http://ocw.mit.edu/resources/res-2-002-finite-element-procedures-for-solids-and-structures-spring-2010/linear 2013-01-12T07:13:18+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> Some basic concepts of engineering analysis</p> <ul> <li style="list-style: disc outside none;">Introduction to the course, objective of lectures</li> <li style="list-style: disc outside none;">Some basic concepts of engineering analysis, discrete and continuous systems, problem types: steady-state, propagation, and eigenvalue problems</li> <li style="list-style: disc outside none;">Analysis of discrete systems: example analysis of a spring system</li> <li style="list-style: disc outside none;">Basic solution requirements</li> <li style="list-style: disc outside none;">Use and explanation of the modern direct stiffness method</li> <li style="list-style: disc outside none;">Variational formulation</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe&nbsp;</p>Thumbnail - <a href= http://img.youtube.com/vi/oNqSzzycRhw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec01_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-1-some-basic-concepts/id443981620?i=94847721>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/oNqSzzycRhw>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/linear/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>&nbsp;Topics:</strong> Analysis of continuous systems; differential and variational formulations</p> <ul> <li style="list-style: disc outside none;">Basic concepts in the analysis of continuous systems</li> <li style="list-style: disc outside none;">Differential and variational formulations</li> <li style="list-style: disc outside none;">Essential and natural boundary conditions</li> <li style="list-style: disc outside none;">Definition of C^m-1 variational problem</li> <li style="list-style: disc outside none;">Principle of virtual displacements</li> <li style="list-style: disc outside none;">Relation between stationarity of total potential, the principle of virtual displacements, and the differential formulation</li> <li style="list-style: disc outside none;">Weighted residual methods, Galerkin, least squares methods</li> <li style="list-style: disc outside none;">Ritz analysis method</li> <li style="list-style: disc outside none;">Properties of the weighted residual and Ritz methods</li> <li style="list-style: disc outside none;">Example analysis of a nonuniform bar, solution accuracy, introduction to the finite element method</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/o2Vlt1avXCs/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec02_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-2-analysis-continuous/id443981620?i=94847719>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/o2Vlt1avXCs>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/linear/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>&nbsp;Topics:</strong> Formulation of the displacement-based finite element method</p> <ul> <li style="list-style: disc outside none;">General effective formulation of the displacement-based finite element method</li> <li style="list-style: disc outside none;">Principle of virtual displacements</li> <li style="list-style: disc outside none;">Discussion of various interpolation and element matrices</li> <li style="list-style: disc outside none;">Physical explanation of derivations and equations</li> <li style="list-style: disc outside none;">Direct stiffness method</li> <li style="list-style: disc outside none;">Static and dynamic conditions</li> <li style="list-style: disc outside none;">Imposition of boundary conditions</li> <li style="list-style: disc outside none;">Example analysis of a nonuniform bar, detailed discussion of element matrices</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/20WSeL4tz2k/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec03_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-3-the-displacement/id443981620?i=94847716>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/20WSeL4tz2k>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/linear/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>&nbsp;Topics:</strong> Generalized coordinate finite element models</p> <ul> <li style="list-style: disc outside none;">Classification of problems: truss, plane stress, plane strain, axisymmetric, beam, plate, and shell conditions; corresponding displacement, strain, and stress variables</li> <li style="list-style: disc outside none;">Derivation of generalized coordinate models</li> <li style="list-style: disc outside none;">One-, two-, three-dimensional elements, plate and shell elements</li> <li style="list-style: disc outside none;">Example analysis of a cantilever plate, detailed derivation of element matrices</li> <li style="list-style: disc outside none;">Lumped and consistent loading</li> <li style="list-style: disc outside none;">Example results</li> <li style="list-style: disc outside none;">Summary of the finite element solution process</li> <li style="list-style: disc outside none;">Solution errors</li> <li style="list-style: disc outside none;">Convergence requirements, physical explanations, the patch test</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/tkU3bM_6YLk/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec04_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-4-generalized-coordinate/id443981620?i=94847720>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/tkU3bM_6YLk>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/linear/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> Implementation of methods in computer programs; examples SAP, ADINA</p> <ul> <li style="list-style: disc outside none;">Implementation of the finite element method</li> <li style="list-style: disc outside none;">The computer programs SAP and ADINA</li> <li style="list-style: disc outside none;">Details of allocation of nodal point degrees of freedom, calculation of matrices, the assemblage process</li> <li style="list-style: disc outside none;">Example analysis of a cantilever plate</li> <li style="list-style: disc outside none;">Out-of-core solution</li> <li style="list-style: disc outside none;">Effective nodal-point numbering</li> <li style="list-style: disc outside none;">Flow chart of total solution process</li> <li style="list-style: disc outside none;">Introduction to different effective finite elements used in one-, two-, three-dimensional, beam, plate, and shell analyses</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/Tf0FDnIUHCI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec05_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-5-implementation-methods/id443981620?i=95105469>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Tf0FDnIUHCI>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/linear/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: Formulation and calculation of isoparametric models</strong></p> <ul> <li style="list-style: disc outside none;">Formulation and calculation of isoparametric continuum elements</li> <li style="list-style: disc outside none;">Truss, plane-stress, plane-strain, axisymmetric, and three-dimensional elements</li> <li style="list-style: disc outside none;">Variable-number-nodes elements, curved elements</li> <li style="list-style: disc outside none;">Derivation of interpolations, displacement and strain interpolation matrices, the Jacobian transformation</li> <li style="list-style: disc outside none;">Various examples; shifting of internal nodes to achieve stress singularities for fracture mechanics analysis</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/E2HglWZcfKw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec06_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-6-formulation-calculation/id443981620?i=94847717>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/E2HglWZcfKw>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/linear/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> Formulation of structural elements</p> <ul> <li style="list-style: disc outside none;">Formulation and calculation of isoparametric structural elements</li> <li style="list-style: disc outside none;">Beam, plate, and shell elements</li> <li style="list-style: disc outside none;">Formulation using Mindlin plate theory and unified general continuum formulation</li> <li style="list-style: disc outside none;">Assumptions used including shear deformations</li> <li style="list-style: disc outside none;">Demonstrative examples: two-dimensional beam, plate elements</li> <li style="list-style: disc outside none;">Discussion of general variable-number-nodes elements</li> <li style="list-style: disc outside none;">Transition elements between structural and continuum elements</li> <li style="list-style: disc outside none;">Low- versus high-order elements</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/Krb1fF2Ycgo/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec07_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-7-formulation-structural/id443981620?i=94847722>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Krb1fF2Ycgo>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/linear/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> Numerical integrations, modeling considerations</p> <ul> <li style="list-style: disc outside none;">Evaluation of isoparametric element matrices</li> <li style="list-style: disc outside none;">Numerical integrations, Gauss, Newton-Cotes formulas</li> <li style="list-style: disc outside none;">Basic concepts used and actual numerical operations performed</li> <li style="list-style: disc outside none;">Practical considerations</li> <li style="list-style: disc outside none;">Required order of integration, simple examples</li> <li style="list-style: disc outside none;">Calculation of stresses</li> <li style="list-style: disc outside none;">Recommended elements and integration orders for one-, two-, three-dimensional analysis, and plate and shell structures</li> <li style="list-style: disc outside none;">Modeling considerations using the elements</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/6pHHh67t6F8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec08_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-8-numerical-integrations/id443981620?i=94847718>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/6pHHh67t6F8>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/linear/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> Solution of finite element equilibrium equations in static analysis</p> <ul> <li style="list-style: disc outside none;">Solution of finite element equations in static analysis</li> <li style="list-style: disc outside none;">Basic Gauss elimination</li> <li style="list-style: disc outside none;">Static condensation</li> <li style="list-style: disc outside none;">Substructuring</li> <li style="list-style: disc outside none;">Multi-level substructuring</li> <li style="list-style: disc outside none;">Frontal solution</li> <li style="list-style: disc outside none;">LDL^T-factorization (column reduction scheme) as used in SAP and ADINA</li> <li style="list-style: disc outside none;">Cholesky factorization</li> <li style="list-style: disc outside none;">Out-of-core solution of large systems</li> <li style="list-style: disc outside none;">Demonstration of basic techniques using simple examples</li> <li style="list-style: disc outside none;">Physical interpretation of the basic operations used</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/L98VIorbFB0/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec09_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-9-solution-equilibrium/id443981620?i=94847723>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/L98VIorbFB0>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/linear/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 finite element equilibrium equations in dynamic analysis</p> <ul> <li style="list-style: disc outside none;">Solution of dynamic response by direct integration</li> <li style="list-style: disc outside none;">Basic concepts used</li> <li style="list-style: disc outside none;">Explicit and implicit techniques</li> <li style="list-style: disc outside none;">Implementation of methods</li> <li style="list-style: disc outside none;">Detailed discussion of central difference and Newmark methods</li> <li style="list-style: disc outside none;">Stability and accuracy considerations</li> <li style="list-style: disc outside none;">Integration errors</li> <li style="list-style: disc outside none;">Modeling of structural vibration and wave propagation problems</li> <li style="list-style: disc outside none;">Selection of element and timestep sizes</li> <li style="list-style: disc outside none;">Recommendations on the use of the methods in practice</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/gzG2p-Su8Vw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec10_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-10-solution-equilibrium/id443981620?i=94847725>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/gzG2p-Su8Vw>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/linear/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> Mode superposition analysis; time history</p> <ul> <li style="list-style: disc outside none;">Solution of dynamic response by mode superposition</li> <li style="list-style: disc outside none;">The basic idea of mode superposition</li> <li style="list-style: disc outside none;">Derivation of decoupled equations</li> <li style="list-style: disc outside none;">Solution with and without damping</li> <li style="list-style: disc outside none;">Caughey and Rayleigh damping</li> <li style="list-style: disc outside none;">Calculation of damping matrix for given damping ratios</li> <li style="list-style: disc outside none;">Selection of number of modal coordinates</li> <li style="list-style: disc outside none;">Errors and use of static correction</li> <li style="list-style: disc outside none;">Practical considerations</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/pSdxdfBnu0I/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec11_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-11-mode-superposition/id443981620?i=94847715>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/pSdxdfBnu0I>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/linear/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> Solution methods for calculations of frequencies and mode shapes</p> <ul> <li style="list-style: disc outside none;">Solution methods for finite element eigenproblems</li> <li style="list-style: disc outside none;">Standard and generalized eigenproblems</li> <li style="list-style: disc outside none;">Basic concepts of vector iteration methods, polynomial iteration techniques, Sturm sequence methods, transformation methods</li> <li style="list-style: disc outside none;">Large eigenproblems</li> <li style="list-style: disc outside none;">Details of the determinant search and subspace iteration methods</li> <li style="list-style: disc outside none;">Selection of appropriate technique, practical considerations</li> </ul> <p><strong>Instructor:</strong> Klaus-Jürgen Bathe</p>Thumbnail - <a href= http://img.youtube.com/vi/ejZtBwLUE3Y/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES2_002S10linear/MITRES2_002S10linear_lec12_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/podcast/lecture-12-solution-methods/id443981620?i=94847714>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ejZtBwLUE3Y>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/linear/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