# A modified uncoupled lower-order theory for FG beams

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## Abstract

Though the higher-order beam theory is variationally consistent, the lower-order beam theory has more definite engineering significance in practical applications. This paper begins with the modified uncoupled higher-order theory of functionally graded (FG) beams. After evaluating the three rigidity coefficients, contribution of the two higher-order generalized stresses to the virtual work is ignored and therefore a modified uncoupled lower-order theory is established for FG beams, including the basic equations and the shear correction factor, so that the lower-order beam theory is theoretically correlated with the high-order beam theory. The cases of pure shearing, pure bending and pure tension are solved, compared and discussed for a FG beam. The analytical solutions validate the accuracy and applicability of the present uncoupled lower-order theory.

## Keywords

FG beam Rigidity coefficients The principle of virtual work Uncoupled higher-order beam theory Uncoupled lower-order beam theory## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundations of China (Grant Nos. 11672221, 11272245, 11321062).

## References

- 1.Reddy, J.N.: Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids
**59**(11), 2382–99 (2011)MathSciNetzbMATHCrossRefGoogle Scholar - 2.Abanto-Bueno, J., Lambros, J.: Parameters controlling fracture resistance in functionally graded materials under mode I loading. Int. J. Solids Struct.
**43**(13), 3920–39 (2006)CrossRefGoogle Scholar - 3.Adámek, V., Valeš, F.: Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load. Eur. J. Mech. A/Solids
**49**, 373–81 (2015)MathSciNetzbMATHCrossRefGoogle Scholar - 4.Wetherhold, R.C., Seelman, S., Wang, J.: The use of functionally graded materials to eliminate or control thermal deformation. Compos. Sci. Technol.
**56**(9), 1099–104 (1996)CrossRefGoogle Scholar - 5.Sankar, B.V.: An elasticity solution for functionally graded beams. Compos. Sci. Technol.
**61**(5), 689–96 (2001)CrossRefGoogle Scholar - 6.Do, V.N.V., Thai, C.H.: A modified Kirchhoff plate theory for analyzing thermo-mechanical static and buckling responses of functionally graded material plates. Thin-Walled Struct.
**117**, 113–26 (2017)CrossRefGoogle Scholar - 7.Li, X.F.: A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J. Sound Vib.
**318**(4), 1210–29 (2008)CrossRefGoogle Scholar - 8.Timoshenko, P.S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag.
**7**(245), 239–50 (1921)Google Scholar - 9.Nguyen, T.K., Vo, T.P., Thai, H.T.: Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Compos. Part B Eng.
**55**(55), 147–57 (2013)CrossRefGoogle Scholar - 10.Wattanasakulpong, N., Mao, Q.: Dynamic response of Timoshenko functionally graded beams with classical and non-classical boundary conditions using Chebyshev collocation method. Compos. Struct.
**119**(119), 346–54 (2015)CrossRefGoogle Scholar - 11.Nguyen, T.K., Bonnet, K.S., Shear, G.: Correction factors for functionally graded plates. Mech. Adv. Mater. Struct.
**14**(8), 567–75 (2007)CrossRefGoogle Scholar - 12.Vu, T.-V., Nguyen, N.-H., Khosravifard, A., Hematiyan, M.R., Tanaka, S., Bui, T.Q.: A simple FSDT-based meshfree method for analysis of functionally graded plates. Eng. Anal. Bound. Elem.
**79**, 1–12 (2017)MathSciNetzbMATHCrossRefGoogle Scholar - 13.Sharma, P., Parashar, S.K.: Free vibration analysis of shear-induced flexural vibration of FGPM annular plate using generalized differential quadrature method. Compos. Struct.
**155**(2), 213–22 (2016)CrossRefGoogle Scholar - 14.Eftekhar, H., Zeynali, H., Nasihatgozar, M.: Electro-magneto temperature-dependent vibration analysis of functionally graded-carbon nanotube-reinforced piezoelectric Mindlin cylindrical shells resting on a temperature-dependent, orthotropic elastic medium. Mech. Adv. Mater. Struct.
**25**, 1–14 (2018)CrossRefGoogle Scholar - 15.Jing, L., Ming, P.-J., Zhang, W.-P., Fu, L.-R., Cao, Y.-P.: Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method. Compos. Struct.
**138**, 192–213 (2016)CrossRefGoogle Scholar - 16.Benatta, M.A., Tounsi, A., Mechab, I., Bouiadjra, M.B.: Mathematical solution for bending of short hybrid composite beams with variable fibers spacing. Appl. Math. Comput.
**212**(2), 337–48 (2009)MathSciNetzbMATHGoogle Scholar - 17.Frikha, A., Hajlaoui, A., Wali, M., Dammak, F.: A new higher order C0 mixed beam element for FGM beams analysis. Compos. Part B: Eng.
**106**, 181–9 (2016)CrossRefGoogle Scholar - 18.Dong, S.B., Alpdogan, C., Taciroglu, E.: Much ado about shear correction factors in Timoshenko beam theory. Int. J. Solids Struct.
**47**(13), 1651–65 (2010)zbMATHCrossRefGoogle Scholar - 19.Romano, G., Barretta, A., Barretta, R.: On torsion and shear of Saint-Venant beams. Eur. J. Mech.
**35**(6), 47–60 (2012)MathSciNetzbMATHCrossRefGoogle Scholar - 20.Faghidian, S.A.: Unified formulations of the shear coefficients in Timoshenko beam theory. ASCE J. Eng. Mech.
**143**(9), 06017013 (2017)CrossRefGoogle Scholar - 21.Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech.
**51**(4), 745–52 (1984)zbMATHCrossRefGoogle Scholar - 22.Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng.
**47**(1–3), 663–84 (2000)zbMATHCrossRefGoogle Scholar - 23.Touratier, M.: An efficient standard plate-theory. Int. J. Eng. Sci.
**29**(8), 901–16 (1991)zbMATHCrossRefGoogle Scholar - 24.Soldatos, K.P.: A transverse-shear deformation theory for homogeneous monoclinic plates. Acta Mech.
**94**(3–4), 195–220 (1992)MathSciNetzbMATHCrossRefGoogle Scholar - 25.Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct.
**89**(1), 94–101 (2009)CrossRefGoogle Scholar - 26.Karama, M., Afaq, K., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct.
**40**(6), 1525–46 (2003)zbMATHCrossRefGoogle Scholar - 27.Soldatos, K., Timarci, T.: A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories. Compos. Struct.
**25**(1–4), 165–71 (1993)CrossRefGoogle Scholar - 28.Aydogdu, M., Taskin, V.: Free vibration analysis of functionally graded beams with simply supported edges. Mater. Des.
**28**(5), 1651–6 (2007)CrossRefGoogle Scholar - 29.Mechab, I., Tounsi, A., Benatta, M.A., Bedia, E.A.A.: Deformation of short composite beam using refined theories. J. Math. Anal. Appl.
**346**(2), 468–79 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - 30.Ben-Oumrane, S., Abedlouahed, T., Ismail, M., Mohamed, B.B., Mustapha, M., El Abbas, A.B.: A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Comput. Mater. Sci.
**44**(4), 1344–50 (2009)CrossRefGoogle Scholar - 31.Şimşek, M.: Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl. Eng. Des.
**240**(4), 697–705 (2010)CrossRefGoogle Scholar - 32.Thai, H.T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci.
**52**(3), 56–64 (2012)MathSciNetzbMATHCrossRefGoogle Scholar - 33.Wali, M., Hajlaoui, A., Dammak, F.: Discrete double directors shell element for the functionally graded material shell structures analysis. Comput. Methods Appl. Mech. Eng.
**278**, 388–403 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 34.Thai, H.T., Kim, S.E.: A review of theories for the modeling and analysis of functionally graded plates and shells. Compos. Struct.
**128**(3), 70–86 (2015)CrossRefGoogle Scholar - 35.Van Do, T., Nguyen, D.K., Duc, N.D., Doan, D.H., Bui, T.Q.: Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory. Thin-Walled Struct.
**119**, 687–99 (2017)CrossRefGoogle Scholar - 36.Srividhya, S., Raghu, P., Rajagopal, A., Reddy, J.N.: Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory. Int. J. Eng. Sci.
**125**, 1–22 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 37.Khorasani, V.S., Bayat, M.: Bending analysis of FG plates using a general third-order plate theory with modified couple stress effect and MLPG method. Eng. Anal. Bound. Elem.
**94**, 159–71 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 38.Morimoto, T., Tanigawa, Y., Kawamura, R.: Thermal buckling of functionally graded rectangular plates subjected to partial heating. Int. J. Mech. Sci.
**48**(9), 926–37 (2006)zbMATHCrossRefGoogle Scholar - 39.Abrate, S.: Functionally graded plates behave like homogeneous plates. Compos. Part B: Eng.
**39**, 151–8 (2008)CrossRefGoogle Scholar - 40.Zhang, D.G., Zhou, Y.H.: A theoretical analysis of FGM thin plates based on physical neutral surface. Comput. Mater. Sci.
**44**(2), 716–20 (2008)CrossRefGoogle Scholar - 41.Zhang, D.G.: Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory. Int. J. Mech. Sci.
**68**(7), 92–104 (2013)CrossRefGoogle Scholar - 42.Levinson, M.: A new rectangular beam theory. J. Sound Vib.
**74**(1), 81–7 (1981)zbMATHCrossRefGoogle Scholar - 43.Murthy, M.V.V.: An improved transverse shear deformation theory for laminated anisotropic plates. NASA Technical Paper, 1–39 (1981)Google Scholar
- 44.Duan, T.C., Li, L.X.: Study on higher-order shear deformation theories of thick-plate. Chin. J. Theor. Appl. Mech.
**48**(5), 1096–113 (2016)Google Scholar - 45.Pei, Y.L., Geng, P.S., Li, L.X.: A modified higher-order theory for FG beams. Eur. J. Mech. A/Solids
**134**, 186–97 (2018)MathSciNetzbMATHCrossRefGoogle Scholar - 46.Huang, Y., Wu, J.X., Li, X.F., Yang, L.E.: Higher-order theory for bending and vibration of beams with circular cross section. J. Eng. Math.
**80**(1), 91–104 (2013)MathSciNetzbMATHCrossRefGoogle Scholar - 47.Barretta, R.: Analogies between Kirchhoff plates and Saint-Venant beams under flexure. Acta Mech.
**225**(7), 2075–83 (2014)MathSciNetzbMATHCrossRefGoogle Scholar - 48.Barretta, R., Feo, L., Luciano, R., Sciarra, F.M.D., Penna, R.: Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation. Compos. Part B: Eng.
**100**, 208–19 (2016)CrossRefGoogle Scholar - 49.Barretta, R.: On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint–Venant beam theory. Int. J. Solids Struct.
**49**(21), 3038–46 (2012)CrossRefGoogle Scholar - 50.Barretta, R.: On Cesàro–Volterra method in orthotropic Saint–Venant beam. J. Elast.
**112**(2), 233–53 (2013)MathSciNetzbMATHCrossRefGoogle Scholar - 51.Thai, H.T., Vo, T.P.: Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int. J. Mech. Sci.
**62**(1), 57–66 (2012)CrossRefGoogle Scholar - 52.Wattanasakulpong, N., Gangadhara Prusty, B., Kelly, D.W.: Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int. J. Mech. Sci.
**53**(9), 734–43 (2011)CrossRefGoogle Scholar