Estimation of Structural Deformed Shapes Using Limited Number of Displacement Measurements

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Received October 15 2012, Revised December 27 2012, Accepted April 26 2013

Copyright ⵑ 2013 by the Korean Society of Civil Engineers

ǣǣȀȀǤǤȀͳͲǤͳʹ͸ͷʹȀǤʹͲͳ͵Ǥ͵͵ǤͶǤͳʹͻͷ ǤǤǤ

㬚ⷓ᧚#ኂ㏟#≾ⳂἺ#ⴲⱧ㬚#ጪ⸮⃺#≾㯓#㯓♿#㍒ⷓ

ౖஜ෹ȵ׌਎ஜȵ෉਎߾ȵԳઽஂ

Choi, Junho*, Kim, Seungjun**, Han, Seungryong***, Kang, Youngjong****

Estimation of Structural Deformed Shapes Using Limited Number of Displacement Measurements

ABSTRACT

The structural deformed shape is important information to structural analysis. If the sufficient measuring points are secured at the structural monitoring system, reasonable and accurate structural deformation shapes can be obtained and structural analysis is possible using this deformation. However, the accurate estimation of the global structural shapes might be difficult if sufficient measuring points are not secure under cost limitations. In this study, SFSM-LS algorithm, the economic and effective estimation method for the structural deformation shapes with limited displacement measuring points is developed and suggested. In the suggested method, the global structural deformation shape is determined by the superposition of the pre-investigated structural deformed shapes obtained by preliminary FE analyses, with their optimum weight factors which lead minimization of the estimate errors. 2-span continuous bridge model is used to verify developed algorithm and parametric studies are performed. By the parametric studies, the characteristics of the estimation results obtained by the suggested method were investigated considering essential parameters such as pre-investigated structural shapes, locations and numbers of displacement measuring points. By quantitative comparison of estimation results with the conventional methods such as polynomial, Lagrange and spline interpolation, the applicability and accuracy of the suggested method was validated.

Key words : Structural deformed shape, Measured displacement, Interpolation, SFSM-LS, Structural health monitoring

Ⅹಾ

Ǎ᳑ྜྷ᮹ᄡ⩶⩶ᔢᮡǍ᳑⧕ᕾᮥ᭥⦽ᵲ᫵⦽ᱶᅕᯕ݅. Ǎ᳑ྜྷ༉ܩ░ย᜽ᜅ▽ᨱᕽࠥ∊ᇥ⦽ᄡ᭥ĥ⊂ᱱᯕ⪶ᅕࡹᨕ໦⪶⦹Ł⧊ญᱢᯙǍ

᳑ྜྷ᮹ᄡ⩶⩶ᔢᯕࠥ⇽ࡽ݅໕ᯕෝᯕᬊ⦽Ǎ᳑⧕ᕾᯕa܆⧁äᯕ݅. ⦹ḡอᝅᱽǍ᳑ྜྷᨱᕽ۵⦽ᱶࡽእᬊᮝಽᯙ⧕∊ᇥ⦽ᄡ᭥ĥ⊂ᱱᯕ

⪶ᅕࡹḡ༜⦹ʑভྙᨱ⧊ญᱢᯙǍ᳑ྜྷᄡ⩶⩶ᔢ᮹ࠥ⇽ᯕᨕಖ݅. ᅙᩑǍᨱᕽ۵ĞᱽᱢᯕŁ⧊ญᱢᯙǍ᳑ྜྷᄡ⩶⩶ᔢ⇵ᱶᮥ᭥⧕↽ᗭ ᮹ᄡ᭥ĥ⊂ߑᯕ░ෝᯕᬊ⦽⬉ŝᱢᯙǍ᳑ྜྷ᮹Ñ࠺⩶ᔢ⇵ᱶʑჶᯙSFSM-LS᦭Łญ᷹ᮥ}ၽ⦹ᩡ݅. ᅙʑჶᮡǍ᳑ྜྷ᮹ᄡ⩶⩶ᔢᮥ⇵

ᱶ⦹ʑ᭥⧕ĥ⊂ݡᔢǍ᳑ྜྷ᮹ᔍᱥᮁ⦽᫵ᗭ⧕ᕾᮥ☖⧕ᩍ్Ǎ᳑Ñ࠺⩶ᔢᮥʑᅙǍ᳑⩶ᔢ⧉ᙹಽᱶ᮹⦹Ł, ᯕॅ⧉ᙹෝ⇵ᱶᄡ᭥᮹᪅

₉ෝ↽ᗭ⪵᜽┅۵b⧉ᙹ᮹aᵲ⊹ಽ៉ᵲℊ⦽݅. 2ĞeᩑᗮƱ༉ߙ᮹ᙹ⊹⧕ᕾᮥ☖⧕}ၽࡽ᦭Łญ᷹ᮥá᷾⦹Łๅ}ᄡᙹᩑǍෝᙹ⧪

⦹ᩡ݅. }ၽࡽ᦭Łญ᷹᮹ๅ}ᄡᙹᯙǍ᳑⩶ᔢ⧉ᙹ, ᄡ᭥ĥ⊂᭥⊹, ᄡ᭥ĥ⊂}ᗭᨱݡ⦽⩶ᔢ⇵ᱶđŝ᮹✚ᖒᮥᇥᕾ⦹ŁPolynomial, Lagrange, Spline ᅕeჶŝ⩶ᔢ⇵ᱶᱶၡࠥෝእƱ⦹ᩍ}ၽࡽʑჶ᮹ᱢᬊᖒᮥá᷾⦹ᩡ݅. ᯕෝ☖⧕ᱢᮡ}ᗭ᮹ᄡ᭥ߑᯕ░ಽᱶၡ⦽⩶

ᔢᮥ⇵ᱶ⦹۵đŝෝࠥ⇽⦹ᩍᱽᦩࡽʑჶ᮹ᬒᖒᮥ᯦᷾⦹ᩡ݅.

áᔪᨕ Ǎ᳑ྜྷᄡ⩶⩶ᔢ, ᄡ᭥ĥ⊂, ᅕeჶ, SFSM-LS, Õᱥࠥ༉ܩ░ย

ĵܓėॡ

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1. ᕽು

Ǎ᳑ྜྷᯕݡ⩶⪵ࡹŁי⬥⪵ࡽǍ᳑ྜྷᯕᱱ₉᷾a⧉ᨱ঑௝

Ǎ᳑ྜྷ᮹ᮁḡšญᨱݡ⦽ᵲ᫵ᖒᯕᱱ₉ᇡbࡹŁᯩᮝ໑, ᯕᨱ

ݡ⦽ እᬊ ੱ⦽ ᱱ₉ ᷾a⦹Ł ᯩ݅.

⬉ᮉᱢᯙǍ᳑ྜྷ᮹ᮁḡšญෝ᭥⧕ᕽ݅᧲⦽ᖝᕽෝ⪽ᬊ⦹ᩍ

Ǎ᳑ྜྷ᮹Ñ࠺ᮥĥ⊂⦹Ł, ĥ⊂ࡽߑᯕ░ෝ⪽ᬊ⦹ᩍǍ᳑Õᱥᖒ

༉ܩ░ยၰÕᱥࠥ⠪aaᯕ൉ᨕḡŁᯩ݅. ᯕ౑݅᧲⦽ᖝᕽෝ

⪽ᬊ⦽ ᮁḡšญ ʑᚁᮡ ĥ⊂ʑᚁᯕ ᱱ₉ ၽᱥ⧉ᨱ ঑௝ ᯕᨱ

ݡ⦽እᬊᯕqᗭ⦹Łᱶၡࠥa⨆ᔢࢉᨱ঑௝a܆⦹íࡹᨩ݅.

ᖝᕽෝᯕᬊ⦹ᩍĥ⊂ࡹ۵Ǎ᳑ྜྷÑ࠺ᱶᅕ۵ᵝಽaᗮࠥ᪡

ᄡ᭥ᯕ݅. Ǎ᳑ྜྷ᮹Ñ࠺ᄡ᭥ᱶᅕ۵Ǎ᳑Õᱥࠥ⠪aෝ᭥⦽

ๅᬑᵲ᫵⦽ᱶᅕᯕ݅. ĥ⊂ᄡ᭥۵Ǎ᳑ྜྷ᮹ᱶᱢ · ᵡᱶᱢ · ࠺ᱢ Ñ࠺᮹ ༉ु Ñ࠺ᮥ ၹᩢ ⧁ ᙹ ᯩᮝ໑, Ǎ᳑ྜྷ᮹ ༉ु ᱩᱱ᮹

ᄡ᭥ෝĥ⊂⦹ᩍǍ᳑ྜྷ᮹ ᄡ⩶⩶ᔢᯕࠥ⇽ࡽ݅໕Ǎ᳑⧕ᕾᮥ

☖⧕Ǎ᳑ྜྷ᮹Õᱥࠥෝ⠪a⧁ᙹᯩ݅. ⦹ḡอ, ᝅᱽǍ᳑ྜྷᨱᕽ ۵aᗮࠥᨱݡ⦽⪽ᬊʑᚁᮡᔢݚᇡᇥၽᱥ(᳑ᙹḥ॒, 2007)⧕

ᯩ۵ၹ໕ᄡ᭥ĥ⊂ᱶᅕaๅᬑᇡ᳒⦹ʑভྙᨱĥ⊂ᄡ᭥᮹

⪽ᬊᙹᵡᯕ ၙၙ⦽ ᝅᱶᯕ݅.

ŝÑᔢ᜽ᄡ᭥ĥ⊂ᮥ᭥⦽ᱢᱩ⦽ᰆእ᮹ᇡᰍಽᯙ⧕ᕽ↽ɝᨱ

ॅᨕᕽ᧝ݡ⩶Ǎ᳑ྜྷᨱᕽᔢ᜽ᄡ᭥ĥ⊂ᯕᯕ൉ᨕḡŁᯩᮝ໑

݅᧲⦽ᰆእෝ⪽ᬊ⦹ᩍǍ᳑ྜྷᨱᕽᄡ᭥ෝĥ⊂⦹Łᯩ݅. Ų❭ʑ, ౩ᯕᲙᄡ᭥ĥ᪡zᮡᄡ᭥ĥ⊂ᰆእᨱᕽ⩥ᰍ᮹ᔍᰆƱ, ⩥ᙹƱ᪡

zᮡ ݡ⩶ Ǎ᳑ྜྷᨱᕽ۵ GNSS(Global Navigation Satellite System)ෝ ᯕᬊ⦽ ᔢ᜽ ᄡ᭥ ĥ⊂ᯕ ᯕ൉ᨕḡŁ ᯩ݅.(Im ॒, 2011) ᝅᱽᬕᩢᵲᯙ⩥ᙹƱᨱᕽGNSSෝᯕᬊ⦽ĥ⊂ᄡ᭥ෝ

౩ᯕᲙᄡ᭥ĥෝᯕᬊ⦽ᄡ᭥᪡á᷾⧉ᮝಽ៉GNSSᨱݡ⦽ᱢᬊ ᖒᮥ ᇥᕾ(႑ᯙ⪹ ॒, 2007)⦹Ł GNSSෝ ☖⧕ ĥ⊂ࡽ ᄡ᭥ෝ

aᗮࠥಽᄡ⪹⦹ᩍǍ᳑ྜྷ᮹༉ऽ⩶ᔢእƱෝ☖⧕࠺✚ᖒᮥᇥᕾ (ၶ᳦⋁॒, 2010)⦹۵॒Ǎ᳑ྜྷᨱᕽĥ⊂ࡽᄡ᭥ߑᯕ░a݅᧲⦹

í⪽ᬊᯕࡹŁᯩ݅. ĥ⊂ࡽᄡ᭥۵⩥ᰍ݅ᙹ᮹GNSSෝᯕᬊ⧕

ĥ⊂ࡽߑᯕ░ෝbb᮹ᄡ᭥ĥ⊂᭥⊹ᨱᕽࠦพᱢᮝಽ⪽ᬊ⦹ᩍ

ĥ⊂ ḡᱱᄥ ⦽ĥs Ⅹŝ Ǎe ᇥᕾ, ᜽eᄥ ᄡ᭥ ᇥᕾ(↽ᄲʙ

॒, 2008)ᯕᯕ൉ᨕḡŁᯩᮝ໑, Ñ࠺⩶ᔢࠥ⇽ᮥ᭥⧕b᭥⊹᮹

sॅᮥSpline ᅕeჶᮥᯕᬊ⦹ᩍᅕe(Meo and Zumpano, 2005)

⦹۵ ᱶࠥಽ ⪽ᬊ⦹Ł ᯩ݅. ᝅᱽ ⩥ᙹƱᨱᕽ ⧕ᕾ༉ߙ ᱽᦩᮥ

᭥⧕⍡ᯕትŝÑ޵ᨱᕽbᇡᰍᄥಽ݅ෙ₉ᙹ᮹݅⧎᜾ᅕeჶᮥ

ᔍᬊ⦹ᩍ⩶ᔢᮥđᱶ(ʡ⪙Ğ॒, 2007)⦹ʑ᭥⦽ᖁ⧪ᩑǍᔍಡa

ᯩḡอᯕ۵݅ᙹ᮹ĥ⊂ᄡ᭥ෝᯕᬊ⧩݅۵ᱱᨱᕽ⦽ĥaᯩ݅.

ᯕ᪡zᯕĥ⊂ᄡ᭥᮹⪽ᬊᖒᯕၙၙ⦽ᯕᮁ۵ᄡ᭥༉ܩ░ย

ᰆእaๅᬑŁaᯕʑভྙᨱ݅ᙹ᮹ĥ⊂ḡᱱᮥĥ⊂ᮥ⧁ᙹ

ᨧᨕᄡ᭥ĥ⊂}ᗭᨱᱽ⦽ᯕᯩ۵ᝅᱽᔢ᜽༉ܩ░ย᜽ᜅ▽᮹

⦽ĥ ভྙᯕ݅. ᯕᨱ ݉ᙽ ᄡ᭥ Ⅹŝప ᇥᕾ እƱෝ ᭥⦽ ĥ⊂

ᯱഭಽ⪽ᬊࡹŁᯩ۵ᝅᱶᯕ݅. ĥ⊂ᄡ᭥ෝ☖⦽Ǎ᳑ྜྷ᮹⩶ᔢ

šญ۵݉ᙽ ᜽bᱢᯙ ⢽⩥ᮥ᭥⦽ ᬊࠥಽ ⪽ᬊࡹŁᯩᨕ ⧕ᕾ

༉ߙᮥ᭥⦽⩶ᔢšญ۵ᯕ൉ᨕḡŁᯩḡᦫᦥ⪽ᬊࠥ׳ᮡᄡ᭥

ĥ⊂ ߑᯕ░᮹ ⪽ᬊᖒᯕ ၙእ⦽ äᯕ ᔍᝅᯕ݅.

ᅙᩑǍᨱᕽ۵ᔢ᜽ᄡ᭥༉ܩ░ย᜽ᜅ▽ᨱᕽᄡ᭥ĥ⊂}ᗭ᮹

ᱽ⦽ᮝಽᯙ⦽⦽ĥෝɚᅖ⦹ʑ᭥⧕↽ᗭ⦽᮹ᄡ᭥ĥ⊂ߑᯕ░ෝ

ᯕᬊ⧕ᝅᱽ Ǎ᳑⧕ᕾ ༉ߙಽᔍᬊ⧁ ᙹ ᯩ۵⩶ᔢᮥ ⇵ᱶ⦹ʑ

᭥⦽⩶ᔢ⇵ᱶ᦭Łญ᷹SFSM-LSᮥ}ၽ⦹ᩡ݅. }ၽࡽ᦭Łญ

᷹ᮡᝅᱽǍ᳑ྜྷᯕÑ࠺⧁ᙹᯩ۵⩶ᔢॅŝᝅᱽĥ⊂ࡽᄡ᭥

ߑᯕ░ෝ᦭Łญ᷹᮹᯦ಆߑᯕ░ಽᖅᱶ⦹Ł, ↽ᱢ⪵ʑჶᮥ᯦ࠥ

⦹ᩍ Ǎ᳑ྜྷ Ñ࠺ ⩶ᔢᮥ ⇵ᱶ⦹ᩡ݅. ᦭Łญ᷹᮹ á᷾ᮥ ᭥⧕

2ĞeᩑᗮƱ༉ߙᮥᖁᱶ⦹ᩍๅ}ᄡᙹ⧕ᕾᩑǍෝᙹ⧪⦹ᩡ݅.

⩶ᔢ⇵ᱶđŝᨱᩢ⨆ᮥၙ⋁ᙹᯩ۵Ǎ᳑Ñ࠺a܆⩶ᔢ, ᄡ᭥

ĥ⊂}ᗭ, ᄡ᭥ĥ⊂᭥⊹ෝๅ}ᄡᙹಽ៉ݡ⦽᦭Łญ᷹᮹⇵ᱶ đŝs✚ᖒᮥᇥᕾ⦹ᩡᮝ໑, ʑ᳕ᨱᔍᬊࡹŁᯩ۵Polynomial, Lagrangian, Splineᅕeჶŝ᮹ ⩶ᔢ ⇵ᱶ ᱶၡࠥ እƱ ᩑǍෝ

☖⧕ }ၽࡽ ᦭Łญ᷹᮹ ᬑᙹᖒᮥ á᷾⦹ᩡ݅.

2. SFSM-LS ᦭Łญ᷹

ᝅᱽǍ᳑ྜྷᨱᕽĥ⊂ࡽߑᯕ░ಽǍ᳑ྜྷ᮹Ñ࠺ᮥ⠪a⦹۵ߑ

ᯩᨕᕽ۵ĥ⊂ ߑᯕ░ ᙹ᮹ᱽ᧞ᮝಽ ᯙ⧕ ᱶၡ⦽⧕ᕾ ༉ߙᯕ

᳕ᰍ⦽݅⦹ᩍࠥᱶၡ⦽⧕ᕾđŝෝʑݡ⦹ʑᨕಖ݅. ĥ⊂ݡᔢ

Ǎ᳑ྜྷᮡྕ⦽᮹ᱩᱱᮥw۵᜽ᜅ▽ᮝಽ༉ߙยࢁᙹᯩḡอ, ᝅᱽĥ⊂༉ߙᨱᕽ۵ᄡ᭥⊂ᱱᯕᱽ⦽ᱢᯕအಽĥ⊂ᄡ᭥ᱶᅕෝ

ၵ┶ᮝಽᱢᱩ⯩ᅕe⦹ᩍၙĥ⊂ᱩᱱ᮹ᄡ᭥ᖒᇥᮥ⇵ᱶ⦹ᩍ

Ǎ᳑ྜྷ᮹⩶ᔢᮥ⇵ᱶ⧕᧝⦽݅. ᱶၡ⦽ၙĥ⊂ᄡ᭥᮹⇵ᱶᮥ

᭥⧕ᕽ۵݉ᙽᅕeჶᯕᦥܭ, ᅕ݅⧊ญᱢᯕŁᨥၡ⦽Ǎ᳑⩶ᔢ᮹

⇵ᱶʑჶᯕ⦥᫵⦹݅. ᯕෝ᭥⧕ᅙᩑǍᨱᕽ۵ᮁ⦽᫵ᗭ⧕ᕾ

༉ߙᮥ ᯕᬊ⦹ᩍ Ǎ᳑ྜྷᯕ ᝅᱽ Ñ࠺ ⧁ ᙹ ᯩ۵ ᩍ్ ⩶ᔢᮥ

Ǎ᳑⩶ᔢ⧉ᙹ௝ᱶ᮹⦹ᩡ݅. ᱶ᮹ࡽǍ᳑⩶ᔢ⧉ᙹᨱᯥ᮹᮹aᵲ

⊹ෝᇡᩍ⦹Łᯕॅ⩶ᔢॅᮥᵲℊ⦹ᩍᵲℊ⩶ᔢSFSM(Structural shape Function Superposition Method)ᮥࠥ⇽⦹ᩡ݅. ࠥ⇽ࡽ

ᵲℊ⩶ᔢ᮹ᄡ᭥᪡ᝅᱽǍ᳑ྜྷᨱᕽĥ⊂ࡽᄡ᭥᪡᮹ᄡ᭥᪅₉ෝ

↽ᗭ⪵⦹۵༊ᱢ⧉ᙹෝǍᖒ⦹ᩡ݅. ༊ᱢ⧉ᙹ۵↽ᗭᱽŒჶ(S.C.

Chapra, 2008)ᮥ ᯕᬊ⦹ᩍ ᪅₉ෝ ↽ᗭ⪵⦹ᩡᮝ໑, ᯕෝ ☖⧕

Ǎ᳑⩶ᔢ⧉ᙹaᵲ⊹ෝĥᔑ⦹ᩡ݅. ĥᔑࡽaᵲ⊹ෝbǍ᳑⩶ᔢ

⧉ᙹᨱݡ᯦⦹ᩍ↽᳦ᱢᮝಽĥ⊂ᄡ᭥ෝᯕᬊ⦽ၙ⊂ᱶḡᱱ᮹

ᄡ᭥ॅᮥ⇵ᱶ⦹ᩍǍ᳑ྜྷ⩶ᔢᮥࠥ⇽⦹ᩡ݅. ᯕ᪡zᮡ᦭Łญ᷹

ᮥSFSM-LS(Least Square)௝໦໦⦹ᩡᮝ໑, ᦭Łญ᷹}᫵۵

Figure 1ŝ z݅.

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Fig. 1. Algorithm Flow Chart of SFSM-LS

2.1 ֜୺෴ঃ෌৤(Structural Shape Function)

Ǎ᳑ྜྷ ᮁ⦽᫵ᗭ༉ߙᮥ ᯕᬊ⦹ᩍ Ǎ᳑ྜྷᯕ ᝅᱽ Ñ࠺ ⧁ ᙹ

ᯩ۵݅᧲⦽⩶ᔢᮥǍ᳑⩶ᔢ⧉ᙹ௝ᱶ᮹⦹ᩡ݅. Ǎ᳑⩶ᔢ⧉ᙹ۵

b ᱩᱱ᮹ ᄡ᭥ᄂ░ ⧪಍ಽ Ǎᖒࡽ݅. ᯕॅ ⧉ᙹ۵ ⧕ᕾ༉ߙᨱ

݅᧲⦽⦹ᵲĞᬑෝᰍ⦹⦹ᩍ᨜ᮡ⩶ᔢŝᯱᮁḥ࠺⧕ᕾᮥ☖⧕

᨜ᮡ༉ऽ⩶ᔢ॒݅᧲⦽⩶ᔢᮝಽǍᖒ⧁ᙹᯩ݅. ᩍ్Ǎ᳑⩶ᔢ⧉

ᙹ(ĳ_Ƈ)ᨱ bb᮹ aᵲ⊹(ķ_Ƈ)ෝ ᇡᩍ⦹Ł ᯕॅᮥ ᵲℊ⦹໕ ݅ᮭ

᜾ (1)ŝ zᮡ ᵲℊ⩶ᔢ¬ ෝ ᨜ᮥ ᙹ ᯩ݅.

¬ á ķ_Îĳ_Îâ ķ_Ïĳ_Ïâ z âķ_ƌĳ_ƌ á

ā

_{Ƈ á Î}^ƌ ^ķ^ƌ^ĳ^ƌ ⁽¹⁾

ᩍʑᕽ,¬ : Ǎ᳑⩶ᔢ⧉ᙹ ᵲℊ⩶ᔢ ĳƇ: Ǎ᳑⩶ᔢ⧉ᙹ

ķ_Ƈ: Ǎ᳑⩶ᔢ⧉ᙹ aᵲ⊹

ƌ : Łಅࡽ Ǎ᳑⩶ᔢ⧉ᙹ᮹ }ᙹ

2.2 ࡧୡ෌৤(Object Function)

ᝅᱽǍ᳑ྜྷ᮹bḡᱱᨱᕽĥ⊂ࡽᄡ᭥ߑᯕ░ෝ_ƈ௝⦹Ł

ⅾƋ}᮹ ĥ⊂ }ᗭᨱᕽ ᄡ᭥ෝ ĥ⊂ ⦹ᩡᮥ Ğᬑƈ۵ ĥ⊂

ḡᱱ᭥⊹ჩ⪙ಽ៉ƈ á ÎìÏìÐ zìƋᯕ݅. ᝅᱽǍ᳑ྜྷᨱᕽĥ⊂ࡽ

ḡᱱ᮹ᄡ᭥ƈ᪡ᯕ᪡࠺ᯝ⦽ḡᱱᨱᕽ᮹Ǎ᳑⩶ᔢ⧉ᙹಽᵲℊࡽ

⩶ᔢ᮹ ᄡ᭥ ᪅₉۵ ᜾ (2)᪡ zᯕ Ǎᖒ⧁ ᙹ ᯩ݅.

ƐƐƍƐá

ā

^Ƌ^Þ^ƈ^{à ¬}^ƈ^ß ⁽²⁾

᜾(2)᮹ᄡ᭥᪅₉ෝ↽ᗭ⪵۵Ǎ᳑⩶ᔢ⧉ᙹaᵲ⊹ෝᔑᱶ⦹ʑ

᭥⧕ᕽ↽ᗭᱽŒჶ(Least Square)ᮥ᯦ࠥ⦹ᩍ᪅₉ᱽŒ᮹⧊ᮥ

↽ᗭ⪵⦹۵༊ᱢ⧉ᙹෝǍᖒ⦹ᩡ݅. ᯕෝ᜾(2)ᨱᱢᬊ⦹ᩍ݅ᮭ

᜾ (3)ŝ zᮡ ༊ᱢ⧉ᙹෝ ↽᳦ᱢᮝಽ Ǎᖒ⦹ᩡ݅.

Ļ á_{ƈ á Î}

ā

^Ƌ^Þ^ƈ^{à ¬}^ƈ^ß^Ï ⁽³⁾

2.3 ֜୺෴ঃ෌৤ԧண౿ծ৤(Weight Factor)

ࠥ⇽ࡽ ༊ᱢ⧉ᙹಽ bb᮹ Ǎ᳑⩶ᔢ⧉ᙹ aᵲ⊹ෝ ᔑᱶ⦹ʑ

᭥⧕ᕽ᜾(3)ᮥbb᮹aᵲ⊹ᨱݡ⧕ᕽ⠙ၙᇥᮥ⦹ᩍ᜾(4)᪡

zᮡ⠙ၙᇥႊᱶ᜾ᮥǍᖒ⦹ᩡ݅. ᩍʑᕽ, Ɖ á ÎìÏìÐ zìƌᯕ݅.

ćŞķŞĻ á Ï_Ɖ

ā

_{ƈ á Î}^Ƌ ^ƙ_Ɯ^ƚ^Þ^ƈ^{à ¬}^ƈ^{ß Þć}Şķ_Ɖ Ş_ƈ

à ćŞķ_Ɖ Ş¬_ƈ

ßƛ Ɲƞ

á Ï

ā

_{ƈ á Î}^Ƌ ^ã^Þ^ƈ^{à ¬}^ƈ^{ß Þà ĳ}^Ɖƈ^ß^ä

á Ï

ā

_{ƈ á Î}^Ƌ ^ƙ_Ɯ^ƚ^Þ_{Ƈ á Î}

ā

^ƌ ^ķ^Ƈ^ĳ^Ƈƈ^à^ƈ^{ß Þĳ}^Ɖƈ^ß^ƛ_Ɲ^ƞ ⁽⁴⁾

Łಅࡽ Ǎ᳑⩶ᔢ⧉ᙹ }ᙹอⓝ᮹ aᵲ⊹ᨱ ݡ⧕ᕽ ⠙ၙᇥᮥ

⦹ᩡʑভྙᨱၙḡᙹ᮹}ᙹอⓝᯙƌ}᮹ႊᱶ᜾ᯕࠥ⇽ࡹ໑,

༊ᱢ⧉ᙹෝ↽ᗭ⪵⦹۵aᵲ⊹ĥᙹෝᔑᱶ⦹ʑ᭥⧕ᕽ᜾(5)᪡

zᯕ b ⠙ၙᇥႊᱶ᜾᮹ sᯕ 0ᯕ อ᳒ࡹࠥಾ ⦹ᩡ݅.

ćŞķŞĻ áÏ_Ɖ

ā

_{ƈ á Î}^Ƌ ^ƙ_Ɯ^ƚ^Þ

ā

_{Ƈ á Î}^ƌ ^ķ^Ƈ^ĳ^Ƈƈ^à^ƈ^{ß Þĳ}^Ɖƈ^ß^ƛ_Ɲ^ƞ^{á ×} ⁽⁵⁾

᭥᮹ ᜾ (5)ᮥ ᱶญ⦹໕ ↽᳦ᱢᮝಽ ݅ᮭŝ zᮡ ᜾ (6)ᯕ

ࠥ⇽ࡽ݅.

r ćŞķƉ

ŞĻ Q

ā

_{ƈ á Î}^Ƌ^ƙ_Ɯ^ƚ^Þ

ā

_{Ƈ á Î}^ƌ ^ķ^Ƈ^ĳ^Ƈƈ^{ß Þĳ}^Ɖƈ^ß^ƛ_Ɲ^ƞ^á

ā

_{ƈ á Î}^Ƌ ^ã^Þ^ƈ^{ß Þĳ}^Ɖƈ^ß^ä⁽⁶⁾

᜾ (6)ŝ zᮡ ⅾ n}᮹ ᩑพႊᱶ᜾ᮥ ⣡ᯕ⦹ᩍ ↽᳦ᱢᮝಽ

༊ᱢ⧉ᙹෝ↽ᗭ⪵⦹۵sᮝಽǍ᳑⩶ᔢ⧉ᙹaᵲ⊹ĥᙹađᱶ

ࡽ݅.

ᱶၡ⦽ ⩶ᔢ ⇵ᱶ đŝෝ ࠥ⇽⦹ʑ ᭥⧕ᕽ۵ Ǎ᳑ྜྷ᮹ Ñ࠺

⩶ᔢᯕ ݅᧲⦹í Łಅࢁ ᙹ ᯩࠥಾ Łಅࡽ Ǎ᳑⩶ᔢ ⧉ᙹa᮹

ᙹa᷾a⧁ᙹಾᮁญ⦹ḡอᝅᱽǍ᳑ྜྷᨱᕽ۵እᬊᱢᯙ⊂໕

ভྙᨱᄡ᭥ĥ⊂}ᗭᨱ۵ᱽ⦽ᯕᯩ݅. ᯕ⃹ౝǍ᳑⩶ᔢ⧉ᙹ᮹

ᙹ(N)a Łಅࡹ۵ ᄡ᭥ĥ⊂ }ᗭ(M)ᅕ݅ ฯᮥ Ğᬑᨱ ᜾(6)ŝ

(4)

Fig. 2. Verification process of developed algorithm

Fig. 3. Numerical model of 2-span continuous bridge

Table 1. Section Properties Elastic Modulus

[ ©]

Moment of Inertia [Ƌ^Ñ]

Area [Ƌ^Ï]

25.7 0.614 0.861

DS1

DS2

DS3

Fig. 4. Assumed real deformed shape cases ᦫᦥྕ⦽ᙹ᮹⧕aӹ᪅۵ᇡᱶ᮹ྙᱽaၽᔾ⦹íࡽ݅. ᯕ᪡

zᮡᇡᱶ᮹ྙᱽෝ⧕đ⦹ʑ᭥⧕ᕽᄡ᭥ĥ⊂}ᗭ᪡Ǎ᳑⩶ᔢ⧉

ᙹෝ እƱ⦹ᩍ Figure 1ŝ zᯕ bb ݅ෙ ⧕ჶᮥ ᱢᬊ⦹ᩡ݅.

N=M, N<M ᯝĞᬑ۵ᯝၹᱢᯙᩑพႊᱶ᜾ᮝಽ⧕ෝᔑᱶ⦹Ł

N>MᯝĞᬑᩍ్⧕ᵲ↽ᗭÑญჶ(Mimnimu Length Solution)

ᮥᱢᬊ(G.R.Liu and X. Han, 2003)⦹ŁQR Decomposition (B.N, Datta, 2007)ಽ⧕ෝᔑᱶ⦹ᩍaᵲ⊹ĥᙹෝđᱶ⦹ᩡ݅.

ᯕ్⦽ႊჶᮝಽđᱶࡽbb᮹aᵲ⊹ĥᙹෝbǍ᳑⩶ᔢ⧉ᙹᨱ

ݡ᯦⦹ᩍᵲℊ⩶ᔢ᮹bᱩᱱ᮹ᄡ᭥ෝࠥ⇽⦹ᩡ݅. ᯕ౑ŝᱶᮥ

☖⧕Ǎ᳑ྜྷᨱᕽĥ⊂ࡽᄡ᭥ߑᯕ░ෝᯕᬊ⧕ၙ⊂ᱱᄡ᭥ෝᱶၡ

⦹í ⇵ᱶ⦹ᩍ↽᳦ᱢᮝಽ Ǎ᳑ྜྷ Ñ࠺⧕ᕾ ༉ߙ ᔑᱶᮥ᭥⦽

SFSM-LS᦭Łญ᷹ᮥ }ၽ⦹ᩡ݅.

3. ᦭Łญ᷹á᷾ၰ✚ᖒᇥᕾ

3.1 Ցஹࡦ܄ࢫՑஹ୺Ս

ᝅᱽǍ᳑ྜྷᨱᕽ۵ ĥ⊂ ᰆእᨱ᮹⧕ᕽ ၽᔾ⦹۵ ĥ⊂᪅₉, ᱽ᯲᪅₉, Ǎ᳑ྜྷⅩʑᄡ⩶, ⧕ᕾ༉ߙŝᝅᱽǍ᳑ྜྷŝ᮹ᇩᯝ⊹

॒᮹ ᇩ⪶ᝅᖒᮝಽ ݅᧲⦽ ᪅₉a ᳕ᰍ⦽݅. ᅙ ᩑǍᨱᕽ۵

SFSM-LS ᦭Łญ᷹᮹á᷾ᮥ᭥⧕ᯕ᪡zᮡ᪅₉ᨱ᮹⦽ᩢ⨆ᮡ

႑ᱽ⦹ᩍ༉ु᳑Õᮥ᪥ᱥ⦽ᔢ┽ಽaᱶ⦹Ł݅ᮭFigure 2ŝ

zᮡ ŝᱶᮝಽ ᦭Łญ᷹ á᷾ᮥ ⦹ᩡ݅.

ᬑᖁá᷾ݡᔢ༉ߙᮥᖁᱶ⦹ᩍFE༉ߙᮥǍᖒ⦹Łᯕෝ☖⧕

Ǎ᳑⩶ᔢ⧉ᙹෝǍᖒ, ᄡ᭥ෝĥ⊂⧁Ǎ᳑ྜྷ⩶ᔢᮥǍᖒ⦹ᩡ݅.

ᯕෝ᭥⧕ᯥ᮹᮹Ǎ᳑ྜྷᄡ⩶⩶ᔢᮥอॅŁĥ⊂᭥⊹᪡}ᗭෝ

ᖁᱶ⦹ᩍĥ⊂ᄡ᭥ෝaᱶ⦹ᩡ݅. ᯕෝ☖⧕ǍᖒࡽǍ᳑⩶ᔢ⧉ᙹ

᪡aᱶࡽĥ⊂ᄡ᭥ಽSFSM-LS᦭Łญ᷹ᮥᯕᬊ⧕Ǎ᳑⩶ᔢ⧉ᙹ

aᵲ⊹ෝĥᔑ⦹Ł↽᳦ᮝಽ⇵ᱶࡽǍ᳑⩶ᔢᮥĥᔑ⦹ᩡ݅. aᱶ

ࡽᝅᱽ⩶ᔢŝ⇵ᱶࡽǍ᳑⩶ᔢe᮹᪅₉ෝ⠪Ɂᱩݡs⟝ᖝ✙᪅

₉(Mean Absolute Percent Error)ಽ⇵ᱶđŝᱶၡࠥෝᇥᕾ⦹ᩡ

݅. Figure 2᮹MAPEᨱᕽƒ : ᱩᱱtᨱᕽ᮹Ǎ᳑ྜྷᝅᱽᄡ᭥,

_ƒ : ᱩá tᨱᕽ᮹ ⇵ᱶ ⩶ᔢ ᄡ᭥, Ǝ:יऽ }ᙹᯕ݅.

݅ᮭFigure 3ŝzᮡ2@75m 2ĞeᩑᗮƱ༉ߙᮥᖁᱶ⦹ᩍ

ⅾ61}᮹ᱩᱱŝ60}᮹᫵ᗭෝw۵FE༉ߙᮥǍᖒ⦹ᩡᮝ໑

݉໕ᱽᬱᮡ Table 1᪡ z݅.

Ǎ᳑ྜྷᮡ⦹ᵲ᮹᯲ᬊ⩶┽ᨱ঑௝݅᧲⦽ᄡ⩶⩶ᔢᯕӹ┡ӽ݅.

݅᧲⦽⩶ᔢᨱݡ⦽ᅙᩑǍ᮹᦭Łญ᷹ʑჶ᮹⬉ᮉᖒᮥ᯦᷾⦹ʑ

᭥⧕Figure 4ŝzᯕDS1, DS2ŝzᮡ2}᮹ݡ⋎⩶ᔢ, DS3᪡

zᮡ1}᮹እݡ⋎⩶ᔢ᮹ⅾ3aḡĞᬑ᮹ݡ⢽ᱢᯙ⮉ᄡ⩶᮹

⩶ᔢᮝಽ Ǎ᳑ྜྷ Ñ࠺ ⩶ᔢᮥ aᱶ⦹ᩡ݅.

Ǎ᳑⩶ᔢ⧉ᙹ۵Ǎ᳑ྜྷ᮹݅᧲⦽Ñ࠺⩶ᔢᮥၹᩢ⦹ʑ᭥⧕ᕽ

ⅾ61}᮹ᱩᱱᵲĞĥ᳑Õᯙ3}᮹ᱩᱱᮥᱽ᫙⦽58}᮹bb᮹

(5)

SF 1 SF 15

SF 29 SF 30

SF 44 SF 58

Fig. 5. Examples of structural shape function

Table 2. Combined structural shape function cases Structural shape function cases Loading Distance (m)

N4 25

N6 18.65

N8 15

N10 15

N12 12.5

N16 10.7

N20 7.5

N30 5

N58 2.5

(a) MP2

(b) MP4

(c) MP6

(d) MP8 * M : Measurement location Fig. 6. Displacement measuring location cases according to the

number of measuring location

ᱩᱱᨱᯥ᮹᮹⦹ᵲᮥ ᯕ࠺᜽⍽a໑ᰍ⦹⦹Łᱶᱢ⧕ᕾᮥ☖⧕

ⅾ58}᮹Ǎ᳑⩶ᔢ⧉ᙹ(SF)ෝࠥ⇽⦹ᩡ݅. SF1ⴇSF29ᮡ᳭⊂Ğ eᨱ ⦹ᵲᨱ ᯲ᬊ⦹ᩡᮥ Ğᬑᯕ໑, SF30ⴇSF58ᮡ ᬑ⊂Ğeᨱ

⦹ᵲᮥ ᰍ⦹ ⦹ᩡᮥ Ğᬑ᮹ Ǎ᳑⩶ᔢ ⧉ᙹಽ ݅ᮭ Figure 5۵

Ǎ᳑⩶ᔢ⧉ᙹᵲ6}᮹Ǎ᳑⩶ᔢ⧉ᙹᨱݡ⦽⩶ᔢᮥӹ┡ԕᨩ݅.

Ǎ᳑⩶ᔢ⧉ᙹ᮹᳑⧊ᯕSFSM-LS᦭Łญ᷹ᮥᱢᬊ⦽⩶ᔢ⇵ᱶ đŝᨱၙ⊹۵ᩢ⨆ᖒᮥᇥᕾ⦹ʑ᭥⧕ᕽTable 2᪡zᯕ⦹ᵲ᮹

ᰍ⦹eĊᨱ঑௝SFSM-LSᨱŁಅࡹ۵Ǎ᳑⩶ᔢ⧉ᙹ᮹}ᙹෝ

đᱶ⦹ᩡ݅. N4᮹Ğᬑ⦹ᵲᮥ25m eĊᮝಽᰍ⦹⦹ᩍⅾ4}᮹

⩶ᔢᮥᔍᬊ⦹ᩡᮝ໑, N58᮹Ğᬑ⦹ᵲᮥ2.5m eĊᮝಽᰍ⦹⦹ᩍ

ⅾ 58}᮹ ⩶ᔢᮥ ᔍᬊ⦹ᩡ݅.

ᄡ᭥ĥ⊂᭥⊹᪡}ᗭ۵݅ᮭFigure 6ŝzᯕⅾ4aḡĞᬑಽ

ĥ⊂ ᭥⊹۵࠺ᯝeĊᮝಽ႑⊹⦹Łĥ⊂}ᗭෝbb2}, 4}, 6}, 8}ಽ aᱶ⦹ᩡ݅.

3.2 ֜୺෴ঃ෌৤૕࣡଍ծ౸Թী

Ǎ᳑ྜྷÑ࠺⩶ᔢᨱ঑௝ŁಅࡽǍ᳑⩶ᔢ⧉ᙹ᪡ĥ⊂}ᗭᨱ

঑௝SFSM-LSಽ⇵ᱶࡽ⩶ᔢᱶၡࠥ۵݅ᮭFigure 7ŝz݅.

Figure 7(a)᮹Ğᬑᄡ᭥ĥ⊂}ᗭ2}ᯙMP2᪡ᄡ᭥ĥ⊂}ᗭ

8}ᯙMP8᮹᪅₉ᮉᯕ↽ݡ2%ᱶࠥᯙၹ໕, Figure 7(c)᮹Ğᬑ۵

MP2᪡MP8᮹᪅₉ᮉᯕ↽ݡ8%ᱶࠥಽÑ࠺⩶ᔢᯕᅖᰂ⧁ᙹಾ

ĥ⊂}ᗭᨱ঑ෙ᪅₉᮹ᩢ⨆ᯕⓍíӹ┡ԍ݅. ᯕ᪡zᮡᬱᯙᮡ

ŁಅࡽǍ᳑⩶ᔢ⧉ᙹॅᯕʑᅙᱢᮝಽDS1ŝᮁᔍ⦽⩶ᔢᯕʑভ

ྙᨱŁಅࡽǍ᳑⩶ᔢ⧉ᙹॅŝᝅᱽÑ࠺⩶ᔢᯕᮁᔍ⧁Ğᬑᙹಕ

᳑Õᮥ޵ᬒᛞíอ᳒᜽┅ʑভྙᯕ௝❱݉ࡽ݅. ᄡ᭥ĥ⊂}ᗭᨱ

ݡ⦽ ⇵ᱶᱶၡࠥ۵ ĥ⊂ }ᗭa ᷾a ⧁ᙹಾ ⇵ᱶ ⨆ᔢࡹḡอ

ᯝᱶ}ᗭ᮹ĥ⊂}ᗭᯕᔢᯕࡹ໕⇵ᱶᱶၡࠥ᮹⨆ᔢ⡎ᯕqᗭ⦹

۵Ğ⨆ᮥӹ┡ԍᮝ໑, ༉ुÑ࠺⩶ᔢᨱᕽ6}ᯕᔢ᮹ĥ⊂}ᗭ᮹

Ğᬑእ᜘⦽ᱶၡࠥ᮹⩶ᔢ⇵ᱶđŝෝӹ┡ԕŁᯩ݅. ᯕෝᯕᬊ⦹

໕ Ǎ᳑ྜྷ ⩶ᔢ ⇵ᱶᮥ ᭥⧕ ⦥᫵⦽ ᄡ᭥ ĥ⊂ }ᗭෝ đᱶ⧁

ᙹ ᯩ۵ ᯱഭಽ ⪽ᬊ ⧁ ᙹ ᯩᮥ äᯕ௝ ❱݉ࡽ݅.

ŁಅࡽǍ᳑⩶ᔢ⧉ᙹ᮹}ᙹᨱݡ⦽⇵ᱶᱶၡࠥ۵Łಅࡽ⩶ᔢ

⧉ᙹaᱢᮥĞᬑ(N4ⴇN16)۵ᱶၡࠥ᮹⨆ᔢᨱᯝᱶ⦽Ğ⨆ᖒᯕ

ӹ┡ӹḡ۵ ᦫḡอ, Łಅࡽ Ǎ᳑⩶ᔢ⧉ᙹ᮹ }ᙹa ᷾a⧁ᙹಾ

(N20ⴇN58) ᯝᱶᱶၡࠥಽᙹಕ⦹۵đŝෝᅕᩡ݅. ᯕ᪡zᮡ

⩥ᔢᮡᝅᱽǍ᳑ྜྷ᮹Ñ࠺⩶ᔢᮥ⇵ᱶ⦹۵ߑᯩᨕŁಅࡽǍ᳑⩶

ᔢ⧉ᙹaᝅᱽǍ᳑ྜྷ᮹⩶ᔢၹᩢᮁྕᨱ঑௝ᩢ⨆ᮥၼʑভྙᯕ

௝❱݉ࡽ݅. ᷪ, Łಅࡽ⩶ᔢ⧉ᙹaᱢᮥĞᬑᝅᱽÑ࠺ᮥၹᩢ⧁

⪶ශᯕԏᦥḡŁฯᮡ⩶ᔢᮥ᳑⧊⧩ᮥĞᬑ۵ᝅᱽÑ࠺⩶ᔢᮥ

ၹᩢ⧁ ⪶ශᯕ ᷾a⦹ʑ ভྙᯕ௝ ❱݉ࡽ݅. ᯕ᪡ zᮡ ⩥ᔢᮡ

(6)

(a) Real deformed shape : DS1

(b) Real deformed shape : DS2

(c) Real deformed shape : DS3

Fig. 7. Estimation error between structural shape function and the number of displacement measuring location using SFSM-LS

Fig. 8. Displacement measuring location ᝅᱽÑ࠺⩥ᔢᯕᅖᰂ⧕ḩᙹಾᩢ⨆ᖒᯕⓍíӹ┡ӽ݅. DS1⩶ᔢ

᮹ĞᬑN12ᯕᔢ᮹Ğᬑᯝᱶᱶၡࠥಽᙹಕ⦹ḡอDS2, DS3⩶ᔢ ᮹ĞᬑbbN16, N20ᯕᔢ᮹Ğᬑᨱᯝᱶᱶၡࠥಽᙹಕ⦹۵

Ğ⨆ᮥӹ┡ԙ݅. ੱ⦽ᄡ᭥ĥ⊂}ᗭa᷾a⧁ᙹಾ⇵ᱶᱶၡࠥa

ᯝᱶ⦹íᙹಕ⦹ʑ᭥⧕ᕽ۵ Łಅࡽ⩶ᔢ⧉ᙹ᮹ᙹaฯᦥḡ۵

Ğ⨆ᯕ ӹ┡ӹ۵ߑ ᯕ۵ SFSM-LS᦭Łญ᷹ ᦩᨱᕽ ᄡ᭥ ĥ⊂

}ᗭ᪡Łಅࡹ۵⩶ᔢ⧉ᙹ᮹}ᙹᨱ঑௝ᩑพၙᇥႊᱶ᜾᮹⣡ᯕ ჶ ₉ᯕᨱ ঑ෙ ᩢ⨆ᯕ௝ ❱݉ࡽ݅.

SFSM-LS᦭Łญ᷹ᮥᯕᬊ⧕⩶ᔢ⇵ᱶᮥ⦹ʑ᭥⧕ᕽ۵✚ᱶ

}ᗭᯕᔢ᮹ᄡ᭥ĥ⊂ߑᯕ░aᯩŁŁಅࡹ۵Ǎ᳑⩶ᔢ⧉ᙹa

Ǎ᳑ྜྷ᮹Ñ࠺ᮥၹᩢ⧁ᙹᯩࠥಾ∊ᇥ⯩ฯᮡ⩶ᔢ⧉ᙹaŁಅࡽ

݅໕ ᝁ഑ࠥ ᯩ۵ ⩶ᔢ ⇵ᱶ đŝෝ ࠥ⇽⧁ äᯕ௝ ❱݉ࡽ݅.

3.3 ֜୺෴ঃ෌৤૕࣡଍ծ౸଍౿

ᝅᱽǍ᳑ྜྷᨱᕽ᮹ᄡ᭥ĥ⊂᭥⊹۵ḢšᱢᯙႊჶᮝಽǍ᳑ᱢ ᮝಽᵲ᫵⦽ḡᱱᯕӹ↽ݡၽᔾᄡ᭥ḡᱱੱ۵ݡ⩶Ǎ᳑ྜྷ᮹

Ğᬑᮁḡšญᔢ᮹ᯕᮁಽᯙ⧕⊂ᱱ᭥⊹aǍ᳑ྜྷᨱݡ⧕ᕽእݡ

⋎ᱢᮝಽ ᖁᱶᯕ ࢁ ᙹ ᯩ݅. ᦿ᮹ 3.2 ᩑǍԕᬊᨱᕽ۵ ॒eĊ

႑⊹ᨱ঑ෙ⇵ᱶᱶၡࠥᇥᕾᮥ☖⧕SFSM-LS✚Ḷᮥᇥᕾ⦹ᩡᮝ ໑, ᅙԕᬊᨱᕽ۵ᦿᖁᰆᨱᕽ⇵ᱶࡽđŝෝၵ┶ᮝಽĥ⊂᭥⊹ᨱ

঑ෙ ⩶ᔢ ⇵ᱶđŝ᮹ ᩢ⨆ᮥ á☁⦹ᩡ݅.

ᦿᖁᩑǍԕᬊᨱᕽᄡ᭥ĥ⊂}ᗭa4}ᯝĞᬑDS1, DS2, DS3⩶ᔢ᮹⇵ᱶᱶၡࠥa5%ԕ᮹đŝෝӹ┡ԕᨩ݅. ᯕᨱ঑௝

↽ݡ4}᮹ĥ⊂ḡᱱᮥaᱶ⦹ᩡᮥĞᬑᝁ഑ࠥᯩ۵⇵ᱶđŝෝ

ӹ┡ԝᙹᯩ݅Ł❱݉ࡹᨕ4}᮹ĥ⊂}ᗭᨱݡ⦽᭥⊹ᖁᱶᨱ

঑ෙ⇵ᱶᩢ⨆ᖒᮥá☁⦹ʑ᭥⧕Figure 8᪡zᯕĞeᵲᦺᨱᕽ

ᯝᱶ eĊ(s)ᮥ ᷾a᜽⍽ a໑ đŝෝ ⇵ᱶ⦹ᩡ݅.

݅ᮭFigure 9᪡zᯕĥ⊂᭥⊹ᨱ঑ෙ⩶ᔢ⇵ᱶđŝDS1⩶ᔢ ᯝĞᬑĥ⊂᭥⊹ᨱ঑ෙ᪅₉᮹ᩢ⨆ᯕ2%ԕ᫙ᯙၹ໕, DS3

⩶ᔢᯙĞᬑ↽ݡ6%ᯕᔢ᪅₉aӹ┡ӹ۵äᮝಽӹ┡ԍ݅. ᯕ్

⦽⩶ᔢᮡ3.2᮹ᩑǍԕᬊŝษ₍aḡಽǍ᳑ྜྷ᮹Ñ࠺⩶ᔢᯕᅖᰂ

⧁ᙹಾĥ⊂᭥⊹ᨱ঑ෙᩢ⨆ᖒᯕⓍíӹ┡ӽ݅۵äᮥ᦭ᙹ

ᯩ݅. ༉ु ⩶ᔢᨱᕽ S/L=0.47᮹ Ğᬑ aᰆ ᱶၡ⦽ ⩶ᔢ ⇵ᱶ

đŝෝࠥ⇽⧁ᙹᯩᨕᅙᩑǍ᪡zᮡǍ᳑ྜྷŝĥ⊂᳑Õᨱᕽ۵

Ǎ᳑ྜྷᨱŁ෕íᄡ᭥ෝĥ⊂⦹۵äᯕaᰆ⬉ᮉᱢᯙĥ⊂᭥⊹

ᖁᱶᯕ௝ ❱݉ࡽ݅.

4. SFSM-LS᪡ʑ᳕ᅕeჶŝ᮹እƱᩑǍ

3ᰆ᮹ԕᬊᨱᕽ۵SFSM-LS᦭Łญ᷹ᨱݡ⦽✚ᖒᇥᕾᩑǍෝ

ᙹ⧪⦹ᩡᮝ໑, ᅙᰆᨱᕽ۵ʑ᳕᮹ᅕeჶŝ᮹እƱᇥᕾᩑǍෝ

☖⧕ᱽ᜽⦽᦭Łญ᷹᮹⬉ᮉᖒᮥᇥᕾ⦹ᩡ݅. ᦿᕽᕽುᨱᕽᖅ໦

(7)

(a) Real deformed shape : DS1

(b) Real deformed shape : DS2

(c) Real deformed shape : DS3

Fig. 9. Estimation error between structural shape function and displacement measuring location using SFSM-LS

Table 3. Accuracy comparison between interpolation methods and SFSM-LS algorithm

Estimation Result of DS1

Polynomial Lagrange Spline SFSM-LS

ł[%] ň ł[%] ň ł[%] ň ł[%] ň

MP2 17.8 18.8 17.8 18.8 17.8 18.8 2.5 1.7 MP4 1.3 1.6 7.1 10.5 6.3 8.6 1.5 1.0

MP6 2.3 4.9 2.3 4.9 0.1 0.1 0.3 0.3

ł[%] ň ł[%] ň ł[%] ň ł[%] ň

MP2 50.5 49.0 50.5 49.0 31.1 31.9 4.9 4.9 MP4 18.5 29.1 18.5 29.1 6.9 6.5 3.2 2.4 MP6 8.5 16.7 7.6 14.5 0.3 0.9 0.8 0.8

ł[%] _ň _ł[%] _ň _ł[%] _ň _ł[%] _ň

MP2 77.2 102.9 77.2 102.9 40.1 47.0 8.0 8.4 MP4 31.5 55.5 29.6 53.1 9.6 11.8 4.0 5.8 MP6 26.6 56.3 11.6 29.4 0.2 0.3 0.8 0.8

ł é ň é

⦹ᩡॐ⩶ᔢđᱶᮥ᭥⧕⊂ᱶᄡ᭥ॅᮥ݉ᙽᅕe⦹۵ႊჶᮝಽ

Polynomial, Lagrangian, Spline ᅕeჶ॒݅᧲⦽ႊჶᯕ⪽ᬊࡹ

Ł ᯩ݅.

Polynomial ᅕeჶ᮹Ğᬑn₉݅⧎᜾ᮥᔍᬊ⦹ʑ᭥⧕ᕽ۵

ⅾ n+1}᮹ ĥ⊂ ᱶᅕෝ ⦥᫵ಽ ⦹໑ Ǎ᳑ྜྷ᮹ ⩶ᔢᯕ ᅖᰂ⧕

ḩᙹಾŁ₉᮹݅⧎᜾ᮥᔍᬊ⧕᧝⦹အಽəอⓝฯᮡᄡ᭥ĥ⊂

ᱶᅕෝ⦥᫵ಽ⦽݅. Splineᅕeჶ᮹Ğᬑĥ⊂ᱶᅕ᮹ᙹᨱšĥᨧ ᯕᯥ᮹᮹ࢱᱱᮥᯨ۵łᖁᮥ3₉݅⧎᜾ᮝಽᔍᬊ⦹ᩍๅҥ౞í

ᩑđ⦹۵ ႊჶᮝಽ ĥ⊂ ᄡ᭥ॅᮥ ݉ᙽ ᅕe⦹۵ ႊჶᯕ݅.

⩶ᔢ⇵ᱶᱶၡࠥእƱෝ᭥⧕SFSM-LS᦭Łญ᷹ᨱ۵ⅾ58}

᮹ Ǎ᳑⩶ᔢ ⧉ᙹෝ ᔍᬊ⦹ᩡᮝ໑, ᄡ᭥ ĥ⊂ ḡᱱŝ Ǎ᳑ྜྷ᮹

Ñ࠺⩶ᔢᮡᦿᕽᱽ᜽ࡽ᳑Õŝ࠺ᯝ⦹ḡอʑ᳕ᅕeჶᮡĞĥ᳑

Õ᮹ ᄡ᭥ෝ ⇵aᱢᮝಽ aᱶ⦹ᩡ݅. Ǎ᳑⩶ᔢ⧉ᙹෝ ᔍᬊ⦹۵

SFSM-LS ᦭Łญ᷹᮹ĞᬑᯕၙĞĥ᳑ÕᯕŁಅࡹᨕᯩᨕĞĥ᳑

Õ᮹ᄡ᭥ĥ⊂ᱶᅕa⦥ᙹᱢᯕḡᦫᮡၹ໕ʑ᳕ᅕeჶᮡĞĥ᳑

Õ᮹ᄡ᭥ෝaᱶ⧕᧝อɝᱲ⦽⩶ᔢᮝಽᅕe⧁ᙹᯩʑভྙᨱ

ᯕᨱ ݡ⦽ ᱶᅕa ⦥ᙹᱢᯕ݅.

ݡ⢽ᱢᮝಽᔍᬊࡹ۵ᅕeჶŝSFSM-LS᦭Łญ᷹ŝ᮹⩶ᔢ

⇵ᱶᱶၡࠥෝMAPE᪡᪅₉᮹ᇥᔑᮝಽእƱ⦹ᩡᮝ໑, əđŝ

Table 3ŝ z݅.

ᄡ᭥ĥ⊂}ᗭa᷾a⧁ᙹಾᨕਅႊჶᮝಽǍ᳑ྜྷ᮹⩶ᔢᮥ

⇵ᱶ⦹޵௝ࠥ ⇵ᱶ ᱶၡࠥa ᷾a⦹ḡอ SFSM-LS᦭Łญ᷹ᮡ

݅ෙᅕeჶᨱእ⧕ᱢᮡᄡ᭥ĥ⊂}ᗭಽᱶၡ⦽⇵ᱶđŝෝ

ࠥ⇽⧉ᮥ᦭ᙹᯩ݅. ᅙᩑǍ᮹༊ᱢᯕᄡ᭥ĥ⊂}ᗭෝ↽ᗭ⪵⦹໕ ᕽᱶၡ⦽⩶ᔢ⇵ᱶđŝෝ᨜ʑ᭥⦽᦭Łญ᷹ᮥᱽ᜽⦹۵äᯕအ ಽᯕᨱSFSM-LS ᦭Łญ᷹ᯕๅᬑ⬉ŝᱢᯙ᦭Łญ᷹ᯕ௝⧁

ᙹ ᯩ݅. ᯕ۵ SFSM-LS᦭Łญ᷹᮹ Ğᬑ ᝅᱽ Ǎ᳑ྜྷ᮹ Ñ࠺

a܆⦽⩶ᔢᯙǍ᳑⩶ᔢ⧉ᙹෝʑᅙᮝಽᔍᬊ⦹ʑ ভྙᨱĥ⊂

᭥⊹ෝ↽ᗭ⪵⦹໕ᕽᱶၡ⦽⇵ᱶđŝෝࠥ⇽⦹۵äᯕ௝❱݉ࡽ

(8)

݅. ᯕ్⦽✚ḶᮡǍ᳑ྜྷ᮹Ñ࠺⩶ᔢᯕᅖᰂ⧁ᙹಾࢱऽ్ḡí

ӹ┡ӽ݅. ੱ⦽⇵ᱶ᪅₉᮹ᇥᔑᮥᔕ⠕ᅕ໕Polynomial ᅕeჶ᮹

Ğᬑ DS1ŝ DS3᮹ ⩶ᔢᨱᕽ ᄡ᭥ ĥ⊂ }ᗭa ᷾a⦹޵௝ࠥ

᪅⯩ಅᇥᔑᯕ᷾a⦹۵⩥ᔢᮥӹ┡ԕŁᯩ۵ߑᯕ۵Ł₉᜾ᮥ

ᔍᬊ⧁Ğᬑ⩶ᔢᯕḥ࠺⦹ᩍ᪅⯩ಅᇩᦩᱶ⦽⩶ᔢᯕӹ┡ӹ۵

Rung⧉ᙹ᮹ྙᱽaၽᔾ⦹۵äᮝಽ❱݉ࡹᨕᱶၡ⦽༉ߙᮥࠥ⇽

⦹ʑ᭥⧕Ł₉݅⧎᜾ᮥᔍᬊ⧁Ğᬑᵝ᮹a⦥᫵⦹݅Ł❱݉ࡽ݅.

SFSM-LS᦭Łญ᷹᮹Ğᬑᄡ᭥ĥ⊂}ᗭa᷾a⧁ᙹಾᇥᔑᯕ

qᗭ⦹ᩍRung⧉ᙹ᮹Ğᬑ᪡zᮡ᪅ඹෝ↽ᗭ⪵⦹ᩍᦩᱶᱢᯙ

đŝෝ ࠥ⇽⧉ᮥ ᦭ ᙹ ᯩ݅.

5. đು

ᅙᩑǍᨱᕽ۵ĥ⊂ᄡ᭥ෝ↽ᗭ⪵⦹໕ᕽᱶၡ⦽Ǎ᳑ྜྷ᮹Ñ࠺

⩶ᔢᮥ⇵ᱶ⦹ʑ᭥⦽SFSM-LS ᦭Łญ᷹ᮥ}ၽ⦹ᩡ݅. }ၽࡽ

᦭Łญ᷹᮹✚Ḷᮥᇥᕾ⦹Ł ʑ᳕ᨱᔍᬊ⦹Łᯩ۵ᅕeჶŝ᮹

እƱᩑǍෝ☖⧕᦭Łญ᷹᮹⬉ᮉᖒᮥ᯦᷾⦹ᩡᮝ໑đŝෝᱶญ

⦹໕ ݅ᮭŝ z݅.

(1) FE ༉ߙᮥᯕᬊ⧕Ǎ᳑ྜྷᯕÑ࠺⧁ᙹᯩ۵ᩍ్⩶ᔢᮥǍ᳑⩶

ᔢ⧉ᙹಽaᱶ⦹ŁᯕॅǍ᳑⩶ᔢ⧉ᙹॅᮥᯥ᮹᮹aᵲ⊹ಽ

ᵲℊ⦹ᩍᵲℊ⩶ᔢᮥࠥ⇽⦹ᩡ݅. aᵲ⊹ෝđᱶ⦹ʑ᭥⧕ᝅ ᱽǍ᳑ྜྷᨱᕽ᮹ĥ⊂ᄡ᭥ෝᯕᬊ⦹ᩍ᪅₉⧉ᙹෝᖁᱶ⦹Ł

᪅₉ෝ↽ᗭ⪵⦹ʑ᭥⧕↽ᗭᱽŒჶᮥᯕᬊ⦹ᩡ݅. ᯕෝ☖⧕

Ǎ᳑ྜྷ ⩶ᔢ ⇵ᱶ ᦭Łญ᷹ SFSM-LSᮥ }ၽ⦹ᩡ݅.

(2) SFSM-LS ᦭Łญ᷹ᮡǍ᳑ྜྷᯕᝅᱽÑ࠺⧁ᙹᯩ۵Ǎ᳑⩶ᔢ

⧉ᙹෝʑၹᮝಽ⩶ᔢᮥ⇵ᱶ⦹အಽ݅ෙᅕeჶᨱእ⧕ᱢᮡ

ᄡ᭥ ĥ⊂ ߑᯕ░ಽ ᱶၡ⦽ ⩶ᔢᮥ ⇵ᱶ ⧁ ᙹ ᯩ݅.

(3) SFSM-LS ᦭Łญ᷹ᮡᝅᱽǍ᳑ྜྷ᮹Ñ࠺⩶ᔢᯕᅖᰂ⧕ḩᙹ

ಾ݅ෙᅕeჶᨱእ⧕୑ᨕӽ⩶ᔢ⇵ᱶᱶၡࠥෝӹ┡ԕ໑

ᦩᱶᱢᯙ⇵ᱶđŝෝࠥ⇽⦹ʑভྙᨱ݅᧲⦽Ñ࠺⩶ᔢᨱ

ݡ⧕ᕽࠥ ᱢᬊ a܆ ⧁ äᮝಽ ❱݉ࡽ݅.

(4) SFSM-LSᨱŁಅࡹ۵Ǎ᳑⩶ᔢ⧉ᙹa᷾a⧁ᙹಾ⇵ᱶᱶၡ

ࠥaᯝᱶ⦹íᙹಕ⦹ᩍᦩᱶᱢᯙ⇵ᱶđŝෝࠥ⇽⦹ʑ᭥⧕

ᕽ۵↽ݡ⦽ฯᮡǍ᳑⩶ᔢᮥŁಅ⦹۵äᯕ┡ݚ⦹ḡอ, ᝅᱽ

Ñ࠺ࡹ۵⩶ᔢᮥၹᩢ⧁ᙹᯩ۵Ǎ᳑⩶ᔢ⧉ᙹaŁಅࡽ݅໕

޵ᬒᱶၡ⦽đŝෝࠥ⇽⧁ᙹᯩʑভྙᨱŁಅࡹ۵Ǎ᳑⩶ᔢ

⧉ᙹᨱ ݡ⦽ ↽ᱢ⪵ ŝᱶᯕ ⇵aಽ ⦥᫵⦹݅Ł ❱݉ࡽ݅.

(5) ᄡ᭥ĥ⊂}ᗭa✚ᱶ}ᗭᯕᔢᯕࢁĞᬑSFMS-LS᮹⇵ᱶ ᱶၡࠥa ᯝᱶ⦹í ᙹಕ⦹۵ Ğ⨆ᯕ ӹ┡ӹʑ ভྙᨱ ᯕෝ

☖⧕Ǎ᳑ྜྷ⩶ᔢ⇵ᱶᨱ⦥᫵⦽↽ᗭ᮹ᄡ᭥ĥ⊂}ᗭᖁᱶᮥ

᭥⦽ ʑⅩ ᯱഭಽ ⪽ᬊ ⧁ ᙹ ᯩᮥ äᮝಽ ❱݉ࡽ݅.

(6) 2Ğe ᩑᗮƱᨱᕽ ᄡ᭥ ĥ⊂ }ᗭෝ 4 Ŕᮝಽ aᱶ⦹ᩡᮥ

ĞᬑǍ᳑ྜྷᨱݡ⧕ᕽ॒eĊᮝಽŁ෕íᄡ᭥ෝĥ⊂⦹۵

äᯕᱶၡ⦽⇵ᱶđŝෝࠥ⇽⦹ᩍ, ᄡ᭥ĥ⊂᭥⊹ᨱ঑௝ᕽࠥ

ᱶၡࠥa݅᧲⦹íӹ┡ӹ۵ၵ݅᧲⦽ĥ⊂᳑ÕŝǍ᳑ྜྷᨱ

ݡ⧕ᄡ᭥ĥ⊂᭥⊹ෝᖁᱶ⧁ᙹᯩ۵ᄡ᭥ĥ⊂᭥⊹ᖁᱶᨱ

ݡ⦽ ᩑǍa ⇵aᱢᮝಽ ⦥᫵⦹݅Ł ❱݉ࡽ݅.

qᔍ᮹ɡ

ᅙᩑǍ۵ǎ☁⧕᧲ᇡÕᖅʑᚁ⩢ᝁᩑǍ}ၽᔍᨦ᮹ᩑǍእḡ ᬱ(ŝᱽჩ⪙ 10ʑᚁ⩢ᝁ E05)ᨱ ᮹⧕ ᙹ⧪ࡹᨩ᜖ܩ݅.

References

Bae, I. H., Ha, K. H., Ham, H. J. and Choi, I. H. (2007).“A study on the long-span bridge behavior using GPS.” Korean Society of Civil Engineers conference, pp. 1764-1767 (in Korean).

Chapra, S. C. (2008). Applied numerical methods with MATLAB for engineers and scientists:2nd Edition, McGraw-Hill.

Cho, S. J., Yi, J. H., Lee, C. G. and Yun, C. B. (2007). “Evaluation of load carrying of bridge using ambient vibration tests.” Journal of the Korean Society of civil Engineers, 27(1A), pp. 79-89 (in Korean).

Choi, B. G., Cho, K. H. and Na, Y. W. (2008). “Development of GPS monitoring for behavior analysis of long span bridge.”

Journal of the Korean Society for Geo-Spatial information System, Vol. 16, No. 3, pp. 111-116 (in Korean).

Datta, B. N. (2009). Numerical linear algebra and applications, SIAM.

Im, S., Hurlebaus, S. and Kang, Y. J. (2011). “A summary review of GPS technology for structural health monitoring.” Journal of Structural Engineering.

Kim, H. K., Lee, H. J., Jang, J. H. and Ro, S. K. (2007).

“Development of field-measured data-based analysis model of a suspension bridge.” Journal of the Korean Society of civil Engineers, 27(6A), pp. 829-836 (in Korean).

Liu, G. R. and Han, X. (2003). Computational inverse techniques in nondestructive evaluation, CRC Press.

Meo, M. and Zumpano, G.(2007). “On the optimal sensor placement techniques for a bridge structure.” Engineering Structures, Vol.

27, pp. 1488-1497.

Park, J. C., Gil, H. B., Kang, S. G. and Lim, C. W. (2010).

“Dynamic characteristics of cable-stayed bridge using global navigation satellite system.” Journal of the Korean Society of civil Engineers, 30(4A), pp. 375-382 (in Korean).