Axial and lateral stiffness of spherical self-balancing fiber reinforced rubber pipes under internal pressure

2.1 Material model of the reinforced layer

The geometric model of the spherical pipe is shown in Figure 3. A curvilinear coordinate system (α, β, γ) and a Cartesian coordinate system (x, y, z) are established. The geometry of the spherical pipe is simply determined by the structure vector H = [L, r, R_α, θ₀]^T, where L, r, R_α and θ₀ represents the axial length, the circumferential radius at the end, the meridional radius, and the initial winding angle, respectively.

Figure 3

The geometric model of the spherical pipe. (a) The curvilinear coordinate system and the Cartesian coordinate system. (b) The structural parameters of the spherical pipe.

Figure 4

The winding angle of the reinforced layer.

The single-layer cord/rubber lamina can be regarded as orthotropic material [15]. The equivalent elastic constants of the lamina can be obtained with the following equations [16]

where Ec and Er are the Elastic modulus of the cord and rubber, respectively; μ_c and μ_r are the Poisson's ratios of the cord and rubber, respectively; G_c and G_r is the shear modulus of the cord and rubber, respectively; V_c is the volume fraction of cord. The stress-strain relations of the kth lamina in the reinforced layer are written as [17]

where Q¯ijk represent the off-axis stiffness coefficients of the k th lamina, which are written as

where Qijk represent the material constants of the kth orthotropic lamina. The fiber coordinates of the lamina are written as 1 and 2, where direction 1 is parallel to the fibers and 2 is perpendicular to them. The Qijk are defined as

where E1k and E2k are elasticity modulus of the kth lamina in the 1 and 2 directions, respectively; G12k is shear modulus and μ12k is the major Poisson's ratio. μ21k is determined by the equation μ21kE2k=μ12kE1k . The transformation matrix T_k is

where winding angle θ is the angle between direction 1 of the fiber and the direction α of the curvilinear coordinate system, as shown in Figure 3.

The anisotropy of the reinforced layer are closely related to the local winding angle of the fabric. Although the initial winding angle is uniform, the winding angle of expanded reinforced layer keeps changing at different positions along the meridional direction. For the curing process, the following two assumptions are proposed:

1. The cords of the fabric is not stretched during the curing process;

2. Both ends of the pipe do not rotate relatively to each other during the curing process.

Supposing the initial winding angle is θ₀, the geometric relationship established based on the assumptions abovementioned are written as

where R_β represents the circumferential radius, as shown in Figure 3. The distribution of the winding angle θ along the meridional direction derived from equations (6) is

The structure vector H has significant influence on the distribution of winding angle. Figure 5 shows that the winding angle of cord fabric will become more non-uniform along the axial direction when the initial winding angle and axial length increase, and when the circumferential radius at the end and the meridional radius decrease.

Figure 5

Distribution of winding angle at different H. The initial structure vector H₀ = [0.1, 0.1, 0.1, 40]: (a) varied initial winding angles θ₀; (b) varied axial lengths L; (c) varied circumferential radius at the ends r; (d) varied meridional radius R_α.

The stress-strain relationship is used to obtain the elastic constant along the axial direction of the rubber pipe:

The distribution of the elastic constants is shown in Figure 6a. At any position of the pipe, the shear modulus is greater than the meridional and latitudinal elastic modulus. At the center of the joint, the meridional elastic modulus reaches the minimum value, while the latitudinal elastic modulus reaches a maximum value. Figure 6b shows the relative variance of the elastic constants with different winding angle. The off-axis elastic constants reduce to the orthotropic properties when the winding angle equals 0 degree and 90 degrees. The shear modulus reaches the maximum when the winding angle equals 45 degrees. The stiffness matrix of the 1500D/3-type aramid cords is used in the theoretical calculation, written as follows [15].

Figure 6

(a) Distribution of elastic constants with the structure vector H = [0.1, 0.1, 0.1, 40]. (b) The elastic constants at variant winding angle.

The rubber pipes are reinforced by multi-layer fabric. Based on the composite Reissner shell theory, the stiffness matrix of the reinforced layer is

where N and M are the inplane force and bending moment, respectively; €⁰ is the strain in the midsurface of the reinforced layer; κ is the curvature and torsion ratio of the midsurface. Each sub-matrix of the stiffness matrix is defined as [18]

2.2 Analysis of the axial stiffness

The inplane force is defined as

where δ is the thickness of the reinforced layer. Under the assumption of thin shell theory, the thickness of the reinforced layer is a small quantity, so the terms related to thickness are considered as high-order small quantities and are negligible in equation (12). Without considering the bending moment and shear force, the geometric equation is simplified as follows [17]

where A, B are lamé parameters and k_α, k_β are principal curvature of the midsurface. Combining equations (12) and (13), yields

The equilibrium equations [17] are

The spherical pipe is an axisymmetric shell of revolution, so the coefficients in equations (14) and (15) are

where p is internal pressure. It is assumed that all the constraints and external loads are axisymmetric, and then for any point of the pipe, the equilibrium equation is established in the meridional direction as follows

The axial stiffness of the deformation of joint body is determined by the relationship between F_y and the axial deformation y. Combining equations (14), (15) and (17), the solutions of internal force and displacement are obtained as follows

where C₁ is an integral constant determined by boundary conditions. The boundary condition is

Substituting equations (18) into equation (19) yields the force-displacement relationship for the axial deformation:

The influences of structural parameters are then computed on the axial stiffness, as shown in Figure 7. The axial stiffness will increase significantly when the initial winding angle increases, the meridional radius increases, the axial length decreases, and the circumferential radius at the end increases.

Figure 7

Axial stiffness at different H, and the initial structure vector H₀ = [0.1, 0.1, 0.1, 40]: (a) varied initial winding angle θ₀; (b) varied meridional radii R_α; (c) varied axial lengths L; (d) varied circumferential radii at the ends r.

Aramid fiber is generally used as the cord material for large-diameter rubber pipes resistant to high pressure. The elastic modulus of aramid fiber has strong nonlinearity even in the low-stress stage, so it is necessary to experimentally obtain the variation of the elastic modulus of aramid cord with the cord stress. The distribution of stress in the reinforced layer is

With the formula of off-axis stress, the stress along the direction of cords σ₁ is obtained as follows:

where θ is the winding angle of cords. The elastic modulus of cords under different stresses can be obtained by processing the measured force-displacement data of cords. Substituting this elastic modulus into the theoretical model of stiffness yields the stiffness characteristics of the rubber pipe under different internal pressures.

2.3 Analysis of the lateral stiffness

Based on the Timoshenko beam theory, a model is established to calculate the lateral stiffness of the spherical fiber reinforced pipes. The deflection equation of beam is [19]

where a is the shear coefficient, q is the load, EI is the bending stiffness, and GA is the shear stiffness. The thickness of the tube is thinner than that of the inner diameter, so the shear stress is approximately uniformly distributed in the thickness direction, and the shear coefficient a equals 1. According to Shadmehri et al. [20], the bending stiffness of a composite pipe is

where R_o,n and R_i,n are the outer diameter and inner diameter of the nth layer of fiber, respectively; En is the axial elastic modulus of the nth layer, and can be calculated as follows:

The influences of the initial winding angle and the meridional radius are analyzed on the spatial distributions of bending stiffness and shear stiffness, as shown in Figure 8. When the initial winding angle is 43 degree, the bending stiffness and shear stiffness reach their minimum and maximum values, respectively.

Figure 8

For an initial structure vector H₀ = [0.1, 0.1, 0.1, 40], a) distribution of bending stiffness at different θ₀; b) distribution of shear stiffness at different θ₀; c) distribution of bending stiffness at different meridional radii; d) distribution of shear stiffness at different meridional radii.

Because of the non-uniformity of the distribution of bending stiffness and shear stiffness, the equation is transformed into a difference equation to obtain the numerical solution of deflection. Based on the definition of central difference, the difference equation deflection curve is obtained:

where h is the length step of difference. In this paper, the influences of structural parameters are analyzed on the lateral stiffness of a rubber pipe with Clamped-Clamped boundary condition, and the calculated results are shown in Figure 9. The result reveals that the lateral stiffness will significantly decrease when the winding angle and axial length increase and the circumferential radius decreases.

Figure 9

Lateral stiffness at different H, and the initial structure vector H₀ = [0.1, 0.1, 0.1, 40]: (a) varied initial winding angles θ₀; (b) varied meridional radii R_α; (c) varied axial lengths L; (d) varied circumferential radii at the ends r.

2.4 The self-balance of spherical fiber reinforced rubber pipes

A rubber pipe used in ship pipeline systems is required to have good self-balance; otherwise, when the internal pressure changes, the pipe will produce additional deformation and may damage the connected pipeline or equipment. Self-balance is evaluated by the following indexes: 1) unbalanced displacement, which is defined as the ratio of length change under the working pressure when both ends are free; 2) unbalanced force: the reaction force under the working pressure when both ends are fixed. The methods are derived for solving the ratio of length change and reaction force:

1. When both ends are free, calculate the ratio of length change under the working pressure. Based on the geometric relationship, the axial deformation y of joint body is calculated as follows:

   (27) y=-usinα|α=π2-ϕ=∫π2-ϕπ2Rα|NαA12NβA22|-Rβ|A11NαA12Nβ|δsinα|A11A12A12A22|dα

   It is combined with equation (20), and let the external force in the axial direction F_y equal 0, and then the ratio of length change can be calculated:

   (28) η=2yL

2. When both ends of the pipe are clamped, the unbalanced force is calculated under the working pressure. In formula (20), let the axial deformation y be equal to 0, and the reaction force can be obtained when both ends of the pipe body are fixed:

   (29) Fy=-πp∫π2-ϕπ2A11Rβ3+2(A12-A11)Rβ2Rα+(A22-2A12)RβRα2(A11A22-A122)sinαdα∫π2-ϕπ2A11Rβ2+A22Rα2+2A12RαRβ(A11A22-A122)Rβsin3αdα

   The approach of stiffness calculation of self-balancing fiber reinforced rubber pipes is summarized in a flow chart in Figure 10.

Figure 10

The approach of stiffness calculation of self-balancing fiber reinforced rubber pipes.

Figure 11

(a) Experimental systems used to measure the axial and lateral stiffness; (b) tensile test of aramid cords; (c) axial stiffness test; (d) lateral stiffness test.

3.1 The elastic modulus of aramid cord

Table 3 shows the properties of the aramid cords used in the research. Based on the measured data of tensile test, the fitting results of the elastic modulus of cord are obtained with the least square method and are shown in Figure 12. When the cord strain is less than 0.01, the elastic modulus increases with the increase of the strain; when the strain is greater than 0.01, the elastic modulus becomes linear.

Figure 12

The stress-strain relationship of 1140D/2 aramid cords.

3.2 Axial and lateral stiffness of the rubber pipes

The aforementioned stiffness model is used to derive the axial and lateral stiffness of rubber pipes with four different sizes, and the results are shown in Figure 13a and Figure 13b. The theoretical results are in good agreement with the experimental results, suggesting that the stiffness model obtained in this paper can accurately describe the axial and lateral stiffness of the rubber pipes.

Figure 13

(a) Calculated and measured axial stiffness vs internal pressure; (b) calculated and measured lateral stiffness vs internal pressure.