ASDShellQ4

This command is used to construct an ASDShellQ4 shell element.

PythonTcl

model.element("ASDShellQ4", tag, nodes, section)

Parameters:

• tag (int) – integer tag identifying element object

• nodes (tuple) – tuple of 4 integers defining the element :ref:`node`s

• section (int) – integer tag identifying a Shell Section

The ASDShellQ4 element is a 4-node general purpose thick shell element with the following features: #. The membrane behavior is enhanced with the AGQ6-I [ChenEtAl2004] formulation, which makes the element almost insensitive to geometry distortion, as opposed to standard iso-parametric elements. #. The drilling DOF is treated with the Hughes-Brezzi [HughesEtAl1989] formulation, with special care to avoid membrane locking, using a 1 point quadrature plus stabilization. This formulation constrains the drilling DOFs to the rigid body rotation via a penalty parameter as a function of the initial in-plane shear modulus. However, when using strain-softening materials, this (elastic) constraint may overstiffen the element as the in-plane shear modulus degrades. As a remedy in such a situation, the user can choose to make this constraint non-linear. #. The plate bending part is treated using the MITC4 [DvorkinEtAl1984] [BatheEtAl1985] formulation, to avoid the well known transverse shear locking behavior of thick plate elements. #. It can be used to model both flat and warped geometries. #. Kinematics can be either linear or corotational. The corotational kinematics is based on the work of Felippa et al., i.e. the EICR [Felippa2000] [FelippaEtAl2005] (Element Independent Corotational formulation). Finite rotations are treated with Quaternions. #. It uses a full 2x2 Gauss quadrature, so it has a total of 4 integration points.

Fig. 31 Nodes, Gauss points, local coordinate system, warped and flat geometry

Valid eleResponse queries to the ASDShellQ4 element are:

• "force", "forces", "globalForce", or "globalForces":

  • Internal forces at the element’s nodes.

  • Orientation: global coordinate system.

  • Size: 24 columns of data, 6 components for each one of the 4 nodes.

• "material $secTag $secArg1 ... $secArgN":

  • Section response at section $secTag

  • $secTag is the 1-based index of the integration point (1 to 4).

  • ‘$secArg1 … $secArgN’ are the arguments required by the SectionDeformationObject at the requested integration point.

Examples

Example 1 - Cantilever Bending Roll-up (corotational)

A Cantilever beam is subjected to a total end-moment about the Y axis \(M_y = n 2 \pi EI/L\), where \(n\) is the number of rotations (2 in this example). The files for this example can be downloaded from https://gallery.stairlab.io/examples/shellcircle/

Example 2

TclPython (RT)

# set up a 3D-6DOFs model
model Basic -ndm 3 -ndf 6
node 1  0.0  0.0 0.0
node 2  1.0  0.0 0.0
node 3  1.0  1.0 0.0
node 4  0.0  1.0 0.0

# create a fiber shell section with 4 layers of material 1
# each layer has a thickness = 0.025
nDMaterial ElasticIsotropic  1  1000.0  0.2
section LayeredShell  11  4   1 0.025   1 0.025   1 0.025   1 0.025

# create the shell element using the small displacements/rotations assumption
element ASDShellQ4  1  1 2 3 4  11
# or you can use the corotational flag for large displacements/rotations (geometric nonlinearity)
element ASDShellQ4  1  1 2 3 4  11 -corotational

# record global forces at element nodes (24 columns, 6 for each node)
recorder Element  -xml  force_out.xml  -ele  1  force
# record local section forces at gauss point 1 (8 columns: | 3 membrane | 3 bending | 2 transverse shear |)
# note: gauss point index is 1-based
recorder Element  -xml  force_gp1_out.xml  -ele  1  material  1  force
# record local stresses at fiber 1 of gauss point 1 (5 columns: Szz is neglected (0) )
# note: fiber index is 1-based (while in beams it is 0-based!)
recorder Element  -xml  stress_gp1_fib0_out.xml  -ele  1  material  1  fiber 1 stress

Code Developed by: Massimo Petracca at ASDEA Software, Italy.

[ChenEtAl2004]

Chen, Xiao-Ming, et al. “Membrane elements insensitive to distortion using the quadrilateral area coordinate method.” Computers & Structures 82.1 (2004): 35-54. (Link to article)

[HughesEtAl1989]

Hughes, Thomas JR, and F. Brezzi. “On drilling degrees of freedom.” Computer methods in applied mechanics and engineering 72.1 (1989): 105-121. (Link to article)

[DvorkinEtAl1984]

Dvorkin, Eduardo N., and Klaus-Jurgen Bathe. “A continuum mechanics based four-node shell element for general non-linear analysis.” Engineering computations (1984). (Link to article)

[BatheEtAl1985]

Bathe, Klaus-Jurgen, and Eduardo N. Dvorkin. “A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation.” International Journal for Numerical Methods in Engineering 21.2 (1985): 367-383. (Link to article)

[Felippa2000]

Felippa, Carlos A. “A systematic approach to the element-independent corotational dynamics of finite elements”. Technical Report CU-CAS-00-03, Center for Aerospace Structures, 2000. (Link to article)

[FelippaEtAl2005]

Felippa, Carlos A., and Bjorn Haugen. “A unified formulation of small-strain corotational finite elements: I. Theory.” Computer Methods in Applied Mechanics and Engineering 194.21-24 (2005): 2285-2335. (Link to article)