DESIGNING A FLEXIBLE BELLOWS COUPLING MADE FROM COMPOSITE

MATERIALS USING NUMERICAL SIMULATIONS

SVOČ – FST 2018

Ing. Frantisek Sedlacek,

University of West Bohemia,

Univerzitni 8, 306 14, Pilsen,

Czech Republic

ABSTRACT

This paper deals with the design of a composite bellows coupling. The mechanical properties of the composite material

were determined using experimental tests according to ASTM standards. A geometric optimization was done in

connection with structural analysis to find the optimal design and layout of the laminate of the composite flexible bellows

coupling. An advanced finite element model was used to verify the coupling using advanced intra-laminar and inter-

laminar failure criterions. The parameters for these numerical simulations were obtained using a series of experimental

tests according to ASTM standards.

KEYWORDS

Flexible coupling, composite materials, numerical simulations, FEM, structural optimization, cohesive elements

INTRODUCTION

The main function of the coupling is to transfer torque from a master mover (e.g. electric or combustion motor, steam

turbine) to a functional machine (e.g. gear, compressor, pump, generator) [1]. In most practical applications, the perfect

alignment of couplings to machines, and/or shafts is impossible [2]. The conventional way of connecting is by using rigid

couplings. However, if there are some misalignments, a rigid coupling can generate high reaction forces. These reaction

forces may cause failure of bearings, induce noisy operation, increase vibrations or even cause breakage. Flexible

couplings exist for these reasons. Elastomer flexible couplings are most widely used if there is a requirement for multiaxial

compliance or transfer of loads. But their construction has many disadvantages and problems, such as very high weight,

low durability (service life), high maintenance requirements, they can be used only at low speeds, rubber degradation,

their stiffness is dependent on temperature, complicated design and they are expensive.

The goal of this paper is to create a highly flexible coupling from composite materials which can transmit the

required torque with the possibility of specific angular and axial deformation and all this with the desired stiffness. It must

also have: low weight (ability to work at high rotational speeds), high eigenvalues, high durability, low maintenance

requirements and a simple design. The basic function schema is shown in Fig. 1.

Fig. 1 Schema of loadcases of flexible bellows coupling.

However, composite material was not only used for reasons such as high strength and low weight (compared to

conventional spring steel up to 80%), low friction, high eigenvalues, acid and weather (corrosion) resistance, high fatigue

strength, and many other positive properties of composite materials. The main factor for its selection is its very high

elastic deformation energy per unit of weight U (kJ/kg). This parameter is the main factor when selecting the material for

a spring, and it can be generally expressed as:

𝑈 = 1

2

𝜎𝑓2

𝜌𝐸 [

𝑘𝐽

𝑘𝑔]. (1)

It means that the material with a lower modulus of elasticity E or density 𝜌 has a relatively higher elastic deformation

energy per unit of mass (EDEPM) within the same maximum allowed stress of the material 𝜎f. Fig. 2 gives the EDEPM

for common engineering materials. Carbon Fibre Reinforced Polymers (CFRP) have more than seven times higher

EDEPM than standard spring steel. Only elastomers are better than CFRP, but they have a significant loss factor and

many other issues mentioned above.

Fig. 2 Schema and main values of elastic deformation energy per unit of weight of engineering materials [3].

The design methodology for the composite flexible bellows coupling (CFBC) was created based on specified goals and

general requirements - see Fig. 3. It comprises four main stages; the first deals with creation of a generic CAD model and

basic FEM model of the coupling. The second is intended to find the required mechanical properties of the coupling (the

required stiffness in the main directions/angles). The third stage verifies the design proposal using advanced numerical

simulation and experimental tests. The last stage deals with the final design and validation of the coupling.

Fig. 3 Flowchart of design of composite flexible bellows coupling.

BASIC DESIGN OF THE COUPLING

First, it was necessary to choose the main shape of the coupling. This was very limited due to the restricted installation

dimensions (especially the maximum width b = 80mm). Furthermore, the manufacturing technology and mounting

options were taken into account. Several variations of the basic geometry of the CFBC were created. The most suitable

design variant was the W-shaped cross-section, see Fig. 1.

A generic 2D CAD model of the CFBC was created based on the main shape and FEM was applied to this model.

The structural analysis was done in Siemens Simcenter NX 12 with non-linear multiphysics solver NX Nastran 12 – SOL

401 (based on first-order shear deformation theory). A 2D mapped mesh with parabolic quad elements (CQUAD8) was

used (see Fig. 4). The orthotropic physical properties of the layup of the composite were entered using the special NX

Laminate Composites module. The solution was solved with six load subcases: the load from max. torque moment (M0 =

12 kNm), centrifugal force (nmax = 4000 rev.min-1), max. required positive axial deformation (Δxa = 5.1 mm) and angular

deformation (Δα0 = 1.5 deg); and the load from max. torque moment, centrifugal force, maximum required negative axial

deformation (Δxa = -5.1 mm) and angular deformation (Δα0 = -1.5 deg), see Fig. 1.

Material Elastic deformation energy per unit

weight U (kJ/kg)

Rubber 18 – 45

CFRP (Carbon Fibre Reinforced Polymer) 3.9 – 6.5

Ti alloys 0.9 – 2.6

Nylon 1.3 – 2.1

GFRP (Glass Fibre Reinforced Polymer) 1.0 – 1.8

Spring steel 0.4 – 0.9

Wood 0.3 – 0.7

Fig. 4 2D FEM model of CFBC.

DETERMINATION OF MECHANICAL PROPERTIES OF COMPOSITE

Before the numerical simulation itself, the correct mechanical and strength parameters of the laminate were determined.

High strength 200gsm 2x2 twill carbon fabric prepreg gg200t (with 3k T700 toray fibres) was chosen for the coupling.

The carbon fibre sheet was fabricated in laboratories at the University of West Bohemia. It was cured in an autoclave

with an 8 hour cycle at 110°C. Eight different series of laminate specimens were made according to ASTM standards (a

total of 118 specimens).

Tensile testing the composite

Steel fittings were glued to the carbon sheets because of the high strength of the specimens (to avoid slipping out of the

jaws). The tensile properties of the laminate were determined according to ASTM standard D3039 to find the tensile

modulus (E1, E2, E3) and stress limits (XT, YT, ZT) of the laminate [4]. Experimental tensile tests were carried out by quasi-

static load (2 mm/min) on the Zwick/Roell Z050 machine. Subsequently, failure modes of the specimens were evaluated

(according to mode code, see Fig. 5) and the individual mechanical parameters were calculated using a sub-routine in

Python. The final values are given in Tab. 1.

Fig. 5 Experimental tensile measurement of the composite.

Compressive test of the composite

The compressive properties of the laminate were determined using experimental testing according to ASTM standard

D3410 to find stress limits (XC, YC, ZC) of the prepreg in compression for all the main directions [5]. Experimental

compressive tests were carried out by quasi-static load (1.5 mm/min) on a Zwick/Roell Z050 machine. Subsequently,

failure modes of the specimens were evaluated (according to mode code, Chyba! Nenalezen zdroj odkazů.) and

parameters were calculated with a subroutine in Python 2.71. The final values are given in Tab. 1.

Fig. 6 Experimental compressive measurement of the composite.

In-plane shear test of the composite

A standard test method for in-plane shear response of polymer matrix composite materials by tensile testing at ±45° was

done according to ASTM standard D3518 [6]. Experimental tests were carried out using quasi-static load (2 mm/min) on

the Zwick/Roell Z050 machine. Final parameters of the in-plane shear modulus (G12, G23, G31) and shear stress limits (S12,

S23, S31) are given in Tab. 1.

Fig. 7 Experimental in-plane shear measurement of the composite.

Mechanical properties Strength properties

ρ (kg/m3) 1820 Laminate density XT (MPa) 693 Tensile strength (0°)

t (mm) 0.208 Ply thickness YT (MPa) 610 Tensile strength (90°)

E1 (GPa) 55.8 Young’s Modulus 0° ZT (MPa) 67 Tensile strength (N90°)

E2 (GPa) 53.7 Young’s Modulus 90° XC (MPa) 552 Compression strength 0°

E3 (GPa) 6.4 Young’s Modulus N90° YC (MPa) 558 Compression strength 90°

G12 (GPa) 3.12 In-plane Shear Modulus 12 ZC (MPa) 268 Compression strength N90°

G23 (GPa) 2.89 In-plane Shear Modulus 23 S12 (MPa) 109.1 In-Plane Shear Strength in 12

G31 (GPa) 2.89 In-plane Shear Modulus 31 S23 (MPa) 121 In-Plane Shear Strength in 23

ν12 (-) 0.28 Poisson´s ratio in plane 12 S31 (MPa) 121 In-Plane Shear Strength in 31

kI (GPa/m) 10e3 Interface stiffness for Mode-I GIC (J/m2) 695.7 Fracture toughness for Mode-I

kII (GPa/m) 14e3 Interface stiffness for Mode-II GIIC (J/m2) 1408 Fracture toughness for Mode-II

Tab. 1 Mechanical and strength properties of CFRP (gg200t prepreg).

FINDING THE MOST SUITABLE GEOMETRY AND COMPOSITE LAYOUT OF THE CFBC

The geometric optimization was combined with the structural analysis to find the best shape and layup of the CFBC that

meets all the required mechanical properties of the coupling (stiffness in main directions/angles). The optimization was

done using NX Optimizer 12.0.1. The optimization algorithm implemented in NX Optimizer belongs to a class of methods

called gradient methods [7], [8]. Finding the minimum weight of the coupling was chosen as the objective function of the

optimization. The cross-section of the 3D model was divided into five main zones. The physical properties for individual

plies (thickness of the plies and main angle of the fibres) were applied on these sections (Fig. 9). These forty-two

parameters with four geometric parameters of the coupling were set as design variables of the geometric optimization.

Fig. 8 Change of objective function across iterations of geometry optimization (left) and change of the geometry of the coupling

across iterations of the geometric optimization of the coupling (right).

The maximum allowed stresses in all main normal (XT, YT, Z T, XC, YC, ZC) and shear (S12, S23, S31) directions

were used as constrains of the optimization. The maximum rotation in the direction of the axis of the coupling (minimum

desired torsional stiffness) was set as the last constrain of the geometry optimization.

The progress of the objective function across iterations of the geometric optimization and change of the geometry

of the CFBC are given in Fig. 8. A design weighing 2.53 kg was found with axial stiffness 410 kN/m, torsional stiffness

11 kN.m/deg and twisting stiffness 34.5 kN.m/deg. The final layup of the composite is given in Fig. 9.

15

25

35

45

55

65

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Ob

ject

ive

fun

ctio

n [

N]

Iteration of geometry otimization [-]

Fig. 9 Final values of layup of the composite (from geometric optimization).

VERIFICATION OF FINAL DESIGN USING ADVANCED NUMERICAL SIMULATION

An advanced numerical model was created to verify the mechanical properties of the coupling and to determine the

strength of the final design of the coupling from geometric optimization. An advanced FEM model was created from a

basic 2D model using the NX Laminate Composites module. The 2D FEM model of the CFBC was filled into a 3D mesh

with respect to the manufacturing technology for all plies, such as; cuts of the individual plies, distortions of the main

directions of the fibres of the plies, resin drops and pockets and overlaps of the individual plies. A total of 168 plies were

manually set using the special draping function based on the Fish-net algorithm, see Fig. 10.

Fig. 10 Advanced FEM model of the coupling with details of the layup.

Intra-laminar (strength properties in plies) and inter-laminar strength (strength properties between plies) was evaluated

using a 3D FEM model, see Fig. 11. The inter-laminar strength of the laminate was determined because the coupling is

highly flexible and works with large displacements.

The numerical simulation was done using NX Nastran 12 – SOL401 Multi-step Non-linear Multiphysics solver

(100 steps). The flanges of the shafts and bolted connections were included with a full contact condition in the simulation.

Fig. 11 Schema of Intra- and Inter-laminar failure of fibre composite.

DETERMINATION OF THE INTRA-LAMINAR STRENGTH

The interactive Tsai-Wu failure criterion was used to determine the intra-laminar strength of the laminate [9]. The Tsai-

Wu failure criterion can be written as:

(1

𝑋𝑇−

1

𝑋𝐶) 𝜎1 + (

1

𝑌𝑇−

1

𝑌𝐶) 𝜎2 +

𝜎12

𝑋𝑇𝑋𝐶+

𝜎22

𝑌𝑇𝑌𝐶+

𝜎122

(𝑆𝐿)2+ 2𝐹12

∗𝜎1𝜎2

𝑋𝑇𝑋𝐶= 1,

(2)

Where 𝐹12∗ is the coefficient of interaction and it can be wrriten as:

𝐹12∗ =

1

2𝜎2{1 − [𝑋𝐶 − 𝑋𝑇 +

𝑋𝑇𝑋𝐶

𝑌𝑇𝑌𝐶(𝑌𝐶 − 𝑌𝑇)] 𝜎 + (1 +

𝑋𝑇𝑋𝐶

𝑌𝑇𝑌𝐶) 𝜎2 }. (3)

The individual ultimate stress parameters were determined using the experimental tests listed earlier and they are given

in Tab. 1.

DETERMINATION OF INTER-LAMINAR STRENGTH

For highly flexible composite parts, it is recommended to determine the inter-laminar strength of the laminate too. A

cohesive elements in the form of 2D layers between all the plies of the laminate were created (a total of 92 layers). Special

cohesive parabolic elements were used with a cohesive damage interface approach according to Cachan, Allix and

Ladevèze. It was necessary to find the fracture toughness (GIC, GIIC) and stiffness (kI, kII) of the interface for the first two

modes of the cohesive failure (Mode-I is normal strength and Mode-II is the shear strength of the inter-laminar interface).

Experimental tests for both modes of inter-laminar failure were carried out.

Determination of Mode-I of the inter-laminar strength

Special specimens for the double cantilever beam test (DCB test) were created from material gg200t according to ASTM

standards D5528-01 [11]. Experimental tensile DCB tests were carried out using a quasi-static load (2 mm/min) on the

Zwick/Roell Z050 machine, see Fig. 12.

Fig. 12 Experimental double cantilever beam test for finding parameters of Mode-I.

The parameters of the fracture toughness (GIC) and stiffness (kI) for Mode-I were found using a special Python 2.73

subroutine in combination with a FEM model of the DCB in Abaqus 6.13 that allows fitting of the parameters. The fitting

process was done with an accuracy of 4%, see Fig. 13 (red line in the chart). The final values are given in Tab. 1.

Fig. 13 Fitting of fracture toughness and stiffness of interface for mode-I

Determination of Mode-II of the inter-laminar strength

Special specimens for end-notched flexure (ENF) were fabricated from material gg200t according to ASTM standards

D7905 [12]. Experimental three point bending ENF tests were carried out using a quasi-static load (0.5 mm/min) on the

Zwick/Roell Z050 machine, see Fig. 14.

Fig. 14 Experimental end-notched flexure test to find parameters of Mode-II.

The parameters of the fracture toughness (GIIC) and stiffness (kII) for Mode-II were found in the same way as the DCB

test. The fitting process was done with an accuracy of 3%, see Fig. 15 (red line in the chart). The final values are given in

Tab. 1.

Fig. 15 Fitting of fracture toughness and stiffness of interface for mode-II.

RESULTS OF NUMERICAL SIMULATION

Fig. 16 shows the results of the displacement and the most critical normal and shear stress for the most critical loadcase

(combination of: the load from max. torque moment, centrifugal force, maximum required positive axial deformation and

angular deformation).

Displacement (mm) Normal stress (MPa) – direction 11 and 22 Shear stress (MPa) – direction 12 and 23

Fig. 16 Results of the displacement and critical normal and shear stress for most critical loadcase.

Fig. 17 shows the results of the intra- and inter-laminar strength of the CFBC for the same most critical loadcase. The

results indicate that the design is satisfactory for both intra- and inter-laminar strength.

Failure index of intra-laminar strength (Tsai-Wu) Failure index of inter-laminar strength (cohesive)

Fig. 17 Results of the intra- and inter-laminar strength of CFBC.

CONCLUSION

The optimal design of the CFBC was created using the proposed methodology (Fig. 3). The main shape and composite

layup was found using geometric optimization and verified using advanced FEM numerical simulation. The intra- and

inter-laminar strengths were determined based on the parameters from eight different experimental tests carried out on

118 specimens. The weight of the coupling (2.51 kg) was found to be in compliance with all the mechanical requirements-

it was more than 70% lighter than a conventional steel/elastomer flexible coupling.

Currently, I am working on the validation of the CFBC. A function sample of the CFBC has already been

laminated. A divided positive mould was used for fabrication. The core of the mould was created using additive

manufacturing technology (FDM 3D printer) from a heat-stable copolymer PA6 with short carbon fibres. The

manufacturing process is shown in Fig. 18. In the future, the validation of the CFBC will be carried out using experimental

testing (modal and multiaxial durability tests).

Fig. 18 Laminating of function sample of CFBC.

REFERENCES

[1] C. M. Johnson, ‘An introduction to flexible couplings’, World Pumps, vol. 1996, no. Volume 1996, pp. 38–43.

[2] Anon, ‘Coupling & U-joints’, Motion System Design, no. Sv. no. 12, pp. 1187–1192.

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[5] ASTM D3410 / D3410M-16, ‘Standard Test Method for Compressive Properties of Polymer Matrix Composite

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[8] P. Goncharov, Engineering analysis with NX advanced simulation, vol. 2014. Raleigh: Lulu Press.

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[10] ‘Calculation of damage-dependent directional failure indices from the Tsai - Wu failure criterion’, ResearchGate.

[Online]. Available: https://www.researchgate.net/publication/234025973

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[12] A. Standard, ‘D7905/D7905M–14’, Standard Test Method for Determination of the Mode II Interlaminar Fracture

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