Browse > Article
http://dx.doi.org/10.12989/sss.2021.27.1.001

2D magneto-mechanical vibration analysis of a micro composite Timoshenko beam resting on orthotropic medium  

Mehrabi, Mojtaba (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan)
Mohammadimehr, Mehdi (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan)
Mousavinejad, Fatemeh S. (Department of Civil Engineering, Faculty of Engineering, University of Kashan)
Publication Information
Smart Structures and Systems / v.27, no.1, 2021 , pp. 1-18 More about this Journal
Abstract
In the present study, the free vibration analysis of a size-dependent micro composite Timoshenko beam model reinforced by various distributions of carbon nanotubes under temperature changes and two-dimensional magnetic field is investigated based on modified strain gradient theory. Also, the effects of environment are simulated by orthotropic elastic foundation and it is assumed that the material properties are temperature-dependent. Mathematical formulations are obtained using Hamilton's principle and the governing equations of motion are derived based on energy approach and variation method. These equations are solved using semi-analytical and numerical methods such as Navier's type solution, finite element method and generalized differential quadrature method for various boundary conditions. The obtained results of this study are compared with the other previous researches and there is a good agreement between them. The main purpose of this work is the comparison of various solution methods on the problem outputs. Thus, the results are compared together and the effects of solution approach on the dimensionless natural frequencies is developed. Moreover, the effects of length-to-thickness ratio, magnetic field, temperature changes, elastic foundation and carbon nanotubes volume fractions on the dimensionless natural frequencies are studied. The results of this article demonstrate that the micro composite Timoshenko beam reinforced by FG-O and FG-X CNTs have lowest and highest dimensionless natural frequency, respectively. It is investigated that the dimensionless natural frequency enhances by increasing the magnetic field in x and z-directions.
Keywords
2D magnetic field; free vibration analysis of Timoshenko micro beam; reinforced by carbon nanotubes; temperature-dependent material properties; orthotropic medium;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Anitescu, C., Atroshchenko, E., Alajlan, N. and Rabczuk, T. (2019), "Artificial neural network methods for the solution of second order boundary value problems". Comput. Mater. Contin., 59(1), 345-359. https://doi.org/10.32604/cmc.2019.06641   DOI
2 Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko micro beams based on the most general strain gradient theory", Compos. Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048   DOI
3 Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami, R. and Sadeghi, F. (2014), "Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams", Compos. Struct., 113, 316-327. https://doi.org/10.1016/j.compstruct.2014.03.015   DOI
4 Guo, H., Zhuang, X. and Rabczuk, T. (2019), "A deep collocation method for the bending analysis of Kirchhoff plate". Comput. Mater. Contin., 59(2), 433-456. https://doi.org/10.32604/cmc.2019.06660   DOI
5 Jelodari, I. and Nilseresht, A.H. (2018), "Effects of Lorentz force and induced electrical field on the thermal performance of a magnetic nanofluid-filled cubic cavity", J. Molec. Liq., 252, 296-310. https://doi.org/10.1016/j.molliq.2017.12.143.   DOI
6 Ke, L.L. and Wang, Y.S. (2011), "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos. Struct., 93(2), 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008   DOI
7 Qing, G., Qiu, J. and Liu, Y. (2018), "Free vibration analysis of Stiffened Laminated Plate using FEM", Mater. Today. Proce., 5(2), 5313-5321. https://doi.org/10.1016/j.matpr.2017.12.115   DOI
8 Rabczuk, T., Ren, H. and Zhuang, X. (2019), "A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem", Comput. Mater. Contin., 59(1), 31-55. https://doi.org/10.32604/cmc.2019.04567   DOI
9 Rajabi, J. and Mohammadimehr, M. (2019), "Bending analysis of a micro sandwich skew plate using extended Kantorovich method based on Eshelby-Mori-Tanaka approach", Comput. Concrete, Int. J., 23(5), 361-376. http://doi.org/10.12989/cac.2019.23.5.361   DOI
10 Romanoff, J., Reddy, J.N. and Jelovica, J. (2016), "Using nonlocal Timoshenko beam theories for prediction of micro- and macro-structural responses", Compos. Struct., 156, 410-420. https://doi.org/10.1016/j.compstruct.2015.07.010   DOI
11 Soni, S., Jain, N.K. and Joshi, P.V. (2018), "Vibration analysis of partially cracked plate submerged in fluid", J. Sound. Vib., 412, 28-57. https://doi.org/10.1016/j.jsv.2017.09.016   DOI
12 Kutlu, A. and Omurtag, M.H. (2012), "Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method", Int. J. Mech. Sci., 65(1), 64-74. https://doi.org/10.1016/j.ijmecsci.2012.09.004   DOI
13 Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X   DOI
14 Lei, Z.X., Zhang, L.W. and Liew, K.M. (2015), "Elastodynamic analysis of carbon nanotube-reinforced functionally graded plates", Int. J. Mech. Sci., 99, 208-217. https://doi.org/10.1016/j.ijmecsci.2015.05.014   DOI
15 Sapountzakis, E.J., Tsipiras, V.J. and Argyridi, A.K. (2015), "Torsional vibration analysis of bars including secondary torsional shear deformation effect by the boundary element method", J. Sound. Vib., 355, 208-231. https://doi.org/10.1016/j.jsv.2015.04.032   DOI
16 Soleimani, I. and Beni, Y.T. (2018), "Vibration analysis of nanotubes based on two-node size dependent axisymmetric shell element", Arch. Civil. Mech. Eng., 18(4), 1345-1358. https://doi.org/10.1016/j.acme.2018.04.009   DOI
17 Shahedi, S. and Mohammadimehr, M. (2019), "Vibration analysis of rotating fully-bonded and delaminated sandwich beam with CNTRC face sheets and AL-foam flexible core in thermal and moisture environments", Mech. Based Des. Struct. Mach., 48(5), 584-614. https://doi.org/10.1080/15397734.2019.1646661   DOI
18 Tabbakh, M. and Nasihatgozar, M. (2018), "Buckling analysis of nanocomposite plates coated by magnetostrictive layer", Smart Struct. Syst., Int. J., 22(6), 743-751. http://.doi.org/10.12989/sss.2018.22.6.743   DOI
19 Mirsalehi, M., Azhari, M. and Amoushahi, H. (2017), "Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method", Europ. J. Mech. A/Solid., 61, 1-13. https://doi.org/10.1016/j.euromechsol.2016.08.008   DOI
20 Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007   DOI
21 Mohammadimehr, M. and Mehrabi, M. (2017), "Stability and free vibration analysis of double-bonded micro composite sandwich cylindrical shells conveying fluid flow", Appl. Math. Model., 47, 685-709. https://doi.org/10.1016/j.apm.2017.03.054.   DOI
22 Mohammadimehr, M., Salemi, M. and Navi, B.R. (2016a), "Bending, buckling, and free vibration analysis of MSGT microcomposite Reddy plate reinforced by FG-SWCNTs with temperature-dependent material properties under hydro-thermomechanical loadings using DQM", Compos. Struct., 138, 361-380. https://doi.org/10.1016/j.compstruct.2015.11.055   DOI
23 Mohammadimehr, M., Afshari, H., Salemi, M., Torabi, K. and Mehrabi, M. (2019), "Free vibration and buckling analyses of functionally graded annular thin sector plate in-plane loads using GDQM", Struct. Eng. Mech., Int. J., 71(5), 525-544. http://doi.org/10.12989/.2019.71.5.525   DOI
24 Ghasemi, H., Park, H.S. and Rabczuk, T. (2018), "A multi-material level set-based topology optimization of flexoelectric composites", Comput. Method. Appl. Mech. Eng., 332, 47-62. https://doi.org/10.1016/j.cma.2017.12.005   DOI
25 Ansari, R., Gholami, R. and Sahmani, S. (2011), "Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory", Compos. Struct., 94(1), 221-228. https://doi.org/10.1016/j.compstruct.2011.06.024   DOI
26 Kashani, B.K. and Sani, A.A. (2016), "Free vibration analysis of horizontal cylindrical shells including sloshing effect utilizing polar finite element", Europ. J. Mech. A/Solids., 58, 187-201. https://doi.org/10.1016/j.euromechsol.2016.02.002   DOI
27 Thai, C.H., Ferreira, A.J.M. and Nguyen-Xuan, H. (2018), "Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory", Compos. Struct., 192, 274-288. https://doi.org/10.1016/j.compstruct.2018.02.060   DOI
28 Tang, X.H., Paluszny, A. and Zimmerman, R.W. (2013), "Energy conservative property of impulse-based methods for collision resolution", Int. J. Numer. Method. Eng., 95(6), 529-540. https://doi.org/10.1002/nme.4537   DOI
29 Tang, X., Paluszny, A. and Zimmerman, R.W. (2014), "An impulse-based energy tracking method for collision resolution", Comput. Method. Appl. Mech. Eng., 278, 160-185. https://doi.org/10.1016/j.cma.2014.05.004   DOI
30 Thai, S., Thai, H.T., Vo, T.P. and Nguyen-Xuan, H. (2017), "Nonlinear static and transient isogeometric analysis of functionally graded microplates based on the modified strain gradient theory", Eng. Struct., 153, 598-612. https://doi.org/10.1016/j.engstruct.2017.10.002   DOI
31 Xu, Y., Qian, Y. and Song, G. (2016), "Stochastic finite element method for free vibration characteristics of random FGM", Appl. Math. Model., 40(23-24), 10238-10253. https://doi.org/10.1016/j.apm.2016.07.025   DOI
32 Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solid. Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X   DOI
33 Yang, J., Xiong, J., Ma, L., Zhang, G., Wang, X. and Wu, L. (2014), "Study on vibration damping of composite sandwich cylindrical shell with pyramidal truss-like cores", Compos. Struct., 117, 362-372. https://doi.org/10.1016/j.compstruct.2014.06.042   DOI
34 Farzampour, A., Eatherton, M.R., Mansouri, I. and Hu, J.W. (2019), "Effect of flexural and shear stresses simultaneously for optimized design of butterfly-shaped dampers: Computational study", Smart Struct. Syst., Int. J., 23(4), 329-335. http://doi.org/10.12989/sss.2019.23.4.329   DOI
35 Chau-Dinh, T., Zi, G., Lee, P., Rubczuk, T. and Song, J. (2012), "Phantom-node method for shell models with arbitrary cracks", Comput. Struct., 92-93, 242-256. https://doi.org/10.1016/j.compstruc.2011.10.021   DOI
36 Chu, L., Dui, G. and Ju, C. (2018), "Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory", Compos. Struct., 186, 39-49. https://doi.org/10.1016/j.compstruct.2017.10.083   DOI
37 Domagalski, L. (2018), "Free and forced large amplitude vibrations of periodically inhomogeneous slender beams", Arch. Civil. Mech. Eng., 18(4), 1506-1519. https://doi.org/10.1016/j.acme.2018.06.005   DOI
38 Furtak, K. and Rodacki, K. (2018), "Experimental investigation of load-bearing capacity of composite timber-glass I-beams", Arch. Cvil. Mech. Engin., 18(3), 956-964. https://doi.org/10.1016/j.acme.2018.02.002   DOI
39 Ghasemi, H., Park, H.S. and Rabczuk, T. (2017), "A level-set based IGA formulation for topology optimization of flexoelectric materials", Comput. Method. Appl. Mech. Engin., 313, 239-258. https://doi.org/10.1016/j.cma.2016.09.029   DOI
40 Hamdia, K.M., Ghasemi, H., Zhuang, X., Alajlan, N. and Rabczuk, T. (2018), "Sensitivity and uncertainty analysis for flexoelectric nanostructures", Comput. Method. Appl. Mech. Engin., 337, 95-109. https://doi.org/10.1016/j.cma.2018.03.016   DOI
41 Areias, P., Rabczuk, T. and Msekh, M. (2016), "Phase-field analysis of finite-strain plates and shells including element subdivision", Comput. Method. Appl. Mech. Engin., 312, 322-350. https://doi.org/10.1016/j.cma.2016.01.020   DOI
42 Arani, A.G., Mobarakeh, M.R., Shams, S. and Mohammadimehr, M. (2012), "The effect of CNT volume fraction on the magnetothermo-electro-mechanical behavior of smart nanocomposite cylinder", J. Mech. Sci. Technol., 26(8), 2565-2572. https://doi.org/10.1007/s12206-012-0639-5   DOI
43 Arani, A.G. and Amir, S. (2013), "Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory", Physica B., 419, 1-6. https://doi.org/10.1016/j.physb.2013.03.010   DOI
44 Ardestani, M.M., Zhang, L.W. and Liew, K.M. (2017), "Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates", Comput. Method. Appl. Mech. Engin., 317, 341-379. https://doi.org/10.1016/j.cma.2016.12.009   DOI
45 Banerjee, J.R. (2001), "Frequency equation and mode shape formulae for composite Timoshenko beam", Compos. Struct., 51(4), 381-388. https://doi.org/10.1016/S0263-8223(00)00153-7   DOI
46 Beni, Y.T., Mehralian, F. and Razavi, H. (2015), "Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory", Compos. Struct., 120, 65-78. https://doi.org/10.1016/j.compstruct.2014.09.065   DOI
47 Canales, F.G. and Mantari, J.L. (2017), "Elasto-plastic vibrational analysis of tapered bars under uniform axial loading considering shear deformation and rotary inertia", Int. J. Lin. Mech., 95, 103-116. https://doi.org/10.1016/j.ijnonlinmec.2017.06.001   DOI
48 Mohammadimehr, M., Mehrabi, M., Hadizadeh, H. and Hadizadeh, H. (2018a), "Surface and size dependent effects on static, buckling, and vibration of micro composite beam under thermo-magnetic fields based on strain gradient theory", Steel Compos. Struct., Int. J., 26(4), 513-531. http://doi.org/10.12989/scs.2018.26.4.513   DOI
49 Hsu, Y.S. (2016), "Enriched finite element methods for Timoshenko beam free vibration analysis", Appl. Math. Model., 40(15-16), 7012-7033. https://doi.org/10.1016/j.apm.2016.02.042   DOI
50 Mohammadimehr, M., Mohammadimehr, M.A. and Dashti, P. (2016b), "Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic microplate based on surface stress and modified couple stress theories using differential quadrature method", Appl. Mathe. Mech., 37(4), 529-554. https://doi.org/10.1007/s10483-016-2045-9   DOI
51 Pradhan, S.C. and Mandal, U. (2013), "Finite element analysis of CNTs based on nonlocal elasticity and Timoshenko beam theory including thermal effect", Physica E., 53, 223-232. https://doi.org/10.1016/j.physe.2013.04.029   DOI
52 Paluszny, A., Tang, X.H. and Zimmerman, R.W. (2013), "Fracture and impulse based finite-discrete element modeling of fragmentation", Comput. Mech., 52(5), 1071-1084. http://doi.org/10.1007/s00466-013-0864-5   DOI
53 Pandey, S. and Pradyumna, S. (2015), "A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells", Compos. Struct., 133, 438-450. https://doi.org/10.1016/j.compstruct.2015.07.087   DOI
54 Plagianakos, T.S. and Papadopoulos, E.G. (2015), "Coupled higher-order layerwise mechanics and finite element for cylindrical composite and sandwich shells with piezoelectric transducers", Europ. J. Mech. A/Solids., 54, 11-23. https://doi.org/10.1016/j.euromechsol.2015.06.003   DOI
55 Barquero-Cabrero, J.D., Luevanos-Rojas, A., Lopez-Chavarria, S., Medina-Elizondo, M., Velazquez-Santillan, F. and Sandoval-Rivas, R. (2018), "Deflections and rotations in rectangular beams with straight haunches under uniformly distributed load considering the shear deformations", Smart. Struct. Syst., Int. J., 22(6), 689-697. https://doi.org/10.12989/sss.2018.22.6.689   DOI
56 Ganesh, S., Kumar, K.S. and Mahato, P.K. (2016), "Free vibration analysis of delaminated composite plates using finite element method", Proce. Eng., 144, 1067-1075. https://doi.org/10.1016/j.proeng.2016.05.061   DOI
57 Lenci, S., Clementi, F. and Rega, G. (2017), "Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or geometric curvature definition", Defin. Proce. IUTAM., 20, 34-41. https://doi.org/10.1016/j.piutam.2017.03.006   DOI
58 Mohammadimehr, M., Mehrabi, M. and Mousavinejad, F.S. (2020), "Magneto-mechanical vibration analysis of single-/three-layered micro-Timoshenko porous beam and graphene platelet as reinforcement based on modified strain gradient theory and differential quadrature method", J. Vib. Control. https://doi.org/10.1177%2F1077546320949083   DOI
59 Mohammadimehr, M., Mohammadi-Dehabadi, A.A., Akhavan Alavi, S.M., Alambeigi, K., Bamdad, M., Yazdani, R. and Hanifehlou, S. (2018c), "Bending, buckling, and free vibration analyses of carbon nanotube reinforced composite beams and experimental tensile test to obtain the mechanical properties of nanocompositem", Steel Compos. Struct., Int. J., 29(3), 405-422. https://doi.org/12989/scs.2018.29.3.405   DOI
60 Mohammadimehr, M., Nejad, E.S. and Mehrabi, M. (2018b), "Buckling and vibration analyses of MGSGT double-bonded micro composite sandwich SSDT plates reinforced by CNTs and BNNTs with isotropic foam & flexible transversely orthotropic cores", Struct. Eng. Mech., Int. J., 65(4), 491-504. http://doi.org/10.12989/sem.2018.65.4.491   DOI
61 Nanthakumar, S.S., Lahmer, T., Zhuang, X., Zi, G. and Rabczuk, T. (2016), "Detection of material interfaces using a regularized level set method in piezoelectric structures", Inverse Prob. Sci. Eng., 24(1), 153-176. https://doi.org/10.1080/17415977.2015.1017485   DOI
62 Lee, J. and Schultz, W.W. (2004), "Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method", J. Sound. Vib., 269(3-5), 609-621. https://doi.org/10.1016/S0022-460X(03)00047-6   DOI
63 Lei, J., He, Y., Zhang, B., Gan, Z. and Zeng, P. (2013a), "Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory", Int. J. Eng. Sci., 72, 36-52. https://doi.org/10.1016/j.ijengsci.2013.06.012   DOI
64 Lei, Z.X., Liew, K.M. and Yu, J.L. (2013b), "Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment", Compos. Struct., 106, 128-138. https://doi.org/10.1016/j.compstruct.2013.06.003   DOI
65 Setoodeh, A.R., Shojaee, M. and Malekzadeh, P. (2018), "Application of transformed differential quadrature to free vibration analysis of FG-CNTRC quadrilateral spherical panel with piezoelectric layers", Comput. Method. Appl. Mech. Eng., 335, 510-537. https://doi.org/10.1016/j.cma.2018.02.022   DOI
66 Mohammadimehr, M. and Mehrabi, M. (2018), "Electro-thermomechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow", Appl. Math. Model., 60, 255-272. https://doi.org/10.1016/j.apm.2018.03.008   DOI
67 Nguyen-Thanh, N., Valizadeh, N., Nguyen, M.N., Nguyen-Xuan, H., Zhuang, X., Areias, P., Zi, G., Bazilevs, Y.D.L.L.R.T., De Lorenzis, L. and Rabczuk, T. (2015), "An extended isogeometric thin shell analysis based on Kirchhof-Love theory", Comput. Method. Appl. Mech. Eng., 284, 265-291. https://doi.org/10.1016/j.cma.2014.08.025   DOI
68 AkhavanAlavi, S.M., Mohammadimehr, M. and Edjtahed, S.H. (2019), "Active control of micro Reddy beam integrated with functionally graded nanocomposite sensor and actuator based on linear quadratic regulator method", Eur. J. Mech.-A/Solids 74, 449-461. https://doi.org/10.1016/j.euromechsol.2018.12.008   DOI
69 Ozdemir, Y.I. (2018), "Using fourth order element for free vibration parametric analysis of thick plate resting on elastic foundation", Struct. Eng. Mech., Int. J., 65(3), 213-222. http://doi.org/10.12989/sem.2018.65.3.213   DOI
70 Yang, Y., Chen, L., Xu, D. and Zheng, H. (2016), "Free and forced vibration analyses using the four-node quadrilateral element with continuous nodal stress", Eng. Anal. Bound. Element., 70, 1-11. https://doi.org/10.1016/j.enganabound.2016.05.005   DOI
71 Vu-Bac, N., Lahmer, T., Zhuang, X., Nguyen-Thoi, T. and Rabczuk, T. (2016), "A software framework for probabilistic sensitivity analysis for computationally expensive models", Comput. Method. Appl. Mech. Eng., 100, 19-31. https://doi.org/10.1016/j.advengsoft.2016.06.005   DOI
72 Zhu, P., Lei, Z.X. and Liew, K.M. (2012), "Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory", Compos. Struct., 94(4), 1450-1460. https://doi.org/10.1016/j.compstruct.2011.11.010   DOI
73 Zhang, L.W., Zhang, Y. and Liew, K.M. (2016b), "Free vibration analysis of triangular nanocomposite plates subjected to in-plane stresses using an element-free method", Compos. Struct., 149, 247-260. https://doi.org/10.1016/j.compstruct.2016.04.019   DOI
74 Song, Z.G., Zhang, L.W. and Liew, K.M. (2016), "Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments", Int. J. Mech. Sci., 115-116, 339-347. https://doi.org/10.1016/j.ijmecsci.2016.06.020   DOI
75 Yue, Y.M., Xu, K.Y. and Chen, T. (2016), "A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects", Compos. Struct., 136, 278-286. https://doi.org/10.1016/j.compstruct.2015.09.046   DOI
76 Zeinkiewicz, O.C. and Taylor, R.L. (2000), The Finite Element Method, (5th edition), Oxford: Butterworth- Heinemann, USA.
77 Zhang, H., Wang, C.M., Ruocco, E. and Challamel, N. (2016a), "Hencky bar-chain model for buckling and vibration analyses of non-uniform beams on variable elastic foundation", Eng. Struct., 126, 252-263. https://doi.org/10.1016/j.engstruct.2016.07.062   DOI