Hello Sampad / Jason
I think I managed to solve the classic fixed beam problem by splitting the
problem in two beams and forming a third beam which contains the solution.
The logic is weird but it works. Hope it is useful for whoever tries to
implement beams with indeterminacy of 1 and above.
At present the beam module is unable to handle indeterminate beams.
Final listing attached.
"""
Fixed Beam of length L with concentrated load at center.
Slopes at both ends = 0
Deflections at both ends = 0
Due to indeterminacy of 2, the problem is split up in two parts
Step 1 : beam b1 is defined with far end free and concentrated load in the
centre
Step 2 : beam b2 is defined with far end free and unit load and unit moment
applied at the far end.
Step 3 : the two solutions are solved as a system of two equations in two
unknowns.
Step 4 : beam b3 is the final solution in which M3 and M4 (fixed end
moments) are input from the calculated values
"""
from sympy.physics.continuum_mechanics.beam import Beam
from sympy import symbols, linsolve, plot
from sympy import SingularityFunction
E, I, x = symbols('E, I, x')
R1, R2, R3, R4 = symbols('R1, R2 R3 R4')
M1, M2 = symbols('M1, M2')
P, L = 6.,4.
uR1, uM1 = symbols('uR1, uM1')
w_unit, m_unit = symbols('w_unit m_unit')
# Step 1
b1 = Beam(L, E, I)
b1.apply_load(R1, 0, -1)
b1.apply_load(P, L*.5, -1)
b1.apply_load(M1, 0, -2)
b1.bc_deflection = [(0, 0)]
b1.bc_slope = [(0, 0)]
b1.solve_for_reaction_loads(R1, M1)
# Step 2
b2 = Beam(L, E, I)
b2.apply_load(uR1, 0, -1)
b2.apply_load(uM1, 0, -2)
b2.apply_load(w_unit, L, -1)
b2.apply_load(m_unit, L, -2)
b2.bc_deflection = [(0, 0)]
b2.bc_slope = [(0, 0)]
b2.solve_for_reaction_loads(uR1, uM1)
# Step 3
slope_constraint =
b1.slope().subs({E:1,I:1,x:L})-b2.slope().subs({E:1,I:1,x:L})
deflection_constraint =
b1.deflection().subs({E:1,I:1,x:L})-b2.deflection().subs({E:1,I:1,x:L})
constraints = [slope_constraint,deflection_constraint]
soln = linsolve(constraints,[w_unit,m_unit])
w_u, m_u = soln.args[0]
M4 = -m_u
mtot = b1.reaction_loads[M1] - b2.reaction_loads[uM1]
M3 = mtot.subs({m_unit : m_u,w_unit:w_u})
# Step 4
b3 = Beam(L, E, I)
b3.apply_load(R3, 0, -1)
b3.apply_load(R4, L, -1)
b3.apply_load(P, L*.5, -1)
b3.apply_load(M3, 0, -2)
b3.apply_load(M4, L, -2)
b3.bc_deflection = [(0, 0),(L,0)]
b3.solve_for_reaction_loads(R3, R4)
Sanjiv
On Thursday, January 12, 2017 at 2:10:21 PM UTC, SAMPAD SAHA wrote:
>
>
>
> ---------- Forwarded message ----------
> From: Durve Sanjiv (External) <[email protected] <javascript:>>
> Date: Thu, Jan 12, 2017 at 6:40 PM
> Subject: RE: SingularityFunction in sympy.physics.continuum_mechanics.beam
> To: SAMPAD SAHA <[email protected] <javascript:>>
>
>
> Sampad,
>
>
>
> Thanks for your reply and interest. You can add Sympy group in this thread
> – I have no problem just send me a link.
>
>
>
> I think my question was not very clear. I am sorry about that.
>
>
>
> My problem is a beam fixed at both ends with a concentrated load of 6
> units in the center. Theoretical answer for M1 and M2 is 6*4/8 = 3.
>
>
>
> When I run the following script,
>
>
>
> from sympy.physics.continuum_mechanics.beam import Beam
>
> from sympy.functions import SingularityFunction
>
> from sympy import symbols, Piecewise, plot
>
> E, I, x = symbols('E, I, x')
>
> R1, R2 = symbols('R1, R2')
>
> M1, M2 = symbols('M1, M2')
>
> b = Beam(4, E, I)
>
> b.apply_load(R1, 0, -1)
>
> b.apply_load(6, 2, -1)
>
> b.apply_load(R2, 4, -1)
>
> b.apply_load(M1, 0, -1)
>
> b.apply_load(M2, 4, -1)
>
> b.bc_deflection = [(0, 0), (4, 0)]
>
> b.bc_slope = [(0, 0), (4, 0)]
>
> print b.boundary_conditions
>
> #b.solve_for_reaction_loads(R1, R2)
>
> b.solve_for_reaction_loads(R1, R2,M1,M2)
>
> print b.load
>
> print b.shear_force()
>
> print b.bending_moment()
>
> print b.slope().subs({E:1,I:1})
>
> print b.deflection().subs({E:1,I:1})
>
>
>
>
>
> I get following output
>
>
>
>
>
>
>
> In [*2*]:
> runfile('C:/UserApps/Python27/Lib/site-packages/xy/problem01.py',
> wdir='C:/UserApps/Python27/Lib/site-packages/xy')
>
> {'slope': [(0, 0), (4, 0)], 'deflection': [(0, 0), (4, 0)]}
>
> M1*SingularityFunction(x, 0, -1) + M2*SingularityFunction(x, 4, -1) + (-M1
> - 3)*SingularityFunction(x, 0, -1) + (-M2 - 3)*SingularityFunction(x, 4,
> -1) + 6*SingularityFunction(x, 2, -1)
>
> M1*SingularityFunction(x, 0, 0) + M2*SingularityFunction(x, 4, 0) + (-M1 -
> 3)*SingularityFunction(x, 0, 0) + (-M2 - 3)*SingularityFunction(x, 4, 0) +
> 6*SingularityFunction(x, 2, 0)
>
> M1*SingularityFunction(x, 0, 1) + M2*SingularityFunction(x, 4, 1) + (-M1 -
> 3)*SingularityFunction(x, 0, 1) + (-M2 - 3)*SingularityFunction(x, 4, 1) +
> 6*SingularityFunction(x, 2, 1)
>
> Traceback (most recent call last):
>
>
>
> File "<ipython-input-2-febc462b31fc>", line 1, in <module>
>
> runfile('C:/UserApps/Python27/Lib/site-packages/xy/problem01.py',
> wdir='C:/UserApps/Python27/Lib/site-packages/xy')
>
>
>
> File
> "C:\UserApps\Python27\lib\site-packages\spyderlib\widgets\externalshell\sitecustomize.py",
>
> line 685, in runfile
>
> execfile(filename, namespace)
>
>
>
> File
> "C:\UserApps\Python27\lib\site-packages\spyderlib\widgets\externalshell\sitecustomize.py",
>
> line 71, in execfile
>
> exec(compile(scripttext, filename, 'exec'), glob, loc)
>
>
>
> File "C:/UserApps/Python27/Lib/site-packages/xy/problem01.py", line 28,
> in <module>
>
> print b.slope().subs({E:1,I:1})
>
>
>
> File
> "C:\UserApps\Python27\lib\site-packages\sympy-1.0.1.dev0-py2.7.egg\sympy\physics\continuum_mechanics\beam.py",
>
> line 455, in slope
>
> slope_curve = slope_curve.subs({C3: constants[0][0]})
>
>
>
> IndexError: list index out of range
>
>
>
> In [*3*]: b.reaction_loads
>
> Out[*3*]: {M2: M2, M1: M1, R2: -M2 - 3, R1: -M1 - 3}
>
> I expect to get something like {R2:-3,R1:-3,M1:3, M1:3}
>
> I can understand that the problem I am trying to solve has indeterminacy
> of 2 and that is why my script is missing some additional statements.
>
> Once this appears in the thread, I will pursue this because it is an
> intersting problem for a hobbyist like me.
>
> Thanks for the help.
>
>
>
> Regards,
>
> Sanjiv Durve
>
> Ex- I I T Kanpur (1976)
>
>
>
> *From:* SAMPAD SAHA [mailto:[email protected] <javascript:>]
> *Sent:* 12 January 2017 12:34
> *To:* Durve Sanjiv (External)
> *Subject:* Re: SingularityFunction in
> sympy.physics.continuum_mechanics.beam
>
>
>
> Sorry for the late reply.
>
>
>
> I tried it in my system. I am getting {R2: -3, R1: -3} as output for the
> reaction forces.
>
> Can you please share us your outputs and also the problem statement which
> you are trying to solve?
>
>
>
> And one more thing, it is absolutely fine contacting me directly at my
> gmail but still it is more preferrable that you post these queries into
> the Sympy group and CC other developers. This way you don't have to wait
> for long and can get very quick response as there are many active
> developers working on Sympy currently.
>
>
>
> If you don't mind Can I add Sympy Group in this thread?
>
>
>
>
> Regards
>
> Sampad Kumar Saha
>
> Mathematics and Computing
>
> I.I.T. Kharagpur
>
>
>
> On Tue, Jan 10, 2017 at 10:29 PM, Durve Sanjiv (External) <
> [email protected] <javascript:>> wrote:
>
> Hello Sampad,
>
>
>
> I have been trying to solve a fixed beam problem with concentrated load at
> the center of the beam using sympy. You seem to have done a wonderful job
> of implementing SingularityFunction. I hope you don’t mind me contacting
> you directly at your gmail.
>
>
>
> I have the following code which does not give the correct bending moment
> at the fixed ends. The line * b.bc_slope = [(0, 0), (4, 0)] *seems
> to have no effect
>
>
>
> Is there a quick fix or let me know if I am missing something. I also
> tried introducing M1, M2 and then commented out (shown in bold lines)
>
>
>
> Your help would be appreciated.
>
>
>
>
>
> from sympy.physics.continuum_mechanics.beam import Beam
>
> from sympy.functions import SingularityFunction
>
> from sympy import symbols, Piecewise, plot
>
> E, I, x = symbols('E, I, x')
>
> R1, R2 = symbols('R1, R2')
>
> M1, M2 = symbols('M1, M2')
>
> b = Beam(4, E, I)
>
> b.apply_load(R1, 0, -1)
>
> b.apply_load(6, 2, -1)
>
> b.apply_load(R2, 4, -1)
>
> *#b.apply_load(M1, 0, -1)*
>
> *#b.apply_load(M2, 4, -1)*
>
> b.bc_deflection = [(0, 0), (4, 0)]
>
> b.bc_slope = [(0, 0), (4, 0)]
>
> print b.boundary_conditions
>
> b.solve_for_reaction_loads(R1, R2)
>
> *#b.solve_for_reaction_loads(R1, R2,M1,M2)*
>
> print b.load
>
> print b.shear_force()
>
> print b.bending_moment()
>
> print b.slope().subs({E:1,I:1})
>
> print b.deflection().subs({E:1,I:1})
>
>
>
> p2 = plot(b.shear_force(),(x,0,4))
>
> p3 = plot(b.bending_moment(),(x,0,4))
>
> p1 = plot(b.slope().subs({E:1,I:1}),(x,0,4))
>
> p4 = plot(b.deflection().subs({E:1,I:1}),(x,0,4))
>
> Regards,
>
> Sanjiv Durve
>
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