Civil Engineering Reference
In-Depth Information
The partial derivative of the moment equation is taken with respect to
the particular force
P
. These are substituted into Equation 4.9 to find the
force
P
, which is the reaction at point
A
. Note the deflection at point
A
is
set to zero in Castigliano's equation.
1
2
x
wx
L
x
wx
L
3
Σ
M
==−−+
0
M Px
∴ =−
M
Px
3
6
∂
∂
M
P
=−
x
L
L
L
==
∂
∂
M
P
dx
EI
wx
L
3
dx
EI
wx
L
4
dx
EI
−
()
=− +
∫
∫
∫
2
∆
0
M
=
−
Px
x
Px
A
6
6
0
0
0
5
3
wL
L
P
L
0
=−
+
30
3
PR
wL
A
==
10
This result can be substituted back into Equations 4.10 and 4.11 to find the
reactions at
A
.
2
wL
R
=
B
5
2
wL
M
=
B
15
4.8
SLOPE-DEfLEctiOn MEtHOD
The slope-deflection method is a stiffness method that includes flexural or
bending stiffness. It was introduced in 1915 by George A. Maney (Maney
1915). In
slope-deflection
, moments at the end of a member are expressed
in terms of the rotations at the ends and the fixed-end moments due to the
loads. Once the expressions for the moments at the member ends are writ-
ten, the joint moments are equated to zero and the unknown moments are
found from the system of equations. The basic slope-deflection equations
are shown in Equations 4.12 and 4.13. These are for the
i
end and
j
end of
a member, respectively.
4
EI
L
2
EI
L
6
EI
L
M FEM
=
+
f
+
f
−
b
(4.12)
i
ij
i
j
