Lecture 7 Flexure

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Lecture 7 - Flexure

• June 16, 2003
• CVEN 444

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Lecture Goals

• Doubly Reinforced beams
• T Beams and L Beams
• Pan Joist

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Analysis of Flanged Section

• Floor systems with slabs and beams are placed in
  monolithic pour.
• Slab acts as a top flange to the beam T-beams,
  and Inverted L(Spandrel) Beams.

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Analysis of Flanged Sections
Positive and Negative Moment Regions in a T-beam
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Analysis of Flanged Sections
If the neutral axis falls within the slab depth
analyze the beam as a rectangular beam, otherwise
as a T-beam.
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Analysis of Flanged Sections
Effective Flange Width Portions near the webs are
more highly stressed than areas away from the web.
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Analysis of Flanged Sections
Effective width (beff) beff is width
that is stressed uniformly to give the same
compression force actually developed in
compression zone of width b(actual)
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ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10.2 T Beam Flange
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ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10.3 Inverted L Shape
Flange
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ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10 Isolated T-Beams
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Various Possible Geometries of T-Beams
Single Tee Twin Tee Box
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Analysis of T-Beam
Case 1 Same as rectangular
section Steel is yielding under
reinforced Check
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Analysis of T-Beam
Case 1 Equilibrium
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Analysis of T-Beam
Case 1 Confirm
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Analysis of T-Beam
Case 1 Calculate Mn
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Analysis of T-Beam
Case 2 Assume steel yields
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Analysis of T-Beam
Case 2 Assume steel yields
The flanges are considered to be equivalent
compression steel.
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Analysis of T-Beam
Case 2 Equilibrium
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Analysis of T-Beam
Case 2 Confirm
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Analysis of T-Beam
Case 2 Confirm
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Analysis of T-Beam
Case 2 Calculate nominal moments
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Analysis of T-Beams
The definition of Mn1 and Mn2 for the T-Beam are
given as
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Analysis of T-Beams
The ultimate moment Mu for the T-Beam are given
as
For a T-Beam with the tension steel yielded using
a function c/d
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Limitations on Reinforcement for Flange Beams

• Lower Limits
• Flange in compression

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Limitations on Reinforcement for Flange Beams

• Lower Limits
• Flange in tension

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Limitations on Reinforcement for Flange Beams

• Lower Limits
• If As(provided) 4/3 As(reqd) based on
  analysis then As(min) is not required
  (i.e.) fMn 4/3Mu for
  As(provided) See ACI 10.5.3

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Example - T-Beam
Find Mn and Mu for T-Beam. beff 54 in. hf 3
in. b 7 ft. d 16.5 in. As 8.5 in2 fy
50 ksi fc 3 ksi bw 12 in L 18 ft
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Example of L-Beam
Confirm beff
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Example - T-Beam
Compute the equivalent c value and check the
strain in the steel, es
Steel will yield in the tension zone.
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Example - T-Beam
Compute the reinforcement r and check to make
sure it is greater than rmin
Section works for minimum reinforcement.
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Example - T-Beam
Compute w and check that the c value is greater
than hf
Analysis the beam as a T-beam.
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Example - T-Beam
Compute w and check that the c value is greater
than hf
Compute a
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Example - T-Beam
Compute nominal moment components
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Example - T-Beam
Compute nominal moment
Compute ultimate moment
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Example of L-Beam
Determine the effective b for the spandrel beam
and do the analysis. Use 4 9 bars and find the
ultimate moment capacity. fy50 ksi, fc 3 ksi
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Example of L-Beam
Compute beff
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Example of L-Beam
Compute beff
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Example of L-Beam
The value beff and As
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Example - L-Beam
Compute the equivalent c value and check the
strain in the steel, es
Steel will yield in the tension zone.
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Example - L-Beam
Compute the reinforcement r and check to make
sure it is greater than rmin
Section works for minimum reinforcement.
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Example - L-Beam
Compute w and check that the c value is greater
than hf
False!
Analysis the beam as a Singly reinforced beam.
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Example - L-Beam
Compute a
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Example - L-Beam
Compute nominal moment
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Example - L-Beam
Compute ultimate moment
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Pan Joist Floor Systems
View of Pan Joist Slab from Below
Walter P. Moore Assoc.
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Pan Joist Floor Systems
Walter P. Moore Assoc.
View of Double Skip Joist Slab from Below
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Pan Joist Floor Systems
Placing Reinforcement for a Pan Joist Slab
Walter P. Moore Assoc.
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Pan Joist Floor Systems
General framing layout of the pan joist system.
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Pan Joist Floor Systems
Pouring a Pan Joist Slab
Walter P. Moore Assoc.
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Pan Joist Floor Systems

• Definition The type of slab is also called a
  ribbed slab. It consists of a floor slab,
  usually 2-4 in. thick, supported by reinforced
  concrete ribs. The ribs are usually tapered and
  uniformly spaced at distances that do not exceed
  30 in. The ribs are supported on girders that
  rest on columns. In some ribbed slabs, the
  space between ribs may be filled with permanent
  fillers to provide a horizontal slab soffit.

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One-Way Joist Construction
Definition Pan joist floor systems are series
of closely spaced cast-in-place T-beams or joists
used for long-span floors with relatively light
loads. Typically removable metal forms (fillers
or pans) are used to form joists.
MacGregor, Fig. 10-28
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One-Way Joist Construction
Details of ribbed floor with removable steel pans.
Ribbed-floor cross sections.
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One-Way Joist Construction
The design of a ribbed floor with steel pan forms
and average weight of the floor.
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One-Way Joist Construction
The design of a ribbed floor with steel pan forms
and average weight of the floor.
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One-Way Joist Construction
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Pan Joist Floor Systems

• ACI Requirements for Joist Construction
• (Sec. 8.11, ACI 318-02)
• Slabs and ribs must be cast monolithically.
• Ribs must be spaced consistently
• Ribs may not be less than 4 inches in width

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Pan Joist Floor Systems

• ACI Requirements for Joist Construction (cont.)
• (Sec. 8.11.2, ACI 318-02)
• Depth of ribs may not be more than 3.5 times the
  minimum rib width
• Clear spacing between ribs shall not exceed 30
  inches.
• Ribbed slabs not meeting these requirements
  are designed as slabs and beams.

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Pan Joist Floor Systems

• Slab Thickness
• (ACI Sec. 8.11.6.1)
• t 2 in. for joints formed with 20 in. wide
  pans
• t 2.5 in. for joints formed with 30 in. wide
  pans (1/12 distance)

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Pan Joist Floor Systems

• Slab Thickness (cont.)
• Building codes give minimum fire resistance
  rating
• 1-hour fire rating ¾ in. cover, 3-3.5 slab
  thickness
• 2-hour fire rating 1 in. cover, 4.5 slab
  thickness

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Pan Joist Floor Systems

• Standard Removable Form Dimensions
• Note the shapes

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Pan Joist Floor Systems

• Standard Removable Form Dimensions
• Standard Widths 20 in. 30 in. (measured at
  bottom of ribs)
• Standard Depths 6, 8, 10, 12, 14, 16 or 20 in.

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Pan Joist Floor Systems

• Standard Removable Form Dimensions (cont.)
• End Forms one end is closed (built-in) to form
  the supporting beam
• Tapered End Forms provide additional shear
  capacity at ends of joists by tapering ends to
  increase rib width.

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Pan Joist Slabs
Standard Pan Joist Form Dimensions Ref. CECO
Concrete Construction Catalog
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Pan Joist Slabs
Standard Pan Joist Form Dimensions Ref. CECO
Concrete Construction Catalog
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Pan Joist Floor Systems

• Laying Out Pan Joist Floors
• Rib/slab thickness
• Governed by strength, fire rating, available
  space
• Overall depth and rib thickness
• Governed by deflections and shear

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Pan Joist Floor Systems

• Laying Out Pan Joist Floors (cont.)
• Typically no stirrups are used in joists
• Reducing Forming Costs
• Use constant joist depth for entire floor
• Use same depth for joists and beams (not always
  possible)

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Pan Joist Floor Systems

• Distribution Ribs
• Placed perpendicular to joists
• Spans lt 20 ft. None
• Spans 20-30 ft. Provided a midspan
• Spans gt 30 ft. Provided at third-points
• At least one continuous 4 bar is provided at top
  and bottom of distribution rib.
• Note not required by ACI Code, but typically
  used in construction

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Member Depth

• ACI provides minimum member depth and slab
  thickness requirements that can be used without a
  deflection calculation (Sec. 9.5 ACI 318)
• Useful for selecting preliminary member sizes

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Member Depth

• ACI 318 - Table 9.5a
• Min. thickness, h (for beams or ribbed one-way
  slab)
• For beams with one end continuous L/18.5
• For beams with both ends continuous L/21
• L is span length in inches
• Table 9.5a usually gives a depth too shallow for
  design, but should be checked as a minimum.

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Member Depth
ACI 318-99 Table 9.5a
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Member Depth

• Rule of Thumb
• hb (in.) L (ft.)
• Ex.) 30 ft. span -gt hb 30 in.
• May be a little large, but okay as a start to
  calc. DL
• Another Rule of Thumb
• wDL (web below slab) 15 (wSDL wLL)
• Note For design, start with maximum moment for
  beam to finalize depth.
• Select b as a function of d
• b (0.45 to 0.65) (d)