SECTION 7 DESIGN OF COMPRESSION MEMBERS
© Dr S R Satish Kumar, IIT Madras
1
INTRODUCTION TO COLUMN BUCKLING
• Introduction
• Elastic buckling of an ideal column
• Strength curve for an ideal column
• Strength of practical column
• Concepts of effective lengths
• Torsional and torsional-flexural buckling
• Conclusions
2
INTRODUCTION
• Compression members: short or long
• Squashing of short column
• Buckling of long column
• Steel members more susceptible to buckling
compared to RC and PSC members
3
ELASTIC BUCKLING OF EULER
COLUMN
Assumptions:
•
Material of strut - homogenous and
linearly elastic
•
No imperfections (perfectly straight)
•
No eccentricity of loading
•
No residual stresss
4
ELASTIC BUCKLING OF EULER COLUMN
Pcr
The governing differential
equation is
y

d 2 y Pcr

.y  0
2
EI
dx
x
Lowest value of the critical load
Pcr  2 E I
 cr 

A
A 2
 2E r2
 2E
 2E
 cr 

 2
2
2

( / r )

Pcr 
 2 EI
2
5
STRENGTH CURVE FOR AN IDEAL
STRUT
axially loaded initially straight pin-ended column
f
1
B
Column fails when the
Plastic yield defined
compressive stress is greater
C
by 
than or equal to the values
fy
fy
f
=
A
A
defined by ACB.
Elastic buckling (  )
cr
defined by  2 E / 
AC  Failure by yielding (Low
slenderness ratios)
CB  Failure by bucking (  c )
2
B
c
 =  /r
6
STRENGTH CURVE FOR AN IDEAL STRUT
f /fy
Plastic yield
Elastic buckling
1.0
1.0
 = (fy / cr )1/2
Strength curve in a non-dimensional form
7
FACTORS AFFECTING STRENGTH
OF A COLUMN
IN PRACTICE:
•
Effect of initial out of straightness
•
Effect of eccentricity of applied
loading
•
Effect of residual stress
•
Effect of a strain hardening and the
absence of clearly defined yield
point
•
Effect of all features taken together
8
Residual
Stresses
Residual stresses
in web
Residual stresses
in flanges
Residual stresses distribution (no applied load)
Residual stresses in an
elastic section subjected
9
to a mean stress a
Effect of all features taken together
a
fy
Data from collapse tests

 

Theoretical elastic buckling

 


  
 

   
 

Lower
bound curve
 (E/fy)1/2
/r
10
SECTION 7 DESIGN OF COMPRESSION MEMBERS
7.1
Design Strength
7.2
Effective Length of Compression Members
7.3
Design Details
7.3.1
Thickness of Plate Elements
7.3.2
Effective Sectional Area
7.3.3
Eccentricity for Stanchions and Columns
7.3.4
Splices
]7.4
Column Bases
7.5
7.4.1
Gusseted Bases
7.4.2
Slab Bases
Angle Struts
7.5.1
Single Angle Struts
7.5.2
Double Angle Struts
7.5.3
Continuous Members
7.5.4
Combined Stresses
© Dr S R Satish Kumar, IIT Madras
Cont...
11
7.6
7.7
7.8
SECTION 7 DESIGN OF COMPRESSION MEMBERS
Laced Columns
7.6.1
General
7.6.2
Design of Lacings
7.6.3
Width of Lacing Bars
7.6.4
Thickness of Lacing Bars
7.6.5
Angle of Inclination
7.6.6
Spacing
7.6.7
Attachment to Main Members
7.6.8
End Tie Plates
Battened Columns
7.7.1
7.7.2
General
Design of Battens
7.7.3
Spacing of Battens
7.7.4
Attachment to Main Members
Compression Members Composed of Two Components
Back-to-Back
© Dr S R Satish Kumar,
IIT Madras
end
12
INTRODUCTION
c
fy
Test data (x) from collapse tests
on practical columns
xxx
x
x
200
Euler curve
xx
x
x
xx x
x
Design curve
x
100
x
x
50
xx
x x
100
150
Slenderness  (/r)
Typical column design curve
© Dr S R Satish Kumar, IIT Madras
13
Cross Section Shapes for
Rolled Steel Compression Members
(a) Single Angle
(d) Channel
(b) Double Angle
(c) Tee
(e) Hollow Circular (f) Rectangular Hollow
Section (RHS)
Section (CHS)
© Dr S R Satish Kumar, IIT Madras
14
Cross Section Shapes for Built - up or
fabricated Compression Members
(a) Box Section
(d) Plated I Section
(b) Box Section
(c) Box Section
(e) Built - up I Section (f) Built-up Box Section
© Dr S R Satish Kumar, IIT Madras
15
7.1 DESIGN STRENGTH
7.1.2 The design compressive strength of a member is given by
Pd  Ae f cd
f y /  m0
f cd 
0.5
2
2


   



 f y /  m0
 f y /  m0
 = 0.5[1+ ( - 0.2)+ 2]
fcd = the design compressive stress,
λ = non-dimensional effective slenderness ratio, f y
fcc = Euler buckling stress = 2E/(KL/r)2
f cc 


2 2
f y KL
 E
r
 = imperfection factor as in Table 7
 = stress reduction factor as in Table 8
© Dr S R Satish Kumar, IIT Madras
16
Table 10 Buckling Class of Cross-sections
Cross Section
Limits
Rolled I-Sections
h/b > 1.2 :
tf 40 mm
40 < tf <100
z-z
y-y
z-z
y-y
a
b
b
c
Welded I-Section
tf <40 mm
z-z
y-y
z-z
y-y
b
c
c
d
tf >40 mm
Buckling about Buckling Curve
axis
Hollow Section
Hot rolled
Cold formed
Any
Any
a
b
Welded Box
Section, built-up
Generally
Any
Any
b
c
Any
c
Channel, Angle, T
and Solid Sections
© Dr S R Satish Kumar, IIT Madras
17
7.1 DESIGN STRENGTH
Buckling Curves
1
0.9
a
0.8
b
c
0.7
fcd/fy
0.6
d
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
Lamda
TABLE 7.1 IMPERFECTION FACTOR, α
Buckling Class
a
b
c
d

0.21
0.34
0.49
0.76
© Dr S R Satish Kumar, IIT Madras
18
7.2 Effective Length of Compression Members (Table 11)
Boundary Conditions
At one end
At the other end
Translation
Rotation
Translation
Rotation
Restrained
Restrained
Free
Free
Effective
Length
Schematic
represen
-tation
2.0L
Free
Restrained
Restrained
Free
Restrained
Free
Restrained
Free
1.0L
Restrained
Restrained
Free
Restrained
1.2L
Restrained
Restrained
Restrained
Free
0.8L
Restrained
Restrained
Restrained
Restrained
0.65 L
© Dr S R Satish Kumar, IIT Madras
19
7.4 COLUMN BASES
7.4.2 Gusseted Bases
7.4.3 Slab Bases
t s  2.5 w (a 2  0.3b 2 ) m0 / f y
 tf
a
© Dr S R Satish Kumar, IIT Madras
b
20
STEPS IN THE DESIGN OF
AXIALLY LOADED COLUMNS
Design steps:
•
Assume a trial section of area A = P/150
•
Make sure the section is at least semi-compact !
•
•
Arrive at the effective length of the column.
Calculate the slenderness ratios.
•
•
Calculate fcd values along both major and minor axes.
Calculate design compressive strength Pd = (fcd A).
•
Check P < Pd
© Dr S R Satish Kumar, IIT Madras
21
BEHAVIOUR OF ANGLE COMPRESSION MEMBERS
U
• Angles under compression
– Concentric loading - Axial force
1. Local buckling
2. Flexural buckling about v-v axis
V
3. Torsional - Flexural buckling about u-u axis
– Eccentric loading - Axial force & bi-axial moments
– Most practical case
U
– May fail by bi-axial bending or FTB
– (Equal 1, 2, 3 & Unequal 1, 3)
V
© Dr S R Satish Kumar, IIT Madras
V
U
V
U
22
7.5 ANGLE STRUTS
Basic compressive strength curve
• Curve C of Eurocode 3
• Slenderness Ratio:
concentric loading
Single leg Connection
kL/r
(kl/r)eq
Equivalent normalised slenderness ratio
  k1  k 2   k 3 
2
e
2
vv
2
Where, k1, k2, k3 are constants to account for different
end conditions and type of angle.
© Dr S R Satish Kumar, IIT Madras
23
vv
 KL 


 r

  vv 
 2E

250
b1  b2 
 

 E
2
250
 2t
Where
L = laterally unsupported length of the member
rvv = radius of gyration about the minor axis
b1, b2 = width of the two legs of the angle
t = thickness of the leg
ε = yield stress ratio ( 250/fy)0.5
© Dr S R Satish Kumar, IIT Madras
24
7.5
ANGLE STRUTS
7.5.1.2 Loaded through one leg
k1, k2, k3 = constants depending upon the end condition (Table 12)
e  k1  k22vv  k32
No. of bolts at
the each end
connection
Gusset/Connec
-ting member
Fixity†
k1
k2
k3
Fixed
0.20
0.35
20
Hinged
0.70
0.60
5
Fixed
0.75
0.35
20
Hinged
1.25
0.50
60
>2
1
Design ?
© Dr S R Satish Kumar, IIT Madras
25
DESIGN CONSIDERATIONS FOR
LACED AND BATTENED COLUMNS
(a) Single Lacing
(b) Double Lacing
(c) Battens
Built-up column members
© Dr S R Satish Kumar, IIT Madras
26
LACED AND BATTENED COLUMNS
7.6.1.5 The effective slenderness ratio, (KL/r)e = 1.05 (KL/r)0,
to account for shear deformation effects.
7.7.1.4 The effective slenderness ratio of battened column, shall be
taken as 1.1 times the (KL/r)0, where (KL/r)0 is the maximum actual
slenderness ratio of the column, to account for shear deformation
effects.
© Dr S R Satish Kumar, IIT Madras
27
Dr S R Satish Kumar
Department of Civil Engineering
IIT Madras Chennai 600 036
sr.satishkumar@gmail.com
© Dr S R Satish Kumar, IIT Madras
28