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: Virtual Work Equation

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 develop the rate of virtual work equation that forms the basis for finite element methods used in solid mechanics  This equation will be developed using the differential equation for equilibrium written in terms of the Cauchy stress tensor.  It will then be transformed into an integral equation of equilibrium using the principle of virtual work.  The divergence theorem will be used to transform the integral equation into a more usable form.  The rate of virtual work equation will then be presented in terms of work conjugate stress and strain tensors that are based on a known reference configuration. Section II – Static Analysis Module 1 - Virtual Work Equation 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 Configurations  Different configurations that a body passes through while deforming under load are important to the development of the equations used in finite element analysis.  All of the configurations that a real body passes through are in equilibrium (i.e. satisfy Newton’s 2 nd Law).  However, some of the configurations that are computed by a finite element program are not in equilibrium.  These non-equilibrium configurations are encountered during the process of searching for a configuration that satisfies the equations of equilibrium. Section II – Static Analysis Module 1 - Virtual Work Equation 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 Original Configurations  The original configuration represents the state the body is in prior to the application of any external disturbances.  The original configuration is in equilibrium. x y z Original Configuration Section II – Static Analysis Module 1 - Virtual Work Equation Page 4

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 Reference Configurations  It is necessary to use configurations having a known surface area and volume during the solution process.  Everything about a reference configuration is known  Stress distributions,  Temperature distributions,  Position,  Surface area and volume.  The original configuration is a reference configuration. x y z Original Configuration Section II – Static Analysis Module 1 - Virtual Work Equation 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 Deformed Configurations  As the body is subjected to external disturbances, it moves through various deformed configurations until it reaches its final configuration.  Each of the deformed configurations satisfies the equations of equilibrium to within a convergence tolerance.  A deformed configuration can also be used as a reference configuration. x y z Original Configuration Deformed Configurations Section II – Static Analysis Module 1 - Virtual Work Equation 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 Desired Configuration  During the process of computing the response of a body to external disturbances, there arises the need to determine a new deformed configuration.  The shape, position, and stresses in this new configuration are not known, and it is the purpose of the analysis to determine them. x y z Previously determined Deformed Configuration Desired configuration Section II – Static Analysis Module 1 - Virtual Work Equation 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 Non-equilibrium Configurations  During the process of determining a desired configuration, a numerical algorithm may develop an estimate of the desired configuration that does not satisfy equilibrium.  The numerical algorithm must have the capability to detect when the equations of equilibrium are not satisfied and a method for converging to the correct solution. x y z Previously determined configuration Desired configuration True path Computed Configuration Section II – Static Analysis Module 1 - Virtual Work Equation 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 Equilibrium at a Point  The equations of equilibrium are the fundamental equations used by finite element programs.  The equations for equilibrium at a point are: Section II – Static Analysis Module 1 - Virtual Work Equation Page 9 Cartesian Components of Cauchy Stress Tensor Cartesian components of the body force per unit volume

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 Equilibrium at a Point  This equation must be satisfied in the each configuration.  The Cauchy stress components can be thought of as true stress components that are based on the area of a differential element in the desired configuration.  The components are defined with respect to the current configuration base vectors. Section II – Static Analysis Module 1 - Virtual Work Equation 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 Rate of Virtual Work at a Point  The rate of virtual work at a point is obtained by multiplying the differential equation of equilibrium by a virtual velocity,  There are no limitations on the virtual velocity except that it satisfy the boundary conditions acting on the body and that it be constant ( ). Section II – Static Analysis Module 1 - Virtual Work Equation Page 11 Cartesian Components of the Virtual Velocity There are no changes in the external loads or internal stresses when a virtual velocity is applied.

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 New Form for Rate of Virtual Work Rate of Virtual Work at a Point Chain Rule for a Product Rearranging Substitution New form for the rate of virtual work at a point. Rearranging Section II – Static Analysis Module 1 - Virtual Work Equation Page 12 Decompose this term using the chain rule for a product

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. www.autodesk.com/edcommunity Education Community Rate of Virtual Work for a Body  The rate of virtual work for a body is obtained by integrating the rate of virtual work at a point over the volume of the body.  The Cauchy stress formula relates the components of a traction vector acting on a surface to the surface normal vector and the components of the Cauchy stress. Section II – Static Analysis Module 1 - Virtual Work Equation Page 13

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. www.autodesk.com/edcommunity Education Community Rate of Virtual Work for a Body  The Cauchy stress formula and the Divergence Theorem allows the rate of virtual work for a body to be written as:  This is an integral form of the equations of equilibrium for a body.  It is sometimes called the weak form of the equations of equilibrium. Section II – Static Analysis Module 1 - Virtual Work Equation Page 14

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. www.autodesk.com/edcommunity Education Community Rate of Deformation and Vorticity  The gradient of the virtual velocity is defined by the equation Section II – Static Analysis Module 1 - Virtual Work Equation Page 15 where is the rate of the virtual deformation gradient.

16. © 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 Rate of Deformation  The rate of the virtual deformation can be broken into symmetric and skew-symmetric components  The first term is the virtual rate of deformation. It is important to this development.  The second term is the virtual vorticity that is not related to material constitutive equations and is dropped. Section II – Static Analysis Module 1 - Virtual Work Equation Page 16

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. www.autodesk.com/edcommunity Education Community Virtual Rate of Deformation  The components of the virtual rate of deformation are  These are also the components of the rate of the small deformation strains due to virtual displacements. Section II – Static Analysis Module 1 - Virtual Work Equation Page 17

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. www.autodesk.com/edcommunity Education Community Updated Equation: Rate of Virtual Work  Combining the rate of virtual work equation with the rate of the virtual deformation equation yields the equation  The problem with this equation is that volume and surface area in the desired configuration are not known. For infinitesimal deformations and rotations, the original configuration volume and surface can be used with minimal error.  When deformations and/or rotations are large, a Lagrangian statement of the rate of virtual work is required. Section II – Static Analysis Module 1 - Virtual Work Equation Page 18

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. www.autodesk.com/edcommunity Education Community Lagrangian Statement: Rate of Virtual Work  The Cauchy stress tensor and the rate of deformation are work conjugate quantities referred to the current configuration  The 2 nd Piola-Kirchhoff and rate of Green’s strain are work conjugate quantities referred to a reference configuration.  Relations exist between the Cauchy and 2 nd Piola-Kirchhoff stress tensors and the rate of deformation, and rate of Green’s strain tensor. Section II – Static Analysis Module 1 - Virtual Work Equation Page 19

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. www.autodesk.com/edcommunity Education Community Lagrangian Statement: Rate of Virtual Work  The rate of virtual work written in terms of the 2 nd Piola-Kirchhoff stress tensor,, and the rate of Green’s strain tensor,, is  One of the important differences in the above equation is that all quantities are defined with respect to a reference configuration that has a known volume and surface area.  This enables the integrations to be carried out for problems involving large deformations. Section II – Static Analysis Module 1 - Virtual Work Equation Page 20

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. www.autodesk.com/edcommunity Education Community Internal and External Rate of Virtual Work  The external surface tractions or internal body forces can be thought of as external causes of the internal stresses in the body.  The Lagrangian statement of the rate of virtual work can be written as Section II – Static Analysis Module 1 - Virtual Work Equation Page 21 Internal rate of virtual work External rate of virtual work  The last equation can alternately be written as  This form of the rate of virtual work equation will be used to develop a basic solution strategy.  Any body that is in equilibrium satisfies this equation.

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. www.autodesk.com/edcommunity Education Community Module Summary  This module has developed an equation for the rate of virtual work.  The differential equation of equilibrium for stresses at a point was used as a starting point.  This equation was converted to an integral equation that relates the virtual work done by internal stresses to the virtual work done by external surface tractions and body forces.  This virtual work equation is the foundation upon which commercial finite element programs used to solve problems in solid mechanics are based.  Module 2 will use the virtual work equation to develop an incremental and iterative solution strategy. Section II – Static Analysis Module 1 - Virtual Work Equation Page 22