Minimization of the Stressed State of a Stringer Plate with a Hole and Rectilinear Cracks
Keywords:
stringer plate, stress minimization, cracks, optimal hole contour, minimax criterionAbstract
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.References
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. https://doi.org/10.1080/14484846.2006.11464495
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. https://doi.org/10.2140/jomms.2006.1.387
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).
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.
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).
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. https://doi.org/10.2140/jomms.2010.5.661
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. https://doi.org/10.1177/1081286511411571
Сherepanov, G. P. (2015). Optimum shapes of elastic bodies: equistrong wings of aircraft and equistrong underground tunnels. Physical Mesomechanics, vol. 18, iss. 4, pp. 391–401. https://doi.org/10.1134/S1029959915040116
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). https://doi.org/10.15407/pmach2017.04.031
Samadi, N, Abolbashari, M. H., & Ghaffarianjam H. R. (2017). An effective approach for optimal hole shape with evolutionary structural optimization [Retrieved from https://search.informit.com.au/documentSummary;dn=389813149728265;res=IELENG]. In the 9th Australasian Congress on Applied Mechanics (ACAM9).Sydney: EngineersAustralia, [1]–[8].
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. https://doi.org/10.1080/15397734.2016.126103623
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. https://doi.org/10.1177/1081286517691807
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. https://doi.org/10.15372/FTPRPI20190102
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. https://doi.org/10.1155/2019/4931489
Mirsalimov, V. M. (2019). Minimizing the stressed state of a plate with a hole and cracks. Engineering Optimization. https://doi.org/10.1080/0305215X.2019.1584619
Muskhelishvili, N. I. (1977). Some basic problem of mathematical theory of elasticity. Dordrecht: Springer, 732 p. https://doi.org/10.1007/978-94-017-3034-1
Kalandija, A. I. (1973). Matematicheskiye metody dvumernoy uprugosti [Mathematical methods of two-dimensional elasticity].Moscow: Nauka, 304 p. (in Russian).
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).
Mirsalimov, V. M. (1986). Some problems of structural arrest of cracks. Physicochemical Mechanics of Materials, vol. 22, iss. 1, pp. 81–85. https://doi.org/10.1007/BF00720871.
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