Expert Tip: Analysis of Suspension Bridges 5

Dr. Hirai was a Japanese engineer, professor, and researcher. In 1940, Dr. Hirai went to a movie theater and watched the news about the failure of the old Tacoma bridge. He was very inspired and started his research on wind engineering for suspension bridges by himself. His papers had been published since 1942, but they were rarely introduced into the Western world because Japan was isolated during World War II.

The basic step of wind engineering is eigenvalue analysis, and his method is as follows.

Model view

Analysis of Suspension Bridge - Vertical vibration-Asymmetrical mode

Assumed shape function

Center span

Analysis of Suspension Bridge - 2^nd Mode

Vertical, Asymmetrical, 2^nd mode, T=6.821 sec

Analysis of Suspension Bridge - 4^th Mode

Vertical, Asymmetrical, 4th mode, T=2.263 sec

Sider span

Analysis of Suspension Bridge - 1^st Mode

Vertical, Asymmetrical, 1st mode, T=3.959 sec

Analysis of Suspension Bridges - Vertical vibration-Symmetrical mode

Frequency equation

Assumed shape function

There are many ways to solve this frequency equation. The author’s favorite is the Stodola-Vianello method.

Vertical vibration-Symmetrical mode - 1^st mode

Vertical, Symmetrical, 1st mode, T=5.8733 sec (dashed line shows the 1^st term)

Vertical vibration-Symmetrical mode - 2^nd mode

Vertical, Symmetrical, 2nd mode, T=3.5699 sec (dashed line shows the 1^st term)

Vertical vibration-Symmetrical mode - 3rd mode

Vertical, Symmetrical, 3rd mode, T=2.9019 sec (dashed line shows the 1^st term)

Torsional vibration - Torsional rigidity

Waring torsional rigidity

Torsional vibration-Asymmetrical mode

Assumed shape function

By energy methods

In the first term, the warping term, and the third term, effects from cable tension are negligible in the lower modes as can be expected.

Torsional, Asymmetrical, 2nd mode, T=1.614 sec (red line shows rotational angle)

Torsional, Asymmetrical, 4th mode, T=0.778 sec (red line shows rotational angle)

Torsional vibration-Symmetrical mode

Assumed shape function

Ritz’s methods, frequency equation is,

Solve frequency equation by Stodola-Vianello method

1^st mode

Torsional, Symmetrical, 1st mode, T=2.5640 sec (dashed line shows the 1^st term)

3rd mode

Torsional, Symmetrical, 3rd mode, T=1.0767 sec (dashed line shows the 1^st term)

5th mode

Torsional, Symmetrical, 5th mode, T=1.0529 sec (dashed line shows the 1^st term)

Transverse vibration

Assumed shape function

Frequency equation

1^st mode

In-phase

Transverse, 1st mode, in-plane, T=13.31 sec (dashed line shows cable)

Out-of-phase

Transverse, 1st mode, out-of-plane, T=3.238 sec (dashed line shows cable)

2nd mode

In-phase

Transverse, 2nd mode, in-plane, T=4.896 sec (dashed line shows cable)

Out-of-phase

Transverse, 2nd mode, out-of-plane, T=3.175 sec (dashed line shows cable)

Eigen value analysis results from classic methods and MIDAS

For lower modes, transverse modes are very well matched, vertical modes have 10% to 23% differences. In the vibration analysis, only some lower modes are important.

From the author’s view, this classic method from Dr Hirai gives very reasonable results and it is an excellent tool to check the computer output. After we make a simple Excel, it is very easy to check for other geometry and/or loadings which takes some time with displacement methods.

Finally results from MIDAS are attached for the lower five modes.