Tri6#

class sectionproperties.analysis.fea.Tri6(el_id: int, coords: npt.NDArray[np.float64], node_ids: list[int], material: Material)[source]#

Bases: object

Class for a six-node quadratic triangular element.

Provides methods for the calculation of section properties based on the finite element method.

Parameters:

el_id (int) – Unique element id
coords (numpy.ndarray) – A 2 x 6 array of the coordinates of the Tri6 nodes. The first three columns relate to the vertices of the triangle and the last three columns correspond to the mid-nodes.
node_ids (list[int]) – A list of the global node ids for the current element
material (Material) – Material object for the current finite element

Methods

`element_stress`	Calculates the stress within an element resulting from a specified loading.
`geometric_properties`	Calculates the geometric properties for the current finite element.
`local_coord`	Maps a global point onto a local point.
`local_element_stress`	Calculates the stress at a point resulting from a specified loading.
`monosymmetry_integrals`	Calculates the integrals used to evaluate the monosymmetry constants.
`point_within_element`	Determines whether a point lies within the current element.
`shear_coefficients`	Calculates the variables used to for the shear deformation coefficients.
`shear_load_vectors`	Calculates the element shear load vectors for evaluating the shear functions.
`shear_warping_integrals`	Calculates the element shear centre and warping integrals for shear analysis.
`torsion_properties`	Calculates the element warping stiffness matrix and the torsion load vector.

Attributes

`el_id`
`coords`
`node_ids`
`material`

geometric_properties() → tuple[float, float, float, float, float, float, float, float, float][source]#

Calculates the geometric properties for the current finite element.

Returns:: Tuple containing the geometric properties, and the elastic and shear moduli of the element: (area, qx, qy, ixx, iyy, ixy, e, g, rho)
Return type:: tuple[float, float, float, float, float, float, float, float, float]

torsion_properties() → tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]][source]#

Calculates the element warping stiffness matrix and the torsion load vector.

Returns:: Element stiffness matrix k_el, element torsion load vector f_el and element constraint vector c_el
Return type:: tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]]

shear_load_vectors(ixx: float, iyy: float, ixy: float, nu: float) → tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]][source]#

Calculates the element shear load vectors for evaluating the shear functions.

Parameters:

ixx (float) – Second moment of area about the centroidal x-axis
iyy (float) – Second moment of area about the centroidal y-axis
ixy (float) – Second moment of area about the centroidal xy-axis
nu (float) – Effective Poisson’s ratio for the cross-section

Returns:

Element shear load vector psi f_psi and phi f_phi

Return type:

tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]]

shear_warping_integrals(ixx: float, iyy: float, ixy: float, omega: ndarray[Any, dtype[float64]]) → tuple[float, float, float, float, float, float][source]#

Calculates the element shear centre and warping integrals for shear analysis.

Parameters:

ixx (float) – Second moment of area about the centroidal x-axis
iyy (float) – Second moment of area about the centroidal y-axis
ixy (float) – Second moment of area about the centroidal xy-axis
omega (ndarray[Any, dtype[float64]]) – Values of the warping function at the element nodes

Returns:

Shear centre integrals about the x and y axes and warping integrals (sc_xint, sc_yint, q_omega, i_omega, i_xomega, i_yomega)

Return type:

tuple[float, float, float, float, float, float]

shear_coefficients(ixx: float, iyy: float, ixy: float, psi_shear: ndarray[Any, dtype[float64]], phi_shear: ndarray[Any, dtype[float64]], nu: float) → tuple[float, float, float][source]#

Calculates the variables used to for the shear deformation coefficients.

Parameters:

ixx (float) – Second moment of area about the centroidal x-axis
iyy (float) – Second moment of area about the centroidal y-axis
ixy (float) – Second moment of area about the centroidal xy-axis
psi_shear (ndarray[Any, dtype[float64]]) – Values of the psi shear function at the element nodes
phi_shear (ndarray[Any, dtype[float64]]) – Values of the phi shear function at the element nodes
nu (float) – Effective Poisson’s ratio for the cross-section

Returns:

Shear deformation variables (kappa_x, kappa_y, kappa_xy)

Return type:

tuple[float, float, float]

monosymmetry_integrals(phi: float) → tuple[float, float, float, float][source]#

Calculates the integrals used to evaluate the monosymmetry constants.

Parameters:: phi (float) – Principal bending axis angle, in degrees
Returns:: Integrals used to evaluate the monosymmetry constants (int_x, int_y, int_11, int_22)
Return type:: tuple[float, float, float, float]

element_stress(n: float, mxx: float, myy: float, m11: float, m22: float, mzz: float, vx: float, vy: float, ea: float, cx: float, cy: float, ixx: float, iyy: float, ixy: float, i11: float, i22: float, phi: float, j: float, nu: float, omega: ndarray[Any, dtype[float64]], psi_shear: ndarray[Any, dtype[float64]], phi_shear: ndarray[Any, dtype[float64]], delta_s: float) → tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]][source]#

Calculates the stress within an element resulting from a specified loading.

Parameters:

n (float) – Axial force
mxx (float) – Bending moment about the centroidal xx-axis
myy (float) – Bending moment about the centroidal yy-axis
m11 (float) – Bending moment about the centroidal 11-axis
m22 (float) – Bending moment about the centroidal 22-axis
mzz (float) – Torsion moment about the centroidal zz-axis
vx (float) – Shear force acting in the x-direction
vy (float) – Shear force acting in the y-direction
ea (float) – Modulus weighted area
cx (float) – x position of the elastic centroid
cy (float) – y position of the elastic centroid
ixx (float) – Second moment of area about the centroidal x-axis
iyy (float) – Second moment of area about the centroidal y-axis
ixy (float) – Second moment of area about the centroidal xy-axis
i11 (float) – Second moment of area about the principal 11-axis
i22 (float) – Second moment of area about the principal 22-axis
phi (float) – Principal bending axis angle
j (float) – St. Venant torsion constant
nu (float) – Effective Poisson’s ratio for the cross-section
omega (ndarray[Any, dtype[float64]]) – Values of the warping function at the element nodes
psi_shear (ndarray[Any, dtype[float64]]) – Values of the psi shear function at the element nodes
phi_shear (ndarray[Any, dtype[float64]]) – Values of the phi shear function at the element nodes
delta_s (float) – Cross-section shear factor

Returns:

Tuple containing element stresses and integration weights (\(\sigma_{zz,n}\), \(\sigma_{zz,mxx}\), \(\sigma_{zz,myy}\), \(\sigma_{zz,m11}\), \(\sigma_{zz,m22}\), \(\sigma_{zx,mzz}\), \(\sigma_{zy,mzz}\), \(\sigma_{zx,vx}\), \(\sigma_{zy,vx}\), \(\sigma_{zx,vy}\), \(\sigma_{zy,vy}\), \(w_i\))

Return type:

tuple[ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]], ndarray[Any, dtype[float64]]]

local_element_stress(p: tuple[float, float], n: float, mxx: float, myy: float, m11: float, m22: float, mzz: float, vx: float, vy: float, ea: float, cx: float, cy: float, ixx: float, iyy: float, ixy: float, i11: float, i22: float, phi: float, j: float, nu: float, omega: ndarray[Any, dtype[float64]], psi_shear: ndarray[Any, dtype[float64]], phi_shear: ndarray[Any, dtype[float64]], delta_s: float) → tuple[float, float, float][source]#

Calculates the stress at a point resulting from a specified loading.

Parameters:

p (tuple[float, float]) – Point (x, y) in the global coordinate system that is within the element
n (float) – Axial force
mxx (float) – Bending moment about the centroidal xx-axis
myy (float) – Bending moment about the centroidal yy-axis
m11 (float) – Bending moment about the centroidal 11-axis
m22 (float) – Bending moment about the centroidal 22-axis
mzz (float) – Torsion moment about the centroidal zz-axis
vx (float) – Shear force acting in the x-direction
vy (float) – Shear force acting in the y-direction
ea (float) – Modulus weighted area
cx (float) – x position of the elastic centroid
cy (float) – y position of the elastic centroid
ixx (float) – Second moment of area about the centroidal x-axis
iyy (float) – Second moment of area about the centroidal y-axis
ixy (float) – Second moment of area about the centroidal xy-axis
i11 (float) – Second moment of area about the principal 11-axis
i22 (float) – Second moment of area about the principal 22-axis
phi (float) – Principal bending axis angle
j (float) – St. Venant torsion constant
nu (float) – Effective Poisson’s ratio for the cross-section
omega (ndarray[Any, dtype[float64]]) – Values of the warping function at the element nodes
psi_shear (ndarray[Any, dtype[float64]]) – Values of the psi shear function at the element nodes
phi_shear (ndarray[Any, dtype[float64]]) – Values of the phi shear function at the element nodes
delta_s (float) – Cross-section shear factor

Returns:

Tuple containing stress values at point p (\(\sigma_{zz}\), \(\sigma_{zx}\), \(\sigma_{zy}\))

Return type:

tuple[float, float, float]

point_within_element(pt: tuple[float, float]) → bool[source]#

Determines whether a point lies within the current element.

Parameters:: pt (tuple[float, float]) – Point to check (x, y)
Returns:: Whether the point lies within an element
Return type:: bool

local_coord(p: tuple[float, float]) → tuple[float, float, float][source]#

Maps a global point onto a local point.

Parameters:: p (tuple[float, float]) – Global coordinate (x, y)
Returns:: Point in local coordinates (eta, xi, zeta)
Return type:: tuple[float, float, float]