Civil Engineering Reference
In-Depth Information
11.2 Exercises
1. The undamped beam shown in Figure 11.22 is initially at rest and subjected to a
suddenly applied moment of one unit at its left support. Using a single finite element,
a time step of 1 s and the Constant Acceleration Method (
β
=
1
/
4,
γ
=
1
/
2), estimate
the rotation at both ends of the beam after 2 s.
(Ans:
θ
1
=
0
.
284,
θ
2
=
0
.
0
.
036)
EI= 1.0
r
A=420.0
1.0
1.0
Figure 11.22
2. Repeat the previous question assuming 5% damping. Use Rayleigh damping by
assuming the mass matrix damping parameter (
f
m
) equals zero. The fundamental
natural frequency of the beam is
ω
1
=
0
.
48.
(Ans: If
f
m
=
0, then
ζ
1
=
ω
1
f
k
/
2, hence with
ζ
1
=
0
.
05 and
ω
1
=
0
.
48,
f
k
=
0
.
208.
θ
1
=
.
θ
2
=
.
.
0
254,
0
0
015)
3. The undamped propped cantilever shown in Figure 11.23 is initially at rest and
subjected to a suddenly applied load at its mid-span. Using two finite elements, a
time step of 1 s and the Linear Acceleration Method (
β
=
1
/
6,
γ
=
1
/
2), estimate
the deflection under the load after 2 s.
(Ans:
u
=
0
.
077)
1.0
EI= 10.0
r
A=210.0
1.0
2.0
Figure 11.23
4. The undamped cantilever shown in Figure 11.24 is initially at rest and subjected
to a suddenly applied load and moment at its tip. Using one finite element, a time
step of 1 s and the Constant Acceleration Method (
β
=
1
/
4,
γ
=
1
/
2), estimate the
deflection and rotation at the tip after 2 s.
(Ans:
u
=
0
.
092,
θ
=
0
.
653)
