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Problems
for Chapter 10
Approximate
Theories for Solids with Special Shapes:
Rods,
Beams, Membranes, Plates and Shells
10.5. Motion and
Deformation of Thin Shells
10.5.1.
A
spherical-polar coordinate system is to be used to describe the deformation
of a spherical shell with radius R. The two angles   illustrated in the figure are to be used as
the coordinate system  for this geometry.
10.5.1.1.
Write down the
position vector  in terms of ,
expressing your answer as components in the basis  shown in the figure.
10.5.1.2.
Calculate the
covariant basis vectors  in terms of   and
10.5.1.3.
Calculate the
contravariant basis vectors  in terms of   and
10.5.1.4.
Calculate the
covariant, contravariant and mixed components of the metric tensor
10.5.1.5.
Calculate the
covariant, contravariant and mixed components of the curvature tensor  for the shell
10.5.1.6.
Find the
components of the Christoffel symbol  for the
coordinate system
10.5.1.7.
Suppose that
under loading the shell simply expands radially to a new radius r.Â
Find the components of the mid-plane Lagrange strain tensor  and the components of the curvature change
tensor
10.5.1.8.
Suppose that
the shell is elastic, with Young’s modulus E and Poisson’s ratio . Calculate the contravariant components of
the internal force  and internal moment
10.5.1.9.
Find the
physical components of the internal force and moment, expressing your answer
as components in the spherical-polar basis of unit vectors
10.5.2. A cylindrical-polar coordinate system is to be used
to describe the deformation of a cylindrical shell with radius r.Â
The angles and axial distance  illustrated in the figure are to be used as
the coordinate system  for this geometry.
10.5.2.1.
Write down the
position vector  in terms of ,
expressing your answer as components in the basis  shown in the figure.
10.5.2.2.
Calculate the
covariant basis vectors  in terms of   and
10.5.2.3.
Calculate the
contravariant basis vectors  in terms of   and
10.5.2.4.
Calculate the
covariant, contravariant and mixed components of the metric tensor
10.5.2.5.
Calculate the
covariant, contravariant and mixed components of the curvature tensor  for the shell
10.5.2.6.
Find the
components of the Christoffel symbol  for the
undeformed shell
10.5.2.7.
Suppose that
under loading the shell simply expands radially to a new radius , without axial stretch. Find the covariant components of the
mid-plane Lagrange strain tensor  and the covariant components of the
curvature change tensor
10.5.2.8.
Suppose that
the shell is elastic, with Young’s modulus E and Poisson’s ratio . Calculate the contravariant components of
the internal force  and internal moment .
10.5.2.9.
Find the
physical components of the internal force and moment, expressing your answer
as components in the spherical-polar basis of unit vectors
10.5.3.
The figure
illustrates a triangular plate, whose geometry can be described by two
vectors a and b parallel to two sides of the triangle. The position vector of a point is to be
characterized using a coordinate system  by setting  where ,
.
10.5.3.1.
Calculate the
covariant basis vectors  in terms of a and b
10.5.3.2.
Calculate the
contravariant basis vectors  in terms of a and b
10.5.3.3.
Calculate the
covariant, contravariant and mixed components of the metric tensor
10.5.3.4.
Suppose that
the plate is subjected to a homogeneous deformation, so that after
deformation its sides lie parallel to vectors  and . Find the mid-plane Lagrange strain tensor ,
in terms of a, b, Â and
10.5.3.5.
Suppose that
the plate is elastic, with Young’s modulus E and Poisson’s ratio . Calculate the contravariant components of
the internal force
10.5.4.
The figure
illustrates a triangular plate. The position points in the plate is to be
characterized using the height  and angle  as the coordinate system .
10.5.4.1.
Calculate the
covariant basis vectors  expressing your answer as components
in the basis  shown in the figure.
10.5.4.2.
Calculate the
contravariant basis vectors   as components in
10.5.4.3.
Calculate the
covariant, contravariant and mixed components of the metric tensor
10.5.4.4.
Find the
components of the Christoffel symbol  for the
undeformed plate
10.5.4.5.
Suppose that
the plate is subjected to a deformation such that the position vector of a point
that lies at  in the undeformed shell has position vector  after deformation Find the mid-plane
Lagrange strain tensor ,
in terms of
10.5.4.6.
Suppose that
the plate is elastic, with Young’s modulus E and Poisson’s ratio . Calculate the contravariant components of
the internal force
10.5.4.7.
Find the
physical components of the internal force T as components in the  basis.
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