shell_element Derived Type

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.

pure function element_shape_function(this, i, s) result(rst) Prototype

Defines the signature of a routine for computing the value of the i-th element shape function at natural coordinate.

Arguments

Return Value real(kind=real64)

The value of the i-th shape function at s.

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).

pure function element_get_node(this, i) result(rst) Prototype

Defines the signature of a function for retrieving the requested node from the element.

Arguments

Return Value type(node)

The node.

pure function element_query(this) result(rst) Prototype

Defines the signature of a function performing a query on an integer-valued property of a element type.

Arguments

Return Value integer(kind=int32)

The resulting value.

pure function element_get_node_natural_coordinates(this, i) result(rst) Prototype

Defines the signature of a function returning the natural coordinates of an element node.

Arguments

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

The natural coordinates of the node.

pure subroutine shell_integration_rule(this, pts, wts) Prototype

Defines the signature of a routine returning the numerical integration rule used by the element. An integral over the element is evaluated as $\int_A f\,dA \approx \sum_i w_i f(\boldsymbol{s}_i) \det J(\boldsymbol{s}_i)$.

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.

pure function shell_natural_gradient(this, s) result(rst) Prototype

Defines the signature of a routine returning the derivatives of the element shape functions with respect to the natural coordinates.

Arguments

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

A 2-by-N matrix, where N is the number of nodes, whose first row contains the derivatives with respect to the first natural coordinate and whose second row contains the derivatives with respect to the second natural coordinate.

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.

pure function element_matrix_function(this, s) result(rst) Prototype

Defines the signature of a routine for returning a matrix associated with the element.

Arguments

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

The resulting 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.