# Finite Plate with Circular and Square Hole under Partial Loading

Author

Department of Mechanical Engineering, Yazd University, Yazd, Iran

Abstract

In this paper a general analytical solution is obtained to find stress distribution in a finite elastic plate with a circular or square hole subjected to arbitrary biaxial partial loading using modified boundary condition by assuming plane stress conditions. The method employed is based on solution of circular hole in finite rectangular plate. This plate is mapped to circular ones and the partial loading is transformed to new boundaries as form of triangular functions. The Airy stress functions are selected according to these triangular functions and the unknown factors of Airy stress functions are derived by applying boundary conditions. The stresses in this plate with circular hole are mapped to plate with square hole using Muskhelishvili’s complex variable method. The results of this method are compared with theoretical solution of infinite plate and finite elemnt method solution of finite plate. The results showed the dimensions of plate and square hole and length of biaxial partial loading affected on Von Mises stress around the square hole. Von Mises stress increases around the square hole by decreasing length of the plate or increasing hole’s area.

Keywords

#### References

1.     Muskhelishvili, N., Some basic problems of the mathematical theory of elasticity. 1966, Nauka, Moscow.
2.     Gao, X.-L., "A general solution of an infinite elastic plate with an elliptic hole under biaxial loading", International Journal of Pressure Vessels and Piping,  Vol. 67, No. 1, (1996), 95-104.
3.     Savin, G.N., "Stress concentration around holes",  New York: Pergamon Press; 1961.
4.     Ukadgaonker, V. and Rao, D.K., "A general solution for stresses around holes in symmetric laminates under inplane loading", Composite Structures,  Vol. 49, No. 3, (2000), 339-354.
5.     Vasil’ev, V. and Fedorov, L., "Geometric elasticity problem of stress concentration in a plate with a circular hole", Mechanics of Solids,  Vol. 43, No. 4, (2008), 528-538.
6.     Kubair, D. and Bhanu-Chandar, B., "Stress concentration factor due to a circular hole in functionally graded panels under uniaxial tension", International Journal of Mechanical Sciences,  Vol. 50, No. 4, (2008), 732-742.
7.     Rao, D.N., Babu, M.R., Reddy, K.R.N. and Sunil, D., "Stress around square and rectangular cutouts in symmetric laminates", Composite Structures,  Vol. 92, No. 12, (2010), 2845-2859.
8.     Mohammadi, M., Dryden, J.R. and Jiang, L., "Stress concentration around a hole in a radially inhomogeneous plate", International Journal of Solids and Structures,  Vol. 48, No. 3-4, (2011), 483-491.
9.     Najimi, M., Aboutalebi, F.H. and Beheshti, H., "Stress intensity factor determination in functionally graded materials, considering continuously varying of material properties",  International Journal of Engineering, Transactions C: Aspects, Vol. 29, (2016), 1741-1746
10.   Louhghalam, A., Igusa, T., Park, C., Choi, S. and Kim, K., "Analysis of stress concentrations in plates with rectangular openings by a combined conformal mapping–finite element approach", International Journal of Solids and Structures,  Vol. 48, No. 13, (2011), 1991-2004.
11.   Jamal-Omidi, M. and Suki, M.M., "A numerical study on aluminum plate response under low velocity impact", International Journal of Engineering-Transactions C: Aspects,  Vol. 30, No. 3, (2017), 439-447.
12.   Sharma, D.S., "Stress distribution around polygonal holes", International Journal of Mechanical Sciences,  Vol. 65, No. 1, (2012), 115-124.
13.   Shahabian Moghadam, F. and Haji Kazemi, H., "A proposed method to evaluate ultimate resistance of plate girders subjected to shear and patch load", International Journal of Engineering,  Vol. 17, (2004), 347-356.
14.   Khechai, A., Tati, A. and Guettala, A., "Finite element analysis of stress concentrations and failure criteria in composite plates with circular holes", Frontiers of Mechanical Engineering,  Vol. 9, No. 3, (2014), 281-294.
15.   Grm, A. and Batista, M., "On the coupling of analytical and fem solution in stress analysis around the polygonal hole shape in a finite two-dimensional domain", International Journal of Mechanical Sciences,  Vol. 118, (2016), 254-267.
16.   Pan, Z., Cheng, Y. and Liu, J., "Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions", International Journal of Mechanical Sciences,  Vol. 75, (2013), 265-277.
17.   Jafari, M. and Ardalani, E., "Stress concentration in finite metallic plates with regular holes", International Journal of Mechanical Sciences,  Vol. 106, (2016), 220-230.
18.   Barber, J.R., "Elasticity, Springer,  Dordrecht, (2002), ISBN: 978-90-481-3809-8.