I-Beam Girder Computions.xls

(1)

Prestressed Precast I-Girder Design for Intermediate Beams - CE767

Geometrical Properties

Material Properties of Concrete

Elastic Modulus - AASHTO LRFD 5.4.2.4-1

Girder Span Length

L

24 m

Descrip.

fc`

Unit W

Ec

Girder Depth

h

182.88 cm

(MPA)

(kg/m3)

(MPA)

Spacing of Girders

S

0.85 m

CIP Deck

30 2500

29440

Beam@transfer

40 2500

33994

Deck Thickness

tdeck

20 cm

Beam@service

50 2500

38007

Haunch Thickness

0 cm

Haunch Width

0 cm

Loads

DW, Dead Load Placed on Structural Components

Please choose the type of the beam cross section (1/2)

1 Thickness of wearing surface

6 cm

Please enter the dimensions of the section in "SectionComposer"

Unit weight of wearing surface

2200

kg/m3

Cross sectional Properties for a Single Beam

DC, Dead Load of Structural Components and non-structural elements

Girder from LARSA Section Composer

Self Weight

1.518 t/m

Area

Istrong

Iweak

bw

yb

Deck Weight

0.425 t/m

(cm2)

(cm4)

(cm)

Haunch

0.000 t/m

6070.9556 2.79E+07 2095094

15.24

92.55 Sum

1.943 t/m

Cross Diaphragms

0.499375 tons per girder at mid-span

width

25 cm

0.499375 tons per girder at each end

height

94 cm

Quantity

3 two at ends and one at mid-span

Barrier

0.100 t/m per beam

Wearing Surface

0.112 t/m per beam

Cross sectional Properties for the Composite Beam

Sum

0.212 t/m per beam

Descript.

Area

yb

A.yb

Istrong

(cm2)

(cm)

(cm3)

(cm4)

LL, Distrubution Factors for LiveLoad H30 truck

Beam

6070.96

92.6 561896

1941330.8 2.79E+07

2.99E+07

Haunch

0.00

0

0 0.00E+00

Distribution Factor for Bending Moment - lane/beam

Table 4.6.2.2.2b-1

Deck

1316.81

192.88 253987

8950186.1 43894

8.99E+06

Sum

7387.77

8.16E+05

3.89E+07

S =

850 mm

NOT OK

ts =

200 mm

OK

Section, ycb =

110.4 cm

L =

24000

mm

OK

Nb =

>=4

OK

Effective Flange Width (AASHTO LRFD 4.6.2.6.1)

Kg

1.15E+12 mm4

OK

1/4 Span =

6 m

12ts + web

2.908 m

Kg =

1.15E+12 mm4

Spacing =

0.85 m

DFM =

0.369 lanes/beam

Use

0.85 m

Distribution Factor for Shear - lane/beam

Table 4.6.2.2.3a-1

Modular Ratio of Deck to Beam = 0.77

Span to Depth Ratio

13 DFS =

0.430 lanes/beam

Prestressing steel

(1/2 in. Dia. Seven wire, low relaxation)

# of strands

22 Prestressing force

Area of 1 strand

Ab

98.71 mm2

Ultimate strength

fpu

1861.65

MPa

Yield strength

fpy

1675.485 MPa

(LRFD Table 5.4.4.1-1)

Spacing for prestressing strands

5 cm

Check for fitting

Initially (=0.75 fpu)

fpi

1396.2

MPa

(LRFD Table 5.9.3-1)

x

y,from bottom

Initial loss

4.3 %

(cm)

Initial loss

60.0 MPa

Layer 1 - # of strands

11

60

5 At Transfer after initial losses

1336.2

MPa

Layer 2 - # of strands

11

60

10 Total Prestressing Force

2901.7

kN

Layer 3 - # of strands

0

5

15 Layer 4 - # of strands

0

5

20 Layer 5 - # of strands

0

5

25 Layer 6 - # of strands

0

5

30 Layer 7 - # of strands

0

5

35 Layer 8 - # of strands

0

5

40 Layer 9 - # of strands

0

5

45 c.g of prestressing tendons from bottom

7.50 cm

Reinforcing Bars

Yield strength

fpy

420 MPa

(2)

STRESSES AT TRANSFER

Moment due to prestressing

Mp at c/g of beam

2468.0

kN-m

Moment due to SW of the beam

Mbeam at c.g of beam

1072.0

kN-m

Stress check at transfer - midspan

Bottom Fiber - Compression

`=-P/A-Mp/Sb+Mb/Sb

-9.405

MPa

<

-24

MPa

OK

Top Fiber - Tension Check

`=-P/A+Mp/St-Mb/St

-0.265

MPa

<

1.581 MPa

OK

without bonded reinf.

Check Total Loss due to Initial Prestressing

Loss = n * elastic shortening stress

n = Es/Ec

5.78 Elastic Shortening Stress

`=(-P/A)-(Mpe)/I+(Mbe)/I

-9.031

MPa

Loss

-52.20

iterate for loss

60.0 MPa

(estimated)

Check Stresses at Transfer Length Section

Transfer Length =

60 dia =

762 mm

LRFD Art. 5.8.2.3

Debonded strands =

22 Mbeam @ end of Transfer Length =

131.82 kN-m

P =

0.00 kN

Mp at c.g of beam =

0.00 kN-m

Bottom Fiber Stresses =

`=-P/A-Mp/Sb+Mb/Sb

0.437 MPa

<

-24

MPa

OK

Top fiber Stresses =

`=-P/A+Mp/St-Mb/St

-0.426

MPa

<

1.581 MPa

OK

(3)

STRESSES AT SERVICE LOADS

Prestress Losses at Service Level

Elastic Shortening

-52.20

MPa

(see above comp.)

fpi

1396.24

MPa

Aps

2171.61

mm2

Ag

607095.56 mm2

gamma-k

0.8 gamma-st

0.74 delta fpr

17 MPa

(AASHTO LRFD Section 5.9.5.3)

from AASHTO Table

delta fpl

96.20 MPa

Live Load Table

H30-S24

Total Prestress Loss at Service

-148.40

MPa

Span

Moment

Total Prestress loss (%)

10.6 m

kN-m

Total Prestress Stress after losses

1247.84

MPa

(iterative)

0.3

16.2 Finding the number of strands

0.6

32.55 Compute stresses using

0.9

48.75 Total Prestress Force

2709.8

kN

non-composite

fbc

8.83 Mpa

1.2

65.1 Mp

2304.8

kN-m

non-composite

fpb

5.30 MPa

1.5

81.3 Mdc

1372.2

kN-m

non-composite

ybs

9.144 cm

1.8

97.65 Mdw

149.9 kN-m

composite

ec

83.41 cm

2.1

113.85 Mll

3464.6

kN-m

from the table in "live loads" sheet

Ppe

1.20E+03 Kn

2.4

130.2 Mll+im

1698.5

kN-m

composite

Final loss 10.6

%

2.7

146.4 #

10

3

162.75 Stresses at Mid-Span

Service I =

1.00(DC+DW) + 1.00 (LL+IM)

3.4

178.95 Check compressive Stresses in prestressed comp.

3.7

195.3 P/A

Mp term

Mdc term Mdw term Mll+im term

Total

4

211.5 (MPa)

(MPa)

4.3

227.7 Beam Top Fiber Stresses =

-4.46

7.45 -4.44

-0.28

-1.73

<

-22.50

OK

4.6

244.05

4.9

260.25 -4.46

7.45 -4.44

-0.28

-3.17

-4.89

<

-22.50

OK

5.2

276.6

5.5

292.8

5.8

309.15 Top of Deck Fiber Stresses

-0.28

-3.13

-3.41

<

-13.5

OK

6.1

325.35

6.4

341.7

6.7

357.9 Service III =

1.00(DC+DW) + 0.80 (LL+IM)

7

374.25 Check tensile stresses in prestressed concrete comp.

7.3

391.95 Beam Bottom Fiber Stresses =

-4.46

-7.64

4.55

0.43 -7.13

<

3.54 OK

7.6

421.8

7.9

451.8 -4.46

-7.64

4.55

0.43

3.86 -3.27

<

3.54 OK

8.2

481.95

8.5

512.4

(4)

FATIGUE CHECK

9.1

573.75 Fatigue is typically checked for one lane load instead of multiple lanes however for simplicity use above Mll+IM

9.4

604.65 Distribution Factor for 1 lane loading

9.8

635.55

10.1

666.6 Bottom Compressive Stress due to permanent loads and prestress =

-7.13

MPa

10.4

698.55 Bottom Tensile Stress due to( 0.75 Mll+IM) =

2.84 MPa

10.7

734.55

11

770.55 Ratio comp/tension =

2.51 >

2 don't check fatigue

LRFD 5.5.3.1

11.3

806.55

11.6

842.55 check of fatigue is not provided in this spreadsheet

DFS

0.290 lanes/beam

11.9

878.7

12.2

914.7 STRENGTH LIMIT STATE

12.8

986.85

13.4 1059.3

Mu1 = 1.25(DC)+1.5(DW)+1.75(LL+IM)

14 1131.75

Mu2 =0.9(DC)+0.65(DW)+1.75(LL+IM)

Mu =

4912.5

kN-m

14.6 1204.05

Mu =

4304.8

kN-m

14.6 1204.05

15.2 1276.05

dp

195.38 cm

15.8 1349.55

16.5 1422.15

β

k

17.1 1494.9

0.75

0.38

17.7 1567.5

18.3 1640.1

c=

23.72 cm

>

20 cm

18.9 1713.15

T-section behaviour

19.5 1785.75

20.1 1858.8

Top flange thickness of the PC beam =

12.7 cm

20.7 1931.4

21.3 2004.3

c=

33.38 >

32.7 cm

22.9 2186.4

24.4 2368.95

Average stress in prestressing tendons

25.9 2551.65

fps =

1775.7592 MPa

27.4 2734.05

Mn =

7145.5724 kN-m

for rectangular

29 2916.45

Mn=

71078.885 kN-m

for T-section

30.5 3099.3

33.5 3464.55

36.6 3829.95

Ø =

1 LRFD 5.5.4.2.1

39.6 4195.35

42.7 4561.05

Mr =

71078.89

>

4912.5

kN-m

OK

45.7 5033.4

48.8 5629.05

51.8 6257.7

SHEAR DESIGN

54.9 6918.6

57.9 7612.05

Vp =

0 kN

no draped tendons exist

61 8337.9

67.1 9887.55

Critical shear section approax. = de = h-ybs =

195.38 cm

73.2 11567.25

79.2 13377.15

Vdc =

193.9 kN

85.3 15317.25

Vdw =

20.9 kN

91.4 17387.55

Vll=

424.35 kN

from the live loads table

Vll+im =

242.6 kN

Vu =

698.3 kN

Vc =

349.5 kN

Vu =

698.3 >

157.3 kN

provide stirrups

Vn >

Vu/phi =

775.8 kN

Req'd Vs =

426.3 kN

θ = 45 deg

Av/s =

0.520 mm2/mm

Say S =

15 mm

Required Av =

7.8 mm2

Bar Dia =

8 mm

2 Bars, Av =

100.48 mm2

OK

Use

8 dia

@

15 mm

stirrups

Check minumum required reinforcement and maximum nominal capacity that can be provided by shear reinforcement

Not done in this spreadsheet

(5)

(6)

(7)

H30-S24

kN

213.45

227.55

240.15

251.55

260.85

269.55

277.5

284.85

290.85

296.85

302.25

307.65

312.3

316.2

320.25

325.65 End shear

and end

reaction

(8)

330.9

335.55

340.2

344.25

348.3

352.35

355.65

358.95

362.25

365.7

368.25

373.65

378.3

382.35

387

390.3

394.35

397.65

400.35

403.05

405.6

408.3

410.4

412.95

414.3

416.4

421.05

424.35

427.65

430.35

433.05

435.75

439.65

442.95

451.05

472.35

493.8

515.1

536.4

557.85

579.15

600.45

643.2

685.95

728.55

771.3

814.05

(9)

GIRDER TYPE

1 Input lenghts (cm)

X1

66.04 A(cm2)

yb (cm)

A*yb

Ix

X2

101.6 A1

1341.9328

10.16 13634.03725 46173.94

X3

15.24 A2

645.16

28.79 18572.00587 23123.97

X4

10.16 A3

387.096

33.02 12781.90992 20811.57

Y1

20.32 A4

1625.8032

99.06 161052.065

1541887.69

Y2

25.4 A5

103.2256

159.17 16430.76284 591.97

Y3

106.68 A6

154.8384

157.48 24383.95123 1331.94

Y4

10.16 A7

251.6124

167.64 42180.30274 811.65

Y5

7.62 A8

270.9672

166.37 45080.81306 1311.13

Y6

12.7 A9

1290.32

176.53 227780.1896 17342.98

Σ 561896.0375 1653386.842

XX -->

A

6070.9556 cm2

Strong axis

yb'

92.55 cm

IXX

27,932,801.336 cm4

6.711E+05

YY -->

A

6070.9556 cm2

Weak Axis

yl'

50.80 cm

IYY

2.095E+06 cm4

GIRDER TYPE

2 Required lenghts (cm)

X1

75 A(cm2)

yb (cm)

A*yb

Ix

X2

75 A1

1125

7.5 8437.5

21093.75

X3

20 A2

275

18.33 5041.666667 1527.78

Y1

15 A3

200

20 4000

1666.67

Y2

10 A4

650

41.25 26812.5

57213.54

Y3

32.5 A5

206.25

62.5 12890.625

644.53 Y4

7.5 A6

150

61.25 9187.5

703.13 Y5

10 A7

750

70 52500

6250.00

Σ 118869.7917 89099.39

XX -->

A

3356.25 cm2

Strong axis

yb'

35.42 cm

IXX

2.264E+06 cm4

YY -->

A

3356.25 cm2

Weak Axis

yl'

37.50 cm

IYY

1.109E+06 cm4

Calculation of I

XX

(10)

A*(yb-yb')^2

A

yl

A*yl

Iy

A*(yl-yl')^2

9110250.10

A1

1341.9328

50.80 68170.19

487712.24

0.00 2623461.79

A2

322.58

34.71 11197.83

11561.98

83477.52

1372019.79

A3

322.58

66.89 21576.30

11561.98

83477.52

68800.29

A4

387.096

50.80 19664.48

7492.17

0.00 458118.30

A5

1625.8032

50.80 82590.80

31467.10

0.00 652687.60

A6

51.6128

39.79 2053.85

295.99 6252.72

1418537.46

A7

51.6128

61.81 3190.02

295.99 6252.72

1476414.87

A8

154.8384

50.80 7865.79

2996.87

0.00 9099124.29

A9

125.8062

22.01 2769.41

7620.50

104252.10

26279414.4938

A10

125.8062

79.59 10012.50

7620.50

104252.10

A11

270.9672

50.80 13765.13

28553.48

0.00 A12

1290.32

50.80 65548.26

1109950.47

0.00 Σ 308404.54

1707129.26 387964.69

A*(yb-yb')^2

A

yl

A*yl

Iy

A*(yl-yl')^2

876806.55

A1

1125

37.50 42187.50

527343.75

0.00 80263.37

A2

137.5

18.33 2520.83

5776.91

50512.15

47539.51

A3

137.5

56.67 7791.67

5776.91

50512.15

22112.17

A4

200

37.5 7500.00

6666.666667 0.00

151277.14

A5

650

37.50 24375.00

21666.67

0.00 100098.15

A6

103.125

18.33 1890.63

4332.68

37884.11

896964.96

A7

103.125

56.67 5843.75

4332.68

37884.11

2175061.85

A8

150

37.5 5625.00

5000

0.00 A9

750

37.50 28125.00

351562.50

0.00 Σ 125859.38

932458.77

176792.53

Calculation of I

YY

Calculation of I

YY