Reliability Analysis of Nanocomposite Beams Reinforced with CNTs under Buckling Forces Using the Conjugate HL-RF

Document Type : Research Paper


1 Department of Civil Engineering, University of Zabol, Zabol, 9861335-856, Iran,

2 Department of Civil Engineering, University of Sistan and Baluchestan, Zahedan, 98798-155, Iran

3 Department of Civil Engineering, Faculty of Engineering, University Malaya, Kuala Lumpur, 50603, Malaysia


In this paper, the nonlinear conjugate map is applied based on the conjugate Hasofer-Lind and Rackwitz- Fiessler (CHL-RF) method to evaluate the reliability index using the first order reliability method of the embedded nanocomposite beam, which is made of a polymer reinforced with carbon nanotubes (CNTs). The structure is simulated with the Timoshenko beam model. The Mori-Tanaka model is applied for calculating the effective material properties of the nanocomposite beam and the surrounding elastic medium is considered as spring and shear constants. The governing equations are derived based on the energy method and the Hamilton's principle. Moreover, using an analytical method, the buckling performance function of the structure is obtained. The effects of the basic random variables including the length-to-thickness ratio of the beam (L/h), the spring constant, and the shear constant of the foundation with respect to the volume fraction of CNTs are investigated based on the reliability index of the nanocomposite beam which is subjected to an axial force of 20 GPa. The results indicate that the failure probabilities of the studied nanocomposite beams are sensitive to the length-to-thickness ratio of the beam (L/h) and the spring constant of the foundation variables.


[1] Engesser, F., Über Die Knickfestigkeit Gerader Stäbe, Z. Archit. Ing. Ver. Hann., Vol. 35, pp. 455–462, 1889.
[2] Shanley, F.R., Inelastic Column Theory, J. Aeronaut. Sci., Vol. 14, Pp. 261–264, 1947.
[3] Mau, S.T., Effect of Tie Spacing Oninelastic Buckling of Reinforcing Bars, ACI Struct. J., Vol. 87, No. 6, pp. 617-677, 1990.
[4] Mau, S.T. and El-Mabsout, M., Inelastic Buckling of Reinforcing Bars, J. Eng. Mech., Vol. 115, No. 1, pp. 1-17, 1989.
[5] Pantazopoulou, S.J., Detailing for Reinforcement Stability in RC Members, J. Struct. Eng., Vol. 124, No. 6, pp. 623-6321998.
[6] Dhakal, R.P. and Maekawa, K., Modeling for Postyield Buckling of Reinforcement, J. Struct. Eng., Vol. 128, No.9, pp. 1139-1147, 2002.
[7] Bae, S., Mieses, A.M. and Bayrak, O., Inelastic Buckling of Reinforcing Bars, J. Struct. Eng., Vol. 131, No. 2, pp. 314-321, 2005.
[8] Dhakal, R.P. and Maekawa, K., Reinforcement Stability and Fracture of Cover Concrete in Reinforced Concrete Members, J. Struct. Eng., Vol. 128, No. 10, pp. 1253-1262, 2002.
[9] Krauberger, N., Saje, M., Planinc, I. and Bratina, S., Exact Buckling Load of a Restrained RC Column, Struct. Eng. Mech., Vol. 27, pp. 293–310, 2007.
[10] Lou, T., Lopes, S.M.R. and Lopes, A.V. (2015), “Numerical Modelling of Nonlinear Behaviour of Prestressed Concrete Continuous Beams”, Comput. Concrete, 15, 391-410.
[11] Bajc, U., Saje, M., Planinc, I. and Bratina, S., Semi-analytical Buckling Analysis of Reinforced Concrete Columns Exposed to Fire, Fire Safety J., Vol. 71, pp. 110–122, 2015.
[12] Vijai, K., Kumutha, R. and Vishnuram, B.G., Flexural Behaviour of Fibre Reinforced Geopolymer Concrete Composite Beams, Comput. Concrete, Vol. 15, pp. 437-459, 2015.
[13] Keshtegar, B. and Miri, M., Reliability Analysis of Corroded Pipes Using Conjugate HL–RF Algorithm Based on Average Shear Stress Yield Criterion, Engineering Failure Analysis, Vol. 46, pp. 104–117, 2014.
[14] Keshtegar B. and Hao P., A Hybrid Loop Approach Using the Sufficient Descent Condition for Accurate, Robust and Efficient Reliability-Based Design Optimization, Journal of Mechanical Design, Vol. 138, No. 12: pp. 121401-11
[15] Keshtegar, B. (2016), A Modified Mean Value of Performance Measure Approach for Reliability-Based Design Optimization, Arab J Sci Eng. 1-9, doi:10.1007/s13369-016-2322-02016
[16] Keshtegar, B., Chaotic Conjugate Stability Transformation Method for Structural Reliability Analysis, Computer Methods in Applied Mechanics and Engineering, Vol. 310, pp. 866-885, 2016.
[17] Keshtegar, B., Stability Iterative Method for Structural Reliability Analysis Using a Chaotic Conjugate Map, Nonlinear Dyn., Vol. 84, No. 4, pp. 2161-2174, 2016.
[18] Keshtegar, B., Limited Conjugate Gradient Method for Structural Reliability Analysis, Engineering with Computers, doi:10.1007/s00366-016-0493-7, pp. 1-9, 2016.
[19] Keshtegar, B. and Miri, M., Introducing Conjugate Gradient Optimization for Modified HL-RF Method, Engineering Computations, Vol. 31, pp. 775-790, 2014.
[20] Fletcher, R. and Reeves, C., Function minimization by conjugate gradients, J. Comput. Vol. 7, pp. 149–154, 1964.
[21] Gong, J.X. and Yi, P., A Robust Iterative Algorithm for Structural Reliability Analysis, Struct. Multidisc. Optim., Vol. 43, pp. 519–527, 2011.
[22] Meng, Z., Li, G., Yang, D. and Zhan, L., A New Directional Stability Transformation Method of Chaos Control for First Order Reliability Analysis, Struct. Multidiscipl. Optim., DOI: 10.1007/s00158-016-1525-z, pp. 1-12, 2016.