Source: https://guthlabs.ai/docs/instruments/beam-deflection-reactions-simply-supported-udl/mechanism/

instrument mechanism / verified-clean

# Beam Deflection & Reactions — Simply Supported, UDL mechanism

The inspectable method record for Beam Deflection & Reactions — Simply Supported, UDL, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The core maximum-value formulas are standard and numerically verified. The specification needs review because it claims reactions and full diagrams without defining their equations, sign conventions, or the exact meaning of Vmax, and because input-unit normalization is not explicitly stated.

- The stated formulas are correct for a prismatic, simply supported beam carrying a constant full-span UDL, with constant E and I, under linear-elastic small-deflection Euler-Bernoulli assumptions.

- R1 and R2 are claimed outputs but their formulas are not stated; for a symmetric full-span UDL they are R1=R2=wL/2.

- Vmax is ambiguous unless defined as maximum absolute shear. The signed shear ranges from +wL/2 at the left support to -wL/2 at the right support.

- The diagram outputs require explicit field equations and sign conventions: V(x)=wL/2-wx, M(x)=wLx/2-wx^2/2, and downward deflection magnitude d(x)=wx(L^3-2Lx^2+x^3)/(24EI) for 0<=x<=L.

- The maximum-deflection formula gives the midspan magnitude only; the location x=L/2 should be stated.

- The method omits shear deformation, support settlement, beam self-weight unless included in w, and any partial, nonuniform, or point loading.

## Units and conventions

- Use one consistent unit system: w in force/length, L in length, E in force/length^2, and I in length^4.

- With consistent units, reactions and shear are force, moment is force-length, and deflection is length.

- The test vectors use SI units unless explicitly marked otherwise; GPa and kN/m must be converted to Pa and N/m before calculation.

- E and I are constant along the beam, and w is a constant load over the entire span.

- Positive reactions are upward, positive sagging moment is used, and reported deflection is downward magnitude.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "w": 1000,
    "w_unit": "N/m",
    "L": 4,
    "L_unit": "m",
    "E": 200000000000,
    "E_unit": "Pa",
    "I": 0.000008,
    "I_unit": "m^4",
    "sample_x": [
      0,
      1,
      2,
      3,
      4
    ]
  },
  "expected_outputs": {
    "R1_N": 2000,
    "R2_N": 2000,
    "Vmax_absolute_N": 2000,
    "Mmax_Nm": 2000,
    "delta_max_m": 0.00208333333333,
    "delta_max_location_m": 2,
    "sampled_points": [
      {
        "x_m": 0,
        "V_N": 2000,
        "M_Nm": 0,
        "deflection_down_m": 0
      },
      {
        "x_m": 1,
        "V_N": 1000,
        "M_Nm": 1500,
        "deflection_down_m": 0.001484375
      },
      {
        "x_m": 2,
        "V_N": 0,
        "M_Nm": 2000,
        "deflection_down_m": 0.00208333333333
      },
      {
        "x_m": 3,
        "V_N": -1000,
        "M_Nm": 1500,
        "deflection_down_m": 0.001484375
      },
      {
        "x_m": 4,
        "V_N": -2000,
        "M_Nm": 0,
        "deflection_down_m": 0
      }
    ],
    "curve_features": {
      "shear_zero_m": 2,
      "moment_roots_m": [
        0,
        4
      ],
      "moment_max_at_m": 2,
      "deflection_max_at_m": 2,
      "deflection_asymptotes": []
    }
  },
  "working": "EI=200000000000*0.000008=1600000 N·m^2. R1=R2=wL/2=1000*4/2=2000 N. Vmax=2000 N. Mmax=wL^2/8=1000*4^2/8=16000/8=2000 N·m. delta_max=5wL^4/(384EI)=5*1000*4^4/(384*1600000)=5*1000*256/(614400000)=1280000/614400000=0.00208333333333 m. For plotting, V(x)=2000-1000x, M(x)=2000x-500x^2, and d(x)=1000x(64-8x^2+x^3)/(24*1600000)=1000x(64-8x^2+x^3)/38400000. At x=1: V=1000, M=2000-500=1500, d=1000(64-8+1)/38400000=57000/38400000=0.001484375 m. At x=2: V=0, M=4000-2000=2000, d=2000(64-32+8)/38400000=80000/38400000=0.00208333333333 m. Symmetry gives the x=3 values; at x=4, V=-2000, M=0, d=0."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The deflection calculation is undefined when I=0 because its denominator 384EI vanishes; an implementation must reject zero or nonpositive E and I instead of returning an infinite or misleading numeric deflection.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/beam-deflection-reactions-simply-supported-udl/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/beam-deflection-reactions-simply-supported-udl/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/beam-deflection-reactions-simply-supported-udl/qualification/>)

## 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`