Civil Engineering Reference
In-Depth Information
holdr
holds element “actions” at convergence
km
element stiffness matrix
kv
global stiffness matrix
global load (displacement) vector
loads
element “mass” matrix
mm
nodal displacements from previous iteration
oldis
holds integration point (local) coordinates
points
element properties matrix
prop
element self-equilibrating “correction” vector
react
nodal load weightings
val
fixed displacements vector
value
holds weighting coefficients for numerical integration
weights
x
-coordinates of mesh layout
x coords
y
-coordinates of mesh layout
y coords
4.3 Exercises
1. A simply supported beam (
L
=
1,
EI
=
1) supports a unit point transverse load
Q
=
(
1) at its mid-span. The beam is also subjected to a compressive axial force
P
which will reduce the bending stiffness of the beam. Using two ordinary beam
elements of equal length, assemble the global matrix equations for this system but do
not attempt to solve them. Take full account of symmetries in the expected deformed
shape of the beam to reduce the number of equations.
U
1
U
2
0
−
(
8
−
P/
15
)
−
24
+
P/
10
)
=
Ans:
(
−
24
+
P/
10
)(
96
−
12
P/
5
)
1
/
2
2. Derive the mass matrix of a 3-noded 1D rod element (one node at each end and one
in the middle) of unit length, cross-sectional area and density, given the following
shape functions:
2
=
−
x
+
N
1
2
(x
1
.
5
0
.
5
)
2
N
2
=−
4
(x
−
x)
2
N
3
=
2
(x
−
0
.
5
x)
42
−
1
1
30
Ans:
216
2
−
12 4
3. A cantilever (
L
=
1,
EI
=
1) rests on an elastic foundation of stiffness
k
=
10. A
transverse point load
1 is applied at the cantilever tip. Using a single finite
element, estimate the transverse deflection under the load.
Ans: 0.188
P
=
