# An Uncertainty Model for Strain Gages Using Monte Carlo Methodology

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Scope of This Paper

#### 1.3. Measurements and Applied Equipment

**Figure 1.**Overview screen of the test bridge in Roding from [21].

**Figure 2.**(

**a**) Strain gage mounted on the superstructure; (

**b**) test results visualized in the data acquisition system.

## 2. State of the Art

#### 2.1. Foil-Type Strain Gages for Measurement on Concrete

#### 2.2. Evaluation of Measurement Uncertainties According to GUM

#### 2.2.1. Analytical Approximation

#### 2.2.2. Monte Carlo Methodology and the GUM Uncertainty Framework

**Formulation**: Determination of the input by its probability function and modeling the functional relation of input and output variables of the indirect measurement procedure;**Propagation**: Starting the Monte Carlo runs until a sufficient number of runs are obtained to ensure convergence;**Summarizing**: Evaluation of the resulting vector of simulation results concerning measures of central tendency and dispersion.

#### 2.3. Variance-Based Sensitivity Analysis

## 3. Considered Measurement Uncertainties

#### 3.1. Overview

#### 3.2. Strain Sensitivity (k-Factor)

#### 3.3. Inaccuracies in the Application Process

#### 3.3.1. Theoretical Background

#### 3.3.2. Experimental Study and Results

#### 3.3.3. Assumed Random Variables

#### 3.4. Data Acquisition System

#### 3.5. Thermal Output

#### 3.6. Summary of Applied Probability Distribution Function

## 4. Probabilistic Model for Uncertainties in Measurement of Concrete Constructions

**Input values**

**Strain sensitivity**

**Measurement amplifier**

**Thermal output**

**Evaluation of strain**

**Simulation**

## 5. Results of the Uncertainty Propagation

#### Monte Carlo Study for Uncertainty Propagation

## 6. Evaluation of Sensitivities

## 7. Conclusions and Outlook

- An experimental study on the expectable misalignment of the gage was conducted, and a probability model for the angle variation in the application process was derived.
- Based on physical relations, expertise in metrology, and engineering estimation, an uncertainty model is developed that comprises the entire measuring chain, but is sufficiently simple to be used in field application and practical engineering by inexperienced users.
- The advantages of Monte Carlo simulation over simple Gaussian error propagation are discussed in terms of correctness and applicability.
- In order to study the presented model for the uncertainty of strain measurement, sensitivity indices according to Sobol are introduced to measure the importance of different contributions to the combined uncertainty at different levels of strain and for different scatter in the external conditions (change in temperature).
- The importance of considering interactions due to the nonlinearity of the model function $f\left(x\right)$ is studied. The results highlight the importance of the thermal expansion characteristics of the measured object by its high activeness in the interaction with temperature uncertainty.
- The presented workflow can serve as a template for the evaluation of uncertainties according to [10] and as a starting point for improving the accuracy of measurement. Sensitivity analysis via Monte Carlo simulation serves as a means to identify the driving factors behind the measurement uncertainty and those that can presumably be disregarded in further considerations. From this, conclusions about similar measurement projects can be drawn concerning the recommendable measurement equipment, the order of tests, and zeroing intervals.

- The application to different measurement materials and test setups;
- Possibilities to further simplify the implemented model based on the conducted sensitivity study;
- Integration of different assumptions in the temperature range, e.g., after improvement on the knowledge due to additional measurements;
- Extension of the presented model to include temperature hysteresis and mechanical hysteresis;
- The statistical modeling and the importance of bonding effects;
- Advancements concerning long-term measurements, e.g., in SHM or IoT applications;
- Implementation as a commercial software package.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Illustration of the measurement project concerning measurement location and load position no. 1, 2 and 3.

**Figure 4.**Components of foil-type strain gage, after [18].

**Figure 5.**Illustration of the procedure proposed in [10] to evaluate the distribution and the dispersion of the possible results.

**Figure 7.**Illustration of the decline in measurement recording for different angles of misalignment for a Poisson’s ratio of $\nu =0.2$.

**Figure 9.**(

**a**) Histogram of the evaluated angle of misalignment from a sample of size 500; (

**b**) normal probability plot of the observed inaccuracy from misalignment.

**Figure 10.**Detail of the thermal output curve and linear approximation of the minimum and maximum gradients according to the manufacturer’s specifications.

**Figure 11.**Illustration of the statistical modeling in the Monte Carlo simulation for the reference measurement and the strain evaluation at the loaded state no. i composed of sensitivity of the sensor $\tilde{k}$, measurement reading MR, error in the voltage metering ${X}_{rec}$, and thermal output ${\u03f5}_{S,X}$.

**Figure 12.**(

**a**) Histogramms and fitted Gaussian PDF for the uncertainty propagation at position nos. 1–3. (

**b**) Probability plot for the uncertainty propagation at position nos. 1–3 (every 2000th sample displayed).

**Figure 15.**Pie chart of the sensitivity analysis of measurement no. 2 using sensitivity indices of first order.

**Figure 17.**Pie chart of the sensitivity analysis of measurement no. 3 using sensitivity indices of first order.

**Table 1.**Components of the assumable maximum error depending on the measurement reading (MR) and the supply voltage (SV) from the manufacturer’s specifications for DS-NET BR8 and the maximum value for nonlinearity according to the calibration certificate.

Uncertainty Component | Maximum Error of Component |
---|---|

Temperature | $\left(\right)open="["\; close="]">0.2\frac{\mathsf{\mu}\mathrm{V}}{\mathrm{V}}\xb7SV+0.05\%\xb7MR\xb7{U}_{0}$ |

Long-term drifting | $0.2\frac{\mathsf{\mu}\mathrm{V}}{\mathrm{V}}\xb7\frac{\Delta {T}_{board}}{24\text{}\mathrm{h}}\xb7SV$ |

Nonlinearity | $5\times {10}^{-4}\mathrm{V}$ |

Random Variable | Probability Distribution | Reference |
---|---|---|

data acquisition system | ${X}_{Rec}\sim N\left(\right)open="("\; close=")">0\phantom{\rule{0.222222em}{0ex}};\phantom{\rule{0.222222em}{0ex}}\frac{{\left(\right)}_{\frac{\Delta U}{SV}}}{i,max}3$ | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ of maximum error from calibration data |

gradient of the thermal output curve | ${X}_{\nabla {\u03f5}_{s}}\sim U\left(\right)open="("\; close=")">1.0\frac{\mathsf{\mu}\mathrm{m}\text{}\xb0\mathrm{C}}{\mathrm{m}};1.8\frac{\mathsf{\mu}\mathrm{m}\text{}\xb0\mathrm{C}}{\mathrm{m}}$ | manufacturer’s specification |

thermal expansion of concrete | ${X}_{{\alpha}_{B}}\sim U\left(\right)open="("\; close=")">7\times {10}^{-6};13\times {10}^{-6}$ | [53] |

strain sensitivity ${k}_{0}$ | ${X}_{{k}_{0}}\sim N\left(\right)open="("\; close=")">2.13;1.09\times {10}^{-2}$ | manufacturer’s specification |

angle of misalignment | ${X}_{\Phi}\sim N\left(\right)open="("\; close=")">0.262\xb0;0.422\xb0$ | experimental estimation |

temp coefficient gage factor ${\alpha}_{K}$ | ${X}_{{\alpha}_{K}}\sim U\left(\right)open="("\; close=")">1.0\times {10}^{-4}\frac{1}{\mathrm{K}};2.0\times {10}^{-4}\frac{1}{\mathrm{K}}$ | manufacturer’s specification |

Poisson’s ration of concrete | ${X}_{\nu}\sim U\left(\right)open="("\; close=")">0.14;0.26$ | [52] |

**Table 3.**Bounds of the 95% confidence intervals of the estimators for the mean and the standard deviation evaluated by bootstrapping using 10,000 resamples.

Measurement | Standard Deviation | Mean Value |
---|---|---|

No. 1 | [0.2338;0.2501] | [−9.6732;−9.6716] |

No. 2 | [0.3884;0.3905] | [45.5700;45.5720] |

No. 3 | [0.3201;0.3219] | [−10.2030;−10.0200] |

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**MDPI and ACS Style**

Haslbeck, M.; Böttcher, J.; Braml, T.
An Uncertainty Model for Strain Gages Using Monte Carlo Methodology. *Sensors* **2023**, *23*, 8965.
https://doi.org/10.3390/s23218965

**AMA Style**

Haslbeck M, Böttcher J, Braml T.
An Uncertainty Model for Strain Gages Using Monte Carlo Methodology. *Sensors*. 2023; 23(21):8965.
https://doi.org/10.3390/s23218965

**Chicago/Turabian Style**

Haslbeck, Matthias, Jörg Böttcher, and Thomas Braml.
2023. "An Uncertainty Model for Strain Gages Using Monte Carlo Methodology" *Sensors* 23, no. 21: 8965.
https://doi.org/10.3390/s23218965