Prestress Loss Distributions along Simply Supported Pretensioned Concrete Beams

Transcription

1 Southern Illinois University Carbondale OpenSIUC Publiations Department o Civil and Environmental Engineering 017 Prestress Loss Distributions along Simply Supported Pretensioned Conrete Beams Jen-kan Kent Hsiao hsiao@engr.siu.edu Follow this and additional works at: Reoended Citation Hsiao, Jen-kan K. "Prestress Loss Distributions along Simply Supported Pretensioned Conrete Beams." Eletroni Journal o Strutural Engineering Volume 16 ( Jan 017): This Artile is brought to you or ree and open aess by the Department o Civil and Environmental Engineering at OpenSIUC. It has been aepted or inlusion in Publiations by an authorized administrator o OpenSIUC. For more inormation, please ontat opensiu@lib.siu.edu.

2 Prestress Loss Distributions along Simply Supported Pretensioned Conrete Beams J. Kent Hsiao Department o Civil and Environmental Engineering, Southern Illinois University at Carbondale, Carbondale, IL 601, USA hsiao@engr.siu.edu ABSTRACT: The prestressing ores in prestressed tendons undergo a proess o redution over a period o time. A oon assumption is that prestress loss is onstantly distributed throughout the span o a simply supported pretensioned onrete beam. The purpose o this work is to investigate the auray o this assumption. The types o prestressed onrete beams investigated in this work inlude the ollowing three typial types o tendon proile: (I) straight strands, (II) single-point depressed, and (III) two-point depressed. The major indings derived rom this work are: (1) The total prestress loss is not onstantly distributed throughout the span o a simply supported pretensioned onrete beam with any o the three types o tendon proiles, () The variation o prestress loss along the span o a pretensioned beam aused by elasti shortening o onrete or reep o onrete is muh more signiiant than that aused by shrinkage o onrete or relaxation o tendons, and (3) The type o tendon proile in a simply supported pretensioned onrete beam has signiiant eets on the pattern o prestress loss distribution along the beam. 1 INTRODUCTION A pretensioned onrete beam is a prestressed onrete beam in whih the tendons are tensioned prior to asting the onrete. Strutural engineers typially use the ollowing three typial tendon proiles or the design o pretensioned onrete beams: (1) straight strands, () single-point depressed, and (3) two-point depressed, as shown in Figures 1(a), (b), and (), respetively (PCI 00). Note that.g.s. shown in Figure 1 represents the enter o gravity o prestressing steel and.g.. represents the enter o gravity o a onrete setion. The prestressing ores in prestressed tendons undergo a proess o redution over a period o time. The our major losses o prestress or pretensioned onrete beams (Lin 163; Nawy 010) are: (1) Elasti shortening o onrete (ES), () Creep o onrete (CR), (3) Shrinkage o onrete (SH), and () Relaxation o tendons (RE). Elasti shortening o onrete (EC) ours when the prestress in tendon is transerred to the onrete beam, whih auses the beam to shorten and the tendon to shorten with it, resulting in a prestress loss in the tendon. Creep o onrete (CR) is deined as time-dependent deormation in the onrete element resulting rom the presene o stress, whih in turn results in a prestress loss in the tendon. Shrinkage o onrete (SH) ours when the onrete element ontrats due to drying shrinkage in onrete (derease in the volume o a onrete element due to drying that is dependent on time and on moisture onditions), whih results in a prestress loss in the tendon. Relaxation o tendons (RE) reers to stress relaxation in prestressing steel, whih is the loss o stress when prestressing steel is prestressed and maintained at a onstant strain or a period o time..g.. (a) Straight strands (b) Single-point depressed.g...g...g.s.g.s. () Two-point depressed.g.s. Figure1. Typial prestressing tendon proiles or simply supported pretensioned onrete beams

3 LOSS OF PRESTRESS The ollowing addresses eah o the our major losses (PCA 008): 1) Elasti shortening o onrete (ES): For members with bonded tendons, ir ES KesEps (1) Ei where Kes = a ator (= 1.0 or pretensioned members); Eps = the modulus o elastiity o prestressing steel; Ei = the modulus o elastiity o onrete at the time o initial prestress; and ir = the net ompressive stress in onrete at the enter o gravity o prestressing steel iediately ater the prestress has been applied to the onrete, that is, ir = Kir pi g () where Kir = a ator (= 0. or pretensioned members); g = the stress in onrete at the enter o gravity o prestressing steel due to the weight o the struture at time the prestress is applied; and pi = the stress in onrete at the enter o gravity o the prestressing steel due to Ppi, that is, pi Ppi Ppie (3) A I where Ppi = prestressing ore in tendons beore redution or elasti shortening o onrete, reep o onrete, shrinkage o onrete, and relaxation o the tendons; A = gross area o the onrete setion, negleting reinorement; I = moment o inertia o the gross onrete setion about the entroidal axis, negleting reinorement; and e = the eentriity o gs with respet to g at the ross-setion onsidered. From Eqs. () and (3), one has: Ppi Ppie Mde ir Kir () A I I where Md = the moment due to the weight o the member at the time o prestressing. ) Creep o onrete (CR): The reep o onrete (CR) an be omputed using Eq. (5): E E ps CR = K r ir ds (5) where Kr = a ator (=.0 or pretensioned members); E = the modulus o elastiity o onrete at 8 days; and ds = the stress in onrete at the enter o gravity o prestressing steel due to all superimposed permanent dead loads that are applied to the member ater it has been prestressed. 3) Shrinkage o onrete (SH): The shrinkage o onrete (SH) an be omputed using Eq. (6): V S (6) SH = Ksh Eps RH where Ksh = a ator (= 1.0 or pretensioned members); V/S = the volume to surae ratio o the onrete member, usually approximately taken as the gross ross-setional area o onrete member divided by its perimeter [note that Eq. (6) uses the unit inh or the omputation o V/S]; and RH = the average relative humidity surrounding the onrete member. ) Relaxation o tendons (RE): The relaxation o tendon stress (RE) an be omputed using Eq. (7): RE = [Kre J(SH + CR + ES)]C (7) where Kre = 3.8 MPa (5000 psi) or 70 Grade low-relaxation strand; J = 0.00 or 70 Grade lowrelaxation strand, and C = a ator whih is related to the ratio o pi/pu (where pi is the prestressing stress in tendons beore redution or elasti shortening o onrete, reep o onrete, shrinkage o onrete, and relaxation o prestressing steel and pu is the speiied tensile strength o prestressing steel) and is used to determine RE. 3 LOSS OF PRESTRESS COMPUTATION EX- AMPLES Three examples are demonstrated or the omputation o prestress losses. These three examples inlude: (1) a onrete beam with straight strands, () a onrete beam with single-point depressed tendons, and (3) a onrete beam with two-point depressed tendons. Example 1: A simply supported onrete beam with a span length o 1. m (0 t.), as shown in Figure, is pretensioned using (0.5 in.) diameter low-relaxation strands with a modulus o elastiity, Eps, o 16,510 MPa (8,500 ksi). Compute the pre-

4 stress losses along the span o the beam. Assume that (1) pu (speiied tensile strength o prestressing steel) = 186 MPa (70 ksi), () pi (prestressing stress in tendons beore redution or elasti shortening o onrete, reep o onrete, shrinkage o onrete, and relaxation o prestressing steel) = 0.7pu, (3) i (the ompressive strength o onrete at the time o initial prestress) = 6.0 MPa (3,00 psi.), () (the speiied 8-day ompressive strength o onrete) = 37.3 Mpa (5,500 psi.), (5) the average relative humidity surrounding the onrete member is 75 (in %), (6) the superimposed permanent dead load that is applied to the girder ater it has been prestressed is.0 kn/m (0.1 kips/t), and (7) the prestress transer point loated within 0.61 m ( t.) rom the support. Sine the area o 1.7 (½") Grade 70 strand, Aps = 8.7 (0.153 in. ) (Nawy 010), Ppi = 0.7puAps = 0.7(186 MPa)(6)(8.7 ) = 816 kn. Also, reerring to Figure (b), A = (305)(660 ) = 01,300, e = (660 / ) - 51 = 7 and I = (305)(660) 3 /1 = From Eq. (), one has: ir 816kN 0. 01, kN m kN M Pa Prestressing steel 660 ('-") For normal-weight onrete, the modulus o elastiity o onrete at time o initial prestress, Ei, an be omputed to be (AASHTO 007): 1. m (0'-0") (a) Elevation E i ,00MPa ES 8.10MPa,00MPa ,510MPa 63.6MPa (½") strand 305 (1") (b) Cross-setion 610 ('-0") 51 (") 660 ('-") Figure. Pretensioned beam with straight strands or Example 1 Loss o Prestress at Midspan (6.1 m rom the support) (1) Computation o ES: Reerring to Figure, the moment at the midspan due to the weight o the beam [the weight o the beam is estimated to be 3.55 kn/m 3 (0.15 kips/t 3 )] at the time o prestressing an be omputed to be: () Computation o CR: The modulus o elastiity o onrete at 8 days an be omputed to be: E =,560MPa The moment at the midspan due to superimposed permanent dead load that is applied to the beam ater it has been prestressed an be omputed to be: M ds Thereore, ds 1.m (.0kN/m) kN m kN m = 1. MPa M d 1.m kN/m 8 = 88.0 kn m 16,510,560 CR = = 88.8 MPa

5 (3) Computation o SH: Reerring to Figure, one has: V S 1in. 6in. 6in. 1in..11 Note that Eq. (6) uses the unit inh or the omputation o V/S. From Eq. (6), one has: SH = (1.0) (16,510MPa) =30.35 MPa () Computation o RE: RE = [ ( )](0.5) = 5.81 MPa Note that C = 0.5 or pi / pu = 0.7 (PCA 008). The omputations shown above result in the total prestress losses at the midspan = ES + CR + SH + RE = = MPa Loss o Prestress at a Distane o 3.05 m (10 t.) rom the Support (1) Computation o ES: Reerring to Figure, the moment at a distane o 3.05 m (10 t.) rom the support due to the weight o the beam an be omputed to be: M d kN/m 6.1m 3.05m 3.05m 1.55m 66.15kN m Substituting Ppi = 816 kn, e = 7, and Md = kn m into Eq. (), one has: ir 816kN 0. 01, kN m kN M Pa 8.5MPa ES ,510MPa = MPa,00MPa () Computation o CR: The moment at a distane o 3.05 m rom the support due to the superimposed permanent dead load that is applied to the beam ater it has been prestressed an be omputed to be: M ds (.0kN/m) 8.66kN m Thereore, ds 6.1m 3.05m 3.05m 1.55m 8.66kN m ,510,560 CR = = MPa = 10. MPa (1) Computation o SH: SH = MPa (SH remains the same or the entire length o the span.) () Computation o RE: RE = [ ( )](0.5) =.5 MPa The omputations shown above result in the total prestress losses at a distane o 3.05 m (10 t.) rom the support = ES + CR + SH + RE = = 30.3 MPa Loss o Prestress at a Distane o 0.61 m ( t.) rom the Support (1) Computation o ES: Reerring to Figure, the moment at a distane o 0.61 m ( t.) rom the support due to the weight o the beam an be omputed to be: M d kN/m 6.1m 0.61m 0.61m 0.305m kN m Substituting Ppi = 816 kn, e = 7 and Md = kn m into Eq. (), one has: ir kN 01, kN m kN M Pa 10.83MPa ES ,510MPa = 85. MPa,00MPa

6 () Computation o CR: The moment at a distane o 0.61 m ( t.) rom the support due to the superimposed permanent dead load that is applied to the beam ater it has been prestressed an be omputed to be: M ds (.0kN/m) 7.11kN m Thereore, ds 6.1m 0.61m 0.61m 0.305m 7.11kN m ,510,560 CR = = 0.75 MPa = MPa (3) Computation o SH: SH = MPa (SH remains the same or the entire length o the span.) () Computation o RE: RE = [ ( )](0.5) = 3.0 MPa Thereore, the total prestress losses at a distane o 3.05 m (10 eet) rom the midspan = ES + CR + SH + RE = = 7. MPa. Using the same omputation proedure, the prestress losses at the ross-setions at an interval o 0.61 m ( t.) along the span o the beam are omputed and are suarized in Table 1. As shown in Table 1, the prestress loss at the setion lose to the ends o the beam is about 3 % larger than that at the midspan. Table 1. Prestress losses at the ross-setions at an interval o 0.61 m ( t.) along the span o the beam in Example 1 loation (distane rom the support) ES CR SH RE total losses ES+CR+SH +RE (m) Example : Repeat o Example 1 or a onrete beam o 1. m (0 t.) simple span with singlepoint depressed tendons, as shown in Figure m (0'-0") (½") strand 305 (1") Prestressing steel (a) Elevation (b) Cross-setion 6.1 m (0'-0") 660 ('-") 660 ('-") 51 ( in.) at midspan 330 (13 in.) at support Figure 3. Pretensioned beam with single-point depressed strands or Example Loss o Prestress at Midspan Reerring to Example 1, the total prestress losses at the midspan = ES + CR + SH + RE = = MPa. Loss o Prestress at a Distane o 3.05 m (10 t.) rom the Support

7 (1) Computation o ES: Reerring to Example 1, the moment at a distane o 3.05 m (10 t.) rom the support due to the weight o the beam an be omputed to be Md = kn m. Substituting Ppi = 816 kn, e = 13.5, and Md = kn m into Eq. (), one has: ir kN 01, kN m kN M Pa.31MPa ES ,510MPa = 3.6 MPa,00MPa () Computation o CR: Reerring to Example 1, the moment at a distane o 3.05 m rom the support due to superimposed permanent dead load that is applied to the beam ater it has been prestressed an be omputed to be Mds = 8.66 kn m. Thereore, ds 8.66kN m ,510,560 CR = = 0.53 MPa = MPa (3) Computation o SH: SH = MPa (SH remains the same or the entire length o the span.) () Computation o RE: RE = [ ( )](0.5) = 8.38 MPa The omputations shown above result in the total prestress losses at a distane o 3.05 m (10 t.) rom the support = ES + CR + SH + RE = = 13. MPa Loss o Prestress at a Distane o 0.61 m ( t.) rom the Support Reerring to Example 1, the moment at a distane o 0.61 m ( t.) rom the support due to the weight o the beam an be omputed to be Md = kn m. Substituting Ppi = 816 kn, e = 7. and Md = kn m into Eq. (), one has: ir kN 01, kN m kN M Pa 3.663MPa ES ,510MPa = 8.1 MPa,00MPa () Computation o CR: Reerring to Example 1, the moment at a distane o 0.61 m ( t.) rom the support due to superimposed permanent dead load that is applied to the beam ater it has been prestressed an be omputed to be Mds = 7.11 kn m. Thereore, ds 7.11kN m ,510,560 CR = = 0.08 MPa = 8.33 MPa (3) Computation o SH: SH = MPa (SH remains the same or the entire length o the span.) () Computation o RE: RE = [ ( )](0.5) = 8.67 MPa Thereore, the total prestress losses at a distane o 3.05 m (10 t.) rom the midspan = ES + CR + SH + RE = = MPa. Following the same omputation proedure, the prestress losses at the ross-setions at an interval o 0.61 m ( t.) along the span o the beam are omputed and are suarized in Table. (1) Computation o ES:

8 As shown in Table, the prestress loss at the setion lose to the ends o the beam is about 35 % smaller than that at the midspan. Table. Prestress losses at the ross-setions at an interval o 0.61 m ( t.) along the span o the beam in Example loation (distane rom the support) ES CR SH RE total losses ES+CR+SH +RE (m) As shown in Table 3, the prestress loss at the setion lose to the ends o the beam is about 3 % smaller than that at the midspan. Table 3. Prestress losses at the ross-setions at an interval o 0.61 m ( t.) along the span o the beam in Example 3 loation (distane rom the support) ES CR SH RE total losses ES+CR+SH +RE (m) Example 3: Repeat o Example 1 or a onrete beam o 1. m (0 t.) simple span with two-point depressed tendons, as shown in Figure. End segment.7 m (1'-0") (½") strand 305 (1") Prestressing steel Midsegment 3.66 m (1'-0") (a) Elevation (b) Cross-setion End segment.7 m (1'-0") 660 ('-") 660 ('-") 51 ( in.) in midsegment 330 (13 in.) at support Figure. Pretensioned beam with two-point depressed strands or Example 3 Using the same omputation proedure shown in Examples 1 and, the prestress losses at the rosssetions at an interval o 0.61 m ( t.) along the span o the beam shown in Example 3 are omputed and are suarized in Table 3. RESULTS AND DISCUSSION Table 1 suarizes the variation o prestress losses along the span o a simply supported pretensioned onrete beam with straight tendons. The table indiates that: (1) The ES loss is the lowest at the midspan and gradually inreases towards the ends o the beam; () The CR loss is the lowest at the midspan and gradually inreases towards the ends o the beam; (3) The SH loss remains the same throughout the entire span o the beam; and () The RE loss is the highest at the midspan and slightly dereases towards the ends o the beam. Table suarizes the variation o prestress losses along the span o a simply supported pretensioned onrete beam with single-point depressed tendons. The table indiates that: (1) The ES loss is the highest at the midspan and gradually dereases towards the ends o the beam; () The CR loss is the highest at the midspan and gradually dereases towards the ends o the beam; (3) The SH loss remains the same throughout the entire span o the beam; and () The RE loss is the lowest at the midspan and slightly inreases towards the ends o the beam. Table 3 suarizes the variation o prestress losses along the span o a simply supported pretensioned onrete beam with two-point depressed tendons. The table indiates that: (1) The ES loss slightly inreases rom the midspan towards both ends in the midsegment and then sharply dereases towards both ends o the beam; () The CR loss slightly inreases rom the midspan towards both ends in the midsegment and then sharply dereases towards both ends o the beam; (3) The SH loss remains the same throughout the entire span o the beam; and () The RE loss slightly dereases rom the midspan towards

9 both ends in the midsegment and then slightly inreases towards both ends o the beam. Suaries o the total prestress loss distributions (ES + CR + SH + RE) or a Type I beam (a prestressioned beam with straight tendons, as shown in Example 1) in Table 1, a Type II beam (a prestressioned beam with single-point depressed tendons, as shown in Example ) in Table, and a Type III beam (a prestressioned beam with two-point depressed tendons, as shown in Example 3) in Table 3 are graphially presented in Figure 5. Figure 5 graphially desribes the variations o the total prestress losses along the spans o the three types o pretentioned onrete beams with a straight tendon proile, a single-depressed tendon proile, and a two-depressed tendon proile, respetively. The igure signiies that: (1) the total prestress loss in a Type I beam (with a straight tendon proile) gradually inreases rom the midspan towards both ends o the beam; () the total prestress loss in a Type II beam (with a single-depressed tendon proile) gradually dereases rom the midspan towards both ends o the beam; and (3) the total prestress loss in a Type III beam (with a two-depressed tendon proile) slightly inreases rom the midspan towards the both ends in the midsegment o the beam and then sharply dereases towards both ends o the beam. Figure 5. Prestress loss distributions along the beams in Examples 1,, and 3 5 CONCLUSIONS The indings derived rom this work are suarized as the ollowing: (1) None o the three typial types o tendon proiles have a prestress loss onstantly distributed throughout the span o the beam; () The variation in the prestress loss along the span o a pretensioned beam aused by the elasti shortening o onrete or the reep o onrete is muh more signiiant than that aused by the shrinkage o onrete or the relaxation o tendons; (3) The type o tendon proile in a simply supported pretensioned onrete beam has signiiant eets on the pattern o prestress loss distribution along the span o the beam; () The total prestress loss lose to the ends o the beam is larger than that at the midspan or a Type I pretensioned beam; and (5) The total prestress loss lose to the ends o the beam is smaller than that at the midspan or both Type II and Type III pretensioned beams. REFERENCES Amerian Assoiation o State Highway and Transportation Oiials, AASHTO LRFD Bridge Design Speiiations, SI Units, th Ed., Amerian Assoiation o State Highway and Transportation Oiials, Washington, DC, 007. Lin, T.Y., Design o Prestressed Conrete Strutures, nd Ed., John Wiley & Sons, In., New York, 163. Nawy, E.G., Prestressed Conrete: A Fundamental Approah, 5 th Ed., Pearson Eduation, In., Upper Saddle River, New Jersey, 010. Portland Cement Assoiation, Notes on ACI Building Code Requirements or Strutural Conrete, Portland Cement Assoiation, Skokie, IL, 008. Preast/Prestressed Conrete Institute, PCI Design Handbook: Preast and Prestressed Conrete, 6th Ed., Preast/Prestressed Conrete Institute, Chiago, IL, 00. A oon assumption is that prestress loss is onstantly distributed throughout the span o a simply supported pretensioned onrete beam. This work investigates the auray the assumption or the ollowing three typial types o tendon proiles: (I) straight strands throughout the beam, (II) singlepoint depressed with the.g.s. and the.g.. loated at the same elevation at the ends o the beam, and (III) two-point depressed with the.g.s. and the.g.. loated at the same elevation at the ends o the beam.