Static Analysis Truss Element Equations 2011 Autodesk Freely

Static Analysis: Truss Element 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

Section II – Static Analysis Objectives Module 3 – Truss Element Equations Page 2 The objective of this module is to show the equations developed in Modules 1 and 2 convert to matrix equations for a typical element. Notation familiar with upper division undergraduate students is used instead of the more compact indicial notation introduced at the graduate level. § The various differentials, variations, and time derivatives of Green’s strain used in the Lagrangian rate of virtual work are developed. § A one-dimensional truss element is used to demonstrate the process § It contains most of the features of multi-dimensional elements, § The integrations can be carried out manually. § © 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

Section II – Static Analysis Configurations Module 3 – Truss Element Equations Page 3 Q* P* Body in Current Configuration y, y* Q P z, z* © 2011 Autodesk Body in Reference Configuration x, x* § An arbitrary line element is defined by points P & Q in the reference configuration. § The same points are defined by P* and Q* in the current configuration. § f, g & h are functions that relate the coordinates of a point in the current configuration to the coordinates in reference configuration. 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

Section II – Static Analysis Deformation Gradient Module 3 – Truss Element Equations Page 4 § A differential change in the reference configuration and current configuration coordinates are related through the deformation gradient. Differential Changes Matrix Form § The Deformation Gradient is defined by the array [F]. § It is the Jacobian of the transformation between the current and reference configurations. © 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

Section II – Static Analysis Deformation Gradient Components Module 3 – Truss Element Equations Page 5 § The displacements ux, uy, and uz in the x, y and z directions can be used to determine the mapping functions f, g and h. y Current Configuration Using these functions, the components of the deformation gradient become Reference Configuration x z © 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

Section II – Static Analysis Green’s Strain Module 3 – Truss Element Equations Page 6 § Green’s strain is defined by the equation Small strain definition § Green’s strain uses the length squared of a differential line element instead of the differential length. § For small displacements and rotations, Green’s strain and the small strain give similar results. © 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

Section II – Static Analysis Green’s Strain and Deformation Gradient Module 3 – Truss Element Equations Page 7 § The Deformation Gradient is the fundamental building block needed to find the components of Green’s strain Current Configuration Reference Configuration Deformation Gradient © 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

Section II – Static Analysis Components of Green’s Strain Module 3 – Truss Element Equations Page 8 § The combination of equations on the previous slide gives the equation Green’s Strain where Components of Green’s Strain This equation is used extensively in subsequent slides. © 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

Section II – Static Analysis Green’s Strain – Displacement Equations Module 3 – Truss Element Equations Page 9 § The Deformation Gradient can be used to find the equations for the component of Green’s strain commonly found in textbooks. Carrying out the matrix operations yields © 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

Section II – Static Analysis Truss Element Geometry Module 3 – Truss Element Equations Page 10 § Element matrices are derived using the element coordinate system. They can be transformed to the global coordinate system at the end. Y X, Y, Z Global Coordinate System x, y, z Node j Element Coordinate System Current Configuration Element Coordinate System y x Node i Node j Reference Configuration Node i z Z © 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. X Global Coordinate System www. autodesk. com/edcommunity Education Community

Section II – Static Analysis Notation Module 3 – Truss Element Equations Page 11 § Quantities that have an over score associated with a node point (i. e. node i or node j). § In the array, © 2011 Autodesk is equal to the x displacement measured at node i. 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

Section II – Static Analysis Truss Interpolation Functions Module 3 – Truss Element Equations Page 12 § The displacement or virtual velocity components at any point along the length of the element can be found using interpolation functions. where Note that at x = L, N 1 = 0 and N 2 = 1, at x = 0, N 1 = 1 and N 2 = 0. © 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

Section II – Static Analysis Tangent Stiffness Matrix Equations Module 3 – Truss Element Equations Page 13 § The Newton-Raphson equations developed in Module 2 are § The left hand side of this equation can be written as where 1 st Integral 2 nd Integral 3 rd Integral 4 th Integral § The first two integrals are evaluated in subsequent slides. The third and fourth integrals are not as important for common problems and are not evaluated. © 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

Section II – Static Analysis 1 st Integral: Stress Increment Module 3 – Truss Element Equations Page 14 § The first integral that contributes to the tangent stiffness matrix is § The differential of the matrix containing the components of the 2 nd Piola stress tensor can be related to the differential of Green’s strain via material constitutive equations. § For a truss element made from a linear elastic material this equation becomes § Where Y is Young’s Modulus. © 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

Section II – Static Analysis 1 st Integral: Differential of Green’s Strain Module 3 – Truss Element Equations Page 15 § The first integral also requires equations for the differential of the virtual rate of Green’s strain. § All six components are given, but only d. Exx is needed for the truss element. © 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

Section II – Static Analysis 1 st Integral: Differential of Green’s Strain Module 3 – Truss Element Equations Page 16 § The truss element interpolation functions can be used to evaluate d. Exx for the truss element. © 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

Section II – Static Analysis 1 st Integral: Differential of Green’s Strain Module 3 – Truss Element Equations Page 17 § Combining the equations from the previous slide and writing them in matrix notation yields 1 x 1 1 x 6 § The array BL contains the linear terms and array BNL contains the displacement dependent non-linear terms. 6 x 1 where © 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

Section II – Static Analysis 1 st Integral: Virtual Rate of Green’s Strain Module 3 – Truss Element Equations Page 18 § The first integral also requires equations for the virtual rate of Green’s strain. § All six components are given, but only is needed for the truss element. © 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

Section II – Static Analysis 1 st Integral: Virtual Rate of Green’s Strain Module 3 – Truss Element Equations Page 19 § In a manner similar to that used for the differential of Green’s strain, these equations can be written as 1 x 1 1 x 6 6 x 1 where © 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

Section II – Static Analysis 1 st Integral: Final Form Module 3 – Truss Element Equations Page 20 § Collecting terms from previous slides and carrying out the integration yields the equation for the 1 st integral or 6 x 6 Contribution to Tangent Stiffness Matrix © 2011 Autodesk where 1 x 6 Relates element strain increments or rates to the node point displacement increments or rates. 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

Section II – Static Analysis 1 st Integral: Material Stiffness Matrix Module 3 – Truss Element Equations Page 21 § This 6 x 6 matrix is a function of Young’s modulus and is thus dependent on the material. It is also a function of the length and cross sectional area. § The matrix [B] contains two contributions. [BL] is linear and leads to the linear stiffness matrix, [KL]. [BNL] is a function of the displacements and changes as the truss element deforms. This gives rise to a non-linear stiffness contribution. This contribution to the tangent stiffness matrix is denoted as [K(u)]. Linear Stiffness Matrix © 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

2 nd Integral: Differential of Virtual Rate of Green’s Strain Section II – Static Analysis Module 3 – Truss Element Equations Page 22 § The second integral is § Carrying out the matrix multiplications yields § This is the only component needed for a truss element. © 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

Section II – Static Analysis 2 nd Integral: Manipulations Module 3 – Truss Element Equations Page 23 § The integrand for a truss element is § The integrand can be written in matrix form as © 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

Section II – Static Analysis 2 nd Integral: Interpolation Functions Module 3 – Truss Element Equations Page 24 § Using the interpolation functions the partial derivatives can be written in terms of node point values § The integrand becomes § and the integral becomes where © 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

Section II – Static Analysis 2 nd Integral: Initial Stress Stiffness Matrix Module 3 – Truss Element Equations Page 25 Truss Element Initial Stress Stiffness Matrix © 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. General Form for other Element Types www. autodesk. com/edcommunity Education Community

Section II – Static Analysis Restoring Force Module 3 – Truss Element Equations Page 26 § The restoring force comes from the internal rate of virtual work term § The virtual rate of Green’s strain for the truss element is given on previous slides as § The truss element internal rate of virtual work becomes © 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

Section II – Static Analysis Restoring Force Module 3 – Truss Element Equations Page 27 § Since Txx and the components of BL and BNL are constant over the volume of the element, the previous equation reduces to or where This is a 6 x 1 array that has units of force. © 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

Section II – Static Analysis Module Summary Module 3 – Truss Element Equations Page 28 § This module has shown how to go from the incremental form of the rate of virtual work to the matrix equations for an element. § The deformation gradient and its variations and derivatives are key ingredients to this process. § A truss element was used because the element level integrations can be carried out by hand. § Matrix notation is used in-lieu of the more common indicial notation. © 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