SUSPNDRS: a numerical simulation tool for the nonlinear transient analysis of cable support bridge structures, part 1: theoretical development Page: 25 of 138
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2 = -Aex. (EQ 81)
3= 1 - (9)2 - 2)2 (EQ 82)
The updated global coordinates of the I" vector is finally given by
k+1
iz" x, 1 y, lz" 1
m Z - k k 1m 2 (EQ83)
nz, nm ny m .4
The element convected coordinate system defined by the x', y', z' coordinate axes must
also be updated for the new element configuration. Based on the new locations of the ele-
ment I and J nodes, as given by EQ. 46 to EQ. 51, the direction cosines of the x' axes are
immediately found as outlined in Section 3.0. The direction cosines of the x' axis are
given by the formulas of EQ. 10 to EQ. 12. There is some flexibility in the definition of the
element y' and z' axes as discussed for a planar problem in Section 3 (Figure 7). These
axes are used to measure the amount of rotational deformation at each end of the element
(note as shown in Figure 8, the displacement at each node will be zero since the element
nodal and convected coordinate systems have the same origin). In light of the fact that the
theory developed here is based on the assumption of small deformations, it is advanta-
geous to define the convected coordinate system in a manner which minimizes the angular
deformation measurement. The element y' axis is defined by a new K node location,
where the K node is based on an average of the nodal 3" and ^" vectors. Specifically, a
directional vector i is defined where,
TI = x1 x (EQ 84)
^j +j'"
and xl is the element length. The new K node position is established from element node I
in the direction of vector 1 as indicated in Figure 10. Thus if the i vector is written,
1= 11i+ TJ+ lz~k (EQ 85)
the coordinates of the new K node are given by,
xK = x + T, (EQ 86)
yK = y+T, (EQ 87)
zK = ZI + l (EQ 88)
The direction cosines of the element x', y', z' axes are then found exactly as outlined in
section 3.0.23
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McCallen, D. & Astaneh-Asl, A. SUSPNDRS: a numerical simulation tool for the nonlinear transient analysis of cable support bridge structures, part 1: theoretical development, report, June 1997; California. (https://digital.library.unt.edu/ark:/67531/metadc691453/m1/25/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.