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Existing methods used to calculate the performance of intumescent coatings on structural steel are not sufficiently accurate, explains Hans van de Weijgert. Here he presents a three dimensional interpolation method which, he believes, takes out any uncertainty.
The 3-D Interpolation Method
The 3D interpolation method is an assessment method for characterising intumescent coatings for the protection of structural steelwork against fire. This method was agreed for inclusion in ISO 834 Part 11 during the ISO Conference in Kyoto in November 2006. It is also proposed that this will be incorporated in ENV 13381-4 for the European Standard for fire protection of structural steel. This article describes how the factual data from fire tests is used and how measured performance times are projected in a three dimensional space. Each of the test specimens is represented by a dot (x,y,z) in three dimensions. Three dots form a plane and the mathematical equation of the plane enables calculation of the performance time t (z-value) for any combination of section factor Hp/A (x-value) and dry film thickness DFT (y-value) within the boundaries of the plane. The combination of a large number of intersecting planes forms a 3-D landscape like rolling hills, identifying the performance time of any intumescent coating with greater precision.
In recent years, many discussions have taken place about how best to characterise the contribution of intumescent coatings to the fire resistance of structural steelwork. These have produced different assessment methods, such as various graphical methods, differential equation methods and linear regression methods. The general feeling is that all of these provide a way of predicting performance times, but that none can do this with great precision. In attempt to accommodate such uncertainty, criteria for acceptability of the assessment results have been put into place in many standards and guidelines. There is therefore a need for a method that provides an exact prediction of the performance times, and one that is based on facts rather than estimation.
The 3-D interpolation method is based on measured performance times obtained from fire tests, and projects them into a three dimensional space. Each test specimen and its performance time to a certain temperature is represented by a dot (x, y, z) in the three dimensional space. Three dots construct a plane and the mathematical equation of the plane enables the prediction of the performance time z for any combination of x and y within the plane boundaries – i.e. the triangle of which the three dots form the corners.
Four dimensional problem
The reason it has taken so long to study the behaviour of intumescent coatings without achieving a satisfactory factual, mathematical, characterisation method is due to the complexity of the subject, which is actually a four dimensional problem. The four dimensions are: section factor (Hp/A); dry film thickness (DFT); performance time (t); and design steel temperature (T). In order to grasp the behaviour, this four dimensional mathematical problem can be reduced to three dimensions in which the x-axis represents the Hp/A, the y-axis represents the DFT, and the z-axis represents the performance time.
There is no room for the w-axis, which would represent temperature in a four dimensional space. But this fourth dimension can be taken into account by visualising a three dimensional space over and over again, separately for each design steel temperature. Hence, it is possible to construct a 3-D space for a steel temperature of 350 degrees C, a separate one for 400 degrees C and so on, up to 750 degrees C, or for any temperature in between.
A test specimen can be identified by a combination of Hp/A and DFT. For example, a steel section with an Hp/A value of 230 per metre and a DFT of 1.23mm would be represented as (x, y) = (230, 1.23). If Hp/A is on the x-axis and DFT is on the y-axis, then every dot (x, y) represents a test specimen in the orthogonal x, y-axis system.
If the test specimen is subjected to the standard fire test, it will take a certain time in order to achieve a specific design temperature of, say, 550 degrees C. This time is the so called performance time t. By putting the performance time t on the z-axis, a three dimensional space is created in which every combination of Hp/A, DFT and performance time t is represented by a dot (x, y ,z) in the space formed by the x, y, and z-axis. This is for one particular temperature only and shows for each of the test specimens the performance time achieved, as illustrated in Figure 1. The position of these dots is the only factual information available to form the input for the assessment.
Forming triangles
It is now possible to identify three dots (x1, y1), (x2, y2) and (x3, y3), each of them representing a test specimen (xi, yi), to create a triangle. For simplicity, these three dots are in the x, y-plane (z = 0). It is also possible to draw straight lines through the dots, intersecting at the corners of the triangle. The line equations have the form of: y = ax + b
The lines enclose an area in the x,y-plane, which is called the domain. The domain forms a collection of (xi, yi) dots for which the plane equation, that will be explained below, will be applicable (see figures 2 and 3). In the three dimensional space, the three dots (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3), can be imagined to lay in one plane. The equation of the plane has the form of: ax + by + cz + d = 0. This is illustrated in figure 4.
By filling in a value for x and for y (x = Hp/A, y = DFT) in the plane equation, the z value can be obtained. In other words, for all dots within the triangle in the x,y-plane (domain) the z can be calculated using the plane equation. This means that the performance time t can now be calculated for all combinations of
(xi, yi), i.e. Hp/A and DFT values that are within the triangle with corners
(x1, y1), (x2, y2) and (x3, y3), as in figure 2 previously. A more understandable form of the plane equation is:
(a x Hp/A) + (b x DFT) + (c x time) + d = 0
Test evidence
In most cases, manufacturers have test evidence which contains many more data points than only three test specimens. The principle of forming triangles can be extended to create more triangles in the (x, y)-plane, i.e. the (Hp/A,DFT)-plane. In the previous section it was demonstrated that three dots form one triangle and so: four dots will form two triangles; five dots will form at least four triangles; six dots will form at least five triangles and so on.
Each of the triangles form the domain for which (xi, yi)-dots within that triangle deliver the z-value by using the plane equation i.e. each of the triangles in the (Hp/A,DFT)-plane forms the domain for which a combination of Hp/A and DFT within the triangle provides a performance time by using the plane equation. The planes intersect at lines that connect the measured performance times in the three dimensional space. An example of a data set and its performance time to 550 degrees C is given in figure 5. This shows that when all of the planes are combined, they form a three dimensional landscape, like the rolling hills of England!
Output
An assessment of the performance of an intumescent coating will involve:
– the performance time as a function of DFT (graphed for different values of Hp/A)
– the performance time as a function of Hp/A (graphed for different values of DFT).
The performance time as a function of DFT is nothing else than a vertical cross section through the ‘landscape’ formed by the rolling hills. A vertical plane for a constant Hp/A produces the cross section. A vertical cross section using plane Hp/A = Constant can be taken for any value of Hp/A. This does not only show the performance time as a function of DFT, but also shows where applicable test evidence is available and where no test evidence exists. In other words, it shows the lower and upper limitations of the DFT in the output directly, as performance time will be zero if there is no applicable test evidence. The plane equation only returns z-values within the domain and therefore performance times will only be provided if Hp/A and DFT are both within the domain of the particular plane equation.
The performance time expressed as a function of Hp/A is also a vertical cross-section through the landscape of rolling hills of a vertical plane for a constant DFT. A vertical cross-section using a plane for a constant DFT can be taken for any value of DFT. This vertical cross section will show the lower and upper limitations for Hp/A for that particular DFT value.
All this means that previously unrevealed information about the behaviour of intumescent coatings can now be visualised. The 3-D interpolation method used to calculate the performance time as a function of Hp/A reveals that in some cases, a higher DFT does not necessarily provide a higher performance time. This is illustrated by local dips in the landscape – it may unexpectedly reveal one or more local deep valleys.
Various cross sections through the landscape combined in one graph may show that lines for different DFTs intersect. This is illustrated in figures 6 and 7 where the lines for DFT of 2.5mm and 3.0mm intersect, and this demonstrates that the behaviour is not as expected. For a certain range of Hp/A-values, the 3.0mm DFT provides less performance than the 2.5mm DFT. This method allows identification of certain areas of both Hp/A and DFT where improvement of the intumescent ‘recipe’ may be possible. This is a great tool for the chemists working on developing these coatings.
Spot on
There is no need for correction techniques in order to satisfy criteria of acceptability. As the method is based on facts, the criteria for acceptability are automatically complied with. The measured data points form the planes (i.e. the planes are hung onto the data points). So the difference between measured time and predicted time is zero, hence by definition the ratio: calculated times/measured times = 1. Over-predictions or under-predictions do not exist, as the calculated times cannot be higher or lower than the measured times. In fact the calculated times are identical to the measured times. Having said this, it should be noted that we are approximating a curved surface as a collection of planes, like a geodesic shape, so the data points need to be carefully selected as they determine the group of three points. It provides total accuracy in the vertices themselves, but only if all test results are sound and reliable. If the size of the triangles is too large, i.e. if the number of data points is limited and spaced wide apart, the accuracy of the prediction of points within the triangle, i.e. the domain, becomes less.
The 3-D interpolation method can visualise the performance as a function of Hp/A, DFT, time and Temperature. Figures 8, 9 and 10 show three dimensional graphs of an intumescent coating in the heating process at different stages at 350 degrees C, 650 degrees C and 750 degrees C steel temperatures.
Hans van de Weijgert MSc MIFireE MBEng Eur Ing is principal engineer at International Fire Consultants Ltd (IFC). He studied the physics of the built environment and building construction in Eindhoven and his fire testing and consultancy career started at TNO in The Netherlands in 1987. He joined IFC in 2000.