spring_element_2d

type, public, extends(spring_element) :: spring_element_2d

Defines a 2D linear, axial spring element with x and y translations at each node.

Variables

direction material node_1 node_2 stiffness use_direction

Type-Bound Procedures

axis constitutive_matrix damping_matrix evaluate_shape_function external_force_vector get_dimensionality get_dof_per_node get_node get_node_count get_node_natural_coordinates jacobian mass_matrix shape_function_matrix stiffness_matrix strain strain_displacement_matrix stress

Components

Type	Visibility	Attributes		Name		Initial
real(kind=real64),	public,	dimension(3)	::	direction	=	0.0d0	The unit vector defining the user-supplied element axis. Only the leading components corresponding to the element dimensionality are used. This value is used only if use_direction is true.
type(material),	public		::	material			The material.
type(node),	public		::	node_1			The first node of the element (s = -1).
type(node),	public		::	node_2			The second node of the element (s = 1).
real(kind=real64),	public		::	stiffness			The spring stiffness.
logical,	public		::	use_direction	=	.false.	True if the element axis is defined by direction; else, false if the axis is defined by the vector from node 1 to node 2.

Constructor

public interface spring_element_2d

private pure function se2d_init(k, nd1, nd2, direction) result(rst)

Initializes a new [[spring_element_2d]].

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in)			::	k	The spring stiffness.
class(node),	intent(in)			::	nd1	The first node of the element.
class(node),	intent(in)			::	nd2	The second node of the element.
real(kind=real64),	intent(in),	optional,	dimension(:)	::	direction	An optional 2-element vector defining the spring axis. The vector must be non-zero and is normalized internally. If not supplied, the axis is defined by the vector from node 1 to node 2, in which case the nodes must not be coincident.

Return Value type(spring_element_2d)

The new [[spring_element_2d]].

Type-Bound Procedures

procedure, public :: axis => tnde_axis

private pure function tnde_axis(this) result(rst)

Gets the unit vector defining the element axis in the global coordinate system.

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.

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

The D-element unit vector, where D is the element dimensionality.

procedure, public :: constitutive_matrix => se_constitutive_matrix

private pure function se_constitutive_matrix(this) result(rst)

Returns the matrix relating spring elongation to spring force.

Arguments

Type	Intent	Optional	Attributes		Name
class(spring_element),	intent(in)			::	this	The spring_element object.

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

The 1-by-1 matrix containing the spring stiffness.

procedure, public :: damping_matrix => de_damping_matrix

private pure function de_damping_matrix(this) result(rst)

Returns the element damping matrix. The default implementation returns a zero-valued matrix.

Arguments

Type	Intent	Optional	Attributes		Name
class(discrete_element),	intent(in)			::	this	The discrete_element object.

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

The N-by-N damping matrix, where N is the total number of element degrees of freedom.

procedure, public :: evaluate_shape_function => tnde_shape_function

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

Evaluates the i-th linear shape function $N_1 = \frac{1}{2}(1 - s), \qquad N_2 = \frac{1}{2}(1 + s).$

Arguments

Type	Intent	Attributes		Name
class(two_node_discrete_element),	intent(in)		::	this	The two_node_discrete_element object.
integer(kind=int32),	intent(in)		::	i	The index of the shape function to evaluate.
real(kind=real64),	intent(in),	dimension(:)	::	s	The natural coordinate at which to evaluate the shape function.

Return Value real(kind=real64)

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

procedure, public :: external_force_vector => de_ext_force_vector

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

Returns the element external force vector. Discrete elements do not support distributed loads; therefore, a zero-valued vector is returned. Concentrated loads should be applied directly to the global force vector.

Arguments

Type	Intent	Optional	Attributes		Name
class(discrete_element),	intent(in)			::	this	The discrete_element object.
real(kind=real64),	intent(in),		dimension(:)	::	q	The distributed load vector. This argument is unused and is present for interface compatibility.
integer(kind=int32),	intent(in),	optional		::	rule	The integration rule. This argument is unused and is present for interface compatibility.

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

The N-element force vector, where N is the total number of element degrees of freedom.

procedure, public :: get_dimensionality => se2d_dimensionality

private pure function se2d_dimensionality(this) result(rst)

Gets the dimensionality of the element.

Arguments

Type	Intent	Optional	Attributes		Name
class(spring_element_2d),	intent(in)			::	this	The spring_element_2d object.

Return Value integer(kind=int32)

The dimensionality.

procedure, public :: get_dof_per_node => tnde_dof_per_node

private pure function tnde_dof_per_node(this) result(rst)

Gets the number of degrees of freedom per node. This is equal to the element dimensionality as only translations are considered.

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.

Return Value integer(kind=int32)

The number of DOF per node.

procedure, public :: get_node => tnde_get_node

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

Gets the requested node from the element.

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.
integer(kind=int32),	intent(in)			::	i	The local index of the node to retrieve. This value must be either 1 or 2.

Return Value type(node)

The requested node.

procedure, public :: get_node_count => tnde_get_node_count

private pure function tnde_get_node_count(this) result(rst)

Gets the number of nodes for the element.

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.

Return Value integer(kind=int32)

The number of nodes.

procedure, public :: get_node_natural_coordinates => de_get_node_natural_coordinates

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

Returns the natural coordinate of the requested node. A single-node element has its node at s = 0; a two-node element has its nodes at s = -1 and s = 1, respectively.

Arguments

Type	Intent	Optional	Attributes		Name
class(discrete_element),	intent(in)			::	this	The discrete_element object.
integer(kind=int32),	intent(in)			::	i	The local index of the node.

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

The natural coordinate of the node.

procedure, public :: jacobian => de_jacobian

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

Returns the element Jacobian. Discrete elements have no spatial extent; therefore, a unit 1-by-1 Jacobian is returned.

Arguments

Type	Intent	Optional	Attributes		Name
class(discrete_element),	intent(in)			::	this	The discrete_element object.
real(kind=real64),	intent(in),		dimension(:)	::	s	The natural coordinates at which to evaluate the Jacobian. This argument is unused and is present for interface compatibility.

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

The 1-by-1 Jacobian matrix.

procedure, public :: mass_matrix => de_mass_matrix

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

Returns the element mass matrix. The default implementation returns a zero-valued matrix.

Arguments

Type	Intent	Optional	Attributes		Name
class(discrete_element),	intent(in)			::	this	The discrete_element object.
integer(kind=int32),	intent(in),	optional		::	rule	The integration rule. This argument is unused and is present for interface compatibility.

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

The N-by-N mass matrix, where N is the total number of element degrees of freedom.

procedure, public :: shape_function_matrix => tnde_shape_function_matrix

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

Computes the translational displacement interpolation matrix $N = \begin{bmatrix} N_1 I & N_2 I \end{bmatrix}.$

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.
real(kind=real64),	intent(in),		dimension(:)	::	s	The natural coordinate at which to evaluate the matrix.

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

The D-by-2D shape function matrix, where D is the element dimensionality.

procedure, public :: stiffness_matrix => se_stiffness_matrix

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

Computes the global spring stiffness matrix $K = k\,\boldsymbol{b}\boldsymbol{b}^T, \qquad \boldsymbol{b}=\begin{bmatrix}-\boldsymbol{a}\\ \boldsymbol{a}\end{bmatrix}.$

Arguments

Type	Intent	Optional	Attributes		Name
class(spring_element),	intent(in)			::	this	The spring_element object.
integer(kind=int32),	intent(in),	optional		::	rule	The integration rule. This argument is unused as the stiffness matrix is exact.

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

The 2D-by-2D global stiffness matrix, where D is the element dimensionality.

procedure, public :: strain => e_strain

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

Computes the element strain at the specified natural coordinate. The strain is $\boldsymbol{\varepsilon}=B\boldsymbol{u}_e.$

Arguments

Type	Intent	Attributes		Name
class(element),	intent(in)		::	this	The element object.
real(kind=real64),	intent(in),	dimension(:)	::	displacement	The element displacement vector in the element coordinate system.
real(kind=real64),	intent(in),	dimension(:)	::	s	The natural coordinates at which to evaluate the strain.

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

The resulting strain vector.

procedure, public :: strain_displacement_matrix => tnde_strain_disp_matrix

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

Computes the matrix relating global nodal displacements to the axial elongation of the element $B = \begin{bmatrix} -\boldsymbol{a}^T & \boldsymbol{a}^T \end{bmatrix},$ where $\boldsymbol{a}$ is the unit element axis.

Arguments

Type	Intent	Optional	Attributes		Name
class(two_node_discrete_element),	intent(in)			::	this	The two_node_discrete_element object.
real(kind=real64),	intent(in),		dimension(:)	::	s	The natural coordinate at which to evaluate the matrix. This argument is unused as the elongation is constant along the element.

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

The 1-by-2D matrix, where D is the element dimensionality.

procedure, public :: stress => e_stress

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

Computes the element stress result at the specified natural coordinate. The stress result is $\boldsymbol{\sigma}=D B\boldsymbol{u}_e.$

Arguments

Type	Intent	Attributes		Name
class(element),	intent(in)		::	this	The element object.
real(kind=real64),	intent(in),	dimension(:)	::	displacement	The element displacement vector in the element coordinate system.
real(kind=real64),	intent(in),	dimension(:)	::	s	The natural coordinates at which to evaluate the stress.

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

The resulting stress vector.

spring_element_2d Derived Type

Contents

Variables

Constructor

Type-Bound Procedures

type, public, extends(spring_element) :: spring_element_2d

Contents

Variables

Constructor

Type-Bound Procedures

Components

Constructor

public interface spring_element_2d

private pure function se2d_init(k, nd1, nd2, direction) result(rst)

Arguments

Return Value type(spring_element_2d)

Type-Bound Procedures

procedure, public :: axis => tnde_axis

private pure function tnde_axis(this) result(rst)

Arguments

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

procedure, public :: constitutive_matrix => se_constitutive_matrix

private pure function se_constitutive_matrix(this) result(rst)

Arguments

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

procedure, public :: damping_matrix => de_damping_matrix

private pure function de_damping_matrix(this) result(rst)

Arguments

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

procedure, public :: evaluate_shape_function => tnde_shape_function

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

Arguments

Return Value real(kind=real64)

procedure, public :: external_force_vector => de_ext_force_vector

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

Arguments

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

procedure, public :: get_dimensionality => se2d_dimensionality

private pure function se2d_dimensionality(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: get_dof_per_node => tnde_dof_per_node

private pure function tnde_dof_per_node(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: get_node => tnde_get_node

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

Arguments

Return Value type(node)

procedure, public :: get_node_count => tnde_get_node_count

private pure function tnde_get_node_count(this) result(rst)

Arguments

Return Value integer(kind=int32)

procedure, public :: get_node_natural_coordinates => de_get_node_natural_coordinates

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

Arguments

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

procedure, public :: jacobian => de_jacobian

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

Arguments

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

procedure, public :: mass_matrix => de_mass_matrix

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

Arguments

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

procedure, public :: shape_function_matrix => tnde_shape_function_matrix

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

Arguments

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

procedure, public :: stiffness_matrix => se_stiffness_matrix

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

Arguments

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

procedure, public :: strain => e_strain

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

Arguments

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

procedure, public :: strain_displacement_matrix => tnde_strain_disp_matrix

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

Arguments