1. Dr S R Satish Kumar, IIT Madras 1 Section 9 Members subjected to Combined Forces (Beam-Columns)

2. Dr S R Satish Kumar, IIT Madras 2 SECTION 9 MEMBER SUBJECTED TO COMBINED FORCES 9.1General 9.2Combined Shear and Bending 9.3 Combined Axial Force and Bending Moment 9.3.1 Section Strength 9.3.2 Overall Member Strength

3. Dr S R Satish Kumar, IIT Madras 3 Secondary effects on beam behaviour Elastic Bending stress Elastic Shear stress Plastic range a b c 9.2Combined Shear and Bending

4. Dr S R Satish Kumar, IIT Madras 4 9.2Combined Shear and Bending Sections subjected to HIGH shear force > 0.6 Vd a) Plastic or Compact Section b) Semi-compact Section M fd = plastic design strength of the area of c/s excluding the shear area and considering partial safety factor V = factored applied shear force; Vd = design shear strength

5. Dr S R Satish Kumar, IIT Madras 5 9.3 Combined Axial Force and Bending Moment DESIGN OF BEAM COLUMNS INTRODUCTION SHORT & LONG BEAM-COLUMNS Modes of failure Ultimate strength BIAXIALLY BENT BEAM-COLUMNS DESIGN STRENGTH EQUATIONS Local Section Flexural Yielding Overall MemberFlexural Buckling STEPS IN ANALYSING BEAM-COLUMNS SUMMARY

6. Dr S R Satish Kumar, IIT Madras 6 INTRODUCTION Occurrence of Beam Columns x y z  Eccentric Compression  Joint Moments in Braced Frames Rigid  Sway Moments in Unbraced Frames  Biaxial Moments in Corner Columns of Frames

7. Dr S R Satish Kumar, IIT Madras 7 SHORT BEAM-COLUMNS P = P y Axial compression M P Bending moment F c M Combined compression and bending, P & M fyfy fyfy fyfy fyfy fyfy fyfy f y fyfy fyfy + M P y = A g *f y M p = Z p *f y

8. Dr S R Satish Kumar, IIT Madras 8 SHORT BEAM-COLUMNS Failure envelope 1.0 O M o /M p M max /M p M/M p Short column loading curve P cl /P y P 0 /P y P/P y M / M P  1.0 P / P y + 0.85 M / M P  1.0 P/P y + M/M p  1.0 (conservative) M = P e

9. Dr S R Satish Kumar, IIT Madras 9 LONG BEAM COLUMNS Linear Non-Linear  00 M0M0 P *  M0M0 Non – Sway Frame M max = M o + P 

10. Dr S R Satish Kumar, IIT Madras 10 LONG BEAM-COLUMNS Sway Frames 00  M0M0 M M = M o + P 

11. Dr S R Satish Kumar, IIT Madras 11 LONG BEAM-COLUMNS B M0M0 0.5 0.8 P/P cr = 0.0 1.0  0.8 O 1.0 P. P cr M 0 /M P = 0.0  A 0.1 0.5 C m accounts for moment gradient effects

12. Dr S R Satish Kumar, IIT Madras 12 LONG BEAM-COLUMNS Failure Envelope M o /M p Long columns loading curve Short column loading curve F cl /P cs F 0 /P cs F c /P cs Eqn. 3 M max /M p M / M P 1.0

13. Dr S R Satish Kumar, IIT Madras 13 SLENDER BEAM-COLUMNS Modified Strength Curves for Linear Analysis After correction for (P-  ) effect F c /P cs F cl /P cs M y /M py Short column failure envelope After correcting for sway and bow (P-  and P-  ) 1.0 P*  P*  Minor axis bending A 1.0 Major axis bending M x /M px F c /P cs F cl /P cs After correction for (P-  ) effect Short column failure envelope After correcting for sway and bow (P-  and P-  ) 1.0 Uniaxial Bending

14. Dr S R Satish Kumar, IIT Madras 14 BEAM-COLUMNS / BIAXIAL BENDING F cl /P cs M y / M py M x /M px Fig. 8 beam-columns under Biaxial Bending /r = 0 /r increases

15. Dr S R Satish Kumar, IIT Madras 15 9.3 Combined Axial Force and Bending Moment 9.3.1 Section Strength 9.3.1.1 Plastic and Compact Sections 9.3.1.3 Semi-compact section f x.  f y /  m0 9.3.2 Overall Member Strength 9.3.2.1 Bending and Axial Tension   M d

16. Dr S R Satish Kumar, IIT Madras 16 9.3.2.2 Bending and Axial Compression C my, C mz = equivalent uniform moment factor as per table 18 Also C mLT

17. Dr S R Satish Kumar, IIT Madras 17 STEPS IN BEAM-COLUMN ANALYSIS Steps in Beam-Column Analysis  Calculate section properties  Evaluate the type of section  Check using interaction equation for section yielding  Check using interction equation for overall buckling Beam-Column Design  using equivalent axial load

18. Dr S R Satish Kumar, IIT Madras 18 SUMMARY Short Beam-Columns Fail by Section Plastification Slender Beam-Columns may Fail By  Section Plstification  Overall Flexural Yielding  Overall Torsional-Flexural Buckling Intetaction Eqs. Conservatively Consider  P-  and P-  Effects Advanced Analysis Methods Account for P-  and P-  Effects, directly & more accuraely