2005 February, 2 Page 1 Finite Element Analysis Basics – Part 2/2 Johannes Steinschaden

2005 February, 2 Page 2 Finite Element Analysis Procedure 1.Preliminary analysis of the system: Perform an approximate calculation to gain some insights about the system 2.Preparation of the finite element model: a Geometric and material information of the system b Prescribe how is the system supported c Determine how the loads are applied to the system 3.Perform the calculation: Solve the system equations and compute displacements, strains and stresses 4.Post-processing of the results: Viewing the stresses and displacements Interpret the results

2005 February, 2 Page 3 Direct Stiffness Method Two-dimensional Truss Elements F1F2K=EA/L N1N2 x1x2

2005 February, 2 Page 4 ad Two-dimensional Truss Elements X Y F2 F1 K N1 N2 Y1 X1 Y2 X2 local stiffness matrix

2005 February, 2 Page 5 ad Two-dimensional Truss Elements Coordinate transformation equation

2005 February, 2 Page 6 ad Two-dimensional Truss Elements

2005 February, 2 Page 7 ad Two-dimensional Truss Elements

2005 February, 2 Page 8 ad Two-dimensional Truss Elements Example Truss A Truss B Node 1 Node 3 Node 2 F

2005 February, 2 Page 9 ad Two-dimensional Truss Elements Example Element A: local stiffness matrix global stiffness matrix

2005 February, 2 Page 10 ad Two-dimensional Truss Elements Example Element B: local stiffness matrix global stiffness matrix

2005 February, 2 Page 11 ad Two-dimensional Truss Elements Example Summing the two seta of global force-displacement equations:

2005 February, 2 Page 12 ad Two-dimensional Truss Elements Example Nodes 1 and 3 are fixed and only load on node 2 in global X direction

2005 February, 2 Page 13 ad Two-dimensional Truss Elements Example Nodes 1 and 3 are fixed and only load on node 2 in global X direction Solve for nodal displacements:

2005 February, 2 Page 14 ad Two-dimensional Truss Elements Example Substitute the known displacements and solve for the reaction forces:

2005 February, 2 Page 15 ad Two-dimensional Truss Elements Truss element A:

2005 February, 2 Page 16 Stress and Momentum Balance 15 unknown variables –3 displacements –6 strains –6 stresses 15 equations –6 displacement-strain equations –6 strain-stress equations –3 equilibrium equations

2005 February, 2 Page 17 Strains

2005 February, 2 Page 18 ad Strains

2005 February, 2 Page 19 Material law

2005 February, 2 Page 20 ad Material law

2005 February, 2 Page 21 Equilibrium Equations

2005 February, 2 Page 22 ad Equilibrium Equations

2005 February, 2 Page 23 ad Equilibrium Equations

2005 February, 2 Page 24 Plain Stress

2005 February, 2 Page 25 Plain Strain

2005 February, 2 Page 26 Principle of Virtual Work Principle of Virtual Displacements

2005 February, 2 Page 27 ad Principle of Virtual Work

2005 February, 2 Page 28 Basis Function Example: two-dimensional beam element Basis function to approximate displacement inside element

2005 February, 2 Page 29 ad Basis Function

2005 February, 2 Page 30 ad Basis Function

2005 February, 2 Page 31 ad Basis Function

2005 February, 2 Page 32 Thank you!