Tutorial 1: Continuous 3-Span 72 Inch Bulb Tee Girder Design (U.S. Units) (LFD)

Concrete Properties

Art.8.7.1

E_{c} = w_{c}^{1.5} (33) ({f `}_{c}) = 150^{1.5} (33) \sqrt{6000} = 4696000 p s i = 4696.0 k s i

E_{c t} = 150^{1.5} (33) \sqrt{3500} = 35866000 p s i = 3586.6 k s i

Girder Line Data

T-Trib=Tributary width of beam 1

Tributary Width of Beams

T-Trib=Tributary width of beam 1

{T - T r i b}_{1} = 4.25 + 8.5 \div 2 = 8.50 f t

Continuity analysis

W₁= Magnitude of total superimposed dead load

W_{1} = (30.5 p s f) (51 f t) = 1555.5 p l f = 1.556 k l f

Model Used by Precast/Prestressed Girderfor SDL Continuity Analysis

Results of independent frame analysis program:

	Independent Program	Precast/Prestressed Girder
M (Span 1, 58.5ft)	1597.0 k-ft	1597.0 k-ft
V (Span 1, 58.5ft)	-18.2k	-18.2k
Recation 1	-72.8k	-72.8k
Reaction 2	-200.2k	-200.2k

Moving Load Analysis

Position of Truck to Cause Maximum Positive Moment at Span 1: 46.9 ft.

Position of Lane Load to Cause Mmax(-) at Span 1: 117.00 ft.

Results of independent frame analysis program:

	Independent Program	Precast/Prestressed Girder
M (Span 1, 58.5ft)	1597.0 k-ft	1597.0 k-ft
V (Span 1, 58.5ft)	-18.2k	-18.2k
Recation 1	-72.8k	-72.8k
Reaction 2	-200.2k	-200.2k

Position of Lane Load to Cause Vmax at Span 1:117.00 ft.

Position of Lane Load to Cause Mmax(-) at Span 1: 117.00 ft.

Position of Lane Load to Cause Vmax at Span 1: 117.00 ft.

Note: Concentrated load is slightly to the left of the support.

Results of independent frame analysis program:

	Independent Program	Precast/Prestressed Girder
Mmax(+) @46.90 ft	1827.7 k-ft	1830.1 k-ft
Mmax(-) @ Support 2	1757.7 k-ft	1757.8 k-ft
Vmax @ Support 2	90.2 k	90.2 k

Effective Width

Effective width Art 8.10.1.1

$L / 4 = 115.0 f t / 4 = 28.75 f t = 345.0 i n$
Beam spacing= 8.50ft= 102.0in(controls)
*Web width + (12)(Flange thickness + Haunch thickness + Topping thickness)=6in+(12)(2 in+ 0.50in+ 8.00in)=132.0in
*Precast flange width + (12)(Topping thickness)=42in+(12)(8in)=138.0in

Note: *See Theory section for details.

Effective width = 102.0 in

Impact Factor

Eq.3-1 $I = \frac{50}{L + 125}$

Impact (moment):

1 + I = 1 + \frac{50}{115.0 + 125} = 1.208

Distribution Factor

Wheel Load Distribution Factor

table 3.23.1 $\frac{S}{5.5} = \frac{8.5}{5.5} = 1.5455$

Precast/Prestressed Girder performs a moving live load analysis using a full truck and lane load (two wheel lines). Therefore, for consistency, the distribution factor must be divided by two.

DF: 1.5455/2=0.773

Section Properties

Self-weight (Precast)=(A)(Density Concrete)

\frac{767 {i n}^{2}}{144 {i n}^{2} / {f t}^{2}} \times (150 p c f) = 799.0 p l f

Self-weight (Composite)

799.0 p l f + \frac{42 \times 8.5 + 60 \times 8}{144 {i n}^{2} / {f t}^{2}} \times (150 p c f) = 1670.0 p l f

Transformed widths of slab parts:

Transformed topping width = (0.7638)(42 in) = 32.078in

Transformed gap width = (0. 7638)(60 in) = 45.825 in

Element	Width	Thick	A	Yb	Ayb	I
Topping	32.087	8.5	272.663	76.25	20791	1642
Gap	45.825	8.5	366.603	76.50	28045	1955
Beam			767.000	36.60	28072	545894
Total			1406		76908	549491

$H_{t} o f c . g . = \frac{{A y}_{b}}{A} = \frac{76908}{1406} = 54.70 i n$

Moment of Inertia:

Σ I + Σ {(y_{b c} - y_{b})}^{2} A = 549491 + (272.663) {(54.70 - 76.25)}^{2^{2}} + (366.603) {(54.70 - 76.25)}^{2^{2}} + (767.0) {(54.70 - 36.6)}^{2^{2}} = 1101619 {i n}^{4}

Tributary Fraction

DL-Comp Trib Frac = Fraction of total moments and shears of SDL continuity analysis

DL-Comp Trib Frac = $\frac{T r i b u t a r y W i d t h}{B r i d g e W i d t h} = \frac{8.5}{51.00} = 0.167$

Tributary Fraction

DL-Comp Trib Frac = Fraction of total moments and shears of SDL continuity analysis

DL-Comp Trib Frac = $\frac{T r i b u t a r y W i d t h}{B r i d g e W i d t h} = \frac{8.5}{51.00} = 0.167$

Miscellaneous

Kern point:

U p p e r = k_{u} = y_{b} + \frac{S_{b}}{A} = 36.60 + \frac{\frac{{545894 i n}^{4}}{36.60 i n}}{767.0 {i n}^{4}} = 56.05 i n

L o w e r = k_{1} = y_{b} + \frac{S_{t}}{A} = 36.60 + \frac{\frac{{545894 i n}^{4}}{(72 i n - 36.60 i n)}}{767.0 {i n}^{4}} = 16.49 i n

Prestressed Steel

Depressed at 0.4L:

0.4 \times 116.0 f t = 46.4 f t f r o m b e a m e n d

46.4 f t - 0.5 f t = 45.9 f t f r o m \sup p o r t

Total strand area = (38)(0.153 in²) = 5.814 in²

Initial pull = (5.814 in²)(0.75)(270 ksi) = 1178.1 k

Trans len= Transfer length Art. 9.20.2.4

= 50 \times (s t r a n d d i a m e t e r s) = 50 \times 0.500 i n = 25.00 i n = 2.083 f t

Dev. len = Development length of strands at ultimate Eq.9-32

= (f_{s u} - \frac{2}{3} f_{s e}) D = f_{s u} = 264.8 k s i

= f_{s e} = 0.75 \times 270 k s i \times (1 - 0.215) = 159.0 k s i

Dev. len

(264.8 - \frac{2}{3} (159.0)) (0.5) = 79.41 i n = 6.62 f t

COMPUTED LOSSES

SERVICE LOAD MOMENTS AND SHEARS

RELEASE STRESSES

FINAL STRESSES - POSITIVE MOMENT ENVELOPE AT MIDSPAN

FINAL STRESSES - NEGATIVE MOMENT ENVELOPE

VERTICAL SHEAR AT H/2

HORIZONTAL SHEAR AT H/2

ULTIMATE CAPACITY AT MIDSPAN

CRACKING LOAD, AT 0.4L

CAMBER AND DEFLECTIONS

DETENSIONING

REINFORCED DESIGN

Parent topic: Hand Calculation for Selected Items