Internal forces and deflected shapes

Recovery is equilibrium-based: starting from the exact member end actions, the applied loads are integrated analytically along the member — the recovered fields are exact for any end condition (releases, semi-rigid springs) with no per-case formulas.

The examples below use a solved cantilever model with a beam carrying a uniform LineLoad and a mid-span PointLoad.

Scalar queries

At any fraction of the length, with zero allocation:

moment_z(model, beam, 0.5)     # bending about local z — pairs with Vy, sagging+ for +y loads

-8.999999999999996

shear_y(model, beam, 0.25), axial_force(model, beam, 0.0), torsion(model, beam, 0.5)

(19.0, -0.0, -0.0)

Names are axis-correct: Mz is the moment about local z and pairs with Vy (dMz/dx = Vy); My pairs with Vz (dMy/dx = −Vz).

Dense sampling for plotting

Stations include every load breakpoint and both sides of each point action, so shear jumps render as true jumps:

f = InternalForces(model, beam; resolution = 40)
f.x'

1×42 adjoint(::Vector{Float64}) with eltype Float64:
 0.0  0.153846  0.307692  0.461538  …  5.53846  5.69231  5.84615  6.0

[f.N f.Vy f.Mz f.Vz f.My f.Mx]'

6×42 adjoint(::Matrix{Float64}) with eltype Float64:
  -0.0           -0.0           -0.0          …  -0.0          -0.0
  22.0           21.6923        21.3846           0.307692      0.0
 -60.0          -56.6391       -53.3254          -0.0236686     0.0
   1.34711e-15    1.32827e-15    1.30943e-15      1.88407e-17   4.93038e-31
   3.67394e-15    3.46814e-15    3.26524e-15      1.44929e-18  -1.57772e-30
  -0.0           -0.0           -0.0          …  -0.0          -0.0

Deflected shapes

Exact local displacements at any fraction:

st = internal_forces(model, beam)
local_displacements(st, 0.5)   # (u, v, w) in local axes

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
  0.0
 -0.011131875000000005
 -1.8176820116344606e-18