

Chapter 10
Approximate theories for solids with special shapes: Â rods, beams, membranes, plates and shells
10.2 Motion and Deformation of slender rods
The figure illustrates the problem to be solved.Â Â 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.Â The initial shape need not necessarily be stress free.Â 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 rodâ€™s crosssection
The figure illustrates a generic crosssection of the (undeformed) rod. Â We will characterize the shape of the crosssection as follows:
Principal moments of area and their directions are listed for a few simple geometries below.Â Recall also that area moments of inertia for hollow sections can be calculated by subtraction.
10.2.2 Coordinate systems and variables characterizing the deformation of a rod
Â The orientation of the straight rod is characterized using the Â basis described in the preceding section. Â The position vector of a material particle in the reference configuration is , where Â corresponds to the centroid of the cross section, and Â is the height above the base of the cylinder. Â After deformation, the axis of the cylinder lies on a smooth curve.Â The point that lies at Â in the undeformed solid moves to a new position Â after deformation. Â The orientation of the crosssection after deformation will be described by introducing a basis of mutually perpendicular unit vectors , chosen so that Â is parallel to the axis of the deformed rod, and is Â parallel to the line of material points that lay along Â in the reference configuration (or, more precisely, parallel to the projection of this line perpendicular to Â ).Â Note that the three basis vectors Â are all functions of , and if the rod is moving, they are also functions of time.
Â The orientation of Â can be specified by three Euler angles , which characterize the rigid rotation that maps Â ontoÂ .Â Â 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 Â about the Â axis.Â Â This results in a new set of vectors ; (ii) Rotate these new vectors through an angle Â about the Â axis.Â This rotates the vectors onto a second configuration ; (iii) finally, rotate these vectors through angle Â about the Â direction, to create the Â vectors.Â Â Â 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 Â and Â can be determined from the shape of the axis.Â Â The angle Â is an independent degree of freedom, and quantifies the rotation of the rodâ€™s crosssection about its axis.Â Â Â We let Â denote the arc length measured along the axis of the rod in the deformed configuration. Â The velocity of the rod is characterized by the velocity vector of its axis, Â The rate of rotation of the rod is characterized by the angular velocity Â of the basis vectors .Â It will be shown in Sect 10.2.3 that the angular velocity can be related to the velocity v of the barâ€™s axis and its twist Â by
Â The acceleration of the rod is characterized by the acceleration vector of its axis, Â The angular acceleration of the rod is characterized by the angular acceleration Â of the basis vectors.Â It will be shown below that the angular acceleration can be related to the acceleration a and velocity v of the barâ€™s axis and its twist Â 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.
Â 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.Â Â These are defined as follows.
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.
Â The two bases , Â 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.Â Note also that since both sets of basis vectors are triads of mutually perpendicular unit vectors, they must be related by
where Â is a proper orthogonal tensor that can be visualized as a rigid rotation.Â Â The rotation tensor can be expressed in several different forms:
Â In further calculations the variation of basis vectors Â with distance s along the deformed rod will play a central role.Â To visualize this quantity, imagine that the basis Â 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 arclength by a vector , analogous to an angular velocity vector.Â The curvature vector can be expressed as components in the basis Â as .Â This vector has the following properties 1. The curvature vector is (by definition) related to the rate of change ofÂ Â with s by , which can be expanded out to show that
2. The components Â quantify the bending of the rod, and are related the curvature Â and the binormal vector Â of the curve traced by the axis of the deformed rod by .Â You can show this result by comparing the formula for Â with the formula for b. 3. The curvature vector can also be expressed in terms of the position vector of the rodâ€™s centroid as
The component of curvature Â cannot in general be expressed in terms of r, because the rotation of the rodâ€™s crosssection about its centroid axis may provide an additional, independent contribution to .Â For the special case where Â and Â are everywhere parallel to the normal vector n and binormal b, respectively, it follows that .Â In this case, Â is equal to the torsion of the curve.Â
Â The rate of change ofÂ Â with distance s can also be expressed in terms of the Euler angles.Â For example, the derivative of Â can be calculated as follows
Similar results for Â and Â are left as exercises.
Â The bending curvatures Â and the twist rate Â are related to the Euler angles by
These results can be derived from the two different formulas for , together with the equations relatingÂ Â and Â in terms of the Euler angles.Â Â Â The arc length s along the rodâ€™s centerline is related to the position vector of the rodâ€™s axis by
Â Some relationships between the time derivatives of these various kinematic quantities are also useful in subsequent calculations.Â The rate of change in shape of the rod can be characterized by the velocity of the axis Â and the time rate of change of the crosssectional rotation .
Â The time derivative of the tangent vector is a convenient way to characterize the rate of change of bending of the rod.Â Â This is related to the velocity of the rodâ€™s centerline by
If we express the velocity in components Â and recall Â we can write this as
It is important to note that the components Â are not equal to the time derivatives of the components of the tangent vector t, because the basis Â varies with time.
Â The time derivatives of the basis vectors can also be quantified by an angular velocity vector , which satisfies .Â Â The components of Â are readily shown to be
Â The time derivatives of the remaining basis vectors follow as
Â The time derivative of the arc length of the centerline is related to its velocity as follows
Â 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 Â to denote its physical significance: it can be interpreted (see Appendix E) as the corotational time derivative of the curvature vector, as follows
Evaluating the derivatives of Â shows that
The corotational time derivative of curvature must be used to quantify bending rate (instead of the time derivative Â ) to correct for the fact that rigid rotations and pure stretching do not change bending.
Â Finally, to solve dynamic problems, we will need to be able to describe the linear and angular acceleration of the bar.Â The linear acceleration is most conveniently characterized by the acceleration of the centerline of the bar
Â The angular acceleration of the barâ€™s crosssection can be characterized by the angular acceleration Â of the basis vectors .Â A straightforward calculation shows that
Â 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 Â in the undeformed rod can be expressed as
This is a completely general expression.Â Â We now introduce a series of approximations that are based on the assumptions that
With this in mind, we assume thatÂ Â can be approximated by a function of the form
where
the Greek indices Â can have values 1 and 2, and Â can be regarded as the first term in a
for all possible choices of .Â The constants Â can be thought of as the components of a homogeneous inplane deformation applied to the cross section, while the function Â describes the warping of the crosssection.Â To decouple the warping from the axial displacement of the rod, we require that
In addition, for small distorsions, the deformation must satisfy Â and , the rod curvatures must satisfy Â for all , and the variation of arclength s along the axis of the deformed rod with Â must satisfy .
The velocity field in the bar can be approximated as
where it has been assumed that Â and Â for all .
Finally, the acceleration field within the bar will be approximated as
Here, all time derivatives of Â and Â have been neglected.Â This is not so much because they are small, but because they represent a crude approximation to the distortion of the crosssection.Â 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.Â
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.Â The second line approximates the distorsion of its crosssection.
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.Â These tensors are most conveniently expressed as components inÂ Â and , respectively Â we can represent this in diadic notation as Â or .Â The components can be expressed in matrix form as
Â 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 thatÂ 2. Recall that 3. Substitute Â and neglect the derivatives of f and Â with respect to
The decomposition Â follows trivially by substituting Â into the dyadic representation of F and rearranging the result.Â A similar approach gives .
It is straightforward to compute additional strain measures from the deformation gradient.Â Only a partial list will be given here.
Substitute , and note that
A tedious set of matrix multiplications shows that the components of D in Â are
to within second order terms in curvature, Â 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 10.2.310.2.5 to calculate strain measures for rods that are not straight in their initial configuration.Â In this case we must start by describing the geometry of the undeformed rod.Â To this end
The deformed shape of the rod is characterized exactly as described in Section 10.2.1, except that the axial distance Â is replaced by the arclength Â of the undeformed rod.
Assuming small distorsions, the deformation gradient can be expressed in dyadic notation as , where the coefficients Â are given below.Â The deformation gradient can also be decomposed into two successive rotations and a small distorsion
where the rotation tensors Â and Â satisfy , and the tensors Â can be expressed in component form asÂ Â .Â 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 Â direction 2. The point at arclength Â in the unstressed rod has coordinates Â in the intermediate configuration. 3. The principal axes of the cross section are parallel to Â in the intermediate configuration 4. The crosssection 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 .Â The components of Â can be calculated as the inverse of the deformation gradient that maps the intermediate straight rod onto the undeformed shape.Â (ii) Deform the rod from the straight configuration to the deformed configuration, with a deformation gradient .Â The total deformation gradient follows as .Â
10.2.8 Representation of forces and moments in slender rods
The figure shows a generic crosssection 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 Â with unit vectors chosen following the scheme described in 10.2.2.Â We define the following vector components in this basis: The body force acting on the rod .Â For simplicity, we shall assume that the body force is uniform within the cross section (but Â 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
Internal forces and moments in the rod are characterized by the following quantities:
The forces Â and moments Â define components of a vector force and moment
10.2.9 Equations of motion and boundary conditions
The internal forces and moments must satisfy the equations of motion Â Â Â Â Â Â Â Â Here, , T and M are the internal forces and moments in the rod; Â are the external force and couple per unit length; Â is the mass density of the rod; A is its crosssectional area, H is the area moment of inertia tensor defined in Sect 10.2.1, while Â are the acceleration, angular velocity and angular acceleration of the rodâ€™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 Â basis, as
Note that:
Boundary Conditions: The internal stresses, forces and moments must satisfy the following boundary conditions
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 Â and compatible stretch rates .Â 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.Â The virtual velocity and stretch rate can therefore be characterized by Â and compatible sets of .Â This has two consequences: Â The virtual work principle can be expressed in terms of the generalized deformation measures and forces defined in the preceding sections as
Â If the virtual work equation is satisfied for all Â 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.Â The Jacobian is approximated as ; the components ofÂ Â follow from the formulas given in Section 10.2.6, and the velocity field is approximated using the formula in 10.2.5.Â Substituting the definitions given in Section 10.2.7 for generalized internal and external forces immediately gives the required result.Â 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 Â and compatible sets of Â into the virtual work equation.
The condition Â follows immediately.Â
Recall that (by definition) Â must be chosen to satisfy
Since the body force is uniform, the term involving Â is zero.Â The first integral can then be integrated by parts as follows
Choosing Â on the boundary yields the equilibrium equation ; choosing any other Â gives the boundary condition .
where we have integrated by parts to obtain the second line.Â Â Choosing Â to vanish on the ends of the rod yields the equation of motion . Any other choice ofÂ Â yields the boundary conditions Â on the ends of the rod.
To proceed, it is necessary to express Â and Â in terms of the virtual velocity components . The algebra and the resulting equilibrium equations are greatly simplified if the tangent vector Â is regarded as an independent kinematic variable.Â Â The relationship between t and Â 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 Â and Â are consistent.Â Â Finally, recall that the curvature rates and stretch rate are related to Â Â 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 Â gives the last five equations of motion, as well as the boundary conditions Â on s=0 and s=L.Â
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.Â Â Constitutive equations must relate the deformation measures to the forces.Â In this section we list the relationships between these quantities for an isotropic, elastic rod subjected to small distorsions.Â For simplicity, the sides of the rod are assumed to be traction free.
The results depend on the geometry of the rodâ€™s crosssection, which is characterized as follows.Â
The warping function depends only on the geometry of the crosssection, and satisfies the following governing equations and boundary conditions
You can easily show that this choice of Â will automatically satisfy the shear stress equilibrium equation Â as well as the boundary condition Â on C.
Calculating the warping function is a nuisance, because it requires the solution to a PDE.Â In desperation, you can take w=0 Â this will overestimate the torsional stiffness of the rod, but in many practical applications the error is not significant.Â Â For a better approximation, warping functions can be estimated by neglecting the terms involving Â in the governing equation.Â A few such approximate warping functions and modified polar moments of area are listed in the table below.Â
The forcedeformation relations for the rod are
The two shear force components Â cannot be related to the deformation Â 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:
where Â are the components of the material stress tensor, and Â are the components of the Lagrange strain tensor. The components of Â in the basis Â can be found using the formulas for Â given in Section 10.2.7, and when substituted into the constitutive laws give expressions for the components of material stress Â in terms of the deformation measures , , Â and .
Substituting the shear stress components from step (2) into this equilibrium equation and setting Â gives the governing equation for w
Substituting these results into the equation defining Â 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:Â The derivation is similar to the procedure used to compute elastic momentcurvature relations.
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.
We need to show that the integral on the right hand side of this expression is zero.
where we have recalled that the warping function w satisfies Â in A as well as Â on C, and have used the divergence theorem.
The sum of (5) and (6) is zero.Â Using this result and (4) gives the expression for the strain energy of the rod.


(c) A.F. Bower, 2008 