PROFESSIONAL DEVELOPMENT SERIES Post-Tensioning for Two-Way Flat Plate Construction By Amy Reineke Trygestad, P.E. October 2005 SE1005PDH.qxp 10/17/05 2:44 PM Page 1 The post-tensioning system The most common post-tensioning system for two-way slab building construction uses mono-strand unbonded tendons. In this type of construction, the prestressing steel is composed generally of high-strength, single-wire steel, wrapped with another six wires to form a seven-wire strand. Th
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  PROFESSIONALDEVELOPMENTSERIES Post-Tensioning for Two-WayFlat Plate Construction By Amy Reineke Trygestad, P.E. October 2005  The post-tensioning system The most common post-tensioning system for two-wayslab building construction uses mono-strand unbondedtendons. In this type of construction, the prestressing steel iscomposed generally of high-strength, single-wire steel,wrapped with another six wires to form a seven-wire strand.The common strand has a specified tensile strength,  f    pu , of 270kips per square inch (ksi), a nominal diameter of 1/2 inch, andan area of steel,  A   ps  , equaling 0.153 square inch (Figure 1). Bydesign, unbonded tendons have a continuous plastic sheath-ing to prevent the strand from bonding with the concretealong its length. This sheathing serves as the bond breaker;provides protection during handling, shipping, and construc-tion; and limits intrusion of corrosive elements. Corrosion-inhibiting grease coats the strands to reduce friction betweenthe strand and the sheathing during stressing.The force in a stressed tendon is transferred to the concretevia serrated wedges that lock into anchor plates provided atits ends. Anchors are classified as either live (stressing) ends or dead ends. Dead end anchors are embedded into theconcrete and will not be stressed. These anchors are mountedto the tendon at the fabrication plant. Live end anchors aremounted and stressed in the field. Each tendon is stressedindividually and has its own anchor plate (thus, mono -strand)with approximate dimensions of 2-1/4 inches by 5-1/4inches. This small, ductile iron casting transfers 33 kips of  force for a seven-wire, 1/2-inch-diameter, 270-ksi tendon in aconcentrated area. Since this involves high local stresses, it isessential to place the anchors accurately, consolidate thesurrounding concrete, limit or eliminate penetrations in theimmediate vicinity, and sufficiently reinforce the anchoragezone to preserve its long-term integrity. Analysis — Computers have increased the speed of post-tensioning design significantly, but it is still important tounderstand the concepts and calculations to arrive at anaccurate output. When performing manual calculations, theEquivalent Frame Method (EFM) of the American ConcreteInstitute’s Building Code Requirements for StructuralConcrete (ACI 318-05) within Section 13.7 (excludingsections often is used for the structural analysis of a post-tensioned, two-way, flat slab structure.EFM models a 3-D slabsystem as a series of equiva-lent 2-D frames along thesupport lines, taken longitu-dinally and transverselythrough the structure. Eachequivalent frame then canbe analyzed individually asan isolated plane frame,consisting of a row of columns or supports and 2PDH Special Advertizing Section — Portland Cement Association Continuing Education The Professional Development Series is a unique oppor-tunity to earn continuing education credit by readingspecially focused, sponsored articles in Structural Engineer . If  you read the following article, display your understandingof the stated learning objectives, and follow the simpleinstructions, you can fulfill a portion of your continuingeducation requirements at no cost to you. This article also isavailable online at Instructions First, review the learning objectives below, then read theProfessional Development Series article. Next, complete thequiz and submit your answers to the ProfessionalDevelopment Series sponsor. Submittal instructions areprovided on the Reporting Form, which follows the quizand is also available for download at Your quiz answers will be graded by the ProfessionalDevelopment Series sponsor. If you answer at least 80percent of the questions correctly, you will receive a certifi-cate of completion from the Professional Development Seriessponsor within 90 days and will be awarded 1.0 professionaldevelopment hour (equivalent to 0.1 continuing educationunit in most states). Note: It is the responsibility of the licensee to determine if this method of continuing education meets his or her governing board(s) of registration’s requirements. Learning Objectives Upon reading this article and completing the quiz, you should be able to understand the design process for post-tensioned two-way slabs and recognizeconstructibility issues involved when using a post-tensioning system. Professional Development Series Sponsor Portland Cement Association E very major metropolitan area is getting a facelift. Old buildings are beingrenovated and converted, while new construction continues to add tothe skyline. Downtown living is making its resurgence, and with thisbooming residential construction, two-way post-tensioned flat plates are the structural system of choice. Professional Development Series Figure 1: Tendon showinganchor, strand, and sheathing  the corresponding tributary slab width, bounded laterally bythe centerlines of the adjacent slab panels (Figure 2). Theanalysis and design of post-tensioned, two-way slabs incorpo-rate the full tributary slab width without the distribution of  forces and reinforcement between column strips and middlestrips, synonymous with mild reinforced two-way slabdesigns. This allowance eases the structural engineer’s designprocess and ultimately simplifies construction. Preliminary sizing — Before design can begin on thestructure, there needs to be a starting point for the slab thick-ness. For common occupancy structures with live load todead load (LL/DL) ratios less than 1.0, a preliminary slabthickness can be estimated using a longest span to thicknessratio, L/  h , of 45 for floors and 48 for roofs. For example, an8-inch-thick slab is typical for a 30-foot-long floor span (30 feet x (12 inches/foot) / 45 = 8 inches).Because of post-tensioning’s ability to balance loads andgreatly reduce service load deflections, there is a 25 percentto 35 percent reduction in slab thickness in post-tensionedstructures compared with mild reinforced structures.Therefore, in addition to the ability to span further, thereduced structural depth economizes material quantities for the slabs and consequently the columns and foundations. Design for prestressing The amount of prestressing in a slab system is guided byparameters and requirements given in ACI 318 as well asnumerous other references, but the engineer has flexibility inadjusting the design for optimization of an individual project.Tendons for building construction usually are placed with aparabolic vertical profile to counteract a portion of the grav-ity loads on the structure (Figure 3). Thisundulating profile places the center of grav-ity, CGS, of the tendon force, P  , eccentric tothe neutral axis of the concrete section creatingthe primary moment, P  x e  . The eccentricity, e  , isthe distance from the tendon CGS to the section’s neutral axisat any examined cross section.The high and low points are governed by several parame-ters. The ends of the tendons usually are located at thesection’s neutral axis (mid-height for a slab), so as to notinduce additional moment at the anchors. In buildingconstruction, minimum cover requirements per ACI 318 aresufficient for slabs not exposed to a corrosive environment.Fire protection often governs the minimum concrete cover of these structures. Fire resistance issues are addressed in modelbuilding codes, assigned at the local level for a given struc-ture, and should be evaluated during the preliminary design. As it pertains to prestressed concrete two-way slabs, fire resist-ance requirements are dependent on the restrained versusunrestrained conditions. For most applications, the interior spans in the direction of frame design are consideredrestrained. That is, they are restrained against moving duringa fire loading. However, the end spans for a flat slab systemare considered unrestrained in the direction of the tendondesign. The fire resistance provisions do not match the mini-mum cover requirements per ACI 318 for reinforcementprotection, so it is necessary to reference the governing build-ing code (most likely Section 720 of the International CodeCouncil’s 2003 International Building Code) for settingconcrete cover parameters of the prestressed tendons. Typicaltwo-hour bottom covers to the tendons for slabs are 3/4 inch for restrained, interior spans and 1-1/2 inches for unre-strained, exterior spans. When the design is performed manually, the post-tension-ing force typically is selected to balance a specified percent-age of the floor self weight. Superimposed dead loads suchas partitions, flooring, mechanical equipment, and live loadsare not included, since they are not present at the time of stressing. Common load-balancing percentages are in the65-percent to 80-percent range and should be kept relativelyconsistent between spans. Codes do not prescribe limita-tions for these percentages, but engineers still need todesign to appropriate balancing loads to limit slab deflec-tions and cracking.The load-balancing effects reduce the amount of flexuralstresses for ultimate requirements, helping economizemember sizes and materials. Another advantage is the signif-icantly reduced deflections. With a percentage of the dead Special Advertizing Section — Portland Cement AssociationPDH 3 Post-Tensioning for Two-Way Flat Plate Construction Figure 2: Tributary slab widths for equivalent frames (Aalami, B.and Bommer, A., Design Fundamentals of Post-Tensioned Concrete Floors  , Post-Tensioning Institute, Phoenix, AZ, 1999)Figure 3: Tendon profile of a continuous post-tensioned beam.  load being balanced by an upward uniformload, the result is little or no dead loaddeflection. The benefits are most noticeablein the long-term deflection calculationsbecause the structure experiences deflection only from the remaining unbalanced dead load and the live load.These advantages are significant, but excessive use of post-tensioning can be detrimental. Note that it is not consideredconservative to over-balance the gravity loads or over-precompress the section because unwanted camber andexcessive cracking may occur. It is imperative to remember that post-tensioning is active reinforcement, exerting its load for the life of the structure.The effect of a prestressing force on a member can be eval-uated by replacing the tendon with equivalent, externallyapplied loads. The designed force in the tendons, P  , will be a function of the designer-specified equivalent balancing load, w b ; the span length,  l ; and its associated maximum drape, a  .For a simply supported determinate span with a parabolictendon profile, the formula for balancing a uniformly distrib-uted load is P  = w b  l  2 / 8 a  .For a multispan indeterminate structure, the designprocess has additional considerations. Since the eccentricityand lengths may vary between spans, the prestressing force, P  , needs to be determined for each span. The greatest force, P  max  , typically is selected for the entire equivalent frame, butcode requirements and other guidelines (discussed later) mayinfluence the final effective force. Post-tensioning design is aniterative process to determine an optimized solution. Fromthe selected equivalent frame force, the resulting balancingload in each span must be checked to ensure the percentagesare within the range selected by the designer. The percentageis determined from w b / w DL , where w b = 8 P  max  a  /  l  2 .If the balanced loads are above the acceptable limits in agiven span, in lieu of adjusting the force, the engineer has theoption to alter the tendon drape, a  , to reduce the balancedload, w b . It is more efficient to alter the drape by raising thebottom tendon ordinate while maintaining the top ordinates.By doing so, the only change in the construction process is toprovide higher supporting chair heights, instead of alteringthe entire top steel configuration at the supports (Figure 4).It can be difficult to balance a long end bay with the same force calculated for the same length or shorter adjacent inte-rior bays. This is because of the reduction in eccentricity fromthe anchors being located at the neutral axis and the increase for bottom cover requirements at the end bay. Many times,the end bays will require additional single-span tendons for an increased force to achieve an appropriate balancing load. Prestress losses  Although 1/2-inch-diameter, 270-ksi unbonded tendonscan be stressed to 33 kips of force per ACI 318 Section 18.5.1(0.8 x 270 ksi x 0.153 square inch = 33 kips), they will notretain this maximum force for the life of the structure. The force calculated from the balancing load, w b , is an effectivevalue,  f   se  x  A   ps  , where  f   se  , the reduced effective stress, is foundafter calculatingprestress losses.Instantaneous prestresslosses arise from theseating of the wedgesinto the anchor, elasticshortening of theconcrete, and frictionalong the length of thetendon. The long-termstress losses are causedby creep and shrinkageof the concrete and relaxation of the prestressing steel.Further explanation of prestress losses can be found in thePost-Tensioning Institute's Post-Tensioning Manual.Common prestress losses range from 15 ksi to 20 ksi.Therefore, an effective force of 26.6 kips commonly is used for one, 1/2-inch-diameter, 270-ksi tendon calculated fromthe equation  A   ps  (0.7  f    pu -losses). By assuming losses of 15 ksi,(0.153 square inch)[(0.7 x 270 ksi) – 15 ksi] = 26.6 kips.Specifying the effective force as a multiple of 26.6 kips willresult in a more efficient and constructible design. Additional design parameters and requirements  With the required effective tendon force calculated, the following code parameters and requirements may influencethe final design value: Limitation of the average prestress — For slabs notexposed to corrosive environments, the minimum averageprestress is 125 pounds per square inch (psi) and the recommended maximum average prestress is 300 psi.Therefore, the prestress force is limited by 125 psi ≤ P  /  A  ≤ 300psi, where  A  is the gross cross-sectional area of the total tributary slab width. Limitation for service load stresses — The ACI 318provisions limit tensile stresses in the concrete to control thedevelopment of flexural cracking. First introduced in the2002 version, ACI 318-05 classifies flexural members basedupon computed extreme fiber stress in tension,  f   t  , at serviceloads as Class U (Uncracked), Class T (Transition), or Class C(Cracked). Depending upon the classification, members areassumed to behave as cracked or uncracked sections for serv-ice load stress and deflection calculations. Per ACI 318-05Section 18.3.3, prestressed two-way slabs are to be designedas Class U with  f   t   ≤ 6 √  f    ′ c  where  f    ′ c  is the specified concretestrength. This requirement permits service load stresses to becomputed using uncracked section properties.The remaining unbalanced service loads, w net  = w DL+LL – w b , create stresses in the concrete. Upon determining theresulting net moments,  M  net  , flexural stresses can be evaluated at the top and bottom fibers of each critical section, typically at the midspans and supports. Therefore,  f   t  = – P  /  A  +  M  net   c  /  I   g. These stresses then are compared against the maximumcode permissible values for compressive stress,  f   c   ≤ 0.45  f    ′ c  4PDH Special Advertizing Section — Portland Cement Association Post-Tensioning for Two-Way Flat Plate Construction Figure 4: Recommended rebar andtendon layout at columns uses No. 4bars for top steel reinforcement tomatch the 1/2-inch tendon diameter.
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