unit8_notes2

# unit8_notes2 - 16.21 Techniques of Structural Analysis and...

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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2004 Unit #8 - Principle of Virtual Forces Ra´ul Radovitzky March 18, 2005 Principle of Virtual Forces (PVF) Consider a compatible displacement field u i in a deforming structure B of volume V and its associated strain field: 1 ij = ( u i,j + u j,i ) (1) 2 The displacement field necessarily satisfies the displacement (essential) bound- ary conditions: ∗ S on i u (2) u i = u where u ∗ i are the imposed displacements on the displacement part of the boundary S u . We multiply ( 1 ) by an somewhat arbitrary virtual stess field ¯ σ ij to be further defined below and integrate over the volume of the material. 1 ¯ ij − ( u i,j + u j,i ) σ ij dV = 0 (3) 2 V In addition, we multiply ( 2 ) by a set of somewhat arbitrary boundary virtual tractions t ¯ i : ¯ ∗ ) i ( u i − u t i dS = 0 (4) S u 1 V Adding up ( 3 ) and ( 4 ) we obtain: 1 ¯ ij − ( u i,j + u j,i ) σ ij dV + ( u i − u t i dS = 2 ¯ ∗ ) i 0 (5) V S u We will require the virtual stress field and the external virtual tractions to...
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unit8_notes2 - 16.21 Techniques of Structural Analysis and...

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