ABSTRACT
The construction cost of a square foot of a building, which can be estimated early in a building’s design (development) stage, to predict fire-induced structural repair (rebuild) cost, which is not known until much later in the program’s life. As more cost information becomes available, more detailed methods (e.g., engineering methods) of costing become feasible. CERs can play a valuable role in estimating the cost of a design approach, especially when conceptual studies and broad configuration trade-offs are being considered. To make an estimate using CERs, the cost estimator must use the statistical information available with respect to fire-induced structural repairs cost (Beitel, 2002). The purpose of this section is to describe the mathematical steps required to construct a CER method in our case, and introduce several related statistics used to evaluate the quality of the CER. Although the discussion in this article is limited to simple CERs (i.e., a single independent and a single dependent variable), the generalization to multiple, independent variables is briefly discussed. The classical CER in our case relates the maximum fire temperature to the construction cost of a fire-induced repair (rebuild) area of a building. This is an example of a simple relationship developed from a set of two-variable data. Suppose two measurements were taken on a building, where Xi denotes the maximum fire temperature T (the range of temperature for all practical purposes is: 100 ◦C < T < 1200 ◦C) and Yi denotes the cost of construction repair work due to fire-induced damages to the building. Then one would obtain a set of ‘‘n’’ pairs of measurements: (X1, Y1); (X2, Y2); (Xn, Yn). The objective in developing a CER is to determine the relationship, if any, between X and Y. If such a relationship is found, it can be used to predict the costs of other fire-induced damage to the building if the cost estimator has some information on the new fire temperature (for example, ultra fast fire) that will be consistent with the nature of the combustible materials that would be found in the space. One way to proceed is to construct a functional relationship between X and Y based on so-called regression analysis. The first step in regression analysis is to hypothesize a relationship, usually involving one or more parameters, between the independent and dependent variables. There are essentially two approaches to hypothesizing a functional relationship between the independent and dependent variables in a regression analysis. The first approach is to hypothesize a relationship on the reasonable to hypothesize that the repair costs increase as a fire temperature increases (at least within a certain range). The cost estimator must review what factors might cause costs to increase and measure them directly or indirectly. The temperature relationship seems reasonable. Other
relationships might be hypothesized for which there is no direct measure. For example, the construction’s technology level could affect costs, but there is no direct measure of technology. Hence, the cost estimator may resort to an indirect measure such as time. Once the cost estimator has a list of hypothetical relationships, the cost estimator should determine what kind of relationship is expected. Is the relationship expected to be positive (as temperature increases repair cost increases) or negative? Determining this before collecting and analyzing the data enables the cost estimator to judge the reasonableness of the estimating relationship. The second approach is to construct and study (a scatter diagram) of the two variables. For example, the relationship between the X and Y variables could be linear or non-linear. In practice, it is best to employ both approaches. That is, after hypothesizing one or more functional relationships between the independent and dependent variables, the cost estimator should plot the data on a scatter diagram. If the scatter diagram does not confirm the hypothesized relationship, the cost estimator should rethink the prior notions and try to explain the discrepancy. There is no simple and direct way of determining a functional relationship; the process requires good judgment and experience that is gained only through repeated use of CERs. Once the relationship has been hypothesized and the data collected and normalized, the cost estimator should use curve-fitting techniques to specify the relationship in mathematical terms. Here are the steps to hypothesize a relationship in our case:
3.1. The simple One-Degree-Of-Freedom (ODOF) system can represent any element of a building structure: beam, truss, girder etc. or the whole structural system: frame system, tubular system, composite system etc. The mass motion (in terms of temperature as a design load) has two components: due to temperature itself, and dynamic effect created by the acceleration (second derivation) of temperature-time relationship and mass. The corresponding differential equation is:
y¨d + ω2yd = −¨t (1)
Where: yd-displacement due to inertial force action; t-displacement due to temperature action; ω -natural frequency of a given structural system (beam, girder, truss etc.); m-total mass.
