Civil Engineering Reference
In-Depth Information
d
d
d
d
M
P
x
=
x
iy
M
M
x
=−
1
iz
Castigliano's second theorem can be applied noting that at the
i-
end the
rotation is zero and the deflection is ∆
iy
.
d
d
M
M
dx
EI
d
d
V
M
dx
GA
dx
EI
dx
GA
(
)
∫
∫
∫
∫
q
==
0
M
x
+
V
x
= −+
Px M
+
0
iz
x
x
iy
iz
iz
z
iz
y
z
y
2
PL
EI
ML
EI
iy
0
=−
+
iz
2
z
z
PL
iy
M
=
iz
2
d
d
M
P
dx
EI
d
d
V
P
dx
GA
Px Mx
dx
EI
dx
GA
(
)
∫
∫
∫
∫
∆
=
M
x
+
V
x
=
2
−
+
P
iy
x
x
iy
iz
iy
iy
z
iy
y
z
y
3
PL
EI
PL
GA
M
L
EI
2
iy
iy
∆
=
−
i
z
+
iy
3
2
z
z
y
Substituting the first equation into the second equation results in the
following:
3
3
3
PL
EI
PL
EI
PL
GA
PL
EI
PL
GA
iy
iy
iy
iy
iy
∆
=
−
+
=
+
iy
3
4
12
z
z
y
z
y
L
3
L
GA
∆
=
+
P
iy
iy
12
EI
z
y
EI AG
LAG EI
12
zy
.
(5.19)
P
=
∆
iy
iy
3
+
12
y
z
Substituting Equation 5.19 into the following equation repeated from ear-
lier, results in Equation 5.20 for the second stiffness value.
PL
iy
M
=
iz
2
EI AG
LAG I
6
zy
(5.20)
M
=
∆
iz
2
iy
+
12
y
z
