Investigation of Dynamic Characteristics of Imperfect FG Beams on the Winkler–Pasternak Foundation under Thermal Loading
Tóm tắt
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.
Tài liệu tham khảo
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/s00707-013-0883-5
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.2013.12.072
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.2015.18.3.659
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.2015.16.1.195
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.2015.19.6.1421
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.2016.57.5.823
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.2016.04.035
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.2016.17.5.837
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.2016.58.3.423
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/10.1080/15376494.2017.1323143
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.2019.02.089
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.2019.17.3.329
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.2020.26.3.361
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.2020.19.2.091
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.2020.22.4.361
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.2020.9.3.265
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-2019-0039
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.51.09
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/
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.2021.78.5.585
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.2022.82.3.401
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.202100287
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.2021.27.3.199
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-y
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)
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.2010.07.006
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.2012.03.006
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.2019.111837
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.2020.04.012
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/s00466-007-0226-2
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.2015.06.014
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/s12517-018-3579-2
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.2019.69.4.427
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/10.1155/2019/1354150
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/10.2514/2.1807
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/10.2514/2.1626
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.2004.08.004
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.2009.03.005
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/s10999-010-9132-4
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.2010.07.011
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.2011.06.016
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/s00707-013-0834-1
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.2011.08.041
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/10.1080/01495739.2015.1073980
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)30272-5
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.2017.12.001
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.2021.3.3.179
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.2020.103730
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/10.1590/1679-78251574
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.2020.76.3.325
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/s00419-022-02159-5
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.2022.43.5.661
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.1994.139
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-H
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.1999.2415
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/S0219455402000403
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.2007.04.019
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.2007.07.004
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.2013.01.007
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.2014.08.011
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/resm2015.03st0107
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/10.1080/01495739.2015.1120627
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.2018.07.046
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.2020.22.1.065
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/S1029959922020047
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/10.1243/09544062jmes1804
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.1943-7889.0000663
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/10.1080/01495739.2013.839768
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.2015.07.052
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.2016.59.2.343
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.2021.39.6.811
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.2021.11.4.405
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.2021.40.1.139
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.2022.43.1.117
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.2019.33.6.865
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.2019.30.6.603
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.2019.17.2.175
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.2019.69.2.231
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.2020.7.1.027
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.2020.26.2.253
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.2020.36.6.631
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.2020.8.4.283
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.2020.8.1.037
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.2020.112044
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.2019.29442.1598
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.2020.36.2.179
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.2020.35328.2628
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.2021.79.4.401
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.2021.79.6.665
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.2021.11.4.437
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/s00366-021-01478-5
Keleshteri, M.M. and Jelovica, J., Nonlinear Vibration Analysis of Bidirectional Porous Beams, Eng. Comp., 2021. https://doi.org/10.1007/s00366-021-01553-x
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-w
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.2005.04.011
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.2005.04.010
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.2004.06.043
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/10.1063/1.3452255
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)00355-2
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.2015.11.016
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/10.1155/2019/7986569
Wattanasakulpong, N., Prusty, B.G., Kelly, D.W., and Hoffman, M., Free Vibration Analysis of Layered Functionally Graded Beams with Experimental Validation, Mater. Design, 2012, vol. 36, pp. 182–190. https://doi.org/10.1016/j.matdes.2011.10.049
Wattanasakulpong, N. and Ungbhakorn, V., Linear and Nonlinear Vibration Analysis of Elastically Restrained Ends FGM Beams with Porosities, Aerospace Sci. Technol., 2014, vol. 32, no. 1, pp. 111–120. https://doi.org/10.1016/j.ast.2013.12.002
Yang, J. and Shen, H.S., Nonlinear Bending Analysis of Shear Deformable Functionally Graded Plates Subjected to Thermo-Mechanical Loads under Various Boundary Conditions, Composites. B. Eng., 2003, vol. 34, no. 2, pp. 103–115. https://doi.org/10.1016/S1359-8368(02)00083-5
Şimşek, M., Fundamental Frequency Analysis of Functionally Graded Beams by Using Different Higher-Order Beam Theories, Nucl. Eng. Design, 2010, vol. 240, no. 4, pp. 697–705. https://doi.org/10.1016/j.nucengdes.2009.12.013
Chen, W.Q., Lü, C.F., and Bian, Z.G., A Mixed Method for Bending and Free Vibration of Beams Resting on a Pasternak Elastic Foundation, Appl. Math. Model., 2004, vol. 28, no. 10, pp. 877–890. https://doi.org/10.1016/j.apm.2004.04.001