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.)

p of the beam de ection equation, Eq. (5.11) depends on the loading, but not the boundary conditions. For the uniformly loaded beam the particular solution is the rst term in Eq. (5.23d). As an illustration, consider the same pin-pin supported beam loaded by the triangular line load q(x) = q 0 2x l, 0 <x< l 2 (5.33) where q

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