Truss (Docstrings)#

Truss#

This module can be used to solve problems related to 2D Trusses.

class sympy.physics.continuum_mechanics.truss.Truss[source]#

A Truss is an assembly of members such as beams, connected by nodes, that create a rigid structure. In engineering, a truss is a structure that consists of two-force members only.

Trusses are extremely important in engineering applications and can be seen in numerous real-world applications like bridges.

Examples

There is a Truss consisting of four nodes and five members connecting the nodes. A force P acts downward on the node D and there also exist pinned and roller joints on the nodes A and B respectively.

../../../_images/truss_example.png
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node("node_1", 0, 0)
>>> t.add_node("node_2", 6, 0)
>>> t.add_node("node_3", 2, 2)
>>> t.add_node("node_4", 2, 0)
>>> t.add_member("member_1", "node_1", "node_4")
>>> t.add_member("member_2", "node_2", "node_4")
>>> t.add_member("member_3", "node_1", "node_3")
>>> t.add_member("member_4", "node_2", "node_3")
>>> t.add_member("member_5", "node_3", "node_4")
>>> t.apply_load("node_4", magnitude=10, direction=270)
>>> t.apply_support("node_1", type="fixed")
>>> t.apply_support("node_2", type="roller")
add_member(label, start, end)[source]#

This method adds a member between any two nodes in the given truss.

Parameters:

label: String or Symbol

The label for a member. It is the only way to identify a particular member.

start: String or Symbol

The label of the starting point/node of the member.

end: String or Symbol

The label of the ending point/node of the member.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.add_node('C', 2, 2)
>>> t.add_member('AB', 'A', 'B')
>>> t.members
{'AB': ['A', 'B']}
add_node(label, x, y)[source]#

This method adds a node to the truss along with its name/label and its location.

Parameters:

label: String or a Symbol

The label for a node. It is the only way to identify a particular node.

x: Sympifyable

The x-coordinate of the position of the node.

y: Sympifyable

The y-coordinate of the position of the node.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.nodes
[('A', 0, 0)]
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
apply_load(location, magnitude, direction)[source]#

This method applies an external load at a particular node

Parameters:

location: String or Symbol

Label of the Node at which load is applied.

magnitude: Sympifyable

Magnitude of the load applied. It must always be positive and any changes in the direction of the load are not reflected here.

direction: Sympifyable

The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> from sympy import symbols
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> P = symbols('P')
>>> t.apply_load('A', P, 90)
>>> t.apply_load('A', P/2, 45)
>>> t.apply_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45], [P/4, 90]]}
apply_support(location, type)[source]#

This method adds a pinned or roller support at a particular node

Parameters:

location: String or Symbol

Label of the Node at which support is added.

type: String

Type of the support being provided at the node.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.apply_support('A', 'pinned')
>>> t.supports
{'A': 'pinned'}
change_member_label(label, new_label)[source]#

This method changes the label of a member.

Parameters:

label: String or Symbol

The label of the member for which the label has to be changed.

new_label: String or Symbol

The new label of the member.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.change_node_label('A', 'C')
>>> t.nodes
[('C', 0, 0), ('B', 3, 0)]
>>> t.add_member('BC', 'B', 'C')
>>> t.members
{'BC': ['B', 'C']}
>>> t.change_member_label('BC', 'BC_new')
>>> t.members
{'BC_new': ['B', 'C']}
change_node_label(label, new_label)[source]#

This method changes the label of a node.

Parameters:

label: String or Symbol

The label of the node for which the label has to be changed.

new_label: String or Symbol

The new label of the node.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.change_node_label('A', 'C')
>>> t.nodes
[('C', 0, 0), ('B', 3, 0)]
property internal_forces#

Returns the internal forces for all members which are all initialized to 0.

property loads#

Returns the loads acting on the truss.

property member_labels#

Returns the members of the truss along with the start and end points.

property members#

Returns the members of the truss along with the start and end points.

property node_labels#

Returns the node labels of the truss.

property node_positions#

Returns the positions of the nodes of the truss.

property nodes#

Returns the nodes of the truss along with their positions.

property reaction_loads#

Returns the reaction forces for all supports which are all initialized to 0.

remove_load(location, magnitude, direction)[source]#

This method removes an already present external load at a particular node

Parameters:

location: String or Symbol

Label of the Node at which load is applied and is to be removed.

magnitude: Sympifyable

Magnitude of the load applied.

direction: Sympifyable

The angle, in degrees, that the load vector makes with the horizontal in the counter-clockwise direction. It takes the values 0 to 360, inclusive.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> from sympy import symbols
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> P = symbols('P')
>>> t.apply_load('A', P, 90)
>>> t.apply_load('A', P/2, 45)
>>> t.apply_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45], [P/4, 90]]}
>>> t.remove_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45]]}
remove_member(label)[source]#

This method removes a member from the given truss.

Parameters:

label: String or Symbol

The label for the member to be removed.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.add_node('C', 2, 2)
>>> t.add_member('AB', 'A', 'B')
>>> t.add_member('AC', 'A', 'C')
>>> t.add_member('BC', 'B', 'C')
>>> t.members
{'AB': ['A', 'B'], 'AC': ['A', 'C'], 'BC': ['B', 'C']}
>>> t.remove_member('AC')
>>> t.members
{'AB': ['A', 'B'], 'BC': ['B', 'C']}
remove_node(label)[source]#

This method removes a node from the truss.

Parameters:

label: String or Symbol

The label of the node to be removed.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.nodes
[('A', 0, 0)]
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.remove_node('A')
>>> t.nodes
[('B', 3, 0)]
remove_support(location)[source]#

This method removes support from a particular node

Parameters:

location: String or Symbol

Label of the Node at which support is to be removed.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.apply_support('A', 'pinned')
>>> t.supports
{'A': 'pinned'}
>>> t.remove_support('A')
>>> t.supports
{}
solve()[source]#

This method solves for all reaction forces of all supports and all internal forces of all the members in the truss, provided the Truss is solvable.

A Truss is solvable if the following condition is met,

2n >= r + m

Where n is the number of nodes, r is the number of reaction forces, where each pinned support has 2 reaction forces and each roller has 1, and m is the number of members.

The given condition is derived from the fact that a system of equations is solvable only when the number of variables is lesser than or equal to the number of equations. Equilibrium Equations in x and y directions give two equations per node giving 2n number equations. However, the truss needs to be stable as well and may be unstable if 2n > r + m. The number of variables is simply the sum of the number of reaction forces and member forces.

Note

The sign convention for the internal forces present in a member revolves around whether each force is compressive or tensile. While forming equations for each node, internal force due to a member on the node is assumed to be away from the node i.e. each force is assumed to be compressive by default. Hence, a positive value for an internal force implies the presence of compressive force in the member and a negative value implies a tensile force.

Examples

>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node("node_1", 0, 0)
>>> t.add_node("node_2", 6, 0)
>>> t.add_node("node_3", 2, 2)
>>> t.add_node("node_4", 2, 0)
>>> t.add_member("member_1", "node_1", "node_4")
>>> t.add_member("member_2", "node_2", "node_4")
>>> t.add_member("member_3", "node_1", "node_3")
>>> t.add_member("member_4", "node_2", "node_3")
>>> t.add_member("member_5", "node_3", "node_4")
>>> t.apply_load("node_4", magnitude=10, direction=270)
>>> t.apply_support("node_1", type="pinned")
>>> t.apply_support("node_2", type="roller")
>>> t.solve()
>>> t.reaction_loads
{'R_node_1_x': 0, 'R_node_1_y': 6.66666666666667, 'R_node_2_y': 3.33333333333333}
>>> t.internal_forces
{'member_1': 6.66666666666666, 'member_2': 6.66666666666667, 'member_3': -6.66666666666667*sqrt(2), 'member_4': -3.33333333333333*sqrt(5), 'member_5': 10.0}
property supports#

Returns the nodes with provided supports along with the kind of support provided i.e. pinned or roller.