Finite Element Primer for Engineers: Part 2 Mike Barton & S. D. Rajan A A B A B B C A A D C B D B C A B C D A E D A A B A B E D D

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C E H Y E C C E A C E C Z X Contents Introduction to the Finite Element Method (FEM) Steps in Using the FEM (an Example from Solid Mechanics) Examples Commercial FEM Software Competing Technologies Future Trends Internet Resources References 2 FEM Applied to Solid Mechanics Problems Create elements of the beam Nodal displacement and forces dxi 1 dyi 1 dxi 2

1 2 4 3 dyi 2 A FEM model in solid mechanics can be thought of as a system of assembled springs. When a load is applied, all elements deform until all forces balance. F = Kd K is dependant upon Youngs modulus and Poissons ratio, as well as the geometry. Equations from discrete elements are assembled together to form the global stiffness matrix. Deflections are obtained by solving the assembled set of linear equations. Stresses and strains are calculated from the deflections. 3 Classification of Solid-Mechanics AnalysisProblems of solids Static Dynamics Advanced Elementary Stress Stiffening Behavior of Solids Geometric Linear Nonlinear Geometric

Classification of solids Trusses Cables Pipes Instability Fracture Material Skeletal Systems 1D Elements Large Displacement Plasticity Viscoplasticity Plates and Shells 2D Elements Solid Blocks 3D Elements Plane Stress Plane Strain Axisymmetric Plate Bending Shells with flat elements Shells with curved elements Brick Elements Tetrahedral Elements General Elements 4 Governing Equation for Solid Mechanics Problems Basic equation for a static analysis is as follows: [K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld} [K] = total stiffness matrix {u} = nodal displacement {Fapp} = applied nodal force load vector

{Fth} = applied element thermal load vector {Fpr} = applied element pressure load vector {Fma} = applied element body force vector {Fpl} = element plastic strain load vector {Fcr} = element creep strain load vector {Fsw} = element swelling strain load vector {Fld} = element large deflection load vector 5 Six Steps in the Finite Element Method Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements. Step 2 - Develop Element Equations: Developed using the physics of the problem, and typically Galerkins Method or variational principles. Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system. Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations. Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes. Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables. 6 Process Flow in a Typical FEM Analysis Start Analysis and design decisions Problem Definition Processor Pre-processor Reads or generates nodes and elements (ex: ANSYS) Reads or generates material property data. Reads or generates boundary conditions (loads and constraints.)

Step 1, Step 4 Generates element shape functions Calculates master element equations Calculates transformation matrices Maps element equations into global system Assembles element equations Introduces boundary conditions Performs solution procedures Stop Post-processor Prints or plots contours of stress components. Prints or plots contours of displacements. Evaluates and prints error bounds. Step 6 Steps 2, 3, 5 7 Step 1: Discretization - Mesh Generation surface model airfoil geometry (from CAD program) 1 2 12 14 13 3

4 511 mesh generator ET,1,SOLID45 N, 1, 183.894081 N, 2, 183.893935 . . TYPE, 1 E, 1, 2, 80, E, 2, 3, 81, . . . , -.770218637 , -.838009645 79, 80, 4, 5, 5, 6, , , 5.30522740 5.29452965 83, 84, 82 83

meshed model 8 Step 4: Boundary Conditions for a Solid Mechanics Problem Displacements DOF constraints usually specified at model boundaries to define rigid supports. Forces and Moments Concentrated loads on nodes usually specified on the model exterior. Pressures Surface loads usually specified on the model exterior. Temperatures Input at nodes to study the effect of thermal expansion or contraction. Inertia Loads Loads that affect the entire structure (ex: acceleration, rotation). 9 Step 4: Applying Boundary Conditions (Thermal Loads) 300 Nodes from FE Modeler 300 275 275 250 250 225 225 Temp mapper 200 200 175 Thermal Soln Files bf, bf, . . . bf,

bf, 1,temp, 2,temp, 149.77 149.78 1637,temp, 1638,temp, 303.64 303.63 150 150 175 10 Step 4: Applying Boundary Conditions (Other Loads) Speed, temperature and hub fixity applied to sample problem. FE Modeler used to apply speed and hub constraint. antype,static omega,10400*3.1416/30 d,1,all,0,0,57,1 Z Y X 11 PRODUCE ELEMENT PLOT I N DSYS = PREP7: 0