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A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory

Year 2022, Volume: 14 Issue: 1, 1 - 14, 02.09.2022
https://doi.org/10.24107/ijeas.1064690

Abstract

In this work, a finite element formulation for a size dependent frame system is presented. Size dependency is discussed via the modified couple stress theory. The nodal displacement and rotation analyses of a frame system with total of three elements, including two columns and one beam element connecting these two columns, are considered. The classical stiffness and size dependent stiffness matrices of frame system are derived. Then, solution procedure for this problem is explained. Lastly, a numerical application is realized and effect of material length scale parameter on nodal displacements and rotations is discussed. To present the numerical application, it is assumed that the elements of the nanoframe are composed of silicon carbide nanotubes.

References

  • Iijima, S., Helical microtubules of graphitic carbon, Nature, 354(6348), 56-58, 1991.
  • Becknell, N., Son, Y., Kim, D., Li, D., Yu, Y., Niu, Z., ... and Yang, P., Control of architecture in rhombic dodecahedral Pt–Ni nanoframe electrocatalysts. Journal of the American Chemical Society, 139(34), 11678-11681, 2017.
  • Mahmoud, M.A., Qian, W., and El-Sayed, M.A., Following charge separation on the nanoscale in Cu2O–Au nanoframe hollow nanoparticles. Nano letters, 11(8), 3285-3289, 2011.
  • Zhu, X., Huang, L., Wei, M., Tsiakaras, P., and Shen, P.K., Highly stable Pt-Co nanodendrite in nanoframe with Pt skin structured catalyst for oxygen reduction electrocatalysis. Applied Catalysis B: Environmental, 281, 119460, 2021.
  • Arefi, M., Firouzeh, S., Bidgoli, E.M.R., and Civalek, Ö., Analysis of porous micro-plates reinforced with FG-GNPs based on Reddy plate theory. Composite Structures, 247, 112391, 2020.
  • Esen, I., Abdelrahman, A.A., and Eltaher, M.A. On vibration of sigmoid/symmetric functionally graded nonlocal strain gradient nanobeams under moving load. International Journal of Mechanics and Materials in Design, 1-22, 2021.
  • Esen, I., Abdelrhmaan, A.A., and Eltaher, M.A. Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields. Engineering with Computers, 1-20, 2021.
  • Jena, S.K., Chakraverty, S., and Malikan, M. Vibration and buckling characteristics of nonlocal beam placed in a magnetic field embedded in Winkler–Pasternak elastic foundation using a new refined beam theory: an analytical approach. The European Physical Journal Plus, 135(2), 1-18, 2020.
  • Akgöz, B., and Civalek, Ö., Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. International Journal of Engineering Science, 85, 90-104, 2014.
  • Akgöz, B., and Civalek, Ö., A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta mechanica, 226(7), 2277-2294, 2015.
  • Rahmani, O., Hosseini, S.A.H., Ghoytasi, I., and Golmohammadi, H., Free vibration of deep curved FG nano-beam based on modified couple stress theory. Steel and Composite Structures, 26(5), 607-20, 2018.
  • Yayli, M.Ö., Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory. Micro & Nano Letters, 13(5), 595-599, 2018.
  • Yaylı, M.Ö., Uzun, B., and Deliktaş, B., Buckling analysis of restrained nanobeams using strain gradient elasticity. Waves in Random and Complex Media, 1-20, 2021.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 100(12), e202000039, 2020.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Stability analysis of restrained nanotubes placed in electromagnetic field. Microsystem Technologies, 26(12), 3725-3736, 2020.
  • Yaylı, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions. Journal of Mechanical Science and Technology, 29(8), 3373-3380, 2015.
  • Yayli, M.Ö., On the axial vibration of carbon nanotubes with different boundary conditions. Micro & Nano Letters, 9(11), 807-811, 2014.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories. Microsystem Technologies, 27(5), 1913-1923, 2021.
  • Yayli, M.O., Stability analysis of a gradient elastic beam using finite element method. International Journal of Physical Science, 6(12), 2844-2851, 2011.
  • Akbaş, Ş.D., Static, Vibration, and Buckling Analysis of Nanobeams (pp. 123-137). InTech, 2017.
  • Numanoğlu, H.M., Thermal Vibration of Zinc Oxide Nanowires by using Nonlocal Finite Element Method. International Journal of Engineering and Applied Sciences, 12(3), 99-110, 2020.
  • Numanoğlu, H.M., Examination of How Size-Effect Modifies the Stiffness and Mass Matrices of Nanotrusses/Nanoframes. International Journal of Engineering and Applied Sciences, 13(4), 155-165, 2021.
  • Uzun, B., and Yaylı, M.Ö., Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences, 13(4), 1-10, 2020.
  • Uzun, B., Yaylı, M.Ö., and Deliktaş, B., Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40, 2020.
  • Uzun, B., and Yayli, M.Ö., A solution method for longitudinal vibrations of functionally graded nanorods. International Journal of Engineering and Applied Sciences, 12(2), 78-87, 2020.
  • Akbaş, Ş.D., Forced vibration analysis of cracked nanobeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(8), 1-11, 2018.
  • Akbas, S.D., Forced vibration analysis of cracked functionally graded microbeams. Advances in Nano Research, 6(1), 39, 2018.
  • Karamanli, A. Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics. European Journal of Mechanics-A/Solids, 89, 104287, 2021.
  • Eltaher, M.A., Khairy, A., Sadoun, A.M., and Omar, F.A. Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229, 283-295, 2014.
  • Akbaş, Ş.D., Axially forced vibration analysis of cracked a nanorod. Journal of Computational Applied Mechanics, 50(1), 63-68, 2019.
  • Numanoğlu, H. M., Akgöz, B., and Civalek, Ö., On dynamic analysis of nanorods. International Journal of Engineering Science, 130, 33-50, 2018.
  • Ebrahimi, F., Shafiei, N., Kazemi, M., and Mousavi Abdollahi, S.M., Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method. Mechanics of Advanced Materials and Structures, 24(15), 1257-1273, 2017.
  • Khaniki, H.B., and Hosseini-Hashemi, S., Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method. The European Physical Journal Plus, 132(11), 1-15, 2017.
  • Najafzadeh, M., Adeli, M.M., Zarezadeh, E., and Hadi, A., Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field. Mechanics Based Design of Structures and Machines, 1-15, 2020.
  • Shafiei, N., and Kazemi, M., Nonlinear buckling of functionally graded nano-/micro-scaled porous beams. Composite Structures, 178, 483-492, 2017.
  • Shojaeefard, M.H., Googarchin, H.S., Ghadiri, M., and Mahinzare, M., Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT. Applied Mathematical Modelling, 50, 633-655, 2017.
  • Xue, Y., Jin, G., Ma, X., Chen, H., Ye, T., Chen, M., and Zhang, Y., Free vibration analysis of porous plates with porosity distributions in the thickness and in-plane directions using isogeometric approach. International Journal of Mechanical Sciences, 152, 346-362, 2019.
  • Akbaş, Ş.D., Stability of a non-homogenous porous plate by using generalized differantial quadrature method. International Journal of Engineering and Applied Sciences, 9(2), 147-155, 2017.
  • Chen, D., Yang, J., and Kitipornchai, S., Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method. Archives of Civil and Mechanical Engineering, 19(1), 157-170, 2019.
  • Jena, S. K., Chakraverty, S., and Malikan, M., Application of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation. Engineering with Computers, 1-21, 2020.
  • Tang, H., Li, L., and Hu, Y., Buckling analysis of two-directionally porous beam. Aerospace Science and Technology, 78, 471-479, 2018.
  • Civalek, Ö., and Kiracioglu, O., Free vibration analysis of Timoshenko beams by DSC method. International Journal for Numerical Methods in Biomedical Engineering, 26(12), 1890-1898, 2010.
  • Civalek, Ö., and Avcar, M., Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Engineering with Computers, 1-33, 2020.
  • Bouazza, M., and Zenkour, A.M., Vibration of carbon nanotube-reinforced plates via refined n th-higher-order theory. Archive of Applied Mechanics, 90(8), 1755-1769, 2020.
  • Chaabane, L. A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F. Z., Tounsi, A., ... and Tounsi, A., Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation. Structural Engineering and Mechanics, 71(2), 185-196, 2019.
  • Li, L., Li, X., and Hu, Y., Nonlinear bending of a two-dimensionally functionally graded beam. Composite Structures, 184, 1049-1061, 2018.
  • Ghayesh, M.H., Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Applied Mathematical Modelling, 59, 583-596, 2018.
  • Kahya, V., and Turan, M., Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Composites Part B: Engineering, 109, 108-115, 2017.
  • Kahya, V., and Turan, M., Vibration and buckling of laminated beams by a multi-layer finite element model. Steel and Composite Structures, 28(4), 415-426, 2018.
  • Akbaş, Ş.D., Free vibration of axially functionally graded beams in thermal environment. International Journal of Engineering and Applied Sciences, 6(3), 37-51, 2014.
  • Civalek, Ö., Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations. International Journal of Pressure Vessels and Piping, 113, 1-9, 2014.
  • Civalek, Ö., and Baltacıoglu, A.K., Free vibration analysis of laminated and FGM composite annular sector plates. Composites Part B: Engineering, 157, 182-194, 2019.
  • Mercan, K., Demir, Ç., and Civalek, Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layered Structures, 3(1), 2016.
  • Civalek, Ö., Akbaş, Ş.D., Akgöz, B., and Dastjerdi, S., Forced vibration analysis of composite beams reinforced by carbon nanotubes. Nanomaterials, 11(3), 571, 2021.
  • Koçyiğit, K., and Akbaş, Ş.D., Çatlak içeren bir çerçeve taşıyıcı sistemin zorlanmış titreşim analizi. Politeknik Dergisi, 23(4), 1059-1071, 2020.
  • Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C., and Tong, P., Couple stress based strain gradient theory for elasticity. International journal of solids and structures, 39(10), 2731-2743, 2002.
  • Logan, D.L., A first course in the finite element method. Cengage Learning, 2016.
  • Latu-Romain, L., and Ollivier, M., Silicon carbide one-dimensional nanostructures. John Wiley & Sons, 2015.
  • Petrushenko, I.K., and Petrushenko, K.B., Mechanical properties of carbon, silicon carbide, and boron nitride nanotubes: effect of ionization. Monatshefte für Chemie-Chemical Monthly, 146(10), 1603-1608, 2015.
Year 2022, Volume: 14 Issue: 1, 1 - 14, 02.09.2022
https://doi.org/10.24107/ijeas.1064690

Abstract

References

  • Iijima, S., Helical microtubules of graphitic carbon, Nature, 354(6348), 56-58, 1991.
  • Becknell, N., Son, Y., Kim, D., Li, D., Yu, Y., Niu, Z., ... and Yang, P., Control of architecture in rhombic dodecahedral Pt–Ni nanoframe electrocatalysts. Journal of the American Chemical Society, 139(34), 11678-11681, 2017.
  • Mahmoud, M.A., Qian, W., and El-Sayed, M.A., Following charge separation on the nanoscale in Cu2O–Au nanoframe hollow nanoparticles. Nano letters, 11(8), 3285-3289, 2011.
  • Zhu, X., Huang, L., Wei, M., Tsiakaras, P., and Shen, P.K., Highly stable Pt-Co nanodendrite in nanoframe with Pt skin structured catalyst for oxygen reduction electrocatalysis. Applied Catalysis B: Environmental, 281, 119460, 2021.
  • Arefi, M., Firouzeh, S., Bidgoli, E.M.R., and Civalek, Ö., Analysis of porous micro-plates reinforced with FG-GNPs based on Reddy plate theory. Composite Structures, 247, 112391, 2020.
  • Esen, I., Abdelrahman, A.A., and Eltaher, M.A. On vibration of sigmoid/symmetric functionally graded nonlocal strain gradient nanobeams under moving load. International Journal of Mechanics and Materials in Design, 1-22, 2021.
  • Esen, I., Abdelrhmaan, A.A., and Eltaher, M.A. Free vibration and buckling stability of FG nanobeams exposed to magnetic and thermal fields. Engineering with Computers, 1-20, 2021.
  • Jena, S.K., Chakraverty, S., and Malikan, M. Vibration and buckling characteristics of nonlocal beam placed in a magnetic field embedded in Winkler–Pasternak elastic foundation using a new refined beam theory: an analytical approach. The European Physical Journal Plus, 135(2), 1-18, 2020.
  • Akgöz, B., and Civalek, Ö., Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium. International Journal of Engineering Science, 85, 90-104, 2014.
  • Akgöz, B., and Civalek, Ö., A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta mechanica, 226(7), 2277-2294, 2015.
  • Rahmani, O., Hosseini, S.A.H., Ghoytasi, I., and Golmohammadi, H., Free vibration of deep curved FG nano-beam based on modified couple stress theory. Steel and Composite Structures, 26(5), 607-20, 2018.
  • Yayli, M.Ö., Torsional vibration analysis of nanorods with elastic torsional restraints using non-local elasticity theory. Micro & Nano Letters, 13(5), 595-599, 2018.
  • Yaylı, M.Ö., Uzun, B., and Deliktaş, B., Buckling analysis of restrained nanobeams using strain gradient elasticity. Waves in Random and Complex Media, 1-20, 2021.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 100(12), e202000039, 2020.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Stability analysis of restrained nanotubes placed in electromagnetic field. Microsystem Technologies, 26(12), 3725-3736, 2020.
  • Yaylı, M.Ö., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions. Journal of Mechanical Science and Technology, 29(8), 3373-3380, 2015.
  • Yayli, M.Ö., On the axial vibration of carbon nanotubes with different boundary conditions. Micro & Nano Letters, 9(11), 807-811, 2014.
  • Uzun, B., Kafkas, U., and Yaylı, M.Ö., Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories. Microsystem Technologies, 27(5), 1913-1923, 2021.
  • Yayli, M.O., Stability analysis of a gradient elastic beam using finite element method. International Journal of Physical Science, 6(12), 2844-2851, 2011.
  • Akbaş, Ş.D., Static, Vibration, and Buckling Analysis of Nanobeams (pp. 123-137). InTech, 2017.
  • Numanoğlu, H.M., Thermal Vibration of Zinc Oxide Nanowires by using Nonlocal Finite Element Method. International Journal of Engineering and Applied Sciences, 12(3), 99-110, 2020.
  • Numanoğlu, H.M., Examination of How Size-Effect Modifies the Stiffness and Mass Matrices of Nanotrusses/Nanoframes. International Journal of Engineering and Applied Sciences, 13(4), 155-165, 2021.
  • Uzun, B., and Yaylı, M.Ö., Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences, 13(4), 1-10, 2020.
  • Uzun, B., Yaylı, M.Ö., and Deliktaş, B., Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40, 2020.
  • Uzun, B., and Yayli, M.Ö., A solution method for longitudinal vibrations of functionally graded nanorods. International Journal of Engineering and Applied Sciences, 12(2), 78-87, 2020.
  • Akbaş, Ş.D., Forced vibration analysis of cracked nanobeams. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(8), 1-11, 2018.
  • Akbas, S.D., Forced vibration analysis of cracked functionally graded microbeams. Advances in Nano Research, 6(1), 39, 2018.
  • Karamanli, A. Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics. European Journal of Mechanics-A/Solids, 89, 104287, 2021.
  • Eltaher, M.A., Khairy, A., Sadoun, A.M., and Omar, F.A. Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229, 283-295, 2014.
  • Akbaş, Ş.D., Axially forced vibration analysis of cracked a nanorod. Journal of Computational Applied Mechanics, 50(1), 63-68, 2019.
  • Numanoğlu, H. M., Akgöz, B., and Civalek, Ö., On dynamic analysis of nanorods. International Journal of Engineering Science, 130, 33-50, 2018.
  • Ebrahimi, F., Shafiei, N., Kazemi, M., and Mousavi Abdollahi, S.M., Thermo-mechanical vibration analysis of rotating nonlocal nanoplates applying generalized differential quadrature method. Mechanics of Advanced Materials and Structures, 24(15), 1257-1273, 2017.
  • Khaniki, H.B., and Hosseini-Hashemi, S., Dynamic transverse vibration characteristics of nonuniform nonlocal strain gradient beams using the generalized differential quadrature method. The European Physical Journal Plus, 132(11), 1-15, 2017.
  • Najafzadeh, M., Adeli, M.M., Zarezadeh, E., and Hadi, A., Torsional vibration of the porous nanotube with an arbitrary cross-section based on couple stress theory under magnetic field. Mechanics Based Design of Structures and Machines, 1-15, 2020.
  • Shafiei, N., and Kazemi, M., Nonlinear buckling of functionally graded nano-/micro-scaled porous beams. Composite Structures, 178, 483-492, 2017.
  • Shojaeefard, M.H., Googarchin, H.S., Ghadiri, M., and Mahinzare, M., Micro temperature-dependent FG porous plate: Free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT. Applied Mathematical Modelling, 50, 633-655, 2017.
  • Xue, Y., Jin, G., Ma, X., Chen, H., Ye, T., Chen, M., and Zhang, Y., Free vibration analysis of porous plates with porosity distributions in the thickness and in-plane directions using isogeometric approach. International Journal of Mechanical Sciences, 152, 346-362, 2019.
  • Akbaş, Ş.D., Stability of a non-homogenous porous plate by using generalized differantial quadrature method. International Journal of Engineering and Applied Sciences, 9(2), 147-155, 2017.
  • Chen, D., Yang, J., and Kitipornchai, S., Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method. Archives of Civil and Mechanical Engineering, 19(1), 157-170, 2019.
  • Jena, S. K., Chakraverty, S., and Malikan, M., Application of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation. Engineering with Computers, 1-21, 2020.
  • Tang, H., Li, L., and Hu, Y., Buckling analysis of two-directionally porous beam. Aerospace Science and Technology, 78, 471-479, 2018.
  • Civalek, Ö., and Kiracioglu, O., Free vibration analysis of Timoshenko beams by DSC method. International Journal for Numerical Methods in Biomedical Engineering, 26(12), 1890-1898, 2010.
  • Civalek, Ö., and Avcar, M., Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method. Engineering with Computers, 1-33, 2020.
  • Bouazza, M., and Zenkour, A.M., Vibration of carbon nanotube-reinforced plates via refined n th-higher-order theory. Archive of Applied Mechanics, 90(8), 1755-1769, 2020.
  • Chaabane, L. A., Bourada, F., Sekkal, M., Zerouati, S., Zaoui, F. Z., Tounsi, A., ... and Tounsi, A., Analytical study of bending and free vibration responses of functionally graded beams resting on elastic foundation. Structural Engineering and Mechanics, 71(2), 185-196, 2019.
  • Li, L., Li, X., and Hu, Y., Nonlinear bending of a two-dimensionally functionally graded beam. Composite Structures, 184, 1049-1061, 2018.
  • Ghayesh, M.H., Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Applied Mathematical Modelling, 59, 583-596, 2018.
  • Kahya, V., and Turan, M., Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Composites Part B: Engineering, 109, 108-115, 2017.
  • Kahya, V., and Turan, M., Vibration and buckling of laminated beams by a multi-layer finite element model. Steel and Composite Structures, 28(4), 415-426, 2018.
  • Akbaş, Ş.D., Free vibration of axially functionally graded beams in thermal environment. International Journal of Engineering and Applied Sciences, 6(3), 37-51, 2014.
  • Civalek, Ö., Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations. International Journal of Pressure Vessels and Piping, 113, 1-9, 2014.
  • Civalek, Ö., and Baltacıoglu, A.K., Free vibration analysis of laminated and FGM composite annular sector plates. Composites Part B: Engineering, 157, 182-194, 2019.
  • Mercan, K., Demir, Ç., and Civalek, Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layered Structures, 3(1), 2016.
  • Civalek, Ö., Akbaş, Ş.D., Akgöz, B., and Dastjerdi, S., Forced vibration analysis of composite beams reinforced by carbon nanotubes. Nanomaterials, 11(3), 571, 2021.
  • Koçyiğit, K., and Akbaş, Ş.D., Çatlak içeren bir çerçeve taşıyıcı sistemin zorlanmış titreşim analizi. Politeknik Dergisi, 23(4), 1059-1071, 2020.
  • Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C., and Tong, P., Couple stress based strain gradient theory for elasticity. International journal of solids and structures, 39(10), 2731-2743, 2002.
  • Logan, D.L., A first course in the finite element method. Cengage Learning, 2016.
  • Latu-Romain, L., and Ollivier, M., Silicon carbide one-dimensional nanostructures. John Wiley & Sons, 2015.
  • Petrushenko, I.K., and Petrushenko, K.B., Mechanical properties of carbon, silicon carbide, and boron nitride nanotubes: effect of ionization. Monatshefte für Chemie-Chemical Monthly, 146(10), 1603-1608, 2015.
There are 59 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Büşra Uzun 0000-0002-7636-7170

Mustafa Özgür Yaylı 0000-0003-2231-170X

Publication Date September 2, 2022
Acceptance Date March 6, 2022
Published in Issue Year 2022 Volume: 14 Issue: 1

Cite

APA Uzun, B., & Yaylı, M. Ö. (2022). A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. International Journal of Engineering and Applied Sciences, 14(1), 1-14. https://doi.org/10.24107/ijeas.1064690
AMA Uzun B, Yaylı MÖ. A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. IJEAS. September 2022;14(1):1-14. doi:10.24107/ijeas.1064690
Chicago Uzun, Büşra, and Mustafa Özgür Yaylı. “A Finite Element Solution for Bending Analysis of a Nanoframe Using Modified Couple Stress Theory”. International Journal of Engineering and Applied Sciences 14, no. 1 (September 2022): 1-14. https://doi.org/10.24107/ijeas.1064690.
EndNote Uzun B, Yaylı MÖ (September 1, 2022) A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. International Journal of Engineering and Applied Sciences 14 1 1–14.
IEEE B. Uzun and M. Ö. Yaylı, “A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory”, IJEAS, vol. 14, no. 1, pp. 1–14, 2022, doi: 10.24107/ijeas.1064690.
ISNAD Uzun, Büşra - Yaylı, Mustafa Özgür. “A Finite Element Solution for Bending Analysis of a Nanoframe Using Modified Couple Stress Theory”. International Journal of Engineering and Applied Sciences 14/1 (September 2022), 1-14. https://doi.org/10.24107/ijeas.1064690.
JAMA Uzun B, Yaylı MÖ. A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. IJEAS. 2022;14:1–14.
MLA Uzun, Büşra and Mustafa Özgür Yaylı. “A Finite Element Solution for Bending Analysis of a Nanoframe Using Modified Couple Stress Theory”. International Journal of Engineering and Applied Sciences, vol. 14, no. 1, 2022, pp. 1-14, doi:10.24107/ijeas.1064690.
Vancouver Uzun B, Yaylı MÖ. A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. IJEAS. 2022;14(1):1-14.

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