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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. Education Community Static Analysis: Newton-Raphson Equations

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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. 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

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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. 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

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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. 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

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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. 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

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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. 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. F1F1 F2F2 F3F3 F4F4 u1u1 u2u2 u3u3 u4u4 Force Disp. Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 6

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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. 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. F1F1 F2F2 F3F3 F4F4 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

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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. 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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