Transcription of 2.080 Structural Mechanics Lecture 5: Solution Method for ...
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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.)
discontinuous are Bending monents [M] = M (5.37a) Shear force [V] = V (5.37b) As an illustration, consider a pin-pin supported beam loaded at mid-span by a point force P. As mentioned earlier, the point load can be considered as a limiting case of a continuous line load with the help of the Dirac delta function q(x) = P (x l 2), where Z (x l 2 ...
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