Transcript
Lecture 7 - Flexure
June 16, 2003CVEN 444
Lecture GoalsLecture Goals
Doubly Reinforced beamsT Beams and L Beams Pan Joist
Analysis of Flanged Analysis of Flanged SectionSection
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.
Analysis of Flanged Analysis of Flanged SectionsSectionsPositive and Negative Moment Regions in a T-beam
Analysis of Flanged Analysis of Flanged SectionsSections
If the neutral axis falls within the slab depth analyze the beam as a rectangular beam, otherwise as a T-beam.
Analysis of Flanged Analysis of Flanged SectionsSectionsEffective Flange Width
Portions near the webs are more highly stressed than areas away from the web.
Analysis of Flanged Analysis of Flanged SectionsSections
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)
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10.2
T Beam Flange:
eff
f w
actual
4
16
Lb
h b
b
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10.3
Inverted L Shape Flange
eff w
f w
actual w
12
6
0.5* clear distance to next web
Lb b
h b
b b
ACI Code Provisions for ACI Code Provisions for Estimating bEstimating beffeff
From ACI 318, Section 8.10
Isolated T-Beams
weff
wf
42
bb
bh
Various Possible Geometries Various Possible Geometries of T-Beamsof T-Beams
Single Tee
Twin Tee
Box
Analysis of T-BeamAnalysis of T-Beam
Case 1: Same as rectangular section
Steel is yielding under reinforced
Check fha
fha
ysys Assume ff
Analysis of T-BeamAnalysis of T-BeamCase 1:
Equilibriumfha
s y
c eff0.85
A fT C a
f b
Analysis of T-BeamAnalysis of T-BeamCase 1: Confirm
fha
ycus
1
ys
c
cd
ac
Analysis of T-BeamAnalysis of T-BeamCase 1:
Calculate Mn
fha
2ysn
adfAM
Analysis of T-BeamAnalysis of T-Beam
Case 2: Assume steel yields fha
ys
wcw
fwcf
85.0
85.0
fAT
abfC
hbbfC
Analysis of T-BeamAnalysis of T-Beam
Case 2: Assume steel yields fha
c w fsf
y
0.85 f b b hA
f
The flanges are considered to be equivalent compression steel.
Analysis of T-BeamAnalysis of T-Beam
Case 2: Equilibriumfha s sf y
f wc w0.85
A A fT C C a
f b
Analysis of T-BeamAnalysis of T-Beam
Case 2: Confirm
fha
f
1
s cu 0.005
a h
ac
d c
c
Analysis of T-BeamAnalysis of T-Beam
Case 2: Confirm
fha
y
c
f f1
1.18 or
f
f
dh h a
Analysis of T-BeamAnalysis of T-BeamCase 2:
Calculate nominal moments
fha
n n1 n2
n1 s sf y
fn2 sf y
2
2
M M M
aM A A f d
hM A f d
Analysis of T-BeamsAnalysis of T-Beams
The definition of Mn1 and Mn2 for the T-Beam are given as:
Analysis of T-BeamsAnalysis of T-Beams
The ultimate moment Mu for the T-Beam are given as:
u n
M M
cf
d
For a T-Beam with the tension steel yielded using a function c/d
Limitations on Limitations on Reinforcement for Flange Reinforcement for Flange BeamsBeams
Lower Limits Flange in compression
c
ysmin
w
y
3
larger of 200
f
fA
b d
f
Limitations on Limitations on Reinforcement for Flange Reinforcement for Flange BeamsBeams
Lower Limits Flange in tension
dbf
dbf
f
dbf
f
A
effy
effy
c
wy
c
s(min)
200
3
oflarger
6
ofsmaller
Limitations on Limitations on Reinforcement for Flange Reinforcement for Flange BeamsBeams
Lower Limits If As(provided) 4/3 As(req’d) based on
analysisthen As(min) is not
required (i.e.) Mn 4/3Mu
for As(provided) See ACI 10.5.3
Example - T-BeamExample - T-BeamFind 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
Example of L-BeamExample of L-BeamConfirm beff
eff
f w
12 in.18 ft
ft. 54 in.
4 4 16 16 3 in. 12 in.=60 in.
12 in. 7 ft. 84 in.
ft.
Lb
h b
b
Example - T-BeamExample - T-Beam
2s y
c eff
1
s
8.5 in 50 ksi3.09 in.
0.85 0.85 3 ksi 54 in.
3.09 in3.63 in.
0.85
16.5 in.1 0.003 1 0.003 0.0106 0.005
3.63 in.
A fa
f b
ac
d
c
Compute the equivalent c value and check the strain in the steel, s
Steel will yield in the tension zone.
Example - T-BeamExample - T-Beam
2
s
w
y
min min
c
y
8.5 in0.0429
12 in. 16.5 in.
200 2000.004
500000.004
3 3 30000.00329
50000
0.0429 0.004
A
b d
f
f
f
Compute the reinforcement and check to make sure it is greater than min
Section works for minimum reinforcement.
Example - T-BeamExample - T-Beam
y
c
f1
f
50 ksi0.0429 0.7155
3 ksi
1.18 0.7155 16.5 in.1.183 in. 16.388
0.85
3 in. 3.09 in.
f
f
dh
h a
Compute and check that the c value is greater than hf
Analysis the beam as a T-beam.
Example - T-BeamExample - T-BeamCompute and check that the c value is greater than hf
Compute a
c eff w fsf
y
2
0.85 0.85 3 ksi 54 in. 12 in. 3 in.
50 ksi
6.426 in
f b b hA
f
2 2s sf y
c w
8.5 in 6.426 in 50 ksi
0.85 0.85 3 ksi 12 in.
3.889 in.
A A fa
f b
Example - T-BeamExample - T-BeamCompute nominal moment components
2fn2 sf y
3 in.6.426 in 50 ksi 16.5 in.
2 2
4819.5 k-in.
hM A f d
n1 s sf y
2 2
2
3.889 in.8.5 in 6.426 in 50 ksi 16.5 in.
2
1535.34 k-in.
aM A A f d
Example - T-BeamExample - T-BeamCompute nominal moment
u n 0.9 529.57 k-ft.
416.6 k-ft.
M M
n n1 n2
1535.34 k-in. 4819.5 k-in.
6354.84 k-in. 529.57 k-ft.
M M M
Compute ultimate moment
Example of L-BeamExample of L-BeamDetermine the effective b for the spandrel beam and do the analysis.
Use 4 #9 bars and find the ultimate moment capacity. fy=50 ksi, fc = 3 ksi
Example of L-BeamExample of L-BeamCompute beff
eff w
f w
actual w
12
6
0.5* clear distance to next web
Lb b
h b
b b
Example of L-BeamExample of L-BeamCompute beff
eff w
f w
actual w
12 in.20 ft
ft 12 in. =32 in.
12 12 6 6 6 in. 12 in. = 48 in.
0.5* clear distance to next web
12 in. 12 in. + 0.5* 7 ft 54 i
ft
Lb b
h b
b b
n.
Example of L-BeamExample of L-BeamThe value beff and As
eff
2 2s
= 32 in.
4 1.0 in 4.0 in
b
A
Example - L-BeamExample - L-Beam
2s y
c eff
1
s
4.0 in 50 ksi2.45 in.
0.85 0.85 3 ksi 32 in.
2.45 in2.88 in.
0.85
24 in.1 0.003 1 0.003 0.0220 0.005
2.88 in.
A fa
f b
ac
d
c
Compute the equivalent c value and check the strain in the steel, s
Steel will yield in the tension zone.
Example - L-BeamExample - L-Beam
2
s
w
y
min min
c
y
4.0 in0.0139
12 in. 24 in.
200 2000.004
500000.004
3 3 30000.00329
50000
0.0139 0.004
A
b d
f
f
f
Compute the reinforcement and check to make sure it is greater than min
Section works for minimum reinforcement.
Example - L-BeamExample - L-Beam
y
c
f1
f
50 ksi0.0139 0.2315
3 ksi
1.18 0.2315 24 in.1.186 in. 7.71 in.
0.85
6 in. 2.45 in.
f
f
dh
h a
Compute and check that the c value is greater than hf
Analysis the beam as a Singly reinforced beam.
False!
Example - L-BeamExample - L-BeamCompute a
2s y
c
4.0 in 50 ksi
0.85 0.85 3 ksi 32 in.
2.451 in.
A fa
f b
Example - L-BeamExample - L-BeamCompute nominal moment
n s y
2
2
2.451 in.4.0 in 50 ksi 24.0 in.
2
4554.9 k-in. 379.58 k-ft.
aM A f d
Example - L-BeamExample - L-Beam
u n 0.9 379.58 k-ft.
341.62 k-ft.
M M
Compute ultimate moment
Pan Joist Floor SystemsPan Joist Floor Systems
View of Pan Joist Slab from Below Walter P. Moore & Assoc.
Pan Joist Floor SystemsPan Joist Floor Systems
View of Double Skip Joist Slab from BelowWalter P. Moore & Assoc.
Pan Joist Floor Pan Joist Floor SystemsSystems
Placing Reinforcement for a Pan Joist Slab
Walter P. Moore & Assoc.
Pan Joist Floor SystemsPan Joist Floor Systems
General framing layout of the pan joist system.
Pan Joist Pan Joist Floor SystemsFloor Systems
Pouring a Pan Joist Slab
Walter P. Moore & Assoc.
Pan Joist Floor SystemsPan 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.
One-Way Joist One-Way Joist ConstructionConstruction
MacGregor, Fig. 10-28
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.
One-Way Joist ConstructionOne-Way Joist Construction
Details of ribbed floor with removable steel pans.
Ribbed-floor cross sections.
One-Way Joist One-Way Joist ConstructionConstruction
The design of a ribbed floor with steel pan forms and average weight of the floor.
One-Way Joist ConstructionOne-Way Joist Construction
The design of a ribbed floor with steel pan forms and average weight of the floor.
One-Way Joist ConstructionOne-Way Joist Construction
Joist Details
Pan Joist Floor SystemsPan 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
Pan Joist Floor SystemsPan 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. **
Pan Joist Floor SystemsPan Joist Floor SystemsSlab 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)
Pan Joist Floor SystemsPan Joist Floor SystemsSlab 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
Pan Joist Pan Joist Floor Floor SystemsSystems
Standard Removable Form Dimensions
Note the shapes
Pan Joist Floor SystemsPan Joist Floor SystemsStandard 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.
Pan Joist Floor SystemsPan Joist Floor SystemsStandard 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.
Pan Joist Slabs
Standard Pan Joist Form Dimensions
Ref. CECO Concrete Construction Catalog
Pan Joist Slabs
Standard Pan Joist Form Dimensions
Ref. CECO Concrete Construction Catalog
Pan Joist Floor SystemsPan Joist Floor SystemsLaying Out Pan Joist Floors
Rib/slab thicknessGoverned by strength, fire rating,
available space
Overall depth and rib thicknessGoverned by deflections and shear
Pan Joist Floor SystemsPan 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
floorUse same depth for joists and
beams (not always possible)
Pan Joist Floor SystemsPan Joist Floor Systems
Distribution Ribs Placed perpendicular to joists* Spans < 20 ft.: None Spans 20-30 ft.: Provided a midspan Spans > 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
Member DepthMember DepthACI 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
Member DepthMember DepthACI 318 - Table 9.5a:
Min. thickness, h (for beams or ribbed one-way slab)For beams with one end continuous: L/18.5For beams with both ends continuous: L/21L is span length in inches
Table 9.5a usually gives a depth too shallow for design, but should be checked as a minimum.
Member Member DepthDepth
ACI 318-99: Table 9.5a
Member DepthMember DepthRule of Thumb: hb (in.) ~ L (ft.) Ex.) 30 ft. span -> hb ~ 30 in. May be a little large, but okay as a
start to calc. DLAnother 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)
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