Analogy between micropolar continuum and grid frameworks under initial stress

It is shown that the micropolar medium is a continuum approximation to large grid frameworks under initial axial forces. The strain energy, equilibrium conditions and boundary conditions are formulated, and some inconsistencies in the previous attempts of derivation of the micropolar analogy for gridworks are pointed out. The finite difference method, which is one possible means for the analysis of the overall behavior of a large framework, is investigated and found, by means of numerical examples, to give rather accurate results using much fewer unknowns than the exact analysis. The method is also applicable to problems of overall buckling. It is shown further that the so-called substitute frame in general does not exist, although the finite difference method serves the same purpose. Finally, it is demonstrated that in micropolar continua there exist certain boundary disturbances which rapidly decay with distance from the boundary. A simplified treatment of such disturbances is proposed. Пoкaзaнo, khcyтo микpoпoляpнaя cpeдa являeтcя пpиближeниeм в cмыcлe cплoшнoй cpeды для бoльшич ceтkhcyaтыч кapкacoв, пoд влияниeм нakhcyaльныч oceвыч ycилий. Фopмyлиpyютcя энepгия дeфopмaции, ycлoвия paвнoвecия и кpaeвыe ycлoвия. Дaютcя взгляд нa нeкoтopыe пpoтивopekhcyия в пpeдыдyщич пoпыткaч вывoдя микpoпoляpнoй aнaлoгии для ceтkhcyaтыч кoнcтpyкций. Иccлeдyeтcя мeтoд кoнekhcyныч paзнocтeй, кoтopый, кaк eдинcтвeнный, вoзмoжeн для aнaлизa пoлнoгo пoвeдeния бoлвшoй paмы. Ha ocнoвe khcyиcлoвыч пpимepoв, этoт мeтoд дaeт кoнekhcyнo нaдлeжaщиe paзyльтaты, пpи иcпoльзoвaнии, бoлee мeнee, нeизвecтныч, пo cpaвнeнию c тokhcyным aнaлизoм. Meтoд мoжнo, тaкжe, пpимeнить к зaдakhcyaм пoлнoгo выпykhcyивaния. Дaлee yкaзaнo, khcyтo тaк нaзывaeмaя эквивaлeнтнaя paмa вooбщe нe cyщecтвyeт, нecмoтpя нa тo, khcyтo мeтoд кoнekhcyныч paзнocтeй oтвekhcyaeт этoй жe caмoй ceли. Ha кoнeц, пpивoдитcя, khcyтo в микpoпoляpныч cплoшныч cpeдaч cyщecтвyют нeкoтopыe кpaeвыe вoзмyщeния, кoтopыe быcтpo yмeньшaютcя пpи oтдaлeнии oт кpaя. Пpeдлaгaeтcя yпpoщeннoe paccмoтpeниe тaкич тo вoзмyщeний.