The Design of Modern Steel Bridges - TEDI

198 The Design of Modern Steel Bridges
Let A 1 and A 2 be the arc lengths of the cable under tension s 1 and s 2. Then
Hence
A1 ¼ C1 þ 8
3
A2 ¼ C2 þ 8
3
gC 2 1
8s1
gC 2 2
8s2
C2 C1 ¼ A2 A1 þ g2 cos 3 y
24
2 cos 3 y
L1
2 cos 3 y
L2
C 4 1
s 2 1 L1
C 4 2
s 2 2 L 2
But A 2 A 1 is the elastic stretch of the cable-stay of initial length A 1 due to
an increase in tensile stress from s 1 to s 2 and is thus equal to (s 2 s 1)A 1/E.
Hence
C2 C1
C1
¼ s2 s1
E
which is approximately equal to
s2 s1
E
A1
C1
þ g2 cos 3 y
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þ g2 L 2
24
1
s 2 1
C 3 1
L1s 2 1
1
s 2 2
C 4 2
C1L2s 2 2
(C 2 C 1)/C 1 is the strain of the chord length of the cable stay due to a change
in stress of (s2 s1). If we define Es as a secant modulus representing the ratio
of the change of stress between s 1 and s 2 to the resultant strain of the chord
length of the cable-stay, then
It should be noted that:
Es ¼ ½s2 s1ŠC1
C2 C1
E
¼ h i ð7:3Þ
1 þ Eg2 L 2
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s1þs2
s 2 1 s2 2
E s is not directly related to the magnitude of the stress change (s 2 s 1),
but to the magnitude of both the stresses s 1 and s 2
Es is less than the elastic modulus E of the material of the cable; and as the
ratio of s 2/s 1 increases, E s increases.
We may define a tangent modulus Et at a stress level s1 such that Et is equal
to Es for a very small stress change from s1. Thus Et is the effective modulus at
the stress level s1 for a small change of stress and is given by
E
E
Et ¼
h i ¼ h i ð7:4Þ
1 þ Eg2 L 2
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2s1
s 4 1
1 þ Eg2 L 2
12s 3 1