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Elements
\begin{array}{l} Z _ {e} \\ = 2 I _ {2 2} / D = \frac {\pi}{3 2 D} (D ^ {4} - (D - 2 t) ^ {4}), \\ Z _ {p} \\ = [ D ^ {3} - (D - 2 t) ^ {3} ] / 6, \\ = \sqrt {I _ {2 2} / A} = \frac {1}{4} \sqrt {D ^ {2} + (D - 2 t) ^ {2}}. \\ \end{array}
The terms in the ISO equation are calculated as follows:
f _ { c }
is the axial compressive stress, f _ { c } = P / A with P the axial force in the element.
f _ { b 1 } , \ f _ { b 2 }
are the maximum bending stresses about the 1 or 2 cross-section axis,
f _ {b 1} = M _ {1} / Z _ {e}, f _ {b 2} = M _ {2} / Z _ {e} \text {with} M _ {1} \text {and} M _ {2}
the bending moments about the 1 and 2 direction.
F _ { y c }
is the characteristic local buckling stress,
F _ {y c} = \sigma^ {0} \text { for } \frac {\sigma^ {0}}{F _ {e}} \leq 0. 1 7 0,
F _ {y c} = \left(c _ {2} - c _ {3} \frac {\sigma^ {0}}{F _ {e}}\right) \sigma^ {0} \text {for} \frac {\sigma^ {0}}{F _ {e}} \leq 1. 9 1 1, \text {where} c _ {2} = 1. 0 4 6 5 4 8 7 3 \text {and}
c _ {3} = 0. 2 7 3 8 1 6 0 6,
F _ {y c} = F _ {e} \mathrm{for} \frac {\sigma^ {0}}{F _ {e}} > 1. 9 1 1,
F _ {e} = 2 C E (t / D),
C = 0 . 3 is a critical buckling coefficient.
F _ { c }
is the characteristic axial compressive stress,
F _ {c} = [ 1. 0 - 0. 2 8 \lambda^ {2} ] F _ {y c} \text { for } \lambda \leq 1. 3 4,
F _ {c} = \frac {c _ {1}}{\lambda^ {2}} F _ {y c} \text {for} \lambda > 1. 3 4, \text {where} c _ {1} = 0. 8 9 2 8 2 9 7 8,
with \lambda = \operatorname* { m a x } ( \lambda _ { 1 } , \lambda _ { 2 } ) ,
\lambda_ {1} = \frac {k _ {1} L _ {1}}{\pi r} \sqrt {\frac {F _ {y c}}{E}}, \lambda_ {2} = \frac {k _ {2} L _ {2}}{\pi r} \sqrt {\frac {F _ {y c}}{E}}.
F _ { b }
is the characteristic bending stress (for PIPE sections \bar { \xi } = \sigma ^ { 0 } D / E t ) ,
F _ {b} = (Z _ {p} / Z _ {e}) \sigma^ {0} \text { for } \bar {\xi} \leq 0. 0 5 1 7,
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F _ {b} = (c _ {4} - 2. 5 8 \bar {\xi}) (Z _ {p} / Z _ {e}) \sigma^ {0} \mathrm{for} 0. 0 5 1 7 < \bar {\xi} \leq 0. 1 0 3 4,
F _ {b} = \left(c _ {5} - 0. 7 6 \bar {\xi}\right) \left(Z _ {p} / Z _ {e}\right) \sigma^ {0} \text {for} 0. 1 0 3 4 < \bar {\xi} \leq 1 2 0 \sigma^ {0} / E,
where c _ { 4 } = 1 . 1 3 3 3 8 6 \ { \mathrm { a n d } } \ c _ { 5 } = 0 . 9 4 5 1 9 8 . .
F _ {e 1}, F _ {e 2}
are Euler buckling stresses corresponding to the 1 or 2 directions,
F _ {e 1} = \frac {F _ {y c}}{\lambda_ {1} ^ {2}} \mathrm{and} F _ {e 2} = \frac {F _ {y c}}{\lambda_ {2} ^ {2}}.
c _ {m 1}, c _ {m 2}
are reduction factors corresponding to the cross-section directions 1 and 2, respectively. These factors are user-defined as functions of the end moments, compression stress, and Euler buckling stresses. The default value for each factor is 0:85.
If switching between standard frame element response and buckling strut response is permitted, the one-time-only switch to buckling strut response occurs when I ( f _ { c } , b _ { b 1 } , f _ { b 2 } ) = 1 . 0 . The ISO equation provides the value of critical load, P _ { c r } , which is defined as the value of axial force f _ { c } A in the element when the ISO equation is satisfied. To prevent switching in cases where negligible axial force exists with large bending moments, an additional inequality is used. This additional check, called the strength equation, takes the following form:
S = \frac {f _ {c}}{F _ {y c}} + \frac {1}{F _ {b}} \sqrt {f _ {b 1} ^ {2} + f _ {b 2} ^ {2}}.
For a frame element to switch to buckling strut behavior, both the ISO equation and the strength equation must be satisfied, I = 1:0 and S \le 1 . 0 . If buckling strut response is requested for the element from the beginning of the analysis, bending moments cannot be supported by the element. In this case the ISO equation becomes the simplified statement that no buckling occurs for
f _ {c} < F _ {c} \text { and } P _ {c r} = F _ {c} A.
Marshall strut envelope
The Marshall strut envelope defines the postbuckling damaged elasticity model and the hysteretic loop response. To define the Marshall strut envelope, the value of P _ { c r } and the following seven constants are needed:
\xi is the coefficient defining P_{y} = \xi \sigma^{0}A ( \xi = 0.95 ), \gamma is the isotropic hardening slope coefficient (0.02), \alpha_{0} is the coefficient defining \alpha = \alpha_{0} + \alpha_{1}\frac{L}{D} , ( \alpha_{0} = 0.03 ),
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\alpha_{1} is the coefficient defining \alpha = \alpha_{0} + \alpha_{1}\frac{L}{D}, (\alpha_{1} = 0.004) , \kappa is the force coefficient (0.28), \beta is the slope coefficient (0.02), and \zeta is the force coefficient (\min(1.0, 5.8(\frac{t}{D})^{0.7}/\xi)) .
The values in parentheses are the default values supplied by ABAQUS, and the value of P _ { c r } is found from the ISO equation as explained above.
The Marshall envelope governs the compressive and tensile response of the strut as shown in Figure 3.9.4-1. The dotted lines in the interior of the envelope indicate the damaged-elastic modulus defining the loading-unloading force versus strain path.
Figure 3.9.4-1 Marshall strut theory buckling envelope.
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force P_y ζP_y γEA EA strain βEA αEA κP_cr P_cr
3.9.5 Tube support elements
These elements are provided for the specific case of modeling the interaction between a tube and a support that is not always in contact with the tube during dynamic events. The tube is assumed to have a circular section and can interact with one of two tube support geometries: a circular hole and an "egg-crate" support. Two interface elements of this type are provided, one for each geometry, as shown in Figure 3.9.5-1 and Figure 3.9.5-2. As indicated in Figure 3.9.5-2, one cylindrical geometry interface is needed to model the interaction of the tube with a circular hole, while Figure 3.9.5-1 shows that
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several unidirectional geometry elements are needed to model the interaction with an egg-crate--one element perpendicular to each pair of egg-crate faces.
Figure 3.9.5-1 ITSUNI elements for tube/"egg-crate" support interaction.
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Tube ITSUNI elements C of tube 1 2 2 n₁ Center of opening in support plates n₂ Parallel support plates for element 2 n₁ Parallel support plates for element 1
Figure 3.9.5-2 ITSCYL element for tube/drilled hole support interaction.
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Tube center Tube C of tube 1 2 Tube support plate Center of hole ITSCYL element
The interface elements themselves consist of a spring and friction link and a dashpot, as shown in
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Figure 3.9.5-3. The spring is assumed to behave as shown in Figure 3.9.5-4: when there is no contact between the tube and the support, no force is transmitted by the spring; when the tube is in contact with the support, the force increases as the tube wall is deformed. This force can be modeled as a linear or a nonlinear function of the relative displacement between the axis of the tube and the center of the hole in the support.
Figure 3.9.5-3 Tube support element behavior.
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Spring (linear or nonlinear) Friction Dashpot (linear or nonlinear) 1 Q P₃ P₃ Q P₃
Figure 3.9.5-4 Nonlinear spring behavior in ITS elements to model clearance and tube flattening.
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ITSUNI P₃ Stiffness associated with tube wall flattening -c₀ c₀ u₃ c₀ = clearance between tube and support side in fully aligned position
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| u₃ | P₃ |
|---|---|
| c₀ | 0 |
| u₃ | >c₀ |
C。= difference between support plate hole radius and tube outside radius
The frictional part of the spring and friction link uses the Coulomb friction model in ABAQUS: that model is described in ``Coulomb friction,'' Section 5.2.3.
The dashpot is provided to model fluid effects in the annulus between the tube and the support plate. Its behavior can be linear or nonlinear. The model assumes that shear forces created by the fluid are negligible, so that the only shear forces transmitted by one of these interface elements are the frictional forces caused by direct contact between the tube and the support.
A major simplification in these elements that saves considerable computational effort in dynamic applications is the assumption that impacts between the tube and its support plates involve no instantaneous transfer of momentum or energy loss: the standard impact algorithm of ABAQUS/Standard used with gap and other interface elements (and described in ``Intermittent contact/impact,'' Section 2.4.2) is not needed. This simplification derives from the assumption that these elements will be used in conjunction with beam element models of the tube, so the tube section is defined by the position and orientation of its axis and local deformation of the cross-section of the tube is neglected. In reality, when the tube hits a support, initially only a small part of the tube wall loses momentum so that there is--instantaneously--only a small loss of kinetic energy. This instantaneous energy loss is neglected when these elements are used. The subsequent flattening of the tube wall is modeled by the spring link in the element, acting between the node on the tube axis and the node
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representing the center of the hole. Thus, the modeling of this local flattening behavior as an equivalent spring provides the simplification that instantaneous impact calculations are not needed. In cases where this approach is not reasonable, gap elements can be used instead of these special interface elements, at the cost of more computational effort.
The remainder of this section discusses the kinematic definitions used in these elements and their contributions to the overall equilibrium equations and to the Jacobian (stiffness) matrix needed in the Newton solution of those equations.
Geometry and kinematics
Each tube support element has two nodes. One node represents the axis of the tube, the other is the center of the hole in the support plate (or midway between a pair of parallel sides of an "egg-crate").
Let \mathbf { a } _ { 2 } be a unit vector along the axis of the tube, and let \mathbf { a } _ { 3 } be a unit vector along the axis of the interface element (that is, perpendicular to the parallel sides of the support in the "unidirectional" interface that is used with egg-crate supports, and parallel to the line joining the two nodes of the element in the "cylindrical" interface that is used with circular holes). It is assumed that \mathbf { a } _ { 3 } is in the cross-section of the tube and, hence, orthogonal to \mathbf { a } _ { 2 } . We define a third basis vector as
\mathbf {a} _ {1} = \mathbf {a} _ {2} \times \mathbf {a} _ {3}.
Let \mathbf { x } ^ { N } be the current position of node N of the element at any point in time (here N is 1 or 2).
Relative displacements in the element are measured from the position when the tube and the hole in the support plate are exactly aligned; that is, when the nodes of the element are at the same location. They are defined as follows:
axial to the interface element,
u _ {3} = \left(\mathbf {x} ^ {1} - \mathbf {x} ^ {2}\right) \cdot \mathbf {a} _ {3};
axial to the tube,
u _ {2} = \left(\mathbf {x} ^ {1} - \mathbf {x} ^ {2}\right) \cdot \mathbf {a} _ {2};
and
tangential, in the plane of the tube's cross-section,
for the unidirectional case:
u _ {1} = \left(\mathbf {x} ^ {1} - \mathbf {x} ^ {2}\right) \cdot \mathbf {a} _ {1},
and, for the cylindrical case:
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u _ {1} = \int \left(d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2}\right) \cdot \mathbf {a} _ {1}.
The basis vector--a2, along the axis of the tube and of the hole in the support plate--is assumed to be fixed. In the unidirectional element the { \bf a } _ { 1 } and { \bf a } _ { 3 } vectors are also fixed. In the cylindrical interface { \bf a } _ { 3 } is parallel to the line joining the nodes of the element, so
\mathbf {a} _ {3} = (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) / l,
where
l = \sqrt {(\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) \cdot (\mathbf {x} ^ {1} - \mathbf {x} ^ {2})}.
Therefore,
\mathbf {a} _ {1} = \mathbf {a} _ {2} \times \mathbf {a} _ {3} = \mathbf {a} _ {2} \times (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) / l.
Thus, for this element
u _ {1} = \int \frac {1}{l} \mathbf {a} _ {2} \times (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) \cdot (d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2})
and
d u _ {1} = \frac {1}{l} \mathbf {a} _ {2} \times (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) \cdot (d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2}).
For simplicity we replace the integral with the backward difference approximation
\Delta u _ {1} = [ \frac {1}{l} \mathbf {a} _ {2} \times (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) ] \bigg | _ {t + \Delta t} \cdot (\Delta \mathbf {x} ^ {1} - \Delta \mathbf {x} ^ {2}).
Forces in the element
The element generates an axial force-- P _ { 3 } , parallel to { \bf a } _ { 3 } --and two shear forces-- { \bf \nabla } \cdot Q _ { 1 } , parallel to { \bf a } _ { 1 } ; and Q _ { 2 } , parallel to \mathbf { a } _ { 2 } . In addition, because the nodes of the element are at the center of the tube and at the center of the hole in the support, while the interaction forces between the tube and its support are transmitted at the point of contact of the tube with the support, these forces also cause moments at the nodes of the element. It is assumed that the moments caused by P _ { 3 } and by Q _ { 2 } are not significant and can be neglected. The only moments considered are Q _ { 1 } d _ { 1 } / 2 at node 1, the center of the tube, and Q _ { 1 } d _ { 2 } / 2 at node 2, the center of the hole. Here d _ { 1 } is the outside diameter of the tube and d _ { 2 } is the diameter of the hole for the cylindrical interface or is the distance between the parallel support plates for the uniaxial interface (see Figure 3.9.5-5).
Figure 3.9.5-5 Contact forces in the cross-section.
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P₁ P₃ Q₁ P₃ d₁/2 d₂/2 +1 +2 Q₁ Q₁ P₃ P₃ d₁/2 d₂/2 Q₁ Q₁
The virtual work contribution of the element is, then,
\delta W ^ {I} = P _ {3} \delta u _ {3} + Q _ {1} \delta u _ {1} + Q _ {2} \delta u _ {2} + Q _ {1} \frac {d _ {1}}{2} \delta \phi_ {2} ^ {1} - Q _ {1} \frac {d _ {2}}{2} \delta \phi_ {2} ^ {2},
where \delta \phi _ { 2 } ^ { N } is the virtual rate of rotation about the \mathbf { a } _ { 2 } \mathbf { - a } \mathbf { x } \mathbf { i } \mathbf { s } at node N .
From this expression the contribution of the element to the Jacobian (stiffness) matrix of the equilibrium equations is immediately available as
\begin{array}{l} d \delta W ^ {I} = \delta u _ {3} d P _ {3} + \left(\delta u _ {1} + \frac {d _ {1}}{2} \delta \phi_ {2} ^ {1} + \frac {d _ {2}}{2} \delta \phi_ {2} ^ {2}\right) d Q _ {1} \\ + \delta u _ {2} d Q _ {2} + P _ {3} d \delta u _ {3} + Q _ {1} d \delta u _ {1}. \\ \end{array}
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The "initial stress" terms,
P _ {3} d \delta u _ {3} + Q _ {1} d \delta u _ {1},
are only nonzero for the cylindrical interface, for which
u _ {3} = \sqrt {(\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) \cdot (\mathbf {x} ^ {1} - \mathbf {x} ^ {2})} = l,
so that
\delta u _ {3} = \mathbf {a} _ {3} \cdot (\delta \mathbf {x} ^ {1} - \delta \mathbf {x} ^ {2}),
and so
d \delta u _ {3} = \frac {1}{u _ {3}} (\delta \mathbf {x} ^ {1} - \delta \mathbf {x} ^ {2}) \cdot [ \mathbf {I} - \mathbf {a} _ {3} \mathbf {a} _ {3} ] \cdot (d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2}).
Also, for this element,
\delta u _ {1} = \frac {1}{u _ {3}} \mathbf {a} _ {2} \times (\mathbf {x} ^ {1} - \mathbf {x} ^ {2}) \cdot (\delta \mathbf {x} ^ {1} - \delta \mathbf {x} ^ {2}),
and so
d \delta u _ {1} = \frac {1}{u _ {3}} (\delta \mathbf {x} ^ {1} - \delta \mathbf {x} ^ {2}) \cdot [ \mathbf {a} _ {3} \mathbf {a} _ {1} - 2 \mathbf {a} _ {1} \mathbf {a} _ {3} ] \cdot (d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2}).
This term is not symmetric.
The "initial stress" terms for the cylindrical interface are, therefore,
P _ {3} d \delta u _ {3} + Q _ {1} d \delta u _ {1} =
\left(\delta \mathbf {x} ^ {1} - \delta \mathbf {x} ^ {2}\right) \cdot \frac {1}{u _ {3}} \bigg [ P _ {3} [ \mathbf {I} - \mathbf {a} _ {3} \mathbf {a} _ {3} ] + Q _ {1} [ \mathbf {a} _ {3} \mathbf {a} _ {1} - 2 \mathbf {a} _ {1} \mathbf {a} _ {3} ] \bigg ] \cdot (d \mathbf {x} ^ {1} - d \mathbf {x} ^ {2}).
The other terms in the stiffness matrix are associated with changes in the forces in the element, d P _ { 3 } , d Q _ { 1 } , and d Q _ { 2 } . We assume P _ { 3 } is made up of a spring force that is a function of u _ { 3 } and a dashpot force that is a function of \dot { u } _ { 3 } :
P _ {3} = P _ {3} ^ {s} + P _ {3} ^ {d},
P _ {3} ^ {s} = P _ {3} ^ {s} (u _ {3}), \quad P _ {3} ^ {d} = P _ {3} ^ {d} (\dot {u} _ {3}),