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. ( ))d2 Mdx2+Nd2wdx2+q= 0( )The derivation of the equilibrium is valid for all types of materials.

Structural Mechanics 2.080 Lecture 5 Semester Yr We now turn our attention to the solution of the beam de ection, Eq. (5.11). This is the fourth-order linear inhomogeneous equation which requires four boundary conditions. There are four types of boundary conditions, de ned by (M M ) w0= 0 (5.22a) (V V ) w= 0 (5.22b)

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