Arabian Journal for Science and Engineering

, Volume 44, Issue 10, pp 8149–8170 | Cite as

Numerical and Analytical Behavior of Beams Prestressed with Unbonded Internal or External Steel Tendons: A State-of-the-Art Review

  • Maha AlqamEmail author
  • Fadi Alkhairi
Review --Civil Engineering


This paper presents a detailed state-of-the-art chronological account of past studies dating back to the late 1980s consisting of an overview of the vast majority of prediction equations for the stress at ultimate in prestressed unbonded internal or external steel tendons of simply supported concrete beams. The paper also details past studies carried out to investigate the complete nonlinear behavior of such beams throughout loading. Past and relevant studies were carefully reviewed in detail and then synthesized in the most reduced, yet accurately depicted, format in an effort to capture the pertinent highlights. For each review, the most relevant factors considered by the authors were highlighted, including span-to-depth ratio, second-order effects, length of the plastic hinge, friction and/or slippage at deviators, and experimental verification. An evaluation was carried out to compare and contrast the various nonlinear analysis studies using a simple tabulated format. Gaps were identified, and recommendations for future research were further proposed by the authors.


External prestressing Unbonded tendons Nonlinear analysis Second-order effects Ultimate strength 

List of Symbols


Area of prestressed unbonded reinforcement


Area of non-prestressed tensile reinforcement


Beam width of a rectangular section


Depth of neutral axis at ultimate nominal strength


Distance from extreme compressive fiber to the centroid of the non-prestressed reinforcement


Depth from the extreme to compressed fiber to the centroid of the pretsressing steel


Eccentricity of the unbonded prestressed reinforcement at any point along the beam


Eccentricity of the prestressing steel at midspan


Eccentricity of the prestressing steel at the support


Modulus of elasticity of the prestressing steel


Stress in the concrete top fiber of a simply supported beam


Effective prestress in unbonded tendons


Stress in the unbonded prestressed reinforcement at nominal strength


Characteristic (ultimate) strength of the prestressed reinforcement


Yield stress of unbonded tendons


Stress in the non-prestressed tensile reinforcement


Yield stress in the non-prestressed reinforcement


Concrete compressive strength


Finite difference method


Finite element method


Fiber-reinforced polymers


Generalized iterative analysis


Cracked moment of inertia


Transformed moment of inertia


Length of the simply supported beam


Length of the region of constant moment [17]


Length of the equivalent plastic region [42]


Length of the equivalent plastic hinge [54, 64]


Length of the undeformed external tendon between end anchorages [64]


Length of the undeformed inclined external tendon between end anchorage and deviator [64]


External moment at any point x along the span


Internal moment at x along the span calculated from force and moment equilibrium


Moment at ultimate nominal strength


Plastic moment calculated excluding unbonded prestressing force at ultimate


Concentrated load at midspan


Concentrated load at the third point


Stress increment reduction factor responsible for second-order effects [58]


Spacing of the deviators [43, 64]


Elastic stiffness matrix


Geometric stiffness matrix


Tangent stiffness matrix


A ps x f ps


Distance measured from simply supported end up to deviator at the third point [64]


Shear span or the distance between the point of maximum moment and the point of contra flexure


Midspan deflection


Stress increase in the unbonded tendons beyond effective prestress


Midspan deflection [54]


Average elongation in the unbonded external tendon at ultimate [64]


Average elongation in the unbonded tendon at a specific loading stage


Strain at the concrete top fiber of a simply supported beam at ultimate


Strain in the prestressed reinforcement obtained from the stress–strain curve


A fraction between 0.65 and 0.85 used by the ACI Code to compute the width of the Whitney stress block


Beam curvature assuming an elastic cracked section


Beam curvature assuming yielding of the prestressed reinforcement


Beam curvature assuming ultimate nominal strength


Beam curvature assuming yielding of the non-prestressed reinforcement


Strain reduction coefficient of unbonded to bonded tendons


Strain reduction coefficient assuming elastic cracked analysis


Strain reduction coefficient assuming ultimate nominal strength


Shape function that varies between 0 and 1 derived from experimental tests [64]


  1. 1.
    Ng, C.K.; Tan, K.H.: Flexural behavior of externally prestressed beams. Part I: analytical model. Eng. Struct 28(4), 609–621 (2006)Google Scholar
  2. 2.
    Collins, M.P.; Mitchell, D.: Prestressed Concrete Basics. Canadian Prestressed Concrete Institute, Ottawa (1987)Google Scholar
  3. 3.
    Harajli, M.H.; Naaman, A.E.: Evaluation of the Inelastic Behavior of Partially Prestressed Concrete Beams. Report No. UMCE 85-02. University of Michigan, Ann Arbor (1985)Google Scholar
  4. 4.
    Naaman, A.E.; Burns, N.; French, C.; Gamble, W.L.; Mattock, A.H.: Stresses in unbonded prestressing tendons at ultimate: recommendation. ACI Struct. J. 99(4), 518–529 (2002)Google Scholar
  5. 5.
    Harajli, M.: On the stress in unbonded tendons at ultimate: critical assessment and proposed changes. ACI Struct. J. 103(6), 803–812 (2006)Google Scholar
  6. 6.
    Naaman, A.E.; Alkhairi, F.M.: Stress at ultimate in unbonded post-tensioning tendons: part 1—evaluation of the state of the art. ACI Struct. J. 88(5), 641–651 (1991)Google Scholar
  7. 7.
    Harajli, M.H.: Effect of span-depth ratio on the ultimate steel stress in unbonded prestressed concrete members. ACI Struct. J. 87(3), 305–312 (1990)Google Scholar
  8. 8.
    Alkhairi, F.M.; Naaman, A.E.: Analysis of beams prestressed with unbonded internal or external tendons. J. Struct. Eng. 119(9), 2680–2700 (1993)Google Scholar
  9. 9.
    Naaman, A.E.: An approximate nonlinear design procedure for partially prestressed concrete beams. Comput. Struct. 17(2), 287–293 (1983)Google Scholar
  10. 10.
    EI-Habr, K.C.: Finite Element Analysis of Externally Prestressed Segmental Construction. M.Sc. Thesis, University of Texas at Austin, Austin (1988)Google Scholar
  11. 11.
    Muller, J.; Gauthier, Y: Ultimate behavior of precast segmental box girders with external tendons. External prestressing in bridges. In: Naaman, A.E., Breen, J.E. (eds.) ACI SP 120-17, Proceedings of International Symposium, pp. 355–373. American Concrete Institute (ACI). Detroit (1989)Google Scholar
  12. 12.
    Naaman, A.E.: A new methodology for the analysis of beams prestressed with external or unbonded tendons. External prestressing in bridges. In: ACI Special Publication SP-120, (pp. 339–354). American Concrete Institute, Detroit (1990)Google Scholar
  13. 13.
    Naaman, A.E.; Alkhairi, F.M.: Stress at ultimate in unbonded post-tensioning tendons: part 2—proposed methodology. ACI Struct. J. 88(6), 683–692 (1991)Google Scholar
  14. 14.
    AASHTO LRFD Bridge design specification. Washington, D.C. (1994)Google Scholar
  15. 15.
    Mutsuyoshi, H.; Tsuchida, K.; Matupayont, S.; Machida, A.: Flexural behavior and proposal of design equation for flexural strength of externally PC members. J. Mater. Concr. Struct. Pavements. Japan Society of Civil Engineers. No. 508/V-26, Feb., 67–76 (1995) (in Japanese) Google Scholar
  16. 16.
    Naaman, A. E.: Stress at ultimate in unbonded prestressing tendons by strain compatibility. Progress in structural engineering. In: Proceedings of an international workshop on progress and advances in structural engineering and mechanics. University of Brescia, Italy (1991)Google Scholar
  17. 17.
    Harajli, M.H.; Hijazi, S.A.: Evaluation of the ultimate steel stress in partially prestressed concrete members. PCI J. 36(1), 62–82 (1991)Google Scholar
  18. 18.
    Harajli, M.H.; Kanj, M.Y.: Service load behavior of concrete members prestressed with unbonded tendons. J. Struct. Eng. 118(9), 2569–2589 (1992)Google Scholar
  19. 19.
    Tan, K.H.; Naaman, A.E.: Strut-and-tie model for externally prestressed concrete beams. Struct. J. 90(6), 683–691 (1993)Google Scholar
  20. 20.
    Harajli, M.: Strengthening of concrete beams by external prestressing. Prestress. Concr. Inst. 38(6), 76–88 (1993)Google Scholar
  21. 21.
    Picard, A.; Massicotte, B.; Bastien, J.: Relative efficiency of external prestressing. J. Struct. Eng. 121(12), 1832–1841 (1995)Google Scholar
  22. 22.
    Rao, P.S.; Mathew, G.: Behavior of externally prestressed concrete beams with multiple deviators. Struct. J. 93(4), 387–396 (1996)Google Scholar
  23. 23.
    Pisani, M.A.: A numerical model for externally prestressed beams. Struct. Eng. Mech. 4(2), 177–190 (1996)Google Scholar
  24. 24.
    Pisani, M.A.; Nicoli, E.: Beams prestressed with unbonded tendons at ultimate. Can. J. Civ. Eng. 23, 1220–1230 (1996)Google Scholar
  25. 25.
    Ramos, G.; Aparicio, A.C.: Ultimate analysis of monolithic and segmental externally prestressed concrete bridges. J. Bridge Eng. 1(1), 10–17 (1996)Google Scholar
  26. 26.
    Moon, J.H.; Burns, N.H.: Flexural behavior of members with unbonded tendons. II: applications. J. Struct. Eng. 123(8), 1095–1101 (1997)Google Scholar
  27. 27.
    Dall’Asta, A.; Dezi, L.: Nonlinear behavior of externally prestressed composite beams: analytical model. J. Struct. Eng. 124(5), 588–597 (1998)Google Scholar
  28. 28.
    Harajli, M.; Khairallah, N.; Nassif, H.: Externally prestressed members: evaluation of second-order effects. J. Struct. Eng. 125(10), 1151–1161 (1999)Google Scholar
  29. 29.
    Oh, B.H.; Yoo, S.W.: Flexural analysis of prestressed concrete bridges with external unbonded tendons. J. Korean Soc. Civ. Eng. 19(15), 761 (1999)Google Scholar
  30. 30.
    Allouche, E.N.; Campbell, T.I.; Green, M.F.; Soudki, K.A.: Tendon stress in continuous unbonded prestressed concrete members part 2: parametric study. PCI J. 44(1), 60–73 (1999)Google Scholar
  31. 31.
    Park, R.; Priestley, M.J.N.; Scott, B.D.: Stress–strain behavior of concrete confined by overlapping hoops at low and high strain rates. ACI Struct. J. 79(1), 13–27 (1982)Google Scholar
  32. 32.
    Menegotto, M.; Pinto, P. E.: Method of analysis for cyclically loaded R.C. plane frames. In: IABSE Preliminary Report for Symposium on Resistance and Ultimate Deforma-bility of Structures Acted on by Well Defined Repeated Loads. Lisbon, pp. 15–22 (1973)Google Scholar
  33. 33.
    Lee, L.H.; Moon, J.H.; Lim, J.H.: Proposed methodology for computing of unbonded tendon stress at flexural failure. ACI Struct J. 96(6), 1040–1048 (1999)Google Scholar
  34. 34.
    Ariyawardena, N.; Ghali, A.: Prestressing with unbonded internal or external tendons: analysis and computer model. J. Struct. Engineering. 128(12), 1493–1501 (2002)Google Scholar
  35. 35.
    Tan, K.-H.; Ng, C.-K.: Effects of DEVIATORS AND TENDON CONfiGURATION ON BEHAVIOR OF EXTERNALLY PRESTRESSED BEAMS. ACI Struct. J. 94(1), 13–22 (1997)Google Scholar
  36. 36.
    Ariyawardena, N.; Ghali, A.: Design of precast prestressed concrete members using external prestressing. Prestress. Concr. Inst. J. 47(2), 84–94 (2002)Google Scholar
  37. 37.
    Harajli, M.H.; Mabsout, M.E.; Al-Hajj, J.A.: Response of externally post-tensioned continuous members. ACI Struct. J. 99(5), 671–680 (2002)Google Scholar
  38. 38.
    Diep, B.K.; Umehara, H.: Non-linear analysis of externally prestressed concrete beams. Electron. J. Struct. Eng. 2, 85–96 (2002)Google Scholar
  39. 39.
    Wu, X.H.; Lu, X.: Tendon model for nonlinear analysis of externally prestressed concrete structures. J. Struct. Eng. 129(1), 96–104 (2003)Google Scholar
  40. 40.
    Ng, C.K.: Tendon stress and flexural strength of externally prestressed beams. ACI Struct. J. 100(5), 644–653 (2003)Google Scholar
  41. 41.
    El-Ariss, B.: Stiffness of reinforced concrete beams with external tendons. Eng. Struct. 26(14), 2047–2051 (2004)Google Scholar
  42. 42.
    Au, F.K.; Du, J.S.: Prediction of ultimate stress in unbonded prestressed tendons. Mag. Concr. Res. 56(1), 1–11 (2004)Google Scholar
  43. 43.
    Ghallab, A.; Beeby, A.W.: Factors affecting the external prestressing stress in externally strengthened prestressed concrete beAMs. Cement Concr. Compos. 27, 945–957 (2005)Google Scholar
  44. 44.
    ACI (American Concrete Institute) ACI 318-14: Building Code Requirements For Structural Concrete And Commentary. Farmington Hills (2014)Google Scholar
  45. 45.
    BSI BS 8110: Structural use of concrete. Part 1: code of practice for design and construction. BSI, London (1997)Google Scholar
  46. 46.
    Abdel Aziz, M.; Abdel-Sayed, G.; Ghrib, F.; Grace, N.F.; Madugula, M.S.: Analysis of concrete beams prestressed and post-tensioned with externally unbonded carbon fiber reinforced polymer tendons. Can. J. Civ. Eng. 32(6), 1138–1151 (2005)Google Scholar
  47. 47.
    Simulia: Abaqus Theory Manual. Version 6.14, Simulia, Providence (2014)Google Scholar
  48. 48.
    Lou, T.; Xiang, Y.: Finite element modeling of concrete beams prestressed with external tendons. Eng. Struct. 28(14), 1919–1926 (2006)Google Scholar
  49. 49.
    Roberts-Wollmann, C.L.; Kreger, M.E.; Rogowsky, D.M.; Breen, J.E.: Stresses in external tendons at ultimate. ACI Struct. J. 102(2), 203–213 (2006)Google Scholar
  50. 50.
    MacGregor, R.J.G.: Strength and Ductility of Externally Post-Tensioned Segmental Box Girders. Ph.D. dissertation, The University of Texas at Austin, Austin (1989)Google Scholar
  51. 51.
    Manisekar, R.; Senthil, R.: Stress at ultimate in unbonded post tensioning tendons for simply supported beams: a state-of-the-art review. Adv. Struct. Eng. 9(3), 321–335 (2006)Google Scholar
  52. 52.
    Dall’Asta, A.; Ragni, L.; Zona, A.: Simplified method for failure analysis of concrete beams prestressed with external tendons. J. Struct. Eng. 133(1), 121–131 (2007)Google Scholar
  53. 53.
    Dall’Asta, A.; Zona, A.: Non-linear analysis of composite beams by a displacement approach. Comput. Struct. 80(27), 2217–2228 (2002)Google Scholar
  54. 54.
    Ozkul, O.; Nassif, H.; Tanchan, P.; Harajli, M.: Rational approach for predicting stress in beams with unbonded tendons. ACI Struct. J. 105(3), 338–347 (2008)Google Scholar
  55. 55.
    Vu, N.; Castel, A.; François, R.: Response of post-tensioned concrete beams with unbonded tendons including serviceability and ultimate state. Eng. Struct. 32(2), 556–569 (2010)Google Scholar
  56. 56.
    Nassif, H.; Ozkul, O.; Harajli, M.H.: Flexural behavior of beams prestressed with bonded and unbonded. PTI J. 1, 60–71 (2003)Google Scholar
  57. 57.
    Au, F.T.K.; Chan, K.H.E.; Kwan, A.K.H.; Du, J.S.: Flexural ductility of prestressed concrete beams with unbonded tendons. Comput. Concr. 6(6), 451–472 (2009)Google Scholar
  58. 58.
    He, Z.Q.; Liu, Z.: Stresses in external and internal unbonded tendons: unified methodology and design equations. J. Struct. Eng. 136(9), 1055–1065 (2010)Google Scholar
  59. 59.
    Chakrabarti, P.R.: Ultimate stress for unbonded post-tensioning tendons in partially prestressed beams. ACI Struct. J. 92(6), 689–697 (1995)Google Scholar
  60. 60.
    Lou, T.; Xiang, Y.: Numerical analysis of second-order effects of externally prestressed concrete beams. Struct. Eng. Mech. 35(5), 631–643 (2010)Google Scholar
  61. 61.
    Lee, D.H.; Kim, K.S.: Flexural strength of prestressed concrete members with unbonded tendons. Struct. Eng. Mech. 38(5), 675–696 (2011)Google Scholar
  62. 62.
    Kim, K.S.; Lee, D.H.: Flexural behavior model for post-tensioned concrete members with unbonded tendons. Comput. Concr. 10(3), 241–258 (2012)Google Scholar
  63. 63.
    Senthil, R.; Manisekar, R.: Ultimate flexural behaviour of externally prestressed new beams and distressed beams. J. Eng. Sci. Technol. 10(4), 461–484 (2015)Google Scholar
  64. 64.
    Peng, H.; Xue, W.; Tan, Y.: Design approach for flexural capacity of prestressed concrete beams with external tendons. J. Struct. Eng. (2018). Google Scholar
  65. 65.
    Harajli, M.H.: Tendon stress at ultimate in continuous unbonded post-tensioned members: proposed modification of ACI Eqs. (18–4) and (18–5). ACI Struct. J. 103(2), 183–192 (2012)Google Scholar
  66. 66.
    Maghsoudi, M.; Maghsoudi, A.A.: Ultimate tendon stress of strengthened and non-strengthened unbonded post-tensioned i-beams. Proc. Inst. Civ. Eng. Struct. Build. 171(12), 946–965 (2018)Google Scholar
  67. 67.
    CSA (Canadian Standards Association) CSA-A23.3-94: Design of concrete structures. CSA, Rexdale, Ontario (1994)Google Scholar
  68. 68.
    SIA (Schweizerischer Inge- nieur- und Architektenverein) SIA 162: Ultimate load behaviour of slabs, Swiss code working party 5, SIA, Zurich (1979)Google Scholar
  69. 69.
    Dundu, M.; Ward, M.: The effects of lubricant and tendon mass variances on the coefficient of friction in unbonded post-tensioning tendons. J. South Afr. Inst. Civ. Eng. 58(1), 62–65 (2016)Google Scholar
  70. 70.
    Moreira, L.S.; Sousa, J.B.M.; Parente, E.: Nonlinear finite element simulation of unbonded prestressed concrete beams. Eng. Struct. 170, 167–177 (2018)Google Scholar
  71. 71.
    Huang, Y.; Kang, T.H.K.: Modeling of sliding behavior of unbonded tendons in post-tensioned concrete members. ACI Struct. J. 115(4), 1153–1164 (2018)MathSciNetGoogle Scholar
  72. 72.
    Tao, X.; Du, G.: Ultimate stress of unbonded tendons in partially prestressed concrete beams. Precast/Prestress. Concr. Inst. 30(6), 72–91 (1985)Google Scholar
  73. 73.
    DIN (Deutsches Institut für Normung) DIN 1045: Reinforced and prestressed concrete structures. Part 1. DIN, Berlin (2001)Google Scholar
  74. 74.
    JPCEA (Japan Prestressed Concrete Engineering Association) JPCEA-97: design specification for precast and externally prestressed structures. JPCEA, Tokyo (1997)Google Scholar
  75. 75.
    NEN (Nederlands Normalisatie Instituut) NEN 3880: Part H: regulations for concrete. NEN, Delft (1984)Google Scholar
  76. 76.
    SNZ (Standards New Zealand) NZS 3101: concrete structures standard. Parts 1 and 2. SNZ, Wellington (2006)Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringThe University of JordanAmmanJordan
  2. 2.Magna MEP Electromechanical LLCDubaiUAE

Personalised recommendations