Beam Deflection — Overhanging Beam mechanism

Model scope

NEEDS_REVIEW because the specification does not uniquely define the support conditions, load model, sign conventions, or numeric-integration algorithm. The four deterministic benchmarks assume a simply supported beam with a free overhang, downward point loads, and exact Euler–Bernoulli/Macaulay calculations. A production implementation needs those assumptions exposed or explicitly fixed, plus validation of the numerical integration method against these closed-form results.

• The stated formula, "Superposition of simple-span and cantilever-overhang equations," is not sufficiently specific to verify. Reactions, V(x), M(x), and deflection depend on the exact support topology; the vectors below assume simple supports at x=0 and x=L, with a free overhang from x=L to x=L+a.
• For arbitrary load positions, deflection must satisfy compatibility conditions δ(0)=δ(L)=0 and requires a fully specified Euler–Bernoulli/Macaulay or equivalent influence-function equation. Separate simple-span and cantilever formulas can be superposed only when their equivalent reactions and boundary conditions are handled consistently.
• The numeric-integration engine is unspecified: no integration method, step size, interpolation rule, or convergence tolerance is given. The benchmark vectors therefore use the exact closed-form point-load solution rather than an unspecified numerical integrator.
• Load types are ambiguous: "loads/positions" does not state whether loads may be point forces, distributed loads, moments, or combinations. The supplied formula is not sufficient for distributed loads or applied couples.
• The sign convention for reactions, shear, moment, and deflection is missing. The vectors use upward reactions positive, sagging moment positive, and downward deflection positive.
• No stated constraints prevent supports from coinciding, L<=0, negative overhang length, loads outside 0<=x<=L+a, E<=0, or I<=0.

Units and conventions

• All quantities in each vector use one consistent unit system.
• For SI vectors, length is in m, force in N, E in Pa=N/m^2, I in m^4, moment in N·m, and deflection in m.
• For the millimetre vector, length is in mm, force is in N, E is in N/mm^2, I is in mm^4, moment is in N·mm, and deflection is in mm.
• Euler–Bernoulli small-deflection, linear-elastic beam theory is assumed; shear deformation and geometric nonlinearity are neglected.
• Assumed geometry is a simply supported beam at x=0 and x=L with a free overhang to x=L+a.
• For downward point loads P_i at z_i, R2=sum(P_i z_i)/L and R1=sum(P_i)-R2. With Macaulay brackets <q>^3=0 for q<0 and q^3 otherwise, M(x)=R1 x-sum(P_i<x-z_i>), and downward-positive deflection is δ(x)=-(R1 x^3/6-sum(P_i<x-z_i>^3/6+C x)/(E I)), where C=(-R1 L^3/6+sum(P_i<L-z_i>^3/6))/L.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "model": "simple supports at x=0 and x=L; free overhang to x=L+a",
    "L_m": 6,
    "a_m": 2,
    "point_loads": [
      {
        "P_N": 10000,
        "z_m": 8
      }
    ],
    "E_Pa": 200000000000,
    "I_m4": 0.000008,
    "sample_x_m": [
      0,
      3,
      6,
      8
    ]
  },
  "expected_outputs": {
    "R1_N": -3333.333333,
    "R2_N": 13333.333333,
    "V_N": {
      "0<x<8": -3333.333333,
      "8<x<8": "0 after the tip load; no beam interval beyond x=8"
    },
    "M_Nm": {
      "x=0": 0,
      "x=3": -10000,
      "x=6": -20000,
      "x=8": 0
    },
    "delta_downward_m": {
      "x=0": 0,
      "x=3": -0.028125,
      "x=6": 0,
      "x=8": 0.0777777778
    },
    "curve_features": {
      "moment_roots_m": [
        0,
        8
      ],
      "moment_extremum": {
        "type": "minimum",
        "x_m": 6,
        "M_Nm": -20000
      },
      "deflection_support_roots_m": [
        0,
        6
      ],
      "asymptotes": []
    }
  },
  "working": "Equilibrium: R2=(10000*8)/6=13333.333333 N; R1=10000-13333.333333=-3333.333333 N. EI=200000000000*0.000008=1600000 N·m^2. For x<=6, M=R1*x, so M(3)=(-3333.333333)(3)=-10000 N·m and M(6)=(-3333.333333)(6)=-20000 N·m. For x=8, M=R1*8-10000*(8-8)=-26666.666664+26666.666664=0. The deflection constant is C=(-R1*6^3/6)/6=20000. At x=3, the bracket is R1*3^3/6+C*3=(-3333.333333*27/6)+60000=-15000+60000=45000; δ=-45000/1600000=-0.028125 m. At x=6, the bracket is (-3333.333333*216/6)+20000*6=-120000+120000=0; δ=0. At x=8, the bracket is (-3333.333333*512/6)+20000*8=-284444.4444+160000=-124444.4444; δ=124444.4444/1600000=0.0777777778 m. The shear is -3333.333333 N before the tip load and jumps by -10000 N at x=8."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The instrument must reject I=0 because deflection formulas divide by E*I=0. It also risks incorrect signed results unless load direction, reaction signs, and deflection-positive direction are explicitly defined.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json