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COLUMN DESIGN

Dr. Izni Syahrizal bin IbrahimFaculty of Civil Engineering

Universiti Teknologi Malaysia

Email: iznisyahrizal@utm.my

mailto:iznisyahrizal@utm.my

Introduction

Column: Subjected to axial compressive forces

Carries load from beams and slabs down to the foundation.

May also resist bending moment due to continuity of structureand loading eccentricity

EC2 Clause 5.3.1(7):a) Compression member where the greater cross sectional

dimension does not exceed 4 times the smaller dimension (h 4b)

b) Height is at least 3 times the section depth

Classification of Columns

Co

lum

n C

lass

ific

atio

n

BracedSlender

Non-slender

UnbracedSlender

Non-slender

Lateral stability to the structure as a whole is provided by walls or bracing resist all lateral forces

Lateral loads are resisted by the bending action of

the column

Classification of Columns

Slender or Non-slender column depending on the sensitivity to second order effect (P- effect)

Use slenderness ratio, to measure column vulnerability by elastic instability or buckling

Non-Slender:a) Design action are not significantly affected by

deformation (P- effect is small)b) P- effect can be ignored if does not exceed a

particular valuec) P- effect can be ignored if 10% of the

corresponding first order moments

Classification of Columns

Short column , crushing at ultimate strength

Slender column , buckling under low compressive load

Compression failure

Buckling failure

Classification of Columns

Major axis

(x-x)

Minor axis

(y-y)

Plane of

bending

Clear height, l Actual height

Slenderness Ratio

==

lo = Effective length of the columni = Radius of gyration and the axis considerI = Section moment of area of the section about the axisA = Cross sectional area of the column

Effective Length of Column

a) lo = l b) lo = 2l c) lo = 0.7l d) lo = 0.5l e) lo = l g) lo 2lf) 0.5 lo l

For constant cross section

Effective Length of Column

Braced Column:

= . +

. + +

. +

Unbraced Column:

= . + +

; +

+ +

+

k1, k2 Relative flexibilities of rotational restraints at ends 1 & 2, respectively =

Rotation of restraining members for bending moment, MEI Bending stiffness of compression memberl Clear height of compression member between end restraints at each end

k1, k2 = ,

,=

2

EC2: Clause 5.8.3.2(3)

Effective Length of Column

Table 3.19 & 3.20, BS 8110: Part 1: 1997

End Condition at TopEnd Condition at Bottom

1 2 3

Braced Column

1 0.75 0.80 0.90

2 0.80 0.85 0.95

3 0.90 0.95 1.00

Unbraced Column

1 1.2 1.3 1.6

2 1.3 1.5 1.8

3 1.6 1.8 -

4 2.2 - -

Simplified Method

Effective Length of Column

End Condition at TopEnd Condition at Bottom

1 2 3

Braced Column

1 0.75 0.80 0.90

2 0.80 0.85 0.95

3 0.90 0.95 1.00

Unbraced Column

1 1.2 1.3 1.6

2 1.3 1.5 1.8

3 1.6 1.8 -

4 2.2 - -

Condition 1 Column connected monolithically to beams on either side which are at least as deep as the overall dimension of the column in the plane considered. Where column connected to a foundation this should be designed to carry moment

Condition 2 Column connected monolithically to beams or slabs on either side which are shallower than the overall dimension of the column in the plane considered

Condition 3 Column connected to members that do not provide more than nominal restraint to rotation

Condition 4 End of column is unrestrained against both lateral movement and rotation

Table 3.19 & 3.20, BS 8110: Part 1: 1997

Limiting Slenderness Ratio

=

A = 1

1+0.2: = Effective creep ratio

B = 1 + 2 : =

C = 1.7 : =1

2

n =

NEd = Design ultimate axial column loadMo1, Mo2 = First order moments at the end of the column with 2 1fyd = Design yield strength of reinforcementfcd = Design compressive strength of concrete

If , & rm is not known, A = 0.7, B = 1.1 & C = 0.7 may be used

EC2: Clause 5.8.3.1

Limiting Slenderness Ratio

Condition apply for C:

(1) If the end moments, Mo1 & Mo2 give rise to tension on the same side of the column, then rm should be taken +ve (follows C 1.7)

(2) If the column is in a state of double curvature, then rm should be taken ve (follows C 1.7)

(3) For braced members in which the first order moment arise only from or predominantly due to imperfections or transverse loading, then rm should be taken as 1.0 (C = 0.7)

(4) For unbraced member, in general rm should be taken as 1.0 (C = 0.7)

If lim: Short (Non-slender) columnIf lim: Slender column. Second order effects must be considered

in design

EC2: Clause 5.8.3.1

Example 1

SLENDERNESS

Example 1: Slenderness

4 m

4 m

4 m

250 500

250 500

250 500

250 500

25

0

40

0

25

0

40

0

25

0

40

0

275 350

A B C

1

2

3

4

Braced column Axial load = 1050 kN Bending moment (major axis) = 40 kNm (top) & 12 kNm (bottom) Bending moment (minor axis) = 15 kNm (top) & 10 kNm (bottom) Concrete grade = C25 Steel = 500 N/mm2

l1 = 6 m l2 = 8 m

Example 1: Slenderness

l1 = 6 m l2 = 8 m

4 m

5 m

1.5 m250 500

250 500

250 500

275 350

A B C

Roof

1st Floor

Ground

Example 1: SlendernessEC2: Clause 5.8.3.2(3) Method

z

z

y y

Secondary beam250 400

L = 4 m 5 m

Main beam250 500

L = 6 m Main beam250 500

L = 8 m

Secondary beam250 400

L = 4 m

-12 kNm

40 kNm

NEd = 1050 kN

Mz

15 kNm

-10 kNmMy

27

5 m

m

Mz

350 mm

My

Moment & Axial Force

Example 1: Slenderness

Dimension & Size

Column:b h = 275 300 mmActual length: lz = 5000 500 = 4500 mm

ly = 5000 400 = 4600 mm

Beam:Main beam, b h = 250 500 mmActual length: l1 = 6000 mm

l2 = 8000 mm

Secondary beam, b h = 250 400 mmActual length: l1 = l2 = 4000 mm

Example 1: Slenderness

Moment of Inertia, I = bh3/12

Column:

=2753003

12= 0.98 109 mm4

=3002753

12= 0.61 109 mm4

Beam:

Main beam, =2505003

12= 2.60 109 mm4

Secondary beam, =2504003

12= 1.33 109 mm4

Example 1: Slenderness

Stiffness, K = EI/l

Column:

=0.98109

4500= 2.18 105 mm3

=0.61109

4600= 1.32 105 mm3

Beam:

Main beam 1 =2.60109

6000= 4.34 105 mm3

2 =2.60109

8000= 3.26 105 mm3

Secondary beam 1 = 2 =1.33109

4000= 3.33 105 mm3

Example 1: Slenderness

Relative Column Stiffness, k = Kcol/2(Kbeam)

z-axis:

Top end: 2 =2.18105

2 4.34105+3.26105= 0.14 > 0.1 k2 = 0.14

Bottom end: 1 =2.18105

2 4.34105+3.26105= 0.14 > 0.1 k1 = 0.14

y-axis:

Top end: 2 =1.32105

2 3.33105+3.33105= 0.10 < 0.1 k2 = 0.10

Bottom end: 1 =1.32105

2 3.33105+3.33105= 0.10 < 0.1 k1 = 0.10

Example 1: Slenderness

Effective Length of Column

= 0.5 1 +1

0.45 + 11 +

20.45 + 2

, = 0.5 1 +0.14

0.45+0.141 +

0.14

0.45+0.14= 2795mm

, = 0.5 1 +0.10

0.45+0.101 +

0.10

0.45+0.10= 2718mm

Example 1: Slenderness

Radius of Gyration, =

==

0.98 109

275 350= 101

=

=

0.61 109

275 350= 79.4

Slenderness Ratio, = lo/i

=2795

101= .

=2718

79.4= .

Example 1: Slenderness

Slenderness Limit, =

A = 0.7 (eff NOT known)B = 1.1 ( NOT known)C = 1.7 rm (where rm = (Mo1/Mo2)

z-axis: , =12

40= 0.30

Cz = 1.7 (0.30) = 2.00

y-axis: , =10

15= 0.67

Cy = 1.7 (0.67) = 2.37

=

=0.85

=0.8525

1.5= 14.17 N/mm2

=1050103

275350 14.17= 0.77

, =200.71.12.00

0.77= 35.1 > = 27.7 Non-slender about z-axis

, =200.71.12.37

0.77= 41.5 > = 34.2 Non-slender about y-axis

Example 1: Slenderness

Effective Length, lo = Factor Clear Height

z-axis: End condition: Top & Bottom = Condition 1Factor = 0.75 , = 0.75 4500 = 3375mm

y-axis: End condition: Top & Bottom = Condition 1Factor = 0.75 , = 0.75 4600 = 3450mm

Slenderness Ratio, = lo/i

=3375

101= 33.4 < , = 35.1 Non-slender about z-axis

=3450

79.4= 43.5 > , = 41.5 Slender about y-axis

Simplified Method

Axial Load & Moment in Column

For analysis without full frame analysis:

a) Axial loads may generally be obtained by increasing the loads obtained by 10% by assumption that beams & slabs are simply supported. Higher percentage may be required when adjacent spans and/or loadings on them are grossly dissimilar.

b) Bending moments may be calculated using the simplified one free-joint sub-frame. The arrangement of the design ultimate variable action should be such as that to cause maximum moment in the column.

Example 2