Section 9 Members subjected to Combined Forces (Beam-Columns) Dr S R Satish Kumar Dept. of Civil Engineering I I T Madras, Chennai-36 sr.satishkumar@gmail.com Dr S R Satish Kumar, IIT Madras

SECTION 9 MEMBER SUBJECTED TO COMBINED FORCES IS 800: 2007 SECTION 9 MEMBER SUBJECTED TO COMBINED FORCES 9.1 General 9.2 Combined Shear and Bending 9.3 Combined Axial Force and Bending Moment 9.3.1 Section Strength 9.3.2 Overall Member Strength Dr S R Satish Kumar, IIT Madras

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

9.2 Combined Shear and Bending Sections subjected to HIGH shear force > 0.6 Vd a) Plastic or Compact Section b) Semi-compact Section Mfd = 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 Dr S R Satish Kumar, IIT Madras

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 Member Flexural Buckling STEPS IN ANALYSING BEAM-COLUMNS SUMMARY Dr S R Satish Kumar, IIT Madras

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

Py = Ag*fy Mp = Zp*fy SHORT BEAM-COLUMNS = Py Axial compression MP Bending moment Fc M Combined compression and bending, P & M fy + M P Py = Ag*fy Mp = Zp*fy Dr S R Satish Kumar, IIT Madras

P/Py + M/Mp  1.0 (conservative) SHORT BEAM-COLUMNS O 1.0 Mo/Mp Mmax/Mp M/Mp Short column loading curve Pcl /Py P0/Py P/Py M / MP  1.0 P / Py + 0.85 M / MP  1.0 P/Py + M/Mp  1.0 (conservative) Failure envelope 1.0 M = P e Dr S R Satish Kumar, IIT Madras

Non – Sway Frame LONG BEAM COLUMNS Mmax = Mo + P  M0 P * 0 Linear Non-Linear Mmax = Mo + P Dr S R Satish Kumar, IIT Madras

Sway Frames 0  M0 M LONG BEAM-COLUMNS M = Mo + P Dr S R Satish Kumar, IIT Madras

Cm accounts for moment gradient effects LONG BEAM-COLUMNS M0 0.5 0.8 P/Pcr = 0.0 1.0  O P . Pcr M0/MP= 0.0  A 0.1 B Cm accounts for moment gradient effects Dr S R Satish Kumar, IIT Madras

Failure Envelope LONG BEAM-COLUMNS 1.0 Fc/Pcs 1.0 M / MP Short column Long columns loading curve Short column Fcl /Pcs F0/Pcs Fc/Pcs Eqn. 3 1.0 Failure Envelope 1.0 Mo/Mp Mmax/Mp M / MP Dr S R Satish Kumar, IIT Madras

SLENDER BEAM-COLUMNS Modified Strength Curves for Linear Analysis Uniaxial Bending A 1.0 Major axis bending Mx/Mpx Fc/Pcs Fcl/Pcs After correction for (P-) effect Short column failure envelope After correcting for sway and bow (P- and P-) After correction for (P-) effect Fc/Pcs Fcl/Pcs My/Mpy Short column failure envelope After correcting for sway and bow (P- and P-) 1.0 P* P*  Minor axis bending Dr S R Satish Kumar, IIT Madras

SLENDER BEAM-COLUMNS Mx  Mpx P/Py < 0.25 P/Py >0.5 Short Column Slender Column Fig. 7 Beam-column Moment Rotation Behaviour O B C C’ F F’ F’’ E D A Dr S R Satish Kumar, IIT Madras

BEAM-COLUMNS / BIAXIAL BENDING Fcl/Pcs My/ Mpy Mx/Mpx Fig. 8 beam-columns under Biaxial Bending /r = 0 /r increases Dr S R Satish Kumar, IIT Madras

IS 800: 2007  Md 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 fx.  fy /m0 9.3.2 Overall Member Strength 9.3.2.1 Bending and Axial Tension   Md 9.3.2.2 Bending and Axial Compression  Dr S R Satish Kumar, IIT Madras

9.3.2.2 Bending and Axial Compression Cmy, Cmz = equivalent uniform moment factor as per table 18 Also CmLT Dr S R Satish Kumar, IIT Madras

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 Dr S R Satish Kumar, IIT Madras

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 Dr S R Satish Kumar, IIT Madras