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© Dr S R Satish Kumar, IIT Madras 1
SECTION 7 DESIGN OF COMPRESSION MEMBERS
2
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
3
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
4
ELASTIC BUCKLING OF EULER COLUMN
Assumptions:
• Material of strut - homogenous and linearly elastic
• No imperfections (perfectly straight)
• No eccentricity of loading
• No residual stresss
5
The governing differential equation is
02
2
yEI
P
dx
yd cr .
x
y
Pcr
ELASTIC BUCKLING OF EULER COLUMN
2
2
EI
Pcr
Lowest value of the critical load
2
2
2
2
2
22
2
2
)/(
E
r
ErE
A
IE
A
P
cr
cr
cr
6
axially loaded initially straight pin-ended column
B1f
f y A
c = /r
Plastic yield defined
by ff = yElastic buckling (
cr )
defined by 2 E / 2
AC
B
STRENGTH CURVE FOR AN IDEAL STRUT
Column fails when the compressive stress is greater than or equal to the values defined by ACB.
AC Failure by yielding (Low slenderness ratios)
CB Failure by bucking ( c )
7
f /fy
1.0
= (fy/cr)1/2
1.0
Elastic buckling
Plastic yield
Strength curve in a non-dimensional form
STRENGTH CURVE FOR AN IDEAL STRUT
8
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
9
Residual Stresses
10
Effect of all features taken together
© Dr S R Satish Kumar, IIT Madras 11
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
7.4.1 Gusseted Bases 7.4.2 Slab Bases
7.5 Angle Struts
7.5.1 Single Angle Struts
7.5.2 Double Angle Struts
7.5.3 Continuous Members
7.5.4 Combined Stresses Cont...
© Dr S R Satish Kumar, IIT Madras 12
SECTION 7 DESIGN OF COMPRESSION MEMBERS 7.6 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
7.7 Battened Columns
7.7.1 General 7.7.2 Design of Battens
7.7.3 Spacing of Battens
7.7.4 Attachment to Main Members
7.8 Compression Members Composed of Two Components
Back-to-Back end
© Dr S R Satish Kumar, IIT Madras 13
INTRODUCTION
Typical column design curve
c
fy
Test data (x) from collapse testson practical columns
Euler curve
Design curve
Slenderness (/r)
x
x x
x xx x
x x
x x
x x x
x x
x x x x
200
100
50 100 150
© Dr S R Satish Kumar, IIT Madras 14
(a) Single Angle (b) Double Angle (c) Tee
(d) Channel (e) Hollow Circular Section (CHS)
(f) Rectangular HollowSection (RHS)
Cross Section Shapes for Rolled Steel Compression Members
© Dr S R Satish Kumar, IIT Madras 15
(b) Box Section (c) Box Section
(d) Plated I Section (e) Built - up I Section (f) Built-up Box Section
(a) Box Section
Cross Section Shapes for Built - up or fabricated Compression Members
© Dr S R Satish Kumar, IIT Madras 16
7.1.2 The design compressive strength of a member is given by
7.1 DESIGN STRENGTH
0/0/5.022
0/myfmyf
myfcdf
cdfeAdP
= 0.5[1+ ( - 0.2)+ 2]
fcd = the design compressive stress, λ = non-dimensional effective slenderness ratio,
fcc = Euler buckling stress = 2E/(KL/r)2
= imperfection factor as in Table 7
= stress reduction factor as in Table 8
ccy ff ErKLyf 22
© Dr S R Satish Kumar, IIT Madras 17
Cross Section Limits Buckling about axis
Buckling Curve
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
tf >40 mm
z-z
y-y
z-z
y-y
b
c
c
d
Hollow Section Hot rolled
Cold formed
Any
Any
a
b
Welded Box Section, built-up
Generally Any
Any
b
c
Channel, Angle, T and Solid Sections
Any c
Table 10 Buckling Class of Cross-sections
© Dr S R Satish Kumar, IIT Madras 18
TABLE 7.1 IMPERFECTION FACTOR, α
Buckling Class a b c d
0.21 0.34 0.49 0.76
7.1 DESIGN STRENGTH
Buckling Curves
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3Lamda
fcd
/fy
a
bc
d
© Dr S R Satish Kumar, IIT Madras 19
7.2 Effective Length of Compression Members (Table 11)
Boundary Conditions
Schematic represen
-tation
Effective Length
At one end At the other end
Translation Rotation Translation Rotation
Restrained Restrained Free Free
2.0L
Free Restrained Restrained Free
Free Restrained Free 1.0L
Restrained Restrained Free Restrained 1.2L
Restrained Restrained Free 0.8L
Restrained Restrained Restrained 0.65 L
Restrained
Restrained
Restrained
© Dr S R Satish Kumar, IIT Madras 20
7.4 COLUMN BASES
fyms tfbawt /)3.0(5.2 022
7.4.2 Gusseted Bases7.4.3 Slab Bases
a b
© Dr S R Satish Kumar, IIT Madras 21
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 22
• Angles under compression – Concentric loading - Axial force
1. Local buckling
2. Flexural buckling about v-v axis
3. Torsional - Flexural buckling about u-u axis– Eccentric loading - Axial force & bi-axial moments
– Most practical case– May fail by bi-axial bending or FTB
– (Equal 1, 2, 3 & Unequal 1, 3)
BEHAVIOUR OF ANGLE COMPRESSION MEMBERS
V
V U
U
V
V U
U
© Dr S R Satish Kumar, IIT Madras 23
7.5 ANGLE STRUTS
Basic compressive strength curve
• Curve C of Eurocode 3
• Slenderness Ratio:
concentric loading kL/r
Single leg Connection (kl/r)eq
Equivalent normalised slenderness ratio
Where, k1, k2, k3 are constants to account for different end conditions and type of angle.
23
221
2 kkk vve
© Dr S R Satish Kumar, IIT Madras 24
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
250
2E
rKL
vvvv
tE
bb
2250
2
21
© Dr S R Satish Kumar, IIT Madras 25
7.5 ANGLE STRUTS
7.5.1.2 Loaded through one leg
k1, k2, k3 = constants depending upon the end condition (Table 12)
23
221 kkk vve
No. of bolts at the each end connection
Gusset/Connec-ting member
Fixity†
k1 k2 k3
> 2Fixed 0.20 0.35 20
Hinged 0.70 0.60 5
1Fixed 0.75 0.35 20
Hinged 1.25 0.50 60
Design ?
© Dr S R Satish Kumar, IIT Madras 26
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 27
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.
LACED AND BATTENED COLUMNS
© Dr S R Satish Kumar, IIT Madras 28
Dr S R Satish KumarDepartment of Civil Engineering
IIT Madras Chennai 600 [email protected]