Introduction to fire resistance
About a year ago my colleagues and I faced somewhat of a challenge on one of our projects: the contractor asked us to determine the critical steel temperature in case of fire for all columns througout the building we were designing at the time.
There's software to do this. One profile at a time. That's quite time consuming as you can understand. So we were looking for a different approach to this problem. And yes, again for me the solution was MathCAD, this time with a slice of Newton-Raphson iteration and a sprinkle of bulk calculation.
The graphs look jagged, because the code gives a number of values at a number of temperatures. In between, you are allowed to interpolate. Fortunately, MathCAD has the capacity to make an interpolated function, which we can then use to iterate towards our solution: the critical steel temperature at a given load in case of fire, by defining all relevant formulas as functions related to the temperature:
Then we use the MathCAD solve block to determine the critical steel temperature, by intersecting the (constant) function of the given load with the (linearly interpolated function) of the capacity at a given steel temperature:
Unfortunately, (I think?) we cannot use a solve block inside a MathCAD function, which we need to run this iteration on a large list of column data. So we need to implement our own solve function
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| Source: https://research.bauforumstahl.de/fileadmin/user_upload/LOCAFIplus_Design_Guide_UK.pdf |
There's software to do this. One profile at a time. That's quite time consuming as you can understand. So we were looking for a different approach to this problem. And yes, again for me the solution was MathCAD, this time with a slice of Newton-Raphson iteration and a sprinkle of bulk calculation.
Critical steel temperature
Determining the critical steel temperature at a given axial load is an iterative procedure. With an increase of the steel temperature, the steel strength and stiffness start to degrade. This degradation is described with (empirical?) formulas by Eurocode (NEN-)EN 1993-1-2, so at a given temperature, you can determine the residual capacity, using reduction factors on the material properties of steel (not going into too much detail, lots to write about!):
The graphs look jagged, because the code gives a number of values at a number of temperatures. In between, you are allowed to interpolate. Fortunately, MathCAD has the capacity to make an interpolated function, which we can then use to iterate towards our solution: the critical steel temperature at a given load in case of fire, by defining all relevant formulas as functions related to the temperature:
Then we use the MathCAD solve block to determine the critical steel temperature, by intersecting the (constant) function of the given load with the (linearly interpolated function) of the capacity at a given steel temperature:
Unfortunately, (I think?) we cannot use a solve block inside a MathCAD function, which we need to run this iteration on a large list of column data. So we need to implement our own solve function
Newton-Raphson saywhat?
There's all kinds of iteration functions in math. The one that I implemented here, is a Newton-Raphson iteration. I'm no mathematician, or programmer, so here's an attempt at describing what this does in normal-folk-language:- You define a function which calculates the difference between the capacity and the given load at a random temperature
- You determine the derivative of this difference function
- You use the ratio between these two function to "travel" along the capacity curve, until you've reached the intersection between both functions, using a loop function
My first attempt at this Newton-Raphson iteration looked like this:
The matrix collects the calculated temperature per step, the difference between capacity and given load and the derivative of this difference. The division of the last two gives the convergence, which can be compared with the given tolerance value. A larger tolerance will result in a quicker calculation, but also a larger inaccuracy of the iteration.
Now all we have to do, after inputting a large Excel sheet with all column data (section properties, buckling lengths, axial loading in case of fire, etc.), is putting this Newton-Raphson iteration to work in a bunch of functions:
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| Crunch a matrix with column profiles and their respective loads and geometries |
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| Newton-Raphson-as-a-function |
Et voilá, a matrix with the critical steel temperatures for all the columns in the input Excel file:
Finally, using the Excel output object in MathCAD, combining the column input sheet and the critical steel temperature output matrix, a nice overview of the relevant data can be generated:
That's it! A tool in MathCAD to automatically perform iterative calculations on large lists of structural data!
