Nonlinear Postbuckling of Eccentrically Oblique-Stiffened Functionally Graded Doubly Curved Shallow Shells Based on Improved Donnell Equations

Mechanics of Composite Materials volume 55, pages727–742(2020)Cite this article

The nonlinear buckling and postbuckling of eccentrically oblique-stiffened doubly curved shallow functionally graded shells is investigated based on improved Donnell equations. The improved Lekhnitskii smeared stiffeners technique is employed to found the stiffness matrix of the stiffened shells. The shells are reinforced by eccentrically oblique stiffeners with an arbitrary inclination angle. Using the Galerkin method, an analytical approximate solution for the deflection of reinforced FGM doubly curved shallow shells is obtained. The influence of geometrical parameters, oblique stiffeners, and temperature on the postbuckling behavior of the shells is analyzed.

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References

  1. 1.

    M. R. Khalil, M. D. Olson, and D. L. Anderson, “Nonlinear dynamic analysis of stiffened plates,” Comput. Struct., 29, No. 6, 929-941 (1988).

    Article  Google Scholar 

  2. 2.

    P. C. Shen and C. Dade, “Dynamic analysis of stiffened plates and shells using the spline Gauss collocation method,” Comput. Struct., 36, No. 4, 623-629 (1990).

    Article  Google Scholar 

  3. 3.

    A. N. Nayak and J. N. Bandyopadhyay, “On the free vibration of stiffened shallow shells,” J. Sound Vib., 255, No. 2, 357-382 (2002).

    Article  Google Scholar 

  4. 4.

    A. H. Sheikh and M. Mukhopadhyay, “Linear and nonlinear transient vibration analysis of stiffened plate structures,” Finite Elem. Anal. Des., 38, No. 6, 477-502 (2002).

    Article  Google Scholar 

  5. 5.

    S. N. Patel, P. K. Datta, and A. H. Sheikh, “Buckling and dynamic instability analysis of stiffened shell panels,” Thin- Walled Struct., 44, No. 03, 321-333 (2006).

    Article  Google Scholar 

  6. 6.

    S. N. Patel, P. K. Datta, and A. H. Sheikh, “Dynamic instability analysis of stiffened shell panels subjected to partial edge loading along the edges,” Int. J. Mech. Sci., 49, No. 12, 1309-1324 (2007).

    Article  Google Scholar 

  7. 7.

    V. Birman, “Buckling of functionally graded hybrid composite plates,” Compos. Eng., 5, No. 7, 913-921 (1995).

    CAS  Article  Google Scholar 

  8. 8.

    E. Feldman and J. Aboudi, “Buckling analysis of functionally graded plates subjected to uniaxial loading,” Compos. Struct., 38, No. 1-4, 29-36 (1997).

    Article  Google Scholar 

  9. 9.

    H. V. Tung and N. D. Duc, “Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads,” Compos. Struct., 92, No. 5, 1184-1191 (2010).

    Article  Google Scholar 

  10. 10.

    H. S. Shen, “Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments,” Int. J. Solids Struct., 39, No. 24, 5991-6010 (2002).

    Article  Google Scholar 

  11. 11.

    H. S. Shen and A. Y. T. Leung, “Postbuckling of pressure-loaded functionally graded cylindrical panels in thermal environments,” J. Eng. Mech. ASCE, 129, No. 4, 414-425 (2003).

    Article  Google Scholar 

  12. 12.

    N. D. Duc and H. V. Tung, “Nonlinear response of pressure-loaded functionally graded cylindrical panels with temperature effects,” Compos. Struct., 92, No. 7, 1664-1672 (2010).

    Article  Google Scholar 

  13. 13.

    Y. Yang, K. M. Liew, Y. F. Wu, and S. Kitipornchai, “Thermo-mechanical postbuckling of FGM cylindrical panels with temperature dependent properties,” Int. J. Solids Struct., 43, No. 2 307-324 (2006).

    Article  Google Scholar 

  14. 14.

    S. I. Li and R. C. Batra, “Buckling of axially compressed thin cylindrical shells with functionally graded middle layer,” Thin-Walled Struct., 44, No. 10, 1039-1047 (2006).

    Article  Google Scholar 

  15. 15.

    H. S. Shen, “Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments,” Engin. Struct., 25, No. 4, 487-497 (2003).

    Article  Google Scholar 

  16. 16.

    H. Huang and Q. Han, “Research on nonlinear postbuckling of functionally graded cylindrical shells under radial loads,” Compos. Struct., 92, No. 6, 1352-1357 (2010).

    Article  Google Scholar 

  17. 17.

    D. H. Bich, “Nonlinear buckling analysis of functionally graded shallow spherical shells,” Vietnam J. Mech., 31, No. 1, 17-30 (2009).

    Article  Google Scholar 

  18. 18.

    R. Naj, M. Sabzikar, and M. R. Eslami, “Thermal and mechanical instability of functionally graded truncated conical shells,” Thin-Walled Struct., 46, No. 1, 65-78 (2008).

    Article  Google Scholar 

  19. 19.

    A. H. Sofiyev, “The stability of compositionally graded ceramic-metal cylindrical shells under aperiodic axial impulsive loading,” Compos. Struct., 69, No. 2, 247-257 (2005).

    Article  Google Scholar 

  20. 20.

    T. Y. Ng, K. Y. Lam, K. M. Liew, and J. N. Reddy, “Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading,” Int. J. Solids Struct., 38, No. 8, 1295-1309 (2001).

    Article  Google Scholar 

  21. 21.

    M. Darabi, M. Darvizeh, and A. Darvizeh, “Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading,” Compos. Struct., 83, No. 2, 201-211 (2008).

    Article  Google Scholar 

  22. 22.

    R. Kolahchi, M. Safari, and M. Esmailpour. “Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium,” Compos. Struct., 150, 255-265 (2016).

    Article  Google Scholar 

  23. 23.

    R. Kolahchi, “A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods,” Aerosp. Sci. Technol., 66, 235-248 (2017).

    Article  Google Scholar 

  24. 24.

    F. Z. Taibi, S. Benyoucef, A. Tounsi, R. B. Bouiadjra, E. A. A. Bedia, and S. R. Mahmoud, “A simple shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations,” J. Sandw. Struct. Mater., 17, No. 2, 99-129 (2015).

    Article  Google Scholar 

  25. 25.

    S. Kamarian, M. Sadighi, M. Shakeri, and M. H. Yas, “Free vibration response of sandwich cylindrical shells with functionally graded materials face sheets resting on Pasternak foundation,” J. Sandw. Struct. Mater., 16, No. 5, 511- 533 (2014).

    Article  Google Scholar 

  26. 26.

    N. D. Duc and H. P. Cong, “Nonlinear postbuckling of an eccentrically stiffened thin FGM plate resting on elastic foundations in thermal environments,” Thin-Walled Struct., 75, 103-112 (2014).

    Article  Google Scholar 

  27. 27.

    N. D. Duc and T. Q. Quan, “Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments,” Compos. Struct., 106, 590-600 (2013).

    Article  Google Scholar 

  28. 28.

    D. H. Bich, D. V. Dung, and V. H. Nam, “Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells,” J. Compos. Struct., 96, 384-395 (2013).

    Article  Google Scholar 

  29. 29.

    N. D. Duc and P. T. Thang, “Nonlinear dynamic response and vibration of shear deformable imperfect eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations,” Aerosp. Sci. Technol., 40, 115-127 (2015).

    Article  Google Scholar 

  30. 30.

    N. D. Duc, J. Lee, T. Nguyen-Thoi, and P. T. Thang, “Static response and free vibration of functionally graded carbon nanotube-reinforced composite rectangular plates resting on Winkler-Pasternak elastic foundations,” Aerosp. Sci. Technol., 68, 391-402 (2017).

    Article  Google Scholar 

  31. 31.

    H. Less and H. Abramovich, “Dynamic buckling of a laminated composite stringer stiffened cylindrical panel,” Compos. Part B: Eng., 43, No. 5, 2348-2358 (2012).

    Article  Google Scholar 

  32. 32.

    Y. V. Satish Kumar and M. Mukhopadhyay, “Transient response analysis of laminated stiffened plates,” Compos. Struct., 58, No. 1, 97-107 (2002).

    Article  Google Scholar 

  33. 33.

    Z. M. Li and P. Qiao, “Nonlinear vibration analysis of geodesically-stiffened laminated composite cylindrical shells in an elastic medium,” Compos. Struct., 111, 473-487 (2014).

    Article  Google Scholar 

  34. 34.

    S. N. Patel, P. K. Datta, and A. H. Sheikh, “Parametric study on the dynamic instability behavior of laminated composite stiffened plate,” J. Eng. Mech. ASCE, 135, No. 11, 1331-1341 (2009).

    Article  Google Scholar 

  35. 35.

    D. D. Brush and B. O. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York (1975).

    Google Scholar 

Download references

Acknowledgment

This research was funded by the National Science and Technology Program of Vietnam for the period of 2016-2020 “Research and development of science education to meet the requirements of fundamental and comprehensive reform education of Vietnam” under the Grant KHGD/16-20.DT.032. The authors are grateful for this support.

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Affiliations

  1. Advanced Materials and Structures Laboratory, VNU, Hanoi - University of Engineering and Technology, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam

    N. D. Duc & N. H. Cuong

  2. Infrastructure Engineering Program, VNU, Hanoi - Vietnam-Japan University My Dinh 1, Tu Liem, Hanoi, Vietnam

    N. D. Duc

  3. National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, 209 Neungdongro, Gwangjin-gu, Seoul, 05006, Korea

    N. D. Duc

  4. University of Transport Technology, 54 Trieu Khuc, Thanh Xuan, Hanoi, Vietnam

    V. H. Nam

Authors
  1. N. D. Duc
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  2. V. H. Nam
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  3. N. H. Cuong
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Correspondence to N. D. Duc.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 55, No. 6, pp. 1059-1080, November- December, 2019.

Appendix

Appendix

$$ {\displaystyle \begin{array}{c}{I}_{11}={A}_{11}-\frac{D_{11}}{R_x^2}{I}_{12}={A}_{66}-\frac{D_{66}}{R_x^2},{I}_{13}={A}_{16}+{A}_{61}-\frac{1}{R_x^2}\left({D}_{61}+{D}_{16}\right),\\ {}{I}_{21}={A}_{16}+{B}_{16}\left(\frac{1}{R_y}-\frac{1}{R_x}\right)-\frac{D_{16}}{R_x{R}_y},{I}_{22}={A}_{62}+{B}_{62}\left(\frac{1}{R_y}-\frac{1}{R_x}\right)-\frac{D_{62}}{R_x{R}_y},\\ {}{I}_{23}={A}_{12}+{A}_{66}+\left({B}_{12}+{B}_{66}\right)\left(\frac{1}{R_y}-\frac{1}{R_x}\right)-\frac{1}{R_x{R}_y}\left({D}_{12}+{D}_{66}\right),\\ {}{I}_{31}=\left({A}_{11}-{B}_{11}\frac{1}{R_x}\right)\frac{1}{R_x}+\left({A}_{12}-\frac{1}{R_x}{B}_{12}\right)\frac{1}{R_y},{I}_{32}=\left({A}_{61}-\frac{1}{R_x}{B}_{61}\right)\frac{1}{R_x}\left({A}_{62}-\frac{D_{62}}{R_x}\right)\frac{1}{R_y},\\ {}{I}_{33}={A}_{16}-\frac{1}{R_x}{B}_{16},{I}_{34}={A}_{66}-\frac{1}{R_x}{B}_{66},{I}_{35}=\left({B}_{11}-\frac{1}{R_x}{D}_{11}\right),\\ {}\begin{array}{l}{I}_{130}=\frac{1}{2{R}_x}{A}_{11}+\frac{1}{2{R}_y}{A}_{21},{I}_{131}=\frac{1}{2{R}_x}{A}_{12}+\frac{1}{2{R}_y}{A}_{22},{I}_{132}=\frac{A_{16}}{R_{,x}}+\frac{A_{26}}{R_y}0,\\ {}\end{array}\\ {}\begin{array}{l}{I}_{133}={A}_{11}\frac{1}{R_x}+{A}_{12}\frac{1}{R_y},{I}_{134}={A}_{11}\frac{1}{R_x}+{A}_{12}\frac{1}{R_y},{I}_{135}={A}_{61}\frac{1}{R_x}+{A}_{62}\frac{1}{R_y},\\ {}\begin{array}{c}{I}_{136}={A}_{61}\frac{1}{R_x}+{A}_{62}\frac{1}{R_y},{I}_{137}={A}_{21}\frac{1}{R_x}+{A}_{22}\frac{1}{R_y},{I}_{138}={A}_{21}\frac{1}{R_x}+{A}_{22}\frac{1}{R_y},\end{array}\end{array}\\ {}{I}_{139}={A}_{61}\frac{1}{R_x}+{A}_{62}\frac{1}{R_y},{I}_{140}\left({A}_{61}\frac{1}{R_x}+{A}_{62}\frac{1}{R_y}\right),{I}_{51}^{\ast }=\frac{8}{9}\left(\frac{3}{2{R}_x}{A}_{11}+\frac{1}{2{R}_y}{A}_{21}+{A}_{12}\frac{1}{R_y}\right),\\ {}{I}_{52}^{\ast }=\frac{8}{9}\left(\frac{1}{2{R}_x}{A}_{12}+\frac{3}{2{R}_y}{A}_{22}+{A}_{21}\frac{1}{R_x}\right),\\ {}{I}_{53}^{\ast }=\frac{16}{9}\frac{{\lambda_m}^3}{\delta_n}{B}_{11}+\frac{4}{9}{B}_{12}{\lambda}_m{\delta}_n+\frac{16}{9}\frac{{\delta_n}^3}{\lambda_m}{B}_{22}-\frac{8}{9}{B}_{66}{\lambda}_m{\delta}_n\\ {}{I}_{36}={B}_{62}-\frac{1}{R_x}{D}_{62},{I}_{37}={B}_{12}+2{B}_{66}-\frac{1}{R_x}\left(2{D}_{66}+{D}_{12}\right),{I}_{38}=2{B}_{16}+{B}_{61}-\frac{1}{R_x}\left({D}_{61}+2{D}_{16}\right),\\ {}{I}_{41}=\frac{1}{2}{A}_{61}-\frac{1}{2{R}_x}{B}_{61},{I}_{42}=\frac{1}{2}{A}_{62}-\frac{1}{2{R}_x}{B}_{62},{I}_{43}=\frac{1}{2}{A}_{11}-\frac{1}{2{R}_x}{B}_{11},{I}_{44}=\frac{1}{2}{A}_{12}-\frac{1}{2{R}_x{B}_{12},}\\ {}{I}_{51}={A}_{61}+{B}_{61}\left(\frac{1}{R_x}-\frac{1}{R_y}\right)-\frac{D_{61}}{R_y{R}_x},{I}_{52}={A}_{26}+{B}_{26}\left(\frac{1}{R_x}-\frac{1}{R_y}\right)-\frac{D_{26}}{R_x{R}_y},\\ {}\end{array}} $$
$$ {\displaystyle \begin{array}{c}{I}_{53}={A}_{21}+{A}_{66}+\left({B}_{21}+{B}_{66}\right)\left(\frac{1}{R_x}-\frac{1}{R_y}\right)-\frac{\left({D}_{66}+{D}_{21}\right)}{R_x{R}_y},\\ {}{I}_{61}=\left({A}_{66}-\frac{D_{66}}{R_y^2}\right),{I}_{62}\left({A}_{22}-\frac{D_{22}}{R_y^2}\right),{I}_{63}\left[{A}_{26}+{A}_{62}-\frac{\left({D}_{62}+{D}_{26}\right)}{R_y^2}\right],\\ {}{I}_{71}=\frac{1}{2}{A}_{21}-\frac{1}{2{R}_y},{I}_{72}=\frac{1}{2}{A}_{22}-\frac{1}{2{R}_y}{B}_{22},{I}_{73}=\frac{1}{2}{A}_{61}-\frac{1}{2{R}_y}{B}_{61},{I}_{74}=\frac{1}{2}{A}_{62}-\frac{1}{2{R}_y}{B}_{62},\\ {}{I}_{81}={A}_{21}\frac{1}{R_x}-\frac{1}{R_y}\left({B}_{21}\frac{1}{R_x}+{B}_{22}\frac{1}{R_y}-{A}_{22}\right),{I}_{82}={B}_{21}+2{B}_{66}-\frac{1}{R_y}\left({D}_{21}+2{D}_{66}\right),\\ {}{I}_{83}={B}_{22}-\frac{1}{R_y}{D}_{22},{I}_{84}=2{B}_{26}+{B}_{62}-\frac{1}{R_y}\left({D}_{62}+2{D}_{26}\right),\\ {}{I}_{85}={A}_{61}\frac{1}{R_x}+{A}_{62}\frac{1}{R_y}-\frac{1}{R_y}\left({B}_{61}\frac{1}{R_x}+{D}_{62}\frac{1}{R_y}\right),{I}_{86}={B}_{61}-\frac{1}{R_y}{D}_{61},\\ {}{I}_{87}={A}_{26}-\frac{1}{R_y}{B}_{26},{I}_{88}={A}_{66}-\frac{1}{R_y}{B}_{66},\\ {}{I}_{91}={B}_{11}+{D}_{11}\frac{1}{R_x},{I}_{92}={B}_{26}+{D}_{26}\frac{1}{R_x},{I}_{93}={B}_{16}+{D}_{16}\frac{1}{R_x}+2\left({B}_{61}+{D}_{61}\frac{1}{R_x}\right),\\ {}{I}_{94}={B}_{21}+{D}_{21}\frac{1}{R_x}+2\left({B}_{66}+{D}_{66}\frac{1}{R_x}\right),{I}_{95}=\frac{1}{R_x}\left({A}_{11}+{B}_{11}\frac{1}{R_x}\right)+\frac{1}{R_y}\left({A}_{21}+{B}_{21}\frac{1}{R_x}\right),\\ {}{I}_{96}=\frac{1}{R_x}\left({A}_{16}+{B}_{16}\frac{1}{R_x}\right)+\frac{1}{R_y}\left({A}_{26}+{B}_{26}\frac{1}{R_x}\right),{I}_{110}={B}_{16}+{D}_{16}\frac{1}{R_y},{I}_{111}={B}_{22}+{D}_{22}\frac{1}{R_y},\\ {}{I}_{112}={B}_{26}+{D}_{26}\frac{1}{R_y}+2\left({B}_{62}+{D}_{62}\frac{1}{R_y}\right),{I}_{113}={B}_{12}+{D}_{12}\frac{1}{R_y}+2\left({B}_{66}+{D}_{66}\frac{1}{R_y}\right),\\ {}{I}_{114}=\frac{1}{R_x}\left({A}_{12}+{B}_{12}\frac{1}{R_y}\right)+\frac{1}{R_y}\left({A}_{22}+{B}_{22}\frac{1}{R_y}\right),{I}_{115}=\frac{1}{R_x}\left({A}_{16}+{B}_{16}\frac{1}{R_y}\right)+\frac{1}{R_y}\left({A}_{26}+{B}_{26}\frac{1}{R_y}\right),\\ {}{P}_3(w)={I}_{130}{\left({w}_{,x}\right)}^2+{I}_{131}{\left({w}_{,y}\right)}^2+\frac{1}{2}{B}_{11}{\left({w}_{,x}^2\right)}_{,x\;x}+\frac{1}{2}{B}_{12}{\left({w}_{,y}^2\right)}_{,x\;x}+\frac{1}{2}{B}_{21}{\left({w}_{,x}^2\right)}_{,y\;y}+\frac{1}{2}{B}_{22}{\left({w}_{,y}^2\right)}_{,y\;y}\\ {}+{B}_{61}{\left({w}_{,x}^2\right)}_{,x\;y}+{B}_{62}{\left({w}_{,y}^2\right)}_{,x\;y}+2{B}_{66}{\left({w}_{,x}{w}_{,y}\right)}_{,x\;y}+{B}_{16}{\left({w}_{,x}{w}_{,y}\right)}_{,x\;x}+{B}_{26}{\left({w}_{,x}{w}_{,y}\right)}_{,y\;y}+{I}_{132}{w}_{,x}{w}_{,y}\\ {}-{w}_{,x}^2{I}_{133}-w{w}_{,x\;x}{I}_{134}+\frac{1}{2}{A}_{11}{\left({w}_{,x}^3\right)}_{,x}+\frac{1}{2}{A}_{12}{\left({w}_{,y}^2\right)}_{,x}+{w}_{,x}+\frac{1}{2}{A}_{12}{w}_{,y}^2{w}_{,x\;x}+{A}_{16}{w}_{,x}^2{w}_{,x\;y}+{A}_{16}{\left({w}_{,x}^2\right)}_{,x}{w}_{,y}\\ {}+{B}_{11}{w}_{, xx x}{w}_{,x}+{B}_{11}{w}_{, xx}^2+{B}_{12}{w}_{,x\;x}{w}_{,y\;y}+{B}_{12}{w}_{,x}{w}_{,x\; yy}+2{B}_{16}{w}_{,x\;x}{w}_{, xy}+2{B}_{16}{w}_{,x}{w}_{, xx y}+2{B}_{66}{w}_{, xy y}{w}_{,x}\\ {}\frac{1}{2}{A}_{61}{\left({w}_{,x}^2\right)}_{,x}{w}_{,y}+\frac{1}{2}{A}_{61}{w}_{,x}^2{w}_{,x\;y}+\frac{1}{2}{A}_{62}{\left({w}_{,y}^3\right)}_{,x}+{A}_{66}{w}_{,x\;x}{w}_{,y}^2+{A}_{66}{w}_{,x}{\left({w}_{,y}^2\right)}_{,x}+2{B}_{66}{w}_{, xy}{w}_{, xy}\\ {}-{w}_{,x}{w}_{,y}{I}_{135}-w{w}_{,x\;y}{I}_{136}+{B}_{61}{w}_{, xx x}{w}_{,y}+{B}_{62}{w}_{,x\; yy}{w}_{,y}+{B}_{62}{w}_{, yy}{w}_{,x\;y}+2{B}_{66}{w}_{, xx y}{w}_{,y}\end{array}} $$
$$ {\displaystyle \begin{array}{c}+2{B}_{66}{w}_{, xy}{w}_{,x\;y}-{w}_{,y}^2{I}_{137}-w{w}_{, yy}{I}_{138}+\frac{1}{2}{A}_{21}{\left({w}_{,x}^2\right)}_{,y}{w}_{,y}+\frac{1}{2}{A}_{21}{w}_{,x}^2{w}_{, yy}+\frac{1}{2}{A}_{22}{\left({w}_{,y}^3\right)}_{,y}+{A}_{26}{w}_{, xy}{w}_{,y}^2\\ {}+{A}_{26}{w}_{,x}{\left({w}_{,y}^2\right)}_{,y}+{B}_{21}{w}_{, xx y}{w}_{,y}+{B}_{21}{w}_{, xx}{w}_{, yy}+{B}_{22}{w}_{, yy y}{w}_{,y}+{B}_{22}{w}_{, yy}^2+2{B}_{26}{w}_{, xy y}{w}_{,y}+2{B}_{26}{w}_{, xy}{w}_{, yy}\\ {}-{w}_{,y}{w}_{,x}{I}_{139}-w{w}_{, xy}{I}_{139}^{\ast }+\frac{1}{2}{A}_{61}{\left({w}_{,x}^3\right)}_{,y}+\frac{1}{2}{A}_{62}{w}_{,x}{\left({w}_{,y}^2\right)}_{,y}+{A}_{66}{\left({w}_{,x}^2\right)}_{,y}{w}_{,y}\\ {}+{A}_{66}{w}_{,x}^2{w}_{, yy}+{B}_{61}{w}_{, xx y}{w}_{,x}+{B}_{61}{w}_{, xx}{w}_{, xy}+{B}_{62}{w}_{, yy y}{w}_{,x}+{B}_{62}{w}_{, yy}{w}_{xy},\\ {}{Q}_3\left(u,w\right)={u}_{,x\;x}{w}_{,x}\left({A}_{11}+{B}_{11}\frac{1}{R_x}\right)+{u}_{,x}{w}_{,x\;x}\left({A}_{11}+{B}_{11}\frac{1}{R_x}\right)\\ {}+{u}_{,y}{w}_{,x\;x}\left({A}_{16}+{B}_{16}\frac{1}{R_x}\right){u}_{,y\;x}{w}_{,y}\left({A}_{21}+{B}_{21}\frac{1}{R_x}\right)+{u}_{,x}{w}_{, yy}\left({A}_{21}+{B}_{21}\frac{1}{R_x}\right)\\ {}+{u}_{yy}{w}_{,y}\left({A}_{26}+{B}_{26}\frac{1}{R_x}\right)+{u}_{,y}{w}_{, yy}\left({A}_{26}+{B}_{26}\frac{1}{R_x}\right)+{u}_{, xx}{w}_{,y}\left({A}_{61}+{B}_{61}\frac{1}{R_x}\right)\\ {}+{u}_{,x}{w}_{, xy}\left({A}_{61}+{B}_{61}\frac{1}{R_x}\right)+{u}_{, xy}{w}_{,y}\left({A}_{66}+{B}_{66}\frac{1}{R_x}\right)+{u}_{,y}{w}_{, xy}\left({A}_{66}+{B}_{66}\frac{1}{R_x}\right)\\ {}+{u}_{xy}{w}_{,x}\left({A}_{61}+{B}_{61}\frac{1}{R_x}\right)+{u}_{,x}{w}_{, xy}\left({A}_{61}+{B}_{61}\frac{1}{R_x}\right)+{u}_{, yy}{w}_{,x}\left({A}_{66}+{B}_{66}\frac{1}{R_x}\right)\\ {}+{u}_{,y}{w}_{xy}\left({A}_{66}+{B}_{66}\frac{1}{R_x}\right)+{u}_{, xy}{w}_{,x}\left({A}_{16}+{B}_{16}\frac{1}{R_x}\right),\\ {}{R}_3\left(v,w\right)={v}_{,y}{w}_{,x\;x}\left({A}_{12}+{B}_{12}\frac{1}{R_y}\right)+{v}_{,x\;y}{w}_{,x}\left({A}_{12}+{B}_{12}\frac{1}{R_y}\right)\\ {}+{v}_{, xx}{w}_{,x}\left({A}_{16}+{B}_{16}\frac{1}{R_y}\right)+{v}_{,x}{w}_{,x\;x}\left({A}_{16}+{B}_{16}\frac{1}{R_y}\right)\\ {}+{v}_{y\;y}{w}_{,y}\left({A}_{22}+{B}_{22}\frac{1}{R_y}\right)+{v}_{,y}{w}_{, yy}\left({A}_{22}+{B}_{22}\frac{1}{R_y}\right)\\ {}+{v}_{, xy}{w}_{,y}\left({A}_{26}+{B}_{26}\frac{1}{R_y}\right)+{v}_{,x}{w}_{, yy}\left({A}_{26}+{B}_{26}\frac{1}{R_y}\right)\\ {}+{v}_{,x\;y}{w}_{,y}\left({A}_{62}+{B}_{62}\frac{1}{R_y}\right)+{v}_{,y}{w}_{,x\;y}\left({A}_{62}+{B}_{62}\frac{1}{R_y}\right)\\ {}+{v}_{, xx}{w}_{,y}\left({A}_{66}+{B}_{66}\frac{1}{R_y}\right)+{v}_{,x}{w}_{,x\;y}\left({A}_{66}+{B}_{66}\frac{1}{R_y}\right)\\ {}+{v}_{,y\;y}{w}_{,x}\left({A}_{62}+{B}_{62}\frac{1}{R_y}\right)+{v}_{,y}{w}_{, yx}\left({A}_{62}+{B}_{62}\frac{1}{R_y}\right)\\ {}+{v}_{xy}{w}_{,x}\left({A}_{66}+{B}_{66}\frac{1}{R_y}\right)+{v}_{,x}{w}_{, xy}\left({A}_{66}+{B}_{66}\frac{1}{R_y}\right).\end{array}} $$

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Duc, N.D., Nam, V.H. & Cuong, N.H. Nonlinear Postbuckling of Eccentrically Oblique-Stiffened Functionally Graded Doubly Curved Shallow Shells Based on Improved Donnell Equations. Mech Compos Mater 55, 727–742 (2020). https://doi.org/10.1007/s11029-020-09847-9

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Keywords

  • nonlinear postbuckling
  • oblique stiffeners
  • doubly curved shallow functionally graded shell
  • improved Donnell equations