Civil Engineering Reference
In-Depth Information
After
h
,
a
and
a
have been determined then the third series of decay tests are carried
out with no physical constraints, such that
c
θ
≠ , in which case the full
version of Eq. C.21 applies. Eliminating
h
from the first real and imaginary parts and
a
from the second real and imaginary parts then the following equations are obtained:
c
≠
0
0
and
ª
º
§
2
·
h
ωω
2
z
z
1
c
1
2
sin
«
¨
ζ
−+ −
ζ ζ
+
ζ
¸
ψ
»
z
¨
r
z
r
r
¸
r
2
ω
ω
«
ω
»
©
¹
r
r
r
«
»
-
½
0
§
·
h
h
ªº
ω
ζζ
«
°
°
»
z
1
2
+
c
2
−
−
cos
ψ
−
c
®
¨
¸
¾
«»
«
»
z
z
r
r
θ
ω
ω
ω
=
°
°
©
¹
¯
¿
«
»
»
«
r
r
r
»
«
0
«
»
¬¼
«
»
2
«
§
·
a
»
ωω
§
ω
·
a
( )
2
1
2
3
c
1
θ
2
θ
2
c
θ
c
1
c
ζ
−+ −
ζ ζ
+
ζ
ζ ζ
− +
ζ
+ −
«
¨
¸
»
¨
¸
θ
¨
r
θ
r
¸
θ
r
θ
r
z
r
θ
2
2
ω
ω
ω
ω
ω
«
©
¹
»
©
¹
¬
r
r
r
¼
r
r
(C.29)
2
2
2
2
c
ζ
+−
1
ωω
+
a
ω
r
r
3
r
c
θ
θ
from which
a
=⋅ ⋅
ω
(C.30)
( )
1
r
2
1sin
ζ
+
ψ
z
r
r
-
2
½
§
·
ª
º
c
ω
ω
ω
h
§
ω
·
h
°
°
zr
2
z
z
1
z
1
h
1
2
sin
2
cos
=
ζ
− + −
ζ ζ
+
ζ
ψ
+
ζ ζ
−
−
ψ
¨
¸
®
«
¨
¸
»
¾
2
r
z
r
r
r
z
r
r
¨
¸
c
θ
2
ω
ω
ω
ω
ω
«
»
°
©
¹
°
©
¹
¬
¼
¿
¯
r
r
r
r
r
(C.31)
Finally,
h
may be determined from the first real part of Eq. C.21, rendering
2
ª
2
º
§
·
ω
c
ωω
h
h
r
z
2
z
z
1
2
(
)
h
1
2
cos
sin
=
⋅
⋅
ζ
− + −
ζ ζ
+
ζ
+ ⋅
ζ
ψ
−
ψ
«
¨
¸
»
3
r
z
r
r
r
r
r
¨
¸
cos
c
θ
2
ψ
ω
ω
ω
ω
«
»
©
¹
r
¬
r
r
r
¼
r
(C.32)
From Eqs. C.17 and C.18
2
a
2
h
2
h
*
1
*
2
*
3
A
,
H
and
H
(C.33)
=⋅
=⋅
=⋅
1
2
3
2
β ω
θ
β ω
θ
β
ω
z
r
z
r
z
r
