Approximate theories for solids with
rods, beams, membranes, plates and shells
10.2 Motion and Deformation of slender rods
figure illustrates the problem to be solved.
We suppose that a long, initially straight rod is subjected to forces
and moments that cause it to stretch, bend and twist into a complex three
dimensional shape, which we wish to determine.
The initial shape need not necessarily be stress free. Consequently, we can solve problems involving
a rod that is bent and twisted in its unloaded configuration (such as a helical
spring) by first mapping it onto an intermediate, straight reference
configuration, and then analyzing the deformation of this shape.
characterizing the geometry of the rod’s cross-section
figure illustrates a generic cross-section of the (undeformed) rod.
will characterize the shape of the cross-section as follows:
introduce three mutually perpendicular, unit basis vectors ,
with pointing parallel to the axis of the
cylinder, and parallel to the principal moments of area
of the (undeformed) cross section.
- We introduce a
coordinate system within the cross section, with origin at
the centroid of the cross-section.
- The cross-sectional
area of the rod is denoted by
- The principal moments
of area of the cross-section are defined as
- We define a moment of
area tensor H for the
cross-section, with components ,
and all other components zero.
- In calculations to
follow, it will be helpful to note that, because of the choice of origin
and coordinate system,
moments of area and their directions are listed for a few simple geometries
below. Recall also that area moments of
inertia for hollow sections can be calculated by subtraction.
Areas and area moments of inertia for simple
10.2.2 Coordinate systems and variables
characterizing the deformation of a rod
orientation of the straight rod is characterized using the basis described in the preceding section.
The position vector of a material particle in
the reference configuration is ,
where corresponds to the centroid of the cross
section, and is the height above the base of the cylinder.
deformation, the axis of the cylinder lies on a smooth curve. The point that
lies at in the undeformed solid moves to a new
position after deformation.
orientation of the cross-section after deformation will be described by
introducing a basis of mutually perpendicular unit vectors ,
chosen so that is parallel to the axis of the deformed rod,
and is parallel to the line of material points that
lay along in the reference configuration (or, more
precisely, parallel to the projection of this line perpendicular to ). Note
that the three basis vectors are all functions of ,
and if the rod is moving, they are also functions of time.
The orientation of can be specified by three Euler angles ,
which characterize the rigid rotation that maps onto . To visualize the significance of the three angles,
note that the rotation can be accomplished in three stages (i) rotate the basis
vectors through an angle about the axis.
This results in a new set of vectors ;
(ii) Rotate these new vectors through an angle about the axis.
This rotates the vectors onto a second configuration ;
(iii) finally, rotate these vectors through angle about the direction, to create the vectors.
Relationships between the Euler angles and the
curve characterizing the axis of the rod will be given shortly: these results
will show that the angles and can be determined from the shape of the
axis. The angle is an independent degree of freedom, and
quantifies the rotation of the rod’s cross-section about its axis.
We let denote the arc length measured along the axis
of the rod in the deformed configuration.
velocity of the rod is characterized by the velocity vector of its axis,
of rotation of the rod is characterized by the angular velocity of the basis vectors . It will be shown in Sect 10.2.3 that the
angular velocity can be related to the velocity v of the bar’s axis and its twist by
acceleration of the rod is characterized by the acceleration vector of its
angular acceleration of the rod is characterized by the angular acceleration of the basis vectors. It will be shown below that the angular
acceleration can be related to the acceleration a and velocity v of the
bar’s axis and its twist by
10.2.3 Additional deformation measures
and useful kinematic relations
this section we introduce some additional measures of the deformation of the
rod, as well as several useful relations between the various deformation
corresponding to the axis of the deformed rod is often characterized by its tangent, normal and binormal vectors,
together with its curvature, and its torsion. These are defined as follows.
- The tangent vector
- The normal vector and curvature
are defined so that ,
where n is a unit vector
- The binormal vector is defined as
- The triad of unit vectors defines the Frenet Basis for the curve
- The torsion
of the curve is defined as . Note that the torsion is simply a
geometric property of the curve it is not necessarily related to the
variables are not sufficient to completely describe the deformation, however,
since the twist of the rod can vary independently of the shape of its axis.
The two bases ,
can be related in terms of the Euler angles as
results can be derived by calculating the effects of the sequence of three
rotations. Note also that since both
sets of basis vectors are triads of mutually perpendicular unit vectors, they
must be related by
is a proper orthogonal tensor that can be
visualized as a rigid rotation. The
rotation tensor can be expressed in several different forms:
- It can be expressed as the sum of three dyadic
- It can be expressed as
components in either or ,
which can be written in dyadic notation as or . Surprisingly, the components both bases
are equal, and are given by .
The components can be expressed in terms of the Euler angles as a matrix
further calculations the variation of basis vectors with distance s along the deformed rod will play a central role. To visualize this quantity, imagine that the
basis travels up the deformed rod. The basis vectors
will then rotate with an angular velocity that depends on the curvature and
twist of the deformed rod, suggesting that we can characterize the rate of
change of orientation with arc-length by a vector ,
analogous to an angular velocity vector.
The curvature vector can be expressed as components in the basis as . This vector has the following properties
1. The curvature vector is (by definition) related to the
rate of change of with s by
which can be expanded out to show that
The components quantify the bending of the rod, and are
related the curvature and the binormal vector of the curve traced by the axis of the deformed
rod by . You can show this result by comparing the
formula for with the formula for b.
curvature vector can also be expressed in
terms of the position vector of the rod’s centroid as
The component of curvature cannot in general be expressed in terms of r, because the rotation of the rod’s
cross-section about its centroid axis may provide an additional, independent
contribution to . For the special case where and are everywhere parallel to the normal vector n and
binormal b, respectively, it follows
that . In this case, is equal to the torsion of the curve.
of change of with distance s can also be expressed in terms of the Euler angles. For example, the derivative of can be calculated as follows
Similar results for and are left as exercises.
bending curvatures and the twist rate are related to the Euler angles by
results can be derived from the two different formulas for ,
together with the equations relating and in terms of the Euler angles.
length s along the rod’s centerline
is related to the position vector of the rod’s axis by
Some relationships between the time
derivatives of these various kinematic quantities are also useful in subsequent
calculations. The rate of change in
shape of the rod can be characterized by the velocity of the axis and the time rate of change of the
cross-sectional rotation .
The time derivative of the tangent vector is a
convenient way to characterize the rate of change of bending of the rod. This is related to the velocity of the rod’s
we express the velocity in components and recall we can write this as
It is important to note that the components are not equal to the time derivatives of the
components of the tangent vector t,
because the basis varies with time.
The time derivatives of the basis vectors can
also be quantified by an angular velocity vector ,
which satisfies . The components of are readily shown to be
The time derivatives of the remaining basis
vectors follow as
The time derivative of the arc length of the
centerline is related to its velocity as follows
We shall also require the gradient of the
angular velocity ,
which quantifies the rate of change of bending. We shall give this vector the
symbol to denote its physical significance: it can be
interpreted (see Appendix E) as the co-rotational time derivative of the
curvature vector, as follows
Evaluating the derivatives of shows that
The co-rotational time derivative of curvature must be used
to quantify bending rate (instead of the time derivative ) to correct for the fact that rigid rotations
and pure stretching do not change bending.
Finally, to solve dynamic problems, we will
need to be able to describe the linear and angular acceleration of the
bar. The linear acceleration is most
conveniently characterized by the acceleration of the centerline of the bar
The angular acceleration of the bar’s
cross-section can be characterized by the angular acceleration of the basis vectors . A straightforward calculation shows that
The second time derivative of the basis
vectors can then be calculated as
10.2.4 Approximating the displacement,
velocity and acceleration in the rod
The position vector after
deformation of the material point that has coordinates in the undeformed rod can be expressed as
is a completely general expression. We
now introduce a series of approximations that are based on the assumptions that
- The rod is thin compared with its length;
- The radius of
curvature of the rod (due to bending) is much larger than the
characteristic dimension of its cross section;
- The rate of change of
twist of the rod has the same order of magnitude as the
bending curvature of the rod.
- The material in the rod
experiences small distorsions i.e. the change in length of any
infinitesimal material fiber in the rod is much less than its undeformed
With this in
mind, we assume that can be approximated by a function of the form
the Greek indices can have values 1 and 2, and can be regarded as the first term in a Taylor expansion of . The definition of requires that . We assume in addition that
all possible choices of . The constants can be thought of as the components of a
homogeneous in-plane deformation applied to the cross section, while the
function describes the warping of the cross-section.
To decouple the warping from the axial displacement of the rod, we
addition, for small distorsions, the deformation must satisfy and ,
the rod curvatures must satisfy for all ,
and the variation of arc-length s along
the axis of the deformed rod with must satisfy .
The velocity field in the bar
can be approximated as
where it has been assumed
that and for all .
Finally, the acceleration
field within the bar will be approximated as
all time derivatives of and have been neglected. This is not so much because they are small,
but because they represent a crude approximation to the distortion of the
cross-section. The time derivatives of
these quantities are associated with short wavelength oscillations in the bar,
which cannot be modeled accurately using the assumed displacement field.
10.2.5 Approximating the deformation
Based on the assumptions
listed in Section 10.2.3, the deformation gradient in the rod can be
The first line of this
expression quantifies the effects of axial stretching, bending and twisting of
the rod. The second line approximates
the distorsion of its cross-section.
The deformation gradient can
also be decomposed as
R is the rigid rotation satisfying ,
and G and H are deformation gradient like tensors that describe the change in
shape of the rod. These tensors are most
conveniently expressed as components in and ,
respectively we can represent this in diadic notation as or . The components can be expressed in matrix
deformation gradient is, by definition, the derivative of the position vector
of material particles with respect to their position in the reference
To reduce this to the
Substitute and neglect the derivatives of f and with respect to
The decomposition follows trivially by substituting into the dyadic representation of F and rearranging the result. A similar approach gives .
10.2.6 Other strain measures
is straightforward to compute additional strain measures from the deformation
gradient. Only a partial list will be
determinant of the deformation gradient follows as
- The components of the
left and right Cauchy-Green tensors can be
computed from and ,
where G and H were defined in 10.2.4. C
and B are most conveniently
expressed as components in and ,
respectively we can represent this in diadic notation
as or .
For small distorsions, the result can be approximated by
Lagrange strain is defined as . Its components follow trivially from the
preceding formula. Note that the
matrix of components for E
resembles the formula for the infinitesimal
strain components in a straight bar subjected to axial stretching, bending
and twist deformation. However, if
the bent rod does not lie in one plane, the twisting measure includes contributions from both the
rotation of the rod’s cross section about its axis, and also from the
bending of the rod.
rate of deformation tensor will also be required. It is simplest
to calculate the velocity gradient by differentiating the expression given
for the velocity vector in the preceding section.
and note that
A tedious set of matrix multiplications shows that the
components of D in are
within second order terms in curvature, and .
10.2.7 Kinematics of rods that are bent
and twisted in the unstressed state
is straightforward to generalize the results in sections 10.2.3-10.2.5 to
calculate strain measures for rods that are not straight in their initial
configuration. In this case we must
start by describing the geometry of the undeformed rod. To this end
- We denote the distance measured along the axis of
the initial, unstressed, twisted rod by
- At each point on the initial rod, we introduce a set of
three mutually perpendicular unit vectors ,
where is chosen to be tangent to the axis of
the undeformed rod; while are parallel to the principal moments of
inertia of the cross-section.
- We also introduce an arbitrary Cartesian
basis where the unit vectors denote three fixed
directions in space.
- The basis vectors and together define a set of three Euler
which completely describe the shape of the undeformed rod.
- We define a rotation
tensor satisfying that characterizes the orientation of with respect to . The components of can be found using the formulas in
- We define three
curvature components that characterize the bending and
twisting of the initial rod, as follows
deformed shape of the rod is characterized exactly as described in Section
10.2.1, except that the axial distance is replaced by the arc-length of the undeformed rod.
small distorsions, the deformation gradient can be expressed in dyadic notation
where the coefficients are given below. The deformation gradient can also be
decomposed into two successive rotations and a small distorsion
the rotation tensors and satisfy ,
and the tensors can be expressed in component form as . Their components are given by
deformation gradient can be written down immediately, by mapping the initial
rod onto a fictitious intermediate configuration in which the rod is straight,
chosen as follows:
The straight rod
has axis parallel to the direction
2. The point at arc-length in the unstressed rod has coordinates in the intermediate configuration.
axes of the cross section are parallel to in the intermediate configuration
4. The cross-section of the rod has the same shape in the
intermediate configuration as in the undeformed configuration.
deformed state can be reached in two steps (i) Deform the rod from the
unstressed configuration to the intermediate configuration, with a deformation
gradient . The components of can be calculated as the inverse of the
deformation gradient that maps the intermediate straight rod onto the
undeformed shape. (ii) Deform the rod
from the straight configuration to the deformed configuration, with a
deformation gradient . The total deformation gradient follows as .
10.2.8 Representation of forces and
moments in slender rods
figure shows a generic cross-section of the rod, in the deformed configuration.
To define measures of internal and external force acting on the rod, we define the
A basis with unit vectors chosen following the scheme
described in 10.2.2. We define the
following vector components in this basis:
The body force acting on the rod . For simplicity, we shall assume that the body
force is uniform within the cross section (but may vary along the length of the rod).
acting on the exterior surface of the rod
The Cauchy stress
within the rod .
External forces and
moments acting on
the rod are characterized by
- The force per unit length acting on the rod, . The force components can be calculated
from the tractions and body force acting on the rod as
- The moment per unit length acting on the rod, . The moment components can be calculated from the tractions acting
on the exterior surface of the rod as as
- The resultant force acting on each end of the
rod. Each force can be expressed as
components as . The components are related to the
tractions acting on the end of the rod by ,
where the area integral is taken over the cross section at the appropriate
end of the rod.
- The resultant moment
acting on each end of the rod. Each
moment can be expressed as components as . The components are related to the
tractions acting on the end of the rod by
Internal forces and moments in the rod are characterized by the following
- The variation of internal shear stress in the
- The average in-plane stress components
- Three components of a vector bending moment,
- The axial force on the cross-section
- Two additional
generalized forces ,
which represent the transverse shear forces acting on the rod’s cross
section. Unlike the axial force,
however, these forces cannot be directly related to the deformation of the
rod. Instead, they are calculated
from the bending moments, using the equilibrium equations listed in the
The forces and moments define components of a vector force and moment
- is the resultant force acting on an
internal cross-section of the rod;
- is the resultant moment (about the
centroid of the cross section) acting on the cross-section.
10.2.9 Equations of motion and boundary
The internal forces and
moments must satisfy the equations of motion
T and M are the internal forces and moments in the rod; are the external force and couple per unit
length; is the mass density of the rod; A is its cross-sectional area, H is the area moment of inertia tensor
defined in Sect 10.2.1, while are the acceleration, angular velocity and
angular acceleration of the rod’s centerline. The two equations of motion for T and M clearly represent linear and angular moment balance for an
infinitesimal segment of the rod.
equations of motion for T and M are
often expressed as components in the basis, as
- If the system is in
static equilibrium, the right hand sides of all the equations of motion
- In addition, in many
dynamic problems, the right hand sides of the angular momentum balance equations
may be taken to be approximately zero, since the area moments of inertia
are small. For example, the
rotational inertia may be ignored when modeling the vibration of a beam. The rotational inertia terms can be
important if the rod is rotating rapidly: examples include a spinning
shaft, or a rotating propeller.
Boundary Conditions: The internal stresses, forces and moments must satisfy
the following boundary conditions
- on C
- The ends of the rod
may be subjected to a prescribed displacement. Alternatively, the transverse or axial
tractions may be prescribed on the ends of the bar: in this case the
internal forces must satisfy for and for s=0.
- The ends of the rod
may be subjected to a prescribed rotation.
Alternatively, if the ends are free to rotate, the internal moments
must satisfy for and for s=0.
Derivation: Measures of internal force and the equilibrium
equations emerge naturally from the principle of virtual work, which states
that the Cauchy stress distribution must satisfy
all virtual velocity fields and compatible stretch rates . The virtual velocity field and virtual
stretch rates in the bar must have the same general form as the actual velocity
and stretch rates, as outlined in Section 10.2.4 and 10.2.5. The virtual velocity and stretch rate can
therefore be characterized by and compatible sets of . This has two consequences:
virtual work principle can be expressed in terms of the generalized deformation
measures and forces defined in the preceding sections as
virtual work equation is satisfied for all and compatible sets of ,
then the internal forces and moments must satisfy the equilibrium equations and
boundary conditions listed above.
is straightforward to derive the first result.
The Jacobian is approximated as ;
the components of follow from the formulas given in Section
10.2.6, and the velocity field is approximated using the formula in
10.2.5. Substituting the definitions
given in Section 10.2.7 for generalized internal and external forces
immediately gives the required result.
The algebra involved is lengthy and tedious and is left as an exercise.
equilibrium equations and boundary conditions are obtained by substituting
various choices of and compatible sets of into the virtual work equation.
- Choosing reduces the virtual work equation to
condition follows immediately.
- Choosing reduces the virtual work equation to
that (by definition) must be chosen to satisfy
the body force is uniform, the term involving is zero.
The first integral can then be integrated by parts as follows
Choosing on the boundary yields the equilibrium
choosing any other gives the boundary condition .
- Choosing ,
using as well as yields
where we have integrated by parts to obtain the second
line. Choosing to vanish on the ends of the rod yields the
equation of motion .
Any other choice of yields the boundary conditions on the ends of the rod.
- Choosing and substituting ,
where are the components of a virtual rate of
change of the tangent vector reduces the virtual work equation to
To proceed, it is necessary to express and in terms of the virtual velocity components .
The algebra and the resulting equilibrium equations are greatly simplified if
the tangent vector is regarded as an independent kinematic
variable. The relationship between t and must be enforced by a vector valued Lagrange
which must satisfy
all variations .
The second integral can be expressed in component form as
This equation can simply be added to the virtual work
equation to ensure that and are consistent. Finally, recall that the curvature rates and
stretch rate are related to by
Substituting these results into the augmented virtual work
This equation must be satisfied for all admissible .
Considering each component in turn, and integrating by parts appropriately and
using gives the last five equations of motion, as
well as the boundary conditions on s=0
10.2.10 Constitutive equations relating
forces to deformation measures in elastic rods
equations must relate the deformation measures defined in Section 10.2.3 to the
forces defined in 10.2.8. In this
section we list the relationships between these quantities for an isotropic,
elastic rod subjected to small distorsions.
For simplicity, the sides of the rod are assumed to be traction free.
The results depend on the
geometry of the rod’s cross-section, which is characterized as follows.
- Introduce a Cartesian
coordinate system as follows: the origin for this coordinate system is
at the centroid of the cross-section, the basis vectors are parallel to the principal axes of
inertia for the cross-section, and is parallel to the rod’s axis.
- We denote the cross-sectional area of the rod by A, and the curve bounding the
cross-section by C, and let denote the three principal moments of
area of the cross-section (see Sect 10.2.1)
- We introduce a warping
function to describe the out-of-plane displacement
component in the cross-section of the rod. The warping function is related to the
out-of-plane displacement by
The warping function depends only on the geometry of
the cross-section, and satisfies the following governing equations and boundary
You can easily show that this choice of will automatically satisfy the shear stress
equilibrium equation as well as the boundary condition on C.
- Finally we define a modified polar moment of
inertia for the cross section as
the warping function is a nuisance, because it requires the solution to a
PDE. In desperation, you can take w=0 this will overestimate the torsional stiffness
of the rod, but in many practical applications the error is not
significant. For a better
approximation, warping functions can be estimated by neglecting the terms
involving in the governing equation. A few such approximate warping functions and
modified polar moments of area are listed in the table below.
Warping functions and modified polar moments of area
for simple cross-sections
relations for the rod are
two shear force components cannot be related to the deformation they are Lagrange multipliers that enforce the
condition that the rod does not experience transverse shear, as discussed in
the preceding section.
results can be derived as follows:
- The elastic
constitutive equations for materials subjected to small distorsions, but
arbitrary rotations, are listed in Section 3.3. They have the form
where are the components of the material stress
tensor, and are the components of the Lagrange strain
tensor. The components of in the basis can be found using the formulas for given in Section 10.2.7, and when substituted
into the constitutive laws give expressions for the components of material
stress in terms of the deformation measures ,
- The Cauchy stress is
related to the material stress by . For small distorsions, but arbitrary
rotations, we may approximate this by ,
so the components of the material stress tensor in can be used as an approximation to the
components of the Cauchy stress tensor in .
- Since we have assumed
that the tractions on the sides of the rod vanish, the in-plane stress
components must satisfy . Substituting the formulas for stresses
from (2) and noting that (since that the origin for the coordinate
system coincides with the centroid of the cross section) shows that ,
- Substituting the
formula for into the definitions of ,
given in Section 10.2.8 and noting that (because the basis vectors coincide with
the principal axes of inertia) yields
- Recall that the shear
stress components must satisfy the equilibrium equation and
Substituting the shear stress components from step (2)
into this equilibrium equation and setting gives the governing equation for w
- The shear stresses now follow as
these results into the equation defining in Section 10.2.6 gives the last equation
10.2.11 Strain energy of an elastic rod
The total strain energy of an
elastic rod can be computed from its curvatures as
derivation is similar to the procedure used to compute elastic moment-curvature
- The strain energy
density in the rod can be computed from the Lagrange strain and the Material Stress as . The material stress can be related to
the Lagrange strain using the formulas in Section 10.2.10, while the
Lagrange strain can be expressed in terms of of the deformation measures ,
and using the formulas for the deformation
gradient listed in Sections 10.2.7.
- The results can be
simplified by recalling that ,
which shows that the strain energy density can be approximated as
where w is
the warping function defined in Section 10.2.9. The two terms in this
expression represent the strain energy density due to stretching and bending
the rod, and twisting the rod, respectively.
- The total strain energy follows by integrating U over the volume of the rod. Using the measures of cross-sectional
geometry listed in Section 10.2.1, it is straightforward to show that
- Some additional algebra is required to calculate
the energy associated with twisting the rod. Begin by noting that
need to show that the integral on the right hand side of this expression is zero.
- To this end, note that
we have recalled that the warping function w
satisfies in A
as well as on C,
and have used the divergence theorem.
- Secondly, note that
sum of (5) and (6) is zero. Using this
result and (4) gives the expression for the strain energy of the rod.