contribute to lateral stability of the structure (safe to include for all members) E= Modulus of Elasticity I= Moment of Inertia about Axis of Bending  b =Reduction Factor for Inelastic Action Direct Analysis Method 19Combined Forces Spec 13th Ed

20. Reduce Flexural Stiffness EI*  b =Reduction Factor for Inelastic Action for Direct Analysis Method P r = Required Axial Compressive Strength P y = AF y = Member Yield Strength  = 1.0 (LRFD), 1.6 (ASD) 20Combined Forces Spec 13th Ed

21. Reduce Axial Stiffness EA* EA*=0.8EA E= Modulus of Elasticity A= Cross Sectional Member Area Direct Analysis Method 21Combined Forces Spec 13th Ed Required for all members who contribute to lateral stability of the structure (safe to include for all members)