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

5.6.2 Basis Of Regression Analysis
To develop a numerical model for the hysteresis behavior of the extended end-plate connections, the
nonlinear regression equations are developed using the data collected from finite element analysis. To
model the hysteresis of the connections, it is important to accurately predicting the value of four
parameters. These parameters are selected as the moment capacity of the connection at yield (My), the
initial stiffness of the connection (E), the post yielding stiffness of the connection (Et), and the maximum
rotation of the end-plate at the fracture of the bolt (θf). These parameters can be used to calculate the
value of the moment capacity of the connection at fracture (Mf), and the rotation of the end-plate at yield
(θy). These parameters are presented in Figures 5-6 and 5-7.
Non-linear regression analysis was performed to obtain prediction equations for dependent
parameters as functions of independent variables. The dependent variables are defined as parameters
defining the outer loop of moment-rotation hysteresis model and the independent variables are geometric
variables of the connection assembly.
Thus, the dependent variables are:
M fb,t,S Efb,t, S
E fb,t, Z
θ fb,t,S,Z 93
(5.1)
Determination of the function f(x) is discussed in general terms as follows. Let
x f X , X , X ,......... ....... X )
(5.2)
( 1 2 3
n
be a function of n independent parameters, intended to fit data collected from a study. A linear (or
summation) regression model for the function is written as
x C0
C1X
1 C2
X 2 C3
X 3 .......... .. Cn
X n
.......... C X X
C23X2X 3 ..........
... Cn
1XnX
1 C123X1X2
X3
..........
......
C X X X .......... .......... ...... X )
(5.3)
123......... n(
1 2 3
n
12
1
2