Chapter 10


Approximate theories for solids with special shapes:

 rods, beams, membranes, plates and shells




10.4 Exact solutions to simple problems involving elastic rods


This section lists solutions to various boundary and initial value problems involving deformable rods, to illustrate representative applications of the equations derived in Sections 10.2.2 and 10.2.3.   Specifically, we derive solutions for:

  1. The natural frequencies and mode shapes for an initially straight vibrating beam;
  2. The buckling load for a vertical rod subjected to graviatational loading;
  3. The full post-buckled shape for a straight rod compressed by axial loads on its ends;
  4. Internal forces and moments in an initially straight rod that is bent and twisted into a helix;
  5. Internal forces, moments, and the deflected shape of a helical spring.




10.4.1 Free vibration of a straight beam without axial force


The figure illustrates the problem to be solved: an initially straight beam, with axis parallel to the  direction and principal axes of inertia parallel to  is free of external force.  The beam has Young’s modulus  and mass density , and its cross-section has area A and principal moments of area .  Its ends may be constrained in various ways, as described in more detail below.  We wish to calculate the natural frequencies and mode shapes of vibration for the beam, and to use these results to write down the displacement  for a beam that is caused to vibrate with initial conditions ,  at time t=0.


Mode shapes and natural frequencies: The physical significance of the mode shapes and natural frequencies of a vibrating beam can be visualized as follows:

  1. Suppose that the beam is made to vibrate by bending it into some (fixed) deformed shape ; and then suddenly releasing it.   In general, the resulting motion of the beam will be very complicated, and may not even appear to be periodic.
  2. However, there exists a set of special initial deflections , which cause every point on the beam to experience simple harmonic motion at some (angular) frequency , so that the deflected shape has the form .
  3. The special frequencies  are called the natural frequencies of the beam, and the special initial deflections  are called the mode shapes.   Each mode shape has a wave number , which characterizes the wavelength of the harmonic vibrations, and is related to the natural frequency by


  1. The mode shapes  have a very useful property (which is proved in Section 5.9.1):




The mode shapes, wave numbers and corresponding natural frequencies depend on the way the beam is supported at its ends.  A few representative results are listed below


Beam with  free ends:

The wave numbers for each mode are given by the roots of the equation  

The mode shapes are


where  are arbitrary constants.


Beam with pinned ends:

The wave numbers for each mode are  

The mode shapes are , where  are arbitrary constants


Cantilever beam (clamped at , free at  :


The wave numbers for each mode are given by the roots of the equation  

The mode shapes are


where  are arbitrary constants.



Vibration of a beam with given initial displacement and velocity


The solution for free vibration of a beam with given initial displacement and velocity can be found by superposing contributions from each mode as follows






Derivation:  We will derive the equations for the natural frequencies and mode shapes of a beam with free ends as a representative example.  This is a small deflection problem and can be modeled using Euler-Bernoulli beam theory summarized in Section 10.3.2.

1.       The deflection of the beam must satisfy the equation of motion given in Sect 10.3.2


2.       The general solution to this equation (found, e.g. by separation of variables, or just by direct substitution) is


where the frequency and wave number must be related by  to satisfy the equation of motion.

3.       The coefficients  and the wave number  must be chosen to satisfy the boundary conditions at the ends of the bar.   For a beam with free ends, the boundary conditions reduce to ,  at .  Substituting the formula from (2) into the four boundary conditions, and writing the resulting equations in matrix form yields


4.       For a nonzero solution, the matrix in this equation must be singular.  This implies that the determinant of the matrix  is  zero, which gives the governing equation for the wave-number


5.       Since the equation system in (3) is now singular, we may discard any one of the four equations and use the other three to determine an equation relating  to .  Choosing to discard the last row of the matrix, and taking the first column to the right hand side shows that


Solving this equation system shows that .  Substituting these values back into the solution in step (2) gives the mode shape.

6.       To understand the formula for the vibration of a beam with given initial conditions, note that the most general solution consists of a linear combination of all possible mode shapes, i.e.


Formulas for  found by substituting , multiplying both sides of the equation by  and integrating over the length of the beam.   We know that


so the result reduces to


The formula for  is found by differentiating the general solution with respect to time to find the velocity, substituting , and then proceeding as before to extract each coefficient .




10.4.2 Buckling of a column subjected to gravitational loading


The problem to be solved is illustrated in the figure.  A straight, vertical elastic cantilever beam with mass density  and elastic modulus  is clamped at its base and subjected to gravitational loading. The beam has length L, cross-sectional area A and principal moments of area .  The straight, vertical rod is always an equilibrium configuration, but this configuration is stable only if .


Our objective is to show that the critical buckling length is  


A number of different techniques can be used to find buckling loads.  One of the simplest procedures (which will be adopted here) is to identify the critical conditions where both the straight configuration (with  ), and also the deflected configuration (with a small transverse deflection  ) are possible equilibrium shapes for the rod.


This problem can be solved using the governing equations for a beam subjected to large axial forces, listed in Section 10.3.3.    For the present case, we note that

1.       The external forces acting on the rod are , , where g is the gravitational acceleration;

2.       The acceleration is zero (because the rod is in static equilibrium)

3.       The equilibrium equations therefore reduce to


4.       These equations must be solved subject to the boundary conditions


5.       Integrating the second equation of (3) and using the boundary condition  at  reduces the first equation of (3) to


6.       Integrating this equation with respect to  and imposing the boundary condition  at  shows that


7.       This equation can be solved for  using a symbolic manipulation program, which yields


where  are special functions called `Airy Wave functions of order zero’

8.       The remaining boundary conditions are  at , and  at .  Substituting (7) into the boundary conditions and writing the results in matrix form gives


where  and  are Airy wave functions of order 1.

9.       For this system of equations to have a nonzero solution, the determinant of the matrix must vanish, which shows that  must satisfy .  This equation can easily be solved (numerically) for .  The smallest value of  that satisfies the equation is .

10.   The buckling length follows as





10.4.3  Post-buckled shape of an initially straight rod subjected to end thrust


The figure illustrates the problem to be solved.  An initially straight, inextensible elastic rod, with Young’s modulus E, length L and principal in-plane moments of area   (with  ) is subjected to end thrust.  The ends of the rod are constrained to travel along a line that is parallel to the undeformed rod, but the ends are free to rotate.   We wish to calculate the deformed shape of the rod.   You are probably familiar with the simple Euler buckling analysis that predicts the critical buckling loads.  Here, we derive the full post-buckling solution.


The rod is assumed to bow away from its straight configuration as shown: the deflected rod lies in the plane perpendicular to .  The basis  and the Euler angle  that characterize the rotation of the rod’s cross sections are shown in the picture; the remaining Euler angles are .


Solution: Several possible equilibrium solutions may exist, depending on the applied load P.

1.       The straight rod, with  is always an equilibrium solution.  It is stable for applied loads .

2.       For applied loads , with n an integer, there are n+1 possible equilibrium solutions.  One of these is the straight rod; the rest are possible buckling modes.  The shape of each buckling mode depends on a parameter  which satisfies


where K denotes a complete elliptic integral of the first kind .  Note that K has a minimum value  at k=0, and increases monotonically to infinity as .  The equation for  has no solutions for , and n solutions for , as expected.  If multiple solutions exist, only the solution with n=1 is stable.

3.       The shape of the deformed rod can be characterized by the Euler angle , which satisfies


where sn(x,k) denotes a Jacobi-elliptic function called the `sine-amplitude:’ its second argument k is called the `modulus’ of the function.

4.       The coordinates of the buckled rod can also be calculated.  They are given by


Here am(x,k) and cn(x,k) denote Jacobi elliptic functions called the `amplitude’ and `cosine amplitude’, and E(x,k) denotes an incomplete elliptic integral of the second kind .  The shape of the deflected rod for the stable buckling mode (n=1) is shown in the figure above. 


Derivation:  This is a large deflection problem and must be treated using the general equations listed in Sections 10.7-10.9.

  1. The equilibrium equation  immediately shows that T=constant along the rod’s length.  The boundary conditions at the end of the rod give , so that the components of T in  follow as
  2. Substituting the expressions for  into the moment balance equations shows that  and  
  3. Finally, note that the curvatures are , and recall that , so that the angle  satisfies  
  4. This is the equation that governs oscillations of a pendulum, and its solution is well known. The equation is satisfied trivially by  (this is the straight configuration), and also by two one-parameter families of functions of the form


Here,  and  are parameters whose values must be determined from the boundary conditions.  The first of these two functions is called an `inflexional’ solution, because the curve has points where .  The second is called `non-inflexional’ because it has no such points.  For the pendulum, inflexional solutions correspond to periodic swinging motion; the non-inflexional solution corresponds to the pendulum whirling around the pivot. 

  1. The bending moment must satisfy  at both ends of the rod, which requires that  at  and .  Only the inflexional solution can satisfy these boundary conditions.   For this case, we have


The cosine amplitude cn is a periodic function (it is a generalized cosine) and satisfies  at .  We may therefore satisfy the boundary conditions by choosing  and  .  This leads to the defining equations for .

  1. Finally, the formula for the coordinates follows by integrating  and  subject to boundary conditions  at  and  at s=0.
  2. Finally, the (global) stability of the various solutions can be checked by comparing their potential energy.




10.4.4 Rod bent and twisted into a helix


We consider an initially straight rod with Young’s modulus E and shear modulus .  The cross-section of the rod has area , principal in-plane moments of inertia  and an effective torsional inertia .  The  rod is initially straight and unstressed, and is then subjected to forces and moments on its ends to bend and twist it into a helical shape.  The geometry of the deformed rod can be characterized by:

1.       The radius r of the cylinder that generates the helix

2.       The number of turns per unit axial length  in the helix

3.       The helix angle , which is related to n by  

4.       The twist curvature , which quantify the distorsion induced by twisting the rod about its deformed axis.  For the rod to be in equilibrium,  must be constant.

5.       The stretch ratio .  For the rod to be in equilibrium,  must be constant.


The geometry and forces in the deformed rod are most conveniently described using a cylindrical-polar coordinate system  and basis  shown in the figure.  In terms of these basis vectors, we may define

  1. The tangent vector to the rod  
  2. The binormal vector is  


In terms of these variables:

  1. The internal moment in the rod is  
  2. The internal force in the rod is  

For the limiting case of an inextensible rod, the quantity  should be replaced by an indeterminate axial force .


The forces acting on the ends of the rod must satisfy  and  at  and   at s=0.


A variety of force and moment systems may deform the rod into a helical shape, depending on the twist and stretch.  An example of particular practical significance consists of a force  and moment  acting at s=L (with equal and opposite forces at s=0), where


This force system is statically equivalent to a wrench with force  and moment  acting at r=0.


Finally note that this analysis merely gives conditions for a helical rod to be in static equilibrium.  The configuration may not be stable.





  1. We take  at s=0, so that the cylindrical polar coordinates are related to arc-length by . Note also that the basis vectors satisfy , , so that


  1. The position vector of a point on the axis of the rod can be expressed as ;
  2. The tangent vector follows as ;
  3. By definition, the curvature vector is , which can be expressed in terms of the binormal vector as ;
  4. The moment-curvature relations then give the internal moment ;
  5. The equilibrium equation  shows that T=constant.  We may express this constant internal force vector in terms of its components as  
  6. The internal forces and moments must satisfy the moment equilibrium equation, which shows that


Taking the dot product of both sides of this equation with  shows that . It then follows that  and


  1. Finally, the force-stretch relation requires that .  This equation can be solved together with the final result of (7) for the components of internal force in the rod.




10.4.5 Helical spring


The behavior of a helical spring can be deduced by means of a simple extension of the results in the preceding section.  We assume that the spring is made from a material with Young’s modulus E and shear modulus .  The cross-section of the rod has principal in-plane moments of inertia  and an effective torsional inertia .  The rod is assumed to be inextensible, for simplicity.  The geometry of the undeformed spring can be characterized as follows

  1. The length of the rod L, radius  of the cylinder that generates the helix; the height  of the spring, the number of turns in the coil , and the helix angle  
  2. The variables characterizing the undeformed spring are related as follows


  1. It is helpful to select a basis  to characterize the orientation of the initial spring. Since  we may select  and  arbitrarily. It is convenient to choose  and  to be parallel to the normal vector n and binormal vector b of the undeformed spring, respectively, which gives  
  2. The initial curvature components can be calculated from the condition that   and follow as .


The end of the spring at s=0 is held fixed (so it cannot move or rotate).  The end of the spring at  is subjected to a combination of a force  and moment  which act at the axis of the helical coil. 


The solution can be calculated in exactly the same way as the derivation in Section 10.3.5.  It can be shown  that

  1. The spring remains helical: its deformed shape can be characterized by new values of r,  and  after deformation.
  2. The spring may tend to twist about the axis of the helix when it is subjected to load.  The twisting can be quantified by the change in cylindrical-polar coordinates of the point at s=L on the spring.  In the undeformed state, these are  ; after deformation, they are .  The twisting can be characterized by the rotation   The point where the load is applied therefore displaces through a distance , and rotates through the angle  about the axis of the cylinder.
  3. The displacement and rotation are related to the rod’s length L, the coil radius r and helix angle  by


  1. The vectors  and  after deformation are given by .
  2. The curvatures after deformation follow as  
  3. The internal moment and force in the spring are related to the curvatures and external force and moment by



  1. The external force and moment applied to the axis of the spring are related to the helix angle and coil radius before and after deformation by


8.       If the spring is subjected a prescribed force and moment, these equations can be solved for  and , and the results can be substituted into (3) to calculate the extension  and rotation  of the spring.  The results cannot be expressed in closed form for large shape changes.  If  and  are small, however, the relations can be linearized to yield


where the spring stiffnesses are





(c) A.F. Bower, 2008
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