Liner Wrinkling in a Lined Pipe – A Finite Element Approach
Figure 3: Deformed Liner with Initial Imperfection of 10-3mm
Figure 5: Stress–Strain Curves of Axially Compressed Liners with Different Initial Imperfections
1.5
1
U, U3
+2.000e+00 +1.833e+00 +1.667e+00 +1.500e+00 +1.333e+00 +1.167e+00 +1.000e+00 +8.333e-01 +6.667e-01 +5.000e-01 +3.333e-01 +1.667e-01 -2.866e-27
Y Z X
Step: step-NLB-4 Increment 8: step time = 1.000 Primary var: U, U3 Deformed var: U deformation scale factor: +5.000e+00
Figure 4: Stress–Strain Curve of a Perfect Axially Compressed Liner and One with an Initial Imperfection of 10-3mm
1.5 B A 1 0.5 0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalised strain [-]
Perfect liner Initial imperfection amplitude 1E-3mm
The onset of wrinkling (A) followed by axisymmetric collapse (B) is shown.
To investigate the sensitivity of the models for the initial imperfection amplitude, buckling behaviour of the liner has been simulated for different initial imperfection amplitudes. The resulting stress–strain curves are given in Figure 5. It was concluded that the pre-buckling behaviour, bifurcation point (defined as the point where a stress–strain curve deviates from the perfect path) and load limit depend solely on the initial amplitude of the imperfection. The load-limit strain and bifurcation strain decrease with increasing initial imperfection amplitude. Furthermore, the model showed that the post-buckling behaviour is highly discretisation- sensitive in terms of the geometry and time. From the literature, it is known that a plastic-buckling paradox exists in terms of the constitutive model used. For general load paths involving plastic strains, analysis and experiments indicate that the correct theory to use is the flow (or incremental) theory, rather than the deformation (or total strain) theory.
initial imperfection corresponding to the critical axisymmetric mode shape,9
as found from linear eigenvalue analysis. In Figure 3, for example, a deformed liner with an initial imperfection of 10-3mm is shown.
The liner modelled with shell elements has a length that equals three critical half-wavelengths and has symmetrical boundary conditions at the ends. Plastic buckling failure of cylinders in axial compression can be described clearly using the resulting stress–strain curves. This curve and the related behaviour of the axially compressed liner model with a small initial imperfection is shown in Figure 4. It corresponds well with the behaviour described by Bardi.9,10
Initially, the cylinder expands uniformly in
a radial direction. At a strain level indicated by the arrow in Figure 4, axisymmetric wrinkles start to develop. The wrinkles, initially small in
70
However, for some plastic-buckling problems such as axially compressed circular cylindrical shells, it is known that test results are in better agreement with deformation theory than flow theory,11–13
the latter
resulting in unrealistically high bifurcation loads. For this reason, all models were developed based on these two different theories. It was found that, although the numerical results for the liner alone, based on deformation theory, match the analytical results best, the results from flow theory give smoother and more consistent behaviour and as a result appear less sensitive to discretisation. For larger initial imperfection amplitudes, the flow theory results move closer to the analytical results. The results presented in this article are all based on flow theory.
Confined Liner
A simplified configuration is used to model the lined pipe. The liner is confined within the outer pipe with a perfect fit, which means that initially
EXPLORATION & PRODUCTION – VOLUME 8 ISSUE 1
0.5
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Normalised strain [-]
Perfect S4R flow
Initial imperfection amplitude 1E-2mm
Initial imperfection amplitude 1E-3mm
Initial imperfection amplitude 1E-1mm
amplitude, gradually grow. For the relatively thick cylinders in this research (D/t = 100), the accompanying reduction in axial rigidity eventually leads to load limit instability (indicated by ^ in Figure 4). Deformation can become localised in one of the wrinkles after reaching the load limit under displacement-controlled loading.
Normalised stress [-]
Normalised stress [-]
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72 |
Page 73 |
Page 74 |
Page 75 |
Page 76 |
Page 77 |
Page 78 |
Page 79 |
Page 80 |
Page 81 |
Page 82 |
Page 83 |
Page 84 |
Page 85 |
Page 86 |
Page 87 |
Page 88 |
Page 89 |
Page 90 |
Page 91 |
Page 92 |
Page 93 |
Page 94 |
Page 95 |
Page 96 |
Page 97 |
Page 98 |
Page 99 |
Page 100 |
Page 101 |
Page 102 |
Page 103 |
Page 104 |
Page 105 |
Page 106 |
Page 107 |
Page 108 |
Page 109 |
Page 110 |
Page 111 |
Page 112 |
Page 113 |
Page 114 |
Page 115 |
Page 116 |
Page 117 |
Page 118 |
Page 119 |
Page 120 |
Page 121 |
Page 122 |
Page 123 |
Page 124 |
Page 125 |
Page 126 |
Page 127 |
Page 128 |
Page 129 |
Page 130 |
Page 131 |
Page 132 |
Page 133 |
Page 134 |
Page 135 |
Page 136 |
Page 137 |
Page 138 |
Page 139 |
Page 140 |
Page 141 |
Page 142 |
Page 143 |
Page 144 |
Page 145 |
Page 146 |
Page 147 |
Page 148