Arabian Journal for Science and Engineering https://doi.org/10.1007/s13369-020-04475-w RESEARCH ARTICLE-CIVIL ENGINEERING An Improved Methodology for the Prediction of the Stress at Ultimate in Unbonded Internal and External Steel Tendons Maha Alqam 1 · Fadi Alkhairi 2 · Antoine Naaman 3 Received: 5 November 2019 / Accepted: 17 March 2020 © King Fahd University of Petroleum & Minerals 2020 Abstract This study involves the analysis of the stress increase beyond effective prestress in beams prestressed with unbonded internal or external tendons at their ultimate limit state, Δf ps , and is divided into two distinct parts. In the first part, twenty-five equations published between 1962 and 2019 to predict Δf ps in unbonded tendons and proposed by various researchers and Codes of Practice are presented. Predicted versus experimental results for Δf ps are compared about the 45° perfect correlation line using an updated database comprising 227 laboratory-tested beams obtained from past studies. An extensive statistical analysis is carried out for each equation using the sum of Least Square Deviation Error and Coefficient of Correlation. The percentage of data points representing conservative predictions is also numerically calculated. Statistical analysis confirmed superior performance of the mechanically derived equation developed in the past research (Naaman in ACI Struct J 88:683–692, 1991b, ACI Struct J 99:518–529, 2002; AASHTO in AASHTO LRFD bridge design specifications, AASHTO, Washington, 1994). That equation was rationally derived from a sectional strain compatibility analysis augmented to a member deflection compatibility through the use of a strain reduction coefficient,  u . In the second part of the study, and using the updated experimental database, the authors derive a new and improved expression for the strain reduction coefficient,  u , that led to the lowest Least Square Deviation Error and highest Coefficient of Correlation. The newly proposed expression for  u offers two alternatives, one for precise prediction and the other for code implementation. Based on their findings, the authors strongly recommend the equation of f ps (Naaman in ACI Struct J 88:683–692, 1991b, ACI Struct J 99:518–529, 2002; AASHTO in AASHTO LRFD bridge design specifications, AASHTO, Washington, 1994) and the new expression for Ω u for code implementation. Keywords Unbonded tendons · External prestressing · Second-order effects · Bond reduction coefficient · Span-to-depth ratio · Eccentricity variations List of Symbols a Depth of equivalent rectangular stress block as defined in the ACI Code 318-19  β 1 c [1, 2] B Maha Alqam m.alqam@ju.edu.jo Fadi Alkhairi fadial@umich.edu Antoine Naaman antoinenaaman@yahoo.com 1 Department of Civil Engineering, The University of Jordan, Amman 11942, Jordan 2 Magna MEP Electromechanical LLC, Dubai, UAE 3 Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, USA A ps Area of prestressed unbonded reinforcement A s Area of non-prestressed tensile reinforcement A’ s Area of non-prestressed compressive reinforcement b Beam width of a rectangular section or flange width of a T-section b w Web width of a T-section c Depth of neutral axis at ultimate nominal strength (Fig. 1) c u Depth of neutral axis at ultimate nominal strength assuming f ps [3] c y Depth of neutral axis at ultimate nominal strength assuming f ps  f py [4] d e Effective depth from the extreme compressed fiber to the centroid of the prestressed and non-prestressed reinforcement using f py [5] 123