# Design Step 5 Design of Superstructure Prestressed Concrete

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## Transcript Of Design Step 5 Design of Superstructure Prestressed Concrete

Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

Design Step 5.6
Design Step 5.6.1
Design Step 5.6.1.1

FLEXURE DESIGN Flexural stress at transfer Stress limits at transfer Compression stress:

The allowable compression stress limit for pretensioned concrete components is calculated according to S5.9.4.1.1.

fCompression

= -0.60(f′ci) = -0.60(4.8 ksi) = -2.88 ksi

Tension stress:

From Table S5.9.4.1.2-1, the stress limit in areas with bonded reinforcement sufficient to resist 120% of the tension force in the cracked concrete computed on the basis of an uncracked section is calculated as:

fTension

= 0.22 fc′i
= 0.22 4.8 = 0.48 ksi

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

Design Step 5.6.1.2

Stress calculations at transfer Table 5.6-1 – Stresses at Top and Bottom of Beam at Transfer

Location
(ft.) (1) 0
1.75 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54.5 55.0 60.5 66.0 71.5 77.0 82.5 88.0 93.5 99.0 104.5 107.25 109.0

Girder
self weight moment (k-ft) (2) 47 153 368 656 909 1,128 1,313 1,464 1,580 1,663 1,711 1,725 1,725 1,705 1,650 1,562 1,439 1,282 1,091 865 606 312 153 47

Fps at transfer (kips) (3) 277.3 924.4 924.4 993.7 1,097.7 1,097.7 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,271.0 1,097.7 1,097.7 924.4 924.4 924.4 277.3

Stress at transfer

Top of beam

Bottom of beam

(ksi) 0.135 0.451 0.326 0.209 0.123 -0.005 -0.009 -0.097 -0.155 -0.203 -0.231 -0.240 -0.240 -0.228 -0.196 -0.144 -0.083 0.009 0.017 0.149 0.197 0.358 0.451 0.135

(ksi) -0.654 -2.183 -2.055 -2.065 -2.171 -2.040 -2.358 -2.269 -2.209 -2.160 -2.132 -2.120 -2.123 -2.135 -2.168 -2.220 -2.284 -2.377 -2.063 -2.197 -1.923 -2.105 -2.200 -0.660

Notes: 1 - Distance measured from the centerline of the bearing of the simple span girder 2 - See Section 5.3, based on 110.5 ft. length 3 - See Section 5.5 for prestressing forces

Sample Calculations for Flexural Stresses at Transfer
Definitions:
Pt = Initial prestressing force taken from Table 5.5-1 (kips) Ag = Gross area of the basic beam (in2) e = Distance between the neutral axis of the noncomposite girder and the
center of gravity of the prestressing steel (in.)

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

St = Section moduli, top of noncomposite beam (in3) Sb = Section moduli, bottom of noncomposite beam (in3) Mg = Moment due to the girder self weight only (k-ft)

See Section 2.2 for section properties.

Sample Calculations at 1 ft. – 9 in. From CL of Bearing (2 ft. – 6 in. From Girder End)

Girder top stress:

ftop

= -Pt/Ag + Pte0’/St – Mg/St

= − 924.4 + 924.4(31.005) − 153(12)

1,085

20,588 20,588

= 0.451 ksi < Stress limit for tension (0.48 ksi) OK Girder bottom stress:

fbottom

= -Pt/Ag – Pte0’/Sb + Mg/Sb

= − 924.4 − 924.4(31.005) + 153(12)

1,085

20,157

20,157

= -2.183 ksi < Stress limit for compression (-2.88 ksi) OK Sample Calculations at 11 ft. From the CL of Bearing (11 ft. – 9 in. From Girder End)

Girder top stress:

ftop

= -Pt/Ag + Pte11’/St – Mg/St

= − 993.7 + 993.7(31.222) − 656(12)

1,085

20,588

20,588

= 0.209 ksi < Stress limit for tension (0.48 ksi) OK Girder bottom stress:

fbottom

= -Pt/Ag – Pte11’/Sb + Mg/Sb

= − 993.7 − 993.7(31.222) + 656(12)

1,085

20,157

20,157

= -2.064 ksi < Stress limit for compression (-2.88 ksi) OK

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

Sample Calculations at 54 ft. – 6 in. From the CL of Bearing (55 ft. – 3 in. From Girder End) – Midspan of Noncomposite Beam

Girder top stress:

ftop

= -Pt/Ag + Pte54.5’/St – Mg/St

= −1,271.0 + 1,271.0(31.38) − 1,725(12)

1,085

20,588

20,588

= -0.239 ksi < Stress limit for compression (-2.88 ksi) OK Girder bottom stress:

fbottom

= -Pt/Ag – Pte54.5’/Sb + Mg/Sb

= −1,271.0 − 1,271.0(31.38) + 1,725(12)

1,085

20,157

20,157

= -2.123 ksi < Stress limit for compression (-2.88 ksi) OK

Design Step 5.6.2

Final flexural stress under Service I limit state
Maximum compression is checked under Service I limit state and maximum tension is checked under Service III limit state. The difference between Service I and Service III limit states is that Service I has a load factor of 1.0 for live load while Service III has a load factor of 0.8.

As indicated in Section 5.3, many jurisdictions do not include creep and shrinkage effects in designing a pretensioned girder bridge. The calculations presented herein do not include creep and shrinkage moments. If creep and shrinkage are required by a specific jurisdiction, then their effects should be included. See Section 5.3 and Appendix C for calculations and values of creep and shrinkage effects for the example bridge.

Design Step Stress limits 5.6.2.1 Compression stress:
From Table S5.9.4.2.1-1, the stress limit due to the sum of the effective prestress, permanent loads, and transient loads and during shipping and handling is taken as 0.6ϕwf′c (where ϕw is equal to 1.0 for solid sections).

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

For prestressed concrete beams (f′c = 6.0 ksi)
fComp, beam1 = -0.6(6.0 ksi) = -3.6 ksi
For deck slab (f′c = 4.0 ksi)
fComp, slab = -0.6(4.0 ksi) = -2.4 ksi
From Table S5.9.4.2.1-1, the stress limit in prestressed concrete at the service limit state after losses for fully prestressed components in bridges other than segmentally constructed due to the sum of effective prestress and permanent loads shall be taken as:
fComp, beam 2 = -0.45(f′c) = -0.45(6.0) = -2.7 ksi
From Table S5.9.4.2.1-1, the stress limit in prestressed concrete at the service limit state after losses for fully prestressed components in bridges other than segmentally constructed due to live load plus one-half the sum of the effective prestress and permanent loads shall be taken as:
fComp, beam 3 = -0.40(f′c) = -0.40(6.0) = -2.4 ksi

Tension stress:

From Table S5.9.4.2.2-1, the stress limit in prestressed concrete at the service limit state after losses for fully prestressed components in bridges other than segmentally constructed, which include bonded prestressing tendons and are subjected to not worse than moderate corrosion conditions shall be taken as the following:

fTensile

= 0.19 fc′
= 0.19 6 = 0.465 ksi

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

Table 5.6-2 – Stresses in the Prestressed Beam

Location
(ft.) (1)
0 1.75 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54.5 55.0 60.5 66.0 71.5 77.0 82.5 88.0 93.5 99.0 104.5 107.25 109.0

Girder noncomposite
moment (k-ft) (2)
0 217 661 1,252 1,776 2,230 2,616 2,933 3,181 3,360 3,471 3,512 3,511 3,456 3,333 3,141 2,880 2,551 2,152 1,686 1,150 546 217
0

Fps after losses (kips) (3)
239.0 797.2 797.0 857.0 946.7 946.7 1,092.1 1,092.1 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 1,096.2 946.7 946.7 797.2 797.2 797.2 239.0

0 36 108 199 276 337 384 414 429 429 414 387 384 338 277 201 108 2 -121 -258 -452 -580 -670 -729

Live load positive moment (k-ft) (2)
0 170 476 886 1,230 1,509 1,724 1,882 1,994 2,047 2,045 2,015 2,010 1,927 1,794 1,613 1,388 1,124 825 524 297 113 58 15

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

Table 5.6-2 – Stresses in the Prestressed Beam (cont.)

Location
(ft.) (1)
0 1.75 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54.5 55.0 60.5 66.0 71.5 77.0 82.5 88.0 93.5 99.0 104.5 107.25 109.0

Final stress under PS & DL

Top of beam (ksi) (4)

Bottom of beam (ksi) (4)

0.140 0.333 0.061 -0.255 -0.521 -0.796 -0.943 -1.133 -1.270 -1.374 -1.436 -1.455 -1.454 -1.414 -1.331 -1.206 -1.046 -0.835 -0.670 -0.374 -0.116 0.250 0.458

-0.588 -1.816 -1.519 -1.283 -1.158 -0.861 -0.969 -0.767 -0.631 -0.525 -0.465 -0.453 -0.455 -0.508 -0.609 -0.757 -0.945 -1.189 -1.112 -1.450 -1.487 -1.910 -2.146

0.269

-0.918

Stress under 1/2 (DL + P/S)
0.070 0.136 -0.054 -0.285 -0.479 -0.666 -0.777 -0.900 -0.988 -1.050 -1.081 -1.085 -1.083 -1.049 -0.984 -0.889 -0.769 -0.617 -0.481 -0.280 -0.111 0.105 0.219 0.132

Top of beam (ksi) (4)

Bottom of beam (ksi) (5)

Top of slab
(ksi) (4)

0.140 0.303 -0.023 -0.412 -0.739 -1.064 -1.249 -1.467 -1.623 -1.737 -1.799 -1.812 -1.810 -1.756 -1.649 -1.492 -1.292 -1.034 -0.816 -0.467 -0.169 0.230 0.448

-0.588 -1.755 -1.349 -0.966 -0.719 -0.321 -0.353 -0.094 0.081 0.207 0.266 0.267 0.263 0.180 0.032 -0.181 -0.449 -0.787 -0.817 -1.263 -1.381 -1.870 -2.125

0.000 -0.041 -0.116 -0.215 -0.298 -0.365 -0.417 -0.454 -0.479 -0.490 -0.487 -0.475 -0.474 -0.448 -0.410 -0.359 -0.296 -0.223 -0.139 -0.053 0.031 0.092 0.121

0.266

-0.913

0.141

Notes: 1 - Distance measured from the centerline of the bearing of the end abutment 2 - See Section 5.3 for load effects 3 - See Section 5.5 for prestressing forces 4 - Service I limit state for compression 5 - Service III limit state for tension
Definitions:
Pt = Final prestressing force taken from Design Step 5.4 (kips) Stc = Section moduli, top of the beam of the composite section – gross
section (in3) Sbc = Section moduli, bottom of the beam of the composite section – gross
section (in3) Stsc = Section moduli, top of slab of the composite beam (in3)

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

MDNC MDC
MLLC

= Moment due to the girder, slab, haunch and interior diaphragm (k-ft) = Total composite dead load moment, includes parapets and future
wearing surface (k-ft) = Live load moment (k-ft)

All tension stresses and allowables use positive sign convention. All compression stresses and allowables use negative sign convention. All loads are factored according to Table 3.4.1-1 in the AASHTO LRFD Specifications for Service I and Service III limit states as applicable.

Design Step Sample Calculations at 11 ft. From the CL of Bearing (11 ft. – 9 in. From Girder End) 5.6.2.2
Girder top stress after losses under sum of all loads (Service I):

ftop = -Pt/Ag + Pte11’/St – MDNC/St – MDC/Stc – MLLC/Stc
= − 857 + 857(31.222) − 1,252(12) − 199(12) − 886(12)
1,085 20,588 20,588 67,672 67,672

= -0.790 + 1.300 – 0.730 – 0.035 – 0.157

= -0.412 ksi < Stress limit for compression under full load (-3.6 ksi) OK

ftop = -Pt/Ag + Pte11’/St – MDNC/St – MDC/Stc
= − 857 + 857(31.222) − 1,252(12) − 199(12)
1,085 20,588 20,588 67,672

= -0.790 + 1.300 – 0.730 – 0.035

= -0.255 ksi < Stress limit for compression under permanent load (-2.7 ksi) OK
Girder top stress under LL + ½(PS + DL) after losses:

ftop = -Pt/Ag + Pte11’/St – MDNC/St – MDC/Stc – MLL/Stc
= 1,−08855(72) + 85270(,5381.82(22)2) − 21,02,5582(81(22)) − 6179,697(122(2) ) − 86876,6(1722)

= -0.395 + 0.650 – 0.365 – 0.018 – 0.157

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

= -0.285 ksi < Stress limit for compression under LL + ½(DL + PS) load (-2.4 ksi) OK

Girder bottom stress under all loads (Service III): fbottom = -Pt/Ag – Pte11’/Sb + MDNC/Sb + MDC/Sbc + MLLC/Sbc

= − 857 − 857(31.222) + 1,252(12) + 199(12) + 0.8(886)(12)

1,085 20,157

20,157 26,855 26,855

= -0.790 – 1.327 + 0.745 + 0.089 + 0.317

= -0.966 ksi < Stress limit for compression under full load (-2.7 ksi) OK
Notice that the gross concrete composite section properties are typically used for the stress calculations due to all load components. However, some jurisdictions use the transformed section properties in calculating the stress due to live load. The transformed section properties are listed in Section 2. In this example, the gross section properties are used for this calculation.

fbottom = -Pt/Ag – Pte11’/Sb + MDNC/Sb + MDC/Sbc

= − 857 − 857(31.222) + 1,252(12) + 199(12)

1,085 20,157

20,157 26,855

= -0.790 – 1.327 + 0.745 + 0.089

= -1.283 ksi < Stress limit for compression under prestress and permanent loads (-2.7 ksi) OK
Sample Calculations at 54 ft. – 6 in. From the CL of Bearing (55 ft. – 3 in. From Girder End) – Midspan of Noncomposite Girder

Girder top stress after losses under sum of all loads (Service I):

ftop = -Pt/Ag + Pte54.5’/St – MDNC/St – MDC/Stc – MLLC/Stc

= −1,096.2 + 1,096.2(31.38) − 3,512(12) − 387(12) − 2,015(12)

1,085

20,588

20,588 67,672 67,672

= -1.010 + 1.671 – 2.047 – 0.069 – 0.357

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Design Step 5 – Design of Superstructure

Prestressed Concrete Bridge Design Example

= -1.812 ksi < Stress limit for compression under full load (-3.6 ksi) OK
Girder top stress after losses under prestress and permanent loads:
ftop = -Pt/Ag + Pte54.5’/St – MDNC/St – MDC/Stc

= −1,096.2 + 1,096.2(31.38) − 3,512(12) − 387(12)

1,085

20,588

20,588 67,672

= -1.010 + 1.671 – 2.047 – 0.069

= -1.455 ksi < Stress limit for compression under prestress and permanent loads (-2.7 ksi) OK
Girder top stress under LL + ½(PS + DL) after losses:

ftop = -Pt/Ag + Pte54.5’/St – MDNC/St – MDC/Stc – MLL/Stc
= −1,10,8059(62.2) + 1,02906,.528(381(2.3)8) − 23,05,5182(81(22)) − 6378,677(122(2) ) − 26,071,65(7122)

= -0.505 + 0.835 – 1.024 – 0.034 – 0.357

= -1.085 ksi < Stress limit for compression under LL + ½(DL + PS) load (-2.4 ksi) OK
Girder bottom stress (Service III):

fbottom

= -Pt/Ag – Pte54.5’/Sb + MDNC/Sb + MDC/Sbc + MLLC/Sbc

= −1,096.2 − 1,096.2(31.38) + 3,512(12) + 387(12) + 0.8(2,015)(12)

1,085

20,157

20,157 26,855

26,855

= -1.010 – 1.707 + 2.091 + 0.173 + 0.720

= 0.267 ksi < Stress limit for tension (0.465 ksi) OK
Notice that the stresses are calculated without including creep and shrinkage. Jurisdictions that do not include creep and shrinkage typically design the girders for a reduced tensile stress limit or for zero tension at final condition. Including creep and shrinkage would normally result in additional tensile stress at the bottom of the beam at the midspan section.