Chapter 10
Approximate
theories for solids with special shapes:
Â rods, beams, membranes, plates and shells
10.3 Simplified versions of the general theory of
deformable rods
In
many practical cases of interest the general equations governing deformation
and deformation of generally curved rods can be vastly simplified.Â In this section, we summarize the governing
equations for a number of special solids, including flexible strings, and
various forms of beam theory.
10.3.1 Stretched flexible string with
small transverse deflections.
This
is the simplest possible version of the general theory outlined in 10.2.Â Â The problem to be solved is illustrated in
the figure.Â A `stringâ€™ with Youngâ€™s
modulus ,
mass density ,
crosssectional area Â and negligible area moments of inertia Â is initially straight and parallel to the Â direction. The ends of the string are
subjected to an axial load Â and are prevented from moving transverse to
the string.Â A force per unit length Â acts on the string, inducing a small, time
dependent, transverse deflection .
The general governing
equations can be simplified to the following form:
 The curvature of the string can be approximated
as
 The stretch of the string can be approximated
as
 The only nonzero internal force is the axial
force Â (the momentcurvature relations show
that the internal moments vanish; the angular momentum balance equations
show Â )
 The equations of motion reduce to
 Combining 14 shows Â and gives the equation of motion for
the stretched string
 The boundary conditions are Â at
Large
deflection equations for a flexible string can also be found by substituting Â into the general equations of motion for a
rod.Â The details are left as an
exercise.
10.3.2 Straight
elastic beam with small deflections and no axial force (EulerBernoulli beam
theory)
The
figure illustrates the problem to be solved: an initially straight beam, with
axis parallel to the Â direction and principal axes of inertia
parallel to Â is subjected to a transverse force per unit
length .Â
The beam has Youngâ€™s modulus Â and mass density ,
and its crosssection has area A and
principal moments of inertia .Â Its ends may be constrained in various
ways, as described in more detail below.Â Â
We suppose that the beam experiences a small transverse displacement ,
and wish to calculate Â as a function of time, given appropriate
initial conditions.
The general equations of
motion for a rod can be used to deduce that:
 The stretch of the bar and the curvature vector
may be approximated byÂ
 The internal forces
can be characterized by the internal force Â and internal moment .Â Â These can be interpreted as the force
and moment acting on an internal crosssection of the beam which has
normal in the Â direction.
 Momentcurvature relations reduce to
 Equations of motion can be reduced to Â Â Â Â
5. Alternatively, the equations in (3) and (4) can be
combined to express the equations of motion in terms of displacement
Boundary
conditions: Elementary beam
theory offers the following boundary conditions:
 The end of the beam
may be clamped, i.e. rotations and displacement
of the end are completely prevented. The transverse displacement must
then satisfy .
 The end of the beam
may be simply supported, i.e.
the end cannot move, but may rotate freely.Â In this case the transverse displacement
must satisfy Â and the internal moment must satisfy
 The end of the beam
may be free, i.e. the end can
move and rotate freely.Â In this
case the internal moment and internal force must satisfy ,
10.3.3 Straight
elastic beam with small transverse deflections and significant axial force
This
version of beam theory is used to model beams that are subjected to
substantial axial loads (usually due to forces applied at their ends).Â The equations can be used to estimate the
effects of axial load on the transverse deflection or vibration of a beam.
The theory can also be used to calculate buckling loads for beams, but does
not accurately model their deformation if the buckling loads are exceeded.
The
problem to be solved is illustrated in the figure.Â Â An initially straight beam, with axis
parallel to the Â direction and principal axes of inertia
parallel to Â is subjected to a force per unit length .Â
The beam has Youngâ€™s modulus Â and mass density ,
and its crosssection has area A and
principal moments of inertia .Â Its ends may be constrained in various
ways, as described in the preceding section. We assume that a large axial
internal force Â is developed in the beam, either by a
horizontal force per unit length Â or horizontal forces Â acting at the ends of the beam.Â We suppose that the beam experiences a
small transverse displacement ,
and wish to calculate Â as a function of time, given appropriate
initial conditions.
The
general equations of motion for a deformable rod can be approximated as
follows:
 The stretch of the bar and the curvature vector
may be approximated byÂ
 The internal forces
can be characterized by the internal force Â and internal moment .Â Â These can be interpreted as the force
and moment acting on an internal crosssection of the beam which has
normal in the Â direction.Â
 Momentcurvature relations reduce to
 Equations of motion may be approximated by
Â
To interpret these equations, note that (i) the
axial force N has been assumed to
be much larger than the transverse force V,
so that nonlinear terms associated with the axial force have been retained
when approximating the equations of motion; (ii) Although the equations of
motion are expressed in terms of components of displacement and external
force in the Â basis, the equations of motion themselves
represent an approximation to the components of the full, nonlinear equations
of motion in the Â basis.
5. The results of 14 can be combined to obtain an
equation for the transverse deflection of the beam
Boundary
conditions: Elementary beam
theory offers the following boundary conditions:
 The end of the beam
may be clamped, i.e. rotations and displacement
of the end are completely prevented. The transverse displacement must
then satisfy .
 The end of the beam
may be simply supported, i.e.
the end cannot move, but may rotate freely.Â In this case the transverse
displacement must satisfy Â and the internal moment must satisfy
 The end of the beam
may be free, i.e. the end can
move and rotate freely.Â In this
case the internal moment and internal force must satisfy ,
while the transverse force must satisfy
4. In addition, the axial force N must satisfyÂ
