CEEN 3304 T3 Flexural Analysis and Design of 3304/Lectures/CEEN 3304 T3 Flexural...Flexural Analysis...

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Transcript of CEEN 3304 T3 Flexural Analysis and Design of 3304/Lectures/CEEN 3304 T3 Flexural...Flexural Analysis...

  • 1

    Page 1

    CEEN 3304 Concrete Design

    Flexural Analysis and Design of Beams

    Francisco AguigaAssistant Professor

    Civil and Architectural Engineering ProgramTexas A&M University Kingsville

    Page 2

    Stress distributions

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    Page 3

    Concrete beam behavior

    Page 4

    Composite beams

    Satisfy Mint = M, and P = 0

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    Page 5

    Composite beamsdF = dA = (E1)dzdydF = dA = (E2)ndzdyEquating dF and dFn = E1/E2The force in material 1 isdF = dA = dAdzdy = ndzdy = n

    Page 6

    Uncracked concrete beams

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    Page 7

    Stresses in uncracked beam

    Page 8

    Stresses in uncracked beam

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    Page 9

    Stresses on cracked beam

    1. Find neutral axis2. Find Icr3. Find stresses

    Page 10

    Stresses on cracked beam

    ( ) ( ) 02

    2

    = kddnAkdb s

    bkdf

    C c2

    =

    ss fAT =

    jdfATjdM ss==

    jdAMfs

    s =

    2

    22kjbd

    fbkdjd

    fCjdM cc ===

    2

    2kjbd

    Mfc =

    bdAs=

    ( ) nnnk += 22

    3kddjd =

    31 kj =

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    Page 11

    Design aids

    Page 12

    Example 3.2

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    Page 13

    Ultimate flexural strength

    Steel fails when fs = fyConcrete fails when c = u = 0.003Knowing c need to know: C and

    Page 14

    Ultimate flexural strength

    0.72

    0.425

    0.325

    0.56

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    Page 15

    Ultimate flexural strength

    Page 16

    Ultimate flexural strength

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    Page 17

    Example 3.3

    Page 18

    Ultimate flexural strength

    Whitneys equivalent rectangular stress block

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    Page 19

    Rectangular stress block

    Page 20

    Rectangular stress block

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    Page 21

    Moment capacity of beams

    Page 22

    Design aids resistance factor

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    Page 23

    Example 3.4

    Solve the same beam using the rectangular stress block

    Page 24

    Limiting reinforcement ratios

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    Page 25

    ACI strength reduction factors

    ACI (9.3.2)Tension-controlled failure

    When c = 0.003, s > 0.005, So = 0.9

    Compression-controlled failureWhen c = 0.003, s < y = 0.002, So = 0.65

    Transition-controlled failureWhen c = 0.003, y < s < 0.005So = A1 + B1t

    y

    yA

    =

    005.09.000325.0

    1y

    B

    =005.0

    25.01

    Page 26

    ACI strength reduction factors

    Tension-controlled failureWhen c = 0.003, s > 0.005So = 0.9

    Compression-controlled failure

    When c = 0.003, s < ySo = 0.65

    Transition-controlled failureWhen c = 0.003We have y < s < 0.005So = A1 + B1t

    max

    tc

    bal

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    Page 27

    ACI transition-controlled failure

    Reason for limiting t > 0.004 in tension-controlled failure (ACI 10.3.5)

    In 2002 ACI codeLimit to max to 0.75balResults in t = 0.00376, so limit t to 0.004End up with max < 0.75bal

    bdAs=

    ty

    cbal f

    f

    +=

    003.0003.085.0 '1

    Page 28

    ACI tension-controlled failure

    From

    For t = 0.004, c/dt = 3/7And cmax = 3/7 dtSo amax = 1cmax = 3/71dtSo max = 0.364 1 fc/fy (dt/d)

    ttdc

    +=

    003.0003.0

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    Page 29

    ACI tension-controlled failure

    From

    For t = 0.005, c/dt = 3/8And cmax = 3/8 dtSo amax = 1cmax = 3/81dtSo max = 0.319 1 fc/fy (dt/d)

    ttdc

    +=

    003.0003.0

    Page 30

    ACI minimum reinforcement

    ACI 10.5.1 requires that

    dbf

    dbff

    A wy

    wy

    cs

    2003 'min, =

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    Page 31

    Example - analysis

    Page 32

    Example - analysis

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    Page 33

    Design of R/C beams

    ACI 9.5.2.1 minimum span/depth ratios for beams

    Page 34

    Design of R/C slabs

    ACI 9.5.2.1 minimum span/depth ratios for one-way slabs

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    Page 35

    Design of R/C beams

    Selection of widthACI 7.6.1, 7.6.2, and 3.3.2Minimum space for single layer bars

    smin = largest of (db or 1 in. or max aggregate size)

    Page 36

    Design of R/C beams

    Minimum space for multiple layers of bars

    Bars in upper layer placed directly above bottom layerClear distance between layers > 1 in.Also satisfy single layer requirements

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    Page 37

    Design of R/C beamsMinimum cover

    Cast in place concrete protection in. for slabs and 1 in. for beams and columns

    Exposed to weather or in contact with soilCover > 2 in.Concrete cast directly on ground - cover >= 3 in.

    Page 38

    Design of R/C beams

    Minimum widthUsually #3 or #4 are used for stirrupsMinimum cover for bars in beam is 1.5 in.bmin = 2 x 1.5 in. + 2 x 1/2in. + 4 x 1 in. + 3 x 1 in. = 11 in.

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    Page 39

    Design example 1

    Page 40

    Design example 1

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    Page 41

    Design example 1

    Page 42

    Design example 1

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    Page 43

    Design example 2

    Page 44

    Design example 2

    As = 0.0124 x 13 x (28 2.5) = 4.11 in.2

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    Page 45

    T-beams

    Effective flange width beams with slab on both sides (ACI 8.10)

    beff < span of beamEff. overhang width on each side < 8 hf and clear distance to next web

    Effective flange width beams with slab on one side only (ACI 8.10)

    Effective overhang width less than1/12 span of beam6 hf and clear distance to next web

    Page 46

    T-beams

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    Page 47

    T-beams

    Page 48

    T-beams

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    Page 49

    T-beams