Design studies of a square ring shaped force sensor - Integrated ...

INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL 

Volume 1, No 4, 2011 

© Copyright 2010 All rights reserved Integrated Publishing Association 

RESEARCH ARTICLE ISSN 09764259 

Design studies of a square ring shaped force sensor 

Madhuban Prasad 1 , Nabi Hasan 2 , Anil Kumar 3 , Harish Kumar 3 

1Department of Mechanical Engineering, Sundeerdeep College of Engineering and 

Technology, Ghaziabad, Uttar Pradesh, India 

2Department of Mechanical Engineering, Al Falah School of Engineering and Technology, 

Faridabad, Haryana, India 

3 National Physical Laboratory (Council of Scientific & Industrial Research), New Delhi. 

kradprasadm@gmail.com 

ABSTRACT 

The present study aims to study the square ring shaped force sensor and role of finite element 

analysis in designing the force sensor. Square ring shaped force sensor has been derived from 

the circular ring shaped force sensor, commonly used. A square ring shaped force sensors has 

been designed and its three dimensional model has been developed using software ABAQUS 

6.7.2. The force sensor has been studied using finite element analysis and the findings are 

presented in form of stress – strain and deflection plots. The study further attempts to suggest 

modifications in the analytical expressions available for finding stress – strain and deflection 

in ring shaped force sensor to use them in case of square ring shaped force sensors. 

Key words: Force sensor, finite element analysis 

Symbols 

b 

width of the cross section of ring (mm) 

t 

thickness of cross section of ring (mm) 

R i 

inner radius of ring (mm) 

R o 

outer radius of ring (mm) 

R 

mean radius (mm) 

F 

applied force (N) 

E young’s modulus of elasticity (N/m 2 ) 

U 

strain energy (J) 

M A moment due to force at A (Nm) 

M 

moment due to force at any position (Nm) 

q 

angle of segment of ring (radian) 

d 

deflection of the ring (mm) 

σ stress, N/m 2 

ε 

strain 

1. Introduction 

Force, being a vital parameter having applications in various functional areas has been 

measured by a variety of sensors of capacities ranging from few newtons to mega newtons. 

The force sensors may be ring shaped force sensors, strain gauged load cells, tuning fork type 

sensors depending upon their suitability for particular applications. Ring shaped force sensors, 

commonly used due to ease in manufacturing, have been always a centre of research for the 

force sensors for many decades. A number of researchers have worked in past and have 

suggested suitable modifications like octagonal, extended octagonal ring shaped force sensors 

727


Volume 1, No 4, 2011 



etc (Chen.B et.al,2007). Various expressions have been developed for stress / strain and axial 

deflection for the modified shapes suggested so far. But, other modifications have not been 

suggested and studied so far like hexagonal ring shape or square ring shape. Besides it, no 

past observations are found about the studies related to square ring shaped force sensors. 

Hence, the present paper attempts to study a square ring shaped force sensor as a 

modifications of the ring shaped force sensors analytically and suggests suitable 

modifications in the analytically derived expressions for stress / strain and deflection under 

action of external forces subjected to a ring shaped force sensor with the help of finite 

element analysis. The software used for finite element analysis is ABAQUS 7.2 standard 

student edition. 

2. Analytical Study 

Ring shaped force sensors have been studied by various researchers and the analytically 

derived expressions for stress / strain and deflection of the force sensor are well know, though 

few deviations has been observed by experimental studies. The ring shaped force sensors and 

the expressions used for stress / strain and deflection have been discussed else where 

(Harishkumar Et.al, 2011). The ring shaped force sensor has been analyzed under action of 

axial forces (Figure 1) and has been modeled as a ring. Taking the advantage of symmetry, a 

quarter may be selected for the analytical study. The free body diagram suggests that the 

moment M is statistically indeterminate, the strain energy U for this quadrant of the ring is 

due only to the bending moment M in the ring. According the castigliano’s theorem 

(Kumar,2010) 

F 

R i 

R 0 

F/2 

θ 

F 

Figure 1: Ring Shaped Force Transducer – Idealizing as a Ring 

U 

M 

= 0 

(1) 

728


Volume 1, No 4, 2011 



( M ds ) 

U = ò (2) 

2 EI 

Where, θ defines any section of the beam and strain energy. 

Hence, 

2 

( M Rd q ) 

U = ò (3) 

2 EI 

And, the radial deflection at B is calculated from the Castigliano’s theorem as, 

2 

U 

d i = ,.....( i = 1, 2,... n ) 

(4) 

F 

For any section, the bending moment may be find as, 

FR (1 - cos q ) 

M = M x - (5) 

2 

Mx 

1 1 

= FR ( - ) 

(6) 

2 p 

FR 2 

M x = ( )(1 - ) 

(7) 

2 p 

3 

æ FR ö æ p 2 ö 

d = ç ÷ ç – ÷ 

(8) 

è EI ø è 4 p ø 

Net deflection for a ring cross section is as follow, 

3 

0.149PR 

d = (9) 

EI 

6M 

Stress, s = (10) 

2 

bt 

6M 

Strain, e = (11) 

2 

Ebt 

For, a 20 kN force sensor made of material EN 24 and modulus of elasticity 210 GPa, having 

inner radii 86 mm, outer radii 96 mm and width 45mm, the following particulars are found 

using above expressions. 

Deflection = 2.85 mm 

Stress = 441 MPa 

Strain = 2.1 x 10 3 

The force sensor has been modified as a square ring shaped force sensor taking the 

dimensions given above and studied using finite element analysis a tool. 

729


Volume 1, No 4, 2011 



3. Finite Element Model and Analysis 

Similar finite element analysis has been done for ring shaped force sensor earlier by various 

researchers. The software ABAQUS standard student edition 6.7.2 has been used for 

modeling and analysis of the force sensors developed. A three dimensional quarter of the 

idealized square ring has been designed and suitable boundary conditions have been defined 

for finite element analysis. A three dimensional solid continuum 8 node element with reduced 

integration is considered and the analysis is of linear type. The material is of isotropic nature. 

The axial force is applied in compression mode as an indication of methodology using 

ABAQUS 6.7.2 software for studying stress – strain and axial deflection pattern. Suitable 

procedure for finite element analysis of the quarter of the ring has been adopted and the stress, 

strain and axial deflection patterns have been plotted. The findings of the finite element 

analysis may be summarized in form of stress, strain and axial deflection patterns (Fig 24). 

4. Results & Discussions 

Figure 2 expresses the stress distribution of the quarter of ring when it is under axial force in 

tension mode. Maximum stress occurs at the upper top point where the force is applied. From 

the point of application of axial force, along the periphery, the stress tends to decrease. 

Figure 2: Stress Distribution of a Quarter Square Ring 

The maximum value of stress is found to be 118.7 MPa in comparison to 449.1 MPa for ring 

shaped force sensor of similar dimensions and capacity (maximum values of stresses have 

730


Volume 1, No 4, 2011 



been rejected due to Saint Venant’s theory). Figure 3 discusses the strain distribution for 

quarter of ring under action of axial force. The strain distribution follows the similar pattern 

to the stress distribution. The strain is 0.51 x 10 3 in comparison to 2.02 x 10 3 for ring 

shaped force sensor of similar dimensions and capacity (maximum values of strains have 

been rejected due to Saint Venant’s theory). 

Figure 3: Strain Distribution of a Quarter Square Ring 

Figure 4 explains the axial deflection of the quarter of the ring when it is an external force is 

applied. The axial deflection is highest at the free end, but it tends to decrease along the 

periphery of the quarter and is almost negligible at the other end of the quarter. One end of 

the quarter of the ring is considered fixed and no rotation has been permitted, hence, the 

axial deflection has been almost negligible there. It is 0.927 mm in comparison to 1.749 mm 

for ring shaped force sensor of similar dimensions and capacity (maximum values of 

deflections have been rejected due to Saint Venant’s theory). 

731


Volume 1, No 4, 2011 



Figure 4: Deflection Pattern of a Quarter of Square Ring 

It has been found that the stress and strain are about 1/4 th of the stress – strain for the ring 

shaped force sensors of similar capacities. Beside it, the deflection is about half of the 

deflection for similar ring shaped force sensor [9]. 

The analytical expressions may be suggested as follow for the square ring shaped force 

sensor. 

3 

0.075PR 

d = 

EI 

(12) 

1.5M 

s = 

2 

bt 

(13) 

1.5M 

e = 

2 

Ebt 

(14) 

5. Conclusions 

The finite element analysis of the square ring shaped force sensor reveals information about 

stress / strain distribution and axial deflection pattern of force sensor under action of axial 

forces. The finite element analysis has been used for obtaining vital informations related to 

stress – strain and has been used here to modify the analytical expressions available for stress 

– strain and deflection for a square ring shaped force sensor. Further, the stress / strain 

732


Volume 1, No 4, 2011 



distribution may be useful for developing strain gauged sensor by locating suitably the strain 

gauges that may be implanted to the sensor. The use of finite element analysis further may be 

extended to study the effect of nominal quantities over the force sensor. 

6. References 

1. B Chen, X Wu, X Peng, 2007, Finite Element Analysis of Ring Strain Sensor, Sensor 

Actuat A Phys, 139, pp 6669. 

2. Harish Kumar, Chitra Sharma, Anil Kumar, 2011, Design studies of Ring Shaped 

Force Transducer, International Journal of Engineering Science and Technology, 

3(2), pp 15361541. 

3. Harish Kumar, S K Jain, Evaluation of Axial Sensitivity Evaluation of Circular 

Shaped Force Proving Ring using Finite Element Method (FEM), 7 th International 

Conference on Advances in Metrology (Admet 2009), National Physical Laboratory, 

New Delhi, pp 118119. 

4. Kumar H, Sharma C, Gupta S, Deflection Studies of a Force Transducer, 10 th 

International Conference on Research & Development in Mechanical Industry 

(RaDMI 2010), Donji Milanovac, Serbia, pp 300303. 

5. Josue Njock Libii, 2006, Design, Design Analysis and Testing of a Force Sensor for 

Use in Teaching and Research, World Transaction on Engineering and Technology 

Education, 5(1) , pp 175178. 

6. M A Rehman, S Rehman, 2007, A Computer Program for Designing Circular Proving 

Rings of Uniform Strength, Journal of the Institution of Engineers (India) Mechanical 

Engineering Division, 88, pp 37. 

7. M J O’Dogherty, 1996, The Design of Octagonal Ring Dynamometers, J. agric. 

Engng Res., 63, pp 918. 

8. Sedat Karabay, 2007, Analysis of Drill Dynamometer with Octagonal Ring Type 

Transducers for Monitoring of Cutting Forces in Drilling and Allied Process, Mater 

Des, 28, pp. 673685. 

9. Sudhir Kumar, Wakkar Ali, Anil Kumar, Harish Kumar, Deflection Studies of a Ring 

Shaped Force Transducer: A Review, 2011, International Journal of Engineering 

Science and Technology, 3(2), pp 13301333. 

733

Design studies of a square ring shaped force sensor - Integrated ...

Create successful ePaper yourself

Delete template?

Save as template?