2.080 Structural Mechanics Lecture 5: Solution Method for ...

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Structural Lecture 5Semester YrLecture 5: Solution Method for Beam Governing EquationsSo far we have established three groups of equations fully characterizing the response ofbeams to different types of loading. In Lecture 2 relations were established to calculatestrains from the displacement field. (x,z) = (x) +z ( )where (x) =dudx+12(dwdx)2, = d2wdx2( )The above geometrical relation are independent on equilibrium and apply to any kind secondset of equations, derived in Lecture 3, is the equilibrium requirementdV dx+q(x) = 0 force equilibrium ( )dMdx V= 0 moment equilibrium ( )whereV =V+Ndwdxis the effective shear.( )dNdx= 0( )EliminatingVandV between the above equations, the beam equilibrium equation wasobtained (See Eq. ( ))d2Mdx2+Nd2wdx2+q= 0( )The derivation of the equilibrium is valid for all types of materials.

The above geometrical relation are independent on equilibrium and apply to any kind of materials. The second set of equations, derived in Lecture 3, is the equilibrium requirement dV dx + q(x) = 0 force equilibrium (5.3) dM dx V = 0 moment equilibrium (5.4) where V = V+ N dw dx is the e ective shear. (5.5) dN dx = 0 (5.6)

  Equilibrium