Static deflection modeling of combined flexible beams using elliptic integral solution


Compliant mechanisms can be realized by serially connecting flexible segments and beams. Generally, they have the advantages of reduced backlash and wear as well as high precision. Unlike rigid-body mechanisms, compliant mechanisms require relatively fewer parts to perform a specific task. Unfortunately, compliant mechanisms exhibit high nonlinearity induced by the large static deflection of the flexible beams that complicate its motion’s description, leading to inaccurate results. Therefore, the development of reliable parametric models for accurately predicting the large static deflection is significant in their design and synthesis.

Numerous methods have been developed to model the static deflection behaviors of flexible beams comprising compliant mechanisms. The finite element model (FEM) based on commercial software is deemed the most accurate and appropriate for evaluating the static deflection behavior of beams under applied loads. However, it fails to accurately describe the force-displacement relationship making it unsuitable for some analytical approaches. Other methods have been proposed to overcome such limitations. They include the integral model, chained beam constraint model (CBCM) and pseudo-rigid-body model. Nevertheless, despite the promising efficiency and reliability of these models, they suffer from different deficiencies that limit their practical applications. In addition, most of the existing studies focus on single beams with continuous or constant curvature.

Recent findings revealed that combined flexible beams (CFBs) comprising two or more beams with multiples curvatures could diversity the component option for designing high-performance compliant mechanisms. However, this is yet to be approved due to insufficient information regarding the static deflection analysis and modeling of CFB, which is sparsely explored in the literature because of complex interactions. To address this problem, Dr. Ke Xu, Professor Haitao Liu and Professor Juliang Xiao from Tianjin University studied the static modeling of CFBs based on the elliptical integration solution. The main objective was to investigate the behavior of the large static deflection of the CFB systematically to provide a guideline for compliant mechanism design. The research work is currently published in the International Journal of Non-Linear Mechanics.

In their approach, the research team proposed modeling approach entailed either circular or straight-arc segments. The large static deflections of CFB subjected to combined loads were obtained through a systematic derivation of the bending moments. The feasibility and accuracy of the proposed model were verified through a numeric comparison with the finite element analysis. The deflections of the fabricated CFB were experimentally tested, and the results were discussed.

Results demonstrated the feasibility and practicability of the proposed model to predict the static deflection of the beam accurately. The approach could easily extend the elliptical integral solutions to more complicated cases. Moreover, the comparison between the numerical simulation results and the FEA’s experimental result also demonstrated the model’s practicability. This indicated the promising application of CFB in the design of reliable and robust compliant mechanisms. Furthermore, it was worth noting that the method can only accurately predict CFB’s static deflection at the moment.

In summary, a more reliable and effective method for modeling the static deflection behavior of CFB subjected to combined load was reported. The model segments were derived based on elliptical integrals by dividing the bema into various flexible segments. The numerical results were consistent with the experimental results, indicating the feasibility of the modeling approach. In a statement to Advances in Engineering, the authors said that the study insights would pave the way for advanced research on the dynamics of CFB to aid the advanced design of compliant mechanisms for different types of beams.

Static deflection modeling of combined flexible beams using elliptic integral solution - Advances in Engineering

About the author

Ke Xu received the M.S. degree in mechanical engineering from Changzhou University, Changzhou, China, in 2018. He is a Ph.D. candidate in mechanical engineering with the Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin, China.

His research interests include compliant mechanisms and robotics. (E-mail address: [email protected])

About the author

Haitao Liu received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Tianjin University, Tianjin, China, in 2004, 2007, and 2010, respectively. From 2013 to 2014, he was hosted at the Chair of Mechanics and Robotics at the University of Duisburg-Essen with a scholarship from the Alexander von Humboldt (AvH) Foundation of Germany. He is currently a Professor with the Department of Mechanical Engineering, Tianjin University, Tianjin, China.

He is the author of more than 100 journal articles and more than 30 inventions. His research interests include robotics and mechanisms, parallel mechanisms, Lie group and Lie algebra, and screw theory. (E-mail address: [email protected]; website:


Xu, K., Liu, H., & Xiao, J. (2021). Static deflection modeling of combined flexible beams using elliptic integral solution. International Journal of Non-Linear Mechanics, 129, 103637.

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