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FIGURE 7: SPRING-SUPPORTED ARCH MODEL Since the surrounded pressure armor is treated as springs, then the concentric ring structure could be simplified as a spring-supported arch model, as shown in Figure 7.

Figure 7 SPRING-SUPPORTED ARCH MODEL Since the surrounded pressure armor is treated as springs, then the concentric ring structure could be simplified as a spring-supported arch model, as shown in Figure 7.

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FIGURE 1: MAIN COMPONENT LAYERS OF A FLEXIBLE RISER [3] they contribute the most to radial stiffness. Once the external pressure exceeds the collapse capacity of a flexible riser, the collapse failure happens, which is called “wet collapse” [5]. Mostly, the wet collapse pressure is higher than the collapse limit of the carcass since the surrounded pressure armor provides a significant support effect [6]. However, this critical pressure may be weakened by two initial imperfections, which are ovality of the carcass, and the radial gap between carcass/liner and pressure armor, respectively. The ovality of the carcass is generated from the manufacture tolerance while the radial gap is caused by the volume change or extrusion of the liner, as shown in Figure 2 [7-8]. Although tensile armors can reduce the radial gaps by squeezing the pressure armor, it is hard to close these gaps completely if the pipe is under a longitudinal tension. According to some _ researchers’ investigation, it indicates that the critical pressure of flexible risers 1s sensitive to those two imperfections [7-10].
FIGURE 1: MAIN COMPONENT LAYERS OF A FLEXIBLE RISER [3] they contribute the most to radial stiffness. Once the external pressure exceeds the collapse capacity of a flexible riser, the collapse failure happens, which is called “wet collapse” [5]. Mostly, the wet collapse pressure is higher than the collapse limit of the carcass since the surrounded pressure armor provides a significant support effect [6]. However, this critical pressure may be weakened by two initial imperfections, which are ovality of the carcass, and the radial gap between carcass/liner and pressure armor, respectively. The ovality of the carcass is generated from the manufacture tolerance while the radial gap is caused by the volume change or extrusion of the liner, as shown in Figure 2 [7-8]. Although tensile armors can reduce the radial gaps by squeezing the pressure armor, it is hard to close these gaps completely if the pipe is under a longitudinal tension. According to some _ researchers’ investigation, it indicates that the critical pressure of flexible risers 1s sensitive to those two imperfections [7-10].
FIGURE 3: BUCKLING OF THE INNER CYLINDER WITH SPRING SUPPORTS [6] The pressure armor is considered as springs in the above model which supports the carcass in the horizontal direction, as shown in Figure 3 [6]. However, this model is developed for the flexible risers with no gap in-between and only provides an elastic solution. Since the flexible risers are more likely to be collapsed in the plastic range in a deep water condition [18], efforts were made to predict the elastic-plastic collapse pressure of the flexible risers with initial imperfections in this paper.
FIGURE 3: BUCKLING OF THE INNER CYLINDER WITH SPRING SUPPORTS [6] The pressure armor is considered as springs in the above model which supports the carcass in the horizontal direction, as shown in Figure 3 [6]. However, this model is developed for the flexible risers with no gap in-between and only provides an elastic solution. Since the flexible risers are more likely to be collapsed in the plastic range in a deep water condition [18], efforts were made to predict the elastic-plastic collapse pressure of the flexible risers with initial imperfections in this paper.
FIGURE 4: PROGRESSIVE BUCKLING PROCESS FOR A SYMMETRICAL COLLAPSE MODEL until the contact happens. Once the contact occurs, the inner carcass starts to be divided into two portions: attached and detached portions. In this post-contact phase, the collapse pressure of the carcass is decided by the buckling strength of the detached portion [19]. The presented analytical scheme adopts different analytical models for these two phases. For t formulae of the ring buckling are used formulae of the arch buckling are emp detached portion of the carcass in the he pre-contact phase, to solve the external pressure P.,, at contact moment. For the post-contact phase, oyed to determine the buckling pressure P,,,., of the detached portion. The shape of the post-contact phase is decided by the initial ovality and ga collapse pressure of a flexible pipe is the p width together. The pressure armor which restrains the deformation of the detached portion at its two ends is regarded as springs. The critical sum Of Poon and Parch.
FIGURE 4: PROGRESSIVE BUCKLING PROCESS FOR A SYMMETRICAL COLLAPSE MODEL until the contact happens. Once the contact occurs, the inner carcass starts to be divided into two portions: attached and detached portions. In this post-contact phase, the collapse pressure of the carcass is decided by the buckling strength of the detached portion [19]. The presented analytical scheme adopts different analytical models for these two phases. For t formulae of the ring buckling are used formulae of the arch buckling are emp detached portion of the carcass in the he pre-contact phase, to solve the external pressure P.,, at contact moment. For the post-contact phase, oyed to determine the buckling pressure P,,,., of the detached portion. The shape of the post-contact phase is decided by the initial ovality and ga collapse pressure of a flexible pipe is the p width together. The pressure armor which restrains the deformation of the detached portion at its two ends is regarded as springs. The critical sum Of Poon and Parch.
FIGURE 5: CONTACT MOMENT OF A CONCENTRIC RING STRUCTURE For a carcass encased in the pressure armor, it might have two different buckling situations, depending on the value of radial gap between the carcass/liner and pressure armor. The inner carcass would be collapsed as a single ring if the gap width is large enough; otherwise, layer contact happens followed by the post-contact phase, as shown in Figure 5. The formula given by Timoshenko and Gere for the plastic collapse of a single ring with an initial deflection @ can be used for the first situation [13]
FIGURE 5: CONTACT MOMENT OF A CONCENTRIC RING STRUCTURE For a carcass encased in the pressure armor, it might have two different buckling situations, depending on the value of radial gap between the carcass/liner and pressure armor. The inner carcass would be collapsed as a single ring if the gap width is large enough; otherwise, layer contact happens followed by the post-contact phase, as shown in Figure 5. The formula given by Timoshenko and Gere for the plastic collapse of a single ring with an initial deflection @ can be used for the first situation [13]
FIGURE 6: PROGRESSIVE BUCKLING PROCESS DURING THE POST-CONTACT PHASE After the carcass contacts with the surrounded pressure armor, the contact point will keep moving upwards until the critical pressure is reached [20], as shown in Figure 6. An assumption from Timoshenko and Gere is used to define the elastic-plastic collapse of the flexible pipes: the collapse occurs at which yielding in the extreme fibers of the carcass begins [13], i.e. the carcass collapses when its maximum compressive stress equals to the material yield stress.
FIGURE 6: PROGRESSIVE BUCKLING PROCESS DURING THE POST-CONTACT PHASE After the carcass contacts with the surrounded pressure armor, the contact point will keep moving upwards until the critical pressure is reached [20], as shown in Figure 6. An assumption from Timoshenko and Gere is used to define the elastic-plastic collapse of the flexible pipes: the collapse occurs at which yielding in the extreme fibers of the carcass begins [13], i.e. the carcass collapses when its maximum compressive stress equals to the material yield stress.
FIGURE 8: EQUILIBRIUM OF A DIFFERENTIAL ELEMENT OF THE ARCH [21] Where M, N, and Q, are the bending moment, hoop force and radial shear force on the differential element; g, and q, are the uniform loads along the radial and circumferential directions, as shown in Figure 8.
FIGURE 8: EQUILIBRIUM OF A DIFFERENTIAL ELEMENT OF THE ARCH [21] Where M, N, and Q, are the bending moment, hoop force and radial shear force on the differential element; g, and q, are the uniform loads along the radial and circumferential directions, as shown in Figure 8.
By substituting those coefficients into Eqs.(19), the external pressure can be written as a function of the maximum compressive stress at the crown point of the arch. Since the plastic collapse is defined by material yielding, therefore, the buckling pressure Pach of this arch can be worked out as and the coefficients in Eq.(23) can be calculated as
By substituting those coefficients into Eqs.(19), the external pressure can be written as a function of the maximum compressive stress at the crown point of the arch. Since the plastic collapse is defined by material yielding, therefore, the buckling pressure Pach of this arch can be worked out as and the coefficients in Eq.(23) can be calculated as
FIGURE 9: BENDING MOMENTS AT THE CONTACT POSITION With the value of R,., that estimated from Eq.(28), the buckling pressure P,,., of the arch can be determined by decreasing the value of f,, (from 7/2 to 0) continually until the bending moments of the attached and detached portions at the contact position equal to each other, as shown in Figure 9. The bending moment for the detached portion can be calculated with Eq.(19). For the attached portion, the bending moment at the contact position is obtained by [13]
FIGURE 9: BENDING MOMENTS AT THE CONTACT POSITION With the value of R,., that estimated from Eq.(28), the buckling pressure P,,., of the arch can be determined by decreasing the value of f,, (from 7/2 to 0) continually until the bending moments of the attached and detached portions at the contact position equal to each other, as shown in Figure 9. The bending moment for the detached portion can be calculated with Eq.(19). For the attached portion, the bending moment at the contact position is obtained by [13]
For verification purposes, a 4” flexible riser model presented by Gay Neto and Martins is used [5], which was also adopted by Edmans in his research [23]. This FE model given in the above-mentioned literature possesses three layers, the carcass, the inner liner and the pressure armor. The carcass was built with a detailed profile while the liner and pressure armor were represented as two homogeneous equivalent layers. Abaqus 6.13 is employed to recreate this FE model, as shown in Figure 11. The critical pressure given by this recreated FE model is 23.60 MPa, which is quite close to results of Gay Neto (25.56 MPa) and Edmans (24.1 MPa), with differences of 7.67% and 2.07%, respectively. Table | lists the geometric and material data of this FE model. Geometric details of the carcass profile are given in the source reference. The material stress-strain curves of the liner
For verification purposes, a 4” flexible riser model presented by Gay Neto and Martins is used [5], which was also adopted by Edmans in his research [23]. This FE model given in the above-mentioned literature possesses three layers, the carcass, the inner liner and the pressure armor. The carcass was built with a detailed profile while the liner and pressure armor were represented as two homogeneous equivalent layers. Abaqus 6.13 is employed to recreate this FE model, as shown in Figure 11. The critical pressure given by this recreated FE model is 23.60 MPa, which is quite close to results of Gay Neto (25.56 MPa) and Edmans (24.1 MPa), with differences of 7.67% and 2.07%, respectively. Table | lists the geometric and material data of this FE model. Geometric details of the carcass profile are given in the source reference. The material stress-strain curves of the liner
Once M and M, are matched, the angle f£., can be determined as well as the buckling pressure P,,., of the arch. A flowchart that shows the whole procedure is given as Figure 10. 4. VERIFICATION
Once M and M, are matched, the angle f£., can be determined as well as the buckling pressure P,,., of the arch. A flowchart that shows the whole procedure is given as Figure 10. 4. VERIFICATION
TABLE 2: COMPARISO OF COLLAPSE PRESSURE BETWEEN FE AND ANALYTICAL MODELS FOR DIFFERENT OVALIZATION
TABLE 2: COMPARISO OF COLLAPSE PRESSURE BETWEEN FE AND ANALYTICAL MODELS FOR DIFFERENT OVALIZATION
TABLE 3: COMPARISO OF COLLAPSE PRESSURE BETWEEN FE AND ANALYTICAL MODELS FOR DIFFERENT GAP WIDTH
TABLE 3: COMPARISO OF COLLAPSE PRESSURE BETWEEN FE AND ANALYTICAL MODELS FOR DIFFERENT GAP WIDTH
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