http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear 2013-01-12T07:13:21+05:00 MIT OpenCourseWare http://ocw.mit.edu en-US Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Introduction to nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">Introduction to the course</li> <li style="list-style: disc outside none;">The importance of nonlinear analysis</li> <li style="list-style: disc outside none;">Four illustrative films depicting actual and potential nonlinear analysis applications</li> <li style="list-style: disc outside none;">General recommendations for nonlinear analysis</li> <li style="list-style: disc outside none;">Modeling of problems</li> <li style="list-style: disc outside none;">Classification of nonlinear analyses</li> <li style="list-style: disc outside none;">Example analysis of a bracket, small and large deformations, elasto-plastic response</li> <li style="list-style: disc outside none;">Two computer-plotted animations <ul> <li style="list-style: circle outside none;">elasto-plastic large deformation response of a plate with a hole</li> <li style="list-style: circle outside none;">large displacement response of a diamond-shaped frame</li> </ul> </li> <li style="list-style: disc outside none;">The basic approach of an incremental solution</li> <li style="list-style: disc outside none;">Time as a variable in static and dynamic solutions</li> <li style="list-style: disc outside none;">The basic incremental/iterative equations</li> <li style="list-style: disc outside none;">A demonstrative static and dynamic nonlinear analysis of a shell</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/TJh7KPABk6I/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec01_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-1-introduction-to/id444084582?i=94870856>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/TJh7KPABk6I>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-1 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Basic considerations in nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">The principle of virtual work in general nonlinear analysis, including all material and geometric nonlinearities</li> <li style="list-style: disc outside none;">A simple instructive example</li> <li style="list-style: disc outside none;">Introduction to the finite element incremental solution, statement and physical explanation of governing finite element equations</li> <li style="list-style: disc outside none;">Requirements of equilibrium, compatibility, and the stress-strain law</li> <li style="list-style: disc outside none;">Nodal point equilibrium versus local equilibrium</li> <li style="list-style: disc outside none;">Assessment of accuracy of a solution</li> <li style="list-style: disc outside none;">Example analysis: Stress concentration factor calculation for a plate with a hole in tension</li> <li style="list-style: disc outside none;">Example analysis: Fracture mechanics stress intensity factor calculation for a plate with an eccentric crack in tension</li> <li style="list-style: disc outside none;">Discussion of mesh evaluation by studying stress jumps along element boundaries and pressure band plots</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/uVavsfJOsKc/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec02_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-2-basic-considerations/id444084582?i=94870842>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/uVavsfJOsKc>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-2 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Lagrangian continuum mechanics variables for general nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">The principle of virtual work in terms of the 2nd Piola-Kirchhoff stress and Green-Lagrange strain tensors</li> <li style="list-style: disc outside none;">Deformation gradient tensor</li> <li style="list-style: disc outside none;">Physical interpretation of the deformation gradient</li> <li style="list-style: disc outside none;">Change of mass density</li> <li style="list-style: disc outside none;">Polar decomposition of deformation gradient</li> <li style="list-style: disc outside none;">Green-Lagrange strain tensor</li> <li style="list-style: disc outside none;">2nd Piola-Kirchhoff stress tensor</li> <li style="list-style: disc outside none;">Important properties of the Green-Lagrange strain and 2nd Piola-Kirchhoff stress tensors</li> <li style="list-style: disc outside none;">Physical explanations of continuum mechanics variables</li> <li style="list-style: disc outside none;">Examples demonstrating the properties of the continuum mechanics variables</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/iOilZsS_cnM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec03_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-3-lagrangian-continuum/id444084582?i=94870843>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/iOilZsS_cnM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-3 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Total Lagrangian formulation for incremental general nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">Review of basic principle of virtual work equation, objective in incremental solution</li> <li style="list-style: disc outside none;">Incremental stress and strain decompositions in the total Lagrangian form of the principle of virtual work</li> <li style="list-style: disc outside none;">Linear and nonlinear strain increments</li> <li style="list-style: disc outside none;">Initial displacement effect</li> <li style="list-style: disc outside none;">Considerations for finite element discretization with continuum elements (isoparametric solids with translational degrees of freedom only) and structural elements (with translational and rotational degrees of freedom)</li> <li style="list-style: disc outside none;">Consistent linearization of terms in the principle of virtual work for the incremental solution</li> <li style="list-style: disc outside none;">The &quot;out-of-balance&quot; virtual work term</li> <li style="list-style: disc outside none;">Derivation of iterative equations</li> <li style="list-style: disc outside none;">The modified Newton-Raphson iteration, flowchart of complete solution</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/4-ehnTIyV0A/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec04_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-6-formulation-finite/id444084582?i=94870844>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/4-ehnTIyV0A>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-4 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Updated Lagrangian formulation for incremental general nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">Principle of virtual work in terms of 2nd Piola-Kirchhoff stresses and Green-Lagrange strains referred to the configuration at time t</li> <li style="list-style: disc outside none;">Incremental stress and strain decompositions in the updated Lagrangian form of the principle of virtual work</li> <li style="list-style: disc outside none;">Linear and nonlinear strain increments</li> <li style="list-style: disc outside none;">Consistent linearization of terms in the principle of virtual work</li> <li style="list-style: disc outside none;">The &quot;out-of-balance&quot; virtual work term</li> <li style="list-style: disc outside none;">Iterative equations for modified Newton-Raphson solution</li> <li style="list-style: disc outside none;">Flowchart of complete solution</li> <li style="list-style: disc outside none;">Comparison to total Lagrangian formulation</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/_d27jyqzoKQ/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec05_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-5-updated-lagrangian/id444084582?i=94870855>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/_d27jyqzoKQ>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-5 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Formulation of finite element matrices</p> <ul> <li style="list-style: disc outside none;">Summary of principle of virtual work equations in total and updated Lagrangian formulations</li> <li style="list-style: disc outside none;">Deformation-independent and deformation-dependent loading</li> <li style="list-style: disc outside none;">Materially-nonlinear-only analysis</li> <li style="list-style: disc outside none;">Dynamic analysis, implicit and explicit time integrations</li> <li style="list-style: disc outside none;">Derivations of finite element matrices for total and updated Lagrangian formulations, materially-nonlinear-only analysis</li> <li style="list-style: disc outside none;">Displacement and strain-displacement interpolation matrices</li> <li style="list-style: disc outside none;">Stress matrices</li> <li style="list-style: disc outside none;">Numerical integration and application of Gauss and Newton-Cotes formulas</li> <li style="list-style: disc outside none;">Example analysis: Elasto-plastic beam in bending</li> <li style="list-style: disc outside none;">Example analysis: A numerical experiment to test for correct element rigid body behavior</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/L27JVpZoz_Y/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec06_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-6-formulation-finite/id444084582?i=94870844>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/L27JVpZoz_Y>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-6 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Two- and three-dimensional solid elements; plane stress, plane strain, and axisymmetric conditions</p> <ul> <li style="list-style: disc outside none;">Isoparametric interpolations of coordinates and displacements</li> <li style="list-style: disc outside none;">Consistency between coordinate and displacement interpolations</li> <li style="list-style: disc outside none;">Meaning of these interpolations in large displacement analysis, motion of a material particle</li> <li style="list-style: disc outside none;">Evaluation of required derivatives</li> <li style="list-style: disc outside none;">The Jacobian transformations</li> <li style="list-style: disc outside none;">Details of strain-displacement matrices for total and updated Lagrangian formulations</li> <li style="list-style: disc outside none;">Example of 4-node two-dimensional element, details of matrices used</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/lsS2NysCVM4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec07_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-7-2d-3d-solid-elements/id444084582?i=94870841>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/lsS2NysCVM4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-7 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> The two-noded truss element - updated Lagrangian formulation</p> <ul> <li style="list-style: disc outside none;">Derivation of updated Lagrangian truss element displacement and strain-displacement matrices from continuum mechanics equations</li> <li style="list-style: disc outside none;">Assumption of large displacements and rotations but small strains</li> <li style="list-style: disc outside none;">Physical explanation of the matrices obtained directly by application of the principle of virtual work</li> <li style="list-style: disc outside none;">Effect of geometric (nonlinear strain) stiffness matrix</li> <li style="list-style: disc outside none;">Example analysis: Prestressed cable</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/ut04RoDL-gk/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec08_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-8-2-node-truss-element/id444084582?i=94870845>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ut04RoDL-gk>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-8 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> The two-noded truss element - total Lagrangian formulation</p> <ul> <li style="list-style: disc outside none;">Derivation of total Lagrangian truss element displacement and strain-displacement matrices from continuum mechanics equations</li> <li style="list-style: disc outside none;">Mathematical and physical explanation that only one component of the 2nd Piola-Kirchhoff stress tensor is nonzero</li> <li style="list-style: disc outside none;">Physical explanation of the matrices obtained directly by application of the principle of virtual work</li> <li style="list-style: disc outside none;">Discussion of initial displacement effect</li> <li style="list-style: disc outside none;">Comparison of updated and total Lagrangian formulations</li> <li style="list-style: disc outside none;">Example analysis: Collapse of a truss structure</li> <li style="list-style: disc outside none;">Example analysis: Large displacements of a cable</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/ChYAqW_MnW0/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec09_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-9-2-node-truss-element/id444084582?i=94870849>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ChYAqW_MnW0>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-9 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Solution of the nonlinear finite element equations in static analysis I</p> <ul> <li style="list-style: disc outside none;">Short review of Newton-Raphson iteration for the root of a single equation</li> <li style="list-style: disc outside none;">Newton-Raphson iteration for multiple degree of freedom systems</li> <li style="list-style: disc outside none;">Derivation of governing equations by Taylor series expansion</li> <li style="list-style: disc outside none;">Initial stress, modified Newton-Raphson and full Newton-Raphson methods</li> <li style="list-style: disc outside none;">Demonstrative simple example</li> <li style="list-style: disc outside none;">Line searches</li> <li style="list-style: disc outside none;">The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method</li> <li style="list-style: disc outside none;">Computations in the BFGS method as an effective scheme</li> <li style="list-style: disc outside none;">Flowcharts of modified Newton-Raphson, BFGS, and full Newton-Raphson methods</li> <li style="list-style: disc outside none;">Convergence criteria and tolerances</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/BekDicq9MdM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec10_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-10-solution-nonlinear/id444084582?i=95105820>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/BekDicq9MdM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-10 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Solution of the nonlinear finite element equations in static analysis II</p> <ul> <li style="list-style: disc outside none;">Automatic load step incrementation for collapse and post-buckling analysis</li> <li style="list-style: disc outside none;">Constant arc-length and constant increment of work constraints</li> <li style="list-style: disc outside none;">Geometrical interpretations</li> <li style="list-style: disc outside none;">An algorithm for automatic load incrementation</li> <li style="list-style: disc outside none;">Linearized buckling analysis, solution of eigenproblem</li> <li style="list-style: disc outside none;">Value of linearized buckling analysis</li> <li style="list-style: disc outside none;">Example analysis: Collapse of an arch - linearized buckling analysis and automatic load step incrementation, effect of initial geometric imperfections</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/N6rt_YxXuoA/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec11_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-11-solution-nonlinear/id444084582?i=95105819>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/N6rt_YxXuoA>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-11 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Demonstrative example solutions in static analysis</p> <ul> <li style="list-style: disc outside none;">Analysis of various problems to demonstrate, study, and evaluate solution methods in statics</li> <li style="list-style: disc outside none;">Example analysis: Snap-through of an arch</li> <li style="list-style: disc outside none;">Example analysis: Collapse analysis of an elastic-plastic cylinder</li> <li style="list-style: disc outside none;">Example analysis: Large displacement response of a shell</li> <li style="list-style: disc outside none;">Example analysis: Large displacements of a cantilever subjected to deformation-independent and deformation-dependent loading</li> <li style="list-style: disc outside none;">Example analysis: Large displacement response of a diamond-shaped frame</li> <li style="list-style: disc outside none;">Computer-plotted animation: Diamond-shaped frame</li> <li style="list-style: disc outside none;">Example analysis: Failure and repair of a beam/cable structure</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/4M-ijbL1gsk/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec12_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-12-demonstrative-example/id444084582?i=94870850>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/4M-ijbL1gsk>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-12 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Solution of nonlinear dynamic response I</p> <ul> <li style="list-style: disc outside none;">Basic procedure of direct integration</li> <li style="list-style: disc outside none;">The explicit central difference method, basic equations, details of computations performed, stability considerations, time step selection, relation of critical time step size to wave speed, modeling of problems</li> <li style="list-style: disc outside none;">Practical observations regarding use of the central difference method</li> <li style="list-style: disc outside none;">The implicit trapezoidal rule, basic equations, details of computations performed, time step selection, convergence of iterations, modeling of problems</li> <li style="list-style: disc outside none;">Practical observations regarding use of the trapezoidal rule</li> <li style="list-style: disc outside none;">Combination of explicit and implicit integrations</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/GyeJwReGKWg/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec13_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-13-solution-nonlinear/id444084582?i=94870860>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GyeJwReGKWg>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-13 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Solution of nonlinear dynamic response II</p> <ul> <li style="list-style: disc outside none;">Mode superposition analysis in nonlinear dynamics</li> <li style="list-style: disc outside none;">Substructuring in nonlinear dynamics, a schematic example of a building on a flexible foundation</li> <li style="list-style: disc outside none;">Study of analyses to demonstrate characteristics of procedures for nonlinear dynamic solutions</li> <li style="list-style: disc outside none;">Example analysis: Wave propagation in a rod</li> <li style="list-style: disc outside none;">Example analysis: Dynamic response of a three degree-of-freedom system using the central difference method</li> <li style="list-style: disc outside none;">Example analysis: Ten-story tapered tower subjected to blast loading</li> <li style="list-style: disc outside none;">Example analysis: Simple pendulum undergoing large displacements</li> <li style="list-style: disc outside none;">Example analysis: Pipe whip solution</li> <li style="list-style: disc outside none;">Example analysis: Control rod drive housing with lower support</li> <li style="list-style: disc outside none;">Example analysis: Spherical cap under uniform pressure loading</li> <li style="list-style: disc outside none;">Example analysis: Solution of fluid-structure interaction problem</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/D_lVfCfGVao/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec14_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-14-solution-nonlinear/id444084582?i=94870851>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/D_lVfCfGVao>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-14 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Use of elastic constitutive relations in total Lagrangian formulation</p> <ul> <li style="list-style: disc outside none;">Basic considerations in modeling material response</li> <li style="list-style: disc outside none;">Linear and nonlinear elasticity</li> <li style="list-style: disc outside none;">Isotropic and orthotropic materials</li> <li style="list-style: disc outside none;">One-dimensional example, large strain conditions</li> <li style="list-style: disc outside none;">The case of large displacement/small strain analysis, discussion of effectiveness using the total Lagrangian formulation</li> <li style="list-style: disc outside none;">Hyperelastic material model (Mooney-Rivlin) for analysis of rubber-type materials</li> <li style="list-style: disc outside none;">Example analysis: Solution of a rubber tensile test specimen</li> <li style="list-style: disc outside none;">Example analysis: Solution of a rubber sheet with a hole</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/BH06RODmHsc/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec15_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-15-elastic-constitutive/id444084582?i=95105821>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/BH06RODmHsc>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-15 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Use of elastic constitutive relations in updated Lagrangian formulation</p> <ul> <li style="list-style: disc outside none;">Use of updated Lagrangian (U.L.) formulation</li> <li style="list-style: disc outside none;">Detailed comparison of expressions used in total Lagrangian (T.L.) and U.L. formulations; strains, stresses, and constitutive relations</li> <li style="list-style: disc outside none;">Study of conditions to obtain in a general incremental analysis the same results as in the T.L. formulation, and vice versa</li> <li style="list-style: disc outside none;">The special case of elasticity</li> <li style="list-style: disc outside none;">The Almansi strain tensor</li> <li style="list-style: disc outside none;">One-dimensional example involving large strains</li> <li style="list-style: disc outside none;">Analysis of large displacement/small strain problems</li> <li style="list-style: disc outside none;">Example analysis: Large displacement solution of frame using updated and total Lagrangian formulations</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/ieV1yZ1l7-c/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec16_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-16-elastic-constitutive/id444084582?i=95107107>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ieV1yZ1l7-c>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-16 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Modeling of elasto-plastic and creep response I</p> <ul> <li style="list-style: disc outside none;">Basic considerations in modeling inelastic response</li> <li style="list-style: disc outside none;">A schematic review of laboratory test results, effects of stress level, temperature, strain rate</li> <li style="list-style: disc outside none;">One-dimensional stress-strain laws for elasto-plasticity, creep, and viscoplasticity</li> <li style="list-style: disc outside none;">Isotropic and kinematic hardening in plasticity</li> <li style="list-style: disc outside none;">General equations of multiaxial plasticity based on a yield condition, flow rule, and hardening rule</li> <li style="list-style: disc outside none;">Example of von Mises yield condition and isotropic hardening, evaluation of stress-strain law for general analysis</li> <li style="list-style: disc outside none;">Use of plastic work, effective stress, effective plastic strain</li> <li style="list-style: disc outside none;">Integration of stresses with subincrementation</li> <li style="list-style: disc outside none;">Example analysis: Plane strain punch problem</li> <li style="list-style: disc outside none;">Example analysis: Elasto-plastic response up to ultimate load of a plate with a hole</li> <li style="list-style: disc outside none;">Computer-plotted animation: Plate with a hole</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/GpV_9EtObvs/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec17_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-17-modeling-elasto/id444084582?i=94870846>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GpV_9EtObvs>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-17 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Modeling of elasto-plastic and creep response II</p> <ul> <li style="list-style: disc outside none;">Strain formulas to model creep strains</li> <li style="list-style: disc outside none;">Assumption of creep strain hardening for varying stress situations</li> <li style="list-style: disc outside none;">Creep in multiaxial stress conditions, use of effective stress and effective creep strain</li> <li style="list-style: disc outside none;">Explicit and implicit integrations of stresses</li> <li style="list-style: disc outside none;">Selection of size of time step in stress integration</li> <li style="list-style: disc outside none;">Thermo-plasticity and creep, temperature-dependency of material constants</li> <li style="list-style: disc outside none;">Example analysis: Numerical uniaxial creep results</li> <li style="list-style: disc outside none;">Example analysis: Collapse analysis of a column with offset load</li> <li style="list-style: disc outside none;">Example analysis: Analysis of cylinder subjected to heat treatment</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/Us2Myb5csu4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec18_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-18-modeling-elasto/id444084582?i=94870861>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Us2Myb5csu4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-18 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Beam, plate, and shell elements I</p> <ul> <li style="list-style: disc outside none;">Brief review of major formulation approaches</li> <li style="list-style: disc outside none;">The degeneration of a three-dimensional continuum to beam and shell behavior</li> <li style="list-style: disc outside none;">Basic kinematic and static assumptions used</li> <li style="list-style: disc outside none;">Formulation of isoparametric (degenerate) general shell elements of variable thickness for large displacements and rotations</li> <li style="list-style: disc outside none;">Geometry and displacement interpolations</li> <li style="list-style: disc outside none;">The nodal director vectors</li> <li style="list-style: disc outside none;">Use of five or six nodal point degrees of freedom, theoretical considerations and practical use</li> <li style="list-style: disc outside none;">The stress-strain law in shell analysis, transformations used at shell element integration points</li> <li style="list-style: disc outside none;">Shell transition elements, modeling of transition zones between solids and shells, shell intersections</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/-BYC6cNSO78/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec19_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-19-beam-plate-shell/id444084582?i=94870847>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/-BYC6cNSO78>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-19 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> Beam, plate, and shell elements II</p> <ul> <li style="list-style: disc outside none;">Formulation of isoparametric (degenerate) beam elements for large displacements and rotations</li> <li style="list-style: disc outside none;">A rectangular cross-section beam element of variable thickness; coordinate and displacement interpolations</li> <li style="list-style: disc outside none;">Use of the nodal director vectors</li> <li style="list-style: disc outside none;">The stress-strain law</li> <li style="list-style: disc outside none;">Introduction of warping displacements</li> <li style="list-style: disc outside none;">Example analysis: 180 degrees, large displacement twisting of a ring</li> <li style="list-style: disc outside none;">Example analysis: Torsion of an elastic-plastic cross-section</li> <li style="list-style: disc outside none;">Recommendations for the use of isoparametric beam and shell elements</li> <li style="list-style: disc outside none;">The phenomena of shear and membrane locking as observed for certain elements</li> <li style="list-style: disc outside none;">Study of solutions of straight and curved cantilevers modeled using various elements</li> <li style="list-style: disc outside none;">An effective 4-node shell element (the MITC4 element) for analysis of general shells</li> <li style="list-style: disc outside none;">The patch test, theoretical and practical considerations</li> <li style="list-style: disc outside none;">Example analysis: Solution of a three-dimensional spherical shell</li> <li style="list-style: disc outside none;">Example analysis: Solution of an open box</li> <li style="list-style: disc outside none;">Example analysis: Solution of a square plate, including use of distorted elements</li> <li style="list-style: disc outside none;">Example analysis: Solution of a 30-degree skew plate</li> <li style="list-style: disc outside none;">Example analysis: Large displacement solution of a cantilever</li> <li style="list-style: disc outside none;">Example analysis: Collapse analysis of an I-beam in torsion</li> <li style="list-style: disc outside none;">Example analysis: Collapse analysis of a cylindrical shell</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/EsiGSf2bt9k/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec20_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-20-beam-plate-shell/id444084582?i=94870859>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/EsiGSf2bt9k>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-20 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> A demonstrative computer session using ADINA - linear analysis</p> <ul> <li style="list-style: disc outside none;">Use of the computer program ADINA for finite element analysis, discussion of data preparation, program solution, and display of results</li> <li style="list-style: disc outside none;">Capabilities of ADINA</li> <li style="list-style: disc outside none;">Computer laboratory demonstration - Part I</li> <li style="list-style: disc outside none;">Linear analysis of a plate with a hole for the stress concentration factor</li> <li style="list-style: disc outside none;">Data input preparation and mesh generation</li> <li style="list-style: disc outside none;">Solution of the model</li> <li style="list-style: disc outside none;">Study and evaluation of results using plots of stresses, stress jumps, and pressure bands</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/Jfibd3L_E_o/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec21_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-21-demonstration-using/id444084582?i=94870862>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Jfibd3L_E_o>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-21 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Topics:</strong> A demonstrative computer session using ADINA - nonlinear analysis</p> <ul> <li style="list-style: disc outside none;">Use of ADINA for elastic-plastic analysis of a plate with a hole</li> <li style="list-style: disc outside none;">Computer laboratory demonstration - Part II</li> <li style="list-style: disc outside none;">Selection of solution parameters and input data preparation</li> <li style="list-style: disc outside none;">Study of the effect of using different kinematic assumptions (small or large strains) in the finite element solution</li> <li style="list-style: disc outside none;">Effect of a shaft in the plate hole, assuming frictionless contact</li> <li style="list-style: disc outside none;">Effect of expanding shaft</li> <li style="list-style: disc outside none;">Study and evaluation of solution results</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/NJUIkyavUD4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10nonlinear/MITRES2_002S10nonlinear_lec22_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-22-demonstration-using/id444084582?i=94870853>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/NJUIkyavUD4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/nonlinear/lecture-22 Bathe, Klaus-Jürgen 2011-06-23T15:56:42+05:00 en-US MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm