Skip to main content
SpringerLink
Log in

Stochastic bending and buckling analysis of laminated composite plates using Latin hypercube sampling

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Deterministic and stochastic bending and buckling characteristics of antisymmetric cross-ply and angle-ply laminated composite plates are thoroughly examined. Partial differential equations for cross-ply and angle-ply laminates are derived using the three variable refined shear deformation theory based on the Hamilton principle. Deterministic Navier’s solutions are obtained for specific boundary conditions and numerical results are validated with the first-order and third-order shear deformation theories. Two stochastic sampling methods, namely Monte Carlo simulation and Latin hypercube sampling, are presented and analyzed to determine the optimal one based on convergence studies and criteria of sampling errors. Comprehensive probability characteristics of stochastic bending deflections and stochastic critical buckling loads of antisymmetric cross-ply and angle-ply laminated composite plates are investigated using the optimal sampling technique. Probability distribution functions of various stochastic cases provide good assessments for the effects of each inevitable source uncertainty on the bending and buckling behaviors of the laminated composites. This study presents a good alternative for the classical and expensive Monte Carlo simulations and provides a fundamental understanding of bending and buckling statistics of laminated composites.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37

Similar content being viewed by others

References

  1. Lu W, Zhang Q, Qin F et al (2021) Hierarchical network structural composites for extraordinary energy dissipation inspired by the cat paw. Appl Mater Today 25:101222

    Google Scholar 

  2. Mouritz AP (2020) Review of z-pinned laminates and sandwich composites. Compos Part A Appl Sci Manuf 139:106128

    Google Scholar 

  3. Nguyen DD, Pham CH (2018) Nonlinear dynamic response and vibration of sandwich composite plates with negative Poisson’s ratio in auxetic honeycombs. J Sandw Struct Mater 20:692–717

    Google Scholar 

  4. Zhang X, Bai C, Qiao Y et al (2021) Porous geopolymer composites: a review. Compos Part A Appl Sci Manuf 150:106629

    Google Scholar 

  5. Bui TQ, Hu X (2021) A review of phase-field models, fundamentals and their applications to composite laminates. Eng Fract Mech 248:107705

    Google Scholar 

  6. Duc ND, Lam PT, Quan TQ et al (2020) Nonlinear post-buckling and vibration of 2D penta-graphene composite plates. Acta Mech 231:539–559

    MathSciNet  MATH  Google Scholar 

  7. Kim SE, Duc ND, Nam VH, Van Sy N (2019) Nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in thermal environment. Thin-Wall Struct 142:287–296

    Google Scholar 

  8. Trinh MC, Nguyen DD, Kim SE (2019) Effects of porosity and thermomechanical loading on free vibration and nonlinear dynamic response of functionally graded sandwich shells with double curvature. Aerosp Sci Technol 87:119–132

    Google Scholar 

  9. Riddle R, Lesuer D, Syn C et al (1996) Application of metal laminates to aircraft structures: prediction of penetration performance. Finite Elem Anal Des 23:173–192

    MATH  Google Scholar 

  10. Bui VP, Thitsartarn W, Liu EX et al (2015) EM performance of conductive composite laminate made of nanostructured materials for aerospace application. IEEE Trans Electromagn Compat 57:1139–1148

    Google Scholar 

  11. Borba NZ, Blaga L, dos Santos JF, Amancio-Filho ST (2018) Direct-friction riveting of polymer composite laminates for aircraft applications. Mater Lett 215:31–34

    Google Scholar 

  12. Subadra SP, Griskevicius P, Yousef S (2020) Low velocity impact and pseudo-ductile behaviour of carbon/glass/epoxy and carbon/glass/PMMA hybrid composite laminates for aircraft application at service temperature. Polym Test 89:106711

    Google Scholar 

  13. Liu H, Zhang Q, Yang X, Ma J (2021) Size-dependent vibration of laminated composite nanoplate with piezo-magnetic face sheets. Eng Comput. https://doi.org/10.1007/s00366-021-01285-y

    Article  Google Scholar 

  14. Friedrich K, Almajid AA (2013) Manufacturing aspects of advanced polymer composites for automotive applications. Appl Compos Mater 20:107–128

    Google Scholar 

  15. Thornton PH, Jeryan RA (1988) Crash energy management in composite automotive structures. Int J Impact Eng 7:167–180

    Google Scholar 

  16. Zain NM, Roslin EN, Ahmad S (2016) Preliminary study on bio-based polyurethane adhesive/aluminum laminated composites for automotive applications. Int J Adhes Adhes 71:1–9

    Google Scholar 

  17. Heggemann T, Homberg W (2019) Deep drawing of fiber metal laminates for automotive lightweight structures. Compos Struct 216:53–57

    Google Scholar 

  18. Kalita K, Ghadai RK, Chakraborty S (2021) A comparative study on the metaheuristic-based optimization of skew composite laminates. Eng Comput. https://doi.org/10.1007/s00366-021-01401-y

    Article  Google Scholar 

  19. Foroutan K, Varedi-Koulaei SM, Duc ND, Ahmadi H (2022) Non-linear static and dynamic buckling analysis of laminated composite cylindrical shell embedded in non-linear elastic foundation using the swarm-based metaheuristic algorithms. Eur J Mech A/Solids 91:104420

    MathSciNet  MATH  Google Scholar 

  20. Van ThuP, Duc ND (2016) Nonlinear stability analysis of imperfect three-phase sandwich laminated polymer nanocomposite panels resting on elastic foundations in thermal environments. VNU J Sci Math 32:20–36

    Google Scholar 

  21. Dinh Nguyen P, Vu Q-V, Papazafeiropoulos G et al (2020) Optimization of laminated composite plates for maximum biaxial buckling load. VNU J Sci Math - Phys 36:1–12

    Google Scholar 

  22. Trinh MC, Mukhopadhyay T, Kim S-E (2020) A semi-analytical stochastic buckling quantification of porous functionally graded plates. Aerosp Sci Technol 105:105928

    Google Scholar 

  23. Trinh MC, Kim SE (2021) Deterministic and stochastic thermomechanical nonlinear dynamic responses of functionally graded sandwich plates. Compos Struct 274:114359

    Google Scholar 

  24. Trinh MC, Mukhopadhyay T (2021) Semi-analytical atomic-level uncertainty quantification for the elastic properties of 2D materials. Mater Today Nano 15:100126

    Google Scholar 

  25. Trinh MC, Jun H (2021) Stochastic vibration analysis of functionally graded beams using artificial neural networks. Struct Eng Mech 78:529–543

    Google Scholar 

  26. Köllner A, Nielsen MWD, Srisuriyachot J et al (2021) Buckle-driven delamination models for laminate strength prediction and damage tolerant design. Thin-Wall Struct 161:107468

    Google Scholar 

  27. Baucke A, Mittelstedt C (2015) Closed-form analysis of the buckling loads of composite laminates under uniaxial compressive load explicitly accounting for bending-twisting-coupling. Compos Struct 128:437–454

    Google Scholar 

  28. Lee HSJ, York CB (2020) Compression and shear buckling performance of finite length plates with bending-twisting coupling. Compos Struct 241:112069

    Google Scholar 

  29. Tran LV, Wahab MA, Kim SE (2017) An isogeometric finite element approach for thermal bending and buckling analyses of laminated composite plates. Compos Struct 179:35–49

    Google Scholar 

  30. Chen Q, Qiao P (2021) Buckling and postbuckling of rotationally-restrained laminated composite plates under shear. Thin-Wall Struct 161:107435

    Google Scholar 

  31. Qin XC, Dong CY, Yang HS (2019) Isogeometric vibration and buckling analyses of curvilinearly stiffened composite laminates. Appl Math Model 73:72–94

    MathSciNet  MATH  Google Scholar 

  32. Manickam G, Bharath A, Das AN et al (2018) Thermal buckling behaviour of variable stiffness laminated composite plates. Mater Today Commun 16:142–151

    Google Scholar 

  33. Kharghani N, Guedes Soares C (2020) Analysis of composite laminates containing through-the-width and embedded delamination under bending using layerwise HSDT. Eur J Mech A/Solids 82:104003

    MATH  Google Scholar 

  34. Kharghani N, Guedes Soares C (2020) Experimental, numerical and analytical study of buckling of rectangular composite laminates. Eur J Mech A/Solids 79:103869

    MathSciNet  MATH  Google Scholar 

  35. Liu T, Zhang W, Mao JJ, Zheng Y (2019) Nonlinear breathing vibrations of eccentric rotating composite laminated circular cylindrical shell subjected to temperature, rotating speed and external excitations. Mech Syst Signal Process 127:463–498

    Google Scholar 

  36. Dey S, Mukhopadhyay T, Naskar S et al (2019) Probabilistic characterisation for dynamics and stability of laminated soft core sandwich plates. J Sandw Struct Mater 21:366–397

    Google Scholar 

  37. Sepahvand K, Marburg S (2015) Non-sampling inverse stochastic numerical-experimental identification of random elastic material parameters in composite plates. Mech Syst Signal Process 54–55:172–181

    Google Scholar 

  38. Noh HC, Park T (2011) Response variability of laminate composite plates due to spatially random material parameter. Comput Methods Appl Mech Eng 200:2397–2406

    MathSciNet  MATH  Google Scholar 

  39. Chen NZ, Guedes Soares C (2008) Spectral stochastic finite element analysis for laminated composite plates. Comput Methods Appl Mech Eng 197:4830–4839

    MATH  Google Scholar 

  40. Dodwell TJ, Kynaston S, Butler R et al (2021) Multilevel Monte Carlo simulations of composite structures with uncertain manufacturing defects. Probab Eng Mech 63:103116

    Google Scholar 

  41. Parviz H, Fakoor M (2020) Free vibration of a composite plate with spatially varying Gaussian properties under uncertain thermal field using assumed mode method. Phys A Stat Mech its Appl 559:125085

    MathSciNet  MATH  Google Scholar 

  42. Gadade AM, Lal A, Singh BN (2020) Stochastic buckling and progressive failure of layered composite plate with random material properties under hygro-thermo-mechanical loading. Mater Today Commun 22:100824

    Google Scholar 

  43. Chen X, Wang X, Wang L et al (2018) Uncertainty quantification of multi-dimensional parameters for composite laminates based on grey mathematical theory. Appl Math Model 55:299–313

    MathSciNet  MATH  Google Scholar 

  44. Mahjudin M, Lardeur P, Druesne F, Katili I (2020) Extension of the Certain Generalized Stresses Method for the stochastic analysis of homogeneous and laminated shells. Comput Methods Appl Mech Eng 365:112945

    MathSciNet  MATH  Google Scholar 

  45. Trinh MC, Nguyen SN, Jun H, Nguyen-Thoi T (2021) Stochastic buckling quantification of laminated composite plates using cell-based smoothed finite elements. Thin-Wall Struct 163:107674

    Google Scholar 

  46. Nguyen HX, Duy Hien T, Lee J, Nguyen-Xuan H (2017) Stochastic buckling behaviour of laminated composite structures with uncertain material properties. Aerosp Sci Technol 66:274–283

    Google Scholar 

  47. Jeon HM, Lee Y, Lee PS, Bathe KJ (2015) The MITC3+ shell element in geometric nonlinear analysis. Comput Struct 146:91–104

    Google Scholar 

  48. Lee Y, Jeon HM, Lee PS, Bathe KJ (2015) The modal behavior of the MITC3+ triangular shell element. Comput Struct 153:148–164

    Google Scholar 

  49. Jun H, Yoon K, Lee PS, Bathe KJ (2018) The MITC3+ shell element enriched in membrane displacements by interpolation covers. Comput Methods Appl Mech Eng 337:458–480

    MathSciNet  MATH  Google Scholar 

  50. Trinh MC, Jun H (2021) A higher-order quadrilateral shell finite element for geometrically nonlinear analysis. Eur J Mech A/Solids 89:104283

    MathSciNet  MATH  Google Scholar 

  51. Petelet M, Iooss B, Asserin O, Loredo A (2010) Latin hypercube sampling with inequality constraints. AStA Adv Stat Anal 94:325–339

    MathSciNet  MATH  Google Scholar 

  52. Reddy JN (2004) Mechanics of laminated composite plates and shells: theory and Analysis, 2nd edn. CRC Press LLC

    MATH  Google Scholar 

  53. Trinh MC, Kim SE (2019) A three variable refined shear deformation theory for porous functionally graded doubly curved shell analysis. Aerosp Sci Technol 94:105356

    Google Scholar 

  54. Lyon A (2014) Why are normal distributions normal? Br J Philos Sci 65:621–649

    MathSciNet  MATH  Google Scholar 

  55. Dunn WL, Shultis JK (2011) Exploring Monte Carlo methods. Elsevier

    MATH  Google Scholar 

  56. Dimov IT (2008) Monte Carlo methods for applied scientists. World Scientific

    MATH  Google Scholar 

  57. Caflisch RE (1998) Monte Carlo and quasi-Monte Carlo methods. Acta Numer 7:1–49

    MathSciNet  MATH  Google Scholar 

  58. Nguyen TMS, Trinh MC, Kim SE (2021) Uncertainty quantification of ultimate compressive strength of CCFST columns using hybrid machine learning model. Eng Comput. https://doi.org/10.1007/s00366-021-01339-1

    Article  Google Scholar 

  59. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–245

    MathSciNet  MATH  Google Scholar 

  60. Denaranjo MVS (1983) Central limit theorems for non-linear functionals of Gaussian fields. J Multivar Anal 13:425–441

    Google Scholar 

Download references

Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2020R1I1A3073577) and by Korea Institute of Planning and Evaluation for Technology in Food, Agriculture and Forestry (IPET) and Korea Smart Farm R&D Foundation through Smart Farm Innovation Technology Development Program, funded by Ministry of Agriculture, Food and Rural Affairs (MAFRA), Ministry of Science and ICT (MSIT), and Rural Development Administration (RDA) (421016041HD030).

Author information

Authors and Affiliations

  1. Division of Mechanical System Engineering, Jeonbuk National University, 567, Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do, 54896, Republic of Korea

    Minh-Chien Trinh & Hyungmin Jun

Authors
  1. Minh-Chien Trinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyungmin Jun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyungmin Jun.

Ethics declarations

Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$a_{11} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{12} = a_{21} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{12}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{13} = b_{11} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z\,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{14} = c_{11} = - \frac{1}{4}a_{13} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{3} \,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{15} = b_{16} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z\,\overline{C}_{16}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{16} = c_{16} = - \frac{1}{4}a_{15} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{3} \,\overline{C}_{16}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{22} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{23} = b_{21} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z\,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{24} = c_{21} = - \frac{1}{4}a_{23} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{3} \,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{25} = b_{26} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z\,\overline{C}_{26}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{26} = c_{26} = - \frac{1}{4}a_{25} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{3} \,\overline{C}_{26}^{\left( i \right)} \,{\text{d}}z} }$$
$$a_{31} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{66}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{12} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{13} = b_{22} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{12}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{14} = c_{12} = - \frac{1}{4}b_{12} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{15} = b_{24} = c_{13} = c_{22} = - \frac{1}{4}b_{13} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{12}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{23} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{25} = c_{23} = - \frac{1}{4}b_{23} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{31} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{66}^{\left( i \right)} \,{\text{d}}z} }$$
$$b_{32} = \,c_{31} = - \frac{1}{4}b_{31} + \frac{5}{{3h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{66}^{\left( i \right)} \,{\text{d}}z} }$$
$$c_{14} = \frac{1}{16}b_{12} - \frac{5}{{6h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{9h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{6} \,\overline{C}_{11}^{\left( i \right)} \,{\text{d}}z} }$$
$$c_{15} = c_{24} = \frac{1}{16}b_{13} - \frac{5}{{6h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{12}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{9h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{6} \,\overline{C}_{12}^{\left( i \right)} \,{\text{d}}z} }$$
$$c_{25} = \frac{1}{16}b_{23} - \frac{5}{{6h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{9h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{6} \,\overline{C}_{22}^{\left( i \right)} \,{\text{d}}z} }$$
$$c_{32} = \frac{1}{16}b_{31} - \frac{5}{{6h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{66}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{9h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{6} \,\overline{C}_{66}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{11} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{55}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{12} = \frac{5}{4}d_{11} - \frac{5}{{h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{55}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{13} = \frac{25}{{16}}d_{11} - \frac{25}{{2h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{55}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{55}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{21} = \sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {\overline{C}_{44}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{22} = \frac{5}{4}d_{21} - \frac{5}{{h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{44}^{\left( i \right)} \,{\text{d}}z} }$$
$$d_{23} = \frac{25}{{16}}d_{21} - \frac{25}{{2h^{2} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{2} \,\overline{C}_{44}^{\left( i \right)} \,{\text{d}}z} } + \frac{25}{{h^{4} }}\sum\limits_{i = 1}^{N} {\int\limits_{{z_{i} }}^{{z_{i + 1} }} {z^{4} \,\overline{C}_{44}^{\left( i \right)} \,{\text{d}}z} }$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trinh, MC., Jun, H. Stochastic bending and buckling analysis of laminated composite plates using Latin hypercube sampling. Engineering with Computers 39, 1459–1497 (2023). https://doi.org/10.1007/s00366-021-01544-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01544-y

Keywords

Search

Navigation