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

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CEEN 3304 Concrete Design

Flexural Analysis and Design of Beams

Francisco AguigaAssistant Professor

Civil and Architectural Engineering ProgramTexas A&M University Kingsville

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Stress distributions

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Concrete beam behavior

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Composite beams

Satisfy Mint = M, and P = 0

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

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Uncracked concrete beams

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Stresses in uncracked beam

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Stresses in uncracked beam

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Stresses on cracked beam

1. Find neutral axis2. Find Icr3. Find stresses

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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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Design aids

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Example 3.2

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Ultimate flexural strength

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

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Ultimate flexural strength

0.72

0.425

0.325

0.56

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Ultimate flexural strength

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Ultimate flexural strength

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Example 3.3

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Ultimate flexural strength

Whitneys equivalent rectangular stress block

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Rectangular stress block

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Rectangular stress block

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Moment capacity of beams

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Design aids resistance factor

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Example 3.4

Solve the same beam using the rectangular stress block

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Limiting reinforcement ratios

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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

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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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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

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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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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

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ACI minimum reinforcement

ACI 10.5.1 requires that

dbf

dbff

A wy

wy

cs

2003 'min, =

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Example - analysis

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Example - analysis

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Design of R/C beams

ACI 9.5.2.1 minimum span/depth ratios for beams

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Design of R/C slabs

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

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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)

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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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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.

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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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Design example 1

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Design example 1

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Design example 1

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Design example 1

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Design example 2

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Design example 2

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

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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

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T-beams

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T-beams

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T-beams

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T-beams