Beam Deflection & Reactions — Simply Supported, Point Load mechanism

Model scope

The reactions, maximum moment, and center-load deflection formula are standard under the stated beam assumptions. The instrument fails verification because the asymmetric deflection formula is presented without its required half-span condition, gives the wrong branch for loads left of midspan, omits the piecewise deflection-curve equation, and leaves b and Vmax ambiguous.

• The general maximum-deflection formula is only valid when the load is on the right half of the span (a >= L/2), with b = L - a and x measured from the left support. For a < L/2, the corresponding maximum occurs at x = L - sqrt((L^2 - a^2)/3), and the formula must use a instead of b: delta_max = P*a*(L^2-a^2)^(3/2)/(9*sqrt(3)*E*I*L).
• The input list includes a but not b, although the formulas use b. The instrument must explicitly define b = L - a and state that a is measured from the left support.
• Vmax is not defined. For a point load, the maximum absolute internal shear is max(R1,R2), while the shear diagram also has a jump of magnitude P at the load.
• The deflection-curve formula is missing even though deflection curve is an output. It should specify a piecewise Euler-Bernoulli formula and a sign convention. With b = L-a and downward deflection reported as positive: y(x)=P*b*x*(L^2-b^2-x^2)/(6*L*E*I) for 0 <= x <= a, and y(x)=P*a*(L-x)*(L^2-a^2-(L-x)^2)/(6*L*E*I) for a <= x <= L.
• The center-load formula delta_max = P*L^3/(48*E*I) is correct only for a = L/2 and a transverse point load on a prismatic, simply supported Euler-Bernoulli beam.

Units and conventions

• Use one consistent unit system: P in force, L and a in length, E in force/length^2, and I in length^4. Then reactions and shear are in force, moments in force-length, and deflections in length.
• In the first three vectors, SI units are used: N, m, Pa, and m^4. In the fourth vector, N, mm, N/mm^2, and mm^4 are used consistently.
• R1 is the vertical reaction at the left support and R2 is the vertical reaction at the right support; b = L-a.
• Deflection values below are downward magnitudes. Positive upward load or signed deflection would require an explicit sign convention.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "P": 1000,
    "L": 4,
    "a": 2,
    "E": 200000000000,
    "I": 0.000008,
    "units": "N,m,Pa,m^4",
    "curve_x": [
      0,
      1,
      2,
      3,
      4
    ]
  },
  "expected_outputs": {
    "R1_N": 500,
    "R2_N": 500,
    "Vmax_N": 500,
    "Mmax_Nm": 1000,
    "delta_max_m": 0.0008333333333333334,
    "deflection_curve_m": {
      "x=0": 0,
      "x=1": 0.0005729166666666667,
      "x=2": 0.0008333333333333334,
      "x=3": 0.0005729166666666667,
      "x=4": 0
    },
    "curve_features": {
      "roots_x_m": [
        0,
        4
      ],
      "maximum_at_x_m": 2,
      "maximum_deflection_m": 0.0008333333333333334,
      "asymptotes": []
    }
  },
  "working": "b = L-a = 4-2 = 2 m. EI = 200000000000*0.000008 = 1600000 N*m^2. R1 = P*b/L = 1000*2/4 = 500 N. R2 = P*a/L = 1000*2/4 = 500 N. Vmax = max(500,500) = 500 N. Mmax = P*a*b/L = 1000*2*2/4 = 1000 N*m. Since the load is centered, delta_max = P*L^3/(48*E*I) = 1000*4^3/(48*1600000) = 64000/76800000 = 0.0008333333333333334 m. For 0 <= x <= 2, y = P*b*x*(L^2-b^2-x^2)/(6*L*E*I) = x*(12-x^2)/19200 m, giving y(0)=0, y(1)=11/19200=0.0005729166666666667, and y(2)=16/19200=0.0008333333333333334. Symmetry gives y(3)=0.0005729166666666667 and y(4)=0. The first derivative vanishes at x=2, so the maximum is at midspan."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A zero-span input makes reaction and moment formulas divide by zero and makes the deflection model physically invalid. The instrument must validate L>0 and reject this case rather than returning a misleading zero or NaN result.

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