SOLVE_BEAM_SYMBOLIC
Solves a one-dimensional beam model with symbolic or numeric inputs and returns closed-form expressions for reactions together with the piecewise shear-force and bending-moment equations over each span segment. The implementation uses symbeam to assemble supports, loads, and stiffness properties, then calls the symbolic solver.
The solver enforces static equilibrium and constitutive compatibility to produce piecewise fields. In symbolic form, these satisfy the standard beam differential relation E I \frac{d^4 v(x)}{dx^4} = q(x) together with support and continuity boundary conditions.
Excel Usage
=SOLVE_BEAM_SYMBOLIC(span, supports, loads, young_modulus, inertia)span(str, required): Length of the beam (m or symbolic string).supports(list[list], required): 2D array of support definitions [[x, type]]. x: position. type: “fixed”, “pin”, “roller”, “hinge”.loads(list[list], required): 2D array of load definitions [[type, val, x1, x2]]. type: “point”, “moment”, “dist”. val: magnitude or expression. x1, x2: coordinates.young_modulus(str, optional, default: “E”): Young’s Modulus E (Pa or symbol). Default is ‘E’.inertia(str, optional, default: “I”): Second moment of area I (m4 or symbol). Default is ‘I’.
Returns (dict): Excel data type containing symbolic reactions and piecewise shear-force and bending-moment equations.
Example 1: Cantilever beam with no applied load
Inputs:
| span | supports | loads | young_modulus | inertia | |||
|---|---|---|---|---|---|---|---|
| 1 | 0 | fixed | point | 0 | 1 | 200000000000 | 0.00001 |
Excel formula:
=SOLVE_BEAM_SYMBOLIC(1, {0,"fixed"}, {"point",0,1}, 200000000000, 0.00001)Expected output:
{"type":"Double","basicValue":2,"properties":{"Reactions":{"type":"Array","elements":[[{"type":"String","basicValue":"Point"},{"type":"String","basicValue":"Type"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Force"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Moment"},{"type":"String","basicValue":"0"}]]},"Equations":{"type":"Array","elements":[[{"type":"String","basicValue":"Span"},{"type":"String","basicValue":"Field"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"[0, 1.00000000000000]"},{"type":"String","basicValue":"V(x)"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"[0, 1.00000000000000]"},{"type":"String","basicValue":"M(x)"},{"type":"String","basicValue":"0"}]]}}}
Example 2: Cantilever beam with zero point load at free end
Inputs:
| span | supports | loads | young_modulus | inertia | |||
|---|---|---|---|---|---|---|---|
| 2 | 0 | fixed | point | 0 | 2 | 200000000000 | 0.00001 |
Excel formula:
=SOLVE_BEAM_SYMBOLIC(2, {0,"fixed"}, {"point",0,2}, 200000000000, 0.00001)Expected output:
{"type":"Double","basicValue":2,"properties":{"Reactions":{"type":"Array","elements":[[{"type":"String","basicValue":"Point"},{"type":"String","basicValue":"Type"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Force"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Moment"},{"type":"String","basicValue":"0"}]]},"Equations":{"type":"Array","elements":[[{"type":"String","basicValue":"Span"},{"type":"String","basicValue":"Field"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"V(x)"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"M(x)"},{"type":"String","basicValue":"0"}]]}}}
Example 3: Cantilever beam with zero end moment
Inputs:
| span | supports | loads | young_modulus | inertia | |||
|---|---|---|---|---|---|---|---|
| 2 | 0 | fixed | moment | 0 | 2 | 200000000000 | 0.00001 |
Excel formula:
=SOLVE_BEAM_SYMBOLIC(2, {0,"fixed"}, {"moment",0,2}, 200000000000, 0.00001)Expected output:
{"type":"Double","basicValue":2,"properties":{"Reactions":{"type":"Array","elements":[[{"type":"String","basicValue":"Point"},{"type":"String","basicValue":"Type"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Force"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Moment"},{"type":"String","basicValue":"0"}]]},"Equations":{"type":"Array","elements":[[{"type":"String","basicValue":"Span"},{"type":"String","basicValue":"Field"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"V(x)"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"M(x)"},{"type":"String","basicValue":"0"}]]}}}
Example 4: Cantilever beam with zero distributed load
Inputs:
| span | supports | loads | young_modulus | inertia | ||||
|---|---|---|---|---|---|---|---|---|
| 2 | 0 | fixed | dist | 0 | 0 | 2 | 200000000000 | 0.00001 |
Excel formula:
=SOLVE_BEAM_SYMBOLIC(2, {0,"fixed"}, {"dist",0,0,2}, 200000000000, 0.00001)Expected output:
{"type":"Double","basicValue":2,"properties":{"Reactions":{"type":"Array","elements":[[{"type":"String","basicValue":"Point"},{"type":"String","basicValue":"Type"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Force"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"0"},{"type":"String","basicValue":"Moment"},{"type":"String","basicValue":"0"}]]},"Equations":{"type":"Array","elements":[[{"type":"String","basicValue":"Span"},{"type":"String","basicValue":"Field"},{"type":"String","basicValue":"Expression"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"V(x)"},{"type":"String","basicValue":"0"}],[{"type":"String","basicValue":"[0, 2.00000000000000]"},{"type":"String","basicValue":"M(x)"},{"type":"String","basicValue":"0"}]]}}}
Python Code
Show Code
from symbeam import beam
def solve_beam_symbolic(span, supports, loads, young_modulus='E', inertia='I'):
"""
Solve a statically determinate beam symbolically and return reactions plus internal-force equations.
See: https://github.com/amcc1996/symbeam
This example function is provided as-is without any representation of accuracy.
Args:
span (str): Length of the beam (m or symbolic string).
supports (list[list]): 2D array of support definitions [[x, type]].
x: position.
type: "fixed", "pin", "roller", "hinge".
loads (list[list]): 2D array of load definitions [[type, val, x1, x2]].
type: "point", "moment", "dist".
val: magnitude or expression.
x1, x2: coordinates.
young_modulus (str, optional): Young's Modulus E (Pa or symbol). Default is 'E'. Default is 'E'.
inertia (str, optional): Second moment of area I (m4 or symbol). Default is 'I'. Default is 'I'.
Returns:
dict: Excel data type containing symbolic reactions and piecewise shear-force and bending-moment equations.
"""
try:
def to2d(value):
if not isinstance(value, list):
return [[value]]
if value and not any(isinstance(row, list) for row in value):
return [value]
return value
def to_text(value):
return str(value)
def to_float_or_raw(value):
try:
return float(value)
except (TypeError, ValueError):
return value
span_value = to_float_or_raw(span)
b = beam(span_value)
supports = to2d(supports)
loads = to2d(loads)
if not isinstance(supports, list) or not all(isinstance(row, list) for row in supports):
return "Error: supports must be a 2D list"
if not isinstance(loads, list) or not all(isinstance(row, list) for row in loads):
return "Error: loads must be a 2D list"
b.set_young(0, span_value, to_float_or_raw(young_modulus))
b.set_inertia(0, span_value, to_float_or_raw(inertia))
for row in supports:
if len(row) < 2:
continue
pos = to_float_or_raw(row[0])
support_type = str(row[1]).lower()
b.add_support(pos, support_type)
for row in loads:
if len(row) < 3:
continue
load_type = str(row[0]).lower()
value = to_float_or_raw(row[1])
x_one = to_float_or_raw(row[2])
x_two = to_float_or_raw(row[3]) if len(row) > 3 else None
if load_type == "point":
b.add_point_load(x_one, value)
elif load_type == "moment":
b.add_point_moment(x_one, value)
elif load_type in ("dist", "udl", "distributed"):
if x_two is None:
return "Error: distributed load requires both x1 and x2"
b.add_distributed_load(x_one, x_two, value)
b.solve(output=False)
reactions = []
for point in b.points:
if point.has_reaction_force():
reactions.append([to_text(point.x_coord), "Force", to_text(point.reaction_force)])
if point.has_reaction_moment():
reactions.append([to_text(point.x_coord), "Moment", to_text(point.reaction_moment)])
equations = []
for segment in b.segments:
span_label = f"[{to_text(segment.x_start)}, {to_text(segment.x_end)}]"
equations.append([span_label, "V(x)", to_text(segment.shear_force)])
equations.append([span_label, "M(x)", to_text(segment.bending_moment)])
return {
"type": "Double",
"basicValue": float(len(reactions)),
"properties": {
"Reactions": {
"type": "Array",
"elements": [
[
{"type": "String", "basicValue": "Point"},
{"type": "String", "basicValue": "Type"},
{"type": "String", "basicValue": "Expression"}
]
] + [
[{"type": "String", "basicValue": cell} for cell in row] for row in reactions
]
},
"Equations": {
"type": "Array",
"elements": [
[
{"type": "String", "basicValue": "Span"},
{"type": "String", "basicValue": "Field"},
{"type": "String", "basicValue": "Expression"}
]
] + [
[{"type": "String", "basicValue": cell} for cell in row] for row in equations
]
}
}
}
except Exception as e:
return f"Error: {str(e)}"