Transcript Document
Section 9 Members subjected to
Combined Forces
(Beam-Columns)
Dr S R Satish Kumar, IIT Madras
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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
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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
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9.2
Combined Shear and Bending
Sections subjected to HIGH shear force > 0.6 Vd
a) Plastic or Compact Section
M dv M d M d M fd 1.2 Ze f y m0
2 V / Vd 1
2
b) Semi-compact Section
M dv Z e f y / m 0
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
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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
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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
y
z
x
Dr S R Satish Kumar, IIT Madras
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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
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SHORT BEAM-COLUMNS
Short column
loading curve M = P e
1.0
P/Py
Failure
envelope
P0/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,yIIT Madras
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LONG BEAM COLUMNS
Non – Sway Frame
M0
M0
P *
0
Mmax = Mo + P
Dr S R Satish Kumar, IIT Madras
Linear
Non-Linear
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LONG BEAM-COLUMNS
Sway Frames
0
M
M0
M = Mo + P
Dr S R Satish Kumar, IIT Madras
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LONG BEAM-COLUMNS
M0/MP= 0.0
P/Pcr = 0.0
M0
A
1.0
0.1
P.
Pcr
0.5
B
0.5
0.8
0.8
1.0
O
0
0
M max
Cm M0
1 P
1 P
1 P
P
P
PE
E
E
Cm accounts for moment gradient effects
Dr S R Satish Kumar, IIT Madras
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LONG BEAM-COLUMNS
1.0
Short column
loading curve
Failure Envelope
Fc/Pcs
F0/Pcs
Long columns
loading curve
Fcl /Pcs
Eqn. 3
Mo/Mp
Mmax/Mp
1.0
M / MP
Dr S R Satish Kumar, IIT Madras
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SLENDER BEAM-COLUMNS
Modified Strength Curves for Linear Analysis
Uniaxial Bending
After correcting for sway
and bow (P- and P-)
After correcting for sway and
bow (P- and P-)
1.0
Short column
failure envelope
Fc/Pcs
Fcl/Pcs
After correction
for (P-) effect
P*
1.0
Short column
failure envelope
Fc/Pcs
A
After
correction
for (P-) effect
Fcl/Pcs
P*
My/Mpy
1.0
Minor axis bending
Mx/Mpx
1.0
Major axis bending
Dr S R Satish Kumar, IIT Madras
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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
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9.3 Combined Axial Force and Bending Moment
1
9.3.1 Section Strength
My
9.3.1.1 Plastic and Compact Sections
M ndy
My Mz
N
1.0
N d M dy M dz
Mz
1.0
M ndz
N My Mz
fy /m0
1.0
N d M dy M dz
fx.
9.3.1.3 Semi-compact section
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9.3.2 Overall Member Strength
9.3.2.1 Bending and Axial Tension
P KyM y KzM z
1.0 Md
Pd
M dy
M dz
M eff M T Z e c / A
Dr S R Satish Kumar, IIT Madras
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9.3.2.2 Bending and Axial Compression
Cmy M y
P
Mz
Ky
K LT
1.0
Pdy
M dy
M dz
Cmy M y
C M
P
0.6 K y
K z mz z 1.0
Pdz
M dy
M dz
Cmy, Cmz = equivalent uniform moment factor as per table 18
Also CmLT
K y / z 1 ( y / z 0.2)n y / z 1 0.8n y / z
K LT 1
0.1LT n y
(CmLT 0.25)
1
0.1n y
(CmLT 0.25)
n y / z P / Pdy / z
Dr S R Satish Kumar, IIT Madras
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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
Py,eq P Cmyu y m y M y Cmz mz M z
Dr S R Satish Kumar, IIT Madras
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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
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