|       Chapter 10   Approximate
  theories for solids with special shapes:  rods, beams, membranes, plates and shells       10.2 Motion and Deformation of slender rods   The
  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.             10.2.1 Variables
  characterizing the geometry of the rod’s cross-section   The
  figure illustrates a generic cross-section of the (undeformed) rod.    We
  will characterize the shape of the cross-section as follows: 
   We
       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.ÂWe 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,    
   Principal
  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
    cross-sections |  
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 |        10.2.2 Coordinate systems and
  variables characterizing the deformation of a rod     The
  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.
   After
  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.
   The
  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.  Â
   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. Â
   We let    denote the arc length measured along the
  axis of the rod in the deformed configuration.
   The
  velocity of the rod is characterized by the velocity vector of its axis,   
   The
  rate 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
   
   The
  acceleration of the rod is characterized by the acceleration vector of its
  axis,   
   The
  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   In
  this section we introduce some additional measures of the deformation of the
  rod, as well as several useful relations between the various deformation
  measures.     The curve
  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 vectorThe binormal vector is defined as
         The triad of unit vectors    defines the Frenet Basis for the curveThe 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
       rod’s twist. These
  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 follows.
   
 These
  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   
 where
     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
       products   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   
     In
  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   
 2.       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. 3.       The 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.     The
  rate 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.     The
  bending curvatures    and the twist rate    are related to the Euler angles by
   
 These
  results can be derived from the two different formulas for   ,
  together with the equations relating    and    in terms of the Euler angles.     The arc
  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 centerline by
   
 If
  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   
 This
  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
       length.    With this
  in mind, we assume that     can be approximated by a function of the
  form   
 where
  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   
 for
  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
  require that   
 In
  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    
 Here,
  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
  gradient   Based on the assumptions
  listed in Section 10.2.3, the deformation gradient in the rod can be
  approximated by   
 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   
 where
  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
  form as   
     Derivation: The
  deformation gradient is, by definition, the derivative of the position vector
  of material particles with respect to their position in the reference
  configuration, i.e.   
 To reduce this to the
  expression given, 1.       Note that     2.       Recall that    3.       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   It
  is straightforward to compute additional strain measures from the deformation
  gradient.  Only a partial list will be
  given here.   
   The
       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   
   
   The
       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.   
   The
       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.   
 Substitute   ,
  and note that   
 A tedious set of matrix multiplications shows that
  the components of D in    are   
 to
  within second order terms in curvature,    and   .       10.2.7 Kinematics of rods that are
  bent and twisted in the unstressed state   It
  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
       angles   ,
       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
       Section 10.2.3.We define three
       curvature components    that characterize the bending and
       twisting of the initial rod, as follows   
   The
  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.   Assuming
  small distorsions, the deformation gradient can be expressed in dyadic
  notation as   ,
  where the coefficients    are given below.  The deformation gradient can also be
  decomposed into two successive rotations and a small distorsion   
 where
  the rotation tensors    and    satisfy   ,
  and the tensors    can be expressed in component form as    .  Their components are given by   
   The
  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: 1.      
  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. 3.      
  The principal
  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.   The
  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   The
  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 following variables  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).
  The tractions 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
  quantities: 
   The variation of internal shear stress in the
       cross section   The average in-plane stress components   
 
   Three components of a vector bending moment,
       defined as   
 
   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 next section.    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 conditions   The internal forces and
  moments must satisfy the equations of motion              Here,
    ,
  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.   The
  equations of motion for T and M are
  often expressed as components in the    basis, as   
   
 Note
  that: 
   If the system is in
       static equilibrium, the right hand sides of all the equations of motion
       are zero.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 CThe 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    
 for
  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:   The
  virtual work principle can be expressed in terms of the generalized
  deformation measures and forces defined in the preceding sections as
   
   If the
  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.
   It
  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.   The
  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   
 The
  condition    follows immediately. 
   Choosing    reduces the virtual work equation to   
 Recall
  that (by definition)    must be chosen to satisfy   
 Since
  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
  equation   ;
  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
  multiplier   ,
  which must satisfy   
 for
  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
  equation gives    
 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
  and s=L.       10.2.10 Constitutive equations
  relating forces to deformation measures in elastic rods   The
  generalized deformation measures and their conjugate generalized forces are
  listed in the table below.  
  Constitutive equations must relate the deformation measures to the
  forces.  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 conditions   
 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   
   Calculating
  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 |  
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 |      The force-deformation
  relations for the rod are   
 The
  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.   Derivation: These
  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   ,   ,    and   . 
   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   . ÂSince 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   ,   ,
       and   
 
   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 boundary condition   
 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   
 Substituting
  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:  The
  derivation is similar to the procedure used to compute elastic
  moment-curvature relations.  
   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.ÂThe 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    
 We
  need to show that the integral on the right hand side of this expression is
  zero. 
   To this end, note that    
 where
  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   
 The
  sum of (5) and (6) is zero.  Using this
  result and (4) gives the expression for the strain energy of the rod.       |