Analysis of elliptical cross-section helical spring with small helix angle under static load


A number of researchers have devoted their research efforts to investigate helical springs under static loads. However, most of these studies have been limited to circular cross-section springs where only a few of them have focused on elliptical cross-section springs.

Studying elliptical cross-section springs has been done previously, but with no elaborate analytical support. However, there has been found that for the same free height, elliptic cross-section helical springs can enable larger stroke than the circular ones. Therefore, it is possible to reduce the mounting height. In addition, for elliptical cross-section helical spring, shear stress distribution can be made uniform along the section by choosing a particular aspect ratio. Moreover, maximum shear stress can be reduced across the wire compared to equivalent circular cross-section helical spring. This would allow for supporting larger loads for the same spring dimensions.

These advantages are particularly important in view of the recent advances in materials and manufacturing methods promoting downsizing of machines. Finite Element Analysis has been adopted in some spring applications including leaf springs and helical springs with round wire. This has been in a move to establish the spring rate and stress distribution in the cross-section. It has also been founded in quest to validate analytical approaches. Unfortunately, no published application of Finite Element Analysis was available at the time this study was conducted.

Majdi Gzal (currently a PhD student) and Professor Oleg Gendelman at Technion, Israel Institute of Technology in collaboration with Professor Morel Groper at University of Haifa in Israel developed an analytical expression for the stresses in an elliptical cross-section helical springs while considering the helix curvature effect as well as the aspect ratio of the elliptical wire cross-section. The proposed expression eased the design process of these useful springs. Their research work is published in International Journal of Mechanical Sciences.

The authors, with an objective of analyzing the shear stresses in an elliptical cross-section helical spring with a small helix angle, they developed an analytical expression based on the theory of elasticity. They validated the accuracy of this expression by comparing their results with existing numerical studies and consistent with circular cross-section springs. In addition, they also conducted finite element as well as experimental analyses in a bid to analyze the spring in terms of shear stresses and spring rate.

The research team observed that the proposed expression of stress distribution based on small helix angles was accurate in view of the observed finite element analysis and experimental results obtained from a real automotive valve spring with a 6° helix angle. The results can then be adopted for the design of elliptical cross-section helical springs with small helix angle as well as circular cross-section helical springs as a unique case.

The outcomes of their study provided an explicit straightforward expression for the actual location of the maximum shear stress as a function of the aspect ratio as well as spring index. The results also enabled the obtaining of maximum shear stress analytically.

Above the analytical expression, the designer can accurately obtain spring’s rate as well as shear stresses by conducting a real experiment implementing strain gauges and applying finite element analysis.

analysis of elliptical cross-section helical spring with small helix angle under static load. Advances in Engineering

About the author

Majdi Gzal is currently a Ph.D. Mechanical Engineering student under the supervision of Prof. Oleg Gendelman at the Technion, Israel Institute of Technology (IIT). Recently he received the 2017/18 Neubauer Doctoral Fellowship Fund for Excellent Minority Students.

Majdi finished his undergraduate studies in Mechanical Engineering (2015) at the Technion in the framework of the NAM excellence program (an acronym in Hebrew for Outstanding Arab Youth). In March 2015, he started his MSc studies in Mechanical Engineering at the Technion majoring in design and testing of nonlinear valve springs. During the master period, he received the Scholarship Fund for Excellent Arab Israeli Masters from the Council for Higher Education in Israel. In January 2017, Majdi was accepted to the direct Ph.D. track.

His research is designed to promote both academic and industry applications by conducting a comprehensive study including analytical, finite element and experimental investigation with potential to capture the non-linear behavior of the progressive valve springs and improve the response of those useful springs.

About the author

Prof. Morel Groper served between 1987 to 2011 in the Israel Navy as a design engineer, Navy Shipyard Chief Marine Engineer and finally as the Navy Head of Naval Architecture and Marine Engineering. Through his career in the Israel Navy he was directly involved in many of the naval architecture and marine engineering research and development efforts, including the development of advanced naval platforms and unique marine systems. Prof.  Groper received his Ph.D. degree in 1999 from the Faculty of Mechanical Engineering at the Technion, Israel Institute of Technology in Haifa, Israel.

His research under the supervision of Professor Izhak Etsion focused in the cavitation phenomena in hydrodynamic bearings. In 2010 Prof. Groper retired from the Navy and launched his own R&D company to provide comprehensive mechanical engineering and naval architecture services to naval, marine, offshore and industrial sectors.

In 2014 Prof. Groper joined the University of Haifa to promote his own research, as well as assist with establishing and developing the new Hatter Department of Marine Technologies

Currently Prof. Groper serves as the Head of this department at the Charney School of Marine Sciences at the University of Haifa. His research focuses on maneuvering and propulsion of Underwater Marine Vehicles and on the engineering design of components for underwater applications.

About the author

Prof. Oleg Gendelman

Faculty of Mechanical Engineering – Technion-Israel Institute of Technology

Email: [email protected]

Oleg Gendelman is a professor in the Department of Mechanical Engineering at the Technion, Israel Institute of Technology (IIT).

 He received his MSc degree from the Department of Molecular and Chemical Physics at Moscow Institute of Physics and Technology in 1992. Then he continued his education with the Institute of Chemical Physics in Moscow, where he obtained his PhD degree in 1995 and his Habilitation degree in 2000.

 His main scientific interests are: nonlinear dynamics, nonlinear oscillations in discrete and continuous systems, nonlinear normal modes, energy transfer, vibration protection and mitigation, dynamics and transport phenomena in low-dimensional systems, and applications to polymer systems and granular materials.  He has published about 240 papers and 2 monographs.

Prof. Gendelman currently holds the position of Vice Dean for Graduate Studies in the Department of Mechanical Engineering at the Technion.


Majdi Gzal, Morel Groper, Oleg Gendelman. Analytical, experimental and finite element analysis of elliptical cross-section helical spring with small helix angle under static load. International Journal of Mechanical Sciences, volume 130 (2017), pages 476–486.

Go To International Journal of Mechanical Sciences 

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