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A213-STPinho-OSA-lecture3

# A213-STPinho-OSA-lecture3 - Overview of last lectures...

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08/02/2009 STPinho 1 1 Structural Mechanics and Dynamics Overall Structural Analysis Lecture 3/5 Dr Silvestre Pinho 2 Overview of last lectures c The set of forces, R , and displacements, r , can be related in the stiffness form Kr R = c Or the flexibility form FR r = c The stiffness matrix of the structure is K and F is the flexibility matrix 3 Overview of last lectures c For constructing all of the terms in the stiffness matrix: r unit displacement r strains r stresses r forces b column of the stiffness matrix 4 Overview of last lectures c Alternatively , stiffness matrix of typical member is found c Then assembled to give stiffness matrix of complete structure 5 Overview of last lectures c Consider the typical member: α ρ 4 Ρ 4 ρ 1 Ρ 1 ρ 2 Ρ 2 ρ 3 Ρ 3 6 Overview of last lectures c The local stiffness matrix for the typical member is: c Need to set up mapping α α α α - α α - α α α α α - α - α - α α - α α α α α - α - α α α = ) ( sin ) sin( ) cos( ) ( sin ) sin( ) cos( ) sin( ) cos( ) ( cos ) sin( ) cos( ) ( cos ) ( sin ) sin( ) cos( ) ( sin ) sin( ) cos( ) sin( ) cos( ) ( cos ) sin( ) cos( ) ( cos L EA 2 2 2 2 2 2 2 2 k

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08/02/2009 STPinho 2 7 Overview of last lectures c Mapping defines how to assemble the local stiffness matrix into the global Mapping for Member 12 Local Global 1 3 2 4 3 1 4 2 8 Overview of last lectures Two processes for including supports: c First process b support DOFs are all mapped to zero and forgotten about c Second process b initially assume no supports and work then out equations 9 This lecture 10 Objectives of this lecture c Learn how to form and use the flexibility matrix 11 Outline of this lecture c Forming and using the flexibility matrix c Example 12 Forming and using the flexibility matrix
08/02/2009 STPinho 3 13 Forming and using the flexibility matrix Forming flexibility matrix: c Much less straightforward c Nowhere near so systematic as stiffness method 14 Forming and using the flexibility matrix c First, and often the most difficult , step is to decide upon how many redundancies there are in the structure c If structure is a mechanism, it cannot be solved (the stiffness method also gives a singular matrix in this case). Why? 15

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A213-STPinho-OSA-lecture3 - Overview of last lectures...

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