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Answer:
32
a
2
p
0
cosh
k
2
u
1
=
3
[2 cosh
2
k
2
+
π
π
+
π
]
7.27 Consider a square thin element with in-plane loading. If the origin of the coordinates is
at the center of the element, then the element covers the area
E
k
2 sinh
k
−
1
≤
x
≤
1
,
−
1
≤
y
≤
1.
The
y
edges are fixed and the
x
edges
(
x
=±
1
)
are loaded with
y
2
N
x
=
(
1
−
)
tE
/(
1
+
ν)
Compute the displacement and stress distribution for the element.
Hint:
Assume the case of plane stress. Choose trial functions that satisfy the
y
edge
y
2
x, x
3
,xy
2
for the
x
direction and
(
y
=±
1
)
conditions, e.g., use
a
(
1
−
)
,
where
a
=
y, x
2
y, y
3
for the
y
direction.
Answer:
Check your result with such evident conditions as
σ
=
E
/(
1
+
ν)
at
x
=±
1
,y
=
0
x
σ
=
0
at
x
=±
1
,y
=
1
x
7.28 Solve the boundary value problem of Problem 7.7 using the finite element method.
7.29 Suppose the domain of interest for the problem of Problem 7.2 is broken up into three
elements. Now solve the problem again using finite element methodology.
7.30 Derive the element stiffness matrix and loading vector for the differential equation
d
2
u
dx
2
+
e
−
x
3
u
=
using the Ritz method.
7.31 Consider a simple supported beam of length
L
subjected to the distributed load
p
z
=
sin
Compute the distribution of deflection using the finite element method
with stiffness matrices developed using Ritz's method. Compare your results with
the exact solution.
7.32 Use the finite element method to find the deflection along a beam of unit length
on
a elastic foundation. Suppose the loading is uniformly distributed of magnitude
|
(π
x
/
L
).
Use Ritz's method to develop an
element stiffness matrix. Compare this stiffness matrix with one developed using the
transfer matrix method. Let the beam be formed of four elements.
7.33 Find the displacement and forces along the beam of Fig. P7.17 using a Ritz-based
finite element method for a three-element model.
p
z
|=
1 and the ends are fixed. Also, let
k
=
EI
=
1
.
Answer:
See Problem 7.17.
7.34 Find the response of the beam of Fig. P7.20 using the finite element method with a
model made of three elements. Use Ritz's method to establish the stiffness matrix.
Compare the results with the exact solution.
Hint:
Care should be taken in forming the loading vector
G
T
1
0
p
i
N
u
p
i
z
=
d
ξ
where
is the length of the element. Approximate
p
z
on the whole span of the
beam by
4
p
0
L
2
4
p
0
L
x
2
p
z
=−
+
x
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