10 Slender Columns

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    Slender Columns

    2

    RCDesign2-Dr.HanyNessim

    Monday,

    November26,

    2012

    Definition Slender columns

    are those subjectto buckling thatproducesadditionalmoment resultingin significant

    reduction ofcolumn capacity.

    P

    P

    e

    Madd = P.

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    2012

    Definition

    If the column isunbraced, i.e.both ends canmove relative toeach other, thenthe additionalmoment is

    greater.

    P

    P

    e

    Unbraced

    P

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    November26,

    2012

    Braced and Unbraced StructuresWhen the structure contains members to resist side-sway (lateralforces) then it is a braced structure.

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    Braced and Unbraced Structures

    When the structure does not contain members to resist side-sway(lateral forces) then it is an unbraced structure.

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    2012

    Braced and Unbraced StructuresAccording to ECP 203Bracing of RC structures is usuallyprovided by shear walls or cores.

    A building without shear walls or coresis considered an unbraced structures.

    A building with a shear wall or a core isconsidered braced only if

    4nfor0.1n0.20

    4nfor0.60

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    Braced and Unbraced Structures

    Hb = Total height of the building (m) N = Sum of all unfactored vertical loadsEI = Sum of EI for all vertical stiffening

    elements n = number of stories E = 4400 fcu

    4nfor0.1n0.20

    4nfor0.60

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    Examples

    Unbra

    cedin

    Y-dir

    ection

    &Braced

    inX-

    direct

    ion

    X

    Y

    Stiff shear walls

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    Examples

    Bracedin

    X&Ydir

    ection

    s

    X

    Y

    Stiff shear walls

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    Examples

    Shear

    wall+

    Core

    System

    Bracedin

    X&Ydir

    ection

    s

    X

    Y

    Stiff shear wallsCore

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    2012

    Slenderness Ratio; Is the ratio between the effective length of

    the column and its cross sectional dimensionsin the direction under consideration.

    He = k.Ho

    Ho = unsupported (clear) length of thecolumn b-column dimension in the direction of study

    i

    H

    b

    H eeb == ior

    g

    g

    A

    Ii =

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    Limitations

    35810Unbraced

    501215Braced

    iDb or tColumn Status

    Columns are considered short columns (not slender) when

    isless than values of the following table

    Columns are considered slender when is greater than values inthe above table BUT less than the values in the following table

    701823Unbraced

    1002530Braced

    iDbColumn Status

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    2012

    Effective Height of ColumnsAccording to the support

    Condition at each end ofThe column, the value of kIs determined, hence, He Ho

    Column end is freeCase4

    Column is connected to an element that cannotprevent rotation.

    Case3

    Column is connected to a slab or a beam shallowerthan the column dimension.

    Case2

    Column is connected to a beam with a depth

    column dimension in this direction

    Case1

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    Values of k

    0.900.800.75Case1

    1.000.950.90Case3

    0.950.850.80Case2

    Case3Case2Case1End condition at top

    End condition at bottom

    k = He/Ho for Braced columns

    k = He/Ho for Unbraced columns

    --1.801.60Case3

    1.601.301.20Case1

    ----2.20Case4

    1.801.501.30Case2

    Case3Case2Case1End condition at top

    End condition at bottom

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    Calculating Additional Moment For Braced Columns

    Calculate column deflection,

    Calculate additional moment, MaddMadd = P.

    columnscircular2000

    .

    columnsrrectangula2000

    .or

    2000

    .

    2

    22

    D

    tb

    D

    tb

    =

    =

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    Design of Slender Braced Columns

    Design for the greatest moment of:M2M1 + Madd/2Mi + Madd/2Pu.emin

    Where:Mi= 0.4M1+0.6M2 0.4M2M2>M1M1 has a negative sign if it is atthe other side of the column.

    M2

    M1

    Mi

    M1

    M2

    Mi

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    Calculating Additional Moment For UnBraced Columns

    Calculate column deflection, Calculate average deflection per story, av

    Where:N; total number of columns in a storyNeglect > 2av

    Calculate additional moment, MaddMadd = P.av

    n

    av

    =

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    RCDesign2-Dr.HanyNessim

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    November26,

    2012

    Design of Slender UnBraced Columns

    Design for the greater moment of:M2 + MaddPu.emin

    M2

    M1

    Mi

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    2012

    Minimum Reinforcement

    Minimum reinforcement ratio is:min = 0.25 + 0.015iormin = 0.25 + 0.052b

    rectangular section

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    Example

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    Example1Design an RC rectangular column

    in a ground floor of a bracedmultistory building, knowingthat:

    Column is connected to a beamwith section (20x60)cm in bothdirections.

    Pu = 1500kN

    b = 250mm & Ho = 4.50m

    fcu = 30N/mm2 & st. 360/520

    t

    250mm

    y

    x

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    Example1

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    Example1

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    Example1