Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 2, 2019 (June)
Pages 59-69
Cited by J. of Mech. Eng., 2019, vol. 22, no. 2, pp. 59-69



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



As is known, thin plates with holes are one of the most common structural ele-ments. To increase their reliability and service life, it is of interest to find such a hole contour that ensures the minimum circumferential stress thereon, and also prevents the growth of possible cracks in the plate. This article deals with the problem of minimizing the stress state on the contour of a hole in an un-bounded isotropic stringer plate weakened by two rectilinear cracks. Crack faces are considered to be free of stress. Determined is the optimal hole con-tour, at which   no crack growth occurs, and the maximum circumferential stress thereon is minimal. The minimax criterion is used. The parameter char-acterizing the stress state in the vicinity of crack tips, according to the Irwin-Oroan theory of quasi-brittle fracture, is the stress intensity factor. The plate undergoes uniform stretching at infinity along the stringers. It is believed that the plate and the stringers are made of various elastic materials. The action of the stringers is replaced by the unknown equivalent concentrated forces ap-plied at the points of their attachment to the plate. To determine these forces, Hooke’s law is used. Applying the small parameter method, the theory of ana-lytic functions and the method of direct solution to singular equations, we con-structed a closed system of algebraic equations. This system depends on the mechanical and geometrical parameters of the plate and stringers, ensures the on-hole contour stress state minimization and equality of stress intensity fac-tors to zero in the vicinity of crack tips. The minimization problem is reduced to a linear programming problem. The simplex method is applied.


Keywords: stringer plate, stress minimization, cracks, optimal hole contour, minimax criterion.


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  1. Waldman, W., & Heller, M. (2006). Shape optimisation of holes for multi-peak stress minimisation. Australian Journal of Mechanical Engineering, vol. 3, iss. 1, pp. 61–71.
  2. Vigdergauz, S. (2006). The stress-minimizing hole in an elastic plate under remote shear. Journal of Mechanics of Materials and Structures, vol. 1, no. 2, pp. 387–406.
  3. 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, Technology and Instruments, no. 12(4), pp. 10–14 (in Russian).
  4. 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.
  5. 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: TGTU. pp. 238– 46 (in Russian).
  6. Vigdergauz, S. (2010). Energy-minimizing openings around a fixed hole in an elastic plate. Journal of Mechanics of Materials and Structures, vol. 5, no. 4, pp. 661–677.
  7. 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.
  8. Сherepanov, G. P. (2015). Optimum shapes of elastic bodies: equistrong wings of aircraft and equistrong underground tunnels. Physical Mesomechanics, vol. 18, iss. 4, pp. 391–
  9. 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).
  10. Samadi, N., Abolbashari, M. H., & Ghaffarianjam H. R. (2017). An effective approach for optimal hole shape with evolutionary structural optimization [Retrieved from;dn=389813149728265;res=IELENG]. In the 9th Australasian Congress on Applied Mechanics (ACAM9). Sydney: Engineers Australia, [1]–[8].
  11. Wang, S. J., Lu, A. Z., Zhang, X. L., & Zhang, N. (2018). Shape optimization of the hole in an orthotropic plate. Mechanics Based Design of Structures and Machines, vol. 46, iss. 1, pp. 23–37.
  12. 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.
  13. Mirsalimov, V. M. (2019). Maksimalnaya prochnost vyrabotki v gornom massive, oslablennom treshchinoy [Maximum strength of a working in a solid rock weakened by a crack]. Fiziko-tekhnicheskiye problemy razrabotki poleznykh iskopayemykh – Journal of Mining Science, vol. 55, iss. 1, pp. 12–21.
  14. 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.
  15. Mirsalimov, V. M. (2019). Minimizing the stressed state of a plate with a hole and cracks. Engineering Optimization.
  16. Muskhelishvili, N. I. (1977). Some basic problem of mathematical theory of elasticity. Dordrecht: Springer, 732 p.
  17. Kalandija, A. I. (1973). Matematicheskiye metody dvumernoy uprugosti [Mathematical methods of two-dimensional elasticity]. Moscow: Nauka, 304 p. (in Russian).
  18. 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]. Kiev: Naukova Dumka, 443 p. (in Russian).
  19. Mirsalimov, V. M. (1986). Some problems of structural arrest of cracks. Physicochemical Mechanics of Materials, vol. 22, iss. 1, pp. 81–85.


Received 12 May 2019

Published 30 June 2019