Elastic Investigation - Structural Period
Elastic Investigation - Structural Period |
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| The fundamental period on the Ji
Lu bridge greatly affects its overall behavior. And the transverse
stiffness of the deck is the most important parameter for determining the
dynamic properties of the system. Using Rayleigh's quotient with
cubic polynomial shape functions, an approximate variation between the
transverse moment of area of the deck and the overall first structural
period of vibration of the Ji Lu can be found. This assumes no
contribution of the cables properties in the transverse direction which
when viewed elastically, is a valid assumption.
The exact deck properties of the Ji Lu are unknown. The
differences between whether deck is cracked or uncracked and whether the
wings are active compression elements or not greatly affect the deck
stiffness. |
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Figure 1 - Transverse moment of area of the deck verses structural period
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| The moment of area in the direction where the missing precast wings
would have been in compression is 17.0 m4 if cracked. If,
through the prestressing and cables axial load into the deck, the section
remained uncracked, then the gross value jumps to 89.7 m4.
Then, if the direction of loading is reversed, and the wings are in
compression, the moment of area jumps again to 254 m4 (Figure
2).
In the elastic modeling, these 3 stiffness values were considered and the system elastic displacements and forces were compared. From the analysis of period verses deck moment of area, we see that below values of around 50 m4, the period only changes slightly, where under 50 m4 the period change is dramatic. The implications of this is that if the deck were to crack to a value of 17 m4 the dynamics in the first mode would cease and the period is compared to the high frequency content of earthquake vibrations. This would in turn create a natural structural fuse against large deformations in the first mode and the overall behavior would change.. Another interesting phenomenon is that the bending stiffness is different in the two directions of transverse bending. The missing precast panels at the most critical juncture of the structure means that the dynamic properties are different depending on which direction the structure in moving. This effect of asymmetric stiffness is one that will be considered further in the Nonlinear Investigation but will be bounded in the elastic analysis. As the period gets softer, the dominant lateral mode for mass participation changes. In the field investigation, if was uncovered that the peak tower displacement in the transverse direction was opposite to the peak lateral direction at the P11 and P13 supports (Figure 3). Given that these largest deformations most likely occurred during the same pulse, the deck stiffness must be in a range where the mode in Figure 3 is the dominant. The analysis with the uncracked deck section without precast wings showed a predominant mode where the ends and the tower move in the same direction (Figure 4). Between the two modes and periods of vibration, the actual period is bounded. The dominant period of vibration of the Ji Lu bridge is between 2.1 sec and 3.8 sec. Effective moments of area are between the uncracked and the uncracked with wings case, 89 m4 and 245 m4 respectively. |
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Figure 2 - Moment of area of the deck with different modeling assumptions |
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Figure 3 - Expected dominant mode of vibration from field damage pattern. Mode shown for I = 245 m4 with a period of 2.1 sec.
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Figure 4 - Incorrect dominant mode shape. Mode shown for I = 89.7 m4 with a period of 3.8 sec.
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