김경종
442 lines
22 KiB
Markdown
442 lines
22 KiB
Markdown
<!-- source-page: 131 -->
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General or shear surface tractions and pressure loads can be applied in Abaqus as element-based or surface-based distributed loads. The units of these loads are force per unit area.
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The distributed surface load types that are available in Abaqus, along with the corresponding load type labels, are listed in Table 34.4.3–5 and Table 34.4.3–6. Part VI, “Elements,” lists the distributed surface load types that are available for particular elements and the Abaqus/CAE load support for each load type. For some element-based loads you must identify the face of the element upon which the load is prescribed in the load type label (for example, Pn or PnNU for continuum elements).
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Table 34.4.3–5 Distributed surface load types.
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<table><tr><td>Load description</td><td>Load type label for element-based loads</td><td>Load type label for surface-based loads</td></tr><tr><td>General surface traction</td><td>TRVECn, TRVEC</td><td>TRVEC</td></tr><tr><td>Shear surface traction</td><td>TRSHRn, TRSHR</td><td>TRSHR</td></tr><tr><td>Nonuniform general surface traction</td><td>TRVECnNU, TRVECNU</td><td>TRVECNU</td></tr><tr><td>Nonuniform shear surface traction</td><td>TRSHRnNU, TRSHRNU</td><td>TRSHRNU</td></tr><tr><td>Pressure</td><td>Pn, P</td><td>P</td></tr><tr><td>Nonuniform pressure</td><td>PnNU, PNU</td><td>PNU</td></tr><tr><td>Hydrostatic pressure (available only in Abaqus/Standard)</td><td>HPn, HP</td><td>HP</td></tr><tr><td>Viscous pressure (available only in Abaqus/Explicit)</td><td>VPn, VP</td><td>VP</td></tr><tr><td>Stagnation pressure (available only in Abaqus/Explicit)</td><td>SPn, SP</td><td>SP</td></tr><tr><td>Pore mechanical pressure (available only in Abaqus/Standard)</td><td>PORMECHn, PORMECH</td><td>PORMECH</td></tr><tr><td>Hydrostatic internal and external pressure (only for PIPE and ELBOW elements)</td><td>HPI, HPE</td><td>N/A</td></tr><tr><td>Uniform internal and external pressure (only for PIPE and ELBOW elements)</td><td>PI, PE</td><td>N/A</td></tr><tr><td>Nonuniform internal and external pressure (only for PIPE and ELBOW elements)</td><td>PINU, PENU</td><td>N/A</td></tr></table>
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<!-- source-page: 132 -->
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Table 34.4.3–6 Distributed surface load types in Abaqus/CAE.
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<table><tr><td>Load description</td><td>Abaqus/CAE load type</td></tr><tr><td>General surface traction</td><td rowspan="2">Surface traction</td></tr><tr><td>Shear surface traction</td></tr><tr><td>Nonuniform general surface traction</td><td rowspan="2">Surface traction (surface-based loads only)</td></tr><tr><td>Nonuniform shear surface traction</td></tr><tr><td>Pressure</td><td>Pressure</td></tr><tr><td>Nonuniform pressure</td><td rowspan="4">Pressure (surface-based loads only)</td></tr><tr><td>Hydrostatic pressure (available only in Abaqus/Standard)</td></tr><tr><td>Viscous pressure (available only in Abaqus/Explicit)</td></tr><tr><td>Stagnation pressure (available only in Abaqus/Explicit)</td></tr><tr><td>Hydrostatic internal and external pressure (only for PIPE and ELBOW elements)</td><td rowspan="3">Pipe pressure</td></tr><tr><td>Uniform internal and external pressure (only for PIPE and ELBOW elements)</td></tr><tr><td>Nonuniform internal and external pressure (only for PIPE and ELBOW elements)</td></tr></table>
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# Follower surface loads
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By definition, the line of action of a follower surface load rotates with the surface in a geometrically nonlinear analysis. This is in contrast to a non-follower load, which always acts in a fixed global direction.
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With the exception of general surface tractions, all the distributed surface loads listed in Table 34.4.3–5 and Table 34.4.3–6 are modeled as follower loads. The hydrostatic and viscous pressures listed in Table 34.4.3–5 and Table 34.4.3–6 always act normal to the surface in the current configuration, the shear tractions always act tangent to the surface in the current configuration, and the internal and external pipe pressures follow the motion of the pipe elements.
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General surface tractions can be specified to be follower or non-follower loads. There is no difference between a follower and a non-follower load in a geometrically linear analysis since the configuration of the body remains fixed. The difference between a follower and non-follower general surface traction is illustrated in the next section through an example.
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<!-- source-page: 133 -->
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Input File Usage: Use one of the following options to define general surface tractions as follower loads (the default):
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```csv
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*DLOAD, FOLLOWER=YES
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*DSLOAD, FOLLOWER=YES
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```
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Use one of the following options to define general surface tractions as nonfollower loads:
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```txt
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*DLOAD, FOLLOWER=NO
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*DSLOAD, FOLLOWER=NO
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```
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Abaqus/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction: General, toggle on or off Follow rotation
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# Specifying general surface tractions
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General surface tractions allow you to specify a surface traction, , acting on a surface S. The resultant load, , is computed by integrating over S:
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$$
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\mathbf {f} = \int_ {S} \mathbf {t} d S = \int_ {S} \alpha \hat {\mathbf {t}} d S,
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$$
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where is the magnitude and is the direction of the load. To define a general surface traction, you must specify both a load magnitude, , and the direction of the load with respect to the reference configuration, $\mathbf { t } _ { u s e r }$ . The magnitude and direction can also be specified in user subroutine UTRACLOAD. The specified traction directions are normalized by Abaqus and, thus, do not contribute to the magnitude of the load:
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$$
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\hat {\mathbf {t}} ^ {o} = \frac {\mathbf {t} _ {u s e r}}{\| \mathbf {t} _ {u s e r} \|}.
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$$
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Input File Usage: Use one of the following options to define a general surface traction:
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```txt
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*DLOAD
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element number or element set, load type label, magnitude,
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direction components
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where load type label is TRVECn, TRVEC, TRVECnNU, or TRVECNU.
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*DSLOAD
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surface name, TRVEC or TRVECNU, magnitude, direction components
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```
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Abaqus/CAE Usage: Use the following input to define an element-based general surface traction:
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Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction: General, Distribution: select an analytical field
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Use the following input to define a surface-based general surface traction:
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<!-- source-page: 134 -->
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# Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction: General, Distribution: Uniform or User-defined
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Nonuniform element-based general surface traction is not supported in Abaqus/CAE.
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Defining the direction vector with respect to a local coordinate system
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By default, the components of the traction vector are specified with respect to the global directions. You can also refer to a local coordinate system (see “Orientations,” Section 2.2.5) for the direction components of these tractions. See “Examples: using a local coordinate system to define shear directions” below for an example of a traction load defined with respect to a local coordinate system. When using local coordinate systems for tractions applied to two-dimensional solid elements, you must ensure that the nonzero components of the loads are applied only in the X- and Y-directions. Tractions loads in the third direction are not supported (Z-direction for plane strain and plane stress elements, -direction for axisymmetric elements).
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Input File Usage: Use one of the following options to specify a local coordinate system:
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\*DLOAD, ORIENTATION=name
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\*DSLOAD, ORIENTATION=name
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Abaqus/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: select CSYS: Picked and click Edit to pick a local coordinate system, or select CSYS: User-defined to enter the name of a user subroutine that defines a local coordinate system
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Rotation of the traction vector direction
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The traction load acts in the fixed direction $\hat { \mathbf { t } } = \hat { \mathbf { t } } ^ { o }$ in a geometrically linear analysis or if a non-follower load is specified in a geometrically nonlinear analysis (which includes a perturbation step about a geometrically nonlinear base state).
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If a follower load is specified in a geometrically nonlinear analysis, the traction load rotates rigidly with the surface using the following algorithm. The reference configuration traction vector, $\mathbf { t } ^ { o } = \alpha \hat { \mathbf { t } } ^ { o }$ , is decomposed by Abaqus into two components: a normal component,
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$$
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\alpha \hat {\mathbf {t}} ^ {o} \cdot \mathbf {N} \mathbf {N} = \alpha_ {n} \mathbf {N},
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$$
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and a tangential component,
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$$
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\alpha \left(\hat {\mathbf {t}} ^ {o} - \hat {\mathbf {t}} ^ {o} \cdot \mathbf {N} \mathbf {N}\right) = \alpha_ {d} \mathbf {D},
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$$
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where is the unit reference surface normal and is the unit projection of onto the reference surface. The applied traction in the current configuration is then computed as
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$$
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\mathbf {t} = \alpha_ {n} \mathbf {n} + \alpha_ {d} \mathbf {d},
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$$
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<!-- source-page: 135 -->
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where is the normal to the surface in the current configuration and is the image of rotated onto the current surface; i.e., , where is the standard rotation tensor obtained from the polar decomposition of the local two-dimensional surface deformation gradient .
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# Examples: follower and non-follower tractions
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The following two examples illustrate the difference between applying follower and non-follower tractions in a geometrically nonlinear analysis. Both examples refer to a single 4-node plane strain element (element 1). In Step 1 of the first example a follower traction load is applied to face 1 of element 1, and a non-follower traction load is applied to face 2 of element 1. The element is rotated rigidly 90° counterclockwise in Step 1 and then another 90° in Step 2. As illustrated in Figure 34.4.3–1, the follower traction rotates with face 1, while the non-follower traction on face 2 always acts in the global x-direction.
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["1"] --> B["2"]
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B --> C["3"]
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C --> D["4"]
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D --> A
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```
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</details>
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(a)
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["1"] --> B["2"]
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B --> C["3"]
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C --> D["4"]
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```
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</details>
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(b)
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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1 --> 2
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2 --> 3
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3 --> 4
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```
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</details>
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A[" follower traction "] --> B[" non-follower traction "]
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```
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</details>
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Figure 34.4.3–1 Follower and non-follower traction loads in a geometrically nonlinear analysis, load applied in Step 1: (a) beginning of Step 1; (b) end of Step 1, beginning of Step 2; (c) end of Step 2.
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```matlab
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*STEP, NLGEOM
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Step 1 - Rotate square 90 degrees
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...
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*DLOAD, FOLLOWER=YES
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1, TRVEC1, 1., 0., -1., 0.
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*DLOAD, FOLLOWER=NO
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1, TRVEC2, 1., 1., 0., 0.
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*END STEP
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*STEP, NLGEOM
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```
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<!-- source-page: 136 -->
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Step 2 - Rotate square another 90 degrees
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\*END STEP
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In the second example the element is rotated 90° counterclockwise with no load applied in Step 1. In Step 2 a follower traction load is applied to face 1, and a non-follower traction load is applied to face 2. The element is then rotated rigidly by another 90°. The direction of the follower load is specified with respect to the original configuration. As illustrated in Figure 34.4.3–2, the follower traction rotates with face 1, while the non-follower traction on face 2 always acts in the global x-direction.
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram of a square with labeled vertices (no text or symbols beyond numbers 1, 2, 3, 4)
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</details>
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(a)
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["1"] --> B["2"]
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B --> C["3"]
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C --> D["4"]
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```
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</details>
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<details>
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<summary>text_image</summary>
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1
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2
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3
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4
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</details>
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(c)
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["→"] --> B["follower traction"]
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C["→"] --> D["non-follower traction"]
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```
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</details>
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Figure 34.4.3–2 Follower and non-follower traction loads in a geometrically nonlinear analysis, load applied in Step 2: (a) beginning of Step 1; (b) end of Step 1, beginning of Step 2; (c) end of Step 2.
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```txt
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*STEP, NLGEOM
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Step 1 - Rotate square 90 degrees
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...
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*END STEP
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*STEP, NLGEOM
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Step 2 - Rotate square another 90 degrees
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*DLOAD, FOLLOWER=YES
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1, TRVEC1, 1., 0., -1., 0.
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*DLOAD, FOLLOWER=NO
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1, TRVEC2, 1., 1., 0., 0.
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...
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*END STEP
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```
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<!-- source-page: 137 -->
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# Specifying shear surface tractions
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Shear surface tractions allow you to specify a surface force per unit area, $\mathbf { t } _ { s }$ , that acts tangent to a surface S. The resultant load, , is computed by integrating $\mathbf { t } _ { s }$ over $\mathbb { S }$
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$$
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\mathbf {f} = \int_ {S} \mathbf {t} _ {s} d S = \int_ {S} \alpha \mathbf {d} d S,
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$$
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where is the magnitude and is a unit vector along the direction of the load. To define a shear surface traction, you must provide both the magnitude, , and a direction, $\mathbf { t } _ { u s e r } $ , for the load. The magnitude and direction vector can also be specified in user subroutine UTRACLOAD.
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Abaqus modifies the traction direction by first projecting the user-specified vector, $\mathbf { t } _ { u s e r }$ , onto the surface in the reference configuration,
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$$
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\mathbf {t} _ {u s e r} ^ {p o} = \mathbf {t} _ {u s e r} - \mathbf {t} _ {u s e r} \cdot \mathbf {N N},
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$$
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where is the reference surface normal. The specified traction is applied along the computed traction direction tangential to the surface:
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$$
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\mathbf {D} = \frac {\mathbf {t} _ {u s e r} ^ {p o}}{\left\| \mathbf {t} _ {u s e r} ^ {p o} \right\|}.
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$$
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Consequently, a shear traction load is not applied at any point where $\mathbf { t } _ { u s e r }$ is normal to the reference surface.
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The shear traction load acts in the fixed direction in a geometrically linear analysis. In a geometrically nonlinear analysis (which includes a perturbation step about a geometrically nonlinear base state), the shear traction vector will rotate rigidly; i.e., , where is the standard rotation tensor obtained from the polar decomposition of the local two-dimensional surface deformation gradient F=RU.
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Input File Usage: Use one of the following options to define a shear surface traction:
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*DLOAD
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element number or element set, load type label, magnitude,
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direction components
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where load type label is TRSHRn, TRSHR, TRSHRnNU, or TRSHRNU.
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*DSLOAD
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surface name, TRSHR or TRSHRNU, magnitude, direction components
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Abaqus/CAE Usage: Use the following input to define an element-based shear surface traction:
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Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction:
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Shear, Distribution: select an analytical field
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Use the following input to define a surface-based general surface traction:
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<!-- source-page: 138 -->
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Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction: Shear, Distribution: Uniform or User-defined
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Nonuniform element-based shear surface traction is not supported in Abaqus/CAE.
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Defining the direction vector with respect to a local coordinate system
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By default, the components of the shear traction vector are specified with respect to the global directions. You can also refer to a local coordinate system (see “Orientations,” Section 2.2.5) for the direction components of these tractions.
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Input File Usage: Use one of the following options to specify a local coordinate system:
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\*DLOAD, ORIENTATION=name
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\*DSLOAD, ORIENTATION=name
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Abaqus/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: select CSYS: Picked and click Edit to pick a local coordinate system, or select CSYS: User-defined to enter the name of a user subroutine that defines a local coordinate system
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Examples: using a local coordinate system to define shear directions
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It is sometimes convenient to give shear and general traction directions with respect to a local coordinate system. The following two examples illustrate the specification of the direction of a shear traction on a cylinder using global coordinates in one case and a local cylindrical coordinate system in the other case. The axis of symmetry of the cylinder coincides with the global z-axis. A surface named SURFA has been defined on the outside of the cylinder.
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In the first example the direction of the shear traction, $\mathbf { t } _ { u s e r } ~ = ~ ( 0 . , 1 . , 0 . )$ , is given in global coordinates. The sense of the resulting shear tractions using global coordinates is shown in Figure 34.4.3–3(a).
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<details>
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<summary>text_image</summary>
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y
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x
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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y
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x
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</details>
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Figure 34.4.3–3 Shear tractions specified using global coordinates (a) and a local cylindrical coordinate system (b).
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<!-- source-page: 139 -->
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```txt
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*STEP
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Step 1 - Specify shear directions in global coordinates
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...
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*DSLOAD
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SURFA, TRSHR, 1., 0., 1., 0.
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...
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*END STEP
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```
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In the second example the direction of the shear traction, $\mathbf { t } _ { u s e r } = ( 0 . , 1 . , 0 . )$ , is given with respect to a local cylindrical coordinate system whose axis coincides with the axis of the cylinder. The sense of the resulting shear tractions using the local cylindrical coordinate system is shown in Figure 34.4.3–3(b).
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```matlab
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*ORIENTATION, NAME=CYLIN, SYSTEM=CYLINDRICAL
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0., 0., 0., 0., 0., 1.
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...
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*STEP
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Step 1 - Specify shear directions in local cylindrical coordinates
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...
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*DSLOAD, ORIENTATION=CYLIN
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SURFA, TRSHR, 1., 0., 1., 0.
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...
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*END STEP
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```
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# Resultant loads due to surface tractions
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You can choose to integrate surface tractions over the current or the reference configuration by specifying whether or not a constant resultant should be maintained.
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In general, the constant resultant method is best suited for cases where the magnitude of the resultant load should not vary with changes in the surface area. However, it is up to you to decide which approach is best for your analysis. An example of an analysis using a constant resultant can be found in “Distributed traction and edge loads,” Section 1.4.18 of the Abaqus Verification Guide.
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# Choosing not to have a constant resultant
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If you choose not to have a constant resultant, the traction vector is integrated over the surface in the current configuration, a surface that in general deforms in a geometrically nonlinear analysis. By default, all surface tractions are integrated over the surface in the current configuration.
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Input File Usage: Use one of the following options:
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```txt
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*DLOAD, CONSTANT RESULTANT=NO
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*DSLOAD, CONSTANT RESULTANT=NO
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```
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Abaqus/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction is defined per unit deformed area
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<!-- source-page: 140 -->
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# Maintaining a constant resultant
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If you choose to have a constant resultant, the traction vector is integrated over the surface in the reference configuration and then held constant.
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Input File Usage: Use one of the following options:
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\*DLOAD, CONSTANT RESULTANT=YES
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\*DSLOAD, CONSTANT RESULTANT=YES
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Abaqus/CAE Usage: Load module: Create Load: choose Mechanical for the Category and Surface traction for the Types for Selected Step: Traction is defined per unit undeformed area
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# Example
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The constant resultant method has certain advantages when a traction is used to model a distributed load with a known constant resultant. Consider the case of modeling a uniform dead load, magnitude p, acting on a flat plate whose normal is in the -direction in a geometrically nonlinear analysis (Figure 34.4.3–4).
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<details>
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<summary>text_image</summary>
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e₂
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P
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e₁
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deformed configuration
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</details>
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Figure 34.4.3–4 Dead load on a flat plate.
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Such a model might be used to simulate a snow load on a flat roof. The snow load could be modeled as a distributed dead traction load $\mathbf { t } = - p \mathbf { e } _ { 2 }$ . Let $S _ { o }$ and S denote the total surface area of the plate in the reference and current configurations, respectively. With no constant resultant, the total integrated load on the plate, , is
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$$
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\mathbf {f} = \int_ {S} \mathbf {t} d S = \int_ {S} - p \mathbf {e} _ {2} d S = - p \mathbf {e} _ {2} S.
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$$
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In this case a uniform traction leads to a resultant load that increases as the surface area of the plate increases, which is not consistent with a fixed snow load. With the constant resultant method, the total integrated load on the plate is
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$$
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\mathbf {f} = \int_ {S _ {o}} \mathbf {t} d S _ {o} = \int_ {S _ {o}} - p \mathbf {e} _ {2} d S _ {o} = - p \mathbf {e} _ {2} S _ {o}.
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$$
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In this case a uniform traction leads to a resultant that is equal to the pressure times the surface area in the reference configuration, which is more consistent with the problem at hand.
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