Part 2

This is the second of two articles discussing high-level design considerations of prestressed concrete girders. Part 1 (STRUCTURE, January ) provided an overview of the post-tensioning and pretensioning processes, described the common materials used in constructing prestressed bridge girders and discussed the time-dependent prestress losses inherent in their design. The discussion continues in Part 2 with fundamental design considerations for internal stress distributions within prestressed concrete girders and the methodology for application.

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Raised, Draped, and Debonded Strands

When straight strands are bonded for the full length of a prestressed girder, the tensile and compressive stresses near the ends of the girder will likely exceed the allowable service limit state stresses. This occurs because the strand pattern is designed for stresses at or near midspan, where the dead load moment is highest and can best balance the effects of the prestress. Near the ends of the girder, this dead load moment approaches zero and is less able to balance the prestress force. This results in tensile stresses at the top of the girder and compressive stresses at the bottom of the girder.

General tendon profiles (Source: Raja, ).

With a raised strand pattern, the center of gravity of the strand pattern is raised slightly and is a constant distance from the bottom of the girder for the girder&#;s entire length. Other strand configurations are available, as shown in the Figure. Draping some of the strands is another available method to decrease stresses from prestress at the ends of the I-beam, where the stress due to applied loads is at a minimum. Note that all the strands that lie within the &#;vertical web zone&#; of the mid-span arrangement are used in the draped group. The designer may also use debonded strands. Partially debonded strands are fabricated by wrapping sleeves around individual strands for a specific length from the ends of the girder, rendering the bond between the strand and the girder concrete ineffective for the wrapped or shielded length. Preference for each of these methods is on a state-by-state basis. For example, Wisconsin&#;s order of preference for strand placement is straight, draped, and then partially debonded. For the Michigan Department of Transportation (MDOT), debonding is the preferred method of controlling stresses at the end of I-beams, and draped strands should be avoided where possible.

Strand Development, Transfer, Anchorage, and Spacing

Development length is the shortest length of a strand in which the strand stress can increase from zero to the yield strength. If the distance from a point where the strand equals the yield strength to the end of the strand is less than the development length, the strand will pull out of the concrete.

Transfer length represents the first portion of the development length over which the prestressing strand should be bonded so that a stress in the prestressing strand at the nominal strength of the member may develop. A short transfer length increases stresses and the risk of cracking by concrete splitting, bursting, or spalling in the end regions. A long transfer length reduces the available member length to resist bending moment and shear and therefore increases member cost. A pretensioned member&#;s design strength is taken as the nominal strength multiplied by the strength reduction factors in sections within the transfer length and the development length. If a critical section occurs within these regions, where the strand is not fully developed, failure may occur by bond-slip (Building Code Requirements for Structural Concrete and Commentary, ACI 318-11).

Bond stresses are derived through a combination of adhesion, friction, and mechanical interlocking. It has been widely believed that a wedging effect, unique to pretensioned strands, creates significant bond stresses in the transfer zone. The effective prestressing force is transferred from the pretensioned strand to the concrete. In those same regions, slip occurs between strand and concrete due to the difference in strain condition. Research has indicated that the strand end slip can be used as a quality control measure for the bond of prestressing strands. Furthermore, the relative slip between strand and concrete virtually ensures that adhesion plays little or no role in transferring prestressing forces to concrete.

Historically, the American Association of State Highway Transportation Officials (AASHTO) limited the strand clear spacing to a minimum of three times the strand diameter. In bridge codes before the AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications, this provision was made an explicit part of the design code. This code provision likely mirrored the standard of placing 0.5-inch strands at 2.0-inch, center-to-center (c/c) spacing. If this provision were extended to the larger diameter 0.6-inch strands, then the 0.6-inch strands would have to be placed at 2.4-inch c/c. Nevertheless, using this strand spacing would cancel out the economic value inherent in using a 0.6-inch strand. It would also cancel out the most compelling reasons to use high-strength concrete in pretensioned girder applications. Russell () showed that 0.6-inch strands must be placed at a spacing of about 2.0-inch c/c to enable designs to take advantage of high-strength concrete. Cousins et al. conducted a series of tests in and concluded that the strand spacing had no effects on the measured transfer and development lengths.

Design Methodology

There are three primary stages to be addressed in prestressed girder design: transfer, service, and ultimate. The California Department of Transportation&#;s (Caltrans&#;) Bridge Design Practice Guide describes them succinctly:

• Transfer refers to the stage at which the tensile force in the strands is transferred to the Precast (PC) girder by cutting or detensioning the strands after a minimum girder concrete strength has been verified. Because the girder is simply supported and only self-weight acts with the prestressing at this stage, the most critical stresses typically occur at the ends of the girder or harping points (also known as drape points). Both tensile and compressive stresses should be checked at these locations against AASHTO LRFD stress limits.
• Service refers to the stage at which the girder and deck self-weight act on the non-composite girder, together with additional dead loads (e.g., barrier and wearing surface) and live loads on the composite section. This stage is checked using the AASHTO LRFD Service I and III load combinations, per Caltrans Amendments, Table 5.9.4.2.2.-1. The girder must also be designed to prevent tension in the precompressed tensile zones (&#;zero tension&#;) due to permanent loads.
• Ultimate refers to the Strength Limit State. Flexural and shear strengths are provided to meet all factored load demands, including the Caltrans P-15 design truck (Strength II load combination).

Generally speaking, the mechanics present in prestressed beams are similar to non-prestressed beams. The principles of flexural, shear, and torsional forces must be understood and accounted for in the design. The inclusion of prestressing strands, however, modifies some of these characteristics. The modifications primarily occur in the flexural/moment realm of element design through beam-end reactions and continuity moments. They can also affect the shear resistance of an element due to bursting considerations. A secondary design consideration is the prestressing forces at release, or transfer, which produce temporary tensile stresses at the top of the concrete member and need to be checked to ensure that they do not exceed the concrete&#;s tensile capacity. Additional longitudinal reinforcement in the bridge deck is generally required near piers to provide adequate moment capacity for negative bending.

The amount of reinforcement is that which is sufficient to resist the total tension force in the concrete based on an uncracked section. For draped designs, the control is at the hold-down point of the girder. At the hold-down point, the initial prestress is acting together with the girder dead load stress. This is where tension due to prestress is still maximum, and compression due to girder dead load decreases. For non-draped designs, the control is at the end of the member where prestress tension exists but where dead load stresses do not occur.

As the bottom fibers of bridge girders encounter prestressing losses (this is where most of the prestressing strands are focused), the girders gradually encounter upward camber and beam-end moments. The beam-end rotations would continue to grow unless restrained. Current design methodology accounts for this phenomenon and restrains the beam-ends by inducing continuity through the spans of a bridge.

Deck-only continuity is when deck joints over the beam-ends are eliminated, and the deck itself acts as a hinge for the composite structure. In this case, the beams are designed as simple spans, and the reinforcing steel within the deck must be designed for compatibility with the girder rotations.

Full-section continuity is similar to deck-only continuity, except that the girders&#; bottom flanges are tied together, and the space between the composite section of the deck and girders is sealed with a concrete closure pour. Full-section continuity allows a series of girders to act as a single girder placed continuously over a bearing.

AASHTO LRFD Article 5.14.1.4.5 defines two joint types: a fully effective joint, which treats the structure as continuous for all limit states, and a partially effective joint, which treats the structure as continuous for the strength limit state only.

Effects on Design for Flexure

Positive moment requires prestressing force below the section centroid; negative moment requires it above the centroid. The required eccentricity of prestressing tendons with the section&#;s centroid increases with increased applied bending moment. Curved tendon profiles that approximately follow the shape of the appropriate bending moment diagram are not easily feasible in pretensioned members, so general tendon profiles have been established for design purposes.

Effects on Design for Shear and Torsion

Orangun et al. () indicated that transverse reinforcement confines the concrete around anchored bars and limits the progression of splitting cracks.

Edge girders encounter torsion due to the eccentricity of their loading of the main deck slab, barrier and parapet walls, and/or cantilevered components. In some cases, edge girders are loaded in torsion by (floor) beams framing into them from the sides between the ends of the girder. The beam&#;s free-end rotation is retrained by the girder&#;s torsional stiffness and is, therefore, a design consideration.

Trends of Prestressed Girders

The AASHTO LRFD Bridge Design Specifications contains restrictions on the use of high-strength concrete due to the lack of previous performance history and empirical data available. These restrictions limit the application of existing and new technology to bridges. Concretes with design compressive strengths above 10 ksi shall be used only when allowed by specific articles of the specifications or when physical tests are made to establish the relationship between the concrete strength and other properties. Revisions in design guidance will continue as data becomes increasingly available, leading to a broader array of high-strength concrete applications.

The lessons learned and knowledge base of prestressed concrete girders are being leveraged into other structural design aspects. Practitioners worldwide are beginning to realize the stakeholder benefits and time-saving possibilities of employing the principles of prestressed concrete in other, more irregular shapes. For example, Accelerated Bridge Construction (ABC) is predicated on the idea of designing elements to be constructed off-site using prestressed technologies and rapidly deploying them on site.

Conclusion

Prestressing losses are characteristic of composite members once they are cast and affect the ultimate utility of the design. Elastic shortening, shrinkage, and creep all affect the design life and efficacy of prestressed concrete beams. The current methodology tries to consider and account for the interaction of these phenomena to develop an effective design. This highly complex process is the result of the bridge designer&#;s iterations over many years.&#;

References

American Concrete Institute (ACI). Building Code Requirements for Structural Concrete and Commentary, ACI 318-11, Detroit, .

Beal, D. NCHRP Report 603: Transfer, Development, and Splice Length for Strand/Reinforcement in High-Strength Concrete, Foreword, Transportation Research Board, Washington D.C., .

Caltrans Bridge Design Practice Guide Chapter 8: Precast Pretensioned Concrete Girders, California Department of Transportation, Sacramento, CA, .

Cousins, T.E., Stallings, J.M., and Simmons, M.B. Effect of Strand Spacing on Development of Prestressing Strands, Alaska Department of Transportation and Public Facilities, Juneau, AK, .

Marti-Vargas, J. and Hale, W.M. Predicting Strand Transfer Length in Pretensioned Concrete: Eurocode versus North American Practice, American Society of Civil Engineers, .

MDOT. Bridge Design Manual Chapter 7: LRFD, Vol. 5, Michigan Department of Transportation, Lansing, MI, .

Orangun, C.O., Jirso, J.O., and Breen, J.E. &#;A Reevaluation of Test Data on Development Length and Splices.&#; ACI Journal of Proceedings, 74(3), 114-122, .

Parsons Brickerhoff. Example No. 1: Prestressed Concrete Girder Bridge Design, New Mexico Department of Transportation, Sante Fe, NM, .

Raja, N. Thesis: Design of Prestressed Concrete Girder Bridge, Edith Cowan University, Perth, Western Australia, .

Ramirez, J. and Russell, B. NCHRP Report 603: Transfer, Development, and Splice Length for Strand/Reinforcement in High-Strength Concrete, Transportation Research Board, Washington D.C., .

Rose, D.R., and Russell, B. &#;Investigation of Standardized Tests to Measure the Bond Performance of Prestressing Strand.&#; PCI Journal, 42(4), 56-80, .

Russell, B.W. &#;Impact of High Strength Concrete on the Design and Construction of Pretensioned Girder Bridges.&#; PCI Journal, 39(4), 76-89, .

Russel, B.W., and Burns, N.H. &#;Measured Transfer Lengths of 0.5 in. and 0.6 in. Strands in Pretensioned Concrete.&#; PCI Journal, 41(5), 44-63, .

Wight, J. Reinforced Concrete Mechanics and Design, 7th Ed., Pearson, Hoboken, NJ, .

WisDOT. WisDOT Bridge Manual Chapter 19 &#; Prestressed Concrete, Wisconsin Department of Transportation, Madison, WI, .

PC strand

PC Strand, or prestressed concrete steel strand, is a twisted steel cable composed of 2, 3, 7 or 19 high strength steel wires and is stress-relieved (stabilized) for prestressed concrete or similar purposes.

 

Classification

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PC strand is classified according to the number of steel wires in a strand: 2 wire strand, 3 wire strand, 7 wire steel strand[1] and 19 wire steel strand. It can be classified according to the surface morphology and can be divided into: smooth steel strand, scoring strand, mold pulling strand (compact), coated epoxy resin steel strand. They can also be classified by diameter, or intensity level, or standard.

 

Specifications

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In the description and list of the table we often see, there are 15-7Φ5, 12-7Φ5, 9-7Φ5 and other specifications of the prestressed steel strand. To 15-7Φ5, for example, 5 said a single diameter 5.0mm of steel, 7Φ5 said seven of the steel wire to form a strand, and 15 that the diameter of each strand of 15mm, the total meaning is "one The beam consists of 7 strands of diameter 15 mm (each having a total diameter of about 15.24 mm, a dimensional deviation +0.40 -0.20; a diameter of about 5.0 mm per filament). The general sectional area is calculated according to 140mm ^ 2. The theoretical breaking value is 140 * = 260.4 kN, which can withstand the tension of 156.24-169.26 kN according to the prestressing standard of 60% -65%.

 

Materials

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Using high-carbon steel wire rod, after surface treatment it is cold drawn into steel wire, and then the strand structure will be a number of steel wires stranded into shares. Next the elimination of stress by way of a stabilization process. In order to extend durability, the wire can have metal or non-metallic coatings, such as galvanized, or epoxy resin coating. In order to increase the bond strength with the concrete, the surface can have nicks and so on. The prestressed strands of the mold are twisted to form a mold compression process, the structure is more compact, and the surface layer is more suitable for anchoring. Production of unbonded prestressed steel strand (unbonded steel strand) using ordinary prestressed steel wire, coated with oil or paraffin after the packaging into high-density polyethylene (HDPE) bags.

 

Features

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The main characteristic of the prestressed steel strand is high strength and relaxation performance is good, the other when the more straight. Common tensile strength levels of MPa, as well as , , , , MPa and the like intensity levels. The yield strength of this steel is also higher.

 

Application

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In most of the post-tensioned and pre-tensioned prestressed project, smooth steel strand is the most widely used prestressed steel. Stranded strand is mainly used to enhance the project, but also for nuclear power and the like works. Galvanized steel strand commonly used in the bridge of the tie rod, cable and external prestressing works. Epoxy coated steel stranded wire is similar to galvanized prestressed steel wire.

 

Standards

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Countries have standards for prestressed strand, such as: China Standard GB / T , American Standard ASTM A416, British standard BS and the Japanese standard JIS G, the Australian standard AS / NZS , Brazilian standard NBR-[2]

 

References

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PGSuper: Permanent Strands

This tab allows you to specify the possible locations of permanent prestressing strands.

General Information

The strand grid defines the possible locations of permanent strands in a girder and the order in which the strand positions are filled. Another way to think of this is a template for strand placement. Actual strands are placed in a project by specifying the number of strands, or by individually selecting locations, to be filled in the template in the Strands Tab of the Girder Editing dialog.

There are two basic types of permanent strands.

Straight Permanent Strands can be placed anywhere within a girder section and are straight along the entire length of the girder.

Adjustable Strands are permanent strands that have the unique capability to be vertically adjusted within the web region of a girder. Adjustable strands can only be placed in the girder's web(s). There are two types of Adjustable Strands: Harped Strands and Adjustable Straight strands.

Harped strands are draped at the girders harping points. Harping (draping) is created by the difference in vertical adjustment between the strand group at the girder ends and harping points.

Adjustable Straight Strands are straight along the entire length of the girder. The entire group can be vertically adjusted. Adjustable Straight Strands located above the girder's upper kern location can be used by the automated design algorithm to alleviate girder end stresses.

Permanent strands are placed and tensioned at casting time and remain in the girder for its lifetime. Adjustable strands are bonded along the entire length of the girder. Individual straight strands can be debonded only if the "allow debonding" option is selected for the strands in question.

Adjustable Strand Settings

Item Description Adjustable Strand Type The type of adjustable strands can be designated as Straight, Harped, or "Harped or Straight". If "Harped or Straight" is selected, the type of strand can be set by the end user when editing the girder. Coerce Odd Number of Adjustable Strands When selected, PGSuper will force (coerce) the highest (last in the fill order) pair of adjustable strands to alternate between a single strand at X=0.0, and two strands at the prescribed +/- X values. This allows a strand grid that contains only pairs of coordinates to place strands one at a time. This feature is uncommonly needed for most users, and is used on some WSDOT I-Beams or Ribbed Girders with an odd number of webs. Use Different Harped Locations at Girder Ends This option allows you to define different locations of a harped strand at the ends of the girder and at the harping points. This option is most often used to describe the "fanned" harped bundles at harping points used on WSDOT I girders.

NOTE: All end strands in the web strand grid must have positive X values (i.e., filled in pairs) when the "Coerce Odd Number of Harped Strands" option is selected.

NOTE: The "Use Different Harped Strand Locations at Girder Ends" option is not needed if the relative distance between harped strands is the same along the entire length of the girder (i.e., strands are always parallel). In this case, harping is achieved by adjusting the vertical offset of strand patterns at the girder ends and harping points.

Strand Grid (Potential Strand Locations)

The strand grid lists the possible strand locations and fill order of permanent straight and adjustable strands in the girder.

Item Description Fill # Fill sequence number Xb,Yb Strand position at the harping point measured from the bottom center of the cross section Type Strand Type Xt,Yb Strand position at the ends of the girder measured from the top center of the cross section [Insert] Insert a strand position at the current location in the grid [Append] Append a strand position at the end of the fill sequence [Edit] Edit the strand position description. See Strand Location. [Delete] Delete the selected strand positions [Move Up] Moves a selected strand position up in the fill sequence [Move Down] Moves a selected strand position down in the fill sequence [Reverse Adjustable Strand Sequence] Reverses the fill sequence of the adjustable strands [Generate Strand Positions] Activates a tool to generate a uniform layout of strand positions

Contact us to discuss your requirements of PC Strand Bonded. Our experienced sales team can help you identify the options that best suit your needs.

NOTE: Double click on a strand position to quickly edit its properties

NOTE: The strand grid shows the strand positions on both sides of the vertical axis of the cross section

NOTE: Strands listed as "Straight" are straight strands and cannot be debonded

NOTE: Strands listed as "Straight-DB" are straight strands that can be debonded

Grid Status

The Grid Status region provides summary information about the strand grid.

Item Description # Debondable Strands Number of strand positions that have been designated as debondable # Straight Strands Number of strand positions used for straight strands # Harped Strands Number of strands positions used for harped strands [View at Ends] Press this button to see the strand positions at the end of the girder [View at Mid-Girder] Press this button to see the strand positions at the center of the girder

Vertical Adjustment of Adjustable Strands

The parameters defined in this group control the vertical range over which adjustable strands can be moved from their default position as defined in the strand grid as well as the adjustment increment used for design. Harped strands can be offset differently at the girder ends than at harping points, thus forming the harped drape. Adjustable Straight Strands use a single offset because the strands are straight and parallel to the top of the girder.

Item Description Allow Check the box to allow vertical adjustment Design Increment In general, adjustable web strands can be adjusted continuously up and down within their adjustment limits through user-input adjustment values in the girder editing dialog. The design increment defines the step size to be used by the automated design algorithm when adjusting strand offsets. The increment must be less than or equal to the maximum adjustment value. If this value is zero, the design algorithm will make continuous adjustments within the offset limits. Lower Strand Limit
Upper Strand Limit These values define the range of possible uppermost and lowermost adjustable strand Y locations. Adjustable strands may not lie above or below these limits. Limits can be measured downward from the top of the girder or upward from the bottom.

Bridges & Structures

Comprehensive Design Example for Prestressed Concrete (PSC) Girder Superstructure Bridge

Design Step 5 Design of Superstructure

 

Design Step 5.5 Stress in Prestressing Strands

 

Design Step 5.5.1 - Stress in prestressing strands at nominal flexural resistance

The stress in prestressing steel at nominal flexural resistance may be determined using stress compatibility analysis. In lieu of such analysis a simplified method is presented in S5.7.3.1.1. This method is applicable to rectangular or flanged sections subjected to flexure about one axis where the Whitney stress block stress distribution specified in S5.7.2.2 is used and for which fpe, the effective prestressing steel stress after losses, is not less than 0.5fpu. The average stress in prestressing steel, fps, may be taken as:

fps    = fpu[1 - k(c/dp)] (S5.7.3.1.1-1) where:   k = 2(1.04 - fpy /fpu) (S5.7.3.1.1-2)

The value of "k" may be calculated using the above equation based on the type and properties of prestressing steel used or it may be obtained from Table SC5.7.3.1.1-1.

The distance from the neutral axis to the compression face of the member may be determined as follows:

for T-section behavior (Eq. S5.7.3.1.1-3):

 

for rectangular section behavior (Eq. S5.7.3.1.1-4):

 

T-sections where the neutral axis lies in the flange, i.e., "c" is less than the slab thickness, are considered rectangular sections.

From Table SC5.7.3.1.1-1:

k = 0.28 for low relaxation strands

Assuming rectangular section behavior with no compression steel or mild tension reinforcement:

c = Apsfpu /[0.85f&#;cβ1b + KAps(fpu /dp)]

For the midspan section

Total section depth, h    = girder depth + structural slab thickness
= 72 + 7.5
= 79.5 in. dp    = h - (distance from bottom of beam to location of P/S steel force)
= 79.5 - 5.0
= 74.5 in. β1 = 0.85 for 4 ksi slab concrete (S5.7.2.2) b = effective flange width (calculated in Section 2 of this example)
= 111 in. c = 6.73(270)/[0.85(4)(0.85)(111) + 0.28(6.73)(270/74.5)]
= 5.55 in. < structural slab thickness = 7.5 in.
The assumption of the section behaving as a rectangular section is correct.

Notice that if "c" from the calculations above was greater than the structural slab thickness (the integral wearing surface is ignored), the calculations for "c" would have to be repeated assuming a T-section behavior following the steps below:

Assume the neutral axis lies within the precast girder flange thickness and calculate "c". For this calculation, the girder flange width and area should be converted to their equivalent in slab concrete by multiplying the girder flange width by the modular ratio between the precast girder concrete and the slab concrete. The web width in the equation for "c" will be substituted for using the effective converted girder flange width. If the calculated value of "c" exceeds the sum of the deck thickness and the precast girder flange thickness, proceed to the next step. Otherwise, use the calculated value of "c".

Assume the neutral axis is below the flange of the precast girder and calculate "c". The term "0.85 f&#;cβ1(b - bw)" in the calculations should be broken into two terms, one refers to the contribution of the deck to the composite section flange and the second refers to the contribution of the precast girder flange to the composite girder flange.

fps    = fpu[1 - k(c/dp)]      (S5.7.3.1.1-1)
= 270[1 - 0.28(5.55/74.5)]
= 264.4 ksi

Design Step 5.5.2

Transfer and development length    = 60(Strand diameter)     (S5.11.4.1)
60(0.5 in.)
= 30 in.

Development Length = ld &#; κ[fps - (2/3)fpe]db      (S5.11.4.2-1)

From earlier calculations:

fps    = 264.4 ksi (Design Step 5.4.8) fpe = 162.83 ksi (Design Step 5.5.1)

From S5.11.4.2, κ = 1.6 for fully bonded strands

From S5.11.4.3, κ = 2.0 for partially debonded strands

For fully bonded strands (32 strands):

ld &#; 1.6[264.4 - (2/3)162.83](0.5) = 124.7 in. (10.39 ft. or 10'-4 11/16")

For partially debonded strands (two groups of 6-strands each):

ld &#; 2.0[264.4 - (2/3)162.83](0.5) = 155.8 in. (12.98 ft. or 12'-11 ¾")

 

Design Step 5.5.3 - Variation in stress in prestressing steel along the length of the girders

According to S5.11.4.1, the prestressing force, fpe, may be assumed to vary linearly from 0.0 at the point where bonding commences to a maximum at the transfer length. Between the transfer length and the development length, the strand force may be assumed to increase in a parabolic manner, reaching the tensile strength of the strand at the development length.

To simplify the calculations, many jurisdictions assume that the stress increases linearly between the transfer and the development lengths. This assumption is used in this example.

As shown in Figures 2-5 and 2-6, each beam contains three groups of strands:

Group 1: 32 strands fully bonded, i.e., bonded length starts 9 in. outside the centerline of bearings of the noncomposite beam
Group 2: 6 strands. Bonded length starts 10 ft. from the centerline of bearings of the noncomposite beam, i.e., 10'-9" from the end of the beam
Group 3: 6 strands. Bonded length starts 22 ft. from the centerline of bearings of the noncomposite beam, i.e., 22'-9" from the end of the beam

For each group, the stress in the prestressing strands is assumed to increase linearly from 0.0 at the point where bonding commences to fpe, over the transfer length, i.e., over 30 inches. The stress is also assumed to increase linearly from fpe at the end of the transfer length to fps at the end of the development length. Table 5.5-1 shows the strand forces at the service limit state (maximum strand stress = fpe) and at the strength limit state (maximum strand stress = fps) at different sections along the length of the beams. To facilitate the calculations, the forces are calculated for each of the three groups of strands separately and sections at the points where bonding commences, end of transfer length and end of development length for each group are included in the tabulated values. Figure 5.5-1 is a graphical representation of Table 5.5-1.

Table 5.5-1 - Prestressing Strand Forces

Dist. from Grdr End
(ft) Dist. from CL of Brg
(ft) Initial Prestressing Force at Transfer Group 1
(k) Group 2
(k) Group 3
(k) Total
(k) 0* -0.75* 0.0     0.0 0.75 0.00 277.3     277.3 2.50 1.75 924.4     924.4 7.75 7.00 924.4     924.4 10.39 9.64 924.4     924.4 10.75** 10.00** 924.4 0.0   924.4 11.75 11.00 924.4 69.3   993.7 13.25 12.50 924.4 173.3   1,097.7 17.25 16.50 924.4 173.3   1,097.7 22.75*** 22.00*** 924.4 173.3 0.0 1,097.7 23.73 22.98 924.4 173.3 67.9 1,165.6 25.25 24.50 924.4 173.3 173.3 1,271.0 28.25 27.50 924.4 173.3 173.3 1,271.0 33.75 33.00 924.4 173.3 173.3 1,271.0 35.73 34.98 924.4 173.3 173.3 1,271.0 39.25 38.50 924.4 173.3 173.3 1,271.0 44.75 44.00 924.4 173.3 173.3 1,271.0 50.25 49.50 924.4 173.3 173.3 1,271.0 55.25 54.50 924.4 173.3 173.3 1,271.0 55.75 55.00 924.4 173.3 173.3 1,271.0 61.25 60.50 924.4 173.3 173.3 1,271.0 66.75 66.00 924.4 173.3 173.3 1,271.0 72.25 71.50 924.4 173.3 173.3 1,271.0 74.77 74.02 924.4 173.3 173.3 1,271.0 77.75 77.00 924.4 173.3 173.3 1,271.0 83.25 82.50 924.4 173.3 173.3 1,271.0 85.25 84.50 924.4 173.3 173.3 1,271.0 86.77 86.02 924.4 173.3 67.9 1,165.6 87.75+++ 87.00+++ 924.4 173.3 0.0 1,097.7 88.75 88.00 924.4 173.3   1,097.7 94.25 93.50 924.4 173.3   1,097.7 97.25 96.50 924.4 173.3   1,097.7 99.75++ 99.00++ 924.4 0.0   924.4 100.11 99.36 924.4     924.4 103.25 102.50 924.4     924.4 108.00 107.25 924.4     924.4 109.75 109.00 277.3     277.3 110.5+ 109.75+ 0.0     0.0

*, **, *** - Point where bonding commences for strand Groups 1, 2, and 3, respectively

+, ++, +++ - Point where bonding ends for strand Groups 1, 2, and 3, respectively

Table 5.5-1 (cont.) - Presstressing Strand Forces

Dist. from Grdr End
(ft) Dist. from CL of Brg
(ft) Prestressing Force After Losses Force at the Nominal Flexural Resistance Group 1
(k) Group 2
(k) Group 3
(k) Total
(k) Group 1
(k) Group 2
(k) Group 3
(k) Total
(k) 0* -0.75* 0.0     0.0 0.0     0.0 0.75 0.00 239.0     239.0 239.0     239.0 2.50 1.75 797.2     797.2 797.2     797.2 7.75 7.00 797.2     797.2 1,128.1     1,128.1 10.39 9.64 797.2     797.2 1,294.5     1,294.5 10.75** 10.00** 797.2 0.0   797.2 1,294.5 0.0   1,294.5 11.75 11.00 797.2 59.8   857.0 1,294.5 59.8   1,354.3 13.25 12.50 797.2 149.5   946.7 1,294.5 149.5   1,444.0 17.25 16.50 797.2 149.5   946.7 1,294.5 185.1   1,479.6 22.75*** 22.00*** 797.2 149.5 0.0 946.7 1,294.5 234.0 0.0 1,528.5 23.73 22.98 797.2 149.5 58.6 1,005.3 1,294.5 242.7 58.6 1,595.8 25.25 24.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 149.5 1,686.7 28.25 27.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 176.2 1,713.4 33.75 33.00 797.2 149.5 149.5 1,096.2 1,294.5 242.7 225.1 1,762.3 35.73 34.98 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 39.25 38.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 44.75 44.00 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 50.25 49.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 55.25 54.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 55.75 55.00 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 61.25 60.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 66.75 66.00 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 72.25 71.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 74.77 74.02 797.2 149.5 149.5 1,096.2 1,294.5 242.7 242.7 1,779.9 77.75 77.00 797.2 149.5 149.5 1,096.2 1,294.5 242.7 216.2 1,753.4 83.25 82.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 167.3 1,704.5 85.25 84.50 797.2 149.5 149.5 1,096.2 1,294.5 242.7 149.5 1,686.7 86.77 86.02 797.2 149.5 58.6 1,005.3 1,294.5 242.7 58.6 1,595.8 87.75+++ 87.00+++ 797.2 149.5 0.0 946.7 1,294.5 234.0 0.0 1,528.5 88.75 88.00 797.2 149.5   946.7 1,294.5 225.1   1,519.6 94.25 93.50 797.2 149.5   946.7 1,294.5 176.2   1,470.7 97.25 96.50 797.2 149.5   946.7 1,294.5 149.5   1,444.0 99.75++ 99.00++ 797.2 0.0   797.2 1,294.5 0.0   1,294.5 100.11 99.36 797.2     797.2 1,294.5     1,294.5 103.25 102.50 797.2     797.2 1,096.6     1,096.6 108.00 107.25 797.2     797.2 797.2     797.2 109.75 109.00 239.0     239.0 239.0     239.0 110.5+ 109.75+ 0.0     0.0 0.0     0.0

*, **, *** - Point where bonding commences for strand Groups 1, 2, and 3, respectively

+, ++, +++ - Point where bonding ends for strand Groups 1, 2, and 3, respectively

 

Figure 5.5-1 - Prestressing Strand Forces Shown Graphically

 

Transfer length = 30 in.
Development length, fully bonded = 124.7 in.
Development length, debonded = 155.8 in.

Figure 5.5-1 (cont.) - Prestressing Strand Forces Shown Graphically

Design Step 5.5.4 - Sample strand stress calculations

 

Prestress force at centerline of end bearing after losses under Service or Strength

Only Group 1 strands are bonded at this section. Ignore Group 2 and 3 strands.

Distance from the point bonding commences for Group 1 strands = 0.75 ft < transfer length

Percent of prestressing force developed in Group 1 strands    = 0.75/transfer length
= (0.75/2.5)(100)
= 30%

Stress in strands = 0.3(162.83) = 48.8 ksi

Force in strands at the section = 32(0.153)(48.8) = 239 kips

Prestress force at a section 11 ft. from the centerline of end bearing after losses under Service conditions

Only strands in Group 1 and 2 are bonded at this section. Ignore Group 3 strands.

The bonded length of Group 1 strands before this section is greater than the transfer length. Therefore, the full prestressing force exists in Group 1 strands.

Force in Group 1 strands = 32(0.153)(162.83) = 797.2 kips

Distance from the point bonding commences for Group 2 strands = 1.0 ft. < transfer length

Percent of prestressing force developed in Group 2 strands =    1.0/transfer length
= (1.0/2.5)(100) = 40%

Stress in Group 2 strands = 0.4(162.83) = 65.1 ksi

Force in Group 2 strands at the section = 6(0.153)(65.1) = 59.8 kips

Total prestressing force at this section = force in Group 1 + force in Group 2 = 797.2 + 59.8 = 857 kips

Strands maximum resistance at nominal flexural capacity at a section 7.0 ft. from the centerline of end bearing

Only Group 1 strands are bonded at this section. Ignore Group 2 and 3 strands.

Distance from the point bonding commences for Group 1 strands, i.e., distance from end of beam = 7.75 ft. (7'- 9")

This distance is greater than the transfer length (2.5 ft.) but less than the development length of the fully bonded strands (10.39 ft.). Therefore, the stress in the strand is assumed to reach fpe, 162.83 ksi, at the transfer length then increases linearly from fpe to fps, 264.4 ksi, between the transfer length and the development length.

Stress in Group 1 strands    = 162.83 + (264.4 - 162.83)[(7.75 - 2.5)/(10.39 - 2.5)]
= 230.41 ksi

Force in Group 1 strands = 32(0.153)(230.41)
= 1,128.1 kips

Strands maximum resistance at nominal flexural capacity at a section 22 ft. from centerline of end bearing

Only strands in Group 1 and 2 are bonded at this section. Ignore Group 3 strands.

The bonded length of Group 1 strands before this section is greater than the development length for Group 1 (fully bonded) strands. Therefore, the full force exists in Group 1 strands.

Force in Group 1 strands = 32(0.153)(264.4) = 1,294.5 kips

The bonded length of Group 2 at this section = 22 - 10 = 12 ft.

Stress in Group 2 strands    = 162.83 + (264.4 - 162.83)[(12 - 2.5)/(12.98 - 2.5)]
= 254.9 ksi

Force in Group 2 strands = 6(0.153)(254.9) = 234.0 kips

Total prestressing force at this section    = force in Group 1 + force in Group 2
= 1,294.5 + 234.0
= 1,528.5 kips

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