ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 10: Solution of Continuous Systems – Fundamental Concepts Mixed Formulations Intrinsic Coordinate Systems

Last Time Weighted Residual Formulations Consider a general representation of a governing equation on a region V L is a differential operator eg. For Axial element

Last Time Weighted Residual Formulations Exact Approximate Objective: Define so that weighted average of Error vanishes NOT THE ERROR ITSELF !!

Last Time Weighted Residual Formulations Set Error relative to a weighting function  Objective: Define so that weighted average of Error vanishes

Weighted Residual Formulations   ERROR

Weighted Residual Formulations   ERROR

Last Time Weighted Residual Formulations  ERROR

Last Time Weighted Residual Formulations Assumption for approximate solution (Recall shape functions) Assumption for weighting function GALERKIN FORMULATION

Last Time Weighted Residual Formulations  i are arbitrary and  0

Last Time Galerkin Formulation Algebraic System of n Equations and n unknowns

Last Time Galerkin’s Method in Elasticity Governing equations Interpolated Displ Field Interpolated Weighting Function

Last Time Galerkin’s Method in Elasticity Integrate by part…

Last Time Galerkin’s Method in Elasticity Virtual Work Compare to Total Potential Energy Virtual Total Potential Energy

Last Time Galerkin’s Formulation More general method Operated directly on Governing Equation Variational Form can be applied to other governing equations Preffered to Rayleigh-Ritz method especially when function to be minimized is not available.

Mixed Formulation Displacement Based FE approximations –Combine subsidiary equations to obtain G.E. –G.E. in terms of displacements –Stresses, Strains etc enter as natural B.C. Mixed Formulation –Apply Galerkin directly on subsidiary relations –Nodal dof contain displacements AND other field quantities

Mixed Formulation Axial Equilibrium… Stress-Displacement…

Mixed Formulation

Galerkin Residual Equations Axial Equilibrium… Stress-Displacement…

Mixed Formulation Axial Equilibrium…

Mixed Formulation A

Stress-Displacement… B

Mixed Formulation kuku kuku k  A B

Mixed Formulation

Application Example

INTRINSIC COORDINATE SYSTEMS

Intrinsic Coordinate System  x1x1 x x2x2 x3x3  1 =-1 1  3  2 =1 2 Global C.S. Local C.S.

Intrinsic Coordinate System  x1x1 x x2x2 x3x3  1 =-1 1  3  2 =1 2 Linear Relationship Between GCS and LCS

Shape Functions wrt LCS  u(-1)=a 0 -a 1 +a 2 =u 1 u(1)=a 0 +a 1 +a 2 =u 2 u(0)=a 0 =u 3 … u(  )=a 0 +a 1  +a 2  2  1 =-1 1   2 =1 32

Shape Functions wrt Intrinsic Coordinate System  N1()N1() N2()N2() N3()N3()

 wrt 

Element Strain-Displacement Matrix Cast in Matrix Form  e  = B u e  e  = E B u e

Linear Stress Axial Element - In Summary  = B u  = E B u  1 =-1 1     2 =1 32

Linear Stress Axial Element - k e Stiffness Matrix

Linear Stress Axial Element - k e Stiffness Matrix  1 =-1 1     2 =

Linear Stress Axial Element – f e,T e  1 =-1 1     2 =1 32 Body Force Uniformly Distributed Force