Arch bridges - Computational Analysis and Design of Bridge Structures - page 286

Approximated displacement function of the first-order vibration mode is built
by using the deflection curve of a fixed-end rebar under the uniform load q .

2 3 4

4

ql
EI

x
l

x
l

x
l

+

( )

( ) = 2 0 1

Y x − ≤ x ≤ (9.8)

1

24

Approximated displacement function of the second-order vibration mode is
built by using deflection curve of a fixed-end rebar under the antisymmetric
uniform load.

2 3 4

x
l

x
l

x
l

3 − 14 + 12

4

ql 1

2

Y x ( ) = 0 ≤ x ≤ (9.9)

2

288 EI

x x

l

95 7 . sinh

+ −

2 l 2

2 3 4

l x
l

−

−

l x
l

−

+

l x
l

−

3 14 12

4

ql 1

2

Y x ( ) = − ≤ x ≤ 1 (9.10)

2

288

EI

l x
l

− l x

l

−

−

+ 95 . 7 sinh

2 2

So, inherent frequencies ω can be calculated:

) ⋅ (

) + (

= (

) ⋅

EI l

2

T

4 5 2 105

0

ω 2

(9.11)

( ) ⋅

2

1 630 ml

( ) +

2 − 2

3 59059 . EI l 2 13849 10 . × T

0

ω 2 = (9.12)

4.44568 10 4

− 2

× ml

where:
ω is the inherent frequency
T 0 is the cable tension
EI is the bending stiffness
l is the length of the rebar
m is the mass per unit length of the rebar

In Equations 9.11 and 9.12, cable tension T 0 has an explicit relationship with
the inherent frequency f , so cable tension can be easily calculated from a
measured frequency. When using a natural frequency f 1 , cable tension T 0 is

2

π EI

l

2 2

T = ⋅ ml f − (9.13)

42 0 1

2

3