䈖䈉䈋䈇䊐䉤䊷䊤䊛╙ 㪈㪈 ภ㩷 㪆㩷㪉㪇㪇㪊㪅㪈㩷㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕↪䈇䈢䊃䊮䊈䊦ⷒᎿ䈱ⶄᢙ䈵䈶ഀ䉏⸃ᨆ
NUMERICAL ANALYSIS OF MULTIPLE CRACKS IN TUNNEL LINING USING THE
DISCRETE MODELING APPROACH㋈ᧁᱜ᮸䋪䊶Ꮷ㩷 ⥄ᶏ䋪䋪䊶ᄢᵤ᡽ᐽ䋪䋪䋪䊶ᣣᲧ㊁ᖝਭ䋪䋪䋪䋪
Masaki SUZUKI, Zihai SHI, Masayasu OHTSU and Yoshihisa HIBINO㩷 The discrete crack approach, which has been extensively used for analyzing single cracks, isextended for numerical analysis of multiple cracks, based on the fictitious crack model and thecrack-tip-controlled numerical algorithm. Assuming each crack as an effective crack while
restraining the growth of others in turn, an explicit formulation of the mode I-Type crack
propagation for multiple cracks is presented, and a minimum load criterion for crack extension
is proposed for its numerical implementation. The accuracy and effectiveness of thisnumerical method in analyzing multiple-crack problems are demonstrated through anexperimental verification with fracture tests on real-size tunnel specimens.

㩷 Key Words : multiple cracks, discrete modeling, fictitious crack, minimum load criterion,
tunnel lining㩷㪈㪅㩷䈲䈛䉄䈮㩷ᄖജ䈅䉎䈇䈲⚻ᐕ䈭䈬䈱ⷐ࿃䈪䈵䈶ഀ䉏៊்䉕ฃ䈔䈢䉮䈩㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕ㆡ↪䈚䈢੐଀䈏䈾䈫䉖䈬䈭䈇䈱䈏⃻⁁䈪䈅䉎䇯䊮䉪䊥䊷䊃᭴ㅧ‛䈪䈲䇮䈠䈱᭴ㅧஜోᕈ䉕␜䈜᦭ല䈭ᜰᮡ䈫䈠䈖䈪ᧄ⎇ⓥ䈪䈲䇮䈜䈪䈮ⶄᢙ䈵䈶ഀ䉏䈏↢䈛䈩䈇䉎ᣢ⸳䈚䈩䇮䈵䈶ഀ䉏㐿ญ᏷䈏䉋䈒૶䉒䉏䉎䇯࿷᧪Ꮏᴺ䈪ᣉᎿ䈘䉏䉮䊮䉪䊥䊷䊃᭴ㅧ‛䈮䈧䈇䈩䇮䈵䈶ഀ䉏㐿ญ᏷䈮ၮ䈨䈇䈩ቯ䈢ዉ᳓〝䊃䊮䊈䊦䈱ᄌ⁁⃻⽎䉕଀䈮䈫䉏䈳䇮࿑䋭㪈 䈮␜䈜䈫㊂⊛䈮ᄌ⁁䊜䉦䊆䉵䊛䉕⸃᣿䈜䉎ᚻᴺ䈫䈚䈩䇮৻ᧄ䈱䈵䈶ഀ䈍䉍䇮ⷒᎿ䉝䊷䉼䈱⢛㕙ⓨᵢ䈮⿠࿃䈚䈩䉝䊷䉼䈎䉌஥ო䉴䊒䉏᜼േ䈱ౣ⃻䈮䉋䈒↪䈇䉌䉏䉎㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕ⶄᢙ䊥䊮䉫䊤䉟䊮䈮䈎䈔䈩㐿ญ᏷ 2䌾5mm ⒟ᐲ䈫Ყセ⊛ᄢ䈐䈭ⶄ䈵䈶ഀ䉏䈱ᢙ୯⸃ᨆ䈻᜛ᒛ䈚䈢 1),2) 䇯䈖䉏䈲䇮ᒁᒛエൻೣ䈮ᢙ䈵䈶ഀ䉏䈱⊒↢⃻⽎䈏䈅䉎䇯䈖䈱䈫䈐䇮ⷒᎿ⴫㕙䈮↢䈛䈩ၮ䈨䈇䈩䇮ⶄᢙ䈵䈶ഀ䉏䈱䈉䈤છᗧ䈱䋱ᧄ䈱䈵䈶ഀ䉏䈏ㅴዷ䈇䉎ⶄᢙ䈵䈶ഀ䉏䈱㐿ญ᏷䈲䇮䈠䈱᭴ㅧ቟ోᕈ䉕⹏ଔ䈜䉎䈜䉎䈱䈮ᔅⷐ䈭ᄖജ䉕䈠䉏એᄖ䈱䈵䈶ഀ䉏䉕᜔᧤䈜䉎ਛ䈪਄䈪㊀ⷐ䈭ᜰᮡ䈫䈭䉎䇯䈠䉏䈡䉏᳞䉄䇮䈠䉏䉌䈱䈉䈤ᦨዊ⩄㊀䉕␜䈜䈵䈶ഀ䉏䊌䉺䊷৻ᣇ䇮䉮䊮䉪䊥䊷䊃᭴ㅧ‛䈱䈵䈶ഀ䉏⸃ᨆ䊝䊂䊦䈮䈍䈇䈩䊮䈏⌀䈱䈵䈶ഀ䉏䊌䉺䊷䊮䈫䈚䈩ㆬቯ䈜䉎⸃ᨆᴺ䈪䈅䉎䇯䈠䈚䈲䇮䉮䊮䉪䊥䊷䊃䈱⎕უജቇ䈮ၮ䈨䈒ಽᏓ䈵䈶ഀ䉏䊝䊂䊦䈫㔌䈩䇮ᧄᢙ୯⸃ᨆ䊝䊂䊦䈱᦭ലᕈ䈮䈧䈇䈩䇮ታⷙᮨ䈱䊃䊮䊈䊦ᢔ䈵䈶ഀ䉏䊝䊂䊦䈏䈅䉎䇯೨⠪䈮䈍䈇䈩䈲䇮䈵䈶ഀ䉏䉕ㅪ⛯ⷒᎿ⎕უታ㛎䈫䈱ᬌ⸽䈮䉋䉍⏕⹺䈚䈢䇯૕䊝䊂䊦䈱ਛ䈪⴫⃻䈚䈩䈍䉍䇮RC ㇱ᧚䈮⷗䉌䉏䉎ಽᢔ䈚䈢ⶄᢙ䈵䈶ഀ䉏䈱᜼േ䉕ౣ⃻䈜䉎䈱䈮䉋䈒૶䉒䉏䈩䈇䉎䇯৻ⓨᵢᣇ䇮ᓟ⠪䈮䈍䈇䈩䈲䇮䈵䈶ഀ䉏䉕⋥ធਇㅪ⛯㕙䈫䈚䈩䊝䊂䊦ൻ䈚䈩䈍䉍䇮ዪᚲ⊛䈭৻ᧄ䈱䈵䈶ഀ䉏᜼േ䉕ౣ⃻䈜䉎䈱䈮䉋䈒૶䉒䉏䈩䈇䉎䇯࿶ᝂ䈚䈢䈏䈦䈩䇮䈵䈶ഀ䉏㐿ญ᏷䈮⌕⋡䈚䈩䇮೨ㅀ䈚䈢ዪᚲ⊛䈭ᄢ䈐䈭ⶄᢙ䈵䈶ഀ䉏䈱ㅴዷ᜼േ䉕ౣ⃻䈜䉎䈮䈲䇮ಽᏓ䈵䈵䈶䉒䉏᏷
2䌾5 䋘⒟ᐲ

䈶ഀ䉏䊝䊂䊦䉋䉍㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䈱ᣇ䈏ㆡ䈚䈩䈇䉎䈫⠨䈋䉌䉏䉎䈏䇮ᣢᓔ⎇ⓥ䈪䈲ⶄᢙ䈵䈶ഀ䉏䈱ㅴዷ᜼േ䈮䈧䈇㩷㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽㿽䋪㩷 㩷 㩷 㩷 㩷 ␠ળⅣႺ䉣䊮䉳䊆䉝䊥䊮䉫੐ᬺㇱ 䍵䍐䍪䍙䍐䍖䍷䍭䍦䍚䍼䍰䍻䍢ㇱ䋪䋪㩷 㩷 㩷 ਛᄩ⎇ⓥᚲ㩷 ✚วᛛⴚ㐿⊒ㇱ䋪䋪䋪㩷 㩷 ᾢᧄᄢቇᄢቇ㒮㩷 ⥄ὼ⑼ቇ⎇ⓥ⑼䋪䋪䋪䋪㩷 ᧲੩㔚ജ䊺㩷 ᑪ⸳ㇱ㩷 ࿯ᧁ䊶ᑪ▽ᛛⴚ䉶䊮䉺䊷࿑䋭㪈㩷 ዉ᳓〝䊃䊮䊈䊦䈱ᄌ⁁੐଀㩷㪈㪇㪐㩷㔌ᢔ߭߮ഀࠇࡕ࠺࡞ࠍ↪޿ߚ࠻ࡦࡀ࡞ⷒᎿߩⶄᢙ߭߮ഀࠇ⸃ᨆ! 㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕↪䈇䈢ⶄᢙ䈵䈶ഀ䉏䈱ᢙ୯㪉㪅㩷䈖䈖䈪䇮 Q la 䈲ᒁᒛᒝᐲ䈮⋧ᒰ䈜䉎▵ὐജ䇮 CR a 䈲ᄖജ䈱⸃ᨆ㩷! ၮᧄ⊛䈭⠨䈋ᣇ૞↪૏⟎䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱 Crack A 䈱వ┵䈪䈵䈶ഀ䉏వ┵䉕೙ᓮ䈜䉎ᚻᴺ䈮ၮ䈨䈇䈩ⶄᢙ䈵䈶ഀ䉏䈱ㅴዷ䊌䉺䊷䊮䉕᳿ቯ䈜䉎႐ว䈮䈲䇮ᰴ䈱䉋䈉䈭ⶄ㔀䈭䈵䈶ഀ!

䉏㑆䈱ᐓᷤ໧㗴䉕⠨ᘦ䈜䉎ᔅⷐ䈏䈅䉎䇯䈠䉏䈲䇮࿑䋭㪉䈮␜䈜䈫䈍䉍䇮㽲ਥ䈵䈶ഀ䉏䈏ㅴዷ䈚䇮ઁ䈱䈵䈶ഀ䉏䈏஗ṛ䈜䉎!

႐ว䇮㽳ਥ䈵䈶ഀ䉏䈏ㅴዷ䈚䇮ઁ䈱䈵䈶ഀ䉏䈏หᤨ䈮ㅴዷ䈜䉎႐ ว 䇮㽴ਥ 䈵䈶ഀ 䉏䈏ㅴ ዷ 䈚䇮ઁ 䈱䈵䈶ഀ 䉏䈏㐽ว 䈜䉎႐ว䇮䈪䈅䉍䇮䈖䉏䉕ⶄᢙ䈱䈵䈶ഀ䉏䈮䈧䈇䈩ో䈩⠨ᘦ 䈜!

䉎䈫㕖Ᏹ䈮ⶄ㔀䈪䈅䉎䈖䈫䈏ኈᤃ䈮ផ᷹䈪䈐䉎䈪䈅䉐䈉䇯䈠䈖䈪䇮ⶄᢙ䈵䈶ഀ䉏䈱ㅴዷ䊌䉺䊷䊮䉕᳿ቯ䈜䉎䈢䉄䈱䊦䈱▵ὐജ䇮 P a 䈲 Crack A 䈱ㅴዷ䈮ᔅⷐ䈭ᄖജ䋨ᧂ⍮ᢙ䋺1
୘䋩䇮 CI a i 䈲 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪
䈘䈞䈢䈫䈐䈱 Crack A 䈱䈵䈶ഀ䉏వ┵䈮䈍䈔䉎▵ὐജ䇮 F a i
䈲 Crack A 䈱 ฦ 䈵 䈶 ഀ 䉏 ▵ ὐ 䈱 ⚿ ว ജ 䋨 ᧂ ⍮ ᢙ 䋺i=1,2,...,N䋩䇮CI ab j 䈲 Crack B 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱 Crack A 䈱䈵䈶ഀ䉏వ┵䈱▵ὐജ䇮

F' b j 䈲 Crack B 䈱 ฦ 䈵 䈶 ഀ 䉏 ▵ ὐ 䈱 ⚿ ว ജ 䋨 ᧂ ⍮ ᢙ 䋺j=1,2,...,M䋩䈪䈅䉎䇯㩷 ᰴ䈮䇮ฦ䈵䈶ഀ䉏▵ὐ䈮䈍䈔䉎ᄌ૏䈱㊒䉍ว䈇ᑼ䉕┙䈩䉎䇯䊷䊦䉕ᰴ䈱䉋䈉䈮⸳ቯ䈚䈢䇯!

䉁䈝䇮ⶄᢙ䈵䈶ഀ䉏䈱䈠䉏䈡䉏䈮ኻ䈚䈩䇮છᗧ䈱৻ᧄ䈱䈵㩷 㩷 W i = BK i × P +aaa䈶ഀ 䉏䋨ਥ 䈵䈶ഀ 䉏䋩䈏ㅴ ዷ 䈜䉎䈫઒ ቯ 䈚䈩䇮䈠䉏એ ᄖ 䈱䈵!

䈶ഀ䉏䈱ㅴዷ 䉕᜔᧤䈜䉎ਛ 䈪䇮ਥ䈵䈶ഀ 䉏䈏ㅴዷ䈜䉎䈱䈮ᔅⷐ䈭ᄖജ䉕䉪䊤䉾䉪ᣇ⒟ᑼ䈮䉋䉍▚ቯ䈜䉎䇯䈠䈚䈩䇮䈠䉏䉌ⶄᢙ䈵䈶ഀ䉏䈱䈠䉏䈡䉏䈏ਥ䈵䈶ഀ䉏䈫䈚!

!

䈩ㅴ ዷ 䈜䉎䈱䈮ᔅ ⷐ 䈭ᄖ ജ䈱䈉䈤䇮ᦨ ዊ⩄㊀ 䉕␜ 䈜䈵䈶ഀ䉏䊌䉺䊷䊮䉕⌀䈱䈵䈶ഀ䉏䊌䉺䊷䊮䈫䈚䈩᳿ቯ䈜䉎䇯!

!

bai =1baa

k =1bb3䈖䈖䈪䇮 W a i 䈲 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈱ᄌ૏㊂䋨ᧂ⍮ᢙ䋺i=1,2,...,N䋩䇮 BK a i 䈲ᄖജ䈱૞↪૏⟎䈮න૏⩄㊀䉕૞↪
䈘䈞䈢䈫䈐䈱 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮䈍䈔䉎ᄌ૏㊂䇮ᧂ⍮ᢙ䋺䈞䈢䈫䈐䈱 Crack B 䈱ฦ 䈵䈶ഀ 䉏▵ ὐ 䈮䈍䈔䉎ᄌ ૏ ㊂ 䇮

AK ba ji 䈲 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱 Crack B 䈱ฦ䈵䈶ഀ 䉏▵ ὐ 䈱ᄌ૏ ㊂䇮 AK bjk 䈲Crack B 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱࿑䋭㪊 䈮䇮䊝䊷䊄㸇䈱ੑᧄ䈱䈵䈶ഀ䉏䋨Crack A䇮Crack!

B䋩䈮䈧䈇䈩䇮䈵䈶ഀ䉏వ┵೙ᓮ䈮䉋䉎ฦ䈵䈶ഀ䉏䈱ฦ䈵䈶ഀ䉏▵ὐ䈮૞↪䈜䉎⚿วജ䈍䉋䈶㐿ญᄌ૏㊂䇮ᄖജ䈱㑐ଥ࿑䉕␜䈜䇯એ ਅ䈮䇮ੑᧄ 䈱䈵䈶ഀ䉏䈮䈍䈔䉎䉪䊤䉾䉪ᣇ ⒟ᑼ䈱ቯᑼൻ䈮䈧䈇䈩␜䈜䇯㩷 䉁䈝䇮Crack A 䈏ਥ䈵䈶ഀ䉏䈫䈚䈩ㅴዷ䈜䉎႐ว䈮䈧䈇䈩⠨䈋䉎䇯੹䇮Crack A 䈱వ┵䈏ᒁᒛᒝᐲ䈮೔㆐䈚䈢䈫઒ቯ

Mj =1㩷 ᦨᓟ䈮䇮ฦ䈵䈶ഀ䉏▵ὐ䈱⚿วജ䈫㐿ญᄌ૏䈱㑐ଥ䈮䈍䈇䈩䇮࿑㧙 䈱ᒁᒛエൻ․ᕈ䉕ㆡ↪䈚䇮(4)(5)ᑼ䉕┙䈩䉎䇯( )㩷 㩷 F ' = f (W ' )ii㩷 㩷 Fa = f Wa
jbjb

45

䈭䈍䇮 F' b j 䈍䉋䈶 W' b j 䈮䉝䊘䉴䊃䊨䊐䉞䉕ઃ⸥䈚䈩䈇䉎䈱䈲䇮඙೎䈚䈢䉅䈱䈪䈅䉎䇯å CI × F + å CI × F 'ia

Crack B 䈱ฦ䈵䈶ഀ䉏▵ὐ䈱ᄌ૏㊂䈪䈅䉎䇯᜔᧤䈘䉏䈩䈇䉎䈵䈶ഀ䉏䉕ᒝ⺞䈚䈢䉅䈱䈪䈅䉍䇮ਥ䈵䈶ഀ䉏䈫䈚䇮Crack A 䈱వ┵䈪䈱ജ䈱㊒䉍ว䈇ᑼ䉕┙䈩䉎䇯

i =1bj=1,2,...,M䋩䇮 BK b j 䈲ᄖജ䈱૞↪૏⟎䈮න૏⩄㊀䉕૞↪䈘! 䉪䊤䉾䉪ᣇ⒟ᑼ䈱ቯᑼൻ㩷 㩷 Qla = CR a × Pa +M㩷 㩷 W ' j = BK j × P + å AK ji × F i + å AK jk × F' k

Crack B 䈱 ฦ 䈵 䈶 ഀ 䉏 ▵ ὐ 䈱 ᄌ ૏ ㊂

Crack BiaN

2

Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈱ᄌ૏㊂䈪䈅䉎䇯䉁䈢䇮 W'bj 䈲࿑䋭㪉㩷 ⶄᢙ䈵䈶ഀ䉏㑆䈱ᐓᷤ໧㗴䋨㪉 ᧄ䈱႐ว䋩㩷

Nj =1

Crack B 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱㽴㐽ว
Crack Aਥ䈵䈶ഀ䉏k =1䈢䈫䈐䈱 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈱ᄌ૏㊂䇮 AK ab ij 䈲㽲஗ṛ!

MAK a ik 䈲 Crack A 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞㽳ㅴዷㅴዷN

å AK aik × Fak + å AK abij × F 'bjjabjb1㩷 ᑼ (1) 㨪 (5) 䉕 䉪 䊤 䉾 䉪 ᣇ ⒟ ᑼ 䈫 ๭ 䈶 䇮 ᣇ ⒟ ᑼ 䈏䋨2N+2M+1䋩୘ 䈪䇮ᧂ ⍮ ᢙ䈏䋨2N+2M+1䋩୘ 䈪䈅䉍䇮䈖䈱ㅪ┙ᣇ⒟ᑼ䉕⸃䈒䈖䈫䈮䉋䈦䈩䇮ೋ䉄䈮઒ቯ䈚䈢 Crack A 䈏ㅴዷ䈜䉎䈱䈮ᔅⷐ䈭ᄖജ䇮ฦ䈵䈶ഀ䉏▵ὐ䈪䈱⚿วജ䈍䉋䈶㐿ญᄌ૏䈏᳞䉄䉌䉏䉎䉅䈱䈪䈅䉎䇯㪈㪈㪇㩷䈖䈉䈋䈇䊐䉤䊷䊤䊛╙ 㪈㪈 ภ㩷 㪆㩷㪉㪇㪇㪊㪅㪈㩷

Tip tensile
force CRb

Tip tensile
force CRa

Tip tensile
force CI kba

Tip tensile
force CI kaka

Tip tensilekforce CIab

Influencejkcoefficient AK bakaන૏ᄖജ㧔2㧕ࠍ૞↪ߐߖߚᤨߩฦ߭߮ഀࠇవ┵ߩ▵ὐജߣฦ߭߮ഀࠇߩ㐿ญᄌ૏Dkb

Influencekjkcoefficient AKbb
Influenceikcoefficient AKabInfluenceik
coefficient AKa

C

Tip tensile
kforce CIb%TCEM #ߦ߅ߌࠆ߭߮ഀࠇ▵ὐߦkන૏⚿วജ㧔‫ޓޓ‬㧕ࠍ૞↪ߐߖߚaᤨߩฦ߭߮ഀࠇ▵ὐߩ▵ὐജߣ㐿ญᄌ૏

E %TCEM $ߦ߅ߌࠆ߭߮ഀࠇ▵ὐߦkන૏⚿วജ㧔‫ޓޓ‬㧕ࠍ૞↪ߐߖߚbᤨߩฦ߭߮ഀࠇ▵ὐߩ▵ὐജߣ㐿ญᄌ૏

Limit nodal
force Q lb

Limit nodal
force Q lajbjbiaiajbiajbia

F %TCEM #߇ㅴዷߔࠆᤨߩ⩄㊀⁁ᘒ

F%TCEM $߇ㅴዷߔࠆᤨߩ⩄㊀⁁ᘒ࿑䋭㪊㩷 ⶄᢙ䈵䈶ഀ䉏䈱వ┵೙ᓮ㩷
ᚻᴺ䈱᭎ᔨ࿑㩷 㩷 㩷 㩷 㩷 㩷㩷
࿑–㪊㩷 ⶄᢙ䈵䈶ഀ䉏䈱వ┵೙ᓮᚻᴺ䈱᭎ᔨ࿑㩷㩷 㩷 Fbftj( )= f Wb j10ᒁᒛᔕജ䈖䈖䈪䇮 P b 䈲 Crack B 䈱ㅴዷ䈮ᔅⷐ䈭ᄖജ䇮 Q lb 䈲ᒁᒛᒝ
Wc=5Gf /ftᐲ䈮⋧ᒰ䈜䉎▵ὐജ䇮 CR b 䈲ᄖജ䈱૞↪૏⟎䈮න૏⩄㊀䉕

W1=0.75Gf /ft૞↪䈘䈞䈢䈫䈐䈱 Crack B 䈱వ┵䈪䈱▵ὐജ䇮 CI ba i 䈲 CrackA 䈱ฦ䈵䈶ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱 Crack Bft/4䈱䈵䈶ഀ䉏వ┵䈮䈍䈔䉎▵ὐജ䇮 CI b j 䈲 Crack B 䈱ฦ䈵䈶
W1ഀ䉏▵ὐ䈮න૏⩄㊀䉕૞↪䈘䈞䈢䈫䈐䈱 Crack B 䈱䈵䈶ഀWc䉏వ┵䈱▵ὐജ䉕␜䈜䇯߭߮ഀࠇ᏷

࿑䋭㪋㩷 ᒁᒛエൻ․ᕈ㩷(3) ⸃ᨆ䊐䊨䊷ห᭽䈮䈚䈩䇮Crack B 䈏ਥ䈵䈶ഀ䉏䈫䈚䈩ㅴዷ䈜䉎႐ว䈱䉪䊤䉾䉪ᣇ⒟ᑼ䈏(6)㨪(10)ᑼ䈱䈫䈍䉍ᓧ䉌䉏䉎䇯㩷 㩷 Qlb = CRb × Pb +NMⶄᢙ䈱䈵䈶ഀ䉏䋨I=1,2,..,N䋩䈮ኻ䈚䈩䇮䈠䉏䈡䉏䈱䈵䈶ഀ䉏

å CI × F ' +å CI × Fiaibai =1࿑䋭㪌 䈮⸃ᨆ䊐䊨䊷䉕␜䈜䈫䈍䉍䇮䉁䈝䈅䉌䈎䈛䉄⸳ቯ䈚䈢jbjb6j =1䈏ਥ䈵䈶ഀ䉏䈫䈚䈩ㅴዷ䈜䉎䈱䈮ᔅⷐ䈭ᄖജ䉕䉪䊤䉾䉪ᣇ⒟ᑼ
䉋䉍᳞䉄䉎䇯䈠䈚䈩䇮ᦨዊ⩄㊀䈱䊦䊷䊦䈮ᓥ䈦䈩⌀䈱䈵䈶ഀ䉏䊌䉺䊷䊮䉕᳿ቯ䈜䉎䇯ᰴ䈮䇮䈠䈱䈵䈶ഀ䉏䊌䉺䊷䊮䈮ၮ䈨䈒Ⴚ㩷 㩷 W 'i = BK i × P +aabN

å AK × F ' +å AK × F
ikakak =1

jjb㩷 㩷 Wb = BK × Pb +Ni( )iij
abjb(7)

j =1jibaiaM⇇᧦ઙ䈱ਅ䈪ᔕജ⸃ᨆ䉕ⴕ䈇䇮䈖䈱䈫䈐ਥ䈵䈶ഀ䉏એᄖ䈱䈵䈶ഀ䉏䈱వ┵䈪ᒁᒛᒝᐲ䉕⿥ㆊ䈜䉎䉋䈉䈭႐ว䈮䈲䇮䈠䈱▵ὐ䉕⸃᡼䈚ౣᐲႺ⇇᧦ઙ䉕⸳ቯ䈚䇮䉪䊤䉾䉪ᣇ⒟ᑼ䉕⸃䈐⋥䈚å AK × F ' + å AK × Fi =1

㩷 㩷 F 'a = f W 'a

Mjkbkb8k =1

䈩ᔕജ⸃ᨆ䉕ⴕ䈉䇯⸃ 䈱᦭ ല ᕈ ್ ቯ 䈪䈲䇮એ ਅ 䈱䉋䈉䈭ੑ ⒳ 㘃 䈱Ⴚ ⇇ ᧦ ઙ 䈱9ୃᱜ䉕ⴕ䈉䉅䈱䈫䈚䈢䇯㩷㽲䉪䊤䉾䉪ᣇ⒟ᑼ䈱⸃䈮䈍䈇䈩䇮䈵䈶ഀ䉏▵ὐ䈪䈱㐿ญ㪈㪈㪈㩷㔌ᢔ߭߮ഀࠇࡕ࠺࡞ࠍ↪޿ߚ࠻ࡦࡀ࡞ⷒᎿߩⶄᢙ߭߮ഀࠇ⸃ᨆ1ᄌ૏䈏䊙䉟䊅䉴䋨W䋼0䋩䈫䈭䉎႐ว䋨䈧䉁䉍䇮䈵䈶ഀ䉏䈏㐽ว䈜䉎႐ว䋩䈮䈲䇮⹥ᒰ䈜䉎䈵䈶ഀ䉏䈱వ┵䉕৻▵ὐಽ㐽ว䈘䈞䉎ᣇะ䈮⒖േ䈘䈞䇮Ⴚ⇇᧦ઙ䉕ౣ⸳ቯ䈜䉎䇯㩷㽳䉪䊤䉾䉪ᣇ⒟ᑼ䈱⸃䈮䈍䈇䈩䇮นㅒ⃻⽎䋨䈵䈶ഀ䉏▵ὐ䈪䈱⚿วജ䈏೨䉴䊁䉾䊒䈪䈱⚿วജ䉋䉍ᄢ䈐䈒䈭䉎䉋䈉䈭⁁ᘒ䋩䈏↢䈛䈢႐ว䈮䈲䇮⹥ᒰ䈜䉎䈵䈶ഀ䉏▵ὐ䈪䈱⚿วജ䉕೨䉴䊁䉾䊒䈪䈱⚿วജ䊧䊔䊦䈮଻ᜬ䈜䉎䇯56#46

߭߮ ഀ ࠇࡄ ࠲࡯ࡦ Σ ߩ઒ ቯ
( I= 1, 2. .., N )ࠢ࡜ ࠶ࠢᣇ ⒟ ᑼ ࠍ⸃ ߊ࿑䋭㪍㩷 ታ㛎䊝䊂䊦㩷

11No9ᢙ୯⸃ᨆ䊝䊂䊦Y esNoᢙ୯⸃ᨆ䊝䊂䊦䈲䇮࿑䋭㪎 䈮␜䈜䈫䈍䉍䇮ታ㛎⚿ᨐ䈪䈱䈵㕖 นㅒ ⃻ ⽎2
I =I +1䈶ഀ䉏⁁ᴫ䈎䉌䇮Crack A䋺ᄤ┵ CL ㇱ䇮Crack B䋺ฝ஥ოY esSL ㇱ䇮Crack C䋺ฝᐩㇱ䇮Crack D䋺Ꮐᐩㇱ䇮Crack E䋺

Ꮐ஥ო SL ㇱ䈮䈠䉏䈡䉏䈵䈶ഀ䉏⚻〝䉕⸳ቯ䈚䇮ೋᦼ䊉䉾䉼

NoI =N䉕ฦ䈵䈶ഀ䉏䈮ኻ䈚䈩 2.5cm䋨1/10H䇮H䋺ⷒᎿᏎෘ䋩䈫⸳ቯ䈚

Y es

䈢䇯

P = M in {P ,P ,, ,P }12Nᔕജ ⸃ ᨆࠨ ࡉ ࠢ࡜࠶ ࠢవ ┵ ߩ⸃᡼Noǻ  HVYe sᰴ ߩ⩄ ㊀ ࡟ࡌ ࡞ ߳㩷࿑䋭㪌㩷 ⸃ᨆ䊐䊨䊷㩷㪊㪅㩷ታⷙᮨ䈱䊃䊮䊈䊦ⷒᎿ⎕უታ㛎䈮䉋䉎ᧄᚻᴺ䈱ᬌ⸽㩷ታ㛎᭎ⷐ࿑䋭㪍 䈮ታ㛎䊝䊂䊦䉕␜䈜䇯䊃䊮䊈䊦ⷒᎿ⹜㛎૕䈲䇮⋥ᓘ

2.5m䇮ⷒᎿ䉮䊮䉪䊥䊷䊃ෘ 25cm 䈱ታⷙᮨ䈱ዉ᳓〝䊃䊮䊈䊦ᮡ࿑䋭㪎㩷 ᢙ୯⸃ᨆ䊝䊂䊦㩷Ḱ㚍かᒻ䈫䈚䈩䉮䊮䉪䊥䊷䊃䈪⵾૞䈘䉏䇮タ⩄᧦ઙ䈲䉝䊷䉼ⷒᎿ䈱⢛㕙ⓨᵢ䉕ᗐቯ䈚䈩䇮䈠䈱ⓨᵢၞએᄖ䈱஥ო䈮╬࿶ タ䉁䈢䇮ⷒᎿ䉮䊮䉪䊥䊷䊃䈱᧚ᢱ‛ᕈ୯䈲⴫䋭㪈 䈮␜䈜䈫䈍䉍⩄䈘䉏䈢䉅䈱䈪䈅䉎䇯ታ 㛎⋡ ⊛䈲䈵䈶ഀ䉏ㅴዷ䊌䉺䊷䊮䈍䉋䈪䈅䉍䇮ᢙ୯⸃ᨆ䈮↪䈇䉎䉮䊮䉪䊥䊷䊃䈱ᒁᒛエൻ․ᕈ䈲࿑䋭䈶ⷒᎿ⠴⩄ജ䈱ᛠី䈪䈅䉎䇯䈭䈍䇮タ⩄䈲⩄㊀೙ᓮ䈮䉋䈦䈩㪋 䈮␜䈜㕖✢ᒻ䊝䊂䊦䈪ㄭૃ䈚䈢䇯ታᣉ䈘䉏䇮⸘᷹䈲タ⩄⩄㊀䇮䈵䈝䉂㊂䇮ౝⓨᄌ૏㊂䈫䈭䈦䈩䈍䉍䇮䈵䈶ഀ䉏㐿ญ᏷䈲⸘᷹䈘䉏䈩䈇䈭䈇䇯㪈㪈㪉㩷䈖䈉䈋䈇䊐䉤䊷䊤䊛╙ 㪈㪈 ภ㩷 㪆㩷㪉㪇㪇㪊㪅㪈㩷⴫䋭㪈㩷 ⷒᎿ䉮䊮䉪䊥䊷䊃‛ᕈ୯㩷⸃ᨆ䉴䊁䉾䊒㽵䋨Crack B వ┵䈏 1/2H 䈮೔㆐䈚䈢ᤨὐ䇮H䋺

E(GPa)

㱙

20.00

0.20

fc
(MPa)

ft(MPa)Gf
Wc(=5Gf /f t)(N/mm)(mm)ⷒᎿᏎෘ䋩䈮䈍䈇䈩䊏䊷䉪⩄㊀䈮೔㆐䈜䉎⚿ᨐ䈏ᓧ䉌䉏䈩䈇

20.00

2.00

0.10ᦨ䉅ථ⿧䈜䉎䈵䈶ഀ䉏䋨ਥ 䈵䈶ഀ䉏䋩䈫䈚䈩ㅴዷ䈚䈩ⴕ 䈐䇮ઁ

0.25

䉎䇯䉁䈢䇮ᢙ୯⸃ᨆ䈪䈲 Crack B 䈏ో⸃ᨆ䉴䊁䉾䊒䈮䈍䈇䈩䈱䈵䈶ഀ䉏䈏䈠䉏䈮ㅊ㓐䈚䈩หᤨㅴዷ䇮஗ṛ䇮㐽ว䉕➅䉍㄰䈚䈭䈏䉌ផ⒖䈚䈩ⴕ䈒⁁ᴫ䈏⹦⚦䈮ᓧ䉌䉏䉎䈖䈫䈏ಽ䈎䉎䇯ᬌ⸽⚿ᨐ࿑䋭㪈㪇 䈮⩄㊀䋭ᄌ૏㑐ଥ䉕␜䈜䈫䈍䉍䇮ታ㛎䈫ᢙ୯⸃ᨆ䈲࿑䋭㪏䇮㪐 䈮ታ㛎䈍䉋䈶⸃ᨆ⚿ᨐ䈮ၮ䈨䈒⎕უ⋥೨䈱䈵䈶⦟ 䈇৻ ⥌ 䉕␜䈚䈩䈇䉎䇯䉁䈢䇮ห ࿑ 䈮ᢙ ୯⸃ ᨆ 䈮䉋䉎⩄㊀ 䋭ഀ䉏ㅴዷ⁁ᴫ䉕␜䈜䇯䈭䈍䇮࿑ਛ䈱䂾ᢙሼ䈲⸃ᨆ䉴䊁䉾䊒䈮䈵䈶ഀ䉏㐿ญ᏷㑐ଥ䉕␜䈜䈫䈍䉍䇮ᦨᄢ䈵䈶ഀ䉏㐿ญ᏷䈲䇮䈍䈔䉎ฦ䈵䈶ഀ䉏వ┵૏⟎䉕␜䈚䈢䉅䈱䈪䈅䉎䇯

Crack B 䈮䈍䈇䈩䇮䈵䈶ഀ䉏⽾ㅢ⋥೨䈪 0.2mm ⒟ᐲ䈫䈭䈦ታ㛎䈪䈲䇮ฝ஥ოㇱ䈪䈱䈵䈶ഀ䉏䈏ᦨ䉅ථ⿧䈚䈩⊒↢䊶ㅴዷ䈚䇮䈧䈇䈪ᐩ䈩䈇䉎䇯ㇱ䈱䈵䈶ഀ䉏䈫䈭䉍䇮ᄤ┵ㇱ䈍䉋䈶Ꮐ஥ ოએ਄䉋䉍䇮ᧄ⎇ⓥ䈪㐿⊒䈚䈢㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕↪䈇䈢ㇱ䈪䈲䈅䉁䉍䈵䈶ഀ䉏䈲ㅴዷ䈞䈝䇮䊏䊷䉪⩄㊀એ㒠䈲⣀ᕈ⊛ⶄᢙ䈵䈶ഀ䉏䈱ᢙ୯⸃ᨆᚻᴺ䈲䇮ታ㛎䉕⦟䈇♖ᐲ䈪ౣ⃻䈚䈭⎕უ᜼േ䉕␜䈚䈢䇯Ꮐฝ㕖ኻ⒓䈭䈵䈶ഀ䉏⊒↢⁁ᴫ䈫䈭䈦䈩䈍䉍䇮᦭ല䈭ᚻᴺ䈪䈅䉎䈖䈫䈏⏕⹺䈪䈐䈢䉅䈱䈫⠨䈋䉌䉏䉎䇯䈢䈱䈲䇮╬ ࿶ タ ⩄ 䉕ච ಽ 䈮೙ ᓮ 䈪䈐䈭䈎䈦䈢䈢䉄䈫⠨ 䈋䉌䉏䉎䇯䈖䉏䈮ኻ䈚䈩䇮ᢙ୯⸃ᨆ⚿ᨐ䈲ታ㛎䉕䉋䈒ౣ⃻䈪䈐䈩䈍䉍䇮࿑䋭㪏㩷 ታ㛎䈮䉋䉎⎕უ⋥೨䈱䈵䈶ഀ䉏ㅴዷ⁁ᴫ㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 ࿑䋭㪐㩷 ⸃ᨆ䈮䉋䉎⎕უ⋥೨䈱䈵䈶ഀ䉏ㅴዷ⁁ᴫ㩷㪈㪈㪊㩷㔌ᢔ߭߮ഀࠇࡕ࠺࡞ࠍ↪޿ߚ࠻ࡦࡀ࡞ⷒᎿߩⶄᢙ߭߮ഀࠇ⸃ᨆ㪇㪅㪇㪎㪣㫆㪸㪻㩷䋨㪤㪧㪸㪀㪣㫆㪸㪻㩷䋨㪤㪧㪸㪀㪇㪅㪇㪍㪇㪅㪇㪌㪇㪅㪇㪋㪇㪅㪇㪏㪇㪅㪇㪏㪇㪅㪇㪎㪇㪅㪇㪎㪇㪅㪇㪍㪇㪅㪇㪍㪇㪅㪇㪌㪇㪅㪇㪌㪇㪅㪇㪋㪇㪅㪇㪊㪇㪅㪇㪊㪇㪅㪇㪉㪇㪅㪇㪉㪇㪅㪇㪈An alysis 㧔㪸㫅㪸㫃㫐㫊㫀㫊䋨㰱㪩䋩ǍR 㧕An alysis㧔㪘㫅㪸㫃㫐㫊㫀㫊䋨㰱㪫䋩ǍT㧕T est ( ǍT)㪫㪼㫊㫋㩿㰱㪫㪀

㪇㪅㪇㪈㪇㪅㪇㪇
㪄㪇㪅㪊 㪇㪅㪇㪇㪅㪊㪇㪅㪍㪇㪅㪐㪛㫀㫊㫇㪅㩷㪸㫋㩷㩷㫋㪿㪼㩷㪺㪼㫀㫃㫀㫅㪾㩷䋨䌭䌭䋩㪇㪅㪇㪇T est ( ǍR )㪫㪼㫊㫋㩿㰱㪩㪀
An alysis 㧔ǍL㧕㪸㫅㪸㫃㫐㫊㫀㫊䋨㰱㪣䋩T est ( ǍL)㪫㪼㫊㫋㩿㰱㪣㪀

㪇㪅㪇

㪇㪅㪉

㪇㪅㪋

㪇㪅㪍㪇㪅㪏㪈㪅㪇㪛㫀㫊㫇㪅㩷㪸㫋㩷㫋㪿㪼㩷㫎㪸㫃㫃㫊㩷䋨䌭䌭䋩㪣㫆㪸㪻㩷䋨㪤㪧㪸㪀㪇㪅㪇㪏㪇㪅㪇㪋
CrackA㪚㫉㪸㪺㫂㩷㪘㪇㪅㪇㪊

CrackB㪚㫉㪸㪺㫂㩷㪙
CrackC㪚㫉㪸㪺㫂㩷㪚㪇㪅㪇㪉

CrackD㪚㫉㪸㪺㫂㩷㪛㪇㪅㪇㪈㪇㪅㪇㪇㪇㪅㪇㪇

CrackE㪚㫉㪸㪺㫂㩷㪜㪇㪅㪇㪌 㪇㪅㪈㪇 㪇㪅㪈㪌㪚㪦㪛㩷㩷䋨䌭䌭䋩㪇㪅㪉㪇࿑䋭㪈㪇㩷 ⩄㊀䋭ᄌ૏䇮⩄㊀䋭㐿ญ᏷㑐ଥ㩷㪋㪅㩷 ⚿⺰㩷ㄭ ᐕ 䇮ᚒ 䈏࿖ 䈱ᣢ ⸳ 䉮䊮䉪䊥䊷䊃᭴ ㅧ ‛ 䈱⠧ ᧎ ൻ 䈅䉎䈇䈲䉮䊮䉪䊥䊷䊃᭴ㅧ‛䈮↢䈛䉎ⶄᢙ䈵䈶ഀ䉏䈱ㅴዷ᜼േ⸃ᨆᄖ⊛ⅣႺ䈱ᄌൻ䈭䈬䈮䉋䉍᭴ㅧ⊛䈭䈵䈶ഀ䉏䈏⊒↢䈚䈩䈇䉎䈫䈚䈩䇮৻ ⥸ ⊛ 䈮䈲㔌 ᢔ 䈵䈶ഀ 䉏䊝䊂䊦䈱ㆡ ↪ 䈏࿎ 㔍 䈭䈢ᄌ⁁੐଀䈏ዋ䈭䈒䈭䈇䇯䉁䈢䇮䈖䉏䈎䉌 2025 ᐕ䉁䈪䈱྾ඨ਎䉄䇮ಽᏓ䈵䈶ഀ䉏䊝䊂䊦䈱ㆡ↪䈏䈭䈘䉏䈩䈐䈢䇯䈚䈎䈚䇮ᧄᚻ♿䈲䉴䊃䉾䉪䊜䊮䊁䊅䊮䉴䈱਎♿䈫䈚䈩૏ ⟎䈨䈔䉌䉏䈩䈍䉍䇮䈜ᴺ 䈱ឭ ᩺ 䈮䉋䈦䈩䇮ⶄ ᢙ 䈱ዪ ᚲ ⊛ 䈭䈵䈶ഀ 䉏䈱⊒↢ ⃻⽎ 䈮䈪䈮䈵䈶ഀ䉏䈏↢䈛䈩䈇䉎ᣢ⸳䉮䊮䉪䊥䊷䊃᭴ㅧ‛䉕วℂ⊛䈮䈧䈇䈩䉅䇮㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕ㆡ↪䈜䉎䈖䈫䈏น⢻䈫䈭䉎䈖⛽ᜬ⵬ୃ䊶⵬ᒝ䈚䈩䈇䈒䈖䈫䈏ᦸ䉁䉏䈩䈇䉎䇯䈖䈱䉋䈉䈭⢛᥊䈱䈫䉕␜䈜䈖䈫䈏䈪䈐䈢䇯ਅ䇮䈜䈪䈮䈵䈶ഀ䉏䈏↢䈛䈩䈇䉎䉮䊮䉪䊥䊷䊃᭴ㅧ‛䈱ቯ㊂⊛৻ᣇ䇮ᧄ⎇ⓥ䈱ᦨ⚳⋡ᮡ䈲䇮䊃䊮䊈䊦᭴ㅧ૕䈮䈍䈇䈩䇮ⷒ䈭᭴ ㅧ ஜ ో ᐲ ⹏ ଔ 䈭䉌䈶䈮ว ℂ ⊛ 䈭⵬ ୃ 䊶⵬ ᒝ ⸳⸘䈮䈍䈇Ꮏ䈮⊒↢䈚䈩䈇䉎䈵䈶ഀ䉏㐿ญ᏷䈎䉌૞↪ᄖജ䉕ផቯ䈜䉎䈖䈩䇮⃻ ⁁ 䉕ቯ ㊂ ⊛ 䈭ᜰ ᮡ 䉕䉅䈦䈩⹏ ଔ 䈜䉎䈖䈫䈏㕖Ᏹ 䈮㊀ⷐ䈫䈫䈚䈩䈇䉎䇯వ䈱䊃䊮䊈䊦ⷒᎿ䈵䈶ഀ䉏⸃ᨆ⚿ᨐ䈱⍮⷗䈮ၮ䈪䈅䉎䈫⠨䈋䉎䇯䈖䈱䉋䈉䈭⷗࿾䈎䉌䇮ᧄ⎇ⓥ䈪㐿⊒䈚䈢ᚻᴺ䈲䈨䈔䈳䇮࿾ጊ䈱ᡰᜬ᧦ઙ䉕⠨ᘦ䈞䈝ⷒᎿන૕䈫䈚䈩᳞䉄䉌䉏䈵䈶ഀ䉏㐿ญ᏷䈮ၮ䈨䈇䈩䇮ⶄᢙ䈵䈶ഀ䉏䈱᜼േ䈍䉋䈶૞䉎ᦨᄢ䈵䈶ഀ䉏㐿ญ᏷䈲䇮䈵䈶ഀ䉏䈏ⷒᎿ䉕⽾ㅢ䈜䉎⋥ ೨↪ᄖജ䈭䈬⃻ᄌ⁁䊜䉦䊆䉵䊛䉕ቯ㊂⊛䈮⹏ଔ䈪䈐䉎䈖䈫䈎䉌䇮䈪 0.2mm ⒟ᐲ䈫䈭䈦䈩䈇䉎䇯䈚䈢䈏䈦䈩䇮ታ㓙䈱⃻႐䈪↢䈛䈩㕖Ᏹ䈮᦭ല䈭৻ᚻᴺ䈫䈭䉍䈋䉎䈪䈅䉐䈉䇯䈇䉎 2䌾5mm ⒟ᐲ䈱ᄢ䈐䈭䈵䈶ഀ䉏㐿ญ᏷䉕ౣ⃻䈜䉎䈮䈲䇮ෳ⠨ᢥ₂䈵䈶ഀ䉏䈏ⷒᎿ䉕⽾ㅢ䈚䈢એ㒠䈮䈧䈇䈩䇮࿾ጊ䈱ᡰᜬ᧦ઙ䉕૗䉌䈎⠨ᘦ䈜䉎ᔅⷐ䈏䈅䉎䇯䈠䈖䈪䇮䈵䈶ഀ䉏䈏ⷒᎿ䉕⽾ㅢ䈚䈢એ㒠䉅⸃ᨆน⢻䈫䈭䉎䇮ⷒᎿ⢛㕙䈱࿾ጊ䈱ᡰᜬ᧦ઙ䉕⠨ᘦ䈚䈢㔌ᢔ䈵䈶ഀ䉏䊝䊂䊦䉕↪䈇䈢ᢙ୯⸃ᨆᚻᴺ䈱㐿⊒ 䉕੹ᓟ䈱⺖㗴䈫䈚䈩䈇䉎䇯䋱䋩 Zihai SHI䋬Masayasu OHTSU䋬Masaki SUZUKI䋬Yoshihisa
HIBINO 䋺 NUMERICAL ANALYSIS OF MULTIPLE
CRACKS IN CONCRETE USING THE DISCRETE
MODELING APPROACH䇮╙ 2 ࿁᭴ㅧ‛䈱⎕უㆊ⒟⸃᣿䈮ၮ䈨䈒࿾㔡㒐ἴᕈะ਄䈮㑐䈜䉎䉲䊮䊘䉳䉡䊛⺰ᢥ㓸䇮pp.219-224䇮
2001.3
䋲䋩 Z. Shi, M. Ohtsu, M. Suzuki and Y. Hibino : Numericalanalysis of multiple cracks in concrete using the discrete
approach, Journal of Structural Engineering , ASCE ,127(9),pp.1085-1091, 2001㩷㪈㪈㪋㩷