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    Lecture 8 - Flexure

    June 18, 2003

    CVEN 444

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    Pan Joist Floor Systems

    ACI Requirements for Joist Construction

    (Sec. 8.11, ACI 318-02)

    Slabs and ribs must be cast monolithically. Ribs must be spaced consistently

    Ribs may not be less than 4 inches in width

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    Pan Joist Floor Systems

    ACI Requirements for Joist Construction (cont.)

    (Sec. 8.11.2, ACI 318-02)

    Depth of ribs may not be more than 3.5

    times the minimum rib width Clear spacing between ribs shall not exceed

    30 inches.

    ** Ribbed slabs not meeting theserequirements are designed as slabs andbeams. **

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    Pan Joist Floor Systems

    Slab Thickness

    (ACI Sec. 8.11.6.1)

    t 2 in. for joints formed with 20 in. widepans

    t 2.5 in. for joints formed with 30 in. widepans (1/12 distance)

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    Pan Joist Floor Systems

    Slab Thickness (cont.)

    Building codes give minimum fire resistancerating:

    1-hour fire rating: in. cover, 3-3.5 slabthickness

    2-hour fire rating: 1 in. cover, 4.5 slabthickness

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    Pan Joist FloorSystems

    StandardRemovable FormDimensions

    Note the shapes

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    Pan Joist Floor Systems

    Standard Removable Form Dimensions

    Standard Widths: 20 in. & 30 in.

    (measured at bottom of ribs)

    Standard Depths: 6, 8, 10, 12, 14, 16 or20 in.

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    Pan Joist Floor Systems

    Standard Removable Form Dimensions(cont.)

    End Forms: one end is closed (built-in) toform the supporting beam

    Tapered End Forms: provide additional shearcapacity at ends of joists by tapering ends toincrease rib width.

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    Pan JoistSlabs

    Standard Pan Joist

    Form Dimensions

    Ref. CECO Concrete

    Construction Catalog

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    Pan Joist

    Slabs

    Standard Pan Joist

    Form DimensionsRef. CECO Concrete Construction

    Catalog

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    Pan Joist Floor Systems

    Laying Out Pan Joist Floors

    Rib/slab thickness

    Governed by strength, fire rating,available space

    Overall depth and rib thickness

    Governed by deflections and shear

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    Pan Joist Floor Systems

    Laying Out Pan Joist Floors (cont.)

    Typically no stirrups are used in joists

    Reducing Forming Costs:

    Use constant joist depth for entire floor

    Use same depth for joists and beams(not always possible)

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    Pan Joist Floor Systems

    Distribution Ribs

    Placed perpendicular to joists*

    Spans < 20 ft.: None

    Spans 20-30 ft.: Provided a midspan

    Spans > 30 ft.: Provided at third-points

    At least one continuous #4 bar is provided at topand bottom of distribution rib.

    *Note: not required by ACI Code, but typically usedin construction

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    Member DepthACI provides minimum member depth andslab thickness requirements that can be usedwithout a deflection calculation (Sec. 9.5 ACI318)

    Useful for selecting preliminary membersizes

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    Member Depth

    ACI 318 - Table 9.5a:

    Min. thickness, h (for beams or ribbed one-way

    slab)For beams with one end continuous: L/18.5

    For beams with both ends continuous: L/21

    L is span length in inches

    Table 9.5a usually gives a depth too shallow fordesign, but should be checked as a minimum.

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    MemberDepth

    ACI 318-99: Table 9.5a

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    Member Depth

    Rule of Thumb:

    hb (in.) ~ L (ft.)

    Ex.) 30 ft. span -> hb ~ 30 in.

    May be a little large, but okay as a start tocalc. DL

    Another Rule of Thumb:

    wDL (web below slab) ~ 15% (wSDL+ wLL)

    Note: For design, start with maximummoment for beam to finalize depth.

    Select b as a function of d

    b ~ (0.45 to 0.65) (d)

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    Approximate Analysis of ContinuousBeam and One-Way Slab Systems

    ACI Moment and Shear Coefficients

    Approximate moments and shearspermitted for design of continuousbeams and one-way slabs

    Section 8.3.3 of ACI Code

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    Approximate Analysis of ContinuousBeam and One-Way Slab Systems

    ACI Moment and Shear Coefficients -Requirements:

    Two or more spans

    Approximately Equal Spans Larger of 2 adjacent spans not greater than

    shorter by > 20%

    Uniform Loads

    LL/DL 3 (unfactored)

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    Approximate Analysis of ContinuousBeam and One-Way Slab Systems

    ACI Moment and Shear Coefficients -Requirements: ( cont.)

    Prismatic members

    Same A, I, E throughout member lengthBeams must be in braced frame withoutsignificant moments due to lateral forces

    Not state in Code, but necessary for

    coefficients to apply.

    ** All these requirements must be met to use thecoefficients!**

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    Approximate Analysis of ContinuousBeam and One-Way Slab Systems

    2

    )(2

    nu

    vu

    numu

    lw

    CV

    lwCMwu = Total factored dead and live

    load per unit length

    Cm = Moment coefficientCv = Shear coefficient

    ln = Clear span length for span inquestion forMu at interior

    face of exterior support, +Muand Vuln = Average of clear span length

    for adjacent spans forMu atinterior supports

    ACI Moment and Shear Coefficients Methodology:

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    Approximate Analysis of ContinuousBeam and One-Way Slab Systems

    ACI Moment andShearCoefficients

    See Section8.3.3 of ACICode

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    Example

    Design the eight-span east west

    in figure. A typical 1-ft wide

    design strip is shaded. A

    partial section through this

    strip is shown. The beams areassumed to be 14 in. wide.

    The concrete strength is 3750

    psi and the reinforcement

    strength is 60 ksi. The liveload is 100 psf and dead load

    of 50 psf.

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    ExampleOne-way Slab

    Use table 9.5(a) to determine the minimum

    thickness of the slab. 12 in15 ft 180 in

    ftl

    180 in.

    min. h = 7.5 in.24 24

    l End bay:

    180 in.min h = 6.43 in.28 28

    l

    Interior bays:

    Use 7.5 in.

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    ExampleOne-way Slab

    Compute the trial factored loads based on thickness.

    D 3 2

    1 ft lb lb7.5 in 150 93.75

    12 in ft ft

    w

    u D L1.2 1.6 1.2 50 psf + 93.75 psf 1.6 100 psf

    332.5 psf

    w w w

    Factored load

    L D3w wCheck ratio for 8.3.3

    OK!

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    ExampleOne-way Slab

    Compute factored external moment.

    22

    UU

    332.5 psf 15 ft6801. lb-ft/ft

    C 1181.61 k-in/ft

    w LM

    UN

    81.61 k-in/ft 90.68k-in/ft0.9

    MM

    Nominal moment

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    ExampleOne-way Slab

    The thickness is 7.5 in. so we will assume that the bar

    is located d = 7.5in1.0 in. = 6.5 in. (From 3.3.2 ACI

    318 0.75 in + ~0.25 in( 0.5*diameter of bar) = 1.0 in

    N s y

    2Ns

    y

    0.92

    90.68 k-in/ft0.258 in /ft0.9 60 ksi 0.9 6.5 in

    aM T d A f d

    MA f d

    Assume that the

    moment arm is 0.9d

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    ExampleOne-way Slab

    Recalculate using As = 0.2 in2

    s y

    c

    NN s y s

    y

    s

    2

    0.258 in. 60 ksi0.405 in.

    0.85 0.85 3.75 ksi 12 in

    2

    2

    90.68 k-in/ft0.405 in.

    60 ksi 6.5 in.2

    0.240 in /ft

    A fa

    f b

    MaM A f d A

    af d

    A

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    ExampleOne-way Slab

    Check the yield of the steel

    1

    t cu

    0.405 in.0.476 in.

    0.85

    6.5 in. 0.476 in.0.003

    0.476 in.

    0.038 0.005

    ac

    d c

    c

    Steel has yielded so

    we can use = 0.9

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    ExampleOne-way Slab

    Check to minimum requirement for every foot

    s

    y

    min min

    c

    y

    0.24 in.0.00301

    12 in. 6.5 in.

    200 200 0.0033360000

    0.003333 3 3750

    0.003160000

    A

    bd

    f

    f

    f

    Problem!

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    ExampleOne-way Slab

    What we can do is rework the spacing between the bars

    by change b Use a #4 bar As = 0.2 in2

    2

    s s 0.2 in 9.23 in.0.00333 6.5 in.

    Use b = 9 in.

    A Ab

    bd d

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    ExampleOne-way Slab

    Check for shrinkage and temperature reinforcement for

    min = 0.0018 As = minbh from 7.12.2.1 ACI

    2s min

    2

    2

    0.0018 12 in. 7.5 in. 0.162 in /ft

    0.2 inspacing = 12 in. =14.8 in.

    0.162 in

    A bd

    Use 1 # 4 bar every 9 in.

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    Pattern Loads

    Using influence lines to determine patternloads

    Largest moments in a continuous beam orframe occur when some spans are loadedand others are not.

    Influence lines are used to determine whichspans to load and which spans not to load.

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    Pattern Loads

    Influence Line: graph of variation ofshear, moment, or other effect at one

    particular point in a structure due to a unitload moving across the structure.

    P tt

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    PatternLoads

    QuantitativeInfluenceLines

    Ordinate arecalculated(exact)

    MacGregor (1997)

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    Pattern Loads

    Qualitative Influence Lines

    Mueller-Breslau Principle

    Used to provide a qualitative guide tothe shape of the influence line

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    Pattern Loads

    Qualitative Influence Lines (cont.)

    For moments

    Insert pin at location of interest

    Twist beam on either side of pinOther supports are unyielding, so

    distorted shape may be easily drawn.

    For frames, joints are assumed free to

    rotate, assume members are rigidlyconnected (angle between membersdoes not change)

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    Qualitative Influence LinesThe Mueller-Breslau principle

    can be stated as follows:

    I f a function at a point on a

    structure, such as reaction, or

    shear, or moment is allowed to

    act without restraint, the

    deflected shape of the structure,

    to some scale, represents the

    inf luence line of the function.

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    Pattern LoadsFrame Example:

    Maximize +M at point B.

    Draw qualitative

    influence lines.

    Resulting pattern load:

    checkerboard pattern

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    Pattern Loads

    Arrangement of Live Loads (ACI 318-02, Sec. 8.9.1)

    It shall be permitted to assume that:

    The live load is applied only to the flooror roof under consideration, and

    The far ends of columns built integrally

    with the structure are considered to befixed.

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    Pattern Loads

    Arrangement of Live Loads ACI 318-99, Sec. 8.9.2:

    It shall be permitted to assume that the

    arrangement of live load is limited tocombinations of:

    Factored dead load on all spans with full

    factored live load on two adjacentspans.

    Factored dead load on all spans with fullfactored live load on alternate spans.