김경종
221 lines
20 KiB
Markdown
221 lines
20 KiB
Markdown
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relative motion. For example, the REVOLUTE connection type has one available component of relative motion, $u r _ { 1 }$ , and two kinematic rotation constraints (equivalent to setting two rotation components, and $u r _ { 3 }$ , to zero). Conjugate to the available rotation component is the kinetic moment $m _ { 1 }$ acting about the local -direction.
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In general, kinetic forces and moments include the effects of connector behaviors, such as elastic springs, viscous damping, friction, and reaction forces and moments due to connector stops and locks. For constitutive response defined as a function of displacement or rotation, the initial position may not correspond with the reference position where constitutive forces and moments are zero. You can define reference lengths and angles (given in degrees) for connector behavior as described in “Defining reference lengths and angles for constitutive response” in “Connector behavior,” Section 31.2.1. These reference quantities define $u _ { i } ^ { m a t }$ and , the connector constitutive displacements and rotations. These constitutive displacements and rotations are used only to define constitutive response and differ from the relative displacements and rotations measured in the connector elements only when you define the reference lengths or angles.
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As an example, if the REVOLUTE connection included linear spring and dashpot behavior combined with a connector stop,
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$$
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m _ {1} = K _ {1} u r _ {1} ^ {m a t} + C _ {1} \dot {u} r _ {1} + R M _ {1},
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$$
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where $K _ { 1 }$ is the spring stiffness, $C _ { 1 }$ is the dashpot coefficient, and $R M _ { 1 }$ is the reaction moment caused by the connector stop. In the REVOLUTE connection there are two constraint moment components, about ${ \bf e } _ { 2 } ^ { a }$ and $m _ { 3 }$ about ${ \bf e } _ { 3 } ^ { a }$ .
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# Interpreting connector forces and moments
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The kinematic constraint and kinetic forces and moments are always computed as work conjugates of the kinematics in the connector (components of relative motion). In most connection types one direct consequence is that the constraint forces (and moments) in connectors are reported as the forces (and moments) applied at the second node but in the local system associated with the first node. Since the kinematics are complex in many of the connection types, the connector forces and moments can be somewhat surprising upon first inspection. For example, consider the case of a HINGE connection defined with the local -direction aligned with the global X-direction and the local -direction aligned with the global Y-direction. Assume that the second connector node is grounded and that the first node is subjected to a concentrated load along the global Y-direction. If the only available relative rotation in the HINGE is constrained with a zero-valued connector motion, the second node does not rotate with respect to the first node and the connector reaction force along the local -direction matches the applied load while the other two connector reaction forces are zero. However, if a nonzero connector motion is specified, the first connector reaction force is still zero while both the second and third connector reaction forces are nonzero and only the vector-norm of these two forces matches the applied load. In both cases the only nonzero nodal reaction force at the second connector node is the one in the global Y-direction, as dictated by the equilibrium in a free body diagram. Hence, the connector reaction forces and nodal reaction forces are not equivalent in most cases.
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# Coulomb-like friction behavior
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Coulomb-like friction behavior is possible for any connection type that has available components of relative motion; see “Connector friction behavior,” Section 31.2.5, for details. Friction behavior requires a “tangent” direction (the direction in which slipping may occur) and a “normal” direction (the direction perpendicular to the contacting surfaces). In the most general case you define the normal force causing friction in the connector. However, Abaqus predefines friction behavior for a limited number of connection types, as discussed in the connection-type library in this section. In these predefined friction cases you do not have to define the contact normal force.
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# Summary table
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Each connection library entry includes a table summarizing the connection type. This summary table indicates whether the connection type is basic, assembled, or complex. It gives the kinematic constraints; constraint force or moment components; available components of relative motion; “kinetic” force or moment components following from the constitutive behavior in the available components of relative motion; which orientation directions are required, optional, or ignored; how connector stops limit the available components of relative motion; what reference lengths and angles are used to define the constitutive behavior; what parameters are used for predefined Coulomb-like friction; and how the contact normal forces are defined by Abaqus in association with predefined Coulomb-like friction.
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# Basic connection components
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Basic connection components are divided into three categories:
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• Translational basic connection components, which affect translational degrees of freedom at both nodes and may affect rotational degrees of freedom at the first node or at both nodes
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• Rotational basic connection components, which affect only rotational degrees of freedom at both nodes
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• Specialized rotational basic connection components, which in addition to rotational degrees of freedom affect other degrees of freedom at the nodes
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Only one translational basic connection component and one rotational or specialized rotational basic connection component can be used in the definition of a connector element. If a more complicated connection requires more basic connection components than this, use multiple connector elements attached to the same nodes.
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# Translational basic connection components
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The following basic connection components affect translational degrees of freedom at both node a and node b. Some of these connector components affect rotational degrees of freedom at node a or at both node a and node b. Any basic connection component from this list can be used to define the translational behavior of a connector element.
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<table><tr><td>ACCELEROMETER</td><td>Provide a connection between two nodes to measure the relative acceleration, velocity, and position of a body in a local coordinate system. This connection type is available only in Abaqus/Explicit. If it is defined in an Abaqus/Standard model, it will be converted internally to a CARTESIAN connector type.</td></tr><tr><td>AXIAL</td><td>Provide a connection between two nodes that acts along the line connecting the nodes.</td></tr><tr><td>CARTESIAN</td><td>Provide a connection between two nodes that allows independent behavior in three local Cartesian directions that follow the system at node a.</td></tr><tr><td>JOIN</td><td>Join the position of two nodes.</td></tr><tr><td>LINK</td><td>Provide a pinned rigid link between two nodes to keep the distance between the two nodes constant.</td></tr><tr><td>PROJECTION CARTESIAN</td><td>Provide a connection between two nodes that allows independent behavior in three local Cartesian directions that follow the system at both nodes a and b.</td></tr><tr><td>RADIAL-THRUST</td><td>Provide a connection between two nodes that allows different behavior for radial and thrust displacements.</td></tr><tr><td>SLIDE-PLANE</td><td>Provide a slide-plane connection to make the position of the second node remain on a plane defined by the orientation of the first node and the initial position of the second node.</td></tr><tr><td>SLOT</td><td>Provide a slot connection to make the position of the second node remain on a line defined by the orientation of the first node and the initial position of the second node.</td></tr></table>
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# Rotational basic connection components
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The following basic connection components affect only rotational degrees of freedom at the nodes in the connection. Any basic connection component from this list can be used to define the rotational behavior of a connector element.
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<table><tr><td>ALIGN</td><td>Provide a connection between two nodes that aligns their local directions.</td></tr><tr><td>CARDAN</td><td>Provide a rotational connection between two nodes parameterized by Cardan (or Bryant) angles.</td></tr><tr><td>CONSTANT VELOCITY</td><td>Provide a constant velocity connection between two nodes.</td></tr><tr><td>EULER</td><td>Provide a rotational connection between two nodes parameterized by Euler angles.</td></tr><tr><td>FLEXION-TORSION</td><td>Provide a connection between two nodes that allows different behavior for flexural and torsional rotations.</td></tr></table>
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<table><tr><td>PROJECTION FLEXION-TORSION</td><td>Provide a connection between two nodes that allows different behavior for two flexural rotations and one torsional rotation.</td></tr><tr><td>REVOLUTE</td><td>Provide a revolute connection between two nodes.</td></tr><tr><td>ROTATION</td><td>Provide a rotational connection between two nodes parameterized by the rotation vector.</td></tr><tr><td>ROTATION-ACCELEROMETER</td><td>Provide a connection between two nodes to measure the relative angular acceleration, velocity, and position of a body in a local coordinate system. This connection type is available only in Abaqus/Explicit. If it is defined in an Abaqus/Standard model, it will be converted internally to a CARDAN connector type.</td></tr><tr><td>UNIVERSAL</td><td>Provide a universal connection between two nodes.</td></tr></table>
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# Specialized rotational basic connection components
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The following basic connection component affects rotational and other non-translational degrees of freedom at the nodes in the connection. The specialized rotational basic connection component can be combined with translational basic connection components.
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FLOW-CONVERTER Provide a means of converting the material flow (degree of freedom 10) at a connector node into a rotation.
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Assembled connections
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<table><tr><td colspan="2">Assembled connections are included for convenience. Each assembled connection is created by combinations of basic connection components. The equivalent basic connection components used for each assembled connection are listed in parentheses.</td></tr><tr><td>BEAM</td><td>Provide a rigid beam connection between two nodes. (JOIN + ALIGN)</td></tr><tr><td>BUSHING</td><td>Provide a connection between two nodes that allows independent behavior in three local Cartesian directions that follow the system at both nodes a and b and that allows different behavior in two flexural rotations and one torsional rotation. (PROJECTION CARTESIAN + PROJECTION FLEXION-TORSION)</td></tr><tr><td>CVJOINT</td><td>Join the position of two nodes, and provide a constant velocity connection between their rotational degrees of freedom. (JOIN + CONSTANT VELOCITY)</td></tr><tr><td>CYLINDRICAL</td><td>Provide a slot connection between two nodes, and constrain the rotations by a revolute connection. (SLOT + REVOLUTE)</td></tr><tr><td>HINGE</td><td>Join the position of two nodes, and provide a revolute connection between their rotational degrees of freedom. (JOIN + REVOLUTE)</td></tr></table>
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<table><tr><td>PLANAR</td><td>Provide a slide-plane connection between two nodes with a revolute connection about the normal direction to the plane. The PLANAR connection creates a local two-dimensional system in three-dimensional analyses. (SLIDE-PLANE + REVOLUTE)</td></tr><tr><td>RETRACTOR</td><td>Join the position of two nodes, and convert material flow into rotation. (JOIN + FLOW-CONVERTER)</td></tr><tr><td>TRANSLATOR</td><td>Provide a slot connection between two nodes, and align their three local axis directions. (SLOT + ALIGN)</td></tr><tr><td>UJOINT</td><td>Join the position of two nodes, and provide a universal connection between their rotational degrees of freedom at the nodes. (JOIN + UNIVERSAL)</td></tr><tr><td>WELD</td><td>Join the position of two nodes, and align their three local axis directions. (JOIN + ALIGN)</td></tr></table>
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# Complex connections
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Complex connections affect a combination of degrees of freedom at the nodes in the connection and cannot be combined with other connection components. They typically model highly coupled physical connections.
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SLIPRING Model material flow and stretching between two points of a belt system (such as an automotive seat belt).
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# Connection-type library
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The following descriptions list all the basic connection components and assembled connections in alphabetical order.
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# ACCELEROMETER
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Connection type ACCELEROMETER provides a convenient way to measure the relative position, velocity, and acceleration of a body in a local coordinate system. All kinematic quantities are measured relative to node a. While position and displacement are reported in the coordinate system of node ${ \pmb a } ,$ velocity and acceleration are reported in the coordinate system of node b. Each node of the connector can translate and rotate independently, although fixing the first of the two nodes to ground is more common. With the first node fixed, connection type ACCELEROMETER provides a convenient way to measure the local components of the velocity and acceleration in a coordinate system fixed to a moving body (for example, an accelerometer).
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Connection type ACCELEROMETER is available only in Abaqus/Explicit. It is the translation counterpart to connection type ROTATION-ACCELEROMETER, which measures relative angular position, velocity, and acceleration. ACCELEROMETER connections cannot be used in two-dimensional and axisymmetric analyses in Abaqus/Explicit.
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<details>
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<summary>text_image</summary>
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</details>
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Figure 31.1.5–2 Connection type ACCELEROMETER.
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# Description
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The ACCELEROMETER connection does not impose kinematic constraints. It defines three local directions at node a and three local directions at node b. The ACCELEROMETER connection’s formulation is similar to that for the CARTESIAN connection. The ACCELEROMETER connection measures the position of node b relative to node a
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$$
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x = \mathbf {e} _ {1} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}); \quad y = \mathbf {e} _ {2} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}); \quad \mathrm{and} \quad z = \mathbf {e} _ {3} ^ {a} \cdot (\mathbf {x} _ {b} - \mathbf {x} _ {a}).
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$$
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There are no available components of relative motion for the ACCELEROMETER connection. The connector displacement components are
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$$
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u _ {1} = x - x _ {0}, \quad u _ {2} = y - y _ {0}, \quad \text {and} \quad u _ {3} = z - z _ {0},
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$$
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where $x _ { 0 } , y _ { 0 }$ , and $z _ { 0 }$ are the initial coordinates of node b relative to node a.
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The ACCELEROMETER connection measures velocity and acceleration in the local directions at node a as if node a were an inertial frame. In contrast to the CARTESIAN connection, the ACCELEROMETER connection reports the computed velocity and acceleration in the local directions at node b. Let $T _ { i j }$ be the transformation from ${ \bf e } _ { i } ^ { a }$ to $\mathbf { e } _ { i } ^ { b } .$ . Then the ACCELEROMETER connection measures velocity and acceleration as
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$$
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v _ {i} = T _ {i j} \frac {\mathrm{d} u _ {j}}{\mathrm{d} t} \quad \mathrm{and} \quad a _ {i} = T _ {i j} \frac {\mathrm{d} ^ {2} u _ {j}}{\mathrm{d} t ^ {2}},
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$$
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where the derivatives above are time derivatives in a system moving with ${ \bf e } _ { i } ^ { a }$ .
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In two-dimensional and axisymmetric analyses $u _ { 3 } = 0$ .
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Summary
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<table><tr><td colspan="2">ACCELEROMETER</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td>None</td></tr><tr><td>Constraint force output:</td><td>None</td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic force output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference lengths:</td><td>None</td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>
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# ALIGN
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Connection type ALIGN provides a connection between two nodes where all three local directions are aligned. If both local axes are given and do not align initially, their initial relative angular position is held constant.
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<details>
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<summary>text_image</summary>
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e₃ᵃ
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e₃ᵇ
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b
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a
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</details>
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Figure 31.1.5–3 Connection type ALIGN.
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# Description
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The ALIGN connection imposes kinematic constraints only. The local directions at node b are set equal to those at node a. If the local directions do not align initially, the ALIGN connection holds fixed the Cardan angles between the local orientation directions at node $b , \ \{ \mathbf { e } _ { 1 } ^ { b } , \mathbf { e } _ { 2 } ^ { b } , \mathbf { e } _ { 3 } ^ { b } \}$ , and those at node ${ \pmb a } ,$ $\{ { \bf e } _ { 1 } ^ { a } , { \bf e } _ { 2 } ^ { a } , { \bf e } _ { 3 } ^ { a } \}$ . These fixed angular positions are the connector position output quantities. See connection type CARDAN for a definition of Cardan angles.
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The constraint moment enforcing the alignment of the local directions is
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$$
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\bar {\mathbf {m}} = m _ {1} \mathbf {e} _ {1} ^ {a} + m _ {2} \mathbf {e} _ {2} ^ {a} + m _ {3} \mathbf {e} _ {3} ^ {a}.
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$$
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In two-dimensional analysis $m _ { 1 } = m _ { 2 } = 0$ .
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# Summary
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<table><tr><td colspan="2">ALIGN</td></tr><tr><td>Basic, assembled, or complex:</td><td>Basic</td></tr><tr><td>Kinematic constraints:</td><td> $ur_1 = 0, ur_2 = 0, ur_3 = 0$ </td></tr><tr><td>Constraint moment output:</td><td> $m_1, m_2, m_3$ </td></tr><tr><td>Available components:</td><td>None</td></tr><tr><td>Kinetic moment output:</td><td>None</td></tr><tr><td>Orientation at a:</td><td>Optional</td></tr></table>
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<table><tr><td colspan="2">ALIGN</td></tr><tr><td>Orientation at b:</td><td>Optional</td></tr><tr><td>Connector stops:</td><td>None</td></tr><tr><td>Constitutive reference angles:</td><td>None</td></tr><tr><td>Predefined friction parameters:</td><td>None</td></tr><tr><td>Contact force for predefined friction:</td><td>None</td></tr></table>
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# AXIAL
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Connection type AXIAL provides a connection between two nodes where the relative displacement is along the line separating the two nodes. It models discrete physical connections such as axial springs, axial dashpots, or node-to-node (gap-like) contact.
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<details>
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<summary>text_image</summary>
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u₁
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</details>
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Figure 31.1.5–4 Connection type AXIAL.
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# Description
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The AXIAL connection does not constrain any component of relative motion. The distance between nodes a and b is
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$$
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l = \left\| \mathbf {x} _ {b} - \mathbf {x} _ {a} \right\|.
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$$
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The available component of relative motion, $u _ { 1 }$ , acts along the line connecting the two nodes, measures the change in distance separating the two nodes, and is defined as
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$$
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u _ {1} = l - l _ {0},
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$$
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where $l _ { 0 }$ is the initial distance from node a to b. The connector constitutive displacement is
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$$
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u _ {1} ^ {m a t} = l - l _ {1} ^ {r e f}.
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$$
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The kinetic force is
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$$
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\mathbf {f} _ {a x i a l} = f _ {1} \mathbf {q}, \quad \text { where } \quad \mathbf {q} = \frac {1}{\| \mathbf {x} _ {b} - \mathbf {x} _ {a} \|} (\mathbf {x} _ {b} - \mathbf {x} _ {a}).
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$$
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In Abaqus/Standard an optional orientation can be provided at one of the nodes in an AXIAL connection to provide direction for the force if the nodes are coincident or when one of the nodes is a “ground node.” If the orientation is provided at both of the coincident nodes, the orientation at the first node in the connectivity will be used. The orientation definitions remain fixed during the analysis
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