rectangular_shell_element Derived Type

private pure function quad_init(mat, thickness, nd1, nd2, nd3, nd4) result(rst)

Constructs a new [[rectangular_shell_element]].

Arguments

Return Value type(rectangular_shell_element)

The new [[rectangular_shell_element]].

private pure function shl_area(this) result(rst)

Computes the area of the element.

Arguments

Return Value real(kind=real64)

The element area.

private pure function shl_constitutive_matrix(this) result(rst)

Computes the 8-by-8 constitutive matrix relating the generalized strains to the stress resultants. $$D = \begin{bmatrix} t C & 0 & 0 \\ 0 & \frac{t^3}{12} C & 0 \\ 0 & 0 & k G t I \end{bmatrix}, \quad C = \frac{E}{1 - \nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1 - \nu}{2} \end{bmatrix},$$ where $k$ is the shear correction factor and $G = E / (2 (1 + \nu))$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 8-by-8 constitutive matrix.

private pure function quad_shape_function(this, i, s) result(rst)

Evaluates the i-th bilinear shape function, $N_i = \frac{1}{4}(1 + r r_i)(1 + s s_i)$.

Arguments

Return Value real(kind=real64)

The value of the i-th shape function.

private pure function shl_ext_force_vector(this, q, rule) result(rst)

Computes the consistent nodal force vector, in the global coordinate system, resulting from a uniform surface traction. $$\boldsymbol{f}_i = \left( \int_A N_i \, dA \right) \boldsymbol{q}$$ Only the translational degrees of freedom receive load.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The 6N-element force vector, where N is the number of nodes.

private pure function shl_dimensionality(this) result(rst)

Gets the dimensionality of the element.

Arguments

Return Value integer(kind=int32)

The dimensionality (always 3).

private pure function shl_dof_per_node(this) result(rst)

Gets the number of degrees of freedom per node.

Arguments

Return Value integer(kind=int32)

The number of degrees of freedom per node (always 6).

private pure function quad_get_node(this, i) result(rst)

Gets the requested node from the element.

Arguments

Return Value type(node)

The requested node.

private pure function quad_get_node_count(this) result(rst)

Gets the number of nodes in the element.

Arguments

Return Value integer(kind=int32)

The number of nodes (always 4).

private pure function quad_get_node_natural_coordinates(this, i) result(rst)

Returns the natural coordinates of the requested node.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The 2-element natural coordinate vector of the node.

private pure subroutine quad_integration_rule(this, pts, wts)

Returns the 2-by-2 Gauss quadrature rule.

Arguments

private pure function shl_jacobian(this, s) result(rst)

Computes the 2-by-2 Jacobian matrix of the mapping from natural to local coordinates. $$J = \begin{bmatrix} \frac{\partial x}{\partial s_1} & \frac{\partial y}{\partial s_1} \\ \frac{\partial x}{\partial s_2} & \frac{\partial y}{\partial s_2} \end{bmatrix}$$

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 2-by-2 Jacobian matrix.

private pure function shl_local_coordinates(this) result(rst)

Computes the in-plane nodal coordinates in the element's local coordinate system. Node 1 is located at the local origin.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

An N-by-2 matrix, where N is the number of nodes, containing the local $x$ and $y$ coordinates of each node.

private pure function shl_local_frame(this) result(rst)

Computes the direction cosine matrix of the element's local coordinate system. The rows of the matrix are the local $x$, $y$, and $z$ unit vectors expressed in global coordinates; therefore, $\boldsymbol{x}_{local} = R \boldsymbol{x}_{global}$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 3-by-3 direction cosine matrix.

private pure function shl_mass_matrix(this, rule) result(rst)

Computes the consistent element mass matrix in the global coordinate system. $$M = T^T \left( \int_A \rho N^T \Lambda N \, dA \right) T, \quad \Lambda = \mathrm{diag}\left(t, t, t, \frac{t^3}{12}, \frac{t^3}{12}, \frac{t^3}{12}\right)$$

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 6N-by-6N mass matrix, where N is the number of nodes.

private pure function shl_rotation_matrix(this) result(rst)

Computes the transformation matrix $T$ relating the global element displacement vector to the local element displacement vector such that $\boldsymbol{u}_{local} = T \boldsymbol{u}_{global}$. The matrix is block diagonal with the 3-by-3 direction cosine matrix repeated for the translations and rotations of each node.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 6N-by-6N transformation matrix, where N is the number of nodes.

private pure function shl_shape_function_gradient(this, s) result(rst)

Computes the derivatives of the shape functions with respect to the local $x$ and $y$ coordinates, $\nabla_{xy} N = J^{-1} \nabla_{s} N$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

A 2-by-N matrix, where N is the number of nodes, containing $\partial N_i / \partial x$ in the first row and $\partial N_i / \partial y$ in the second row.

private pure function shl_shape_function_matrix(this, s) result(rst)

Computes the 6-by-6N shape function matrix interpolating the local nodal quantities $[u, v, w, \theta_x, \theta_y, \theta_z]$ with the element's linear (triangle) or bilinear (quadrilateral) shape functions. This interpolation is used to form the mass matrix and the external force vector.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 6-by-6N shape function matrix, where N is the number of nodes.

private pure function quad_shape_function_natural_gradient(this, s) result(rst)

Computes the derivatives of the bilinear shape functions with respect to the natural coordinates.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 2-by-4 matrix of derivatives.

private pure function shl_stiffness_matrix(this, rule) result(rst)

Computes the element stiffness matrix in the global coordinate system. The local stiffness matrix is $$K_{local} = \int_A B^T D B \, dA + \int_A \alpha G t \, \boldsymbol{b}_d \boldsymbol{b}_d^T dA,$$ where $\boldsymbol{b}_d$ is the drilling-rotation operator. The global matrix is $K = T^T K_{local} T$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 6N-by-6N stiffness matrix, where N is the number of nodes.

private pure function shl_strain(this, displacement, s) result(rst)

Computes the generalized strain vector, in the local coordinate system, at the specified natural coordinate. The components are ordered as $[\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy}, \kappa_{xx}, \kappa_{yy}, \kappa_{xy}, \gamma_{xz}, \gamma_{yz}]$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The 8-element generalized strain vector.

private pure function quad_strain_disp_matrix(this, s) result(rst)

Computes the 8-by-24 generalized strain-displacement matrix in the local coordinate system. Rows 1-3 contain the bilinear membrane terms, rows 4-6 the Mindlin-Reissner curvature terms, and rows 7-8 the MITC4 assumed transverse shear strains.

The covariant transverse shear strains are sampled at the edge midpoints $A = (0, 1)$, $B = (-1, 0)$, $C = (0, -1)$, and $D = (1, 0)$ and interpolated as $$\tilde{\gamma}_r = \frac{1}{2}(1 + s)\gamma_r^A + \frac{1}{2}(1 - s)\gamma_r^C, \quad \tilde{\gamma}_s = \frac{1}{2}(1 + r)\gamma_s^D + \frac{1}{2}(1 - r)\gamma_s^B,$$ and then transformed to local Cartesian components with $[\gamma_{xz}, \gamma_{yz}]^T = J^{-1}[\tilde{\gamma}_r, \tilde{\gamma}_s]^T$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The 8-by-24 strain-displacement matrix.

private pure function shl_stress(this, displacement, s) result(rst)

Computes the stress-resultant vector, in the local coordinate system, at the specified natural coordinate. The components are ordered as $[N_{xx}, N_{yy}, N_{xy}, M_{xx}, M_{yy}, M_{xy}, Q_x, Q_y]$. The surface stresses may be recovered as $\sigma = N / t \pm 6 M / t^2$.

Arguments

Return Value real(kind=real64), allocatable, dimension(:)

The 8-element stress-resultant vector.