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Problems
for Chapter 10
Approximate
Theories for Solids with Special Shapes:
Rods,
Beams, Membranes, Plates and Shells
10.2. Motion and
Deformation of Slender Rods
10.2.1. The figure shows an inextensible rod that is bent into
a helical shape. The shape of the
helix can be characterized by the radius r of the generating cylinder,
and the number of turns n in the helix per unit axial length. Consider a point on the axis of the rod
specified by the polar coordinates  .
10.2.1.1.
Write down an
expression for in terms of r, n and .
10.2.1.2.
Write down the
position vector of the point as components in the basis.
10.2.1.3.
Calculate an
expression for the unit vector that is tangent to the rod, in terms of the
basis vectors and appropriate coordinates.
10.2.1.4.
Assume that is perpendicular to the axis of the
cylinder. Use this and the solution to
1.3 to find expressions for the basis vectors in terms of .
10.2.1.5.
Calculate the
normal and binormal vectors to the curve and hence deduce an expression for
the torsion of the curve.
10.2.1.6.
Deduce an
expression for the curvature vector of the rod.
10.2.1.7.
Suppose that the stress state in
the deformed rod is a simple axial distribution . Calculate the stress components in the basis.
10.2.2. An initially straight, inextensible slender bar with
length L and circular cross-section
with radius a is bent into a circle
with radius R by terminal couples,
as shown in the figure. Assume that
cross-sections of the rod remain circles with radius a and remain transverse to the axis of the rod after deformation.
10.2.2.1.
Write down an
expression for the position vector of a position vector on the axis of the
deformed rod, expressing your answer as components in the basis
10.2.2.2.
Find an expression for the basis vectors as a function of arc-length s, expressing each unit vector as
components in . Hence find an expression for the orthogonal
tensor R that maps onto .
10.2.2.3.
Write down the
curvature vector for the deformed rod, and verify that
10.2.2.4.
Write down
expressions for the deformation gradient in the rod, expressing your answer
as both components in and
10.2.2.5.
Find an
expression for the Lagrange strain tensor in the rod, expressing your answer
as both components in and . Neglect second-order terms.
10.2.2.6.
Hence deduce
expressions for the Material stress and Cauchy stress in the rod.
10.2.2.7.
Calculate the
resultant internal moment and force acting on a generic internal
cross-section of the rod.
10.2.2.8.
Show that the
internal moment satisfies the equations of equilibrium.
10.2.3. Consider a deformable rod, as shown in the
figure. Let denote the arc-length of a point on the axis
un-deformed rod from some arbitrary origin, and let denote the twist of the rod, as defined in
Section 10.2.2. In addition, let and denote the position vector and velocity of
this point on the deformed rod, let denote its arc-length after deformation, and
let denote the curvature vector of the rod,
where are basis vectors aligned with the deformed
rod as discussed in Section 10.2.2
Show that
10.2.3.1.
The time derivative of a unit vector tangent to the rod can be computed
as
10.2.3.2.
The time derivative of the the rate of stretching of the rod’s
centerline is related to its velocity by
10.2.3.3.
The angular velocity of the basis vectors
can be calculated as
10.2.3.4.
The angular acceleration of the
basis vectors can be calculated as
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