B31.3 Stress equations:

Sustained stress:
S_L = (P*D)/(4*t) + (0.75*i*M_A)/Z <= S_h

Expansion stress:
S_E = sqrt(S_b^2 + 4*S_t^2) <= S_A

Occasional stress:
S_L + S_occ <= k*S_h

Where:
P = pressure
D = outside diameter
t = wall thickness
i = stress intensification factor (SIF)
M_A = sustained moment
Z = section modulus
S_h = hot allowable stress
S_A = allowable stress range
k = occasional load factor

Linear expansion:
delta_L = alpha * L * (T2 - T1)

Where:
alpha = coefficient of thermal expansion
L = pipe length
T2 - T1 = temperature change

Typical alpha values (in/in/F):
Carbon steel: 6.5 x 10^-6
Stainless steel: 9.5 x 10^-6
Copper: 9.3 x 10^-6

Key principles:
1. Piping expands when heated
2. Expansion induces stress if restrained
3. Flexibility (bends, loops) reduces stress
4. Over-constrained systems have high stress
5. Under-constrained systems have excessive movement

Flexibility characteristic:
h = t*R/(r^2)

Where:
t = wall thickness
R = bend radius
r = mean radius of pipe

Suggested maximum spans (B31.1):

| Pipe Size | Water (ft) | Steam/Gas (ft) |
|-----------|------------|----------------|
| 1" | 7 | 9 |
| 2" | 10 | 13 |
| 4" | 14 | 17 |
| 6" | 17 | 21 |
| 8" | 19 | 24 |
| 12" | 23 | 30 |

Nozzle load limits:
- Equipment vendor provides allowables
- Common standards: API 610, API 617, NEMA SM23
- Consider sustained and thermal loads separately
- Combined loads may use interaction formula

Typical check:
sqrt((F_x^2 + F_y^2 + F_z^2)/(F_allow^2) +
     (M_x^2 + M_y^2 + M_z^2)/(M_allow^2)) <= 1.0

Operating case:
W + P + T + D

Hydrotest case:
W + H + D

Where:
W = Weight
P = Pressure
T = Thermal
D = Displacement
H = Hydrotest pressure

ASME B16.5 flange rating:
- Check P-T rating at operating conditions
- Include pressure equivalent from moments

Equivalent pressure method:
P_eq = P + (16*M)/(pi*G^3)

Where:
M = bending moment at flange
G = flange gasket diameter

NC(T)MF method:
Uses ASME VIII Appendix 2 calculations
More accurate for high moment cases

Key elements:
1. Include all pipe runs
2. Model equipment properly (rigid/flexible)
3. Define support locations accurately
4. Include all branch connections
5. Apply correct operating conditions
6. Model spring hangers if used

{
  "piping_system": {
    "line_number": "string",
    "code": "B31.1|B31.3",
    "material": "string",
    "size": "string (NPS)",
    "schedule": "string"
  },
  "operating_conditions": {
    "design_pressure": "number (psig)",
    "design_temperature": "number (F)",
    "operating_pressure": "number (psig)",
    "operating_temperature": "number (F)"
  },
  "geometry": {
    "isometric": "file reference",
    "length": "number (ft)",
    "elevation_change": "number (ft)"
  },
  "equipment_connections": [
    {
      "equipment": "string",
      "nozzle": "string",
      "allowable_loads": "object"
    }
  ]
}

{
  "stress_results": {
    "code_compliance": "pass|fail",
    "sustained_stress": {
      "max_value": "number (psi)",
      "allowable": "number (psi)",
      "location": "string",
      "ratio": "number"
    },
    "expansion_stress": {
      "max_value": "number (psi)",
      "allowable": "number (psi)",
      "location": "string",
      "ratio": "number"
    }
  },
  "nozzle_loads": [
    {
      "equipment": "string",
      "forces": "array [Fx, Fy, Fz]",
      "moments": "array [Mx, My, Mz]",
      "compliance": "pass|fail"
    }
  ],
  "support_schedule": [
    {
      "location": "string",
      "type": "string",
      "load": "number (lb)"
    }
  ],
  "thermal_movements": {
    "max_displacement": "number (in)",
    "location": "string"
  },
  "recommendations": "array"
}