Deflection Analysis of Straight FGM Beam with Different Parameters

Abstract

This paper deals with the deflection analysis of Straight functionally Graded Beam (FGM) considering various parameters. It is analysed with different boundary conditions like simply supported, clamped-clamped. The FGM beam is analysed for different L/h ratio and different volume fraction index. is analysed with different L/h ratios using higher order shear deformation theory. It was found that for a fixed value of L/h ratio as the volume fraction index is increased the maximum deflection of FGM beam also increases. Again, if the value of volume fraction index is kept constant and the L/h ratio is increased the maximum deflection also decreases. The dependence of support conditions on maximum deflection is highlighted. Comparison and convergence study has been performed to validate the present formulation. Thus, we observe that for both clamped -clamped and simply supported condition, for a fixed value of of L/h ratio as the volume fraction index is increased the maximum deflection is increased. Again, if the value of volume fraction index is kept constant and the L/h ratio is increased the maximum deflection increases.

Introduction

Leaf springs are crucial automotive components, especially for heavy vehicles, as they absorb vibrations and provide a comfortable ride. Their unique advantage lies in guiding deflections along a definite path, serving both as structural members and energy absorbers.

Significant research efforts have focused on reducing leaf spring vibrations to enhance ride comfort. This study models the master leaf spring as a curved beam and investigates its vibration behavior under varying materials, length-to-radius (L/R) ratios, and boundary conditions.

Key literature contributions include:

• Mahmood et al. used ANSYS to optimize composite leaf spring weight.

• Abdul Rahim Abu Talib et al. optimized elliptic spring parameters for trucks.

• Vinkel Arora assessed fatigue life of 65Si7 leaf springs via analytical and experimental methods.

• Mesut Simsek studied vibrations of straight beams with third-order shear deformation theory.

• Other researchers have explored nonlinear vibrations, thermo-elastic effects, microstructural defects, and modal analysis of beams and curved structures using various analytical and numerical methods.

The paper also presents a detailed mathematical formulation:

• Displacement fields describe bending and shear deformation using higher-order terms.

• Strain-displacement relations incorporate beam curvature effects (curved vs. straight beams).

• Constitutive relations define stresses for isotropic and functionally graded material (FGM) beams.

• Finite element formulation discretizes the curved beam using two-node elements with four degrees of freedom per node for numerical vibration analysis.

This comprehensive approach aims to predict and improve the vibration characteristics of leaf springs under realistic conditions, contributing to the design of more comfortable and durable suspension systems.

Conclusion

The deflection analysis straight FGM was done under various conditions. MATLAB was used for the analysis. The effect of end condition on the vibration of straight FGM beam was studied for L/ h =5,20,50 and 100. The effect of change in volume fraction index and support condition is also studied. The volume fraction index was taken as 0.5,2,5 and 10. The result was also validated. During the analysis it was found that as the L/h ratio is increased the value of maximum deflection in both the clamped-clamped condition and simply supported condition increases. We also tried to observe the effect on maximum deflection on volume fraction index and it was found that it increases with increase in volume fraction index for both clamped and simply supported condition.

References

[1] Mahmood M. Shokrieh, Davood Rezaei, 2003, Analysis and optimization of composite leaf spring, Composite Structure, 60,317-325. [2] Abdul Rahim Abu Talib, Aidy Ali, G Goudah, Nur Azida Che Lah, A.F Golestaneh, 2010, Developing a composite based elliptic spring for automotive applications, Material and Design 31,475-484. [3] Vinkel Arora, Gian Bhushan,M.L. Aggarwal, 2014, Fatigue Life Assessment of 65Si7 Springs: A comparative Study, International Scholarly Research Notices, Volume 2014, Article ID 607272 [4] MESUT.S and KOCATURK. T, 2007, Free vibration analysis of beams by using a third-order shear deformation theory, Sadhana Vol. 32, Part 3, pp. 167–179. [5] Gupta. A, Talha. M , 2017,Large amplitude free flexural vibration analysis of finite element modeled FGM plates using new hyperbolic shear and normal deformation theory, Aerospace Science and Technology 67, 287–308. [6] Raveendranath. P, Singh. G, Pradhan. B., 2000, Free vibration of arches using a curved beam element based on a coupled polynomial displacement, Computers and Structures 78 , 583-590 . [7] Kawakami.M, Sakiyama .T, Matsuda. H, Morita.C, 1995, In-plane and out-of-plane free vibrations of curved beams with variable sections, Journal of Sound and Vibration 187(3), 381–401. [8] M. Ibrahim. S, P. Patel. B, Nath .Y, 2009, Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams, International Journal of Non-Linear Mechanics 44 , 1073 -1084. [9] Amir.M & Talha. M , 2018, Thermo-elastic Vibration Of Shear Deformable Functionally Graded Curved Beams with Microstructural Defects ,International Journal of Structural Stability and Dynamics. [10] Amir.M &Talha. M , 2018,Imperfection sensitivity in the vibration behavior of functionally graded arches by considering microstructural defects, J Mechanical Engineering Science , 1–15.. [11] Ghodge. V, P. Bhattu .A , B.Patil. S, January 2018, Vibration Analysis of Beams , International Journal of Engineering Trends and Technology (IJETT)2018 – Volume 55 Number 2, pp.81-86. [12] Fu. Y, Wang. J, Mao. Y, 2012, Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment, Applied Mathematical Modeling 36, 4324–4340. [13] Yuan. J and. M. Dickinson.S, 1992, on the use of artificial springs in the study of the free vibrations of systems comprised of straight and curved beams Journal of Sound and Vibration 153(2), 203-216. [14] Y. Yang. S, C. Sin. H, 1995, Curvature-based beam elements for the analysis of Timoshenko and shear-deformable curved beams, Journal of Sound and Vibration 187(4), 569–584. [15] Hajianmaleki. M, S. Qatu. M, 2013, Vibrations of straight and curved composite beams: A review, Composite Structures 100, 218–232. [16] Chidamparam. P and W. Leissa. A, 1993, Vibrations of planar curved beams, rings, and arches, Applied Mechanics Reviews vol 46(9), 467-483.

Copyright

Copyright © 2025 Md Rashid Akhtar, Aas Mohammad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.