The paper presents a review on large deflection behavior of curved beams, as manifested through the responses under static loading. The term large deflection behavior refers to the inherent nonlinearity present in the analysis of such beam system response. The analysis leads to the field of geometric nonlinearity, in which equation of equilibrium is generally written in deformed configuration. Hence the domain of large deflection analysis treats beam of any initial configuration as curved beam. The term curved designates the geometry of center line of beam, distinguishing it from the usual straight or circular arc configuration. Different methods adopted by researchers, to analyze large deflection behavior of beam bending, have been taken into consideration. The methods have been categorized based on their application in various formats of problems. The nonlinear response of a beam under static loading is also a function of different parameters of the particular problem. These include boundary condition, loading pattern, initial geometry of the beam, etc. In addition, another class of nonlinearity is commonly encountered in structural analysis, which is associated with nonlinear stress-strain relations and known as material nonlinearity. However the present paper mainly focuses on geometric nonlinear analysis of beam, and analysis associated with nonlinear material behavior is covered briefly as it belongs to another class of study. Research works on bifurcation instability and vibration responses of curved beams under large deflection is also excluded from the scope of the present review paper.

Periodical:

International Journal of Engineering and Technologies (Volume 11)

Pages:

13-39

DOI:

10.18052/www.scipress.com/IJET.11.13

Citation:

S. Ghuku and K. N. Saha, "A Review on Stress and Deformation Analysis of Curved Beams under Large Deflection", International Journal of Engineering and Technologies, Vol. 11, pp. 13-39, 2017

Online since:

Jul 2017

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

[1] A.P. Boresi, R.J. Schmidt, Advanced mechanics of materials, sixth ed., John Wiley and Sons Incorporated, New York, (2003).

[2] S.P. Timoshenko, D.H. Young, Elements of strength of materials, fifth ed., Van Nostrand Reinhold Company, New York, (1968).

[3] Y.B. Yang, S.R. Kuo, Theory and analysis of nonlinear framed structures, Prentice Hall, Singapore, (1994).

[4] I.H. Shames, S. Dym, Energy and finite element methods in structural mechanics, Hemisphere Publishing Corporation, New York, (1985).

[5] C.S. Jog, Foundations and applications of mechanics, volume I: Continuum mechanics, second ed., Narosa publishing house private limited, New Delhi, (2007).

[6] E. Reissner, On one-dimensional finite-strain beam theory: the plane problem, Z. Angew. Math. Phys. 23 (1972) 795-804.

[7] E. Reissner, On finite deformations of space-curved beams, Z. Angew. Math. Phys. 32 (1981) 734-744.

DOI: 10.1007/bf00946983[8] S.S. Antman, Kirchhoff's problem for nonlinearly elastic rods, Q. Appl. Math. 32 (1974) 221-240.

[9] E. Cosserat, F. Cosserat, Theory of deformable bodies (Translated by D.H. Delphenich), volume 6, Scientific Library A Herman and Sons, Rue De La Sorbonne, Paris, (1909).

[10] A.M. Wahrhaftig, R.M.L.R.F. Brasil, Representative experimental and computational analysis of the initial resonant frequency of largely deformed cantilevered beams, Int. J. Solids. Struct. 102 (2016) 44-55.

[11] A.M. Wahrhaftig, R.M.L.R.F. Brasil, J.M. Balthazar, The first frequency of cantilevered bars with geometric effect: a mathematical and experimental evaluation, J. Brazil. Soc. Mech. Sci. Eng. 35 (2013) 457-467.

[12] S.P. Timoshenko, History of strength of materials, McGraw-Hill Book Company, New York, (1953).

[13] X.F. Li, K.Y. Lee, Effect of horizontal reaction force on the deflection of short simply supported beams under transverse loadings, Int. J. Mech. Sci. 99 (2015) 121-129.

[14] A. Mohyeddin, A. Fereidoon, An analytical solution for the large deflection problem of Timoshenko beams under three-point bending, Int. J. Mech. Sci. 78 (2014) 135-139.

[15] K.E. Bisshopp, D.C. Drucker, Large deflection of cantilever beams, Q. Appl. Math. 3 (1945) 272-275.

[16] W. Lacarbonara, H. Yabuno, Refined models of elastic beams undergoing large in-plane motions: theory and experiment, Int. J. Solids Struct. 43 (2006) 5066-5084.

[17] S. Eugster, Geometric continuum mechanics and induced beam theories, Lecture notes in applied and computational mechanics, Vol. 75, Springer, (2015).

DOI: 10.1007/978-3-319-16495-3_2[18] C. Meier, W.A. Wall, A. Popp, Geometrically exact finite element formulations for curved slender beams: Kirchhoff-Love theory vs. Simo-Reissner theory, arXiv preprint, arXiv: 1609. 00119, (2016).

DOI: 10.1007/s11831-017-9232-5[19] E.C. Da Lozzo, F. Auricchio, Geometrically exact three-dimensional beam theory: modeling and FEM implementation for statics and dynamics analysis, Master degree thesis, Earthquake Engineering, Istituto Universitario di Studi Superiori di Pavia, Pavia, Italy, (2010).

[20] S.S. Antman, Problems in nonlinear elasticity, in: Nonlinear problems of elasticity, Springer, 2005, pp.513-584.

[21] J.C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput. Method. Appl. M. 49 (1985) 55-70.

[22] J.C. Simo, L. Vu-Quoc, A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Struct. 27 (1991) 371-393.

DOI: 10.1016/0020-7683(91)90089-x[23] M.A. Crisfield, G. Jelenić, Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation, P. Roy. Soc. Lond. A Mat. 455 (1999) 1125-1147.

DOI: 10.1098/rspa.1999.0352[24] K. Washizu, Some considerations on a naturally curved and twisted slender beam, Stud. Appl. Math. 43 (1964) 111-116.

DOI: 10.1002/sapm1964431111[25] O.A. Bauchau, C.H. Hong, Large displacement analysis of naturally curved and twisted composite beams, AIAA J. 25(11) (1987) 1469-1475.

DOI: 10.2514/3.9806[26] K. Pan, J. Liu, Geometric nonlinear formulation for curved beams with varying curvature, Theor. Appl. Mech. Lett. 2 (2012) 063006.

[27] R.K. Kapania, J. Li, On a geometrically exact curved/twisted beam theory under rigid cross-section assumption, Comput. Mech. 30 (2003) 428-443.

[28] D.H. Hodges, Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams, AIAA J. 41 (2003) 1131-1137.

DOI: 10.2514/2.2054[29] H. Kurtaran, Large displacement static and transient analysis of functionally graded deep curved beams with generalized differential quadrature method, Compos. Struct. 131 (2015) 821-831.

[30] R.C. Batra, J. Xiao, Finite deformations of curved laminated St. Venant–Kirchhoff beam using layer-wise third order shear and normal deformable beam theory (TSNDT), Compos. Struct. 97 (2013) 147-164.

DOI: 10.1016/j.compstruct.2012.09.039[31] M. Cetraro, W. Lacarbonara, G. Formica, Nonlinear dynamic response of carbon nanotube nanocomposite microbeams, J. Comput. Nonlin. Dyn. 12(3) (2017) 031007.

[32] F. Daneshmand, Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory, Appl. Math. Comput. 243 (2014) 856-869.

DOI: 10.1016/j.amc.2014.05.094[33] F. Kaviani, H.R. Mirdamadi, Snap-through and bifurcation of nano-arches on elastic foundation by the strain gradient and nonlocal theories, Int. J. Struct. Stab. Dy. 13 (2013) 1350022-1-1350022-21.

[34] A.W. McFarland, J.S. Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, J. Micromech. Microeng. 15 (2005) 1060-1067.

DOI: 10.1088/0960-1317/15/5/024[35] N.A. Fleck et al., Strain gradient plasticity: theory and experiment, Acta Metall. Mater. 42 (1994) 475-487.

[36] D.C. Lam et al., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids. 51 (2003) 1477-1508.

[37] R. Ansari, R. Gholami, H. Rouhi, Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Compos. Part B-Eng. 43 (2012) 2985-2989.

DOI: 10.1016/j.compositesb.2012.05.049[38] R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. An. 11 (1962) 415-448.

DOI: 10.1007/bf00253946[39] F.A. Yang et al., Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct. 39 (2002) 2731-2743.

DOI: 10.1016/s0020-7683(02)00152-x[40] R.D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. An. 16 (1964) 51-78.

[41] A.C. Eringen, D.G. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972) 233-248.

[42] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703-4710.

DOI: 10.1063/1.332803[43] N.A. Fleck, J.W. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids. 41 (1993) 1825-1857.

[44] B. Akgöz, Ö. Civalek, A new trigonometric beam model for buckling of strain gradient microbeams, Int. J. Mech. Sci. 81 (2014) 88-94.

[45] B. Wang, J. Zhao, S. Zhou, A micro scale Timoshenko beam model based on strain gradient elasticity theory, Eur. J. Mech. A-Solids. 29 (2010) 591-599.

[46] B. Akgöz, Ö. Civalek, Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Arch. Appl. Mech. 82 (2012) 423-443.

DOI: 10.1007/s00419-011-0565-5[47] B. Akgöz, Ö. Civalek, Deflection of a hyperbolic shear deformable microbeam under a concentrated load, J. Appl. Comput. Mech. 2 (2016) 65-73.

[48] R. Artan, A. Tepe, The initial values method for buckling of nonlocal bars with application in nanotechnology, Eur. J. Mech. A-Solids. 27 (2008) 469-477.

DOI: 10.1016/j.euromechsol.2007.09.004[49] S.C. Pradhan, T. Murmu, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E. 42 (2010) 1944-(1949).

[50] M. Aydogdu, S. Filiz, Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity, Physica E. 43 (2011) 1229-1234.

[51] M.M. Shokrieh, I. Zibaei, Determination of the appropriate gradient elasticity theory for bending analysis of nano-beams by considering boundary conditions effect, Lat. Am. J. Solids Struct. 12 (2015) 2208-2230.

[52] R. Ansari, M.F. Oskouie, R. Gholami, Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory, Physica E. 75 (2016) 266-271.

[53] M.E. Gurtin, A.I. Murdoch, Surface stress in solids, Int. J. Solids Struct. 14 (1978) 431-440.

[54] M. Safarabadi et al., Effect of surface energy on the vibration analysis of rotating nanobeam, J. Solid Mech. 7 (2015) 299-311.

[55] K. Kiani, Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy, Int. J. Eng. Sci. 106 (2016) 57-76.

[56] K. Mercan, Ö. Civalek, Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Compos. Part B-Eng. 114 (2017) 34-45.

DOI: 10.1016/j.compositesb.2017.01.067[57] M.F. Oskouie, R. Ansari, Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects, Appl. Math. Model. 43 (2017) 337-350.

DOI: 10.1016/j.apm.2016.11.036[58] M. Saje, Finite element formulation of finite planar deformation of curved elastic beams, Comput. Struct. 39 (1991) 327-337.

DOI: 10.1016/0045-7949(91)90030-p[59] J. Zhao et al., Post-buckling and snap-through behavior of inclined slender beams, J. Appl. Mech. -T. ASME. 75 (2008) 041020-1-041020-7.

[60] M. Batista, Large deflections of a beam subject to three-point bending, Int. J. Nonlin. Mech. 69 (2015) 84-92.

[61] X.T. He et al., Nonlinear large deflection problems of beams with gradient: A biparametric perturbation method, Appl. Math. Comput. 219 (2013) 7493-7513.

[62] M. Maleki, S.A. Tonekaboni, S. Abbasbandy, A homotopy analysis solution to large deformation of beams under static arbitrary distributed load, Appl. Math. Model. 38 (2014) 355-368.

[63] A. Banerjee, B. Bhattacharya, A.K. Mallik, Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches, Int. J. Nonlin. Mech. 43 (2008) 366-376.

[64] H. Tari, On the parametric large deflection study of Euler–Bernoulli cantilever beams subjected to combined tip point loading, Int. J. Nonlin. Mech. 49 (2013) 90-99.

[65] G.D. Angel et al., Chord line force versus displacement for thin shallow arc pre-curved bimetallic strip, P. I. Mech. Eng. C-J. Mec. 229 (2015) 116-124.

[66] P.V. Sarma, B.R. Rao, S. Gopalacharyulu, Eigenfunction solution for the plane stress problems of curved beams, Int. J. Eng. Sci. 13 (1975) 149-159.

DOI: 10.1016/0020-7225(75)90025-7[67] J.M. Segura, G. Armengaud, Analytical formulation of stresses in curved composite beams, Arch. Appl. Mech. 68 (1998) 206-213.

[68] S.R. Ahmed, A.A. Mamun, P. Modak, Analysis of stresses in a simply-supported composite beam with stiffened lateral ends using displacement-potential field, Int. J. Mech. Sci. 78 (2014) 140-153.

[69] M. Wang, Y. Liu, Elasticity solutions for orthotropic functionally graded curved beams, Eur. J. Mech. A-Solids. 37 (2013) 8-16.

[70] P. Chu et al., Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending, Acta Mech. 226 (2015) 2235-2253.

[71] L. Chen, An integral approach for large deflection cantilever beams, Int. J. Nonlin. Mech. 45 (2010) 301-305.

[72] E. Solano-Carrillo, Semi-exact solutions for large deflections of cantilever beams of non-linear elastic behaviour, Int. J. Nonlin. Mech. 44 (2009) 253-256.

[73] H. Ahuett-Garza et al., Studies about the use of semicircular beams as hinges in large deflection planar compliant mechanisms, Precis. Eng. 38 (2014) 711-727.

[74] T. Beléndez, C. Neipp, A. Beléndez, Large and small deflections of a cantilever beam, Eur. J. Phys. 23 (2002) 371-379.

DOI: 10.1088/0143-0807/23/3/317[75] S. Ghuku, K.N. Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Eng. Sci. Technol. Int. J. 19 (2016) 135-146.

[76] C.M. Wang et al., Large deflections of an end supported beam subjected to a point load, Int. J. Nonlin. Mech. 32 (1997) 63-72.

[77] M. Mutyalarao, D. Bharathi, B.N. Rao, On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load, Int. J. Nonlin. Mech. 45 (2010) 433-441.

[78] A.K. Nallathambi, C.L. Rao, S.M. Srinivasan, Large deflection of constant curvature cantilever beam under follower load, Int. J. Mech. Sci. 52 (2010) 440-445.

[79] B.S. Shvartsman, Large deflections of a cantilever beam subjected to a follower force, J. Sound Vib. 304 (2007) 969-973.

DOI: 10.1016/j.jsv.2007.03.010[80] B.S. Shvartsman, Direct method for analysis of flexible cantilever beam subjected to two follower forces, Int. J. Nonlin. Mech. 44 (2009) 249-252.

[81] B.S. Shvartsman, Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force, Int. J. Nonlin. Mech. 50 (2013) 75-80.

[82] A.N. Eraslan, E. Arslan, A computational study on the nonlinear hardening curved beam problem, Int. J. Pure Appl. Math. 43 (2008) 129-143.

[83] M. Sitar, F. Kosel, M. Brojan, Large deflections of nonlinearly elastic functionally graded composite beams, Arch. Civ. Mech. Eng. 14 (2014) 700-709.

[84] A.G. Rodríguez et al., Design of an adjustable-stiffness spring: Mathematical modeling and simulation, fabrication and experimental validation, Mech. Mach. Theory. 46 (2011) 1970-(1979).

[85] I. Eren, Determining large deflections in rectangular combined loaded cantilever beams made of non-linear Ludwick type material by means of different arc length assumptions, Sadhana. 33 (2008) 45-55.

DOI: 10.1007/s12046-008-0004-7[86] M. Mutyalarao, D. Bharathi, B.N. Rao, Large deflections of a cantilever beam under an inclined end load, Appl. Math. Comput. 217 (2010) 3607-3613.

[87] H.A. Santos, P.M. Pimenta, J.M. De Almeida, Hybrid and multi-field variational principles for geometrically exact three-dimensional beams, Int. J. Nonlin. Mech. 45 (2010) 809-820.

[88] P. Hansbo, M.G. Larson, K. Larsson, Variational formulation of curved beams in global coordinates, Comput. Mech. 53 (2014) 611-623.

[89] K. Satō, Large deflection of a circular cantilever beam with uniformly distributed load, Ing. Arch. 27 (1959) 195-200.

DOI: 10.1007/bf00536388[90] M. Dado, S. Al-Sadder, A new technique for large deflection analysis of non-prismatic cantilever beams, Mech. Res. Commun. 32 (2005) 692-703.

[91] H. Niknam, A. Fallah, M.M. Aghdam, Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading, Int. J. Nonlin. Mech. 65 (2014) 141-147.

[92] M. Cannarozzi, L. Molari, Stress-based formulation for non-linear analysis of planar elastic curved beams, Int. J. Nonlin. Mech. 55 (2013) 35-47.

[93] M. Afshin, F. Taheri-Behrooz, Interlaminar stresses of laminated composite beams resting on elastic foundation subjected to transverse loading, Comp. Mater. Sci. 96 (2015) 439-447.

[94] R. Kumar, L.S. Ramachandra, D. Roy, Techniques based on genetic algorithms for large deflection analysis of beams, Sadhana. 29 (2004) 589-604.

[95] J.Q. Tarn, W.D. Tseng, Exact analysis of curved beams and arches with arbitrary end conditions: a Hamiltonian state space approach, J. Elasticity. 107 (2012) 39-63.

[96] M.Z. Aşık et al., A mathematical model for the behavior of laminated uniformly curved glass beams, Compos. Part B-Eng. 58 (2014) 593-604.

[97] A. Mitra, P. Sahoo, K. Saha, Large displacement of crossbeam structure through energy method, Int. J. Automot. Mech. Eng. 5 (2012) 520-544.

[98] A. Majumdar, D. Das, A study on thermal buckling load of clamped functionally graded beams under linear and nonlinear thermal gradient across thickness, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 2016. doi: 10. 1177/1464420716649213.

[99] A. Muliana, Large deformations of nonlinear viscoelastic and multi-responsive beams, Int. J. Nonlin. Mech. 71 (2015) 152-164.

[100] Y.J. Liu, Y.X. Li, Slow convergence of the BEM with constant elements in solving beam bending problems, Eng. Anal. Bound. Elem. 39 (2014) 1-4.

[101] R. Poya, A.J. Gil, P.D. Ledger, A computational framework for the analysis of linear piezoelectric beams using hp-FEM, Comput. Struct. 152 (2015) 155-172.

DOI: 10.1016/j.compstruc.2015.01.012[102] H. Sugiyama et al., Development of nonlinear elastic leaf spring model for multibody vehicle systems, Comput. Method. Appl. M. 195 (2006) 6925-6941.

[103] D.A. Miller, A.N. Palazotto, Nonlinear finite element analysis of composite beams and arches using a large rotation theory, Finite Elem. Anal. Des. 19 (1995) 131-152.

[104] G. Wu, X. He, P.F. Pai, Geometrically exact 3D beam element for arbitrary large rigid-elastic deformation analysis of aerospace structures, Finite Elem. Anal. Des. 47 (2011) 402-412.

[105] A.A. Correia, J.P. Almeida, R. Pinho, Force-based higher-order beam element with flexural–shear–torsional interaction in 3D frames, Part I: Theory, Eng. Struct. 89 (2015) 204-217.

[106] T. Beléndez, C. Neipp, A. Beléndez, Numerical and experimental analysis of a cantilever beam: a laboratory project to introduce geometric nonlinearity in mechanics of materials, Int. J. Eng. Ed. 19 (2003) 885-892.

[107] G.S. Shankar, S. Vijayarangan, Mono composite leaf spring for light weight vehicle–design, end joint analysis and testing, Mater. Sci. 12 (2006) 220-225.

[108] A. Dorogoy, D. Rittel, Transverse impact of free–free square aluminum beams: An experimental–numerical investigation, Int. J. Impact Eng. 35 (2008) 569-577.

DOI: 10.1016/j.ijimpeng.2007.05.004[109] P.F. Pai, T.J. Anderson, E.A. Wheater, Large-deformation tests and total-Lagrangian finite-element analyses of flexible beams, Int. J. Solids Struct. 37 (2000) 2951-2980.

[110] L.N. Gummadi, A.N. Palazotto, Large strain analysis of beams and arches undergoing large rotations, Int. J. Nonlin. Mech. 33 (1998) 615-645.

[111] C.A. Almeida et al., Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation, Mech. Res. Commun. 38 (2011) 553-559.

[112] L.N. Gummadi, A.N. Palazotto, Finite element analysis of arches undergoing large rotations—I: Theoretical comparison, Finite Elem. Anal. Des. 24 (1997) 213-235.

[113] K. Yoon, P.S. Lee, Nonlinear performance of continuum mechanics based beam elements focusing on large twisting behaviors, Comput. Methods Appl. M. 281 (2014) 106-130.

DOI: 10.1016/j.cma.2014.07.023[114] R.E. Erkmen, M.A. Bradford, Nonlinear elastic analysis of composite beams curved in-plan, Eng. Struct. 31 (2009) 1613-1624.

[115] K.J. Bathe, S. Bolourchi, Large displacement analysis of three-dimensional beam structures, Int. J. Numer. Meth. Eng. 14 (1979) 961-986.

[116] P.D. Gosling, L. Liu, Total Lagrangian perspectives on analytical sensitivities for flexible beams, Int. J. Eng. Sci. 40 (2002) 1363-1379.

DOI: 10.1016/s0020-7225(02)00025-3[117] Z.Q. Chen, T.J. Agar, Geometric nonlinear analysis of flexible spatial beam structures, Comput. Struct. 49 (1993) 1083-1094.

DOI: 10.1016/0045-7949(93)90019-a[118] S. Ghuku, K.N. Saha, An experimental study on stress concentration around a hole under combined bending and stretching stress field, Procedia Technol. 23 (2016) 20-27.

DOI: 10.1016/j.protcy.2016.03.068[119] M. Vangbo, An analytical analysis of a compressed bistable buckled beam, Sensors Actuat. A-Phys. 69 (1998) 212-216.

DOI: 10.1016/s0924-4247(98)00097-1[120] D. Pandit, S.M. Srinivasan, A simplified approach to solve quasi-statically moving load problems of elastica using end loaded elastica solution, Sadhana. 41 (2016) 707-712.

[121] F. Mujika, On the effect of shear and local deformation in three-point bending tests, Polym. Test. 26 (2007) 869-877.

[122] A.B. Pippard, The elastic arch and its modes of instability, Eur. J. Phys. 11 (1990) 359-365.

[123] D. Ieşan, On the thermal stresses in beams, J. Eng. Math. 6 (1972) 155-163.

[124] J.J. Ryan, L.J. Fischer, Photoelastic analysis of stress concentration for beams in pure bending with a central hole, J. Franklin I. 225 (1938) 513-526.

[125] L.M. Kachanov, Foundations of the theory of plasticity, North-Holland Publishing Company, Amsterdam, London, (1971).

[126] E. Mahdi, A.M. Hamouda, An experimental investigation into mechanical behavior of hybrid and nonhybrid composite semi-elliptical springs, Mater. Des. 52 (2013) 504-513.

[127] M.M. Rasheedat et al., Functionally graded material: An overview, in Proceedings of the World Congress on Engineering, London, U.K. 2012, Vol. 3, (2012).

[128] K. Sarkar, R. Ganguli, Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed–fixed boundary condition, Compos. Part B-Eng. 58 (2014) 361-370.

[129] U. Eroglu, Large deflection analysis of planar curved beams made of functionally graded materials using variational iterational method, Compos. Struct. 136 (2016) 204-216.

DOI: 10.1016/j.compstruct.2017.08.100[130] A. Pydah, R.C. Batra, Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams, Compos. Struct. 172 (2017) 45-60.

[131] A. Pydah, A. Sabale, Static analysis of bi-directional functionally graded curved beams, Compos. Struct. 160 (2017) 867-876.

[132] G. Nie, Z. Zhong, Closed-form solutions for elastoplastic pure bending of a curved beam with material inhomogeneity, Acta Mech. Solida Sinica. 27 (2014) 54-64.

[133] B. Štok, M. Halilovič, Analytical solutions in elasto-plastic bending of beams with rectangular cross section, Appl. Math. Model. 33 (2009) 1749-1760.

[134] D. Pandit, S.M. Srinivasan, Numerical analysis of large elasto-plastic deflection of constant curvature beam under follower load, Int. J. Nonlin. Mech. 84 (2016) 46-55.

DOI: 10.1016/j.ijnonlinmec.2016.04.013[135] K.Z. Ding, Q.H. Qin, M. Cardew-Hall, Substepping algorithms with stress correction for the simulation of sheet metal forming process, Int. J. Mech. Sci. 49 (2007) 1289-1308.

[136] H.B. Motra, J. Hildebrand, A. Dimmig-Osburg, Assessment of strain measurement techniques to characterise mechanical properties of structural steel, Eng. Sci. Technol. Int. J. 17 (2014) 260-269.

[137] J. Politch, Methods of strain measurement and their comparison, Opt. Lasers Eng. 6 (1985) 55-66.