김경종
384 lines
18 KiB
Markdown
384 lines
18 KiB
Markdown
<!-- source-page: 1031 -->
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1. For a flat-based spud can:
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$$
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D = D _ {o}.
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$$
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2. For a conical-based spud can:
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a. Cone portion partially penetrating $( \nu _ { m } < \nu _ { c } ) \cdot$ :
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$$
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D = 2 \nu_ {m} \tan {\frac {\theta}{2}}
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$$
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b. Penetration beyond cone-cylinder transition $( \nu _ { m } \ge \nu _ { c } )$ :
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$$
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D = D _ {o}.
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$$
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The current spud can area at the soil surface, A, is defined through $A = \pi D ^ { 2 } / 4$ . The effective diameter can vary throughout the analysis only for a conical spud can with plasticity.
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The embedment has no effect and is not required if the spud can is cylindrical and spud can plasticity is not defined.
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# Specifying the embedment directly
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The embedment value can be prescribed directly using initial conditions (see “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1).
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Input File Usage: \*INITIAL CONDITIONS, TYPE=SPUD EMBEDMENT
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# Specifying the spud can preload
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If spud can plasticity is defined, you can specify the initial compressive capacity (“preload”), $V _ { c } ^ { ( i ) }$ , instead of the embedment. In this case Abaqus/Aqua will use the hardening law to calculate the plastic embedment that follows when the preload is applied vertically.
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The preload initial condition is used only to calculate the initial plastic embedment; the spud can starts the analysis in a zero strain and stress state at this initial plastic embedment, and the preload is assumed to be removed. You must apply any operational vertical load through loading within the history definition.
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Input File Usage: \*INITIAL CONDITIONS, TYPE=SPUD PRELOAD
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# Embedment in an elastic spud can analysis
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If the spud can model is purely elastic, the spud can geometry is needed only for calculating the embedded diameter of the spud can for spud can elastic moduli. The embedment is required for this calculation only if the spud can is conical.
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# Output
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Force and moment output in the element local system is available through the “stress” output variable S. Extension and relative rotation are available through the “strain” output variable E. Elastic and plastic
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<!-- source-page: 1032 -->
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strains are available through the output variables EE and PE. For spud cans the plastic embedment since the start of the analysis is available through the vertical component of plastic strain, PE11, and will usually be negative, indicating compression; the total vertical embedment, $\nu _ { m }$ , is available through output variable PEEQ. Element nodal force (the force the element places on its nodes, in the global system) is available through element variable NFORC.
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# Joint elasticity models
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The elastic load-displacement behavior of the JOINT2D and JOINT3D elements is characterized by elastic spring stiffnesses, which are assembled to form the elastic element stiffness matrix. A special diagonal modulus for spud cans can be specified or, alternatively, a fully populated (general) elastic modulus can be specified.
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# Spud can moduli
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Spud can moduli can be prescribed for either two-dimensional or three-dimensional elements.
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Two-dimensional spud can moduli
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The elastic stiffness for a two-dimensional spud can is
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$$
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\left\{ \begin{array}{c} \sigma_ {1 1} \\ \sigma_ {2 2} \\ \sigma_ {1 2} \end{array} \right\} = \left[ \begin{array}{c c c} k _ {1 1 1 1} & 0 & 0 \\ 0 & k _ {2 2 2 2} & 0 \\ 0 & 0 & k _ {1 2 1 2} \end{array} \right] \left\{ \begin{array}{c} \varepsilon_ {1 1} \\ \varepsilon_ {2 2} \\ \varepsilon_ {1 2} \end{array} \right\},
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$$
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where
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$k _ { 1 1 1 1 }$ is the vertical elastic spring stiffness, $2 D G _ { v v } / ( 1 - \nu )$ ;
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$k _ { 2 2 2 2 }$ is the horizontal elastic spring stiffness, $1 6 ( 1 - \nu ) D G _ { h h } / ( 7 - 8 \nu )$ ;
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$k _ { 1 2 1 2 }$ is the elastic spring stiffness in bending, $D ^ { 3 } G _ { r r } / 3 ( 1 - \nu )$ ;
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in which $G _ { v v } , G _ { h h }$ , and $G _ { r r }$ are equivalent elastic shear moduli for vertical, horizontal, and rotational displacements, respectively; is the Poisson’s ratio of the soil (suggested value: 0.2 for sand and 0.5 for clay).
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Input File Usage: \*JOINT ELASTICITY, MODULI=SPUD CAN, NDIM=2
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Three-dimensional spud can moduli
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For a three-dimensional spud can the moduli are
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$$
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\left\{ \begin{array}{l} \sigma_ {1 1} \\ \sigma_ {2 2} \\ \sigma_ {3 3} \\ \sigma_ {1 2} \\ \sigma_ {1 3} \\ \sigma_ {2 3} \end{array} \right\} = \left[ \begin{array}{c c c c c c} k _ {1 1 1 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & k _ {2 2 2 2} & 0 & 0 & 0 & 0 \\ 0 & 0 & k _ {3 3 3 3} & 0 & 0 & 0 \\ 0 & 0 & 0 & k _ {1 2 1 2} & 0 & 0 \\ 0 & 0 & 0 & 0 & k _ {1 3 1 3} & 0 \\ 0 & 0 & 0 & 0 & 0 & k _ {2 3 2 3} \end{array} \right] \left\{ \begin{array}{l} \varepsilon_ {1 1} \\ \varepsilon_ {2 2} \\ \varepsilon_ {3 3} \\ \varepsilon_ {1 2} \\ \varepsilon_ {1 3} \\ \varepsilon_ {2 3} \end{array} \right\},
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$$
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<!-- source-page: 1033 -->
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where
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$k _ { 1 1 1 1 }$ is the vertical elastic spring stiffness, $2 D G _ { v v } / ( 1 - \nu )$ ;
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$k _ { 2 2 2 2 }$ is a horizontal elastic spring stiffness, $1 6 ( 1 - \nu ) D G _ { h h } / ( 7 - 8 \nu )$
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$k _ { 3 3 3 3 }$ is a horizontal elastic spring stiffness, $1 6 ( 1 - \nu ) D G _ { h h } / ( 7 - 8 \nu )$ ;
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$k _ { 1 2 1 2 }$ is an elastic spring stiffness in bending, $D ^ { 3 } G _ { r r } / 3 ( 1 - \nu )$
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$k _ { 1 3 1 3 }$ is an elastic spring stiffness in bending, $D ^ { 3 } G _ { r r } / 3 ( 1 - \nu )$
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$k _ { 2 3 2 3 }$ is the torsional elastic spring stiffness, $k _ { t } ;$
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in which $G _ { v v } , G _ { h h } , G _ { r r }$ , and are as before and $k _ { t }$ is a user-specified torsional stiffness value.
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Straining out of the 1–2 plane through the strains $\varepsilon _ { 3 3 } , \varepsilon _ { 1 3 }$ , and $\varepsilon _ { 2 3 }$ produces purely elastic response in the three-dimensional model regardless of plasticity. The moduli related to these strains are assumed not to be affected by the plasticity so that $k _ { 3 3 3 3 } , k _ { 1 3 1 3 }$ , and $k _ { 2 3 2 3 }$ are based on the initial embedded diameter, while the other moduli depend on the current embedded diameter.
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Input File Usage: \*JOINT ELASTICITY, MODULI=SPUD CAN, NDIM=3
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# General moduli
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General moduli can be specified for either two-dimensional or three-dimensional elements.
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# Two-dimensional general moduli
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For the two-dimensional case six independent elastic moduli are needed. The stress-strain relations are as follows:
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$$
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\left\{ \begin{array}{l} \sigma_ {1 1} \\ \sigma_ {2 2} \\ \sigma_ {1 2} \end{array} \right\} = \left[ \begin{array}{c c c} k _ {1 1 1 1} & k _ {1 1 2 2} & k _ {1 1 1 2} \\ & k _ {2 2 2 2} & k _ {2 2 1 2} \\ s y m & & k _ {1 2 1 2} \end{array} \right] \left\{ \begin{array}{l} \varepsilon_ {1 1} \\ \varepsilon_ {2 2} \\ \varepsilon_ {1 2} \end{array} \right\}
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$$
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Input File Usage: \*JOINT ELASTICITY, MODULI=GENERAL, NDIM=2
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# Three-dimensional general moduli
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For the three-dimensional case 21 independent elastic moduli are needed. The stress-strain relations are as follows:
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$$
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\left\{ \begin{array}{l} \sigma_ {1 1} \\ \sigma_ {2 2} \\ \sigma_ {3 3} \\ \sigma_ {1 2} \\ \sigma_ {1 3} \\ \sigma_ {2 3} \end{array} \right\} = \left[ \begin{array}{l l l l l l} k _ {1 1 1 1} & k _ {1 1 2 2} & k _ {1 1 3 3} & k _ {1 1 1 2} & k _ {1 1 1 3} & k _ {1 1 2 3} \\ & k _ {2 2 2 2} & k _ {2 2 3 3} & k _ {2 2 1 2} & k _ {2 2 1 3} & k _ {2 2 2 3} \\ & & k _ {3 3 3 3} & k _ {3 3 1 2} & k _ {3 3 1 3} & k _ {3 3 2 3} \\ & & & k _ {1 2 1 2} & k _ {1 2 1 3} & k _ {1 2 2 3} \\ & s y m & & & k _ {1 3 1 3} & k _ {1 3 2 3} \\ & & & & & k _ {2 3 2 3} \end{array} \right] \left\{ \begin{array}{l} \varepsilon_ {1 1} \\ \varepsilon_ {2 2} \\ \varepsilon_ {3 3} \\ \varepsilon_ {1 2} \\ \varepsilon_ {1 3} \\ \varepsilon_ {2 3} \end{array} \right\} = [ D ^ {e l} ] \left\{ \begin{array}{l} \varepsilon_ {1 1} \\ \varepsilon_ {2 2} \\ \varepsilon_ {3 3} \\ \varepsilon_ {1 2} \\ \varepsilon_ {1 3} \\ \varepsilon_ {2 3} \end{array} \right\}.
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$$
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Input File Usage: \*JOINT ELASTICITY, MODULI=GENERAL, NDIM=3
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<!-- source-page: 1034 -->
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# Joint plasticity
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In what follows $V ~ = ~ - \sigma _ { 1 1 } , H ~ = ~ \sigma _ { 2 2 }$ , and $M \ : = \ : \sigma _ { 1 2 }$ represent the vertical compressive load, the horizontal load in the 1–2 plane, and the bending moment in the local 1–2 plane, respectively.
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If plasticity is defined, the joint can yield axially, horizontally, or rotationally. The stress depends linearly on the elastic strain. The elastic moduli can depend on the plasticity in the case of a conical spud can, through the diameter at the surface, D.
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The models are rate independent, with a yield equation of the form
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$$
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f (\pmb {\sigma}, \mathbf {H}) \leq 0,
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$$
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where $f$ is the yield function and is a set of hardening parameters, which in these models depend on total vertical plastic embedment, $\nu _ { m }$ ; the form of f and the definition of defines the type of plasticity model.
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The flow rule requires that the plastic flow direction is normal to the contours of the flow potential, $\pmb { g } .$ Associated flow is assumed in all of these models (except at vertices in the yield surface, as discussed below).
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# Yield surface
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The three available plasticity models all use parabolic yield surfaces. Each has a compressive and a tensile limit for the stress in the 1-direction, which are termed $V _ { c }$ and $V _ { t } .$ , respectively; $V _ { t }$ is zero for the clay model. The sign convention for $V _ { c }$ and $V _ { t }$ is such that they are always positive; thus, $V = - \sigma _ { 1 1 }$ always obeys
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$$
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- V _ {t} \leq V \leq V _ {c}.
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$$
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The yield surface is most conveniently drawn in $( { \bar { V } } , { \bar { R } } )$ -space, where $\bar { V }$ is normalized compressive vertical load and is defined as
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$$
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\bar {V} = \frac {V - V _ {o}}{V _ {u}},
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$$
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where $\begin{array} { r } { V _ { o } = \frac { 1 } { 2 } ( V _ { c } - V _ { t } ) } \end{array}$ is the middle value of the limiting elastic range for $V ,$ and $\begin{array} { r } { V _ { u } = \frac { 1 } { 2 } ( V _ { c } + V _ { t } ) } \end{array}$ is the length of the limiting range for V. The normalized load is, therefore, always within the range
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$$
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- 1 \leq \bar {V} \leq 1,
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$$
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with $\bar { V } = - 1$ representing the tensile limit $V = - V _ { t }$ and $\bar { V } = 1$ representing the compressive limit $V = V _ { c }$ . is the normalized equivalent horizontal load and is defined through
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$$
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\bar {R} = \sqrt {\left(\frac {M}{M _ {m}}\right) ^ {2} + \left(\frac {H}{H _ {m}}\right) ^ {2}},
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$$
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<!-- source-page: 1035 -->
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where $M _ { m }$ and $H _ { m }$ are the moment and horizontal yield stresses. The normalized moment and normalized horizontal force are defined through $\bar { M } = M / M _ { m }$ and $\bar { H } = H / H _ { m }$ .
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The normalized yield function in $( { \bar { V } } , { \bar { R } } )$ -space for each model is defined through
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$$
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f = \bar {R} + \bar {V} ^ {2} - 1
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$$
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and is a parabola as plotted in Figure 32.10.1–2. The yield surface in the space of the three normalized stresses $\bar { ( V , M , H ) }$ is the surface of revolution of this parabola.
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<details>
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<summary>text_image</summary>
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"tensile" yield
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(softening)
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f, g = 0
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compressive yield
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(hardening)
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-1
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1
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V̄
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g = 0
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f = 0
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.95 1
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</details>
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Figure 32.10.1–2 Yield surface and flow potential contour.
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# Flow potential
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The flow potential is the same as the yield function (associated flow) except that some smoothing is done to the flow potential where the yield function has corners.
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The yield surface has corners and, therefore, nonunique normals at points where it is intersected by the -axis.
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<!-- source-page: 1036 -->
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To avoid problems with the indeterminate flow directions at these corners, Abaqus/Standard uses a flow potential whose contours are rounded in the region of the vertex, as indicated in the detail of a vertex shown in Figure 32.10.1–2. This rounding is achieved by fitting an elliptical segment to the flow potential contour for $| V | \geq 0 . 9 5$ .
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# Integration of the plasticity equations
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Abaqus/Aqua uses fully implicit integration for the plasticity equations. The corresponding tangent stiffness is unsymmetric for these plasticity models. By default, the symmetrized tangent is used in the global Newton loop. If the convergence rate seems to be poor, you may get some benefit out of using the unsymmetric matrix storage and solution scheme for the step (see “Defining an analysis,” Section 6.1.2).
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# Joint plasticity models
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The three models differ only in the definitions of $V _ { c } , V _ { t } , M _ { m }$ , and $H _ { m }$ and in the hardening definitions. We present the yield function for each model as it is presented in the literature rather than in normalized form. The equivalent normalized form can be obtained by identifying $M _ { m }$ and $H _ { m }$ , which are explicit in the given yield functions for clay and member plasticity; for the sand model they are provided for reference.
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# Sand model
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A. Yield function:
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$$
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f = \sqrt {\left(\frac {M}{D V _ {c}}\right) ^ {2} + \Lambda_ {1} \left(\frac {H}{V _ {c}}\right) ^ {2}} + \Lambda_ {2} [ (\frac {V}{V _ {c}}) ^ {2} - (1 - \frac {V _ {t}}{V _ {c}}) \frac {V}{V _ {c}} - \frac {V _ {t}}{V _ {c}} ] = 0,
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$$
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where $\Lambda _ { 1 }$ and $\Lambda _ { 2 }$ are constant coefficients that determine the geometric shape of the yield function. The special case of $\Lambda _ { 1 } = 1 . 0 , \Lambda _ { 2 } = 0 . 5$ and $V _ { t } = 0 . 0$ gives the yield function as proposed by Osborne, et al.
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B. Work hardening equations:
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i. Flat-base spud can:
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$$
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\frac {V _ {c}}{A D _ {o} \gamma} = 0. 3 N _ {\gamma} (1 - e ^ {- \alpha \nu_ {m} / D _ {o}}) + N _ {q} \nu_ {m} / D _ {o},
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$$
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where $\gamma$ is soil unit weight; is an experimentally determined constant; and $N _ { \gamma }$ and $N _ { q }$ are classical bearing capacity factors, which can be calculated as:
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$$
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N _ {q} = e ^ {\pi \tan \phi} \tan^ {2} \left(4 5 + \frac {\phi}{2}\right),
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$$
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$$
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N _ {\gamma} = 2 (N _ {q} + 1) \tan \phi ,
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$$
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where $\phi$ is the soil friction angle.
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ii. Conical-base spud can:
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a. Cone portion partially penetrating:
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<!-- source-page: 1037 -->
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$$
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\frac {V _ {c}}{A D \gamma} = 0. 3 N _ {\gamma} (1 - e ^ {- \alpha \beta \nu_ {m} / D}) + N _ {q} \beta \nu_ {m} / D.
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$$
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b. Penetration beyond cone-cylinder transition:
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$$
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\frac {V _ {c}}{A D _ {o} \gamma} = 0. 3 N _ {\gamma} (1 - e ^ {- \alpha (\nu_ {m} - \nu_ {c} + \beta \nu_ {c}) / D _ {o}}) + N _ {q} (\nu_ {m} - \nu_ {c} + \beta \nu_ {c}) / D _ {o},
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$$
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where $\beta$ is a “cone equivalency coefficient.”
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The constants and $\beta$ are based on the following empirical relation, which has been derived from centrifuge data:
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$$
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\alpha = 1. 9 5 4 \times 1 0 ^ {- 9} \phi^ {6. 1 2 9}
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$$
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$$
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\beta = 0. 7 1 - 0. 0 1 4 \phi
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$$
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in which the soil friction angle $\phi$ is in degrees.
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The sand model yield function can be put in normalized form by using $M _ { m } = \kappa D V _ { c }$ and $H _ { m } =$ $\kappa V _ { c } \Lambda _ { 1 } ^ { - . 5 }$ where $\kappa = \Lambda _ { 2 } ( 1 + V _ { t } / V _ { c } ) ^ { 2 } / 4$ . For the model of Osborne et al. $\kappa = 1 / 8$ .
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This model requires a nonzero initial embedment or equivalent preload.
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Input File Usage: \*JOINT PLASTICITY, TYPE=SAND
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# Clay model
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A. Yield function:
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$$
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f = \sqrt {\left(\frac {M}{8 M _ {m}}\right) ^ {2} + \left(\frac {H}{8 H _ {m}}\right) ^ {2}} - 0. 5 (\frac {V}{V _ {c}}) (1 - \frac {V}{V _ {c}}) = 0,
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$$
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where
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$$
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M _ {m} = \frac {V _ {c} D}{3 \pi},
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$$
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$$
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H _ {m} = s _ {u} (A + 2 A _ {h}).
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$$
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$s _ { u }$ is the undrained shear strength of clay; and $A _ { h }$ is the elevation area of the embedded portion of the spud can, defined through:
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i. Flat-base spud can:
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$$
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A _ {h} = D _ {o} \nu_ {m}
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$$
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ii. Conical-base spud can:
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a. Cone portion penetrating:
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<!-- source-page: 1038 -->
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$$
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A _ {h} = 0. 5 D \nu_ {m} = \nu_ {m} ^ {2} \tan \frac {\theta}{2}
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$$
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b. Penetration beyond cone-cylinder transition:
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$$
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A _ {h} = D _ {o} \Big (\nu_ {m} - \frac {0 . 2 5 D _ {o}}{\tan \frac {\theta}{2}} \Big)
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$$
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B. Work hardening equations:
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i. Flat-base spud can:
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$$
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V _ {c} = a + b \nu_ {m}
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$$
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ii. Conical-base spud can:
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$$
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V _ {c} = \frac {\nu_ {m} - c}{a + b \nu_ {m}}
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$$
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where $a , b ,$ and c are user-defined empirical constants.
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This model has zero yield strength in tension $( V _ { t } = 0 )$ and requires a nonzero initial embedment or equivalent preload.
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Input File Usage: \*JOINT PLASTICITY, TYPE=CLAY
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Parabolic model for structural joints/members
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A. Yield function:
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$$
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f = \sqrt {\left(\frac {M}{M _ {m}}\right) ^ {2} + \left(\frac {H}{H _ {m}}\right) ^ {2}} ] + \left(\frac {V - V _ {o}}{V _ {u}}\right) ^ {2} - 1 = 0,
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$$
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where $M _ { m } , H _ { m }$ are horizontal and moment capacities, respectively.
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B. Work hardening: no work hardening is assumed (the model is perfectly plastic).
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Input File Usage: \*JOINT PLASTICITY, TYPE=MEMBER
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# Plasticity analysis issues
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Because associated flow is assumed in the spud can plasticity models, tensile vertical plastic strain can occur whenever the yield surface is encountered with $\bar { V } < 0$ . It is not required that the vertical force itself be tensile for tensile plastic yield to occur; tensile plastic yield can occur on any part of the yield surface where $V < V _ { o }$ . The spud can models soften during this tensile plastic yield; if there is insufficient support from the rest of the model, an instability can occur and the analysis may fail to converge. When this happens, the spud can is likely to be lifting out of the sea floor.
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To make it easier to diagnose analysis problems that may arise due to these issues, a message is printed to the message file in the following cases: if tensile plastic yield occurs for a spud can, if yield occurs near the top of the parabolic yield surface $( \bar { V } < 0 . 1 )$ where there is very little hardening, or if
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<!-- source-page: 1039 -->
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the embedment of a spud can becomes less than 10% of the initial embedment. These messages are not printed more than once in a given step.
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The plasticity algorithm can fail in an iteration if the strain increment is excessively large. Some details that may be of help in diagnosing failure in joint elements can be obtained by requesting detailed printout to the message file of problems with the plasticity algorithms (see “The Abaqus/Standard message file” in “Output,” Section 4.1.1).
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