Civil Engineering Reference
In-Depth Information
51
y
−
8
y
+
24
y
−
14
y
+
3
y
i
i
−
1
i
−
2
i
−
3
i
−
4
y
′′′ =
i
2
h
3
3 4
y
−
y
+
26
y
−
24
y
+
11
y
−
2
y
i
i
−
1
i
−
2
i
−
3
i
−
4
i
−
5
y
′′′ ′=
i
h
4
i
−−−−−
−
5
i
4
i
3
i
2
i
1
i
hD
hD
hD
hD
2
1 43
14 52
22
−
−
2
33
3
−
14
24
−
18 5
44
−
2112
−
442643
−
3.9
nuMERic MODELing WitH DiffEREncE
OPERAtORS
The difference operators, sometimes referred to as finite difference operators,
can be used to solve many structural engineering problems involving differ-
ential equation relationships. One common relationship is that of the equa-
tion of the elastic curve. The equation of the elastic curve relates the deflected
shape of a beam to the rotation, moment, shear, and load on the beam. This is
a typical strength of materials topic and the following are the basic relation-
ships based on the deflection equation in
y
and
θ
,
M
,
V
, and
q
:
′
=
y
q
i
M
EI
y
′′ =−
i
V
EI
y
′′′ =−
i
q
EI
y
′′ ′′=−
i
Example 3.10
Simple beam with difference operator
Calculate the shear, moment, rotation, and deflection for a 25 foot long,
simply supported beam with a uniformly distributed load of 4 k/ft using
central difference operator of order of error
h
2
at 1/6
th
points. The beam has
E
= 40,000 ksi (modulus of elasticity) and
I
= 1000 in
4
(moment of inertia).
To solve the problem, a sketch of the beam and the assumed deflected
shape is created. To use central difference operators, the model must go
beyond the boundaries of the physical beam. The deflected shape must
