Title: Lecture 7 Flexure
1Lecture 7 - Flexure
2Lecture Goals
- Doubly Reinforced beams
- T Beams and L Beams
- Pan Joist
3Analysis 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.
4Analysis of Flanged Sections
Positive and Negative Moment Regions in a T-beam
5Analysis of Flanged Sections
If the neutral axis falls within the slab depth
analyze the beam as a rectangular beam, otherwise
as a T-beam.
6Analysis of Flanged Sections
Effective Flange Width Portions near the webs are
more highly stressed than areas away from the web.
7Analysis 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)
8ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10.2 T Beam Flange
9ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10.3 Inverted L Shape
Flange
10ACI Code Provisions for Estimating beff
From ACI 318, Section 8.10 Isolated T-Beams
11Various Possible Geometries of T-Beams
Single Tee Twin Tee Box
12Analysis of T-Beam
Case 1 Same as rectangular
section Steel is yielding under
reinforced Check
13Analysis of T-Beam
Case 1 Equilibrium
14Analysis of T-Beam
Case 1 Confirm
15Analysis of T-Beam
Case 1 Calculate Mn
16Analysis of T-Beam
Case 2 Assume steel yields
17Analysis of T-Beam
Case 2 Assume steel yields
The flanges are considered to be equivalent
compression steel.
18Analysis of T-Beam
Case 2 Equilibrium
19Analysis of T-Beam
Case 2 Confirm
20Analysis of T-Beam
Case 2 Confirm
21Analysis of T-Beam
Case 2 Calculate nominal moments
22Analysis of T-Beams
The definition of Mn1 and Mn2 for the T-Beam are
given as
23Analysis 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
24Limitations on Reinforcement for Flange Beams
- Lower Limits
- Flange in compression
25Limitations on Reinforcement for Flange Beams
- Lower Limits
- Flange in tension
26Limitations 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
27Example - 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
28Example of L-Beam
Confirm beff
29Example - T-Beam
Compute the equivalent c value and check the
strain in the steel, es
Steel will yield in the tension zone.
30Example - T-Beam
Compute the reinforcement r and check to make
sure it is greater than rmin
Section works for minimum reinforcement.
31Example - T-Beam
Compute w and check that the c value is greater
than hf
Analysis the beam as a T-beam.
32Example - T-Beam
Compute w and check that the c value is greater
than hf
Compute a
33Example - T-Beam
Compute nominal moment components
34Example - T-Beam
Compute nominal moment
Compute ultimate moment
35Example 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
36Example of L-Beam
Compute beff
37Example of L-Beam
Compute beff
38Example of L-Beam
The value beff and As
39Example - L-Beam
Compute the equivalent c value and check the
strain in the steel, es
Steel will yield in the tension zone.
40Example - L-Beam
Compute the reinforcement r and check to make
sure it is greater than rmin
Section works for minimum reinforcement.
41Example - L-Beam
Compute w and check that the c value is greater
than hf
False!
Analysis the beam as a Singly reinforced beam.
42Example - L-Beam
Compute a
43Example - L-Beam
Compute nominal moment
44Example - L-Beam
Compute ultimate moment
45Pan Joist Floor Systems
View of Pan Joist Slab from Below
Walter P. Moore Assoc.
46Pan Joist Floor Systems
Walter P. Moore Assoc.
View of Double Skip Joist Slab from Below
47Pan Joist Floor Systems
Placing Reinforcement for a Pan Joist Slab
Walter P. Moore Assoc.
48Pan Joist Floor Systems
General framing layout of the pan joist system.
49Pan Joist Floor Systems
Pouring a Pan Joist Slab
Walter P. Moore Assoc.
50Pan 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.
51One-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
52One-Way Joist Construction
Details of ribbed floor with removable steel pans.
Ribbed-floor cross sections.
53One-Way Joist Construction
The design of a ribbed floor with steel pan forms
and average weight of the floor.
54One-Way Joist Construction
The design of a ribbed floor with steel pan forms
and average weight of the floor.
55One-Way Joist Construction
56Pan 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
57Pan 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.
58Pan 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)
59Pan 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
60Pan Joist Floor Systems
- Standard Removable Form Dimensions
- Note the shapes
61Pan 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.
62Pan 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.
63Pan Joist Slabs
Standard Pan Joist Form Dimensions Ref. CECO
Concrete Construction Catalog
64Pan Joist Slabs
Standard Pan Joist Form Dimensions Ref. CECO
Concrete Construction Catalog
65Pan 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
66Pan 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)
67Pan 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
68Member 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
69Member 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.
70Member Depth
ACI 318-99 Table 9.5a
71Member 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)