Analytical Solutions for the Flexural Behavior of Metal Beams Strengthened with Prestressed Unbonded CFRP Plates

Document Type: Research Papers


1 Arizona State University, School of Sustainable Engineering and the Built Environment, Tempe AZ 85287, USA

2 Empa, Swiss Federal Laboratories for Materials Science and Technology, Structural Engineering Research Laboratory, Dübendorf, Switzerland

3 University of Tehran, School of Civil Engineering, Tehran, Iran


Trapezoidal prestressed unbonded retrofit (TPUR) systems have been recently developed and tested. The authors have already developed a comprehensive and accurate analytical solution for the TPUR system that takes many system parameters into account. The main aim of this paper is to develop a simplified analytical solution for predicting the behavior of metal beams that have been strengthened with the TPUR system. The developed analysis method can be useful to engineers because of its simplicity. An energy approach based on Castigliano’s theorems is used to study the flexural behavior of a steel beam retrofitted with the TPUR system. A parametric study was performed and the comparative results showed that the results using Castigliano’s first theorem are in agreement with the results using the flexibility approach.


Main Subjects


Al-Emrani, M. and Kliger, R. (2006). "Analysis of interfacial shear stresses in beams strengthened with bonded prestressed laminates", Composites Part B: Engineering, 37(4), 265-272.

Aljabar, N.J., Zhao, X.L., Al-Mahaidi, R., Ghafoori, E., Motavalli, M. and Koay, Y.C. (2017). "Fatigue tests on UHM-CFRP strengthened steel plates with central inclined cracks under different damage levels", Composite Structures, 160, 995-1001.

Aljabar, N.J., Zhao, X.L., Al-Mahaidi, R., Ghafoori, E., Motavalli, M. and Powers, N. (2016). "Effect of Crack Orientation on Fatigue Behaviour of CFRP- Strengthened Steel Plates", Composite Structures, 152, 295-305.

Benachour, A., Benyoucef, S. and Tounsi, A. (2008). "Interfacial stress analysis of steel beams reinforced with bonded prestressed FRP plate", Engineering Structures, 30(11), 3305-3315.

Boresi, A.P., Schmidt, R.J. and Sidebottom, O.M. (1993). Advanced mechanics of materials, Wiley New York.

Fernando, D., Schumacher, A., Motavalli, M., Teng, J. G., Yu, T. and Ghafoori, E. (2010). "Fatigue strengthening of cracked steel beams with CFRP plates", ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 271-276.

Ghafoori, E. (2013). "Interfacial stresses in beams strengthened with bonded prestressed plates", Engineering Structures, 46, 508-510.

Ghafoori, E. (2015). "Fatigue strengthening of metallic members using un-bonded and bonded CFRP laminates." PhD Thesis, ETH-Zurich, ( ).

Ghafoori, E. and Motavalli, M. (2011). "Analytical calculation of stress intensity factor of cracked steel I-beams with experimental analysis and 3D digital image correlation measurements", Engineering Fracture Mechanics, 78(18), 3226-3242.

Ghafoori, E. and Motavalli, M. (2013). "Flexural and interfacial behavior of metallic beams strengthened by prestressed bonded plates", Composite Structures, 101, 22-34.

Ghafoori, E. and Motavalli, M. (2015a). "Innovative CFRP-prestressing system for strengthening metallic structures", Journal of Composites for Construction, 19(6), 04015006.

Ghafoori, E. and Motavalli, M. (2015b). "Lateral-torsional buckling of steel I-beams retrofitted by bonded and un-bonded CFRP laminates with different pre-stress levels: Experimental and numerical study", Construction and Building Materials, 76, 194-206.

Ghafoori, E. and Motavalli, M. (2015c). "Normal, high and ultra-high modulus CFRP laminates for bonded and un-bonded strengthening of steel beams", Materials and Design, 67, 232-243.

Ghafoori, E. and Motavalli, M. (2016). "A retrofit theory to prevent fatigue crack initiation in aging riveted bridges using carbon fiber-reinforced polymer materials", Polymers, 8, 308.

Ghafoori, E., Motavalli, M., Botsis, J., Herwig, A. and Galli, M. (2012). "Fatigue strengthening of damaged metallic beams using prestressed unbonded and bonded CFRP plates", International Journal of Fatigue, 44, 303-315.

Ghafoori, E., Motavalli, M., Nussbaumer, A., Herwig, A., Prinz, G. and Fontana, M. (2015a). "Determination of minimum CFRP pre-stress levels for fatigue crack prevention in retrofitted metallic beams", Engineering Structures, 84, 29–41.

Ghafoori, E., Motavalli, M., Nussbaumer, A., Herwig, A., Prinz, G.S. and Fontana, M. (2015b). "Design criterion for fatigue strengthening of riveted beams in a 120-year-old railway metallic bridge using pre-stressed CFRP plates", Composites Part B: Engineering, 68, 1-13.

Ghafoori, E., Motavalli, M., Zhao, X. L., Nussbaumer, A. and Fontana, M. (2015c). "Fatigue design criteria for strengthening metallic beams with bonded CFRP plates." Engineering Structures, 101, 542-557.

Ghafoori, E. and Motavalli, M. (2015d). "Innovative CFRP-prestressing system for strengthening metallic structures",  Journal of Composites for Construction, 19(6), 04015006.

Ghafoori, E., Prinz, G.S., Mayor, E., Nussbaumer, A., Motavalli, M., Herwig, A. and Fontana, M. (2014). "Finittione element analysis for fatigue damage reduction in metallic riveted bridges using pre-stressed CFRP plates", Polymers, 6(4), 1096-1118.

Ghafoori, E., Schumacher, A. and Motavalli, M. (2012). "Fatigue behavior of notched steel beams reinforced with bonded CFRP plates: Determination of prestressing level for crack arrest", Engineering Structures, 45, 270-283.

Hibbeler, R.C. (2014). Structural analysis, Pearson Education, Inc., Upper Saddle River, New Jersey 07458.

Hosseini, A., Ghafoori, E., Motavalli, M. and Nussbaumer, A. (2016). "Stress analysis of unbonded and bonded prestressed CFRP-strengthened steel plates", 8th International Conference on Fiber Reinforced Polymer (FRP) Composites in Civil Engineering (CICE2016), 14-16 December 2016, Hong Kong, China.

Huawen, Y., König, C., Ummenhofer, T., Shizhong, Q. and Plum, R. (2010). "Fatigue performance of tension steel plates strengthened with prestressed CFRP laminates", Journal of Composites for Construction, 14(5), 609-615.

Kerboua, B. and Benmoussat, A. (2011). "Strengthening of damaged structures with bonded prestressed FRP composites plates: An improved theoretical solution", Journal of Composite Materials, 45(5), 499-512.

Kianmofrad, F., Ghafoori, E., Elyasi, M.M., Motavalli, M. and Rahimian, M. (2017). "Strengthening of metallic beams with different types of pre-stressed un-bonded retrofit systems", Composite Structures, 159, 81-95.

Park, S., Kim, T., Kim, K. and Hong, S.-N. (2010). "Flexural behavior of steel I-beam prestressed with externally unbonded tendons", Journal of Constructional Steel Research, 66(1), 125-132.

Ponnada, M.R. and Vipparthy, R. (2013). "Improved Method of Estimating Deflection in Prestressed Steel I-Beams", Asian Journal of Civil Engineering (BHRC), 14(5), 765-772.

Rahimian, M. and Ghorbani Tanha, A.K. (2002). Strutural analysis, Sanjesh, Tehran, Iran, (in Persian).

Rashid Dadash, P. and Ramezanianpour, A.A. (2014). "Hybrid fiber reinforced concrete containing pumice and metakaolin", Civil Engineering Infrastructures Journal, 47(2), 229-238.

Saleh Jalali, R. and Shadafza, E. (2016). "The elastic modulus of steel fiber reinforced concrete (SFRC) with random distribution of aggregate and fiber", Civil Engineering Infrastructures Journal, 49(1), 21-32.

Schnerch, D. and Rizkalla, S. (2008). "Flexural strengthening of steel bridges with high modulus CFRP strips", Journal of Bridge Engineering, 13(2), 192-201.

Soranakom, C. and Mobasher, B. (2007). "Closed-form solutions for flexural response of fiber-reinforced concrete beams", Journal of Engineering Mechanics, 133(8), 933-941.

Täljsten, B., Hansen, C.S. and Schmidt, J.W. (2009). "Strengthening of old metallic structures in fatigue with prestressed and non-prestressed CFRP laminates", Construction and Building Materials, 23(4), 1665-1677.

Tavakkolizadeh, M. and Saadatmanesh, H. (2003). "Fatigue strength of steel girders strengthened with carbon fiber reinforced polymer patch", Journal of Structural Engineering, 129(2), 186-196.

Timoshenko, S.P. and Young, D.H. (1965). Theory of structures, McGraw-Hill, New York.

U.S. Department of Transportation. (1986). "Highway bridge replacement and rehabilitation program", Federal Highway Administration, Bridge Division Office of Engineering.