Analysis of T-Beam

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


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


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


Calculate Mn

fha

2ysn

adfAM


Case 2: Assume steel yields fha

ys

wcw

fwcf

85.0

85.0

fAT

abfC

hbbfC


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.


Case 2: Equilibriumfha s sf y

f wc w0.85

A A fT C C a

f b


Case 2: Confirm

fha

f

1

s cu 0.005

a h

ac

d c

c


Case 2: Confirm

fha

y

c

f f1

1.18 or

f

f

dh h a


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


Lower Limits Flange in tension

dbf

dbf

f

dbf

f

A

effy

effy

c

wy

c

s(min)

200

3

oflarger

6

ofsmaller


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.


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.


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


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.


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.


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.


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


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.


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.


General framing layout of the pan joist system.

Pan Joist Pan Joist Floor SystemsFloor Systems

Pouring a Pan Joist Slab

Walter P. Moore & Assoc.


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.


The design of a ribbed floor with steel pan forms and average weight of the floor.


Joist Details


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


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


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)


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)

Analysis of T-Beam

Engineering

beamanalysis of t

beam t beams

f t c c

t beam flange

beamsanalysis of t

reinforced beams t beams

fha y c f f

f b b h