# © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.

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© 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: Newton-Raphson Equations

© 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 use the rate of virtual work equation to develop a solution strategy based on the iterative Newton-Raphson method.  The resulting equations will have an increment in displacement needed to satisfy equilibrium as the unknown.  The incremental method allows the load to be applied in small increments to improve convergence. Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Virtual Work: Incremental Form  The virtual work equation developed in Module 1 is  If the stresses are in equilibrium with the surface tractions and body forces, then  If the stresses are not in equilibrium with the surface tractions and body forces, then  Consider a Taylor’s series expansion of the virtual work equation about a state which is not in equilibrium (subscript 1) Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Incremental Equations  If configuration 2 (left hand side of previous equation) is in equilibrium, then the previous equation becomes  The right hand side of this equation represents the amount that configuration 1 is out of equilibrium and is an unbalanced force.  The left hand side of this equation represents a system of equations with unknown incremental displacements,.  An improved estimate of the displacements that will bring configuration 1 into equilibrium is obtained by the equation where i is an iteration number. Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Newton-Raphson Equations  The equations represent an iterative solution method known as the Newton- Raphson method.  The left hand side of the equation leads to a “Tangent Stiffness Matrix”.  The right hand side of the equation leads to an unbalanced load vector that goes to zero as the system approaches equilibrium.  The Newton-Raphson method is used extensively to solve both static and dynamic problems. Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Incremental Loading  In many problems, it is necessary to apply the external loads in small increments.  At each load increment, the Newton-Raphson equations are applied until a converged solution is obtained. F1F1 F2F2 F3F3 F4F4 u1u1 u2u2 u3u3 u4u4 Force Disp. Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Unbalanced Load  Convergence of the Newton- Raphson method is controlled by the unbalanced load that arises from the right hand side of the equation  The unbalanced load is shown graphically at the beginning of a load increment. F1F1 F2F2 F3F3 F4F4 u1u1 u2u2 u3u3 u4u4 Force Disp. 1 External force from Internal restoring force from Unbalanced load from Section II – Static Analysis Module 2 – Newton-Raphson Equations 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. www.autodesk.com/edcommunity Education Community Module Summary  The rate of virtual work equation was used to establish a solution strategy based on the Newton-Raphson Method.  The methodology can be used to solve non-linear equations that result from either material or geometric non-linearities.  The equations from this module are used to develop the equations for an element in the next module.  The equations for all elements are added up to give a mathematical representation of the system. Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 8

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