Recommendations for Future Study

CHAPTER 5 CONCLUSIONS

5.3 Recommendations for Future Study

CHAPTER 5

b) Development of design aids for continuous post-tensioned PC girders and box- girders considering the similar parameters as adopted for simply supported post- tensioned PC girder in this study.

c) Development of design aids for simply supported and continuous post-tensioned PC girders, box girders, voided slab etc. considering the different loading conditions for different countries of the world.

d) Development of a more effective span range of Simply Supported PC I-girder for selecting efficient girders type.

REFERENCES

[1] AASHTO LRFD (2017), American Society of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD) Bridge Design Specification, 8^th Edition, Washington, DC.

[2] Ahsan, R., Rana, S., and Ghani, S. (2012), Cost Optimum Design of Post- tensioned I-Girder Bridge Using Global Optimization Algorithm, Journal of Structural Engineering, ASCE, Volume 138, Issue 2, pp 273-284.

[3] Chen, L., and Graybeal, B. A. (2012), Modeling Structural Performance of Ultrahigh Performance Concrete I-Girders, Journal of Bridge Engineering, ASCE, Volume 17, Issue 5, pp 754-764.

[4] Chehab, A.I., and Eamon, C.D. (2018), Regression-Based Adjustment Factor to Better Estimate Shear Capacity for Load-Rating Simple Span PC Girders, Journal of Bridge Engineering, ASCE, Volume 23, Issue 5.

[5] Collins, M. P., and Mitchell, D. (1991). Prestressed concrete structures, Prentice- Hall, Englewood Cliffs, NJ, pp. 61-65.

[6] Devalupura, R. K, and Tadros, M. K. (1992), Critical assessment of ACI 318 Eq. (18-3) for Prestressing Steel Stress at Ultimate Flexure, ACI Structural Journal, Volume 89, Issue 5, pp 538-546.

[7] FHWA (2015a), Load and Resistance Factor Design (LRFD) for Highway Bridge Superstructures, Reference Manual, Federal Highway Administration, U.S. Department of Transportation, Washington, DC.

[8] FHWA (2015b), Post-tensioned Box Girder Design Manual, Federal Highway Administration, U.S. Department of Transportation, Washington, DC.

[9] Fereig, S.M. (1996), Economic Preliminary Design of Bridges with Prestressed I-Girders, Journal of Bridge Engineering, ASCE, Volume 1, Issue 1, pp 18-25.

[10] Freyssinet Inc. (1999), The C Range post-tensioning system, www.freyssinet.com, downloaded date May 10, 2010.

[11] Harries, K.A., Shahrooz, B.M., Ross, B.E., Ball, P., and Hamilton H.R.T.

(2019), Modeling and Detailing Pre-post-tensioned Concrete Bridge Girder End Regions Using the Strut-and-Tie Approach, Journal of Bridge Engineering, ASCE, Volume 24, Issue 3.

[12] Jacques, F. J. (1971), Study of long-span Prestressed concrete Bridge Girders, Journal of the Prestressed Concrete Institute, pp 24-42.

[13] Lounis, Z., Mirza, M.S., and Cohn, M. Z. (1997), Segmental and Conventional Precast Prestressed Concrete I-Bridge Girder, Journal of Bridge Engineering, ASCE, Volume 2, Issue 3, pp 73-82.

[14] Lu, P., Zhuang, Y., Nabizadeh, A., and Tabatabai, H. (2020), Analytical and Experimental Evaluation of Repairs to Continuous PC Girder Bridge, Journal of Performance of Constructed Facilities, ASCE, Volume 34, Issue 1.

[15] Marquez, J., Jauregui, D.V., Brad D, Weldon, B. D., and Newtson, C. M.

(2016), Simplified Procedure to Obtain LRFD Preliminary Design Charts for Simple-Span Prestressed Concrete Bridge Girders, Practice Periodical on Structural Design and Construction, ASCE, Volume 21, Issue 1.

[16] Morris, D. (1978), Prestressed Concrete Design by Linear Programming, Journal of the Structural Division, ASEC, Volume 104, Issue 3, pp 439 - 452.

[17] Naaman, A. E. (1982), Prestressed Concrete Analysis and Design Fundamentals.

Mc-Graw-Hill Book Co., Inc., New York, N.Y.

[18] PCI (2003), Bridge Design Manual, 2^nd Edition, Precast/Prestressed Concrete Institute, Chicago.

[19] PCI (2011), Bridge Design Manual, 3^rd Edition, Precast/Prestressed Concrete Institute, Chicago.

[20]

Ross, B.E., Hamilton, H. R., and Consolazio, G. R. (2015), Experimental Study of End Region Detailing and Shear Behavior of Concrete I-Girders, Journal of Bridge Engineering, ASCE, Volume 20, Issue 6.

This means must input This means may input 1 - General Arrangement of P.C. I-Girder Bridge:

Length of girder, L_g = 39.6 m = 130 ft.

Width of bridge, W_b = 7.38 m = 24.2 ft.

Footpath width of bridge, w_f = 0.65 m = 2.1 ft.

Number of girder, N_g = 3.0 nos.

Overhanging width from C/L of facia girder = 1.25 m = 4.1 ft.

Center to center spacing of girder, S_g = 8.0 ft.

Consider max^m number of duct in a single row = 3.0 nos.

Min^m clear horizontal distance between two ducts, d_h = 3.0 in.

(S 5.9.5.1.1, AASHTO-2017, Page; 5-145)

Concrete clear cover for unprocted reinforcing steel, c_c = 3.0 in.

(T 5.10.1-1, AASHTO-2017, Page; 5-165)

Consider the diameter of post-tensioning duct, d_p = 4.0 in.

1.17 128 span of noncomposite loads

Pile Abutment

Figure 1-1: Sectional Elevation of P.C. I-Girder Bridge

24 total width 8

2.1 18 roadway width 2

8 R.C.C. deck 3 W.C.

1.1 Figure 1-2: X-Section of P.C. I-Girder Bridge

APPENDIX - A, TYPICAL POST-TENSIONED I - GIRDER DESIGN

C/L of bearing C/L of bearing

ft.

ft.

ft.

in.

ft.

ft.

in.

in.

in.

1.1 - Overhanging width of deck slab: (C4.6.2.6.1, AASHTO-2017, Page; 4-55)

1. Overhang width is less or equal to 0.5S_g = 4.0 ft

Applied overhanging width from C/L of facia girder = 4.1 ft 1.2 - Web thickness of P.C. I-girder: (S 5.12.5.3.11b, AASHTO-2017, Page; 5-233) 1. Min^m web thickness is 8.0 in. without prestressing ducts = 8.0 in.

2. Minm web thickness is 12.0 in. with only longitudinal or vertical ducts = 12.0 in.

Applied web thickness for girder, t_w = 12.0 in.

1.3 - Top flange width of P.C. I-girder:

3. Top flange width of girder as per AASHTO I‐Beams type V-VI, w_tf = 42.0 in.

Applied girder top flange width, w_tf = 42.0 in.

1.4 - Top flange thickness of P.C. I-girder: (S 5.12.3.5.1a, AASHTO-2017, Page; 5-219) 1. Min^m top flange thick. of girder, t_tf > clear span betⁿ webs divided by 20 = 4.2 in.

2. Min^m top flange thick. of girder as per AASHTO I‐Beams type V-VI, t_tf = 5.0 in.

Applied girder top flange thickness, t_tf = 5.0 in.

1.5 - Bottom flange width of P.C. I-girder:

1. Girder bottom flange width as per tendon requirement, w_bf = 24.0 in.

2. Girder bottom flange width as per AASHTO I‐Beams type V-VI, w_bf = 28.0 in.

Applied girder bottom flange width, w_bf = 28.0 in.

1.6 - Bottom flange thickness of P.C. I-girder:

(S 5.12.3.5.1b, AASHTO-2017, Page; 5-219)

1. Min^m bottom flange thickness of girder, t_bf > 5.5 in. = 5.5 in.

2. Min^m bottom flange thick. of girder, tbf > clear span betⁿ webs divided by 30 = 2.8 in.

3. Min^m bottom flange thick. of girder as per AASHTO I‐Beams type V-VI, t_bf = 8.0 in.

Applied girder bottom flange thickness, t_bf = 8.0 in.

1.7 - Height of P.C. I-girder: (T 2.5.2.6.3-1, AASHTO-2017, Page; 2-14)

1. Min^m height of precast I-girder including deck, hg with deck = 0.045L_g-brg = 68.9 in.

2. Min^m height of P.C. I-girder as per AASHTO I‐Beams type V-VI, t_bf = 72.0 in.

Applied height of non-composite precast I-girder, h_g = 72.0 in.

42

20 11 5

3 4 4

12

72 42 56

8 10

8

7 28

Figure 1-3: X-Section of Non Composite P.C. I-Girder 1.8 - Property of P.C. I-girder:

Moment of inertia, I_g = bh³/12 + ad²

Section modulus from top of girder, S_t = I_g/Y_t Section modulus from bottom of girder, S_b = I_g/Y_b

Table 1-1 Noncomposite P.C. I-girder section properties along X-X or strong axis:

Y_b Y_t S_t S_b

a 224 1195 896 -33

b 80 444 907 -26

c 672 175616 24192 -0.8

d 16 14.2 1003 25.8

e 60 45.0 3930 28.7

f 33 16.5 2178 29.2

g 210 438 14595 32.7

1295 177768 47700

Part

Gross area, A_g (in.²)

y of each part from bottom (in.)

I_x of each part (in.⁴)

AY (in.³)

Neutral axis, N.A

(in.)

y - Y_b (in.)

A×(y - Y_b)² (in.⁴)

Moment of Inertia, I_g

(in.⁴)

Section Modulus, S

(in.³) 4.0

36.8 35.2

241491

783884 22291.1 21281

11.3 52024

36.0 468

62.7 10677

65.5 49304

66.0 28071

69.5 224081

606116 in.

in.

in.

in.

in. in.

in.

in.

in.

(a) (b)

(c) (d) (f) (g) (e)

in.

1.9 - X-sectional area, surface area & volume of P.C. I-girder:

X-sectional of non-composite girder at end, A_g-end = 2107 in.²

Surface area of girder, Sa_g = 2877 in.²/ft

Volume of girder, v_g = 15540in.³/ft

Table 1-2 Noncomposite P.C. I-girder section properties along Y-Y or weak axis:

x_left x_right S_left S_right

a 224 14635 4704 0.0

b 40 142 493 -8.7

b' 40 142 1187 8.7

c 672 8064 14112 0.0

d 8 7.1 109 -7.3

d' 8 7.1 227 7.3

e 60 2000 1260 0.0

f 16.5 111 121 -14

f' 16.5 111 572 13.7

g 210 30870 4410 0.0

1295 56089 27195 0.0

1.10 - Effective flange width of interior P.C. I-girder:

(S 4.6.2.6.1, AASHTO-2017, Page; 4-54)

Effective flange width of composite girder is 1/2 distance to adjacent girder on each side = 48.0 in.

The effective flange width for an interior girder, b_f = 95.9 in.

1.11 - Effective flange width of exterior P.C. I-girder:

Effective flange width of composite girder is 1/2 distance to adjacent girder on each side = 48.0 in.

Overhanging width from C/L of facia girder = 49.2 in.

The effective flange width for an exterior girder, b_f = 97.2 in.

Part

Gross area, A_g (in.²)

x of each part from bottom (in.)

I_x of each part (in.⁴)

Ax (in.³)

Neutral axis, N.A

(in.)

x - X_left (in.)

A×(x - X_left)² (in.⁴)

Moment of Inertia, I_lat

(in.⁴)

Section Modulus, S

(in.³) 21.0

21.0 21.0

0

69122 3292 3292

12.3 3004

29.7 3004

21.0 0

13.7 430

28.3 430

21.0 0

7.3 3082

34.7 3082

21.0 0

13033

96 42

8

20 11 5

3 4 4

12

72 42 56 80

8 10

8

28

Figure 1-4: X-Section of Composite Interior P.C. I-Girder

Table 1-3 Interior composite P.C. I-girder section properties along X-X or strong axis:

Y_bc Y_tc Y_tsc S_tsc S_tc S_bc

a 224 4.0 1195 896 -47 503463

b 80 11.3 444 907 -40 128484

c 672 36.0 175616 24192 -15 159555

d 16 62.7 14.2 1003 11.3 2028

e 60 65.5 45.0 3930 14.1 11914

f 33 66.0 16.5 2178 14.6 7026

g 210 69.5 438 14595 18.1 68731

h 768 76.0 4093 58332 24.6 464137

2063 181862 106032 1345337

Part Gros

s area,

A_c (in.²)

y of each part from bottom

(in.)

I_x of each part (in.⁴)

AY (in.³)

Neutral axis, N.A (in.)

y - Y_b (in.)

A×(y - Y_b)² (in.⁴)

Moment of Inertia, I_gc

(in.⁴)

Section Modulus, S (in.³)

51.4 20.6 28.6 1527199 53415 74167.9 29707

in.

in.

in.

in.

in.

in. in. in. in.

in.

(a) (b)

(c) (d) (f) (g) (e)

(h) in.

97 42

8

20 11 5

3 4 4

12

72 42 56 80

8 10

8

28

Figure 1-5: X-Section of Composite Exterior P.C. I-Girder

Table 1-4 Exterior composite P.C. I-girder section properties along X-X or strong axis:

Y_bc Y_tc Y_tsc S_tsc S_tc S_bc

a 224 4.0 1195 896 -48 505946

b 80 11.3 444 907 -40 129234

c 672 36.0 175616 24192 -16 161983

d 16 62.7 14.2 1003 11.1 1986

e 60 65.5 45.0 3930 14.0 11717

f 33 66.0 16.5 2178 14.5 6914

g 210 69.5 438 14595 18.0 67846

h 777 76.0 4146 59079 24.5 465634

2072 181914 106780 1351259

Part

Gross area, A_c (in.²)

y of each part from botto m (in.)

I_x of each part (in.⁴)

AY (in.³)

Neutral axis, N.A (in.)

y - Y_b (in.)

A×(y - Y_b)² (in.⁴)

Moment of Inertia, I_c

(in.⁴)

Section Modulus, S (in.³)

51.5 20.5 28.5 1533173 53844 74882.6 29756

in.

in.

in.

in. in.

in.

in.

in.

in.

(h) (f) (g) (e)

(d)

(c)

(a) (b)

2 - Live Load Distribution Factors:

Multiple presence factor of single lane, m₁ = 1.2

(T 3.6.1.1.2-1, AASHTO-2017, Page; 3-21)

Multiple presence factor of two lane, m₂ = 1.0

Correction factor of source aggregate, k₁ = 1.0

(S 5.4.2.4, AASHTO-2017, Page; 5-20)

Number of girder, N_g = 3.0 nos.

Skew angle, θ = 1.0 deg

Concrete strength (28-days) of girder, f'_cg = 7.0 ksi

Concrete strength (28-days) of deck slab, f'_cs = 4.4 ksi

Moment of inertia of non-composite-girder, I_g = 8.E+05in.⁴ Cross sectional area of non-composite-girder, A_g = 1295 in.² Length of girder betⁿ bearing center to center, L_g-brg = 128 ft.

Center to center spacing of girder, S_g = 8.0 ft.

Distance from C/L of exterior girder to gutter, d_e < 3 = 1.1 ft.

(S 4.6.2.2.1, AASHTO-2017, Page; 4-30)

Slab thickness excluding wearing surface, t_s = 8.0 in.

(T 4.6.2.2.2b-1, AASHTO-2017, Page; 4-37)

Distance betⁿ C.G. of non-composite girder and deck, e_g = Y_t+t_s/2 = 39.2 in.

2.1 - Modular ratio between P.C. I-girder and deck:

Unit weight of concrete, γ_c (5.0 < f'_cg < 15 ksi) = 0.14+0.001×f'_cg = 0.15 kcf (T 3.5.1-1, AASHTO-2017, Page; 3-19)

Modulus of elasticity of girder, E_G =33000×K₁×γ_c^1.5×√f'_cg = 4921 ksi (EQ C5.4.2.4-2, AASHTO-2017, Page; 5-20)

Modulus of elasticity of deck slab, E_s = 3879 ksi

Modular ratio betⁿ girder and deck slab, n = E_G/E_s = 1.27 (EQ 4.6.2.2.1-2, AASHTO-2017, Page; 4-32)

2.2 - Longitudinal stiffness parameter:

Longitudinal stiffness parameter, K_g = n(I_g+A_g×e_g²) in.⁴ (EQ 4.6.2.2.1-1, AASHTO-2017, Page; 4-32) Longitudinal stiffness is ok 2.3 - Moment distribution factor of an interior P.C. I-girder with two or more design lanes:

More design lanes, D_M = 0.075+(S_g/9.5)^0.6×(S_g/L_g-brg)^0.2×{K_g/(12×L_g-brg×t_s³)}^0.1 = 0.68 (T 4.6.2.2.2b-1, AASHTO-2017, Page; 4-37)

2.4 - Moment distribution factor of an interior P.C. I-girder with one design lane:

One design lane, D_M = 0.06+(S_g/14)^0.4×(S_g/L_g-brg)^0.3×{K_g/(12×L_g-brg×t_s³)}^0.1 = 0.46 2.5 - Skew correction factor of shear: (T 4.6.2.2.3c-1, AASHTO-2017, Page; 4-46) Skew correction factor, S_C = 1+0.2×(12×L_g-brg×t_s³/K_g)^0.3×tanθ = 1.0

3514312

=

X₁

2 6

4.3 X₂

8.0

Figure 2-1: Lever Rule of an Interior P.C. I-girder

2.6 - Shear distribution factor of an interior P.C. I-girder with one, two or more design lanes:

For fatigue limit state, D_V = (x₁+x₂)/(2×S_g) = 0.62 (T 4.6.2.2.3a-1, AASHTO-2017, Page; 4-43)

For S.L.S. & U.L.S., D_V will be multiplied by multilple presence factor = 0.75

Applying skew correction factor, D_V = 0.75

X₁

3.0 2 6 1.1

X₂

8.0

Figure 2-2: Lever rule of an exterior P.C. I-girder

2.7 - Moment distribution factor of an exterior P.C. I-girder with two or more design lanes:

Ecentricity of a design truck, e = 0.77+d_e/9.1 = 0.89

Two or more design lanes, D_M = e×D_MInterior = 0.61

(T 4.6.2.2.2d-1, AASHTO-2017, Page; 4-40)

2.8 - Moment distribution factor of an exterior P.C. I-girder with one design lane using lever rule:

Fatigue limit state, D_M = (X₁+X₂)/(2×S_g) = 0.52 Service & strength limit state, D_M will be multiplied by multilple presence factor = 0.62

ft. ft. ft. ft.

ft.

ft.

ft.

ft.

ft.

2.9 - Shear distribution factor of an exterior P.C. I-girder with two or more design lanes:

Ecentricity of a design truck, e = 0.6+d_e/10 = 0.71

(T 4.6.2.2.3b-1, AASHTO-2017, Page; 4-45)

Two or more design lanes, D_V = e×D_VInterior = 0.54

2.10 - Shear distribution factor of an exterior P.C. I-girder with one design lane using lever rule:

Service and strength limit state, D_V = D_MExterior×S_C = 0.62 C/L

7.1

2 4 4

8.0

Figure 2-3: Additional Check of Rigidly Connected Girder

2.11 - Distribution factor of an exterion girder with one design lane by additional check:

Number of lane, N_L = 1.0 nos.

Horizontal distance from C.G. of pattern to exterior girder, X_ext = 8.0 ft.

Ecentricity of a design truck from C.G. of pattern of girders, Σe = 4.1 ft.

Fatigue limit state reaction on exterior beam in terms of lanes, R = N_L/N_g+X_ext(Σe)/Σx² = 0.59 Strength limit state applying multiple presence factor, R = 0.71

(EQ C4.6.2.2.2d-1, AASHTO-2017, Page; 4-39)

2.12 - Distribution factor of an exterion girder with two or more design lane by additional check:

Number of lane, N_L = 2.0 nos.

Ecentricity of a design truck from C.G. of pattern of girders, Σe = 0.0 ft.

Reaction on exterior beam in terms of lanes, R = N_L/N_g+X_ext(Σe)/Σx² = 0.67 Strength limit state applying multiple presence factor, R = 0.67 Table 2-1 Summary of service and strength limit state distribution factors:

Load case

D_M for interior girder

D_M for exterior

girder

D_V for interior

girder

D_V for exterior

girder Distribution factors

Multiple lane load 0.68 0.61 0.75 0.54

Single lane load 0.46 0.62 0.75 0.62

Additional check for rigidly connected girders

Multiple lane load NA 0.67 NA 0.67

Single lane load NA 0.71 NA 0.71

Design value 0.68 0.71 0.75 0.71

P1 R1

P1 P2 R2

P2 ft.

ft.

ft. ft.

3 - Dead Load & Live Load:

Dynamic load allowance, IM (T 3.6.2.1-1, AASHTO-2017, Page; 3-31) = 33.0 % Moment distribution factor of an interior girder, D_M(I) = 0.68 Moment distribution factor of an exterior girder, D_M(E) = 0.71 Shear distribution factor of an interior girder, D_V(I) = 0.75 Shear distribution factor of an exterior girder, D_V(E) = 0.71

Number of girder, N_g = 3.0 nos.

Unit weight of concrete, γ_c = 0.15 kcf

Unit weight of bituminous wearing course, γ_wc = 0.14 kcf (T 3.5.1-1, AASHTO-2017, Page; 3-19)

Cross sectional area of non-composite girder at end, A_g-end = 2107 in.² Cross sectional area of non-composite girder at middle, A_g-mid = 1295 in.² X-sectional area of non-composite girder at transformed section, A_g-trns. = 1701 in.²

Length of girder, L_g = 130 ft.

Length of girder betⁿ bearing center to center, L_g-brg = 128 ft.

Total length of girder for end section, L_g-end = 13.3 ft.

Total length of girder for transformed section, L_g-trns. = 6.7 ft.

Total length of girder for mid section, L_g-mid. = 110 ft.

Center to center spacing of girder, S_g = 8.0 ft.

Overhanging width of slab from C/L of exterior girder = 4.1 ft.

Distance from C/L of exterior girder to gutter, d_e = 1.1 ft.

Nos. of diaphragm, n_d = 5.0 nos.

Thickness of diaphragm, t_d = 1.1 ft.

Height of diaphragm, h_d = 4.5 ft.

Web thickness of girder, t_w = 12.0 in.

Thickness of wearing course, t_wc = 3.0 in.

Slab thickness excluding wearing surface, t_s = 8.0 in.

Spacing of rail post = 4.1 ft.

No. of rail post = 33.0 no.

No. of rail bar = 3.0 no.

Height of rail post = 4.3 ft.

Cross sectional area of rail post = 64 in.²

Cross sectional area of rail bar = 36 in.²

Cross sectional area of kerb = 120 in.²

Footpath width,w_f = 2.1 ft.

Cross sectional area of pedestrain slab = 0.54 ft.²

3.1 - Interior P.C. I-girder weight:

Girder weight, DC_girder(I) = A_g×γ_c = 1.43 k/ft./gir.

Deck slab weight, DC_slab(I) = 0.78 k/ft./gir.

3.2 - Exterior P.C. I-girder weight:

Girder weight, DC_girder(E) = A_g × γ_c = 1.43 k/ft./gir.

Deck slab weight, DC_slab(E) = 0.79 k/ft./gir.

3.3 - Diaphragm weight:

Diaphragm weight, DCdiaphragm(I) = 0.19 k/ft./gir.

Diaphragm weight, DCdiaphragm(E) = 0.09 k/ft./gir.

3.4 - Parapet weight:

Rail post weight, DC_{rail post} = 0.05 k/ft./gir.

Rail bar weight, DC_{rail bar} = 0.07 k/ft./gir.

Pedestrain weight on railing, DCpeds.(rail) = 0.03 k/ft./gir.

Kerb weight, DC_kerb (S 13.8.2, AASHTO-2017, Page; 13-10) = 0.08 k/ft./gir.

Pedestrain slab weight, DC_{p, slab} = 0.05 k/ft./gir.

Pedestrain weight, DCpeds.(walkway) (S 3.6.1.6, AASHTO-2017, Page; 3-30) = 0.11 k/ft./gir.

Total parapet & footpath weight, DCpar.&foot. = 0.40 k/ft./gir.

3.5 - Wearing surface weight:

Wearing surface weight, DW_FWS(I) = 0.28 k/ft./gir.

Wearing surface weight, DW_FWS(E) = 0.18 k/ft./gir.

8 32 32

14 14

Figure 3-1: Design Truck of HL-93K 3.6 - Live load weight:

Total axel load of HL-93K (S 3.6.1.2.2, AASHTO-2017, Page; 3-22) = 72.0 kip The design lane load, W_lane (S 3.6.1.2.4, AASHTO-2017, Page; 3-23) = 0.64 k/ft.

3.7 - Load combinations: (T 3.4.1-1, AASHTO-2017, Page; 3-15) Strength-I = 1.25(DC) + 1.5(DW) + 1.75(LL + IM)

Service-I = 1.0[DC + DW + (LL + IM)]

Service-III = 1.0(DC + DW) + 0.8(LL + IM)

kip kip kip

ft. ft.

Table 3-1 Summary of unfactored & factored moments of interior P.C. I-girder:

kip-ft.kip-ft. kip-ft.

0 0 0

223 123 44

438 240 86

1048 575 205

1862 1022 365

2444 1341 479

2793 1532 547

2910 1596 570

2793 1532 547

2444 1341 479

1862 1022 365

1048 575 205

438 240 86

223 123 44

0 0 0

Table 3-2 Summary of unfactored & factored shear of interior P.C. I-girder:

kip kip kip

92.8 50.9 18.2

89.3 49.0 17.5

85.7 47.0 16.8

74.3 40.7 14.6

55.7 30.6 10.9

37.1 20.4 7.3

18.6 10.2 3.6

0.0 0.0 0.0

-19 -10 -3.6

-37 -20 -7.3

-56 -31 -11

-74 -41 -15

-86 -47 -17

-89 -49 -17

-93 -51 -18

Live load +

IM Factored moment as

strength-I Girder Slab DiaphragmTotal noncompParapet &

Footpath FWS Positive HL-93

ft. kip-ft. kip-ft. kip-ft. kip-ft.

Location

Noncomposite Composite

kip-ft.

0.0 0 0 0 0 0

2.5 30 376 62 227 1009

5.0 58 736 121 444 1978

12.8 139 1761 290 1063 4732

25.5 248 3131 516 1889 8412

38.3 325 4110 677 2479 11041

51.1 371 4697 774 2833 12618

63.8 387 4893 806 2952 13144

76.6 371 4697 774 2833 12618

89.4 325 4110 677 2479 11041

102.1 248 3131 516 1889 8412

114.9 139 1761 290 1063 4732

122.7 58 736 121 444 1978

125.2 30 376 62 227 1009

127.7 0 0 0 0 0

Live load +

IM Factored shear as strength-I Girder Slab DiaphragmTotal noncompParapet &

Footpath FWS Positive HL-93

ft. kip kip kip kip

Location

Noncomposite Composite

kip

0.0 12.3 156.1 25.7 103.2 435.2

2.5 11.9 150.1 24.7 100.6 420.9

5.0 11.4 144.1 23.7 98.0 406.5

13.0 9.9 124.9 20.6 89.8 360.7

26.0 7.4 93.7 15.4 76.3 286.3

39.0 4.9 62.4 10.3 62.9 211.8

52.0 2.5 31.2 5.1 49.4 137.4

65.0 0.0 0.0 0.0 36.0 63.0

78.0 -2.5 -31.2 -5.1 -49.4 -137.4

91.0 -4.9 -62.4 -10.3 -62.9 -211.8

104.0 -7.4 -93.7 -15.4 -76.3 -286.3

117.0 -9.9 -124.9 -20.6 -89.8 -360.7

125.0 -11.4 -144.1 -23.7 -98.0 -406.5

127.5 -11.9 -150.1 -24.7 -100.6 -420.9

130.0 -12.3 -156.1 -25.7 -103.2 -435.2

Table 3-3 Summary of unfactored & factored moments of exterior P.C. I-girder:

kip-ft.kip-ft. kip-ft.

0 0 0

223 124 28

438 243 55

1048 582 132

1862 1035 234

2444 1358 307

2793 1552 351

2910 1617 366

2793 1552 351

2444 1358 307

1862 1035 234

1048 582 132

438 243 55

223 124 28

0 0 0

Table 3-4 Summary of unfactored & factored shear of exterior P.C. I-girder:

kip kip kip

92.8 51.6 11.7

89.3 49.6 11.2

85.7 47.6 10.8

74.3 41.3 9.3

55.7 30.9 7.0

37.1 20.6 4.7

18.6 10.3 2.3

0.0 0.0 0.0

-19 -10 -2.3

-37 -21 -4.7

-56 -31 -7.0

-74 -41 -9.3

-86 -48 -11

-89 -50 -11

-93 -52 -12

Live load +

IM Factored moment as

strength-I Girder Slab DiaphragmTotal noncompParapet &

Footpath FWS Positive HL-93

ft. kip-ft. kip-ft. kip-ft. kip-ft.

Location

Noncomposite Composite

kip-ft.

0.0 0 0 0 0 0

2.5 15 362 62 238 989

5.0 29 710 121 466 1938

12.8 70 1699 290 1114 4634

25.5 124 3021 516 1981 8239

38.3 162 3965 677 2600 10813

51.1 186 4531 774 2972 12358

63.8 193 4720 806 3095 12873

76.6 186 4531 774 2972 12358

89.4 162 3965 677 2600 10813

102.1 124 3021 516 1981 8239

114.9 70 1699 290 1114 4634

122.7 29 710 121 466 1938

125.2 15 362 62 238 989

127.7 0 0 0 0 0

Live load +

IM Factored shear as strength-I Girder Slab DiaphragmTotal noncompParapet &

Footpath FWS Positive HL-93

ft. kip kip kip kip

Location

Noncomposite Composite

kip

0.0 6.2 150.6 25.7 97.5 408.5

2.5 5.9 144.8 24.7 95.1 395.1

5.0 5.7 139.0 23.7 92.6 381.7

13.0 4.9 120.5 20.6 84.8 338.7

26.0 3.7 90.4 15.4 72.1 268.9

39.0 2.5 60.2 10.3 59.4 199.1

52.0 1.2 30.1 5.1 46.7 129.3

65.0 0.0 0.0 0.0 34.0 59.5

78.0 -1.2 -30.1 -5.1 -46.7 -129.3

91.0 -2.5 -60.2 -10.3 -59.4 -199.1

104.0 -3.7 -90.4 -15.4 -72.1 -268.9

117.0 -4.9 -120.5 -20.6 -84.8 -338.7

125.0 -5.7 -139.0 -23.7 -92.6 -381.7

127.5 -5.9 -144.8 -24.7 -95.1 -395.1

130.0 -6.2 -150.6 -25.7 -97.5 -408.5

4 - Loss of Prestress:

Wobble friction coefficient, k = 0.0002/ft

Coefficient of friction, μ (T 5.9.3.2.2b-1, AASHTO-2017, Page; 5-130) = 0.2

Average annual ambient relative humidity, H = 70 %

(FIG 5.4.2.3.3-1, AASHTO-2017, Page; 5-19)

Factor accounting for low relaxation strands, K_L = 30.0 (S 5.9.3.4.2c, AASHTO-2017, Page; 5-136)

Consider total nos. of provided strand, Nstrands(total) = 50 nos.

Consider total nos. of provided cable, Ncable(total) = 3 nos.

Nos. of strand in each cable = 16.7 nos.

Modulus of elasticity of prestressing strand, E_ps = 28500ksi (S 5.4.4.2, AASHTO-2017, Page; 5-23)

Ultimate stress of prestressing strand, f_pu = 270 ksi

(T 5.4.4.1-1, AASHTO-2017, Page; 5-23)

Yield stress of prestressing strand, f_py (90% of f_pu) = 243 ksi Strength of concrete at time of initial loading/at transfer, f'_ci/f_ct (80% of f'_cg) = 5.6 ksi

(S 5.4.2.3.2, AASHTO-2017, Page; 5-18)

Modulus of elasticity of girder concrete at transfer, E_ci/E_ct = 33000×K₁×γ_c^1.5×√f'_ci = 4401 ksi Midspan moment due to P.C. girder self weight, M_g = 34918k-in.

Strand area for0.6 a_ps (as per manufacturing chart) = 0.22 in². Cross-sectional area of non-composite girder, A_g = 1295 in.² Moment of inertia of non-composite girder, I_g = 783884in.⁴

Length of girder, L_g = 130 ft.

Neutral axis of non-composite girder from bottom, Y_b = 36.8 in.

Min^m clear vertical distance betⁿ two ducts, d_v = 2.0 in.

(S 5.9.5.1.1, AASHTO-2017, Page; 5-145)

The anchorage set value, δ_s (C5.9.3.2.1, AASHTO-2017, Page; 5-128) = 0.38 in.

Assume, 16 dia. bar use as longitudinal reinf. at bottom, d_s-long. = 0.6 in.

Assume, 12 dia. bar are used as stirrup, d_s-stir. = 0.5 in.

4.1 - Cable arrangement of P.C. I-girder at middle:

6.0 6.0 6.1

Figure 4-1: X-Section of P.C. I-Girder Showing 3nos.Cable in.

mm mm

Table 4-1 C.G. of prestressing strands and ecentricity at65.0 ft. middle of P.C.-I girder:

1 2 3 0 0 0 0

Table 4-2 Horizontal & vertical location of prestressing cable at middle:

4.2 - Initial stress in the tendons immediately prior to transfer:

(T 5.9.2.2-1, AASHTO-2017, Page; 5-121)

Stress in prestressing steel immediately prior to transfer, f_pbt = 0.9f_py = 208 ksi 77.1 81.0 % 4.3 - Instantaneous losses:

Elastic shortening (EQ C5.9.3.2.3b-1, AASHTO-2017, Page; 5-132)

Total area of strand, A_ps = Nstrand(total)×a_ps = 10.9 in.² Loss due to elastic shortening after transfer, Δf_pES

Δf_pES =(N-1)/2N×(A_psf_pbt(I_g + e²_midA_g) - e_midM_gA_g)/(A_ps(I_g+e²_midA_g)+A_gI_gE_ci/E_ps) = 5.9 ksi 4.4 - Prestressing stress at transfer:

Stress in prestressing strands at transfer, f_pt = f_pbt - Δf_pES = 202 ksi Force in prestressing strands at transfer, P_t = A_ps×f_pt = 2196 kip

Initial loss of prestressing force = 2.8 %

Cable no. Direction 'X' (in.)

0 780 Emergence

angle, θ

36 6.1 4.4

0

Strand area (in.²)

C.G. of strand at

middle from bottom (in.)

P/S force ecentricity at

middle (in.)

Direction 'Y' (in.)

18 6.1 1.7 3.6

6.1 30.7

3.6

54 6.1 7.0 3.6

0 0.0 0.0 0.0

0.0 0.0 0.0

0 0 0.0 0.0

0 0 0.0 0.0

Cable No. Horizontal distance from C/L of girder (in.)

Vertical distance from bottom of girder (in.)

1 -6.0 6.1

2 0.0 6.1

3 6.0 6.1

0 0.0

0 0.0

0 0.0

0 0.0

<

4.5 - Jacking stress:

Loss due to relaxation of prestressing strands at transfer, ΔfpR1 = fpt/KL(fpt/fpy-0.55) = 1.9 ksi

Jacking stress, f_pj = f_pbt + Δf_pR1 = 210 ksi

(EQ 5.9.3.4.2c-1, AASHTO-2017, Page; 5-136)

Jaking force, P_j = A_ps×f_pj = 2281 kip

Jaking force at each strand, p_j(each) = 45.6 kip

= 203 kN Friction losses (EQ 5.9.3.2.2b-1, AASHTO-2017, Page; 5-128)

Sum of absolute values of angular change of P.S. steel path from jacking end, α = 4.4 ⁰

Loss due to friction betⁿ internal prestressing tendons & duct, ΔfpF = fpj(1- e− (kx + μα)) = 8.5 ksi Anchorage slip losses

Influence length of anchor set, x_pA = √(E_P×δ_s×L_g/(12×Δf_pF)) = 404.1 in.

Anchorage slip loss, Δf_pAS = 2×Δf_pF×x_pA/L_g = 4.4 ksi 4.6 - Time-dependent losses after transfer:

Creep losses (S 5.9.3.3, AASHTO-2017, Page; 5-133)

Loss due to creep of concrete, Δf_pCR = 10×f_pbt×A_ps/A_g×(1.7 - 0.01H)×5/(1+f'_ci) = 13.2 ksi Shrinkage losses

Loss due to shrinkage of concrete, Δf_pSR =12×(1.7 - 0.01H)×5/(1+f'_ci) = 9.1 ksi Relaxation losses

Loss due to relaxation after transfer, ΔfpR2 = 6.0 - 0.12×ΔfpES - 0.06×(ΔfpSR + ΔfpCR) = 2.4 ksi 4.7 - Total loss after transfer:

Total loss after transfer, ΔfpT = ΔfpES + ΔfpSR + ΔfpCR + ΔfpR2 + ΔfpAS + ΔfpF = 43.5 ksi 4.8 - Final effective prestress responses: (T 5.9.2.2-1, AASHTO-2017, Page; 5-121) At S.L.S. after all losses allowable stress in prestressing steel, f_pa = 0.8f_py = 194.4 ksi Effective stress in prestressing strands after all losses, f_pe = f_pbt - Δ_fpT = 165 ksi

Ok Actual force in prestressing strands after all losses, P_e = A_ps×f_pe = 1788 kip

Total loss of stress, Δ_total = 20.9 %

4.9 - Elongation:

Elongation = p_j(each)×L_g/a_ps(each)×E_ps (as per 1^st formula) = 11.5 in.

Again, Strain, £_ps = f_pj/E_ps = 0.01

Elongation/change in length(considering both end), δl = £_ps×(L_g/2) = 5.8 in.

Elongation(considering one end), δl = £_ps×L_g (as per 2^nd formula) = 11.5 in.

Elongation at each strand per feet, δl_(each) = 2.25 mm

5 - Stress in Prestressing Strands:

Total nos. of provided strand, Nstrands(total) = 50 nos.

Consider nos. of provided cable at 1^st row = 3 nos.

Nos. of strand for 1^st row, Nstrands(1st) = 50 nos.

Consider nos. of provided cable at 2^nd row = 0 nos.

Nos. of strand for second row, Nstrands(2nd) = 0 nos.

Consider nos. of provided cable at 3^rd row (Please adjust here) = 0 nos.

Nos. of strand for third row, Nstrands(3rd) = 0 nos.

Consider nos. of provided cable at 4^th row (Please adjust here) = 0 nos.

Nos. of strand for fourth row, Nstrands(4th) = 0 nos.

Consider nos. of provided cable at 5^th row = 0 nos.

Nos. of strand for fifth row, Nstrands(5th) = 0 nos.

Total nos. of provided cable, Ncable(total) = 3 nos.

Concrete strength (28-days) of girder, f'_cg = 7.0 ksi

Strength of concrete at time of initial loading/at transfer, f'_ci = 5.6 ksi

Concrete strength of deck slab, f'_cs = 4.4 ksi

Stress block factor of slab concrete (f'_c < 4 _ksi), β₁ = 0.80 (S 5.6.2.2, AASHTO-2017, Page; 5-36)

Stress block factor of slab concrete (f'_c < 10 _ksi), α₁ 0.85

As per table of low relaxation strand factor, k = 0.28

(T C5.6.3.1.1-1, AASHTO-2017, Page; 5-37)

Ultimate stress of prestressing strand, f_pu = 270 ksi

Yield stress of prestressing strand, f_py = 243 ksi

Force in prestressing strands at transfer, P_t = 2196 kip

Effective force in prestressing strands after all losses, P_e = 1788 kip

Modular ratio between girder and deck, n = 1.3

The effective flange width of an interior girder, b_f = 95.9 in.

Top flange width of girder, w_tf = 42.0 in.

Slab thickness excluding wearing surface, t_s = 8.0 in.

Distance from extreme compression fiber to C.G. of P/S strands, dp = h-C.G. of strand from bottom= 73.9 in.

Diameter of post-tensioning duct = 4.0 in.

Total area of strand, A_ps = 10.9 in.²

5.1 - Stress in prestressing strands at nominal flexural resistance:

For the midspan section (EQ 5.6.3.1.1-2, AASHTO-2017, Page; 5-36)

Correction factor as closely spaced anchors for low relaxation strand, k=2(1.04-fpy/fpu) = 0.28

Allowable / provided strand factor, k = 0.28

For rectangular section behaviour, distance between neutral axis and compressive face, c c = A_psf_pu/(α₁f'_csβ₁b_f + kA_ps×f_pu/d_p) = 9.9 in.

(EQ 5.6.3.1.1-4, AASHTO-2017, Page; 5-37)

Girder is behaving as a T - section so please calculate again