FORMULATION OF MATHEMATICAL MODEL FOR MAXIMUM SHEAR STRAIN (DISTORTION) ENERGY THEORY OF YIELD FOR PLANE CONTINUUM

E.I. Adah, T.J. Ukya

Abstract


 

The failure of engineering structures can have catastrophic consequences, leading to significant economic losses—both material and financial—and, in some cases, loss of human lives. Such failures often occur due to inadequate determination of structural loads or when materials exceed their yield strength, causing excessive deformation. This study revisits von Mises' theory and presents a newly formulated general applied stress mathematical model. Polynomial displacement shape profiles were employed to evaluate n-values for different plate types, leading to the derivation of n-value equations. Additionally, new general applied stress equations, Stress Factor (Fss), equations for shear strain energy theory for various boundary conditions, and allowable stress equations were developed. Validation of the newly formulated equations revealed that the applied stress values exceeded the yield stress of structural steel for plates with one free edge. To mitigate potential failure, a safety factor of 1.15 was introduced for the plate types considered which reduces the applied stress of 281N/mm2 to allowable stress of 244N/mm2 below the material yield stress of 250N/mm2. The results obtained align with findings from existing literature, confirming the reliability of the developed equations. As such, the new equations provide an effective means of predicting the allowable stress of plane materials based on the maximum shear strain energy theory.

Keywords: Structural Failure, Applied Stress, Shear Strain Energy Theory, Polynomial Displacement Profiles, Factor of Safety


Full Text:

PDF

References


Adah, E.I., Anyin, P. B., Oludire, O. O., and Ukya, T. J. (2025). Simplified Octahedral Shear Stress Theory for Plane, NAU Journal of Civil Engineering, 3(1), pp 23-29.

Bhat, S., Adarsha, H., Pattanaik, A., & Ravinarayan, V. (2018). On distortion energy theory in high cycle multi-axial fatigue. International Journal of Mechanical Engineering and Technology (IJMET), 9(7), 1240–1254.

Emuakpor O.S., George T., Cross C. and Shen M.H.H. (2010). Multi-axial fatigue life prediction via a strain energy method, AIAA Journal, 48 (1), 2010, 63-72.

Hosford, W. F. (2005). Mechanical Behavior of Materials, 2nd edition. Cambridge. 9- 11.

Ibearugbulem, O. M., Opara, H. E., Ibearugbulem, C. N., and Nwanchukwu, U. C. (2019). Closed form buckling analysis of thin rectangular plates, IOSR Journal of Mechanical and Civil Engineering, 16 (1), PP 83-90, DOI: 10.9790/1684-1601028390, www.iosrjournals.org.

Ibearugbulem, O. M, Ezeh, J. C. & Ettu, L. O. (2014). Energy Methods in Theory of Rectangular Plates: Use of Polynomial Shape Functions. Liu House of Excellence Ventures, Owerri.

Jin, L., Zhang, B., Chen, F., Yu, W., Lei, Y., Miao, L., and Du, X. (2022). Numerical investigations on the strain-rate-dependent mechanical behavior and size effect of RC shear walls. International Journal of Impact Engineering, 167.

Karmankar, R. G. (2017). Analysis of von-Mises stress for interference fit and pull- out states by using finite element method. International Research Journal of Engineering and Technology (IRJET), 1367-1374.

Kosaroglu, E. S., and Khalikov, F. (2009). Lecture Notes.

Liu, B., Guo, D., Jiang, C., Li, G., & Huang, X. (2019). Stress optimization of smooth continuum structures based on the distortion strain energy density. Computer Methods in Applied Mechanics and Engineering, 343, 276-296.

Meyers M. A. and Chawla K. K. (2009). Mechanical Behavior of Materials, Prentice- Hall. 71.

Okajima K., Tanaka T., and Mori H. (2001). Elasto-Plastic Finite Element Collapse Analysis of Retaining Wall by Excavation. Computational Mechanics–New Frontiers for the New Millennium1, 439-444.

Onaka, S. (2010). Equivalent strain in simple shear deformation described by using the Hencky strain. Philosophical Magazine Letters, 90(9), 633–639.

Pardis, N., Ebrahimi, R., & Kim, H. S. (2017). Equivalent strain at large shear deformation: Theoretical, numerical and finite element analysis. Journal of Applied Research and Technology, 15, 442–448.

Ross, C. T. F. (1987). Advanced Applied Stress Analysis. Ellis Horwood Limited, England.

Shrivastava, S., Ghosh, C., & Jonas, J. J. (2012). A comparison of the von Mises and Hencky equivalent strains for use in simple shear experiments. Philosophical Magazine, 92(7), 779–786.

Sutar, K. M. (2025). Lecture Notes on Machine Design II, Mechanical Engineering Department, VSSUT Burla, pp. 7.

von Mises, R. (1913). Mechanik der festen Körper im plastischen Zustand.

Zhang, J., Gao, N., and Starink, M. J. (2011). Microstructure development and hardening during high pressure torsion of commercially pure aluminium: Strain reversal experiments and a dislocation based model. Materials Science and Engineering: A, 528(6), 2581–2591.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2025 JOURNAL OF INVENTIVE ENGINEERING AND TECHNOLOGY (JIET)



Copyright  2020-2024. Journal of Inventive Engineering (JIET). All rights reserved. Nigerian Society of Engineers (NSE), Awka Branch.ISSN: 2705-3865

Powered by Myrasoft Systems Ltd.