Week 4: Solving hinged beams

This week I worked on adding support for beams connected via hinge in #14773. Support for axially fixed beams and its API was implemented last week.

_solve_hinge_beams was added as a helper function to solve such Beams. 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.

So the final API looks like:

>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> l = symbols('l', positive=True)
>>> R1, M1, R2, R3, P = symbols('R1 M1 R2 R3 P')
>>> b1 = Beam(2*l, E, I)
>>> b2 = Beam(2*l, E, I)
>>> b = b1.join(b2,"hinge")
>>> b.apply_load(M1, 0, -2)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, l, -1)
>>> b.apply_load(R3, 4*l, -1
>>> b.apply_load(P, 3*l, -1)
>>> b.bc_slope = [(0, 0)]
>>> b.bc_deflection = [(0, 0), (l, 0), (4*l, 0)]
>>> b.solve_for_reaction_loads(M1, R1, R2, R3)
>>> b.reaction_loads
{R3: -P/2, R2: -5*P/4, M1: -P*l/4, R1: 3*P/4}
>>> b.slope().subs(x, 3*l)
-7*P*l**2/(48*E*I)
>>> b.deflection().subs(x, 2*l)
7*P*l**3/(24*E*I)

Next Week

• See for changes in #14786 to get it merged.
• Add support for non-horizontal beams.
• See for any remaining implementation from first two stages of my proposal.