def test_simply_supported_beam_offset_load():
"""simple beam - concentrated load at arbitrary points
Load case 8
"""
locations = [2, 3, 5, 7, 8]
for location in locations:
# Exact setup
a = location
b = L - a
R1 = -P * b / L
R2 = -P * a / L
M_loc = -P * a * b / L # moment at load
d_loc = P * a**2 * b**2 / (3 * EI * L) # deflection at load
# numerical result
beam = Beam(
L,
loads=[PointLoad(P, location)],
reactions=[PinnedReaction(x) for x in [0, L]],
E=E,
Ixx=Ixx,
)
beam.solve()
# verify reactions
validate(beam,
loc=location,
R=[(R1, 0), (R2, 0)],
M_loc=M_loc,
d_loc=d_loc)
def test_simply_supported_beam_unequal_non_symmetric_loads():
"""Simple beam: Two unequal concentrated loads non-symmetrically placed
load case 11
"""
P1 = -900
P2 = -1200
a = L / 4 # location of first load from right
b = L / 5 # location of second load from left
R1 = -(P1 * (L - a) + P2 * b) / L
R2 = -(P1 * a + P2 * (L - b)) / L
M1 = R1 * a
x = L / 2
Mx = R1 * x + P1 * (x - a)
M2 = R2 * b
p = [
PointLoad(magnitude=P1, location=a),
PointLoad(magnitude=P2, location=L - b)
]
r = [PinnedReaction(x) for x in [0, L]]
beam = Beam(length=L, loads=p, reactions=r, E=E, Ixx=Ixx)
beam.solve()
# verify reactions
for m, loc in zip((M1, Mx, M2), (a, L / 2, L - b)):
# assert approx(beam.reactions[0].value[0], rel=1e-4) == R1*1.25
validate(beam, loc=loc, R=[(R1, 0), (R2, 0)], M_loc=m, d_loc=None)
def test_solve_method():
beam = Beam(25, [PointLoad(-100, 25)], [FixedReaction(0)], 29e6, 345)
reaction = beam.reactions[0]
assert reaction.force is None, "Reaction force was not None before being solved"
assert reaction.moment is None, "Reaction moment was not None before being solved"
beam.solve()
reaction = beam.reactions[0]
assert reaction.force == 100, "Reaction force must be equal to and opposite load"
assert (
reaction.moment == 100 *
25), "Reaction moment must be equal to the load times the moment arm"
def test_simply_supported_beam_equal_symmetric_loads():
"""Simple beam: Two equal concentrated loads symmetrically placed
load case 9
"""
R = -P # both reactions are equal
a = L / 4
M_loc = -P * a # max moment (at center between loads)
d_loc = P * a / (24 * EI) * (3 * L**2 - 4 * a**2
) # max deflection (at center)
p = [PointLoad(magnitude=P, location=x) for x in [a, L - a]]
r = [PinnedReaction(x) for x in [0, L]]
beam = Beam(length=L, loads=p, reactions=r, E=E, Ixx=Ixx)
beam.solve()
# verify reactions
validate(beam, loc=L / 2, R=[(R, 0), (R, 0)], M_loc=M_loc, d_loc=d_loc)
def example_2():
"""
Cantilevered Beam with 3 Pinned Supports and End Loading
"""
print("=" * 79)
print("Example 2")
print("Show an example with 3 Pinned Supports and End Loading\n")
beam_len = 10
# Note that both the reaction and load are both lists. They must always be
# given to Beam as a list,
r = [PinnedReaction(0), PinnedReaction(2), PinnedReaction(6)] # define reactions
p = [PointLoad(magnitude=-2, location=beam_len)] # define loads
b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
# an explicit solve is required to calculate the reaction values
b.solve()
print(b)
def test_simply_supported_beam_center_load():
"""simple beam - concentrated load at center
Load case 7
"""
# Exact setup
R = -P / 2 # lbs, reactions
M_max = -P * L / 4 # psi, maximum moment
d_max = P * L**3 / (48 * EI) # max displacement
# Numerical setup
beam = Beam(
length=L,
loads=[PointLoad(magnitude=P, location=L / 2)],
reactions=[PinnedReaction(x) for x in [0, L]],
E=E,
Ixx=Ixx,
)
beam.solve()
validate(beam, loc=L / 2, R=[(R, 0), (R, 0)], M_loc=M_max, d_loc=d_max)
def example_1():
"""
Cantilevered Beam with Fixed Support and End Loading
"""
print("=" * 79)
print("Example 1")
print(
"Show an example with a cantilevered beam with a fixed support and "
"point load at the end\n"
)
beam_len = 10
# Note that both the reaction and load are both lists. They must always be
# given to Beam as a list,
r = [FixedReaction(0)] # define reactions as list
p = [PointLoad(magnitude=-2, location=beam_len)] # define loads as list
b = Beam(beam_len, loads=p, reactions=r, E=29e6, Ixx=125)
# an explicit solve is required to calculate the reaction values
b.solve()
print(b)
def test_cantilevered_beam_load_at_end():
"""fixed beam with concentrated load at free end
case 13
"""
R = -P
M_max = P * L # at fixed end
d_max = P * L**3 / (3 * EI) # at free end
beam = Beam(
length=L,
loads=[PointLoad(magnitude=P, location=0)],
reactions=[FixedReaction(L)],
E=E,
Ixx=Ixx,
)
beam.solve()
validate(beam, loc=0, R=[(R, M_max)], M_loc=0, d_loc=d_max)
assert pytest.approx(beam.moment(L), rel=TOL) == M_max
def test_simply_supported_beam_equal_non_symmetric_loads():
"""Simple beam: Two equal concentrated loads non-symmetrically placed
load case 10
"""
a = L / 4 # location of first load from right
b = L / 5 # location of second load from left
R1 = -P / L * (L - a + b)
R2 = -P / L * (L + a - b)
M1 = R1 * a # moment at first load
x = L / 2
Mx = R1 * x + P * (x - a) # moment at center
M2 = R2 * b # moment at second load
p = [PointLoad(magnitude=P, location=x) for x in [a, L - b]]
r = [PinnedReaction(x) for x in [0, L]]
beam = Beam(length=L, loads=p, reactions=r, E=E, Ixx=Ixx)
beam.solve()
# verify reactions
for m, loc in zip((M1, Mx, M2), (a, L / 2, L - b)):
validate(beam, loc=loc, R=[(R1, 0), (R2, 0)], M_loc=m, d_loc=None)