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. ( ))d2 Mdx2+Nd2wdx2+q= 0( )The derivation of the equilibrium is valid for all types of materials. In the theory ofmoderately large deflections, the equilibrium is coupled with the third groupof equation define the material behavior and relates the generalizedstrains to generalized forcesN=EA ( )M=EI ( )Independence of geometry and equilibrium on constitutive equation allows to develop thegeneral framework of a solver in the Finite Element codes.
Structural Mechanics 2.080 Lecture 5 Semester Yr The solution of the above system is C 1 = ql 2 (5.26a) C 2 = 0 (5.26b) C 3 = ql3 12 (5.26c) C 4 = 0 (5.26d) …
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