📋 Summary
Rigorous calibration is the cornerstone of any reliable mechanical testing laboratory. This article presents the calculation and verification methods applicable to tensile testing machines in accordance with international NF EN ISO, ASTM standards, and internal procedures.
1. Purpose
This document aims to verify the calculation notes from calibration body reports. It allows for the validation of calibration and the monitoring of deviation history.
An e-book is available with this presentation, complete with equations and an Excel file for calculation notes.
2. Scope
These calculations concern the following mechanical tests:
- Tensile test
Equipment concerned — Tensile tests
- Tensile testing machine (Force measurement)
- Extensometer (Extension measurement)
- Micrometer (Specimen diameter measurement)
- Calipers (Elongation length measurement)
- An Excel file or automatic data acquisition and calculation software
3. Verification
Verification of the tensile testing machine consists of comparing the values displayed by the machine with the reference values provided by the accredited calibration body. The verified parameters are:
- q — Relative error of trueness (repeatability of readings)
- b — Relative error of repeatability
- a — Relative error of resolution
- f₀ — Relative error at zero load
Applicable Standards
| Standard | Subject | Classes |
|---|---|---|
| NF EN ISO 7500-1 | Tensile Testing Machines — Verification and Calibration | 0.5 / 1 / 2 / 3 |
| ASTM E4 | Standard Practices for Force Verification of Testing Machines | A / B |
| NF EN ISO 9513 | Calibration of Extensometers | 0.2 / 0.5 / 1 / 2 |
| ASTM E83 | Standard Practice for Verification and Classification of Extensometer Systems | A to E |
| ASTM E2309 | Standard Practices for Verification of Displacement Measuring Systems | — |
| ASTM E1012 | Standard Practice for Verification of Testing Frame and Specimen Alignment | — |
| ASTM E2658 | Standard Practices for Verification of Speed for Material Testing Machines | — |
4. Explanation of Formulas
4.1 — ISO 7500 : Verification Parameters
b = [(Fmax − Fmin) / Fref] × 100 [repeatability, %]
a = [r / (2 × Fref)] × 100 [resolution, %]
f₀ = [F₀ / Fmax_nominal] × 100 [zero load indication, %]
4.2 — ASTM E4 : Verification Parameters
R% = [(Fmax − Fmin) / Favg] × 100 [ASTM repeatability, %]
4.3 — Accuracy Classes
| Parameter | Class 0.5 | Class 1 | Class 2 | Class 3 |
|---|---|---|---|---|
| |q̄| max (%) | ±0.5 | ±1.0 | ±2.0 | ±3.0 |
| b max (%) | 0.75 | 1.5 | 3.0 | 4.5 |
| a max (%) | 0.25 | 0.5 | 1.0 | 1.5 |
| f₀ max (%) | 0.05 | 0.1 | 0.2 | 0.3 |
5. Validation by Uncertainty Calculation
In accordance with ISO/IEC 17025 and GUM (Guide to the Expression of Uncertainty in Measurement), the combined uncertainty of the measured force is calculated from 5 components:
u2 = b / (2 × 100) [repeatability, normal distribution k=2]
u3 = U_standard / (k × 100) [standard uncertainty, certificate]
u4 = drift / (√3 × 100) [inter-calibration drift, uniform distribution]
u5 = max_residual / √3 [linear regression residual, in N]
uc(F) = √(u1² + u2² + u3² + u4² + u5²) [combined uncertainty]
U(F) = 2 × uc(F) [expanded uncertainty, k=2, ~95%]
5.1 — Correction Factor C(F) = a×Steps + b — 300 kN Machine
📈 Model Deviation = 0.000868 × F + 126.3 N — R² = 0.979 ✅ — Points: 2024 model values
5.2 — Modeled EC Uncertainty: U(F) = 0.00133×F + 48.3 N
📊 📊 U(F) = 0.00133×F + 48.3 N — R² ≈ 1 ✅ — X-axis in N (30k to 300k) (R² ≈ 1) — X-axis in daN
6. Definition of Parameter Uncertainties
| Component | Source | Distribution | Formula |
|---|---|---|---|
| u1 — Resolution | Machine display | Uniform | r / (2√3 × F) |
| u2 — Repeatability | Calibration certificate (b%) | Normal k=2 | b / 2 |
| u3 — Standard | Accredited body certificate | Normal k=2 | U_cert / k |
| u4 — Drift | Calibration history | Uniform | drift / √3 |
| u5 — Regression Residual | Regression C(F) = a×F + b | Uniform | max_residual / √3 |
7.4 — Permissible Deviations for Rm and Rp0.2% — ISO 7500-1 & ASTM E4
300 kN Machine — Ø10 mm (S₀ = 78.54 mm²) and Ø12.5 mm (S₀ = 122.72 mm²) Specimens
| Standard | Class | Max Deviation (%) | U(F) at 300 kN (N) | U(Rm) Ø10 mm (MPa) | U(Rm) Ø12.5 mm (MPa) |
|---|---|---|---|---|---|
| ISO 7500-1 | 0.5 | ±0.5 % | ±1,500 N | ±19.1 MPa | ±12.2 MPa |
| ISO 7500-1 | 1 | ±1.0 % | ±3,000 N | ±38.2 MPa | ±24.4 MPa |
| ISO 7500-1 | 2 | ±2.0 % | ±6,000 N | ±76.4 MPa | ±48.9 MPa |
| ASTM E4 | A | ±1.0 % | ±3,000 N | ±38.2 MPa | ±24.4 MPa |
| ASTM E4 | B | ±2.0 % | ±6,000 N | ±76.4 MPa | ±48.9 MPa |
* Rp0.2% has the same permissible deviations as Rm for the same specimen diameter.
8. Calculate Your Machine's Deviations — Practical Guide
This chapter allows you to calculate the measurement uncertainties and deviations in MPa for your own tensile testing machine, following the ISO/IEC 17025 and GUM methods. Enter your data in the tables below.
8.1 — Your Machine's Data (to be filled in)
| Parameter | Symbol | Your Value | Unit | Source |
|---|---|---|---|---|
| Max nominal force | Fmax | ___________ | N | Machine Plate |
| Indicator resolution | r | ___________ | N | Machine Display |
| Max repeatability (b) | b | ___________ | % | Calibration Certificate |
| Standard uncertainty (u3) | u3 | ___________ | % | Accreditation Body Certificate |
| Inter-calibration drift (u4) | u4 | ___________ | % | Calibration History |
| Specimen diameter | Ø | ___________ | mm | Micrometer Measurement |
| Maximum measured force | Fm | ___________ | N | Test Report |
| Force at 0.2% (Rp) | Fp0,2% | ___________ | N | Test Report |
8.2 — Calculating the uncertainty budget u(F)
u2 = b / (2 × 100) [repeatability, normal distribution k=2]
u3 = U_standard / (k × 100) [standard uncertainty, certificate]
u4 = drift / (√3 × 100) [drift, uniform distribution]
u5 = max_residual / √3 [regression residual, in N]
uc(F) = √(u1² + u2² + u3² + u4² + u5²)
U(F) = 2 × uc(F) [k=2, ~95%]
8.3 — Calculating deviations in MPa for Rm and Rp0.2%
Rm = Fm / S₀ U(Rm) = 2 × √[ (1/S₀)² × u(F)² + (Fm/S₀²)² × u(S₀)² ]
Rp0.2% = Fp0.2% / S₀ U(Rp) = 2 × √[ (1/S₀)² × u(F)² + (Fp0.2%/S₀²)² × u(S₀)² ]
🧮 Calculate your testing machine's deviations
Interactive tool compliant with ISO 7500-1 & ASTM E4