Section 9 Members subjected to Combined Forces BeamColumns
Section 9 Members subjected to Combined Forces (Beam-Columns) Dr S R Satish Kumar, IIT Madras 1
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 2
9. 2 Combined Shear and Bending Elastic Shear stress Elastic Bending stress a c b Plastic range Secondary effects on beam behaviour Dr S R Satish Kumar, IIT Madras 3
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 4
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 Overall Member Flexural Yielding Flexural Buckling STEPS IN ANALYSING BEAM-COLUMNS SUMMARY Dr S R Satish Kumar, IIT Madras 5
INTRODUCTION Occurrence of Beam Columns u. Eccentric Compression u. Joint Moments in Braced Frames Rigid u. Sway Moments in Unbraced Frames u. Biaxial Moments in Corner Columns of Frames y z x Dr S R Satish Kumar, IIT Madras 6
SHORT BEAM-COLUMNS fy fy = PM fy Py Axial compression Py = Ag*fy Dr S R Satish Kumar, IIT Madras fy fy fy MP Bending moment fy + fy fy Fc M Combined compression and bending, P&M Mp = Zp*fy 7
SHORT BEAM-COLUMNS Short column loading curve 1. 0 P/Py M=Pe Failure envelope P 0/Py Pcl /Py O Mo/Mp Mmax/Mp 1. 0 M/Mp M / MP 1. 0 P / Py + 0. 85 M / MP 1. 0 P/P + M/Mp 1. 0 (conservative) Dr S R Satish Kumar, y. IIT Madras 8
LONG BEAM COLUMNS Non – Sway Frame M 0 P * 0 Mmax = Mo + P Dr S R Satish Kumar, IIT Madras Linear Non-Linear 9
LONG BEAM-COLUMNS Sway Frames 0 M M 0 M = Mo + P Dr S R Satish Kumar, IIT Madras 10
LONG BEAM-COLUMNS M 0/MP= 0. 0 M 0 P/Pcr = 0. 0 A 1. 0 0. 1 P. Pcr 0. 5 B 0. 5 0. 8 1. 0 O Cm accounts for moment gradient effects Dr S R Satish Kumar, IIT Madras 11
LONG BEAM-COLUMNS 1. 0 Short column loading curve Failure Envelope Fc/Pcs F 0/Pcs Long columns loading curve Fcl /Pcs Eqn. 3 Mo/Mp Mmax/Mp M / MP Dr S R Satish Kumar, IIT Madras 1. 0 12
SLENDER BEAM-COLUMNS Modified Strength Curves for Linear Analysis Uniaxial Bending 1. 0 After correcting for sway and bow (P- and P- ) Short column failure envelope Fc/Pcs Fcl/Pcs After correction for (P- ) effect P* P* 1. 0 Short column failure envelope Fc/Pcs A After correction for (P- ) effect Fcl/Pcs My/Mpy 1. 0 Minor axis bending Mx/Mpx 1. 0 Major axis bending Dr S R Satish Kumar, IIT Madras 13
BEAM-COLUMNS / BIAXIAL BENDING Fcl/Pcs /r = 0 /r increases My/ Mpy Mx/Mpx Fig. 8 beam-columns under Biaxial Bending Dr S R Satish Kumar, IIT Madras 14
9. 3 Combined Axial Force and Bending Moment 9. 3. 1 Section Strength 9. 3. 1. 1 Plastic and Compact Sections fx. fy / m 0 9. 3. 1. 3 Semi-compact section 9. 3. 2 Overall Member Strength 9. 3. 2. 1 Bending and Axial Tension Md Dr S R Satish Kumar, IIT Madras 15
9. 3. 2. 2 Bending and Axial Compression n n Cmy, Cmz = equivalent uniform moment factor as per table 18 Also Cm. LT Dr S R Satish Kumar, IIT Madras 16
STEPS IN BEAM-COLUMN ANALYSIS Steps in Beam-Column Analysis u. Calculate section properties u. Evaluate the type of section u. Check using interaction equation for section yielding u. Check using interction equation for overall buckling Beam-Column Design u using equivalent axial load Dr S R Satish Kumar, IIT Madras 17
SUMMARY n n Short Beam-Columns Fail by Section Plastification Slender Beam-Columns may Fail By u Section Plstification u Overall Flexural Yielding u Overall Torsional-Flexural Buckling Intetaction Eqs. Conservatively Consider u P- and P- Effects Advanced Analysis Methods Account for P- and P- Effects, directly & more accuraely Dr S R Satish Kumar, IIT Madras 18
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