Chapter 10

 

Approximate theories for solids with special shapes:

 rods, beams, membranes, plates and shells

 

 

 

10.3 Simplified versions of the general theory of deformable rods

 

In many practical cases of interest the general equations governing deformation and deformation of generally curved rods can be vastly simplified.  In this section, we summarize the governing equations for a number of special solids, including flexible strings, and various forms of beam theory.

 

 

10.3.1 Stretched flexible string with small transverse deflections.

 

This is the simplest possible version of the general theory outlined in 10.2.   The problem to be solved is illustrated in the figure.  A `string’ with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3152@ , mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@ , cross-sectional area A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeaaaa@314E@  and negligible area moments of inertia I 1 = I 2 = I 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamysamaa BaaaleaacaaIZaaabeaakiabg2da9iaaicdaaaa@3994@  is initially straight and parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@325F@  direction. The ends of the string are subjected to an axial load T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@3247@  and are prevented from moving transverse to the string.  A force per unit length p= p 1 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacqGH9aqpcaWGWbWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@3642@  acts on the string, inducing a small, time dependent, transverse deflection u= u 1 ( x 3 ) e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWG1bWaaSbaaSqaai aaigdaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGG PaGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@3995@ .

 

The general governing equations can be simplified to the following form:

  1. The curvature of the string can be approximated as κ d 2 u 1 d x 3 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahQ7acqGHijYUdaWcaaqaaiaadsgada ahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaaGc baGaamizaiaadIhadaqhaaWcbaGaaG4maaqaaiaaikdaaaaaaOGaaC yzamaaBaaaleaacaaIYaaabeaaaaa@3CC2@
  2. The stretch of the string can be approximated as ds d x 3 =1+ T 0 /EA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadohaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGH9aqpcaaIXaGaey4k aSIaamivamaaBaaaleaacaaIWaaabeaakiaac+cacaWGfbGaamyqaa aa@3C01@
  3. The only nonzero internal force is the axial force T 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaG4maaqabaaaaa@324A@  (the moment-curvature relations show that the internal moments vanish; the angular momentum balance equations show T 1 = T 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGubWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGimaaaa @36E3@  )
  4. The equations of motion reduce to κ 2 T 3 + p 1 =ρA a 1 d T 3 ds =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIYaaabeaaki aadsfadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGWbWaaSbaaSqa aiaaigdaaeqaaOGaeyypa0JaeqyWdiNaamyqaiaadggadaWgaaWcba GaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVpaalaaabaGaamizaiaadsfadaWgaaWcbaGaaG4maa qabaaakeaacaWGKbGaam4CaaaacqGH9aqpcaaIWaaaaa@4FE1@
  5. Combining 1-4 shows T 3 = T 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGubWaaSbaaSqaaiaaicdaaeqaaaaa@3519@  and gives the equation of motion for the stretched string

T 0 d 2 u 1 d x 3 2 + p 1 =ρA d 2 u 1 d t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaGcda WcaaqaaiaadsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqa aiaaigdaaeqaaaGcbaGaamizaiaadIhadaqhaaWcbaGaaG4maaqaai aaikdaaaaaaOGaey4kaSIaamiCamaaBaaaleaacaaIXaaabeaakiab g2da9iabeg8aYjaadgeadaWcaaqaaiaadsgadaahaaWcbeqaaiaaik daaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadsha daahaaWcbeqaaiaaikdaaaaaaaaa@46B4@

  1. The boundary conditions are u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3433@  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@

 

 

Large deflection equations for a flexible string can also be found by substituting M 1 = M 2 = M 3 = T 1 = T 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGnbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaamytamaa BaaaleaacaaIZaaabeaakiabg2da9iaadsfadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaWGubWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Ja aGimaaaa@3F41@  into the general equations of motion for a rod.  The details are left as an exercise.

 

 

 

10.3.2 Straight elastic beam with small deflections and no axial force (Euler-Bernoulli beam theory)

 

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 subjected to a transverse force per unit length p= p 1 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacqGH9aqpcaWGWbWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@3642@ .  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 inertia 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 suppose that the beam experiences a small transverse displacement u= u 1 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWG1bWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@364C@ , and wish to calculate u 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaaaaa@3269@  as a function of time, given appropriate initial conditions.

 

The general equations of motion for a rod can be used to deduce that:

  1. The stretch of the bar and the curvature vector may be approximated by  ds d x 3 1κ d 2 u 1 d x 3 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadohaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHijYUcaaIXaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaCOUdiabgIKi7oaalaaabaGaamizamaa CaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaaqabaaake aacaWGKbGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaaaaGccaWH LbWaaSbaaSqaaiaaikdaaeqaaaaa@567C@
  2. The internal forces can be characterized by the internal force T T 1 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGHijYUcaWGubWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@36B5@  and internal moment M= M 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWGnbWaaSbaaSqaai aaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@35FE@ .   These can be interpreted as the force and moment acting on an internal cross-section of the beam which has normal in the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@325F@  direction.
  3. Moment-curvature relations reduce to M 2 =E I 2 d 2 u 1 d x 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakmaalaaabaGa amizamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaa qabaaakeaacaWGKbGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaa aaaaaa@3D3F@
  4. Equations of motion can be reduced to d T 1 d x 3 + p 1 =ρA a 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadsfadaWgaaWcba GaaGymaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa aaGccqGHRaWkcaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaeq yWdiNaamyqaiaadggadaWgaaWcbaGaaGymaaqabaaaaa@3E45@     d M 2 d x 3 + T 1 =ρ I 2 α 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaad2eadaWgaaWcba GaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa aaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaeq yWdiNaamysamaaBaaaleaacaaIYaaabeaakiabeg7aHnaaBaaaleaa caaIYaaabeaakiabgIKi7kaaicdaaaa@424C@  

5.      Alternatively, the equations in (3) and (4) can be combined to express the equations of motion in terms of displacement

E I 2 d 4 u 1 d x 3 4 +ρA d 2 u 1 d t 2 = p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaI0aaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaI0aaaaaaakiabgUcaRiabeg8aYjaadgeadaWcaaqaaiaa dsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaae qaaaGcbaGaamizaiaadshadaahaaWcbeqaaiaaikdaaaaaaOGaeyyp a0JaamiCamaaBaaaleaacaaIXaaabeaaaaa@4779@

 

Boundary conditions: Elementary beam theory offers the following boundary conditions:

  1. The end of the beam may be clamped, i.e. rotations and displacement of the end are completely prevented. The transverse displacement must then satisfy u 1 = u 1 /d x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaa dsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaaa@3C16@ .
  2. The end of the beam may be simply supported, i.e. the end cannot move, but may rotate freely.  In this case the transverse displacement must satisfy u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3433@  and the internal moment must satisfy M 2 =E I 2 d 2 u 1 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaadsgadaah aaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai 4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeyyp a0JaaGimaaaa@3FAC@
  3. The end of the beam may be free, i.e. the end can move and rotate freely.  In this case the internal moment and internal force must satisfy M 2 =E I 2 d 2 u 1 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaadsgadaah aaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai 4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeyyp a0JaaGimaaaa@3FAC@ , T 1 =E I 2 d 3 u 1 /d x 3 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqGHsislcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaa dsgadaahaaWcbeqaaiaaiodaaaGccaWG1bWaaSbaaSqaaiaaigdaae qaaOGaai4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIZaaa aOGaeyypa0JaaGimaaaa@40A1@

 

 

 

 

10.3.3 Straight elastic beam with small transverse deflections and significant axial force

 

This version of beam theory is used to model beams that are subjected to substantial axial loads (usually due to forces applied at their ends).  The equations can be used to estimate the effects of axial load on the transverse deflection or vibration of a beam. The theory can also be used to calculate buckling loads for beams, but does not accurately model their deformation if the buckling loads are exceeded.

 

The problem to be solved is illustrated in the figure.   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 subjected to a force per unit length p= p 1 e 1 + p 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacqGH9aqpcaWGWbWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa dchadaWgaaWcbaGaaG4maaqabaGccaWHLbWaaSbaaSqaaiaaiodaae qaaaaa@3AED@ .  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 inertia 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 the preceding section. We assume that a large axial internal force N m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacaWHTbWaaSbaaSqaaiaaiodaae qaaaaa@333A@  is developed in the beam, either by a horizontal force per unit length p 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaG4maaqabaaaaa@3266@  or horizontal forces P 3 (0) e 3 , P 3 (L) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfadaqhaaWcbaGaaG4maaqaaiaacI cacaaIWaGaaiykaaaakiaahwgadaWgaaWcbaGaaG4maaqabaGccaGG SaGaamiuamaaDaaaleaacaaIZaaabaGaaiikaiaadYeacaGGPaaaaO GaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3CBF@  acting at the ends of the beam.  We suppose that the beam experiences a small transverse displacement u= u 1 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWG1bWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaaaaa@364C@ , and wish to calculate u 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaaaaa@3269@  as a function of time, given appropriate initial conditions.

 

The general equations of motion for a deformable rod can be approximated as follows:

  1. The stretch of the bar and the curvature vector may be approximated by  ds d x 3 1κ d 2 u 1 d x 3 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadohaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHijYUcaaIXaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaCOUdiabgIKi7oaalaaabaGaamizamaa CaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaaqabaaake aacaWGKbGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaaaaGccaWH LbWaaSbaaSqaaiaaikdaaeqaaaaa@567C@
  2. The internal forces can be characterized by the internal force T=V m 1 +N m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcaWGwbGaaCyBamaaBa aaleaacaaIXaaabeaakiabgUcaRiaad6eacaWHTbWaaSbaaSqaaiaa iodaaeqaaaaa@38C1@  and internal moment M= M 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWGnbWaaSbaaSqaai aaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@35FE@ .   These can be interpreted as the force and moment acting on an internal cross-section of the beam which has normal in the m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaG4maaqabaaaaa@3267@  direction. 
  3. Moment-curvature relations reduce to M 2 =E I 2 d 2 u 1 d x 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakmaalaaabaGa amizamaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaaGymaa qabaaakeaacaWGKbGaamiEamaaDaaaleaacaaIZaaabaGaaGOmaaaa aaaaaa@3D3F@
  4. Equations of motion may be approximated by

dV d x 3 +N d 2 u 1 d x 3 2 p 3 d u 1 d x 3 + p 1 =ρA a 1 dN d x 3 + p 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadAfaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHRaWkcaWGobWaaSaa aeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaamyDamaaBaaaleaaca aIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaI YaaaaaaakiabgkHiTiaadchadaWgaaWcbaGaaG4maaqabaGcdaWcaa qaaiaadsgacaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaa dIhadaWgaaWcbaGaaG4maaqabaaaaOGaey4kaSIaamiCamaaBaaale aacaaIXaaabeaakiabg2da9iabeg8aYjaadgeacaWGHbWaaSbaaSqa aiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8+aaSaaaeaacaWGKbGaamOtaaqaaiaadsga caWG4bWaaSbaaSqaaiaaiodaaeqaaaaakiabgUcaRiaadchadaWgaa WcbaGaaG4maaqabaGccqGH9aqpcaaIWaaaaa@6625@   d M 2 d x 3 +V=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaad2eadaWgaaWcba GaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa aaGccqGHRaWkcaWGwbGaeyypa0JaaGimaaaa@399B@

To interpret these equations, note that (i) the axial force N has been assumed to be much larger than the transverse force V, so that nonlinear terms associated with the axial force have been retained when approximating the equations of motion; (ii) Although the equations of motion are expressed in terms of components of displacement and external force in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3988@  basis, the equations of motion themselves represent an approximation to the components of the full, nonlinear equations of motion in the { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@39A0@  basis.

5.      The results of 1-4 can be combined to obtain an equation for the transverse deflection of the beam

E I 2 d 4 u 1 d x 3 4 +ρA d 2 u 1 d t 2 + p 3 d u 1 d x 3 =N d 2 u 1 d x 3 2 + p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaWGjbWaaSbaaSqaaiaaikdaae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaI0aaaaOGaamyDamaa BaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaaio daaeaacaaI0aaaaaaakiabgUcaRiabeg8aYjaadgeadaWcaaqaaiaa dsgadaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaae qaaaGcbaGaamizaiaadshadaahaaWcbeqaaiaaikdaaaaaaOGaey4k aSIaamiCamaaBaaaleaacaaIZaaabeaakmaalaaabaGaamizaiaadw hadaWgaaWcbaGaaGymaaqabaaakeaacaWGKbGaamiEamaaBaaaleaa caaIZaaabeaaaaGccqGH9aqpcaWGobWaaSaaaeaacaWGKbWaaWbaaS qabeaacaaIYaaaaOGaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiaa dsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaaaakiabgUcaRi aadchadaWgaaWcbaGaaGymaaqabaaaaa@5922@

 

 

 

Boundary conditions: Elementary beam theory offers the following boundary conditions:

  1. The end of the beam may be clamped, i.e. rotations and displacement of the end are completely prevented. The transverse displacement must then satisfy u 1 = u 1 /d x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaa dsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaaa@3C16@ .
  2. The end of the beam may be simply supported, i.e. the end cannot move, but may rotate freely.  In this case the transverse displacement must satisfy u 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3433@  and the internal moment must satisfy M 2 =E I 2 d 2 u 1 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaadsgadaah aaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai 4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeyyp a0JaaGimaaaa@3FAC@
  3. The end of the beam may be free, i.e. the end can move and rotate freely.  In this case the internal moment and internal force must satisfy M 2 =E I 2 d 2 u 1 /d x 3 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaWGfbGaamysamaaBaaaleaacaaIYaaabeaakiaadsgadaah aaWcbeqaaiaaikdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaai 4laiaadsgacaWG4bWaa0baaSqaaiaaiodaaeaacaaIYaaaaOGaeyyp a0JaaGimaaaa@3FAC@ , while the transverse force must satisfy

V={ P 3 (0) (d u 1 /d x 3 ) x 3 =0 P 3 (L) (d u 1 /d x 3 ) x 3 =L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfacqGH9aqpdaGabaqaauaabeqace aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaamiuamaaDaaaleaacaaI ZaaabaGaaiikaiaaicdacaGGPaaaaOGaaiikaiaadsgacaWG1bWaaS baaSqaaiaaigdaaeqaaOGaai4laiaadsgacaWG4bWaaSbaaSqaaiaa iodaaeqaaOGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaamiEamaaBaaaleaacaaIZaaabeaakiabg2da9iaaicdaaeaacq GHsislcaWGqbWaa0baaSqaaiaaiodaaeaacaGGOaGaamitaiaacMca aaGccaGGOaGaamizaiaadwhadaWgaaWcbaGaaGymaaqabaGccaGGVa GaamizaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadIhadaWg aaWcbaGaaG4maaqabaGccqGH9aqpcaWGmbaaaaGaay5Eaaaaaa@87AB@

4.      In addition, the axial force N must satisfy 

N= P 3 (0) ( x 3 =0)N= P 3 (L) ( x 3 =L) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacqGH9aqpcqGHsislcaWGqbWaa0 baaSqaaiaaiodaaeaacaGGOaGaaGimaiaacMcaaaGccaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaiikaiaadIhadaWgaa WcbaGaaG4maaqabaGccqGH9aqpcaaIWaGaaiykaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaad6eacqGH9aqpca WGqbWaa0baaSqaaiaaiodaaeaacaGGOaGaamitaiaacMcaaaGccaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaiikaiaadI hadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGmbGaaiykaaaa@81C1@