ON THE EFFECTS OF CIRCULAR BOLT PATTERNS ON THE ...

These techniques yield information on the relative significance of not only the main parameters
X X ,..., X , but also the interactions between the same parameters X X ( X X ... X )
1, 2 n
94
X .
1 2 3,...,
1 2 n
However, in most practical problems, such as the one studied, many of the higher-order interactions may
be eliminated on the basis of physical and intuitive considerations. Probable interactions must, however,
be included in the model. The behavior of the extended end-plate connection seems to be a simple
solution considering the cantilever profile of the member, but there are many more parameters that can
be considered in an analytical study and regression analysis. For example, bolt diameter, end-plate
thickness, column flange thickness, beam depth, and many other parameters can be contributing in the
connection behavior. This possibility makes this type of an analytical study and regression analysis a
complex and interesting study, but does not facilitate the complete defining of all the interactions.
If a linear regression model is not found satisfactory, an alternative method is the product regression
model of the form:
C1
1
C2
2
Cn
n
x C X X ... X
(5.4)
0
This nonlinear regression method was used in this project because of the complexity of the
interactions involved. This may be reduced to a linear regression model if the logarithms are taken off
from both sides as shown below:
ln 0 1 1 2 2
x lnC
C
ln X C
X ...
C
ln X
(5.5)
n
Denoting the logarithms of the various parameters by prime superscripts, Equation 5.5 becomes:
' ' '
'
x C0
C1X
1 C2X
2
'
... Cn
X n
(5.6)
This is similar to the first group of terms in Equation 5.3. It should be noted that in Equation 5.6,
'
1
'
2
'
3
product terms of the form X , X , X , etc., do not occur, so no interactions are present.
In this study, the coefficient 0 C and the exponents n C C C ,..., , 1 2 in Equation 5.5 are determined by
multiple regression analysis, so as to obtain the best least square fit to the data. With this method, the
best fit regression equation is taken as the one which minimizes the sum of the squares of the deviations
of the data points from the equation fit to the data. To demonstrate the basic principles, say that the value
n