Lec DSM II.pptx - Finite Element Methods THE DIRECT STIFFNESS METHOD II INSTRUCTOR DR SOHAIL IQBAL COURSE MSME F-19 WEEK NO 5 D AT E 10 OCTOBER 2018

# Lec DSM II.pptx - Finite Element Methods THE DIRECT...

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I N S T R U C TO R : D R S O H A I L I QB A L C OU R S E : M S M E F -1 9 W E E K N O : 5 D AT E : 1 0 O C TOB E R 2 0 1 8 Finite Element Methods THE DIRECT STIFFNESS METHOD II
SEQUENCE The Remaining DSM Assembly: Merge Governing Rules Assembly by Hand Using Augment-and-Add 3.3 Solution Applying Boundary Conditions by Reduction Solving for Displacements Post Processing Recovery of Reaction Forces Recovery of Internal Forces and Stresses Reaction Recovery: General Case Computer Oriented Assembly and Solution Assembly by Freedom Pointers Applying DBC by Modification Prescribed Nonzero Displacements Application of Nonzero-DBCs by Reduction Application of Nonzero-DBCs by Modification Matrix Forms of Nonzero-DBC Application Methods

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The Remaining DSM Steps The three breakdown steps: Disconnection Localization Formation of member stiffness The highest level of fragmentation. This is followed by the assembly and solution stage.
The Remaining DSM Steps Assembly involves merging the stiffness equations of each member into the global stiffness equations. For this to make sense, the member equations must be referred to a common coordinate system, which for a plane truss is the global Cartesian system { x, y}. This is done through the globalization process On the computer the formation, globalization and merge steps are done concurrently, member by member. After all members are processed we have the free-free master stiffness equations.

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The Remaining DSM Steps The next step is solution. This process embodies two steps: Applying boundary conditions (BC) Solving for the unknown joint displacements. Application of BC is done by modifying the free-free master stiffness equations to take into account which components of the joint displacements and forces are given and which are unknown
The Remaining DSM Steps The modified equations are submitted to a linear equation solver, which returns the unknown joint (node) displacements. Postprocessing steps may follow, in which derived quantities such as internal forces and stresses are recovered from the displacement solution.

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Assembly: Merge
Governing Rules Merge logic is mathematically governed by two rules of structural mechanics 1. Compatibility of displacements : The displacements of all members that meet at a joint are the same. 2 . Force equilibrium: The sum of internal forces exerted by all members that meet at a joint balances the external force applied to that joint.

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Governing Rules
Governing Rules But appending those terms enables us to write the compact matrix relation for the complete structure:

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Assembly by Hand Using Augment-and-Add To directly visualize how the two rules translate to merging logic, we first augment the global stiffness equations by adding zero rows and columns as appropriate to complete the force and displacement vectors.
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