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Inside_the_code_02

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Inside_the_code_02

  1. 1. INSIDE THE CODE THE OBSCURE WORK OF ROTATION MATRICES (1st part) Structural simulation codes make an extensive use of Rotation Matrices R. These matrices are embedded into the codes. User does not see them at all but –transparently- makes a great use of them. For a 3D beam (red segment in the picture J->K overlapped to local XM), R has 12 rows and 12 columns. The core of R is given by four diagonal repeated sub-matrices r having as vectors rows cosines direction of local axes XM, YM, ZM. with respect to the global axes XS, YS, ZS. To transform local stiffness matrix Sloc in global reference system, the basic transformation is given by: r = 𝒓 𝟏,𝟏 𝒓 𝟏,𝟐 𝒓 𝟏,𝟑 𝒓 𝟐,𝟏 𝒓 𝟐,𝟐 𝒓 𝟐,𝟑 𝒓 𝟑,𝟏 𝒓 𝟑,𝟐 𝒓 𝟑,𝟑  3x3 𝑺 𝒈𝒍𝒐 = 𝑹 𝑇 𝑺𝑙𝑜𝑐 𝑹 R is an orthogonal matrix having Det=1, being RT the transpose of orthogonal matrix it follows that RT= R-1. Since the generic members position in the space can be thought also as the sum of 3 rigid rotations, R can be built applying a triple matrix product considering three single rotations in this order (see figure for angles): R = Rα Rγ Rβ. When beam cross section hasn’t principal inertia axes or their space orientation is immaterial (i.e space truss), Rα is simply the Identity matrix I (3x3). (to be continued..) Original content by https://it.linkedin.com/in/fabrizio-sperduto R= 𝒓 𝟎 𝟎 𝟎 𝟎 𝒓 𝟎 𝟎 𝟎 𝟎 𝒓 𝟎 𝟎 𝟎 𝟎 𝒓  12 x12 Node J XM Node k

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