Open Access

DOI: 10.1186/2196-4351-1-9

Accepted: 30 October 2013

Published: 30 December 2013

## Abstract

### Keywords

Adhesive bonded joint Singular stress field Riser pipe Sea environment

## Background

Due to low manufacturing costs, low stress concentration and ease of maintenance, adhesive joints are most frequently used in numerous industrial sectors such as automobile, shipbuilding, aeronautical, etc., replacing or supplementing traditional joining technologies, such as welding or riveting. Adhesive bonded joints are also widely used in riser pipes because of their light weight, high strength, and high corrosion resistance[1]. With the wide use of adhesive joints, many research works have been done to evaluate their strength including experimental and analytical methods [28].

### Methods

#### Theoretical analysis

At the end of the interface, as shown in Figure 2, it is known that the interface stress σ ij (ij = rr, θθ, ) goes to infinity at the edge of the joint and has a singularity of σ ij  1/r1 − λ when a(a-2β) > 0. Kσ is the parameter used to evaluate the intensity of singular stresses, where ${K}_{\sigma }=\underset{r\to 0}{lim}\left({r}^{1-\lambda }×{\sigma }_{\theta }\right)$. Besides, when θ = π/2, the singularity of the stress λ at the joint interface can be expressed by the following equation [1820].
${\left[{sin}^{2}\left(\frac{\pi }{2}\lambda \right)-{\lambda }^{2}\right]}^{2}{\beta }^{2}+2{\lambda }^{2}\left[{sin}^{2}\left(\frac{\pi }{2}\lambda \right)-{\lambda }^{2}\right]\phantom{\rule{0.12em}{0ex}}\mathit{a\beta }+{\lambda }^{2}\left({\lambda }^{2}-1\right)\phantom{\rule{0.12em}{0ex}}{a}^{2}+\frac{{sin}^{2}\left(\mathit{\lambda \pi }\right)}{4}=0$
(1)
Where
$a=\frac{{G}_{1}\phantom{\rule{0.12em}{0ex}}\left({k}_{2}+1\right)-{G}_{2}\phantom{\rule{0.12em}{0ex}}\left({K}_{1}+1\right)}{{G}_{1}\phantom{\rule{0.12em}{0ex}}\left({k}_{2}+1\right)-{G}_{2}\phantom{\rule{0.12em}{0ex}}\left({K}_{1}+1\right)}\phantom{\rule{1.08em}{0ex}}\beta =\frac{{G}_{1}\phantom{\rule{0.12em}{0ex}}\left({k}_{2}-1\right)-{G}_{2}\phantom{\rule{0.12em}{0ex}}\left({K}_{1}-1\right)}{{G}_{1}\phantom{\rule{0.12em}{0ex}}\left({k}_{2}+1\right)-{G}_{2}\phantom{\rule{0.12em}{0ex}}\left({K}_{1}+1\right)}$
${k}_{j}=\left\{\begin{array}{l}\frac{3-{v}_{j}}{1+{v}_{j}}\phantom{\rule{0.12em}{0ex}}\left(\mathit{plane}\phantom{\rule{0.24em}{0ex}}\mathit{stress}\right)\\ 3-4{v}_{j}\left(\mathit{plane}\phantom{\rule{0.36em}{0ex}}\mathit{strain}\right)\end{array}\right\,{k}_{j}=\left(j=1,2\right)$
(2)

Here r, θ are the polar coordinates around the interface edge, a, β are Dunders’ parameters which are expressed by Possion’s ratio v and shear modulus G.

For the 3D model, as shown in Figure 3, there are also several interface ends, and in this paper, the singular stress fields for the interface end in the 3D adhesive joint model was studied under several kinds of deep sea loadings.

### Numerical model

In this paper, the scaling model with the following dimensions was used: adhesive thickness h0 = 6 mm, cover thickness h1 =10 mm, total length 2 L = 200 mm and coupling length 2c = 25 mm. The outer diameter of the adherend was 140 mm and the inner diameter was 110 mm. The adherend steel elastic properties were E = 70,000 MPa, v = 0.33. The adhesive material elastic properties were E = 3500 MPa, v = 0.30, which means that the existence of singular stress fields condition a(a-2β) > 0 is satisfied at the edge of interface between adhesive and adherend.

The commercial software ABAQUS was used to perform the analysis. The finite element method model was constructed using 3-D solid elements (Figure 3). Two mesh densities were used to conduct the analysis. A coarse mesh with the mesh density of 32 rows of elements circumferentially 2 rows (radially) · 30 rows (axially) was used to model the pipe region, while the mesh density was doubled to model the joint region. Moreover, the mesh of the joint region was graded along the axial direction of the pipe, finer toward the free edges of the joint.

## Results and discussion

In this paper, a riser pipe under a depth of 1500 meters water was considered, so the out pressure P op = ρgH =14.7 MPa was applied the adhesive joint. The boundary conditions for this case were : ur = 0, at x = 0; ux = 0, x = 0; and us = 0, x = 0. According to the theroretical analysis, a singular stress field exists around the end of the interface between the adhesive and the adherends. Here, one example of the stress distribution at the interface of the joint subjected to the external pressure is shown in Figure 4. It is found that stresses at the end of interface tend to infinity, which indicates that a singular stress field exists around the end of the interface.

Generally, the internal pressure of a riser pipe is higher than the external pressure to prevent buckling of the pipe, and in this research, an internal pressure P ip  = 15.0 MPa was chosen. The boundary conditions for this case were: u r  = 0, at x = 0; u x  = 0, x = 0; and u s  = 0, x = 0.

Figure 5 shows the stress distribution at the interface between the adhesive and the adherends of the joint subjected to internal pressure. The stress values at the end of the interface are related to element sizes, but Figure 5 indicates that the stresses around the end of the interface tend to infinity with the element sizes in this paper, which means that a singular stress field exists around the end of the interface. From the comparion with the stress distribution under external pressure, it can be concluded that stresses increase more quickly under internal pressure than external pressure at the end of the interface, which means that the riser pipe under internal pressure is more dangerous than under external pressure if the same increase of stresses happens.

The riser pipe in the deep sea also suffers tension loading. The tension loading T = 20.0 MPa was considered in this paper, and the boundary conditions are the same as those used for out pressure and inner pressure, which are ur = 0, at x = 0; ux = 0, x = 0; and us = 0, x = 0. Figure 6 shows the stress distribution at the interface between the adhesive and the adherends. Stress values at the end of the interface are related to the elment sizes, but it can be found that stresses at the end of the interface tend to infinity with the element sizes in this paper, which indicates that a singular stress field exists at the end of the interface.

The bending loading was applied by a load P = 200 N prependicular to the pipe axis, and the left end of the riser pipe was fixed as: u r  = 0, θr = 0,at x = 0;ux = 0, θ x = 0, at x= 0; and u s  = 0, θ s  = 0 at x = 0 Figure 7 shows the stress distribution at the interface. It indicates that stress distribution is unsymmetric, and stresses decrease from on end of the interface, which is because moments along the pipe are not symmetric. However, the stress distribution indicates that both ends of the interface have singular stress fields.

A torque loading was applied by a load P = 200 N which was tangent to the circumference of the riser pipe section, so the torque is equal to 200 × 140 = 28000 M mm. Figure 8 shows the stress distribution at the interface, and it indicates that the stress also trends to infinity.

### Comparison of stress intensity fields between the cases of external pressure, internal pressure, tension, bending and torque loading

Figure 9 shows that the gradients of stress lines for external pressure, internal pressure and tension loading cases are larger than that for bending and torque loading cases, which means that stresses go to infinity more quickly for cases of external pressure, internal pressure and tension compared to cases of bending and torque. Therefore, failure happens more easily around the end of the interface, which means that the riser pipes under external pressure, internal pressure and tension loading are more dangerous than under bending and torque loading in deep sea environment.

## Declarations

### Acknowledgements

The authors appreciate the reviewer’s constructive comments and detailed feedback for improving the manuscript. This research was supported by Science Foundation of China University of Petroleum, Beijing (No. 2462013YJRC44), and the National Basic Research Program of China (973 Program) Grant No. 2011CB013702.

## Authors’ Affiliations

(1)
China University of Petroleum
(2)
China Agricultural University
(3)
Kyusyu Institute of Technology

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