© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Static Analysis: Static Analysis

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Objective The objective of this module is to introduce the methods used to solve static problems where inertia or time-dependent material effects are not important.  The solution methods will build on material presented in Modules 1 through 3.  The methods are based on the Newton-Raphson method and are applicable to the solution of non-linear geometric or material problems.  The solution of problems governed by linear equations is treated as a special case of the more general non-linear methods. Section II – Static Analysis Module 4 – Static Analysis Page 2

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Governing Equations  The governing equations for a static finite element analysis can be written as  The tangent stiffness matrix,, has three components  Where are the linear, displacement, and stress dependent contributions. are the displacement increments. Section II – Static Analysis Module 4 – Static Analysis Page 3

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Governing Equations is the unbalanced load array. It is the difference between two arrays. is an array of external forces acting on the nodes. This array is obtained from the external virtual work term. is an array of node forces associated with the stresses inside the body. This array is obtained from the internal virtual work term. At equilibrium the two arrays are equal and is zero. Section II – Static Analysis Module 4 – Static Analysis Page 4

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Graphical Illustration  The solution of this equation can be illustrated graphically for a single degree-of-freedom system.  Point 1 lies on the solution path and is in equilibrium.  Point 1 can be at any configuration that is in equilibrium.  Point 2 is the desired solution point and is also in equilibrium.  Point A is an estimate for point 2 based on the tangent stiffness and displacement increment,  u. Slope = K T Desired Solution Point 1 2 A Section II – Static Analysis Module 4 – Static Analysis Page 5

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Graphical Illustration  The displacement increment,  u, can be found by inverting the tangent stiffness matrix  The total displacement for point A is  If the solution path is linear, points A and 2 will be coincident and point 2 would be in equilibrium. Slope = K T Desired Solution Point 1 2 A Section II – Static Analysis Module 4 – Static Analysis Page 6

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Iterative Solution  In the case of a material or geometric non-linearity, Point A will only provide an approximation to the equilibrium configuration at Point 2.  A numerical method is necessary that will take the information available and obtain an improved estimate that is closer to the true equilibrium configuration at Point 2. Slope = K T Desired Solution Point 1 2 A Section II – Static Analysis Module 4 – Static Analysis Page 7

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Newton-Raphson Method  The derivation of the governing equation was based on the Newton-Raphson method.  There are two fundamental iteration methods that can be used with this method:  First is a full Newton-Raphson iteration,  Second is a modified Newton-Raphson iteration.  These two methods can be used individually or in combination.  Each iteration method can also be used in combination with a line search algorithm based on the method of steepest descent used in optimization theory. Section II – Static Analysis Module 4 – Static Analysis Page 8

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Full Newton-Raphson Iteration  A full Newton-Raphson iteration uses a new tangent stiffness matrix based on the latest estimate of the stresses, displacements, and material properties along with an updated internal restoring force.  A sequence of new estimates is obtained until the error is determined to be acceptable. Slope = K T 1 2 A B Section II – Static Analysis Module 4 – Static Analysis Page 9

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Modified Newton-Raphson Iteration  A modified Newton-Raphson method uses a previously factored tangent stiffness matrix along with an updated internal restoring force.  A sequence of new estimates is obtained until the error is determined to be acceptable.  This method uses reduced computational effort associated with forming and factoring the tangent stiffness matrix, but generally requires more iterations. Slope = K T 1 2 A Section II – Static Analysis Module 4 – Static Analysis Page 10

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Convergence  Both the full and modified Newton-Raphson iterations can be applied repeatedly until convergence is achieved.  The driver behind both methods is the unbalanced load that is the error between the desired equilibrium point and the current estimate.  Either the equilibrium error or displacement change can be used to determine convergence.  For example, an error tolerance based on the ratio of the most recently computed displacement increment to the sum of all displacement increments for the current load increment is Section II – Static Analysis Module 4 – Static Analysis Page 11

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Simulation Iteration Controls Newton-Raphson Iterations Modified Newton- Raphson Iterations Combination of full and modified Newton- Raphson iterations  Simulation enables the user to select the type of equilibrium iteration to be used in an analysis. Simulation also provides a line search option for each type of iteration. Control parameters used with the Combined Newton Option Section II – Static Analysis Module 4 – Static Analysis Page 12

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Simulation Convergence Tolerance User can select type of convergence criteria to use Default displacement convergence tolerance Use default convergence tolerance if checked  Simulation allows the user to change the type of convergence criterion used and the associated convergence tolerance. Section II – Static Analysis Module 4 – Static Analysis Page 13

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Solution Methods  Both of the Newton-Raphson iteration methods requires the solution of the equation  Matrix inversion of the tangent stiffness matrix is not efficient and finite element programs rely on factorization methods or iteration methods.  Factorization methods decompose the matrix into multiplicative components.  For example, the Cholesky factorization method decomposes the tangent stiffness matrix into lower and upper triangular matrices  The lower triangular matrix has only non-zero elements on or below the diagonal, while the upper triangular matrix only has non-zero terms on or above the diagonal. Section II – Static Analysis Module 4 – Static Analysis Page 14

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Iterative Methods  Iterative methods are based on an additive decomposition of the stiffness matrix  The governing equation then becomes or  If an initial guess is made for the displacement increment on the right hand side of the equation, an improved estimate can be found by solving the left hand side.  The additive decomposition of the tangent stiffness matrix takes less time than the multiplicative decomposition.  However, iterations are required as a trade-off. Section II – Static Analysis Module 4 – Static Analysis Page 15

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example Problem  The iteration and convergence character of nonlinear solutions will be demonstrated with a cantilevered beam subjected to gravity and a pressure load. The pressure load will stay normal to the surface as it deforms. The beam is inch thick, 1 inch wide, and 12 inches long. It uses brick elements with mid-side nodes to improve the bending response of the brick elements. The elements are generated with a 1/16 inch absolute mesh size. It is subjected to gravity and a 2 psi pressure on its top surface. The material is 6061-T6. Section II – Static Analysis Module 4 – Static Analysis Page 16 Close up of mesh without pressure.

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 1 – Analysis Parameters Load is applied in five increments A maximum of 10 iterations per load increment will be performed A displacement-based tolerance ratio of will indicate that equilibrium has been achieved Section II – Static Analysis Module 4 – Static Analysis Page 17

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 1 – Analysis Log Iteration Number Convergence parameter for each iteration This iteration converged in 5 iterations Each load increment required five iterations. Section II – Static Analysis Module 4 – Static Analysis Page 18

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 1 - Results Contour plot of Von Mises stress superimposed on deformed shape of the structure. The maximum stress is 58.2 ksi. Note the neutral axis running down the side of the beam. Section II – Static Analysis Module 4 – Static Analysis Page 19

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 2 – Analysis Parameters Load is applied in one increment. A maximum of 10 iterations per load increment will be performed. A displacement-based tolerance ratio of will indicate that equilibrium has been achieved. Section II – Static Analysis Module 4 – Static Analysis Page 20

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 2 – Analysis Log Iteration Number Convergence parameter for each iteration Note that only six iterations were required with one load increment. Section II – Static Analysis Module 4 – Static Analysis Page 21

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Run 2 - Results Contour plot of Von Mises stress superimposed on a deformed shape of the structure. The maximum stress is 58.6 ksi which compares well with 58.2 ksi obtained from Run 1. Section II – Static Analysis Module 4 – Static Analysis Page 22

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example Summary  Both of the runs presented obtained similar answers for different combinations of load increments and iterations.  Both runs used a full Newton- Raphson iteration.  A modified Newton-Raphson iteration had trouble converging for this problem.  Although not shown, a full Newton-Raphson iteration with Line Search required more iterations than the standard full Newton-Raphson iteration.  The type of iteration and its performance depends on the problem.  Experience and trial and error is required to determine the best method for a particular problem. Section II – Static Analysis Module 4 – Static Analysis Page 23

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Module Summary  This module has provided an introduction to the solution methods used in static analysis.  Full and modified Newton- Raphson equations are presented and illustrated.  The driver behind static solution methods is the unbalanced load vector that approaches zero as the solution approaches equilibrium.  The methods presented are applicable to linear and non- linear problems involving either material or geometric non- linearities.  The solution for a linear system simply converges in one iteration whereas the solution for a non- linear system requires multiple iterations. Section II – Static Analysis Module 4 – Static Analysis Page 24