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TABLE 4.2
Loading Vector p i 0
of a Beam Stiffness Matrix for Applied Distributed Loading (Sign Convention 2)
Definitions
Force and Displacement Vectors
0
p 0 /
0
0
m 0
0
0
0
u a
w a
θ u b
w b
θ k
N V a
M a
N b
V M b
(
7 p a +
3 p b )/
2
20
(
m a +
m b )/
2
2
2
p 0
/
12
(
p a /
20
+
p b /
30
)
(
m a
m b )/
12
p i 0
=
p i
0
p 0 /
0
0
v i =
=
2
m 0
(
3 p a +
7 p b )/
20
(
m a +
m b )/
2
2
0
2
(
m a
m b )/
12
p 0
/
12
(
p a /
30
+
p b /
20
)
p p b
m m b
Normalised Force and
Displacement Vectors
0
0
1
0
0
0
0
0
0
1
/
2
7
/
20
3
/
20
1
/
2
1
/
2
u a
w
N a
V a
M a /
N b
V b
M b /
1
/
12
1
/
20
1
/
30
1
/
12
1
/
12
p i 0
a
θ a
u b
w b
θ b
=
p 0
m 0
0
0
0
0
0
v i
p i
=
=
1
/
2
1
0
3
/
20
7
/
20
1
/
2
1
/
2
1
/
12
1
/
30
1
/
20
1
/
12
1
/
12
Uniform Temperature
Change ∆T
M T b
M T a
M T
=
M T = A TzdA
α =
Thermal Coefficient
0
M T
p x
/
2
(
2 p xa
+
p xb )/
6
(/
2
EA
T
0
0
p x /
0
0
0
0
M T a
0
2
(
p xa +
2 p xb )/
6
(/
2
EA
T
M T
M T b
0
0
0
0
0
0
p xa
0
M T a
1
0
0
1
0
0
21
00
00
12
00
00
1
0
0
1
0
0
M T b
p xb
M T a
0
M T b
1
px
2
6
2 α
EA
T
M T a
M T b
Note: This table applies to the ith element of a simple Euler-Bernoulli beam. For more complex beam elements, see Table 4.4.
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