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Design computations according to ACI 318-14 [2] Step 1: Determine the factored bending moment, shear force, and torsional moment at the face of Bent 2 of the box-girder based on the load

PRESTRESSED CONCRETE GIRDER BRIDGE

Kevin S Benítez C

February 2019

Keywords: prestressed concrete; torsion; shear; box girder bridges

This report will present the design of a cast-in-place, post-tensioned concrete, multi-cell box girder bridge under combined torsion, shear and flexure, based on the geometry of an example from the California Department of Transportation [1] Figure 1 shows the elevation view of the bridge used for this example The total length of the three spans is 412’ (125.6 m): the first span is 126’ (38.4 m), the center span is 168’ (51.2 m), and the third span

is 118’ (36.0 m)

Figure 1––Elevation view of the bridge

Figure 2 presents the cross-section of the bridge The total width of the bridge is 58’-10’’ (17.93 m) The bridge carries three 12’ (3.7 m) traffic lanes, two 10’ (3.0 m) shoulders and two 1’-5’’ (0.45 m) concrete edge barriers

Figure 2––Cross-section of the bridge

The loads that act on the bridge include the self-weight, load of the asphalt concrete (A.C.) wearing surface with

a thickness of 3 in (75 mm), and the live load in accordance with AASHTO LRFD 2017 [3] HL-93 (design truck plus design lane load)

Bending Moment, shear and torsional moment envelopes for bridge example

Figure 5––Torsional moment diagrams for separate load cases Conversion: 1 kip-ft = 1.356 kNm, 1 ft =

Design computations for flexure according to AASHTO LRFD 2017 [3]

The following steps will be used for the design of the example bridge in flexure:

Step 1 Determine the tendon profile

The tendon profile should be chosen so that the points of maximum eccentricities will occur at the points of maximum moments The tendon profile should include the radius of the parabolas, points of inflexion along the cross section and distances from the C.G of the cross section to points of maximum eccentricities

Figure 6––Prestressing tendon profile along bridge length

Step 2 Determine the balancing forces and balancing bending moments

The balancing bending moments for each one of the sections of the cable path is determined

2

8 j

P

P f q

l

First parabola: (From 0 ft to 110 ft)

2.02 10110

These forces should be distributed over the whole width of the bridge, for this calculation the overhanging flanges are not taken into account and a mean value of the width of 50.5 ft, between the upper and lower flanges

is used, to count for the skewed exterior girders

5

2 1

5

2 2

5

2 3

5

2 4

5

2 5

ft kip q

ft kip q

ft kip q

ft kip q

Figure 7––Balancing moments from prestressing Conversion: 1 kip-ft = 1.356 kNm, 1 ft = 0.31 m

Step 3 Calculate the prestressing losses according with Section 5.9.3

Calculation of Friction Losses:

For the calculation of friction losses, the friction coefficient is calculated based on the following equation:

 Then the cumulative coefficient is calculated adding the values of angular coefficients obtained for each section

Table––1, presents the cumulative angular force coefficients for Two-End stressing

Table 1––Cumulative angular force coefficients for two-end stressing

LOCATION LEFT END

STRESSING

RIGHT END STRESSING

Losses due to Elastic shortening:

The elastic shortening is computed based on a value of Δf pES = 3 ksi As used by the California Department of Transportation [1]

It is then converted into a force coefficient:

3

0.015202.5

pES pES ps

Losses due to Anchor set:

For the losses due to Anchor set the following value is used:

0.097

pA FC

Time-dependant losses: Concrete Shrinkage and Creep

To calculate the losses due to concrete shrinkage, the strain due to shrinkage is calculated using the following equation:

The total time-dependant losses converted into a force coefficient are:

Step 4 Check stresses at Service Flexure (considering just DC and PT)

Service flexure will just consider the effects of the Dead Load (DC) and prestressing force (PT), that are the force acting as soon as the bridge is terminated, it not includes the effects produced by the wearing surface (DW) and the Live Load (LL)

The allowable stresses are the following:

Table 2––Allowable stresses at Service Flexure

Compression 0.60 f’ci = 2.1(ksi) 14.5(MPa) Tension No tension allowed

The stresses will be computed at different positions along the bridge length:

Table 3––Calculated stresses at Service Flexure

Length (ft)

Stresses

at top of section (ksi)

Length (ft)

Stresses at bottom of section (ksi)

The allowable stresses are the following:

Table 4––Allowable stresses at Service III (without losses)

Compression 0.60 f’c = 3.0(ksi) 30.7(MPa)

Stresses

at top of section (ksi)

Length (ft)

Stresses at bottom of section (ksi)

Table 6––Allowable stresses at Service III (with Losses)

Compression 0.60 f’c = 3.0(ksi) 30.7(MPa)

Stresses

at top of section (ksi)

Length (ft)

Stresses at bottom of section (ksi)

Design computations according to ACI 318-14 [2]

Step 1: Determine the factored bending moment, shear force, and torsional moment at the face of Bent 2 of the box-girder based on the load combinations of Article 5.3.1

56804.2 (77016 ) at the face of Bent 144485.1 (60314 ) at a distance of the face of Bent 1

3035 (13500.5 ) at a distance of the face of Bent 1

  5kNm) at a distance of the face of Bent 1d

Step 2: Determine the section properties and material properties

Table 8––Section properties (ACI318-14)

A A

62959.2 (85361.2 )

n

This area of reinforcement will correspond to 39 #8 (25 mm) bars

Step 5: Check if torsion can be neglected based on Article 22.7.4

Since 0.866ksi (5.97MPa)0.690ksi (4.76MPa) , the size of the cross-section is not adequate and the

dimensions should be increased By an iterative procedure, it is determined that the new b w = 86 in (2184 mm)

All vertical girders will be flared Exterior girders will now have a thickness t = 19 in (483 mm) and interior girders t = 16 in (407 mm)

44470 (60293.3 ) at a distance of the face of Bent 1

3035 (13500.5 ) at a distance of the face of Bent 1

Step 7: Calculate the required area of transverse reinforcement for shear based on Article 22.5.10.5

For shear, the required transverse reinforcement is #5 (16 mm) bars at 5.50 in (140 mm) on center

Step 8: Calculate the required area of transverse reinforcement for shear in the exterior webs based on Article 22.5.10

The factored shear force for the left exterior girder considering the additional self-weight of the larger section is

Step 9: Calculate the required area of transverse reinforcement for torsion based on Article 22.7.6

Step 10: Calculate the required area of transverse reinforcement for exterior girders considering the effects of shear and torsion

For the exterior (skewed) webs, the provided transverse reinforcement becomes two-legged #5 (16 mm) stirrups

at 3 in (75 mm) on center For the internal (vertical) webs, the required transverse reinforcement remains legged #5 (16 mm) stirrups at 5 in (125 mm) on center

two-Step 11: Calculate the transverse reinforcement in the flanges for torsion only based on Article 22.7.6.1

§9.7.6.2.2 states that the spacing for transverse reinforcement provided for shear shall not be larger than the

minimum between d = 64.8 in (1646 mm) or 12 in (305 mm) §9.7.6.3.3 limits the spacing for transverse torsional reinforcement to a value no larger than the minimum of p h = 1250 in (31750 mm) or 12 in (305 mm) For the interior webs, a spacing of 5.50 in (140 mm) was used, which fulfils this requirement For the exterior webs, a spacing of 3 in (75 mm) was used, which fulfils this requirement

For the flanges, a spacing of 6.25 in (160 mm) was used, which fulfils this requirement

Step 13: Calculate the required area of longitudinal reinforcement for torsion based on Article 22.7.6.1

Step 14: Check minimum requiered longitudinal and transverse reinforcement

Design computations according to EN 1992-1-1:2004 [4]

Step 1: Determine the factored bending moment, shear force, and torsional moment at the face of Bent 2 of the box-girder based on the load combinations

61102.3 (82844 ) at the face of Bent 147741.3 (64715 ) at a distance of the face of Bent 13162.2 (14066.2 ) at a distance of the face of Bent 1

  4kNm) at a distance of the face of Bent 1d

Step 2: Determine the section properties and material properties

Table 9––Section and material properties EN 1992-1-1:2004

f p,ud 235 ksi 1620.3 Mpa

f y,d 52.2 ksi 360 Mpa

f´ c,d 3.33 ksi 23 Mpa

f p0.1k 243 ksi 1676 Mpa

f p,d 211.3 ksi 1457 Mpa Step 3: Check the flexural resistance for the tendon layout shown in Fig 6, based on the factored bending moment produced at the face of Bent 2

This area of reinforcement will correspond to 51 #8 (25 mm) bars

Step 5: Verify if transverse reinforcement must be provided

P

ksi A

Since V u > V rdc transverse reinforcement for shear must be provided

Step 6: Compute the maximum and minimum shear reinforcement

,min

0.08 '

0.000951.2

Step 7: Compute the maximum shear resistance

The calculations will be developed for angles of 45°, 35° and 22°

For θ = 45°

,max

'3631.8 (16155.1 )cot( ) tan( )

Step 8: Compute the design torsional resistance moment

The calculations will be developed for angles of 45°, 35° and 22°

Step 9: Verify if the cross section is adequate

The calculations will be developed for angles of 45°, 35° and 22°

Therefore when assuming that θ = 35° and 22° the inequality is not fulfilled so the section should be enlarged

This process could be done by iteration, finally the sections are changed and the inequality is checked one more time

For θ = 35° and b w = 62in and t f = 13in

,max

'3526.5 (15687 )cot( ) tan( )

For θ = 22° and b w = 84in and t f = 18in

,max

'

3532 (15711.1 )cot( ) tan( )

Step 10 Compute the required shear reinforcement

The calculations will be developed for angles of 45°, 35° and 22°

Step 11.3 Check if cross-sections are adequate

The calculations will be developed for angles of 45°, 35° and 22°

For θ = 45°

,max

'726.4 (3231.2 )cot( ) tan( )

For θ = 35° and t ext = 14in

,max

'796.3 (3542.1 )cot( ) tan( )

Step 11.4 Calculate the required shear reinforcement on exterior webs

The calculations will be developed for angles of 45°, 35° and 22°

Step 12 Calculate the required torsion reinforcement

The calculations will be developed for angles of 45°, 35° and 22° The angle of inclination of exterior webs

should be considered, therefore α = 27°

T

Step 16 Compute the required spacing for stirrups on exterior and interior webs

As stirrups, #5 (16mm) bars with an area of 0.31in2 (198mm2)

Design computations according to AASHTO LRFD 2017 [3]

Step 1: Determine the factored bending moment, shear force, and torsional moment at the face of Bent 2 of the box-girder based on the load combinations

60865.2 (82522 ) at the face of Bent 147683.3 (64650 ) at a distance of the face of Bent 13268.3 (14538.2 ) at a distance of the face of Bent 1

  954kNm) at a distance of the face of Bent 1d

Step 2: Determine the section properties and material properties

Table 10––Section and material properties AASHTO LRFD 2017

Step 4: Compute the nominal resistance of the section based on the resistance provided by the prestressing steel

P j = 9300kip (41369kN)

1

6.85 (174 )0.85 '

Step 9 Check if longitudinal reinforcement can resist the required tension

Step 13 Determine the amount of transverse reinforcement to be used on the exterior webs

Design computations according to Model Code MC-2010 [5]

Step 1: Determine the factored bending moment, shear force, and torsional moment at the face of Bent 2 of the box-girder based on the load combinations

61102.3 (82844 ) at the face of Bent 147741.3 (64715 ) at a distance of the face of Bent 13162.2 (14066.2 ) at a distance of the face of Bent 1

  4kNm) at a distance of the face of Bent 1d

Step 2: Determine the section properties and material properties

Table 10––Section and material properties MC-2010

f y,d 52.2 ksi 360 Mpa

f´ c,d 3.33 ksi 23 Mpa

f p0.1k 243 ksi 1676 Mpa

f p,d 211.3 ksi 1457 Mpa Step 3: Check the flexural resistance for the tendon layout shown in Fig 6, based on the factored bending moment produced at the face of Bent 2

This area of reinforcement will correspond to 51 #8 (25 mm) bars

Step 5 Determine the net longitudinal strain in the cross section

P j = 9300kips (41368.5kN) e p = d p – 45.6in = 19.2in

63.9 (1623 )(0.9 ) (0.9 )

0.92

Step 6 Compute the maximum shear resistance and torsional resistance

For Level of Approximation 1

k e1 = 0.55

1

30

0.96'

Step 7 Check if the dimensions of the cross section are adequate

For θ = 25°, b w = 74in (1880mm), t f = 16in(406mm)

Level of Approximation III

Level of Approximation III

0.92

T

Step 13 Determine the amount of transverse reinforcement for torsion on flanges Level of Approximation I

Level of Approximation III

1cot( )

Level of Approximation III

Level of Approximation III

Final results comparison

Table 11––Results comparison of gross concrete area and required transverse reinforcement

Design Code

A g

disregarding overhanging flanges (ft2)

Required transverse reinforce-ment for exterior web (shear and torsion) (in2/in)

Required transverse reinforce- ment for interior web (shear) (in2/in)

Required transverse reinforce-ment for flange (torsion) (in2/in)

Required longitudinal reinforcement for flexure (in2)

Required longitudinal reinforcement for torsion (in2)

REFERENCES

[1] California Department of Transportation 2015 Bridge Design Practice, 7-33

[2] ACI Committee 318 2014 Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary, Farmington Hills, MI

[3] American Association of Highway and Transportation Officials 2017 AASHTO LRFD Bridge Design Specifications, 8th Edition, Washington, D.C

[4] European Committee for Standardization 2004 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, Brussels, Belgium

[5] The International Federation for Structural Concrete 2010 fib Model Code for Concrete Structures 2010, Lausanne, Switzerland