Beam (Docstrings)¶
Beam¶
This module can be used to solve 2D beam bending problems with singularity functions in mechanics.

class
sympy.physics.continuum_mechanics.beam.
Beam
(length, elastic_modulus, second_moment, variable=x, base_char='C')[source]¶ A Beam is a structural element that is capable of withstanding load primarily by resisting against bending. Beams are characterized by their cross sectional profile(Second moment of area), their length and their material.
Note
While solving a beam bending problem, a user should choose its own sign convention and should stick to it. The results will automatically follow the chosen sign convention.
Examples
There is a beam of length 4 meters. A constant distributed load of 6 N/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. The deflection of the beam at the end is restricted.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols, Piecewise >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(4, E, I) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(6, 2, 0) >>> b.apply_load(R2, 4, 1) >>> b.bc_deflection = [(0, 0), (4, 0)] >>> b.boundary_conditions {'deflection': [(0, 0), (4, 0)], 'slope': []} >>> b.load R1*SingularityFunction(x, 0, 1) + R2*SingularityFunction(x, 4, 1) + 6*SingularityFunction(x, 2, 0) >>> b.solve_for_reaction_loads(R1, R2) >>> b.load 3*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 2, 0)  9*SingularityFunction(x, 4, 1) >>> b.shear_force() 3*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 2, 1)  9*SingularityFunction(x, 4, 0) >>> b.bending_moment() 3*SingularityFunction(x, 0, 1) + 3*SingularityFunction(x, 2, 2)  9*SingularityFunction(x, 4, 1) >>> b.slope() (3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3)  9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I) >>> b.deflection() (7*x  SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4  3*SingularityFunction(x, 4, 3)/2)/(E*I) >>> b.deflection().rewrite(Piecewise) (7*x  Piecewise((x**3, x > 0), (0, True))/2  3*Piecewise(((x  4)**3, x  4 > 0), (0, True))/2 + Piecewise(((x  2)**4, x  2 > 0), (0, True))/4)/(E*I)

applied_loads
¶ Returns a list of all loads applied on the beam object. Each load in the list is a tuple of form (value, start, order, end).
Examples
There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A pointload of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point. Another pointload of magnitude 5 N is applied at same position.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(3, 0, 2) >>> b.apply_load(4, 2, 1) >>> b.apply_load(5, 2, 1) >>> b.load 3*SingularityFunction(x, 0, 2) + 9*SingularityFunction(x, 2, 1) >>> b.applied_loads [(3, 0, 2, None), (4, 2, 1, None), (5, 2, 1, None)]

apply_load
(value, start, order, end=None)[source]¶ This method adds up the loads given to a particular beam object.
Parameters: value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and point forces this is the location of application.
order : Integer
The order of the applied load.
 For moments, order = 2
 For point loads, order =1
 For constant distributed load, order = 0
 For ramp loads, order = 1
 For parabolic ramp loads, order = 2
 … so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point within the length of the beam.
Examples
There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A point load of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 2 meters to 3 meters away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(3, 0, 2) >>> b.apply_load(4, 2, 1) >>> b.apply_load(2, 2, 2, end=3) >>> b.load 3*SingularityFunction(x, 0, 2) + 4*SingularityFunction(x, 2, 1)  2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)

apply_support
(loc, type='fixed')[source]¶ This method applies support to a particular beam object.
Parameters: loc : Sympifyable
Location of point at which support is applied.
type : String
Determines type of Beam support applied. To apply support structure with  zero degree of freedom, type = “fixed”  one degree of freedom, type = “pin”  two degrees of freedom, type = “roller”
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(30, E, I) >>> b.apply_support(10, 'roller') >>> b.apply_support(30, 'roller') >>> b.apply_load(8, 0, 1) >>> b.apply_load(120, 30, 2) >>> R_10, R_30 = symbols('R_10, R_30') >>> b.solve_for_reaction_loads(R_10, R_30) >>> b.load 8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 2) + 2*SingularityFunction(x, 30, 1) >>> b.slope() (4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)

bending_moment
()[source]¶ Returns a Singularity Function expression which represents the bending moment curve of the Beam object.
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(8, 0, 1) >>> b.apply_load(R1, 10, 1) >>> b.apply_load(R2, 30, 1) >>> b.apply_load(120, 30, 2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.bending_moment() 8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)

boundary_conditions
¶ Returns a dictionary of boundary conditions applied on the beam. The dictionary has three kewwords namely moment, slope and deflection. The value of each keyword is a list of tuple, where each tuple contains loaction and value of a boundary condition in the format (location, value).
Examples
There is a beam of length 4 meters. The bending moment at 0 should be 4 and at 4 it should be 0. The slope of the beam should be 1 at 0. The deflection should be 2 at 0.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.bc_deflection = [(0, 2)] >>> b.bc_slope = [(0, 1)] >>> b.boundary_conditions {'deflection': [(0, 2)], 'slope': [(0, 1)]}
Here the deflection of the beam should be
2
at0
. Similarly, the slope of the beam should be1
at0
.

deflection
()[source]¶ Returns a Singularity Function expression which represents the elastic curve or deflection of the Beam object.
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(8, 0, 1) >>> b.apply_load(R1, 10, 1) >>> b.apply_load(R2, 30, 1) >>> b.apply_load(120, 30, 2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.deflection() (4000*x/3  4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3) + 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3  12000)/(E*I)

elastic_modulus
¶ Young’s Modulus of the Beam.

join
(beam, via='fixed')[source]¶ This method joins two beams to make a new composite beam system. Passed Beam class instance is attached to the right end of calling object. This method can be used to form beams having Discontinuous values of Elastic modulus or Second moment.
Parameters: beam : Beam class object
The Beam object which would be connected to the right of calling object.
via : String
States the way two Beam object would get connected  For axially fixed Beams, via=”fixed”  For Beams connected via hinge, via=”hinge”
Examples
There is a cantilever beam of length 4 meters. For first 2 meters its moment of inertia is \(1.5*I\) and \(I\) for the other end. A pointload of magnitude 4 N is applied from the top at its free end.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b1 = Beam(2, E, 1.5*I) >>> b2 = Beam(2, E, I) >>> b = b1.join(b2, "fixed") >>> b.apply_load(20, 4, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 0, 2) >>> b.bc_slope = [(0, 0)] >>> b.bc_deflection = [(0, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.load 80*SingularityFunction(x, 0, 2)  20*SingularityFunction(x, 0, 1) + 20*SingularityFunction(x, 4, 1) >>> b.slope() (((80*SingularityFunction(x, 0, 1)  10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))/I  120/I)/E + 80.0/(E*I))*SingularityFunction(x, 2, 0) + 0.666666666666667*(80*SingularityFunction(x, 0, 1)  10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 0, 0)/(E*I)  0.666666666666667*(80*SingularityFunction(x, 0, 1)  10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 2, 0)/(E*I)

length
¶ Length of the Beam.

load
¶ Returns a Singularity Function expression which represents the load distribution curve of the Beam object.
Examples
There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A point load of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 3 meters away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(3, 0, 2) >>> b.apply_load(4, 2, 1) >>> b.apply_load(2, 3, 2) >>> b.load 3*SingularityFunction(x, 0, 2) + 4*SingularityFunction(x, 2, 1)  2*SingularityFunction(x, 3, 2)

max_deflection
()[source]¶ Returns point of max deflection and its coresponding deflection value in a Beam object.

plot_bending_moment
(subs=None)[source]¶ Returns a plot for Bending moment present in the Beam object.
Parameters: subs : dictionary
Python dictionary containing Symbols as key and their corresponding values.
Examples
There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**6)) >>> b.apply_load(5000, 2, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 8, 1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_bending_moment() Plot object containing: [0]: cartesian line: 13750*SingularityFunction(x, 0, 1) + 5000*SingularityFunction(x, 2, 1) + 5000*SingularityFunction(x, 4, 2)  31250*SingularityFunction(x, 8, 1)  5000*SingularityFunction(x, 8, 2) for x over (0.0, 8.0)

plot_deflection
(subs=None)[source]¶ Returns a plot for deflection curve of the Beam object.
Parameters: subs : dictionary
Python dictionary containing Symbols as key and their corresponding values.
Examples
There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**6)) >>> b.apply_load(5000, 2, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 8, 1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_deflection() Plot object containing: [0]: cartesian line: 0.00138541666666667*x  2.86458333333333e5*SingularityFunction(x, 0, 3) + 1.04166666666667e5*SingularityFunction(x, 2, 3) + 5.20833333333333e6*SingularityFunction(x, 4, 4)  6.51041666666667e5*SingularityFunction(x, 8, 3)  5.20833333333333e6*SingularityFunction(x, 8, 4) for x over (0.0, 8.0)

plot_loading_results
(subs=None)[source]¶ Returns Axes object containing subplots of Shear Force, Bending Moment, Slope and Deflection of the Beam object.
Parameters: subs : dictionary
Python dictionary containing Symbols as key and their corresponding values.
.. note::
This method only works if numpy and matplotlib libraries are installed on the system.
Examples
There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**6)) >>> b.apply_load(5000, 2, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 8, 1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> axes = b.plot_loading_results()

plot_shear_force
(subs=None)[source]¶ Returns a plot for Shear force present in the Beam object.
Parameters: subs : dictionary
Python dictionary containing Symbols as key and their corresponding values.
Examples
There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**6)) >>> b.apply_load(5000, 2, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 8, 1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_shear_force() Plot object containing: [0]: cartesian line: 13750*SingularityFunction(x, 0, 0) + 5000*SingularityFunction(x, 2, 0) + 10000*SingularityFunction(x, 4, 1)  31250*SingularityFunction(x, 8, 0)  10000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0)

plot_slope
(subs=None)[source]¶ Returns a plot for slope of deflection curve of the Beam object.
Parameters: subs : dictionary
Python dictionary containing Symbols as key and their corresponding values.
Examples
There is a beam of length 8 meters. A constant distributed load of 10 KN/m is applied from half of the beam till the end. There are two simple supports below the beam, one at the starting point and another at the ending point of the beam. A pointload of magnitude 5 KN is also applied from top of the beam, at a distance of 4 meters from the starting point. Take E = 200 GPa and I = 400*(10**6) meter**4.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> R1, R2 = symbols('R1, R2') >>> b = Beam(8, 200*(10**9), 400*(10**6)) >>> b.apply_load(5000, 2, 1) >>> b.apply_load(R1, 0, 1) >>> b.apply_load(R2, 8, 1) >>> b.apply_load(10000, 4, 0, end=8) >>> b.bc_deflection = [(0, 0), (8, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.plot_slope() Plot object containing: [0]: cartesian line: 8.59375e5*SingularityFunction(x, 0, 2) + 3.125e5*SingularityFunction(x, 2, 2) + 2.08333333333333e5*SingularityFunction(x, 4, 3)  0.0001953125*SingularityFunction(x, 8, 2)  2.08333333333333e5*SingularityFunction(x, 8, 3) + 0.00138541666666667 for x over (0.0, 8.0)

point_cflexure
()[source]¶ Returns a Set of point(s) with zero bending moment and where bending moment curve of the beam object changes its sign from negative to positive or vice versa.
Examples
There is is 10 meter long overhanging beam. There are two simple supports below the beam. One at the start and another one at a distance of 6 meters from the start. Point loads of magnitude 10KN and 20KN are applied at 2 meters and 4 meters from start respectively. A Uniformly distribute load of magnitude of magnitude 3KN/m is also applied on top starting from 6 meters away from starting point till end. Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(10, E, I) >>> b.apply_load(4, 0, 1) >>> b.apply_load(46, 6, 1) >>> b.apply_load(10, 2, 1) >>> b.apply_load(20, 4, 1) >>> b.apply_load(3, 6, 0) >>> b.point_cflexure() [10/3]

reaction_loads
¶ Returns the reaction forces in a dictionary.

remove_load
(value, start, order, end=None)[source]¶ This method removes a particular load present on the beam object. Returns a ValueError if the load passed as an argument is not present on the beam.
Parameters: value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and point forces this is the location of application.
order : Integer
The order of the applied load.  For moments, order= 2  For point loads, order=1  For constant distributed load, order=0  For ramp loads, order=1  For parabolic ramp loads, order=2  … so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point within the length of the beam.
Examples
There is a beam of length 4 meters. A moment of magnitude 3 Nm is applied in the clockwise direction at the starting point of the beam. A pointload of magnitude 4 N is applied from the top of the beam at 2 meters from the starting point and a parabolic ramp load of magnitude 2 N/m is applied below the beam starting from 2 meters to 3 meters away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> b = Beam(4, E, I) >>> b.apply_load(3, 0, 2) >>> b.apply_load(4, 2, 1) >>> b.apply_load(2, 2, 2, end=3) >>> b.load 3*SingularityFunction(x, 0, 2) + 4*SingularityFunction(x, 2, 1)  2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2) >>> b.remove_load(2, 2, 2, end = 3) >>> b.load 3*SingularityFunction(x, 0, 2) + 4*SingularityFunction(x, 2, 1)

second_moment
¶ Second moment of area of the Beam.

shear_force
()[source]¶ Returns a Singularity Function expression which represents the shear force curve of the Beam object.
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(8, 0, 1) >>> b.apply_load(R1, 10, 1) >>> b.apply_load(R2, 30, 1) >>> b.apply_load(120, 30, 2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.shear_force() 8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, 1) + 2*SingularityFunction(x, 30, 0)

slope
()[source]¶ Returns a Singularity Function expression which represents the slope the elastic curve of the Beam object.
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(8, 0, 1) >>> b.apply_load(R1, 10, 1) >>> b.apply_load(R2, 30, 1) >>> b.apply_load(120, 30, 2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.solve_for_reaction_loads(R1, R2) >>> b.slope() (4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) + 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)

solve_for_reaction_loads
(*reactions)[source]¶ Solves for the reaction forces.
Examples
There is a beam of length 30 meters. A moment of magnitude 120 Nm is applied in the clockwise direction at the end of the beam. A pointload of magnitude 8 N is applied from the top of the beam at the starting point. There are two simple supports below the beam. One at the end and another one at a distance of 10 meters from the start. The deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols, linsolve, limit >>> E, I = symbols('E, I') >>> R1, R2 = symbols('R1, R2') >>> b = Beam(30, E, I) >>> b.apply_load(8, 0, 1) >>> b.apply_load(R1, 10, 1) # Reaction force at x = 10 >>> b.apply_load(R2, 30, 1) # Reaction force at x = 30 >>> b.apply_load(120, 30, 2) >>> b.bc_deflection = [(10, 0), (30, 0)] >>> b.load R1*SingularityFunction(x, 10, 1) + R2*SingularityFunction(x, 30, 1)  8*SingularityFunction(x, 0, 1) + 120*SingularityFunction(x, 30, 2) >>> b.solve_for_reaction_loads(R1, R2) >>> b.reaction_loads {R1: 6, R2: 2} >>> b.load 8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 2) + 2*SingularityFunction(x, 30, 1)

variable
¶ A symbol that can be used as a variable along the length of the beam while representing load distribution, shear force curve, bending moment, slope curve and the deflection curve. By default, it is set to
Symbol('x')
, but this property is mutable.Examples
>>> from sympy.physics.continuum_mechanics.beam import Beam >>> from sympy import symbols >>> E, I = symbols('E, I') >>> x, y, z = symbols('x, y, z') >>> b = Beam(4, E, I) >>> b.variable x >>> b.variable = y >>> b.variable y >>> b = Beam(4, E, I, z) >>> b.variable z


class
sympy.physics.continuum_mechanics.beam.
Beam3D
(length, elastic_modulus, shear_modulus, second_moment, area, variable=x)[source]¶ This class handles loads applied in any direction of a 3D space along with unequal values of Second moment along different axes.
Note
While solving a beam bending problem, a user should choose its own sign convention and should stick to it. The results will automatically follow the chosen sign convention. This class assumes that any kind of distributed load/moment is applied through out the span of a beam.
Examples
There is a beam of l meters long. A constant distributed load of magnitude q is applied along yaxis from start till the end of beam. A constant distributed moment of magnitude m is also applied along zaxis from start till the end of beam. Beam is fixed at both of its end. So, deflection of the beam at the both ends is restricted.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols, simplify >>> l, E, G, I, A = symbols('l, E, G, I, A') >>> b = Beam3D(l, E, G, I, A) >>> x, q, m = symbols('x, q, m') >>> b.apply_load(q, 0, 0, dir="y") >>> b.apply_moment_load(m, 0, 1, dir="z") >>> b.shear_force() [0, q*x, 0] >>> b.bending_moment() [0, 0, m*x + q*x**2/2] >>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])] >>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])] >>> b.solve_slope_deflection() >>> b.slope() [0, 0, l*x*(l*q + 3*l*(A*G*l*(l*q  2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I)) + 3*m)/(6*E*I) + q*x**3/(6*E*I) + x**2*(l*(A*G*l*(l*q  2*m) + 12*E*I*q)/(2*(A*G*l**2 + 12*E*I))  m)/(2*E*I)] >>> dx, dy, dz = b.deflection() >>> dx 0 >>> dz 0 >>> expectedy = ( ... l**2*q*x**2/(12*E*I) + l**2*x**2*(A*G*l*(l*q  2*m) + 12*E*I*q)/(8*E*I*(A*G*l**2 + 12*E*I)) ... + l*m*x**2/(4*E*I)  l*x**3*(A*G*l*(l*q  2*m) + 12*E*I*q)/(12*E*I*(A*G*l**2 + 12*E*I))  m*x**3/(6*E*I) ... + q*x**4/(24*E*I) + l*x*(A*G*l*(l*q  2*m) + 12*E*I*q)/(2*A*G*(A*G*l**2 + 12*E*I))  q*x**2/(2*A*G) ... ) >>> simplify(dy  expectedy) 0
References
[R501] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf 
apply_load
(value, start, order, dir='y')[source]¶ This method adds up the force load to a particular beam object.
Parameters: value : Sympifyable
The magnitude of an applied load.
dir : String
Axis along which load is applied.
order : Integer
The order of the applied load.  For point loads, order=1  For constant distributed load, order=0  For ramp loads, order=1  For parabolic ramp loads, order=2  … so on.

apply_moment_load
(value, start, order, dir='y')[source]¶ This method adds up the moment loads to a particular beam object.
Parameters: value : Sympifyable
The magnitude of an applied moment.
dir : String
Axis along which moment is applied.
order : Integer
The order of the applied load.  For point moments, order=2  For constant distributed moment, order=1  For ramp moments, order=0  For parabolic ramp moments, order=1  … so on.

area
¶ Crosssectional area of the Beam.

bending_moment
()[source]¶ Returns a list of three expressions which represents the bending moment curve of the Beam object along all three axes.

boundary_conditions
¶ Returns a dictionary of boundary conditions applied on the beam. The dictionary has two keywords namely slope and deflection. The value of each keyword is a list of tuple, where each tuple contains loaction and value of a boundary condition in the format (location, value). Further each value is a list corresponding to slope or deflection(s) values along three axes at that location.
Examples
There is a beam of length 4 meters. The slope at 0 should be 4 along the xaxis and 0 along others. At the other end of beam, deflection along all the three axes should be zero.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols >>> l, E, G, I, A, x = symbols('l, E, G, I, A, x') >>> b = Beam3D(30, E, G, I, A, x) >>> b.bc_slope = [(0, (4, 0, 0))] >>> b.bc_deflection = [(4, [0, 0, 0])] >>> b.boundary_conditions {'deflection': [(4, [0, 0, 0])], 'slope': [(0, (4, 0, 0))]}
Here the deflection of the beam should be
0
along all the three axes at4
. Similarly, the slope of the beam should be4
along xaxis and0
along y and z axis at0
.

deflection
()[source]¶ Returns a three element list representing deflection curve along all the three axes.

load_vector
¶ Returns a three element list representing the load vector.

moment_load_vector
¶ Returns a three element list representing moment loads on Beam.

second_moment
¶ Second moment of area of the Beam.

shear_force
()[source]¶ Returns a list of three expressions which represents the shear force curve of the Beam object along all three axes.

shear_modulus
¶ Young’s Modulus of the Beam.

slope
()[source]¶ Returns a three element list representing slope of deflection curve along all the three axes.

solve_for_reaction_loads
(*reaction)[source]¶ Solves for the reaction forces.
Examples
There is a beam of length 30 meters. It it supported by rollers at of its end. A constant distributed load of magnitude 8 N is applied from start till its end along yaxis. Another linear load having slope equal to 9 is applied along zaxis.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D >>> from sympy import symbols >>> l, E, G, I, A, x = symbols('l, E, G, I, A, x') >>> b = Beam3D(30, E, G, I, A, x) >>> b.apply_load(8, start=0, order=0, dir="y") >>> b.apply_load(9*x, start=0, order=0, dir="z") >>> b.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])] >>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4') >>> b.apply_load(R1, start=0, order=1, dir="y") >>> b.apply_load(R2, start=30, order=1, dir="y") >>> b.apply_load(R3, start=0, order=1, dir="z") >>> b.apply_load(R4, start=30, order=1, dir="z") >>> b.solve_for_reaction_loads(R1, R2, R3, R4) >>> b.reaction_loads {R1: 120, R2: 120, R3: 1350, R4: 2700}
