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 gravitational 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 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@325F@  direction and principal axes of inertia parallel to e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@34ED@  is free of external force.  The beam has Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  and mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ , and its cross-section has area A and principal moments of area I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamysamaaBaaaleaacaaIYaaabeaakiaacYcacaWGjbWaaSba aSqaaiaaiodaaeqaaaaa@371E@ .  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 u 1 ( x 3 ,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIZaaabeaakiaacYcacaWG0bGaaiyk aaaa@3765@  for a beam that is caused to vibrate with initial conditions u 1 = u 0 ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWG1bWaaWbaaSqabeaacaaIWaaaaOGaaiikaiaadIhadaWg aaWcbaGaaG4maaqabaGccaGGPaaaaa@38AD@ , d u 1 /dt= v 0 ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG1bWaaSbaaSqaaiaaigdaae qaaOGaai4laiaadsgacaWG0bGaeyypa0JaamODamaaCaaaleqabaGa aGimaaaakiaacIcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaa aa@3C2C@  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 u 1 = u 0 ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWG1bWaaWbaaSqabeaacaaIWaaaaOGaaiikaiaadIhadaWg aaWcbaGaaG4maaqabaGccaGGPaaaaa@38AD@ ; 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 u 0 = U n ( k n x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaahaaWcbeqaaiaaicdaaaGccq GH9aqpcaWGvbWaaSbaaSqaaiaad6gaaeqaaOGaaiikaiaadUgadaWg aaWcbaGaamOBaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaai ykaaaa@3ADE@ , which cause every point on the beam to experience simple harmonic motion at some (angular) frequency ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGUbaabeaaaa a@3374@ , so that the deflected shape has the form u 1 (x,t)= U n ( k n x 3 )cos( ω n t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamiEaiaacYcacaWG0bGaaiykaiabg2da9iaadwfadaWgaaWc baGaamOBaaqabaGccaGGOaGaam4AamaaBaaaleaacaWGUbaabeaaki aadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaci4yaiaac+gacaGG ZbGaaiikaiabeM8a3naaBaaaleaacaWGUbaabeaakiaadshacaGGPa aaaa@46F8@ .
  3. The special frequencies ω n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGUbaabeaaaa a@3374@  are called the natural frequencies of the beam, and the special initial deflections U n ( k n x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOBaaqabaGcca GGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaadIhadaWgaaWcbaGa aG4maaqabaGccaGGPaaaaa@37ED@  are called the mode shapes.   Each mode shape has a wave number k n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaaaaa@3297@ , which characterizes the wavelength of the harmonic vibrations, and is related to the natural frequency by

ω n = k n 2 E I 2 ρA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGUbaabeaaki abg2da9iaadUgadaqhaaWcbaGaamOBaaqaaiaaikdaaaGcdaGcaaqa amaalaaabaGaamyraiaadMeadaWgaaWcbaGaaGOmaaqabaaakeaacq aHbpGCcaWGbbaaaaWcbeaaaaa@3C95@

  1. The mode shapes U n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOBaaqabaaaaa@3281@  have a very useful property (which is proved in Section 5.9.1):

0 L U i ( k i x 3 ) U j ( k j x 3 )=0ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaGaamyvamaaBaaaleaacaWGPb aabeaakiaacIcacaWGRbWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaa BaaaleaacaaIZaaabeaakiaacMcaaSqaaiaaicdaaeaacaWGmbaani abgUIiYdGccaWGvbWaaSbaaSqaaiaadQgaaeqaaOGaaiikaiaadUga daWgaaWcbaGaamOAaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaO Gaaiykaiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaadMgacqGHGjsUcaWGQbaaaa@5E3C@

 

 

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 cos( k n L)cosh( k n L )1=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiaacIcacaWGRb WaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaciGGJbGaai4Baiaa cohacaGGObWaaeWaaeaacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaam itaaGaayjkaiaawMcaaiabgkHiTiaaigdacqGH9aqpcaaIWaaaaa@4338@

The mode shapes are

U n = A n ( sinh( k n x 3 )+sin( k n x 3 )+ cosh( k n L)cos( k n L) sinh( k n L)+sin( k n L) [ cosh( k n x 3 )+cos( k n x 3 ) ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOBaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaaciGGZbGa aiyAaiaac6gacaGGObGaaiikaiaadUgadaWgaaWcbaGaamOBaaqaba GccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiabgUcaRiGacoha caGGPbGaaiOBaiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaam iEamaaBaaaleaacaaIZaaabeaakiaacMcacqGHRaWkdaWcaaqaaiGa cogacaGGVbGaai4CaiaacIgacaGGOaGaam4AamaaBaaaleaacaWGUb aabeaakiaadYeacaGGPaGaeyOeI0Iaci4yaiaac+gacaGGZbGaaiik aiaadUgadaWgaaWcbaGaamOBaaqabaGccaWGmbGaaiykaaqaaiGaco hacaGGPbGaaiOBaiaacIgacaGGOaGaam4AamaaBaaaleaacaWGUbaa beaakiaadYeacaGGPaGaey4kaSIaci4CaiaacMgacaGGUbGaaiikai aadUgadaWgaaWcbaGaamOBaaqabaGccaWGmbGaaiykaaaadaWadaqa aiGacogacaGGVbGaai4CaiaacIgacaGGOaGaam4AamaaBaaaleaaca WGUbaabeaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaey4k aSIaci4yaiaac+gacaGGZbGaaiikaiaadUgadaWgaaWcbaGaamOBaa qabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaGaay5waiaa w2faaaGaayjkaiaawMcaaaaa@7E6F@

where A n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamOBaaqabaaaaa@326D@  are arbitrary constants.

 

Beam with pinned ends:

The wave numbers for each mode are k n = nπ L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaGccq GH9aqpdaWcaaqaaiaad6gacqaHapaCaeaacaWGmbaaaaaa@3738@

The mode shapes are U n = A n sin( k n x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOBaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaad6gaaeqaaOGaci4CaiaacMgacaGG UbGaaiikaiaadUgadaWgaaWcbaGaamOBaaqabaGccaWG4bWaaSbaaS qaaiaaiodaaeqaaOGaaiykaaaa@3DB9@ , where A n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamOBaaqabaaaaa@326D@  are arbitrary constants

 

Cantilever beam (clamped at x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3438@ , free at  x 3 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGmbaaaa@344F@ :

 

The wave numbers for each mode are given by the roots of the equation cos( k n L)cosh( k n L )+1=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiaacIcacaWGRb WaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaciGGJbGaai4Baiaa cohacaGGObWaaeWaaeaacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaam itaaGaayjkaiaawMcaaiabgUcaRiaaigdacqGH9aqpcaaIWaaaaa@432D@

The mode shapes are

U n = A n ( sinh( k n x 3 )sin( k n x 3 )+ cosh( k n L)+cos( k n L) sin( k n L)sinh( k n L) [ cosh( k n x 3 )cos( k n x 3 ) ] ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOBaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaaciGGZbGa aiyAaiaac6gacaGGObGaaiikaiaadUgadaWgaaWcbaGaamOBaaqaba GccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiabgkHiTiGacoha caGGPbGaaiOBaiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaam iEamaaBaaaleaacaaIZaaabeaakiaacMcacqGHRaWkdaWcaaqaaiGa cogacaGGVbGaai4CaiaacIgacaGGOaGaam4AamaaBaaaleaacaWGUb aabeaakiaadYeacaGGPaGaey4kaSIaci4yaiaac+gacaGGZbGaaiik aiaadUgadaWgaaWcbaGaamOBaaqabaGccaWGmbGaaiykaaqaaiGaco hacaGGPbGaaiOBaiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGa amitaiaacMcacqGHsislciGGZbGaaiyAaiaac6gacaGGObGaaiikai aadUgadaWgaaWcbaGaamOBaaqabaGccaWGmbGaaiykaaaadaWadaqa aiGacogacaGGVbGaai4CaiaacIgacaGGOaGaam4AamaaBaaaleaaca WGUbaabeaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaeyOe I0Iaci4yaiaac+gacaGGZbGaaiikaiaadUgadaWgaaWcbaGaamOBaa qabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaGaay5waiaa w2faaaGaayjkaiaawMcaaaaa@7E85@

where A n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaamOBaaqabaaaaa@326D@  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

u 1 ( x 3 ,t)= n=1 C n U n ( k n x 3 )cos( ω n t)+ n=1 D n U n ( k n x 3 )sin( ω n t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIZaaabeaakiaacYcacaWG0bGaaiyk aiabg2da9maaqahabaGaam4qamaaBaaaleaacaWGUbaabeaakiaadw fadaWgaaWcbaGaamOBaaqabaGccaGGOaGaam4AamaaBaaaleaacaWG UbaabeaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaci4yai aac+gacaGGZbGaaiikaiabeM8a3naaBaaaleaacaWGUbaabeaakiaa dshacaGGPaGaey4kaScaleaacaWGUbGaeyypa0JaaGymaaqaaiabg6 HiLcqdcqGHris5aOWaaabCaeaacaWGebWaaSbaaSqaaiaad6gaaeqa aOGaamyvamaaBaaaleaacaWGUbaabeaakiaacIcacaWGRbWaaSbaaS qaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIZaaabeaakiaacMca ciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaad6gaae qaaOGaamiDaiaacMcaaSqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOh IukaniabggHiLdaaaa@690D@

where

C n = 0 L u 0 ( x 3 ) U n ( k n x 3 )d x 3 0 L { U n ( k n x 3 ) } 2 d x 3 D n = 0 L v 0 ( x 3 ) U n ( k n x 3 )d x 3 ω n 0 L { U n ( k n x 3 ) } 2 d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamOBaaqabaGccq GH9aqpdaWcaaqaamaapehabaGaamyDamaaCaaaleqabaGaaGimaaaa kiaacIcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaadwfada WgaaWcbaGaamOBaaqabaGccaGGOaGaam4AamaaBaaaleaacaWGUbaa beaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaamizaiaadI hadaWgaaWcbaGaaG4maaqabaaabaGaaGimaaqaaiaadYeaa0Gaey4k IipaaOqaamaapehabaWaaiWaaeaacaWGvbWaaSbaaSqaaiaad6gaae qaaOGaaiikaiaadUgadaWgaaWcbaGaamOBaaqabaGccaWG4bWaaSba aSqaaiaaiodaaeqaaOGaaiykaaGaay5Eaiaaw2haamaaCaaaleqaba GaaGOmaaaakiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaqaaiaa icdaaeaacaWGmbaaniabgUIiYdaaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamiramaaBa aaleaacaWGUbaabeaakiabg2da9maalaaabaWaa8qCaeaacaWG2bWa aWbaaSqabeaacaaIWaaaaOGaaiikaiaadIhadaWgaaWcbaGaaG4maa qabaGccaGGPaGaamyvamaaBaaaleaacaWGUbaabeaakiaacIcacaWG RbWaaSbaaSqaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIZaaabe aakiaacMcacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaeaacaaI WaaabaGaamitaaqdcqGHRiI8aaGcbaGaeqyYdC3aaSbaaSqaaiaad6 gaaeqaaOWaa8qCaeaadaGadaqaaiaadwfadaWgaaWcbaGaamOBaaqa baGccaGGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaadIhadaWgaa WcbaGaaG4maaqabaGccaGGPaaacaGL7bGaayzFaaWaaWbaaSqabeaa caaIYaaaaOGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaabaGaaG imaaqaaiaadYeaa0Gaey4kIipaaaaaaa@9E96@

 

 

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

E I 2 d 4 u 1 d x 3 4 +ρA d 2 u 1 d t 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaI0aaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaI0aaaaaaakiabgUcaRiabeg8aYjaadgeadaWcaaqaaiaa dsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaae qaaaGcbaGaamizaiaadshadaahaaWcbeqaaiaaikdaaaaaaOGaeyyp a0JaaGimaaaa@4657@

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

u 1 ={ A 1 sinh( k n x 3 )+ A 2 cosh( k n x 3 )+ A 3 sin( k n x 3 )+ A 4 cos( k n x 3 ) }cos( ω n t+ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpdaGadaqaaiaadgeadaWgaaWcbaGaaGymaaqabaGcciGGZbGa aiyAaiaac6gacaGGObGaaiikaiaadUgadaWgaaWcbaGaamOBaaqaba GccaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiabgUcaRiaadgea daWgaaWcbaGaaGOmaaqabaGcciGGJbGaai4BaiaacohacaGGObGaai ikaiaadUgadaWgaaWcbaGaamOBaaqabaGccaWG4bWaaSbaaSqaaiaa iodaaeqaaOGaaiykaiabgUcaRiaadgeadaWgaaWcbaGaaG4maaqaba GcciGGZbGaaiyAaiaac6gacaGGOaGaam4AamaaBaaaleaacaWGUbaa beaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaey4kaSIaam yqamaaBaaaleaacaaI0aaabeaakiGacogacaGGVbGaai4CaiaacIca caWGRbWaaSbaaSqaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIZa aabeaakiaacMcaaiaawUhacaGL9baaciGGJbGaai4BaiaacohacaGG OaGaeqyYdC3aaSbaaSqaaiaad6gaaeqaaOGaamiDaiabgUcaRiabew 9aMjaacMcaaaa@6CAD@

where the frequency and wave number must be related by k n 4 =ρA ω n 2 /E I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaqhaaWcbaGaamOBaaqaaiaais daaaGccqGH9aqpcqaHbpGCcaWGbbGaeqyYdC3aa0baaSqaaiaad6ga aeaacaaIYaaaaOGaai4laiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaaaa@3DD2@  to satisfy the equation of motion.

3.      The coefficients A 1 ... A 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaaGymaaqabaGcca GGUaGaaiOlaiaac6cacaWGbbWaaSbaaSqaaiaaisdaaeqaaaaa@3605@  and the wave number k n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaaaaa@3297@  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 d 2 u 1 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgadaahaaWcbeqaaiaaikdaaaGcca WG1bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaadsgacaWG4bWaa0ba aSqaaiaaiodaaeaacaaIYaaaaOGaeyypa0JaaGimaaaa@3A58@ , d 3 u 1 /d x 3 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgadaahaaWcbeqaaiaaiodaaaGcca WG1bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaadsgacaWG4bWaa0ba aSqaaiaaiodaaeaacaaIZaaaaOGaeyypa0JaaGimaaaa@3A5A@  at x 3 =0, x 3 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaGaaiilaiaadIhadaWgaaWcbaGaaG4maaqabaGccqGH 9aqpcaWGmbaaaa@38AF@ .  Substituting the formula from (2) into the four boundary conditions, and writing the resulting equations in matrix form yields

[ 1 0 1 0 0 1 0 1 cosh( k n L) sinh( k n L) cos( k n L) sin( k n L) sinh( k n L) cosh( k n L) sin( k n L) cos( k n L) ][ A 1 A 2 A 3 A 4 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabqabaaaaaeaacaaIXa aabaGaaGimaaqaaiabgkHiTiaaigdaaeaacaaIWaaabaGaaGimaaqa aiaaigdaaeaacaaIWaaabaGaeyOeI0IaaGymaaqaaiGacogacaGGVb Gaai4CaiaacIgacaGGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaa dYeacaGGPaaabaGaci4CaiaacMgacaGGUbGaaiiAaiaacIcacaWGRb WaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaaeaacqGHsislciGG JbGaai4BaiaacohacaGGOaGaam4AamaaBaaaleaacaWGUbaabeaaki aadYeacaGGPaaabaGaci4CaiaacMgacaGGUbGaaiikaiaadUgadaWg aaWcbaGaamOBaaqabaGccaWGmbGaaiykaaqaaiGacohacaGGPbGaai OBaiaacIgacaGGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaadYea caGGPaaabaGaci4yaiaac+gacaGGZbGaaiiAaiaacIcacaWGRbWaaS baaSqaaiaad6gaaeqaaOGaamitaiaacMcaaeaacqGHsislciGGZbGa aiyAaiaac6gacaGGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaadY eacaGGPaaabaGaeyOeI0Iaci4yaiaac+gacaGGZbGaaiikaiaadUga daWgaaWcbaGaamOBaaqabaGccaWGmbGaaiykaaaaaiaawUfacaGLDb aadaWadaqaauaabeqaeeaaaaqaaiaadgeadaWgaaWcbaGaaGymaaqa baaakeaacaWGbbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyqamaaBa aaleaacaaIZaaabeaaaOqaaiaadgeadaWgaaWcbaGaaGinaaqabaaa aaGccaGLBbGaayzxaaGaeyypa0JaaGimaaaa@8428@

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

cos( k n L)cosh( k n L )1=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiaacIcacaWGRb WaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaciGGJbGaai4Baiaa cohacaGGObWaaeWaaeaacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaam itaaGaayjkaiaawMcaaiabgkHiTiaaigdacqGH9aqpcaaIWaaaaa@4338@

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 A 2 , A 3 , A 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaaGOmaaqabaGcca GGSaGaamyqamaaBaaaleaacaaIZaaabeaakiaacYcacaWGbbWaaSba aSqaaiaaisdaaeqaaaaa@3709@  to A 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaaGymaaqabaaaaa@3235@ .  Choosing to discard the last row of the matrix, and taking the first column to the right hand side shows that

[ 0 1 0 1 0 1 sinh( k n L) cos( k n L) sin( k n L) ][ A 2 A 3 A 4 ]= A 1 [ 1 0 cosh( k n L) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmWaaaqaaiaaicdaae aacqGHsislcaaIXaaabaGaaGimaaqaaiaaigdaaeaacaaIWaaabaGa eyOeI0IaaGymaaqaaiGacohacaGGPbGaaiOBaiaacIgacaGGOaGaam 4AamaaBaaaleaacaWGUbaabeaakiaadYeacaGGPaaabaGaeyOeI0Ia ci4yaiaac+gacaGGZbGaaiikaiaadUgadaWgaaWcbaGaamOBaaqaba GccaWGmbGaaiykaaqaaiGacohacaGGPbGaaiOBaiaacIcacaWGRbWa aSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaaaaacaGLBbGaayzxaa WaamWaaeaafaqabeWabaaabaGaamyqamaaBaaaleaacaaIYaaabeaa aOqaaiaadgeadaWgaaWcbaGaaG4maaqabaaakeaacaWGbbWaaSbaaS qaaiaaisdaaeqaaaaaaOGaay5waiaaw2faaiabg2da9iabgkHiTiaa dgeadaWgaaWcbaGaaGymaaqabaGcdaWadaqaauaabeqadeaaaeaaca aIXaaabaGaaGimaaqaaiGacogacaGGVbGaai4CaiaacIgacaGGOaGa am4AamaaBaaaleaacaWGUbaabeaakiaadYeacaGGPaaaaaGaay5wai aaw2faaaaa@663C@

Solving this equation system shows that A 2 = A 4 = cosh( k n L)cos( k n L) sinh( k n L)+sin( k n L) A 1 A 3 = A 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaSaaaeaa ciGGJbGaai4BaiaacohacaGGObGaaiikaiaadUgadaWgaaWcbaGaam OBaaqabaGccaWGmbGaaiykaiabgkHiTiGacogacaGGVbGaai4Caiaa cIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaaeaaci GGZbGaaiyAaiaac6gacaGGObGaaiikaiaadUgadaWgaaWcbaGaamOB aaqabaGccaWGmbGaaiykaiabgUcaRiGacohacaGGPbGaaiOBaiaacI cacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaamitaiaacMcaaaGaamyq amaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadgeadaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaaigdaaeqaaOGaaGPaVl aaykW7aaa@6D45@ .  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.

u 1 ( x 3 ,t)= n=1 C n U n ( k n x 3 )cos( ω n t)+ n=1 D n U n ( k n x 3 )sin( ω n t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamiEamaaBaaaleaacaaIZaaabeaakiaacYcacaWG0bGaaiyk aiabg2da9maaqahabaGaam4qamaaBaaaleaacaWGUbaabeaakiaadw fadaWgaaWcbaGaamOBaaqabaGccaGGOaGaam4AamaaBaaaleaacaWG UbaabeaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaci4yai aac+gacaGGZbGaaiikaiabeM8a3naaBaaaleaacaWGUbaabeaakiaa dshacaGGPaGaey4kaScaleaacaWGUbGaeyypa0JaaGymaaqaaiabg6 HiLcqdcqGHris5aOWaaabCaeaacaWGebWaaSbaaSqaaiaad6gaaeqa aOGaamyvamaaBaaaleaacaWGUbaabeaakiaacIcacaWGRbWaaSbaaS qaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIZaaabeaakiaacMca ciGGZbGaaiyAaiaac6gacaGGOaGaeqyYdC3aaSbaaSqaaiaad6gaae qaaOGaamiDaiaacMcaaSqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOh IukaniabggHiLdaaaa@690D@

Formulas for C n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamOBaaqabaaaaa@326F@  found by substituting t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaIWaaaaa@3341@ , multiplying both sides of the equation by U j ( k n x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaamOAaaqabaGcca GGOaGaam4AamaaBaaaleaacaWGUbaabeaakiaadIhadaWgaaWcbaGa aG4maaqabaGccaGGPaaaaa@37E9@  and integrating over the length of the beam.   We know that

0 L U n ( k n x 3 ) U j ( k j x 3 )=0nj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaGaamyvamaaBaaaleaacaWGUb aabeaakiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaamiEamaa BaaaleaacaaIZaaabeaakiaacMcaaSqaaiaaicdaaeaacaWGmbaani abgUIiYdGccaWGvbWaaSbaaSqaaiaadQgaaeqaaOGaaiikaiaadUga daWgaaWcbaGaamOAaaqabaGccaWG4bWaaSbaaSqaaiaaiodaaeqaaO Gaaiykaiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaad6gacqGHGjsUcaWGQbaaaa@5E4B@

so the result reduces to

0 L u 0 ( x 3 ) U j ( k n x 3 )d x 3 = C j 0 L { U j ( k n x 3 ) } 2 d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaGaamyDamaaCaaaleqabaGaaG imaaaakiaacIcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaa dwfadaWgaaWcbaGaamOAaaqabaGccaGGOaGaam4AamaaBaaaleaaca WGUbaabeaakiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaamiz aiaadIhadaWgaaWcbaGaaG4maaqabaaabaGaaGimaaqaaiaadYeaa0 Gaey4kIipakiabg2da9iaadoeadaWgaaWcbaGaamOAaaqabaGcdaWd XbqaamaacmaabaGaamyvamaaBaaaleaacaWGQbaabeaakiaacIcaca WGRbWaaSbaaSqaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIZaaa beaakiaacMcaaiaawUhacaGL9baadaahaaWcbeqaaiaaikdaaaGcca WGKbGaamiEamaaBaaaleaacaaIZaaabeaaaeaacaaIWaaabaGaamit aaqdcqGHRiI8aOGaaGPaVlaaykW7caaMc8oaaa@5CBA@

The formula for D n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamOBaaqabaaaaa@3270@  is found by differentiating the general solution with respect to time to find the velocity, substituting t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaIWaaaaa@3341@ , and then proceeding as before to extract each coefficient D n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamOBaaqabaaaaa@3270@ .

 

 

 

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 ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@  and elastic modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@  is clamped at its base and subjected to gravitational loading. The beam has length L, cross-sectional area A and principal moments of area I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamysamaaBaaaleaacaaIYaaabeaakiaacYcacaWGjbWaaSba aSqaaiaaiodaaeqaaaaa@371E@ .  The straight, vertical rod is always an equilibrium configuration, but this configuration is stable only if L< L crit MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeacqGH8aapcaWGmbWaaSbaaSqaai aadogacaWGYbGaamyAaiaadshaaeqaaaaa@3720@ .

 

Our objective is to show that the critical buckling length is L crit 2 ( E I 2 ρAg ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaam4yaiaadkhaca WGPbGaamiDaaqabaGccqGHijYUcaaIYaWaaeWaaeaadaWcaaqaaiaa dweacaWGjbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeqyWdiNaamyqai aadEgaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaa iodaaaaaaa@41AF@

 

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 u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3433@  ), and also the deflected configuration (with a small transverse deflection u 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GHGjsUcaaIWaaaaa@34F4@  ) 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 p 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@342E@ , p 3 =ρgA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqaHbpGCcaWGNbGaamyqaaaa@36E8@ , 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

E I 2 d 4 u 1 d x 3 4 +ρAg d u 1 d x 3 =N d 2 u 1 d x 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaI0aaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaI0aaaaaaakiabgUcaRiabeg8aYjaadgeacaWGNbWaaSaa aeaacaWGKbGaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgaca WG4bWaaSbaaSqaaiaaiodaaeqaaaaakiabg2da9iaad6eadaWcaaqa aiaadsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaig daaeqaaaGcbaGaamizaiaadIhadaqhaaWcbaGaaG4maaqaaiaaikda aaaaaaaa@4DD0@           dN d x 3 +ρAg=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaad6eaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHRaWkcqaHbpGCcaWG bbGaam4zaiabg2da9iaaicdaaaa@3B40@

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

N=0, d 2 u 1 d x 3 2 =0, d 3 u 1 d x 3 3 =0 x 3 =0; u 1 =0, d u 1 d x 3 =0 x 3 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacqGH9aqpcaaIWaGaaiilaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaa dsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaae qaaaGcbaGaamizaiaadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaaa aOGaeyypa0JaaGimaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7daWcaaqaaiaadsgadaahaaWcbeqaaiaaiodaaaGccaWG 1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadIhadaqhaaWcba GaaG4maaqaaiaaiodaaaaaaOGaeyypa0JaaGimaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaamiEamaaBaaaleaacaaIZaaabeaakiabg2da9iaaicdacaGG 7aGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyD amaaBaaaleaacaaIXaaabeaakiabg2da9iaaicdacaGGSaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaSaaaeaacaWGKbGaamyD amaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaai aaiodaaeqaaaaakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaadIhadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGmb GaaGPaVlaaykW7caaMc8oaaa@ACAC@

5.      Integrating the second equation of (3) and using the boundary condition N=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacqGH9aqpcaaIWaaaaa@331B@  at x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3438@  reduces the first equation of (3) to

E I 2 d 4 u 1 d x 3 4 +ρAg d u 1 d x 3 +ρAg x 3 d 2 u 1 d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaI0aaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaI0aaaaaaakiabgUcaRiabeg8aYjaadgeacaWGNbWaaSaa aeaacaWGKbGaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgaca WG4bWaaSbaaSqaaiaaiodaaeqaaaaakiabgUcaRiabeg8aYjaadgea caWGNbGaamiEamaaBaaaleaacaaIZaaabeaakmaalaaabaGaamizam aaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaaqabaaa keaacaWGKbGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaaaaGccq GH9aqpcaaIWaaaaa@5405@

6.      Integrating this equation with respect to x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@326E@  and imposing the boundary condition d 3 u 1 /d x 3 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgadaahaaWcbeqaaiaaiodaaaGcca WG1bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaadsgacaWG4bWaa0ba aSqaaiaaiodaaeaacaaIZaaaaOGaeyypa0JaaGimaaaa@3A5A@  at x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3438@  shows that

E I 2 d 3 u 1 d x 3 3 +ρAg x 3 d u 1 d x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIZaaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaIZaaaaaaakiabgUcaRiabeg8aYjaadgeacaWGNbGaamiE amaaBaaaleaacaaIZaaabeaakmaalaaabaGaamizaiaadwhadaWgaa WcbaGaaGymaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaa beaaaaGccqGH9aqpcaaIWaaaaa@4842@

7.      This equation can be solved for d u 1 /d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG1bWaaSbaaSqaaiaaigdaae qaaOGaai4laiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@36DE@  using a symbolic manipulation program, which yields

d u 1 d x 3 = C 1 Ai 0 [ x 3 ( ρAg E I 2 ) 1/3 ]+ C 2 Bi 0 [ x 3 ( ρAg E I 2 ) 1/3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadwhadaWgaaWcba GaaGymaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa aaGccqGH9aqpcaWGdbWaaSbaaSqaaiaaigdaaeqaaOGaciyqaiaacM gadaWgaaWcbaGaaGimaaqabaGcdaWadaqaaiabgkHiTiaadIhadaWg aaWcbaGaaG4maaqabaGcdaqadaqaamaalaaabaGaeqyWdiNaamyqai aadEgaaeaacaWGfbGaamysamaaBaaaleaacaaIYaaabeaaaaaakiaa wIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaG4maaaaaOGaay 5waiaaw2faaiabgUcaRiaadoeadaWgaaWcbaGaaGOmaaqabaGcciGG cbGaaiyAamaaBaaaleaacaaIWaaabeaakmaadmaabaGaeyOeI0Iaam iEamaaBaaaleaacaaIZaaabeaakmaabmaabaWaaSaaaeaacqaHbpGC caWGbbGaam4zaaqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaaeqaaa aaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+cacaaIZaaa aaGccaGLBbGaayzxaaaaaa@5E75@

where Ai 0 (x), Bi 0 (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacgeacaGGPbWaaSbaaSqaaiaaicdaae qaaOGaciikaiaadIhacaGGPaGaaiilaiGackeacaGGPbWaaSbaaSqa aiaaicdaaeqaaOGaciikaiaadIhacaGGPaaaaa@3B31@  are special functions called `Airy Wave functions of order zero’

8.      The remaining boundary conditions are d u 1 2 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG1bWaa0baaSqaaiaaigdaae aacaaIYaaaaOGaai4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaa caaIYaaaaOGaeyypa0JaaGimaaaa@3A22@  at x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3438@ , and d u 1 /d x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG1bWaaSbaaSqaaiaaigdaae qaaOGaai4laiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaeyyp a0JaaGimaaaa@38A8@  at x 3 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGmbaaaa@344F@ .  Substituting (7) into the boundary conditions and writing the results in matrix form gives

[ Ai 1 (0) Bi 1 (0) Ai 0 (λ) Bi 0 (λ) ][ C 1 C 2 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabiGaaaqaaiGacgeaca GGPbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaaicdacaGGPaaabaGa ciOqaiaacMgadaWgaaWcbaGaaGymaaqabaGccaGGOaGaaGimaiaacM caaeaaciGGbbGaaiyAamaaBaaaleaacaaIWaaabeaakiaacIcacqGH sislcqaH7oaBcaGGPaaabaGaciOqaiaacMgadaWgaaWcbaGaaGimaa qabaGccaGGOaGaeyOeI0Iaeq4UdWMaaiykaaaaaiaawUfacaGLDbaa daWadaqaauaabeqaceaaaeaacaWGdbWaaSbaaSqaaiaaigdaaeqaaa GcbaGaam4qamaaBaaaleaacaaIYaaabeaaaaaakiaawUfacaGLDbaa cqGH9aqpcaaIWaaaaa@506A@

where λ=L (ρAg/E I 2 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iaadYeacaGGOaGaeq yWdiNaamyqaiaadEgacaGGVaGaamyraiaadMeadaWgaaWcbaGaaGOm aaqabaGccaGGPaWaaWbaaSqabeaacaaIXaGaai4laiaaiodaaaaaaa@3E73@  and Ai 1 (x), Bi 1 (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacgeacaGGPbWaaSbaaSqaaiaaigdaae qaaOGaciikaiaadIhacaGGPaGaaiilaiGackeacaGGPbWaaSbaaSqa aiaaigdaaeqaaOGaciikaiaadIhacaGGPaaaaa@3B33@  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 λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@323C@  must satisfy Ai 1 (0) Bi 0 (λ) Bi 1 (0) Ai 0 (λ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacgeacaGGPbWaaSbaaSqaaiaaigdaae qaaOGaaiikaiaaicdacaGGPaGaciOqaiaacMgadaWgaaWcbaGaaGim aaqabaGccaGGOaGaeyOeI0Iaeq4UdWMaaiykaiabgkHiTiGackeaca GGPbWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaaicdacaGGPaGaciyq aiaacMgadaWgaaWcbaGaaGimaaqabaGccaGGOaGaeyOeI0Iaeq4UdW Maaiykaiabg2da9iaaicdaaaa@49E3@ .  This equation can easily be solved (numerically) for λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@323C@ .  The smallest value of λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@323C@  that satisfies the equation is λ2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabgIKi7kaaikdaaaa@34A9@ .

10.  The buckling length follows as

L crit 2 ( E I 2 ρAg ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaam4yaiaadkhaca WGPbGaamiDaaqabaGccqGHijYUcaaIYaWaaeWaaeaadaWcaaqaaiaa dweacaWGjbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeqyWdiNaamyqai aadEgaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaa iodaaaaaaa@41AF@

 

 

 

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 I 1 , I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamysamaaBaaaleaacaaIYaaabeaaaaa@34AD@   (with I 2 < I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaikdaaeqaaO GaeyipaWJaamysamaaBaaaleaacaaIXaaabeaaaaa@373F@  ) 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 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@325E@ .  The basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@39A0@  and the Euler angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  that characterize the rotation of the rod’s cross sections are shown in the picture; the remaining Euler angles are ψ=ϕ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabew9aMjabg2da9i aaicdaaaa@36E4@ .

 

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

1.      The straight rod, with θ= y 2 =0, x 3 =s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaadMhadaWgaaWcba GaaGOmaaqabaGccqGH9aqpcaaIWaGaaiilaiaadIhadaWgaaWcbaGa aG4maaqabaGccqGH9aqpcaWGZbaaaa@3B92@  is always an equilibrium solution.  It is stable for applied loads P< π 2 E I 2 / L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH8aapcqaHapaCdaahaaWcbe qaaiaaikdaaaGccaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaa c+cacaWGmbWaaWbaaSqabeaacaaIYaaaaaaa@3A08@ .

2.      For applied loads P> n 2 π 2 E I 2 / L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH+aGpcaWGUbWaaWbaaSqabe aacaaIYaaaaOGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGaamyraiaa dMeadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamitamaaCaaaleqaba GaaGOmaaaaaaa@3BF2@ , 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 k n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaaaaa@3297@  which satisfies

L P/E I 2 =2nK( k n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaGcaaqaaiaadcfacaGGVaGaam yraiaadMeadaWgaaWcbaGaaGOmaaqabaaabeaakiabg2da9iaaikda caWGUbGaam4saiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaai ykaaaa@3C72@

where K denotes a complete elliptic integral of the first kind K(k)= 0 π/2 (1 k 2 sin 2 x) 1/2 dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacaGGOaGaam4AaiaacMcacqGH9a qpdaWdXbqaaiaacIcacaaIXaGaeyOeI0Iaam4AamaaCaaaleqabaGa aGOmaaaakiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaaki aadIhacaGGPaWaaWbaaSqabeaacqGHsislcaaIXaGaai4laiaaikda aaGccaWGKbGaamiEaaWcbaGaaGimaaqaaiabec8aWjaac+cacaaIYa aaniabgUIiYdaaaa@49D6@ .  Note that K has a minimum value K=π/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacqGH9aqpcqaHapaCcaGGVaGaaG Omaaaa@358A@  at k=0, and increases monotonically to infinity as k1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGHsgIRcaaIXaaaaa@3420@ .  The equation for k n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaaaaa@3297@  has no solutions for P< π 2 E I 2 / L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH8aapcqaHapaCdaahaaWcbe qaaiaaikdaaaGccaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaa c+cacaWGmbWaaWbaaSqabeaacaaIYaaaaaaa@3A08@ , and n solutions for P> n 2 π 2 E I 2 / L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH+aGpcaWGUbWaaWbaaSqabe aacaaIYaaaaOGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGaamyraiaa dMeadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamitamaaCaaaleqaba GaaGOmaaaaaaa@3BF2@ , 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 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@ , which satisfies

θ=2 sin 1 { k n sn( s (P/E I 2 ) +K( k n ); k n ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaikdaciGGZbGaai yAaiaac6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaGadaqaaiaa dUgadaWgaaWcbaGaamOBaaqabaGccaGGZbGaaiOBamaabmaabaGaam 4CamaakaaabaGaaiikaiaadcfacaGGVaGaamyraiaadMeadaWgaaWc baGaaGOmaaqabaGccaGGPaaaleqaaOGaaGPaVlaaykW7cqGHRaWkca WGlbGaaiikaiaadUgadaWgaaWcbaGaamOBaaqabaGccaGGPaGaai4o aiaadUgadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaiaawU hacaGL9baacaaMc8oaaa@5398@

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

y 3 =s2 E I 2 /P [ Ε( am(s P/E I 2 +K( k n ); k n ); k n )Ε( am(K( k n ); k n ); k n ) ] y 2 =2 k n E I 2 /P cn(s P/E I 2 +K( k n ); k n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyEamaaBaaaleaacaaIZaaabe aakiabg2da9iabgkHiTiaadohacqGHsislcaaIYaWaaOaaaeaacaWG fbGaamysamaaBaaaleaacaaIYaaabeaakiaac+cacaWGqbaaleqaaO WaamWaaeaacqqHvoqrdaqadaqaaiaabggacaqGTbGaaeikaiaadoha daGcaaqaaiaadcfacaGGVaGaamyraiaadMeadaWgaaWcbaGaaGOmaa qabaaabeaakiabgUcaRiaadUeacaGGOaGaam4AamaaBaaaleaacaWG UbaabeaakiaacMcacaGG7aGaam4AamaaBaaaleaacaWGUbaabeaaki aacMcacaGG7aGaam4AamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaa wMcaaiabgkHiTiabfw5afnaabmaabaGaaeyyaiaab2gacaqGOaGaam 4saiaacIcacaWGRbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaacUda caWGRbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaacUdacaWGRbWaaS baaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaa baGaamyEamaaBaaaleaacaaIYaaabeaakiabg2da9iabgkHiTiaaik dacaWGRbWaaSbaaSqaaiaad6gaaeqaaOWaaOaaaeaacaWGfbGaamys amaaBaaaleaacaaIYaaabeaakiaac+cacaWGqbaaleqaaOGaae4yai aab6gacaqGOaGaam4CamaakaaabaGaamiuaiaac+cacaWGfbGaamys amaaBaaaleaacaaIYaaabeaaaeqaaOGaey4kaSIaam4saiaacIcaca WGRbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaacUdacaWGRbWaaSba aSqaaiaad6gaaeqaaOGaaiykaaaaaa@7F78@

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 Ε(x,k)= 0 x (1 k 2 sin 2 t) 1/2 dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfw5afjaacIcacaWG4bGaaiilaiaadU gacaGGPaGaeyypa0Zaa8qCaeaacaGGOaGaaGymaiabgkHiTiaadUga daahaaWcbeqaaiaaikdaaaGcciGGZbGaaiyAaiaac6gadaahaaWcbe qaaiaaikdaaaGccaWG0bGaaiykamaaCaaaleqabaGaaGymaiaac+ca caaIYaaaaOGaamizaiaadshaaSqaaiaaicdaaeaacaWG4baaniabgU IiYdaaaa@48F7@ .  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 dT/ds=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHubGaai4laiaadsgacaWGZb Gaeyypa0JaaGimaaaa@36A2@  immediately shows that T=constant along the rod’s length.  The boundary conditions at the end of the rod give T=P e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcqGHsislcaWGqbGaaC yzamaaBaaaleaacaaIZaaabeaaaaa@3604@ , so that the components of T in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@39A0@  follow as T 1 =0, T 2 =Psinθ, T 3 =Pcosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaGaaiilaiaadsfadaWgaaWcbaGaaGOmaaqabaGccqGH 9aqpcaWGqbGaci4CaiaacMgacaGGUbGaeqiUdeNaaiilaiaadsfada WgaaWcbaGaaG4maaqabaGccqGH9aqpcqGHsislcaWGqbGaci4yaiaa c+gacaGGZbGaeqiUdehaaa@46C3@
  2. Substituting the expressions for T i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaamyAaaqabaaaaa@327B@  into the moment balance equations shows that M 1 = M 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGnbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaaa @36D6@  and d M 2 /ds+Psinθ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGnbWaaSbaaSqaaiaaikdaae qaaOGaai4laiaadsgacaWGZbGaey4kaSIaamiuaiGacohacaGGPbGa aiOBaiabeI7aXjabg2da9iaaicdaaaa@3DCE@
  3. Finally, note that the curvatures are κ 2 =dθ/ds, κ 1 = κ 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIYaaabeaaki abg2da9iaadsgacqaH4oqCcaGGVaGaamizaiaadohacaGGSaGaaGPa VlaaykW7caaMc8UaeqOUdS2aaSbaaSqaaiaaigdaaeqaaOGaeyypa0 JaeqOUdS2aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaaa@46C4@ , and recall that M 2 =E I 2 κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiabeQ7aRnaa BaaaleaacaaIYaaabeaaaaa@3876@ , so that the angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  satisfies E I 2 d 2 θ d s 2 +Psinθ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaeqiUdeha baGaamizaiaadohadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaam iuaiGacohacaGGPbGaaiOBaiabeI7aXjabg2da9iaaicdaaaa@418D@
  4. This is the equation that governs oscillations of a pendulum, and its solution is well known. The equation is satisfied trivially by θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaicdaaaa@33FE@  (this is the straight configuration), and also by two one-parameter families of functions of the form

θ=2 sin 1 { ksn( (s s 0 ) (P/E I 2 ) ;k ) }θ=2 sin 1 { sn( (s s 0 ) (P/ k 2 E I 2 ) ;k ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaikdaciGGZbGaai yAaiaac6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaGadaqaaiaa dUgacaGGZbGaaiOBamaabmaabaGaaiikaiaadohacqGHsislcaWGZb WaaSbaaSqaaiaaicdaaeqaaOGaaiykamaakaaabaGaaiikaiaadcfa caGGVaGaamyraiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGPaaale qaaOGaai4oaiaadUgaaiaawIcacaGLPaaaaiaawUhacaGL9baacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cqaH4oqCcqGH9aqpcaaIYaGaci4Caiaa cMgacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaiWaaeaaca GGZbGaaiOBamaabmaabaGaaiikaiaadohacqGHsislcaWGZbWaaSba aSqaaiaaicdaaeqaaOGaaiykamaakaaabaGaaiikaiaadcfacaGGVa Gaam4AamaaCaaaleqabaGaaGOmaaaakiaadweacaWGjbWaaSbaaSqa aiaaikdaaeqaaOGaaiykaaWcbeaakiaacUdacaWGRbaacaGLOaGaay zkaaaacaGL7bGaayzFaaaaaa@7A4D@

Here, s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohadaWgaaWcbaGaaGimaaqabaaaaa@3266@  and 0<k<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacqGH8aapcaWGRbGaeyipaWJaaG ymaaaa@34F5@  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 dθ/ds=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH4oqCcaGGVaGaamizaiaado hacqGH9aqpcaaIWaaaaa@377B@ .  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 M 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaaIWaaaaa@340C@  at both ends of the rod, which requires that dθ/ds=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH4oqCcaGGVaGaamizaiaado hacqGH9aqpcaaIWaaaaa@377B@  at s=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaaIWaaaaa@3340@  and s=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaWGmbaaaa@3357@ .  Only the inflexional solution can satisfy these boundary conditions.   For this case, we have

dθ ds =2k I 2 P cn( (s s 0 ) (P/E I 2 ) ;k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiabeI7aXbqaaiaads gacaWGZbaaaiabg2da9iaaikdacaWGRbWaaOaaaeaadaWcaaqaaiaa dMeadaWgaaWcbaGaaGOmaaqabaaakeaacaWGqbaaaaWcbeaakiaaco gacaGGUbWaaeWaaeaacaGGOaGaam4CaiabgkHiTiaadohadaWgaaWc baGaaGimaaqabaGccaGGPaWaaOaaaeaacaGGOaGaamiuaiaac+caca WGfbGaamysamaaBaaaleaacaaIYaaabeaakiaacMcaaSqabaGccaGG 7aGaam4AaaGaayjkaiaawMcaaaaa@4A5B@

The cosine amplitude cn is a periodic function (it is a generalized cosine) and satisfies cn(x,k)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacaWGUbGaaiikaiaadIhacaGGSa Gaam4AaiaacMcacqGH9aqpcaaIWaaaaa@3819@  at x=(2n+1)K(k) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhacqGH9aqpcaGGOaGaaGOmaiaad6 gacqGHRaWkcaaIXaGaaiykaiaadUeacaGGOaGaam4AaiaacMcaaaa@3A49@ .  We may therefore satisfy the boundary conditions by choosing s 0 P/E I 2 =K(k) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaadohadaWgaaWcbaGaaGimaa qabaGcdaGcaaqaaiaadcfacaGGVaGaamyraiaadMeadaWgaaWcbaGa aGOmaaqabaaabeaakiabg2da9iaadUeacaGGOaGaam4AaiaacMcaaa a@3B9E@  and  L P/E I 2 =2nK(k) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaGcaaqaaiaadcfacaGGVaGaam yraiaadMeadaWgaaWcbaGaaGOmaaqabaaabeaakiabg2da9iaaikda caWGUbGaam4saiaacIcacaWGRbGaaiykaaaa@3B49@ .  This leads to the defining equations for k n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaaaaa@3297@ .

  1. Finally, the formula for the coordinates follows by integrating d y 3 /ds=cosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaaiodaae qaaOGaai4laiaadsgacaWGZbGaeyypa0Jaci4yaiaac+gacaGGZbGa eqiUdehaaa@3B85@  and d y 2 /ds=sinθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaaikdaae qaaOGaai4laiaadsgacaWGZbGaeyypa0Jaci4CaiaacMgacaGGUbGa eqiUdehaaa@3B89@  subject to boundary conditions y 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3437@  at s=0,L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaaIWaGaaiilaiaadY eaaaa@34C1@  and y 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaaIWaaaaa@3439@  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 μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@323E@ .  The cross-section of the rod has area A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeaaaa@314E@ , principal in-plane moments of inertia I 1 = I 2 =I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamysaaaa @36E1@  and an effective torsional inertia J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaG4maaqabaaaaa@3240@ .  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 n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gaaaa@317B@  in the helix

3.      The helix angle α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@ , which is related to n by tanα=1/(2πrn) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacshacaGGHbGaaiOBaiabeg7aHjabg2 da9iaaigdacaGGVaGaaiikaiaaikdacqaHapaCcaWGYbGaamOBaiaa cMcaaaa@3D28@

4.      The twist curvature κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@ , which quantify the distorsion induced by twisting the rod about its deformed axis.  For the rod to be in equilibrium, κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@  must be constant.

5.      The stretch ratio λ=ds/d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iaadsgacaWGZbGaai 4laiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@38A5@ .  For the rod to be in equilibrium, λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@323C@  must be constant.

 

The geometry and forces in the deformed rod are most conveniently described using a cylindrical-polar coordinate system (r, θ ^ ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGYbGaaiilaiqbeI7aXzaaja GaaiilaiaadQhacaGGPaaaaa@36FD@  and basis { e r , e θ , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaacmaabaGaaCyzamaaBaaaleaacaWGYb aabeaakiaacYcacaWHLbWaaSbaaSqaaiabeI7aXbqabaGccaGGSaGa aCyzamaaBaaaleaacaaIZaaabeaaaOGaay5Eaiaaw2haaaaa@3AEF@  shown in the figure.  In terms of these basis vectors, we may define

  1. The tangent vector to the rod t= m 3 =cosα e θ +sinα e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshacqGH9aqpcaWHTbWaaSbaaSqaai aaiodaaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqySdeMaaCyz amaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaci4CaiaacMgacaGGUb GaeqySdeMaaCyzamaaBaaaleaacaaIZaaabeaaaaa@43F6@
  2. The binormal vector is b=sinα e θ +cosα e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqGH9aqpcqGHsislciGGZbGaai yAaiaac6gacqaHXoqycaWHLbWaaSbaaSqaaiabeI7aXbqabaGccqGH RaWkciGGJbGaai4BaiaacohacqaHXoqycaWHLbWaaSbaaSqaaiaaio daaeqaaaaa@41E2@

 

In terms of these variables:

  1. The internal moment in the rod is M=EI cos 2 α r b+μ J 3 κ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWGfbGaamysamaala aabaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqyS degabaGaamOCaaaacaWHIbGaey4kaSIaeqiVd0MaamOsamaaBaaale aacaaIZaaabeaakiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaah2ga daWgaaWcbaGaaG4maaqabaaaaa@4431@
  2. The internal force in the rod is T= cos 2 α r ( μ J 3 κ 3 EI r cosαsinα )b+EA(λ1) m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpdaWcaaqaaiGacogaca GGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHbqaaiaadkha aaWaaeWaaeaacqaH8oqBcaWGkbWaaSbaaSqaaiaaiodaaeqaaOGaeq OUdS2aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0YaaSaaaeaacaWGfbGa amysaaqaaiaadkhaaaGaci4yaiaac+gacaGGZbGaeqySdeMaci4Cai aacMgacaGGUbGaeqySdegacaGLOaGaayzkaaGaaCOyaiabgUcaRiaa dweacaWGbbGaaiikaiabeU7aSjabgkHiTiaaigdacaGGPaGaaCyBam aaBaaaleaacaaIZaaabeaaaaa@56E3@

For the limiting case of an inextensible rod, the quantity EA(λ1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGbbGaaiikaiabeU7aSjabgk HiTiaaigdacaGGPaaaaa@36CD@  should be replaced by an indeterminate axial force T 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaG4maaqabaaaaa@324A@ .

 

The forces acting on the ends of the rod must satisfy P=T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahcfacqGH9aqpcaWHubaaaa@3344@  and Q=M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacqGH9aqpcaWHnbaaaa@333E@  at s=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaWGmbaaaa@3357@  and P=T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahcfacqGH9aqpcqGHsislcaWHubaaaa@3431@   Q=M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacqGH9aqpcqGHsislcaWHnbaaaa@342B@  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 P= F z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahcfacqGH9aqpcaWGgbWaaSbaaSqaai aadQhaaeqaaOGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@363E@  and moment Q=r F z e θ + Q z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacqGH9aqpcaWGYbGaamOramaaBa aaleaacaWG6baabeaakiaahwgadaWgaaWcbaGaeqiUdehabeaakiab gUcaRiaadgfadaWgaaWcbaGaamOEaaqabaGccaWHLbWaaSbaaSqaai aadQhaaeqaaaaa@3D3F@  acting at s=L (with equal and opposite forces at s=0), where

F z = cosα r (μ J 3 κ 3 EI r cosαsinα) Q z =μ J 3 κ 3 sinα+ EI r cos 3 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamOEaaqabaGccq GH9aqpdaWcaaqaaiGacogacaGGVbGaai4Caiabeg7aHbqaaiaadkha aaGaaiikaiabeY7aTjaadQeadaWgaaWcbaGaaG4maaqabaGccqaH6o WAdaWgaaWcbaGaaG4maaqabaGccqGHsisldaWcaaqaaiaadweacaWG jbaabaGaamOCaaaaciGGJbGaai4BaiaacohacqaHXoqyciGGZbGaai yAaiaac6gacqaHXoqycaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWGrbWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0JaeqiVd0MaamOsamaaBaaaleaacaaIZaaabeaakiabeQ7a RnaaBaaaleaacaaIZaaabeaakiGacohacaGGPbGaaiOBaiabeg7aHj abgUcaRmaalaaabaGaamyraiaadMeaaeaacaWGYbaaaiGacogacaGG VbGaai4CamaaCaaaleqabaGaaG4maaaakiabeg7aHbaa@6FD6@

This force system is statically equivalent to a wrench with force F z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamOEaaqabaGcca WHLbWaaSbaaSqaaiaadQhaaeqaaaaa@34A1@  and moment Q z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfadaWgaaWcbaGaamOEaaqabaGcca WHLbWaaSbaaSqaaiaadQhaaeqaaaaa@34AC@  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.

 

 

Derivation

 

  1. We take θ ^ =z=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeI7aXzaajaGaeyypa0JaamOEaiabg2 da9iaaicdaaaa@3613@  at s=0, so that the cylindrical polar coordinates are related to arc-length by θ ^ = s r cosαz=ssinα MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeI7aXzaajaGaeyypa0ZaaSaaaeaaca WGZbaabaGaamOCaaaaciGGJbGaai4BaiaacohacqaHXoqycaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadQhacq GH9aqpcaWGZbGaci4CaiaacMgacaGGUbGaeqySdegaaa@4D91@ . Note also that the basis vectors satisfy d e r /d θ ^ = e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHLbWaaSbaaSqaaiaadkhaae qaaOGaai4laiaadsgacuaH4oqCgaqcaiabg2da9iaahwgadaWgaaWc baGaeqiUdehabeaaaaa@3AC4@ , d e θ /d θ ^ = e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHLbWaaSbaaSqaaiabeI7aXb qabaGccaGGVaGaamizaiqbeI7aXzaajaGaeyypa0JaeyOeI0IaaCyz amaaBaaaleaacaWGYbaabeaaaaa@3BB1@ , so that

d e r ds = cosα r e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaahwgadaWgaaWcba GaamOCaaqabaaakeaacaWGKbGaam4CaaaacqGH9aqpdaWcaaqaaiGa cogacaGGVbGaai4Caiabeg7aHbqaaiaadkhaaaGaaCyzamaaBaaale aacqaH4oqCaeqaaaaa@3ECC@        d e θ ds = cosα r e r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaahwgadaWgaaWcba GaeqiUdehabeaaaOqaaiaadsgacaWGZbaaaiabg2da9iabgkHiTmaa laaabaGaci4yaiaac+gacaGGZbGaeqySdegabaGaamOCaaaacaWHLb WaaSbaaSqaaiaadkhaaeqaaaaa@3FB9@

  1. The position vector of a point on the axis of the rod can be expressed as r=r e r +z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkhacqGH9aqpcaWGYbGaaCyzamaaBa aaleaacaWGYbaabeaakiabgUcaRiaadQhacaWHLbWaaSbaaSqaaiaa iodaaeqaaaaa@3953@ ;
  2. The tangent vector follows as m 3 = dr ds =cosα e θ +sinα e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaG4maaqabaGccq GH9aqpdaWcaaqaaiaadsgacaWHYbaabaGaamizaiaadohaaaGaeyyp a0Jaci4yaiaac+gacaGGZbGaeqySdeMaaCyzamaaBaaaleaacqaH4o qCaeqaaOGaey4kaSIaci4CaiaacMgacaGGUbGaeqySdeMaaCyzamaa BaaaleaacaaIZaaabeaaaaa@46CE@ ;
  3. By definition, the curvature vector is κ= m 3 × d m 3 ds + κ 3 m 3 = cos 2 αsinα r e θ + cos 3 α r e 3 + κ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahQ7acqGH9aqpcaWHTbWaaSbaaSqaai aaiodaaeqaaOGaey41aq7aaSaaaeaacaWGKbGaaCyBamaaBaaaleaa caaIZaaabeaaaOqaaiaadsgacaWGZbaaaiabgUcaRiabeQ7aRnaaBa aaleaacaaIZaaabeaakiaah2gadaWgaaWcbaGaaG4maaqabaGccqGH 9aqpcqGHsisldaWcaaqaaiGacogacaGGVbGaai4CamaaCaaaleqaba GaaGOmaaaakiabeg7aHjGacohacaGGPbGaaiOBaiabeg7aHbqaaiaa dkhaaaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSYaaSaaae aaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaiodaaaGccqaHXoqy aeaacaWGYbaaaiaahwgadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcq aH6oWAdaWgaaWcbaGaaG4maaqabaGccaWHTbWaaSbaaSqaaiaaioda aeqaaaaa@5F4D@ , which can be expressed in terms of the binormal vector as κ=( cos 2 α/r)b+ κ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahQ7acqGH9aqpcaGGOaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaai4laiaadkha caGGPaGaaCOyaiabgUcaRiabeQ7aRnaaBaaaleaacaaIZaaabeaaki aah2gadaWgaaWcbaGaaG4maaqabaaaaa@418D@ ;
  4. The moment-curvature relations then give the internal moment M=EI( cos 2 α/r)b+μ J 3 κ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWGfbGaamysaiaacI caciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaHXoqy caGGVaGaamOCaiaacMcacaWHIbGaey4kaSIaeqiVd0MaamOsamaaBa aaleaacaaIZaaabeaakiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaa h2gadaWgaaWcbaGaaG4maaqabaaaaa@462D@ ;
  5. The equilibrium equation dT/ds+p=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHubGaai4laiaadsgacaWGZb Gaey4kaSIaaCiCaiabg2da9iaahcdaaaa@387C@  shows that T=constant.  We may express this constant internal force vector in terms of its components as T r e r + T θ e θ + T z e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaamOCaaqabaGcca WHLbWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaamivamaaBaaaleaa cqaH4oqCaeqaaOGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaS IaamivamaaBaaaleaacaWG6baabeaakiaahwgadaWgaaWcbaGaaG4m aaqabaaaaa@3FF1@
  6. The internal forces and moments must satisfy the moment equilibrium equation, which shows that

dM ds + m 3 ×T= cos 2 α r ( EI r cosαsinα J 3 κ 3 ) e r +μ J 3 d κ 3 ds (cosα e θ +sinα e 3 ) +( T z cosα T θ sinα ) e r + T r sinα e θ T r cosα e 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbGaaCytaaqaai aadsgacaWGZbaaaiabgUcaRiaah2gadaWgaaWcbaGaaG4maaqabaGc cqGHxdaTcaWHubGaeyypa0ZaaSaaaeaaciGGJbGaai4Baiaacohada ahaaWcbeqaaiaaikdaaaGccqaHXoqyaeaacaWGYbaaamaabmaabaWa aSaaaeaacaWGfbGaamysaaqaaiaadkhaaaGaci4yaiaac+gacaGGZb GaeqySdeMaci4CaiaacMgacaGGUbGaeqySdeMaeyOeI0IaamOsamaa BaaaleaacaaIZaaabeaakiabeQ7aRnaaBaaaleaacaaIZaaabeaaaO GaayjkaiaawMcaaiaahwgadaWgaaWcbaGaamOCaaqabaGccqGHRaWk cqaH8oqBcaWGkbWaaSbaaSqaaiaaiodaaeqaaOWaaSaaaeaacaWGKb GaeqOUdS2aaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadohaaaGa aiikaiGacogacaGGVbGaai4Caiabeg7aHjaahwgadaWgaaWcbaGaeq iUdehabeaakiabgUcaRiGacohacaGGPbGaaiOBaiabeg7aHjaahwga daWgaaWcbaGaaG4maaqabaGccaGGPaaabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkdaqadaqa aiaadsfadaWgaaWcbaGaamOEaaqabaGcciGGJbGaai4Baiaacohacq aHXoqycqGHsislcaWGubWaaSbaaSqaaiabeI7aXbqabaGcciGGZbGa aiyAaiaac6gacqaHXoqyaiaawIcacaGLPaaacaWHLbWaaSbaaSqaai aadkhaaeqaaOGaey4kaSIaamivamaaBaaaleaacaWGYbaabeaakiGa cohacaGGPbGaaiOBaiabeg7aHjaahwgadaWgaaWcbaGaeqiUdehabe aakiabgkHiTiaadsfadaWgaaWcbaGaamOCaaqabaGcciGGJbGaai4B aiaacohacqaHXoqycaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0 JaaCimaaaaaa@D758@

Taking the dot product of both sides of this equation with m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaG4maaqabaaaaa@3267@  shows that d κ 3 /ds=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH6oWAdaWgaaWcbaGaaG4maa qabaGccaGGVaGaamizaiaadohacqGH9aqpcaaIWaaaaa@386A@ . It then follows that T r =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaamOCaaqabaGccq GH9aqpcaaIWaaaaa@344E@  and

cos 2 α r ( μ J 3 κ 3 EI r cosαsinα )=( T z cosα T θ sinα ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaci4yaiaac+gacaGGZbWaaW baaSqabeaacaaIYaaaaOGaeqySdegabaGaamOCaaaadaqadaqaaiab eY7aTjaadQeadaWgaaWcbaGaaG4maaqabaGccqaH6oWAdaWgaaWcba GaaG4maaqabaGccqGHsisldaWcaaqaaiaadweacaWGjbaabaGaamOC aaaaciGGJbGaai4BaiaacohacqaHXoqyciGGZbGaaiyAaiaac6gacq aHXoqyaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadsfadaWgaaWc baGaamOEaaqabaGcciGGJbGaai4BaiaacohacqaHXoqycqGHsislca WGubWaaSbaaSqaaiabeI7aXbqabaGcciGGZbGaaiyAaiaac6gacqaH XoqyaiaawIcacaGLPaaaaaa@5C47@

  1. Finally, the force-stretch relation requires that T m 3 = T θ cosα+ T z sinα=EA(λ1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGHflY1caWHTbWaaSbaaSqaai aaiodaaeqaaOGaeyypa0JaamivamaaBaaaleaacqaH4oqCaeqaaOGa ci4yaiaac+gacaGGZbGaeqySdeMaey4kaSIaamivamaaBaaaleaaca WG6baabeaakiGacohacaGGPbGaaiOBaiabeg7aHjabg2da9iaadwea caWGbbGaaiikaiabeU7aSjabgkHiTiaaigdacaGGPaaaaa@4C87@ .  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 μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@323E@ .  The cross-section of the rod has principal in-plane moments of inertia I 1 = I 2 =I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamysaaaa @36E1@  and an effective torsional inertia J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaG4maaqabaaaaa@3240@ .  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 r ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadkhagaqeaaaa@3197@  of the cylinder that generates the helix; the height h ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadIgagaqeaaaa@318D@  of the spring, the number of turns in the coil N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eaaaa@315B@ , and the helix angle α ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg7aHzaaraaaaa@323F@
  2. The variables characterizing the undeformed spring are related as follows

tan α ¯ = h ¯ 2π r ¯ N h ¯ =Lsin α ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacshacaGGHbGaaiOBaiqbeg7aHzaara Gaeyypa0ZaaSaaaeaaceWGObGbaebaaeaacaaIYaGaeqiWdaNabmOC ayaaraGaamOtaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UabmiAayaaraGaeyypa0JaamitaiGacohacaGGPbGaaiOB aiqbeg7aHzaaraaaaa@4DBE@

  1. It is helpful to select a basis { m ¯ 1 , m ¯ 2 , m ¯ 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaacmaabaGabCyBayaaraWaaSbaaSqaai aaigdaaeqaaOGaaiilaiqah2gagaqeamaaBaaaleaacaaIYaaabeaa kiaacYcaceWHTbGbaebadaWgaaWcbaGaaG4maaqabaaakiaawUhaca GL9baaaaa@3A19@  to characterize the orientation of the initial spring. Since I 1 = I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaikdaaeqaaaaa@3503@  we may select m ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIXaaabe aaaaa@327D@  and m ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIYaaabe aaaaa@327E@  arbitrarily. It is convenient to choose m ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIXaaabe aaaaa@327D@  and m ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIYaaabe aaaaa@327E@  to be parallel to the normal vector n and binormal vector b of the undeformed spring, respectively, which gives m ¯ 1 = e r , m ¯ 2 =sin α ¯ e θ +cos α ¯ e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaqeamaaBaaaleaacaaIXaaabe aakiabg2da9iabgkHiTiaahwgadaWgaaWcbaGaamOCaaqabaGccaGG SaGaaGPaVlaaykW7caaMc8UaaGPaVlqah2gagaqeamaaBaaaleaaca aIYaaabeaakiabg2da9iabgkHiTiGacohacaGGPbGaaiOBaiqbeg7a HzaaraGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaey4kaSIaci4yai aac+gacaGGZbGafqySdeMbaebacaWHLbWaaSbaaSqaaiaaiodaaeqa aaaa@5010@
  2. The initial curvature components can be calculated from the condition that  d m i /ds=κ× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHTbWaaSbaaSqaaiaadMgaae qaaOGaai4laiaadsgacaWGZbGaeyypa0JaaCOUdiabgEna0kaah2ga daWgaaWcbaGaamyAaaqabaaaaa@3C92@  and follow as κ ¯ 1 =0, κ ¯ 2 = cos 2 α ¯ / r ¯ , κ ¯ 3 =sin α ¯ cos α ¯ / r ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeQ7aRzaaraWaaSbaaSqaaiaaigdaae qaaOGaeyypa0JaaGimaiaacYcacaaMc8UaaGPaVlaaykW7cuaH6oWA gaqeamaaBaaaleaacaaIYaaabeaakiabg2da9iGacogacaGGVbGaai 4CamaaCaaaleqabaGaaGOmaaaakiqbeg7aHzaaraGaai4laiqadkha gaqeaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UafqOUdSMbaebada WgaaWcbaGaaG4maaqabaGccqGH9aqpciGGZbGaaiyAaiaac6gacuaH XoqygaqeaiGacogacaGGVbGaai4Caiqbeg7aHzaaraGaai4laiqadk hagaqeaaaa@5ACF@ .

 

The end of the spring at s=0 is held fixed (so it cannot move or rotate).  The end of the spring at s=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaWGmbaaaa@3357@  is subjected to a combination of a force F z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamOEaaqabaGcca WHLbWaaSbaaSqaaiaadQhaaeqaaaaa@34A1@  and moment Q z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfadaWgaaWcbaGaamOEaaqabaGcca WHLbWaaSbaaSqaaiaadQhaaeqaaaaa@34AC@  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, α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  and h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIgaaaa@3175@  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  ( r ¯ , θ ^ 0 , h ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcaceWGYbGbaebacaGGSaGafqiUde NbaKaadaWgaaWcbaGaaGimaaqabaGccaGGSaGabmiAayaaraGaaiyk aaaa@380B@ ; after deformation, they are (r, θ ^ ,h) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGYbGaaiilaiqbeI7aXzaaja GaaiilaiaadIgacaGGPaaaaa@36EB@ .  The twisting can be characterized by the rotation Δω= θ ^ θ ^ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3jabg2da9iqbeI7aXz aajaGaeyOeI0IafqiUdeNbaKaadaWgaaWcbaGaaGimaaqabaaaaa@3A20@   The point where the load is applied therefore displaces through a distance Δh=h h ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadIgacqGH9aqpcaWGObGaey OeI0IabmiAayaaraaaaa@36C0@ , and rotates through the angle Δω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3baa@33BB@  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 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  by

Δω=L( cosα r cos α ¯ r ¯ )Δh=L(sinαsin α ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3jabg2da9iaadYeada qadaqaamaalaaabaGaci4yaiaac+gacaGGZbGaeqySdegabaGaamOC aaaacqGHsisldaWcaaqaaiGacogacaGGVbGaai4Caiqbeg7aHzaara aabaGabmOCayaaraaaaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqqHuoarcaWGObGaeyyp a0JaamitaiaacIcaciGGZbGaaiyAaiaac6gacqaHXoqycqGHsislci GGZbGaaiyAaiaac6gacuaHXoqygaqeaiaacMcaaaa@6B50@

  1. The vectors m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaaaaa@3265@  and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaaaaa@3266@  after deformation are given by m 1 = e r , m 2 =sinα e θ +cosα e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqGHsislcaWHLbWaaSbaaSqaaiaadkhaaeqaaOGaaiilaiaa ykW7caaMc8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaeyOeI0Iaci4CaiaacMgacaGGUbGaeqySdeMaaCyzamaa BaaaleaacqaH4oqCaeqaaOGaey4kaSIaci4yaiaac+gacaGGZbGaeq ySdeMaaCyzamaaBaaaleaacaaIZaaabeaaaaa@4FB0@ .
  2. The curvatures after deformation follow as κ 1 =0, κ 2 = cos 2 α/r, κ 3 =sinαcosα/r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGSaGaaGPaVlaaykW7caaMc8UaeqOUdS2aaSba aSqaaiaaikdaaeqaaOGaeyypa0Jaci4yaiaac+gacaGGZbWaaWbaaS qabeaacaaIYaaaaOGaeqySdeMaai4laiaadkhacaGGSaGaaGPaVlaa ykW7caaMc8UaaGPaVlabeQ7aRnaaBaaaleaacaaIZaaabeaakiabg2 da9iGacohacaGGPbGaaiOBaiabeg7aHjGacogacaGGVbGaai4Caiab eg7aHjaac+cacaWGYbaaaa@5A0F@
  3. The internal moment and force in the spring are related to the curvatures and external force and moment by

M=EI( κ 2 κ ¯ 2 ) m 2 +μ J 3 ( κ 3 κ ¯ 3 ) m 3 = Q z e z +r F z e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWGfbGaamysaiaacI cacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcuaH6oWAgaqe amaaBaaaleaacaaIYaaabeaakiaacMcacaWHTbWaaSbaaSqaaiaaik daaeqaaOGaey4kaSIaeqiVd0MaamOsamaaBaaaleaacaaIZaaabeaa kiaacIcacqaH6oWAdaWgaaWcbaGaaG4maaqabaGccqGHsislcuaH6o WAgaqeamaaBaaaleaacaaIZaaabeaakiaacMcacaWHTbWaaSbaaSqa aiaaiodaaeqaaOGaeyypa0JaamyuamaaBaaaleaacaWG6baabeaaki aahwgadaWgaaWcbaGaamOEaaqabaGccqGHRaWkcaWGYbGaamOramaa BaaaleaacaWG6baabeaakiaahwgadaWgaaWcbaGaeqiUdehabeaaaa a@5752@

T={ μ J 3 cosα r ( κ 3 κ ¯ 3 )EI sinα r ( κ 2 κ ¯ 2 ) } e z = F z e z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpdaGadaqaaiabeY7aTj aadQeadaWgaaWcbaGaaG4maaqabaGcdaWcaaqaaiGacogacaGGVbGa ai4Caiabeg7aHbqaaiaadkhaaaGaaiikaiabeQ7aRnaaBaaaleaaca aIZaaabeaakiabgkHiTiqbeQ7aRzaaraWaaSbaaSqaaiaaiodaaeqa aOGaaiykaiabgkHiTiaadweacaWGjbWaaSaaaeaaciGGZbGaaiyAai aac6gacqaHXoqyaeaacaWGYbaaaiaacIcacqaH6oWAdaWgaaWcbaGa aGOmaaqabaGccqGHsislcuaH6oWAgaqeamaaBaaaleaacaaIYaaabe aakiaacMcaaiaawUhacaGL9baacaWHLbWaaSbaaSqaaiaadQhaaeqa aOGaeyypa0JaamOramaaBaaaleaacaWG6baabeaakiaahwgadaWgaa WcbaGaamOEaaqabaaaaa@5C20@

  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

F z =μ J 3 cosα r ( κ 3 κ ¯ 3 )EI sinα r ( κ 2 κ ¯ 2 ) Q z =μ J 3 sinα( κ 3 κ ¯ 3 )+EIcosα( κ 2 κ ¯ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamOEaaqabaGccq GH9aqpcqaH8oqBcaWGkbWaaSbaaSqaaiaaiodaaeqaaOWaaSaaaeaa ciGGJbGaai4BaiaacohacqaHXoqyaeaacaWGYbaaaiaacIcacqaH6o WAdaWgaaWcbaGaaG4maaqabaGccqGHsislcuaH6oWAgaqeamaaBaaa leaacaaIZaaabeaakiaacMcacqGHsislcaWGfbGaamysamaalaaaba Gaci4CaiaacMgacaGGUbGaeqySdegabaGaamOCaaaacaGGOaGaeqOU dS2aaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaa WcbaGaaGOmaaqabaGccaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caWGrbWaaSbaaSqaaiaadQhaaeqaaO Gaeyypa0JaeqiVd0MaamOsamaaBaaaleaacaaIZaaabeaakiGacoha caGGPbGaaiOBaiabeg7aHjaacIcacqaH6oWAdaWgaaWcbaGaaG4maa qabaGccqGHsislcuaH6oWAgaqeamaaBaaaleaacaaIZaaabeaakiaa cMcacqGHRaWkcaWGfbGaamysaiGacogacaGGVbGaai4Caiabeg7aHj aacIcacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcuaH6oWA gaqeamaaBaaaleaacaaIYaaabeaakiaacMcaaaa@8162@

8.      If the spring is subjected a prescribed force and moment, these equations can be solved for α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  and r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhaaaa@317F@ , and the results can be substituted into (3) to calculate the extension Δh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadIgaaaa@32DB@  and rotation Δω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3baa@33BB@  of the spring.  The results cannot be expressed in closed form for large shape changes.  If Δh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadIgaaaa@32DB@  and Δω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3baa@33BB@  are small, however, the relations can be linearized to yield

F z = k FF Δh+ k FQ Δω Q z = k QQ Δω+ k QF Δh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamOEaaqabaGccq GH9aqpcaWGRbWaaSbaaSqaaiaadAeacaWGgbaabeaakiabfs5aejaa dIgacqGHRaWkcaWGRbWaaSbaaSqaaiaadAeacaWGrbaabeaakiabfs 5aejabeM8a3jaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam yuamaaBaaaleaacaWG6baabeaakiabg2da9iaadUgadaWgaaWcbaGa amyuaiaadgfaaeqaaOGaeuiLdqKaeqyYdCNaey4kaSIaam4AamaaBa aaleaacaWGrbGaamOraaqabaGccqqHuoarcaWGObaaaa@6425@

where the spring stiffnesses are

k FF = sin α ¯ h ¯ r ¯ 2 ( μ J 3 cos 2 α ¯ +EI sin 2 α ¯ ) k QQ = sin α ¯ h ¯ ( μ J 3 sin 2 α ¯ +EI cos 2 α ¯ ) k FQ = k QF = sin 3 α ¯ cos α ¯ h ¯ r ¯ ( μ J 3 EI ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4AamaaBaaaleaacaWGgbGaam OraaqabaGccqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiqbeg7a HzaaraaabaGabmiAayaaraGabmOCayaaraWaaWbaaSqabeaacaaIYa aaaaaakmaabmaabaGaeqiVd0MaamOsamaaBaaaleaacaaIZaaabeaa kiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiqbeg7aHz aaraGaey4kaSIaamyraiaadMeaciGGZbGaaiyAaiaac6gadaahaaWc beqaaiaaikdaaaGccuaHXoqygaqeaaGaayjkaiaawMcaaiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadUgadaWgaaWcba GaamyuaiaadgfaaeqaaOGaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaa c6gacuaHXoqygaqeaaqaaiqadIgagaqeaaaadaqadaqaaiabeY7aTj aadQeadaWgaaWcbaGaaG4maaqabaGcciGGZbGaaiyAaiaac6gadaah aaWcbeqaaiaaikdaaaGccuaHXoqygaqeaiabgUcaRiaadweacaWGjb Gaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGafqySdeMb aebaaiaawIcacaGLPaaaaeaacaWGRbWaaSbaaSqaaiaadAeacaWGrb aabeaakiabg2da9iaadUgadaWgaaWcbaGaamyuaiaadAeaaeqaaOGa eyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaio daaaGccuaHXoqygaqeaiGacogacaGGVbGaai4Caiqbeg7aHzaaraaa baGabmiAayaaraGabmOCayaaraaaamaabmaabaGaeqiVd0MaamOsam aaBaaaleaacaaIZaaabeaakiabgkHiTiaadweacaWGjbaacaGLOaGa ayzkaaGaaGPaVdaaaa@9E68@