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

Structural Lecture 5 Semester 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.)

The general solution for u(x) is du dx = D 1, u= D 1x+ D 0 (5.13) There are two integration constants, and two boundary conditions are needed. There are only four combinations of boundary conditions: 1. Beam restricted from axial motion, see Fig. (5.1). u(x= 0) = u(x= l) = 0 (5.14) This gives rise to the solution of two algebraic equation 0 = D ...

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