Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness

Büşra Uzun

Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness

Abstract

In this study, the buckling of perforated columns supported by foundation springs with variable stiffness is presented, taking the size effect into account. To account for the size effect, a simple strain gradient theory incorporating a single length scale parameter is used. The perforation pattern in the column consists of square holes arranged at regular intervals along its length. The variation in stiffness of the foundation springs supporting the column is considered in two forms: linear and parabolic. In this study, the equivalent bending stiffness of the perforated column and the foundation properties with variable stiffness are first introduced and subsequently integrated into the total minimum potential energy based on the strain gradient theory. Then, the buckling analysis of perforated columns with both ends clamped, based on the strain gradient theory, is performed using the energy-based Rayleigh-Ritz method. In this study, which investigates the critical buckling loads of perforated strain gradient columns, the effects of the number of holes, the foundation stiffness coefficient, the non-uniform parameter, the filling ratio, and the material length scale parameter are presented and discussed through a series of tables and figures for both constant and variable foundation stiffness.

Keywords

References

Turan, M., Yaylacı, E. U., & Yaylacı, M. (2023). Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Archive of Applied Mechanics, 93(4), 1351–1372. https://doi.org/10.1007/s00419-022-02332-w
Mesbah, A., Belabed, Z., Amara, K., Tounsi, A., Bousahla, A. A., & Bourada, F. (2023). Formulation and evaluation a finite element model for free vibration and buckling behaviours of functionally graded porous (FGP) beams. Structural Engineering and Mechanics, An Int’l Journal, 86(3), 291–309. https://doi.org/10.12989/sem.2023.86.3.291
Patil, H. B., Pitchaimani, J., & Mailan Chinnapandi, L. B. (2021). Buckling and free vibration of porous functionally graded metal ceramic beams under thermal and mechanical loading: a comparative study. Journal of The Institution of Engineers (India): Series C, 102(5), 1107–1117. https://doi.org/10.1007/s40032-021-00742-3
Turan, M., & Adiyaman, G. (2024). Free vibration and buckling analysis of porous two-directional functionally graded beams using a higher-order finite element model. Journal of Vibration Engineering & Technologies, 12(1), 1133–1152. https://doi.org/10.1007/s42417-023-00898-5
Nazir, A., & Jeng, J.-Y. (2020). Buckling behavior of additively manufactured cellular columns: Experimental and simulation validation. Materials & Design, 186, 108349. https://doi.org/10.1016/j.matdes.2019.108349
Tu, T. M., Hoa, L. K., Hung, D. X., & Hai, L. T. (2020). Nonlinear buckling and post-buckling analysis of imperfect porous plates under mechanical loads. Journal of Sandwich Structures & Materials, 22(6), 1910–1930. https://doi.org/10.1177/1099636218789612
Kaddari, M., Balubaid, M., Mahmoud, S. R., Kaci, A., Bousahla, A. A., Tounsi, A., Al-Osta, M. A., & Tounsi, A. (2025). Influence of porosity distribution on buckling behavior of FG sandwich plates using a quasi-3D refined plate theory. Mechanics Based Design of Structures and Machines, 53(5), 3635–3662. https://doi.org/10.1080/15397734.2024.2434552
Sitli, Y., Mhada, K., Bourihane, O., & Rhanim, H. (2021). Buckling and post-buckling analysis of a functionally graded material (FGM) plate by the asymptotic numerical method. Structures, 31, 1031–1040. https://doi.org/10.1016/j.istruc.2021.01.100

Hadji, L., Zouatnia, N., & İkinci, B. (2025). Buckling analysis of porous power-law and sigmoid functionally graded sandwich plates. Journal of Ceramics and Composites, 1(2), 1–10.
Liu, X., Liu, X., & Zhou, W. (2020). An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions. Applied Mathematical Modelling, 86, 36–53. https://doi.org/10.1016/j.apm.2020.05.010
Papargyri-Beskou, S., Tsepoura, K. G., Polyzos, D., & Beskos, D. (2003). Bending and stability analysis of gradient elastic beams. International Journal of Solids and Structures, 40(2), 385–400. https://doi.org/10.1016/S0020-7683(02)00522-X
Siddique, M. U. M., Islam, M. N., & Devnath, I. (2023). Analytical solutions for bending, buckling and vibration of nanobeams on Winkler–Pasternak foundation. International Journal of Computational Materials Science and Engineering, 12(02), 2250020. https://doi.org/10.1142/S2047684122500208
Gözübüyük, I. S., & Uzun, B. (2026). Free vibration of short-fiber-reinforced tapered micro column via the modified couple stress theory. The Journal of Strain Analysis for Engineering Design, 03093247261455906. https://doi.org/10.1177/03093247261455906
Civalek, Ö., & Akgöz, B. (2025). Size-dependent stability of embedded beams with variable cross section. International Journal of Engineering Science, 208, 104210. https://doi.org/10.1016/j.ijengsci.2024.104210
Soltani, M., & Gholamizadeh, A. (2018). Size-dependent buckling analysis of non-prismatic Timoshenko nanobeams made of FGMs rested on Winkler foundation. Numerical Methods in Civil Engineering, 3(2), 35–46. https://doi.org/10.29252/nmce.3.2.35
Artan, R., & Toksöz, A. (2013). Stability analysis of gradient elastic beams by the method of initial value. Archive of Applied Mechanics, 83(8), 1129–1144. https://doi.org/10.1007/s00419-013-0739-4
Aifantis, E. C. (1999). Strain gradient interpretation of size effects. International Journal of Fracture, 95(1–4), 299–314. https://doi.org/10.1023/A:1018625006804
Ru, C. Q., & Aifantis, E. C. (1993). A simple approach to solve boundary-value problems in gradient elasticity. Acta Mechanica, 101(1–4), 59–68. https://doi.org/10.1007/BF01175597
Eringen, A. C., & Edelen, D. (1972). On nonlocal elasticity. International Journal of Engineering Science, 10(3), 233–248. https://doi.org/10.1016/0020-7225(72)90039-0
Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X
Lam, D. C., Yang, F., Chong, A. C. M., Wang, J., & Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508. https://doi.org/10.1016/S0022-5096(03)00053-X
Ferrari, M., Granik, V. T., Imam, A., & Nadeau, J. C. (2008). Advances in doublet mechanics (Vol. 45). Springer Science & Business Media.
Almitani, K. H., Abdelrahman, A. A., & Eltaher, M. A. (2022). On forced and free vibrations of cutout squared beams. Steel and Composite Structures, 643–655. https://doi.org/10.12989/scs.2019.32.5.643
Abdelrahman, A. A., & Eltaher, M. A. (2022). On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Engineering with Computers, 38(3), 2385–2411. https://doi.org/10.1007/s00366-020-01211-8
Eltaher, M. A., Kabeel, A. M., Almitani, K. H., & Abdraboh, A. M. (2018). Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsystem Technologies, 24(12), 4881–4893. https://doi.org/10.1007/s00542-018-3905-3
Kadioglu, H. G., & Yayli, M. O. (2026). Size dependent dynamics of a perforated Timoshenko nanobeam made of viscoelastic material with rotationally restrained ends. Mechanics of Time-Dependent Materials, 30(2), 40. https://doi.org/10.1007/s11043-026-09885-1
Eltaher, M. A., Shanab, R. A., & Mohamed, N. A. (2023). Analytical solution of free vibration of viscoelastic perforated nanobeam. Archive of Applied Mechanics, 93(1), 221–243. https://doi.org/10.1007/s00419-022-02184-4
Uzun, B., Akpınar, M., & Yaylı, M. Ö. (2025). Modeling the thermal dynamic behavior of a perforated Timoshenko nanobeam with deformable ends based on nonlocal elasticity. Journal of Thermal Stresses, 48(10), 1229–1252. https://doi.org/10.1080/01495739.2025.2545405
Kafkas, U. (2025). Size-dependent bending response of perforated nanobeams on Winkler-Pasternak foundation. International Journal of Engineering and Applied Sciences, 17(1), 1–16. https://doi.org/10.24107/ijeas.1590000
Kafkas, U. (2025). A coupled MCST-FEM investigation of size-dependent buckling of perforated nanobeams on Winkler-Pasternak foundation. Konya Journal of Engineering Sciences, 13(2), 368–383. https://doi.org/10.36306/konjes.1587217
Mohamed, N. A., Shanab, R. A., Eltaher, M. A., & Abdelrahman, A. A. (2024). Vibration response of viscoelastic perforated higher-order nanobeams rested on an elastic substrate under moving load. Acta Mechanica, 235(2), 1213–1233.https://doi.org/10.1007/s00707-023-03776-z
Güçlü, G. (2026). Nonlocal free vibration analysis of perforated nanobeams resting on kerr-type elastic foundation. Mathematics, 14(5), 749. https://doi.org/10.3390/math14050749
Siam, O. A., Shanab, R. A., Eltaher, M. A., & Mohamed, N. A. (2023). Free vibration analysis of nonlocal viscoelastic nanobeam with holes and elastic foundations by Navier analytical method. Advances in Aircraft and Spacecraft Science, 10(3), 257. https://doi.org/10.12989/aas.2023.10.3.257
Mirzabeigy, A., Haghpanahi, M., & Madoliat, R. (2017). A new finite element formulation for buckling and free vibration analysis of Timoshenko beams on variable elastic foundation. Journal of Solid Mechanics, 9(2), 276-290.
Kacar, A., Tan, H. T., & Kaya, M. O. (2011). Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method. Mathematical and Computational Applications, 16(3), 773–783. https://doi.org/10.3390/mca16030773
Jena, S. K., & Chakraverty, S. (2018). Free vibration of single walled carbon nanotube resting on exponentially varying elastic foundation. Curved and Layered Structures, 5(1), 260–272. https://doi.org/10.1515/cls-2018-0019
Dynamical behavior of nanobeam embedded in constant, linear, parabolic, and sinusoidal types of Winkler elastic foundation using first-Order nonlocal strain gradient model. Materials Research Express, 6(8), 0850f2. https://doi.org/10.1088/2053-1591/ab2779
Uzun, B., & Çağlayan, A. B. (2025). Parabolically varying elastic foundation effect on vibration of a rayleigh strain gradient beam having cutouts. International Multidisciplinary Scientific Research Congress, 437-450. Sivas, Türkiye.
Çağlayan, A. B., & Uzun, B. (2025). Linearly varying elastic foundation effect on nonlocal vibration of nanobeams including cutouts. 2nd International Conference on Engineering, Natural Sciences, and Technological Developments (ICENSTED 2025), 646–654. Bayburt, Türkiye.
Uzun, B., Çağlayan, A. B., & Karadeniz, Ö. (2025). Rijitliği değişken elastik zemin yayları ile etkileşimdeki perfore mikro kolonların değiştirilmiş gerilme çifti teorisi ile burkulması. V. Bilsel Uluslararası Aspendos Bilimsel Araştırmalar Kongresi, 278–291. Antalya, Türkiye.
Luschi, L., & Pieri, F. (2014). An analytical model for the determination of resonance frequencies of perforated beams. Journal of Micromechanics and Microengineering, 24(5), 055004. https://doi.org/10.1088/0960-1317/24/5/055004

Details

Primary Language

English

Subjects

Materials Engineering (Other)

Journal Section

Research Article

Authors

Büşra Uzun ^*
0000-0002-7636-7170
Türkiye

Publication Date

August 27, 2026

Submission Date

July 31, 2026

Acceptance Date

August 13, 2026

Published in Issue

Year 2026 Volume: 2 Number: 2

IZ

https://izlik.org/JA29SD56GZ

Cite

RIS / Bibtex

APA

Uzun, B. (2026). Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness. Journal of Ceramics and Composites, 2(2), 17-27. https://izlik.org/JA29SD56GZ

AMA

1.Uzun B. Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness. jcc. 2026;2(2):17-27. https://izlik.org/JA29SD56GZ

Chicago

Uzun, Büşra. 2026. “Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs With Variable Stiffness”. Journal of Ceramics and Composites 2 (2): 17-27. https://izlik.org/JA29SD56GZ.

EndNote

Uzun B (August 1, 2026) Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness. Journal of Ceramics and Composites 2 2 17–27.

IEEE

[1]B. Uzun, “Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness”, jcc, vol. 2, no. 2, pp. 17–27, Aug. 2026, [Online]. Available: https://izlik.org/JA29SD56GZ

ISNAD

Uzun, Büşra. “Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs With Variable Stiffness”. Journal of Ceramics and Composites 2/2 (August 1, 2026): 17-27. https://izlik.org/JA29SD56GZ.

JAMA

1.Uzun B. Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness. jcc. 2026;2:17–27.

MLA

Uzun, Büşra. “Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs With Variable Stiffness”. Journal of Ceramics and Composites, vol. 2, no. 2, Aug. 2026, pp. 17-27, https://izlik.org/JA29SD56GZ.

Vancouver

1.Büşra Uzun. Critical Buckling Loads of Perforated Strain Gradient Columns Supported by Foundation Springs with Variable Stiffness. jcc [Internet]. 2026 Aug. 1;2(2):17-2. Available from: https://izlik.org/JA29SD56GZ