ª T z T y T x a a a A (3) ߎߎߢ㧘ax, ay, az㧦ߘࠇߙࠇዪᚲᐳᮡߩx, y, zᣇะߩᣇะ ૛ᒏࡌࠢ࠻࡞ߢ޽ࠆ㧚 ᆫ൓ࡑ࠻࡝ࠢࠬߪᣇะ૛ᒏࡌࠢ࠻࡞ߦࠃߞߡ᭴ᚑߐࠇ ߡ޿ࠆߩߢ㧘න૏ࡌࠢ࠻࡞㐳ࠍ଻ߟߚ߼ߩⷙቯൻࠍㅙᰴ ⴕߞߚ8)㧚 㓸ว૕ⷐ⚛ߪ࿑2 ߦ␜ߔࠃ߁ߥⶄᢙ୘ߦࠃߞߡᒻᚑ ߐࠇࠆ೰૕ࠍ৻ⷐ⚛ߣߔࠆ㧚࿑3ߦߪ㧘4୘ߩⷐ⚛ߩ㓸 ว૕ࠍ␜ߒߡ޿ࠆ߇㧘ߎࠇࠄߪ೰⚿ߐࠇߚ߽ߩߣ઒ቯߔ ࠆߣ㧘ߘߩ⋧ኻ૏⟎ᐳᮡLˆ jߪਇᄌߢ޽ࠆߩߢ㓸ว૕ⷐ ⚛ߩ㊀ᔃࠍᐳᮡේὐߣߔࠆዪᚲᐳᮡ♽(

╙10࿁ ᭴ㅧ‛ߩⴣ᠄໧㗴ߦ㑐ߔࠆࠪࡦࡐࠫ࠙ࡓ⺰ᢥ㓸( 2010ᐕ12᦬) ࿯ᧁቇળ

୘೎ⷐ⚛ᴺߦࠃࠆ␕᧚ߩ቟ᕷⷺ⸃ᨆ

Angle of repose on analysis in gravels for distinct element method simulation

ၳญବⴕ*㧘ằ⼱ ৻**㧘㚅᦬ ᥓ***

Toshiyuki Horiguchi, Hajime Shibuya, Satoshi Katsuki

*㒐ⴡᄢቇᩞℂᎿቇ⎇ⓥ⑼ቇ↢, ᑪ⸳ⅣႺᎿቇ⑼㧔ޥ239-8686␹ᄹᎹ⋵ᮮ㗇⾐Ꮢ⿛᳓1-10-20㧕

**㒐ⴡᄢቇᩞℂᎿቇ⎇ⓥ⑼ቇ↢, ᑪ⸳ⅣႺᎿቇ⑼㧔ޥ239-8686␹ᄹᎹ⋵ᮮ㗇⾐Ꮢ⿛᳓1-10-20㧕

***Ꮏඳ㧘㒐ⴡᄢቇᩞᢎ᝼, ᑪ⸳ⅣႺᎿቇ⑼㧔ޥ239-8686␹ᄹᎹ⋵ᮮ㗇⾐Ꮢ⿛᳓1-10-20㧕 Key Words: distinct element method, aggregate element, angle of repose

ࠠ࡯ࡢ࡯࠼㧦୘೎ⷐ⚛ᴺ, 㓸ว૕ⷐ⚛, ቟ᕷⷺ

㧚 ✜ ⸒

ᚒ߇࿖ߪ㧘࿖࿯ߩᄢㇱಽࠍጊ࿾߿ਐ㒺࿾߇භ߼ࠆߣߣ

߽ߦ᪢㔎߿บ㘑ߩᤨᦼߦ㓸ਛ⽕㔎߇ᄙ޿ߎߣ߆ࠄ㧘࿯⍹

ᵹ࡮࿾ߔߴࠅ╬ߩ࿯⍾ἴኂߦࠃࠆⵍኂ߇ฦ࿾ߢ↢ߓߡ޿

ࠆ㧚ߎࠇࠄߩ࿯⍾ἴኂ߆ࠄੱฬ߿⽷↥ߩ៊ᄬࠍ㒐ߋߚ߼

ߩઍ⴫⊛ߥ㒐⼔᭴ㅧ‛ߣߒߡฦ⒳ߩ⍾㒐ႍႇ߇޽ࠆ㧚ߎ ߩ⍾㒐ႍႇߩౝㇱ᭴ㅧߦߪ㧘ਛ⹣᧚ߣ޿߁߽ߩ߇૶ࠊࠇ ߡ޿ࠆ㧚ߎߩ␕ਛ⹣᧚ߩᛶ᛫ജ⹏ଔࠍߔࠆ⎇ⓥߪ㧘ਇ᣿

⏕ߥ߹߹⃻࿷ߦ⥋ߞߡ޿ࠆ㧚ߎߩ⹏ଔᣇᴺߩ㧝ߟߦ୘೎

ⷐ⚛ᴺ¹⁾߇ᵴ↪ߐࠇߡ޿ࠆ㧚

୘೎ⷐ⚛ᴺߪ㧘☸⁁૕⸃ᨆߦㆡ↪ߒ㧘ߘߩ☸ሶ㑆ߩ᜼

േ࡮ᄌᒻߩࡔࠞ࠾࠭ࡓࠍᓸⷞ⊛ⷰὐ߆ࠄ᣿ࠄ߆ߦߒࠃ߁ ߣߔࠆ⹜ߺ߇ᦨㄭⴕࠊࠇߡ޿ࠆ㧚ߒ߆ߒ㧘⸃ᨆࡄ࡜ࡔ࡯

࠲ߩ᳿ቯᣇᴺ㧘․ߦ␕᧚ߩ㓸ว૕․ᕈࠍ⴫ߔߚ߼ߩᣇᴺ ߪ⏕┙ߐࠇߡ޿ߥ޿㧚ߎߩߚ߼㧘␕ਛ⹣᧚ߩᛶ᛫୯ ^{2), 3)} ߩ⹏ଔࠍ್ᢿߔࠆ㓙ߦ㧘࿯⾰᧚ᢱߩ૕♽ࠍᒁ↪ߒߡ☼⌕

ജ߿ౝㇱ៺ᡂⷺߥߤࠍ↪޿ࠆ߇㧘ߘ߽ߘ߽୘૕㑆ߩ☼⌕

ജ߿៺ᡂଥᢙߣ㓸ว૕ߩౝㇱ៺ᡂⷺߥߤߣߩ⋧㑐ᕈߪ ਇ᣿ߢ޽ࠆߛߌߢߥߊ㧘1ߟߩ␕߇10㨪30cmߦߥࠆਛ⹣

᧚ߩౝㇱ៺ᡂⷺࠍ᷹ቯߔࠆታ㛎ᣇᴺߐ߃ߥ޿ߣ޿߁໧

㗴߇޽ࠆ㧚ߔߥࠊߜ㧘ᄢ߈ߥࠨࠗ࠭ߩ☸⁁૕ߩၮ␆⊛ߥ ᕈ⾰ߣߒߡ⸘᷹น⢻ߥߩߪ㧘቟ᕷⷺߊࠄ޿ߣ⠨߃ࠄࠇࠆ㧚 ߎߩࠃ߁ߦ㊀ⷐߥᗧ⟵ࠍ߽ߟ቟ᕷⷺߩ⎇ⓥߪᭂ߼ߡ ዋߥߊVan Bulkalow⁴⁾㧘Statham⁵⁾╬ߩᢙ߃ࠆ߶ߤߒ߆⷗ᒰ ߚࠄߥ޿㧚ߐࠄߦ㧘୘೎ⷐ⚛ᴺߦࠃߞߡ቟ᕷⷺࠍ⸃ᨆߒ ߚ߽ߩ߽ዋߥߊ㧘ታ㓙ߩᒻ⁁ߦⷐ⚛ࠍㄭߠ߆ߖ቟ᕷⷺߩ ᵹേᕈࠍࠪࡒࡘ࡟࡯࡚ࠪࡦߒߚ଀ ⁶⁾߿㧘⍱ᒻኈེߦ⹣߼

ㄟࠎߛ☸⁁૕ߦኻߒ㧘ߘߩ஥㕙ࠍ㐿޿ߚ႐วߩ☸ሶߩ᜼

േࠍᬌ⸛ߒߚ଀⁷⁾ߥߤ߇⷗ᒰߚࠆ⒟ᐲߢ޽ࠆ㧚

ߘߎߢᧄ⎇ⓥߪ㧘ਛ⹣᧚⸃ᨆ߳ߩㆡ↪ߦ⾗ߔࠆၮ␆Ბ 㓏ߣߒߡ୘೎ⷐ⚛ᴺߦࠃࠆ቟ᕷⷺ⸃ᨆᴺߦߟ޿ߡᬌ⸛

ߒߚ߽ߩߢ޽ࠆ㧚ߘߩ㓙㧘቟ᕷⷺߩਥⷐߥᡰ㈩࿃ሶߣߥ ࠅᓧࠆⷐ⚛ߩಳಲᒻ⁁ߦ⌕⋡ߒߡ㧘6 ୘ߩ╬⃿ᒻⷐ⚛ߩ 㓸ว૕ⷐ⚛ࠍ↪޿ߚ୘೎ⷐ⚛ᴺࠍ㐿⊒ߒ㧘㓸ว૕ⷐ⚛ߩ ಳಲᐲ߿ⷐ⚛㑆߫ߨߩ៺ᡂⷺߩᄌൻ߇⸃ᨆ⚿ᨐߩ቟ᕷ

ⷺߦ෸߷ߔᓇ㗀ߦߟ޿ߡ⠨ኤߔࠆߣߣ߽ߦ㧘੐೨ታ㛎ߣ

߽Ყセ⠨ኤߔࠆ߽ߩߢ޽ࠆ㧚 ⸃ᨆᚻᴺ

୘೎ⷐ⚛ᴺߪ㧘߹ߕฦⷐ⚛߇ធ⸅⁁ᘒߦ޽ࠆߎߣࠍ್

ቯߒ㧘ⷐ⚛㑆ߦ⸳ቯߒߚⷐ⚛㑆߫ߨߩធ⸅ജࠍ▚ቯߒ㧘

ᤨޘೞޘ㧘୘ޘߩⷐ⚛ߏߣߦㆇേᣇ⒟ᑼࠍ⸃ߊߎߣߦࠃ ࠅᄌ૏ࠍ᳞߼㧘୘ޘߩⷐ⚛ߩㆇേࠍㅊ〔ߔࠆ߽ߩߢ޽ࠆ㧚 ᧄ⎇ⓥߦ߅޿ߡߪ㧘ⶄᢙ୘ߩ⃿ᒻⷐ⚛ࠍ೰⚿ߒ㧘৻ߟ ߩ㓸ว૕ⷐ⚛ߣߔࠆߎߣߢ୘ޘߩ␕ࠍࡕ࠺࡞ൻߔࠆ㧚ߔ ߥࠊߜ㧘ធ⸅್ቯ߅ࠃ߮ធ⸅ജߩ⸘▚ߪ୘ޘߩⷐ⚛ߏߣ ⴕ޿㧘ⷐ⚛ߩㆇേࠍ⸃ߊ㓙ߦ㧘㓸ว૕ࠍ᭴ᚑߔࠆ୘ޘߩ ⷐ⚛㧔એਅ㧘᭴ᚑⷐ⚛ߣ๭߱㧕ߦ߆߆ࠆജࠍ㓸ว૕㊀ᔃ ὐߦ㓸ᚑߒߡ㓸ว૕ߩㆇേࠍ⸃߈㧘ߎࠇࠍฦ᭴ᚑⷐ⚛ߩ ㆇേߦ⟎߈឵߃ߡ㧘ߘߩ᜼േࠍㅊ〔ߔࠆ߽ߩߣߔࠆ㧚 એਅߦߪ㧘ᧄ⎇ⓥߢឭ᩺ߔࠆ㓸ว૕ⷐ⚛ߩㆇേࠍ⴫ߔ ᄌ૏ߣᆫ൓㧘߫ߨജߩ⸘▚ᣇᴺࠍ␜ߔ㧚

ⷐ⚛ߩᄌ૏ߣᆫ൓

3ᰴరⓨ㑆ߦ߅޿ߡ㧘୘ޘߩⷐ⚛ߪ࿑1ߦ␜ߔࠃ߁ߦ

⸘▚㐿ᆎᤨ߆ࠄߩᄌ૏࡮࿁ォⷺߦࠃߞߡⷐ⚛ߩ૏⟎㧘ᆫ

൓߇ᄌൻߔࠆ㧚ߎߩⷐ⚛ߩ૏⟎ߣᄌ૏㧘ᆫ൓ࠍᰴߩࠃ߁ ߦ⴫ߔ⁸⁾㧚

u L

L ⁰ (1)

A A

A ⁰' (2)

㪋㪊

(8)

ߎߎߢ㧘L㧦ⷐ⚛㊀ᔃߩో૕ᐳᮡ♽ߦኻߔࠆ૏⟎ࡌࠢ࠻

࡞㧘L⁰㧦ೋᦼߩ૏⟎ࡌࠢ࠻࡞㧘u㧦ᄌ૏ࡌࠢ࠻࡞㧔u^T = [ ux, uy, uz ]㧕㧘A㧦ⷐ⚛ߩ㊀ᔃࠍਛᔃߣߔࠆో૕ᐳᮡ♽߳ߩ࿁

ォⷺࠍ⴫ߔᆫ൓ࡑ࠻࡝ࠢࠬ㧘A⁰㧦ೋᦼߩᆫ൓ࡑ࠻࡝ࠢࠬ㧘 A㧦ᆫ൓ࡑ࠻࡝ࠢࠬߩᄌൻ㊂㧘㧦࿁ォⷺࡌࠢ࠻࡞㧔^T = [ x, y, z ]㧕ߢ޽ࠆ㧚

ᆫ൓ࡑ࠻࡝ࠢࠬߪ㧘㊀ᔃࠍේὐߣߔࠆᣇะ૛ᒏࡌࠢ࠻

࡞ߦࠃߞߡਈ߃ࠄࠇࠆ㧚

»»

»

¼ º

««

«

¬ ª

T z T y T x

a a a

A (3)

ߎߎߢ㧘ax, ay, az㧦ߘࠇߙࠇዪᚲᐳᮡߩx, y, zᣇะߩᣇะ

૛ᒏࡌࠢ࠻࡞ߢ޽ࠆ㧚

ᆫ൓ࡑ࠻࡝ࠢࠬߪᣇะ૛ᒏࡌࠢ࠻࡞ߦࠃߞߡ᭴ᚑߐࠇ ߡ޿ࠆߩߢ㧘න૏ࡌࠢ࠻࡞㐳ࠍ଻ߟߚ߼ߩⷙቯൻࠍㅙᰴ

ⴕߞߚ⁸⁾㧚

㓸ว૕ⷐ⚛ߪ࿑2 ߦ␜ߔࠃ߁ߥⶄᢙ୘ߦࠃߞߡᒻᚑ ߐࠇࠆ೰૕ࠍ৻ⷐ⚛ߣߔࠆ㧚࿑3ߦߪ㧘4୘ߩⷐ⚛ߩ㓸 ว૕ࠍ␜ߒߡ޿ࠆ߇㧘ߎࠇࠄߪ೰⚿ߐࠇߚ߽ߩߣ઒ቯߔ ࠆߣ㧘ߘߩ⋧ኻ૏⟎ᐳᮡLˆ jߪਇᄌߢ޽ࠆߩߢ㓸ว૕ⷐ

⚛ߩ㊀ᔃࠍᐳᮡේὐߣߔࠆዪᚲᐳᮡ♽^{(xˆ,yˆ,zˆ)}ࠍ৻ᣤ᳿

ቯߔࠆߣਇᄌߢ޽ࠆ㧚ߔߥࠊߜ㧘

j j j

j const j n

z y x

, 1 . ˆ

ˆ ˆ ˆ

»»

»

¼ º

««

«

¬ ª

L (4)

ߎߎߢ㧘ⁿa㧦㓸ว૕ⷐ⚛ߩ᭴ᚑⷐ⚛ᢙߢ޽ࠆ㧚

ࠃߞߡ㧘ᤨޘೞޘߩ㓸ว૕ⷐ⚛^kߩ૏⟎ᐳᮡLkߣᆫ൓

ࡑ࠻࡝ࠢࠬ Ak߇ਈ߃ࠄࠇࠆߣ㧘᭴ᚑⷐ⚛^j ߩ૏⟎ᐳᮡ

Ljߣᆫ൓ࡑ࠻࡝ࠢࠬAj߇ᰴᑼߩࠃ߁ߦਈ߃ࠄࠇࠆ㧚

j k k

j L A L

L ^Tˆ (5)

k j

j AA

A ˆ (6)

ߎߎߢ㧘Aˆ j㧦㓸ว૕ⷐ⚛ߩዪᚲᐳᮡ♽ߦ߅ߌࠆ᭴ᚑⷐ

⚛^jߩᆫ൓ࡑ࠻࡝ࠢࠬߢ޽ࠆ㧚 ⷐ⚛㑆߫ߨߣ߫ߨജ

ⷐ⚛㑆߫ߨߪ㧘࿑4ߦ␜ߔࠃ߁ߦ㧘߫ߨߩᆫ൓ߦࠃߞ ߡቯ߹ࠆዪᚲᐳᮡ♽㧔ᴺ✢ᣇะ㧦^xs㧘ធ✢ᣇะ㧦^ys, zs㧕ߦ ኻᔕߒߚ3ᰴరߩ߫ߨߢ޽ࠅ㧘߹ߚ㧘ߘࠇߙࠇߦਗⴕߒ ߚ࠳࠶ࠪࡘࡐ࠶࠻ࠍ⸳⟎ߒߡ޿ࠆ㧚ߎߩ࠳࠶ࠪࡘࡐ࠶࠻

ߦࠃࠅ↢ߓࠆਗㅴജfD߅ࠃ߮ࡕ࡯ࡔࡦ࠻MDߦߟ޿ߡߪ㧘 એਅߢㅀߴࠆ߫ߨജߦࠃࠅ↢ߓࠆਗㅴജf^K߅ࠃ߮ࡕ࡯ࡔ ࡦ࠻ MKߩ▚ቯߣห᭽ߩᚻ⛯߈ߦࠃࠅ᳞߼ࠆߎߣ߇ߢ ߈ࠆ㧚

࿑4 ߩዪᚲᐳᮡ♽ߦ߅ߌࠆ߫ߨജߣⷐ⚛ߦ૞↪ߔࠆ ജߩߟࠅว޿ᑼߪ㧘ᰴᑼߦࠃߞߡ᳞߹ࠆ㧚

1) ਗㅴᣇะ

K

K S

f (7-a)

2) ࿁ォᣇะ

K s T s G

K TC TS

M (7-b)

ߎߎߢ㧘CsT

㧦ⷐ⚛㑆߫ߨߩዪᚲᐳᮡ♽ߦ߅ߌࠆߟࠅว޿

ࡑ࠻࡝ࠢࠬ㧘SK㧦ⷐ⚛㑆߫ߨߩዪᚲߩᆫ൓ࡑ࠻࡝ࠢࠬ

Asߦᓥ߁ౝജࡌࠢ࠻࡞㧔SK T

= [ SKx, SKy, SKz ]㧕㧘Ts㧦ⷐ⚛

㑆߫ߨߩᆫ൓ࡑ࠻࡝ࠢࠬ Asߦᓥ߁ዪᚲᐳᮡ♽߆ࠄⷐ⚛

ߩዪᚲߩᆫ൓ࡑ࠻࡝ࠢࠬAߦᓥ߁ዪᚲᐳᮡ♽߳ᄌ឵ߔࠆ ᐳᮡᄌ឵ࡑ࠻࡝ࠢࠬ㧘TG㧦ⷐ⚛ߩዪᚲߩᆫ൓ࡑ࠻࡝ࠢࠬ

Aߦᓥ߁ዪᚲᐳᮡ♽߆ࠄో૕ᐳᮡ♽߳ᄌ឵ߔࠆᐳᮡᄌ឵

ࡑ࠻࡝ࠢࠬߢ޽ࠆ㧚

ⷐ⚛㑆߫ߨߩౝജߪ㧘ᰴᑼߦࠃߞߡ᳞߹ࠆ㧚 q

k S

S_K(t) _K(t't) ' (8) ߎߎߢ㧘SK (t)㧦ᤨೞ^tߦ߅ߌࠆⷐ⚛㑆߫ߨߩዪᚲߩᆫ൓

ࡑ࠻࡝ࠢࠬ Asߦᓥ߁ౝജࡌࠢ࠻࡞㧘q㧦SK(t)ࡌࠢ࠻࡞

ߦኻᔕߔࠆⷐ⚛㑆߫ߨߩዪᚲߩᆫ൓ࡑ࠻࡝ࠢࠬ Asߦᓥ ߁t㑆ߩჇಽᄌᒻ㊂ࡌࠢ࠻࡞㧘k㧦Ⴧಽ㑆ߩ೰ᕈࡑ࠻࡝

ࠢࠬߢ޽ࠆ㧚

ߥ߅㧘ធ✢ᣇะߩ߫ߨߣ࠳࠶ࠪࡘࡐ࠶࠻ߩวജP^sߪᰴ

ᑼߦ␜ߔࠃ߁ߦࡕ࡯࡞࡮ࠢ࡯ࡠࡦߩ⎕უၮḰࠍ⸳ߌߚ㧚

x x

n S S

P _{K D} (9)

࿑1 ⷐ⚛ߩᄌ૏ߣᆫ൓

㫑

㫏 㫐 㪱

㪯 㪰

㪣_㫁 㫑

㫏 㫐

࿑2 㓸ว૕ⷐ⚛ߩዪᚲᐳᮡ ࿑3 㓸ว૕ⷐ⚛ߩዪᚲᐳᮡ

࿑4 ⷐ⚛㑆߫ߨ

㪋㪋

I

ntan

s_cr C P

P (10)

°¯

°®

­

! d

) (

) (

) (

) (

s_cr s s_cr

s_cr s 2 D K 2 D K

s P P P

P P S

S S

P S^{y y z z} (11)

ߎߎߢ㧘Pn㧘Ps㧦ߘࠇߙࠇߩᴺ✢ᣇะ㧘ធ✢ᣇะߩ߫ߨ ߣ࠳࠶ࠪࡘࡐ࠶࠻ߩวജ㧘SDx㧘SDy㧘SDz㧦ߘࠇߙࠇߩ࠳

࠶ࠪࡘࡐ࠶࠻ߩౝജߩx, y, zᚑಽ㧘Ps_cr㧦ធ✢ᣇะวജ PsߩṖࠅ㒢⇇㧘C㧦☼⌕ജ㧘^tanI㧦៺ᡂଥᢙߢ޽ࠆ㧚 㧚੐೨ታ㛎

ᧄ⎇ⓥߦ߅޿ߡߪᵈ౉ᴺ ⁹⁾ࠍ↪޿ߡ㧘ᒻ⁁ߩ⇣ߥࠆ³

⒳㘃ߩ␕ࠍ௑ᢳ᧼ߩࠬ࡝࠶࠻ㇱ߇␕ߢ㐽Ⴇߒߥ޿▸࿐

ߢ㧘␕ࠍ^100~150୘^/⑽ߩㅦߐߢ௑ᢳ᧼ߩਛᄩઃㄭߩᛩ౉

࿾ὐ㧔࿑^5(b)㧕ߦᵹਅߐߖ㧘቟ᕷⷺߩ᷹ቯታ㛎ࠍⴕ޿㧘

␕ߩᒻ⁁ߣ቟ᕷⷺߩ㑐ଥߦߟ޿ߡၮ␆⊛ߥᬌ⸛ࠍⴕߞ ߚ㧚

ታ㛎ⵝ⟎

౮⌀¹߅ࠃ߮࿑⁵ߦታ㛎ౕེߩ᭎ⷐࠍ␜ߔ㧚ߎߩታ㛎

ౕེߪ㧘ᛩ౉↪ߩ௑ᢳ᧼ߣၸⓍㇱ߆ࠄߥࠆ৻ㄝ^30cmߩ

┙ᣇ૕ߩ▫ߢ޽ࠆ㧚௑ᢳ᧼ߩ௑ᢳⷺᐲߪ ²⁵ ᐲߢ޽ࠅ㧘

ᦨ߽ૐ޿ਃⷺࠦ࡯࠽࡯ߦ৻ㄝ^{4 2cm}ߩࠬ࡝࠶࠻߇ⓨߌ ࠄࠇߡ߅ࠅ㧘ߎߩࠬ࡝࠶࠻߆ࠄ␕ߪ⪭ਅ㜞ߐ^15cmߢ⪭

ਅߔࠆ㧚▫ߩ஥㕙ߪㅘ᣿ߥࠕࠢ࡝࡞᧼ߢ㧘ၸⓍ⁁ᴫࠍⷰ

ኤߢ߈ࠆ㧚

␕⹜ᢱ

౮⌀² ߦታ㛎ߦ↪޿ߚ⹜ᢱࠍ␜ߔ㧚⍹὇Ἧ㧘₹⍾೑㧘

☻㛽᧚ߩ³⒳㘃ߩ⇣ߥࠆ␕ࠍ↪޿ߚ㧚ߥ߅ߘߩᒻ⁁․ᕈ ߪ㧘⍹὇ἯߪਣߺࠍᏪ߮ߡ߅ࠅಳಲ߇ዋߥߊ㧘₹⍾೑㧘

☻㛽᧚ߣߥࠆߦᓥ޿ಳಲ߇ᄢ߈޿⹜ᢱߣߥߞߡ޿ࠆ㧚ߘ ߩ☸ᓘߪ㧘߰ࠆ޿ಽߌߦࠃࠅ^5mm㨪^10mmߩ৻᭽☸ᓘߣ ߒߚ㧚߹ߚ㧘ߘࠇߙࠇߩ⚛᧚ߩᲧ㊀ࠍ⴫¹ߦ␜ߔ㧚ߘࠇ ߙࠇ㧘⍹὇Ἧߪ^1.8㧘₹⍾೑߅ࠃ߮☻㛽᧚ߪ^2.6ߣߥߞߡ

޿ࠆ㧚

቟ᕷⷺߩ᷹ቯ

቟ᕷⷺߩ⸘᷹ᣇᴺߦߪ⒳ޘߩᣇᴺ߇޽ࠆ߇¹⁰⁾㧘ᧄታ㛎 ߢߪએਅߩᚻ㗅ߦࠃࠅ᷹ቯߒߚ㧚

㪌䌾㪈㪇㫄㫄 㪌䌾㪈㪇㫄㫄

㪌䌾㪈㪇㫄㫄

(a) ⍹὇Ἧಳಲዊ (b) ₹⍾೑ಳಲਛ (c) ☻㛽᧚ಳಲᄢ

౮⌀¹ ቟ᕷ᷹ⷺቯ ౮⌀² ␕⹜ᢱ

(a) ୄ⍑࿑ (b) ਄㕙࿑

࿑⁵ ቟ᕷ᷹ⷺቯౕེ

⴫¹ ␕⹜ᢱߩ⻉ర

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(a) ஥㕙࿑ (b) ਄㕙࿑

࿑⁶ ቟ᕷ᷹ⷺቯߩၮᧄ୯

⴫² ⸃ᨆࡄ࡜ࡔ࡯࠲

㗄㗄 ⋡⋡ ୯୯ ⷐ

ⷐ⚛ᢙ⚛ᢙ㧔㧔නන૕૕㧕㧕 1120200000 ⷐ

ⷐ⚛ᢙ⚛ᢙ㧔㧔㓸㓸วว૕૕㧕㧕 22000000 ⷐ⚛ⷐ⚛

ኒኒᐲᐲ 19190000kkgg//mm³³ ᴺ

ᴺ✢ᣇ✢ᣇะะKK_㨚㨚 11..00××1100⁶⁶N/N/mm

߫

߫ߨߨ

ቯቯᢙᢙ ធធ✢ᣇ✢ᣇะะKK_SS 11..55××1100⁵⁵N/N/mm

ᷫᷫ⴮⴮ቯቯᢙᢙ hh 00..88

☼☼⌕ജ⌕ജ CC 00NN ⷐ

ⷐ⚛⚛㑆㑆

߫

߫ߨߨ

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៺ᡂⷺᡂⷺ ttaann 00..555454((2299°)°)

⸘⸘▚᧦▚᧦ઙઙ ᤨᤨ㑆ೞ㑆ೞߺߺ tt 5.5.00××1100⁷⁷ss

㪋㪌

(1) ஥ᣇ߆ࠄࠕࠢ࡝࡞᧼ࠍㅢߒߡ᠟ᓇߒߚ౮⌀ࠃࠅ㧘 ၸⓍߒߚ␕ߩᦨ߽㜞޿૏⟎ࠍ㜞ߐhߣߔࠆ㧚

(2) ਄ᣇ߆ࠄ᠟ᓇߒߚ౮⌀ࠃࠅ㧘࿑^6(b)ߦ␜ߔࠃ߁ߦ ၸⓍߒߚ␕ߩඨᓘࠍ⁵ᣇะ᷹ቯߒ㧘ߘߩᐔဋඨᓘ"ࠍ᳞

߼ࠆ㧚ߎߩ㓙㧘ᐩ㕙ߦ޽ߞߡઁߩ߽ߩߣធ⸅ߒߥ޿␕ߪ 㒰ᄖߔࠆ^.

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(3) 㜞ߐh㧘ඨᓘ"ࠍ↪޿ߡᰴᑼࠃࠅ቟ᕷⷺࠍ᳞߼ࠆ㧚

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ታ㛎ࠤ࡯ࠬ

␕߇ၸⓍߒߡ቟ᕷⷺࠍᒻᚑߔࠆ႐ว㧘␕ᢙ߇ዋߥ޿ߣ

቟ᕷⷺߩ߫ࠄߟ߈߇ᄢ߈ߊߥࠆ㧚ߎߩߚ߼㧘߹ߕ␕ᢙ߇

቟ᕷⷺߦਈ߃ࠆᓇ㗀ࠍⷰኤߒߚ㧚ߘߩ⚿ᨐ㧘ߤߩ⹜ᢱߦ ߅޿ߡ߽␕ᢙ߇ ¹⁵⁰⁰ એਅߩ▸࿐ߢߪ቟ᕷⷺߦ߫ࠄߟ߈ ߇⷗ฃߌࠄࠇࠆ߇㧘␕ᢙ߇ ²⁰⁰⁰ ࠍ⿧߃ࠆߣ቟ᕷⷺߩ߫

ࠄߟ߈߇߶ߣࠎߤ⷗ฃߌࠄࠇߥ޿ߎߣ߇ࠊ߆ߞߚ㧚ᓥߞ ߡ㧘ታ㛎ߢߪ㧘਄ㅀߩᬌ⸛⚿ᨐࠍ〯߹߃ߡ␕ᢙࠍ ²⁰⁰⁰

୘ߣߒߡ㧘ฦ⹜ᢱߦኻߒ⁵࿁ߕߟታ㛎ࠍⴕߞߚ㧚 ታ㛎⚿ᨐ

࿑⁷ߦ²⁰⁰⁰୘ߩ␕ࠍ↪޿ߚోࠤ࡯ࠬߦ߅ߌࠆฦ⹜ᢱ ߩ቟ᕷⷺࠍ␜ߔ㧚ߎࠇࠃࠅ㧘⍹὇Ἧ㧘₹⍾೑㧘☻㛽᧚ߣ ಳಲ߇ᄢ߈ߊߥࠆߦߟࠇߡ㧘቟ᕷⷺ߇ᄢ߈ߊߥࠆߎߣ߇ ࠊ߆ࠆ㧚

୘೎ⷐ⚛ᴺ⸃ᨆ

␕ᒻ⁁ࡕ࠺࡞

ᧄ⸃ᨆߪ␕⹜ᢱߩಳಲࠍ⴫⃻ߔࠆߚ߼ߦ㧘࿑⁸ߦ␜ߔ㧘 ห৻ඨᓘ⃿ࠍ⁶୘↪޿ߚ㓸ว૕ⷐ⚛ࠍ↪޿ߡ቟ᕷⷺࠍ᳞

߼ࠆ㧚ߘߩၮᧄᒻ⁁ߪ㧘࿑^8(a)ߦ␜ߔࠃ߁ߦ੕޿ߦኒߦ ធ⸅ߒߚᱜ౎㕙૕ߩࠃ߁ߥᒻ⁁ߢ޽ࠆ㧚ߎߩၮᧄᒻ⁁ࠍ

ฦ᭴ᚑⷐ⚛߇ห⒟ᐲߦ⋧੕㊀ⶄߐߖߚ߽ߩࠍ૞ᚑߔࠆ ߣ㧘࿑^8(b)ߦ␜ߔࠃ߁ߦ㧘ߘߩ㊀ⶄ⒟ᐲߦࠃߞߡಳಲߩ ᐲว޿߇⴫⃻ߢ߈ࠆ㧚ᧄ⎇ⓥߢߪ㧘㊀ⶄߒߡ޿ߥ޿ㇱಽ ߇⃿ߩ⋥ᓘDߦኻߒߡ㧘ߤߩ⒟ᐲಲㇱࠍᒻᚑߒߡ޿ࠆ߆ ࠍ⴫ߔࡄ࡜ࡔ࡯࠲ߣߒߡ㧘ಲᐲࠍᰴᑼߢ⴫ߔ㧚

ED ⁽¹⁴⁾

ߎߎߢ㧘㧦ಲᐲࠍ⴫ߔࡄ࡜ࡔ࡯࠲㧔^0.0㨪^1.0㧕㧘^D㧦᭴ᚑ

⃿ᒻⷐ⚛ߩ⋥ᓘߢ޽ࠆ㧚

⸃ᨆࠤ࡯ࠬ߅ࠃ߮ࡄ࡜ࡔ࡯࠲

⸃ᨆߪ㧘ಲᐲࠍ^0.20㨪^0.50߹ߢߩ⸘⁷⒳㘃ߦߟ޿ߡ㧘 ฦࠤ࡯ࠬ²࿁ߕߟ⸃ᨆࠍⴕߞߚ㧚

⴫²ߦ⸃ᨆ᧦ઙࠍ␜ߔ㧚ߎߎߢ㧘ⷐ⚛㑆߫ߨߩ៺ᡂⷺ

ߪ㧘ฦ⹜ᢱߦኻߔࠆ੐೨ታ㛎ࠍⴕߞߚ⚿ᨐ߆ࠄ^29°ߣߒ㧘

⸘▚ߦ↪޿ߚߘߩઁߩቯᢙ╬ߪ㧘ታ㛎᧦ઙࠍෳ⠨ߣߒߡ㧘

੐೨ߩࡄ࡜ࡔ࠻࡝࠶ࠢ⸃ᨆߦࠃࠅ᳿ቯߒߚ㧚

⸃ᨆ⚿ᨐ ၸⓍㆊ⒟

࿑⁹ߦ㧘ಲᐲ^=0.40ߣߒߚ႐วߩ⸃ᨆㆊ⒟ࠍታ㛎ߣ

⚻ㆊᤨ㑆ࠍၮḰߣߒߡᲧセߒߡ␜ߔ㧚࿑^9(b)ߩ⁴⑽ᓟߢ ߪ㧘ታ㛎ߢߪᣢߦ⚂ ^30°ߩ቟ᕷⷺߩၸⓍࠍᒻᚑߒ㧘⪭ਅ ὐࠃࠅ㆙ߊߦᢔੂߒߚ␕ߪዋߥ޿ߩߦኻߒ㧘⸃ᨆߢߪ㆙

ߊߦᢔੂߒߚⷐ⚛߇Ყセ⊛ᄙߊߥࠅ⪭ਅὐߦၸⓍߒߚ ⷐ⚛߇ዋߥ޿ߚ߼㧘቟ᕷⷺߪ ^20°ࠃࠅዊߐߊ㧘✭߿߆ߢ

޽ࠆ㧚ߒ߆ߒ㧘࿑^{9 (c)}ߩ⁸⑽ㆊ⒟ᤨߦߪ㧘⸃ᨆߩᢔੂⷐ

⚛ߪ࿑^{9 (b)}ߦᲧߒߡߐ߶ߤჇ߃ߡ߅ࠄߕ㧘ၸⓍߦነਈߔ

ࠆⷐ⚛ᢙߩഀว߇ᄙߊߥࠅ㧘ᦨ㜞ὐߩ㜞ߐ߽ታ㛎ߩ߽ߩ

࿑⁷ ቟ᕷⷺߦ෸߷ߔᒻ⁁ߩᓇ㗀

⍹὇Ἧ ₹⍾೑ ☻㛽᧚

␕⹜ᢱ

቟ᕷⷺ㧔q㧕

(a) 㓸ว૕ⷐ⚛ၮᧄᒻ A-A´ᢿ㕙

࿑8 㓸ว૕ⷐ⚛ࡕ࠺࡞

(b) ಲᐲ B-B´ᢿ㕙

㪘

㪙

㪙㫪

㪘㫪

㪋㪍

ߣ߶߷หߓߦߥߞߡ޿ࠆ㧚ߘߩᓟߪ㧘࿑^{9 (d)}ߩ¹²⑽ᓟ㧘

࿑^{9 (e)}ߩ¹⁶⑽ᓟߦ߅޿ߡ߽ߘߩ቟ᕷⷺߪ߶߷หߓⷺᐲ

ߢ⛽ᜬߐࠇߡ⋧ૃᒻߢၸⓍⷐ⚛ᢙ߇Ⴧടߔࠆ㧚ߟ߹ࠅ㧘 10 ⑽޽ߚࠅߢߩ቟ᕷⷺߪ቟ቯ⁁ᘒߦ㆐ߒߡ޿ࠆὐߦ߅

޿ߡ㧘ታ㛎⚿ᨐߣ৻⥌ߒߡ޿ࠆߎߣ߇ಽ߆ࠆ㧚 ቟ᕷⷺߣಲᐲߩ㑐ଥ

࿑¹⁰ߦ⸃ᨆߢᓧࠄࠇߚ቟ᕷⷺ㨪ಲᐲ㑐ଥࠍ␜ߔ㧚 ߎࠇࠃࠅ㧘ឭ᩺㓸ว૕ⷐ⚛ࡕ࠺࡞ߩಲᐲߣ቟ᕷⷺߦߪ

᣿⍎ߥ⋧㑐㑐ଥ߇ሽ࿷ߔࠆ㧚⹜ߺߦߘߩ㑐ଥࠍࡈࠖ࠶࠹

ࠖࡦࠣߔࠆߣᰴᑼߣߥࠅ㧘࿑¹⁰ ਛߦ⎕✢ߢ␜ߒߡ޿ࠆ ࠃ߁ߦ⸃ᨆ⚿ᨐߣࠃߊ৻⥌ߔࠆ㧚

60 .

0

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T

቟ᕷⷺߣⷐ⚛㑆߫ߨߩ៺ᡂⷺߩ㑐ଥ

࿑¹¹ߦಲᐲ^=0.30ߣߒߚ႐วߩ㧘ⷐ⚛㑆߫ߨߩ៺ᡂⷺ

ߩᄌൻ߇቟ᕷⷺߦਈ߃ࠆᓇ㗀ߦߟ޿ߡ␜ߔ㧚߹ߕⷐ⚛

㑆߫ߨߩ៺ᡂⷺ߇^0°~24°ߦ߅޿ߡߪ៺ᡂⷺߩᄌൻ߇቟

ᕷⷺߦਈ߃ࠆᓇ㗀߇㕖Ᏹߦᄢ߈޿ߎߣ߇ࠊ߆ࠆ㧚৻ᣇ㧘 ⷐ⚛㑆߫ߨߩ៺ᡂⷺ߇^24°એ਄ߦߥࠆߣ቟ᕷⷺߪ⚂^28°

ߢᄌൻ߇ߥߊ㧘߹ߚ୘ޘߩⷐ⚛ߩേ߈ߦ߽ᄢ߈ߥᄌൻ߇

⷗ࠄࠇߥߊߥߞߡ޿ࠆ㧚 ᩿ࠄߦࠃࠆ㓸ว૕ⷐ⚛ࠍ↪޿

ߚ቟ᕷⷺߦ㑐ߔࠆ⎇ⓥ¹¹⁾ߦ߅޿ߡ߽㧘ⷐ⚛㑆߫ߨߩ៺ᡂ

ⷺ߇ ^20°એਅߢߪ቟ᕷⷺߦ㕖Ᏹߦᄢ߈ߥᓇ㗀ࠍਈ߃ߡ޿

ࠆ߽ߩߩ㧘^20°એ਄ߢߪ߶ߣࠎߤᄌൻ߇ߥ޿ߎߣࠍ⺒ߺߣ ࠆߎߣ߇ߢ߈ࠆ㧚ߎࠇࠃࠅ㧘ⷐ⚛㑆߫ߨߩ៺ᡂⷺߪ ^24°

ࠃࠅዊߐ޿႐วߦߪ቟ᕷⷺߦኻߒ㕖Ᏹߦᄢ߈ߥᓇ㗀ࠍ ਈ߃ࠆ߽ߩߩ㧘^24°એ਄ߢߪ቟ᕷⷺߦኻߒߡ߶ߣࠎߤᓇ㗀 ࠍਈ߃ߥ޿ߎߣ߇ࠊ߆ߞߚ㧚

(a) t=t0

(e) t=t0+16s (b) ^t=t0+4s

(d) t=t0+12s (c) t=t0+8s

࿑^{9 =0.40}ߣ☻㛽᧚ߩᤨೞᱧᲧセ Ꮐ㧦⸃ᨆ⚿ᨐ ฝ㧦ታ㛎⚿ᨐ

࿑¹⁰ ቟ᕷⷺ㨪ಲᐲ㑐ଥ

㪇 㪌 㪈㪇 㪈㪌 㪉㪇 㪉㪌 㪊㪇

㪇 㪈㪇 㪉㪇 㪊㪇 㪋㪇 㪌㪇

ⷐ⚛㑆䈳䈰䈱៺ᡂⷺ䋨㫦䋩

቟ᕷⷺ䋨㫦䋩

⍹὇Ἧ

₹⍾೑

☻㛽᧚

㪇 㪌 㪈㪇 㪈㪌 㪉㪇 㪉㪌 㪊㪇 㪊㪌 㪋㪇 㪋㪌

㪇 㪇㪅㪈 㪇㪅㪉 㪇㪅㪊 㪇㪅㪋 㪇㪅㪌 㪇㪅㪍 ಲᐲ㱎

቟ᕷⷺ㩿㫦㪀

࿑¹¹ ቟ᕷⷺ㨪ⷐ⚛㑆៺ᡂ㑐ଥ

㪋㪎

⚿ ⸒

ᧄ⎇ⓥߪ㧘୘೎ⷐ⚛ᴺߦ೰⚿㓸ว૕ⷐ⚛ࠍዉ౉ߔࠆߎ ߣߢ㧘␕ⷐ⚛ߩಳಲᒻ⁁߿ⷐ⚛㑆߫ߨߩ៺ᡂⷺߩᄌൻ߇

቟ᕷⷺᒻᚑߦਈ߃ࠆᓇ㗀ߦߟ޿ߡၮ␆⊛ߥᬌ⸛ࠍⴕߞ ߚ߽ߩߢ޽ࠆ㧚ᧄ⎇ⓥߩᚑᨐࠍ߹ߣ߼ࠆߣ㧘એਅߩࠃ߁ ߦߥࠆ㧚

(1) ห৻⃿ᓘ⁶୘ࠍᱜ౎㕙૕⁁ߦ㈩⟎ߒߚ㓸ว૕ⷐ⚛ࠍ

↪޿ߡ㧘቟ᕷⷺᒻᚑㆊ⒟ߩࠪࡒࡘ࡟࡯࡚ࠪࡦࠍⴕߞߚ㧚 ߎࠇࠃࠅ㧘㓸ว૕ⷐ⚛ߩಲᐲߦᔕߓߡ቟ᕷⷺߪᄌൻߒ㧘

੐೨ታ㛎ߩᦨ⚳቟ᕷⷺ߿቟ᕷⷺߩᒻᚑㆊ⒟ࠍ᭎ߨౣ⃻

ߔࠆߎߣ߇ߢ߈ߚ㧚

(2) ⷐ⚛㑆߫ߨߩ៺ᡂⷺߪ㧘^24°એਅߢߪ቟ᕷⷺߦᄢ߈ ߥᓇ㗀ࠍਈ߃ࠆ߽ߩߩ㧘^24°એ਄ߢߪ቟ᕷⷺߦ޽߹ࠅᄌൻ ߇⷗ࠄࠇߥ޿ߎߣ߇ࠊ߆ߞߚ㧚

(3) ቟ᕷⷺߩᡰ㈩࿃ሶߦߪ㧘⚛᧚ߩᒻ⁁એᄖߩⷐ࿃㧘଀

߃߫㧘Ყ㊀㧘߹ߚߪ෻⊒ଥᢙߥߤ߇⠨߃ࠄࠇࠆ㧚ߘߩὐ ߦߟ޿ߡ೎ㅜᬌ⸛ࠍㅴ߼ࠆᔅⷐ߇޽ࠆ

ෳ⠨ᢥ₂

1)Cundall, P.A. and Strack, O.D.L.㧦A Discrete Numerical Model for Granular Assemblies, Geotechnique, No.29, pp.47-65, 1979

2) 㚅᦬ᥓ㧦㍑⵾ᨒ⍾㒐࠳ࡓߩ␕ਛ⹣᧚࿶ߦ㑐ߔࠆ⎇ⓥ^,

ᑪ▽ᐕႎ, pp.155, 1991

3)ᧃፒ዁ม, ࿦↰૫Ꮒ^, ᄢ㓈ਭ㧦␕ਛ⹣᧚ߦࠃࠆⴣ᠄✭

ⴣലᨐߦ㑐ߔࠆၮ␆⊛⎇ⓥ^,࿯ᧁቇળ╙⁶⁴࿁ᐕᰴቇⴚ

⻠Ṷળ⻠Ṷ᭎ⷐ㓸, pp.1143-1144, 2009.9

4)Van Burkalow Anastasia㧦Angle of repose and angle of sliding friction: An experimental study, Bulletin, Geological Society of America, No.56, pp.669-708, 1945 5)Statham Ian㧦The relationship of porosity and angle of

repose to mixture proportions in assemblages of different sized materials, Sedimentology, No.21, pp.149-162, 1974 6) ᩿ᷕ^, ᧻ፉਗ਼ᔒ^, ጊ↰ᕶᄩ㧦ਇⷙೣᒻ⁁ࡕ࠺࡞☸ሶ

ࠍ↪޿ߚ቟ᕷⷺᵹേࠪࡒࡘ࡟࡯࡚ࠪࡦ^, ╙⁵⁷࿁ℂ⺰

ᔕ↪ജቇ⻠Ṷળ, Vol.57, No.3B, pp.511-512, 2008.6 7)ศ↰㗅㧦☸⁁૕ߩ቟ᕷⷺᒻᚑߦ㑐ߔࠆᢙ୯⸃ᨆ⊛ᬌ⸛㧘

࿯ᧁቇળᐕᰴቇⴚ⻠Ṷળ⻠Ṷ᭎ⷐ㓸╙³ㇱ(B), Vol.51, No.3B, pp.790-791, 1996.9

8)ේᧁᄢテ^, 㚅᦬ᥓ^, ↰ઍరม㧦౞ᩇᒻⷐ⚛ࠍ↪޿ߚ୘

೎ⷐ⚛ᴺߦࠃࠆ⪭⍹㒐⼔✂ߩⴣ᠄ᔕ╵⸃ᨆ^, ࿯ᧁቇ ળ⺰ᢥ㓸A, Vol.65, No.2, pp.536-552, 2009.6

9)ᣣ㜞㊀ഥ^, ਃベ⨃㓶㧦☳૕Ꮏቇታ㛎ࡑ࠾ࡘࠕ࡞㧘ᣣೀ

Ꮏᬺᣂ⡞␠, pp.129-135, 1984.7

10)᧻ୖ౏ᙗ^, ᕲ↰⵨৻㧦቟ᕷⷺ㧦ቯ⟵ߣ᷹ቯᴺߦ߹ߟࠊ ࠆ⻉໧㗴^, ╳ᵄᄢቇ᳓ℂታ㛎࠮ࡦ࠲࡯ႎ๔^{, No.13}㧘 pp.27~35, 1989

11) ᩿ ᷕ^, ᧻ፉਗ਼ᔒ㧦᦬ߩ⍾ߩ☸ሶࡕ࠺࡝ࡦࠣߣᢙ୯

⸃ᨆ^, ╳ᵄᄢቇୃ჻⺰ᢥ, pp.14, 2008.3

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