1. © 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. www.autodesk.com/edcommunity Education Community Static Analysis: Load Factor Analysis

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. www.autodesk.com/edcommunity Education Community Objectives The objective of this module is to introduce the equations and numerical methods used to determine if a system is stable.  Buckling is a failure mode, due to compressive components becoming unstable.  The type of analysis presented in this module is frequently called load factor, linear buckling, or eigenbuckling analysis. Section II - Static Analysis Module 8 - Load Factor Analysis Page 2

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. www.autodesk.com/edcommunity Education Community Load Deflection Curve  The figure shows the force versus deflection curve of a system.  As the load is increased, the slope (tangent stiffness) progresses to a point at which it is zero.  When the stiffness is zero, the application of an infinitesimally small load will cause a very large displacement.  This type of behavior is typical of a system subjected to compressive loading. F u Section II - Static Analysis Module 8 - Load Factor Analysis Page 3

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. www.autodesk.com/edcommunity Education Community Governing Equations Section II - Static Analysis Module 8 - Load Factor Analysis Page 4  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.

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. www.autodesk.com/edcommunity 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. Section II - Static Analysis Module 8 - Load Factor Analysis Page 5

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. www.autodesk.com/edcommunity Education Community Zero Tangent Stiffness Matrix  When the system is in equilibrium, the unbalanced load vector is equal to zero and the equations reduce to  There is the possibility that the system is unstable and that another equilibrium configuration exists that the system could move to without an increase in the external loads  The stress levels at which unstable configurations exist can be determined by introducing a Load Factor ( ) into the above equation Section II - Static Analysis Module 8 - Load Factor Analysis Page 6

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. www.autodesk.com/edcommunity Education Community Eigenvalue Problem  This equation is a homogeneous set of equations that has a solution only when the determinant of the coefficient matrix is zero.  The determinant is a polynomial that will be equal to zero only for specific values of  The order of the polynomial is equal to the number of equations. det    Characteristic Polynomial Section II - Static Analysis Module 8 - Load Factor Analysis Page 7

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. www.autodesk.com/edcommunity Education Community Load Factor Interpretation  If is equal to 1, then the system is unstable as loaded.  If is between 0 and 1, then the system is unstable at a load level less than the applied load.  If is greater than 1, then the load can be increased by a multiple of before it becomes unstable.  If is equal to -1, the system is unstable if the load is reversed.  If is between 0 and -1 then the system is unstable at a load level less than the applied load if the load is reversed.  If is greater than, -1 then the system becomes unstable at a multiple of when the load is reversed. Only the lowest value of is of practical interest. Section II - Static Analysis Module 8 - Load Factor Analysis Page 8

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. www.autodesk.com/edcommunity Education Community Example Section II - Static Analysis Module 8 - Load Factor Analysis Page 9 1 inch wide x 12 inch long x 1/8 inch thick. Material - 6061-T6 aluminum. Brick elements with mid-side nodes are used to improve bending accuracy through the thin section. 0.0625 inch element size.  Simulation is used to compute the load factor and buckled shape of the cantilevered beam that has an axial load of 5 lbs. located at its free end. Fixed end boundary condition 5 lbs.

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. www.autodesk.com/edcommunity Education Community Example – Analysis Parameters Use Sparse matrix solver Use BCS-LIB parallel sparse solver Compute first five load factors (lowest is only one of practical interest) Use all threads/cores during the solution (fastest) Section II - Static Analysis Module 8 - Load Factor Analysis Page 10

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. www.autodesk.com/edcommunity Education Community Example - Results Buckled shape for lowest load factor. Beam will buckle at a load that is 5.6 times that applied. The buckling load is 5 lbs. x 5.6 load factor = 28 lbs. Section II - Static Analysis Module 8 - Load Factor Analysis Page 11 Caution – initial distortions in the shape of the beam could make the buckling load be smaller.

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. www.autodesk.com/edcommunity Education Community Summary  A method for determining the stability of a system based on the tangent stiffness matrix was introduced.  A load factor was introduced that acts like a safety factor to tell how far from being unstable the system is at an applied load level.  An example showing results for a simple system were presented. Section II - Static Analysis Module 8 - Load Factor Analysis Page 12