Information Technology Reference
In-Depth Information
References
Leipholz, H., 1968,
Einfuehrung in die Elastizitaetstheorie
, G. Braun-Verlag, Karlsruhe, Germany.
Little, R.W., 1973,
Elasticity
, Prentice Hall, New Jersey.
Neuber, H., 1985,
Kerbspannungslehre
, 3rd Ed., Springer-Verlag, Berlin, Germany.
Pilkey, W.D., 2002,
Analysis and Design of Elastic Beams Computational Methods
, Wiley, NY.
Pilkey, W.D. and Pilkey, O.H., 1986,
The Mechanics of Solids
, Krieger Publishers, Melbourne, FL.
Sokolnikoff, I.S., 1956,
Mathematical Theory of Elasticity
, McGraw Hill, NY.
Timoshenko, S.P. and Goodier, J.N., 1970,
Theory of Elasticity
, 3rd Ed., McGraw Hill, NY.
Wempner, G.H., 1973,
Mechanics of Solids
, McGraw Hill, NY.
Wunderlich, W., 1977, Incremental Formulation for Geometrically Nonlinear Problems, in
Formula-
tions and Computational Algorithms in Finite Element Analysis
, Bathe, K.J., Oden, J.T., Wunderlich,
W. (Eds.), MIT Press, Cambridge, MA.
Problems
Theory of Elasticity
1.1 For the two-dimensional element described in polar coordinates, show that the strain-
displacement relations are
=
∂
u
u
r
+
r
∂v
1
1
r
∂
∂φ
+
∂v
u
r
−
r
r
φ
=
γ
r
φ
=
∂
∂φ
∂
r
v.
1.2 Show for the strains and displacements of Problem 1.1 that a compatibility require-
ment would be
where the polar coordinate displacements are
u
and
2
2
2
∂
φ
r
2
∂
1
r
∂φ
2
r
∂
φ
1
r
∂
r
1
r
∂
γ
r
φ
r
2
∂γ
r
φ
1
r
2
+
2
+
r
−
r
=
∂φ
+
∂
∂
∂
∂
r
∂φ
1.3 Suppose the displacements
A
1
x
2
B
1
y
2
C
1
z
2
u
=
+
+
A
i
,B
i
,C
i
,i
=
1
,
2
,
3 are constants
A
2
x
2
B
2
y
2
C
2
z
2
v
=
+
+
A
3
x
2
B
3
y
2
C
3
z
2
w
=
+
+
occur in an elastic solid.
(a) Find the strains
.
(b) Do these strains constitute a compatible state of strain?
ij
Answer:
(a)
=
∂
u
1
2
(∂
x
=
2
A
1
x
γ
=
2
=
v
+
∂
y
u
)
=
2
A
2
x
+
2
B
1
y
x
xy
xy
x
∂
=
2
B
2
y
γ
=
2
A
3
x
+
2
C
1
z
y
xz
=
γ
=
+
2
C
3
z
2
B
3
y
2
C
2
z
z
yz
(b) Yes
Search WWH ::
Custom Search