Civil Engineering Reference
In-Depth Information
quantities. From basic theory of elasticity (Hook's law and Navier's hypothesis, see e.g.
Chen & Atsuta [27])
s
and δ
ε
are given by
T
⎫
⎡
Er
′
Eyr
′′
Ezr
′′
Gr r
′
⎤
s
=
−
−
⎣
⎦
⎪
⎬
⎪
x
y
z
p
θ
el
tot
(9.13)
T
r
y r
z r
r r
δ
ε
=
⎡
δ
′
−
δ
′′
−
δ
′′
′
⎤
⎣
⎦
⎭
x
y
z
p
θ
where primes indicate derivation with respect to
x
and where
E
is the modulus of
elasticity,
G
is the corresponding shear modulus and
r
is a cross sectional coordinate
used to identify the St Venant torsion constant (
r
should be perceived as a symbolic
representative as it strictly spoken is only applicable to a circular cross section). Defining
N
′
00000
N
′
00 0 0 0
⎡
⎤
1
7
⎢
⎥
0
N
′′
000
N
′′
0
N
′′
0 0 0
N
′′
⎢
⎥
()
2
6
8
12
x
N
=
⎢
0
0
N
′′
0
N
′′
0
0
0
N
′′
0
N
′′
0
⎥
3
5
9
11
⎢
⎥
000
N
′
00000
N
′
0 0
⎣
⎦
4
10
(9.14)
and
[
]
diagEEEG
e
f
=
⎫
⎪
⎬
(9.15)
=−−
⎡
1
yzr
⎤
⎪
⎣
⎦
⎭
p
then the internal work is given by
L
L
(
)
T
(
)
(
)
T
(
)
W
∫∫
fN d
feNd
dA dx
∫
N d
c Nd
dx
=
⋅
⋅
δ
⋅
⋅
⋅
⋅
⋅
⋅
+
⋅
δ
⋅
⋅
⋅
⋅
int
tot
0
0
A
0
(9.16)
L
L
⎛
⎞
T
T
T
T
T
∫ ∫
∫
=
δ
dN f
⋅
⋅
⋅
f
⋅
dA
⋅
eN d
⋅
⋅
dx
⋅
+
δ
dNcN d
⋅
⋅
⋅
dx
⋅
⎜
⎟
tot
0
⎜
⎟
⎝
⎠
0
0
A
Defining
2
2
2
A
=
∫
,
dA
I
=
∫
,
ydA
I
=
∫
,
zdA
I
=
∫
and
rdA
z
y
t
p
A
A
A
A
T
∫
dA
diag EA
⎡
EI
EI
GI
⎤
k
=
f
⋅
f
⋅
⋅
e
=
(9.17)
⎣
⎦
0
z
y
t
A
then the internal work is given by
(
)
dkd+cd
T
W
=
δ
⋅
⋅
⋅
(9.18)
int
tot
