COLUMN DESIGN - civil.utm. Classification of Columns tion Braced Slender Non-slender Unbraced Slender

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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