Q4/E4

Q4/E4#

Model.element("Q4/E4", tag, nodes, section[, pressure, rho, b1, b2])

Parameters:

tag: integer

unique Element tag

nodes

tuple, a tuple of four element nodes in counter-clockwise order

section

tuple or int. If int, it is the tag of a previously defined PlaneSection. If tuple, it is a tuple of the form (thick, type, material) where

`thick` float	element thickness
`type` str	string representing material behavior. The type parameter can be either `"PlaneStrain"` or `"PlaneStress"`
`material` integer	tag of an Multiaxial

pressure

float, surface pressure (optional, default = 0.0)

rho

float, element mass density (per unit volume) from which a lumped element mass matrix is computed (optional, default=0.0)

b1: float

constant body forces defined in the domain (optional, default=0.0)

b2: float

constant body forces defined in the domain (optional, default=0.0)

element quad $eleTag $iNode $jNode $kNode $lNode $thick $type $matTag <$pressure $rho $b1 $b2>

Argument	Type	Description
$eleTag	integer	unique Element tag
$iNode $jNode $kNode $lNode	integer four nodes defining element boundaries	input in counter-clockwise order around the element.
$thick	float	element thickness
$type	string	string representing material behavior. The type parameter can be either “PlaneStrain” or “PlaneStress.”
$matTag	integer	tag of nDMaterial
$pressure	float	surface pressure (optional: default = 0.0)
$rho	float	element mass density (per unit volume) from which a lumped element mass matrix is computed (optional: default=0.0)
$b1 $b2	float	constant body forces defined in the isoparametric domain (optional: default=0.0)

Theory#

../../../../../_images/Q4.svg — *EnhancedQuad* element node numbering#

This element implements the Q1/E4 assumed strain interpolation [4]. The formulation is generally credited to Taylor, Beresford, and Wilson (1976) [1]. A variational basis for the formulation is given by Simo and Rifai (1990) [2].

\[\begin{split}\mathbf{M}_{\xi}=\left[\begin{array}{llll} \xi & 0 & 0 & 0 \\ 0 & \eta & 0 & 0 \\ 0 & 0 & \xi & \eta \end{array}\right]\end{split}\]

For linear-elastic response, the formulation is equivalent to a Hellinger-Reissner element [3] with interpolation

\[\begin{split}\mathbf{P}_{\xi}=\left[\begin{array}{cccccccc} 1 & 0 & 0 & \eta & 0 & \xi \eta & 0 & 0 \\ 0 & 1 & 0 & 0 & \xi & 0 & \xi \eta & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & \xi \eta \end{array}\right]\end{split}\]

Q4/E4

Contents

Q4/E4#

Theory#

Examples#

References#