DataVXPhysical MesomechanicsPaper
journal article·2023-10

Investigation of Dynamic Characteristics of Imperfect FG Beams on the Winkler–Pasternak Foundation under Thermal Loading

S. Abdelbari, A. Attia, F. Bourada, A. A. Bousahla, A. Tounsi, M. H. Ghazwani

Physical Mesomechanics· Volume 26· Issue 5
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Source abstract

Abstract

Abstract The interest of the present paper is the analysis of free vibration of imperfect functionally graded (FG) beams resting on foundations (with two elastic parameters). The FG beam is made of temperature-dependent metal (Al)/ceramic (Al2O3) material, which is graded in the thickness direction and subjected to various thermal loads (uniform and nonuniform). The appearance of microvoids is considered as porosity in the body structure. Two models of the symmetric porosity distribution are examined. The studied one-dimensional (1D) structure is modeled by employing simple three-variable higher-order integral formulations. Zero traction on the free surface of the 1D structure is gained by using the sinusoidal warping function in the current model, which avoids correction factors. Analytical modeling of structures is carried out using the Hamilton principle and Navier approach to derive the equations of motion and the analytical solution of the current model. Several examples of the free vibration analysis are presented in graphical and tabular forms. A detailed parametric analysis is performed to illustrate the impact of several beam parameters, such as dimensions, inhomogeneity and porosity indices, as well as of the foundation reaction on the fundamental frequency of imperfect FG beams.
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Research context

Ceramics & glasses

OpenAlex topics: Composite Structure Analysis and Optimization · Vibration and Dynamic Analysis · Structural Load-Bearing Analysis

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References (100+)

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1. Akgöz, B. and Civalek, Ö., Buckling Analysis of Functionally Graded Microbeams Based on the Strain Gradient Theory, Acta Mech., 2013, vol. 224, no. 9, pp. 2185–2201. https://doi.org/10.1007/s00707-013-0883-510.1007/s00707-013-0883-5 ↗
2. Eltaher, M.A., Khairy, A., Sadoun, A.M., and Omar, F.A., Static and Buckling Analysis of Functionally Graded Timoshenko Nanobeams, Appl. Math. Comput., 2014, vol. 229, pp. 283–295. https://doi.org/10.1016/j.amc.2013.12.07210.1016/j.amc.2013.12.072 ↗
3. Arefi, M., Elastic Solution of a Curved Beam Made of Functionally Graded Materials with Different Cross Sections, Steel Compos. Struct., 2015, vol. 18, no. 3, pp. 659–672. https://doi.org/10.12989/scs.2015.18.3.65910.12989/scs.2015.18.3.659 ↗
4. Arefi, M., Nonlinear Electromechanical Analysis of a Functionally Graded Square Plate Integrated with Smart Layers Resting on Winkler–Pasternak Foundation, Smart Struct. Syst., 2015, vol. 16, no. 1, pp. 195–211. https://doi.org/10.12989/sss.2015.16.1.19510.12989/sss.2015.16.1.195 ↗
5. Akbaş, Ş.D., Wave Propagation of a Functionally Graded Beam in Thermal Environments, Steel Compos. Struct., 2015, vol. 19, no. 6, pp. 1421–1447. https://doi.org/10.12989/scs.2015.19.6.142110.12989/scs.2015.19.6.1421 ↗
6. Celebi, K., Yarimpabuc, D., and Keles, I., A Unified Method for Stresses in FGM Sphere with Exponentially-Varying Properties, Struct. Eng. Mech., 2016, vol. 57, no. 5, pp. 823–835. https://doi.org/10.12989/sem.2016.57.5.82310.12989/sem.2016.57.5.823 ↗
7. Akavci, S.S., Mechanical Behavior of Functionally Graded Sandwich Plates on Elastic Foundation, Composites. B. Eng., 2016, vol. 96, pp. 136–152. https://doi.org/10.1016/j.compositesb.2016.04.03510.1016/j.compositesb.2016.04.035 ↗
8. Ebrahimi, F. and Shafiei, N., Application of Eringen’s Nonlocal Elasticity Theory for Vibration Analysis of Rotating Functionally Graded Nanobeams, Smart Struct. Syst., 2016, vol. 17, no. 5, pp. 837–857. https://doi.org/10.12989/sss.2016.17.5.83710.12989/sss.2016.17.5.837 ↗
9. Turan, M., Adiyaman, G., Kahya, V., and Birinci, A., Axisymmetric Analysis of a Functionally Graded Layer Resting on Elastic Substrate, Struct. Eng. Mech., 2016, vol. 58, no. 3, pp. 423–442. https://doi.org/10.12989/sem.2016.58.3.42310.12989/sem.2016.58.3.423 ↗
10. Karami, B., Shahsavari, D., and Janghorban, M., Wave Propagation Analysis in Functionally Graded (FG) Nanoplates under In-Plane Magnetic Field Based on Nonlocal Strain Gradient Theory and Four Variable Refined Plate Theory, Mech. Adv. Mater. Struct., 2018, vol. 25, no. 12, pp. 1047–1057. https://doi.org/10.1080/15376494.2017.132314310.1080/15376494.2017.1323143 ↗
11. Karami, B., Shahsavari, D., Janghorban, M., and Li, L., Influence of Homogenization Schemes on Vibration of Functionally Graded Curved Microbeams, Compos. Struct., 2019, vol. 216, pp. 67–79. https://doi.org/10.1016/j.compstruct.2019.02.08910.1016/j.compstruct.2019.02.089 ↗
12. Safa, A., Hadji, L., Bourada, M., and Zouatnia, N., Thermal Vibration Analysis of FGM Beams Using an Efficient Shear Deformation Beam Theory, Earthq. Struct., 2019, vol. 17, no. 3, pp. 329–336. https://doi.org/10.12989/eas.2019.17.3.32910.12989/eas.2019.17.3.329 ↗
13. Selmi, A., Exact Solution for Nonlinear Vibration of Clamped-Clamped Functionally Graded Buckled Beam, Smart Struct. Syst., 2020, vol. 26, no. 3, pp. 361–371. https://doi.org/10.12989/sss.2020.26.3.36110.12989/sss.2020.26.3.361 ↗
14. Chami, K., Messafer, T., and Hadji, L., Analytical Modeling of Bending and Free Vibration of Thick Advanced Composite Beams Resting on Winkler–Pasternak Elastic Foundation, Earthq. Struct., 2020, vol. 19, no. 2, pp. 91–101. https://doi.org/10.12989/eas.2020.19.2.09110.12989/eas.2020.19.2.091 ↗
15. Merzoug, M., Bourada, M., Sekkal, M., Ali Chaibdra, A., Belmokhtar, C., Benyoucef, S., and Benachour, A., 2D and Quasi 3D Computational Models for Thermoelastic Bending of FG Beams on Variable Elastic Foundation: Effect of the Micromechanical Models, Geomech. Eng., 2020, vol. 22, no. 4, pp. 361–374. https://doi.org/10.12989/gae.2020.22.4.36110.12989/gae.2020.22.4.361 ↗
16. Hadji, L., Vibration Analysis of FGM Beam: Effect of the Micromechanical Models, Coupl. Syst. Mech., 2020, vol. 9, no. 3, pp. 265–280. https://doi.org/10.12989/csm.2020.9.3.26510.12989/csm.2020.9.3.265 ↗
17. Chikh, A., Free Vibration Analysis of Simply Supported P-FGM Nanoplate Using a Nonlocal Four Variables Shear Deformation Plate Theory, Strojníckyčasopis. J. Mech. Eng., 2019, vol. 69, no. 4, pp. 9–24. https://doi.org/10.2478/scjme-2019-003910.2478/scjme-2019-0039 ↗
18. Chikh, A., Investigations in Static Response and Free Vibration of a Functionally Graded Beam Resting on Elastic Foundations, Fratt. Integr. Strutt., 2020, vol. 14, no. 51, pp. 115–126. https://doi.org/10.3221/IGF-ESIS.51.0910.3221/IGF-ESIS.51.09 ↗
19. Ton-That, H.L., Finite Element Analysis of Functionally Graded Skew Plates in Thermal Environment Based on the New Third-Order Shear Deformation Theory, J. Appl. Computat. Mech., 2020, vol. 6, no. 4, pp. 1044–1057. https://doi.org/
20. Yaylaci, M., Yayli, M., Yaylaci, E.U., Olmez, H., and Birinci, A., Analyzing the Contact Problem of a Functionally Graded Layer Resting on an Elastic Half Plane with Theory of Elasticity, Finite Element Method and Multilayer Perceptron, Struct. Eng. Mech., 2021, vol. 78, no. 5, pp. 585–597. https://doi.org/10.12989/sem.2021.78.5.58510.12989/sem.2021.78.5.585 ↗
21. Yaylaci, M., Sabano, B.S., Ozdemir, M.E., and Birinci, A., Solving the Contact Problem of Functionally Graded Layers Resting on a HP and Pressed with a Uniformly Distributed Load by Analytical and Numerical Methods, Struct. Eng. Mech., 2022, vol. 82, no. 3, pp. 401–416. https://doi.org/10.12989/SEM.2022.82.3.40110.12989/SEM.2022.82.3.401 ↗
22. Öner, E., Şabano, B.Ş., Yaylacı, E.U., Adıyaman, G., Yaylacı, M., and Birinci, A., On the Plane Receding Contact between Two Functionally Graded Layers Using Computational, Finite Element and Artificial Neural Network Methods, J. Appl. Math. Mech., 2022, vol. 102, no. 2. https://doi.org/10.1002/zamm.20210028710.1002/zamm.202100287 ↗
23. Yaylaci, M., Adiyaman, G., Oner, E., and Birinci, A., Investigation of Continuous and Discontinuous Contact Cases in the Contact Mechanics of Graded Materials Using Analytical Method and FEM, Comp. Concr., 2021, vol. 27, no. 3, pp. 199–210. https://doi.org/10.12989/CAC.2021.27.3.19910.12989/CAC.2021.27.3.199 ↗
24. Adıyaman, G., Birinci, A., Öner, E., and Yaylacı, M., A Receding Contact Problem between a Functionally Graded Layer and Two Homogeneous Quarter Planes, Acta Mech., 2016, vol. 227, no. 6, pp. 1753–1766. https://doi.org/10.1007/s00707-016-1580-y10.1007/s00707-016-1580-y ↗
25. Reddy, J., Analysis of Functionally Graded Plates, Int. J. Numer. Meth. Eng., 2000, vol. 47, no. 1–3, pp. 663–684. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)10.1002/(SICI)1097-0207(20000110/30) ↗
26. Alshorbagy, A.E., Eltaher, M.A., and Mahmoud, F., Free Vibration Characteristics of a Functionally Graded Beam by Finite Element Method, Appl. Math. Model., 2011, vol. 35, no. 1, pp. 412–425. http://dx.doi.org/10.1016/j.apm.2010.07.00610.1016/j.apm.2010.07.006 ↗
27. Natarajan, S. and Manickam, G., Bending and Vibration of Functionally Graded Material Sandwich Plates Using an Accurate Theory, Finite Elem. Anal. Design, 2012, vol. 57, pp. 32–42. http://dx.doi.org/10.1016/j.finel.2012.03.00610.1016/j.finel.2012.03.006 ↗
28. Ghatage, P.S., Kar, V.R., and Sudhagar, P.E., On the Numerical Modelling and Analysis of Multi-Directional Functionally Graded Composite Structures: A Review, Composite Struct., 2020, vol. 236, p. 111837. http://dx.doi.org/10.1016/j.compstruct.2019.11183710.1016/j.compstruct.2019.111837 ↗
29. Melaibari, A., Abo-bakr, R.M., Mohamed, S.A., and Eltaher, M.A., Static Stability of Higher Order Functionally Graded Beam under Variable Axial Load, Alex. Eng. J., 2020, vol. 59, no. 3, pp. 1661–1675. http://dx.doi.org/10.1016/j.aej.2020.04.01210.1016/j.aej.2020.04.012 ↗
30. Shaker, A., Abdelrahman, W., Tawfik, M., and Sadek, E., Stochastic Finite Element Analysis of the Free Vibration of Functionally Graded Material Plates, Comput. Mech., 2008, vol. 41, no. 5, pp. 707–714. https://doi.org/10.1007/s00466-007-0226-210.1007/s00466-007-0226-2 ↗
31. Ebrahimi, F. and Zia, M., Large Amplitude Nonlinear Vibration Analysis of Functionally Graded Timoshenko Beams with Porosities, Acta Astronaut., 2015, vol. 116, pp. 117–125. https://doi.org/10.1016/j.actaastro.2015.06.01410.1016/j.actaastro.2015.06.014 ↗
32. Avcar, M. and Mohammed, W.K.M., Free Vibration of Functionally Graded Beams Resting on Winkler–Pasternak Foundation, Arabian J. Geosci., 2018, vol. 11, no. 232. https://doi.org/10.1007/s12517-018-3579-210.1007/s12517-018-3579-2 ↗
33. Madenci, E., A Refined Functional and Mixed Formulation to Static Analyses of FGM Beams, Struct. Eng. Mech., 2019, vol. 69, no. 4, pp. 427–437. https://doi.org/10.12989/sem.2019.69.4.42710.12989/sem.2019.69.4.427 ↗
34. Zhang, N., Khan, T., Guo, H., Shi, S., Zhong, W., and Zhang, W., Functionally Graded Materials: An Overview of Stability, Buckling, and Free Vibration Analysis, Adv. Mater. Sci. Eng., 2019. https://doi.org/10.1155/2019/135415010.1155/2019/1354150 ↗
35. Najafizadeh, M.M. and Eslami, M.R., First-Order-Theory-Based Thermo Elastic Stability of Functionally Graded Material Circular Plates, AIAA J., 2002, vol. 40, no. 7, pp. 1444–1450. https://doi.org/10.2514/2.180710.2514/2.1807 ↗
36. Javaheri, R. and Eslami, M.R., Thermal Buckling of Functionally Graded Plates, AIAA J., 2002, vol. 40, no. 1, pp. 162–169. https://doi.org/10.2514/2.162610.2514/2.1626 ↗
37. Najafizadeh, M.M. and Heydari, H.R., Thermal Buckling of Functionally Graded Circular Plates Based on Higher Order Shear Deformation Plate Theory, Eur. J. Mech. A. Solids, 2004, vol. 23, no. 6, pp. 1085–1100. https://doi.org/10.1016/j.euromechsol.2004.08.00410.1016/j.euromechsol.2004.08.004 ↗
38. Zhao, X., Lee, Y.Y., and Liew, K.M., Mechanical and Thermal Buckling Analysis of Functionally Graded Plates, Compos. Struct., 2009, vol. 90, no. 2, pp. 161–171. https://doi.org/10.1016/j.compstruct.2009.03.00510.1016/j.compstruct.2009.03.005 ↗
39. Kiani, Y. and Eslami, M.R., Thermal Buckling Analysis of Functionally Graded Material Beams, Int. J. Mech. Mater. Des., 2010, vol. 6, no. 3, pp. 229–238. https://doi.org/10.1007/s10999-010-9132-410.1007/s10999-010-9132-4 ↗
40. Ma, L.S. and Lee, D.W., A Further Discussion of Nonlinear Mechanical Behavior for FGM Beams under In-Plane Thermal Loading, Compos. Struct., 2011, vol. 93, no. 2, pp. 831–842. https://doi.org/10.1016/j.compstruct.2010.07.01110.1016/j.compstruct.2010.07.011 ↗
41. Ma, L.S. and Lee, D.W., Exact Solutions for Nonlinear Static Responses of a Shear Deformable FGM Beam under an In-Plane Thermal Loading, Eur. J. Mech. A. Solids, 2012, vol. 31, no. 1, pp. 13–20. https://doi.org/10.1016/j.euromechsol.2011.06.01610.1016/j.euromechsol.2011.06.016 ↗
42. Levyakov, S.V., Elastic Solution for Thermal Bending of a Functionally Graded Beam, Acta Mech., 2013, vol. 224, no. 8, pp. 1731–1740. https://doi.org/10.1007/s00707-013-0834-110.1007/s00707-013-0834-1 ↗
43. Fallah, A. and Aghdam, M.M., Thermo-Mechanical Buckling and Nonlinear Free Vibration Analysis of Functionally Graded Beams on Nonlinear Elastic Foundation, Composites. B. Eng., 2012, vol. 43, no. 3, pp. 1523–1530. https://doi.org/10.1016/j.compositesb.2011.08.04110.1016/j.compositesb.2011.08.041 ↗
44. Ebrahimi, F., Salari, E., and Hosseini, S.A.H., Thermomechanical Vibration Behavior of FG Nanobeam Subjected to Linear and Non-Linear Temperature Distributions, J. Therm. Stress., 2015, vol. 38, no. 12, pp. 1360–1386. https://doi.org/10.1080/01495739.2015.107398010.1080/01495739.2015.1073980 ↗
45. Ebrahimi, F. and Barati, M.R., Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory, Acta Mech. Sol. Sin., 2016, vol. 29, no. 5, pp. 547–554. https://doi.org/10.1016/s0894-9166(16)30272-510.1016/s0894-9166(16)30272-5 ↗
46. Dehrouyeh-Semnani, A.M., On the Thermally Induced Non-Linear Response of Functionally Graded Beams, Int. J. Eng. Sci., 2018, vol. 125, pp. 53–74. https://doi.org/10.1016/J.IJENGSCI.2017.12.00110.1016/J.IJENGSCI.2017.12.001 ↗
47. Yahea, H.T. and Majeed, W.I., Free Vibration of Laminated Composite Plates in Thermal Environment Using a Simple Four Variable Plate Theory, Compos. Mater. Eng., 2021, vol. 3, no. 3, pp. 179–199. https://doi.org/10.12989/cme.2021.3.3.17910.12989/cme.2021.3.3.179 ↗
48. Yaylaci, M., Eyüboğlu, A., Adıyaman, G., Yaylaci, E.U., Öner, E., and Birinci, A., Assessment of Different Solution Methods for Receding Contact Problems in Functionally Graded Layered Mediums, Mech. Mater., 2021, vol. 154, p. 103730. https://doi.org/10.1016/j.mechmat.2020.10373010.1016/j.mechmat.2020.103730 ↗
49. Birinci, A., Adıyaman, G., Yaylacı, M., and Öner, E., Analysis of Continuous and Discontinuous Cases of a Contact Problem Using Analytical Method and FEM, Lat. Am. J. Solids Struct., 2015, vol. 12, no. 9, pp. 1771–1789. https://doi.org/10.1590/1679-7825157410.1590/1679-78251574 ↗
50. Yaylaci, M., Adıyaman, G., Öner, E., and Birinci, A., Examination of Analytical and Finite Element Solutions Regarding Contact of a Functionally Graded Layer, Struct. Eng. Mech., 2020, vol. 76, no. 3, pp. 325–336. https://doi.org/10.12989/SEM.2020.76.3.32510.12989/SEM.2020.76.3.325 ↗
51. Yaylaci, M., Abanoz, M., Yaylaci, E.U., Ölmez, H., Sekban, D.M., and Birinci, A., Evaluation of the Contact Problem of Functionally Graded Layer Resting on Rigid Foundation Pressed Via Rigid Punch by Analytical and Numerical (FEM and MLP) Methods, Arch. Appl. Mech., 2022, vol. 92, pp. 1953–1971. https://doi.org/10.1007/s00419-022-02159-510.1007/s00419-022-02159-5 ↗
52. Yaylaci, M., Abanoz, M., Yaylaci, E.U., Olmez, H., Sekban, D.M., and Birinci, A., The Contact Problem of the Functionally Graded Layer Resting on Rigid Foundation Pressed Via Rigid Punch, Steel Compos. Struct., 2022, vol. 43, no. 5, pp. 661–672. https://doi.org/10.12989/scs.2022.43.5.66110.12989/scs.2022.43.5.661 ↗
53. Eisenberger, M., Vibration Frequencies for Beams on Variable One- and Two-Parameter Elastic Foundations, J. Sound Vibr., 1994, vol. 176, no. 5, pp. 577–584. https://doi.org/10.1006/jsvi.1994.13910.1006/jsvi.1994.139 ↗
54. Zhou, D., A General Solution to Vibrations of Beams on Variable Winkler Elastic Foundation, Comp. Struct., 1993, vol. 47, no. 1, pp. 83–90. https://doi.org/10.1016/0045-7949(93)90281-H10.1016/0045-7949(93)90281-H ↗
55. Matsunaga, H., Vibration and Buckling of Deep Beam-Columns on Two-Parameter Elastic Foundations, J. Sound Vibr., 1999, vol. 228, no. 2, pp. 359–376. https://doi.org/10.1006/jsvi.1999.241510.1006/jsvi.1999.2415 ↗
56. Chen, C.N., DQEM Vibration Analyses of Nonprismatic Beams Resting on Elastic Foundations, Int. J. Struct. Stability Dyn., 2002, vol. 2, no. 1, pp. 99–115. https://doi.org/10.1142/S021945540200040310.1142/S0219455402000403 ↗
57. Malekzadeh, P. and Karami, G., A Mixed Differential Quadrature and Finite Element Free Vibration and Buckling Analysis of Thick Beams on Two-Parameter Elastic Foundations, Appl. Math. Model., 2008, vol. 32, no. 7, pp. 1381–1394. https://doi.org/10.1016/j.apm.2007.04.01910.1016/j.apm.2007.04.019 ↗
58. Ying, J., Lü, C.F., and Chen, W.Q., Two-Dimensional Elasticity Solutions for Functionally Graded Beams Resting on Elastic Foundations, Compos. Struct., 2008, vol. 84, no. 3, pp. 209–219. https://doi.org/10.1016/j.compstruct.2007.07.00410.1016/j.compstruct.2007.07.004 ↗
59. Esfahani, S.E., Kiani, Y., and Eslami, M.R., Non-Linear Thermal Stability Analysis of Temperature Dependent FGM Beams Supported on Non-Linear Hardening Elastic Foundations, Int. J. Mech. Sci., 2013, vol. 69, pp. 10–20. https://doi.org/10.1016/j.ijmecsci.2013.01.00710.1016/j.ijmecsci.2013.01.007 ↗
60. Akgöz, B. and Civalek, Ö., Thermo-Mechanical Buckling Behavior of Functionally Graded Microbeams Embedded in Elastic Medium, Int. J. Eng. Sci., 2014, vol. 85, pp. 90–104. https://doi.org/10.1016/j.ijengsci.2014.08.01110.1016/j.ijengsci.2014.08.011 ↗
61. Akbaş, Ş.D., Free Vibration and Bending of Functionally Graded Beams Resting on Elastic Foundation, Res. Eng. Struct. Mater., 2015, vol. 1, no. 1, pp. 25–37. http://dx.doi.org/10.17515/resm2015.03st010710.17515/resm2015.03st0107 ↗
62. Sun, Y., Li, S.R., and Batra, R.C., Thermal Buckling and Post-Buckling of FGM Timoshenko Beams on Nonlinear Elastic Foundation, J. Therm. Stress., 2016, vol. 39, no. 1, pp. 11–26. https://doi.org/10.1080/01495739.2015.112062710.1080/01495739.2015.1120627 ↗
63. Robinson, M.T.A. and Adali, S., Buckling of Nonuniform and Axially Functionally Graded Nonlocal Timoshenko Nanobeams on Winkler–Pasternak Foundation, Compos. Struct., 2018, vol. 206, pp. 95–103. https://doi.org/10.1016/j.compstruct.2018.07.04610.1016/j.compstruct.2018.07.046 ↗
64. Rachedi, M.A., Benyoucef, S., Bouhadra, A., Bachir Bouiadjra, R., Sekkal, M., and Benachour, A., Impact of the Homogenization Models on the Thermoelastic Response of FG Plates on Variable Elastic Foundation, Geomech. Eng., 2020, vol. 22, no. 1, pp. 65–80. https://doi.org/10.12989/gae.2020.22.1.06510.12989/gae.2020.22.1.065 ↗
65. Timesli, A., Buckling Behavior of SWCNTs and MWCNTs Resting on Elastic Foundations Using an Optimization Technique, Phys. Mesomech., 2022, vol. 25, no. 2, pp. 129–141. https://doi.org/10.1134/S102995992202004710.1134/S1029959922020047 ↗
66. Mohammadi, M., Saidi, A.R., and Jomehzadeh, E., A Novel Analytical Approach for the Buckling Analysis of Moderately Thick Functionally Graded Rectangular Plates with Two Simply-Supported Opposite Edges, Mech. Eng. Sci., 2010, vol. 224, pp. 1831–1841. https://doi.org/10.1243/09544062jmes180410.1243/09544062jmes1804 ↗
67. Jabbari, M., Mojahedin, A., Khorshidvand, A.R., and Eslami, M.R., Buckling Analysis of a Functionally Graded Thin Circular Plate Made of Saturated Porous Materials, J. Eng. Mech., 2013, vol. 140, no. 2, pp. 287–295. https://doi.org/10.1061/(asce)em.1943-7889.000066310.1061/(asce)em.1943-7889.0000663 ↗
68. Jabbari, M., Hashemitaheri, M., Mojahedin, A., and Eslami, M.R., Thermal Buckling Analysis of Functionally Graded Thin Circular Plate Made of Saturated Porous Materials, J. Therm. Stress., 2014, vol. 37, no. 2, pp. 202–220. https://doi.org/10.1080/01495739.2013.83976810.1080/01495739.2013.839768 ↗
69. Chen, D., Yang, J., and Kitipornchai, S., Elastic Buckling and Static Bending of Shear Deformable Functionally Graded Porous Beam, Compos. Struct., 2015, vol. 133, pp. 54–61. https://doi.org/10.1016/j.compstruct.2015.07.05210.1016/j.compstruct.2015.07.052 ↗
70. Ebrahimi, F. and Jafari, A., Thermo-Mechanical Vibration Analysis of Temperature-Dependent Porous FG Beams Based on Timoshenko Beam Theory, Struct. Eng. Mech., 2016, vol. 59, no. 2, pp. 343–371. https://doi.org/10.12989/sem.2016.59.2.34310.12989/sem.2016.59.2.343 ↗
71. Akbaş, Ş.D., Dynamic Analysis of Axially Functionally Graded Porous Beams under a Moving Load, Steel Compos. Struct., 2021, vol. 39, no. 6, pp. 811–821. https://doi.org/10.12989/SCS.2021.39.6.81110.12989/SCS.2021.39.6.811 ↗
72. Ghandourah, E.E., Ahmed, H.M., Eltaher, M.A., Attia, M.A., and Abdraboh, A.M., Free Vibration of Porous FG Nonlocal Modified Couple Nanobeams via a Modified Porosity Model, Adv. Nano Res., 2021, vol. 11, no. 4, pp. 405–422. https://doi.org/10.12989/ANR.2021.11.4.40510.12989/ANR.2021.11.4.405 ↗
73. Huang, W. and Tahouneh, V., Frequency Study of Porous FGPM Beam on Two-Parameter Elastic Foundations via Timoshenko Theory, Steel Compos. Struct., 2021, vol. 40, no. 1, pp. 139–156. https://doi.org/10.12989/SCS.2021.40.1.13910.12989/SCS.2021.40.1.139 ↗
74. Al-Osta, M.A., Wave Propagation Investigation of a Porous Sandwich FG Plate under Hygrothermal Environments via a New First-Order Shear Deformation Theory, Steel Compos. Struct., 2022, vol. 43, no. 1, pp. 117–127. https://doi.org/10.12989/SCS.2022.43.1.11710.12989/SCS.2022.43.1.117 ↗
75. Ramteke, P.M., Panda, S.K., and Sharma, N., Effect of Grading Pattern and Porosity on the Eigen Characteristics of Porous Functionally Graded Structure, Steel Compos. Struct., 2019, vol. 33, no. 6, pp. 865–875. https://doi.org/10.12989/scs.2019.33.6.86510.12989/scs.2019.33.6.865 ↗
76. Avcar, M., Free Vibration of Imperfect Sigmoid and Power Law Functionally Graded Beams, Steel Compos. Struct., 2019, vol. 30, no. 6, pp. 603–615. https://doi.org/10.12989/scs.2019.30.6.60310.12989/scs.2019.30.6.603 ↗
77. Ahmed, R.A., Fenjan, R.M., and Faleh, N.M., Analyzing Post-Buckling Behavior of Continuously Graded FG Nanobeams with Geometrical Imperfections, Geomech. Eng., 2019, vol. 17, no. 2, pp. 175–180. https://doi.org/10.12989/gae.2019.17.2.17510.12989/gae.2019.17.2.175 ↗
78. Hadji, L., Zouatnia, N., and Bernard, F., An Analytical Solution for Bending and Free Vibration Responses of Functionally Graded Beams with Porosities: Effect of the Micromechanical Models, Struct. Eng. Mech., 2019, vol. 69, no. 2, pp. 231–241. https://doi.org/10.12989/sem.2019.69.2.23110.12989/sem.2019.69.2.231 ↗
79. Abdulrazzaq, M.A. Kadhim, Z.D., Faleh, N.M., and Moustafa, N.M., A Numerical Method for Dynamic Characteristics of Nonlocal Porous Metal-Ceramic Plates under Periodic Dynamic Loads, Struct. Monitor. Maint., 2020, vol. 7, no. 1, pp. 27–42. https://doi.org/10.12989/smm.2020.7.1.02710.12989/smm.2020.7.1.027 ↗
80. Hadji, L., Influence of the Distribution Shape of Porosity on the Bending of FGM Beam Using a New Higher Order Shear Deformation Model, Smart Struct. Syst., 2020, vol. 26, no. 2, pp. 253–262. https://doi.org/10.12989/sss.2020.26.2.25310.12989/sss.2020.26.2.253 ↗
81. Fenjan, R.M., Faleh, N.M., and Ridha, A.A., Strain Gradient Based Static Stability Analysis of Composite Crystalline Shell Structures Having Porosities, Steel Compos. Struct., 2020, vol. 36, no. 6, pp. 631–642. https://doi.org/10.12989/SCS.2020.36.6.63110.12989/SCS.2020.36.6.631 ↗
82. Fenjan, R.M., Moustafa, N.M., and Faleh, N.M., Scale Dependent Thermal Vibration Analysis of FG Beams Having Porosities Based on DQM, Adv. Nano Res., 2020, vol. 8, no. 4, pp. 283–292. https://doi.org/10.12989/anr.2020.8.4.28310.12989/anr.2020.8.4.283 ↗
83. Gafour, Y., Hamidi, A., Benahmed, A., Zidour, M., and Bensattalah, T., Porosity-Dependent Free Vibration Analysis of FG Nanobeam Using Non-Local Shear Deformation and Energy Principle, Adv. Nano Res., 2020, vol. 8, no. 1, pp. 37–47. https://doi.org/10.12989/anr.2020.8.1.03710.12989/anr.2020.8.1.037 ↗
84. Vinyas, M., On Frequency Response of Porous Functionally Graded Magneto-Electro-Elastic Circular and Annular Plates with Different Electro-Magnetic Conditions Using HSDT, Compos. Struct., 2020, vol. 240, p. 112044. https://doi.org/10.1016/j.compstruct.2020.11204410.1016/j.compstruct.2020.112044 ↗
85. Rahmani, M., Mohammadi, Y., Kakavand, F., and Raeisifard, H., Vibration Analysis of Different Types of Porous FG Conical Sandwich Shells in Various Thermal Surroundings, J. Appl. Comput. Mech., 2020, vol. 6, no. 3, pp. 416–432. https://doi.org/10.22055/jacm.2019.29442.159810.22055/jacm.2019.29442.1598 ↗
86. She, G.-L., Liu, H.-B., and Karami, B., On Resonance Behavior of Porous FG Curved Nanobeams, Steel Compos. Struct., 2020, vol. 36, no. 2, pp. 179–186. https://doi.org/10.12989/scs.2020.36.2.17910.12989/scs.2020.36.2.179 ↗
87. Hadji, L. and Avcar, M., Free Vibration Analysis of FG Porous Sandwich Plates under Various Boundary Conditions, J. Appl. Comput. Mech., 2021, vol. 7, no. 2, pp. 505–519. https://doi.org/10.22055/JACM.2020.35328.262810.22055/JACM.2020.35328.2628 ↗
88. Mohsen Rahmani, Y.M., Vibration of Two Types of Porous FG Sandwich Conical Shell with Different Boundary Conditions, Struct. Eng. Mech., 2021, vol. 79, no. 4, pp. 401–413. https://doi.org/10.12989/SEM.2021.79.4.40110.12989/SEM.2021.79.4.401 ↗
89. Xu, X., Zhang, C., Musharavati, F., Sebaey, T.A., and Khan, A., Wave Propagation Analysis of Porous Functionally Graded Curved Beams in the Thermal Environment, Struct. Eng. Mech., 2021, vol. 79, no. 6, pp. 665–675. https://doi.org/10.12989/SEM.2021.79.6.66510.12989/SEM.2021.79.6.665 ↗
90. Li, X., Wang, T., Liu, F., and Zhu, Z., Computer Simulation of the Nonlinear Static Behavior of Axially Functionally Graded Microtube with Porosity, Adv. Nano Res., 2021, vol. 11, no. 4, pp. 437–451. https://doi.org/10.12989/ANR.2021.11.4.43710.12989/ANR.2021.11.4.437 ↗
91. Priyanka, R., Twinkle, C.M., and Pitchaimani, J., Stability and Dynamic Behavior of Porous FGM Beam: Influence of Graded Porosity, Graphene Platelets, and Axially Varying Loads, Eng. Comp., 2021. https://doi.org/10.1007/s00366-021-01478-510.1007/s00366-021-01478-5 ↗
92. Keleshteri, M.M. and Jelovica, J., Nonlinear Vibration Analysis of Bidirectional Porous Beams, Eng. Comp., 2021. https://doi.org/10.1007/s00366-021-01553-x10.1007/s00366-021-01553-x ↗
93. Chen, S., Zhang, Q., and Liu, H., Dynamic Response of Double-FG Porous Beam System Subjected to Moving Load, Eng. Comp., 2021. https://doi.org/10.1007/s00366-021-01376-w10.1007/s00366-021-01376-w ↗
94. Chi, S.-H. and Chung, Y.-L., Mechanical Behavior of Functionally Graded Material Plates under Transverse Load—Part I: Analysis, Int. J. Solids Struct., 2006, vol. 43, no. 13, pp. 3657–3674. https://doi.org/10.1016/j.ijsolstr.2005.04.01110.1016/j.ijsolstr.2005.04.011 ↗
95. Chi, S.-H. and Chung, Y.-L., Mechanical Behavior of Functionally Graded Material Plates under Transverse Load—Part II: Numerical Results, Int. J. Solids Struct., 2006, vol. 43, no. 13, pp. 3675–3691. https://doi.org/10.1016/j.ijsolstr.2005.04.01010.1016/j.ijsolstr.2005.04.010 ↗
96. Kim, Y.-W., Temperature Dependent Vibration Analysis of Functionally Graded Rectangular Plates, J. Sound Vibr., 2005, vol. 284, no. 3–5, pp. 531–549. https://doi.org/10.1016/j.jsv.2004.06.04310.1016/j.jsv.2004.06.043 ↗
97. Li, Q. and Iu, V.P., Three-Dimensional Free Vibration of Functionally Graded Material Plates on Different Boundary Conditions, Mech. Adv. Mater. Struct., 2011, vol. 18, pp. 597–601. https://doi.org/10.1063/1.345225510.1063/1.3452255 ↗
98. Zhu, J., Lai, Z., Yin, Z., Jeon, J., and Lee, S., Fabrication of ZrO2-NiCr Functionally Graded Material by Powder Metallurgy, Mater. Chem. Phys., 2001, vol. 68, no. 1–3, pp. 130–135. https://doi.org/10.1016/S0254-0584(00)00355-210.1016/S0254-0584(00)00355-2 ↗
99. Boutahar, L. and Benamar, R., A Homogenization Procedure for Geometrically Non-Linear Free Vibration Analysis of Functionally Graded Annular Plates with Porosities, Resting on Elastic Foundations, Ain Shams Eng. J., 2016, vol. 7, no. 1, pp. 313–333. https://doi.org/10.1016/j.asej.2015.11.01610.1016/j.asej.2015.11.016 ↗
100. Ibnorachid, Z., Boutahar, L., EL Bikri, K., and Benamar, R., Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment, Adv. Acoust. Vibr., 2019. https://doi.org/10.1155/2019/798656910.1155/2019/7986569 ↗