Suspension bridges - Computational Analysis and Design of Bridge Structures - page 381

ε = H

(12.6)

L d

2

Equation 12.4 can then be rewritten as

2

=

EAH
L

2

δ Π = δ d − Pd

0

(12.7)

3

From Equation 12.7, the same result of the displacement as Equation 12.3
can be obtained.
However, in the earlier approach to the displacement under external load,
it is assumed that the two truss elements have no stress at all when the
external load is applied. What it would be if there was an initial stress and
strain when the external load is applied? The principle of minimum poten-
tial energy is still valid. However, the stress in Equation 12.5 will become

σ = E ε σ +

0

(12.8)

where σ 0 = T / is the initial stress in the truss elements. By substituting

Equation 12.8 into the equation of total strain energy, Equation 12.4 can
then be rewritten as

0

2

=

EAH
L

H
L T d Pd

2

δ Π = δ d + −

0

(12.9)

0

3

From Equation 12.9, the displacement at node B can be derived as

3

L
EAH

H
L T

d = P − (12.10)

0

2

2

where T 0 is the initial tension force when the external load is applied.
Equation 12.10 reveals that the displacement under external loads will be
reduced due to initial stress, and the higher the initial stress, the more it is
reduced. In general, the vertical stiffness is enhanced by initial stresses in cables.
Note that the strain and displacement relationship in Equation 12.6 is
obtained with the assumption that d is very small compared with L or H .
As illustrated in Chapter 3, large displacement can also be considered in
this simple truss structure as follows:

2 2

L = d + Hd L + (12.11)

2 1

where L 1 is the truss element length after deformed. Using Maclaurin series
to expand Equation 12.11 and taking only the second order, Equation
12.11 becomes