Determination of the Equi-stress Hole Shape for a Stringer Plate Weakened by a Surface Crack

Journal Journal of Mechanical Engineering
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 23, no. 3, 2020 (September)
Pages 16-26
Cited by J. of Mech. Eng., 2020, vol. 23, no. 3, pp. 16-26



Minavar V. Mir-Salim-zade, Institute of Mathematics and Mechanics of the NAS of Azerbaijan (9, Vahabzade St., Baku, AZ1141, Azerbaijan), e-mail:, ORCID: 0000-0003-4237-0352



On the basis of the principle of equal stress, a solution is given to the inverse problem of determining the optimal shape of the hole contour for a plate weakened by a surface rectilinear crack. The plate is reinforced by a regular system of elastic stiffeners (stringers). The crack originates from the hole contour perpendicular to the riveted stringers. The plate is subjected to uniform tension at infinity along the stiffeners. The plate under consideration is assumed to be either elastic or elastic-plastic. The criterion that determines the optimal shape of the hole is the condition that there is no stress concentration on the hole surface and the requirement that the stress intensity factor in the vicinity of the crack tip be equal to zero. In the case of an elastic-plastic plate, the plastic region at the moment of nucleation should encompass the entire hole contour at once, without deep penetration. The problem posed is to determine the hole shape at which the tangential normal stress acting on the contour is constant, and the stress intensity factor in the vicinity of the crack tip is zero, as well as to determine the magnitudes of the concentrated forces that replace both the action of the stringers and the stress-strain state of the reinforced plate. The method of a small parameter, the theory of analytic functions, and the method for direct solution of singular integral equations were used. The problem posed is reduced to the problem of finding a conditional extremum. The method of Lagrange indefinite multipliers was used. The obtained solution to the inverse problem allows increasing the bearing capacity of the stringer plate.


Keywords: plate, stringers, equal strength hole, crack.


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  1. Cherepanov, G. P. (1963). Obratnaya uprugoplasticheskaya zadacha v usloviyakh ploskoy deformatsii [Inverse elastic-plastic problem under plane deformation]. Izv. AN SSSR. Mekhanika i mashinostroyeniye News of the USSR Academy of Sciences. Mechanics and mechanical engineering, no. 2. pp. 57–60 (in Russian).
  2. Cherepanov, G. P. (1974). Inverse problems of the plane theory of elasticity. Journal of Applied Mathematics and Mechanics, vol. 38, iss. 6, pp. 915–931.
  3. Mirsalimov, V. M. (1974). On the optimum shape of apertures for a perforated plate subject to bending. Journal of Applied Mechanics and Technical Physics, vol. 15, pp. 842–845.
  4. Mirsalimov, V. M. (1975). Converse problem of elasticity theory for an anisotropic medium. Journal of Applied Mechanics and Technical Physics, vol. 16, pp. 645–648.
  5. Vigdergauz, S. B. (1976). Integral equations of the inverse problem of the theory of elasticity. Journal of Applied Mathematics and Mechanics, vol. 40, iss. 3, pp. 518–522.
  6. Vigdergauz, S. B. (1977). On a case of the inverse problem of two-dimensional theory of elasticity. Journal of Applied Mathematics and Mechanics, vol. 41, iss. 5, pp. 927–933.
  7. Mirsalimov, V. M. (1977). Inverse doubly periodic problem of thermoelasticity. Mechanics of Solids, vol. 12, iss. 4, pp. 147–154.
  8. Mirsalimov, V. M. (1979). A working of uniform strength in the solid rock. Soviet Mining, vol. 15, pp 327–330.
  9. Banichuk, N. V. (1980). Optimizatsiya form uprugikh tel [Shape optimization for elastic solids]. Moscow: Nauka, 255 p. (in Russian).
  10. Ostrosablin, N. I. (1981). Equal-strength hole in a plate in an inhomogeneous stress state. Journal of Applied Mechanics and Technical Physics, vol. 22, pp. 271–277.
  11. Bondar, V. D. (1996). A full-strength orifice under conditions of geometric nonlinearity Journal of Applied Mechanics and Technical Physics, vol. 37, pp. 898–904.
  12. Savruk, M. P. & Kravets, V. S. (2002). Application of the method of singular integral equations to the determination of the contours of equistrong holes in plates. Materials Science. vol. 38, pp. 34–46.
  13. Mir-Salim-zada, M. V. (2007). Obratnaya uprugoplasticheskaya zadacha dlya klepanoy perforirovannoy plastiny [Inverse elastoplastic problem for riveted perforated plate]. Sbornik statey “Sovremennye problemy prochnosti, plastichnosti i ustoychivosti” Collected papers “Modern problems of strength, plasticity and stability”. Tver: Tver State Technical University, pp. 238–246 (in Russian).
  14. Bantsuri, R. & Mzhavanadze, Sh. (2007). The mixed problem of the theory of elasticity for a rectangle weakened by unknown equi-strong holes. Proceedings of A. Razmadze Mathematical Institute, vol. 145, pp. 23–34.
  15. Mir-Salim-zada, M. V. (2007). Opredeleniye formy ravnoprochnogo otverstiya v izotropnoy srede, usilennoy regulyarnoy sistemoy stringerov [Determination of equistrong hole shape in isotropic medium, reinforced by regular system of stringers]. Materialy, tehnologii, instrumenty Materials, technologies, tools, no. 12 (4), pp. 10–14 (in Russian).
  16. Vigdergauz, S. (2012). Stress-smoothing holes in an elastic plate: From the square lattice to the checkerboard. Mathematics and Mechanics of Solids, vol. 17, iss. 3, pp. 289–299.
  17. Сherepanov, G. P. (2015). Optimum shapes of elastic bodies: Equistrong wings of aircraft and equistrong underground tunnels. Physical Mesomechanics, vol. 18, pp. 391–401.
  18. Vigdergauz, S. (2018). Simply and doubly periodic arrangements of the equi-stress holes in a perforated elastic plane: The single-layer potential approach. Mathematics and Mechanics of Solids, vol. 23, iss. 5, pp. 805–819.
  19. Zeng, X., Lu, A. & Wang, S. (2020). Shape optimization of two equal holes in an infinite elastic plate. Mechanics Based Design of Structures and Machines, vol. 48, iss. 2, pp. 133–145.
  20. Kalantarly, N. M. (2017). Ravnoprochnaya forma otverstiya dlya tormozheniya rosta treshchiny prodolnogo sdviga [Equal strength hole to inhibit longitudinal shear crack growth]. Problemy Mashinostroyeniya – Journal of Mechanical Engineering, vol. 20, no. 4, pp. 31–37 (in Russian).
  21. Mirsalimov, V. M. (2019). Maximum strength of opening in crack-weakened rock mass. Journal of Mining Science, vol. 55, pp. 9–17.
  22. Mirsalimov, V. M. (2019). Inverse problem of elasticity for a plate weakened by hole and cracks. Mathematical Problems in Engineering, vol. 2019, Article ID 4931489, 11 pages.
  23. Mir-Salim-zade, M. V. (2019). Minimization of the stressed state of a stringer plate with a hole and rectilinear cracks. Journal of Mechanical Engineering, vol. 22, no. 2, pp. 59–69.
  24. Mirsalimov, V. M. (2020). Minimizing the stressed state of a plate with a hole and cracks. Engineering Optimization, vol. 52, iss. 2, pp. 288–302.
  25. Mir-Salim-zada, M. V. (2020). Ravnoprochnaya forma otverstiya dlya stringernoy plastiny s treshchinami [An equi-stress hole for a stringer plate with cracks]. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics, iss. 64. pp. 121–135.
  26. Ishlinsky, A. Yu. & Ivlev, D. D. (2001). Matematicheskaya teoriya plastichnosti [Mathematical theory of plasticity]. Moscow: Fizmatlit, 704 p. (in Russian).
  27. Muskhelishvili, N. I. (1977). Some basic problems of mathematical theory of elasticity. Dordrecht: Springer, 732 p.
  28. Kalandiya, A. I. (1973). Matematicheskiye metody dvumernoy uprugosti [Mathematical methods of two-dimensional elasticity]. Moscow: Nauka, 304 p. (in Russian).
  29. Panasyuk, V. V., Savruk, M. P., & Datsyshin, A. P. (1976). Raspredeleniye napryazheniy okolo treshchin v plastinakh i obolochkakh [Stress distribution around cracks in plates and shells]. Kiyev: Naukova Dumka, 443 p. (in Russian).
  30. Mirsalimov, V. M. (1987). Neodnomernyye uprugoplasticheskiye zadachi [Non-one-dimensional elastoplastic problems]. Moscow: Nauka, 255 p. (in Russian).
  31. Mirsalimov, V. M. (1986). Some problems of structural arrest of cracks. Soviet materials science, vol. 22, pp. 81–85.


Received 25 April 2020