Stochastic analysis of lattice, nonlocal continuous beams in vibration


  • Yuchen Li Université Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France
  • Noël Challamel Université Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France
  • Isaac Elishakoff Department of Ocean and Mechanical Engineering, Florida Atlantic University, 777 Glades Road – EG 36/Rm 106, Boca Raton, FL 33431, United States



Vibration, Lattice elasticity, Discrete beams, multibody system dynamics, Nonlocal beams


In this paper, we study the stochastic behavior of some lattice beams, called Hencky bar-chain model and their non-local continuous beam approximations. Hencky bar-chain model is a beam lattice composed of rigid segments, connected by some homogeneous rotational elastic links. In the present stochastic analysis, the stiffness of these elastic links is treated as a continuous random variable, with given probability density function. The fundamental eigenfrequency of the linear difference eigenvalue problem is also a random variable in this context. The reliability is defined as the probability that this fundamental frequency is less than an excitation frequency. This reliability function is exactly calculated for the lattice beam in conjunction with various boundary conditions. An exponential distribution is considered for the random stiffness of the elastic links. The stochastic lattice model is then compared to a stochastic nonlocal beam model, based on the continualization of the difference equation of the lattice model. The efficiency of the nonlocal beam model to approximate the lattice beam model is shown in presence of rotational elastic link randomness. We also compare such stochastic function with the one of a continuous local Euler-Bernoulli beam, with a special emphasis on scale effect in presence of randomness. Scale effect is captured both in deterministic and non-deterministic frameworks.


Download data is not yet available.


Metrics Loading ...


A. K. Noor. Continuum modeling for repetitive lattice structures. Applied Mechanics Reviews, 41, (1988), pp. 285–296. DOI:

M. Ostoja-Starzewski. Lattice models in micromechanics. Applied Mechanics Reviews, 55, (2002), pp. 35–60. DOI:

H. Hencky. Über die angenäherte Lösung von Stabilitätsproblemen im Raum mittels der elastischen Gelenkkette. Der Eisenbau, 11, (1920), pp. 437–452.

M. G. Salvadori. Numerical computation of buckling loads by finite differences. Transactions of the American Society of Civil Engineers, 116, (1951), pp. 590–624.

F. A. Leckie and G. M. Lindberg. The effect of lumped parameters on beam frequencies. Aeronautical Quarterly, 14, (1963), pp. 224–240. DOI:

M. S. El Naschie and S. Stress. Chaos in structural engineering: An energy approach. McGraw Hil, New York, (1990).

H. Zhang, C. M. Wang, E. Ruocco, and N. Challamel. Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation. Engineering Structures, 126, (2016), pp. 252–263. DOI:

R. K. Livesley. The equivalence of continuous and discrete mass distributions in certain vibration problems. The Quarterly Journal of Mechanics and Applied Mathematics, 8, (3), (1955), pp. 353–360. DOI:

M. G. Salvadori. Numerical computation of buckling loads by finite differences. Transactions of the American Society of Civil Engineers, 116, (1951), pp. 590–624. DOI:

T. Wahand, L. R. Calcote. Structural analysis by finite difference calculus. Van Nostrand Reinhold Company, (1970).

C. M. Wang, H. Zhang, R. P. Gao, W. H. Duan, and N. Challamel. Henckybar-chain model for buckling and vibration of beams with elastic end restraints. International Journal of Structural Stability and Dynamics, 15, (2015), p. 1540007. DOI:

I. Elishakoff and R. Santoro. Error in the finite difference based probabilistic dynamic analysis: analytical evaluation. Journal of Sound and Vibration, 281, (2005), pp. 1195–1206. DOI:

R. Santoro and I. Elishakoff. Accuracy of the finite difference method in stochastic setting. Journal of Sound and Vibration, 291, (2006), pp. 275–284. DOI:

N. Challamel, Z. Zhang, C. M. Wang, J. N. Reddy, Q. Wang, T. Michelitsch, and B. Collet. On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Archive of Applied Mechanics, 84, (2014), pp. 1275–1292. DOI:

Z. Zhang, C. M. Wang, N. Challamel, and I. Elishakoff. Obtaining Eringen’s length scale coefficient for vibrating nonlocal beams via continualization method. Journal of Sound and Vibration, 333, (2014), pp. 4977–4990. DOI:

I. Elishakoff and C. Soize. Nondeterministic mechanics. International Centre for Mechanical Sciences, Courses and Lectures - No. 539, SpringerWienNew York, (2012). DOI:

E. Vanmarcke and M. Grigoriu. Stochastic finite element analysis of simple beams. Journal of Engineering Mechanics, 109, (1983), pp. 1203–1214. DOI:

K. Gao, W. Gao, D. Wu, and C. Song. Nonlinear dynamic stability analysis of Euler–Bernoulli beam–columns with damping effects under thermal environment. Nonlinear Dynamics, 90, (2017), pp. 2423–2444. DOI:

K. Gao, W. Gao, B. Wu, and C. Song. Nondeterministic dynamic stability assessment of Euler–Bernoulli beams using Chebyshev surrogate model. Applied Mathematical Modelling, 66, (2019), pp. 1–25. DOI:

K. Gao, R. Li, and J. Yang. Dynamic characteristics of functionally graded porous beams with interval material properties. Engineering Structures, 197, (2019), p. 109441. DOI:

S. Kukla and A. Owczarek. Stochastic vibration of a Bernoulli–Euler beam under random excitation. Scientific Research of the Institute of Mathematics and Computer Science, 4, (1), (2005), pp. 71–78.

C. R. A. Silva Junior, A. T. Beck, and E. Rosa. Solution of the stochastic beam bending problem by Galerkin method and the Askey-Wiener scheme. Latin American Journal of Solids and Structures, 6, (2009), pp. 51–72.

A. C. Eringen. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, (1983), pp. 4703–4710. DOI:

J. A. D. Wattis. Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation. Journal of Physics A: Mathematical and General, 33, (2000), pp. 5925–5944. DOI:

I. V. Andrianov, J. Awrejcewicz, and D. Weichert. Improved continuous models for discrete media. Mathematical Problems in Engineering, 2010, (2010), pp. 1–35. DOI:

W. H. Duan, N. Challamel, C. M. Wang, and Z. Ding. Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams. Journal of Applied Physics, 114, (2013). DOI:

N. Challamel, Z. Zhang, and C. M. Wang. Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams. Journal of Nanomechanics and Micromechanics, 5, (2015). DOI:

J. N. Reddy. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, (2007), pp. 288–307. DOI:

N. Challamel, C. M. Wang, and I. Elishakoff. Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis. European Journal of Mechanics - A/Solids, 44, (2014), pp. 125–135. DOI:

V. D. Potapov. Stability via nonlocal continuum mechanics. International Journal of Solids and Structures, 50, (2013), pp. 637–641. DOI:

G. Alotta, M. D. Paola, G. Failla, and F. P. Pinnola. On the dynamics of non-local fractional viscoelastic beams under stochastic agencies. Composites Part B: Engineering, 137, (2018), pp. 102–110. DOI:

C. Y. Wang and C. M. Wang. Structural vibration: exact solutions for strings, membranes, beams, and plates. CRC Press, (2019).

R. D. Blevins. Formulas for dynamics, acoustics and vibration, chapter 4, pp. P139–P140. John Wiley & Sons, Ltd, (2015). DOI:

C.M.Wang,H.Zhang,N.Challamel,andW.H.Duan.Onboundaryconditionsforbuckling and vibration of nonlocal beams. European Journal of Mechanics-A/Solids, 61, (2017), pp. 73–81. DOI:

C.M.Wang,H.Zhang,N.Challamel,andW.Pan.HenckyBar-chain/netforStructuralAnalysis. World Scientific, (2020). DOI:

I. Elishakoff. Prologue. In Safety factors and reliability: Friends or foes?, pp. 1–11. Springer Netherlands, (2004). DOI:

N. Challamel, V. Picandet, B. Collet, T. Michelitsch, I. Elishakoff, and C. M. Wang. Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua. European Journal of Mechanics - A/Solids, 53, (2015), pp. 107–120. DOI:




How to Cite

Li, Y., Challamel, N., & Elishakoff, I. (2021). Stochastic analysis of lattice, nonlocal continuous beams in vibration. Vietnam Journal of Mechanics, 43(2), 139–170.