Contents as of 12/8/2017. Preface. 1. Overview...1

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Brianna Pearson
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1 Contents as of 12/8/2017 Preface 1. Overview Introduction Finite element data Matrix notation Matrix partitions Special finite element matrix notations Finite element analysis sequence Layout of this book Summary and notation Polynomial interpolation Types of interpolation: Lagrange one-dimensional interpolation Natural coordinates Hermite one-dimensional interpolation Lagrangian quadrilateral elements Lagrangian triangular elements Serendipity quadrilaterals* Hierarchical interpolation Summary and notation Exercises Numerical integration One-dimensional quadratures Two- and three-dimensional quadratures Exact integrals for straight sided triangles Reduced and selected integration Summary Exercises Calculus review Parametric geometry Jacobian matrix Inverse Jacobian Parametric substitution in integrals Integration by parts Integral of point sources Axisymmetric integrals...16

2 4.8 Summary Exercises Terminology from differential equations Definitions* Boundary conditions Adjoint operator* Three classes of PDEs* Eigen-problems Model elliptic PDE Directionally dependent data Point singularities* Summary Exercises Equivalent integral forms Variational calculus*: Method of weighted residuals Common weighting methods Eigen-problem analysis Summary Exercises Matrix procedures for finite elements Introduction Equation numbers for gather and scatter Vector subscripts Partitioning the system equations Numerically equivalent process EBC by a penalty method* Multiple point constraints* Wilson's static condensation algorithm Equation factorization Skyline sparse storage Summary Exercises Applications of 1-d Lagrange elements Introduction Variable source terms Mixed boundary conditions Automating Lagrange element solutions...43

3 8.6 Numerically integrated elements Symbolic solutions Symmetry and anti-symmetry Patch test* Creating exact solutions* Summary Exercises Truss analysis Planar truss Space truss Summary Exercises Applications of 1-d Hermite elements Introduction General case fourth-order beam equation Integral form Element arrays C1 element models Classic beams Structural symmetry The Rayleigh quotient Multiple span beams BOEF without axial load BOEF with axial load BOEF with axial load and end transverse force Summary Exercises Frame analysis Planar frames Frame member reactions Enhanced frame post-processing* Space frames Numerically integrated frame members Summary Exercises Thermal and scalar field analysis Introduction General field problem...3

4 12.3 Common flux components Galerkin integral form Galerkin integral form* Orthotropic two-dimensional fields Corresponding element and boundary matrices Numerical evaluation of the Jacobian matrix Jacobian matrix for equal space dimensions Physical space > parametric space: Field flux vector at a point Evaluation of integrals with constant Jacobian Symmetry and anti-symmetry Viscous fluid flow in a channel Axisymmetric fields Summary Elasticity Introduction Linear springs Mechanical work Strain energy Material properties Simplified elasticity models* Interpolating displacement vectors Mechanical work in matrix form The strain-displacement matrix Stiffness matrix Work done by initial strains Matrix equilibrium equations Symmetry and anti-symmetry Plane stress analysis Axisymmetric stress analysis Solid stress analysis Kinetic energy Summary Exercises Eigen-analysis Introduction Finite element eigen-problems Spring-mass systems Vibrating string Torsional vibrations Beam vibrations Membrane vibration...15

5 14.8 Beam-column buckling Beam frequency with an axial load Plane-frame modes and frequencies: Modes and frequencies of 2-d continua Acoustical vibrations: Principal stresses Mohr's circle for eigenvalues* Time independent Schrödinger equation* Summary Exercises Transient and dynamic solutions Introduction to transient systems Generalized trapezoidal algorithms Accuracy and control Introduction to dynamic solutions Wilson method Summary Exercises Vector field elements (draft) Introduction Parametric coordinates Edge based (vector) finite elements Whitney vector elements Ainsworth vector elements Weak form Vector element matrices Summary Exercises Continuous flux and error estimates Introduction Optimal gradient locations Continuous nodal flux recovery Additional data arrays for flux averaging* A one-dimensional flux smoothing example Error estimates Mesh h-adaptivity* Summary Exercises...19 Appendix

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation