Lecture 11 Analysis and Design

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Lecture 11 Analysis and Design

• February 7, 2003
• 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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Pattern Loads

• Using influence lines to determine pattern loads
• Largest moments in a continuous beam or frame
  occur when some spans are loaded and others are
  not.
• Influence lines are used to determine which spans
  to load and which spans not to load.

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Pattern Loads

• Influence Line graph of variation of shear,
  moment, or other effect at one particular point
  in a structure due to a unit load moving across
  the structure.

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Pattern Loads

• Quantitative Influence Lines
• Ordinate are calculated (exact)

MacGregor (1997)
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Pattern Loads

• Qualitative Influence Lines
• Mueller-Breslau Principle
• Used to provide a qualitative guide to the 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 pin
• Other supports are unyielding, so distorted shape
  may be easily drawn.
• For frames, joints are assumed free to rotate,
  assume members are rigidly connected (angle
  between members does not change)

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Qualitative Influence Lines
The Mueller-Breslau principle can be stated as
follows If 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
influence line of the function.
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Pattern Loads
Qualitative Influence Lines
Fig. 10-7 (b,f) from MacGregor (1997)
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Pattern Loads

• Frame 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 floor or
  roof under consideration, and
• The far ends of columns built integrally with the
  structure are considered to be fixed.

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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 to
  combinations of
• Factored dead load on all spans with full
  factored live load on two adjacent spans.
• Factored dead load on all spans with full
  factored live load on alternate spans.

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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
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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Approximate Analysis of Continuous Beam and
One-Way Slab Systems

• ACI Moment and Shear Coefficients
• Approximate moments and shears permitted for
  design of continuous beams and one-way slabs
• Section 8.3.3 of ACI Code

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Approximate Analysis of Continuous Beam 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 gt 20
• Uniform Loads
• LL/DL 3 (unfactored)

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Approximate Analysis of Continuous Beam and
One-Way Slab Systems

• ACI Moment and Shear Coefficients - Requirements
  ( cont.)
• Prismatic members
• Same A, I, E throughout member length
• Beams must be in braced frame without significant
  moments due to lateral forces
• Not state in Code, but necessary for coefficients
  to apply.
• All these requirements must be met to use the
  coefficients!

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Approximate Analysis of Continuous Beam and
One-Way Slab Systems
ACI Moment and Shear Coefficients Methodology
wu Total factored dead and live load per
unit length Cm Moment coefficient Cv Shear
coefficient ln Clear span length for span in
question for Mu at interior face of exterior
support, Mu and Vu ln Average of clear span
length for adjacent spans for Mu at interior
supports
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Approximate Analysis of Continuous Beam and
One-Way Slab Systems

• ACI Moment and Shear Coefficients
• See Section 8.3.3 of ACI Code

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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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Homework
Homework problems on the web