Lecture 9 - Flexure

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Lecture 9 - Flexure

• June 20, 2003
• CVEN 444

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Lecture Goals

• Load Envelopes
• Resistance Factors and Loads
• Design of Singly Reinforced Rectangular Beam
• Unknown section dimensions
• Known section dimensions

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MomentEnvelopes
The moment envelope curve defines the extreme
boundary values of bending moment along the beam
due to critical placements of design live loading.
Fig. 10-10 MacGregor (1997)
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MomentEnvelopes Example
Given following beam with a dead load of 1 k/ft
and live load 2 k/ft obtain the shear and bending
moment envelopes
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MomentEnvelopes Example
Use a series of shear and bending moment
diagrams Wu 1.2wD 1.6wL
Moment Diagram
Shear Diagram
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MomentEnvelopes Example
Use a series of shear and bending moment
diagrams Wu 1.2wD 1.6wL
(Dead Load Only)
Moment Diagram
Shear Diagram
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MomentEnvelopes Example
Use a series of shear and bending moment
diagrams Wu 1.2wD 1.6wL
Moment Diagram
Shear Diagram
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MomentEnvelopes Example
The shear envelope
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MomentEnvelopes Example
The moment envelope
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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
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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
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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
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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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Example - Singly Reinforced Beam
Design a singly reinforced beam, which has a
moment capacity, Mu 225 k-ft, fc 3 ksi, fy
40 ksi and c/d 0.275 Use a b 12 in. and
determine whether or not it is sufficient space
for the chosen tension steel.
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Example - Singly Reinforced Beam
From the calculation of Mn
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Example - Singly Reinforced Beam
Select c/d 0.275 so that f 0.9. Compute k and
determine Ru
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Example - Singly Reinforced Beam
Calculate the bd 2
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Example - Singly Reinforced Beam
Calculate d, if b 12 in.
Use d 22.5 in., so that h 25 in.
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Example - Singly Reinforced Beam
Calculate As for the beam
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Example - Singly Reinforced Beam
Chose one layer of 4 9 bars Compute r
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Example - Singly Reinforced Beam
Calculate rmin for the beam
The beam is OK for the minimum r
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Example - Singly Reinforced Beam
Check whether or not the bars will fit into the
beam. The diameter of the 9 1.128 in.
So b 12 in. works.
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Example - Singly Reinforced Beam
Check the height of the beam.
Use h 25 in.
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Example - Singly Reinforced Beam
Find a
Find c
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Example - Singly Reinforced Beam
Check the strain in the steel
Therefore, f is 0.9
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Example - Singly Reinforced Beam
Compute the Mn for the beam
Calculate Mu
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Example - Singly Reinforced Beam
Check the beam Mu 225 k-ft12 in/ft 2700 k-in
Over-designed the beam by 6
Use a smaller c/d ratio