Lecture 13 Analysis and Design

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Lecture 13 Analysis and Design

• September 30, 2002
• CVEN 444

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Lecture Goals

• One-way Slab design
• Resistance Factors and Loads
• Design of Singly Reinforced Rectangular Beam
• Unknown section dimensions
• Known section dimensions

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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 are assumed to be 14 in. wide. The
concrete strength is 3750 psi and the
reinforcement strength is 60 ksi. The live load
is 100 psf and dead load of 50 psf.
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Example One-way Slab
Use table 9.5(a) to determine the minimum
thickness of the slab.
End bay
Interior bays
Use 7.5 in.
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Example One-way Slab
Compute the trial factored loads based on
thickness.
Factored load
Check ratio for 8.3.3
OK!
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Example One-way Slab
Compute factored external moment.
Nominal moment
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Example One-way Slab
The thickness is 7.5 in. so we will assume that
the bar is located d 7.5in 1.0 in. 6.5 in.
(From 3.3.2 ACI 318 0.75 in 0.25 in(
0.5diameter of bar) 1.0 in
Assume that the moment arm is 0.9d
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Example One-way Slab
Recalculate using As 0.2 in2
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Example One-way Slab
Check the yield of the steel
Steel has yielded so we can use f 0.9
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Example One-way Slab
Check to minimum requirement for every foot
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Example One-way Slab
What we can do is rework the spacing between the
bars by change b Use a 4 bar As 0.2 in2
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Example One-way Slab
Check for shrinkage and temperature reinforcement
for rmin 0.0018 As rminbh from 7.12.2.1 ACI
Use 1 4 bar every 9 in.
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
Analysis Versus Design
Analysis Given a cross-section, fc ,
reinforcement sizes, location, fy compute
resistance or capacity Design Given factored
load effect (such as Mu) select suitable
section(dimensions, fc, fy, reinforcement,
etc.)
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
ACI Code Requirements for Strength Design
Basic Equation factored resistance
factored load effect
Ex.
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ACI Code Requirements for Strength Design
Mu Moment due to factored loads (required
ultimate moment) Mn Nominal moment capacity
of the cross-section using nominal dimensions
and specified material strengths. f
Strength reduction factor (Accounts for
variability in dimensions, material strengths,
approximations in strength equations.
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
Required Strength (ACI 318, sec 9.2)
U Required Strength to resist factored
loads D Dead Loads L Live loads W Wind
Loads E Earthquake Loads
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Flexural Design of Reinforced Concrete Beams and
Slab Sections
Required Strength (ACI 318, sec 9.2)
H Pressure or Weight Loads due to
soil,ground water,etc. F Pressure or weight
Loads due to fluids with well defined densities
and controllable maximum heights. T Effect
of temperature, creep, shrinkage, differential
settlement, shrinkage compensating.
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Factored Load Combinations
U 1.2 D 1.6 L Always check even if other
load types are present. U 1.2(D F
T) 1.6(L H) 0.5 (Lr or S or R) U 1.2D
1.6 (Lr or S or R) (L or 0.8W) U 1.2D 1.6
W 1.0L 0.5(Lr or S or R) U 0.9 D 1.6W
1.6H U 0.9 D 1.0E 1.6H
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Resistance Factors, f - ACI Sec 9.3.2 Strength
Reduction Factors
1 Flexure w/ or w/o axial tension The strength
reduction factor, f, will come into the
calculation of the strength of the beam.
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Resistance Factors, f - ACI Sec 9.3.2 Strength
Reduction Factors
2 Axial Tension f 0.90 3 Axial
Compression w or w/o flexure (a) Member w/
spiral reinforcement f 0.70 (b) Other
reinforcement members f 0.65 (may increase
for very small axial loads)
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Resistance Factors, f - ACI Sec 9.3.2 Strength
Reduction Factors
4 Shear and Torsion f 0.75 5 Bearing on
Concrete f 0.65 ACI Sec 9.3.4 f
factors for regions of high seismic risk
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Background Information for Designing Beam Sections
1.
Location of Reinforcement locate reinforcement
where cracking occurs (tension region) Tensile
stresses may be due to a ) Flexure b )
Axial Loads c ) Shrinkage effects
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Background Information for Designing Beam Sections
2.
Construction formwork is expensive - try to
reuse at several floors
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Background Information for Designing Beam Sections
3.

• Beam Depths
• ACI 318 - Table 9.5(a) min. h based on l
  (span) (slab beams)
• Rule of thumb hb (in) l (ft)
• Design for max. moment over a support to set
  depth of a continuous beam.

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Background Information for Designing Beam Sections
4.
Concrete Cover Cover Dimension between the
surface of the slab or beam and the
reinforcement
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Background Information for Designing Beam Sections
Concrete Cover Why is cover needed? a
Bonds reinforcement to concrete b Protect
reinforcement against corrosion c Protect
reinforcement from fire (over heating
causes strength loss) d Additional cover used
in garages, factories, etc. to account for
abrasion and wear.
4.
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Background Information for Designing Beam Sections

• Minimum Cover Dimensions (ACI 318 Sec 7.7)
• Sample values for cast in-place concrete
• Concrete cast against exposed to earth - 3 in.
• Concrete (formed) exposed to earth weather
  No. 6 to No. 18 bars - 2 in. No. 5 and
  smaller - 1.5 in

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Background Information for Designing Beam Sections

• Minimum Cover Dimensions (ACI 318 Sec 7.7)
• Concrete not exposed to earth or weather - Slab,
  walls, joists No. 14 and No. 18 bars - 1.5
  in No. 11 bar and smaller - 0.75 in - Beams,
  Columns - 1.5 in

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Background Information for Designing Beam Sections
5.
Bar Spacing Limits (ACI 318 Sec. 7.6) -
Minimum spacing of bars - Maximum spacing of
flexural reinforcement in walls slabs
Max. space smaller of
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Minimum Cover Dimension
Interior beam.
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Minimum Cover Dimension
Reinforcement bar arrangement for two layers.
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Minimum Cover Dimension
ACI 3.3.3 Nominal maximum aggregate size. 3/4
clear space., 1/3 slab depth, 1/5 narrowest dim.
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
1) For design moment Substitute
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Let
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Let
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)
Assume that the material properties, loads, and
span length are all known. Estimate the
dimensions of self-weight using the following
rules of thumb a. The depth, h, may be taken as
approximate 8 to 10 of the span (1in deep per
foot of span) and estimate the width, b, as
about one-half of h. b. The weight of a
rectangular beam will be about 15 of the
superimposed loads (dead, live, etc.). Assume
b is about one-half of h. Immediate values of h
and b from these two procedures should be
selected. Calculate self-weight and Mu.
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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Select a reasonable value for r based on
  experience or start with a value of about 45 to
  55 of rbal.
• Calculate the reinforcement index,

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Calculate the coefficient
• Calculate the required value of

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Select b as a function of d. b (0.45d to
  0.65d)
• Solve for d. Typically round d to nearest 0.5
  inch value to get a whole inch value for h, which
  is approximately d h - 2.5 in.

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Solve for the width, b, using selected d value.
  Round b to nearest whole inch value.
• Re-calculate the beam self-weight and Mu based on
  the selected b and h dimensions. Go back to step
  1 only if the new self-weight results in
  significant change in Mu.

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Calculate required As rbd. Use the selected
  value of d from Step 6. And the calculated (not
  rounded) value of b from step 7 to avoid errors
  from rounding.

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Select steel reinforcing bars to provide As
  (As required from step 9). Confirm that the
  bars will fit within the cross-section. It may
  be necessary to change bar sizes to fit the steel
  in one layer. If you need to use two layers of
  steel, the value of h should be adjusted
  accordingly.

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Design Procedure for section dimensions are
unknown (singly Reinforced Beams)

• Calculate the actual Mn for the section
  dimensions and reinforcement selected. Check
  strength, (keep over-design within
  10)

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Design Procedure for section dimensions are known
(singly Reinforced Beams)
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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Calculate controlling value for the design
  moment, Mu.
• Calculate d, since h is known. d h -
  2.5in. for one layer of reinforcement. d
  h - 3.5in. for two layers of reinforcement.

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Solve for required area of tension reinforcement,
  As , based on the following equation.
  

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Rewrite the equation

Assume (d-a/2) 0.9d to 0.95d and solve for
As(reqd) Note f 0.9 for flexure without
axial load (ACI 318-02, Sec. 9.3)
Assume that the steel will yield et gt 0.005
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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Select reinforcing bars so As(provided)
  As(reqd) Confirm bars will fit within the
  cross-section. It may be necessary to change bar
  sizes to fit the steel in one layer or even to go
  to two layers of steel.

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Calculate the actual Mn for the section
  dimensions and reinforcement selected. Verify
  . Check strength
  (keep over-design with 10)

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Design Procedure for section dimensions are known
(singly Reinforced Beams)

• Check whether As(provided) is within the
  allowable limits. As(provided)
  As(min)

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Homework
Problem 5.4