# ⚛  L1 Principle — Timoshenko Beam — shear-deformation thick-beam

**ID:** `L1-209` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Thick-beam bending with shear*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** beam w and phi 1D
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554010

---

## 🧠 1. Introduction

**Timoshenko Beam — shear-deformation thick-beam** is a **linear inverse problem** whose unknown lives in **beam w and phi 1D** space, within the **Thick-beam bending with shear** sub-domain of **Structural Mechanics**.

Measurements consist of mechanical vibrations or strain signals via a **strain gauges lvdt** sensing mechanism.

The forward operator applies, in order: L · shear stiffness operator; L · bending operator; L · coupled w phi operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · shear stiffness → L · bending → L · coupled w phi → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.coupled_w_phi` `L.bending` `L.shear_stiffness` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.shear_stiffness` | L · shear stiffness operator |
| `L.bending` | L · bending operator |
| `L.coupled_w_phi` | L · coupled w phi operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Thick-beam bending with shear |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | beam_w_and_phi_1D |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `timoshenko` · **Forward operator:** `timoshenko_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| Ei error | — | 0 |
| Ga error | — | 0 |
| Bc error | — | 0 |
| N sensors | — | 16 |
| N elements | — | 50 |
| K shear error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| Ei error | — | 0.0 – 0.1 |
| Ga error | — | 0.0 – 0.1 |
| Bc error | — | 0.0 – 0.2 |
| N sensors | — | 2 – 256 |
| N elements | — | 5 – 10000 |
| K shear error | — | 0.0 – 0.1 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 27.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x589cf5d720367181b83396157800b98f39525ff8a8d631f30fa88ced7e795864`
- **Chain tx hash:** `0x853802db4a9b5c030b52e3d5e83ceebfc0dff4c22c10c34494b0ae893e40b3dc`
- **Chain block:** `41554010`

---

## File Mapping

This bundle consists of: `L1-209.md`, `L1-209.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-209.md` | Source of truth — edit this | Human or LLM |
| `L1-209.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-209`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._