Difference between revisions of "751.22 Prestressed Concrete I Girders"

Revision as of 08:24, 1 June 2022

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751.22.1 General

EPG 751.22 illustrates the general design procedure for prestressed concrete I girders (Type 2, 3, 4 and 6), bulb-tee girders (Type 7 and 8) and NU girders (NU 35, 43, 53, 63, 70 and 78) using AASHTO LRFD Bridge Design Specifications except as noted.

751.22.1.1 Material Properties

Increasing Girder Capacity

The following allowable modification of material properties listed in order of increasing costs may be considered if required by design.

1. Increase concrete strength up to 8.0 ksi (f′_c = 6.5 ksi) (readily producible by fabricator)

2. Increase concrete strength greater than 8.0 ksi (readily producible by fabricator)

3. Modify geometric properties (in order of increasing costs)

a. Increase top flange height (and overall height accordingly)

b. Use modified Type 2, 3, 4 and 6 girders (most costly and inconvenient due to required forming bed modifications)

Class A-1 Concrete

Conventional concrete with the following compression strengths shall be used as required for I girders and bulb-tee girders.

f′_c = 6.0 ksi (f′_ci = 4.5 ksi) → f′_c = 7.0 ksi (f′_ci = 5.0 ksi) → f′_c = 8.0 ksi (f′_ci = 6.5 ksi)

The lowest concrete strength satisfying the demand shall be used.

The NU girders shall use conventional concrete with the following compression strength.

f′_c = 8.0 ksi (f′_ci = 6.5 ksi)

High strength concrete with compressive strengths up to f′_c =10 ksi (f′_ci = 7.0 ksi) may be used with the permission of the Structural Project Manager or Structural Liaison Engineer. High strength concrete may increase costs due to production modifications necessary to obtain the required strength.

Modulus of Elasticity

E_c = 33,000K₁w_c^1.5√f′_c (f′_c in ksi)

where:

K₁ = correction factor for source of aggregate = 1.0 unless determined by physical testing

w_c = 0.140 + 0.001f′_c (f′_c in ksi)

Prestressing Strands

Prestressing strands shall be Grade 270 uncoated, low relaxation, seven-wire strands in accordance with AASHTO M 203 with the following design properties.

Ultimate tensile strength, f_pu = 270 ksi		1/2-inch strand: diameter, d_ps = 0.5 in. and area, A_ps = 0.153 in.²
Yield strength, f_py = 0.9f_pu = 243 ksi		0.6-inch strand: diameter, d_ps = 0.6 in. and area, A_ps = 0.217 in.²
Maximum allowed force per strand, f_pbt A_ps = 30.98 kips (1/2-inch strands) f_pbt A_ps = 43.94 kips (0.6-inch strands)		Modulus of elasticity, E_p = 28,500 ksi Maximum allowed stress prior to transfer, f_pbt = 0.75f_pu = 202.5 ksi

Total initial prestress force equals the number of strands multiplied by the required initial prestress force per strand.

Typically, the required initial prestress force per strand is the maximum allowed force per strand.

Report on the plans the required number of strands by design and the total initial prestress force using EPG 751.50 Standard Detailing Notes H2c1.3.

Reinforcing Steel

Deformed bars shall be Grade 60 in accordance with AASHTO M 31 with the following design properties.

Yield strength, f_y = 60.0 ksi

Modulus of elasticity, E_s = 29,000 ksi

Welded wire reinforcement shall be in accordance with AASHTO M 336 with the following design properties.

Yield strength, f_y = 70.0 ksi

Modulus of elasticity, E_s = 29,000 ksi

All bars extending into slab shall be epoxy coated. Welded wired reinforcement shall not be epoxy coated.

751.22.1.2 Geometric Properties

The following sections, except those specified as modified, shall be preferred because of their familiarity to local precast plants. These sections have been entered into the beam section libraries of in-house design software. The top flange height and overall height may be increased if required but any deviation from the standard sections shown shall be discussed with Structural Project Manager or Structural Liaison Engineer. The use of the modified girders shall be discussed with Structural Project Manager or Structural Liaison Engineer.

I Girders:

Bulb-Tee Girders:

NU Girders:

751.22.1.3 Typical Span Ranges

The following charts provide span ranges (limits) for P/S I-girders based on girder spacing and standard roadway widths.

Limitations of the Charts:

A. Standard Concrete Charts Only

Criteria used in determining maximum span lengths for lower conventional concrete strength:

1) Low-relaxation strand with 0.5” strand diameter

2) Concrete strengths,

\,f'_{ci}

= 4.5 ksi and

\,f'_{c}

= 6.0 ksi

3) 3-span bridge consisting of 3 equal length girders made continuous and composite

B. Optional Concrete Charts Only

Criteria used in determining maximum span lengths for greater conventional concrete strength:

1) Low-relaxation strand with 0.6” strand diameter

2) Concrete strengths,

\,f'_{ci}

= 5.0 ksi and

\,f'_{c}

= 7.0 ksi

3) 3-span bridge consisting of 3 equal length girders made continuous and composite

C. Both Standard Concrete and Optional Concrete Charts

Criteria used in determining span ranges for both Standard and Optional Concrete conventional strengths.

1) Minimum span lengths were determined by the positive moment capacity of the smallest strand arrangement per beam shape. Shorter span lengths are possible.

2) Based on 10 ft. design lanes. (Current design practice meets AASHTO LRFD and uses 12 ft. design lanes.)

3) Based on unrefined prestress loss equations. (Current design practice meets AASHTO LRFD and uses refined losses.)

Recommended Adjustments for Using the Charts:

Because the span limit charts were developed using older design criteria as noted above, increased span lengths are probable.

1) Span limits given in all charts should be increased 10 percent to account for current design practice. Ten percent can safely be used without a preliminary girder analysis.

2) Span limits given in all charts shall be increased when a preliminary girder analysis based on actual design conditions is performed which shall be noted on the Design Layout.

Span range charts are planned for future replacement. Use the recommended adjustments until implemented.

Standard Concrete ( $\,f'_{c}$ = 6 ksi) P/S I Beam Span Ranges for
Given Roadway Widths and Girder Spacing

751.22 standard conc PSI span ranges.gif

Optional Concrete ( $\,f'_{c}$ = 7 ksi) P/S I Beam Span Ranges for
Given Roadway Widths and Girder Spacing

751.22 optional conc PSI span ranges.gif

751.22.1.4 Span and Structure Lengths

751.22.1.4.1 Limits

Span Lengths

Designs using MoDOT standard Type 6 girders shall be limited to 105 feet maximum length to ensure stability during fabrication, shipping and erection.

No limits are set for other types of prestressed girders however the Structural Project Manager or Structural Liaison Engineer shall be consulted prior to the design of any unusually long prestressed girder.

Continuous Structure Lengths

751.22.1.4.2 Girder Length and Geometric Layout

Tangent Bridges

Girder lengths of exterior spans (i.e., end spans) and interior spans shall be computed using the requirements shown below.

The layout length for single span shall be measured from centerline of bearing to centerline of bearing. If the difference between layout length of the end span and interior span is within one foot, then layout length should be adjusted if possible so the girder lengths are equal for end span and interior span.

(1) Minimum dimension from edge of bearing pad to end of girder equals one inch.

(2) Design layout lengths are horizontal lengths. Girder lengths should be adjusted according to grade and shall be specified to the nearest 1/8 inch.

(3) For large skews, end bent beam caps may need to be larger to provide edge distance.

(4) Horizontal distance along certerline of girder.

(5) = 1ʺ (minimum) + ½ bearing pad length which equals:

5ʺ (minimum) for I-girders and squared-end adjacent beams,

3 ½ʺ (minimum) for NU girders and spread beams with squared ends,

3⅝ʺ (minimum) for skewed-end spread beams, 3½ʺSEC(15°).

Curved Bridges

Layout of any curved structure may be done using any coordinate geometry programs available. To layout the bridge, use the following steps:

Start out by laying in the centerline (CL) of the survey curve.
Locate the tie point of the bridge. This point will usually be on the CL of the survey curve but may be on a baseline which is offset a certain distance to the CL of the survey curve.
A second tie point may be required if the skew is not measured to the CL of roadway at the bridge tie point. If this is the case, establish the tie point at the specified station and plot the skew line at the required angle.
Next, on the centerline of structure or baseline curve, locate the station of the CL of bent for each intermediate bent and the fill face for the end bents. Once these points are located, plot lines through these stations parallel to skew line. Normally the layout file will specify that all bents are parallel to the skew line; however, there may be times when the bents are radial or have varying skews.
When locating the stations in the preceding step, the distance between CL of intermediate bents are exactly the layout lengths specified on the file. However, the end spans need to follow the procedure for calculating length set forth in Tangent Bridges.
When the CL of the intermediate bents and the fill face lines have been added, chords should be drawn connecting these points sequentially. For example, if you have a three-span bridge, chords should be drawn from the fill face of bent 1 to CL of bent 2, CL bent 2 to CL bent 3, and CL bent 3 to fill face bent 4.
When all the chords are in, offset each girder in each span parallel to this chord. The perpendicular distance between girders will be the same for all spans, but the skew distance between girders along the bent will vary from bent to bent depending on the skew to the CL at that point. The designer needs to be aware of the fact that at an intermediate bent the distance between bearings on the approaching and leaving span sides will be different distances. These bearings will not line up across the bent and will actually diverge more the farther away they are from the CL of the survey.
When establishing the CL of bearing points, the designer needs to allow for a minimum of seven (7) inches between ends of girders at the bents while keeping in mind that the girders will be offset and at different skews. If the offset is greater than half the girder bottom flange width, see Structural Project Manager. The distance from the end of girder to CL of bearing point should be half of the bearing length plus one inch minimum clearance. Once the distance for CL bent to CL of bearing is calculated, the designer should offset lines by that dimension on either side of the CL of bent. These lines will then be intersected with each of the girder lines to create the bearing points on each bent.
Between the bearing points at the ends of the girders, quarter points or tenth points need to be established, depending on the girder span. These points will be used in calculating the haunch and bottom of slab elevations for the bridge deck.
The bridge deck and barrier or railing can be laid in by offsetting the centerline of roadway to each side by the proper distance. Curves should be laid in to designate both the inside and outside edges of the barrier or railing. These will later be useful in laying in the wings and end bents.
After the outside edge of slab curves are plotted, the curve offsets need to be found. The intersection points of the outside edge of slab and the centerline of each bent or fill face can be connected with chords. The distance between these chords and their partner curves need to be calculated at five-foot intervals beginning at the center point of each chord.
Joints are placed in the barrier or railing at each bent. These joints are placed perpendicular to the centerline of the roadway through the intersection point of the centerline bent and the inside edge of barrier or railing.
Wing layout length is given on the profile sheets in the layout file. An arc should be struck so as to intersect the inside edge of barrier or railing the specified length from a point at the intersection of the fill face and the inside edge of barrier or railing. This point will mark the end of the wing which is perpendicular to the centerline of the roadway.

The vertical curve information needs to be added so a program can calculate the elevations at the desired stations. After this is done, the designer can request any of the following information which will be needed:

Stations and elevations of all points
Offset distances to the chords
Lengths of girders
Distances between bearings
Angles between girders and each bent
Lengths of bents
Lengths of barrier or railing between joints
Minimum vertical clearance.

751.22.1.4.3 Coping of Girder Ends

Non-Integral end bents with skews greater than 40 degrees shall always have girder ends coped. Skews less than 40 degrees shall have girder ends coped on case by case basis. It is preferable to not cope across the web.

Check clearance from fill face of integral end bents to bottom flanges of NU girders. Maintain 3-inch minimum clearance. Coping may be permitted with approval of the Structural Project Manager or Structural Liaison Engineer.

PART PLAN SHOWING COPING DETAIL (I Girder shown; Bulb-Tee and NU Girder similar)

751.22.1.5 Constant and Varied Joint Filler Loads

Varied joint filler load

Girders shall be first designed assuming that the contractor will vary the joint filler supporting the panels on the girder flange. This assumption will maintain the minimum slab/panel combination thickness of 8 1/2 inches, and will eliminate the possibility of increased load due to varying slab thickness.

Constant joint filler load

With the girder designed and the camber and haunching dimensions calculated, the girder should be checked assuming the contractor will use a constant 1” joint filler. This will cause the slab thickness to vary due to camber of the girder, increasing load. This additional load shall be placed as a concentrated load at 1/8 point from each end of the girder.

An example of how this concentrated load could be calculated is shown as follows:

Load

\,w=(A)(0.15kips/ft.^{3})

Determine the concentrated load* to girders by distributing w transversely across the girders. If the minimum haunch is greater than 1” joint filler, the additional haunch shall be included in the slab thickness as a uniform load. If the use of these loads causes the girder design to change, it shall be the responsibility of the designer to determine if the camber and haunching should be recalculated.

$*$ This load shall be positioned at the 1/8 point from centerline of bearing pad.

The girder and bearing designs should be checked for the constant joint filler option and constant joint filler load. However, camber, haunching and beam seat elevations shown on the plans should be based on the variable joint filler option.

JOINT FILLER LOADS

751.22.2 Design

751.22.2.1 Load Combinations

In general, each component shall satisfy the following equation:

$\,Q=\textstyle \sum$ $\,\eta _{i}\gamma _{i}Q_{i}\leq \phi R_{n}=R_{r}$

Where:

$\,Q$	= Total factored force effect
$\,Q_{i}$	= Force effect
$\,\eta _{i}$	= Load modifier
$\,\gamma _{i}$	= Load factor
$\,\phi$	= Resistance factor
$\,R_{n}$	= Nominal resistance
$\,R_{r}$	= Factored resistance

Limit States

The following limit states shall be considered for P/S Girder design:

SERVICE I - for compressive stress

SERVICE III - for tensile stress

STRENGTH I

See LRFD Table 3.4.1-1 for Loads and Load Factors applied at each given limit state.

Resistance factors, $\,\phi$

STRENGTH limit states, see LRFD Article 6.5.4.2 & 5.5.4.2
For all other limit states, $\,\phi$ = 1.00

See EPG 751.2.3.1 Load Modifiers.

751.22.2.2 Prestressing Strands

Transfer Length of Prestressing Strands

The prestressing force may be assumed to vary linearly from zero at the point where bonding commences to a maximum at the transfer length. The transfer length may be taken as 60 times the strand diameter.

Development Length of Prestressing Strands

The development length for prestressing strands shall be taken as:

$\,l_{d}\geq 1.6{\Bigg (}f_{ps}-{\frac {2}{3}}f_{pe}{\Bigg )}d_{ps}$

Where: $\,d_{ps}$ = Nominal diameter of strand, (in.) $\,f_{ps}$ = Average stress in prestressing strand at the time for which the nominal resistance of the girder is required, (ksi)

Stress limits for prestressing strands

Strand stress at service limit state shall not exceed the following:

At jacking:

\,f_{pj}\leq 0.75f_{pu}

ksi

(For typical girders and fabrication economy,

\,f_{pj}=0.75f_{pu}

)

At service limit state after all losses:

\,f_{pe}\leq 0.80f_{py}

ksi

Where:

$\,f_{pj}$	= Stress in prestressing strand at jacking, (ksi)
$\,f_{pe}$	= Effective stress of strand after all losses, (ksi)
$\,f_{py}$	= Yield strength of strand, (ksi)
$\,f_{pu}$	= Ultimate tensile strength of strand, (ksi)

Prestress Losses

Refined estimates of time-dependent losses are used, based on AASHTO LRFD Article 5.9.3.4, as opposed to approximate lump sum estimate of losses in AASHTO LRFD Article 5.9.3.3.

The prestress losses shall be calculated to investigate concrete stresses at two different stages.

Temporary stresses immediately after transfer:
Final stresses

SERVICE I and SERVICE III Limit states shall be investigated at each stage.

Harped Strands

Harped strands, although they add to the shear strength of the girder, are primarily used to keep the girder stresses (both top and bottom) within allowable limits while developing the full capacity of the girder at midspan.

Harped strands should be held down at points of 0.4 of the distance from each end of the girder. Distances along girder to hold-down devices and between hold-down devices should be reported on the plans to the nearest inch. Per Sec 1029, precaster may position hold-down devices +/- 6 in. longitudinally from position shown on the plans.

Example Harped Strand Layout

The jacking force applied to prestress strands produces an excessive vertical uplift in short spans on tall girders resulting in failure of harped strand hold-downs. The allowable limits for hold-downs are as follows:

5 kip/strand
10 kip/bolt
42 kip/hold-down

Hold-Down Device

If necessary lower harped strand end location to meet criteria or use straight strands only. Investigate the possibility of using all straight strands when strength check of a hold-down device exceeds allowable.

Straight Strands.

Short spans (<40 ft.) are to use straight strands only for all girders greater than 2'-8" tall. Use at least two straight strands at the top of the girder when straight strands are used. Where straight strands only will not work a single hold-down point may be used. Note: A single point hold-down has twice the uplift force.

Strand Arrangement Optimizing

Using all straight strands for girder lengths less than 70 feet shall be investigated for Type 6, 7 and 8 girders and all NU girders in order to reduce risk of strand or hold-down breakage, increase safety by reducing risk of injury during fabrication and reduce cost.

Consider using the same section for all spans. This permits the use of shorter girders in the casting bed with longer girders, even if straight strands are needed, in the top flanges of the girders. They can be placed at either end of the bed and still optimize the usage of the bed.

Consider using the same number of draped strands for all spans and debond where needed. Strand patterns should be similar between long and short spans. For example, the designer should not use a single column of draped strands on the short spans and two columns of draped strands on the long spans. This will prevent optimization of the bed.

When using straight strands in the top flange of NU Girders and harped strands, lower (drop) the harped strand end locations and vertically align straight strands directly over harped strands to facilitate top flange blockout fabrication by removing interference created between straight strands placed to the outside of the harped strands and the flange blockout forms. If for any reason this is not possible, then place straight strands to the outside of the harped strands.

Debonding Strands

Debonding at girder ends may be used to reduce concrete compressive forces and shall be used if required to reduce the prestress force at transfer to meet bursting/splitting requirements. Debonding strands shall be avoided at girder ends located underneath expansion joints.

In all debonding operations the prestressing forces must be in such a manner as to prevent any sudden or shock loading.

Debonding a strand consists of wrapping the unnecessary strand(s) with a polyethylene plastic sleeve that prevents interaction of the strand with the concrete during casting and release which prevents any prestress force transfer.

751.22.2.3 Flexure

Flexure capacity of girders shall be determined as the following.

Flexural resistance at strength limit state

$\,M_{r}=\phi M_{n}\geq M_{u}$

Where:

$\,M_{r}$	=	Flexural resistance
$\,M_{n}$	=	Nominal flexural resistance
$\,M_{u}$	=	Total factored moment from Strength I load combination
$\,\phi$	=	Flexural resistance factor as calculated in LRFD 5.5.4.2

Negative moment reinforcement design

P/S I-girder shall be designed as a reinforced concrete section at regions of negative flexures (i.e., negative moments).

At least one-third of the total tensile reinforcement provided for negative moment at the support shall have an embedment length beyond the point of inflection not less than the specified development length of the bars used.

Slab longitudinal reinforcement that contributes to making the precast beam continuous over an intermediate bent shall be anchored in regions of the slab that can be shown to be crack-free at strength limit states. This reinforcement anchorage shall be staggered. Regular longitudinal slab reinforcement may be utilized as part of the total longitudinal reinforcement required.

Effective Slab Thickness

An effective slab thickness shall be used for design by deducting from the actual slab thickness a 1” integral, sacrificial wearing surface.

Design A1 reinforcement in the top flange

The A1 reinforcement shall resist the tensile force in a cracked section computed on the basis of an uncracked section.

For I girders and bulb-tee girders, A1 reinforcement shall consist of deformed bars (minimum #5 for Type 2, 3 and 4 and minimum #6 for Type 6, 7 and 8).

For NU girders, A1 reinforcement shall consist of the four 3/8-inch diameter reinforcement support strands with deformed bars added only as needed. The WWR in the top flange shall not be used for A1 reinforcement because there is insufficient clearance to splice the WWR.

Reinforcement shall be designed and spliced using f’_ci in accordance with EPG 751.5.9.2.8 Development and Lap Splices.

Required steel area is equal to:

$\,A1={\frac {T_{t}}{f_{s}}}$

Where:

$\,f_{s}$	= $\,0.5f_{y}\leq 30KSI$ , allowable tensile stress of mild steel, (ksi)
$T_{t}$	= Resultant of total tensile force computed on the basis of an uncracked section, (kips)

Limits for reinforcement

The following criteria shall be considered only at composite stage.

Minimum amount of prestressed and non-prestressed tensile reinforcement shall be so that the factored flexural resistance, M_r, is at least equal to the lesser of:

1) M_cr LRFD Eq. 5.6.3.3-1

2) 1.33M_u

Where:

M_cr	=	Cracking moment, (kip-in.)
M_u	=	Total factored moment from Strength I load combination, (kip-in.)

751.22.2.4 Shear

Shear capacity of girders shall be checked along girder length and girder-slab interface.

Shear resistance at strength limit state

$\,V_{r}=\phi V_{n}\geq V_{u}$

Where:

$\,V_{r}$	=	Shear resistance
$\,V_{n}$	=	Nominal shear resistance
$\,V_{u}$	=	Total factored shear from Strength I load combination
$\,\phi$	=	Shear resistance factor

Nominal shear resistance

The nominal shear resistance, $\,V_{n}$ , shall be lesser of:

$\,V_{c}+V_{s}+V_{p}$ , or
$\,0.25f'_{c}b_{v}d_{v}+V_{p}$

Where:

$\,V_{c}=0.0316\beta b_{v}d_{v}{\sqrt {f'_{c}}}$

$\,V_{s}={\frac {A_{v}f_{y}d_{v}(cot\theta +cot\alpha )sin\alpha }{s}}$

Where:

$\,V_{c}$	=	Nominal concrete shear resistance, (kips)
$\,V_{s}$	=	Nominal shear reinforcement resistance, (kips)
$\,V_{p}$	=	Component of prestressing force in the direction of shear force, (kips)
$\,b_{v}$	=	Thickness of web, (in.)
$\,d_{v}$	=	Effective shear depth taken as the distance measured perpendicular to the neutral axis, between the resultants of tensile and compressive forces due to flexure, (in.)
$\,s$	=	Spacing of shear reinforcement, (in.)
$\,\beta$	=	Factor indicating ability of diagonally cracked concrete to transmit tension
$\,\theta$	=	Angle of inclination of diagonal compressive stress, (degree)
$\,\alpha$	=	90.0, Angle of inclination of shear reinforcement to a longitudinal axis, (degree)
$\,A_{v}$	=	Area of shear reinforcement, (in.²)
$\,f_{y}$	=	Minimum yield strength of tension shear reinforcement, (ksi)

Design sections near supports

Where a reaction force in the direction of the applied shear introduces compression into the end region of girder, the location of the critical section for shear is measured from the internal face of support a distance, d_v. Otherwise, the design section shall be taken at the internal face of the support.

Where:

$\,d_{v}$ = effective shear depth taken as the distance, measured perpendicular to the neutral axis, between the resultants of the tensile and compressive forces due to flexure; it need not be taken to be less than the greater of 0.9d_e and 0.72h.

Girder regions requiring shear reinforcement

Girder shear reinforcement, usually consisting of stirrups, shall be provided where:

$\,V_{u}>0.50\phi (V_{c}+V_{p})$

Where:

$\,V_{u}$	=	Factored shear force from Strength I load combination, (kips)
$\,V_{c}$	=	Nominal concrete shear resistance, (kips)
$\,V_{p}$	=	Component of prestressing force in the direction of shear force, (kips)
$\,\phi$	= =	Shear resistance factor 0.9 for normal weight concrete

Shear Reinforcement Limits

Minimum reinforcement

Area of shear reinforcement shall not be less than:

$\,A_{v}\geq 0.0316{\Bigg (}{\frac {b_{v}s}{f_{y}}}{\Bigg )}{\sqrt {f'_{c}}}$

Where:

$\,A_{v}$	=	Area of shear reinforcement, (in.²)
$\,b_{v}$	=	Thickness of web, (in.)
$\,s$	=	Spacing of shear reinforcement, (in.)
$\,f'_{c}$	=	Final concrete compressive strength, (ksi)

Maximum spacing

Maximum spacing of shear reinforcement shall be determined as:
If $\,v_{u}<0.125f'_{c}$ , then $\,s_{max}=0.8d_{v}\leq 24.0^{\prime \prime }$

If $\,v_{u}\geq 0.125f'_{c}$ , then $\,s_{max}=0.4d_{v}\leq 12.0^{\prime \prime }$

Where:

$\,d_{v}$	=	Effective shear depth taken as the distance measured perpendicular to the neutral axis, between the resultants of tensile and compressive forces due to flexure, (in.)
$\,v_{u}$	=	Shear stress on concrete, (ksi)
$s_{max}$	=	Maximum spacing of shear reinforcement, (in.)

Shear stress on concrete shall be determined as:

$\,v_{u}={\frac {V_{u}-\phi V_{p}}{\phi b_{v}d_{v}}}$

$\,d_{v}={\Bigg (}d_{e}-{\frac {a}{2}}{\Bigg )}\geq largerof{\begin{cases}0.9d_{e}\\0.72h\end{cases}}$

Where:

$\,v_{u}$	=	Shear stress on concrete, (ksi)
$\,V_{u}$	=	Factored shear from Strength I load combination, (kips)
$\,\phi$	= =	Shear resistance factor 0.9 for normal weight concrete
$\,b_{v}$	=	Thickness of web, (in.)
$\,V_{p}$	=	Component of prestressing force in the direction of shear force, (kips)
$\,d_{v}$	=	Effective shear depth taken as the distance measured perpendicular to the neutral axis, between the resultants of tensile and compressive forces due to flexure, (in.)
	=	$\,{\frac {M_{n}}{A_{s}f_{y}+A_{ps}f_{ps}}}$
$\,d_{e}$	=	Distance from extreme compression fiber to the centroid of tensile force in the tensile reinforcement, (in.)
$\,h$	=	Total height of girder including slab thickness, (in.)

Girder-Slab Interface

The horizontal shear between the girder and slab shall be determined as specified in LRFD 5.7.4.4. The nominal horizontal shear resistance of the interface plane shall be taken as specified in LRFD 5.7.4.3. Minimum interface shear reinforcement shall be provided as specified in LRFD 5.7.4.2. The parameters used in determining the nominal horizontal shear resistance shall be taken as specified for a “cast-in-place concrete slab on clean concrete girder surfaces, free of laitance with surface roughened to an amplitude of 0.25 inch.”

The interface shear shall be resisted by extending and anchoring the vertical shear reinforcement into the slab. If the resistance provided by extending the vertical shear reinforcement is inadequate then, in lieu of increasing shear reinforcement, additional U bars may be provided as shown for a Type 7 girder in EPG 751.22.3.4 Girder Reinforcement.

For NU girders and spread beams the top flange shall be debonded at the edges using a smooth finish and an applied bond breaker to help aid with future deck removal and minimize stress concerns with the thin flange of the NU girders. The debonded regions shall not be included when determining the nominal horizontal shear resistance. The minimum debonded width shown below may be increased in lieu of adding additional U bars in order to reduce the minimum interface shear reinforcement.

The debonding regions shall be indicated on the plans by specifying the required smooth finish and applied bond breaker in the dimensions detail on the beam or girder sheet. Omit underlined portion of footnote (1) if prestressed panels are not used.

Similarly, for all other prestressed girders and beams, the joint filler width supporting precast panels shall be considered debonded and excluded when determining the interface resistance.

751.22.2.5 Pretensioned Anchorage Zones

(LRFD 5.9.4.4)

Bursting Resistance (AASHTO Splitting Resistance)

The bursting resistance of anchorage zones provided by vertical reinforcement (i.e., B2 bars and the D31 wires of WWR6) in the ends of prestressed girders at the service limit state shall be taken as:

\,P_{r}=f_{s}A_{s}\geq 0.04f_{pbt}

The required vertical reinforcement shall be provided within the following end regions:

I Girders, NU 35, 43 & 53 Girders:

- Within h/3, in accordance with research by Davis, Buckner and Ozyildirimon (Dunkman et al. 2009).

Bulb-Tee Girders and NU 63, 70 & 78 Girders:

- Within h/4

Where:

$\,f_{s}$	=	Stress in mild steel not exceeding 20 ksi
$\,A_{s}$	=	Total area of vertical reinforcement located within the specified minimum distance from the end of the girder where h equals the overall depth of precast member as shown below.
$\,f_{pbt}$	=	Prestressing force immediately prior to transfer

Anchorage Zone (Bursting and Confinement) for I Girders
(Bursting zone or reinforcement may differ for other girder types)

The number of strands bonded in the anchorage zone is limited by the standard bursting reinforcement of EPG 751.22.3.4.3 Anchorage Zone Reinforcement and the 20 ksi resistance stress limit.

Maximum Number of Bonded Strands at Girder Ends, N_s


I Girders & Bulb-Tee Girders^a						NU Girders
Type	h/3 or h/4 (in.)	#6-B2 Pairs	A_s (in.²)	N_s		Type	h/3 or h/4 (in.)	D31 Pairs	A_s (in.²)	N_s
Type	h/3 or h/4 (in.)	#6-B2 Pairs	A_s (in.²)	(0.5")	(0.6")	Type	h/3 or h/4 (in.)	D31 Pairs	A_s (in.²)	(0.5")	(0.6")
Type 2	10.67	3.0	2.65	20	20	NU 35	11.81	5.5	3.41	54	38
Type 3	13.00	4.0	3.53	24	24	NU 43	14.44	7.0	4.34	62^a	48
Type 4	15.00	4.0	3.53	32	32	NU 53	17.72	8.5	5.27	62^a	52^b
Type 6	18.00	4.0	3.53	38	38	NU 63	15.75	7.5	4.65	62^a	52
Type 7	18.13	4.0	3.53	40	40	NU 70	17.72	8.5	5.27	62^a	58
Type 8	15.88	4.0	3.53	40	40	NU 78	19.69	9.0	5.58	62^a	62^a
Area of one #6-B2 bar = 0.4418 in.²						Area of one D31 wire = 0.31 in.²
^a Maximum number capped by strand arrangements shown in this article and on the Bridge Standard Drawings.
^b Maximum number capped not to exceed the maximum allowed for the NU 63 girder.

N_s ≤ A_s /A_s(req) (Rounded down to nearest even number for the purpose of symmetry.)

Where:

A_s(req) = Required bursting reinforcement for one prestressing strand.

= 0.04f_pbtA_ps /f_s

= (0.04)(202.5 ksi)(0.153 in.²)/(20 ksi) = 0.0620 in.² (1/2-inch strand)

= (0.04)(202.5 ksi)(0.217 in.²)/(20 ksi) = 0.0879 in.² (0.6-inch strand)

Confinement reinforcement

Confinement reinforcement (i.e., D1 bars or G1 bars spaced with WWR4) shown in the figure above shall be placed to confine the prestressing strands in the bottom flange for a minimum distance of 1.5d from the end of beam.

The reinforcement shall not be less than #3 deformed bar, with spacing not exceeding 6.0 inches and shaped to enclose the strands.

The use of D1 bars shall be extended for the full length of I girders, bulb-tee girders and alternate bar-reinforced NU girders.

The use of G1 bars at 6-inch spacing shall be extended for the first 10 feet and then at 12-inch spacing for the remaining length of welded wire-reinforced NU girders.

751.22.2.6 Deformations

Criteria for deflection

For investigating maximum absolute deflection, all design lanes shall be loaded, and all supporting components should be assumed to deflect equally.

For composite design, the design cross-section should include the entire width of the roadway and the structurally continuous portions of railings, sidewalks, and median barriers. Note that barrier and railing are usually discontinuous over the bents. For skewed bridges, a right cross-section may be used.

Service I load combination shall be used. Dynamic load allowance shall be applied.

See EPG 751.2.4.2 Live Load Deflection Limits.

Calculation of deflection and camber

Deflection and camber calculations shall consider all internal loads (i.e., prestressing, concrete creep, and shrinkage) and external loads such as dead loads and live loads.

Camber is an upward displacement caused by moment due to prestressing forces. Deflection is a downward displacement due to external loads. Therefore, both camber and deflection shall be considered in making an appropriate adjustment for final profile grade on the bridge.

Initial camber at transfer at midspan

Total initial camber at transfer due to self-weight of girder and prestressing forces shall be determined as:

$\,\Delta _{IC}=\Delta _{g}+\Delta _{SS}+\Delta _{HS}$

Where:

$\,\Delta _{IC}$	= Initial camber at transfer
$\,\Delta _{g}$	= Deflection due to self-weight of girder
$\,\Delta _{SS}$	= Camber due to prestressing straight strands
$\,\Delta _{HS}$	= Camber due to prestressing harped strands

Note: Positive and negative values indicate downward and upward displacements, respectively.

Camber at midspan after strand release (Estimated at 7 days)

Theoretical camber of girder after strand release due to self-weight of girder and prestressing forces shall be determined at 7 days as:

$\,\Delta _{7}=\Delta _{IC}+\Delta _{CR\ at\ 7\ days}$

Where:

$\,\Delta _{7}$	= Camber at 7 days after strand release with creep
$\,\Delta _{CR\ at\ 7\ days}$	= Time - dependent camber due to creep at 7 days

Note: Camber is calculated 7 days after strand release to allow sufficient time for inspection. See EPG 1029 Fabricating Prestressed Concrete Members for Bridges.

Camber at midspan after erection (Estimated at 90 days)

Theoretical camber of girder after erection due to self-weight of girder and prestressing forces shall be determined at 90 days as:

$\,\Delta _{90}=\Delta _{IC}+\Delta _{CR\ at\ 90\ days}$

Where:

$\,\Delta _{90}$	= Camber at 90 days after strand release with creep
$\,\Delta _{CR\ at\ 90\ days}$	= Time - dependent camber due to creep at 90 days

Final camber at midspan after slab is poured

Total deformation after slab is poured can be determined as the sum of theoretical camber of girder after erection (90 days) and deflections due to slab and concentrated loads (haunch, diaphragms, etc.) before composite action between slab and girder.

$\,\Delta _{FC}=\Delta _{90}+\Delta _{S}+\sum \Delta _{C}$

Where:

$\,\Delta _{FC}$	=	Final camber after slab is poured
$\,\Delta _{s}$	=	Deflection due to weight of slab
$\,\sum \Delta _{c}$	=	Deflection due to concentrated loads (haunch, diaphragms, etc.)

Final camber along span length

Deformations along the span length can be approximately determined as a product of final camber at midspan times correction factors.

$\,\Delta _{0.10}$	=	0.3140 $\,\Delta _{FC}$ at span fraction of 0.10
$\,\Delta _{0.20}$	=	0.5930 $\,\Delta _{FC}$ at span fraction of 0.20
$\,\Delta _{0.25}$	=	0.7125 $\,\Delta _{FC}$ at span fraction of 0.25
$\,\Delta _{0.30}$	=	0.8130 $\,\Delta _{FC}$ at span fraction of 0.30
$\,\Delta _{0.40}$	=	0.9520 $\,\Delta _{FC}$ at span fraction of 0.40
$\,\Delta _{0.50}$	=	1.0000 $\,\Delta _{FC}$ at span fraction of 0.50

Calculation of camber (upward) using transformed properties

Camber at midspan due to strand forces is determined by the following:

For straight strands (groups determined by debonding lengths),

$\,\Delta _{SS}=\Delta _{ss-j}+\Delta _{ss-l}$

Where: $\,\Delta _{ss-j}=\sum {\frac {F_{1-j}e_{1}}{8E_{ci}I_{tri}}}(L^{2}-4l_{0}^{2})$

\,\Delta _{ss-l}=\Delta _{ss-j}{\frac {Initial\ Loss}{f_{Pj}}}

Where:

$\,F_{1-j}$	= Total prestressing force of straight strand group just prior to transfer, (kips)
$\,L$	= Distance between centerlines of bearing pads, (in.)
$\,l_{0}$	= Debond length of straight strand group from end of girder, (in.)
$\,E_{ci}$	= Initial concrete modulus of elasticity based on $\,f'_{ci}$ , (ksi)
$\,I_{tri}$	= Moment of inertia of transformed non-composite section computed based on $\,E_{ci}$ , (in.⁴)
$\,e_{1}$	= Eccentricity between centroid of straight strand group (CSS) and center of gravity of transformed non-composite section (CGB) as shown in Figure below, (in.)
$\,f_{Pj}$	= Prestressing force in the strand just prior to transfer, (ksi)
$\,Initial\ Loss$	= Summation of the time dependent losses (7 or 90 day). Losses include relaxation, creep and shrinkage, but exclude elastic shortening.

Gross properties may be used to calculate losses and is consistent with AASHTO LRFD 5.9.3.4.

For two-point harped strands,

$\,\Delta _{HS}=\Delta _{HS-j}+\Delta _{HS-l}$

Where: $\,\Delta _{HS-j}={\frac {F_{2-j}e_{2}L^{2}}{8E_{ci}I_{tri}}}-{\frac {F_{2-j}(e_{2}+e_{3})a^{2}}{6E_{ci}I_{tri}}}$

\,\Delta _{HS-l}={\frac {F_{2-l}e_{2}L^{2}}{8E_{c}I_{tr}}}-{\frac {F_{2-l}(e_{2}+e_{3})a^{2}}{6E_{c}I_{tr}}}

\,\Delta _{HS-l}=\Delta _{HS-j}{\frac {Initial\ Loss}{f_{Pj}}}

a=(L-b)/2

Where:

$\,F_{2-j}$	= Total prestressing force of harped strands just prior to transfer, (kips)
$\,b$	= Length between harped points, (in.)
$\,e_{2}$	= Eccentricity between centroid of harped strands (CHS) and center of gravity of transformed non-composite section (CGB) at midspan as shown in Figure below, (in.)
$\,e_{3}$	= Eccentricity between centroid of harped strands (CHS) and center of gravity of transformed non-composite section (CGB) at the end of girder as shown in Figure below, (in.)

Details of girder showing distances and eccentricities used in camber calculations

Calculations of deflections (downward)

Deflections at midspan due to dead loads are determined as the following: For self-weight of girder,

$\,\Delta _{g}={\frac {5W_{g}L^{4}}{384E_{ci}I_{tri}}}$

Where:

W_{g}

= Uniform load due to self-weight of girder, (kip/in.)

For self-weight of slab,

\,\Delta _{s}={\frac {5W_{s}L^{4}}{384E_{c}I'_{tr}}}

Where:

\,W_{s}

= Uniform load due to self-weight of slab, (kip/in.)

\,E_{c}

= Final concrete modulus of elasticity based on f'_c, (ksi)

\,I'_{tr}

= Moment of inertia of transformed non-composite section based on E_c, (in.⁴)

Weight of additional slab haunch may be treated as uniform or concentrated load as appropriate. Diaphragm weight should be treated as concentrated load.

For one concentrated load at midspan,

$\,\Delta _{c}={\frac {PL^{3}}{48E_{c}I'_{tr}}}$

For two equal concentrated loads,

$\,\Delta _{c}={\frac {Px}{24E_{c}I'_{tr}}}(3L^{2}-4x^{2})$

Where:

$\,P$	= Concentrated load due to diaphragm and/or additional slab haunch, (kips)
$\,x$	= Distance from the centerline of bearing pad to the applied load, P, (in.)

Creep coefficient

LRFD 5.4.2.3.2

Research has indicated that high strength concrete (HSC) undergoes less ultimate creep and shrinkage than conventional concrete.

Creep is a time-dependent phenomenon in which deformation increases under a constant stress. Creep coefficient is a ratio of creep strain over elastic strain, and it can be estimated as follows:

$\,\Psi (t,t_{i})$	= $\,1.9k_{s}k_{hc}k_{f}k_{td}t_{i}^{-0.118}$
$\,k_{s}$	= $\,1.45-0.13(v/s)>=1.0$
$\,k_{hc}$	= $\,1.56-0.008H$
$\,k_{f}$	= $\,5/(1+f'_{ci})$
$\,k_{td}$	= $\,t/{\Big [}{\frac {12\ (100-4f'_{ci})}{f'_{ci}+20}}+t{\Big ]}$

Where:

$\,\Psi$	= Creep coefficient.
$\,H$	= 70, Average annual ambient relative humidity
$\,t$	= Maturity of concrete, (days) Use 7 days for camber design after strand release Use 90 days for camber design after erection
$\,t_{i}$	= Age of concrete when a load is initially applied, (days) Use 0.75 days for camber design.
$\,v/s$	= Volume-to-surface area ratio, (in.)
$\,f'_{ci}$	= Initial girder concrete compressive strength, (ksi)

$\,\Delta _{CR}=(\Delta _{SS-j}+\Delta _{HS-j}+\Delta _{g})\Psi +(\Delta _{SS-I}+\Delta _{HS-I})0.7\Psi$

751.22.2.7 Dowel Bars

PART ELEVATION
(FIXED BENT)

SECTION A-A

Dowel bars shall be used for all fixed intermediate bents under prestressed superstructures. Generally, for typical bridges that require seismic details only (strength limit states), shear resistance from shear key is not considered.

Dowel bars connect standard concrete diaphragms and beams on concrete girder bridges (standard fixed diaphragms are those with beam stirrups NOT extending up into the diaphragm). For a calculated seismic vertical reaction or an anticipated foundation settlement resulting in a net tensile reaction, use the development length of dowel bars into beam and into diaphragm based on dowel bar size. If the dowel bars are not exposed to net tension a 15-inch embedment shall be used regardless of bar size. Dowel bars size and spacing shall be determined by shear design of the bars. (Minimum #6 Bars @ 12" cts.). Dowel bars should be designed for a minimum horizontal force equal to 25% of the maximum dead load applied to the bearing. Live load is ignored in horizontal force computation.

The number of dowels must also fit into the space available on the key:

min. bar size = #6; max. bar size = #11

min. spacing = 6"; max. spacing = 12"

min. end distance = 3"; max. end distance = 6" (≤ half the spacing)

For seismic details only (strength limit states)

Horizontal factored shear force, $F_{H}={\sqrt {(F_{T})^{2}+(F_{L})^{2}}}$ in kips

For expansion bearings, transverse F_T = 0.25(DL) & longitudinal F_L = 0.

Where DL = unfactored dead load reaction at the bent, kips

For fixed bearings, Transverse F_T = 0.25(DL) and Longitudinal F_L = (0.25)(segment weight) at bent.

Segment weight includes the full width of superstructure and should be distributed appropriately among fixed bents.

For complete seismic analysis

Dowel bar designs must meet requirements for strength limit states from above as well as seismic force demand from seismic analysis.

At Intermediate bent,

F_{H}={\sqrt {\sum (V_{L})^{2}+\sum (V_{T})^{2}}}

where:

F_H = horizontal seismic force per bent, kips

If columns are designed for plastic hinging, use the plastic hinging shear.

∑V_L = summation of top of column longitudinal shears at the bent

∑V_T = summation of top of column transverse shears at the bent

P_{u}={\frac {F_{H}}{n_{d}}}

P_u = Horizontal factored shear force per dowel bar, kips

n_d = number of dowel bars

Shear Resistance

Factored shear force shall be less than or equal to the nominal shear resistance.

P_u ≤ ∅_s x R_n

where:

∅_s = 0.75 resistance for seismic details only (strength limit states) and 1.0 for complete seismic analysis

Nominal shear resistance of the dowel bar, R_n = 0.625 A_bF_ub, kips

Note: Since there is no reduced areas as seen in bolts and there is no reduction for bolted connection length, use 0.625 instead of 0.5.

A_b =

{\frac {\pi D^{3}}{4}}

= area of the dowel bar, square inches

F_ub = minimum tensile strength of the dowel bar, ksi

F_ub = 80 ksi for Grade 60

D = diameter of the dowel bar, inch

Tensile Resistance

Factored tensile force shall be less than or equal to the nominal tensile resistance.

T = the maximum seismic tensile (uplift) force (DL ± EQ) from the seismic analysis, kips. If (DL+EQ) and (DL-EQ) are both compressive, then there is no need to design the dowel for tensile force.

{\frac {T}{n_{d}}}\leq {\emptyset _{t}}T_{n}

where:

∅_t= 0.8 resistance factor for seismic details only (strength limit states) and 1.0 for complete seismic analysis

n_d = the number of dowel bars

Nominal tensile resistance of the dowel bar, T_n = A_bF_ub Kips

Note: Since there is no pretension or reduced areas as seen for bolts, the 0.76 factor is not warranted.

A_b = area of the dowel bar, square inches

F_ub = minimum tensile strength of the dowel bar, ksi

F_ub = 80 ksi for Grade 60

Combined Tension and Shear Resistance

The resistance of dowel bars for combined tension and shear force shall be determined in accordance with LRFD 6.13.2.11.

Note: Since there is no pretension or reduced areas as seen for bolts, the 0.76 factor is not warranted.

If

{\frac {P_{u}}{R_{n}}}\leq 0.33

, then T_n = A_bF_ub

Otherwise

T_{n}=A_{b}F_{ub}{\Big [}1-{\Big (}{\frac {p_{u}}{\emptyset _{s}R_{n}}}{\Big )}^{2}{\Big ]}^{0.5}

751.22.3 Details

751.22.3.1 Reinforcement Criteria

Minimum Concrete Cover

2.0" (Min.) to centerline of strands
1.0" for stirrups

Minimum Bend Diameter for Stirrups

#3 through #5 bars = 4.0 x Nominal Bar Diameter.
Deformed wire larger than D6 = 4.0 x Nominal Wire Diameter

Minimum Spacing of Reinforcement Bars and Wires For precast concrete, the clear distance between parallel bars in a layer shall not be lesser than:

Nominal Bar Diameter or Nominal Wire Diameter
1.33 x Maximum Aggregate Size
1.0"

Minimum Spacing of Prestressing Strands Spacing between each pretressing strand shall not be less than the larger of:

A clear distance of 1.33 x Maximum Aggregate Size
Center-to-center spacing of 2" for 0.6" strand diameter
Center-to-center spacing of 1.75" for 0.5" strand diameter

751.22.3.2 Strand Arrangements

Designers shall first attempt to use one of the strand arrangements specified in EPG 751.22.3.2.1 through EPG 751.22.3.2.5. The strand arrangement number shall be specified in the design. Bridge standard drawings for prestressed I-girders include strand details for each of these arrangements, by number, in the reference files for quick insertion by the technician.

For Group 1 arrangements, all strands in the center two columns are harped. For Group 2 arrangements, the bottom two center strands are straight (two less draped strands). Group 2 arrangements are not provided in diagrams below for Type 6, 7 and 8 girders, but may be derived similarly to how specified for the smaller girders.

Designers shall include an equivalent detail in the design computations when strand arrangements other than those shown are required.

The use of all straight strands (none harped) may be considered when strength check of a hold-down device exceeds allowable.

How Strand Arrangements are Detailed from Tables

1. For strand locations at mid-span (centerline of girder): Find the “#” designation that corresponds with the number of total strands (T) needed. The strands are to be placed at locations labeled up to and including that number. Example: For 14 total strands, the strands will be placed at all locations labeled 8 thru 14 and are designated as arrangement #14. (See Fig. 751.22.3.2.)

2. For harped strand locations at end of girder: Harped strands will be placed at locations labeled up to and including the number in the “H” column. Example: For 6 harped strands, the strands will be placed at all locations labeled 2 thru 6. (See Fig. 751.22.3.2.)

Fig. 751.22.3.2	Where:
	# = Strand Arrangement Number T = Total Number of Strands H = Number of Harped Strands S = Number of Straight Strands