© Dr S R Satish Kumar, IIT Madras1 SECTION 7 DESIGN OF COMPRESSION MEMBERS.

© 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


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...


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


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


(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


(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


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


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


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


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


7.4 COLUMN BASES

fyms tfbawt /)3.0(5.2 022

7.4.2 Gusseted Bases7.4.3 Slab Bases

a b


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


• 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


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


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


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 ?


DESIGN CONSIDERATIONS FOR LACED AND BATTENED COLUMNS

(a) Single Lacing (b) Double Lacing (c) Battens

Built-up column members


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 KumarDepartment of Civil Engineering

IIT Madras Chennai 600 [email protected]

mailto:[email protected]

© Dr S R Satish Kumar, IIT Madras1 SECTION 7 DESIGN OF COMPRESSION MEMBERS.

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Transcript of © Dr S R Satish Kumar, IIT Madras1 SECTION 7 DESIGN OF COMPRESSION MEMBERS.