BeamLR - Linear Timoshenko 3D beam element with reduced integration (`pyfe3d.beamlr`)#

class pyfe3d.beamlr.BeamLR#

Timoshenko 3D beam element with linear shape functions

Formulation based on reference, replacing the consistent shape functions by linear functions and performing numerical integration with just 1 point at the beam center:

Luo, Y., 2008, “An Efficient 3D Timoshenko Beam Element with Consistent Shape Functions,” Adv. Theor. Appl. Mech., 1(3), pp. 95–106.

Nodal connectivity for the beam element:

^ y axis
|
|
______   --> x axis
1    2

Attributes:

eid,int: Element identification number.
pid,int: Property identification number.
length,double: Element length.
r11, r12, r13, r21, r22, r23, r31, r32, r33double: Rotation matrix from local to global coordinates.
vxyi, vxyj, vxykdouble: Components of a vector on the \(XY\) plane of the element coordinate system, defined using global coordinates.
c1, c2int: Position of each node in the global stiffness matrix.
n1, n2int: Node identification number.
init_k_KC0, init_k_KG, init_k_Mint: Position in the arrays storing the sparse data for the structural matrices.
probeBeamLRProbe object: probe: pyfe3d.beamlr.BeamLRProbe

Methods

`update_KC0`(self, long[, long[, double[, ...)	Update sparse vectors for linear constitutive stiffness matrix KC0
`update_KG`(self, long[, long[, double[, ...)	Update sparse vectors for geometric stiffness matrix KG
`update_M`(self, long[, long[, double[, ...)	Update sparse vectors for mass matrix M
`update_length`(self)	Update element length
`update_probe_ue`(self, double[)	Update the local displacement vector of the probe of the element
`update_probe_xe`(self, double[)	Update the 3D coordinates of the probe of the element
`update_rotation_matrix`(self, double vxyi, ...)	Update the rotation matrix of the element

c1#: c1: ‘int’

c2#: c2: ‘int’

eid#: eid: ‘int’

init_k_KC0#: init_k_KC0: ‘int’

init_k_KG#: init_k_KG: ‘int’

init_k_M#: init_k_M: ‘int’

length#: length: ‘double’

n1#: n1: ‘int’

n2#: n2: ‘int’

pid#: pid: ‘int’

probe#: probe: pyfe3d.beamlr.BeamLRProbe

r11#: r11: ‘double’

r12#: r12: ‘double’

r13#: r13: ‘double’

r21#: r21: ‘double’

r22#: r22: ‘double’

r23#: r23: ‘double’

r31#: r31: ‘double’

r32#: r32: ‘double’

r33#: r33: ‘double’

update_KC0(self, long[::1] KC0r, long[::1] KC0c, double[::1] KC0v, BeamProp prop, int update_KC0v_only=0) → void#

Update sparse vectors for linear constitutive stiffness matrix KC0

Parameters:

KC0rnp.array: Array to store row positions of sparse values
KC0cnp.array: Array to store column positions of sparse values
KC0vnp.array: Array to store sparse values
propBeamProp object: Beam property object from where the stiffness and mass attributes are read from.
update_KC0v_onlyint: The default 0 means that the row and column indices KC0r and KC0c should also be updated. Any other value will only update the stiffness matrix values KC0v.

update_KG(self, long[::1] KGr, long[::1] KGc, double[::1] KGv, BeamProp prop, int update_KGv_only=0) → void#

Update sparse vectors for geometric stiffness matrix KG

Parameters:

KGrnp.array: Array to store row positions of sparse values
KGcnp.array: Array to store column positions of sparse values
KGvnp.array: Array to store sparse values
propBeamProp object: Beam property object from where the stiffness and mass attributes are read from.
update_KGv_onlyint: The default \(0\) means that only \(KGv\) is updated. Any other value will lead to \(KGr\) and \(KGc\) also being updated.

update_M(self, long[::1] Mr, long[::1] Mc, double[::1] Mv, BeamProp prop, int mtype=0) → void#

Update sparse vectors for mass matrix M

Parameters:

Mrnp.array: Array to store row positions of sparse values
Mcnp.array: Array to store column positions of sparse values
Mvnp.array: Array to store sparse values
mtypeint, optional: 0 for consistent mass matrix using method from Brockman 1987 1 for reduced integration mass matrix using method from Brockman 1987 2 for lumped mass matrix using method from Brockman 1987

update_length(self) → void#: Update element length

update_probe_ue(self, double[::1] u) → void#

Update the local displacement vector of the probe of the element

Note

The probe attribute object BeamLRProbe is updated, not the element object.

Parameters:

uarray-like: Array with global displacements, for a total of \(M\) nodes in the model, this array will be arranged as: \(u_1, v_1, w_1, {r_x}_1, {r_y}_1, {r_z}_1, u_2, v_2, w_2, {r_x}_2, {r_y}_2, {r_z}_2, ..., u_M, v_M, w_M, {r_x}_M, {r_y}_M, {r_z}_M\).

update_probe_xe(self, double[::1] x) → void#

Update the 3D coordinates of the probe of the element

Note

The probe attribute object BeamLRProbe is updated, not the element object.

Parameters:

xarray-like: Array with global nodal coordinates, for a total of \(M\) nodes in the model, this array will be arranged as: \(x_1, y_1, z_1, x_2, y_2, z_2, ..., x_M, y_M, z_M\).

update_rotation_matrix(self, double vxyi, double vxyj, double vxyk, double[::1] x) → void#

Update the rotation matrix of the element

Attributes r11,r12,r13,r21,r22,r23,r31,r32,r33 are updated, corresponding to the rotation matrix from local to global coordinates.

The element attributes vxyi, vxyj and vxyk are also updated when this function is called.

The element coordinate system is determined, identifying the \(ijk\) components of each axis: \({x_e}_i, {x_e}_j, {x_e}_k\); \({y_e}_i, {y_e}_j, {y_e}_k\); \({z_e}_i, {z_e}_j, {z_e}_k\).

Parameters:

vxyi, vxyj, vxykdouble: Components of a vector on the \(XY\) plane of the element coordinate system, defined using global coordinates.
xarray-like: Array with global nodal coordinates, for a total of \(M\) nodes in the model, this array will be arranged as: \(x_1, y_1, z_1, x_2, y_2, z_2, ..., x_M, y_M, z_M\).

vxyi#: vxyi: ‘double’

vxyj#: vxyj: ‘double’

vxyk#: vxyk: ‘double’

class pyfe3d.beamlr.BeamLRData#

Used to allocate memory for the sparse matrices.

Attributes:

KC0_SPARSE_SIZE,int: KC0_SPARSE_SIZE = 144
KG_SPARSE_SIZE,int: KG_SPARSE_SIZE = 36
M_SPARSE_SIZE,int: M_SPARSE_SIZE = 144

KC0_SPARSE_SIZE#: KC0_SPARSE_SIZE: ‘int’

KG_SPARSE_SIZE#: KG_SPARSE_SIZE: ‘int’

M_SPARSE_SIZE#: M_SPARSE_SIZE: ‘int’

class pyfe3d.beamlr.BeamLRProbe#

Probe used for local coordinates, local displacements, local stresses etc

Attributes:

xe,array-like: Array of size NUM_NODES*DOF//2=6 containing the nodal coordinates in the element coordinate system, in the following order \({x_e}_1, {y_e}_1, {z_e}_1, {x_e}_2, {y_e}_2, {z_e}_2\).
ue,array-like: Array of size NUM_NODES*DOF=12 containing the element displacements in the following order \({u_e}_1, {v_e}_1, {w_e}_1, {{r_x}_e}_1, {{r_y}_e}_1, {{r_z}_e}_1, {u_e}_2, {v_e}_2, {w_e}_2, {{r_x}_e}_2, {{r_y}_e}_2, {{r_z}_e}_2\).

ue#: ue: ‘double[::1]’

xe#: xe: ‘double[::1]’

BeamLR - Linear Timoshenko 3D beam element with reduced integration (pyfe3d.beamlr)#

BeamLR - Linear Timoshenko 3D beam element with reduced integration (`pyfe3d.beamlr`)#