Question

Comment on sources on uncertainty in culvert design and how would an engineer manage these uncertainties

Answer 1

Uncertainties in engineering structure such as culvert and bridges, mainly
can be attributed to ambiguity and vagueness in defining the architecture, parameters and
governing prediction models for the systems. The ambiguity component is generally
attributed to non-cognitive sources. These sources include (1) physical randomness; (2)
statistical uncertainty due to the use of limited information in estimating the
characteristics of these variables. Vagueness related uncertainty is due to
cognitive sources, which include (1) variables such as structural performance,
deterioration, and quality; (2) human error and other human factors; and (3)
interrelationships among variables of a problem, especially for a complex system. uncertainty is broadly classified
into two types, cognitive and non-cognitive. Cognitive uncertainty is subjective and cannot be easily quantified. The non-cognitive type is the uncertainty in predicting the
behavior and design of structural systems.   

To evaluvate the uncertainty following steps should be taken:

1. Specify the Process and Equation

Before you dive in and begin calculating uncertainty, it is best to have a plan. The first part of your plan should be to identify the measurement process or system that you wish to evaluate.

If you are performing indirect measurements that require you to calculate your measurement results, you should specify the equation used. This is because each variable in the equation will have its own uncertainty, which will directly affect the uncertainty associated with the calculated result.

2. Identify and Characterize the Uncertainty Sources

Now that you know your measurement process, you need to identify the factors that influence uncertainty in measurement results. This process is not always easy and can get frustrating. So, stay calm, be patient, and keep researching. You may be surprised by how many influences can affect your measurement result.

3. Quantify the Magnitude of Uncertainty Components

Before calculating measurement uncertainty, you must first determine the magnitude of each contributing factor. To accomplish this, you may need to perform some data reduction and analysis.

To get started, you may want to gather the following items. They will be helpful when analyzing data.

-Calibration Reports
-Repeatability and Reproducibility (R&R) Studies
-Experiment Results
-Manufacture Manuals and Specifications
-Technical Documents and Guides
-Published papers, studies, journal articles, etc.

Using the items listed above, you should be able determine how much uncertainty is contributed from each source. If you need help, you can contact me for additional guidance or hire me to analyze the data for you.

4. Convert Uncertainty Components to Standard Deviation Equivalents

The hardest part of the uncertainty analysis is complete, now you can focus on the easier process of calculating measurement uncertainty. To begin, you need to reduce each uncertainty contributor to a standard deviation equivalent.

This process starts with identifying a probability distribution that characterizes or resembles the dispersion of your data set. You should choose a probability distribution for each uncertainty contributor. If you are not sure which probability distribution best describes your data, most people choose the Rectangular distribution by default.

Some of the most common probability distributions used are;
-Normal or Gaussian
-Rectangular or Uniform
-Triangular
-Log-Normal
-Quadratic
-U-shaped

Use the following chart to help you select the appropriate probability distribution.

Once you select a probability distribution, you can identify the equation to reduce each uncertainty contributor to a standard deviation.

5. Calculate the Combined Standard Uncertainty

After you have reduced your uncertainty sources to standard deviation equivalents, it is time to combine them. To accomplish this, you will use the root sum of squares method or RSS. This will mathematically combine your uncertainty sources in quadrature. So, keep reading to learn how to combine uncertainty.

To begin, simply square the value of each uncertainty source. Next, add them all together to calculate the sum (i.e. the sum of squares). Then, calculate the square-root of the summed value (i.e. the root sum of squares). The result will be your Combined Uncertainty.

If you are a more visual learner, like me, take a look at the process below to see if it makes more sense.

Both equations yield the same result. So, use the equation that is best for you. If you are using an excel spreadsheet calculator, you may find this function beneficial.

=SQRT(SUMSQ(cell1,cell2,…)

When you combine uncertainty sources, you are also combining the probability distributions. According to the Central Limit Theorem, the sum of the set of independent random variables (i.e. uncertainty sources) will approach a normal distribution regardless of the individual variable’s distribution. Simply, the probability distribution associated with your combined uncertainty will now be normal.

If you need visual guidance, use the image below to see if it helps you understand.

6. Calculate the Expanded Uncertainty

You are now entering the home stretch, so stay with me. I am going to show you how to calculate expanded uncertainty. In this step, you are going to expand your combined uncertainty to an acceptable level of confidence.

To satisfy ISO/IEC 17025:2005 requirements, you must expand uncertainty to approximately 95%. Most people use an expansion factor (k) of 2 to achieve a confidence interval of 95.45%. However, you can also use an expansion factor of 1.96 for a confidence interval of exactly 95.00%.

The choice is yours. Just make sure to select an expansion factor that you will consistently use in each of your uncertainty analyses. Also, it helps to know why you chose your expansion factor so you can justify it to assessors (if they ask).

Use the image below to see the breakdown of the normal probability distribution when expand your estimate to 2-sigma or 95.45% confidence.

7. Evaluate Uncertainty for Appropriateness

Once you have calculated the expanded uncertainty, it is best to evaluate your uncertainty estimate for appropriateness. Essentially, you want to make sure that your measurement uncertainty estimate adequately represents your measurement process and is not overestimated or underestimated.