The spherical flexible joint is extensively used in engineering. It is designed to provide flexibility in rotation while bearing vertical compression load. The linear rotational stiffness of the flexible joint is formulated. The rotational stiffness of the bonded rubber layer is related to inner radius, thickness and two edge angles. FEM is used to verify the analytical solution and analyze the stiffness. The Mooney–Rivlin, Neo Hooke and Yeoh constitutive models are used in the simulation. The experiment is taken to obtain the material coefficient and validate the analytical and FEM results. The Yeoh model can reflect the deformation trend more accurately, but the error in the nearly linear district is bigger than the Mooney–Rivlin model. The Mooney–Rivlin model can fit the test result very well and the analytical solution can also be used when the rubber deformation in the flexible joint is small. The increase of Poisson's ratio of the rubber layers will enhance the vertical compression stiffness but barely have effect on the rotational stiffness.

The spherical flexible joints are widely used as non-rigid connection
in aerospace and offshore oil and gas industry. The flexible joint is
the critical part of the flexible bearing nozzle in the solid rocket
boosters (SRB). It allows nozzle to deflect in given directions for
booster thrust vector control (Kumar et al., 2015). Elastomeric
bearings are flexible joints used in helicopter rotor hubs. These
bearings are spherical hinges to withstand centrifugal and lateral
forces while tolerating rotational, flap and lead-lag motions (Donguy,
2015). In the offshore oil and gas industry, the flex joints, also
known as FlexJoints^{®}, are used to connect TLP
(tension leg platform) to the subsea foundation (Kumar,
2000). A typical flexible joint consists of alternately laminated
spherical rubber and metal layers. For different application, the
configuration of the metal layers may be slightly different. The metal
layers of rocket solid booster flexible bearings may protrude outside
the rubber layers to resist to higher temperature (Lampani et al.,
2012). On the contrary, to protect the metal from seawater corrosion,
the metal layers of FlexJoints^{®} are wrapped in the
rubber layer (Gunderson et al., 1992).

Due to near incompressibility of rubber, the bearing load capacity and
stiffness will increase dramatically when the lateral edge motion of
the rubber is constrained. Meanwhile the relatively low shear modulus
may facilitate the rotation and release the bending moment produced
stress at the connection. Therefore, the stiffness is one of the most
important factors in the flexible joint design. Typically, the
vertical compression stiffness is designed sufficiently high to bear
compression loading and the rotational stiffness needs to be
relatively low. The effect factors of both stiffnesses are necessary
to be analysed. Moreover the stiffness also plays a part in failure
analysis (Stevenson and Harris, 1992). Thus an analytical solution of
the stiffness is necessary. The effective compression modulus is
defined to formulate the vertical compression stiffness (Tsai,
2012). Gent and Lindley (1959) proposed an approximation to
calculate the effective compression modulus of the bonded rubber. That
method was called “mean pressure” method. The solution are obtained
by superposing two stages. In the first stage, the elastomer is
compressed between unbonded rigid plates; in the second stage, the
shear stresses are then applied to the bonded surface. This
approximation was used for rubber discs and infinite long rubber
strips based on two kinematic assumptions and one stress assumption:

planes parallel to the rigid surface will keep plane and parallel during or after deformation;

lines normal to the rigid surface will be changed to parabolic after deformation produced by the compression;

normal stress components in three directions are equal to the mean pressure.

Typical flexible joint.

The flexible joint undergoes pure shear deformation under the torsional moment. Some studies have been taken on the shear deformation of the rubber bearings with circular and square shapes (He et al., 2012; Mishra and Igarashi, 2013). Exact closed-form expressions are derived for the torsional stiffness of the spherical rubber bush mountings (Horton and Tupholme, 2005). Zhang et al. (2012) used nonlinear FEM to simulate the SRB flexible joint structural behaviour and conducted experiments to validate the analysis. But the stiffness was not studied. Chen and Yang (2015) proposed an analytical method to calculate the compression and torsional stiffness of the helicopter rotor elastomeric bearings with incompressible assumption. An experiment was conducted to validate the analytical solution. But the calculation error is too big.

The linear vertical stiffness of the spherical bonded rubber layer has been presented in the previous work (Wang et al., 2017). A closed-form expression of the linear rotational stiffness of the bonded rubber layer is proposed. The influence factors on the rotational stiffness are studied. FEM is used to verify the analytical result and analyse the stiffness of the flexible joint. The experiment is taken to validate the simulation and analytical results.

The parabolic form of rubber boundary.

The sectional view of a typical flexible joint is shown in
Fig. 1. It is spherical and consists of
laminated rubber and metal layers. The top plate is fixed and loads
are applied to the bottom plate. Generally, the spherical rubber and
metal layers are equivalent to be in series and have a coincident
centre. Thus the total linear stiffness,

Some assumptions are given to calculate the linear vertical
compression stiffness and rotational stiffness of the flexible joint:

the metal layer is rigid;

planes parallel to the metal layer will remain plane and parallel during or after compression deformation;

lines normal to the metal layer will be changed to parabolic after the deformation produced by the compression loads, as shown in Fig. 2.

A single rubber layer of the flexible joint.

Figure 3 shows the cross section of a single spherical rubber
layer. According to Wang et al. (2017), the vertical stiffness can be
expressed as

The boundary conditions and loading of rotational rubber layer.

Because the rubber layer is axisymmetric about

To make the results more general, define a parameter

Substitute Eqs. (

The vertical compression stiffness of the bonded rubber layer has been
analyzed in Wang et al. (2017), here the rotational stiffness will be
analyzed. According to Eqs. (

Rotational stiffness variation along with edge angles.

Figure 5 shows the relation between the rotational stiffness and the
edge angles. When the outer edge angle is constant the rotational
stiffness decreases if the inner edge angle increases and when the
inner angle is constant it increases if the outer edge angle
increases. The trend is approximately linear when edge angles are
between 0.3 and 1.3

Figure 6a shows the relation between the rotational stiffness and the
thickness of the rubber layer. The curve is a hyperbola. When the
thickness approaches to zero, the stiffness approaches to infinite. As
the thickness increases from 0 to 0.5

Geometric parameters of rubber layers.

Coefficients of constitutive model.

Rotational stiffness variation along with thickness and inner radius.

A prototype of the flexible joint is manufactured, as shown in
Fig. 7. The flexible joint consists of four rubber layers. Three metal
layers are evenly arranged and fully wrapped by the rubber. The
thickness of the inner and outer rubber on the edge of the flexible
joint is 3

Detailed dimension of the flexible joint.

The linear elastic material model is applied to the metal layers. The
property coefficients of the metal layers are Young's modulus

The Nitrile rubber reinforced by carbon-black is used in the flexible
joint prototype. Figure 8a shows the test data of the uniaxial tension
test of the rubber. The test was performed on dumbbell shaped
specimens prepared according to Type 1A GB/T 528-2009. The test was
carried out on the tensile testing machine. The interested strain
range is 0–200 % in the rubber of the flexible joint. The rubber
material is nearly incompressible, which means its bulk modulus is far
larger than shear modulus and Poisson's ratio is close to 0.5. In the
simulation of the rubber with tensile and shear deformation, Poisson's
ratio is always assumed to be 0.5 and then bulk modulus goes to
infinite. But that will overestimate the compression capacity of the
rubber component when it is highly constrained and compressed. Thus
the measuring of the incompressible parameter

Test data of the flexible joint rubber.

The increment is fixed as 0.1 in all steps in the simulation. The
nlgeom option is on. As shown in Fig. 9, the top plate of the flexible
joint is fixed. A reference point RP-1 is set at the coincident
center. The reference point is coupled with the bottom plate of the
flexible joint in the interaction module. A displacement of
2.5

FE mesh of the flexible joint models.

The meshes of models are shown in Fig. 9. For nearly incompressible materials, hourglassing and volume locking phenomena may occur during simulation. Thus the hybrid and second-order elements are used to solve these problems. The 2-D axisymmetric element, CAX8R, is used for the metal part and CAX8RH is used for the rubber part in 2-D model. The 3-D solid element, C3D20R, is used for the metal part and C3D20RH is used for the rubber part in 3-D model.

To validate the theoretical and simulation results, a test device was
designed to carry out the compression and rotation test for the
prototype, as shown in Fig. 10. The compression and rotation loading
was provided by a 2000 kN compression tester. Four 500 kN pressure
sensors were used in parallel to measure the loading. A laser
displacement sensor and an angle sensor were respectively used in
compression and rotation test. In the compression test, the flexible
joint was directly compressed by the pressure tester. The maximum
compression loading is set as 1000 kN. The loading speed is about
40

Figure 11a shows the results of the compression simulation and
test. It can be seen that the curve of the test data is nearly linear
when the compression displacement is smaller than 1.5

Figure 11b shows the results of the rotation simulation and test. It
can be seen that the test curve is almost linear when the rotation
angle is smaller than 2

After analyzing Fig. 11, it can be considered that the Mooney–Rivlin model can fit the test result very well when the rubber deformation in the flexible joint is small. The Yeoh model can reflect the deformation trend more accurately, but the error in the nearly linear district is bigger than the Mooney–Rivlin model. The analytical solutions are accurate in the nearly linear district.

The typical unfilled rubber has Poisson's ratio in the range of 0.4995
to 0.499995 and filled rubber has Poisson' ratio in the range of 0.49
to 0.497 (Hibbitt et al., 2016). In order to
study the effect of Poisson's ratio on the flexible joint under
compression or torsional moment, six FE models with different
Poisson's ratios are created. These Poisson's ratios are 0.49, 0.495,
0.499, 0.4995, 0.4999 and 0.49995 respectively. The corresponding
incompressible parameters are 0.028, 0.014, 0.0028, 0.0014, 0.00028
and 0.00014

Figure 12a shows the vertical compression stiffnesses of the flexible
joints with different Poisson's ratios vs. the displacement. The
analysis is taken in the nearly linear district, which means the
maximum displacement is 1.5

The compression and rotation test device.

The FEM simulation and test results.

Vertical/rotational stiffness variation along with vertical displacement/angle.

Figure 12b shows the rotational stiffnesses of the flexible joints
with different Poisson's ratios vs. the angle. The analysis is taken
in the nearly linear district, which means the maximum angle is
2

By comparing two figures in Fig. 12, it can be found that Poisson's ratio has a more considerable effect on the vertical compression stiffness than the rotational stiffness. The vertical stiffness increases by 6.2 times as Poisson's ratio increases from 0.49 to 0.49995. On the contrary, the rotational stiffness only increases by 1.47 % as Poisson's ratio increases from 0.49 to 0.49995.

The analytical formulae of the linear rotational stiffness are derived for the flexible joint. The rotational stiffness of rubber layer is related to the inner radius, the thickness and two edge angles. It will decrease when the inner edge angle increases and increase when the outer edge angle increases. The increase of the rubber thickness will reduce the rotational stiffness. The increase of the inner radius will enhance the rotational stiffness.

The FEM is used to verify the analytical method and analyze the stiffness of the flexible joint. The Mooney–Rivlin, Neo Hooke and Yeoh constitutive models are used in the simulation. The experiment is taken to obtain the material coefficient and validate the analytical and simulation results. The Yeoh model can reflect the deformation trend more accurately, but the error in the nearly linear district is bigger than the Mooney–Rivlin model. The Mooney–Rivlin model can fit the test result very well when the rubber deformation in the flexible joint is small. The error of two analytical solutions are respectively 5.5 and 3.5 %. That's usable in the flexible joint design. The increase of Poisson's ratio of the rubber layers will enhance the vertical compression stiffness but barely have effect on the rotational stiffness. The vertical stiffness increases by 6.2 times and the rotational stiffness only increases by 1.47 % as Poisson's ratio increases from 0.49 to 0.49995.

The data generated during this study are available from the corresponding author on reasonable request.

The authors declare that they have no conflict of interest.

This research is supported by the National Natural Science Foundation of China (No. 51305088). Edited by: Lotfi Romdhane Reviewed by: two anonymous referees