Static Analysis: Newton-Raphson Equations

1. Static Analysis: Newton-Raphson Equations

2. Objectives Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 2 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.

3. Virtual Work: Incremental Form Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 3 • 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)

4. Incremental Equations Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 4 • 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.

5. Newton-Raphson Equations Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 5 • 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.

6. Incremental Loading Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 6 Force • 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. DF4 DF3 DF2 DF1 Disp. u3 u1 u2 u4

7. Unbalanced Load Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 7 Force • 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. 1 Unbalanced load from DF4 DF3 External force from DF2 Internal restoring force from DF1 Disp. u3 u1 u2 u4

8. Module Summary Section II – Static Analysis Module 2 – Newton-Raphson Equations Page 8 • 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.