# Source code for sympy.physics.continuum_mechanics.beam

"""
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.

"""

from __future__ import print_function, division

from sympy.core import S, Symbol, diff
from sympy.solvers import linsolve
from sympy.printing import sstr
from sympy.functions import SingularityFunction, Piecewise
from sympy.core import sympify
from sympy.integrals import integrate
from sympy.series import limit

[docs]class Beam(object):
"""
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.bc_deflection = [(0, 0), (4, 0)]
>>> b.boundary_conditions
{'deflection': [(0, 0), (4, 0)], 'slope': []}
R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0)
-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)
"""

def __init__(self, length, elastic_modulus, second_moment, variable=Symbol('x')):
"""Initializes the class.

Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material.
second_moment : Sympifyable
A SymPy expression representing the Beam's Second moment of area.
It is a geometrical property of an area which reflects how its
points are distributed with respect to its neutral axis.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to Symbol('x').
"""
self.length = length
self.elastic_modulus = elastic_modulus
self.second_moment = second_moment
self.variable = variable
self._boundary_conditions = {'deflection': [], 'slope': []}
self._composite_type = None
self._hinge_position = None

def __str__(self):
str_sol = 'Beam({}, {}, {})'.format(sstr(self._length), sstr(self._elastic_modulus), sstr(self._second_moment))
return str_sol

@property
""" Returns the reaction forces in a dictionary."""

@property
def length(self):
"""Length of the Beam."""
return self._length

@length.setter
def length(self, l):
self._length = sympify(l)

@property
def variable(self):
"""
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
"""
return self._variable

@variable.setter
def variable(self, v):
if isinstance(v, Symbol):
self._variable = v
else:
raise TypeError("""The variable should be a Symbol object.""")

@property
def elastic_modulus(self):
"""Young's Modulus of the Beam. """
return self._elastic_modulus

@elastic_modulus.setter
def elastic_modulus(self, e):
self._elastic_modulus = sympify(e)

@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment

@second_moment.setter
def second_moment(self, i):
self._second_moment = sympify(i)

@property
def boundary_conditions(self):
"""
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 at 0.
Similarly, the slope of the beam should be 1 at 0.
"""
return self._boundary_conditions

@property
def bc_slope(self):
return self._boundary_conditions['slope']

@bc_slope.setter
def bc_slope(self, s_bcs):
self._boundary_conditions['slope'] = s_bcs

@property
def bc_deflection(self):
return self._boundary_conditions['deflection']

@bc_deflection.setter
def bc_deflection(self, d_bcs):
self._boundary_conditions['deflection'] = d_bcs

[docs]    def join(self, beam, via="fixed"):
"""
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.

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.bc_slope = [(0, 0)]
>>> b.bc_deflection = [(0, 0)]
80*SingularityFunction(x, 0, -2) - 20*SingularityFunction(x, 0, -1) + 20*SingularityFunction(x, 4, -1)
>>> b.slope()
Piecewise((0.666666666666667*(80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2) +
10*SingularityFunction(x, 4, 2))/(E*I), x <= 2), (((80*SingularityFunction(x, 0, 1) -
10*SingularityFunction(x, 0, 2) + 10*SingularityFunction(x, 4, 2))/I - 120/I)/E + 80.0/(E*I), x <= 4))
"""
x = self.variable
E = self.elastic_modulus
new_length = self.length + beam.length
if self.second_moment != beam.second_moment:
new_second_moment = Piecewise((self.second_moment, x<=self.length),
(beam.second_moment, x<=new_length))
else:
new_second_moment = self.second_moment

if via == "fixed":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "fixed"
return new_beam

if via == "hinge":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "hinge"
new_beam._hinge_position = self.length
return new_beam

[docs]    def apply_support(self, loc, type="fixed"):
"""
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')
>>> R_10, R_30 = symbols('R_10, R_30')
-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)
"""
if type == "pin" or type == "roller":
self.bc_deflection.append((loc, 0))
else:
reaction_moment = Symbol('M_'+str(loc))
self.bc_deflection = [(loc, 0)]
self.bc_slope.append((loc, 0))

[docs]    def apply_load(self, value, start, order, end=None):
"""
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 constant distributed load, order=0
- 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(-2, 2, 2, end = 3)
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2)
+ 2*SingularityFunction(x, 3, 0) + 2*SingularityFunction(x, 3, 2)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)

if end:
if order == 0:
elif order.is_positive:
self._load -= value*SingularityFunction(x, end, order) + value*SingularityFunction(x, end, 0)
else:
raise ValueError("""Order of the load should be positive.""")

[docs]    def remove_load(self, value, start, order, end=None):
"""
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 constant distributed load, order=0
- 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(-2, 2, 2, end = 3)
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2)
+ 2*SingularityFunction(x, 3, 0) + 2*SingularityFunction(x, 3, 2)
>>> b.remove_load(-2, 2, 2, end = 3)
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)

if (value, start, order, end) in self._applied_loads:
else:
raise ValueError("""No such load distribution exists on the beam object.""")

if end:
if order == 0:
elif order.is_positive:
self._load += value*SingularityFunction(x, end, order) + value*SingularityFunction(x, end, 0)
else:
raise ValueError("""Order of the load should be positive.""")

@property
"""
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 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 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)
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2)
"""

@property
"""
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)
-3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1)
[(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)]
"""

def _solve_hinge_beams(self, *reactions):
"""Method to find integration constants and reactional variables in a
composite beam connected via hinge.
This method resolves the composite Beam into its sub-beams and then
equations of shear force, bending moment, slope and deflection are
evaluated for both of them separately. These equations are then solved
for unknown reactions and integration constants using the boundary
conditions applied on the Beam. Equal deflection of both sub-beams
at the hinge joint gives us another equation to solve the system.

Examples
========
A combined beam, with constant fkexural rigidity E*I, is formed by joining
a Beam of length 2*l to the right of another Beam of length l. The whole beam
is fixed at both of its both end. A point load of magnitude P is also applied
from the top at a distance of 2*l from starting point.

>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> l=symbols('l', positive=True)
>>> b1=Beam(l ,E,I)
>>> b2=Beam(2*l ,E,I)
>>> b=b1.join(b2,"hinge")
>>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
>>> b.bc_slope=[(0,0), (3*l, 0)]
>>> b.bc_deflection=[(0,0), (3*l, 0)]
{A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
>>> b.slope()
Piecewise(((5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36
+ 5*P*SingularityFunction(x, l, 2)/36)/(E*I), l >= x), ((P*l**2/18 - 4*P*l*SingularityFunction(-l +
x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
- 13*P*SingularityFunction(-l + x, 2*l, 2)/36)/(E*I), x < 3*l))
>>> b.deflection()
Piecewise(((5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108
+ 5*P*SingularityFunction(x, l, 3)/108)/(E*I), l >= x), ((5*P*l**3/54 + P*l**2*(-l + x)/18
- 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108
+ P*SingularityFunction(-l + x, l, 3)/6 - 13*P*SingularityFunction(-l + x, 2*l, 3)/108)/(E*I), x < 3*l))
"""
x = self.variable
l = self._hinge_position
E = self._elastic_modulus
I = self._second_moment

if isinstance(I, Piecewise):
I1 = I.args[0][0]
I2 = I.args[1][0]
else:
I1 = I2 = I

load_1 = 0       # Load equation on first segment of composite beam
load_2 = 0       # Load equation on second segment of composite beam

# Distributing load on both segments

h = Symbol('h')     # Force due to hinge

eq = []
shear_curve_1 = limit(shear_1, x, l)
eq.append(shear_curve_1)
bending_1 = integrate(shear_1, x)
moment_curve_1 = limit(bending_1, x, l)
eq.append(moment_curve_1)

shear_curve_2 = limit(shear_2, x, self.length - l)
eq.append(shear_curve_2)
bending_2 = integrate(shear_2, x)
moment_curve_2 = limit(bending_2, x, self.length - l)
eq.append(moment_curve_2)

C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_1 = S(1)/(E*I1)*(integrate(bending_1, x) + C1)
def_1 = S(1)/(E*I1)*(integrate((E*I)*slope_1, x) + C1*x + C2)
slope_2 = S(1)/(E*I2)*(integrate(integrate(integrate(load_2, x), x), x) + C3)
def_2 = S(1)/(E*I2)*(integrate((E*I)*slope_2, x) + C4)

for position, value in self.bc_slope:
if position<l:
eq.append(slope_1.subs(x, position) - value)
else:
eq.append(slope_2.subs(x, position - l) - value)

for position, value in self.bc_deflection:
if position<l:
eq.append(def_1.subs(x, position) - value)
else:
eq.append(def_2.subs(x, position - l) - value)

eq.append(def_1.subs(x, l) - def_2.subs(x, 0)) # Deflection of both the segments at hinge would be equal

constants = list(linsolve(eq, C1, C2, C3, C4, h, *reactions))
reaction_values = list(constants[0])[5:]

# Substituting constants and reactional load and moments with their corresponding values
def_1 = def_1.subs({C1: constants[0][0], C2: constants[0][1], h:constants[0][4]}).subs(self._reaction_loads)
slope_2 = slope_2.subs({x: x-l, C3: constants[0][2], h:constants[0][4]}).subs(self._reaction_loads)
def_2 = def_2.subs({x: x-l,C3: constants[0][2], C4: constants[0][3], h:constants[0][4]}).subs(self._reaction_loads)

self._hinge_beam_slope = Piecewise((slope_1, x<=l), (slope_2, x<self.length))
self._hinge_beam_deflection = Piecewise((def_1, x<=l), (def_2, x<self.length))

"""
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(R1, 10, -1)  # Reaction force at x = 10
>>> b.apply_load(R2, 30, -1)  # Reaction force at x = 30
>>> b.bc_deflection = [(10, 0), (30, 0)]
R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1)
- 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2)
{R1: 6, R2: 2}
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
"""
if self._composite_type == "hinge":
return self._solve_hinge_beams(*reactions)

x = self.variable
l = self.length
shear_curve = limit(self.shear_force(), x, l)
moment_curve = limit(self.bending_moment(), x, l)
reaction_values = linsolve([shear_curve, moment_curve], reactions).args

[docs]    def shear_force(self):
"""
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.bc_deflection = [(10, 0), (30, 0)]
>>> b.shear_force()
-8*SingularityFunction(x, 0, 0) + 6*SingularityFunction(x, 10, 0) + 120*SingularityFunction(x, 30, -1) + 2*SingularityFunction(x, 30, 0)
"""
x = self.variable

[docs]    def bending_moment(self):
"""
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.bc_deflection = [(10, 0), (30, 0)]
>>> b.bending_moment()
-8*SingularityFunction(x, 0, 1) + 6*SingularityFunction(x, 10, 1) + 120*SingularityFunction(x, 30, 0) + 2*SingularityFunction(x, 30, 1)
"""
x = self.variable
return integrate(self.shear_force(), x)

[docs]    def point_cflexure(self):
"""
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.point_cflexure()
[10/3]
"""
from sympy import solve, Piecewise

# To restrict the range within length of the Beam
moment_curve = Piecewise((float("nan"), self.variable<=0),
(self.bending_moment(), self.variable<self.length),
(float("nan"), True))

points = solve(moment_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
return points

[docs]    def slope(self):
"""
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.bc_deflection = [(10, 0), (30, 0)]
>>> 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)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment

if self._composite_type == "hinge":
return self._hinge_beam_slope
if not self._boundary_conditions['slope']:
return diff(self.deflection(), x)
if self._composite_type == "fixed":
args = I.args
conditions = []
prev_slope = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S(1)/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
conditions.append((prev_slope + slope_value, args[i][1]))
prev_slope = slope_value.subs(x, args[i][1].args[1])
return Piecewise(*conditions)

C3 = Symbol('C3')
slope_curve = integrate(self.bending_moment(), x) + C3

bc_eqs = []
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
bc_eqs.append(eqs)

constants = list(linsolve(bc_eqs, C3))
slope_curve = slope_curve.subs({C3: constants[0][0]})
return S(1)/(E*I)*slope_curve

[docs]    def deflection(self):
"""
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.bc_deflection = [(10, 0), (30, 0)]
>>> 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)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_deflection
if not self._boundary_conditions['deflection'] and not self._boundary_conditions['slope']:
if self._composite_type == "fixed":
args = I.args
conditions = []
prev_def = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
conditions.append(((prev_def + deflection_value), args[i][1]))
prev_def = deflection_value.subs(x, args[i][1].args[1])
return Piecewise(*conditions)
return S(1)/(E*I)*integrate(integrate(self.bending_moment(), x), x)
elif not self._boundary_conditions['deflection']:
return integrate(self.slope(), x)
elif not self._boundary_conditions['slope'] and self._boundary_conditions['deflection']:
if self._composite_type == "fixed":
args = I.args
conditions = []
prev_def = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
conditions.append(((prev_def + deflection_value), args[i][1]))
prev_def = deflection_value.subs(x, args[i][1].args[1])
return Piecewise(*conditions)
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_curve = integrate(self.bending_moment(), x) + C3
deflection_curve = integrate(slope_curve, x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, (C3, C4)))
deflection_curve = deflection_curve.subs({C3: constants[0][0], C4: constants[0][1]})
return S(1)/(E*I)*deflection_curve

if self._composite_type == "fixed":
args = I.args
conditions = []
prev_def = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
deflection_value = integrate(self.slope().args[i][0], (x, prev_end, x))
conditions.append(((prev_def + deflection_value), args[i][1]))
prev_def = deflection_value.subs(x, args[i][1].args[1])
return Piecewise(*conditions)

C4 = Symbol('C4')
deflection_curve = integrate((E*I)*self.slope(), x) + C4

bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)

constants = list(linsolve(bc_eqs, C4))
deflection_curve = deflection_curve.subs({C4: constants[0][0]})
return S(1)/(E*I)*deflection_curve

[docs]    def max_deflection(self):
"""
Returns point of max deflection and its coresponding deflection value
in a Beam object.
"""
from sympy import solve, Piecewise

# To restrict the range within length of the Beam
slope_curve = Piecewise((float("nan"), self.variable<=0),
(self.slope(), self.variable<self.length),
(float("nan"), True))

points = solve(slope_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
deflection_curve = self.deflection()
deflections = [deflection_curve.subs(self.variable, x) for x in points]
deflections = list(map(abs, deflections))
if len(deflections) != 0:
max_def = max(deflections)
return (points[deflections.index(max_def)], max_def)
else:
return None