hand book.pdf
TRANSCRIPT
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Монгол Улсын Шинжлэх Ухаан Технологийн Их Сургууль
Математикийн Сургууль
Н.Төмөрбаатар
А.Гөлгөө
Р.Найдандорж
Т.Мөнхбат
Д.Ганбат
Н.Пүрэвдорж
МАТЕРИАЛЫН ЭСЭРГҮҮЦЛИЙН
ГРАФИК ТООЦООНЫ АЖИЛ
Дахин боловсруулж, нэмсэн гуравдугаар хэвлэл
ШУТИС – ийн Математикийн сургуулийн сургалт арга зүй,
мэргэжлийн зөвлөлийн шийдвэрээр хэвлэв.
Редактор Н.Төмөрбаатар
Улаанбаатар 2009 он
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Энэхүү гарын авлага ШУТИС – ийн 2003 оны “Шилдэг гарын авлага” – аар шалгарсан болно
Зохиогчид: Нямринчингийн Төмөрбаатар Арилдийн Гөлгөө Раднаагийн Найдандорж
Төмөрбаатарын Мөнхбат Данаагийн Ганбат Нямсүрэнгийн Пүрэвдорж
Редактор Нямринчингийн Төмөрбаатар
Н.Төмөрбаатар, А.Гөлгөө, Р.Найдандорж,
Т.Мөнхбат, Д.Ганбат, Н.Пүрэвдорж.
Материалын
эсэргүүцлийн
график
тооцооны
ажил :
Их, дээд сургуулийн номын санд. – УБ.: 2009. –212 х.,
Энэхүү гарын авлагад Материалын эсэргүүцлийн үндсэн бүлгүүдийг хамаарсан даалгаврууд
түүнийг гүйцэтгэхэд зайлшгүй шаардлагатай товч тодорхойлолт, томъёоны хамт оруулж,
даалгавруудыг хийхэд тус дөхөм болгох зорилгоор бүлэг бүрт хэд хэдэн жишээ бодлогуудыг
зүйлчлэн бодов. Мөн даалгаврыг гүйцэтгэхэд зайлшгүй шаардлагатай лавлах материалуудыг
хавсаргасан болно.
Энэ гарын авлагыг их, дээд сургууль, коллежуудад инженер техникийн мэргэжлээр
суралцагсад болон инженер, магистрантууд ашиглах боломжтой. Энэ номыг Шинжлэх Ухаан
Технологийн Их Сургуулийн Техникийн механикийн салбараас эрхлэн гаргав.
Монгол Улсын Шинжлэх Ухаан Технологийн Их сургууль
Техникийн механикийн салбар
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ªÃÕª’ Ø √
»ÌÊÂÌÂðËÈÌ ·ÓÎÓ‚ÒÓÌ ı ø˜ÌËÈ„ · ̋ÎÚ„ ̋ ı ̋‰ Ï˝ð„ ̋ ÊÎËÈÌ ÒÛÛð¸
ı ˘˝ ̋Îøø‰ËÈÌ Ì˝„ ·ÓÎÓ ı ÇÚÂðˇÎ˚Ì ˝Ò˝ð„øøˆ ̋ Î ˜Û ı ‡Î øøð„ËÈ„
„øȈ ̋ Ú„ ̋ ‰ ̋ „.
ÇÚÂðˇÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ øÌ‰Ò˝Ì ·Ó‰ÎÓ„Ó ·ÓÎÓ ı ·‡Ú ·º ı ,
ı º¯øøÌ ˜‡Ì‡ð˚Ì ÚÓÓˆÓÓ ËÌÊÂÌÂðËÈÌ Ôð‡ÍÚËÍ øÈÎ ‡ÊË焇‡Ì‰
·‡ÈÌ„‡ ÚÓ ı ËÓΉÓÌÓ. ÇÚÂðˇÎ˚Ì ˝Ò˝ð„øøˆÎËÈ„ Òۉ· ı ˇ‚ˆ‡‰
ÒÛð„‡ÎÚ˚Ì ı ‡Ï„ËÈÌ øð ‡¯Ë„Ú‡È ‡ð„‡ ·ÓÎÓ ı ·Ë ‰‡‡Ê ·Ó‰ÎÓ„Ó ·Ó‰Ó ı ,
„ð‡ÙËÍñÚÓÓˆÓÓÌ˚ ‡ÊËÎ „øȈ ̋ Ú„ ̋ ı ËÈ„ ˜Û ı ‡Î˜ËΉ‡„. ›Ì˝ „‡ð˚Ì
‡‚·„ ̊ „ ÇÚÂðˇÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ øÌ‰Ò˝Ì ·ø΄øø‰ËÈ„ ı ‡Ï‡‡ðÒ‡Ì
‰‡‡Î„‡‚ð˚Ì ‡ÊÎ˚„ „øȈ ̋ Ú„ ̋ ı ̋‰ ÁÓðËÛ·‚. “Û ı ‡ÈÌ ·øν„Ú ı ‡Ï‡‡ð‡„‰‡ ı ‰‡‡Î„‡‚ð˚Ì ‡ÊÎ˚„ „øȈ ̋ Ú„ ̋ ı ̋‰ ¯‡‡ð‰Î‡„‡Ú‡È øÌ‰Ò˝Ì ÚÓÏ ˙∏Ó,
ÚÓ‰Óð ı ÓÈÎÓÎÚ, Ê˯˝˝ ·Ó‰ÎÓ„ÛÛ‰ ̊ Ì ·Ó‰ÓÎÚ˚„ Ú˝ð ·ø΄ËÈÌ ˝ı˝Ì‰
ÌËν ̋‰ ÚÓ‰Óð ı ÓÈ ı˝ÏÊ ̋ ˝„ ̋ ˝ð ·Ë˜Î˝˝. Œ ̨ÛÚ‡Ì ̋‰„ ̋ ˝ðËÈ„ ÒÛ‰‡ÎÒÌ˚
‰‡ð‡‡ ‰‡‡Î„‡‚ð‡‡ „øȈ ̋ Ú„ ̋ ı ̋‰ Úº‰ËÈÎºÌ ·˝ð ı ̄ ˝ ̋Î „‡ð‡ ı „øÈ ·‡È ı
·ÓÎÌÓ.
ƒ‡‡Î„‡‚ð˚„ „øȈ ̋ Ú„ ̋ ı ‰‡ð‡‡ÎÎ˚„ Ê˯˝˝ ·Ó΄ÓÒÓÌ ·Ó‰ÎÓ„ ̊ Ì
·Ó‰ÓÎÚÓÓÒ Ï ̋‰ ̋ ı ·øð˝Ì ·ÓÎÓÏÊÚÓÈ. ƒ‡‡Î„‡‚ð˚„ „øȈ ̋ Ú„ ̋ ı ̋‰
¯‡‡ð‰Î‡„‡Ú‡È ÓÌÓÎ˚Ì Ï ̋‰Î˝„ËÈ„ „øÌÁ„ËÈðøøνı ‰ ̋ ˝ ·‡„¯ËÈÌ ÎÂ͈, ı ˘˝ ̋ÎËÈÌ Òڇ̉‡ðÚ‡‰ ‰Ûð¸‰Ò‡Ì ÒÛð‡ ı ·Ë˜Ë„, „‡ð˚Ì ‡‚·„ÛÛ‰ ̊ „
‡¯Ë„· ı ¯‡‡ð‰Î‡„‡Ú‡È. ƒ‡‡Î„‡‚ð˚„ „øȈ ̋ Ú„ ̋ ı ‰ ̋ ˝ ÚÓÓˆÓÓÌ˚ ·ø‰øø‚˜,
ÚÓÓÌ º„º„‰ÎËÈ„ ·‡„¯‡‡Ò º„ÒºÌ ‚‡ðˇÌÚ‡‡ð ÒÓÌ„ÓÊ ‡‚‡ ı ı˝ð˝„Ú˝È.
¬‡ðˇÌÚ Ì¸ ı Ó∏ð ÓðÓÌÚÓÈ ÚÓÓ ·‡È ı ·º„ºº‰ ˝ı ÌËÈ ˆËÙð ÚÓÓˆÓÓÌ˚
·ø‰øø‚˜ËÈÌ ‰Û„‡‡ð˚„; ı Ó∏ð‰Û„‡‡ð ˆËÙð ÚÓÓÌ º„º„‰ÎËÈÌ ı øÒÌ˝„ÚËÈÌ
Ϻð ·Û ̨ Û ·‡„‡Ì‡Ì˚ ‰Û„‡‡ð˚„ Á‡‡Ì‡. Œ ̨ÛÚ‡Ì „ð‡ÙËÍñÚÓÓˆÓÓÌ˚ ‡ÊÎ˚„
Òڇ̉‡ðÚ˚Ì ¿4 ˆ‡‡Ò‡Ì ‰ ̋ ˝ð ı ËÈ ı ¯‡‡ð‰Î‡„‡Ú‡È ·º„ºº‰ ·Ë˜‚˝ð,
ÚÓÓˆÓÓÌ˚ ı˝Ò„ËÈ„ ·˝ı˝˝ð ·Û ̨ Û ÚÓÒÓÌ ·‡Î‡‡ð ·Ë˜ËÊ, ÁÛð‡„ ·ø‰øø‚˜ËÈ„
ËÌÊÂÌÂðËÈÌ ÁÛð‡„ ÁøÈÌ ¯‡‡ð‰Î‡„ ̊ Ì ‰‡„ÛÛ ı ‡ð‡Ì‰‡‡„‡‡ð ÁÛð‚‡Î
ÁÓ ı ËÌÓ. ÃºÌ ÍÓÏÔ¸ ̨ÚÂðËÈÌ Ú ı ÌËÍ ı˝ð˝„Ò ̋Î ‡¯Ë„Î‡Ê ·ÓÎÌÓ. ¡ø ı
ÚÓÓˆÓÓ̉ –» ÒËÒÚÂÏËÈÌ Ì˝„Êøø‰ËÈ„ ı˝ð˝„νı ı˝ð˝„Ú˝È. ¡Ó‰ÎÓ„Ó
·øðËÈ„ ¯ËÌ˝ ı ÛÛ‰‡Ò̇‡Ò ˝ı ̋ÎÊ ·Ó‰ÎÓ„ ̊ Ì Ì˝ð, ̺ ı ̂ ºÎ, ÚÓÓÌ º„º„‰ÎËÈ„
·øð˝Ì ·Ë˜ËÊ ÚÓÓˆÓÓÌ˚ ·ø‰øø‚˜ËÈ„ ÁÛðÒ‡Ì ·‡È‚‡Î ÁÓ ı ËÌÓ. «Ûð‡„ ‰ ̋ ˝ð
ÚÓÓˆÓÓ̉ ‡¯Ë„·„‰Ò‡Ì øÒ„ ̋ Ì Ú˝Ï‰ ̋ „Î ̋Î, ÚÓÓÌ ı˝ÏÊ ̋ ˝„ ·øð˝Ì Á‡‡Ò‡Ì
·‡È ı ¯‡‡ð‰Î‡„‡Ú‡È. ¡Ó‰ÎÓ„ ̊ „ ‡Î„·ð ı ̋η˝ð˝˝ð ·Ó‰ÓÊ ˝ˆÒËÈÌ
ÚÓÏ ˙∏Ó̉ ÚÓÓÌ ÛÚ„ ̊ „ ÓðÎÛÛÎÊ ı ‡ðËÛ̉ ÁÓ ı Ë ı Ì˝„ÊËÈ„ ·Ë˜‚ ̋Î ÁÓ ı ËÌÓ.
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¡‡„¯ËÈÌ º„ÒºÌ Á‡‡‚ð˚Ì ‰‡„ÛÛ ‰‡‡Î„‡‚‡ð ÚÛÒ ·øðËÈ„ ·Û ̨ Û ·ø ı
‰‡‡Î„‡‚ð˚„ ı ‡‚Ú‡ÒÎ‡Ì ( ı ‡‚Ú‡ÒÌ˚ Á‡„‚‡ð˚„ øÁ) ı ‡Ï„‡‡ÎÊ ÚÛ¯‡‡Ì‡.
√ ‡ð˚Ì ‡‚·„ ̊ Ì I, II, VI ·ø΄ËÈ„ Õ.“ºÏºð·‡‡Ú‡ð, III, IV ·ø΄ËÈ„
“.ÃºÌ ı ·‡Ú, V ·ø΄ËÈ„ ƒ.√ ‡Ì·‡Ú, VII, VIII, IX ·ø΄ËÈ„ ¿.√ ºÎ„ºº ̇ð
·Ë˜Ë‚.›Ì˝ „‡ð˚Ì ‡‚·„ ̊ „ øÌ‰Ò˝Ì‰ ̋ ˝ Ï ı ‡ÌËÍ, ·‡ðË΄ ̊ Ì ËÌÊÂÌÂðËÈÌ
Ï˝ð„ ̋ Êν˝ð ÒÛð‡Îˆ‡Ê ·‡È„‡‡ Ó ̨ÛÚÌÛÛ‰‡‰ ÁÓðËÛÎÒ‡Ì ·ÓÎÓ‚˜ ·ÛÒ‡‰
ËÌÊÂÌÂðËÈÌ Ï˝ð„ ̋ Êν˝ð ÒÛð‡Îˆ‡Ê ·ÛÈ Ó ̨ÛÚÌÛÛ‰ ˜ ‡¯Ë„· ı ·øð˝Ì
·ÓÎÓÏÊÚÓÈ.
Ã˝ð„ ̋ ÊÎËÈÌ Ó̈ÎÓ„ÓÓÒ ı ‡Ï‡‡ðÛÛÎ‡Ì Ó ̨Ûڇ̉ º„º ı ‰‡‡Î„‡‚ð˚Ì
ÒÓÌ„ÓÎÚ˚„ ı ˘˝ ̋Î Á‡‡Ê ·‡È„‡‡ ·‡„¯ ı ËÈÌ˝.
›Ì˝ı øø „Ûð‡‚‰Û„‡‡ð ı˝‚Î ̋ÎËÈ„ · ̋ÎÚ„ ̋ ı ‰ ̋ ˝ ºÏ̺ ı ı˝‚Î ̋Ή
‡Ì ı ‡‡ð‡„‰Ò‡Ì ‡Î‰‡‡ χ‰‡„Ú‡È ÁøÈÎËÈ„ Á‡ÎðÛÛÎÊ ‰‡ ı ËÌ ·ÓÎÓ‚ÒðÛÛη‡. “øøÌ˝˝Ò „‡‰Ì‡ Á‡Ò‡Ê Ò‡ÈÊðÛÛÎχ‡ð ÁøÈÎ ‡Ì ı ‡‡ð‡„‰‚‡Î ÿ”“»– ñ
ËÈÌ “ ı ÌËÍËÈÌ Ï ı ‡ÌËÍËÈÌ Ò‡Î·‡ðÚ „ ̋ Ò˝Ì ı ‡ ˇ„‡‡ð ËðøøÎÌ˝ øø.
—≈ ƒ¿ “Œ—
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Õ›√ ƒØ √ ››— ¡ØÀ›√
–”Õ√ ¿À“, ÿ¿’¿À“
1.1. –“¿“» “ŒƒŒ—’Œ… ÿ»À¡›Õ»… “ª¬»…Õ –”Õ√ ¿À“, ÿ¿’¿À“
≈ðºÌ ı ËÈ ÓÈ΄ÓÎÚ
ÿËη˝ÌËÈ Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ ˜Ë„Î˝Ò˝Ì ı ø˜Ìøø‰ËÈÌ øÈΘΠ̋ν˝ðÒÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ øøÒÌ˝.
–ÛÌ„‡ÎÚ˚Ì ø‰ ‰‡„ÛÛ ı ø˜ Ó„ÚÎÓÎ˚Ì „‡‰‡‡‰ ÌÓðχÎËÈÌ ‰‡„ÛÛ,
¯‡ ı ‡ÎÚ˚Ì ø‰ ‰ÓÚÓÓ‰ ÌÓðχÎËÈÌ ‰‡„ÛÛ ˜Ë„Î˝Ì˝.–ÛÌ„‡ ı ‰‡„ÛÛ ı ø˜ËÈ„ ˝Âð˝„, ¯‡ ı ‡ ı ‰‡„ÛÛ ı ø˜ËÈ„ Һ𺄠„ ̋ Ê øÁÌ˝.
ÿËη˝ÌËÈ Ó„ÚÎÓΉ ‰‡„ÛÛ ı ø˜ËÈ„ Ó„ÚÎÓÎ˚Ì ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎÌÓ.ÿËη˝ÌËÈ Ó̈ÎÓ„ Ó„ÚÎÓÎÛÛ‰‡‰ ‰‡„ÛÛ ı ø˜ÌËÈ ÛÚ„ ̊ „ ÓÎÊ ‰‡„ÛÛ ı ø˜ÌËÈ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ̇. ƒ‡„ÛÛ N ı ø˜ÌËÈ ˝Ô˛ð ̸ ¯Ëη˝ÌËÈ ÛðÚ˚Ì ‰‡„ÛÛ‰ ÚøøÌËÈ ººð˜ÎºÎÚËÈ„ ı ‡ðÛÛÎÒ‡Ì „ð‡ÙËÍ ˛Ï.
ÿËη˝ÌËÈ Ó„ÚÎÓΉ ·ÓÒÓÓ ı ø˜‰ ̋ Î ÊË„‰ Ú‡ð ı Ò‡Ì „ ̋ Ê øÁ ̋Ê ÚøøÌËÈ ÛÚ„ ̊ „
A
N =σ (1.1)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ A - ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È, N -‰‡„ÛÛ ı ø˜. ’ ̋ ð˝‚ ¯Ëη˝ Ìø ı Ú˝È ·‡È‚‡Î ı ø˜‰ ̋ ÎËÈ„ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ˆ ̋ ‚˝ð
ڇη‡È„‡‡ð (Ìø ı ÌËÈ Ú‡Î·‡È„ ı ‡Ò‡Ê) ·Ó‰ÌÓ.√ ‡‰Ì˚ ı ø˜ÌËÈ øÈΘΠ̋ν˝ð ¯Ëη˝ÌËÈ ‡Ì ı Ì˚ l ÛðÚ ººð˜Îº„‰ºÌº.
ÿËη˝ÌËÈ ‡·ÒÓÎ ̨Ú ÛðÚÒ‡ÎÚ˚„ (·Ó„ËÌÓÒÓÎÚ˚„)l l l −=Δ 1
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ 1l - ‡˜‡‡ øÈΘËÎÒÌËÈ ‰‡ð‡‡ ı ¯Ëη˝ÌËÈ ÛðÚ,
l ñ ¯Ëη˝
ÌËÈ ‡Ì ı
Ì˚
ÛðÚ.ÿËη˝ÌËÈ ı ‡ð¸ˆ‡Ì„ÛÈ ÛðÚÒ‡ÎÚ (·Ó„ËÌÓÒÓÎÚ)
l
l Δ=ε (1.2)
ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰Ó„‰ÓÌÓ. ’‡ðËÏ ı ‡È ı˝‚ „‡ÊËÎÚ˚Ì ø‰ ı ø˜‰ ̋ Î, ı˝‚ „‡ÊËÎÚ ¯Û„‡Ï‡Ì ı ‡Ï‡‡ð‡ÎÚ‡È
·‡È̇.ε σ ⋅= E (1.3)
ØøÌËÈ„ √ ÛÍËÈÌ ı ÛÛθ „ ̋ Ì˝. Øø̉ E - χÚÂðˇÎ˚Ì Û ˇÌ ı ‡ðËÏ ı ‡ÈÌ ÏÓ‰Ûθ.
(1.1), (1.2) Ëνð ı ËÈÎÎøø‰ËÈ„ (1.3) ÚÓÏ ˙∏Ó̉ ÓðÎÛÛη‡Î ¯Ëη˝ÌËÈ ÛðÚ˚Ì ººð˜ÎºÎÚËÈ„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî6
A E
l N l
⋅
⋅=Δ (1.4)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ ı ·ÓÎÓÏÊÚÓÈ.
ÿËη˝ , A , N E ı˝ÏÊË„‰ ̋ ı øøÌøø‰ ÚÓ„ÚÏÓÎ ·‡È ı ı˝Ò„øø‰ ̋ ˝Ò ·øÚ ̋Ê
·‡È‚‡Î ¯Ëη˝ÌËÈ ·øÚ˝Ì ÛðÚÒ‡ÎÚ (·Ó„ËÌÓÒÓÎÚ) ı˝Ò˝„ ÚÛÒ ·øðËÈÌ ÛðÚÒ‡ÎÚ,
·Ó„ËÌÓÒÓÎÚ˚Ì ‡Î„·ð ÌËÈη˝ðÚ˝È Ú˝Ìˆøø ·‡È̇.
∑= ⋅
⋅=Δ
n
i ii
ii
A E
l N l
1
(1.5)
’ ̋ ð˝‚ , N A ı˝ÏÊË„‰ ̋ ı øøÌøø‰ ¯Ëη˝ÌËÈ ı˝Ò„øø‰ ̋ ‰ ı Û‚¸Ò‡Ê ·‡È‚‡Î (1.5)
ÚÓÏ ˙∏Ó
∑ ∫==Δ
n
i
dz z A z N
E l
1 )( )( 1 (1.6)
ı ̋η˝ðÚ˝È ·ÓÎÌÓ.
(1.6) ÚÓÏ ˙∏Ó̉ ËÌÚ„ð‡Î˚„ ı˝Ò˝„ ·øðËÈÌ ÛðÚ‡‡ð ‡‚‡ ı ·‡ ÌËÈη˝ðÊøøÎ ̋ÎÚ
̸ ¯Ëη˝ÌËÈ ·ø ı ı˝Ò˝„Ú ı ‡Ï‡‡ð̇. ÿËη˝ÌËÈ ÒÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ˚Ì ø‰ ‰‡„ÛÛ
ı˝‚ „‡ÊËÎÚ˚Ì Á˝ð˝„ˆ ̋ ˝ øøÒ˝ı ı ºÌ‰ÎºÌ ı˝‚ „‡ÊËÎÚ˚„
1bbb −=Δ
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ 1b - ¯Ëη˝ÌËÈ ı˝‚ „‡ÊËÎÚ˚Ì ‰‡ð‡‡ ı ºð„ºÌ,
bñ ¯Ëη˝ÌËÈ ‡Ì ı Ì˚ ºð„ºÌ. “ ̋ „‚ ̋Î ı ºÌ‰ÎºÌ ı ‡ð¸ˆ‡Ì„ÛÈ ı˝‚ „‡ÊËÎÚ
b
bΔ='ε
·ÓÎÌÓ. b ñ ̄ Ëη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ‡Ì ı Ì˚ ºð„ºÌ.
’‡ð¸ˆ‡Ì„ÛÈ ı ºÌ‰ÎºÌ ı˝‚ „‡ÊËÎÚ, ı ‡ð¸ˆ‡Ì„ÛÈ ‰‡„ÛÛ ı˝‚ „‡ÊËÎÚ˚Ì
ı ‡ð¸ˆ‡‡Ì˚ ‡·ÒÓÎ ̨Ú ı˝ÏÊ ̋ ˝„ œÛ‡ÒÒÓÌ˚ ÍÓ˝ÙÙˈËÂÌÚ „ ̋ Ì˝.
ε
ε ν
'= (1.7)
»ÁÓÚðÓÔ Ï‡ÚÂðˇÎÛÛ‰ ̊ Ì ı Û‚¸‰ ˝Ì˝ ÍÓ˝ÙÙˈËÂÌÚ 5.00 ≤≤ν ıˇÁ„‡‡ðÚ
ººð˜Îº„‰ºÌº. œÛ‡ÒÒÓÌ˚ ÍÓ˝ÙÙˈËÂÌÚ Ï ̋‰ ̋ „‰ ̋ Ê ·‡È‚‡Î ÒÛÌ„‡ÎÚ,
¯‡ ı ‡ÎÚ˚Ì øÂËÈÌ ˝Á ̋Î ı øøÌËÈ ‡·ÒÓÎ ̨Ú, ı ‡ð¸ˆ‡Ì„ÛÈ ººð˜ÎºÎÚËÈ„ )21( ν ε −=Δ l AV
)21( ν ε ε −=Δ
=V
V V (1.8)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ.
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7
∆ ˯˝˝ 1.1. √ ‡‰Ì˚ F ı ø˜ ·ÓÎÓÌ ººðËÈÌ ÊËÌ„ËÈÌ øÈΘΠ̋Ή Óð¯Ë ı
¯Ëη˝ÌËÈ (1.1añð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜, ı ø˜‰ ̋ Î, ¯ËÎÊËÎÚËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.ÿËη˝ÌËÈ Ì˝„Ê ˝Á ̋Î ı øø̉ ÌÓÓ„‰Ó ı ÊËÌ γ ·ÓÎÌÓ.
1 l
1
(z) N
z
F+Al
N
γ
F
(z)G
EA
σ
γ
F/A
F/A+l
δ F l ----- + ------ EA 2 E
l γ 2
a). a).
1.1ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ÿËη˝ÌËÈ Ú˝Ì ı ν„ ‰ ̋ ˝ð ·ø ı ı ø˜Ìøø‰ËÈ„ ÔðÓÂ͈ÎÓÌ R
ð‡͈ ̊ „ ÓηÓÎ
0=+−− R l A F γ ⇒ γ l A F R += ·ÓÎÌÓ. ÿËη˝ Ì˝„ ı˝Ò˝„Ú˝È ·‡È„‡‡ Û˜Ëð ‰ÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı Ó‰
Áº‚ ı ºÌ 1-1 Ó„ÚÎÓÎ ‡‚‡ ı ‡‰ ı øð ̋Έ ̋ ̋Ú˝È. 1-1 Ó„ÚÎÓÎÓÓÒ ‰ ̋ ˝¯ Óð¯Ë ı z
ÛðÚÚ‡È ı˝Ò„ËÈ„ (1.1·ñð ÁÛð‡„) ҇΄‡Ì ‡‚˜ øΉ ̋ ı ‰ÓÓ‰ ı˝Ò„ËÈÌ øÈΘΠ̋ÎËÈ„
N øÒ„ ̋ ˝ð Ú˝Ï‰ ̋ „Î ̋Ê Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ºÎ ·Ë˜‚ ̋Î
0=∑ z , 0 )( )( =−−− z G F z N
·ÓÎÌÓ. Øø̉ γ z A z G = )( ̸ ̄Ëη˝ÌËÈ z ÛðÚÚ‡È ı˝Ò„ËÈÌ ÊËÌ.
ƒ‡„ÛÛ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈηÓÎγ z A F z G F z N −−=−−= )( )( , )0( l z ≤≤
·ÓÎÌÓ. ’ ̋ Ò„ËÈÌ ˝ı , Úº„Ò„ºÎ‰ ı ø˜ËÈ„ ÓηÓÎ0= z ø‰ F z N −= )(
l z = ø‰ γ l A F z N −−= )(
·ÓÎÌÓ. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ÌËÈ ˝Ô˛ð (1.1‚ñð ÁÛð‡„) ·‡È„ÛÛ·‚.
›Ì˝ ÚÓ ı ËÓΉÓΉ ı ø˜‰ ̋ ÎËÈ„ ÓÎÓ ı ÚÓÏ ˙∏Ó
A
z A F
A
z N γ σ
−−==
)( , )0( l z ≤≤
ı ̋η˝ðÚ˝È ·‡È ı Û˜Ëð ı˝Ò„ËÈÌ ˝ı , Úº„Ò„ºÎ‰ ı ø˜‰ ̋ ÎËÈ„ ÓηÓÎ
· . ‚). „). .‡ .
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî8
0= z ø‰ A
F −=σ
l z = ø‰ l A F
Al A F γ γ σ −−=+−=
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ¯Ëη˝Ì‰ øøÒ˝ı ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ (1.1„ñð ÁÛð‡„)
·‡È„ÛÛ·‚. ÿËη˝ÌËÈ ı˝‚ „‡ÊËÎÚ˚„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ Úº‚κðÒºÌ
F ı ø˜Ì˝˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„
A E
l F l F ⋅
⋅=Δ
ÚÓÏ ˙∏Ó„ÓÓð, ººðËÈÌ ÊËÌ„ ̋ ˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„
A E
l Gl
⋅
⋅=Δ
2γ ; ( γ l AG = )
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈηÓÎ
E
l
A E
l F
A E
l l A
A E
l F l
22
)( 2γ γ
−−=−−=Δ
·ÓÎÌÓ. ›Ì˝ ÛÚ„ ̊ „ ‡¯Ë„Î‡Ì ¯ËÎÊËÎÚËÈÌ ˝Ô˛ðËÈ„ (1.1‰ ñð ÁÛð‡„)
·‡È„ÛÛ·‚. ÿËÎÊËÎÚËÈÌ ˝Ô˛ðËÈ„ ·‡È„ÛÛ· ı ‰‡‡ ˇÏ‡„Ú ·˝ı ̋΄ ̋ ˝Ì˝˝Ò
˝ı ̋ÎÌ˝.
∆ ˯˝˝ 1.2. q ˝ð˜ËÏÚ˝È ÊË„‰ Ú‡ðÒ‡Ì ı ø˜˝˝ð ‡˜‡‡Î‡„‰Ò‡Ì, ÚÓ„ÚÏÓÎ
ı ºÌ‰ÎºÌ Ó„ÚÎÓÎÚÓÈ ¯Ëη˝ÌËÈ (1.2‡ñð ÁÛð‡„) ÛðÚÒ‡ÎÚ˚„ ÚÓ‰Óð ı ÓÈÎ.
¡Ó‰ÓÎÚ. ƒÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı˚Ì ÚÛΉ ÒÛÎ øÁøøð˝˝Ò z Á‡È‰ 1-1
Ó„ÚÎÓÎ ı ËÈÊ ‡‚Ò‡Ì ‰ÓÓ‰ ı˝Ò„ËÈÌ (1.2·ñð ÁÛð‡„) Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ÎËÈ„
·Ë˜‚ ̋Î0 )( =− qz z N , qz z N = )( )0( l z ≤≤
·ÓÎÌÓ. ’ ̋ Ò„ËÈÌ ˝ı , Úº„Ò„ºÎ‰ ‰‡„ÛÛ ı ø˜ÌËÈ ÛÚ„ ̊ „ ·Ó‰‚ÓÎ0= z ø‰ 0 )( = z N
l z = ø‰ l q z N = )(
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ÌËÈ ˝Ô˛ð (1.2‚ñð ÁÛð‡„) ·‡È„ÛÛ·‚.
ƒ‡„ÛÛ ı ø˜ ¯Ëη˝ÌËÈ ÛðÚ˚Ì ‰‡„ÛÛ‰ ¯Û„‡Ï‡Ì ı ÛÛΡð ı Û‚¸Ò‡Ê ·‡È„‡‡
Û˜Ëð ¯Ëη˝ÌËÈ ÛðÚÒ‡ÎÚ˚„
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9
A E
ql
A E
dz qz
A E
dz z N l
l l
2
)( 2
00
===Δ ∫∫
ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰ÌÓ.
‡.) ·.) ‚.) „.)1.2ñð ÁÛð‡„
ÿËη˝ÌËÈ ÒÛÎ øÁøøð˝˝Ò ‰Ûð˚Ì u Á‡È‰ Óð¯Ë ı mn ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚̯ËÎÊËÎÚ ˝Ì˝ Ó„ÚÎÓÎÓÓÒ ÁÓÓÎÚ ı øðÚ ̋Î ı ı˝Ò„ËÈÌ ̄ ËÎÊËÎÚÚ˝È Ú˝Ìˆøø
)( 22
222
ul A E
q
A E
qz
A E
dz qz l
u
l
u
−=== ∫δ
·‡È̇. ØøÌ˝˝Ò øÁ‚ ̋Î ¯Ëη˝ÌËÈ ¯ËÎÊËÎÚ Í‚‡‰ð‡Ú Ô‡ð‡·ÓÎ ı ÛÛΡðººð˜Îº„‰ºÊ ·‡È̇. ÿËÎÊËÎÚËÈÌ ˝Ô˛ðËÈ„ 1.2„ñð ÁÛð‡„Ú øÁøøν‚.
∆ ˯˝˝ 1.3. ÿ‡Ú‡ÎÒ‡Ì ¯Ëη˝ÌËÈ (1.3‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜, ı ø˜‰ ̋ Î,
¯ËÎÊËÎÚËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
º„ÒºÌ Ì¸ : 5.01 =l Ï, 132 == l l Ï, 6.04 =l Ï, 41 = A ÒÏ2
, 22 = A ÒÏ2
5102 ⋅= E Ãœ‡, 20=q ÍÕ/Ï, 30= F ÍÕ
¡Ó‰ÓÎÚ.1. ƒ‡„ÛÛ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı
ƒ‡„ÛÛ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ Ó„ÚÎÓÎ˚Ì ‡ð„‡ ı˝ð˝„νı ·º„ºº‰ ı Ó∏ð ı ø˜ÌËÈ ı ÓÓðÓ̉ Ì˝„ Ó„ÚÎÓÎ ‡‚‰‡„. “ ̋ „‚ ̋Î ˝Ì˝ ·Ó‰ÎÓ„ ̊ Ì ı Û‚¸‰ ‰‡„ÛÛ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı Ó‰ „Ûð‚‡Ì Ó„ÚÎÓÎ ‡‚‡ ı ‡‰ ı øð ̋Έ ̋ ̋Ú˝È. ’ ̋ Ò˝„ ·øðÚ Ó„ÚÎÓÎ ı ËÈÊ
ÁøøÌ ı˝Ò„ËÈ„ Óð ı ËÌ øÎ‰Ò˝Ì ·‡ðÛÛÌ ı˝Ò„ËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ κºÒ, ººðººð ı ̋η ̋Î „‡‰‡‡‰, ‰ÓÚÓÓ‰ ·ø ı ı ø˜Ìøø‰ËÈÌ ¯Ëη˝ÌËÈ Ú˝Ì ı ν„ ( z Ú˝Ì ı ν„) ‰ ̋ ˝ð ı
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî10
ÔðÓÂ͈ÛÛ‰ ̊ Ì ‡Î„·ð ÌËÈη˝ð Ú˝„Ú˝È Ú˝Ìˆøø ·‡È ı ̺ ı ̂ ºÎººÒ ÚÛ ı ‡ÈÌÓ„ÚÎÓΉ øÈÎ˜Î˝ı ‰ÓÚÓÓ‰ ı ø˜ËÈ„ ÓÎÌÓ. “Û ı ‡ÈÌ Ó„ÚÎÓΉ øÈÎ˜Î˝ı ‰ÓÚÓÓ‰ ı ø˜ËÈ„ Ó„ÚÎÓÎÓÓÒ „‡‰‡„¯ ˜Ë„ÎøøÎÊ ‡‚‡ ı ·º„ºº‰ ÚøøÌËÈ ÛÚ„‡ ˝Âð˝„ ·‡È‚‡Î
ÚÛ ı ‡ÈÌ ı˝Ò˝„Ú ÒÛÌ„‡ÎÚ, Һ𺄠·‡È‚‡Î Ú˝ð ı˝Ò˝„Ú ¯‡ ı ‡ÎÚ øøÒÌ˝. ƒ ̋ ˝ð ‰Ûð¸‰Ò‡Ì ÊÛðÏ˚„ ·‡ðËÏÚÎ‡Ì ı˝Ò˝„ ·øðËÈÌ ‰‡„ÛÛ ı ø˜ËÈ„
ÚÓ‰Óð ı ÓÈÎ ˙∏.1-1 Ó„ÚÎÓÎ (1.3·ñð ÁÛð‡„)
0=∑ z , 0 )( 1 = z N . )0( 11 l z ≤≤
2-2 Ó„ÚÎÓÎ (1.3‚ñð ÁÛð‡„)
0=∑ z , 0 )( )( 122 =−+− l z q z N ⇒ )( )( 122 l z q z N −= . )( 2121 l l z l +≤≤
12 l z = ø‰ 0 )( 2 = z N
212 l l z += ø‰ 20120 )( )( 1212 =⋅=−+= l l l q z N ÍÕ
3-3 Ó„ÚÎÓÎ (1.3‚ñð ÁÛð‡„)
0=∑ z , 0 )( 23 =+−− l q F z N ⇒ 1012030 )( 23 −=⋅+−=+−= l q F z N ÍÕ.
)( 321321 l l l z l l ++≤≤+
4-4 Ó„ÚÎÓÎ (1.3‰ñð ÁÛð‡„)
0=
∑ z , 0 )(
24 =+−− l q F z N ⇒ 1012030 )(
24 −=⋅+−=+−= l q F z N ÍÕ.
+ + ≤ ≤ + + +1 2 3 4 1 2 3 4
( )l l l z l l l l
ƒ‡„ÛÛ ı ø˜ÌËÈ ı Û‚¸‰ 4 ñ 4 Ó„ÚÎÓÎ˚„ ı ËÈ ı ¯‡‡ð‰Î‡„‡„øÈ (‰‡„ÛÛ ı ø˜Ó„ÚÎÓÎ˚Ì ı ̋η˝ð˝˝Ò ı ‡Ï‡‡ð‡ ı „øÈ) ·ÓÎÓ‚˜ ı ø˜‰ ̋ Î ÓÎÓ ı Ó‰ ¯‡‡ð‰Î‡„‡Ú‡È Û˜Ëð ˝Ì˝ Ó„ÚÎÓÎ˚„ ı ËÈ‚. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ÌËÈ ˝Ô˛ðËÈ„(1.3ÂñðÁÛð‡„) ·‡È„ÛÛ·‚.
2. ’ø˜‰ ̋ÎËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı
’ø˜‰ ̋ ÎËÈ„ ¯Ëη˝ÌËÈ ı˝Ò˝„ ·øðÚ ÚÓ‰Óð ı ÓÈÎÌÓ. ’ø˜‰ ̋ Î Ó„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝Ì˝˝Ò ı ‡Ï‡‡ð‡ ı Û˜Ëð 4-4 Ó„ÚÎÓΉ ı ø˜‰ ̋ ÎËÈ„ Á‡‡‚‡Î ·Ó‰Ó ı ¯‡‡ð‰Î‡„‡Ú‡È.
1-1
Ó„ÚÎÓÎ .0 )(
1
11 ==
A
z N σ )0( 1 = N
2-2 Ó„ÚÎÓÎ1
12
1
22
)( )(
A
l z q
A
z N −==σ
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
11
‡).
4
4
3
3
2
2
1
1 z 2 z
3 z
4 z
l 4 3l l 2 1l
A A 12
1
F
(z) N
z 1
q
1
·).
‚).
q
z 2
q
3 z
F q
F
z 4
(z) N 2
(z) N 3
(z) N 4
„).
‰).
Â).
20
10 10
50
50
25
0.15
0.275
0.15
Ê).
Á).
1.3 ñð ÁÛð‡„
N , ÍÕ
σ , Ãœ‡
δ , ÏÏ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî12
12 l z = ø‰ σ =2
0
212 l l z += ø‰ σ −
⋅ ⋅= =
⋅
32
2 4
1
20 10 1
4 10
q l
A
Õ/Ï2 50= Ãœ‡
3 ñ 3 Ó„ÚÎÓÎ
4
33
1
2
1
33
104
110201030 )( −⋅
⋅⋅+⋅−=
+−==
A
l q F
A
z N σ Õ/Ï2 25−= Ãœ‡
4 ñ 4 Ó„ÚÎÓÎ
4
33
1
2
2
44
102
110201030 )( −⋅
⋅⋅+⋅−=
+−==
A
l q F
A
z N σ Õ/Ï2 50−= Ãœ‡
›‰„ ̋ ˝ð ÛÚ„‡‡ð ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„(1.3Êñð ÁÛð‡„) ·‡È„ÛÛ·‚.
3. ’ ˝‚ „‡ÊËÎÚ˚„ ÚÓ‰Óð ı ÓÈÎÓ ı
“º‚κðÒºÌ ı ø˜Ì˝˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„ A E
l F l F ⋅
⋅=Δ ÚÓÏ ˙∏Ó„ÓÓð, ÊË„‰
Ú‡ð ı Ò‡Ì ı ø˜Ì˝˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„ A E
l ql q
2
2
=Δ ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰‰Ó„ Û˜Ëð
ı˝Ò˝„ ·øðËÈÌ ı˝‚ „‡ÊËÎÚ01 =Δl ( 01 = N )
3
411
3
1
22
2 10125.01041022
111020
2
−−
⋅=⋅⋅⋅⋅⋅⋅⋅==Δ
A E
l ql Ï 125.0= ÏÏ
( ) 3
411
3
411
3
1
3
1
323 10125.0
104102
11030
102102
111020 −−−
⋅−=⋅⋅⋅
⋅⋅−
⋅⋅⋅
⋅⋅⋅=
⋅−
⋅=Δ
A E
l F
A E
l l ql Ï =
125.0−= ÏÏ
( ) 3
411
3
411
3
2
4
2
424 1015.0
102102
6.01030
102102
6.011020 −−−
⋅−=⋅⋅⋅
⋅⋅−
⋅⋅⋅
⋅⋅⋅=
⋅−
⋅=Δ
A E
l F
A E
l l ql Ï =
15.0−= ÏÏ·ÓÎÌÓ. ’ ̋ Ò˝„ ·øðËÈÌ ı˝‚ „‡ÊËÎÚ˚„ ·Ó‰Ó ı ‰ÓÓ ÚÛ ı ‡ÈÌ ı˝Ò˝„ ÁøøÌ øÁøøðÚ˝˝ ·˝ı ̋΄ ̋ ̋Ú˝È (¯Ëη˝ ÁøøÌ øÁøøðÚ˝˝ ·˝ı ̋΄ ̋ ̋Ú˝È Û˜Ëð) „ ̋ Ê øÁ˝ ̋‰ ÚøøÌ˝˝Ò·‡ðÛÛÌ Ú‡Î‰ øÈÎ˜Î˝ı ı ø˜Ìøø‰ ̋ ˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„ ÓÎÌÓ. ¡øÚ˝Ì ı˝‚„‡ÊËÎÚ˚„ ÓηÓÎ
15.015.0125.0125.004321 −=−−+=Δ+Δ+Δ+Δ=Δ l l l l l ÏÏ
·‡È̇. ÿËÎÊËÎÚËÈÌ ˝Ô˛ðËÈ„ (1.3Áñð ÁÛð‡„) Á‡‡‚‡Î ÁÓÓÎÚÓÓÒ( ı º‰ºÎ„ººÌ„øÈ ̂ ̋ „ ̋ ˝Ò) ˝ı ̋ÎÊ ·‡È„ÛÛΉ‡„.
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1.2. –”Õ√ ¿À“, ÿ¿’¿À“¤Õ –“¿“» “ŒƒŒ—’Œ… ¡»ÿ ¡ŒƒÀŒ√ Œ
≈ðºÌ ı ËÈ ÓÈ΄ÓÎÚ
CÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ı˝ð˝„νı ·ÓÎÓ‚˜ Úøø„ ̋ ˝ð ·øð˝Ì·Ó‰Ó„‰Ó ı „øÈ ·Ó‰ÎÓ„ ̊ „ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó „ ̋ Ì˝.
–Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈÌ ÚÓÓÌÓÓÒ ËÎøø øÎÏ ̋‰ ̋ „‰ ̋ „˜‰ËÈÌ ÚÓÓ„ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ËÈÌ Á˝ð˝„ „ ̋ Ì˝.
–Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„ ̊ „ ·Ó‰Ó ı ‰ÓÓ ººðººð ı ̋η ̋Î ËÎøøÏ ̋‰ ̋ „‰ ̋ „˜‰ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈÌÁ˝ð˝„ˆ ̋ ˝„ ̋ ˝ð ÒËÒÚÂÏËÈÌ ı˝‚ „‡ÊËÎÚ˚Ì Ìº ı ̂ ºÎººÒ ÁÓ ı ËÓÒÓÌ Ì˝Ï ̋ÎÚ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ‡¯Ë„·̇.
–Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„ ̊ „ ·Ó‰Ó ı ÂðºÌ ı ËÈ ‰‡ð‡‡ÎÎ˚„ Ê˯˝˝„ ̋ ˝ð øÁ¸Â.
∆ ˯˝˝ 1.4. ÿËη˝Ì‰ (1.4añð ÁÛð‡„) øøÒ˝ı ‰ÓÚÓÓ‰ ı ø˜, ı ø˜‰ ̋ Î,
¯ËÎÊËÎÚËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
º„ÒºÌ Ì¸: 1001 = F ÍÕ, 1402 = F ÍÕ, 101 = A ÒÏ2
, 82 = A ÒÏ2
,
401 =l ÒÏ, 702 =l ÒÏ, 352/23 == l l ÒÏ, 5102 ⋅= E Ãœ‡.
1.4ñð ÁÛð‡„¡Ó‰ÓÎÚ. –Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„ ̊ „ ‰ºð‚ºÌ ¯‡ÚÚ‡È ·Ó‰‰Ó„.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî14
–Ú‡ÚËÍ Ú‡Î. ›Ì˝ ı˝Ò˝„Ú ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ, ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ËÈÌ Á˝ð„ËÈ„ ÚÓ„ÚÓÓÌÓ. ÿËη˝ÌËÈ Ú˝Ì ı ν„ ‰ ̋ ˝ð ·ø ı ı ø˜ËÈ„ÔðÓÂ͈ÎÓ‚ÓÎ ‡Î„·ð ÌËÈη˝ð ̸ Ú˝„ ·‡È ı ∏ÒÚÓÈ.
0=∑ z , 021 =+−+ B A R F F R (1.9)
’Ó∏ð øÎ Ï ̋‰ ̋ „‰ ̋ „˜Ú˝È Ì˝„ Ú˝„¯ËÚ„ ̋ Î ·‡È„‡‡ ·º„ºº‰, ººðººð ı ̋η ̋ÎÌ˝„‰ø„ ̋ ˝ð Á˝ð„ËÈÌ (Ì˝„ ËÎøø øÎ Ï ̋‰ ̋ „‰ ̋ „˜Ú˝È) ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯·Ó‰ÎÓ„Ó ·‡È̇. √ ÂÓÏÂÚð Ú‡Î. ›Ì˝ ı˝Ò˝„Ú ·Ó‰ÎÓ„ ̊ Ì Ìº ı ̂ ºÎººÒ ø̉ ̋ ÒÎ˝Ì ¯Ëη˝ÌËÈ ı˝Ò„øø‰ËÈÌ ı˝‚ „‡ÊËÎÚÛÛ‰ ̊ Ì ı ÓÓðÓ̉ ̊ Ì ı ÓηÓÓ„ ÚÓ„ÚÓÓÌÓ. ›Ì˝ ·Ó‰ÎÓ„ ̊ ÌÓ̈ÎÓ„ ̺ ı ̂ ºÎ ̸ ı Ó∏ð øÁøøðËÈÌ ÁÓÓÎÚ ˛Ï. ’Ó∏ð øÁøøð ÁÓÓÎÚÚÓÈ Û˜Ëð ı ø˜Ìøø‰ËÈÌ øÈΘΠ̋ÎËÈÌ ‰ø̉ ¯Ëη˝ÌËÈ ÛðÚ ººð˜Îº„‰º ı „øÈ. ÿËη˝ÌËÈ ‰ÓÓ‰ ÁÓÓÎÚ˚„ Óð ı ËÓ‰ ÚøøÌËÈ„
B R ð‡͈ˇð ÒÓÎ¸Ê ı˝‚ „‡ÊËÎÚ˚Ì
̺ ı ̂ ºÎËÈ„ ·Ë˜‚ ̋Î0=Δl , 0=Δ+Δ=Δ
B R F l l l (1.10)
·ÓÎÌÓ. Øø̉: F l Δ - Áº‚ ı ºÌ º„º„‰ÒºÌ ı ø˜Ì˝˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ, B R l Δ - Áº‚ ı ºÌ
‰ÓÓ‰ ÁÓÓÎÚÓ̉ øøÒ˝ı B R ð‡͈‡‡Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ.
‘ËÁËÍ Ú‡Î. ›Ì˝ ı˝Ò˝„Ú ı˝‚ „‡ÊËÎÚ˚„ ı ø˜˝˝ð Ëνð ı ËÈÎÌ˝.
⎪⎪
⎭
⎪⎪
⎬
⎫
⋅−
⋅−=Δ
−+
⋅+
⋅−=Δ
1
1
2
2
2
322
1
12
1
11 )(
A E
l R
A E
l R l
A E
l l F
A E
l F
A E
l F l
B B R
F
B
(1.11)
CËÌÚÂÁ.›Ì˝ ı˝Ò˝„Ú ˝ı ÌËÈ „Ûð‚‡Ì ı˝Ò„ËÈ„ Ì˝„Ú„ ̋ Ì ·Ó‰ÓÊ ËÎøø øÎ Ï ̋‰ ̋ „‰ ̋ „˜ËÈ„
( B R ) ÓÎÌÓ. (1.11)-˚„ (1.10) Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛη‡Î
0 )(
1
1
2
2
2
322
1
12
1
11 =⋅
−⋅
−−
+⋅
+⋅
−
A E
l R
A E
l R
A E
l l F
A E
l F
A E
l F B B
0108102
7.0
1010102
4.0
108102
35.010140
1010102
4.010140
1010102
4.010100411411411
3
411
3
411
3
=⋅⋅⋅
⋅−
⋅⋅⋅
⋅−
⋅⋅⋅
⋅⋅+
⋅⋅⋅
⋅⋅+
⋅⋅⋅
⋅⋅−−−−−−
B B R R
55.60= B R ÍÕ
·ÓÎÌÓ. (1.9) Ú˝„¯ËÚ„ ̋ ν˝Ò = − − = − − = −2 1
140 100 60.55 20.55 A B
R F F R ÍÕ
·ÓÎÌÓ. B R ð‡͈ËÈ„ ÓÎÒÌÓÓð ·Ó‰ÎÓ„ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ڇȷ„‰‡Ê
ˆ‡‡¯‰ ̊ Ì ·Ó‰ÓÎÚ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„ ̊ Ì ‡‰ËÎ ·ÓÎÌÓ.
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15
ÿËη˝Ì‰ øøÒ˝ı ‰ÓÚÓÓ‰ ı ø˜ ̸ :1 ñ 1 Ó„ÚÎÓΉ 55.201 −== A R N ÍÕ;
2 ñ 2 Ó„ÚÎÓΉ = + = − + =2 1
20.55 100 79.45 A
N R F ÍÕ;
3 ñ 3 Ó„ÚÎÓΉ 55.60213 −=−+= F F R N A ÍÕ·‡È̇. ƒ‡„ÛÛ ı ø˜ÌËÈ ˝Ô˛ðËÈ„ ̋‰„ ̋ ˝ð ÛÚ„‡‡ð 1.4·ñð ÁÛð‡„Ú ·‡È„ÛÛ·‚.
ÿËη˝Ì‰ øøÒ˝ı ı ø˜‰ ̋ Î
1 ñ 1 Ó„ÚÎÓΉ 55.201010
1055.204
3
1
11 −=
⋅
⋅−==
− A
N σ Ãœ‡;
2 ñ 2 Ó„ÚÎÓΉ 31.99108
1045.794
3
2
22 =
⋅
⋅==
− A
N σ Ãœ‡;
3 ñ 3 Ó„ÚÎÓΉ σ −
⋅= = − = −⋅
3
33 4
2
60.55 1075.698 10
N
A Ãœ‡
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ 1.4‚ñð ÁÛð‡„Ú ·‡È„ÛÛ·‚.ÿËη˝ÌËÈ Ó„ÚÎÓÎÛÛ‰ ̊ Ì ¯ËÎÊËÎÚËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı˚Ì ÚÛΉ ÚøøÌËÈ ı˝Ò˝„·øðËÈÌ ı˝‚ „‡ÊËÎÚ˚„ ÚÓ‰Óð ı ÓÈηÓÎ
3
411
3
1
111 10041.0
1010102
4.01055.20 −
− ⋅−=
⋅⋅⋅
⋅⋅−==Δ
A E
l N l Ï 041.0−= ÏÏ
3
411
3
2
3222 10173.0
108102
35.01045.79 )( −
− ⋅=
⋅⋅⋅
⋅⋅=
−=Δ
A E
l l N l Ï 173.0= ÏÏ
3
411
3
2
333 10132.0
108102
35.01055.60 −
− ⋅−=
⋅⋅⋅
⋅⋅−==Δ
A E
l N l Ï 132.0−= ÏÏ
·‡È̇. ÿËÎÊËÎÚËÈÌ ˝Ô˛ðËÈ„ ̋‰„ ̋ ˝ð ÛÚ„‡‡ð 1.4„ñð ÁÛð‡„Ú ·‡È„ÛÛ·‚.
∆ ˯˝˝ 1.5. ƒ ̋ ̋‰ øÁøøð ̸ ÁÓÓÎÚÚÓÈ, ‰ÓÓ‰ øÁøøð ̸ ‰ÓÓ‰ ÒÛÛðˇÒ
3.0=Δ ÏÏ Á‡ÈÚ‡È „‡Ì ¯Ëη˝ÌËÈ (1.5ñð ÁÛð‡„) ‰Û̉ 200= F ÍÕ ı ø˜ ‰ÓÓ¯˜Ë„Î˝Ì øÈÎ˜Î˝ı ̋‰ ¯Ëη˝ÌËÈ ‰ ̋ ̋‰, ‰ÓÓ‰ ı˝Ò˝„Ú øøÒ˝ı ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎ.
ª„ÒºÌ Ì¸ :4
1025 −
⋅= A Ï2
, 3=l Ï,5
102 ⋅= E Ãœ‡.¡Ó‰ÓÎÚ. ÿËη˝ÌËÈ ‰ ̋ ̋‰ ı˝Ò˝„Ú F ı ø˜ÌËÈ øÈΘΠ̋ν˝ð øøÒ˝ı ÒÛ̇ÎÚ
3.0=Δ ÏÏñ˝˝Ò Ë ı ·‡È‚‡Î ¯Ëη˝ ‰ÓÓ‰ ÒÛÛðË̉ ı øð˜ ð‡͈ øøÒ˝ı Û˜ËðÒÚ‡ÚËÍËÈÌ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ. ’‡ðËÌ ¯Ëη˝ÌËÈ ‰ÓÓ‰ øÁøøðÒÛÛðË̉ ı øð˝ı „øÈ ÚÓ ı ËÓΉÓΉ ·Ó‰ÎÓ„Ó ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·ÓÎÌÓ. F ı ø˜Ì˝˝Ò øøÒ˝ı ı˝‚ „‡ÊËÎÚ˚„ ÓηÓÎ
−
⋅ ⋅ ⋅Δ = = =
⋅ ⋅ ⋅ ⋅ ⋅
3
5 6 4
/ 2 200 10 1.50.6
2 10 10 25 10 F
F l l
E AÏÏ > Δ ·ÓÎÌÓ.
›Ì˝ ·Ó‰ÎÓ„ ̊ Ì ı Û‚¸‰ Δ>Δ F l ·‡È„‡‡ Û˜Ëð ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó
·ÓÎÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî16
l
R B
F
l / 2 A
A R
Δ
1.5ñð ÁÛð‡„
“ ̋ ̈‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ÁÓ ı ËÓ‚ÓÎ
∑ = 0 z , 0=+− B A R F R (1.12)
·ÓÎÌÓ. ØøÌ˝˝Ò øÁ‚ ̋Î ˝Ì˝ ·Ó‰ÎÓ„Ó Ì˝„‰ø„ ̋ ˝ð Á˝ð„ËÈÌ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯·‡È̇. Õ˝Ï ̋ÎÚ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ¯Ëη˝ÌËÈ ı˝‚ „‡ÊËÎÚ‡‡Ò ÁÓ ı ËÓÊ ·ÓÎÌÓ. “Û ı ‡Èη‡Î
Δ=Δ+Δ 21 l l (1.13)
·‡È̇. Øø̉ 1l Δ - ¯Ëη˝ÌËÈ ‰ ̋ ̋‰ ı˝Ò„ËÈÌ ı˝‚ „‡ÊËÎÚ, 2l Δ - ¯Ëη˝ÌËÈ ‰ÓÓ‰
ı˝Ò„ËÈÌ ı˝‚ „‡ÊËÎÚ.ÿËη˝ÌËÈ ‰ ̋ ̋‰ ı˝Ò˝„ A R ı ø˜˝˝ð ÒÛ̇Ê, ‰ÓÓ‰ ı˝Ò˝„ B R ı ø˜ÌËÈ
øÈΘΠ̋ν˝ð ‡„¯Ë̇. ›‰„ ̋ ˝ð ı˝‚ „‡ÊËÎÚ˚„ ÓηÓÎ
⎪⎪
⎭
⎪⎪
⎬
⎫
⋅−−=
⋅−=Δ
⋅=Δ
A E
l R F
A E
l R l
A E
l R l
A B
A
2/ )( 2/
2/
2
1
(1.14)
·‡È̇.(1.14)-˚„ (1.13) ÚÓÏ ˙∏Ó̉ ÓðÎÛÛη‡Î
Δ=⋅−
−⋅
A E
l R F
A E
l R A A 2/ )( 2/
·ÓÎÓ ı ·‡ øøÌ˝˝Ò A R ð‡͈ËÈ„ ÓηÓÎ
− −⋅ − ⋅ − ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅3 4 11 41.5 (200 10 ) 1.5 3 10 2 10 25 10 A A
R R
310150 ⋅= A
R Õ 150= ÍÕ
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·‡È̇. (1.12) Ú˝„¯ËÚ„ ̋ ν˝Ò333 10501015010200 ⋅=⋅−⋅=−= A B R F R Õ 50= ÍÕ
·ÓÎÌÓ. “ ̋ „‚ ̋Î ¯Ëη˝ÌËÈ ‰ ̋ ̋‰ ı˝Ò˝„Ú øøÒ˝ı ı ø˜‰ ̋ Î
6
4
31 1060
102510150 ⋅=
⋅⋅==
− A R Aσ Õ/Ï2 60= Ãœ‡ ,
‰ÓÓ‰ ı˝Ò˝„Ú øøÒ˝ı ı ø˜‰ ̋ Î
6
4
3
2 10201025
1050⋅−=
⋅
⋅−=−=
− A
R B σ Õ/Ï2 20−= Ãœ‡
·‡È̇.
∆ ˯˝˝ 1.6. ª„º„‰ÒºÌ ¯Ëη˝ÌËÈ ÒËÒÚÂÏËÈÌ (1.6‡ñð ÁÛð‡„) BD ,CL
ı ‡ðËÏ ı ‡È ¯Ëη˝Ìøø‰ËÈÌ Ó„ÚÎÓÎ˚„ ÒÓÌ„Ó.ª„ÒºÌ Ì¸: 200= F ÍÕ, 160=σ Ãœ‡, 030=α , 1=a Ï, 2.2=b Ï,
8.0=c Ï, 5.1: 21 = A A
¡Ó‰ÓÎÚ. –ËÒÚÂÏËÈÌ ı ‡ðËÏ ı ‡È ¯Ëη˝ ·øðÚ Ì˝„, ÚÛ΄ÛÛð ‰ ̋ ˝ð ı Ó∏ð(·ÓÒÓÓ, ı˝‚Ú˝ ̋) øÎ Ï ̋‰ ̋ „‰ ̋ „˜ ı ø˜ øÈÎ˜Î˝ı Û˜Ëð ˝Ì˝ ·Ó‰ÎÓ„Ó Ì˝„‰ø„ ̋ ˝ðÁ˝ð„ËÈÌ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ.
‡).
F
A1 N 1
B
2
C
2 N
2
1 A
D L
l
α
ac
b
30
B C
B
C
C
0
l Δ 2 1l Δ K 1
2
300
K
·).
1.6ñð ÁÛð‡„ –Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ν˝Ò ÚÛ΄ÛÛð ‰ ̋ ˝ð ÏÓÏÂÌÚ ·Ë˜‚ ̋Î
‡¯Ë„Ú‡È. ”˜Ëð ̸ ˝Ì˝ Ú˝„¯ËÚ„ ̋ Ή ˆ ̋ „Ú øøÒ˝ı ·ÓÒÓÓ, ı˝‚Ú˝˝ ð‡͈ËÛ‰ÓðÓÎˆÓ ı „øÈ.
0=∑ K M , 0sin21 =⋅+−− c F b N a N α (1.15)
–ËÒÚÂÏËÈÌ ı˝‚ „‡ÊËÎÚ˚Ì ÁÛð„‡‡Ò (1.6·ñð ÁÛð‡„) ¯Ëη˝Ìøø‰ËÈÌ ı˝‚
„‡ÊËÎÚ˚
Ì ı
ÓÓðÓ̉ ˚
Ì ı
ÓηÓÓ„ ÚÓ„ÚÓÓ‚ÓÎ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî18
b
a
CC
BB =
1
1 ; α sin/21 l CC Δ= ;α sin
21
l
b
al
Δ⋅=Δ (1.16)
·ÓÎÌÓ. ’ ̋ ‚ „‡ÊËÎÚÛÛ‰ ̊ „ ‰ÓÚÓÓ‰ ı ø˜˝˝ð Ëνð ı ËÈη ̋Î
1
11
A E
l N l
⋅
⋅=Δ ;
2
22
sin/
A E
l N l
⋅
⋅=Δ
α
(1.17)
·‡È̇. (1.17), (1.16) ÚÓÏ ˙∏ÓÌÓÓÒ
22
2
1
1
sin EA
l N
b
a
A E
l N
⋅⋅=
⋅
⋅
α
⇒ 22
1
21sin
N A
A
b
a N ⋅⋅
⋅=
α
⇒ 21 73.2 N N =
·ÓÎÓ ı Û˜Ëð (1.15) ÚÓÏ ˙∏Ó„ ‡¯Ë„·‚‡Î 77.412 = N ÍÕ, 05.1141 = N ·ÓÎÌÓ.
21 N N > , =max 2
N N ·‡È„‡‡ Û˜Ëð ·‡Ú ·˝ı ËÈÌ
σ σ = = ≤ ⎡ ⎤⎣ ⎦max 1
max
1
N N A A
̺ ı ̂ κºÒ
σ
− −⋅= = = ⋅ ⋅
⋅⎡ ⎤⎣ ⎦
34 41
1 6
114.05 107.128 10 10
160 10
N A Ï2 = 7.128 ÒÏ2
= = =12
7.1284.752
1.5 1.5
A A ÒÏ2
·ÓÎÌÓ.
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’Œ®—ƒ” √ ¿¿— ¡ØÀ›√
÷›√ »…Õ ’Ø ◊ ƒ›À“ “ªÀª¬
2.1. ŒÕŒÀ¤Õ ØÕƒ–›Õ Œ…À√ ŒÀ“
÷ ̋ „ËÈÌ ı ø˜‰ ̋ ÎÚ ÚºÎº‚ „ ̋ Ê Ú˝ð ˆ ̋ „ËÈ„ ‰‡È𘠷ÓÎÓ ı ·ø ı ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı ı ø˜‰ ̋ Îøø‰ËÈÌ ÓÎÓÌÎÓ„ ˛Ï.
“Û ı ‡ÈÌ ˆ ̋ „ËÈÌ Óð˜Ë̉ Ú‡ÎÒÛÛ‰‡‰ ̸ ı˝‚Ú˝˝ ı ø˜‰ ̋ Î øÈÎ˜Î˝˝„øÈ ÊËÊË„Ô‡ð‡ÎÎÂθÔËÔ‰ ·‡È„ÛÛÎÊ ·ÓÎÌÓ. ’ ̋ ‚Ú˝˝ ı ø˜‰ ̋ ΄øÈ Ú‡Î·‡È„ „ÓΠڇη‡È„ ̋ Ì˝. √ ÓΠڇη‡È‰ øÈÎ˜Î˝ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„ „ÓÎ ı ø˜‰ ̋ Î „ ̋ Ì˝.
›Ì˝ Ô‡ð‡ÎÎÂθÔËÔ‰ ̊ Ì Ú‡ÎÒÛÛ‰‡‰ øÈÎ˜Î˝ı „ÓÎ ı ø˜‰ ̋ Îøø‰ËÈÌ ‡Î„·ð
ı˝ÏÊ ˝˝„ ˝˝ð˝˝ ı ‡Ï„ËÈÌ Ë ı ËÈ„ 1σ , ı ‡Ï„ËÈÌ ·‡„ ˚„ 3σ , Á‡‚Òð˚Ì ÛÚ„ ˚„ 2σ „ ̋ Ê Ú˝Ï‰ ̋ „Î˝Ì˝. “ ̋ „‚ ̋Î 321 σ σ σ >> ̺ ı ̂ ºÎ ·ËÂÎÌ˝.
«º‚ ı ºÌ Ì˝„ „ÓÎ ı ø˜‰ ̋ Î Ú˝„ ̋ ˝Ò ̌΄‡‡Ú‡È ·‡È‚‡Î ¯Û„‡Ï‡Ì, ı Ó∏ð „ÓÎ ı ø˜‰ ̋ Î Ú˝„ ̋ ˝Ò ̌΄‡‡Ú‡È ·‡È‚‡Î ı ‡‚Ú„‡ÈÌ, „Ûð‚‡Ì „ÓÎ ı ø˜‰ ̋ Î Ú˝„ ̋ ˝Ò ̌΄‡‡Ú‡È·‡È‚‡Î ˝Á ̋Î ı øøÌËÈ ı ø˜‰ ̋ ÎÚ ÚºÎº‚ „ ̋ Ì˝.
“ ˝Ì ı Î ˝„ËÈÌ ÒÛÌ„‡ÎÚ˚Ì ø‰, ººðººð ı ̋η ̋Î ¯Û„‡Ï‡Ì ںκ‚ËÈÌ ( 01 >σ ,
032 ==σ σ ) ø‰ ̇ÎÛÛ Ó„ÚÎÓΉ øøÒ˝ı ı ø˜‰ ̋ ÎËÈ„
⎪⎭
⎪⎬
⎫
⋅=
⋅=
α σ
τ
α σ σ
α
α
2sin2
cos2
(2.1)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. ’‡‚Ú„‡ÈÌ ı ø˜‰ ̋ÎÚ ÚºÎº‚ËÈÌ ø‰ ̇ÎÛÛ Ó„ÚÎÓΉ øøÒ˝ı ı ø˜‰ ̋ ÎËÈ„
⎪⎪
⎭
⎪⎪
⎬
⎫
+−
=
−−
++
=
α τ α σ σ
τ
α τ α σ σ σ σ
σ
α
α
2cos2sin
2
2sin2cos22
yx y x
yx y x y x
(2.2)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ α - ÚÛ ı ‡ÈÌ ı ‡‚Ú„‡ÈÌ ÌÓðχÎËÈÌ ¯Ëη˝ÌËÈÚ˝Ì ı ν„Ú˝È øøÒ„ ̋ ı ºÌˆº„.
√ ÓΠڇη‡ÈÌ ·‡Èð·Î˚„
y x
yx tg
σ σ
τ α
−−=
22 0 (2.3)
ÚÓÏ ˙∏Ó„ÓÓð, „ÓÎ ı ø˜‰ ̋ ÎËÈ„
22
minmax 22 yx
y x y x
τ
σ σ σ σ
σ +⎟⎟ ⎠
⎞
⎜⎜⎝
⎛ −
±
+
= (2.4)
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ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈΉӄ.›ÁÎ ˝ı øøÌËÈ ı ø˜‰ ̋ÎÚ ÚºÎº‚ËÈÌ ø‰ √ ÛÍËÈÌ ÂðºÌ ı ËÈ ı ÛÛθ
( )[ ]
( )[ ]
( )[ ] ⎪⎪⎪
⎭
⎪⎪
⎪
⎬
⎫
+−=
+−=
+−=
1233
3122
3211
1
1
1
σ σ ν σ ε
σ σ ν σ ε
σ σ ν σ ε
E
E
E
(2.5)
ı ̋η˝ðÚ˝È ·‡È‰‡„. ›‰„ ̋ ˝ð ı˝‚ „‡ÊËÎÚ‡‡ð ˝Áνı øøÌËÈ ı ‡ð¸ˆ‡Ì„ÛȺºð˜ÎºÎÚËÈ„ ÚÓ‰Óð ı ÓÈηÓÎ
321 ε ε ε θ ++= (2.6)
·ÓÎÓ ı ·‡ ı ø˜‰ ˝ ν˝ð Ëνð ı ËÈη˝ Î( )321
21σ σ σ
ν θ ++
−=
E (2.7)
·‡È̇.α
F α
n
F d
σ
τ
α
α
∆ ˯˝˝ 2.1. 50=d ÏÏ „ÓΘÚÓÈ
¯ËηËÈ„ (2.1ñð ÁÛð‡„)Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ
100= F ÍÕ ı ø˜˝˝ð
ÒÛÌ„‡ ı ‡‰ ÌÓðχθ ̸¯Ëη˝ÌËÈ Ú˝Ì ı ν„Ú˝È
60 ºÌˆº„ øøÒ„ ̋ ı 2.1ñð ÁÛð‡„
Ó„ÚÎÓΉ øøÒ˝ı ·ÓÒÓÓ, ı˝‚Ú˝˝, ·øÚ˝Ì ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎ. ¡Ó‰ÓÎÚ. ÿËη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È
4222
106.194
)105( 14.3
4−
−
⋅=⋅⋅
==d
A π
Ï2
Û˜Ëð ¯Ëη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ øÈÎ˜Î˝ı ·ÓÒÓÓ ı ø˜‰ ̋ Î
64
3
1051106.1910100 ⋅=⋅⋅==
− A F σ Õ/Ï2 51= Ãœ‡
·‡È̇. Œ‰ÓÓ Ì‡ÎÛÛ ı ‡‚Ú„‡È‰ øøÒ˝ı ·ÓÒÓÓ, ı˝‚Ú˝˝, ·øÚ˝Ì ı ø˜‰ ̋ ÎËÈ„ ÓηÓÎ662 108.1260cos1051cos ⋅=⋅⋅==
α σ σ α
Õ/Ï2 8.12= Ãœ‡
66
101.22120sin2
10512sin
2 ⋅=⋅
⋅==
α σ
τ α
Õ/Ï2 1.22= Ãœ‡
5.251.228.12 2222 =+=+= τ σ α
P Ãœ‡
·ÓÎÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî20
∆ ˯˝˝ 2.2. ›ÎÂÏÂÌÚËÈÌ (2.2‡ñð ÁÛð‡„) „ÓΠڇη‡ÈÌÛÛ‰‡‰=1σ 120Ãœ‡, =2σ 60Ãœ‡ ı ø˜‰ ̋ Î øÈΘËÎÊ ·‡È ı ‡‰ ÌÓðχθ ̸ ı˝‚Ú˝˝
˜Ë„Î ̋ÎÚ˝È 030=α ºÌˆº„ øøÒ„ ̋ ı ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı ·ÓÒÓÓ, ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ÓÎ.
σ1 1
σ 2
ασ
σ 2
αn
σ α
ατ
B A
τ
σC
O
Dα
600
60
120
=105σα
=26ατ
‡). ·).2.2ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ’‡‚Ú„‡ÈÌ ı ø˜‰ ̋ ÎÚ ÚºÎº‚ËÈÌ ¯ÛÛ‰ ·‡ Ûð‚ÛÛ ·Ó‰ÎÓ„Ó ·‡È‰‡„. “ ̋ „‚ ̋Î ˝Ì˝ ·Ó‰ÎÓ„Ó Ì¸ ¯ÛÛ‰ ·Ó‰ÎÓ„Ó ˛Ï. ªºðººð ı ̋η ̋Î „ÓÎ ı ø˜‰ ̋ Îøø‰º„º„‰ÒºÌ ·‡È ı ‡‰ ̇ÎÛÛ ı ‡‚Ú„‡È‰ øøÒ˝ı ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ·Ó‰ÎÓ„Ó ˛Ï.
›Ì˝ ·Ó‰ÎÓ„ ˚„ ÚÓÏ ˙∏Ó, „ð‡ÙË͇‡ð ·Ó‰ÓÊ ·ÓÎÌÓ.› ı ν ̋‰ ˝Ì˝ ·Ó‰ÎÓ„ ̊ „ ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰ ˙∏. Õ‡ÎÛÛ ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı
ı ø˜‰ ̋ ÎËÈ„ (2.2) ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. “ ̋ „ ̋ ı ‰ ̋ ˝ ˝Ì˝ ÚÓ ı ËÓΉÓΉ
1σ σ = x , 2σ σ = y , 0= yx τ ·‡È„‡‡„ ‡Ì ı ‡‡ð‚‡Î
1055.0309060cos22
2121 =⋅+=−
++
= σ σ σ σ
σ α
Ãœ‡
26866.0302sin2
21 ≈⋅=−
= α σ σ
τ α
Ãœ‡
·ÓÎÌÓ.›Ì˝ ·Ó‰ÎÓ„ ̊ „ „ð‡ÙË͇‡ð ·Ó‰Ó ı ‰ÓÓ ÃÓð˚Ì ‰Û„ÛÈ (2.2·ñð ÁÛð‡„)
‡¯Ë„·̇. ƒÛ„ÛÈ ·‡È„ÛÛ· ı˚Ì ÚÛΉ ˝ı ν ̋‰ Ú˝„¯ ºÌˆº„Ú ÍÓÓð‰Ë̇Ú˚ÌÒËÒÚÂÏ ‡‚˜ ı˝‚Ú˝˝ Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„ ( σ ), ·ÓÒÓÓÚ˝Ì ı ν„ËÈÌ ‰‡„ÛÛ ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ ( τ ) ı˝ÏÊËÌ˝. ’ø˜‰ ̋ ÎËÈÌ ÚÓ‰Óð ı ÓÈ
χүڇ· ÒÓÌ„ÓÊ ı˝‚Ú˝˝ Ú˝Ì ı ν„ ‰ ̋ ˝ð 1σ =OA , 2σ =OB ı˝ð˜ÏËÈ„
Ú˝Ï‰ ̋ „Î ̋Ê ( ı ø˜‰ ̋ ÎËÈÌ Ú˝Ï‰„ËÈ„ ı ‡ð„‡ÎÁ‡Ì) AB „ÓΘÚÓÈ, C Úº‚Ú˝È ÚÓÈðÓ„·‡È„ÛÛÎ̇. ¡‡È„ÛÛÎÒ‡Ì ˝Ì˝ ‰Û„ÛÈ„ ÃÓð˚Ì ‰Û„ÛÈ ·Û ̨ Û ı ø˜‰ ̋ÎËÈÌ ‰Û„ÛÈ „ ̋ Ì˝.
ƒÛ„ÛÈÌ ˆ ˝
„ËÈÌ ÍÓÓð‰Ë̇ÚÛÛ‰ ˇ
ÌÁ ·øðËÈÌ ı
‡‚Ú„‡È‰ øÈÎ˜Î˝ı
·ÓÒÓÓ, ı˝
‚Ú˝˝
ı ø˜‰ ̋ ÎËÈ„ Ëνð ı ËÈÎÌ˝.
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ÕÓðχθ ̸ ı˝‚Ú˝˝ Ú˝Ì ı ν„Ú˝È 30=α ºÌˆº„ øøÒ„ ̋ ı ı ‡‚Ú„‡ÈÌ ı ø˜‰ ̋ ÎËÈ„ ˝Ì˝
‰Û„ÛÈ„‡‡ð ÚÓ‰Óð ı ÓÈÎÓ ı˚Ì ÚÛΉ ‰Û„ÛÈÌ C Úº‚‰ ÓðÓÈÚÓÈ 602 ==∠ α α
ACD
ºÌˆ„ËÈ„ ·‡È„ÛÛÎ̇. ›Âð˝„ ºÌˆ„ËÈ„ ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„ ı˝ÏÊËÌ˝. ›Ì˝ ºÌˆ„ººð ÚÓ‰Óð ı ÓÈÎÓ„‰Ó ı α D ˆ ˝„ËÈÌ ‡·ÒˆËÒÒ Ì¸ ̇ÎÛÛ ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı 105=
α σ Ãœ‡ ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„, Óð‰ËÌ‡Ú Ì¸ ÏºÌ ı ‡‚Ú„‡È‰
øÈÎ˜Î˝ı 26=α
τ Ãœ‡ ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎÌÓ.
∆ ˯˝˝ 2.3. “‡ÎÒÛÛ‰‡‰ ̸ 100= x σ Ãœ‡, 40−= y σ Ãœ‡,
60−= yx τ Ãœ‡ ı ø˜‰ ̋ Î øÈÎ˜Î˝ı „‡Ì ̋ÎÂÏÂÌÚËÈÌ (2.3ñð ÁÛð‡„) „ÓÎ ı ø˜‰ ̋ Î,
„ÓΠڇη‡ÈÌ ·‡Èð·Î, ı ‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ Î ·ÓÎÓÌ ÌÓðχθ ̸ ı˝‚Ú˝˝
˜Ë„ν ÎÚ˝È
60−=α ºÌˆº„ øøÒ„ ˝ı ̇ÎÛÛ ı ‡‚Ú„‡È‰ øøÒ˝ı ı ø˜‰ ˝ ÎËÈ„ ÚÓÏ ˙∏Ó,„ð‡ÙËÍËÈÌ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎ.
τ
y σ
yx
x σ
y σ
x σ
xy τ
σmin
σmax
σ
max
minτ
σ
τ
α
α
m a x
τ
m i n
τ
maxσσmin
ασ
τ α
P 060
1
2
4
3
5
6
A B
7
C α0O
‡). ·).
2.3ñð ÁÛð‡„¡Ó‰ÓÎÚ. › ı ν ̋‰ ı ø˜‰ ̋ Îøø‰ËÈ„ ÚÓ‰Óð ı ÓÈÎ ˙∏.
‡. “ÓÏ˙ ∏ ÓÌ˚ ‡ð„‡ √ ÓÎ ı ø˜‰ ̋ ÎËÈ„ ÓηÓÎ
( ) 195.9230602
40100
2
40100
222
22
2
min
max ±=−+⎟ ⎠
⎞⎜⎝
⎛ +±
−=+⎟
⎟ ⎠
⎞⎜⎜⎝
⎛ −±
+= yx
y x y x τ
σ σ σ σ σ
195.122195.9230max =+=σ Ãœ‡, 195.62195.9230min −=−=σ Ãœ‡
·‡È̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî22
√ ÓΠڇη‡ÈÌ ·‡Èð·Î
75.0 )40( 100
)60( 222 0 =
−−
−⋅−=
−−=
y x
yx tg
σ σ
τ α ⇒ 8.362 0 =α ⇒ 4.180 =α
ºÌˆ„ººð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. ªºðººð ı ̋η ̋Î „ÓΠڇη‡ÈÌ ÌÓðχθ ı˝‚Ú˝˝
˜Ë„Î ̋ÎÚ˝È 4.180 =α ºÌˆº„ (2.3ñð ÁÛð‡„) øøÒ„ ̋ Ì˝.
’‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ Î
195.922
)195.62( 195.122
2minmax
min
max ±=−−
±=−
±= σ σ
τ Ãœ‡
·ÓÎÌÓ. ÕÓðχθ ̸ ı˝‚Ú˝˝ ˜Ë„Î ̋ÎÚ˝È 60−=α ºÌˆº„ øøÒ„ ̋ ı ̇ÎÛÛ ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı ı ø˜‰ ̋ Î
=−−++= α τ α σ σ σ σ
σ α
2sin2cos22 yx
y x y x
( ) 96.56 )120sin( 60 )120cos( 2
)40( 100
2
40100−=−−−−
−−+
−= Ãœ‡
=+−
= α τ α σ σ
τ α
2cos2sin2 yx
y x
62.30 )120cos( )60( )120sin( 2
)40( 100−=−−+−
−−= Ãœ‡
·‡È̇.
·. √ ð‡ÙËÍ ‡ð„‡ √ ð‡ÙË͇‡ð ·Ó‰Ó ı˚Ì ÚÛΉ ˝ı ν ̋‰ ı˝‚Ú˝˝ Ú˝Ì ı ν„ ‰ ̋ ˝ð 100= x σ Ãœ‡
ı ø˜‰ ̋ ÎËÈ„ ı˝ÏÊËÊ 1 ˆ ̋ „ËÈ„; 40−= y σ Ãœ‡ ı ø˜‰ ̋ ÎËÈ„ ı˝ÏÊËÊ 2 ˆ ̋ „ËÈ„; 1
ˆ ̋ „ ̋ ˝Ò 60−= yx τ Ãœ‡ ı ø˜‰ ̋ ÎËÈ„ ı˝ÏÊËÊ 3 ˆ ̋ „ËÈ„; 2 ˆ ̋ „ ̋ ˝Ò 60= yx τ Ãœ‡
ı ø˜‰ ̋ ÎËÈ„ ı˝ÏÊËÊ 4 ˆ ̋ „ËÈ„ ÓÎÌÓ. ƒ‡ð‡‡ ̸ 3, 4 ˆ ̋ „øø‰ËÈ„ ¯ÛÎÛÛ̇‡ð ı ÓÎ·Ó ı Ó‰ ‡·ÒˆËÒÒ˚„ Ó„ÚÓÎÊ ·‡È„‡‡ C ˆ ̋ „ ÃÓð˚Ì ‰Û„ÛÈÌ Úº‚ (2.3ñð ÁÛð‡„)
·ÓÎÌÓ. – ˆ ˝„Ú Úº‚Ú˝È ‰Û„ÛÈ ·‡È„ÛÛÎÒÌ˚ ‰‡ð‡‡ 3 ˆ ˝„ ˝˝Ò ‡·ÒˆËÒÒÚ‡ÈÔ‡ð‡ÎÎÂθ ¯ÛÎÛÛÌ Ú‡Ú‡Ê ÚÓÈðÓ„ÚÓÈ Ó„ÚÎÓΈÛÛÎ‡Ì P ÚÛÈÎ ˆ ̋ „ËÈ„ ÓÎÌÓ. P ÚÛÈÎ ˆ ̋ „ËÈ„ ¿ ˆ ̋ „Ú˝È ı ÓÎ·Ó ı Ó‰ maxσ ı ø˜‰ ̋ ÎËÈÌ ˜Ë„Î ̋Î „‡ð‡ ı ·º„ºº‰ maxσ
ı ø˜‰ ̋ Î ˝Âð˝„(ÒÛÌ„‡ ı ) ·‡È„‡‡ Û˜Ëð øÈÎ˜Î˝ı „ÓÎ ı ‡‚Ú„‡È„‡‡Ò „‡‰‡„¯ ˜Ë„Î˝Ì˝.
›Ì˝ ÚÓ ı ËÓΉÓΉ OA ı˝ð˜ËÏ 19.122max =σ Ãœ‡ ı ø˜‰ ̋ ÎËÈÌ ı˝ÏÊ ̋ ˝„ Á‡‡Ì‡.
ØøÌËÈ ‡‰Ë·‡ð 19.62min −=σ Ãœ‡ ı ø˜‰ ̋ ÎËÈÌ ı˝ÏÊ ̋ ˝, ˜Ë„Î ̋ÎËÈ„ (¯‡ ı ‡ ı )
ÓÎÌÓ. 5C ð‡‰ËÛÒ 19.92max =τ Ãœ‡ ı ø˜‰ ̋ ÎËÈÌ ı˝ÏÊ ̋ ˝„ Á‡‡ ı ·‡ 0max >τ
Û˜Ëð øÈÎ˜Î˝ı ı ‡‚Ú„‡È„ ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ ˝ð„øøνı
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˝ðÏ ̋ÎÁÎ ̋ÎÚ˝È ·‡È̇. ÃºÌ 6C ð‡‰ËÛÒ 19.92min −=τ Ãœ‡ ı ø˜‰ ̋ ÎËÈÌ ı˝ÏÊ ̋ ˝„
Á‡‡ ı ·‡ 0min <τ Û˜Ë𠈇„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„ ˝ð„ ̋ ı ˜Ë„Î ̋ÎÚ˝È
·‡È̇. P ÚÛÈÎ ˆ ˝„ËÈ„ 5 ˆ ˝„Ú˝È ı ÓηÓÒÓÌ ˜Ë„ν Ή (·ÓÒÓÓ ı ø˜‰ ˝ ÎËÈÌ ˜Ë„)
ÔÂðÔẨËÍÛÎ ˇð ˜Ë„Î ̋Ή maxτ ı ø˜‰ ̋ Î, ÏºÌ 6 ˆ ̋ „Ú˝È ı ÓηÓÒÓÌ ˜Ë„Î ̋Ή
ÔÂðÔẨËÍÛÎ ˇð ˜Ë„Î ̋Ή minτ ı ø˜‰ ̋ Î øÈÎ˜Î˝Ì˝.
Õ‡ÎÛÛ ı ‡‚Ú„‡È‰ øøÒ˝ı ı ø˜‰ ̋ ÎËÈ„ ÓÎÓ ı˚Ì ÚÛΉ P ˆ ̋ „Ú ÓðÓÈÚÓÈ, Ì˝„ Ú‡Î
̸ 3 P ·‡È ı 60−=α (ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ) ºÌˆ„ËÈ„·‡È„ÛÛ· ı ‡‰ 7 ˆ ̋ „Ú ÚÓÈð„ËÈ„ Ó„ÚÓÎÊ ·‡È̇. 7 ˆ ̋ „ËÈÌ ‡·ÒˆËÒÒ
9.56−=α
σ Ãœ‡, Óð‰ËÌ‡Ú 6.30−=α
τ Ãœ‡ ·ÓÎÌÓ. ¡ÛÒ‡‰ ı ø˜‰ ̋ ÎËÈÌ ı˝ÏÊ ̋ ˝,
˜Ë„Î ̋ÎËÈ„ ÁÛð‡„Ú øÁøøν‚. ›Ì˝ ·Ó‰ÎÓ„ ̊ Ì ı Û‚¸‰ y x σ σ > ·‡È„‡‡ Û˜Ëð 0α
ºÌˆº„ maxσ ı ø˜‰ ̋ ÎËÈÌ ˜Ë„ËÈ„ ( ¿ ˆ ̋ „Ú) Á‡‡Ì‡.
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ƒ¿¿À√ ¿¬¿— ¹1. ƒ‡‡Î„‡‚‡ð „Ûð‚‡Ì ·Ó‰ÎÓ„ÓÓÒ ·øð‰ ̋ Ì˝.
1. –ÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„Ó
ÿ‡Ú‡ÎÒ‡Ì „‡Ì ¯Ëη˝Ì‰ (2.4ñð ÁÛð‡„) Úº‚κðÒºÌ, Ú‡ð ı χΠı ø˜( 20=q ÍÕ/Ï) øÈΘËÎÊ ·‡È ı ‡‰
1. ‰‡„ÛÛ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ ;2. ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ ;3. ̄ ËÎÊËÎÚËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
’Óηӄ‰Ó ı ÚÓÓÌ º„º„‰ÎËÈ„ 2.1ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
1
ba
=50 =40
F 12 F
0.4 0.6
q A A1 2
2
ab
1 F 2 F
1 0.5
q
=50 A1
=40 A2
3
a b0.8 0.4
1 F 2 F
q
=50 A1 =40 A2
4
ab
F 1 F 2
1 0.5
q
=50 A1 =40 A2
5
a b
F 1 F 2
1.2 0.4
q
=50 A1 =40 A2
=40 A2
6
ba
1 F F 2
1.4 0.5
q
=50 A1 =40 A2
7
b a
2 F 1 F
0.8 0.4
q
=50 A1 =40 A28
a b
1 F
2 F
1 0.5
q
=50 A1 =40 A2
2.4-ð ÁÛð‡„
ÒÏ
2
ÒÏ2
Ï Ï
ÒÏ2
ÒÏ
2
Ï Ï
ÒÏ2
ÒÏ2
Ï Ï Ï Ï
ÒÏ2
ÒÏ2
ÏÏ
ÒÏ2ÒÏ2ÒÏ2 ÒÏ2 ÒÏ2
Ï
Ï
Ï Ï Ï Ï
ÒÏ2ÒÏ2ÒÏ2 ÒÏ2
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2.1ñð ı øÒÌ˝„Ú
ª „ º „ ‰ Î Ë È Ì
‰ Û „ ‡ ‡
ð
’ ˝ Ò „ ø ø ‰ Ë È Ì
Û ð Ú
, Ï
“ º ‚ Î º
ð Ò º Ì
ı ø ˜ , Í Õ
ª „ º „ ‰ Î Ë È Ì
‰ Û „ ‡ ‡
ð
’ ˝ Ò „ ø ø ‰ Ë È Ì
Û ð Ú
, Ï
“ º ‚ Î º
ð Ò º Ì
ı ø ˜
, Í Õ
‡ b 1 F 2 F ‡ b 1 F
2 F
1 0.40 0.80 60 180 11 0.62 0.36 120 100
2 0.50 0.46 120 80 12 0.64 0.40 140 100
3 0.80 0.40 80 140 13 0.74 0.48 40 120
4 0.42 0.60 100 140 14 0.54 0.68 180 200
5 0.52 0.42 60 120 15 0.40 0.64 60 40
6 0.78 0.50 120 90 16 0.36 0.80 140 100
7 0.30 0.80 90 100 17 0.50 0.40 50 160
8 0.42 0.62 120 140 18 0.80 0.80 180 100
9 0.60 0.30 140 80 19 0.56 0.46 60 160
10 0.70 0.50 100 120 20 0.60 0.74 130 100
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2. ÿËη˝ÌËÈ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ÒËÒÚÂÏËÈÌ ·Ó‰ÎÓ„Ó
F ı ø˜ÌËÈ øÈΘΠ̋Ή Óð¯Ë ı ı º¯øøÌ ̋ÎÂÏÂÌÚ (2.5ñð ÁÛð‡„) ı Ó∏𠄇̯Ëη˝˝ð ı Óηӄ‰ÒÓÌ ·º„ºº‰ „‡Ì„ËÈÌ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î 160][ =σ Ãœ‡,
ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î 210= R Ãœ‡ ·‡È ı ‡‰1. „‡Ì ¯Ëη˝Ìøø‰ ̋ ‰ øøÒ˝ı ‰ÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎ ;2. ¯Ëη˝Ìøø‰ËÈÌ Ú‡Î·‡ÈÌ º„º„‰ÒºÌ ı ‡ð¸ˆ‡‡„‡‡ð ¯Ëη˝Ìøø‰ËÈÌ
ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„ ÓÎ.ª„º„‰ÎËÈ„ 2.2ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
F
A b
cba
2A
1
b
2A
a b c
b
A
F
2
A
2A
a b c
b
F
3
2A
a b c
b
A
F
4
b
b
A
2A F
ca
5
F
A
b
cab
2A
6
b
F
A b
cba
2A
7
A
c a b
2A F
8
b
2.5ñð ÁÛð‡„
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ca
A F
b
2A
b
17
b ac
F
18
A 2A
b
F
bc a
19
2A b
F A
ac b
20
2.5ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
2.2ñð ı øÒÌ˝„Ú
¹ F ‡ b Ò
¹ F ‡ b –
ÍÕ Ï ÍÕ Ï
1 500 0.7 1.1 0.2 11 250 0.4 1.1 0.3
2 300 0.6 1.0 0.2 12 500 0.5 0.8 0.3
3 120 0.5 0.9 0.3 13 160 0.8 1.0 0.2
4 100 0.6 0.9 0.4 14 450 0.7 1.0 0.3
5 250 0.5 1.0 0.3 15 140 0.6 0.8 0.2
6 200 0.5 0.8 0.2 16 800 0.5 1.1 0.3
7 500 0.4 1.0 0.2 17 180 0.6 0.7 0.4
8 120 0.6 0.8 0.3 18 400 0.4 0.9 0.5
9 100 0.5 1.2 0.4 19 600 0.8 1.2 0.2
10 400 0.3 0.9 0.2 20 180 0.6 1.1 0.4
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3. ’‡‚Ú„‡ÈÌ ı ø˜‰ ̋ ÎÚ ÚºÎº‚ËÈÌ ·Ó‰ÎÓ„Ó
’‡ðËÏ ı ‡È „‡ÊËÊ ·ÛÈ ı ËȈËÈÌ ‡ ̨ ÛÎÚ‡È ˆ ̋ „Ú º˜øø ı˝Ì ·‡„‡ Ú˝„¯ ºÌˆº„Ú ̋ÎÂÏÂÌÚ (2.6-ð ÁÛð‡„) ‡‚‡‚. ›ÎÂÏÂÌÚËÈÌ Ú‡ÎÒÛÛ‰‡‰ ·ÓÒÓÓ, ı˝‚Ú˝˝ ı ø˜‰ ̋ Î
øÈΘËÎÊ ·‡È ı ‡‰ ÚÓÏ ˙∏Ó, „ð‡ÙËÍËÈÌ ‡ð„‡‡ð1. „ÓÎ ı ø˜‰ ̋ ÎËÈ„ ÓÎ ;2. „ÓΠڇη‡ÈÌ ·‡Èð·Î˚„ ÓÎ ;3. „ÓÎ ı ø˜‰ ̋ ÎËÈÌ ˜Ë„Î ̋Î, „ÓΠڇη‡ÈÌ ·‡Èð·Î˚„ ÁÛð‡„Ú Á‡‡ ;4. ı ‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎÊ ˜Ë„Î ̋ÎËÈ„ ÁÛð ;5. ̇ÎÛÛ ı ‡‚Ú„‡È‰ øÈÎ˜Î˝ı ·ÓÒÓÓ, ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ ÚÓ‰Óð ı ÓÈÎ ;6. x , y , z Ú˝Ì ı ν„ËÈÌ ˜Ë„Î ̋Ή øøÒ˝ı ı ‡ð¸ˆ‡Ì„ÛÈ ¯Û„‡Ï‡Ì ı˝‚ „‡ÊËÎÚ˚„
ÚÓ‰Óð ı ÓÈÎ ;7. ̋ÎÂÏÂÌÚËÈÌ ˝Á ̋Î ı øøÌËÈ ı ‡ð¸ˆ‡Ì„ÛÈ ººð˜ÎºÎÚËÈ„ ÓÎ.
ª„º„‰ÎËÈ„ 2.3ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
2.6ñð ÁÛð‡„
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2.1ñð ı øÒÌ˝„Ú
¹ x σ y σ yx τ α
¹ x σ y σ yx τ α
Ãœ‡ „ð‡‰ Ãœ‡ „ð‡‰
1 40 130 40 600 11 130 60 30 -600
2 70 100 80 300 12 80 70 60 450
3 30 120 40 -450 13 120 40 50 -300
4 80 50 70 300 14 150 30 40 600
5 90 40 60 300 15 100 20 60 300
6 40 140 40 450 16 130 50 40 -450
7 50 120 60 -300 17 40 120 60 -600
8 110 30 70 -600 18 120 70 20 600
9 60 40 60 600 19 110 50 40 450
10 90 40 60 -450 20 70 80 60 300
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√ ”—¿¬ƒ” √ ¿¿— ¡ØÀ›√
Œ√ “ÀŒÀ, ’ŒÀ√ ŒÀ“¤Õ “ŒŒ÷ŒŒ
3.1.¡ŒŒÀ“ŒÕ ’ŒÀ¡ŒŒ–Õ¤ “ŒŒ÷ŒŒ
ÃÂÚ‡ÎÎ ‰‡ÏÌÛÛð, ·‡„‡Ì‡ Á˝ð˝„ ı ËȈËÈÌ ̋ÎÂÏÂÌÚËÈÌ ı Ó∏ð ·ÓÎÓÌ ı ̋‰ ı ̋‰ ̋ Ì
ı˝Ò„ËÈ„ ı ÓÓðÓ̉ ̸ ı ÓÎ·Ó ı Ó‰ ·ÓÓÎÚÓÌ ı ÓηÓÓÒ˚„ ‡¯Ë„·‰‡„. ¡ÓÓÎÚÓÌ ı ÓηÓÓÒ˚Ì
·ÓÓÎÚÓ̉ Ó„ÚÎÓÎÚ ı Ó΄ÓÎÚ˚Ì ı˝‚ „‡ÊËÎÚÛÛ‰ øøÒ‰ ̋ „.
’ËȈËÈÌ ̋ÎÂÏÂÌÚøø‰ËÈ„ ı ÓÎ·Ó ı Ó‰ ı˝ÏÊ ̋ ˝„ ̋ ˝ð˝˝ Ú˝Ìˆøø ˜Ë„Î ̋ν˝ð˝˝ ˝Òð˝„
ı ÓÒ ı ø˜ ·ÓÓÎÚÓ̉ øÈΘËÎÊ ·ÓÓÎÚÌ˚ ¯ËηËÈ„ ‡b Ó„ÚÎÓÎÓÓð (3.1ñð ÁÛð‡„) Ó„ÚÓΉӄ.
√ Ûð‚‡Ì ı˝Ò„ËÈ„ ı ÓηÓÒÓÌ ÚÓ ı ËÓΉÓΉ ·ÓÓÎÚ˚„ ı Ó∏ð ı ‡‚Ú„‡È„‡‡ð (3.2ñð ÁÛð‡„)
Ó„ÚÓÎÌÓ.
3.1ñð ÁÛð‡„ 3.2ñð ÁÛð‡„
Œ„ÚÎÓÎ˚Ì ø‰ ·ÓÓÎÚÌ˚ ¯Ëη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ ÊË„‰ Ú‡ð ı Ò‡Ì ı˝‚Ú˝˝ ı ø˜‰ ̋ Î
øøÒÌ˝. ¡ÓÓÎÚÌ˚ Ó„ÚÎÓÎ˚„ Ú˝Ò‚˝ðνı ˜‡‰‚‡ð˚„ ‰‡ð‡‡ ı ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººð
¯‡Î„‡Ì‡.
[ ] s sd
mn
F τ≤
π⋅
=τ
4
2
(3.1)
Øø̉ : m ñ ·ÓÓÎÚÌ˚ Ó„ÚÎÓ„‰Ó ı ÚÓÓ
[ ] sτ ñ ·ÓÓÎÚÌ˚ χÚÂðˇÎ˚Ì Ó„ÚÎÓΉ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
n ñ ·ÓÓÎÚÌ˚ ÚÓÓ
d ñ ·ÓÓÎÚÌ˚ ¯Ëη˝ÌËÈ ‰Ë‡ÏÂÚð
F ı ø˜ÌËÈ øÈΘΠ̋ν˝ð ı ÓÎ·Ó ı ı˝Ò„ËÈÌ ı ‡Ì‡ ·ÓÓÎÚÌ˚ ¯Ëη˝ÌËÈ ¯øð„ ̋ Έ ̋ ı „‡‰‡ð„ÛÛ‰ ‰‡ð‡ÎÚ øøÒ˜ ı Ó΄ÓÎÚ˚Ì ı˝‚ „‡ÊËÎÚ ·ËÈ ·Ó΄ÓÌÓ. F ı ø˜ÌËÈ øÈΘΠ̋ν˝ð
ı Óηӄ‰ÓÊ ·‡È„‡‡ ̋‰ ‡Ì„ËÛ‰ ̊ Ì (2.3ñð ÁÛð‡„) ‰ ̋ ̋‰ ı˝Ò˝„ ̸ ·ÓÓÎÚ˚„ ·‡ðÛÛÌ, ‰ÓÓ‰
ı˝Ò˝„ ̸ ÁøøÌ ÚËȯ ¯‡ ı ̇.
a). ·). ‚).3.3ñð ÁÛð‡„
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33
ÿøð„ ̋ Έ ̋ ı „‡‰‡ð„ÛÛ„‡‡ð (3.3·ñð ÁÛð‡„) ı ø˜‰ ̋ Î ÊË„‰ ·Ë¯ Ú‡ð ı ‡ ı ·ÓÎÓ‚˜
Ôð‡ÍÚËÍÚ (3.3‚ñð ÁÛð‡„) „ÓΘ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È‰ ı ø˜‰ ̋ ÎËÈ„ øøÒÌ˝ „ ̋ Ê ÚÓÓˆ‰Ó„.
’Ó΄ÓÎÚ˚Ì ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎ ‰‡ð‡‡ ı ı ̋η˝ðÚ˝È ·‡È̇.
[ ]ÒÓÔÒÓÔnd
F
σ δ
σ ≤⋅⋅=min
( 3.2)
Øø̉ :
[ ]ÒÓÔ
σ ñ·ÓÓÎÚÌ˚ ı Ó΄ÓÎÚ˚Ì Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
n ñ ·ÓÓÎÚÌ˚ ÚÓÓ
d ñ ·ÓÓÎÚÌ˚ ¯Ëη˝ÌËÈ ‰Ë‡ÏÂÚð
minδ ñ ı ÛÛ‰Ò‡Ì ÚºÏðËÈÌ ÁÛÁ‡‡Ì
“ÓÏ ˙∏Ó̉ δ χÚÂðˇÎ˚Ì ÁÛÁ‡‡Ì˚„ ÚÓÓˆÓ ı ‰ÓÓ ı ø˜ÌËÈ Ì˝„ ˜Ë„Î ̋Ή ‡ÊËη ı
ÁÛÁ‡‡Ì˚ ÌËÈη˝ðËÈÌ ·‡„‡ ı˝ÏÊ ̋ ˝„ ‡‚‰‡„.
¡ÓÓÎÚÓÌ ı ÓηÓÓÒÌ˚ ø‰ ·ÓÓÎÚÌ˚ Ó„ÚÎÓÎÚ, ı Ó΄ÓÎÚ˚Ì ÚÓÓˆÓÓÌÓÓÒ „‡‰Ì‡
ı ÓηÓÊ ·‡È„‡‡ ̋ÎÂÏÂÌÚËÈÌ ÒÛÌ„‡ÎÚ˚Ì ÚÓÓˆÓÓ„ ı ËÈ ı ¯‡‡ð‰Î‡„‡Ú‡È. ”˜Ëð ̸
·ÓÓÎÚÓÌ ı ÓηÓÓÒ˚Ì ø‰ ·ÓÓÎÚÌÓÓÒ „‡‰Ì‡ ı ËȈËÈÌ ̋ÎÂÏÂÌÚËÈÌ ı˝Ò„øø‰ º„º‰ÒºÌ
‡˜‡‡Î‡Î‰ ̇ȉ‚‡ðÚ‡È ‡ÊËÎÎ‡Ê ·‡È ı ∏ÒÚÓÈ. ›ÎÂÏÂÌÚøø‰ËÈÌ ÒÛÌ„‡ÎÚ˚Ì ·‡Ú ·º ı ËÈÌ
̺ ı ̂ ÎËÈ„ ‰‡ð‡‡ ı ı ̋η˝ðÚ˝È ·Ë˜Ì˝.
[ ]t
nt A
F σ≤=σ ( 3.3)
Øø̉ : nt A ñ ̋ÎÂÏÂÌÚËÈÌ ı ÓηÓÓÒ ı ËÈ ı ı˝Ò„ËÈÌ ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È. ªºðººð ı ̋η ̋Î ·ÓÓÎÚÌ˚ Ìø ı ËÈ„ ÚÓÓˆÒÓÌ Ú‡Î·‡È
[ ]t σ ñ ’Óηӄ‰Ó ı ̋ÎÂÏÂÌÚ, ı˝Ò„ËÈ„ ı ËÈÒ˝Ì Ï‡ÚÂðˇÎ˚Ì ÒÛÌ„‡ÎÚ˚Ì
Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
∆ ˯˝˝ 3.1. ’ËȈ ̋ ‰ 470= F ÍÕ ‡˜‡‡ øÈΘËÎÌ˝. ’ËȈËÈÌ ı˝Ò„øø‰ËÈ„ 3.4ñð
ÁÛð‡„Ú øÁøøÎÒÌ˝˝ð ·ÓÓÎÚÓÌ ı ÓηÓÓÒÓÓð ı ÓÎ·Ó ı ∏ÒÚÓÈ ·‡ÈÊ ̋ ˝. ’ÓηÓÓÒÌ˚ 16 ÏÏ
„ÓΘÚÓÈ ·ÓÓÎÚ ‡¯Ë„Î‡Ì ı ËÈ ı ·ÓÎ ·ÓÓÎÚÌ˚ ÚÓÓ, ı ËȈËÈÌ ˆÛ‚ËχΠӄÚÎÓÎ˚Ì ·ÛÒ‡‰
ı˝ÏÊ ̋ ˝Òøø‰ËÈ„ ÚÓ‰Óð ı ÓÈθ∏.
3.4ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî34
›‰ ‡Ì„ËÛ‰ ̊ „ „‡Ì„‡‡ð ı ËÈÒ˝Ì ·º„ºº‰ ÒÛÌ„‡Îڇ̉ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
[ ] 160=σt Ãœ‡, Ó„ÚÎÓÎÚÓ̉ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î [ ] 120=τ s Ãœ‡, ı Ó΄ÓÎÚÓ̉
Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î [ ] 250=ÒÓÔ
σ Ãœ‡ ·‡È̇.
¡Ó‰ÓÎÚ. (3.1) ÚÓÏ ˙∏ÓÌÓÓÒ ı ËȈ ̋ ‰ ¯‡‡ð‰Î‡„‡Ú‡È ·ÓÓÎÚÌ˚ ÚÓÓ„ ÚÓ‰Óð ı ÓÈηÓÎ
[ ]7.9
101204
016.014.32
10470
4
62
3
2 =
⋅⋅
⋅=
τπ
≥
s
d m
F n
10=n ·ÓÎÌÓ. ›Ì‰ ·ÓÓÎÚ ı Ó∏ð ı ‡‚Ú„‡È„‡‡ð Ó„ÚÎÓ„‰Ó ı ÚÛÎ 2=m „ ̋ Ê ‡‚Ò‡Ì.
(3.2) ÚÓÏ ˙∏ÓÌÓÓÒ F ı ø˜ÌËÈ Ì˝„ ˜Ë„Î ̋Ή ‡ÊËη ı ˝Ì‰ ‡Ì„ËÈÌ ı ‡Ï„ËÈÌ ·‡„‡
ÁÛÁ‡‡Ì˚„ ÓηÓÎ :
[ ] 01175.0016.01010250
104706
3
min =⋅⋅⋅
⋅=⋅⋅≥
ÒÓÔd n
F
σ δ Ï=11,75Ï
’ËȈ ̋ ‰ ı Ó∏𠯂ÂÎÎÂ𠷇Ȅ‡‡ ÚÛÎ ¯‚ÂÎÎÂðËÈÌ ÁÛÁ‡‡Ì ̸ 87.52min =
δÏÏ Óð˜ËÏ
·‡È̇. ØøÌËÈ„ ‡Ì ı ‡‡ð˜ ÒÓðÚ‡ÏÂÌÚ˝˝Ò ¯‚ÂÎÎÂðËÈÌ ı ‡Ì‡Ì˚ ÁÛÁ‡‡Ì ̸ =1δ 6 ÏÏ
·‡È ı 27 ÌÓÏÂðËÈÌ ¯‚ÂÎÎÂðËÈ„ ÒÓÌ„ÓÊ ‡‚ ˙ˇ. ÿ‚ÂÎÎÂðËÈÌ Ú‡Î·‡È 2.35= A ÒÏ2.
ÃºÌ 1 ı ÛÛ‰Ò‡Ì ÚºÏðËÈÌ ÁÛÁ‡‡Ì˚„ 12=δ ÏÏ „ ̋ Ê Òڇ̉‡ðÚ‡‡ð ÒÓÌ„ÓÊ ‡‚ ˙ˇ. 3.5ñð
ÁÛð‡„Ú øÁøøÎÒÌ˝˝ð ·ÓÓÎÚÛÛ‰ ̊ „ ·‡ÈðÎÛÛÎ̇ „ ̋ ‚ ̋Î ı ÛÛ‰Ò‡Ì ÚºÏðËÈÌ ı ‡Ï„ËÈÌ Ë ı
‡˜‡‡Î‡Î‰ ‡ÊËη ı
ڇη‡È ( )t t b n d b n d δ δ δ = ⋅ − ⋅ ⋅ = −
·ÓÎÓ ı˚
„ ‡Ì ı
‡‡ð˜ (3.3) ÚÓÏ ˙∏Ó„ÓÓð 1 ı ÛÛ‰Ò‡Ì ÚºÏðËÈÌ b ºð„ºÌËÈ„ ÚÓ‰Óð ı ÓÈÎÊ º„º„‰ÒºÌ ‡˜‡‡Î‡Î‰
¯‚ÂÎÎÂðËÈÌ Ì‡È‰‚‡ðÚ‡È ‡ÊËη ı ˝Ò˝ı ËÈ„ ¯‡Î„‡ ˇ.
t
t
F b n d
δ σ ≥ − ⋅ =
⎡ ⎤⎣ ⎦
=⋅−⋅⋅
⋅= 016.03
10160012.0
104706
3
196.0= Ï
20=b ÒÏ „ ̋ Ê ÒÓÌ„Ó∏.ÿ‚ÂÎÎÂðËÈÌ ·Ó‰ËÚ Ú‡Î·‡È
( ) ( ) 34
1 1046.6016.0006.03102.3522 −− ⋅=⋅⋅−⋅=⋅⋅−= d n A At nt
δ Ï2
3t
n = ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓÎ ‰‡ ı ̧ ı ‡‰‡‡ÒÌ˚ ÚÓÓ
ÿ‚ÂÎÎÂðÚ øøÒ˝ı ı ø˜‰ ̋ Î
6
3
3
1075.721046.6
10470⋅=
⋅
⋅==
−nt
A
F σ œ‡=72.75Ãœ‡
=σ 72.75 Ãœ‡ [ ] 160=σ< t Ãœ‡ ÚÛÎ ¯‚ÂÎÎÂ𠇘‡‡Î‡Î‰ ̇ȉ‚‡ðÚ‡È
‡ÊËη̇.
3.5ñð ÁÛð‡„
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3.2. ÿ»—››À“›Õ ’ŒÀ¡ŒŒ–Õ¤ “ŒŒ÷ŒŒ
ÿËð˝ ̋ÎÚ˝Ì ı ÓηÓÓÒ Ì¸ Á‡‰‡ð‰‡„„øÈ ı ÓηÓÓÒÌ˚ Ì˝„ ÚºðºÎ ˛Ï. ÿËð˝ ̋ÎÚ˝Ì
ı ÓηÓÓÒ˚„ ̋‰ ‡Ì„ËÛ‰ ̊ „ ı ÓηÓÊ ¯Ëð˝˝ı ·‡È‰Î‡‡ð ̸ ÚÛ΄‡Ê, ÁºðøøÎÊ (3.6·,‚,„ñð
ÁÛð‡„) ¯Ëð˝˝ı „ ̋ Ê ‡Ì„ËÎ̇. ›‰ ‡Ì„ËÛ‰ ̊ „ ÁºðøøÎÊ ¯Ëð˝ ̋ÎÚ ı ËÈ ı ø‰ ¯Ëð˝ ̋ÎÚÌËÈ
Ó∏ ‰Î˚Ì ·‡Èð··‡Ò ̸ ı ‡Ï‡‡ðÛÛÎÊ ı ‡ÊÛÛ„ËÈÌ, ·Û·̄ËÈÌ, Ú‡¯ÛÛ ¯Ëð˝ ̋ÎÚÚ˝È
(3.6·,‚,„ñð ÁÛð‡„) „ ̋ Ê ‡Ì„ËÎ̇.
3.6ñð ÁÛð‡„
“Û΄‡Ê ¯Ëð˝ ̋ÎÚ ı ËÈ ı ø‰ Ó∏ ‰Î˚Ì ·‡Ú ·˝ı ËÈ„ ÒÛÌ„‡ÎÚ ¯‡ ı ‡ÎÚ˚Ì ·‡Ú ·˝ı ËÈÌ
̺ ı ̂ κºð ÚÓÓˆÌÓ.
[ ]σ≤δ⋅
==σl
F
A
F ( 3.4)
Øø̉ : δ - ̋ÎÂÏÂÌÚËÈÌ ÁÛÁ‡‡Ì ( Ó∏ ‰Î˚Ì ºð„ºÌ )
l - Ó∏ ‰Î˚Ì ÛðÚ
b - ̋ÎÂÏÂÌÚËÈÌ ºð„ºÌ
σ - ›ÎÂÚðÓ‰ ̊ Ì Ï‡ÚÂðˇÎ˚Ì Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
›‰ ‡Ì„Ë ̋ÎÂÏÂÌÚøø‰ËÈ„ ÁºðøøÎÊ ¯Ëð˝ ̋ÎÚ ı ËÈ ı ̋‰ ¯Ëð˝ ̋ÎÚËÈÌ Ó∏ ‰ÓΠ̸
·ËÒÒÂÍÚðËÒÒ˝˝ð ̸ ‰‡Èð‡ ı ı ‡‚Ú„‡È„‡‡ð ˝‚‰ ̋ ðÌ˝ „ ̋ Ê øÁ‰ ̋ „. (3.7· ñð ÁÛð‡„)
«ºðøøÎÊ ¯Ëð˝ ̋ÎÚ ı ËÈ ı øÂËÈÌ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎ ̸ ‰‡ð‡‡ ı ı ̋η˝ðÚ˝È
[ ] ˝
hl
F
A
F τ≤
⋅⋅==τ
7.0 ( 3.5)
Øø̉ : l - Ó∏ ‰Î˚Ì ÌËÈÚ ÛðÚ
h - Ó∏ ‰Î˚Ì ºÌ‰ºð
‡)
·)
‚)
„)
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî36
[ ] ˝τ - ̋ÎÂÍÚðÓ‰ ̊ Ì Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
Œ∏ ‰Î˚Ì ÚÓÓˆÓÓÌ˚ ÛðÚ
[ ] ˝h
F
l τ⋅⋅≥ 7.0 ( 3.5)
œð‡ÍÚËÍÚ Ó∏ ‰Î˚Ì ÛðÚ˚„ ÚÓÓˆÓ ı ‰ÓÓ ¯Ëð˝ ̋ÎÚËÈÌ ø‰ ı ÓÓÒÓÌ Ó∏ ‰ÓÎ Ú‡‚Ë„‰‰‡„ ̊ „
ı ‡ð„‡ÎÁ‡Ì ÚÓÓˆÓÓÌ˚ ÛðÚ ‰ ̋ ˝ð 10 ÏÏ Ì˝ÏÊ ÚÓÓˆ‰Ó„. ›‰ ‡Ì„ËÛ‰ ̊ „ ÁºðøøÎÊ ¯Ëð˝ ̋ÎÚ
ı ËÈ ı ‰ ̋ ˝ ı ‡ÊÛÛ„ ̊ Ì ·Û·̄ ̊ Ì (3.7ñð ÁÛð‡„), ı ‡ÊÛÛ„ ̊ Ì Ú‡¯ÛÛ Ó∏ ‰Î˚Ì ı ‡ÏÚ ı ËÈÊ
¯Ëð˝ ̋ÎÚËÈ„ ı ËÈÊ ·ÓΉӄ .
3.7ñð ÁÛð‡„
3.7ñð ÁÛð‡„Ú ÁÛð‡„‰Ò‡Ì ¯Ëð˝ ̋ÎÚ˝Ì ı ÓηÓÓÒ˚Ì Ó∏ ‰Î˚Ì ÌËÈÚ ÛðÚ Ì¸ bl L += 2 ·‡È̇.
∆ ˯˝˝ 3.2. 3.8ñð ÁÛð‡„Ú øÁøøÎÒÌ˝˝ð ı º‰ºÎ„ººÌ„øÈ ·˝ı ̋ÎÒ˝Ì ı ÛÛ‰Ò‡Ì
ںϺðÚ˝È ‡‰ËÎ ·ÛÒ Ú‡ÎÚ 14 × 9×0.8 ·ÛÎ‡Ì ÚºÏðËÈ„ ¯Ëð˝ ̋ÎÚ˝Ì ı ÓηÓÓÒÓÓð ı ÓÎ·Ó ı
∏ÒÚÓÈ ·‡ÈÊ ̋ ˝. ’ËȈ ̋ ‰ 300= F ÍÕ ı ø˜ øÈÎ˜Î˝ı ·º„ºº‰ Ó∏ ‰Î˚Ì ºÌ‰ºð ̸ Ú· d h .= ,
̋ÎÂÍÚðÓ‰ ̊ Ì Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î [ ] 100=τ ˝
Ãœ‡ ·ÓÎ Ó∏ ‰Î˚Ì ÌËÈÚ ÛðÚ˚„
ÚÓ‰Óð ı ÓÈθ∏.
3.8ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ¿‰ËÎ ı ‡ÊÛÛÚ ·Ë¯ ·ÛÎ‡Ì ÚºÏðËÈÌ ÁÛÁ‡‡Ì 8.0=d ÒÏ ÚÛÎ Ó∏ ‰Î˚Ì ºÌ‰ºð
8.0=h ÒÏ 008.0= Ï ·‡È̇. 3.5′ ÚÓÏ ˙∏Ó„ÓÓð Ó∏ ‰Î˚Ì ÚÓÓˆÓÓÌ˚ ÛðÚ
[ ]535.0
10100008.07.0
10300
7.0 6
3
=
⋅⋅⋅
⋅=
τ≥
˝h
F l Ï
Œ∏ ‰Î˚Ì ÌËÈÚ ÛðÚ 545.001.0 =+= l L Ï 5.54= ÒÏ .
a )
· )
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37
Œ∏ ‰Î˚„ ·ÛÎ‡Ì ÚºÏðËÈÌ 2 ڇ·‡ð ı ËÈÌ˝ „ ̋ ‚ ̋Î Ì˝„ ڇΉ ı ËÈ„‰ ̋ ı Ó∏ ‰Î˚Ì ÛðÚ
25.272:1 == L l ÒÏ.
ƒ‡‡Î„‡‚‡ðπ2. ƒ‡‡Î„‡‚‡ð
ı
Ó∏
ð ·Ó‰ÎÓ„ÓÚÓÈ.
1. ¡ÓÓÎÚÓÌ ı ÓηÓÓÒ˚Ì ÚÓÓˆÓÓ
F ı ø˜ÌËÈ øÈΘΠ̋Ή ·ÓÓÎÚÓÌ ı ÓηÓÓÒÚ (3.9ñð ÁÛð‡„) ·‡È ı ¯‡‡ð‰Î‡„‡Ú‡È
ı˝ÏÊ ̋ ˝Ìøø‰ËÈ„ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººÒ 3.1ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ÚÓ‰Óð ı ÓÈÎ. ›‰
‡Ì„ËÛ‰ ̊ „ „‡Ì„‡‡ð ı ËÈÒ˝Ì ·º„ºº‰ ÒÛÌ„‡ÎÚ ¯‡ ı ‡Îڇ̉ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
[ ] 160=σt Ãœ‡, Ó„ÚÎÓΉ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î [ ] 120=τ s Ãœ‡, ı Ó΄ÓÎÚÓ̉
Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î 250=σ p Ãœ‡ ·ÓÎÌÓ.
1.
¡ÓÓÎÚÓÌ ı ÓηÓÓÒÓ̉ ¯‡‡ð‰Î‡„‡Ú‡È ·ÓÓÎÚÌ˚ ÚÓÓ„ ÚÓ‰Óð ı ÓÈÎÓ ı
2.
–ڇ̉‡ðÚ˚Ì ˆÛ‚ËχΠӄÚÎÓÎ, ı ÛÛ‰Ò‡Ì ·ÓÎÓÌ ı ÓηÓÓÒ ÚºÏðøø‰ËÈÌ ÁÛÁ‡‡Ì˚„ÒÓÌ„Ó ı . ’ÛÛ‰Ò‡Ì ÚºÏðËÈÌ ÁÛÁ‡‡Ì Òڇ̉‡ðÚ‡‡ð =δ 4; 5; 6; 7; 8; 9; 10; 11; 12;
14; 16; 18; 20; 22; 25; 28; 30 ÏÏ ·‡È̇.
3.
’ÓηÓÓÒÓ̉ ‡¯Ë„· ı ·ÓÓÎÚÛÛ‰ ̊ Ì ·‡Èð·Î˚„ ÒÓÌ„ÓÊ ˆÛ‚ËχΠӄÚÎÓÎ˚Ì ·‡Ú
·˝ı ËÈ„ ¯‡Î„‡ ı ¡ÓÓÎÚÌ˚ ·‡Èð·Î˚„ ÒÓÌ„ÓÊ ‡‚‡ ı ‰‡‡ ı ‡Ï„ËÈÌ Ë ı ̸ ˆÛ‚ËχÎ
Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„ 10% , ı ÛÛ‰Ò‡Ì ÚºÏðËÈÌ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„ 20%
·ÛÛðÛÛ· ı ‡‡ð ÚÓÓˆÓÊ ‡‚.
4.
’ÛÛ‰Ò‡Ì ÚºÏºð ˝Ò‚ ̋Î ı ÓηÓÓÒ ÚºÏðËÈÌ ºð„ºÌËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı
5.
¡ÓÓÎÚÌ˚ ·‡Èð·Π̸ ı ‡Ï„ËÈÌ ·‡„‡ ̸ ̋‰ ‡Ì„ËÈÌ Á‡ ı ‡‡Ò 1.5d Á‡È‰
·ÓÓÎÚÌÛÛ‰ ̊ Ì Úº‚ ı ÓÓðÓ̉ ̊ Ì Á‡È„ 3d ñ˝˝Ò ‰ ̋ ˝¯ Á‡È‰ ·‡È ı ‡‡ð ·Ó‰ÓÊ
ı ÓηÓÓÒ˚Ì ÁÛð„ËÈ„ χүڇ·Î‡Ì ÁÛð‡ ı
3.9ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî38
3.9ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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39
3.9ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
3.1ñð ı øÒÌ˝„Ú
¹ F ,
ÍÕ
d ,
ÏÏ
¹ F ,
ÍÕ
d ,
ÏÏ
¹ F ,
ÍÕ
d ,
ÏÏ
¹ F ,
ÍÕ
d ,
ÏÏ1 470 16 6 320 27 11 360 27 16 410 24
2 360 18 7 400 30 12 400 42 17 380 16
3 410 22 8 500 48 13 470 30 18 440 30
4 350 20 9 320 20 14 380 22 19 340 18
5 350 24 10 280 24 15 360 20 20 460 42
2. ÿËð˝ ̋ÎÚ ̋ Ì ı ÓηÓÓÒ˚Ì ÚÓÓˆÓÓ
– ı ÂÏøø‰ ̋ ˝ð (3.10ñð ÁÛð‡„) º„º‰ÒºÌ ¯Ëð˝ ̋ÎÚ˝Ì ı ÓηÓÓÒ˚Ì Ó∏ ‰Î˚Ì ÛðÚ˚„ 3.2ñð
ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ÚÓ‰Óð ı ÓÈÎÓ ı . ÕËÈÚ Ò ı ÂÏËÈÌ ı Û‚¸‰ ̋ÎÂÍÚðÓ‰ ̊ Ì
Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î [ ] 100=τ ˝
Ãœ‡, Ó∏ ‰Î˚Ì ºÌ‰ðËÈ„ ÁÛð‡„Ú º„Ò̺ºð ÚÓÓˆÌÓ.
1.
ª„º„‰ÎººÒ Ò ı Âω ÁÛð‡„‰Ò‡Ì ˆÛ‚ËχÎ, ı ÛÛ‰Ò‡Ì ÚºÏºð, ı ÓηÓÓÒ ÚºÏðËÈÌ
ı˝ÏÊ ̋ ˝Òøø‰Ëȯ ̌΄‡Ì ‡‚̇.
2.
Œ∏ ‰Î˚Ì ÚÓÓˆÓÓÌ˚ ·ÓÎÓÌ ·Ó‰ËÚ ÛðÚ˚„ ÓÎÌÓ.
3.
Œ∏ ‰Î˚Ì ·‡Èð¯Î‡‡Ò ı ‡Ï‡‡ðÛÛÎÊ Ó∏ ‰Î˚Ì ÛðÚ˚„ ı ۂˇð·̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî40
3.10ñð ÁÛð‡„
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41
3.10ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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3.10ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
3.2ñð ı øÒÌ˝„Ú
¹
’ ̋ ÏÊ ̋ ˝Òøø‰
ÿ‚ÂÎÎÂðËÈÌ
‰Û„‡‡ð
ƒ‚ÛÚ‡‚ð˚Ì
‰Û„‡‡ð
F,
ÍÕ
’ÛÛ‰Ò‡Ì
ںϺð ¡ÛÎ‡Ì ÚºÏºð
b δ
1 140 16 16 × 10×0.9 18 20 246
2 150 20 14 × 9×0.8 18 22 300
3 130 14 12.5 × 8×1 14‡ 20 200
4 140 14 14 × 9×1 16 20‡ 250
5 150 18 16 × 10×1 18 22 350
6 160 16 14 × 9×1 20 20 250
7 140 12 16 × 10×1 18‡ 27 320
8 150 14 12.5 × 8×1.2 20 18 2809 140 12 16 × 10×1 20a 20 300
10 160 14 14× 9×1 18 18 260
11 140 12 16 ×10×1.2 18a 20 280
12 160 16 18× 11×1 20 22 300
13 160 14 16 × 10×1.2 20‡ 24 320
14 150 12 18× 11×1 20a 20 250
15 140 10 16 × 10×1 18 18 250
16 150 12 16 × 10×1.2 20 22 240
17 150 12 18 × 11×1.2 18a 20 300
18 160 14 16 × 10×0.9 18 22 250
19 140 12 12.5 × 8×1 20 18 260
20 150 12 16 × 10×1 20‡ 20 280
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ƒª—ª¬ƒØ √ ››— ¡ØÀ›√
’¿¬“ √ ¿… ƒØ—–»…Õ √≈ŒÃ≈ “— ÿ»Õ ∆
4.1. ŒÌÓÎ˚Ì ÚÓ‚˜ ø̉ ˝Òν ÎÿËη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ( 4.1ñð ÁÛð‡„)
∫= A
x ydAS ∫= A
y xdAS (4.1)
ËÌÚ„ð‡ÎÛÛ‰ ̊ „ Œ’, ŒØ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÒÚ‡ÚËÍ ÏÓÏÂÌÚ „ ̋ Ê Ì˝ðÎ˝Ì˝.ÕËÈÎÏ ̋Î ‰øðÒËÈÌ ˇÏ‡ð Ì˝„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÒÚ‡ÚËÍ ÏÓÏÂÌÚ Ì¸
‰øðÒËÈÌ ·øð‰øøÎÊ ·‡È„‡‡ ı ̌η‡ð ‰øðÒ ÚÛÒ ·øðËÈÌ ÚÛ ı ‡ÈÌ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÒÚ‡ÚËÍ ÏÓÏÂÌÚÛÛ‰ ̊ Ì ‡Î„·ð ÌËÈη˝ðÚ˝È Ú˝Ìˆøø.
∑= ⋅=+++=
n
iii x x x x y AS S S S n
1...21
(4.2)
∑=
⋅=+++=n
iii y y y y x AS S S S
n1
...21
Œ„ÚÎÓÎ˚Ì Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ·‡Èð··‡Ò ı ‡Ï‡‡ð˜ ÒÚ‡ÚËÍ ÏÓÏÂÌÚ˝Òð˝„, Òºðº„, Ú˝„ ÛÚ„‡Ú‡È ·‡È‰‡„. Œ„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈ„ ‰‡ÈðÒ‡Ì Ú˝Ì ı ν„ËÈ„ Úº‚ Ú˝Ì ı ν„ „ ̋ Ì˝. “º‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ Ó„ÚÎÓÎ˚Ì ÒÚ‡ÚËÍÏÓÏÂÌÚ Ú˝„Ú˝È Ú˝Ìˆøø ·‡È‰‡„.
0=⋅= c x
y AS 0=⋅= c y
x AS ( 4.3)
( 4.3) ÚÓÏ ˙∏ÓÌÓÓÒ ı ‡‚Ú„‡È Ó„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰Ë̇ÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈΉӄ.
∑
∑
∑
∑
=
=
=
=⋅
==⋅
==n
ii
n
iii
x cn
ii
n
iii
y c
A
y A
A
S y
A
x A
A
S x
1
1
1
1 (4.4)
ÓÓð‰Ë̇Ú˚Ì ÒËÒÚÂÏËÈÌ ÚÛ ı ‡ÈÌ Ú˝Ì ı ν„ ̋ ˝Ò (4.1ñð ÁÛð‡„) Ó„ÚÎÓÎ ˚Ì ̋ÎÂÏÂÌÚ‡ð ڇη‡È dA ı øðÚ ̋Î ı Á‡ÈÌÍ‚‡‰ð‡Ú˚„ ̋ÎÂÏÂÌÚ‡ð ڇη‡È„‡‡ð øðÊøøÎÊ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„‡‡ð‡‚Ò‡Ì ËÌÚ„ð‡Î˚„ ı ‡‚Ú„‡ÈÓ„ÚÎÓÎ˚Ì Ú˝ð Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ „ ̋ Ì˝.
4.1ñð ÁÛð‡„
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∫∫ == A
y A
x dA x I dA y I 22 (4.5)
Œ„ÚÎÓÎ˚Ì ̋ÎÂÏÂÌÚ‡ð ڇη‡È dA ÍÓÓð‰Ë̇Ú˚Ì ÚÛÈÎ ˆ ̋ „ ̋ ˝Ò Ó„ÚÎÓÎ˚Ì
º˜øø ı˝Ì ·‡„‡ ڇη‡È ı øðÚ ̋Î ı Á‡ÈÌ Í‚‡‰ð‡Ú‡‡ð øðÊøøÎÊ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„‡‡ð ‡‚Ò‡Ì ËÌÚ„ð‡Î˚„ Ó„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ ˆ ̋ „Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÚÛÈÎ˚ÌËÌÂðˆËÈÌ ÏÓÏÂÌÚ „ ̋ Ì˝.
∫ρ=ρ A
dA I 2 (4.6)
Œ„ÚÎÓÎ˚Ì ̋ÎÂÏÂÌÚ‡ð dA ڇη‡È„ ÚøøÌ˝˝Ò ÍÓÓð‰Ë̇Ú˚Ì ’; Ø ı øðÚ ̋Î ı Á‡ÈÌÛÛ‰‡‡ð øðÊøøÎÊ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„‡‡ð ‡‚Ò‡Ì ËÌÚ„ð‡Î˚„ Úº‚ººÒ ÁÛ„Ú‡‡ ı ËÌÂðˆËÈÌ ÏÓÏÂÌÚ „ ̋ Ì˝.
∫= A xy xydA D (4.7)
»ÌÂðˆËÈÌ Ú˝Ì ı ν„ËÈÌ ·‡ ÚÛÈÎ˚Ì ÏÓÏÂÌÚÛÛ‰ ˇÏ‡„Ú ˝Âð˝„, ı ‡ðËÌ Úº‚ººÒÁÛ„Ú‡‡ ı ËÌÂðˆËÈÌ ÏÓÏÂÌÚ ˝Âð˝„ , Òºðº„, Ú˝„ ÛÚ„‡Ú‡È ·‡ÈÊ ·ÓÎÌÓ.
–Ú‡ÚËÍ ÏÓÏÂÌÚ Ì¸ ÛðÚ˚Ì Ì˝„ÊËÈÌ ÍÛ· Á˝ð˝„Ú (ÒÏ3 , Ï3 ), ËÌÂðˆËÈÌÏÓÏÂÌÚÛÛ‰ ÛðÚ˚Ì Ì˝„ÊËÈÌ ‰ºð‚ºÌ (ÒÏ4 , Ï4 ) Á˝ð˝„Ú Ì˝„ÊÚ˝È ·‡È̇.
ÕËÈÎÏ ̋Î ‰øðÒËÈ„ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̸ ı ̌η‡ð ÚÓ‰Óð ı ÓÈÎÓ„‰Ó ı ı˝Ò„øø‰ ̋ ‰ ı Û‚‡‡Ê Ú ̋‰„ ̋ ˝ð ı ̌η‡ð ‰øðÒøø‰ËÈÌ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ ̇΄·ð ÌËÈη˝ð˝˝ð ËÌÂðˆËÈÌ ÏÓÏÂÌÚËÈ„ ̸ ÚÓ‰Óð ı ÓÈΉӄ.
∑=
=+++=n
iin I I I I I
121 ... (4.8)
“ÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ Ì¸ ÍÓÓð‰Ë̇Ú˚Ì Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ Ì ÌËÈη˝ðÚ˝È Ú˝Ìˆøø.
y x I I I +=ρ (4.9)
’‡‚Ú„‡È Ó„ÚÎÓÎ˚Ì Ô‡ð‡ÎÎÂθ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌÏÓÏÂÌÚÛÛ‰ (4.2ñð ÁÛð‡„ ) ‰‡ð‡‡ ı ı ‡Ï‡‡ð‡ÎÚ‡È ·‡È‰‡„.
ba A D D
b A I I
a A I I
C C
C
C
y x y x
y y
x x
⋅⋅+=
⋅+=
⋅+=
11
1
1
2
2
(4.10)
4.2ñð ÁÛð‡„
Øø̉ ‡ ̸ Ô‡ð‡ÎÎÂθ ı˝
‚Ú˝˝
Ú˝
Ì ı
ν
„øø‰ËÈÌ , b ̸ Ô‡ð‡ÎÎÂθ ·ÓÒÓÓ Ú˝Ì ı ν„øø‰ËÈÌ ı ÓÓðÓ̉ ̊ Ì Á‡È ·º„ºº‰ ‰ ̋ ˝ð ı ÚÓÏ ˙∏Ó̉ ÓðÎÛÛ· ı ‰‡‡
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45
̋‰„ ̋ ˝ðËÈ„ ÁÓ ı Ë ı Ú˝Ï‰ ̋ „Ú˝È Ì¸ ‡‚̇. 4.10 ÚÓÏ ˙∏ÓÌÓÓÒ Ó„ÚÎÓÎ˚Ì Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÓηÓÎ :
⎪⎪⎭
⎪⎪⎬
⎫
⋅−=
−=
−=
bAa D D
Ab I I
Aa I I
y x y x
y y
x x
C C
C
C
11
1
1
2
2
(4.10`)
’‡‚Ú„‡È Ó„ÚÎÓÎ˚Ì ’,Ø Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰Ï ̋‰ ̋ „‰ ̋ Ê ·‡È ı ‡‰ ’,Ø Ú˝Ì ı ν„ËÈ„ α ºÌˆ„ººð ˝ð„øøνı ̋‰ øøÒ˝ı 11 ,Y X
Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ Ó„ÚÎÓÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ‰‡ð‡‡ ı ÚÓÏ ˙∏ÓÌÛÛ‰‡‡ð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ.
α α
α α α
α α α
2cos2sin2
2sincossin
2sinsincos
11
1
1
22
22
xy y x
y x
xy y x y
xy y x x
D I I
D
D I I I
D I I I
+−
=
−+=
−+=
(4.11)
4.3ñð ÁÛð‡„»ÌÂðˆËÈÌ Úº‚ººÒ ÁÛ„Ú‡‡ ı ÏÓÏÂÌÚ Ì¸ Ú˝„Ú˝È Ú˝Ìˆøø ·‡È ı Ú˝Ì ı ν„øø‰ËÈ„
ËÌÂðˆËÈÌ „ÓÎ Ú˝Ì ı ν„ „ ̋ Ì˝. Œ„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈ„ ‰‡ÈðÒ‡Ì „ÓÎ Ú˝Ì ı ν„ËÈ„ Úº‚ „ÓÎ Ú˝Ì ı ν„ „ ̋ Ê Ì˝ðÎ ̋‰ ̋ „. Œ„ÚÎÓÎ˚Ì Úº‚ Ú˝Ì ı ν„ËÈ„
C C Y X , „ÓÎ Ú˝Ì ı ν„ËÈ„ U, V øÒ„ ̋ ˝ð Ú˝Ï‰ ̋ „νÂ. C C Y X , Ú˝Ì ı ν„øø‰ËÈ„ „ÓÎ
Ú˝Ì ı ν„øø‰Ú˝È ‰‡‚ ı ̂ ÛÛ· ı˚Ì ÚÛΉ ˝ð„øøνı 0α ºÌˆ„ËÈ„
C C
C C
C C
C C
x y
y x
y x
y x
I I
D
I I
Dtg
−=
−−=α
222 0 (4.12)
„ ̋ Ê ÚÓ‰Óð ı ÓÈΉӄ. C X Ú˝Ì ı ν„ËÈ„ ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„
˜Ë„Î ̋Ή ˝ð„øøνı 0α ºÌˆ„ËÈ„ ˝Âð˝„ ÛÚ„‡Ú‡È ‡‚̇.
»ÌÂðˆËÈÌ Úº‚ „ÓÎ ÏÓÏÂÌÚ˚„ ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ.
2
2
min
max22 C C
C C C C
y x y x y x
D I I I I
I +⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −±
+= (4.13)
»ÌÂðˆËÈÌ „ÓÎ U, V Ú˝Ì ı ν„øø‰ËÈ„ º„º„‰ÒºÌ Ú˝Ì ı ν„ „ ̋ Ê øÁ‚ ̋Î Ú ̋‰„ ̋ ˝ðËÈ„ ı ‡ð„‡ÎÁ‡ ı Úº‚ººÒ ÁÛ„‡Ú‡‡ ı ÏÓÏÂÌÚ 0=UV D ·ÓÎÓ ı Û˜Ëð
˝ð„øøÎ ̋ÎÚËÈÌ (IV.11) ÚÓÏ ˙∏Ó ‰‡ð‡‡ ı ·‡È‰Î‡‡ð ı ̌η‡ð˜Î‡„‰‡Ì‡.
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α−=
α+α=
α+α=
2sin2
cossin
sincos
22
22
V U xy
V U y
V U x
I I D
I I I
I I I
(4.14)
xy y x D I I ,, ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰,
minmax , I I „ÓÎ ÏÓÏÂÌÚÛÛ‰ ̊ Ì
ı ÓηÓÓ„ 4.4 ÁÛð‡„Ú ‰øðÒ ̋ÎÒÌ˝˝ðËÌÂðˆËÈÌ ‰Û„ÛÈ ·Û ̨ Û ÃÓð˚Ì ‰Û„ÛÈ„‡‡ð ı ‡ðÛÛÎÊ ·ÓΉӄ.
»ÌÂðˆËÈÌ ‰Û„ÛÈ„ ‡¯Ë„Î‡Ì Ó„ÚÎÓÎ˚ÌËÌÂðˆËÈÌ „ÓÎ Ú˝Ì ı ν„øø‰ËÈÌ ˜Ë„Î ̋Î,„ÓÎ ÏÓÏÂÌÚÛÛ‰ ̊ Ì ı˝ÏÊ ̋ ˝
V U I I , Ï ̋‰ ̋ „‰ ̋ Ê ·‡È ı ‡‰ ˇÏ‡ð Ì˝„ ’,
Ø Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ xy y x D I I ,,
ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎÓ ı ‡ð„ ̊ „¯ÛÛ‰ ·Ó‰ÎÓ„Ó „ ̋ Ì˝. œÂðÔẨËÍÛÎ ˇð
’, Ø Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ xy y x D I I ,, ÏÓÏÂÌÚÛÛ‰ Ï ̋‰ ̋ „‰ ̋ Ê
·‡È ı ‡‰ ËÌÂðˆËÈÌ „ÓÎ ÏÓÏÂÌÚ , „ÓÎ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ËÌÂðˆËÈÌ ‰Û„Ûȇ¯Ë„Î‡Ì ÚÓ‰Óð ı ÓÈÎÊ ·‡È‚‡Î Ûð‚ÛÛ ·Ó‰ÎÓ„Ó „ ̋ Ì˝. ”ð‚ÛÛ ·Ó‰ÎÓ„Ó Ì¸Ôð‡ÍÚËÍËÈÌ ı Û‚¸‰ ËÎøø ‡˜ ı Óηӄ‰ÓÎÚÓÈ ·‡È‰‡„.
Œ„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È‰ ı ‡ð¸ˆÛÛÎÊ Í‚‡‰ð‡Ú ˇÁ„ÛÛ𠄇ð„‡Ò‡ÌÚ‡È Ú˝Ìˆ ̋ ı ı˝ÏÊË„‰ ̋ ı øøÌËÈ„ ÚÛ ı ‡ÈÌ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ð‡‰ËÛÒ „ ̋ Ì˝.
A
I i
A
I i
y y
x x == (4.14)
√ ÓÎ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ð‡‰ËÛÒÛÛ‰ ̊ „ „ÓÎ ð‡‰ËÛÒÛÛ‰ „ ̋ Ì˝.
A
I i
A
I i min
minmax
max == (4.15)
»ÌÂðˆËÈÌ „ÓÎ Ú˝Ì ı ν„øø‰ ‰ ̋ ˝ð ËÌÂðˆËÈÌ „ÓÎ ð‡‰ËÛÒÛÛ‰‡‡ð Ë ı ·‡„‡ Ú˝Ì ı ν„ËÈÌ ı ‡„‡Ò‡‡ ı ËÈÒ˝Ì ̋ÎÎËÔÒËÈ„ ËÌÂðˆËÈÌ ̋ÎÎËÔÒ „ ̋ Ì˝.
4.4ñð ÁÛð‡„
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»ÌÂðˆËÈÌ ̋ÎÎËÔÒËÈ„ ‡¯Ë„·ÌÓ„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚËÈ„ ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ. 4.5ñð ÁÛð‡„Ú ı Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈËÌÂðˆËÈÌ ð‡‰ËÛÒ Ì¸ ̋ÎÎËÔÒËÈÌ ı Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ ¯øð„ ̋ „˜˝˝Ò ̋ÎÎËÔÒËÈÌ Úº‚ ı øðÚ ̋Î ı Œ¬ ı˝ð˜Ï˝˝ð
ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ.
22 OB Ai A I x x ⋅=⋅= (4.16)
ı Œy ÍÓÓð‰Ë̇Ú˚Ì ÒËÒÚÂÏËÈÌ ı Û‚¸‰ ˆ ̋ „ËÈÌ ‡·ˆËÒÒËÈ„ Ï ̋‰Ò˝Ì˝˝ð ËÌÂðˆËÈÌ Úº‚ººÒ ÁÛ„‡Ú‡‡ ı ÏÓÏÂÌÚ˚„
x i A D x xy ⋅⋅= (4.17)
„ ̋ Ê ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ.Œ„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÚÛ ı ‡ÈÌ
Ú˝Ì ı ν„ ̋ ˝Ò Ó„ÚÎÓÎ˚Ì ı ‡Ï„ËÈÌ ı ÓÎ Á‡ ı˚Ì ˆ ̋ „ ı øðÚ ̋Î ı Á‡È̉ ı ‡ð¸ˆÛÛÎÒ‡Ì ı ‡ð¸ˆ‡‡„ Ú˝ð Ú˝Ì ı ν„ËÈÌ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ „ ̋ Ì˝.
maxmax x
I Wy
y
I W
y
x
x
x ==
(4.18)
›Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ ÒÏ3, Ï3 Ì˝„ÊÚ˝È ·‡È̇.
∆ ˯˝˝ 6.1. 4.6ñð ÁÛð‡„Ú º„º„‰ÒºÌ Ó„ÚÎÓÎ˚Ì Úº‚ Ú˝Ì ı ν„Ú˝È
ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎ ˙∏.¡Ó‰ÓÎÚ. Œ„ÚÎÓÎ 3 ˝Ì„ËÈÌ ‰øðÒ˝˝Ò ·øð‰Ò˝Ì ·º„ºº‰ Á‡ðËÏ º„º„‰ÎËÈ„
ÚÓ‰ðÛÛη‡Î:1 ñð ‰øðÒ ( Ú˝„¯ ºÌˆº„Ú)
5.1335.41
=⋅= A Ï2
78.2212
5.43 3
1=
⋅= x I Ï4
12.1012
5.433
1=
⋅= y I Ï4
2 ñð ‰øðÒ ( ڇΠ‰Û„ÛÈ)
57.18
214.3 2
2 =⋅
= A Ï2
ڇΠ‰Û„ÛÈ ı ø̉ËÈÌ Úº‚
4.5ñð ÁÛð‡„
4.6ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî48
42.014.33
14
3
40 =
⋅
⋅=
π=
r a Ï Á‡È‰ Óð¯Ë ı ÚÛÎ 4.10` ÚÓÏ ˙∏Ó„ÓÓð
11.057.142.0128
214.3
128
24
2
2
0
4
2 =⋅−⋅
=−= Aad
I x
π
Ï4
39.0128
214.3
128
44
2=
⋅==
d I x
π
Ï4
3 ñ ð ‰øðÒ ( „Ûð‚‡ÎÊËÌ)
25.22
5.133 =
⋅= A Ï 28.0
36
5.13 3
3=
⋅= x I Ï4 84.0
48
5.133
3=
⋅= y I Ï4
Œ„ÚÎÓÎ y Ú˝Ì ı ν„ËÈÌ ı Û‚¸‰ Ú˝„¯ ı˝ÏÚ˝È Û˜Ëð ı ø̉ËÈÌ Úº‚ ̸ ˝Ì˝ Ú˝Ì ı ν„ ‰ ̋ ˝ð Óð¯ËÌÓ. ’ø̉ËÈÌ Úº‚ËÈ„
1C ı ø̉ËÈÌ Úº‚Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÚÓ‰Óð ı ÓÈÎ ˙∏.
111 ,, y C x ÒËÒÚÂÏËÈÌ ı Û‚¸‰ 1C (0;0); 2C (0;1.83); 3C (0;-1.75) ·‡È̇.
( )1.0
25.257.15.13
75.125.283.157.10
321
332211 =−−
−−⋅−=
−−
−−=
A A A
y A y A y A y C Ï
C C y C x ,, ÒËÒÚÂÏËÈÌ ı Û‚¸‰ ‰øðÒøø‰ËÈÌ ı ø̉ËÈÌ Úº‚ 1C (0;-10); 2C (0;1.73);
3C (0;-1.85) ·‡È„‡‡„ ‡Ì ı ‡‡ð˜ ı ø̉ËÈÌ Úº‚ Áº‚ ÓΉÒÓÌ ˝Ò˝ı ËÈ„ ¯‡Î„‡ ı ‡‰
( ) ( ) 085.125.273.157.11.05.13332211 ≈−−⋅−−=−−= a Aa Aa AS C X
·‡È„‡‡ ̸ ı ø̉ËÈÌ Úº‚ËÈ„ Áº‚ ÓÎÊ ̋ ˝.2. C C y x , Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÓÎ ˙∏
1.01 −=a Ï 73.12 =a 85.13 −=a
13.10 )3( )2( )1( =−−=C C C C x x x x I I I I Ï4
( ) 91.225.131.078.222
121
)1(
1=⋅−+=+= Aa I I x x C
Ï4
8.457.173.111.0 22
22
)2(
2=⋅+=+= Aa I I x x C
Ï4
( ) 98.725.285.128.02
323
)3(
3=⋅−+=+= Aa I I x x C
Ï4
›Ì„ËÈÌ ‰øðÒøø‰ËÈÌ ı Û‚¸‰ 321 y y y y C === Ú˝Ì ı ν„øø‰ ‰‡‚ ı ̂ Ò‡Ì Ú˝„¯
ı˝ÏËÈÌ Ú˝Ì ı ν„ ·ÓÎÓ ı ÚÛÎ 0321 === bbb ·ÓÎÌÓ.
»Èω:84.039.012.10
321−−=−−= y y y y I I I I
C =8.89 Ï4 ·ÓÎÌÓ.
3.
“º‚ Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ 5.1max = x Ï
35.2max = y Ï 31.435.2
13.10
max
=== y
I W C
C
x
x Ï3
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49
92.55.1
89.8
max
=== x
I W C
C
y
y Ï3
·ÓÎÌÓ.
∆ ˯˝˝ 6.2 4.7ñð ÁÛð„‡‡ð º„º„‰ÒºÌ ÌËÈÎÏ ̋Î ‰øðÒËÈÌ ËÌÂðˆËÈÌ „ÓÎÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓÏ ˙∏Ó, „ð‡ÙËÍËÈÌ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎÊ, ËÌÂðˆËÈÌ ̋ÎÎËÔÒËÈ„·‡È„ÛÛÎ. – ı Âω ‰øðÒν„‰Ò˝Ì Ó„ÚÎÓÎÛÛ‰ ‰‡ð‡‡ ı „ÂÓÏÂÚð ¯ËÌÊ ˜‡Ì‡ðÚ‡È ·‡È„.
1ñð ‰øðÒ ‰‚ÛÚ‡‚ð¹ 20
20=h ÒÏ, 10=b ÒÏ, 8.261 = A ÒÏ2
18401= x I ÒÏ4, 115
1= y I ÒÏ4
2ñð ‰øðÒ “ ̋ „¯ ºÌˆº„Ú 16=h ÒÏ,4.2=b ÒÏ, 4.382 = A ÒÏ2
3ñð ‰øðÒ ¿‰ËÎ ı ‡ÊÛÛÚ ·Ë¯ ·ÛÎ‡Ì 100×63×710= B ÒÏ, 3.6=b ÒÏ, 7.0=d ÒÏ, 1.113 = A ÒÏ2,
1133= y I ÒÏ4, 35
3= x I ÒÏ4,
28.30 = x ÒÏ, 46.10 = y ÒÏ, )0421( 392.0 ′=
α
¡Ó‰ÓÎÚ. 1. ª„º„‰ÒºÌ Ó„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈ„ ı Ó∏ð‰Û„‡‡ð ‰øðÒËÈÌ 2C ı ø̉ËÈÌ
Úº‚Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ (ÁÛð‡„ 4.8) ÚÓ‰Óð ı ÓÈÎ ˙∏.
4.8ñð ÁÛð‡„
4.7ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî50
222 ,, y C x ÓÓð‰Ë̇Ú˚Ì ÒËÒÚÂÏËÈÌ ı Û‚¸‰ )0;0( )2.11;3( 21 C C −−
)66.2;72.4( 3C ·‡È̇. 4.4 ÚÓÏ ˙∏Ó„ÓÓð ı ø̉ËÈÌ Úº‚ ̸
36.01.114.388.26
72.41.110 )3( 8.26
321332211 −=++ ⋅++−⋅=++ ++= A A A
x A x A x A x C ÒÏ
54.31.114.388.26
66.21.110 )2.11( 8.26
321
332211 −=++
⋅++−⋅=
++
++=
A A A
y A y A y A y C ÒÏ
’ø̉ËÈÌ Úº‚ )54.3;36.0( −−C ˆ ̋ „ Áº‚ ÓΉÒÓÌ ˝Ò˝ı ËÈ„ ¯‡Î„‡ ˇ . x, y
Ú˝Ì ı ν„ ̸ Úº‚ Ú˝Ì ı ν„ Û˜Ëð (4.3) ÚÓÏ ˙∏Ó ∏ÒÓÓð 0,0 == y x S S ·‡È ı
∏ÒÚÓÈ. “º‚ Ú˝Ì ı ν„ËÈÌ ı Û‚¸‰ ı ̌η‡ð ‰øðÒËÈÌ ı ø̉ËÈÌ Úº‚ËÈÌÍÓÓð‰Ë̇ÚÛÛ‰ ̸: 54.211
−== b x C Ï, 36.022== b x C Ï, 08.533
== b x C ,
66.711−== a y C Ï, 54.322
== b y C Ï , 66.733−== b y C Ï
( ) 002.61.1154.34.3866.78.26332211 ≈⋅+⋅+−⋅=⋅+⋅+⋅= a Aa Aa AS x
( ) 008.51.1136.04.3838.26332211 ≈⋅+⋅+−⋅=⋅+⋅+⋅= b Ab Ab AS y
2. “º‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎ ˙∏.
4.317245.32117.82478.2026 )3( )2( )1( =++=++= x x x x I I I I ÒÏ4
( ) 78.20268.2664.218402
121
)1( 1
=−+=⋅+= Aa I I x x ÒÏ4
17.8244.3836.012
164.2 232
22
)2( 2
=⋅+⋅=⋅+= Aa I I x x ÒÏ4
45.3211.1108.535 23
23
)3( 3
=⋅+=⋅+= Aa I I x x ÒÏ4
82.272668.53954.4995.1687 )3( )2( )1( =++=++= y y y y I I I I ÒÏ4
( ) 5.16878.2666.71152
121
)1( 1
=−+=⋅+= Ab I I y y ÒÏ4
64.4994.3854.3
12
164.2 23
222
)2( 2
=⋅+⋅
=⋅+= Ab I I y y ÒÏ4
68.5391.112.6113 23
23
)3( 3
=⋅+=⋅+= Ab I I y y ÒÏ4
13.66424.7393.4896.541 )3( )2( )1( =++=++= xy xy xy xy D D D D ÒÏ4
96.5418.26 )66.7( )64.2( 0111 )1(
11=⋅−⋅−+=+= Aba D D y x xy ÒÏ4
96.5418.26 )66.7( )64.2( 0111 )2(
22=⋅−⋅−+=+= Aba D D y x xy ÒÏ4
)3( xy D ÏÓÏÂÌÚ˚„ ÓÎÓ ı ‰ÓÓ 33 , y x Ú˝Ì ı ν„øø‰ËÈ„ u, v „ÓÎ Ú˝Ì ı ν„ ðøø α
ºÌˆ„ººð ˝ð„øøÎÒ˝Ì „ ̋ Ê øÁ‚ ̋Î 6.11 ÚÓÏ ˙∏Ó
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51
α+α−
= 2cos2sin2 33
33
y x y x
uv D I I
D ·ÓÎÓ ı ·‡ 0=uv D „ ̋ ‰„ËÈ„ ‡Ì ı ‡‡ð‚‡Î
24.7302432
11335
22
33
33 =′
−
−=α
−
−=
tg tg
I I
D
y x
y x ÒÏ
4
1; 2 ñð ‰øðÒ Ú˝„¯ ı˝ÏÚ˝È ÚÛÎ 00
2211== y x y x D D ·ÓÎÌÓ.
3. »ÌÂðˆËÈÌ Úº‚ „ÓÎ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ÚÓ‰Óð ı ÓÈηÓÎ :
894.05.16874.3172
13.664222 0 =
−
⋅−=
−−=α
y x
xy
I I
Dtg 412 0 ≈α 03200 ′≈α
0α ºÌˆº„ Һ𺄠·‡È„‡‡ Û˜Ëð ÚøøÌËÈ„ ’ Ú˝Ì ı ν„ ̋ ˝Ò ˆ‡„ËÈÌ ÁøøÌËÈ
ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ ı˝ÏÊËÌ ËÌÂðˆËÈÌ Úº‚ „ÓÎ u, v Ú˝Ì ı ν„øø‰ËÈ„ ·‡È„ÛÛÎ̇
.4. »ÌÂðˆËÈÌ Úº‚ „ÓÎ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈηÓÎ :
=+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −±
+== 2
2
min
max22 xy
y x y x
v
u D I I I I
I I
=+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−±
+= 2
2
13.6642
82.27264.3172
2
26.274.3172 y I
⇒±= 50.70061.2949 11.3650max
= I ÒÏ2, 11.49.22min
= I
5. “º‚ „ÓÎ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÃÓð˚Ì ‰Û„ÛÈ„‡‡ð ÚÓ‰Óð ı ÓÈÎ ˙∏.»ÌÂðˆËÈÌ Úº‚ „ÓÎ ÏÓÏÂÌÚ˚„ „ð‡ÙË͇‡ð ÓÎÓ ı˚Ì ÚÛΉ ı ‡ðËΈ‡Ì
ÔÂðÔẨËÍÛÎ ˇð ı Ó∏ð Ú˝Ì ı ν„ ‡‚˜ ı˝‚Ú˝˝ Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ ËÌÂðˆËÈÌ Ú˝Ì ı ν„ËÈÌ ÏÓÏÂÌÚ˚„ ( I ) , ·ÓÒÓÓ Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ ËÌÂðˆËÈÌ Úº‚ººÒÁÛ„‡Ú‡‡ ı ÏÓÏÂÌÚËÈ„ ( D ) ı˝ÏÊËÌ˝.‡. 59.5174= x I ÒÏ4, 27.731= y I ÒÏ4 ÏÓÏÂÌÚÛÛ‰ ̊ „ ı˝ÏÊËÊ 1, 2 ˆ ̋ „ËÈ„ ÓÎÌÓ.
· . 96.541= xy D ÒÏ4 > 0 ÏÓÏÂÌÚ˚„ 1 ˆ ̋ „ ̋ ˝Ò ‰ ̋ ˝¯ ı˝ÏÊËÊ 3 ˆ ̋ „ËÈ„ ÓÎÌÓ.
‚. 2 ˆ ̋ „ ̋ ˝Ò 3 ˆ ̋ „ËÈÌ ˝Òð˝„ (‰ÓÓ¯) ·ÓÒÓÓ ˜Ë„Î ̋Ή 13.664= xy D ÒÏ3
ÏÓÏÂÌÚ˚„ ı˝ÏÊËÌ 4 ˆ ̋ „ËÈ„ ÓÎÌÓ.„. 3, 4 ˆ ̋ „øø‰ËÈ„ ¯ÛÎÛÛ̇‡ð ı ÓÎ·Ó ı Ó‰ ı˝‚Ú˝˝ Ú˝Ì ı ν„ËÈ„ Ó„ÚÎÓ ı – ˆ ̋ „Ú Úº‚Ú˝È ÚÓÈðÓ„ ÁÛð̇.‰ . 3 ˆ ̋ „ ̋ ˝Ò ı˝‚Ú˝˝ Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ ¯ÛÎÛÛÌ Ú‡Ú‡Ê ÚÓÈðÓ„ÚÓÈÓ„ÚÎÓΈÛÛÎ‡Ì — ˆ ̋ „ËÈ„ ÓÎÌÓ.Â. — ˆ ̋ „ ̋ ˝Ò ÚÓÈðÓ„ ı˝‚Ú˝˝ Ú˝Ì ı˝„ËÈ„ Ó„ÚÎÓÒÓÌ ¿¬ ˆ ̋ „øø‰ ðøø ˆ‡ˆð‡„ Ú‡Ú‡ ı ‡‰ Ú ̋‰„ ̋ ˝ð ̸ „ÓÎ Ú˝Ì ı ν„øø‰ËÈÌ ˜Ë„Î ̋ÎËÈ„ Á‡‡Ì‡.∏ . ÃºÌ ¿ ˆ ̋ „ËÈÌ ‡·ˆËÒÒ Ì¸ 94.5394max ≈ I ÒÏ4 ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ , ¬ ˆ ̋ „ËÈÌ
‡·ˆËÒÒ 92.510min ≈ I ÒÏ4 ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ Á‡‡Ì‡.
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Ê. ¬ ˆ ̋ „ËÈ„ ‰‡Èð‡Ì „‡ð‡ ı ˆ‡ˆð‡„ËÈÌ ı˝‚Ú˝˝ Ú˝Ì ı ν„Ú˝È øøÒ„ ̋ ı ºÌˆº„ ̸
0320 ′=α ·‡È ı ·º„ºº‰ ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ ˜Ë„Î ̋ÎÚ˝È ·‡È̇.
4.9ñð ÁÛð‡„
6. »ÌÂðˆËÈÌ „ÓÎ ð‡‰ËÛÒÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ ̋ÎÎËÔÒ (4.8ñð ÁÛð‡„) ·‡È„ÛÛ· ı - ‰‡‡ u Ú˝Ì ı ν„ ‰ ̋ ˝ð vi , v Ú˝Ì ı ν„ ‰ ̋ ˝ð ui ËÌÂðˆËÈÌ ð‡‰ËÛÒ˚„ ı˝ÏÊËÊ
·‡È„ÛÛÎ̇ .4.8
3.76
94.5394max ====
A
I ii uu ÒÏ
58.23.76
92.510min ====
A
I ii vv ÒÏ
ƒ‡‡Î„‡‚‡ð¹3. ƒ‡‡Î„‡‚‡ð ı Ó∏ð ·Ó‰ÎÓ„ÓÚÓÈ.
1. Œ„ÚÎÓÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ ÓÎÓ ı
ª„º„‰ÒºÌ Ó„ÚÎÓÎ˚Ì (4.10ñð ÁÛð‡„) Úº‚ „ÓÎ Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ( b=0.8a d=0.2a) „ ̋ Ê ÚÓÓˆÓÌ 4.1ñð ı øÒÌ˝„Ú˝Ì‰ ‰Ûð¸‰Ò‡Ì º„º„‰Îøø‰ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎ.
1.
º„º„‰ÒºÌ Ó„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈ„ ÚÓ‰Óð ı ÓÈÎ;2. Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎ;3. Úº‚ „ÓÎ Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ÓÎÌÓ.
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4.10 ñð ÁÛð‡„
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4.10ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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4.10ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
4.1ñð ı øÒÌ˝„Ú¹
1 2 3 4 5 6 7 8 9 10,a ÒÏ 6 7.5 9.0 10.5 12.0 13.5 15.0 16.5 18.0 19.5
k 0.3 0.35 0.4 0.3 0.4 0.3 0.4 0.35 0.4 0.25
¹ 11 12 13 14 15 16 17 18 19 20
,a ÒÏ 4.0 4.5 5.0 5.5 6.0 7.0 8.5 9.0 9.5 10.0
k 0.2 0.25 0.3 0.35 0.4 0.25 0.3 0.4 0.3 0.35
2. Œ„ÚÎÓÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ ÚÓ‰Óð ı ÓÈÎÓ ı
ª„º„‰ÒºÌ Ó„ÚÎÓÎ˚Ì (4.11ñð ÁÛð‡„)Úº‚ „ÓÎ Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ 4.2ñð ı øÒÌ˝„Ú˝Ì‰ ‰Ûð¸‰Ò‡Ì º„º„‰Îøø‰ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎ.1. ª„º„‰ÒºÌ Ó„ÚÎÓÎÛÛ‰ ̊ „ ·ø ı Ú˝Ì ı ν„øø‰Ú˝È ̸ ÁÛðÊ øÌ‰Ò˝Ì
ı˝ÏÊ ̋ ˝Òøø‰ËÈ„ Ú˝Ï‰ ̋ „νı 2.
Œ„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈ„ ÚÓ‰Óð ı ÓÈÎ3.
ÕËÈÎÏ ̋Î ‰øðÒËÈ„ ·øð‰øøÎÊ ·‡È„‡‡ ˝Ì„ËÈÌ ‰øðÒ ÚÛÒ ·øðËÈÌ Úº‚ Ú˝Ì ı ν„øø‰Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎÓ ı
4. Œ„ÚÎÓÎ˚Ì Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎ
5.
»ÌÂðˆËÈÌ „ÓÎ Ú˝Ì ı ν„øø‰ËÈÌ ·‡Èð·Î˚„ ÓÎ.6. Œ„ÚÎÓÎ˚Ì Úº‚ „ÓÎ ÏÓÏÂÌÚÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎ7.
»ÌÂðˆËÈÌ Úº‚ „ÓÎ ÏÓÏÂÌÚÛÛ‰ ̊ „ „ð‡ÙË͇‡ð (ËÌÂðˆËÈÌ ‰Û„ÛÈ„‡‡ð) ÓÎ8.
»ÌÂðˆËÈÌ „ÓÎ ð‡‰ËÛÒÛÛ‰ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ ËÌÂðˆËÈÌ ̋ÎÎËÔÒ ·‡È„ÛÛÎ.
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4.11ñð ÁÛð‡„
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4.11ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
4.2ñð ı øÒÌ˝„Ú
¹ ÿ‚ÂÎÎÂ𠃂ÛÚ‡‚ð ¡ÛÎ‡Ì ÚºÏºð “ ̋ „¯ ºÌˆº„Ú
( h ×b )1 16 10 40×25×4 120×8
2 22 14 50×32×4 160×8
3 24 18 70×45×5 200×10
4 30 22 80×50×5 250×8
5 33 24‡ 90×56×6 280×12
6 33 33 100×63×10 300×12
7 36 45 125×80×10 320×14
8 36 55 140×90×10 340×14
9 40 60 160×100×10 360×16
10 40 55 180×110×10 380×18
11 14 12 40×25×4 140×812 22 16 50×32×3 180×8
13 22‡ 18‡ 70×45×4.5 200×8
14 24 20 80×50×5 240×10
15 24‡ 20‡ 90×56×6 280×10
16 27 24 100×63×10 320×10
17 27 27 125×80×10 340×12
18 30 30 140×90×10 380×12
19 36 40 160×100×10 420×12
20 40 50 180×110×10 460×12
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“¿¬ƒ” √ ¿¿— ¡ØÀ›√
ƒ” √ ”… Œ√ “ÀŒÀ“Œ… ÿ»À¡›Õ»… Ôÿ√ »—¿À“
5.1. Ôÿ√ »—¿À“¤Õ –“¿“» “ŒƒŒ—’Œ… ¡ŒƒÀŒ√ Œ
≈ðºÌ ı ËÈ ÓÈ΄ÓÎÚ
ÃÛ¯„Ëð‡ÎÚ˚Ì ø‰ „ÓÎ˚Ì ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ „‡Ìˆ ÚºðÎËÈÌ ‰ÓÚÓÓ‰ ı ø˜ -ÏÛ¯„Ë ı ÏÓÏÂÌÚ øøÒÌ˝.
√ ÓÎ˚Ì ·‡Ú ·˝ı , ı º¯øøÌ ˜‡Ì‡ð˚Ì ÚÓÓˆÓÓ ı ËÈ ı ̋‰ „ÓÎ˚Ì ÛðÚ˚Ì ‰‡„ÛÛ‰
ÏÛ¯„Ë ı
ÏÓÏÂÌÚ˚
Ì ººð˜Îº„‰º ı
ı
ÛÛÎËÈ„ ÚÓ„ÚÓÓ ı
¯‡‡ð‰Î‡„‡Ú‡È. ÃÛ¯„Ë ı
ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ Ó„ÚÎÓÎ˚Ì ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎÌÓ. √ ‡‰‡‡‰ ÏÛ¯„Ë ı ÏÓÏÂÌÚ·øðËÈÌ ı ÓÓðÓ̉ Ì˝„ Ó„ÚÎÓÎ ı ËÈ ı ·º„ºº‰ ÚÛ ı ‡ÈÌ Ó„ÚÎÓÎ˚Ì ÏÛ¯„Ë ı ÏÓÏÂÌÚ Ú˝ð ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ì˝„ ڇΉ Óð¯Ë ı „‡‰‡‡‰ ÏÓÏÂÌÚÛÛ‰ ̊ Ì ‡Î„·ðÌËÈη˝ðÚ˝È Ú˝Ìˆøø ·‡È̇.
∑= eT T (5.1)
ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì Ú˝Ï‰ ̋ „ Ó̈ ‡˜ ı Óηӄ‰Ó΄øÈ ·ÓÎÓ‚˜ Ó„ÚÎÓÎÓÓÒ ı ‡ð‡ ı ‡‰ Ó„ÚÎÓ„‰ÒÓÌ ı˝Ò˝„Ú ˆ‡„ËÈÌ ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„ ˜Ë„Î ̋Ê øÈΘËÎÊ ·ÛÈ ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚„ ˝Âð˝„ „ ̋ Ê øÁÌ˝.
«‡ðËÏ ÚÓ ı ËÓΉÓΉ „ÓÎ˚Ì ‰‡ÏÊÛÛÎÊ ·ÛÈ ÏÛ¯„Ë ı ÏÓÏÂÌÚ º„º„‰ºÎ„øÈ,„ÓÎ˚Ì ‰‡ÏÊÛÛ· ı ˜‡‰‡Î º„º„‰ÒºÌ ·‡È‰‡„. ›Ì˝ ø‰ P ˜‡‰‡Î [ͬÚ] Ì˝„Ê ̋ ˝ðº„º„‰ÒºÌ ø‰ ÏÛ¯„Ë ı ÏÓÏÂÌÚ
n
P T ⋅= 55.9 , [ÍÕÏ] (5.2)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. Øø̉ : n - Ì˝„ ÏËÌÛڇ̉ „ÓÎ˚Ì ˝ð„ ̋ ı ˝ð„ ̋ ÎÚËÈÌ ÚÓÓ, [˝ð„/ÏËÌ]
ÃºÌ „ÓÎ˚Ì Ì˝„ ÏËÌÛڇ̉ ˝ð„ ̋ ı ˝ð„ ̋ ÎÚËÈÌ ÚÓÓ, „ÓÎ˚Ì30
π ω
n= ºÌˆº„
ı Ûð‰Ì˚ ı ‡Ï‡‡ðÎ˚„ ‡¯Ë„·‚‡Î ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚„
ω
P T = , [ÍÕÏ]
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ. Øø̉ : - „ÓÎ˚Ì ºÌˆº„ ı Ûð‰, [ð‡‰/ÒÂÍ]
ÃÛ¯„Ëð‡ÎÚ˚Ì ø‰ „ÓÎ˚Ì ÚÛ ı ‡ÈÌ Ó„ÚÎÓΉ øøÒ˝ı ı ‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ Î
ρ
τ W
T =
(5.3)
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·‡È̇. Øø̉ : T - ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝, [ÕÏ]
pW - ÚÛÈÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ, [Ï3]
ƒÛ„ÛÈ Ó„ÚÎÓÎÚÓÈ „ÓÎ˚Ì ÚÛÈÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ „ÓΘÓÓðËνð ı ËÈη ̋Î
16
3d W p
⋅= π
(5.4)
·ÓÎÌÓ.ÃÛ¯„Ëð‡ÎÚ˚Ì øÂËÈÌ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎ
][τ τ ρ
≤=W
T (5.5)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. Øø̉ ][τ - ÏÛ¯„Ëð‡ÎÚ˚Ì øÂËÈÌ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î, „ÓÎ˚„ · ̋ÎÚ„ ̋ Ò˝Ì Ï‡ÚÂðˇ·‡Ò ı ‡Ï‡‡ð̇. ÷ ̋ ‚˝ð ÏÛ¯„Ëð‡ÎÚ˚Ì ø‰ ÚÓÓˆÓÓ̉
][ )6.0...5.0( ][ σ τ =
„ ̋ Ê ‡‚˜ ·ÓÎÌÓ.(5.5) ÚÓÏ ˙∏ÓÌ˚ ÒøøÎËÈÌ Ú˝Ìˆ ̋ Ú„ ̋ Î ·Ë¯ËÈ„ ‡¯Ë„Î‡Ì „ÓÎ˚Ì ı ºÌ‰ÎºÌ
Ó„ÚÎÓÎ˚Ì ¯‡‡ð‰Î‡„‡Ú‡È „ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ.
3
][
16
τ π ⋅
⋅≥
T d (5.6)
√ ÓÎ˚Ì ÚÓ„ÚÏÓÎ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎÚÓÈ l ÛðÚÚ‡È ı˝Ò˝„ ·øðÚ ÚÓ„ÚÏÓÎ( =T const) ÏÛ¯„Ë ı ÏÓÏÂÌÚ øÈΘËÎÊ ·‡È‚‡Î ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈ„
p I G
l T
⋅
⋅=θ (5.7)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ : G - χÚÂðˇÎ˚Ì ¯ËÎÊËÎÚËÈÌ ÏÓ‰Ûθ, [œ‡]
ρ I - ÚÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ, [Ï4]
ƒÛ„ÛÈ Ó„ÚÎÓÎÚÓÈ „ÓÎ˚Ì ÚÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ
32
4d I p
π = (5.8)
·‡È̇.√ ÓÎ˚Ì ı º¯øøÌ ˜‡Ì‡ð˚„ øÌ ̋νı ‰ ̋ ˝ ÏÛ¯„ËðÎ˚Ì ı ‡ð¸ˆ‡Ì„ÛÈ γ ºÌˆ„ËÈ„
‡¯Ë„·‰‡„. ›Ì˝ ̸ „ÓÎ˚Ì Ì˝„Ê ÛðÚ‡‰ ı ‡ð„‡ÎÁ‡ ı ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ (Ê˯˝ ̋η ̋Î 1 Ï ÛðÚ‡‰ ı ‡ð„‡ÎÁ‡ ı ) ·º„ºº‰
l
θ γ = , [ð‡‰/Ï] (5.9)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî60
ÃÛ¯„ËðÎ˚Ì ı ‡ð¸ˆ‡Ì„ÛÈ ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ „ÓÎ˚Ì ÁÓðËÛ·ÎÚ ·‡ ı ‡ðËۈ·„ ̊ „ ÚÓÓˆÒÓÌ Áº‚¯ººðº„‰º ı ][γ ÛÚ„‡‡Ò ı ̋Úð˝ı „øÈ ·‡È ı
¯‡‡ð‰Î‡„‡Ú‡È „ ̋ ‚ ̋Î ÏÛ¯„Ëð‡ÎÚ˚Ì øÂËÈÌ „ÓÎ˚Ì ı º¯øøÌ ˜‡Ì‡ð˚Ì Ìº ı ̂ ºÎ
][γ γ ρ
≤⋅
= I G
T (5.10)
·ÓÎÌÓ. “ºÒÎËÈÌ ÚÓÓˆÓÓÌ˚ ø‰ (5.10) ÚÓÏ ˙∏ÓÌÓÓÒ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ÚÛÈÎ˚Ì
ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÓÌ ÛÎχ‡ð ¯‡‡ð‰Î‡„‡Ú‡È „ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ.
∆ ˯˝˝ 5.1. √ ‡Ì (5.1‡ñð ÁÛð‡„) ¯Ëη˝Ì‰ øÈÎ˜Î˝ı 31=eT ÍÕÏ,
62 =eT ÍÕÏ, 13 =eT ÍÕÏ ÏÛ¯„Ë ı ÏÓÏÂÌÚÛÛ‰ ·‡ 5.0=a Ï Á‡È º„º„‰ÒºÌ ø‰ 1. ¯Ëη˝ÌËÈ ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ;2. ÏÛ¯„Ëð‡ÎÚ˚Ì øÂËÈÌ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î 60][ =τ Ãœ‡ ·‡È ı ‡‰ ·‡Ú
·˝ı ËÈÌ Ìº ı ̂ κºÒ ‰Û„ÛÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ ÒÓÌ„Ó;3. ¯Ëη˝ÌËÈ ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
¡Ó‰ÓÎÚ1. ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı ‰‡‡ Ó„ÚÎÓÎ˚Ì ‡ð„‡ ı˝ð˝„Î˝Ì˝.
–ÛÎ øÁøøðÚ˝
È ı˝
Ò„ ˝˝
Ò˝ı
ν
Ì 1-1, 2-2, 3-3 „ ˝
Ò˝
Ì Ó„ÚÎÓÎÛÛ‰ (5.1·,‚,„ñð ÁÛð‡„)‡‚ ˙ˇ. ›Ì˝ ̸ ÁÓÓÎÚÓÌ ‰ ̋ ˝ð øøÒ˝ı ð‡͈ËÈ„ ÓÎÓ΄øÈ„ ̋ ˝ð ·Ó‰ÎÓ„ ̊ „ ·Ó‰Ó ı ·ÓÎÓÏÊËÈ„ Ó΄ÓÌÓ. ƒ‡ð‡‡ ̸ Ó„ÚÎÓÎÓÓÒ ÁøøÌ Ú‡Î‰ Óð¯Ë ı ¯Ëη˝ÌËÈ ı˝Ò„ËÈ„Óð ı ËÊ ·‡ðÛÛÌ ı˝Ò„ËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ºÎËÈ„ ‡‚˜ øÁÌ˝. Œ„ÚÎÓÎ ·øð‰ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ ÚÓ‰Óð ı ÓÈηÓÎ ‰‡ð‡‡ ı ÛÚ„ÛÛ‰ ̊ „ ‡‚‡ ı ·ÓÎÌÓ.
1ñ1 Ó„ÚÎÓΉ (5.1·ñð ÁÛð‡„)311 −=−= eT T ÍÕÏ, ( 5.00 1 ≤≤ z Ï)
2ñ2 Ó„ÚÎÓΉ3212 =+−= ee T T T ÍÕÏ, ( 0.15.0 2 ≤≤ z Ï)
3ñ3 Ó„ÚÎÓΉ (5.1„ñð ÁÛð‡„) 43213 =++−= eee T T T T ÍÕÏ, ( 5.10.1 3 ≤≤ z Ï)
“Ó‰Óð ı ÓÈÎÒÓÌ ÛÚ„ÛÛ‰‡‡ð ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ (5.1‰ñð ÁÛð‡„)·‡È„ÛÛη‡.
ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð Áº‚ ·‡È„ÛÛ·„‰Ò‡Ì ˝Ò˝ı ËÈ„ Úº‚κðÒºÌÏÓÏÂÌÚ øÈΘËÎÊ ·‡È„‡‡ Ó„ÚÎÓΉ Ú˝ð ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ ̋ ˝ð øÒð ̋ÎÚÚ˝È·‡È ı ¯ËÌÊ ̋ ˝ð ¯‡Î„‡Ì‡.
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61
‡ ).
· ).
‚ ).
„ ).
‰).
 ).
5.1-ð ÁÛð‡„
2. ÿËη˝ÌËÈ ı˝Ò˝„ ÚÛÒ ·øðËÈÌ ¯‡‡ð‰Î‡„‡Ú‡È „ÓΘËÈ„ (5.6) ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈηÓÎ ‰‡ð‡‡ ı ÛÚ„ÛÛ‰ „‡ð‡ ı ·ÓÎÌÓ.
1ñ1 Ó„ÚÎÓΉ 063.0106014.3
10 )3( 16
][
163
6
3
3 11 =
⋅⋅
⋅−⋅=
⋅
⋅≥
τ π
T d Ï 65≈ ÏÏ
2ñ2 Ó„ÚÎÓΉ 063.0106014.3
10316
][
163
6
3
3 22 =
⋅⋅
⋅⋅=
⋅
⋅≥
τ π
T d Ï 65≈ ÏÏ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî62
3ñ3 Ó„ÚÎÓΉ 070.0106014.3
10416
][
163
6
3
3 33 =
⋅⋅
⋅⋅=
⋅
⋅≥
τ π
T d Ï 70= ÏÏ
›Ì‰ 1d , 2d „ÓΘËÈ„ √ Œ–“ 6636-60 Òڇ̉‡ðÚ˚Ì ÓÈðÓΈÓÓ ÛÚ„‡Ì‰¯ËÎÊøøÎÒ˝Ì ·ÓÎÌÓ.
3. √ ÓÎ˚Ì ı˝Ò˝„ ·øðËÈÌ ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝„ (5.7), (5.8) ÚÓÏ ˙∏Ó„ ‡¯Ë„·Ì
4
32
d G
l T
⋅⋅
⋅⋅=
π θ
ÚÓÏ ˙∏Ó„ÓÓð ÓηÓÎ
1ñ1 Ó„ÚÎÓΉ 034.0065.01080
5.0 )103( 323249
3
41
111 −=
⋅⋅⋅⋅−⋅=
⋅⋅⋅⋅=d Gl T
π θ ð‡‰
2ñ2 Ó„ÚÎÓΉ 034.0065.01080
5.0103323249
3
42
222 =
⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
3ñ3 Ó„ÚÎÓΉ 033.0070.01080
5.0104323249
3
43
333 =
⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
·ÓÎÌÓ.
ƒ ̋ ˝ð ÓÎÒÓÌ ÛÚ„ÛÛ‰‡‡ð ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ ÁÓÓÎÚÚÓÈ ı ‡ð¸ˆ‡Ì„ÛÈ ( ı º‰ºÎ„ººÌ„øÈ ·˝ı ̋΄ ̋ ˝Ì ‰ ̋ ˝ð ¯ËÎÊËÎÚ Ú˝„Ú˝È Ú˝Ìˆøø 0= Aθ
·‡È ı ̺ ı ̂ ÎËÈ„ ‡¯Ë„·Ì) ·‡È„ÛÛÎ ˙ˇ. ’ ̋ Ò˝„ ·øðÚ ºÌˆ„ËÈÌ ˝Ô˛ð ¯ÛÎÛÛ̇‡ðÁÛð‡„‰‡ ı Û˜Ëð ı˝Ò˝„ ·øðËÈÌ ˝ı Î ̋Î, Úº„Ò„ºÎ‰ ºÌˆ„ËÈÌ ÛÚ„ÛÛ‰ ̊ „ ÓÎÓ ı Ó‰ ı ‡Ì„‡ÎÚÚ‡È.
¿ Ó„ÚÎÓÎÚÓÈ ı ‡ð¸ˆ‡Ì„ÛÈ B Ó„ÚÎÓÎ˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝
033.03 ==θ θ B ð‡‰,
– Ó„ÚÎÓÎ˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ 067.0034.0033.023 =+=+= θ θ θ C ð‡‰,
D Ó„ÚÎÓÎ˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ 033.0034.0034.0033.0123 =−+=++= θ θ θ θ D ð‡‰
·‡È̇.›‰„ ̋ ˝ð ÛÚ„ÛÛ‰‡‡ð ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ (5.1Âñð ÁÛð‡„)
·‡È„ÛÛ·‚.
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∆ ˯˝˝ 5.2. ÃËÌÛڇ̉ 500=n Û‰‡‡ ˝ð„ ̋ ı „ÓÎ ‰ ̋ ˝ð ÒÛÛÒ‡Ì ‰‡ÏðÛÛ‰
(5.2‡ñð ÁÛð‡„) ÚÛÒ ·øð 401 = P ͬÚ, 702 = P ͬÚ, 603 = P ͬÚ, 504 = P ͬÚ
˜‡‰Î˚„ ‡ÊÎ˚Ì Ï‡¯Ë̉ ‰‡ÏÊÛÛ· ı ¯‡‡ð‰Î‡„‡Ú‡È ø‰ „ÓÎ, ‰‡ÏðÛÛ‰ ̊ Ì ı Û‚ËÈÌ ÊËÌ, ı ÓÎ ı Ë‚˜ËÈÌ øð ̋ÎÚËÈ„ ÚÓÓˆÓ΄øÈ ˆ‡ ı Ë΄‡‡Ì ı º‰ºÎ„øøð˝˝Ò˝ð„ ̋ ÎÚ ‡‚‡ ı ‰‡Ï‡ðÚ ¯‡‡ð‰Î‡„‡Ú‡È ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚„ ÓÎ. ’ ̋ ð˝‚ „ÓÎ˚„
30][ =τ Ãœ‡ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ ÎÚ˝È „‡Ì„‡‡ð ( 80=G √ œ‡) ı ËÈ ı
¯‡‡ð‰Î‡„‡Ú‡È ·ÓÎ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººÒ „ÓÎ˚Ì ı˝Ò˝„ ÚÛÒ ·øðËÈÌ „ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎÊ, ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ.
¡Ó‰ÓÎÚ1. ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô ̨𠷇ȄÛÛ· ı
ƒ‡Ï‡ð ÚÛÒ ·øðËÈÌ ‰‡ÏÊÛÛ· ı ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚„ (5.2) ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈηÓÎ
764.0500
4055.955.9 1
1=⋅=⋅=
n
P T e ÍÕÏ
337.1500
7055.955.9 2
2=⋅=⋅=
n
P T e ÍÕÏ
146.1500
6055.955.9 3
3=⋅=⋅=
n
P T e ÍÕÏ
573.05003055.955.9 4
4 =⋅=⋅=n
P T e ÍÕÏ
·ÓÎÌÓ. ’ÓÎ ı Ë‚˜Ë̉ øøÒ˝ı øð ̋ÎÚËÈÌ ‡Î‰‡„‰Î˚„ ÚÓÓˆÓ ı „øÈ Û˜Ëð ÒÚ‡ÚËÍËÈÌ
Ú˝Ìˆ‚˝ðËÈÌ ‰‡ð‡‡ ı Ú˝„¯ËÚ„ ̋ ν˝Ò 0eT ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝,
˜Ë„Î ̋ÎËÈ„ ÚÓ‰Óð ı ÓÈÎÌÓ.0=∑ z T ; 0
43021=++−+ eeeee T T T T T ⇒ 82.3
0=eT ÍÕÏ
ƒ ̋ ˝ð ı ·‡È‰Î‡‡ð0eT ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝ Ì˝Ï˝ı „‡ðÒ‡Ì Ì¸ ÚøøÌËÈ
Û𸉘ËÎÒ‡Ì ·‡È‰Î‡‡ð ‡‚Ò‡Ì ˜Ë„Î ̋Î (5.2‡ñð ÁÛð‡„Ú ÒÛχ‡ð øÁøøÎÒ˝Ì) Áº‚·‡È„‡‡„ ı ‡ðÛÛÎÊ ·‡È̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî64
‡).
¡).
¬).
„).
ƒ).
≈ ).
∆ ).
Á).
5.2ñð ÁÛð‡„
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ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ Ó„ÚÎÓÎ˚Ì ‡ð„‡ ı˝ð˝„Î˝Ì˝.›Ì˝ ·Ó‰ÎÓ„ ̊ Ì ı Û‚¸‰ ÓÌÓÎ˚Ì ı˝Ò˝„Ú ‰Ûð‰Ò‡Ì ÊÛðÏ˚„ ‡¯Ë„Î‡Ì ‰ºð‚ºÌÓ„ÚÎÓÎ ı ËÈ ı ̋‰ ı øð ̋Έ ̋ ̋Ú˝È. Œ„ÚÎÓÎ ·øð‰ „ÓÎ˚Ì ·‡ðÛÛÌ ı˝Ò„ËÈ„ Óð ı ËÌ ÁøøÌ ı˝Ò„ËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ºÎººÒ ‰ÓÚÓÓ‰ ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ ÚÓ‰Óð ı ÓÈÎÌÓ.1-1 Ó„ÚÎÓΉ 2.10 1 ≤≤ z Ï ø‰ (5.2·ñð ÁÛð‡„ )
0=∑ z T ; 764.011 == eT T ÍÕÏ.
2-2 Ó„ÚÎÓΉ 2.22.1 2 ≤≤ z Ï ø‰ (5.2‚ñð ÁÛð‡„)
0=∑ z T ; 101.2337.1764.0212 =+=+= ee T T T ÍÕÏ.
3-3 Ó„ÚÎÓΉ 2.32.2 3 ≤≤ z Ï ø‰ (5.2„ñð ÁÛð‡„)
0=∑ z T ; 719.182.3337.1764.00213 −=−+=−+= eee T T T T ÍÕÏ.
4-4 Ó„ÚÎÓΉ 4.42.3 3 ≤≤ z Ï ø‰ (5.2‰ñð ÁÛð‡„)
0=∑ z T ; 573.0146.182.3337.1764.030214 −=+−+=+−+= eeee T T T T T ÍÕÏ.
ƒ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎÒÓÌ ÛÚ„ÛÛ‰‡‡ð ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ (5.2Âñð ÁÛð‡„)·‡È„ÛÛ·‚.
2. √ ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı √ ÓÎ˚Ì ı˝Ò˝„ ÚÛÒ ·øðËÈÌ ¯‡‡ð‰Î‡„‡Ú‡È „ÓΘËÈ„ (5.6) ÚÓÏ ˙∏Ó„ÓÓð
ÚÓ‰Óð ı ÓÈÎÌÓ.
1-1 Ó„ÚÎÓΉ 050.0103014.3
10764.016
][
163
6
3
3 11 =
⋅⋅
⋅⋅=
⋅
⋅=
τ π
T d Ï 50= ÏÏ
2-2 Ó„ÚÎÓΉ 070.0103014.3
10101.216
][
163
6
3
3 22 =
⋅⋅
⋅⋅=
⋅
⋅=
τ π
T d Ï 70= ÏÏ
3-3 Ó„ÚÎÓΉ 066.0103014.3
10719.116
][
163
6
3
33
3 =⋅⋅
⋅⋅=
⋅
⋅=
τ π
T d Ï 66= ÏÏ
4-4 Ó„ÚÎÓΉ 046.0103014.3
10573.016
][
163
6
3
3 44 =
⋅⋅
⋅⋅=
⋅
⋅=
τ π
T d Ï 46= ÏÏ
ƒ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎÒÓÌ „ÓΘÛÛ‰ ̊ „ ı ‡ð„‡ÎÁ‡ ı Òڇ̉‡ðÚ˚Ì ÓÈðÓΈÓÓ ÛÚ„ÛÛ‰‡‰ ¯ËÎÊøøη ̋Î 1d , 2d ‰Ë‡ÏÂÚðøø‰ Òڇ̉‡ðÚ˚Ì ÛÚ„‡Ú‡È ÚÓ ı Ëð˜
·‡È„‡‡ ·‡ 703 ≈d ÏÏ, 504 ≈d ÏÏ ·ÓÎÌÓ. ›‰„ ̋ ˝ð ÛÚ„‡‡ð „ÓÎ˚Ì ¯‡Ú·Î˚Ì
·ø‰øø‚˜ ( ̋ ÒÍËÁ) ÁÛð„ËÈ„ (5.2Êñð ÁÛð‡„) øÁøøν‚.
3. ÃÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô ̨𠷇ȄÛÛ· ı
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî66
(5.7), (5.8) ÚÓÏ ˙∏Ó„ ‡¯Ë„·‚‡Î „ÓÎ˚Ì ı˝Ò˝„ ·øðËÈÌ ÏÛ¯„ËðÎ˚̺̈„ËÈ„
4
32
d G
l T
⋅⋅
⋅⋅=
π θ
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÊ ·ÓÎÌÓ.
1-1 Ó„ÚÎÓΉ 019.0 )050.0( 14.31080
2.110764.0323249
3
41
111 =
⋅⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
2-2 Ó„ÚÎÓΉ 011.0 )070.0( 14.31080
110101.2323249
3
42
222 =
⋅⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
3-3 Ó„ÚÎÓΉ 009.0 )070.0( 14.31080
110 )719.1( 323249
3
43
333 −=
⋅⋅⋅⋅⋅−⋅=
⋅⋅⋅⋅=d G
l T
π θ ð‡‰
4-4 Ó„ÚÎÓΉ 014.0 )050.0( 14.31080
2.110 )573.0( 323249
3
44
444 −=
⋅⋅⋅
⋅⋅−⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
ƒ ̋ ˝ð ÓÎÒÓÌ ÛÚ„ÛÛ‰‡‡ð ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ 1 ‰‡Ï‡ðÚ‡È (ÁøøÌ øÁøøðÚ˝È) ı ‡ð¸ˆ‡Ì„ÛÈ ·‡È„ÛÛÎ ˙ˇ. ’ ̋ Ò˝„ ·øðÚ ˝Ô˛ð ¯ÛÎÛÛ̇‡ð ÁÛð‡„‰‡ ı Û˜Ëð ı˝Ò˝„ ·øðËÈÌ ˝ı Î ̋Î, Úº„Ò„ºÎ‰ ºÌˆ„ËÈÌ ÛÚ„ÛÛ‰ ̊ „ ÓÎÓ ı Ó‰ ı ‡Ì„‡ÎÚÚ‡È.
1 ‰‡Ï‡ðÚ‡È ı ‡ð¸ˆ‡Ì„ÛÈ2 ‰‡Ïð˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝
019.01 =θ ð‡‰ ,
ı ºÚκ ı ‰‡Ïð˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ 03.0011.0019.021 =+=+θ θ ð‡‰ ,
3 ‰‡Ïð˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝ 021.0009.0011.0019.0321 =−+=++ θ θ θ ð‡‰ ,
4 ‰‡Ïð˚Ì ÏÛ¯„ËðÒ‡Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝
007.0014.0009.0011.0019.04321 =−−+=+++ θ θ θ θ ð‡‰
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ (5.2Áñð ÁÛð‡„)·‡È„ÛÛ·‚.
5.2. Ôÿ√ »—¿À“¤Õ –“¿“» “ŒƒŒ—’Œ… ¡»ÿ ¡ŒƒÀŒ√ Œ
–Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ı˝ð˝„νı ·ÓÎÓ‚˜ Úøø„ ̋ ˝ð ·øð˝Ì·Ó‰Ó„‰Ó ı „øÈ ·Ó‰ÎÓ„ ̊ „ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó „ ̋ Ì˝. –Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó Ì¸ ÚøøÌËÈ ı˝‚ „‡ÊËÎÚ˚Ì Ì˝Ï ̋ÎÚ Ìº ı ̂ ºÎËÈ„ ‡¯Ë„·Ì
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ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ËÈ„ Ú‡ÈÎÒÌ˚ ‰‡ð‡‡ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„ ̊ Ì ‡‰ËηӉӄ‰ÓÌÓ.
∆ ˯˝˝ 5.3. 1=e
T ÍÕÏ ·‡È ı ‡‰ ı Ó∏ð øÁøøð ̸ ı º‰ºÎ„ººÌ„øÈ
·˝ı ν„‰Ò˝Ì „‡Ì „ÓÎ˚Ì (5.3‡ñð ÁÛð‡„) ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
‡).
·).
‚).
„).
5.3ñð ÁÛð‡„
¡Ó‰ÓÎÚ.ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ ˝ı ν ̋‰ ·˝ı ̋΄ ̋ ˝Ìøø‰ ̋ ‰ øøÒ˝ı ð‡͈ ÏÓÏÂÌÚÛÛ‰ ̊ „ (5.3·ñð ÁÛð‡„) ÚÓ‰Óð ı ÓÈÎÓ ı ı˝ð˝„Ú˝È.
–Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ·Ë˜‚ ̋Î0=∑ z T , 02 =−++− B ee A T T T T ⇒ e B A T T T 3=+
·ÓÎÌÓ. Õ˝„ Ú˝„¯ËÚ„ ̋ Î ı Ó∏ð øÎ Ï ̋‰ ̋ „‰ ̋ „˜Ú˝È ·‡È„‡‡ Û˜Ëð ˝Ì˝ ·Ó‰ÎÓ„ÓÌ˝„‰ø„ ̋ ˝ð Á˝ð„ËÈÌ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ.
’ ̋ ‚ „‡ÊËÎÚ˚Ì Ì˝Ï ̋ÎÚ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÁÓ ı ËÓ ı ‰ÓÓ ¬ Ó„ÚÎÓÎ˚ÌÏÛ¯„ËðÎ˚Ì ºÌˆº„ Ú˝„Ú˝È Ú˝Ìˆøø ( 0= B θ ) ·‡È ı ̺ ı ̂ ºÎËÈ„ ‡¯Ë„· ˇ. ØøÌËÈ
ÚÛΉ „ÓÎ˚Ì ·‡ðÛÛÌ øÁøøðËÈÌ ·˝ı ̋΄ ̋ ˝„ ˜ºÎººÎÊ ÚøøÌËÈ øÈΘΠ̋ÎËÈ„ B T
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ð‡͈ ÏÓÏÂÌÚÓÓð (5.3‚ñð ÁÛð‡„) ÒÓÎËÌÓ. B Ó„ÚÎÓÎ˚Ì ı Û‚¸‰ ı ø˜ÌËÈ øÈΘΠ̋ÎËÈÌ ı ‡Ï‡‡ð‡Î„øÈÌ Á‡ð˜Ï˚„ ‡¯Ë„·‚‡Î ÏÛ¯„Ë ı ÏÓÏÂÌÚ ·øð˝˝Ò¯ËÎÊËÎÚ øøÒ˝ı ·‡ ¯ËÎÊËÎÚËÈÌ Ú˝„¯ËÚ„ ̋ Π̸ ‰‡ð‡‡ ı ı ̋η˝ðÚ˝È ·‡È̇.
0,2,, =++= B ee T B T B T B B θ θ θ θ
Øø̉ : ¬ Ó„ÚÎÓΉ eT ÏÛ¯„Ë ı ÏÓÏÂÌÚÓÓÒ øøÒ˝ı ÏÛ¯„ËðÎ˚Ì ºÌˆº„
1
, p
eT B
I G
aT e ⋅
⋅=θ , ¬ Ó„ÚÎÓΉ eT 2 ÏÛ¯„Ë ı ÏÓÏÂÌÚÓÓÒ øøÒ˝ı ÏÛ¯„ËðÎ˚Ì ºÌˆº„
1
222,
p
eT B
I G
aT e ⋅
⋅=θ , ¬ Ó„ÚÎÓΉ B T ÏÛ¯„Ë ı ÏÓÏÂÌÚÓÓÒ øøÒ˝ı ÏÛ¯„ËðÎ˚Ì ºÌˆº„
21
2,
p
B
p
B
T B I G
aT
I G
aT B ⋅
⋅
−⋅
⋅
−=θ ·‡È̇.
( )32;32
8.0
32
444
21
d
I
d d
I
π π π ρ ρ =−= ·‡ÈÒÌ˚„
‡Ì ı ‡‡ð‡Ì ¯ËÎÊËÎÚËÈÌ Ú˝„¯ËÚ„ ̋ Ή ̋‰„ ̋ ˝ð ÛÚ„ ̊ „ ÓðÎÛÛη‡Î
0222
2111
=⋅
⋅−
⋅
⋅−
⋅
⋅+
⋅
⋅
p
B
p
B
p
e
p
e
I G
aT
I G
aT
I G
aT
I G
aT ⇒
4444444 )8.0(
2
)8.0(
4
)8.0( d
T
d d
T
d d
T
d d
T B B ee +−
=−
+−
⇒ e ¬ T T 93.1=
·ÓÎÌÓ. –Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Ή ¬ T ð‡͈ ÏÓÏÂÌÚ˚Ì ÛÚ„ ̊ „
ÓðÎÛÛη‡Î e A T T 07.1= ·ÓÎÌÓ. AT , ¬ T ÏÓÏÂÌÚÛÛ‰ ̊ Ì ÛÚ„ ̊ „ ÓÎÒÌÓÓð
·Ó‰ÎÓ„ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ڇȷ„‰‡Ê, ˆ‡‡¯Ë‰ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ·Ó‰ÎÓ„ ̊ Ì ‡‰Ë·‡ð ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð (5.3„ñð ÁÛð‡„) ı ̌η‡ð ı ‡Ì·‡È„ÛÛ·„‰‡Ì‡.
∆ ˯˝˝ 5.4. √ ‡Ì „ÓÎ ·‡ ı ÓÓÎÓÈÌ (5.4‡ñð ÁÛð‡„) Ì˝„ øÁøøðËÈ„
ı º‰ºÎ„ººÌ„øÈ ·˝ı ̋ÎÊ, ̺„ºº øÁøøðËÈ„ ı ÓÓðÓ̉ ̸ ı ÓηÓÊ ̋ ˝. √ ÓÎ ·‡ ı ÓÓÎÓÈÌ ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
¡Ó‰ÓÎÚ.√ ÓÎ ·‡ ı ÓÓÎÓÈÌ ı ‡ðËΈ‡Ì øÈΘΠ̋ÎËÈ„ X ÏÛ¯„Ë ı ÏÓÏÂÌÚÓÓð ÒÓθÊ
·ÓÎÌÓ. √ ÓÎ ·‡ ı ÓÓÎÓÈÌ ‡˜‡‡Î‡„‰‡ ı ·ø‰øø‚˜ËÈ„ (5.4·,‚ñð ÁÛð‡„) ÚÛÒ ÚÛÒ‡‰ ̸ÁÛðÊ ·ÓÎÌÓ.
5.4·ñð ÁÛð„‡‡Ò ¬ Ó„ÚÎÓΉ „ÓÎ˚Ì ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝
11
2,
p p
e„ÓÎ B
I G
a X
I G
aT
⋅
⋅−
⋅
⋅=θ , 5.4‚ñð ÁÛð„‡‡Ò ¬ Ó„ÚÎÓΉ ı ÓÓÎÓÈÌ ÏÛ¯„ËðÎ˚Ì
ºÌˆ„ËÈÌ ı˝
ÏÊ ˝˝
2
2
, p
ı ÓÓÎÓÈ B I G
a X
⋅
⋅
=θ ·‡È̇.
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69
’ÓηÓÓÒ˚Ì B Ó„ÚÎÓΉ „ÓÎ ·‡ ı ÓÓÎÓÈÌ ÏÛ¯„ËðÎ˚Ì ºÌˆ„øø‰ Ú˝Ìˆøø
ı ÓÓÎÓÈ B „ÓÎ B ,, θ θ = ·‡È ı ̺ ı ̂ ºÎººÒ X ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÌÓ.
Õº ı ̂ ºÎËÈ„ ·Ë˜‚ ̋Î
211
22 p p p
e
I Ga X
I Ga X
I GaT
⋅⋅=
⋅⋅−
⋅⋅
·‡È̇. Øø̉ :32
4
1
d I p
⋅= π
- „ÓÎ˚Ì ÚÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ,
32
)3.1(
32
)5.1( 44
2
d d I p
⋅−
⋅=
π π - ı ÓÓÎÓÈÌ ÚÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ.
ƒ ̋ ˝ð ı Ú˝„¯ËÚ„ ̋ ν˝Ò X ñËÈ„ ÓηÓÎ eT ’ 344.0= ·‡È̇. √ ÓÎ ·‡ ı ÓÓÎÓÈÌ
ÏÛ¯„Ë ı
ÏÓÏÂÌÚ˚
Ì˝
Ô˛
ðËÈ„ (5.4·,‚ñð ÁÛð‡„) ÚÛÒ ÚÛÒ‡‰ ̸ ÁÛð·‡.
‡).
·).
‚).
5.4ñð ÁÛð‡„
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∆ ˯˝˝ 5.5. ’Ó∏ð øÁøøð ̸ ı º‰ºÎ„ººÌ„øÈ ·˝ı ν„‰Ò˝Ì 2.1=eT ÍÕÏ
ÏÓÏÂÌÚÓÓð ÏÛ¯„Ë𘠷‡È„‡‡ ¯‡Ú‡ÎÒ‡Ì „‡Ì „ÓÎ˚Ì (5.5‡ñð ÁÛð‡„) 1.
ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
2.
80][ =τ Ãœ‡ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ ÎÚ˝È ø‰ 12 2.1 d d = ı ‡ð¸ˆ‡‡„‡‡ð
„ÓÎ˚Ì ‰Ë‡ÏÂÚðËÈ„ ÚÓ‰Óð ı ÓÈÎ.3. ’‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.4. ÃÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
‡).
·).
‚).
„).
‰).
Â).
5.5ñð ÁÛð‡„
¡Ó‰ÓÎÚ.1. ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ ¿, ¬ ·˝ı ̋΄ ̋ ˝Ìøø‰ ̋ ‰ øøÒ˝ı
A M , B M ð‡͈ ÏÓÏÂÌÚÛÛ‰ ̊ „ (5.5·ñð ÁÛð‡„) ÚÓ‰Óð ı ÓÈÎÓ ı ¯‡‡ð‰Î‡„‡Ú‡È.
–Ú‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ÁÓ ı ËÓ‚ÓÎ0=∑ z T , 0=−+− B e A T T T ⇒ e B A T T T =+
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·ÓÎÌÓ. ’Ó∏ð ð‡͈ ÏÓÏÂÌÚ˚„ Ì˝„ Ú˝„¯ËÚ„ ̋ ν˝Ò ÚÓ‰Óð ı ÓÈÎÊ ˜‡‰‡ ı „øÈ.»Èω ˝Ì˝ ·Ó‰ÎÓ„Ó Ì˝„‰ø„ ̋ ˝ð Á˝ð„ËÈÌ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ.
Õ˝Ï ̋ÎÚ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ¿ ·‡ ¬ Ó„ÚÎÓÎÛÛ‰ ÁÓÓÎÚÚÓÈ ·‡È„‡‡ Û˜Ëð 0= Aθ ,
0= B θ ·‡È ı
ı˝
‚ „‡ÊËÎÚ˚
Ì ı
Ó∏
ð ̺ ı
ˆºÎËÈÌ ‡Î¸ Ì˝
„ËÈ„ ‡¯Ë„Î‡Ì ÁÓ ı
ËÓÌÓ. ØøÌËÈ ÚÛΉ „ÓÎ˚Ì ·‡ðÛÛÌ øÁøøðËÈÌ ·˝ı ̋΄ ̋ ˝„ øÎ Ï ̋‰ ̋ „‰ ̋ ı B T ð‡͈
ÏÓÏÂÌÚÓÓð (5.5‚ñð ÁÛð‡„) ÒÓÎ¸Ê ·ÓÎÌÓ. ’ø˜ÌËÈ øÈΘΠ̋ÎËÈÌ ı ‡Ï‡‡ð‡Î„øÈÌÁ‡ð˜Ï˚„ ‡¯Ë„Î‡Ì ¯ËÎÊËÎÚËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„
0,, =+= B e T B T B B θ θ θ
ı ̋η˝ðÚ˝È ·Ë˜ËÊ ·ÓÎÌÓ. Øø̉ : ¬ Ó„ÚÎÓΉ Áº‚ ı ºÌ eT ÏÓÏÂÌÚÓÓÒ øøÒ˝ı
ÏÛ¯„Ëð‡ÎÚ˚Ì ºÌˆº„
21
15.0,
p
e
p
eT B
I G
T
I G
T e ⋅
⋅+
⋅
⋅=θ , ¬ Ó„ÚÎÓΉ Áº‚ ı ºÌ B T
ÏÓÏÂÌÚÓÓÒ øøÒ˝ı ÏÛ¯„Ëð‡ÎÚ˚Ì ºÌˆº„
21
11,
p
B
p
B T B
I G
T
I G
T B ⋅
⋅−
⋅
⋅−=θ ·‡È̇.
√ ÓÎ˚Ì ı˝Ò˝„ ·øðËÈÌ ÚÛÈÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ 1d „ÓΘÓÓð
Ëνð ı ËÈη ̋Î
41
41 098.0
321d
d I p =
⋅= π
, 41
41
42 203.0
32
)2.1(
322d
d d I p =
⋅=
⋅=
π π ·‡È̇.
eT B ,θ , B T B ,θ ÏÛ¯„Ëð‡ÎÚ˚Ì ºÌˆ„ËÈÌ ÛÚ„ ̊ „ ¯ËÎÊËÎÚËÈÌ Ú˝„¯ËÚ„ ̋ Ή
ÓðÎÛÛη‡Î
01115.0
2121
=⋅
⋅−
⋅
⋅−
⋅
⋅+
⋅
⋅
p
B
p
B
p
e
p
e
I G
T
I G
T
I G
T
I G
T
·Û ̨ Û
0203.0
1
098.0
1
203.0
1
098.0
5.041
41
41
41
=⋅
−⋅
−⋅
+⋅
d
T
d
T
d
T
d
T B B ee ⇒ 795.0663.0 == e B T T ÍÕÏ
·ÓÎÌÓ. ƒ ̋ ˝ð ı ÛÚ„ ̊ „ ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛη‡Î
405.0795.02.1 =−=−=
B e A T T T ÍÕÏ·ÓÎÌÓ. B T ð‡͈ ÏÓÏÂÌÚ˚„ ÓÎÒÌÓÓð ·Ó‰ÎÓ„ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ̸
ڇȷ„‰‡Ê ˆ‡‡¯‰ ̊ Ì ·Ó‰ÓÎÚ Ì¸ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„ ̊ Ì ‡‰ËÎ ·Ó‰Ó„‰ÓÌÓ.ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ÛÚ„ÛÛ‰1-1 Ó„ÚÎÓΉ 5.00 1 ≤≤ z Ï ø‰ 795.01 == B T T ÍÕÏ
2-2 Ó„ÚÎÓΉ 15.0 2 ≤≤ z Ï ø‰ 405.02.1795.02 −=−=−= e B T T T ÍÕÏ
3-3 Ó„ÚÎÓΉ 21 3 ≤≤ z Ï ø‰ 405.02.1795.03 −=−=−= e B T T T ÍÕÏ
·‡È̇. ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ÛÚ„‡ Ó„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝Ì˝˝Ò ı ‡Ï‡‡ð‡ ı „øÈ ÚÛÎ
ÏÛ¯„Ë ı ÏÓÏÂÌÚ ÚÓ‰Óð ı ÓÈÎÓ ı Ó‰ 3-3 Ó„ÚÎÓÎ˚„ ı ËÈ ı ¯‡‡ð‰Î‡„‡„øÈ ·ÓÎÓ‚˜
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî72
ÏÛ¯„ËðÎ˚Ì ºÌˆº„ ÚÓ‰Óð ı ÓÈÎÓ ı Ó‰ ¯‡‡ð‰Î‡„‡Ú‡È Û˜Ëð ˝Ì˝ Ó„ÚÎÓÎ˚„ ı ËÈ‚.›‰„ ̋ ˝ð ÛÚ„‡‡ð ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ (5.5„ñð ÁÛð‡„) ·‡È„ÛÛ·‚.2. √ ÓÎ˚Ì ¯‡‡ð‰Î‡„‡Ú‡È „ÓΘËÈ„ ı ‡Ï„ËÈÌ Ë ı ÏÛ¯„Ë ı ÏÓÏÂÌÚ øÈΘËÎÊ ·ÛÈ
1-1 Ó„ÚÎÓΉ (5.5„ñð ÁÛð‡„) ·‡Ú ·˝ı ËÈÌ (5.5) ̺ ı ̂ ºÎººÒ ÚÓ‰Óð ı ÓÈηÓÎ
037.0108014.3
10795.016
][
16
][
163
6
3
3 13 max1 =
⋅⋅
⋅⋅=
⋅
⋅=
⋅
⋅=
τ π τ π
T T d Ï 37= ÏÏ
·ÓÎÌÓ.¡Ó‰ÎÓ„ ̊ Ì º„º„‰ºÎ ∏ÒÓÓð 12 2.1 d d = (5.5‡ñð ÁÛð‡„) ı ‡ð¸ˆ‡‡„ ı ‡Ì„‡ ı
¯‡‡ð‰Î‡„‡Ú‡È ÚÛÎ 1-1 ·‡ 2-2 Ó„ÚÎÓÎ˚Ì „ÓΘËÈ„ ı ‡ð„‡ÎÁ‡ ı Òڇ̉‡ðÚ˚ÌÓÈðÓΈÓÓ ÛÚ„‡‰ ¯ËÎÊøøη ̋Î 381 =d ÏÏ, 3-3 Ó„ÚÎÓÎ˚Ì ‰Ë‡ÏÂÚð
45382.12.112
≈⋅== d d ÏÏ ·ÓÎÌÓ.
3. ’‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ (5.3) ÚÓÏ ˙∏Ó„ÓÓð „ÓÎ˚Ì ı˝Ò˝„ ·øðÚ ÚÓ‰Óð ı ÓÈÎÌÓ.
1-1 Ó„ÚÎÓΉ 6
3
3
31
111 108.73
)038.0( 14.3
10795.01616
1
⋅=⋅
⋅⋅=
⋅
⋅==
d
T
W
T
p π τ Õ/Ï2
2-2 Ó„ÚÎÓΉ 6
3
3
31
222 106.37
)038.0( 14.3
10 )405.0( 1616
2
⋅−=⋅
⋅−⋅=
⋅
⋅==
d
T
W
T
p π τ Õ/Ï2
3-3 Ó„ÚÎÓΉ 6
3
3
32
333 106.22
)045.0( 14.3
10 )405.0( 1616
3
⋅−=⋅
⋅−⋅=⋅⋅==d
T
W
T
p π τ Õ/Ï2
›‰„ ̋ ˝ð ÛÚ„‡‡ð ı ‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ 5.5‰ñð ÁÛð‡„Ú·‡È„ÛÛ·‚.
4. ÃÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ı˝ÏÊ ̋ ˝„ ÚÓ‰Óð ı ÓÈηÓÎ
1-1 Ó„ÚÎÓΉ 024.0 )038.0( 14.31080
5.010795.0323249
3
41
111 =
⋅⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
2-2 Ó„ÚÎÓΉ 012.0 )038.0( 14.31080
5.010 )405.0( 323249
3
41
222 −=
⋅⋅⋅
⋅⋅−⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
3-3 Ó„ÚÎÓΉ 012.0 )045.0( 14.31080
110 )405.0( 323249
3
42
333 −=
⋅⋅⋅
⋅⋅−⋅=
⋅⋅
⋅⋅=
d G
l T
π θ ð‡‰
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ðËÈ„ 5.5Âñð ÁÛð‡„Ú·‡È„ÛÛ·‚.
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73
5.3. ¡¿ √ ¿ ¿À’¿Ã“¿… ÷»À»Õƒ— œØ—ÿÕ»… “ŒŒ÷ŒŒ
÷ËÎË̉ð Ôøð¯ χ¯ËÌ øÈΉ‚˝ðÎ ̋Î, ·‡„‡ÊÌ˚ øÈΉ‚˝ðÎ ̋Î ·ÓÎÓÌ Ú ı ÌËÍËÈÌ·ÛÒ‡‰ ҇η‡ðÛÛ‰‡‰ ºð„ºÌ ı˝ð˝„턉‰ ̋ „. œøð¯ËÈ„ Ë ı˝‚˜Î˝Ì Ï ı ‡ÌËͯËÌÊ ˜‡Ì‡ð˚Ì ºÌ‰ºð øÁøøÎ ̋ÎÚÚ˝È ººðººð ı ̋η ̋Î Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Π̸ºÌ‰ºð [ ] 800...200=τ Ãœ‡ ·‡È ı „‡Ì„‡‡ð ı Ëȉ ̋ „.
œøð¯ÌËÈ (5.6ñð ÁÛð‡„) ̇·ÎÚ˚Ì ºÌˆº„̸ ·‡„‡ (10 . . . 120ñ‡‡Ò Ë ı „øÈ) „ ̋ Ê øÁ‚ ̋ÎÔøð¯ÌËÈ ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ ı ºÌ‰ÎºÌÓ„ÚÎÓΉ ı Ó∏ð ÚºðÎËÈÌ ‰ÓÚÓÓ‰ ı ø˜ ñ
ÏÛ¯„Ë ı ÏÓÏÂÌÚ (
2
D F M z = ), ı ºÌ‰ÎºÌ
ı ø˜ ( F Q y = ) øøÒ‰ ̋ „. ƒ ̋ ˝ð ı ı Ó∏ð ‰ÓÚÓÓ‰
ı ø˜ ÚÛÒ ·øð˝˝Ò ı ø˜‰ ̋ Î øøÒ˝ı ·º„ºº‰Ôøð¯ÌËÈ ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ ı ºÌ‰ÎºÌÓ„ÚÎÓΉ øøÒ˝ı ı ‡Ï„ËÈÌ Ë ı ı ø˜‰ ̋ ÎËÈ„ ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈΉӄ.
5.6ñð ÁÛð‡„
⎟ ⎠
⎞⎜⎝
⎛ +=+=+=
D
d
d
FD
d
F
d
FD y z Q M
21
848323max
π π π τ τ τ
Øø̉ F ñ ‰‡„ÛÛ ı ø˜, D ñ Ôøð¯ÌËÈ ‰Û̉‡Ê „ÓΘd ñ ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ „ÓΘ
» ı˝‚˜Î˝Ì Ôøð¯ÌËÈ ‰Û̉‡Ê „ÓΘ ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ „ÓΘÓÓÒ ÓÎÓÌ ‰‡ ı ËÌË ı ·‡È‰‡„ ÚÛÎ ı ‡‡Îڇ̉ ·‡È„‡‡ ı Ó∏ð‰Û„‡‡ð Ì˝Ï˝„‰ ̋ ı øøÌËÈ„ ÚÓÓˆ‰Ó„„øÈ·º„ºº‰ Ôøð¯ÌËÈ Ì‡Î‡ÎÚ, ÏÛðÛÈÎÚ, ı ºÌ‰ÎºÌ ı ø˜ÌËÈ ÌºÎºº„ ÚÓÓˆÓ ı
44
14
−
+=
c
ck
ÍÓ˝ÙÙˈËÂÌÚ˚„ ÓðÛÛÎÊ º„‚ºÎ Ôøð¯ÌËÈ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎ
[ ]τ π
τ ≤=3max
8
d
FDk
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. Øø̉d
Dc = ı ‡ð¸ˆ‡‡ ˛Ï.
œøð¯ÌËÈ ÒÛÛÎÚ˚„ (ÛðÚÒ‡ÎÚ ·Û ̨ Û ·Ó„ËÌÓÒÓÎÚ)
4
38
d G
n FD
⋅=λ
ÚÓÏ ˙∏
Ó„ÓÓð ÚÓ‰Óð ı
ÓÈÎÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî74
Øø̉ : n - Ôøð¯ÌËÈ ‡ÊÎ˚Ì ÓðÓÓ‰Ò˚Ì ÚÓÓG - ¯ËÎÊËÎÚËÈÌ ÏÓ‰Ûθ
œøð¯ÌËÈ ı˝
‚ „‡ÊËÎÚ˚
Ì ·øÚ˝
Ì ÔÓÚÂÌˆË‡Î˝
ÌÂð„ËÈ„ 2
λ ⋅=
F
U ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ.
∆ ˯˝˝ 5.6. ÿ‡ ı ‡Îڇ̉ ‡ÊËη ı Ôøð¯ÌËÈ
(5.6ñð ÁÛð‡„) ¯ËÎÊËÎÚËÈÌ ÏÓ‰Ûθ ( 4108 ⋅=G Ãœ‡),Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î ( 250=τ Ãœ‡) º„º„‰ÒºÌ ø‰
5.7ñð ÁÛð‡„Ú øÁøøÎÒ˝Ì ı ø˜, ÒÛÛÎÚ˚Ì ı ‡Ï‡‡ðÎ˚„ ı ‡Ì„‡Ê ˜‡‰‡ ı Ôøð¯ÌËÈ ‰Û̉‡Ê „ÓΘ, ÓðÓÓ‰ÓÒ
ÛÚ‡ÒÌ˚
„ÓΘ, ÓðÓÓ‰Ò˚
Ì ÚÓÓ„ ÚÓ‰Óð ı
ÓÈÎ.5.7ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ”𸉘ËÎÒ‡Ì ·‡È‰Î‡‡ð Ôøð¯ÌËÈ ‰Û̉‡Ê „ÓΘ, ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ „ÓΘËÈÌ
ı ‡ð¸ˆ‡‡„ 5==d
Dc ·‡È ı ‡‡ð (Ë ı˝‚˜Î˝Ì 10...5=c ·‡È‰‡„) ‡‚ ˙ˇ.
ƒ ̋ ˝ð ı ı ‡ð¸ˆ‡‡„ ÚÓÓˆÓÌ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººÒ ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ „ÓΘËÈ„ÓηÓÎ
[ ]τ π
τ ≤⋅⋅
=2
maxmax
8
d
F ck ⇒
[ ]τ π ⋅
⋅⋅⋅≥
c F kd max8
·ÓÎÌÓ. Øø̉ : 5=c ø‰ 31.1454
154
44
14=
−⋅
+⋅=
−
+=
c
ck ·‡È ı ÚÛÎ 4000max = F Õ
„ ̋ ‰„ËÈ„ Ôøð¯Ë̉ øÈÎ˜Î˝ı ı ø˜, ÒÛÛÎÚ˚Ì ı ‡Ï‡‡ð·‡Ò(5.7ñð ÁÛð‡„) ‡‚˜ ÚÓÓÌ ÛÚ„ÛÛ‰ ̊ „ ÓðÎÛÛη‡Î
3
610163.16
1025014.3
54000831.1 −⋅≈=
⋅⋅
⋅⋅⋅≥d Ï 16= ÏÏ
·ÓÎÌÓ. 16=d ÏÏ ø‰ 80516 =⋅=⋅= cd D ÏÏ ·‡È̇.4000max = F Õ ‡˜‡‡ÎÎ˚Ì ø‰ Ôøð¯ÌËÈ ÒÛÛÎÚ 40=λ ÏÏ ·‡È ı ̺ ı ̂ ºÎËÈ„
‡¯Ë„Î‡Ì ‰ÓÓð ı ÚÓÏ ˙∏ÓÌÓÓÒ ÓðÓÓ‰Ò˚Ì ÚÓÓ„ ÚÓ‰Óð ı ÓÈÎÌÓ.
4
3max8
d G
n D F
⋅=λ ⇒ 138.12
08.040008
016.010804.0
8 3
410
3max
4
≈=⋅⋅
⋅⋅⋅==
D F
d Gn
λ
’ ̋ ð˝‚ 10==d
Dc ·‡È ı ‡‡ð ‡‚·‡Î ‰ ̋ ˝ð ı Ú˝È ËÊËÎ ‡ð„‡‡ð ·Ó‰Ó ı Ó‰
22=d ÏÏ, 220= D ÏÏ, 2≈n ·‡È̇.
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75
ƒ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎÒÓÌ ı˝ÏÊ ̋ ˝Òøø‰Ú˝È ı Ó∏ð Ôøð¯ÌËÈ ‡ÎËÈ„ ̸ ˜ ‡‚˜ ·ÓÎÓ ı
·‡ ¯ËÌ˝˝ðd
Dc = ı ‡ð¸ˆ‡‡„ ººð˜ÎºÌ ººð ººð ı˝ÏÊ ̋ ˝Òøø‰Ú˝È Ôøð¯ ÒÓÌ„ÓÒÓÌ
˜ ·ÓÎÌÓ.
∆ ˯˝˝ 5.7. 2021 == d d ÏÏ „ÓΘÚÓÈ „‡Ì
ÛÚÒ‡‡ð ı ËÈÒ˝Ì 10021 == D D ÏÏ ‰Û̉‡Ê „ÓΘÚÓÈ 1
·‡ 2 Ôøð¯ÌËÈ (5.8ñð ÁÛð‡„) ÓðÓÓ‰Ò˚Ì ÚÓÓ Ì¸ ÚÛÒ ·øð81 =n , 62 =n ·ÓÎ 5= F ÍÕ ı ø˜ øÈÎ˜Î˝ı ø‰ –
ˆ ̋ „ËÈÌ ¯ËÎÊËÎÚËÈ„ ÚÓ‰Óð ı ÓÈÎ. ¿˜‡‡Î‡Î„øÈ ø‰ ı ‡‚Ú‡Ì ·‡ 2 Ôøð¯ÌËÈ ı ÓÓðÓ̉ 10=Δ ÏÏ Á‡‚Ò‡ðڇȷ‡È‚.
5.8ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ’ ̋ ð˝‚ 1 Ôøð¯ÌËÈ ÒÛ̇ÎÚ Δ Á‡È̇‡Ò Ë ı ·ÓÎ ˝Ì˝ ·Ó‰ÎÓ„Ó Ì¸ ÒÚ‡ÚËÍËÈÌ
ı Û‚¸‰ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ˛Ï. »Èω ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈÌ ı ‡ÏÚ ı˝‚ „‡ÊËÎÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ı˝ð˝„νı ¯‡‡ð‰Î‡„‡Ú‡È.
1 Ôøð¯Ë̉ øÈÎ˜Î˝ı ı ø˜ËÈ„ 1 N -˝˝ð, 2 Ôøð¯Ë̉ øÈÎ˜Î˝ı ı ø˜ËÈ„ 2 N -ÓÓð
Ú˝Ï‰ ̋ „Î˝Ì ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ·Ë˜‚ ̋Î F N N 221 =+
·ÓÎÌÓ. ’Ó∏ð øÎ Ï ̋‰ ̋ „‰ ̋ „˜Ú˝È Ì˝„ Ú˝„¯ËÚ„ ̋ Î ·‡È„‡‡ ÚÛÎ Ì˝„‰ø„ ̋ ˝ð Á˝ð„ËÈÌÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ.
› ı ν ̋‰ 1 Ôøð¯ 10=Δ ÏÏ ÒÛ̇ÒÌ˚ ‰‡ð‡‡ 2 Ôø即‡ ı ‡„‰‡Ê ˝ı νı Û˜Ëð ı˝‚ „‡ÊËÎÚ˚Ì Ú˝„¯ËÚ„ ̋ Î
1021 =−λ λ
·ÓÎÌÓ. ØøÌËÈ„ ı ø˜˝˝ð Ëνð ı ËÈη ̋Î
1088
42
2322
41
1311 =
⋅
⋅⋅−
⋅
⋅⋅
d G
n D N
d G
n D N ⇒ 102211 =⋅−⋅ n N n N ⇒ 1068 21 =− N N
·ÓÎÌÓ. ›Ì˝ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎÚ˝È ı ‡ÏÚ‡Ú„‡Ì·Ó‰‚ÓÎ
106 )10( 8 22 =−− N N ⇒ 52 = N ÍÕ, 551010 21 =−=−= N N ÍÕ
·ÓÎÌÓ. »Ì„ ̋ Ê Ôøð¯ ÚÛÒ ·øðÚ øÈÎ˜Î˝ı ‰ÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÒÌÓÓðÒÚ‡ÚËÍËÈÌ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„Ó ·ÓÎÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî76
– ˆ ̋ „ËÈÌ ¯ËÎÊËÎÚ 1 Ôøð¯ÌËÈ ÒÛÛÎÚÚ‡È Ú˝Ìˆøø ·‡È̇.
025.0 )02.0( 1080
8 )1.0( 1058849
33
4
1
1311
1 =⋅⋅
⋅⋅⋅⋅=
⋅
⋅⋅==
d G
n D N C λ δ Ï 25= ÏÏ
∆ ˯˝˝ 5.8. œøð¯ÌËÈ (5.9‡ñð ÁÛð‡„) ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ÒËÒÚÂÏËÈÌ
Ôøð¯ ÚÛÒ ·øð‰ øøÒ˝ı ı ø˜‰ ̋ Î, 3 Ôøð¯ÌËÈ ‰Û̉‡Ê – ˆ ̋ „ËÈÌ ÒÛÛÎÚ˚„ ‰‡ð‡‡ ı º„º„‰Îøø‰ ̋ ˝ð ÚÓ‰Óð ı ÓÈÎ.
ª„ÒºÌ Ì¸:
2= F ÍÕ, 10108 ⋅=G œ‡,51 = D ÒÏ, 102 = D ÒÏ,
153 = D ÒÏ, 41 =d ÏÏ,
82 =d ÏÏ, 123 =d ÏÏ,
101 =n , 82 =n , 63 =n
5.9ñð ÁÛð‡„
¡Ó‰ÓÎÚ.
1. –Ú‡ÚËÍ Ú‡Î. ’‡‚ڇ̄ËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ºÎ (5.9·ñð ÁÛð‡„) ·ÓÎÓÌ 2 Ôøð¯Ë̉ øÈÎ˜Î˝ı 2 N ı ø˜, 1 Ôøð¯Ë̉ øÈÎ˜Î˝ı 1 N ı ø˜Ú˝È Ú˝Ìˆøø ·‡È ı
̺ ı ̂ ºÎËÈ„ (5.9‚ñð ÁÛð‡„) ‡¯Ë„Î‡Ì ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ Î ·Ë˜‚ ̋Î
⎩⎨⎧
=
=+
21
31
N N
F N N
·ÓÎÌÓ. √ Ûð‚‡Ì øÎ Ï ̋‰ ̋ „‰ ̋ „˜Ú˝È, ı Ó∏ð Ú˝„¯ËÚ„ ̋ Î ·Ë˜Ë„‰Ò˝Ì Û˜Ëð Ì˝„‰ø„ ̋ ˝ðÁ˝ð„ËÈÌ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó ˛Ï. 2. √ ÂÓÏÂÚð Ú‡Î. ’ ̋ ‚ „‡ÊËÎÚ˚Ì Ìº ı ̂ ºÎ ̸ 1 ·‡ 2 Ôøð¯ÌËÈ ÒÛÛÎÚ˚Ì ÌËÈη˝ð
̸ 3 Ôøð¯ÌËÈ ÒÛÛÎÚÚ‡È Ú˝
̈øø ·‡È̇.321 λ λ λ =+
3. ‘ËÁËÍ Ú‡Î. ›Ì˝ ı˝Ò˝„Ú ı˝‚ „‡ÊËÎÚ˚„ (Ôøð¯ÌËÈ ÒÛÛÎÚ˚„) ‰ÓÚÓÓ‰ ı ø˜˝˝ðËνð ı ËÈη ̋Î
41
1311
1
8
d G
n D N
⋅
⋅⋅=λ
42
2322
2
8
d G
n D N
⋅
⋅⋅=λ
·ÓÎÌÓ.
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
77
43
3333
3
8
d G
n D N
⋅
⋅⋅=λ
4. –ËÌÚÂÁ. ›Ì˝ ı˝Ò˝„Ú ˝ı ÌËÈ „Ûð‚‡Ì ı˝Ò„ËÈÌ Ú˝„¯ËÚ„ ̋ Îøø‰ËÈ„ Ì˝„Ú„ ̋ Ì
·Ó‰ÓÊ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ËÈ„ ڇȷ ı ∏ÒÚÓÈ. ØøÌËÈ ÚÛΉ ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ν˝Ò 2 ·‡ 3 Ôøð¯Ë̉ øÈÎ˜Î˝ı ‰ÓÚÓÓ‰ ı ø˜ËÈ„ 1 Ôøð¯Ë̉ øÈÎ˜Î˝ı ı ø˜˝˝ð 13 N N N −= , 12 N N = „ ̋ Ê Ëνð ı ËÈÎ˝Ì ı˝‚
„‡ÊËÎÚ˚Ì Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛÎÊ ˝Ï ı ̋Ú„ ̋ ‚ ̋Î
( )42
232
142
232
41
131
1d
n D N N
d
n D
d
n D N
⋅⋅−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ⋅+
⋅⋅
·ÓÎÌÓ. ÷‡‡¯Ë‰ º„º„‰ÒºÌ ÚÓÓÌ ÛÚ„ ̊ „ ÓðÎÛÛÎ‡Ì ·Ó‰‚ÓÎ
( ) 4
3
14
3
4
3
112
615020008
81004
1050 ⋅⋅−=⎟⎟ ⎠
⎞
⎜⎜⎝
⎛ ⋅+⋅⋅ N N ⇒ 2501 = N Õ
·ÓÎÌÓ. ÃºÌ 25012 == N N Õ, 1750250200013 =−=−= N N N Õ ·ÓÎÊ
·Ó‰ÎÓ„ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ڇȷ„‰‡Ê ˆ‡‡¯Ë‰ ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ·Ó‰ÎÓ„ ̊ Ì ‡‰ËÎ ·Ó‰Ó„‰ÓÌÓ.
œøð¯ ÚÛÒ ·øð‰ øøÒ˝ı ı ø˜‰ ̋ Î
6
331
111 103.552
)004.0( 14.3
05.0250811.1
8⋅=
⋅
⋅⋅⋅=
⋅
⋅⋅⋅=
d
D N k
π τ œ‡ 3.552= Ãœ‡
6
332
222 101.138
)008.0( 14.3
1.0250811.18 ⋅=⋅
⋅⋅⋅=⋅⋅⋅⋅=
d
D N k
π τ œ‡ 1.138= Ãœ‡
6
333
333 106.429
)012.0( 14.3
15.01750811.1
8⋅=
⋅
⋅⋅⋅=
⋅
⋅⋅⋅=
d
D N k
π τ œ‡ 6.429= Ãœ‡
·‡È̇. Øø̉ 11.145.124
15.124
14
14321 =
−⋅
+⋅=
−
+====
c
ckkkk ·‡È̇. ”˜Ëð ̸ ·ø ı
Ôøð¯ÌËÈ ‰Û̉‡Ê „ÓΘ, ÓðÓÓ‰ÓÒ ÛÚ‡ÒÌ˚ „ÓΘËÈÌ ı ‡ð¸ˆ‡‡ Ú˝Ìˆøø
5.124
501
11 === d Dc , 5.12
8100
2
22 === d Dc , 5.12
12150
3
33 === d Dc ·‡È„‡‡ ˛Ï.
C ˆ ̋ „ 3 Ôøð¯ÌËÈ ‰Û̉‡Ê ˆ ̋ „ ÚÛÎ ÚøøÌËÈ ¯ËÎÊËÎÚ Ì¸ 3 Ôøð¯ÌËÈÒÛÛÎÚ˚Ì ı ‡„‡ÒÚ‡È Ú˝Ìˆøø ·‡È̇.
085.0 )012.0( 1082
6 )15.0( 17508
2
8
2 410
3
43
33333 =
⋅⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅==
d G
n D N C
λ δ Ï 85= ÏÏ
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ƒ¿¿À√ ¿¬¿— ¹4. ƒ‡‡Î„‡‚‡ð „Ûð‚‡Ì ·Ó‰ÎÓ„ÓÓÒ ·øð‰ ̋ Ì˝.
1.
ƒÛ„ÛÈ Ó„ÚÎÓÎÚÓÈ ¯Ëη˝ÌËÈ ÏÛ¯„Ëð‡ÎÚ ̊ Ì øÂËÈÌ ·‡Ú ·˝ı , ı º¯øøÌ ˜‡Ì‡ð˚Ì ÚÓÓˆÓÓ
ª„º„‰ÒºÌ „‡Ì „ÓÎ˚Ì (5.10ñð ÁÛð‡„) 80][ =τ Ãœ‡ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
·‡ 5.0=‡ Ï Á‡È º„º„‰ÒºÌ ø‰ 5.1ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð1. „ÓÎ˚Ì ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ;2. ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººÒ „ÓÎ˚Ì „ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎ;3. ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
5.10ñð ÁÛð‡„
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5.10ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî80
5.1ñð ı øÒÌ˝„Ú
¹,1 P
ͬÚ
,2 P
ͬÚ
,3 P
ͬÚ
,4 P
ͬÚ
,
ð‡‰/Ò¹
,1 P
ͬÚ
,2 P
ͬÚ
,3 P
ͬÚ
,4 P
ͬÚ
,
ð‡‰/Ò
1 1.1 2.2 3.3 4.4 4 11 0.5 0.6 0.7 0.8 15
2 2.2 3.3 4.4 1.1 50 12 0.6 0.7 0.8 0.5 20
3 3.3 4.4 1.1 2.2 6 13 0.7 0.8 0.5 0.6 25
4 4.4 3.3 2.2 1.1 70 14 0.8 0.7 0.6 0.5 30
5 1.0 2.0 3.0 4.0 8 15 4.0 5.0 6.0 7.0 12
6 2.0 3.0 4.0 1.0 90 16 5.0 6.0 7.0 4.0 40
7 3.0 4.0 1.0 2.0 10 17 6.0 7.0 4.0 5.0 35
8 4.0 1.0 2.0 3.0 150 18 7.0 4.0 5.0 6.0 120
9 6.0 7.0 8.0 10.0 100 19 9.0 10.0 12.0 11.0 18010 10.0 20.0 30.0 40.0 200 20 15.0 25.0 35.0 45.0 220
2. ÃÛ¯„Ëð‡ÎÚ ̊ Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ÒËÒÚÂÏËÈÌ ÚÓÓˆÓÓ
ª„º„‰ÒºÌ „ÓÎ˚Ì (5.11ñð ÁÛð‡„) Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î 80][ =τ Ãœ‡,
¯ËÎÊËÎÚËÈÌ ÏÓ‰Ûθ 4108 ⋅=G Ãœ‡ ·‡È ı ‡‰ 5.2ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð 1. ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎººÒ „ÓÎ˚Ì ı˝Ò˝„ ·øðËÈÌ „ÓΘËÈ„ ÚÓ‰Óð ı ÓÈÎ;
2. ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ;3. ÏÛ¯„ËðÎ˚Ì ºÌˆ„ËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ;4. ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓΉ ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
5.11ñð ÁÛð‡„
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81
5.11ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî82
5.11ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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83
5.2ñð ı øÒÌ˝„Ú
¹,a
Ï
,b
Ï
,c
Ï 1
2
d
d
1
3
d
d ,1e
T
ÍÕÏ
,1e
T
ÍÕϹ
,a
Ï
,b
Ï
,c
Ï 1
2
d
d
1
3
d
d ,
1eT
ÍÕÏ
,2e
T
ÍÕÏ
1 0.2 0.1 0.5 1.2 3.2 3.4 1.0 11 0.3 0.4 0.6 1.3 3.6 3.2 0.8
2 0.2 0.4 1.1 1.4 4.0 2.0 -5.0 12 0.3 0.2 0.7 1.5 3.8 -4.0 1.8
3 0.4 0.8 0.1 1.6 4.2 -1.0 2.0 13 0.5 0.6 0.1 1.7 4.2 1.2 2.4
4 0.6 0.3 0.6 1.8 3.1 4.0 -6.0 14 0.7 0.8 1.1 1.9 4.5 3.0 5.0
5 0.8 0.4 0.9 2.0 3.4 -1.0 3.0 15 0.8 0.3 0.9 2.1 4.2 1.5 2.5
6 0.7 0.9 1.0 2.4 3.0 2.0 -3.0 16 0.2 0.7 0.8 2.3 4.6 -3.0 1.0
7 0.6 0.8 0.1 2.6 3.2 -3.0 4.0 17 0.4 1.0 0.3 2.5 4.4 5.5 -2.0
8 0.8 1.0 0.3 2.8 3.6 4.0 2.0 18 0.2 0.5 0.2 2.7 4.1 6.0 -3.0
9 0.2 0.5 0.2 3.0 4.0 5.0 -5.0 19 0.8 0.9 0.5 2.9 3.5 -3.5 2.5
10 0.4 0.6 0.9 1.3 3.8 -6.0 1.0 20 0.5 0.1 0.6 1.3 3.9 2.0 -4.0
3. ¡‡„‡ ‡Î ı ‡ÏÚ‡È ˆËÎË̉ð Ôøð¯ÌËÈ ÚÓÓˆÓÓ
œøð¯ÌËÈ ÒÚ‡ÚËÍ ÚÓ‰Óð ı
ÓÈ ·Ë¯ ÒËÒÚÂÏËÈÌ (5.12ñð ÁÛð‡„) Á‡‡Ò‡Ì ˆ ˝
„Ú(ÚÛ ı ‡ÈÌ Ôøð¯ÌËÈ ‰Û̉ Óð¯Ë ı ¿, B , C ˆ ̋ „) ¯ËÎÊËÎÚ, Ôø叟øðÚ øøÒ˝ı maxτ
ı ø˜‰ ̋ ÎËÈ„ 5.3ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ÚÓ‰Óð ı ÓÈÎ.œøð¯Ìøø‰ËÈÌ ‰Û̉‡Ê „ÓΘ 51 = D ÒÏ, 102 = D ÒÏ, 153 = D ÒÏ,
ÓðÓÓ‰Ò˚Ì „ÓΘ 41 =d ÏÏ, 82 =d ÏÏ, 123 =d ÏÏ, ¯ËÎÊËÎÚËÈÌ ÏÓ‰Ûθ
4108 ⋅=G Ãœ‡ ·ÓÎÌÓ.
5.12ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî84
5.12ñð ÁÛð‡„ ( øð„ ̋ ÎÊÎ ̋Î)
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85
5.3ñð ı øÒÌ˝„Ú
¹ , F ÍÕ 1n 2n 3n ÿËÎÊËÎÚÓÎÓ ı ˆ ̋ „
¹ , F ÍÕ 1n 2n 3n ÿËÎÊËÎÚÓÎÓ ı ˆ ̋ „
1 1.0 100 80 60 ¿ 11 1.2 120 80 60 B2 2.0 10 8 6 B 12 2.4 90 60 30 C
3 0.5 20 16 12 C 13 0.8 24 18 14 ¿
4 3.0 30 20 10 ¿ 14 3.2 40 20 10 B
5 1.5 40 40 20 B 15 1.8 60 40 20 C
6 1.0 50 40 30 C 16 1.6 50 30 30 ¿
7 4.0 60 40 40 ¿ 17 3.6 40 20 20 B
8 5.0 70 60 40 B 18 4.8 90 70 40 C
9 2.0 80 70 60 C 19 3.5 110 60 50 ¿
10 4.0 90 70 50 ¿ 20 4.5 80 60 40 B
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«”—√ ¿¿ƒ” √ ¿¿— ¡ØÀ›√
ƒ¿ÃÕ””—¤Õ √ ”À«¿…À“
6.1. ƒ‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð
√ ÛÎÁ‡ÈÎڇ̉ ‡ÊËη ı ¯ÛÎÛÛÌ ¯ËηËÈ„ ‰‡ÏÌÛÛð „ ̋ Ì˝. ƒ‡ÏÌÛÛð˚Ì Ú˝Ì ı ν„ËÈÌ ÏÛðÛÈÎÚ˚„ ººð˜Îº ı ı˝‚ „‡ÊËÎÚ˚„ „ÛÎÁ‡ÈÎÚ „ ̋ Ì˝. √ ÛÎÁ‡ÈÎÚ øøÒ„ ̋ ı „‡‰‡‡‰ ı ø˜Ìøø‰ ¯Ëη˝ÌËÈ Ú˝Ì ı ν„ËÈ„ ‰‡ÈðÒ‡Ì Ì˝„ ı ‡‚Ú„‡È‰·‡Èð·̇. »ÌÂðˆËÈÌ „ÓÎ Ú˝Ì ı ν„ËÈ„ ‰‡ÈðÒ‡Ì ı ‡‚Ú„‡È„ ËÌÂðˆËÈÌ „ÓÎ ı ‡‚Ú„‡È„ ̋ ‰ ̋ „. ’ø˜ÌËÈ øÈΘΠ̋ÎËÈÌ ı ‡‚Ú„‡È ËÌÂðˆËÈÌ „ÓÎ ı ‡‚Ú„‡ÈÚ‡È ‰‡‚ ı ̂ ‡ ı ø‰ ı ‡‚Ú„‡ÈÌ „ÛÎÁ‡ÈÎÚ øøÒÌ˝. ƒ‡ÏÌÛÛð˚Ì ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ ı ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ „ ̋ Ò˝Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ı Ó∏ð Ù‡ÍÚÓð øÈÎ˜Î˝Ì˝. «º‚ ı ºÌ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì øÈΘΠ̋ν˝ð øøÒ˝ı „ÛÎÁ‡ÈÎÚ˚„ ˆ ̋ ‚˝ð „ÛÎÁ‡ÈÎÚ, ı ºÌ‰ÎºÌ ı ø˜ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ı ‡ÏÚ˚Ì øÈΘΠ̋ν˝ð øøÒ˝ı „ÛÎÁ‡ÈÎÚ˚„ ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ „ ̋ Ì˝.
ƒ‡ÏÌÛÛð˚Ì ·‡Ú ·˝ı , ı º¯øøÌ ˜‡Ì‡ð˚„ ÚÓÓˆÓ ı˚Ì ÚÛΉ ̨ÛÌ˚ ºÏ̺ ‰ÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ı˝ð˝„Ú˝È. ƒÓÚÓÓ‰ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ Ó„ÚÎÓÎ˚Ì
‡ð„˚„ ı˝ð˝„Î˝Ì˝. ƒ‡ÏÌÛÛð˚Ì ÚÛ ı ‡ÈÌ ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ øÈÎ˜Î˝ı ı ºÌ‰ÎºÌ ı ø˜ Ú˝ð
Ó„ÚÎÓÎ˚Ì ‡Î¸ Ì˝„ ڇΉ Óð¯Ë ı ·ø ı ı ø˜Ìøø‰ËÈÌ Ú˝ð Ó„ÚÎÓÎ ‰ ̋ ˝ð ı ÔðÓÂ͈ÛÛ‰ ̊ Ì ‡Î„·ð ÌËÈη˝ðÚ˝È Ú˝Ìˆøø; „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ Ú˝ð Ó„ÚÎÓÎ˚Ì
‡Î¸ Ì˝„ ڇΉ Óð¯Ë ı ·ø ı ı ø˜Ìøø‰ ̋ ˝Ò Ó„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ øøÒ„ ̋ ı ÏÓÏÂÌÚÛÛ‰ ̊ Ì ‡Î„·ð ÌËÈη˝ðÚ˝È Ú˝Ìˆøø ·‡È̇.
’ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝ ‰‡ÏÌÛÛð˚Ì Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ‰ ººð˜Îº„‰º ı ı ÛÛÎËÈ„ ı ‡ðÛÛÎÒ‡Ì „ð‡ÙËÍËÈ„ Ú ̋‰„ ̋ ˝ðËÈÌ ˝Ô˛ð „ ̋ Ì˝.›‰„ ̋ ˝ðËÈÌ ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ºÏ̺ Ú˝Ï‰„ËÈÌ ‰øð˝Ï ÚÓ„ÚÓÓ ı ¯‡‡ð‰Î‡„‡Ú‡È.
ƒ‡ÏÌÛÛð˚Ì ˇÏ‡ð Ì˝„ mn Ó„ÚÎÓΉ ı ºÌ‰ÎºÌ ı ø˜ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓÓ„ÚÎÓÎÓÓÒ ÁøøÌ Ú‡Î‰ ‰ ̋ ˝¯˝˝ ˜Ë„Î˝Ò˝Ì, ·‡ðÛÛÌ Ú‡Î‰ ‰ÓÓ¯ÓÓ ˜Ë„Î˝Ò˝Ì ı ºÌ‰ÎºÌ ı ø˜ËÈ„ ˝Âð˝„ (6.1‡ñð ÁÛð‡„), ˝Òð˝„ (6.1·ñð ÁÛð‡„) ÚÓ ı ËÓΉÓΉ Һ𺄄 ̋ Ê ÚÓÓˆÌÓ.
‡). ·). ‚). „).6.1ñð ÁÛð‡„
ÃºÌ Ó„ÚÎÓΉ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ Ó„ÚÎÓÎ˚Ì ÁøøÌ
ڇ·‡Ò ˆ‡„ËÈÌ ÁøøÌËÈ ı
º‰ºÎ„ººÌËÈ ‰‡„ÛÛ ˜Ë„ν
ÎÚ˝
È, ·‡ðÛÛÌ Ú‡Î‡‡Ò ˆ‡„ËÈÌ
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ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„ ˜Ë„Î ̋ÎÚ˝È øÈÎ˜Î˝ı ÏÓÏÂÌÚ˚„ ˝Âð˝„ (6.1‚ñðÁÛð‡„); ˝Òð˝„ ÚÓ ı ËÓΉÓΉ Һ𺄠(6.1„ ñð ÁÛð‡„) „ ̋ Ê ÚÓÓˆÌÓ.
’ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð Áº‚ ·‡È„ÛÛ·„‰Ò‡Ì ˝Ò˝ı ËÈ„ ‰‡ð‡‡ ı ¯ËÌÊøø‰ ̋ ˝ð ¯‡Î„‡Ê ·ÓÎÌÓ. Øø̉ :1. “º‚κðÒºÌ ı ø˜ øÈΘËÎÊ ·‡È„‡‡ Ó„ÚÎÓΉ ı ºÌ‰ÎºÌ ı ø˜ÌËÈ ˝Ô˛ð Ú˝ð ı ø˜ÌËÈ ı˝ÏÊ ̋ ˝„ ̋ ˝ð øÒð ̋ÎÚÚ˝È, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ı ø˜ÌËÈ ˝Òð˝„ ı ‡Ì‰Ò‡Ì ı Û„‡ð‡ÎÚ‡È ·‡È̇.2. “º‚κðÒºÌ ÏÓÏÂÌÚ øÈΘËÎÊ ·‡È„‡‡ Ó„ÚÎÓΉ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð Ú˝ð ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝„ ̋ ˝ð øÒð ̋ÎÚÚ˝È ·‡È̇. “º‚κðÒºÌ ÏÓÏÂÌÚ ı ºÌ‰ÎºÌ ı ø˜ÌËÈ ˝Ô˛ðËÈÌ ı ̋η˝ðÚ ÌºÎººÎº ı „øÈ.3. ∆ Ë„‰ Ú‡ð ı Ò‡Ì ı ø˜ øÈΘËÎÊ ·‡È„‡‡ ı˝Ò˝„Ú ı ºÌ‰ÎºÌ ı ø˜ÌËÈ ˝Ô˛ð Ú‡ð ı Ò‡Ì ı ø˜ÌËÈ ˜Ë„Î ̋Î ‰‡„ÛÛ Ì‡ÎÛÛ ¯ÛÎÛÛ̇‡ð, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð „ø‰„ ̋ ð ڇ·‡ð‡‡ ‡˜‡‡Ì˚ ˝Òð˝„ ı ‡Ì‰Ò‡Ì Í‚‡‰ð‡Ú Ô‡ð‡·ÓÎÓÓð ÁÛð‡„‰‡Ì‡.4. ’ºÌ‰ÎºÌ ı ø˜ Ú˝„ ·‡È ı Ó„ÚÎÓΉ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ ˝ÍÒÚðÂÏÛÏ ÛÚ„‡Ú‡È·‡È̇.5. ’ºÌ‰ÎºÌ ı ø˜ ˝Âð˝„ ·‡È ı ı˝Ò˝„Ú „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ ºÒºÊ, Һ𺄠·‡È ı ı˝Ò˝„Ú ·ÛÛð̇.
∆ ˯˝˝ 6.1. ª„º„‰ÒºÌ ‰‡ÏÌÛÛð˚Ì (6.2‡ñð ÁÛð‡„) ı ºÌ‰ÎºÌ ı ø˜,
„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
‡).
ª„ÒºÌ Ì¸ :3= F ÍÕ
2=q ÍÕ/Ï
4= M ÍÕÏ1=a Ï
¡Ó‰ÓÎÚ. “Û΄ÛÛð˚Ì
ð‡͈ ̊ „ ÓÎÓ΄øÈ, ÁøøÌ øÁøøð˝˝Ò ˝ı Î˝Ì ·Ó‰ ˙∏.1-1 Ó„ÚÎÓÎ ( a z ≤≤ 10 )
3 )( 1 == F z Q ÍÕ
11 )( z F z M −=
01 = z ø‰ 0 )( 1 = z M ;
a z =1 ø‰ 3 )( 1 −= z M ÍÕÏ.
2-2 Ó„ÚÎÓÎ ( a z a 2.22 ≤≤ )
)( )( 22 a z q F z Q −+=
2
)( )(
22
22
a z q z F z M
−−⋅−=
·).
‚).
6.2ñð ÁÛð‡„
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a z =2 ø‰ 3 )( 2 = z Q ÍÕ, 3 )( 2 −= z M ÍÕÏ
a z 2.22 = ø‰ 4.5 )( 2 = z Q ÍÕ, 04.8 )( 2 −= z M ÍÕÏ
3-3 Ó„ÚÎÓÎ ( a z a 2.32.23 ≤≤ )
4.52.1 )( 3 =⋅+= aq F z Q ÍÕ
M a z aq z F z M +−⋅⋅−⋅−= )6.1( 2.1 )( 333
a z 2.23 = ø‰ 04.4 )( 3 −= z M ÍÕÏ
a z 2.33 = ø‰ 44.9 )( 3 −= z M ÍÕÏ
›‰„ ̋ ˝ð ÛÚ„‡‡ð ı ºÌ‰ÎºÌ ı ø˜ (6.2·ñð ÁÛð‡„), „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.2‚ñðÁÛð‡„) ˝Ô˛ðËÈ„ ·‡È„ÛÛ·‚. ƒ ̋ ˝ð ı ¯ËÌÊ ̋ ˝ð ÁÓÓÎÚÓÌ ‰ ̋ ˝ð øøÒ˝ı ·ÓÒÓÓ ð‡͈5.4ÍÕ, ÏÓÏÂÌÚ˚Ì ð‡͈ 9.44ÍÕÏ ·‡È̇.
∆ ˯˝˝ 6.2. ª„º„‰ÒºÌ ‰‡ÏÌÛÛð˚Ì (6.3‡ñð ÁÛð‡„) ı ºÌ‰ÎºÌ ı ø˜,
„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
a).
ª„ÒºÌ Ì¸ :4= F ÍÕ
3=q ÍÕ/Ï
1=a Ï
¡Ó‰ÓÎÚ. ›Ì˝ ·Ó‰ÎÓ„ ̊ „ Á‡‚Òð˚Ì
– ÌÛ„‡Ò‡‡ð ҇΄‡Ì ·Ó‰ÌÓ. “ ̋ „‚ ̋Î ı Ó∏ð ı ̌η‡ð ‰‡ÏÌÛÛð (6.3·ñð ÁÛð‡„)·ÓÎÌÓ.
› ı ν ̋‰ ·‡ðÛÛÌ CB ‰‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ ˙ˇ.
2
F R R B C == ·‡È ı ̸ Ëνð ı ËÈ. ›Ì˝
‰‡ÏÌÛÛð ı Ó∏ð ı˝Ò˝„Ú˝È ·‡È̇.1-1 Ó„ÚÎÓÎ ( a z ≤≤ 10 )
22
)( 1 === F R z Q C ÍÕ
11 )( z R z M C ⋅=
01 = z ø‰ 0 )( 1 = z M ;
a z =1 ø‰ 2 )( 1 =⋅= a R z M C ÍÕÏ.
·).
‚).
„).
6.3ñð ÁÛð‡„
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2-2 Ó„ÚÎÓÎ ( a z ≤≤ 20 )
22
)( 2 −=−=−= F
R z Q B ÍÕ
22 )( z R z M B ⋅= 02 = z ø‰ 0 )( 2 = z M ;
a z 2.22 = ø‰ 2 )( 2 =⋅= a R z M B ÍÕÏ.
ƒ‡ð‡‡ ̸ ÁøøÌ ¿– ‰‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ ˙ˇ.3-3 Ó„ÚÎÓÎ ( a z ≤≤ 30 )
33 )( z q R z Q C ⋅+=
2
)( 23
33
z q z R z M C −−=
03 = z ø‰ 2 )( 3 == C R z Q ÍÕ, 0 )( 3 = z M
a z =3 ø‰ 5132 )( 3 =⋅+=+= qa R z Q C ÍÕ,
5.32
)( 2
3 −=−⋅−= a
qa R z M C ÍÕÏ
›‰„ ̋ ˝ð ÛÚ„‡‡ð ı ºÌ‰ÎºÌ ı ø˜ (6.3‚ñð ÁÛð‡„), „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.3„ñðÁÛð‡„) ˝Ô˛ð ·‡È„ÛÛ·‚.
∆ ˯˝˝ 6.3. ª„º„‰ÒºÌ ð‡Ï˚Ì (6.4‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ.
ª„ÒºÌ Ì¸ : 6=q ÍÕ/Ï, 1=a Ï
‡). ·). ‚).
6.4ñð ÁÛð‡„
„).
¡Ó‰ÓÎÚ. ’º¯øøÌ ı Óηӄ‰ÒÓÌ ¯Ëη˝ÌËÈ ÒËÒÚÂÏËÈ„ ð‡Ï „ ̋ ‰ ̋ „. ›Ì˝ ð‡Ï
„Ûð‚‡Ì ¯Ëη˝Ì˝˝Ò ·øÚÊ ̋ ˝. —‡Ï˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛ· ı ̸
‰‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛ· ı ‡‡Ò Á‡ð˜Ï˚Ì ı Û‚¸‰ Úº‰ËÈκÌ
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̌΄‡‡„øÈ. ’‡ðËÌ ı ‡‚Ú„‡ÈÌ ð‡Ï‰ ‰‡„ÛÛ ı ø˜, ı ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ øøÒ‰ ̋ „. Œ‰ÓÓ Ú ̋‰„ ̋ ˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÁÓ ı ËÓÊ ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ ˙ˇ.
1-1 Ó„ÚÎÓÎ ( a z ≤≤ 10 )
0 )( 1 = z N
11 )( z q z Q =
2 )(
21
1
z q z M −=
01 = z ø‰ 0 )( 1 = z Q , 0 )( 1 = z M
a z =1 ø‰ 6 )( 1 == aq z Q ÍÕ, 32
)( 2
1 −=−= aq z M ÍÕÏ
2-2 Ó„ÚÎÓÎ ( a z ≤≤ 20 )6 )( 2 == qa z N ÍÕ
0 )( 2 = z Q
32
)( 2
2 −=−= aq z M ÍÕÏ
›Ì˝ ı˝Ò˝„Ú ·ø ı ‰ÓÚÓÓ‰ ı ø˜ ÚÓ„ÚÏÓÎ ·‡È̇.3-3 Ó„ÚÎÓÎ ( a z 20 3 ≤≤ )
0 )( 3 = z N
6 )( 3 == qa z Q ÍÕ
)2
( )( 33
a z qa z M +−=
03 = z ø‰ 32
)( 2
3 −=−= aq z M ÍÕÏ
a z 23 = ø‰ 152
5 )( 2
3 −=−= qa z M ÍÕÏ
›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ (6.4·ñð ÁÛð‡„), ı ºÌ‰ÎºÌ ı ø˜ (6.4‚ñð ÁÛð‡„),„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.4„ñð ÁÛð‡„) ˝Ô˛ðËÈ„ ·‡È„ÛÛη‡.
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ ¯Ëη˝ÌËÈ ¯‡ ı ‡„‰‡Ê ·‡È„‡‡ ڇΉ·‡È„ÛÛÎÊ ·‡È„‡‡ ·º„ºº‰ ı ‡ðËÌ ˝Ô˛ðÚ Ú˝Ï‰ ̋ „ Ú‡‚¸‰‡„„øÈ Û·ÏÊ·Π·ËÈ.
∆ ˯˝˝ 6.4. —‡Ï˚Ì (6.5‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ.
ª„ÒºÌ Ì¸ : 2== qa F ÍÕ, 220 == qa M ÍÕÏ, 2=q ÍÕ/Ï, 1=a Ï
¡Ó‰ÓÎÚ. › ı ν ̋‰ Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ ºÎººÒ ð‡͈ËÛ‰ ̊ „ ÓÎÌÓ.
0=∑ X , 02 =⋅− aq H A ⇒ 42 == aq H A ÍÕ ;
0=∑ A M , 0220 =⋅−⋅⋅−+− a R aaqa F M B ⇒ 2−= B R ÍÕ ;
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0=∑ B M , 022 0 =⋅−−−⋅ aqaa F M a R A ⇒ 4= A R ÍÕ .
—‡͈ÛÛ‰ ̊ „ Áº‚ ÓÎÒÓÌ ˝Ò˝ı ËÈ„ ·ÓÒÓÓ Ú˝Ì ı ν„ ‰ ̋ ˝ð ı ø˜Ìøø‰ËÈ„ÔðÓÂ͈ÎÓÌ ¯‡Î„‡ ˇ.
0=∑Y , 0=+− B A R F R ⇒ 0224 =−− .
‡).
—‡Ï Ú‡‚‡Ì ı˝Ò˝„Ú˝È ·‡È„‡‡„‡Ì ı ‡‡ð‡Ì ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ Ú˝„¯ËÚ„ ̋ η˘¸Â. ’ ̋ Ò˝„ ·øðÚ z Ú˝Ì ı ν„ËÈ„¯Ëη˝ÌËÈ Ú˝Ì ı ν„ ‰‡„ÛÛ ‡‚̇.1-1 Ó„ÚÎÓÎ ( a z ≤≤ 10 )
4 )( 1 −=−= A R z N ÍÕ ;
4 )( 1 −=−= A H z Q ÍÕ ;
11 )( z H z M A ⋅−= 01 = z ø‰ 0 )( 1 = z M ;
11 == a z Ï ø‰ 4 )( 1 −= z M ÍÕÏ ;
2-2 Ó„ÚÎÓÎ ( a z a 22 ≤≤ )
4 )( 2 −=−= A R z N ÍÕ ;
4 )( 2 −=−= A H z Q ÍÕ ;
022 )( M z H z M A −⋅−=
12 ==a z Ï ø‰ 6 )( 2
−= z M ÍÕÏ ;
222 == a z Ï ø‰ 10 )( 2 −= z M ÍÕÏ ;
3-3 Ó„ÚÎÓÎ ( a z ≤≤ 30 )
4 )( 3 −=−= A H z N ÍÕ ;
4 )( 3 == A R z Q ÍÕ ;
033 2 )( M a H z R z M A A −⋅−⋅=
03 = z ø‰ 10 )( 3 −= z M ÍÕÏ ;
13
== a z Ï ø‰ 6 )( 3
−= z M ÍÕÏ ;
4-4 Ó„ÚÎÓÎ ( a z 20 4 ≤≤ )
2 )( 4 =−= B R z N ÍÕ ;
44 )( qz z Q = ;
2 )(
2
44
z q z M −=
04 = z ø‰ 0 )( 4 = z Q ; 0 )( 4 = z M ;
224 == a z Ï ø‰ 4 )( 4 = z Q ÍÕ;
4 )( 4 −= z M ÍÕÏ.
·).
‚).
„).
6.5ñð ÁÛð‡„
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5-5 Ó„ÚÎÓÎ ( a z ≤≤ 50 )
42 )( 5 −=⋅−= aq z N ÍÕ ;
2 )( 5
=−= B
R z Q ÍÕ ;
aaq z R z M B ⋅⋅−⋅= 2 )( 55
05 = z ø‰ 4 )( 5 −= z M ÍÕÏ ;
15 == a z Ï ø‰ 6 )( 5 −= z M ÍÕÏ ;
›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ (6.5·ñð ÁÛð‡„), ı ºÌ‰ÎºÌ ı ø˜ (6.5‚ñð ÁÛð‡„),„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.5„ñð ÁÛð‡„) ˝Ô˛ðËÈ„ ·‡È„ÛÛη‡. 6.1ñð ÁøÈΉ ‰Ûð¸‰Ò‡Ì ¯ËÌÊøø‰ËÈ„ ‡¯Ë„Î‡Ì ̋‰„ ̋ ˝ð ˝Ô˛ð Áº‚ ·‡È„ÛÛ·„‰Ò‡Ì ˝Ò˝ı ËÈ„¯‡Î„‡‡ð‡È.
∆ ˯˝˝ 6.5. ÃÛðÛÈ ¯Ëη˝ÌËÈ (6.6‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ.ª„ÒºÌ Ì¸ : F , r .
‡). ·). ‚). „).
6.6ñð ÁÛð‡„¡Ó‰ÓÎÚ. ÃÛðÛÈ ¯Ëη˝ÌËÈ ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛ· ı ‰‡‡ ÒÛÌ„‡ ı
‰‡„ÛÛ ı ø˜ËÈ„ ˝Âð˝„, ÒÛÌ„‡ ı ‰‡„ÛÛ ı ø˜ÌËÈ ˜Ë„Î ̋ν˝Ò ˆ‡„ËÈÌ ÁøøÌËÈ
ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ 090 Á‡È‰ Óð¯Ë ı ı ºÌ‰ÎºÌ ı ø˜ËÈ„ ˝Âð˝„, ¯Ëη˝ÌËÈÏÛðÛÈÎÚ˚„ Ë ı˝Ò„ ̋ ı øÈΘΠ̋ÎÚ˝È „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚„ ˝Âð˝„ „ ̋ Ê ÚÓÓˆÌÓ.›Òð˝„ ÚÓ ı ËÓΉÓΉ Һ𺄠„ ̋ Ê øÁÌ˝.
ÃÛðÛÈ ¯Ëη˝ÌËÈ Ó„ÚÎÓÎ ºÌˆ„ººð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. ›Ì˝ Ê˯˝˝ÌËÈ ı Û‚¸‰ C Ó„ÚÎÓÎ ϕ ºÌˆ„ººð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. – Ó„ÚÎÓΉ ‰ÓÚÓÓ‰ ı ø˜ÌËÈ
Ú˝„¯ËÚ„ ̋ Î ·Ë˜Ë ı ËÈÌ ÚÛΉ F ı ø˜Ú˝È Ú˝Ìˆøø ·º„ºº‰ Ô‡ð‡ÎÎÂθ, ˝Òð˝„˜Ë„Î ̋ÎÚ˝È ı Ó∏ð F ı ø˜ËÈ„ C Ó„ÚÎÓΉ øÈΘÎøøÎÌ˝. “ ̋ „‚ ̋Î Ì˝„ ÁÛð‡‡ÒÚ‡È F ı ø˜Ìøø‰ ı ÓÒ ·ÓÎÓ ı ·‡ ı Ó∏ð ÁÛð‡‡ÒÚ‡È F ı ø˜ËÈ„ ·‡È„ÛÛ·„˜Ë‰ Á‡‰‡Î·‡Î Q ı ºÌ‰ÎºÌ ı ø˜, N ‰‡„ÛÛ ı ø˜ ÓΉÓÌÓ.
ϕ ϕ sin )( F N =
ϕ ϕ cos )( F Q =
ϕ ϕ sin )( r F h F M ⋅−=⋅−=
›Ì˝ ¯Ëη˝ Ì˝„ ı˝Ò˝„Ú˝È, ı ‡„‡Ò ‰Û„ÛÈ ı ̋η˝ðÚ˝È ·‡È„‡‡ Û˜Ëð01800 ≤≤ϕ ıˇÁ„‡‡ðÚ ‰ ̋ ˝ð ı Ú˝„¯ËÚ„ ̋ Îøø‰ Áº‚ ·‡È̇.
»Èω 0=ϕ ø‰ 0 )( =ϕ N , F Q = )( ϕ , 0 )( =ϕ M ;
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090=ϕ ø‰ F N = )( ϕ , 0 )( =ϕ Q , r F M ⋅= )( ϕ ;
0180=ϕ ø‰ 0 )( =ϕ N , F Q −= )( ϕ , 0 )( =ϕ M .
›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ (6.6·ñð ÁÛð‡„), ı ºÌ‰ÎºÌ ı ø˜ (6.6‚ñð ÁÛð‡„),„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.6„ñð ÁÛð‡„) ˝Ô˛ð ·‡È„ÛÛ·‚. √ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì˝Ô˛ðËÈ„ ÏºÌ ¯‡ ı ‡„‰‡Ê ·‡È„‡‡ ڇΉ ÁÛð̇.
∆ ˯˝˝ 6.6. ÃÛðÛÈ ¯Ëη˝ÌËÈ (6.7‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ.ª„ÒºÌ Ì¸ : q , r
‡). ·). ‚). „).
6.7ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ϕ ºÌˆ„ººð ÚÓ‰Óð ı ÓÈÎÓ„‰Ó ı – Ó„ÚÎÓÎ ‡‚ ˙ˇ. ƒÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ
Ú˝„¯ËÚ„ ̋ Î ·Ë˜Ë ı ËÈÌ ÚÛΉ Ú‡ð ı χΠı ø˜ÌËÈ Ú˝Ìˆøø øÈÎ˜Î˝„˜ËÈ„ ÓÎÓ ı ı˝ð˝„Ú˝È. ¿– ı˝Ò˝„Ú øÈÎ˜Î˝ı Ú‡ð ı χΠı ø˜ÌËÈ Ú˝Ìˆøø øÈÎ˜Î˝„˜ R ı ø˜ ¿–
ı˝ð˜ÏËÈÌ ‰Û̉ ÏºÌ ı˝ð˜Ëω ÔÂðÔẨËÍÛÎ ˇð ·º„ºº‰ AC q R ⋅= ı˝ÏÊ ̋ ̋Ú˝È
·‡È‰‡„. “ ̋ ̈øø øÈÎ˜Î˝„˜ ı ø˜ËÈ„ ÓÎÒÌ˚ ‰‡ð‡‡ 6.5ñð Ê˯˝˝ÌËÈ ‡‰Ë·‡ð ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÁÓ ı ËÓ‚ÓÎ
2sin2
2sin )( 2 ϕ ϕ
ϕ qr R N −=−=
ϕ ϕ ϕ ϕ
ϕ sin2
cos2
sin22
cos )( qr qr R Q −=−=−=
2sin2
2sin )( 22 ϕ ϕ
ϕ qr r R M −=⋅−=
·ÓÎÌÓ. Øø̉2
sin2 ϕ
r AC ⋅= ,2
sin2 ϕ
qr AC q R =⋅= ·‡È̇. R Ú˝Ìˆøø
øÈÎ˜Î˝„˜ AC ı˝ð˜ÏËÈÌ ‰Û̉ øÈΘËΉ ̋ „.
ÿËη˝ ı ‡„‡Ò ‰Û„ÛÈ ı ̋η˝ðÚ˝È ·‡È„‡‡ Û˜Ëð 01800 ≤≤ϕ ıˇÁ„‡‡ðÚ ‰ ̋ ˝ð ı
Ú˝„¯ËÚ„ ̋ Îøø‰ Áº‚ ·‡È̇.»Èω 0=ϕ ø‰ 0 )( =ϕ N , 0 )( =ϕ Q , 0 )( =ϕ M ;
090=ϕ ø‰ qr N −= )( ϕ , qr Q −= )( ϕ , 2 )( qr M −=ϕ ;
0180=ϕ ø‰ qr N 2 )( −=ϕ , 0 )( =ϕ Q , 22 )( qr M −=ϕ .
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›‰„ ̋ ˝ð ÛÚ„‡‡ð ‰‡„ÛÛ ı ø˜ (6.7·ñð ÁÛð‡„), ı ºÌ‰ÎºÌ ı ø˜ (6.7‚ñð ÁÛð‡„),„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì(6.7„ñð ÁÛð‡„) ˝Ô˛ð ·‡È„ÛÛ·‚.
∆ ˯˝˝ 6.7. Œ„ÚÓð„ÛÈÌ ð‡Ï˚Ì (6.8‡ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ ˝Ô˛ð·‡È„ÛÛÎ.
ª„ÒºÌ Ì¸ : F , q , a
‡). ·). ‚). „).6.8ñð ÁÛð‡„
¡Ó‰ÓÎÚ. Œ„ÚÓð„ÛÈÌ ð‡Ï˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð
·‡È„ÛÛ· ı˚Ì ÚÛΉ ÒËÒÚÂÏËÈ„ Ì˝„ ¯Ëη˝Ì˝˝Ò ̺„ºº‰¯ËÎÊË ı ‰ ̋ ˝ øÁ ̋Ê ·ÛÈ ¯Ëη˝ÌËÈ Ú˝Ì ı ν„Ú˝È z Ú˝Ì ı ν„ËÈ„ ‰‡‚ ı ̂ ÛÛÎÊ, z Ú˝Ì ı ν„ËÈÌ ˝Òð˝„
˜Ë„ν
ν˝
Ò ı
‡ð‡ ı
‡‰ Ú˝
Ì ı
ν
„ËÈÌ˝
Âð˝
„ ˜Ë„ν
ν˝
Ò x Ú˝Ì ı ν„ËÈÌ ˝Âð˝„ ˜Ë„Î ̋Î øðøø ¯ËÎÊË ı ̋‰ ˆ‡„ËÈÌ
ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ˝Òð˝„ 090 ºÌˆº„ øøÒ„ ̋ Ê ·‡È ı ‡‡ðÍÓÓð‰Ë̇Ú˚Ì ÒËÒÚÂÏËÈ„ (6.8·ñð ÁÛð‡„) ÒÓÌ„Ó∏. “ ̋ „ ̋ ̋‰ ð‡Ï˚Ì ı˝Ò˝„ ·øð ‰ ̋ ˝ð ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ Ú˝„¯ËÚ„ ̋ Î ÁÓ ı ËÓ‚ÓÎ
1-1 Ó„ÚÎÓÎ ( a z ≤≤ 10 )
0 )( 1 = z N , 11 )( z q z Q x = , F z Q y = )( 1
11 )( z F z M x = ,2
)( 21
1 z q z M y = , 0 )( 1 = z T
2-2 Ó„ÚÎÓÎ ( a z ≤≤ 20 )
F z N = )( 2 , aq z Q x = )( 2 , 0 )( 2 = z Q y
a F z M x = )( 2 , 22 )( z aq z M y = ,2
)( 2
aaq z T −=
‰).
Â).
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3-3 Ó„ÚÎÓÎ ( a z ≤≤ 30 )
aq z N = )( 3 , F z Q x −= )( 3 , 0 )( 3 = z Q y
0 )( 3 = z M x , 33 )( z F z M y = , a F z T = )( 3
·ÓÎÌÓ. ›‰„ ̋ ˝ð Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÁÓ ı ËÓ ı ‰ÓÓ Ú˝Ì ı ν„ËÈÌ ˝Âð˝„ ˜Ë„Î ̋Î ‰‡„ÛÛ˜Ë„Î˝Ò˝Ì ı ºÌ‰ÎºÌ ı ø˜ËÈ„ ˝Âð˝„ „ ̋ Ê ÚÓÓˆÒÓÌ ·‡ ÏÓÏÂÌÚ˚Ì ı Û‚¸‰ Ú˝Ï‰„ËÈÌ ÚÓ‰Óð ı ÓÈ ‰øð˝Ï ·‡ðËÏÚ·‡„øÈ ·ÓÎÓ‚˜ ˝Ô˛ðËÈ„ ̸ ¯Ëη˝ÌËÈ ¯‡ ı ‡„‰‡Ê·‡È„‡‡ ڇΉ ·‡È„ÛÛη‡. ¡ÛÒ‡‰ ı ø˜ÌËÈ ı Û‚¸‰ ºÏ̺ ·‡ðËÏÚ‡ÎÊ ·‡ÈÒ‡Ì Ú˝Ï‰„ËÈÌ ‰øðÏËÈ„ ı˝ð˝„ν‚.
ƒ ̋ ˝ð ı Ú˝„¯ËÚ„ ̋ Îøø‰ËÈ„ ÁÓ ı ËÓÊ ı˝Ò„øø‰ËÈÌ ˝ı , Úº„Ò„ºÎ‰ ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ ÛÚ„ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ ‰‡„ÛÛ ı ø˜ (6.8‚ñð ÁÛð‡„), ı ºÌ‰ÎºÌ ı ø˜(6.8‰ñð ÁÛð‡„), „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ (6.8„ñð ÁÛð‡„), ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì(6.8Âñð ÁÛð‡„) ˝Ô˛ðËÈ„ ·‡È„ÛÛ·‚.
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ƒ¿¿À√ ¿¬¿— ¹5. ƒ‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛ·ÎÚ
ƒ‡‡Î„‡‚‡ð ‰ÓÎÓÓÌ ·Ó‰ÎÓ„ÓÓÒ ·øð‰ ̋ Ì˝.
1. ª„º„‰ÒºÌ ÒÛÎ øÁøøðÚ˝È ‰‡ÏÌÛÛð˚Ì (6.18ñð ÁÛð‡„) ı ºÌ‰ÎºÌ ı ø˜,„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 6.1ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ƒ‡ÏÌÛÛð˚Ì ÛðÚ 3=l Ï.
6.18ñð ÁÛð‡„
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6.18ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.1ñð ı øÒÌ˝„Ú
¹
q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ k n ¹
q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ n k
1 10 0.9 ql 0.05 2ql 0.30 0.60 11 12 0.95 ql 0.09 2ql 0.60 0.30
2 15 0.8 ql 0.03 2ql 0.30 0.70 12 14 0.85 ql 0.09 2ql 0.70 0.30
3 18 0.7 ql 0.03 2ql 0.60 0.85 13 15 0.75 ql 0.08 2ql 0.85 0.60
4 20 0.6 ql 0.02 2ql 0.25 0.65 14 16 0.65 ql 0.08 2ql 0.65 0.25
5 25 0.5 ql 0.022ql 0.35 0.70 15 18 0.55 ql 0.07
2ql 0.70 0.35
6 22 0.4 ql 0.02 2ql 0.40 0.75 16 20 0.45 ql 0.07 2ql 0.75 0.40
7 25 0.3 ql 0.0152ql 0.35 0.70 17 22 0.35 ql 0.06
2ql 0.70 0.35
8 22 0.2 ql 0.01 2ql 0.30 0.65 18 24 0.25 ql 0.06 2ql 0.65 0.30
9 30 0.2 ql 0.01 2ql 0.45 0.55 19 26 0.15 ql 0.04 2ql 0.85 0.75
10 35 0.1 ql 0.01 2ql 0.40 0.65 20 28 0.15 ql 0.04 2ql 0.65 0.40
2. ÕÛ„‡Ò‡Ì ÚÛ΄ÛÛðÚ‡È ‰‡ÏÌÛÛð˚Ì (6.19ñð ÁÛð‡„) ı ºÌ‰ÎºÌ ı ø˜,„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 6.2ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ƒ‡ÏÌÛÛð˚Ì ÛðÚ 5=l Ï.
6.19ñð ÁÛð‡„
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6.19ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.19ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î) 6.2ñð ı øÒÌ˝„Ú
¹q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ
k n ¹ q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ
k n
1 15 0.20 ql 0.07 2ql 0.20 0.50 11 12 0.19 ql 0.09 2ql 0.50 0.30
2 22 0.20 ql 0.07 2ql 0.30 0.70 12 14 0.19 ql 0.09 2ql 0.70 0.30
3 20 0.18 ql 0.06 2ql 0.50 0.35 13 15 0.17 ql 0.08 2ql 0.35 0.50
4 24 0.18 ql 0.06 2ql 0.65 0.85 14 16 0.17 ql 0.08 2ql 0.65 0.25
5 25 0.16 ql 0.05 2ql 0.85 0.30 15 18 0.15 ql 0.07 2ql 0.70 0.35
6 26 0.16 ql 0.05 2ql 0.70 0.20 16 20 0.15 ql 0.07 2ql 0.75 0.30
7 30 0.14 ql 0.04 2ql 0.35 0.20 17 22 0.13 ql 0.06 2ql 0.70 0.35
8 28 0.14 ql 0.042ql 0.50 0.80 18 24 0.13 ql 0.06
2ql 0.65 0.30
9 25 0.12 ql 0.032ql 0.70 0.30 19 26 0.11 ql 0.04
2ql 0.35 0.75
10 30 0.12 ql 0.032
ql 0.85 0.65 20 28 0.11 ql 0.042
ql 0.65 0.40
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3. «‡‚Òð˚Ì ÌÛ„‡ÒÚ‡È ‰‡ÏÌÛÛð˚Ì (6.20ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ
˝Ô˛ðËÈ„ 6.3ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ( 8=l Ï)
6.20ñð ÁÛð‡„
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6.20ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.3ñð ı øÒÌ˝„Ú
¹q ,
ÍÕ/Ï
F ,
ÍÕ
M ,
ÍÕÏ
n k ¹ q ,
ÍÕ/Ï
F ,
ÍÕ
M ,
ÍÕÏ
n k
1 15 5.0 1.5 0.25 0.45 11 12 6.0 1.8 0.65 0.35
2 15 4.5 1.4 0.38 0.55 12 12 5.5 1.7 0.75 0.35
3 20 4.0 1.3 0.25 0.35 13 16 5.0 1.6 0.35 0.56
4 20 4.0 1.3 0.65 0.45 14 16 4.5 1.5 0.65 0.25
5 25 3.5 1.2 0.65 0.25 15 18 4.5 1.4 0.75 0.35
6 25 3.5 1.2 0.25 0.56 16 18 3.5 1.4 0.75 0.36
7 30 2.5 1.0 0.85 0.35 17 22 2.5 1.3 0.75 0.35
8 30 2.5 1.0 1.00 0.36 18 22 2.5 1.2 0.65 0.37
9 35 3.0 1.1 0.70 0.25 19 28 6.0 1.1 0.35 0.75
10 35 3.0 1.1 0.75 0.65 20 28 5.0 1.1 0.65 0.45
4. –ÛÎ øÁøøðÚ˝È, ı ‡‚Ú„‡ÈÌ ð‡Ï˚Ì (6.21ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ˝Ô˛ðËÈ„ 6.4ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ( 4=a Ï)
6.21ñð ÁÛð‡„
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6.21ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.21ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.4ñð ı øÒÌ˝„Ú
π q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ
n k π q ,
ÍÕ/Ï F ,ÍÕ
M ,ÍÕÏ
n k
1 15 1.5 qa 0.1 2qa 0.80 0.80 11 12 2.2 qa 0.1 2qa 0.75 0.30
2 15 1.2 qa 0.1 2qa 0.60 0.60 12 12 2.0 qa 0.1 2qa 0.60 0.40
3 20 qa 0.1 2qa 1.00 0.50 13 16 qa 0.1 2qa 0.35 0.60
4 20 0.8 qa 0.1 2qa 1.00 0.40 14 16 0.8 qa 0.1 2qa 0.65 0.25
5 25 0.6 qa 0.1 2qa 2.00 0.25 15 18 0.6 qa 0.1 2qa 0.70 0.35
6 25 0.4 qa 0.1 2qa 1.20 0.40 16 18 0.4 qa 0.1 2qa 0.65 0.40
7 10 2 qa 0.12qa 1.00 0.80 17 22 1.8 qa 0.1
2qa 0.60 0.35
8 10 1.5 qa 0.12qa 0.80 1.00 18 22 1.5 qa 0.1
2qa 0.85 0.30
9 15 qa 0.12qa 0.50 0.60 19 15 1.6 qa 0.1
2qa 0.35 0.75
10 20 0.5 qa 0.1 2qa 0.90 0.60 20 20 1.4 qa 0.1 2qa 1.00 0.40
5. ÕÛ„‡Ò‡Ì ÚÛ΄ÛÛðÚ‡È, ı ‡‚Ú„‡ÈÌ ð‡Ï˚Ì (6.22ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ˝Ô˛ðËÈ„ 6.5ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ( 4=a Ï)
6.22ñð ÁÛð‡„
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6.22ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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6.22ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
6.5ñð ı øÒÌ˝„Ú
π ,q ÍÕ/Ï , F ÍÕ , M ÍÕÏ k π ,q ÍÕ/Ï , F ÍÕ , M ÍÕÏ k
1 20 10.0 3.0 0.7 11 25 10.0 4.0 0.32 35 5.0 2.5 0.2 12 30 8.0 3.5 0.83 35 7.5 2.0 0.5 13 30 8.0 3.0 0.54 25 10.0 3.0 0.6 14 40 7.0 4.5 0.45 25 5.0 4.0 0.4 15 40 6.5 5.5 0.66 20 12.5 5.0 0.5 16 45 6.0 5.5 0.57 15 15.0 5.0 0.7 17 10 5.5 6.0 0.38 20 7.5 2.5 0.4 18 10 11.0 4.0 0.69 25 7.5 2.0 0.7 19 15 9.0 3.5 0.310 30 15.0 3.0 0.3 20 15 12.0 6.5 0.7
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6. ÃÛðÛÈ ¯Ëη˝ÌËÈ (6.23ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ðËÈ„ 6.6ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ( 3= R Ï)
1
ϕ
kϕ
F 1
2 F
M
2
k ϕ
M
1 F
F 2
3
M k ϕ
2 F
1 F
1 F
4 M
2 F
kϕ
ϕ
5
ϕ
kϕ
M
F 12 F
6
F 2 M
ϕkϕ
F 1
7
2 F
M
ϕ
kϕ
1 F
F 2
1 F
8
ϕ
kϕ
M
6.23ñð ÁÛð‡„
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9
1 F ϕ
kϕ
M F 2
ϕ
kϕ M
2 F
10
1 F
11
2 F
F
kϕ
ϕ
2
F 1
M 12
M
kϕϕ
2 F
F 1
13
F 1
M F
ϕ
kϕ
2
14 F 2
M
ϕ
kϕ
15
ϕ kϕ
2 F 1 F
M
16
ϕ
kϕ1 F
2
M
6.23ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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111
F 1
2 F ϕ
kϕ
M
17 1 F
F 2
k ϕ
ϕ
M
18
1 F
M
kϕ
ϕ
F 2
19
M ϕ
kϕ F 1
20 F 2
6.23ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
6.6ñð ı øÒÌ˝„Ú
π ,1 F ÍÕ ,2 F ÍÕ , M ÍÕÏ k π ,1 F ÍÕ ,2 F ÍÕ , M ÍÕÏ k
1 10 14 50 0.50 11 15 32 40 0.45
2 40 20 10 0.35 12 18 36 60 0.55
3 60 15 15 0.25 13 24 40 25 0.30
4 20 25 20 0.75 14 32 35 20 0.65
5 50 30 25 0.80 15 45 40 35 0.85
6 10 16 30 0.60 16 75 45 30 0.40
7 30 18 40 0.45 17 25 15 50 0.25
8 12 20 45 0.20 18 55 70 45 0.70
9 14 22 35 0.55 19 60 40 25 0.50
10 16 24 10 0.90 20 70 25 55 0.80
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7. Œ„ÚÓð„ÛÈÌ ð‡Ï˚Ì (6.24ñð ÁÛð‡„) ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ðËÈ„ 6.7ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ. ( 2=l Ï)
F
l
l
1 l
q
M
F
l
l
l
2
q
M
l q M
l
l
3 l
F
ll
q l
l
4
M F
l l
q
M
5 F l
F
l
l
l
l
6
M
q
l
M
l
7
q
l
F l
l
l
8
l l
q M
F
6.24ñð ÁÛð‡„
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l
17
l
l
q
M
F
18 M
l
F
q
l
l
l
19
l
l
F q
M
lq
M
l
F
20
l
6.24ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
6.7ñð ı øÒÌ˝„Ú
π ,q ÍÕ/Ï , F ÍÕ , M ÍÕÏ π ,q ÍÕ/Ï , F ÍÕ , M ÍÕÏ
1 20 1.0 ql 1.1 2ql 11 20 1.2 ql 0.9 2ql
2 22 0.7 ql 1.0 2ql 12 22 1.1 ql 1.2 2ql
3 40 0.8 ql 0.8 2ql 13 25 0.9 ql 0.7 2ql
4 26 0.4 ql 0.6 2ql 14 28 0.5 ql 0.4 2ql
5 30 0.7 ql 0.5 2ql 15 30 0.8 ql 0.5 2ql
6 32 1.0 ql 0.4 2ql 16 32 0.6 ql 0.6 2ql
7 35 0.8 ql 0.3 2ql 17 35 0.8 ql 0.7 2ql
8 37 0.4 ql 0.4 2ql 18 40 0.4 ql 0.6 2ql
9 42 0.9 ql 0.5 2ql 19 25 0.3 ql 0.5 2ql
10 18 0.6 ql 0.6 2ql 20 42 0.7 ql 0.2 2ql
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6.2. ƒ¿ÃÕ””—¤Õ ¡¿“ ¡›’»…Õ ¡Ø—›Õ “ŒŒ÷ŒŒ
≈ðºÌ ı ËÈ ÓÈ΄ÓÎÚ
ƒ‡ÏÌÛÛð˚Ì ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ˚Ì ø‰ ÚøøÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ ·ÓÒÓÓ, ı˝‚Ú˝˝ ı ø˜‰ ̋ Î øøÒ‰ ̋ „. ’ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ ˆ ̋ „Ú øøÒ˝ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„
x I
y M ⋅=σ (6.1)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. Øø̉ M ñ ÚÛ ı ‡ÈÌ ı ºÌ‰ÎºÌ Ó„ÚÎÓΉ øÈÎ˜Î˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ
x I ñ Á‡‡„ x Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ
ñ Á‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ø˜‰ ̋ ÎËÈ„ ·Ó‰ÓÊ ·‡È„‡‡ ˆ ̋ „ ı øð˝ı Á‡È
¡ÓÒÓÓ ı ø˜‰ ̋ Î Ó„ÚÎÓÎ˚Ì ºÌ‰ºð ‰‡„ÛÛ‰ ¯Û„‡Ï‡Ì ı ÛÛΡ𠺺ð˜Îº„‰º ı ·º„ºº‰ Á‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı ˆ ̋ „Ú ı ‡Ï„ËÈÌ Ë ı ÛÚ„‡Ú‡È ·‡È̇.
“Ó„ÚÏÓÎ Ó„ÚÎÓÎÚÓÈ ‰‡ÏÌÛÛð˚Ì ı ‡Ï„ËÈÌ Ë ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ øÈÎ˜Î˝ı Ó„ÚÎÓΉ ı ‡Ï„ËÈÌ Ë ı ·ÓÒÓÓ ı ø˜‰ ̋ Î øøÒ˝ı ·º„ºº‰
x I
y M maxmaxmax
⋅=σ (6.2)
ÚÓÏ ˙∏Ó„ÓÓð ÓÎÌÓ.«‡‡„ Ú˝Ì ı ν„ËÈÌ ı Û‚¸‰ Ú˝„¯ ı˝ÏÚ˝È Ó„ÚÎÓÎ˚Ì ı Û‚¸‰
x W
M maxmax =σ (6.3)
·ÓÎÌÓ. Øø̉2/h
I W x
x = ı˝ÏÊË„‰ ̋ ı øøÌËÈ„ x Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ
˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ „ ̋ Ì˝. ›Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ Ì¸ Ú˝„¯ ºÌˆº„Ú Ó„ÚÎÓΉ( b - Á‡‡„ Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ Ú‡Î)
6
2bhW x = ,
‰Û„ÛÈ Ó„ÚÎÓΉ
33
1.032
d d
W x ≈= π
,
ˆ‡„Ëð‡„ Ó„ÚÎÓΉ ( D
d c = )
)1( 1.0 )1( 32
4343
cd cd
W x −≈−=
π
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·‡È̇. ¡ÓÒÓÓ ı ø˜‰ ̋ ν˝ð ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎËÈ„ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ ÎËÈÌ‡ð„‡‡ð ·Ë˜‚ ̋Î
][
maxmaxσ σ ≤
⋅=
x I
y M
, (6.4)
ıˇÁ„‡‡ð˚Ì ÚºÎº‚ËÈÌ ‡ð„‡‡ð ·Ë˜‚ ̋Î
R I
y M
x
≤⋅
= maxmaxσ (6.5)
·ÓÎÌÓ. “ ̋ „¯ ı˝ÏÚ˝È Ó„ÚÎÓÎ˚Ì ı Û‚¸‰ ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ºÎËÈ„ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ ÎËÈÌ ‡ð„‡‡ð ·Ë˜‚ ̋Î
][maxσ σ ≤=
x W
M , (6.6)
ıˇÁ„‡‡ð˚Ì ÚºÎº‚ËÈÌ ‡ð„‡‡ð ·Ë˜‚ ̋Î
R W
M
x
≤= maxσ (6.7)
·ÓÎÌÓ. ØøÌ˝˝Ò ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„
][
max
σ
M W x ≥ ·Û ̨ Û
R
M W x
max≥
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÊ ‰‡ð‡‡ ̸ ÒÓÌ„ÓÒÓÌ ı ̋η˝ðËÈÌ Ó„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝„ÓÎÊ ·ÓÎÌÓ.
’ ̋ ‚Ú˝˝ ı ø˜‰ ̋ ν˝ð ·‡Ú ·˝ı ËÈÌ Ìº ı ̂ ÎËÈ„ ·Ë˜‚ ̋Î
][max τ τ ≤⋅
⋅=
b I
S Q
x
o x ·Û ̨ Û s
x
o x R b I
S Q≤
⋅
⋅=maxτ (6.8)
·‡È̇. Øø̉ s R ñ ¯ËÎÊËÎÚËÈÌ øÂËÈÌ ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î,
][τ ñ Áº‚¯ººðº„‰º ı ı˝‚Ú˝˝ ı ø˜‰ ̋ Î,
b ñ Ó„ÚÎÓÎ˚Ì ºð„ºÌ,
x I ñ Ó„ÚÎÓÎ˚Ì Á‡‡„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ,
o x S ñ Ó„ÚÎÓÎ˚Ì Ó„ÚÎÓ„‰ÒÓÌ ı˝Ò„ËÈÌ ı Á‡‡„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ
ÒÚ‡ÚËÍ ÏÓÏÂÌÚ. ƒ‡ÏÌÛÛð˚Ì Ó„ÚÎÓÎ˚Ì ‰Ûð˚Ì ˆ ̋ „Ú øøÒ˝ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„
x I
y M ⋅=σ (6.9)
ÚÓÏ ˙∏Ó„ÓÓð, ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„
b I
S Q
x
o x
⋅
⋅=τ
(6.10)
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ÚÓÏ ˙∏Ó„ÓÓð ÚÛÒ ÚÛÒ ÚÓ‰Óð ı ÓÈÎÌÓ. ›‰„ ̋ ˝ðËÈ„ ÓÎÒÌ˚ øÌ‰Ò˝Ì ‰ ̋ ˝ð „ÓÎ ı ø˜‰ ̋ ÎËÈ„
22
minmax 22
τ
σ σ
σ +⎟ ⎠
⎞⎜⎝
⎛ ±= (6.11)
ÚÓÏ ˙∏Ó„ÓÓð, „ÓΠڇη‡ÈÌ ·‡Èð·Î˚„
σ
τ
α
22tan 0 −= (6.12)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈΉӄ.
∆ ˯˝˝ 6.8. ƒ‡ÏÌÛÛðÚ (6.9ñð ÁÛð‡„) øøÒ˝ı ı ‡Ï„ËÈÌ Ë ı ·ÓÒÓÓ ı ø˜‰ ̋ Î,
ÏºÌ Ó„ÚÎÓÎ˚Ì – ˆ ̋ „Ú øøÒ˝ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„ ÓÎ.ª„ÒºÌ Ì¸ : 15= F ÍÕ
‡). ·).
6.9ñð ÁÛð‡„
¡Ó‰ÓÎÚ. ’ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì (6.9·ñð ÁÛð‡„) Á‡‡„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ
ËÌÂðˆËÈÌ ·ÓÎÓÌ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ÓηÓÎ
49012
42
12
106 33
=⋅
−⋅
= x I ÒÏ
4
, 985
490
2/===
h
I
W
x
x ÒÏ
3
·ÓÎÌÓ. “ ̋ „‚ ̋Î ‰‡ÏÌÛÛðÚ øøÒ˝ı ı ‡Ï„ËÈÌ Ë ı ı ø˜‰ ̋ Î (6.9·ñð ÁÛð‡„)
6
6
3max
max 101531098
1015⋅=
⋅
⋅==
− x W
M σ Õ/Ï2 153= Ãœ‡,
Ó„ÚÎÓÎ˚Ì – ˆ ̋ „Ú øøÒ˝ı ı ø˜‰ ̋ Î
6
8
23max 1022.61
10490
1021015⋅=
⋅
⋅⋅⋅=
⋅=
−
−
x
C C
I
y M σ Õ/Ï2 22.61= Ãœ‡
·‡È̇.
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∆ ˯˝˝ 6.9. ƒ‡ÏÌÛÛð˚Ì (6.10ñð ÁÛð‡„) Ó„ÚÎÓÎ˚Ì – ˆ ̋ „Ú øøÒ˝ı ı˝‚Ú˝˝
ı ø˜‰ ̋ ÎËÈ„ ÓÎ.ª„ÒºÌ Ì¸ : 20= F ÍÕ
6.10ñð ÁÛð‡„
¡Ó‰ÓÎÚ. Œ„ÚÎÓ„‰ÒÓÌ ı˝Ò„ËÈÌ o x S ÒÚ‡ÚËÍ ÏÓÏÂÌÚ˚„ ÓÎÓ ı˚Ì ÚÛΉ
ÚøøÌËÈ 0 A ڇη‡È„ ı ø̉ËÈÌ Úº‚ËÈÌ 0 y ÍÓÓð‰Ë̇ڇ‡ð øðÊøøη ̋Î
5.9375.12 )1015( 1500 =⋅−⋅=⋅= y AS o x ÒÏ3
·ÓÎÌÓ. Œ„ÚÎÓÎ˚Ì x Á‡‡„ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ
3375012
3015
12
33
=⋅
==bh
I x
ÒÏ4
Û˜Ëð – ˆ ̋ „ËÈÌ ı˝‚Ú˝˝ ı ø˜‰ ̋ Î
6
28
63
1037.010151033750
105.9371020⋅=
⋅⋅⋅
⋅⋅⋅=
⋅
⋅=
−−
−
b I
S Q
x
o x
C τ Õ/Ï2 37.0= Ãœ‡
·‡È̇.
∆ ˯˝˝ 6.10. ƒ‡ÏÌÛÛð˚Ì (6.11ñð ÁÛð‡„) Ú˝„¯ ºÌˆº„Ú, ‰Û„ÛÈ, ‰‚ÛÚ‡‚ð
Ó„ÚÎÓÎ ÒÓÌ„ÓÊ ‰‚ÛÚ‡‚ð Ó„ÚÎÓÎÚÓÈ ‰‡ÏÌÛÛð˚Ì ·‡Ú ·˝ı ËÈÌ ·øð˝Ì ÚÓÓˆÓÓ
ı ËÈ. ƒ‡ÏÌÛÛð˚Ì Ï‡ÚÂðˇΠ–Ú3. «º‚¯ººðº„‰º ı ı ø˜‰ ̋ Î 160][ =σ Ãœ‡.ª„ÒºÌ Ì¸: 1=a Ï, 20=q ÍÕ/Ï, 401 = F ÍÕ, 40= à ÍÕÏ, 202 = F ÍÕ.
¡Ó‰ÓÎÚ.1. “Û΄ÛÛðÛÛ‰ ̊ Ì ð‡͈˚„ ÚÓ‰Óð ı ÓÈÎÓ ı
0=∑ A M ; 065424 21 =⋅+⋅−+⋅+⋅ a F a R M a F aqa B ⇒ 96= B R ÍÕ.
0=∑ B M ; 0345 21 =⋅++⋅−⋅⋅−⋅ a F M a F aaqa R A ⇒ 44= A R ÍÕ.
—‡͈ËÛ‰ ̊ „ Áº‚ ÓÎÒÓÌ ˝Ò˝ı ËÈ„ ¯‡Î„‡ ı˚Ì ÚÛΉ ·ÓÒÓÓ Ú˝Ì ı ν„ ‰ ̋ ˝ð ·ø ı ı ø˜ËÈ„ ÔðÓÂ͈ÎÓ∏.
0=∑ y ; 04 21 =−+−⋅− F R F aq R B A
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
119
0209640142044 =−+−⋅⋅−
6.11ñð ÁÛð‡„
2. ’ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô ̨𠷇ȄÛÛ· ı ƒ‡ÏÌÛÛð „Ûð‚‡Ì ı˝Ò˝„Ú˝È ·‡È„‡‡ Û˜Ëð „Ûð‚‡Ì Ó„ÚÎÓÎ ‡‚‡ ı
¯‡‡ð‰Î‡„‡Ú‡È.1-1 Ó„ÚÎÓÎ ( a z 40 1 ≤≤ )
11 )( z q R z Q A −=
2 )(
21
11
z q z R z M A −=
01 = z ø‰ 44 )( 1 == A R z Q ÍÕ ; 0 )( 1 = z M ,
41 = z Ï ø‰ 36 )( 1 −= z Q ÍÕÏ ; 16 )( 1 = z M ÍÕÏ
›Ì˝ ı˝Ò˝„Ú „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ ˝ÍÒÚðÂÏÛÏ ÛÚ„‡Ú‡È ·‡È̇.
0 )( 1 = z Q ; 02044 01 =− z ⇒ 2.20
1 = z Ï
2.2011 == z z Ï ø‰ 4.48 )( 1 = z M ÍÕÏ
2-2 Ó„ÚÎÓÎ ( a z a 54 2 ≤≤ )
76401420444 )( 12 −=−⋅⋅−=−⋅−= F aq R z Q A ÍÕ ;
)4( )2( 4 )( 21222 a z F a z aq z R z M A −−−⋅⋅−⋅=
42 = z Ï ø‰ 16 )( 2 = z M ÍÕÏ ;
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî120
52 = z Ï ø‰ 60 )( 2 −= z M ÍÕÏ ;
3-3 Ó„ÚÎÓÎ ( a z ≤≤ 30 )
20 )( 23
== F z Q ÍÕ ;
323 )( z F z M ⋅−=
03 = z ø‰ 0 )( 3 = z M ;
13 = z Ï ø‰ 20 )( 3 = z M ÍÕÏ.
3. Œ„ÚÎÓÎ ÒÓÌ„Ó ı Œ„ÚÎÓÎ˚„ ·‡Ú ·˝ı ËÈÌ
][maxmax σ σ ≤=
x W
M
̺ ı ̂ ºÎººÒ ÒÓÌ„ÓÌÓ. ªºðººð ı ̋η ̋Î ˝Ì‰ ̋ ˝Ò ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ÓηÓÎ
3
6
3max 10375.0
10160
1060
][
−⋅=⋅
⋅==
σ
M W x Ï3 375= ÒÏ3
·‡È̇. ØøÌ˝˝Ò ‰Ûð¸‰Ò‡Ì Ó„ÚÎÓÎÛÛ‰ ̊ „ ÒÓÌ„Ó∏.
‡). “ ̋ „¯ ºÌˆº„Ú ( 5.1=b
h ):
322
375.06
)5.1(
6 b
bbbh
W x =
⋅
== , ][375.0
max3
σ
M
b = ⇒ 375375.0
3
=b
10375.0
3753 ==b ÒÏ ⇒ 15105.15.1 =⋅== bh ÒÏ.
·). ƒÛ„ÛÈ :
32
3d W x
π
= ,][32
max3
σ
π M d = , 375
32
3
=d π
⇒ 63.1514.3
323753 =
⋅=d ÒÏ.
‚). ƒ‚ÛÚ‡‚ð :
¡‡Ú ·˝ı
ËÈÌ Ìº ı
ˆÎººÒ ÓÎÒÓÌ 375=
x W ÒÏ
3
˝
Ò˝
ð„øøˆÎËÈÌ ÏÓÏÂÌÚÚÓÈ ‰‚ÛÚ‡‚ð˚„ ÒÓðÚ‡ÏÂÌÚ‡‡Ò( √ Œ–“ 8239ñ72) ı ‡È‚‡Î 371= x W ÒÏ3 ˝Ò˝ð„øøˆÎËÈÌ
ÏÓÏÂÌÚÚÓÈ ‰‚ÛÚ‡‚ð π27 Ï‡Ì‡È ·Ó‰ÎÓ„ ̊ Ì ı ‡ðËÛ ·ÓÎÌÓ. ØøÌËÈ „ÂÓÏÂÚðËÈÌ øÁøøÎ ̋ÎÚøø‰ËÈ„ ‡‚·‡Î 27=h ÒÏ, 5.12=b ÒÏ, 6.0=d ÒÏ, 98.0=t ÒÏ,
5010= x I ÒÏ4, 210= x S ÒÏ4 ·‡È̇.
›Ì ˝ ÒÓÌ„ÓÒÓÌ ‰‚ÛÚ‡‚ð˚Ì (6.12‡-ð ÁÛð‡„) ı Û‚¸‰ ·‡Ú · ˝ı ËÈÌ ‰‡ð‡‡ ı ·øð ˝Ì
ÚÓÓˆÓÓ„ ı ËÈÂ.
4. ’‡Ï„ËÈÌ Ë ı ·ÓÒÓÓ ı ø˜‰ ̋ÎËÈÌ ˝Ô ̨𠷇ȄÛÛ· ı
7.161371
1060
5
maxmax =⋅
== x W
M σ Ãœ‡
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
121
¡ÛÒ‡‰ ˆ ̋ „øø‰ ‰ ̋ ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈ„ x I
y M ⋅= max
σ ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰Ó‚. ›Ì˝
ÛÚ„‡‡ð ı ‡Ï„ËÈÌ Ë ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„(6.12·ñð ÁÛð‡„) ·‡È„ÛÛ·‚.
‡). ·). ‚). „). ‰).
6.12ñð ÁÛð‡„
5. ’‡Ï„ËÈÌ Ë ı ı˝‚Ú ˝˝ ı ø˜‰ ̋ÎËÈÌ ˝Ô ̨𠷇ȄÛÛ· ı
’ ̋ ‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„ ∆ Ûð‡‚ÒÍËÈÌ
)( 10015.0
)( 105010
1076
)(
11
8
3max
max y b
S
y b
S
y b I
S Q o x
o x
x
o x ⋅⋅−=⋅
⋅
⋅−=
⋅
⋅=
−τ
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ. ØøÌ˝˝Ò øÁ‚ ̋Î ı˝‚Ú˝˝ ı ø˜‰ ̋ Î ı ø˜‰ ̋ ÎËÈ„ ·Ó‰ÓÊ·‡È„‡‡ ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡ ı b ºð„ºÌººÒ ı ‡Ï‡‡ð˜ ·‡È„‡‡ ÚÛÎ Ó„ÚÎÓÎ˚Ì Ó̈ÎÓ„ˆ ̋ „øø‰ ̋ ‰ ı ø˜‰ ̋ ÎËÈ„ ·Ó‰ÓÊ ˝Ô˛ð ÁÛð‡ ı ¯‡‡ð‰Î‡„‡Ú‡È.
1 ˆ ̋ „. 5.121 == bb ÒÏ, 01, =o
x S (1 ˆ ̋ „ ̋ ˝Ò ‰ ̋ ˝¯ ڇη‡È„øÈ)
01,
1 =⋅
⋅=
b I
S Q
x
o x
τ .
2 ˆ ̋ „. 5.122 == bb ÒÏ, 25.1532
98.0
2
2798.05.12
222, =⎟
⎠
⎞⎜⎝
⎛ −⋅=⎟
⎠
⎞⎜⎝
⎛ −⋅⋅=
t ht bS o x ÒÏ3
83.1
105.12
1025.15310015.0
2
611
2 −=
⋅
⋅⋅⋅−=
−
−
τ Ãœ‡.
2í ˆ ̋ „. 6.0'2 == d b ÒÏ, 25.1532,'2, == o x
o x S S ÒÏ3
3.38106.0
1025.15310015.0
2
611
'2 −=⋅
⋅⋅⋅−=
−
−
τ Ãœ‡.
3 ˆ ̋ „. 6.03 == d b ÒÏ, 33.19632
276.0210
84
2
3, =⋅−=⋅⋅−=h
d h
S S x o
x ÒÏ3.
08.49
106.0
1033.19610015.0
2
611
3 −=
⋅
⋅⋅⋅−=
−
−
τ Ãœ‡
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî122
4 ˆ ̋ „. 6.04 == d b ÒÏ, 2104, == x o
x S S ÒÏ3
5.52
106.0
1021010015.0
2
611
4 −=
⋅
⋅⋅⋅−=
−
−
τ Ãœ‡
Œ„ÚÎÓÎ Á‡‡„ Ú˝Ì ı ν„ËÈÌ ı Û‚¸‰ Ú˝„¯ ı˝ÏÚ˝È Û˜Ëð ˝Ô˛ð Ú˝„¯ ı˝ÏÚ˝È·‡È̇. ƒ ̋ ˝ð ı ÛÚ„ÛÛ‰‡‡ð ı ‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ (6.12‚ñðÁÛð‡„) ·‡È„ÛÛ·‚.
6. √ ÓÎ ı ø˜‰ ̋ÎËÈÌ ˝Ô ̨𠷇ȄÛÛ· ı
√ ÓÎ ı ø˜‰ ̋ ÎËÈ„ „ÛÎÁ‡ÈÎÚ˚Ì ø‰
22
min
max
22
τ
σ σ
σ +⎟
⎠
⎞⎜
⎝
⎛ ±=
ÚÓÏ ˙∏Ó„ÓÓð ·Ó‰‰Ó„. ›Ì˝ ÚÓÏ ˙∏ÓÌÓÓÒ øÁ‚ ̋Î „ÓÎ ı ø˜‰ ̋ Î ·ÓÒÓÓ, ı˝‚Ú˝˝ ı ø˜‰ ̋ Îøø‰ ̋ ˝Ò ı ‡Ï‡‡ð˜ ·‡È̇. ¡ÓÒÓÓ ı ø˜‰ ̋ Î „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚÓÓÒ, ı˝‚Ú˝˝ ı ø˜‰ ̋ Î ı ºÌ‰ÎºÌ ı ø˜Ì˝˝Ò ı ‡Ï‡‡ð‰‡„ Û˜Ëð ˝ˆ ̋ ÒÚ˝˝ „ÓÎ ı ø˜‰ ̋ Î „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ, ı ºÌ‰ÎºÌ ı ø˜Ì˝˝Ò Ì˝„ ̋ Ì Á˝ð˝„ ı ‡Ï‡‡ð‡ ı ˛Ï. »Èω „ÓÎ ı ø˜‰ ̋ ÎËÈÌ ı Û‚¸‰ ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓΠ̸ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ, ı ºÌ‰ÎºÌ ı ø˜ Ì˝„ ̋ Ì Á˝ð˝„ Ë ı ÛÚ„‡Ú‡È ·‡È ı Ó„ÚÎÓÎ ( ¬ Ó„ÚÎÓÎ) ˛Ï. ›Ì˝ Ê˯˝˝ÌËÈ ı Û‚¸‰ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ, ı ºÌ‰ÎºÌ ı ø˜ ı Ó∏ ÛÎ Ì˝„ ̋ Ì Á˝ð˝„ ¬ Ó„ÚÎÓΉ ı ‡Ï„ËÈÌ Ë ı ÛÚ„‡Ú‡È·‡È̇. √ ̋ ı ‰ ̋ ˝ ̋‰„ ̋ ˝ð ‰ÓÚÓÓ‰ ı ø˜Ìøø‰ËÈÌ ÛÚ„‡ Á‡‡‚‡Î Ì˝„ Ó„ÚÎÓΉ Á˝ð˝„ Ë ı
ÛÚ„‡Ú‡È ·‡È ı ‡Î·‡„øÈ. ’ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ ººð ººð Ó„ÚÎÓΉ ı ‡Ï„ËÈÌ Ë ı ÛÚ„‡Ú‡È ·‡È„‡‡ ÚÓ ı ËÓΉÓΉ ı ‡ð¸ˆ‡Ì„ÛÈ ı Ó∏ ÛÎ Ë ı ÛÚ„‡Ú‡È ·‡È ı Ó„ÚÎÓÎ˚„ „ÓÎ ı ø˜‰ ̋ ÎËÈÌ ı Û‚¸‰ ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓÎ „ ̋ Ê øÁÌ˝. “ ̋ „ ̋ ı ‰ ̋ ˝ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ ÌÓ∏ ÎÓ ı øøð˝„Ú˝È ·‡È‰„ËÈ„ ‡Ì ı ‡‡ð˜ ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓÎÒÓÌ„Ó ı ı˝ð˝„Ú˝È.
√ ÓÎ ı ø˜‰ ̋ ÎËÈ„ Ó„ÚÎÓÎ˚Ì ˆ ̋ „øø‰ ‰ ̋ ˝ð ·Ó‰ ˙∏.1 ˆ ̋ „. 7.1611 =σ Ãœ‡, 01 =τ ;
2
7.161
2
7.161
02
7.161
2
7.1612
)1(
minmax ±=+⎟ ⎠
⎞
⎜⎝
⎛
±=σ
7.16185.8085.80 )1( max =+=σ Ãœ‡, 085.8085.80 )1(
min =−=σ .
2 ˆ ̋ „. 9.1492 =σ Ãœ‡, 83.12 −=τ Ãœ‡ ;
( ) 97.7495.7483.12
9.149
2
9.149 22
)2(
min
max ±=−+⎟ ⎠
⎞⎜⎝
⎛ ±=σ
92.14997.7495.74 )2( max =+=σ Ãœ‡,
02.097.7495.74 )2( min −=−=σ Ãœ‡.
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
123
2í ˆ ̋ „. 9.149'2 =σ Ãœ‡, 3.38'2 −=τ Ãœ‡ ;
( ) 17.8495.743.382
9.149
2
9.149 22
)'2(
min
max ±=−+⎟ ⎠
⎞⎜⎝
⎛ ±=σ
12.15917.8495.74 )'2( max =+=σ Ãœ‡,
22.917.8495.74 )'2( min −=−=σ Ãœ‡.
3 ˆ ̋ „. 85.803 =σ Ãœ‡, 08.493 −=τ Ãœ‡ ;
( ) 58.6343.4008.492
85.80
2
85.80 22
)3(
min
max ±=−+⎟
⎠
⎞⎜⎝
⎛ ±=σ
01.10458.6343.40 )3(
max =+=σ Ãœ‡,15.2358.6343.40 )3(
min −=−=σ Ãœ‡.
4 ˆ ̋ „. 04 =σ , 5.524 −=τ Ãœ‡ ;
( ) 5.525.52002 )4(
min
max ±=−+±=σ
5.52 )4( max =σ Ãœ‡, 5.52 )4(
min −=σ Ãœ‡.
›‰„ ̋ ˝ð ÛÚ„‡‡ð „ÓÎ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ (6.12„ñð ÁÛð‡„) ·‡È„ÛÛ·‚.
7. ›ÍÒÚðÂÏÛÏ ı˝‚Ú ˝˝ ı ø˜‰ ̋ÎËÈÌ ˝Ô ̨𠷇ȄÛÛ· ı
›ÍÒÚðÂÏÛÏ ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈ„
2minmax
min
max
σ σ
τ
−±=
ÚÓÏ ˙∏Ó„ÓÓð Ó„ÚÎÓÎ˚Ì Ú˝Ï‰ ̋ „ν„‰Ò˝Ì ˆ ̋ „øø‰ ̋ ‰ ·Ó‰ ˙∏.
1 ˆ ̋ „. 7.161 )1( max =σ Ãœ‡, 0 )1(
min =σ ;
85.802
07.161 )1(
min
max ±=−
±=τ Ãœ‡.
2 ˆ ̋ „. 9.149 )2( max =σ Ãœ‡, 02.0 )2(
min −=σ Ãœ‡ ;
96.742
)02.0( 9.149 )2(
min
max ±=−−
±=τ Ãœ‡.
2í ˆ ̋ „. 12.159 )'2( max =σ Ãœ‡, 22.9 )'2(
min −=σ Ãœ‡ ;
17.842
)22.9( 12.159 )'2(
minmax ±=
−−
±=τ
Ãœ‡.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî124
3 ˆ ̋ „. 104 )3( max =σ Ãœ‡, 15.23 )3(
min −=σ Ãœ‡ ;
57.63
2
)15.23( 104 )3(
min
max ±=−−
±=τ Ãœ‡.
4 ˆ ̋ „. 5.52 )4( max =σ Ãœ‡, 5.52 )4(
min −=σ Ãœ‡ ;
5.522
)5.52( 5.52 )4(
min
max ±=−−
±=τ Ãœ‡.
›‰„ ̋ ˝ð ÛÚ„‡‡ð
min
maxτ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ (6.12‰ñð ÁÛð‡„) ·‡È„ÛÛ·‚.
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6.3. ƒ¿ÃÕ””—¤Õ ’ªÿØØÕ ◊ ¿Õ¿—¤Õ “ŒŒ÷ŒŒ
≈ðºÌ ı ËÈ ÓÈ΄ÓÎÚ
ƒ‡ÏÌÛÛð ·‡Ú ·˝ı ·‡È‚˜ ı º¯øøÌ ˜‡Ì‡ð˚Ì ‰ÛÚ‡„‰Î‡‡Ò ¯‡ÎÚ„‡‡Îʇ¯Ë„· ı ‡‰ ÚÓ ı ËðÓÏÊ„øÈ ·ÓÎÓ ı ˇ‚‰‡Î ·ËÈ. »Èω ‰‡ÏÌÛÛð˚Ì Ï‡ÚÂðˇÎ˚Ì·‡Ú ·˝ı ËÈ„ ¯‡Î„‡ ı˚Ì Á˝ð˝„ˆ ̋ ˝ ÚøøÌËÈ ı º¯øøÌ ˜‡Ì‡ð˚„ ÚÓÓˆÓ ı ı˝ð˝„Ú˝È. ØøÌËÈ ÚÛΉ ‰‡ÏÌÛÛð˚Ì Ú‡ ı ËÈÎÚ („ÛÎÁ‡ÈÎÚ), Ó„ÚÎÓÎ˚Ì ˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈ„ ÚÓ‰Óð ı ÓÈΉӄ.
Œ„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈÌ ‰‡ÏÌÛÛð˚Ì Ú˝Ì ı ν„Ú ÔÂðÔẨËÍÛÎ ˇð˜Ë„Î ̋Î ‰ ̋ ı ¯ËÎÊËÎÚËÈ„ ‰‡ÏÌÛÛð˚Ì ÚÛ ı ‡ÈÌ ˆ ̋ „ ‰ ̋ ı Ú‡ ı ËÈÎÚ „ ̋ Ì˝.
ƒ‡ÏÌÛÛð˚Ì Ó„ÚÎÓÎ ººðËÈÌ ‡Ì ı Ì˚ ·‡Èð·ÎÚ‡È ı ‡ð¸ˆ‡Ì„ÛÈ ˝ð„ ̋ Ò˝ÌºÌˆ„ËÈ„ Ú˝ð Ó„ÚÎÓÎ˚Ì ̋ð„ ̋ÎÚËÈÌ ºÌˆº„ „ ̋ Ì˝.
ƒ‡ÏÌÛÛðÚ øøÒ˝ı ̋‰„ ̋ ˝ð ¯ËÎÊËÎÚËÈ„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ, „ð‡Ùñ‡Ì‡ÎËÚËÍ, ÃÓð-¬Â𢠇„ËÌ˚ Á˝ð˝„ ÓÎÓÌ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈΉӄ. ›Ì‰ Áº‚ ı ºÌ˝ı ÌËÈ ı Ó∏ð ‡ð„ ̊ „ ÚÓ‚˜ Ú‡Èη‡ðÎ‡Ê Ê˯˝˝ ·Ó‰Ó‚.
¿Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ ÂðºÌ ı ËÈ Ú˝„¯ËÚ„ ̋ Î
⎪⎪
⎭
⎪⎪
⎬
⎫
+−
−+
−+
−+
−++=
+−
−+
−+
−+−+=
∑∑∑ ∑
∑∑∑ ∑
... )( 120
)( *
24
)(
6
)(
2
)(
... )( 24
)( *
6
)(
2
)( )(
5432
00
432
0
d l
d z qc z qb z F a z M z EI EIv EIv
d l
d z qc z qb z F a z M EI EI
θ
θ θ
(6.13)
ı ̋η˝ðÚ˝È ·‡È̇. Øø̉ a, b, c, d ñ ÍÓÓð‰Ë̇Ú˚Ì ˝ı Ì˝˝Ò ÏÓÏÂÌÚ, Úº‚κðÒºÌ ı ø˜ øÈΘËÎÊ
·ÛÈ Ó„ÚÎÓÎ ı øðÚ ̋Î Á‡È ·Û ̨ Û Ú‡ð ı χΠ‡˜‡‡Ì˚ øÈΘËÎÊ ˝ı νı Ó„ÚÎÓÎ ı øðÚ ̋ÎÁ‡È (6.13ñð ÁÛð‡„)
0θ ñ ÍÓÓð‰Ë̇Ú˚Ì ˝ı ÌËÈ ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„
0v ñ ÍÓÓð‰Ë̇Ú˚Ì ˝ı ÌËÈ Ú‡ ı ËÈÎÚ
6.13ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî126
0θ , 0v ñ ËÈ„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚð „ ̋ Ê Ì˝ðνı ·º„ºº‰ ̋‰„ ̋ ˝ðËÈ„ ‰‡ÏÌÛÛð˚Ì
·˝ı ̋΄ ̋ ˝Ì˝˝Ò ø̉ ̋ ÒÎ˝Ì ÁÓ ı ËÓÒÓÌ ‡Ì ı Ì˚ ̺ ı ̂ ºÎøø‰ ̋ ˝Ò ÓÎÌÓ. “Û ı ‡Èη‡Î ‰‡ÏÌÛÛð˚Ì ÁÓÓÎÚ ÍÓÓð‰Ë̇Ú˚Ì ˝ı Ú˝È ‰‡‚ ı ̂ ‡Ê ·‡È‚‡Î 0θ , 0v ‡Ì ı Ì˚
Ô‡ð‡ÏÂÚð Ú˝„ ·‡È̇. ÓÓð‰Ë̇Ú˚Ì ˝ı ‰‡ÏÌÛÛð˚Ì ÌÛ„‡Ò‡Ì ÚÛ΄ÛÛðÚ‡È ‰‡‚ ı ̂ ‡Ê ·‡È‚‡Î 0v Ô‡ð‡ÏÂÚð (Ú‡ ı ËÈÎÚ) Ú˝„Ú˝È Ú˝Ìˆøø, 0θ Ô‡ð‡ÏÂÚð Ú˝„ ̋ ˝Ò
̌΄‡‡Ú‡È ·‡È̇. »ÈÏ ÚÓ ı ËÓΉÓΉ ‰‡ÏÌÛÛð˚Ì Ìº„ºº ÌÛ„‡Ò‡Ì ÚÛ΄ÛÛð ‰ ̋ ˝ð Ú‡ ı ËÈÎÚ ( v ) Ú˝„Ú˝È Ú˝Ìˆøø ·‡È ı ̺ ı ̂ κºÒ 0θ Ô‡ð‡ÏÂÚðËÈ„ ÚÓ‰Óð ı ÓÈÎÌÓ.
0θ , 0v ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈ„ ÓÎÊ (6.13) Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛÎÒÌ˚ ‰‡ð‡‡ ˝Ì˝
Ú˝„¯ËÚ„ ̋ ν˝Ò ‡Î¸ ˜ Ó„ÚÎÓÎ˚Ì ¯ËÎÊËÎÚËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ·ÓÎÓÏÊÚÓÈ.
ƒ‡ÏÌÛÛð˚Ì ¯ËÎÊËÎÚËÈ„ „ð‡Ùñ‡Ì‡ÎËÚËÍ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ
⎪⎪
⎭
⎪⎪
⎬
⎫
=
=
EI
M v
EI
Q
x
x
θ
(6.14)
ÚÓÏ ˙∏Ó„ ‡¯Ë„·̇.
Øø̉ x Q ñ ı ÛÛðχ„ ı ºÌ‰ÎºÌ ı ø˜ ,
x M ñ ı ÛÛðχ„ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ.
∆ ˯˝˝ 6.11. ƒ‡ÏÌÛÛð˚Ì (6.14ñð ÁÛð‡„) ÒÛÎ øÁøøðËÈÌ Ú‡ ı ËÈÎÚ,
˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈ„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ ‡ð„‡‡ð ÓÎ.ª„ÒºÌ Ì¸ : F , M , l .
6.14ñð ÁÛð‡„
¡Ó‰ÓÎÚ. «ÓÓÎÚÌ˚ ð‡͈ÛÛ‰ ̊ „ ÒÚ‡ÚËÍËÈÌ Ú˝Ìˆ‚˝ðËÈÌ Ú˝„¯ËÚ„ ̋ ν˝Ò
ÚÓ‰Óð ı ÓÈÎ ˙∏.
0=∑ y , 0=− F R A ⇒ F R A = ;
0=∑ A M , 02
=+⋅+− M l
F M A ⇒ 2
l F M M A ⋅+= .
›Ì˝ ‰‡ÏÌÛÛð˚Ì ı Û‚¸‰ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ (6.13) Ú˝„¯ËÚ„ ̋ Î
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127
⎪⎪⎪⎭
⎪⎪⎪
⎬
⎫
−−−+−−⋅+=
−−
−+−−=
II I
A A
II I
A A x
l z F z R z M z EI EIv EIv
l z F
z R z M EI EI
6 )2/(
6 )0(
2 )0(
2
)2/(
2
)0( )0(
332
00
22
0
θ
θ θ
ı ̋η˝ðÚ˝È ·‡È̇. «ÓÓÎÚ ‰ ̋ ˝ð ˇÏ‡ð ˜ ¯ËÎÊËÎÚ øøÒ˝ı ·ÓÎÓÏÊ„øÈ Û˜Ëð00 =θ , 00 =v (‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðøø‰) ·‡È ı˚„ ‡Ì ı ‡‡ð˜ ð‡͈ÛÛ‰ ̊ Ì ‰ ̋ ˝ð
ÚÓ‰Óð ı ÓÈÎÒÓÌ ÛÚ„ ̊ „ ÓðÎÛÛη‡Î
⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪
⎪⎪⎪
⎬
⎫
⎟ ⎠
⎞⎜⎝
⎛ −
−⋅
+⋅+
−=
⎟ ⎠
⎞⎜⎝
⎛ −
−⋅
+⋅+−=
II
I
II
I
l z F
z F z
l F Ã
EIv
l z F
z F z
l F Ã EI
6
2
62
)2
(
2
2
2
)
2
(
3
32
2
2
θ
·ÓÎÌÓ.
ØøÌ˝˝Ò ˝Ì˝ ‰‡ÏÌÛÛð˚Ì ˇÏ‡ð ˜ Ó„ÚÎÓÎ˚Ì Ú‡ ı ËÈÎÚ, ˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ·ÓÎÓÏÊÚÓÈ. “Û ı ‡Èη‡Î ÒÛÎ øÁøøðËÈÌ B θ ˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈ„
ÓÎÓ ı˚Ì ÚÛΉ l z = ø‰ ˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈÌ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ·Ó‰‚ÓÎ
2
2
2 )
2(
2
2 ⎟ ⎠
⎞⎜⎝
⎛ −
−⋅
+⋅+−=
l l F
l F l
l F M EI B θ ⇒
x x B
EI
l F
I E
l M
8
2⋅−
⋅−=θ
·ÓÎÓ ı ·‡ ÒÛÎ øÁøøðËÈÌ B f Ú‡ ı ËÈÎÚ˚„ ÓÎÓ ı˚Ì ÚÛΉ l z = ÛÚ„ ̊ „ Ú‡ ı ËÈÎÚ˚Ì
Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛη‡Î
6
2
62
2
2
32 ⎟
⎠
⎞⎜⎝
⎛ −
−+
⋅⎟ ⎠
⎞⎜⎝
⎛ +
−==
l l F
l F l
l F M
f EI v EI B x B x ⇒ x x
B EI
Fl
EI
l M f
48
5
2
32
−−=
·ÓÎÌÓ. ¿Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ Ú˝„¯ËÚ„ ̋ Ή ·ÓÒÓÓ ÁÛð‡‡ÒÌ˚ ‡ð‰ ·Ë˜Ò˝Ì I , II „ ̋ ‰ ̋ „ ÚÓÓ Ì¸ ‰‡ÏÌÛÛð˚Ì ı˝Ò„ËÈÌ ‰Û„‡‡ð ˛Ï. ƒ‡ÏÌÛÛð˚Ì ÒÛÎ øÁøøð II ı˝Ò˝„Ú ı ‡Ï‡‡ð˜ ·‡È„‡‡ Û˜Ëð ÒÛÎ øÁøøðËÈÌ ¯ËÎÊËÎÚËÈ„ ÓÎÓ ı ‰ÓÓ II „ ̋ ‰ ̋ „ ÚÓÓÌÓÓÒ ºÏ̺ ı „˯øø‰ËÈ„ ‡¯Ë„Î‡Ò‡Ì ·ÓÎÌÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî128
∆ ˯˝˝ 6.12. ƒ‡ÏÌÛÛð˚Ì (6.15‡ñð ÁÛð‡„) ¿ ÚÛ΄ÛÛðÚ øøÒ˝ı
˝ð„ ̋ ÎÚËÈÌ ºÌˆº„, – øÁøøðÚ øøÒ˝ı Ú‡ ı ËÈÎÚ˚„ „ð‡Ù-‡Ì‡ÎËÚËÍ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎ.
ª„ÒºÌ Ì¸ : 10=q ÍÕ/Ï, 30= F ÍÕ.
‡).
·).
‚).
„).
6.15ñð ÁÛð‡„
¡Ó‰ÓÎÚ. “Û΄ÛÛðÛÛ‰ ̊ Ì ð‡͈ ̊ „ ÓηÓÎ
0=∑ B M , 012
66
2
=⋅+−⋅ F q R A ⇒ 25= A R ÍÕ ;
0=∑ A
M , 0762
62
=⋅+⋅− F R q B
⇒ 65= B
R ÍÕ
·‡È̇.
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129
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ Ó„ÚÎÓÎ ·øðÚ ÚøøÌËÈ Ú˝„¯ËÚ„ ̋ ÎËÈ„ ÁÓ ı ËÓ∏.
1-1 Ó„ÚÎÓÎ )60( 1 ≤≤ z
2 )(
21
11 z q z R z M A −⋅=
01 = z ø‰ 0 )( 1 = z M ,
61 = z Ï ø‰ 30 )( 1 −= z M ÍÕÏ ;
2-2 Ó„ÚÎÓÎ )10( 2 ≤≤ z
22 )( z F z M ⋅−=
02 = z ø‰ 0 )( 2 = z M ,
12
= z Ï ø‰ 30 )( 2
−= z M ÍÕÏ ;
·‡È̇. ›‰„ ̋ ˝ð ÛÚ„‡‡ð „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð (6.15·ñð ÁÛð‡„)·‡È„ÛÛ·‚. ’ÛÛðχ„ ‰‡ÏÌÛÛð˚„ ‡˜‡‡Ú‡È ̸ 6.15·ñð ÁÛð‡„Ú øÁøøν‚. ›Ì˝ ‰‡ÏÌÛÛð˚„ ˆ‡‡¯Ë‰ ¬ ÌÛ„‡Ò‡‡ð ҇΄‡Ì ·Ó‰ÌÓ. ¿¬ ‰‡ÏÌÛÛðÚ øÈÎ˜Î˝ı ‡˜‡‡„„Ûð‚‡ÎÊËÌ, Ô‡ð‡·ÓÎ (6.15‚ñð ÁÛð‡„) ı Ó∏ð ‰øðÒÚ Á‡‰Î‡ ı ‡‰ ڇη‡È ̸
906302
11 =⋅⋅=ω ÍÕÏ2, 180645
3
22 =⋅⋅=ω ÍÕÏ2 ·ÓÎÌÓ. ›Ì˝ ‰‡ÏÌÛÛð˚Ì
(6.15„ñð ÁÛð‡„) ı ÛÛðχ„ ð‡͈ÛÛ‰ ̊ „ ÓηÓÎ
0=∑ x B M , 0236 12 =⋅−⋅+⋅− ω ω
x A R ⇒ 60= x
A R ÍÕÏ2 ;
0=∑ x A M , 0643 12 =⋅+⋅+⋅− x
B R ω ω ⇒ 30= x B R ÍÕÏ2
·ÓÎÌÓ. ƒ‡ð‡‡ ̸ ¬– ‰‡ÏÌÛÛð˚Ì (6.15‰ñð ÁÛð‡„) – ÚÛ΄ÛÛðÚ øøÒ˝ı x C M
ÏÓÏÂÌÚ˚Ì ð‡͈ ̊ „ ÓηÓÎ
0=∑ x C M , 0
3
21 3 =+⋅−⋅ x
C x B M R ω ⇒
201301302
1
3
21
3
23 −=⋅−⋅⋅⋅=⋅−= x
B x C R M ω ÍÕÏ3
·‡È̇. ƒ‡ÏÌÛÛð˚Ì ¯ËÎÊËÎÚËÈ„ ÚÓ‰Óð ı ÓÈÎÓ ı ‰ÓÓ (6.14) ÚÓÏ ˙∏Ó„ ‡¯Ë„·‚‡Î
¿ ÚÛ΄ÛÛðÚ øøÒ˝ı ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„ x x
x A
x A
A EI EI
R
EI
Q 60−=
−==θ , – øÁøøðÚ øøÒ˝ı
Ú‡ ı ËÈÎÚ x x
x C
C EI EI
M v
20=
−= ·ÓÎÌÓ. ›‰„ ̋ ˝ð ı ‡ðËÛ̇‡Ò øÁ‚ ̋Î ¿ Ó„ÚÎÓÎ ˆ‡„ËÈÌ
ÁøøÌËÈ ı º‰ºÎ„ººÌËÈ ‰‡„ÛÛ (Òºðº„) ˝ð„ ̋ Ê, – Ó„ÚÎÓÎ ‰ ̋ ˝¯˝˝ ¯ËÎÊË ı ˛Ï.›ð„ ̋ ÎÚËÈÌ ºÌˆº„, Ú‡ ı ËÈÎÚ˚„ ÓÎÓ ı ‰ÓÓ ı ÛÛðχ„ ı ºÌ‰ÎºÌ ı ø˜, ı ÛÛðχ„„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì Ú˝Ï‰„ËÈ„ ‰ ̋ ˝ð ÚÓ„ÚÒÓÌ ·Ó‰ËÚ ı ø˜, ÏÓÏÂÌÚ˚Ì
Ú˝Ï‰„ËÈÌ ‰øðÏ˝˝ð ÒÓÌ„ÓÎÓÓ.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî130
∆ ˯˝˝ 6.13. ƒ‡ÏÌÛÛð˚Ì (6.16ñð ÁÛð‡„, Ê˯˝˝ 6.10) ÚÛ΄ÛÛðÛÛ‰
·ÓÎÓÌ ÒÛÎ øÁøøðÚ øøÒ˝ı ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„; ‡ÎÒ·Î˚Ì ‰Û̉ ( – ˆ ̋ „) ·ÓÎÓÌ ÒÛÎ øÁøøðÚ øøÒ˝ı Ú‡ ı ËÈÎÚ˚„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚð, „ð‡Ù-‡Ì‡ÎËÚËÍ ‡ð„‡‡ð
ÚÓ‰Óð ı ÓÈÎ.ª„ÒºÌ Ì¸ : 1=a Ï, 20=q ÍÕ/Ï, 401 = F ÍÕ, 40= M ÍÕÏ, 202 = F ÍÕ.
6.16ñð ÁÛð‡„
¡Ó‰ÓÎÚ.
‡). ¿Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ ‡ð„‡. 44= A R ÍÕ, 96= B R ÍÕ ·ÓÎÓ ı˚„ ºÏ̺
(Ê˯˝˝ 6.10) ·Ó‰ÒÓÌ. ›Ì˝ ‰‡ÏÌÛÛðÚ ÚÓ ı ËðÓ ı ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ Ú˝„¯ËÚ„ ̋ Î ÁÓ ı ËÓ‚ÓÎ
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131
+−
+−
−−
−−
+=
II I
A a z qa z F z q z R EI EI
6
)4(
2
)4(
6
)0(
2
)0( 321
32
0θ θ
III
B a z M a z R 1
)5( 2
)5(
2
−+−+
+−
+−
−−
−−
+⋅+=
II I
A a z qa z F z q z R z EI EIv EIv
24
)4(
6
)4(
24
)0(
6
)0( 431
43
00 θ
III
B a z M a z R
2
)5(
6
)5( 23 −+
−+
·‡È̇. 0θ , 0v ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈ„
0= z ø‰ 0=v ( ¿ ÚÛ΄ÛÛð )a z 5= ø‰ 0=v ( ¬ ÚÛ΄ÛÛð ) ·‡È ı ̺ ı ̂ κºÒ ÚÓ‰Óð ı ÓÈÎÌÓ. › ı ÌËÈ
̺ ı ̂ κºÒ 00 EIv= ·‡È̇. ’Ó∏ð‰Û„‡‡ð ̺ ı ̂ κºÒ
24624
)5(
6
)5( 50
431
43
0
aqa F aqa R a EI A ⋅
+⋅
−⋅
−⋅
+⋅= θ ⇒ 18.780 −=θ EI
·ÓÎÌÓ.
0= z ø‰ ¿ Ó„ÚÎÓÎ˚Ì ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„ x A EI
18.780 −==θ θ ,
a z 5= ø‰ ¬ Ó„ÚÎÓÎ˚Ì ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„
6
120
2
140
6
520
2
54418.78
3232 ⋅+
⋅−
⋅−
⋅+−= B θ ⇒
x B
EI
45.38=θ ,
a z 6= ø‰ D Ó„ÚÎÓÎ˚Ì ˝ð„ ̋ ÎÚËÈÌ ºÌˆº„
1
140
2
196
6
220
3
240
6
620
2
64418.78
23232 ⋅+
⋅+
⋅+
⋅−
⋅−
⋅+−= D EI θ ⇒
x D
EI 52.28=θ ,
a z 6= ø‰ D Ó„ÚÎÓÎ˚Ì Ú‡ ı ËÈÎÚ
2
140
6
196
24
220
6
240
24
620
6
644618.78
234343 ⋅+
⋅+
⋅+
⋅−
⋅−
⋅+⋅−= D EIv ⇒
x D
EI v
92.30=
a z 5.2= ø‰ ‡ÎÒ·Î˚Ì ‰Û̉ C Ó„ÚÎÓÎ˚Ì Ú‡ ı ËÈÎÚ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî132
24
5.220
6
5.2445.218.78
43 ⋅−
⋅+⋅−=C EIv ⇒
x C
EI v
42.113−=
·‡È̇.
·). √ ð‡Ù ñ ‡Ì‡ÎËÚËÍ ‡ð„‡. ƒ‡ÏÌÛÛð˚Ì ¯ËÎÊËÎÚËÈ„ ˝Ì˝ ‡ð„‡‡ð ÓÎÓ ı˚Ì ÚÛΉ ı ÛÛðχ„ ‰‡ÏÌÛÛð˚„
(6.16·ñð ÁÛð‡„) ·‡È„ÛÛÎ̇. ƒ‡ð‡‡ ̸ ı ÛÛðχ„ ‰‡ÏÌÛÛð˚„ (6.16·ñð ÁÛð‡„)º„º„‰ÒºÌ ·Ó‰ËÚ ‰‡ÏÌÛÛð˚Ì „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð˝˝ð ‡˜‡‡Î̇. “ ̋ ı ‰ ̋ ˝ Ú˝ð „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ (6.16‡ñð ÁÛð‡„) ı ̌η‡ð ‰øðÒøø‰ ̋ ‰ ı Û‚‡‡Ê ڇη‡È ·Û ̨ Û Ú˝Ìˆøø øÈÎ˜Î˝„˜ ı ø˜ËÈ„ øÒ„ ̋ ˝ð Ú˝Ï‰ ̋ „ν‚.
’ÛÛðχ„ ð‡͈ÛÛ‰ ̊ „ ÓÎÓ ı˚Ì ÚÛΉ ¬ ÌÛ„‡Ò‡‡ð ҇΄‡Ê (6.16‚ñð ÁÛð‡„)
·Ó‰ÌÓ. ’ ̌η‡ð ‰øðÒøø‰ËÈÌ Ú‡Î·‡È„ ÓηÓÎ
7.10612
420
12
)4( 33
1 =⋅
=⋅
= aq
ω ÍÕÏ2, 324162
12 =⋅⋅= aω ÍÕÏ2,
8162
13 =⋅⋅= aω ÍÕÏ2, 3060
2
14 =⋅⋅= aω ÍÕÏ2, 1020
2
15 =⋅⋅= aω ÍÕÏ2,
·‡È̇. ’ÛÛðχ„ ‰‡ÏÌÛÛð˚Ì Ú˝Ìˆ‚˝ðËÈÌ Ìº ı ̂ κºÒ ı ÛÛðχ„ ð‡͈ÛÛ‰ ̊ „ÓηÓÎ
0=∑ x
B M ; 03
1
3
2 )3
85( 35 4321 =⋅−⋅+−⋅+⋅+⋅− aaaaaa R
x A ω ω ω ω ⇒
02.78= x A R ÍÕÏ2,
0=∑ x A M ; 05 )
3
15( )
3
25(
3
82 4321 =⋅+−+−⋅−⋅−⋅− a R aaaaaa x
B ω ω ω ω ⇒
68.38= x ¬ R ÍÕÏ2,
0=∑ x y (·‡ðÛÛÌ ‰‡ÏÌÛÛðÚ); 05 =+− x D
x ¬ R R ω ⇒ 68.28−= x
D R ÍÕÏ2,
0=∑ x D M (·‡ðÛÛÌ ‰‡ÏÌÛÛðÚ); 0325 =+⋅−⋅ x D x ¬ M aa R ω ⇒
01.32−= x D M ÍÕÏ3
·‡È̇. ›Ì˝ ÚÓ‰Óð ı ÓÈÎÒÓÌ ð‡͈ÛÛ‰‡‡ð ¯ËÎÊËÎÚøø‰ËÈ„ ÚÓ‰Óð ı ÓÈηÓÎ
x x
x A
x
x A
A EI EI
R
EI
Q 02.78−=−==θ ,
x x
x B
x
x B
¬ EI EI
R
EI
Q 68.38=+==θ ,
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
133
x x
x D
x
x D
D EI EI
R
EI
Q 68.28=−==θ ,
x x
x D
D EI EI
M
v
01.32
=
−
=
·‡È̇. ¿ÎÒ·Î˚Ì ‰Û̉ – ˆ ̋ „Ú øøÒ˝ı Ú‡ ı ËÈÎÚ˚„ ÚÓ‰Óð ı ÓÈÎÓ ı˚Ì ÚÛΉ
„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈÌ – Ó„ÚÎÓÎÓÓÒ ºÏ̺ ı ı˝Ò˝„Ú ‰‡ ı ËÌ ı Û‚‡‡ÎÚ ı ËÈ ı (6.17ñð ÁÛð‡„) ı˝ð˝„Ú˝È. ØøÌËÈ ÚÛΉ ˝ı ν ̋‰ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ ν˝Ò – ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡ ı Óð‰Ë̇Ú
5.472
)5.2( 5.2 )(
2
1 =⋅
−⋅= aqa R z M A ÍÕÏ
·‡È ı˚„ ÓÎÌÓ. ƒ‡ð‡‡ ̸ ı ÛÛðχ„ ‰‡ÏÌÛÛð˚Ì – ˆ ̋ „Ú ÏÓÏÂÌÚ
·Ë˜‚ ̋Î
+⋅−=⋅++⋅+⋅−= 5.202.7883.0'25.1'5.2 21 ω ω x A
x C R M
22.11383.0375.5925.104.26 −=⋅+⋅+ ÍÕÏ3 ·‡È̇.
Øø̉ 04.2612
)5.2( '
3
1 == q
ω ÍÕÏ2,
375.595.25.472
1'2 −=⋅⋅=ω ÍÕÏ2.
6.17ñð ÁÛð‡„
“ ̋ „‚ ̋Î – Ó„ÚÎÓÎ˚Ì Ú‡ ı ËÈÎÚ
x x
x C
C EI EI
M v
22.113−==
·ÓÎÌÓ. ’Ó∏ð ‡ð„‡‡ð ·Ó‰ÒÓÌ ı ‡ðËÛÌÛÛ‰ ı ÓÓðÓ̉ÓÓ Ú‡‚‡Ì ı Û‚Ë‡Ò Ë ı „øÈÁºðººÚ˝È ·‡È‚‡Î ÌËÈˆÚ˝È „ ̋ Ê øÁ‰ ̋ „. ›Ì˝ ·Ó‰ÎÓ„ ̊ Ì ı Û‚¸‰ Áºðøø ̸ Ëνð ı ËÈ·‡„‡ ·‡È̇.
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ƒ¿¿À√ ¿¬¿— ¹6. ƒ‡ÏÌÛÛð˚Ì ·‡Ú ·˝ı , ı º¯øøÌ ˜‡Ì‡ð˚Ì ÚÓÓˆÓÓ
ƒ‡‡Î„‡‚‡ð ı Ó∏ð ·Ó‰ÎÓ„ÓÓÒ ·øð‰ ̋ Ì˝.
1. ª„º„‰ÒºÌ „‡Ì ( 160][ =σ Ãœ‡, 210= R Ãœ‡) ‰‡ÏÌÛÛð˚Ì (6.19ñð ÁÛð‡„)
1. ’ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ;
2. ƒ‡ÏÌÛÛð˚Ì ‰‚ÛÚ‡‚ð, Ú˝„¯ ºÌˆº„Ú ⎟ ⎠
⎞⎜⎝
⎛ = 5.1
b
h, ‰Û„ÛÈ Ó„ÚÎÓÎ ÒÓÌ„ÓÊ
ˆ‡‡¯Ë‰ ‰‚ÛÚ‡‚ð Ó„ÚÎÓΉ ÚÓÓˆÓÓ ı ËÈ;3. ’‡Ï„ËÈÌ Ë ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ;4. ’‡Ï„ËÈÌ Ë ı ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ;5. √ ÓÎ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ;6. ›ÍÒÚðÂÏÛÏ ı˝‚Ú˝˝ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.
ª„º„‰ÎËÈ„ 6.2ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
2. ª„º„‰ÒºÌ ‰‡ÏÌÛÛð˚Ì (6.19ñð ÁÛð‡„) ÒÛÎ øÁøøð, ‡ÎÒ·Î˚Ì „ÓΉ Ú‡ ı ËÈÎÚ,ÒÛÎ øÁøøð ·ÓÎÓÌ ÌÛ„‡Ò‡Ì ÚÛ΄ÛÛðÛÛ‰ ‰ ̋ ˝ð øøÒ˝ı ˝ð„ ̋ ÎÚËÈÌ ºÌˆ„ËÈ„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ ·ÓÎÓÌ „ð‡Ù-‡Ì‡ÎËÚËÍ ‡ð„‡‡ð ÚÓ‰Óð ı ÓÈÎ.ª„º„‰ÎËÈ„ 6.2ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
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ƒŒÀŒŒƒ” √ ¿¿— ¡ØÀ›√
Õ»…ÀÛÀ ›–›—√ ØØ÷›À
7.1. ŒÕŒÀ¤Õ “Œ¬◊ ØÕƒ›–
∆ ˯øø „ÛÎÁ‡ÈÎÚ
ÿËη˝ÌËÈ Ê˯øø „ÛÎÁ‡ÈÎÚ˚Ì ø‰ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ‰Ûð˚Ì ˆ ̋ „Ú·ÓÒÓÓ ı ø˜‰ ̋ Î ÚÓÓˆÓ ı ÚÓÏ ˙∏Ó
x I
M y
I
M
y
y
x
x ⋅+⋅=σ ( 7.1)
·‡È‰‡„. Øø̉ : y x I I , - Ó„ÚÎÓÎ˚Ì Úº‚ „ÓÎ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ
ÏÓÏÂÌÚ; x , - ˆ ̋ „ËÈÌ Úº‚ „ÓÎ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ÍÓÓð‰Ë̇Ú; y x M M , ñ y x , Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ.
Œ„ÚÎÓÎ˚Ì Á‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Π( β ºÌˆº„)
ϕ β tantan y
x y
I
I M ⋅−= ( 7.2)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ.– ‡ Ì ‡ Ï Ê. β ϕ , ºÌˆ„øø‰ ËÊËÎ Ú˝Ï‰ ̋ „Ú˝È ·‡È̇.
«‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡ ı ÒÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ÎËÈÌ ÛÚ„‡‡ð ¯Ëη˝ÌËÈ ·º ı ·‡Ú˚„ ¯‡Î„‡Ì‡.
cdan
y
y
dan
x
x
t dan
y
y
dan
x
x
R y I
M y
I
M
R x I
M y
I
M
≤⋅+⋅=
≤⋅+⋅=
σ
σ
( 7.3)
dandan y x , - «‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı ˆ ̋ „ËÈÌ ÍÓÓð‰Ë̇Ú
“º‚ „ÓÎ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆÛÛ· ı ‡‰, Ó„ÚÎÓΠ̸ Ú˝„¯ ı˝ÏÚ˝È ‰øðÒËÈÌ ı Û‚¸‰
·º ı ·‡Ú˚Ì Ìº ı ̂ ºÎËÈ„
c
y
y
x
x
t
y
y
x
x
R W
M
W
M
R W
M
W
M
≤+=
≤+=
σ
σ
( 7.4)
ÚÓÏ ˙∏Ó„ÓÓð Ëνð ı ËÈÎÌ˝.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî136
“º‚ËÈÌ ·Ë¯ ¯‡ ı ‡ÎÚ (ÒÛÌ„‡ÎÚ)
“º‚ËÈÌ ·Ë¯ ¯‡ ı ‡ÎÚ, ÒÛÌ„‡ÎÚ˚Ì ø‰ ¯Ëη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ‰Ûð˚̈ ̋ „Ú ı ø˜‰ ̋ Î ÚÓÓˆÓ ı ÚÓÏ ˙∏Ó
x I
M y
I
M
A
N
y
y
x
x ⋅+⋅+=σ ( 7.5)
·‡È̇.«‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó„ÓÓð ÓÎÌÓ.
F
x
F
y
y
ib
x
ia
2
0
2
0 ; −=−= ( 7.6)
Øø̉ :0a
,0b
- Á‡‡„ Ú˝Ì ı ν„, Ó„ÚÎÓÎ˚Ì „ÓÎ Ú˝Ì ı ν„øø‰ËÈ„ Ó„ÚÎÓ ı ˆ ̋ „ËÈÌ
ÍÓÓð‰Ë̇Ú; y x ii , - x , Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ð‡‰ËÛÒ; F F y x , ñ
‰‡„ÛÛ ı ø˜ÌËÈ øÈΘΠ̋ÎËÈÌ ˆ ̋ „ËÈÌ ÍÓÓð‰ËÌ‡Ú (7.1ñð ÁÛð„ËÈ„ øÁ). «‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı ˆ ̋ „øø‰ ̋ ‰ ( A, B ) ÒÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ÎËÈÌË ı ÛÚ„‡ ı ‡ð„‡ÎÁ‡Ì‡. ƒ ̋ ˝ð ı ˆ ̋ „øø‰ ̋ ‰ χÚÂðˇÎ˚Ì ·º ı ·‡Ú˚„ ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó„ÓÓ𠯇΄‡Ì‡.
[ ]
[ ]⎪⎪
⎭
⎪
⎪
⎬
⎫
≤⋅+⋅+=
≤⋅+⋅+=
C B
y
y
B
x
x B
t K
y
y
K
x
x K
x I
M y
I
M
A
N
x I
M y
I
M
A
N
σ σ
σ σ
( 7.7)
7.1ñð ÁÛð‡„
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137
’ø˜ÌËÈ øÈΘΠ̋ÎËÈÌ ˆ ̋ „ËÈÌ ÍÓÓð‰Ë̇Ú˚„
0
2
0
2
; y
i y
x
i x x
F y
F −=−= ( 7.8)
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈΉӄ.
ÃÛ¯„Ëð‡ÎÚÚ‡È „ÛÎÁ‡ÈÎÚ
“Û ı ‡ÈÌ Ó„ÚÎÓΉ ı ‡ð„‡ÎÁ‡ ı „ÛÎÁ‡È΄‡ ı ·øÚ˝Ì ÏÓÏÂÌÚ .
22 x y M M M += ( 7.9)
›Ì‰: x M , − y M x , Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ·‡È„ÛÛÎÒ‡Ì „ÛÎÁ‡È΄‡ ı
ÏÓÏÂÌÚ˚Ì ÛÚ„‡.
¡º ı ·‡Ú˚Ì IV ÓÌÓÎÓÓð ÚÓÓˆÓÓÌ˚ ÏÓÏÂÌÚ ÓÎÓ ı ÚÓÏ ˙∏Ó22 75.0 “ M M
“ += ( 7.10)
›Ì‰: à - ·øÚ˝Ì „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ; “ ñÏÛ¯„Ë ı ÏÓÏÂÌÚÃÛ¯„Ëð‡ÎÚÚ‡È „ÛÎÁ‡ÈÎÚ˚Ì ·º ı ·‡Ú˚Ì Ìº ı ̂ ºÎËÈ„ ¯‡Î„‡ ı ÚÓÏ ˙∏Ó
[ ]σ σ ≤= x
˝Í‚W
M T ( 7.11)
›Ì‰: ˝Í‚σ - ı ø˜‰ ̋ ÎËÈÌ ˝Í‚Ë‚‡ÎÂÌÚ ÛÚ„‡; [ ]σ - Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î;
32
3
d W x π = - ı Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ
ƒÛ„ÛÈ Ó„ÚÎÓÎÚÓÈ „ÓÎ˚Ì Ó„ÚÎÓÎ ÒÓÌ„Ó ı ÚÓÏ ˙∏Ó
[ ]3
32
σ⋅π
⋅= T
M d ( 7.12)
·ÓÎÌÓ.
7.2. ∆ ˯øø „ÛÎÁ‡ÈÎÚ ̊ Ì ÚÓÓˆÓÓÌ˚ ‰‡ð‡‡Î‡Î
1.
ª„º„‰ÒºÌ ÚÓÓˆÓÓÌ˚ ÁÛð„ËÈ„ z x z y 0,0 ı ‡‚Ú„‡È ‰ ̋ ˝ð, øÈΘËÎÊ ·ÛÈ
ı ø˜ÌËÈ ı ‡ÏÚ ÁÛðÊ øÁøøνı 2.
– ı ÂÏ ÚÛÒ ·øð ‰ ̋ ˝ð ÚÓÓˆÓÓ ı ËÈÊ, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð·‡È„ÛÛÎÊ ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈÌ Ë ı ÛÚ„ ̊ „ ÏÓ‰ÛΡð ̸ ‡‚̇.
3. Œ„ÚÎÓÎ˚Ì Úº‚ „ÓÎ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ y x I I , ËÌÂðˆËÈÌ
ÏÓÏÂÌÚ˚„ ÓÎÌÓ.4. Œ„ÚÎÓÎ˚Ì ˝Ò˝ð„øøˆÎËÈÌ y x W W , ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÌÓ.
5. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ÚÓ‰Óð ı ÓÈÎÌÓ.6.
ƒ‡ÏÌÛÛð˚Ì ·º ı ·‡Ú˚„ ¯‡Î„‡Ì‡.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî138
∆ ˯˝˝ 7.1. ∆ ˯øø „ÛÎÁ‡ÈÎڇ̉ ÓðÊ ·ÛÈ ‰‡ÏÌÛÛð ‰ ̋ ˝ð øÈÎ˜Î˝ı
„‡‰‡‡‰ ı ø˜, Á‡È, Ó„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝„ 7.2ñð ÁÛð‡„Ú øÁøøν‚.
F (ÍÕ)
q
(ÍÕ/Ï )
M
(ÍÕ⋅Ï)a (Ï)
c (Ï)
d (Ï)
h (ÒÏ)
b (ÒÏ)
R Ãœ‡
20 20 20 1 3 2 30 15 20
M F
a
x
y z
q
d c
F M 0
b
0.6b
h
x
y
‡). ·).
7.2ñð ÁÛð‡„
“ÓÓˆÓÓ„ÓÓð ÓÎÓ ı ÁøÈÎ:
1.
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ( y x M M , ) ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ2. “ÓÓˆÓÓÌ˚ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÊ, ‰‡ÏÌÛÛð˚Ì ·‡Ú ·º ı ËÈ„ ¯‡Î„‡.3. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ÚÓ‰Óð ı ÓÈÎÊ, ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓΉ
ı ‡ð„‡ÎÁ‡ ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ
¡Ó‰ÓÎÚ1. ª„º„‰ÒºÌ ‰‡ÏÌÛÛð˚Ì ÚÓÓˆÓÓÌ˚ Ò ı ÂÏËÈ„ (7.3‡, ·ñð ÁÛð‡„) z x z y 0,0
ı ‡‚Ú„‡È ‰ ̋ ˝ð ҇΄‡Ê ÁÛð̇.
2. – ı
ÂÏ ÚÛÒ ·øð ‰ ˝˝
ð ÚÓÓˆÓÓ ı
ËÈÊ „ÛÎÁ‡È΄‡ ı
ÏÓÏÂÌÚ˚
Ì˝
Ô˛
ðËÈ„·‡È„ÛÛÎ̇.7.3 ‡ñð ÁÛð„ËÈÌ ı Û‚¸‰:
( ) 00 =+⋅−⋅+−=∑ d c B c F M M y A
85
32020=
⋅+−=
+
⋅+−=
d c
c F M B y ÍÕ
( ) 00 =+⋅+⋅−−=∑ d c Ad F M M y B
12
5
22020=
⋅+=
+
⋅+=
d c
d F M A y ÍÕ
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
139
—‡͈ËÈ„ ¯‡Î„‡ ˇ.
∑ =+−=+−= 0122080 y y B F A y
a).
·).
M
a
y
c d
F A B y A B
z 1 z 2 3 z
x A x x B
M
4 z
1
1
2
2 3
3
4
4
5
5
5 z
q
3
3
z 3a c d
‚).
„).
-20
x M
M y
10
58.4
52
7.3ñð ÁÛð‡„
Œ„ÚÎÓÎ ·øð ‰ ̋ ˝ð ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ, ÚÓÓÌ ı Óηӄ‰Î˚„ ÓÎÌÓ.1ñ1 Ó„ÚÎÓÎ Œ„ÚÎÓÎ˚Ì ı ËÎ a z ≤≤ 10
20 )( 1 −=−= M z M x ÍÕ⋅Ï=const 2ñ2 Ó„ÚÎÓÎ. Œ„ÚÎÓÎ˚Ì ı ËÎ ca z a +≤≤ 2
( )a z A M z M y x −+−= 22 )( 20 )1( 2−=′ = z M ÍÕ⋅Ï
16 )14( 1220 )4( 2=−+−=′′ = z M ÍÕ⋅Ï
3ñ3 Ó„ÚÎÓÎ. Œ„ÚÎÓÎ˚Ì ı ËÎ d z ≤≤ 30
33 )( z B z M y x ⋅= 0 )0( 3=′ = z M ÍÕ⋅Ï
1628 )2( 3=⋅=′′ = z M ÍÕ⋅Ï
y Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ øÈΘËÎÊ ·ÛÈ ı ø˜Ì˝˝Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì˝Ô˛ðËÈ„ 7.3‚ñð ÁÛð‡„Ú øÁøøν‚.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî140
7.3·ñð ÁÛð„ËÈÌ ı Û‚¸‰:
0 )( 22
022
=−+⋅−⋅+⋅+⋅−=∑ M d c B c
qa F a
q M x A
165
202
3201202
120
)(
22
2222
=−⋅+⋅+⋅−
=+
−⋅+⋅+⋅−=
d c
M cqa F aq B x ÍÕ
∑ =−++++
⋅+⋅−++⋅= 0 )( )2
( )( )( 0 M d c Ad ca
caqd ca F M x B
445
201620620 )
2( )( )(
=+⋅+⋅−
=+
+++
⋅+⋅+++⋅−=
d c
M d ca
caqd ca F A x ÍÕ
—‡͈ Áº‚ ˝Ò˝ı ËÈ„ ¯‡Î„‡ ˇ.∑ =++−=+++⋅−= 016448020 )( 0 x x B Acaq F x
Œ„ÚÎÓÎ ·øð ‰ ̋ ˝ð ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ , ÚÓÓÌ ı Óηӄ‰Î˚„ ÓÎ ˙∏.4ñ4 Ó„ÚÎÓÎ. Œ„ÚÎÓÎ˚Ì ı ËÎ a z ≤≤ 40
2 )(
24
44
qz z F z M y −⋅=
0 )0( 4== z M
102
120120
2
)1( 4=−⋅=′′ = z M ÍÕ⋅Ï
5ñ5 Ó„ÚÎÓÎ. Œ„ÚÎÓÎ˚Ì ı ËÎ ca z a +≤≤ 5
)( 2
)( 5
25
55 a z A z
q z F z M x y −⋅+⋅−⋅=10 )1( 5
=′ = z M ÍÕ⋅Ï
52 )4( 5=′′ = z M ÍÕ⋅Ï
)( 5 z M y -ËÈÌ ˝ÍÒÚðÂÏÛÏ ÛÚ„ ̊ „ ÓÎ ˙∏.
2.320
640 555 ===+⋅−= z A z q F Q x Ï
( )( ) 4.5812.3442
2.3
202.320
2
)2.3( 5 =−+−⋅== z M ÍÕ⋅Ï6ñ6 Ó„ÚÎÓÎ. Œ„ÚÎÓÎ˚Ì ı ËÎ d z ≤≤ 60
M z B z M x y +⋅= 66 )( 20 )0( 6=′ = z M ÍÕ⋅Ï
5220216 )2( 6=+⋅=′′ = z M ÍÕ⋅Ï
x Ú˝Ì ı ν„ ‰‡„ÛÛ øÈΘËÎÊ ·ÛÈ ı ø˜Ì˝˝Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„7.3„-ð ÁÛð‡„Ú øÁøøν‚. 3. Œ„ÚÎÓÎ˚Ì ı , y Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ ÏÓÏÂÌÚ˚„ ÓÎ ˙∏.
(7.2· ñð ÁÛð„ËÈ„ ı ‡ð)
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141
( )
5.2228305.0
64
156.014.3
12
3015
5.064
)6.0(
12
4343
=⋅
⋅⋅−
⋅
=⋅
⋅−
=h
bbh
W x
π
ÒÏ3
( )1.1082
155.064
914.3121530
5.0646.0
12
434
3
=⋅
⋅−⋅=
⋅
⋅π−=
b
bhb
W y ÒÏ3
4. ›Ô˛ð˝˝Ò ÏÓÏÂÌÚ˚Ì Ë ı ÛÚ„ ̊ „ 20−= x M ÍÕ⋅Ï, 4.58= y M ÍÕ⋅Ï
(ÏÓ‰ÛΡð ) ‡‚˜, ÚÓÓˆÓÓÌ˚ ÏÓÏÂÌÚ˚„ ÓÎ ˙∏.
7.614.5820 2222 =+=+= y x M M M T
ÍÕ⋅Ï
5. ƒ‡ÏÌÛÛð˚Ì ·‡Ú ·º ı ËÈ„ ¯‡Î„‡ ı ‰‡‡
[ ]σ≤+=σ y
y
x
x W
M
W
M max ÚÓÏ ˙∏Ó„ ‡¯Ë„· ı ·º„ºº‰ x,y Ú˝Ì ı ν„ ‰ ̋ ˝ð
„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚÛÛ‰ ̊ Ì ÛÚ„‡ Ì˝„ ̋ Ì Á˝ð˝„ Ë ı ·‡È ı ˆ ̋ „ËÈ„ ÒÓÌ„ÓÌ ‡‚̇.
23.55101.1082
1052
105.2228
10166
3
6
3
max =⋅
⋅+
⋅
⋅=
−−σ Ãœ‡< 210Ãœ‡
6. ÿËη˝ÌËÈ ı ‡Ï„ËÈÌ ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓΉ Á‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î
y x
x y
I M
I M
⋅
⋅= β tan ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. ØøÌËÈ ÚÛΉ º„º„‰ÒºÌ
Ó„ÚÎÓÎ˚Ì ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÓÎÌÓ.
34.04.58
20tan ===
x
y
M
M ϕ
x y
y x
y
x
M I
M I
I
I
⋅
⋅−=⋅−= ϕ β tantan
( )33428
64
156.014.3
12
3015
6412
4343
=⋅
−⋅
=π
−= d bh
I x ÒÏ4
( )5.8115
64
156.014.3
12
1530
6412
4343
=⋅
−⋅
=π
−= d hb
I y ÒÏ4
y
x
I
I ⋅= ϕ β tantan 4.1
5.8115
3342834.0tan =⋅= β 6454 ′=
β
«‡‡„ Ú˝Ì ı ν„ Ó„ÚÎÓÎ˚Ì Ì˝„, „Ûð‡‚‰Û„‡‡ð Ϻ˜ËÈ„ ‰‡Èð˜ Ú‡Ú‡„‰‡Ì‡. ›Ì˝ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı B A, ˆ ̋ „Ú ÒÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ˚Ì ı ø˜‰ ̋ ÎËÈÌ
Ë ı ÛÚ„‡ ı ‡ð„‡ÎÁ‡Ì‡.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî142
B
A
σ A = 6 2 .9
B = 6 2
.9
σ
0
β
c a a a
o y i
o e y a
x
y
7.4 ñð ÁÛð‡„
7. B A, ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡ ı ·ÓÒÓÓ ı ø˜‰ ̋ ÎËÈÌ ÛÚ„ ̊ „ ÓÎÊ ˝Ô˛ðËÈ„ (7.4ñð ÁÛð)
·‡È„ÛÛÎ̇. ÓÓð‰ËÌ‡Ú 5.7−= A x ÒÏ; 15= A y ÒÏ ø‰
( ) ( )9.62
105.8115
105.7104.58
1033428
101510208
23
8
23
−=⋅
⋅−⋅+
⋅
⋅⋅⋅−=+=σ
−−
−
A y
y A
x
x A x
I
M y
I
M Ãœ‡
5.7= B x ÒÏ, 15−= B y ÒÏ ø‰
( )9.62
105.8115
105.7104.58
1033428
101510208
23
8
23
=⋅
⋅⋅⋅+
⋅
⋅−⋅⋅−=⋅+⋅=
−−
−
B
y
y B
x
x B x
I
M y
I
M σ Ãœ‡
·ÓÎÌÓ.
7.3. “º‚ËÈÌ ·Ë¯ ¯‡ ı ‡ÎÚÚ‡È ¯Ëη˝ÌËÈ ÚÓÓˆÓÓÌ˚ ‰‡ð‡‡Î‡Î
1. ª„º„‰ÒºÌ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚„ „ÂÓÏÂÚðËÈÌ ı ̌η‡ð ‰øðÒøø‰ ̋ ‰ ı Û‚‡‡Ê, ÚÛÒ ·øðËÈÌ ı ø̉ËÈÌ Úº‚, ڇη‡È„ ÓÎÌÓ.
2.
Œ„ÚÎÓÎ˚Ì ı ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰Ë̇Ú˚„ ÚÓ‰Óð ı ÓÈÎÌÓ.3. √ ÓÎ Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎ.4. »ÌÂðˆËÈÌ ð‡‰ËÛÒ˚„ ÓÎÌÓ.5.
Œ„ÚÎÓÎ ‰ ̋ ˝ð º„º„‰ÒºÌ ı ø˜ øÈÎ˜Î˝ı ˆ ̋ „ËÈÌ ÍÓÓð‰Ë̇ڇ‡ð Á‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ÚÓ‰Óð ı ÓÈÎÌÓ. (7.6ñð ÚÓÏ ˙∏Ó„ øÁÌ˝ øø)
6. Œ„ÚÎÓÎ˚Ì º„º„‰ÒºÌ ı˝ÏÊ ̋ ˝„ ‡¯Ë„Î‡Ì Áº‚¯ººðº„‰º ı ‡˜‡‡„ ( adm F )
ÚÓ‰Óð ı ÓÈÎÌÓ7. –ÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ÎËÈÌ Ë ı ÛÚ„ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ , ÚøøÌËÈ ˝Ô˛ðËÈ„
·‡È„ÛÛÎ̇.
8.
Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ̇ 7.8ñð ÚÓÏ ˙∏Ó„ ‡¯Ë„·̇.
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∆ ˯˝˝ 7.2 7.5ñð ÁÛð‡„Ú ·‡„‡Ì‡Ì˚ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚„ øÁøøν‚.
Œ„ÚÎÓÎ˚Ì P ˆ ̋ „Ú ·‡„‡Ì‡Ì˚ Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ F ı ø˜ ¯‡ ı ̇.
ÇÚÂðˇÎ˚Ì ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î ÒÛÌ„‡Îڇ̉ 3=t R Ãœ‡, ¯‡ ı ‡Îڇ̉
30=c R Ãœ‡ ·ÓÎÌÓ.ª„º„‰ºÎ: F=220ÍÕ a=30 ÒÏ, b=14ÒÏ
“ÓÓˆÓÓ„ÓÓð ÓÎÓ ı ÁøÈÎ1. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î2.
«º‚¯ººðº„‰º ı ‡˜‡‡Ì˚ ı˝ÏÊ ̋ ˝„ ÓÎÌÓ3. ÿ‡ ı ‡ ı , ÒÛÌ„‡ ı ı ø˜‰ ̋ ÎËÈÌ Ë ı ÛÚ„ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ , ÚøøÌËÈ ˝Ô˛ðËÈ„
ÁÛð4.
Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ̇.
¡Ó‰ÓÎÚ 1. Œ„ÚÎÓÎ˚„ I, II, III, IV, V ı ̌η‡ð ‰øðÒ ·Ó΄ÓÌ ı Û‚‡‡‚. “ÛÒ ·øðËÈÌ Ú‡Î·‡È„
ÓÎ ˙∏ I, III Ú˝„¯ ºÌˆº„Ú : 420143013 =⋅=⋅== ba A A ÒÏ2
II Ú˝„¯ ºÌˆº„Ú : 25202890232 =⋅=⋅= ba A ÒÏ2
IV, V Ú˝„¯ ºÌˆº„Ú „Ûð‚‡ÎÊËÌ : 21030142
1
2
154 =⋅⋅=⋅⋅== ab A A ÒÏ4
2. ’ ̌η‡ð ‰øðÒ ·øðËÈÌ 54321 ,,, x x x x x Úº‚ Ú˝Ì ı ν„ËÈ„ 7.5ñð ÁÛð‡„ ‰ ̋ ˝ð
ÁÛðÊ øÁøøν‚.3. ÕËÈÎÏ ̋Î ‰øðÒËÈÌ ı ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰ËÌ‡Ú ÓΉӄ ÚÓÏ ˙∏Ó„ ‡¯Ë„· ˇ.
›Ì˝ ÁÓðË΄ÓÓð Ó„ÚÎÓÎ˚Ì ÒÛÛðËÈ„ ‰‡ÈðÛÛÎ‡Ì ‰Ûð˚Ì ı˝‚Ú˝˝ ı Ú˝Ì ı ν„ ‡‚˜, ÚøøÌÚ˝È ı ‡ð¸ˆÛÛÎ‡Ì ı ̌η‡ð ‰øðÒ ·øðËÈÌ Úº‚ Ú˝Ì ı ν„ ı øðÚ ̋Î ı Á‡È„( ...,, 21 y y „.Ï) ÓÎÌÓ. “ ̋ „‚ ̋Î ÌËÈÎÏ ̋Î ‰øðÒËÈÌ ı ø̉ËÈÌ Úº‚ËÈ„
( )=
++++
⋅⋅+⋅+⋅+⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛ = ∑
54321
44332211 2
A A A A A
y A y A y A y A
A
S y x
C
92.254204202520420
143
2
2102742028252049420≅
+++
⎟ ⎠
⎞⎜⎝
⎛ ⋅⋅⋅+⋅+⋅+⋅
= ÒÏ
ı˝Ï˝˝Ì ÓÎÊ ·ÓÎÌÓ.ŒÎÒÓÌ ÍÓÓð‰Ë̇Ú˚Ì Áº‚ ˝Ò˝ı ËÈ„
00 554433221100=⋅+⋅+⋅+⋅+⋅==∑ a Aa Aa Aa Aa AS S x x
00 554433221100=⋅+⋅+⋅+⋅+⋅==∑ b Ab Ab Ab Ab AS S y y
ÚÓÏ ˙∏ÓÌÛÛ‰‡‡ð ¯‡Î„‡Ì‡. ›Ì‰ C ii y y a −= , C ii x x b −= ·ÓÎÌÓ.
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y
y
x 3
C
B
a
I I I
a
3
a
B
x
σ = − 3 . 7 6
A
I V
A I I
« ‡ ‡ „ Ú ˝ Ì ı
Î ˝ „
x
b
0
C
y C
4
1
4
2 y
y
C 2
C
I
1
1
C
x
a5
3
C 5
a2
4a
2a
1a
C x
V
b
x 4
2
2b b
1 x
y C
a
5
Ã
œ ‡
σ = 2
à œ ‡
7 . 5 ñ ð
Á Û ð ‡„
√ ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎ 144
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145
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
−=−⋅==
−=−=
=−=
=−=
596.16926.25143
2
926.18926.257
074.2926.2528
074.23926.2549
54
3
2
1
aa
a
a
a
̋‰„ ̋ ˝ðËÈ„ ‰ ̋ ˝ð ı Ëνð ı ËÈÎ ̋Ή
ÓðÎÛÛÎ ˙ˇ.0596.162102926.18420074.2252008.23420
0≅⋅⋅−⋅−⋅+⋅= x S
C y -ÌËÈÎÏ ̋Î ‰øðÒËÈÌ ı ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰Ë̇Ú˚„ Áº‚ ÓÎÊ ̋ ˝.
4. œ‡ð‡ÎÎÂθ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ‰‡ð‡‡ ı
ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ.
∑∑ +=+= )( )( ; )(
22
ii y y ii x x Ab I I a Aa I I iC iC ( · ) “ÓÏ ˙∏Ó„ ( ‡, · ) Á‡‰Î‡Ê ·Ë˜¸Â.
( )4213
232
221
21 4321
2 Aa I Aa I Aa I Aa I I x x x x x C ⋅+⋅+⋅++⋅++⋅+= ( a′ )
43212 y y y y y I I I I I
C +++= ( · ′ )
›Ì‰ :
686012
1430
12
33
13=
⋅=
⋅==
ba I I x x ÒÏ4, 31500
12
3014
12
33
13=
⋅=
⋅==
ab I I y y Ï4,
16464012
289012
)2( 333
2 =⋅=⋅= ba I x ÒÏ4, ( ) 1701000129028
1232
33
2=⋅=⋅= ab I y ÒÏ4
,
7.228636
1430
36
33
45=
⋅=
⋅==
ba I I x x ÒÏ4
,
787548
3014
48
33
45=
⋅=
⋅==
ab I I y y ÒÏ4,
( ) ( )· a ′′ , Ëνð ı ËÈÎ ̋Ή ÚÓÓÌ ı Óηӄ‰Î˚„ ÓðÎÛÛÎ ˙ˇ.
+⋅++⋅++⋅+= 42093.186860252008.216464042008.236860 222C x I
( ) 26.6838042106.167.22862 2 =⋅++ ÒÏ4,
17797507875231500170100031500 =⋅+++=C y I ÒÏ4,
5. »ÌÂðˆËÈÌ ð‡‰ËÛÒ˚„ ( y X ii , ) ‰‡ð‡‡ ı ̧ ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ∏.
;2
A
I i C x
x = ;2
A
I i C y
y =
›Ì‰C C
y x I I , -Úº‚ËÈÌ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ; ¿ ڇη‡È-
ÌÛÛ‰ ̊ Ì ÌËÈη˝ð
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“ÓÓÌ ı Óηӄ‰Î˚„ ÓðÎÛÛÎ ˙ˇ.
9.1803780
26.6838042== x i ÒÏ2 8.470
3780
17797502== y i ÒÏ2
6. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ (7.6) ÚÓÏ ˙∏Ó„ÓÓð ÓÎÌÓ.
F
x
F
y
y
ib
x
ia
2
0
2
0 ; −=−=
’ø˜ 1 ˆ ̋ „ ‰ ̋ ˝ð øÈΘËÎÒ˝Ì Û˜ð‡‡Ò 07.3093.2556;15 =−=−= F F y x „ ̋ Ê
‡‚̇. “ ̋ „‚ ̋Î:
607.30
9.180;4.31
15
8.47000 =
−−==
−−= ba ÒÏ ·ÓÎÌÓ.
ŒÎÒÓÌ ÚÓÓÌ ÛÚ„‡‡ð Á‡‡„ Ú˝
Ì ı
ν
„ËÈ„ ÁÛð‡„ ‰ ˝˝
𠷇ȄÛÛ·‚ (7.5-ð ÁÛð‡„).7. –ÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ÎËÈÌ Ë ı ÛÚ„ ̊ „ 7.7-ð ÚÓÏ ˙∏Ó„ÓÓð ÓÎ ˙∏. ’ø˜‰ ̋ ÎËÈÌ Ë ı
ÛÚ„‡ Á‡‡„ Ú˝Ì ı ν„ ̋ ˝Ò ı ‡Ï„ËÈÌ ı ÓÎ Óð¯Ë ı ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡ ı Û˜ð‡‡ÒÍÓÓð‰Ë̇Ú˚„ ¿(-15;30); B (15;-24) „ ̋ Ê ‡‚̇. ’‡ðËÌ
( )3.0;3.0 −⋅=⋅= F M F M y x ·ÓÎÌÓ.
“ÓÓÌ Ëνð ı ËÈÎ ̋ÎËÈ„ ÚÓÏ ˙∏Ó̉ ÓðÎÛÛ΂‡Î:
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ++=σ
221
y
A F
x
A F
Ai
x x
i
y y
A
N
( )76.3
10470
10151015
109.180
103007.301
103780
102204
22
4
4
4
3
−=⎟⎟
⎠
⎞⎜⎜
⎝
⎛
⋅
⋅−⋅⋅−+
⋅
⋅⋅+
⋅
⋅−=
−
−−
−
−
−Ãœ‡.
( )=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅⋅⋅−+
⋅
⋅−⋅⋅+
⋅
⋅−=σ
−
−−
−
−−
− 4
22
4
22
4
3
10470
10151015
109.180
102410301
103780
10220 B 2 Ãœ‡.
B A σσ , Áº‚¯ººðº„‰ÒºÌ ı˝ÏÊ ̋ ˝Ì˝˝Ò ·‡„‡ Û˜ð‡‡Ò ·‡„‡ÌÌ˚ ·º ı ·‡Ú˚Ì
̺ ı ̂ ºÎ ·ËÂÎÊ ·‡È̇. ’ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ 7.5-ð ÁÛð‡„Ú øÁøøν‚.8. Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ ˙ˇ. ›Ì˝ ÁÓðË΄ÓÓð Ó„ÚÎÓÎ˚Ì ı øð˝˝Ìøø‰ ̋ ‰
¯øð„ ̋ „˜ Ú‡Ú‡Ê, ¯øð„ ̋ „˜ËÈÌ Úº‚ Ú˝Ì ı ν„Ú˝È Ó„ÚÎÓÎˆÓ ı ı˝ð˜ÏËÈÌ ÛðÚ‡‡ðÍÓÓð‰Ë̇Ú˚Ì ÛÚ„ ̊ „ ÓÎÊ, ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó̉ ÓðÎÛÛÎ‡Ì ·Ó‰ÌÓ.
b
i y
a
i x x
F y
F
22
; −=−=
1ñ1 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ (ˆºÏ ·‡È„ÛÛ· ı ‰‡‡ 1 „ ̋ Ê Ú˝Ï‰ ̋ „ν‚):93.25; 11 −=∞= ba ÒÏ
98.693.25
9.180
0 11 =−−== F F y x ÒÏ
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2 ñ2 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰ËÌ‡Ú (ˆºÏ ‰ ̋ ˝ð 2 „ ̋ Ê Ú˝Ï‰ ̋ „ν‚).692 =a ÒÏ; 34−=b ÒÏ;
3.5
34
9.180ÒÏ8.6
69
470
22
=
−
=−=−= F F y x ÒÏ ·ÓÎÌÓ
3 ñ3 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰ËÌ‡Ú (ˆºÏ ‰ ̋ ˝ð 3 „ ̋ Ê ·Ë˜Ë‚):ÒÏ;443 =a ∞=3b
0ÒÏ68.1044
47033=−=−= F F y x
4 ñ4 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰ËÌ‡Ú (ˆºÏ ‰ ̋ ˝ð 4 „ ̋ Ê ·Ë˜Ë‚):804 =a ÒÏ 374 =b ÒÏ
89.4
37
9.180ÒÏ88.5
80
470
44
−=−=−=−= F F y x ÒÏ
5 ñ 5 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰ËÌ‡Ú (ˆºÏ ‰ ̋ ˝ð 5 „ ̋ Ê Ú˝Ï‰ ̋ „ν‚):∞=5a 305 =b ÒÏ
03.630
9.180;0
55−=−== F F y x ÒÏ ·ÓÎÌÓ.
ª„º„‰ÒºÌ Ó„ÚÎÓÎ Û Ú˝Ì ı ν„ËÈÌ ı Û‚¸‰ Ú˝„¯ ı˝ÏÚ˝È Û˜ð‡‡Ò 6 ñ6 , 7 ñ7 ,8 ñ8 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ 2 ñ2 , 3 ñ3 , 4 ñ4 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰Ë̇ÚÚ‡È ı˝ÏÊ ̋ ˝ Ú˝Ìˆøø, ˝Òð˝„ Ú˝Ï‰ ̋ „Ú˝È ÚÓÓÌ ÛÚ„‡ „‡ð̇. Œ„ÚÎÓÎ˚Ì ˆºÏËÈÌ ı ̋η˝ðËÈ„
7.6 ñð ÁÛð„‡‡Ò øÁ.
7.6-ð ÁÛð‡„
∆ ˯˝˝ 7.4. 7.7-ð ÁÛð‡„Ú ·‡„‡Ì‡Ì˚ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚„ øÁøøν‚.
Œ„ÚÎÓÎ˚Ì 1-ð ˆ ̋ „Ú ·‡„‡Ì‡Ì˚ Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ F ı ø˜ ¯‡ ı ̇.ÇÚÂðˇÎ˚Ì ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î ÒÛÌ„‡Îڇ̉ : R t =5 Ãœ‡, ¯‡ ı ‡Îڇ̉
R C =25 Ãœ‡ ·ÓÎÌÓª„º„‰ºÎ F =200ÍH
a=40 ÒÏ‚=20 ÒÏ
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“ÓÓˆÓÓ„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ ı ÁøÈÎ:
1. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î;
2.
’ø˜‰ ̋ ÎËÈÌ Ë ı ÛÚ„ ̊ „ ÚÓ‰Óð ı ÓÈÎÊ, ÚøøÌËÈ ˝Ô˛ðËÈ„ ·‡È„ÛÛ· ı .3.
Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ̇.
¡Ó‰ÓÎÚ.
1. √ ÂÓÏÂÚðËÈÌ ı ̌η‡ð ‰øðÒøø‰ (1,2,3,4) ·Ó΄ÓÌ ı Û‚‡‡Ê, Ú ̋‰„ ̋ ˝ðËÈÌ
ڇη‡È„ ÓÎ ˙∏.
I ñ ı ‡„‡Ò ÚÓÈðÓ„: 222
1 ÒÏ25122
4014.3
2=
⋅=
π=
R A
II ñ ı ‡„‡Ò ÚÓÈðÓ„: 2
22
2 ÒÏ6282
2014.32 =
⋅=
π= r A
IIIñÚ˝„¯ ºÌˆº„Ú: 23 ÒÏ6400808042 =⋅=⋅⋅= ba A
IV ñÚ˝„¯ ºÌˆº„Ú: 24 ÒÏ1600202402 =⋅⋅=⋅= ba A
2. √ ÂÓÏÂÚðËÈÌ ‰øðÒ ·øðËÈÌ Úº‚ Ú˝Ì ı ν„ËÈ„ ÁÛð‡„ ‰ ̋ ˝ð ÁÛðÊ ( 4,3,2,1 y y y y )
øÁøøν‚ (7.7-ð ÁÛð‡„).3. ÕËÈÎÏ ̋Î ‰øðÒËÈÌ ı ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰Ë̇Ú˚„ ÓÎÓ ı ‰ÓÓ
∑ ⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
= A
S
x y
c ÚÓÏ ˙∏Ó„ ‡¯Ë„·̇.
ƒÛð˚Ì Û Ú˝Ì ı ν„ËÈ„ III ‰øðÒËÈÌ Úº‚ Ú˝Ì ı ν„Ú˝È ‰‡‚ ı ̂ ÛÛÎÊ ‡‚·‡. “ ̋ „‚ ̋Î ‰øðÒ ·øðËÈÌ Úº‚ Ú˝Ì ı ν„ ı øðÚ ̋Î ı Á‡È ̸:
( ) ( ) ÒÏ96.564040424.02424.011 −=+⋅−=+⋅⋅−=+= abac x
( ) ( ) ÒÏ5.484020424.0424.022 −=+⋅−=+⋅−=+= abac x
03 = x
ÒÏ60204024 =+=+=
a
a x ·ÓÎÌÓ. “ÓÓÌ ı Óηӄ‰Î˚„ ÚÓÏ ˙∏Ó̉ ÓðÎÛÛÎ ˙ˇ.
=++−
++−=
4321
44332211
A A A A
X A X A X A X A x c
( ) ( )ÒÏ7.1
9884
60160005.4862896.562512−=
⋅++−⋅−−=
4. ’ø̉ËÈÌ Úº‚ËÈÌ ÍÓÓð‰Ë̇Ú˚Ì Áº‚ ˝Ò˝ı ËÈ„ Úº‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ
ÒÚ‡ÚËÍ ÏÓÏÂÌÚ˚Ì ÌËÈη˝ð Ú˝„Ú˝È Ú˝Ìˆøø ( )∑ = 0SxC „ ̋ Ê ¯‡Î„‡Ì‡.
∑ =++⋅−⋅= 044332211 b Ab Ab Ab ASx C
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cii x x b −= ÚÓÏ ˙∏Ó„ÓÓð b-Á‡È„ ÓÎÌÓ.
8.462 −=b ÒÏ
7.13 =b ÒÏ
7.614 =b ÒÏ
›Ì˝ ÚÓÓÌ ı Óηӄ‰Î˚„ ‰ ̋ ˝ð ı Ëνð ı ËÈÎ ̋Ή ÓðÎÛÛÎ ˙ˇ.( ) ( ) 07.6116007.164008.486283.552512SxC =⋅+⋅+−−−−=∑
– ‡ Ì ‡ Ï Ê. Œ„ÚÎÓÎ˚Ì ‰ÓÚÓð ı ÓÓÒÓÌ Á‡È (ڇη‡È) ·‡È‚‡Î ÚøøÌËÈ„ ı ‡Ò‡Ê ÚÓÓˆÌÓ.5. œ‡ð‡ÎÎÂθ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎ ˙∏.
›Ì‰ 7.3 ·Ó‰ÎÓ„ ̊ Ì ÚÓÏ ˙∏Ó„ ‡¯Ë„·̇.√ ÂÓÏÂÚðËÈÌ ı ̌η‡ð ‰øðÒËÈÌ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ‰‡ð‡‡ ı ̧ ÚÓÏ ˙∏Ó„ÓÓð
ÓÎÌÓ. I, II ı ‡„‡Ò ÚÓÈð„ËÈÌ ı Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„
128128
44
1
d D I x
π−
π= ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÓ∏.
( ) ( ) ( )[ ] [ ] 4444444 ÒÏ94204080128
14.324
1281281=−=−
π=−
π= bbd D I x
’‡ðËÌ ÏºÌ ‰øðÒËÈÌ Û Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ‰‡ð‡‡ ı ̧ Ëνð ı ËÈÎ ̋ν˝ð ÓÎÌÓ.I ñð ‰øðÒËÈÌ ı Û‚¸‰:
( ) 444121
4
1121 ÒÏ6.2809858000686.000686.0424.01281 =⋅=⋅=⋅−π
=⋅−= D Ar D Ac I I y y
II ñð ‰øðÒËÈÌ ı Û‚¸‰ (7.7 ñð ÁÛð):
4442
22 17561.6ÒÏ400.0068600686.0
2=⋅=⋅=⋅−= d Ac I I y y
ÒÏ402424.0c
ÒÏ804424.0
11
22
===
===
bd r
b Dr c
III ñð ‰øðÒËÈÌ ı Û‚¸‰:
( )3.3413333
12
8080
12
42 33
3=
⋅==
ba I x ÒÏ4
( )3.3413333
12
8080
12
24 33
3=
⋅==
ab I y
ÒÏ4 IV-ð ‰øðÒËÈÌ ı Û‚¸‰:
( ) 3.85333312
4040122
33
44 =⋅=== ba I I x y ÒÏ4 ·ÓÎÌÓ.
7.7ñð ÁÛð‡„
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“ÓÓÌ ı Óηӄ‰Î˚„ ÚÓÏ ˙∏Ó̉ ÓðÎÛÛÎ ˙ˇ.
( ) =+++++−+= 4243
232
221
21 4321
Ab I Ab I Ab I Ab I I y y y y y C
( ) ( ) +⋅++⋅+−+ 64007.13.34133336288.466.1756125123.556.280985222
9.1694606116007.613.853333 2 =⋅++ ÒÏ4 42760683.8533333.34133339420
431=++=++= x x x x I I I I
C ÒÏ4
·ÓÎÌÓ.6. »ÌÂðˆËÈÌ ð‡‰ËÛÒ˚„ ÓÎ ˙∏.
6.4329884
42760862 === A
I i C x
x ÒÏ2
5.17149884
9.169460612
=== A
I
ic y
y ÒÏ2
7. «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ Óθ∏.
8.443.38
5.17142
0 =−
−=−= F
y
x
ia ÒÏ
ÓÓð‰ËÌ‡Ú Ì¸ 3.38−= F x ÒÏ
8.1040
6.4322
0 =−
−=−= F
x
y
ib ÒÏ
40−= F y ÒÏ ·ÓÎÌÓ.
ŒÎÒÓÌ ÚÓÓÌ ÛÚ„‡‡ð ÁÛð‡„ ‰ ̋ ˝ð Á‡‡„ Ú˝Ì ı ν„ËÈ„ ·‡È„ÛÛ·‚ (7.8ñð ÁÛð‡„).
8. –ÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ÎËÈÌ Ë ı ÛÚ„ ̊ „ ÓÎ. ( F N = )
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ⋅+
⋅+=σ
221
y
A F
x
A F A
i
x x
i
y y
A
N
( ) ( )2.1
105.1714
103.38103.38
106.432
104010401
109884
302004
22
4
22
4
3
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅−⋅−+
⋅
⋅−⋅−+
⋅
⋅−=
−
−−
−
−−
−Ãœ
‡
( 1.2Ãœ‡< 25Ãœ‡)
73.0105.1714
107.41103.38
106.432
104010401
109984
102004
22
4
22
4
3
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅⋅⋅−+
⋅
⋅⋅⋅−+
⋅
⋅−=σ
−
−−
−
−−
− B Ãœ‡
( 0.73Ãœ‡< 5Ãœ‡ )ÿËη˝ÌËÈ ·º ı ·‡Ú˚Ì Ìº ı ̂ ºÎ ·ËÂÎÊ ·‡È̇. ’ø˜‰ ̋ ÎËÈÌ ˝Ô˛ðËÈ„ 7.8ñðÁÛð„‡‡Ò øÁ.
9. Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ ˙ˇ.a
i x
y F
2
−= ;b
i y x
F
2
−=
1 ñ1 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰ËÌ‡Ú 40; 1 −=∞= ba F Û˜ð‡‡Ò
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8.1040
6.432;0
11=
−−== F F y x ÒÏ ·ÓÎÌÓ.
2 ñ 2 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰: 1232 =a ÒÏ, 602 −=b ÒÏ
9.13123
5.17142
−=−= F x ÒÏ; 2.760
6.4322
=−
−= F y ÒÏ
3 ñ3 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ : 7.813 =a ÒÏ ∞=3b ·Û ̨ Û
0;217.81
5.171433=−≅−= F F y x
4 ñ4 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ 1254 =a ÒÏ 604 =b ÒÏ ·Û ̨ Û
7.13125
5.17144
−=−= F x ÒÏ 2.760
6.4324
−=−
= F y ÒÏ ·ÓÎÌÓ.
5 ñ5 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ 40; 55 =∞= ba ÒÏ
05= F x ; 8.10
40
6.4325
−=−
= F y ÒÏ
6 ñ 6 “ÓÈð„ËÈÌ Úº‚ººÒ ·ÓÒÓÓ Ú˝Ì ı Îν„Ú˝È 45 „ð‡‰ÛÒ‡‡ð ˆ‡ˆð‡„ Ú‡Ú‡Ê ÚÓÈðÓ„ÚÓÈ Ó„ÚÎÓΈÛÛÎÊ, Ó„ÚÎÓΈÒÓÌ D ˆ ̋ „Ú 6ñ 6 ¯øð„ ̋ „˜ËÈ„ ·‡È„ÛÛ·‚.
ƒ‡ð‡‡ ̸ OKC Δ -˝˝Ò OK a =6 -„ ÓÎÌÓ.
4.953.387.0
403.3845cos
406 −=−−=−−= a ÒÏ 18
4.955.17146 =−−= F x ÒÏ
7.8ñð ÁÛð‡„
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ib Á‡È„ ı˝ð ı˝Ì ÓÎÓ ı˚„ 7.9ñð ÁÛð‡„Ú øÁøøν‚. ODL Δ -˝˝Ò
‡) DL=OD⋅ sin45 =40⋅0.7=28 ÒÏ
·) OL=OD⋅ cos45 = 28 ÒÏ
‚) 684028 =+=+= LM DL DM ÒÏ„) MN CM y F +=
6
‰) DNM Δ -˝˝Ò
686845 =⋅= tg MN ÒÏ
“ ̋ „‚ ̋Î 9668286 =+=b ·Û ̨ Û
5.496
6.4326
−=−= F y ÒÏ ·ÓÎÌÓ.
7 ñ7 ¯øð„ ̋ „˜ËÈÌ 3.787 −=a ÒÏ , ∞=5b
9.213.78
5.17147
=−
−= F x ÒÏ 07= F y
·ÓÎÌÓ.8 ñ8 ¯øð„ ̋ „˜ËÈÌ ı Û‚¸‰ 6 ñ6 ¯øð„ ̋ „˜ËÈÌ ÍÓÓð‰Ë̇Ú˚„ ÓÎÒÓÌ ‡‰Ë·‡ð (7.9ñðÁÛð„ËÈ„ øÁ) „øȈ ̋ Ú„ ̋ Ì˝.
7.5
ÃÛ¯„Ëð‡ÎÚÚ‡È „ÛÎÁ‡ÈÎÚ ̊ Ì ÚÓÓˆÓÓÌ˚ ‰‡ð‡‡Î‡Î
1. √ ÓÎ˚Ì ÚÓÓˆÓÓÌ˚ Ò ı ÂÏËÈ„ øÈΘËÎÊ ·ÛÈ ı ø˜ÌËÈ ı ‡ÏÚ ı , Û Ú˝Ì ı ν„ ‰ ̋ ˝ð ÁÓ ı ËÓÊ ÁÛð̇.
2. √ ÓÎ ‰ ̋ ˝ð øÈÎ˜Î˝ı ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚„ ÓÎÌÓ.
3. √ ÓÎ˚Ì ÚÛ΄ÛÛð˚Ì ð‡͈ËÈ„ ÓÎÌÓ.
4. √ ÓÎ ‰ ̋ ˝ð øÈÎ˜Î˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÚÓÓˆÓÓ ı ËÈÊ , ˝Ô˛ðËÈ„ ̸·‡È„ÛÛÎ̇.
5. √ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÌËÈη˝ð ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ̇.
6. √ ÓÎ ‰ ̋ ˝ð øÈÎ˜Î˝ı ÏÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ÚÓÓˆÓÓ ı ËÈÊ , ÚøøÌËÈ
˝Ô˛ðËÈ„ ·‡È„ÛÛÎ̇.7.
“ÓÓˆÓÓÌ˚ ÏÓÏÂÌÚ˚Ì ÛÚ„ ̊ „ ÓÎÌÓ.8. ¡‡Ú ·º ı ËÈÌ Ìº ı ̂ ºÎººÒ „ÓÎ˚Ì Ó„ÚÎÓÎ˚„ ÒÓÌ„ÓÌÓ.
∆ ˯˝˝ 7.5. ØðÎ˝Ì ı ÓÎ ı Ë‚˜ ÒÛÛ΄‡Ò‡Ì „ÓÎ ‰ ̋ ˝ð 301 =d ÒÏ,
152 =d ÒÏ „ÓΘÚÓÈ ı Ó∏ð ‡ð‡‡ ÒÛÛ΄‡Ê ̋ ˝. ¿ð‡‡Ì˚ ÚÓÈðÓ„ ı ø˜ 51=t F ÍÕ,
102=t F , Áº‚¯ººð„º‰º ı ı ø˜‰ ̋ Î [ ] 100=σ Ãœ‡ ·ÓÎÌÓ. 15=a , 30=l ·ÓÎÌÓ.
¡Ó‰ÓÎÚ.
1. ’ÓÎ ı Ë‚˜ËÈ„ ı º‰ºÎ„ººÌ„øÈ, ı º‰ºÎ„ººÌÚ˝È ÌÛ„‡ÒÚ‡È ÚÛ΄ÛÛð‡‡ð
7.9ñð ÁÛð‡„
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153
ÒÓÎ¸Ê , „ÓÎ˚„ Ú˝Ì ı ν„ËÈÌ ¯Û„‡Ï‡‡ð ÚÓÓˆÓÓÌ˚ Ò ı ÂÏ ‰ ̋ ˝ð ÁÛð̇. (7.9ñËÈÌ ·,
‚ ñð ÁÛð‡„) ÃºÌ ÚÓÈðÓ„ ı ø˜ËÈ„ ÚÓÓˆÓÓÌ˚ Ò ı ÂÏ ‰ ̋ ˝ð ÁººÌº.
2. “ÓÈðÓ„ ı ø˜øø‰ËÈ„ ¬, – ˆ ̋ „ ‰ ̋ ˝ð øÈΘËÎÊ ·ÛÈ „‡‰‡‡‰ ÏÛ¯„Ë ı
ÏÓÏÂÌÚÓÓð ( eT ) ÒÓÎËÌÓ.
75.02
3.05
2
1
11=⋅=⋅=
d F T t e ÍÕ
75.02
15.010
2
2
22=⋅=⋅=
d F T t e ÍÕ
21 ee T T = ˝Òð˝„ ˜Ë„Î ̋ÎÚ˝È Û˜ð‡‡Ò „Ó˚Ì ¬,– Ó„ÚÎÓÎ˚Ì ı ÓÓðÓ̉
ÏÛ¯„Ëð‡ÎÚ ·ÓÎÌÓ.
3. “Û΄ÛÛð˚
Ì ð‡͈ËÈ„ ÓÎÊ, (7.10· ñð ÁÛð‡„) „ÛÎÁ‡È΄‡ ı
ÏÓÏÂÌÚ˚
Ì˝
Ô˛
ð·‡È„ÛÛÎ̇. (7.10„ ñð ÁÛð‡„)
( ) 25.16.0
15.05
2020 1
1=
⋅=
+
⋅==⋅++=∑
al
a F Da F al D M
t y t y A ÍÕ
( ) ( ) ( )
75.36.0
45.05
20220 1
1=
⋅=
+
+⋅==+⋅−+=∑
al
al F Aal F al A M
t y t y D ÍÕ
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÚÓÓˆÓÓ1 ñ1 Ó„ÚÎÓΉ ( )15.000 11 ≤≤≤≤ z a z
11 z A M y z ⋅=
⎪⎩
⎪⎨⎧
=⋅=
=
=
=
56.015.075.3
0
15.0
0
1
1
z
z
M
M
ÍÕ⋅Ï
2 ñ2 Ó„ÚÎÓΉ ( )6.015.0 1 ≤≤ z
( )a z F z A M t y z −⋅−⋅= 22 12 ⎪⎩
⎪⎨⎧
=⋅−⋅=
=⋅=
=
=
045.056.075.3
56.015.075.3
6.0
15.0
1
2
z
z
M
M
ÍÕ⋅Ï
x x D A , ð‡͈ËÈ„ ÓÎ ˙∏. ( 7.10‚ ñð ÁÛð‡„)
( ) ( ) 5.76.0
45.01002;0
2=
⋅==+−+⋅=∑ x x t A Dal Dal F M ÍÕ
( ) 5.26.0
15.01002;0
2=
⋅==⋅−+⋅=∑ x t x D Aa F al A M ÍÕ
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÚÓÓˆÓÓ3 ñ3 Ó„ÚÎÓΉ ( )45.00 3 ≤≤ z
33 z A M x z ⋅= ⎪⎩
⎪⎨
⎧
=⋅=
=
=
=
13.145.05.2
0
45.0
0
3
3
z
z
M
M
ÍÕ⋅Ï
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî 154
4 ñ4 Ó„ÚÎÓΉ ( )15.00 1 ≤≤ z
43 z D M x z ⋅=
⎪⎩
⎪⎨⎧
=⋅=
=
=
=
13.115.05.7
0
15.0
0
3
4
z
z
M
M
ÍÕ⋅ÏÃÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 7.10 ‰ ñð ÁÛð„‡‡Ò ı ‡ð.
‡
·
‚
„
‰
Â
Ê
7.10 ñð ÁÛð‡„
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4. √ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÌËÈη˝ð ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ ˙ˇ.
ØøÌËÈ ÚÛΉ „ÓÎ˚Ì ¬, – Ó„ÚÎÓΉ ı ‡ð„‡ÎÁ‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈÌ (7.10 „, ‰ ñðÁÛð‡„ ) ÛÚ„ ̊ „ ÓÎÊ, ı ÓÓðÓ̉ Ì˝Ï˝ ̋‰ Í‚‡‰ð‡Ú ˇÁ„ÛÛ𠄇ð„‡Ì‡.
38.015.05.2 =⋅=⋅= a A M x
x
B ÍÕ⋅Ï19.015.025.1 =⋅=⋅= a D M y
x C ÍÕ⋅Ï ·ÓÎÌÓ.
“ ̋ „‚ ̋Î ¬, – Ó„ÚÎÓÎ ‰ ̋ ˝ð ı ·øÚ˝Ì „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ:
( ) ( ) 68.038.056.0 2222=+=+= x
B y B B M M M ÍÕ⋅Ï
( ) ( ) 15.113.119.0 2222=+=+= x
C y C C M M M ÍÕ⋅Ï
5. ÃÛ¯„Ë ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ̇(7.10 Êñð ÁÛð). √ ÓÎ˚Ì ¬, –
Ó„ÚÎÓÎ˚
Ì ı
ÓÓðÓ̉ ÏÛ¯„Ëð‡ÎÚ ·ÓÎÌÓ. ›Ì˝
ı˝
Ò„ËÈÌ ÏÛ¯„Ë ı
ÏÓÏÂÌÚ ( T ) ̸75.01== eT T ÍÕ⋅Ï ·ÓÎÌÓ.
6. √ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ÌËÈη˝ð ˝Ô˛ð˝˝Ò (7.10Âñð ÁÛð‡„ ) ı ‡ð‡ ı ‡‰,
„ÓÎ˚Ì ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓΠ̸ – ˆ ̋ „Ú ı ‡ð„‡ÎÁ‡Ì‡. ( 15.1max = M ÍÕ⋅Ï). ¡º ı
·‡Ú˚Ì IV ñð ÓÌÓÎÓÓð ÚÓÓˆÓÓÌ˚ ÏÓÏÂÌÚ˚„ ( Ã “ ) ÚÓ‰Óð ı ÓÈÎÓ∏.
32.115.175.075.075.0 222max
2 =+⋅=+⋅= M T M T ÍÕ⋅Ï
7. ¿ ̨ ÛÎÚ‡È Ó„ÚÎÓΉ ı ‡ð„‡ÎÁ‡ ı „ÓÎ˚Ì ‰Ë‡ÏÂÚðËÈ„ ÓÎ ˙∏.
[ ] 1.51010014.3
1032.132323
6
3
3 ≅⋅⋅
⋅⋅=σ⋅π
⋅= T
M d ÒÏ.
ДААЛГАВАР №7. Нийлмэл эсэргүүцэл
›Ì˝ ‰‡‡Î„‡‚‡ð „Ûð‚‡Ì ·Ó‰ÎÓ„ÓÓÒ ·øð‰ ̋ Ì˝.
1.
∆ ˯øø „ÛÎÁ‡ÈÎÚ
ª„º„‰ÒºÌ ‰‡ÏÌÛÛð˚Ì (7.11ñð ÁÛð‡„) ‰‚ÛÚ‡‚ð Ó„ÚÎÓÎ ÒÓÌ„Óʇ ) ËÌÂðˆËÈÌ „ÓÎ ı ‡‚Ú„‡ÈÌÛÛ‰ øÈÎ˜Î˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð
·‡È„ÛÛ· ı · ) ‡ ̨ ÛÎÚ‡È Ó„ÚÎÓÎ˚Ì Á‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ ÓÎÊ ·ÓÒÓÓ
ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛÎ.‚ ) ÒÛÎ øÁøøðËÈÌ ·øÚ˝Ì Ú‡ ı ËÈÎÚ˚„ ÓÎ.
ª„º„‰ÎËÈ„ 7.1ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî 156
7.11ñð ÁÛð‡„
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7.11– р зургийн үргэлжлэл
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî 158
7.1-ð ı øÒÌ˝„Ú
π ‡,
Ï
c,
Ï
d ,
Ï
F
ÍÕ
Q
ÍÕ/Ï
Ã
ÍÕ⋅Ï
h
ÒÏ
b,
ÒÏ
ƒÛ„‡‡ð
‰‚ÛÚ‡‚𠯂ÂÎÎÂð
1 1 3 2 30 20 10 20 15 18 14
2 2 2 2 10 20 30 24 12 16 16
3 1 3 2 20 20 20 30 15 36 14‡
4 2 2 2 20 20 30 24 18 18 16‡
5 1 3 2 10 10 10 28 12 18‡ 18
6 2 2 2 10 30 30 18 16 16 20
7 1 3 2 20 10 10 24 12 20 18‡
8 2 2 2 20 10 20 20 12 30 20‡9 1 3 2 20 30 20 27 16 27 22
10 2 2 2 10 30 10 21 12 27‡ 22‡
11 1 3 2 10 30 10 20 12 24 24
12 2 2 2 10 10 10 18 12 24‡ 24‡
13 1 3 2 30 10 30 24 16 27 14
14 2 2 2 30 30 30 24 16 24 16
15 1 3 2 30 30 30 30 18 24‡ 16‡
16 2 2 2 10 30 10 30 18 30 27
17 12 3 2 10 20 20 28 12 30‡ 18
18 1 2 2 10 30 10 28 15 20 16
19 2 3 2 20 30 20 24 18 20‡ 20
20 1 2 2 20 10 30 30 18 18 22
2. “º‚ËÈÌ ·Ë¯ ¯‡ ı ‡ÎÚ
¡‡„‡Ì‡Ì˚ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ˇÏ‡ð Ì˝„ ˆ ̋ „Ú (7.12ñð ÁÛð‡„) ÚøøÌËÈ Ú˝Ì ı ν„Ú˝È Ô‡ð‡ÎÎÂθ F ı ø˜ øÈÎ˜Î˝ı ø‰ ÒÛÌ„‡ÎÚ˚Ì ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î
3=t R Ãœ‡ ÒÛÌ„‡ÎÚ˚Ì øÂËÈÌ ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î 30=c R Ãœ‡ ·ÓÎ
‡ ) «‡‡„ Ú˝Ì ı ν„ËÈÌ ·‡Èð·Î˚„ Óη ) «º‚¯ººðº„‰º ı ı ø˜‰ ̋ ΂ ) ’‡Ï„ËÈÌ Ë ı ÒÛÌ„‡ ı , ¯‡ ı ‡ ı ı ø˜‰ ̋ ËÈ„ ÓÎÊ ı ø˜‰ ̋ ÎËÈÌ ˝Ô˛ð ·‡È„ÛÛ΄ ) Œ„ÚÎÓÎ˚Ì ˆºÏ ·‡È„ÛÛÎ
ª„º„‰ÎËÈ„ 7.2ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
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7.12ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî 160
7.2ñð ı øÒÌ˝„Ú
π F ,
ÍÕ
‡,
Ï
b,
Ï
’ø˜ÌËÈ
øÈΘΠ̋ÎËÈÌ ˆ ̋ „
1 120 30 12 1
2 100 24 10 2
3 160 30 12 3
4 100 24 14 1
5 180 36 15 2
6 200 40 18 3
7 320 50 20 1
8 220 28 16 2
9 240 36 18 3
10 260 40 16 1
11 280 30 14 2
12 100 24 10 3
13 120 30 12 1
14 160 40 16 2
15 120 34 12 3
16 160 36 14 1
17 200 40 16 2
18 240 34 18 3
19 260 40 20 1
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3. ÃÛ¯„Ëð‡ÎÚÚ‡È „ÛÎÁ‡ÈÎÚ
ÃËÌÛڇ̉ n Û‰‡‡ ˝ð„ ̋ ı ‰Û„ÛÈ Ó„ÚÎÓÎÚÓÈ „ÓÎ˚Ì (7.13ñð ÁÛð‡„) ı ºÚκ„˜2 ‰‡Ï‡ð P 2 ˜‡‰‡Î ‰‡ÏÊÛÛ· ı ‡‰ 80=σ Ãœ‡ ı ø˜‰ ̋ Î Áº‚¯ººðº„‰ºÊ
·‡È„‡‡ ·ÓÎ „ÓÎ˚Ì „ÓΘËÈÌ ËÌÂð„ËÈÌ ÓÌÓÎÓÓð ÚÓ‰Óð ı ÓÈÎ. ª„º„‰ÎËÈ„ 7.3ñð ı øÒÌ˝„Ú˝˝Ò ‡‚.
7.13ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî 162
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7.3 ñð ı øÒÌ˝„Ú
π K 2 D1
Ï D 2
Ïn
˝ð„/ÏËÌ A
Ïα
„ð‡‰
1 60 0.30 0.35 800 0.6 30
2 70 0.35 0.40 900 0.5 40
3 80 0.40 0.45 1250 0.9 30
4 90 0.45 0.50 1100 1.5 40
5 10 0.50 0.55 1000 1.4 30
6 110 0.55 0.60 950 1.3 40
7 120 0.60 0.65 900 1.2 30
8 130 0.55 0.60 800 1.1 40
9 140 0.50 0.55 1050 1.0 30
10 150 0.45 0.50 1100 0.9 40
11 65 0.40 0.45 900 0.8 30
12 75 0.35 0.40 1100 0.7 40
13 85 0.30 0.35 1050 0.6 30
14 95 0.50 0.55 1000 0.5 40
15 105 0.55 0.60 800 1 30
16 115 0.60 0.65 850 1.1 40
17 125 0.30 0.35 900 1.2 30
18 135 0.35 0.40 1050 1.3 40
19 145 0.40 0.45 800 1.4 30
20 155 0.45 0.50 850 1.5 40
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Õ¿…Ô √ ¿¿— ¡ØÀ›√
√ ”—¬¿Õ ÃŒÃ≈Õ“¤Õ “›√ ÿ»“ √ ›À
8.1. ≈—ªÕ’»… Œ…À√ ŒÀ“
ƒÛ̉‡‡ ·øÚ˝Ì ÌÛ„‡Ò„øÈ, ı Ó∏ðÓÓÒ ‰ ̋ ˝¯ ÚÛ΄ÛÛðÚ‡È ¯ÛÎÛÛÌ ‰‡ÏÌÛÛð˚Ìð‡͈ ÓÎÓ ı Ó‰ „Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚÚ„ ̋ ÎËÈ„ ı˝ð˝„Î˝Ì˝.
ŒÎÓÌ ‡ÎÒ·ÎÚ‡È ‰‡ÏÌÛÛð‡‡Ò ı Ó∏ð ‡ÎÒ·Î˚„ ( 1
, +nn l l ) (8.1ñð ÁÛð‡„)
º„º„‰ÒºÌ ‡˜‡‡, øÎ Ï ̋‰ ̋ „‰ ̋ ı ÏÓÏÂÌÚ˚Ì ı ‡ÏÚ Ò‡Î„‡Ê øÁøøν‚.
−+1
,nn
ww ÏÓÏÂÌÚ˚Ì
˝Ô˛ðËÈÌ Ú‡Î·‡È;−+1, nn C C ڇη‡ÈÌ
ı ø̉ËÈÌ Úº‚;−+1, nn aa ˝Ô˛ðËÈÌ
ÁøøÌ øÁøøð˝˝Ò ı ø̉ËÈÌ Úº‚ ı øðÚ ̋Î ı Á‡È;
−+1, nn bb ˝Ô˛ðËÈÌ
·‡ðÛÛÌ øÁøøð˝˝Ò ı ø̉ËÈÌ Úº‚ ı øðÚ ̋Î ı Á‡È;8.1-ð ÁÛð„ËÈÌ ı Û‚¸‰·Ë˜Ò˝Ì Ú˝„¯ËÚ„ ̋ Î :
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ⋅+
⋅−=⋅++⋅+
+
+++++−
1
111111 6 )( 2
n
nn
n
nnnnnnnnn
l
bw
l
awl M l l M l M ; (8.1)
ØøÌËÈ„ „Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì ÂðºÌ ı ËÈ Ú˝„¯ËÚ„ ̋ Î „ ̋ Ê Ì˝ðÎ˝Ì˝.
›Ì˝ ÚÓÏ ˙∏Ó„ ‰‡ÏÌÛÛð˚Ì ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ ÚÓ„ÚÏÓÎ ø‰ ( EI =const)‡¯Ë„·̇.
√ Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ ÎËÈÌ ¯ËȉËÈ„ ÓÎÒÓÌ˚ ‰‡ð‡‡ ÚÛ΄ÛÛðÛÛ‰ ̊ Ìð‡͈ËÈ„ ‰‡ð‡‡ ı ÚÓÏ ˙∏Ó„ÓÓð Ëνð ı ËÈÎÌ˝.
M
l
n-1
n
n
n A M n-1 B n F
n
n
n B
R n An+1
l n l n+1
n+1nl n+1
n+1n M An+1 n+1 B
M q
8.2ñð ÁÛð‡„
n-1
b nna
nC
nω
l n
b n+1an+1
n+1C
n+1ω
l n+1
n+1
M n-1 M F q n+1 M nn
8.1ñð ÁÛð‡„
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n
n
n
nnn
l
M
l
M A A +−= −10 (8.2)
0 1n nn n
n n
M M B B
l l
−= + − (8.3)
1++= nnn A B R (8.4)
n A , n B ̸ l n ‡ÎÒ·ÎÚ‡È ı˝Ò„ËÈÌ ·‡ðÛÛÌ, ÁøøÌ ÚÛ΄ÛÛð˚Ì ð‡͈;
n R ̸ n ÚÛ΄ÛÛð˚Ì ·øÚ˝Ì ð‡͈; 1+n A ̸ l n+1 ‡ÎÒ·ÎÚ‡È ı˝Ò„ËÈÌ ÁøøÌ
ÚÛ΄ÛÛð˚Ì ð‡͈; 0 0,n n B ̸ l n ‡ÎÒ·ÎÚ‡È ‰‡ÏÌÛÛð ‰ ̋ ˝ð øÈΘËÎÊ ·ÛÈ
Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜Ì˝˝Ò øøÒ˝ı ð‡͈.
– ‡ Ì ‡ Ï Ê: R øÒ˝„ÌËÈ ·‡ðÛÛÌ ‰ ̋ ̋‰ øÒ˝„ ̸ ‡ÎÒ·Î˚Ì ‰Û„‡‡ð˚„, ·‡ðÛÛÌ ‰ÓÓ‰ øÒ˝„ ÚÛ΄ÛÛð˚Ì ‰Û„‡‡ð˚„ ÚÛÒ ÚÛÒ Ëνð ı ËÈÎÌ˝.
8.2. ¡Ó‰ÎÓ„Ó ·Ó‰Ó ı ‰‡ð‡‡Î‡Î1.
«øøÌ Ú‡Î‡‡Ò ÚÛ΄ÛÛð˚„ Ú˝„ ̋ ˝Ò, ‡ÎÒ·Î˚„ Ì˝„ ̋ ˝Ò ˝ı Î˝Ì ‰Û„‡‡ð·̇.2.
ª„º„‰ÒºÌ ‡˜‡‡ ·ÓÎÓÌ ÚÛ΄ÛÛð˚Ì øÎ Ï ̋‰ ̋ „‰ ̋ ı ÏÓÏÂÌÚÓÓð‡˜‡‡Î‡„‰Ò‡Ì øÌ‰Ò˝Ì ÒËÒÚÂÏËÈ„ ·‡È„ÛÛ· ı .
3.
ª„º„‰ÒºÌ ı ø˜Ì˝˝Ò øøÒ˝ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ ‰‡ÏÌÛÛð ÚÛÒ ·øð ‰ ̋ ˝ð·‡È„ÛÛ· ı .
4.
ª„º„‰ÒºÌ ‡˜‡‡ ·ÓÎÓÌ ÚÛ΄ÛÛð˚Ì øÎ Ï ̋‰ ̋ „‰ ̋ ı ÏÓÏÂÌÚÓÓð‡˜‡‡Î‡„‰Ò‡Ì øÌ‰Ò˝Ì ÒËÒÚÂÏËÈ„ ·‡È„ÛÛ· ı .5.
ƒÛ̉‡ ı ÚÛ΄ÛÛð˚Ì ÚÓÓ„ÓÓð „Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ·Ë˜Ë ı .6.
“ ̋ „¯ËÚ„ ̋ ÎËÈ„ ÒËÒÚÂÏ˜Î˝Ì ·Ó‰ÓÊ øÎ Ï ̋‰ ̋ „‰ ̋ ı ÏÓÏÂÌÚ˚„( ......,, 210 M M M „.Ï) ÓÎÓ ı .
7.
“Û΄ÛÛð˚Ì ð‡͈ËÈ„ (8.2), (8.3), (8.4)ÚÓÏ ˙∏Ó„ÓÓð ÚÓ‰Óð ı ÓÈÎÌÓ.8.
¿ÎÒ·Π·øð ‰ ̋ ˝ð Ó„ÚÎÓÎ ı ËÈÊ ı ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚ÌÚ˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ, ÚÓÓÌ ı Óηӄ‰Î˚„ ÓÎÓ ı .
9.
ŒÎÒÓÌ ÚÓÓÌ ÛÚ„‡‡ð ı ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð
·‡È„ÛÛ· ı .
∆ ˯˝˝ 8.1. √ Ûð‚‡Ì ‡ÎÒ·ÎÚ‡È, ÁøøÌ øÁøøð ̸ ÍÓÌÒÓθ ‰‡ÏÌÛÛð(8.3ñð ÁÛð‡„) ‰ ̋ ˝ð F, M, q ı ø˜ øÈΘËÎÌ˝. √ Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ ÎËÈ„‡¯Ë„Î‡Ì ÚÛ΄ÛÛð˚Ì ð‡ÍËÈ„ ÓÎÊ, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ, ı ºÌ‰ÎºÌ ı ø˜ÌËÈ˝Ô˛ð ·‡È„ÛÛÎ.ª„º„‰ºÎ:
‡(Ï)
b(Ï)
c(Ï)
l 2(Ï)
l 3(Ï)
F(ÍH)
M(ÍHÏ)
q(ÍH/Ï)
2 4 2 6 6 30 60 10
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî166
¡Ó‰ÓÎÚ.1. “Û΄ÛÛð˚„ Ú˝„ ̋ ˝Ò ˝ı Î˝Ì ÁøøÌ øÁøøð˝˝Ò (0,1,2,3 „.Ï) ‰Û„‡‡ð·̇ (8.3añð
ÁÛð‡„).
2. ØÌ‰Ò˝
Ì ÒËÒÚÂÏËÈ„ ·‡È„ÛÛÎ̇. (8.3añð ÁÛð„ËÈ„ øÁ). Œ‰ÓÓ„ÓÓð à 1, à 2 ÏÓÏÂÌÚ˚Ì ı˝ÏÊ ̋ ˝, ˜Ë„Î ̋Î Ï ̋‰ ̋ „‰ ̋ ı „øÈ ·‡È„‡‡.3. «‡ ı˚Ì ÚÛ΄ÛÛð ‰ ̋ ˝ð ı ÏÓÏÂÌÚ º„º„‰ºº„øÈ ·ÓÎ Ú˝„Ú˝È Ú˝Ìˆøø „ ̋ Ê ‡‚̇.
¡Ë‰ÌËÈ Ê˯˝˝ÌËÈ ı Û‚¸‰ 602300 −=⋅−=⋅−= a F M ÍH⋅Ï; à 3=0 ·ÓÎÌÓ.
4. ƒÛ̉‡ ı (1,2) ÚÛ΄ÛÛð˚Ì ı Û‚¸‰ „Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ·Ë˜¸Â.
n=1. ( ) ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +
′′⋅′′+
′⋅′−=+++ 062
1
11
1
112221110
l
aw
l
awl M l l M l M (1)
n=2. ( )⎟⎟ ⎠
⎞
⎜⎜⎝
⎛ ⋅+−=+++
3
333332221 062
l
bwl M l l M l M (2)
M q
a 2l cb l 3
a.
·.ql=45
1 M M 0
ω =022
40
a
a
1ω1
8
120
1
0
ωl 1
1
2l cb
3 M 2 M
b 3
32
ω3
l 3
q
‚.
60
28.92
31.0830
16.62
7.22
26.64
25.56
34.4422.77
32.65
Q
(ÍÕ)
„. M
(ÍÕÏ)
8.3ñð ÁÛð‡„
5. ØÌ‰Ò˝Ì ÒËÒÚÂÏËÈ„ (‰‡ÏÌÛÛð ÚÛÒ ·øðËÈ„ ҇΄‡Ì ÁÛð‡ ı 8.4‡, 8.4·ñð ÁÛð‡„)Áº‚ ı ºÌ º„º„‰ÒºÌ „‡‰‡‡‰ ı ø˜ÌËÈ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð (8.3·ñð ÁÛð‡„)
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167
·‡È„ÛÛÎÊ ˝Ô˛ðËÈÌ Ú‡Î·‡È ÚÛ΄ÛÛð‡‡Ò ı ø̉ËÈÌ Úº‚ ı øðÚνı ÚÓÓÌ ÛÚ„ ̊ „ÓÎ ˙∏.
1l b c
M A11 B
0
1 z 1
1
2
2
z 1
0
1
3l
1
12
3 A q
z 1
3 B
3
0 0
8.4‡ñð ÁÛð‡„ 8.4·ñð ÁÛð‡„
ƒ‡ÏÌÛÛð˚Ì 1ñð ‡ÎÒ·Ή (8.4‡ñð ÁÛð‡„) 0 01 1, B ñÁº‚ ı ºÌ º„º„‰ÒºÌ „‡‰‡‡‰
ı ø˜ÌËÈ øÈΘΠ̋ν˝Ò øøÒ˝ı ð‡͈
( )
0
0 10 0 M M B b c= + + =
∑
( )
0
1
6010
6
M B
b c= − = − = −
+ÍÕ
( )01 10 0 M M A b c= − + =∑
( )01
6010
6
M A
b c= = =
+ ÍÕ
ƒ‡ÏÌÛÛð˚Ì 1ñð ‡ÎÒ·Ή Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜Ì˝˝Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ ı Ó∏ð Ó„ÚÎÓÎ ı ËÈÌ˝.1ñ1 Ó„ÚÎÓΉ ( )1 10 0 4 z b z ≤ ≤ ≤ ≤
( ) 01 1 1 z A z = ⋅ 1 0 z = ( )1 0 M z =
1 4 z = Ï ( )1 40 M z = ÍÕ∑Ï2ñ2 Ó„ÚÎÓΉ ( )2 20 0 2 z c z ≤ ≤ ≤ ≤
( ) 02 1 2 z B z = ⋅ 2 0 z = ( )2 0 M z =
2 2 z = Ï ( )2 20 M z = − ÍÕ∑Ï
ƒ‡ÏÌÛÛð˚Ì 2ñð ‡ÎÒ·Ή „‡‰‡‡‰ ı ø˜ øÈÎ˜Î˝ı „øÈ ·‡È„‡‡ ÚÛÎ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛ·„‰‡ ı „øÈ.
ƒ‡ÏÌÛÛð˚Ì 3ñð ‡ÎÒ·Ή (8.4·ñð ÁÛð‡„) 0 03 3, A B -Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜ÌËÈ
øÈΘΠ̋ν˝Ò øøÒ˝ı ð‡͈ ̸
0 0 33 3
10 6030
2 2
ql A B
⋅= = = = ÍÕ
3ñð ‡ÎÒ·Ή Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜Ì˝˝Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð·‡È„ÛÛ· ı˚Ì ÚÛΉ Ì˝„ Ó„ÚÎÓÎ ı ËÈÌ˝.1ñ1 Ó„ÚÎÓΉ ( )1 3 10 0 6 z l z ≤ ≤ ≤ ≤
( )2
0 11 3 1 2
qz M z A z = ⋅ − 1 0 z = ( )1 0 M z =
1 6 z = Ï ( )1 0 M z =
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî168
1 3 z = Ï ( )2
110 3
30 3 452
M z ⋅
= ⋅ − = ÍÕ∑Ï
ƒ ̋ ˝ð ı ‰‡ÏÌÛÛð˚Ì ‡ÎÒ·ΠÚÛÒ ·øð ‰ ̋ ˝ð ı Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜ÌËÈ
øÈΘΠ̋ν˝Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 8.3·ñð ÁÛð‡„Ú ‰øðÒν‚.›Ô˛ðËÈÌ Ú‡Î·‡È, ÚÛ΄ÛÛð‡‡Ò ı ø̉ËÈÌ Úº‚ ı øðÚ ̋Î ı Á‡ÈÌÛÛ‰ ̸:
80402
11 =⋅⋅=′
bw ÍÕ⋅Ï2;3
8
3
21 =⋅=′
ba Ï
20202
11 −=⋅⋅=″
cw ÍÕ⋅Ï2;3
142
3
14
3
11 =⋅+=⋅+=″
cba Ï
18012
610
12
333
3 =⋅
==ql
w ÍÕ⋅Ï2; 323
3 ==l
b Ï
1.
ŒÎÒÓÌ ÚÓÓÌ Ëνð ı ËÈÎν˝ (1), (2)ñð Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛÎÊ, à 1, à 2 ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎ ˙∏.
( )0 1 2
8 1480 20
3 36 2 6 6 6 6 06 6
M M M
⎛ ⎞⋅ ⋅⎜ ⎟
⋅ + ⋅ + + ⋅ = − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
( )1 2180 3
6 2 6 6 0 6 06
M M ⋅⎛ ⎞
⋅ + ⋅ + + = − +⎜ ⎟⎝ ⎠
1 2
1 2
24 6 240
6 24 540
M M
M M
+ = ⎫⎪⎬
+ = − ⎪⎭›Ì‰ ̋ ˝Ò: à 1= 16.66 ÍÕ⋅Ï M 1=ñ26.66ÍÕ⋅Ï
·ÓÎÌÓ.2.
1 2, M M ñÏÓÏÂÌÚÛÛ‰ ̊ „ ‡¯Ë„Î‡Ê ‰‡ÏÌÛÛð˚Ì ·Ó‰ËÚ ð‡͈ÛÛ‰ ̊ „
ÚÓ‰Óð ı ÓÈÎ ˙∏ .(8.2), (8.3) ÚÓÏ ˙∏Ó„ÓÓð 1ñð ‡ÎÒ·Ή
0 0 1
1 1 1 1
60 16.66
10 22.776 6
M M
A A l l
⎛ ⎞
= − + = − − + =⎜ ⎟⎝ ⎠ ÍÕ
0 0 11 1
1 1
60 16.6610 22.77
6 6
M M B B
l l
⎛ ⎞= + − = − + − − = −⎜ ⎟
⎝ ⎠ ÍÕ
2ñð ‡ÎÒ·Ή 0 02 20 0 A B = = ·ÓÎÓ ı˚„ ‡Ì ı ‡‡ð‚‡Î
0 1 22 2
2 2
16.66 26.660 7.22
6 6
M M A A
l l
⎛ ⎞= − + = − + − = −⎜ ⎟
⎝ ⎠ÍÕ
0 1 22 2
2 2
16.66 26.66
0 7.226 6
M M
B B l l
⎛ ⎞= + − = + − − =⎜ ⎟⎝ ⎠ ÍÕ
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169
3ñð ‡ÎÒ·Ή
0 2 33 3
3 3
26.6630 0 34.44
6
M M A A
l l
⎛ ⎞= − + = − − + =⎜ ⎟
⎝ ⎠ÍÕ
0 2 33 33 3
26.6630 0 25.566
M M B B l l
⎛ ⎞= + − = + − + =⎜ ⎟⎝ ⎠
ÍÕ
·ÓÎÌÓ.Õ˝„‰ø„ ̋ ˝ð ÚÛ΄ÛÛð˚Ì ·øÚ˝Ì ð‡͈ ̸
A21 B
l 1 l 2
1
1
( )1 1 2 22.77 7.22 29.99 R B A= + = − + − = − ÍÕ
·ÓÎÌÓ.
’Ó∏ð‰Û„‡‡ð ÚÛ΄ÛÛð˚Ì ·øÚ˝Ì ð‡͈
A32 B
l 2 l 3
R 2
2
2 2 3 7.22 34.44 41.66 R B A= + = + = ÍÕ
·ÓÎÌÓ.
ƒ‡ÏÌÛÛð˚Ì ÁøøÌ (0) ÚÛ΄ÛÛðÚ øøÒ˝ı ·øÚ˝Ì ð‡͈ ̸ ÚÛ΄ÛÛð˚Ì Ó„ÚÎÓÎ ‰‡ ı ̧ ·‡ðÛÛÌ ÁøøÌ ı˝Ò„ËÈÌ ‰ÓÚÓÓ‰ ı ºÌ‰ÎºÌ ı ø˜Ìøø‰ËÈÌ ÌËÈη˝ð˝˝ðÚÓ‰Óð ı ÓÈÎÓ„‰ÓÌÓ. ¡‡ðÛÛÌ ı˝Ò„ËÈÌ ‰ÓÚÓÓ‰ ı ºÌ‰ÎºÌ ı ø˜ (ð‡͈)ñËÈ„ ·Ë‰
1 A „ ̋ Ê ÚÓ‰Óð ı ÓÈÎÒÓÌ ·º„ºº‰ ÁøøÌ ı˝Ò„ ̋ ˝Ò øøÒ˝ı ‰ÓÚÓÓ‰ ı ºÌ‰ÎºÌ ı ø˜
(ð‡͈) ̸
A10 B
a l 1
R 0
0
0 0 1 30 22.77 52.77 R B A= + = + = ÍÕ
·ÓÎÌÓ.
¡‡ðÛÛÌ (4) ÚÛ΄ÛÛðÚ øøÒ˝ı ð‡͈ ̸
3 3 25.56 R B = = ÍÕ ·‡È ı˚„ ‡Ì ı ‡‡ð˜ ÌËÈÚ ‰‡ÏÌÛÛð˚Ì ı Û‚¸‰ ·ÓÒÓÓÚ˝Ì ı ν„ ‰ ̋ ˝ð ı ı ø˜Ìøø‰ËÈÌ ÔðÓÂ͈ ̊ „ ·Ë˜¸Â.
0 y =∑
( )0 1 2 3 3 30 52.77 29.99 41.66 10 6 25.56 0 F R R R ql R − + + + − + = − + + − + − ⋅ + =
·‡È̇. ’ø˜ÌËÈ ÔðÓÂ͈ÛÛ‰ ̊ Ì ÌËÈη˝ð Ú˝„ „‡ð˜ ·‡È„‡‡ ̸ ð‡͈ ̊ Ì ı ø˜Ìøø‰ËÈ„
Áº‚ ÓÎÒÓÌ ·‡È̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî170
ƒ‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛ· ı˚Ì ÚÛΉ Ú‡‚‡Ì Ó„ÚÎÓÎ ı ËÈÌ˝. › ı ÌËÈ 3 Ó„ÚÎÓÎ˚„ ̸ ÚÓÓÎÓÎ˚Ì ˝ı ËÈ„ ‰‡ÏÌÛÛð˚Ì ÁøøÌ øÁøøðÚ,ÒøøÎËÈÌ 2 Ó„ÚÎÓÎ˚„ ̸ ‰‡ÏÌÛÛð˚Ì ·‡ðÛÛÌ øÁøøðÚ ·‡È ı ‡‡ð ı ËÈÂ.
1ñ1 Œ„ÚÎÓΉ 10 z a≤ ≤ ( 10 z ≤ ≤ 4Ï)( )1 30Q z F = − = − ÍÕ ( )1 1 M z F z = − ⋅
( )1 10 0 z M z = = 1 2 z = Ï ( )1 60 M z = − ÍÕ∑Ï
2ñ2 Œ„ÚÎÓΉ 2a z a b≤ ≤ + (2Ï 2 z ≤ ≤ 6Ï)
( )2 0 30 52.77 22.77Q z F R = − + = − + = ÍÕ
( ) ( )2 2 0 2 2 M z F z R z = ⋅ + −
2 2 z = Ï ( )2 60 M z = − ÍÕ∑Ï
2 6 z = Ï ( )2 30 6 52.77 4 31.08 M z = − ⋅ + ⋅ = ÍÕ∑Ï
3ñ3 Œ„ÚÎÓΉ 3a b z a b c+ ≤ ≤ + + (6Ï 1 z ≤ ≤ 8Ï)
( )3 0 22.77Q z F R = − + = ÍÕ ( ) ( )3 3 0 3 2 M z F z R z M = − ⋅ + − −
3 6 z = Ï ( )3 28.92 M z = − ÍÕ∑Ï
3 8 z = Ï ( )3 30 8 52.77 6 60 16.62 M z = − ⋅ + ⋅ − = ÍÕ∑Ï
4ñ4 Œ„ÚÎÓΉ 4 30 z l ≤ ≤ (0 1 z ≤ ≤ 6Ï)
( )4 3 4Q z R q z = − + ⋅ ( )2
44 3 4 2q z M z R z ⋅= ⋅ −
4 0 z = ( )4 25.56Q z = − ÍÕ ( )4 0 M z = ÍÕ∑Ï
4 6 z = Ï ( )4 25.56 10 6 34.44Q z = − + ⋅ = ÍÕ ( )4 26.64 M z = ÍÕ∑Ï
3ñð ‡ÎÒ·Ή Ú‡ð ı χΠ‡˜‡‡Î‡Î øÈÎ˜Î˝ı ÚÛÎ „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì˝ÍÒÚðÂÏÛÏ ÛÚ„‡ ̸
( ) 30 3 0 0
25.560 2.55
10
R Q z R qz z
q= − + = = = = Ï 0 2.55 z = Ï
( )2 20
0 3 010 2.5525.56 2.55 32.55
2 2
q z M z R z ⋅ ⋅= ⋅ − = ⋅ − = ÍÕ∑Ï
5ñ5 Œ„ÚÎÓΉ 3 5 3 2l z l l ≤ ≤ + (6Ï 5 z ≤ ≤ 12Ï)
( )5 3 3 2 25.56 10 6 41.66 7.22Q z R q l R = − + ⋅ − = − + ⋅ − =
( ) ( ) ( )5 3 5 3 5 2 53 6 M z R z ql z R z = ⋅ − − + −
5 6 z = Ï ( )5 25.56 6 10 6 3 26.64 M z = ⋅ − ⋅ ⋅ = − ÍÕ∑Ï
5 12 z = Ï ( )5 25.56 12 10 6 9 41.66 16.68 M z = ⋅ − ⋅ ⋅ + = ÍÕ∑Ï
ƒ ̋ ˝ð ı ÛÚ„ÛÛ‰‡‡ð ·‡È„ÛÛÎÒ‡Ì ˝Ô˛ðËÈ„ 8.3‚, 8.3„ ñð ÁÛð‡„Ú ı ‡ðÛÛη‡.
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171
∆ ˯˝˝ 8.2. «øøÌ øÁøøð ̸ ı º‰ºÎ„ººÌ„øÈ ·˝ı ̋΄ ̋ ˝Ú˝È, ·‡ðÛÛÌ øÁøøð̸ ÍÓÌÒÓθ, ı Ó∏ð ‡ÎÒ·ÎÚÚ‡È ‰‡ÏÌÛÛð(8.10añð ÁÛð‡„) º„º„‰Ê ̋ ˝.
“ÓÓÌ º„º„‰ºÎ: F, (ÍH) q, ÍH/Ï a, (Ï) b, (Ï) 21 l l = , (Ï) 3l , ( Ï)
50 15 3 2 5 3ŒÎÓ ı ̸:
1.
“Û΄ÛÛð˚Ì ð‡͈ 321 ,, R R R -ËÈ„ ÓÎ.
2.
’ºÌ‰ÎºÌ ı ø˜ÌËÈ ˝Ô˛ð ·‡È„ÛÛÎ.3.
√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ·‡È„ÛÛÎ.
‡.
·.
l 3l l
l l 0 1
0 M
0 1
1 M
1
l 2
2
2 M
2
M 3
3
a
q F
b
F q
‚.
1a 2a1b b2
ω =02
1a
1ω
1b
1
M =
4 6 . 9
m a x
„.
‰.
30.54
28.49
21.51
22.95
33.99 46.41
28.59
4.45
8.10ñð ÁÛð‡„
¡Ó‰ÓÎÚ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî172
1.
«ÓÓÎÚ˚„ Úº„Ò„ºÎ„øÈ ·‡„‡ ÛðÚÚ‡È ( )0=ol Úº„Ò„ºÎ„øÈ Ë ı ı º¯øøÌ
˜‡Ì‡ðÚ‡È ‡ÎÒ··‡ð, ÏºÌ ÒÛÎ øÁøøðÚ øÈÎ˜Î˝ı ı ø˜ËÈ„ ÚÛ΄ÛÛð ‰ ̋ ˝ð
øÈÎ˜Î˝ı
ÏÓÏÂÌÚÓÓð (
2 23
0 3
15 3
0; 67.52 2
ql
M M
⋅
= = = = − ÍÕ⋅Ï00 =l ) ÚÛÒ ÚÛÒ ÒÓÎËÌÓ (8.10·ñð ÁÛð‡„).
2.
“Û΄ÛÛðÛÛ‰ ̊ „ 0ñ˝˝Ò ˝ı Î˝Ì ‰Û„‡‡ð·Ê, ÚÛ΄ÛÛð ÚÛÒ ·øð ‰ ̋ ˝ðÏÓÏÂÌÚ ........., 10 M M øÈΘÎøøÎÌ˝.
3.
Øð„ ̋ ÎÊ ‰‡ÏÌÛÛð˚„ ‡ÎÒ·Π·øð˝˝ð ̸ øÈΘËÎÊ ·ÛÈ „‡‰‡‡‰ ı ø˜ÌËÈ ı ‡ÏÚ Ò‡Î„‡Ì ÁÛðÊ (8.11; 8.12ñð ÁÛð‡„), Ú ̋‰„ ̋ ˝ðËÈÌ ð‡͈ËÈ„ ÓÎÓÓ‰,„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ ·‡È„ÛÛÎ̇ ( 8.9‚ñð ÁÛð‡„).
ba
A
1 F
2 z
21
1 z 1 2
B
2
0 02 2
2l
3
32
Aq
z 3
B
3
0 0
3 3
8.11-ð ÁÛð‡„ 8.12-ð ÁÛð‡„
( )02 20; 0 M A a b F b= ⋅ + − ⋅ =∑
0250 2
205
F b A
a b⋅ ⋅= = =+ ÍÕ
( )01 20; 0 M B a b F a= − ⋅ + + ⋅ =∑
02
50 330
5
F a B
a b
⋅ ⋅= = =
+ ÍÕ
0 0 23 3
15 537.5
2 2
ql A B
⋅= = = = ÍÕ
1,2ñð ‡ÎÒ·Î˚Ì Áº‚ ı ºÌ „‡‰‡‡‰ ı ø˜ÌËÈ ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 8.10‚ñð ÁÛð‡„Ú øÁøøν‚.
4. √
ÛÎÁ‡È΄‡ ı
ÏÓÏÂÌÚ˚
Ì Ú˝„¯ËÚ„
˝ Î ·Ë˜¸Â (8.11, 8.12ñð ÁÛð).1ñ1 Ó„ÚÎÓΉ ( );300 11 ≤≤≤≤ z a z
( ) ( ) ( )01 2 1 1 1; 0; 20 3 60 M z A z M z M z ′ ′′= ⋅ = = ⋅ = ÍÕ⋅Ï
2ñ2 Ó„ÚÎÓΉ ( );200 22 ≤≤≤≤ z b z
( ) ( ) ( )02 2 2 2 2; 0; 30 2 60 M z B z M z M z ′ ′′= ⋅ = = ⋅ = ÍÕ⋅Ï
3ñ3 Ó„ÚÎÓΉ: ( );500 323 ≤≤≤≤ z l z
( ) ( ) ( ) 0;0;
233
23
323 =′′=′−⋅= z M z M z
q z A z M
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ÃÓÏÂÌÚ˚Ì ˝ÍÒÚðÂÏÛÏ ÛÚ„‡ ̸:2 22
max15 5
46.878 8
ql M
⋅= = = ÍÕ⋅Ï ·ÓÎÌÓ.
1, 2ñð ÚÛ΄ÛÛð˚Ì ı Û‚¸‰ „Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ Ã 1
, Ã 2
ñ˚„ÓÎ ˙∏.n=1
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ″″+
′′+−=+++
1
11
1
111210100 062
l
bw
l
bwl M l l M l M
n=2
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +
″″+
′′−=+++
2
22
1
11
1
112321211 62
l
bw
l
aw
l
awl M l l M l M
“ ̋ „¯ËÚ„ ̋ ÎËÈÌ øÒ„ ̋ Ì Ëνð ı ËÈÎÎËÈ„ ÚÓÓ„ÓÓð ÚÓ‰Óð ı ÓÈÎ ˙∏.
903602
11 =⋅⋅=′
w ÍÕ⋅Ï2 23
3
2
3
21 =⋅=⋅=′
aa Ï
333
121 =⋅+=′
b Ï
60260
2
160
2
11 =⋅⋅=⋅⋅=″
bw ÍÕ⋅Ï2 3
112
3
13
3
11 =⋅+=⋅+=″
baa Ï
342
321 =⋅=″b Ï
25.15612
515
12
333
2 =⋅
==ql
w ÍÕ⋅Ï2 Ï5.22
5
23
22 ====l
ba
›‰„ ̋ ˝ð ÚÓÓÌ ÛÚ„ ̊ „ ‰ ̋ ˝ð ı Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛÎÊ, à 1, à 2-˚„ ÚÓ‰Óð ı ÓÈÎÌÓ.
⎟ ⎠
⎞⎜⎝
⎛ ⋅⋅
+⋅
+−=++35
460
5
39006510M0 21 M
( ) ⎟ ⎠
⎞
⎜⎝
⎛
⋅
⋅
+⋅
⋅
+
⋅
−=−++ 25
53.156
35
1160
5
290
69.465205 21 M M
⎫⎪⎪⎬⎪⎪⎭
1 30.54 M = − ÍÕ⋅Ï 2 22.92 M = − ÍÕ⋅Ï ·ÓÎÌÓ.
5. Øð„ ̋ ÎÊ ‰‡ÏÌÛÛð˚„ ‡ÎÒ·Î˚Ì ÚÓÓ„ÓÓð ҇΄‡Ì ÁÛðÊ, ð‡͈ËÈ„ (8.2), (8.3)ÚÓÏ ˙∏Ó„ÓÓð Óθ∏ (8.14ñð ÁÛð).– ‡ Ì ‡ Ï Ê: ’º‰ºÎ„ººÌÚ˝È ÌÛ„‡ÒÚ‡È ı Ó∏ð ÚÛ΄ÛÛð‡‡ð ÒÓθÒÓÌ ı˝Ò„ËÈ̇ÎÒ·Î˚„ l 0=0 „ ̋ Ê ‡‚̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî174
1l
b1 a z 2
z
1 2
1 M 1 A
1
F B 1
21 M 2
2
B M 2
3
2l
2
z 3
3 q A23 M
3
2
8.13ñð ÁÛð‡„
0 1 22 2
1 2
M M A A
l l = − + =
30.54 22.9220 21.51
5 5
⎛ ⎞ ⎛ ⎞− − + − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ÍÕ
0 1 22 2
1 1
M M B B
l l = + − =
30.54 22.9230 28.48
5 5
⎛ ⎞ ⎛ ⎞+ − + =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
ÍÕ
0 2 33 3
2 2
M M A A
l l = − + =
22.92 67.537.5 28.58
5 5
⎛ ⎞ ⎛ ⎞= − − + − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ÍÕ
0 2 23 3
2 2
M M B B
l l = + − =
22.92 67.537.5 46.41
5 5
⎛ ⎞ ⎛ ⎞+ − − − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ÍÕ
“Û΄ÛÛðÛÛ‰ ̊ Ì ·øÚ˝Ì ð‡͈ËÈ„ ÓÎ ˙∏.1ñð ÚÛ΄ÛÛð ‰ ̋ ˝ð 1 2 21.51 R A= = ÍÕ
2ñð ÚÛ΄ÛÛð ‰ ̋ ˝ð2 2 3 28.48 28.58 57.06 R B A= + = + = ÍÕ
A B
l 2 l 3
R
2
3
3 4
3ñð ÚÛ΄ÛÛð ‰ ̋ ˝ð 4 3 15 3 45 A ql = = ⋅ = ÍÕ
3 3 4 46.41 45 91.41 R B A= + = + = ÍÕ
·ÓÎÌÓ.
ÕËÈÚ ‰‡ÏÌÛÛð˚Ì ı Û‚¸‰ ·ÓÒÓÓ Ú˝Ì ı ν„ ‰ ̋ ˝ð ı ı ø˜Ìøø‰ËÈÌ ÔðÓÂ͈ ̸
0 y =∑ 1 2 38 21.51 50 57.06 8 15 91.41 0 R F R q R − + − + = − + − ⋅ + ≈
“ ̋ „ „‡ð˜ ·‡È„‡‡ ̸ ð‡͈ÛÛ‰ ̊ „ Áº‚ ÓÎÒÓÌ ·‡È̇.6. ƒ‡ÏÌÛÛð ÚÛÒ ·øðËÈÌ Ó„ÚÎÓÎ ‰ ̋ ˝ð ı ºÌ‰ÎºÌ ı ø˜, „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì
Ú˝„¯ËÚ„ ̋ Î ·Ë˜ËÊ ÚÓÓÌ ÛÚ„ ̊ „ ÓÎ ˙∏ (8.13ñð ÁÛð‡„).
A32 B
l 2 l 3
R 2
2
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1ñ1 Ó„ÚÎÓΉ ( a z ≤≤ 10 ) 10 3 z ≤ ≤ Ï
( )1 1 21.51Q z R = = ÍÕ
( )1 1 1 1
M z R z M = ⋅ +
1 z 0 3
1Q 21.51 21.51
1 z M -30.54 33.99
2ñ2 Ó„ÚÎÓΉ ba z a +≤≤ 2 (3Ï 2 5 z ≤ ≤ Ï)
( )2 1 28.49Q z R F = − = − ÍÕ
( ) ( )2 1 2 2 1 M z R z F z a M = ⋅ − − +
2 z 3 5
2Q --28.49 -28.49
( )2 z 33.99 -22.99
3ñ3 Ó„ÚÎÓΉ ( )3 30 z l ≤ ≤ ( 30 3 z ≤ ≤ Ï)
( )3 3Q z qz =
( )23
3 2
z M z q= −
3 z 0 3
3Q 0 45
3 z M 0 -67.5
4ñ4 Ó„ÚÎÓΉ 3 4 3 4l z l l ≤ ≤ + (3Ï 4 8 z ≤ ≤ Ï)
( )4 4 3Q z qz R = −
( ) ( )24
4 3 4 32
qz M z R z = − + −
4 z 3 8
( )4Q z -46.41 28.59( )4 z -67.5 -22.95
›ÍÒÚðÂÏÛÏ ÛÚ„‡ ÓÎ ˙∏.
( )0 4 3 0Q z q z R = ⋅ − = 34
91.416.09
15
R z
q= = = Ï
( ) ( )2
415 6.09
91.41 6.09 3 4.45
2
M z ⋅
= − + ⋅ − = ÍÕ⋅Ï
’ºÌ‰ÎºÌ ı ø˜, ÏÓÏÂÌÚ˚Ì ·øÚ˝Ì ˝Ô˛ðËÈ„ 8.9 „,‰ñð ÁÛð‡„Ú øÁøøν‚.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî176
≈–ƒØ √ ››— ¡ØÀ›√
ÿ¿’¿ √ ƒ–¿Õ ÿ»À¡›Õ»… “Œ√ “¬Œ— ∆ »À
9.1 ŒÕŒÀ¤Õ “Œ¬◊ ØÕƒ›–À›À
Œ„ÚÎÓÎ ÒÓÌ„Ó ı ÚÓÏ ˙∏Ó:
[ ]σ⋅ϕ≥
N A (9.1)
›Ì‰ ¿ ñ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È; −ϕ øÌ‰Ò˝Ì Áº‚¯ººðº„‰º ı
ı ø˜‰ ̋ ÎËÈ„ ·ÛÛðÛÛ· ı ÍÓ˝ÙÙˈËÂÌÚ (‰‡„ÛÛ „ÛÎÁ‡ÈÎÚ˚Ì ÍÓ˝ÙÙˈËÂÌÚ); N ñ¯Ëη˝ÌËÈ Ú˝Ì ı ν„ ‰‡„ÛÛ ¯‡ ı ‡ ı ı ø˜ ; −σ Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î.
ƒº ı øøνı ‡ð„‡‡ð Ó„ÚÎÓÎ ÒÓÌ„Ó ı ‰ÓÓ ϕ = 0.4...0.6 ıˇÁ„‡‡ðÚ Û𸉘ËÎ̷҇‡È‰Î‡‡ð ‡‚̇.
»ÌÂðˆËÈÌ ð‡‰ËÛÒ˚Ì ·‡„‡ ÛÚ„ ̊ „ ÓÎÓ ı ÚÓÏ ˙∏Ó
A
I i minmin = (9.2)
−min I ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚Ì ÏËÌËÏÛÏ ÛÚ„‡;
ÇÚÂðˇÎ˚Ì Û ˇÌ ˜‡Ì‡ð˚„ Ëνð ı ËÈνı ÚÓÏ ˙∏Ó
minil ⋅μ=λ (9.3)
“Ó„Ú‚ÓðÊÎ˚Ì Ìº ı ̂ ºÎËÈ„ ¯‡Î„‡ ı ÚÓÏ ˙∏Ó
[ ]T A
N σ σ ≤= (9.4)
[ ]−T σ ÚÓ„Ú‚ÓðÊÎ˚Ì Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î.
˝Ò‚ ̋Î R
T
⋅= ϕ σ (9.5)·‡ÈÊ ·ÓÎÌÓ.
’ ̋ ð˝‚ ÚÓ„Ú‚ÓðÊÎ˚Ì Ìº ı ̂ ºÎ ·ËÂνı „øÈ ·ÓÎ ‰º ı øøνı ‡ð„‡‡ð Ó„ÚÎÓÎ˚„ ‰‡ ı ËÌ ÒÓÌ„ÓÌÓ. ƒ‡ð‡‡„ËÈÌ ‰º ı øøÎ ̋ÎÚËÈ„ ı ËÈ ı ‰ ̋ ˝ ϕ ÍÓ˝ÙÙˈËÂÌÚ˚Ì
‰Û̉‡Ê ÛÚ„ ̊ „ ‡‚̇.
211
2ϕ′+ϕ
=ϕ (9.6)
›Ì‰ −ϕ1 ‡Ì ı Ì˚ ‡‚Ò‡Ì ÓÈðÓΈÓÓ ı ÛÚ„‡ ( )5.01 =ϕ ; −ϕ′1 ÚÛ ı ‡ÈÌ
χÚÂðˇÎ˚Ì ı øÒÌ˝„Ú˝˝Ò( −ϕλ, ËÈÌ ı ‡Ï‡‡ð‡Î ) ‡‚Ò‡Ì ÛÚ„‡ .
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›Ì˝ ÚÓÓÌ˚ ·Ó‰ËÚ ÛÚ„ ̊ „ ÓÎÓ ı ‰ÓÓ ÚÓÓˆÓÓ„ÓÓð ÓÎÒÓÌ −ϕ ËÈÌ ÚÓÓÌ
ı Óηӄ‰Î˚Ì ı ‡Ï„ËÈÌ ÓÈð ÛÚ„ÛÛ‰ ̊ „ ËÌÚÂðÔÓÎ ˇˆ˜ËÎ̇. −ϕ2 ı Ó∏ð‰Û„‡‡ð
‰º ı øøÎ ̋ÎÚ˝Ì‰ ‡¯Ë„· ı ·Ó‰ËÚ ÛÚ„‡.ÿ‡ ı ‡„‰Ò‡Ì ¯Ëη˝ÌËÈ ·º ı ·‡Ú˚„ ¯‡Î„‡ ı ÚÓÏ ˙∏Ó
σ σ ⋅
= ± ± ≤ ⎡ ⎤⎣ ⎦o
x x
N M F v
A W W (9.7)
Øø̉ o M -«º‚ ı ºÌ ı ºÌ‰ÎºÌ ‡˜‡‡Î·‡Ò øøÒ˝ı „ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ
flÌÁ ·øðËÈÌ Ï‡ÚÂðˇÎ˚Ì ϕ ÍÓ˝ÙÙˈËÂÌÚ˚Ì ı Óηӄ‰ÓÎ
9.1ñð ı øÒÌ˝„Ú
” ˇÌ˜‡Ì‡ð
i
l μ=λ
−ϕ ËÈÌ ı Óηӄ‰ÓÎ
–Ú0,–Ú2,–Ú3,–Ú4
–Ú5
ÿËð˝Ï–˜-15-30,–˜-15-15–˜15-36,–˜41-40
ÿËð˝Ï
–˜21-14–˜28-48
¡ø ı ÚºðÎËÈÌ
ÏÓ‰
¡ÂÚÓÌ
’ø̉ ’ºÌ„ºÌ
0 1.00 1.00 1.00 1.00 1.00 1.00 1.0010 0.99 0.98 0.97 0.95 0.99 1.00 1.0020 0.96 0.95 0.91 0.87 0.97 0.96 0.9630 0.97 0.92 0.81 0.75 0.93 0.90 0.8640 0.92 0.89 0.69 0.60 0.87 0.84 0.7350 0.89 0.86 0.57 0.43 0.80 0.76 0.6860 0.86 0.86 0.44 0.32 0.71 0.70 0.5970 0.81 0.82 0.34 0.23 0.60 0.63 0.5280 0.75 0.76 0.26 0.18 0.48 0.57 0.4690 0.69 0.70 0.20 0.14 0.38 0.51 -100 0.60 0.62 0.16 0.12 0.31 0.45 -
110 0.52 0.51 - - 0.25 - -120 0.45 0.43 - - 0.22 - -130 0.40 0.36 - - 0.18 - -140 0.36 0.33 - - 0.16 - -150 0.32 0.29 - - 0.14 - -160 0.29 0.26 - - 0.12 - -170 0.26 0.24 - - 0.11 - -180 0.23 0.19 - - 0.10 - -190 0.21 0.17 - - 0.09 - -
200 0.19 0.16 - - 0.08 - -
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ðËÚËÍ ı ø˜ÌËÈ ÚÓÏ ˙∏Ó
( )2min
2
l
I E F cr
μ
π ⋅= (9.8)
›Ì‰ −μ ·˝ı ̋΄ ̋ ˝Ì˝˝Ò ı ‡Ï‡‡ð‡ ı ÍÓ˝ÙÙˈËÂÌÚ; −l ¯Ëη˝ÌËÈ ÛðÚ; ƒ‡„ÛÛ ñ ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ˚Ì Ú˝„¯ËÚ„ ̋ Î
k F
N l
z v
v
−
⋅π⋅
=1
sin0 (9.9)
−0v ı ºÌ‰ÎºÌ ‡˜‡‡Î·‡Ò øøÒ˝ı „ÛÎÁ‡ÈÎÚ; − z ¯Ëη˝ÌËÈ Ú˝Ì ı ν„ËÈÌ ‰‡„ÛÛ
ÍÓÓð‰Ë̇Ú;
’ ̋ ð˝‚ ı ‡Ï„ËÈÌ Ë ı „ÛÎÁ‡ÈÎÚ2l z = ÛÚ„‡Ì‰ ı ‡ð„‡ÎÁ‚‡Î (9.9) ÚÓÏ ˙∏Ó
‰‡ð‡‡ ı ı ̋η˝ðÚ˝È ·ÓÎÌÓ.
k F
N
vv
−=
1
0max
(9.10)
∆ ˯˝˝ 8.3
ƒ‡„ÛÛ- ı ºÌ‰ÎºÌ ı ø˜ÌËÈ øÈΘΠ̋Ή ÓðÊ ·ÛÈ „‡Ì ¯Ëη˝(9.1ñð ÁÛð)º„º„‰Ê ̋ ˝. ÇÚÂðˇÎ˚Ì ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î 160= R Ãœ‡ ·ÓÎÌÓ.ª„º„‰ºÎ : 400= N ÍÕ, 10= F ÍÕ, 5.2=l Ï
“ÓÓˆÓÓ„ÓÓð ÓÎÓ ı ÁøÈÎ1.
ÿËη˝ÌËÈ Ó„ÚÎÓÎ˚„ ‰º ı øøνı ‡ð„‡‡ð ÒÓÌ„Ó.2.
ƒ‡„ÛÛ , ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ˚Ì ÚÓÓˆÓÓ ı ËÈ.3.
ª„º„‰ÒºÌ ¯Ëη˝ÌËÈ ÍðËÚËÍ ı ø˜ËÈ„ ÓÎ.4.
ÿËη˝ÌËÈ ·º ı ·‡Ú˚„ ¯‡Î„‡.
5.
“Ó„Ú‚ÓðÊÎ˚Ì ÌººˆËÈ„ ÓÎ
9.1ñð ÁÛð‡„
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179
¡Ó‰ÓÎÚ .1. “Ó„Ú‚ÓðÊÎ˚Ì Ìº ı ̂ ºÎººÒ ¯Ëη˝ÌËÈ ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚„ ‰º ı øøνı ‡ð„‡‡ðÓÎ ˙∏. ›Ì‰ 4.0=ϕ -‡‡ð ‡‚‡‚.
Õ ˝„‰ø„ ˝ ̋𠉺 ı øøÎ ̋ÎÚ:
0063.0101604.0
104006
3=
⋅⋅⋅=
⋅ϕ≥
R
N A Ï2=63ÒÏ2
Ï‡Ì‡È Ê˯˝˝Ì‰ ·Ó‰ËÚ Ú‡Î·‡È
22
22
22.34
44
22 aa
ad
aa A =π
−=π
−⋅⋅= ·ÓÎÌÓ.
“º‚ Ú˝Ì ı ν„Ú˝È ı ‡ð¸ˆ‡Ì„ÛÈ ˝Ò˝ð„øøˆÎËÈÌ, ËÌÂðˆËÈÌ ÏÓÏÂÌÚ˚„ ÓÎ ˙∏
( )3
4343
28.125.064
14.3
12
22
5.06412
aa
aaa
h
d bh
W x =⋅⋅
−⋅
=
π−
=
( ) 4444343
min 28.064
14.3
12
16
6412
22
6412a
aaaaad hb I =
⋅−=
π−
⋅=
π−=
·ÓÎÌÓ.»ÌÂðˆËÈÌ ð‡‰ËÛÒ˚Ì ·‡„‡ ÛÚ„ ̊ „ ÓÎ ˙∏.
aa
a
A
I i 3.0
22.3
28.02
4min
min ≅==
” ˇ
Ì ˜‡Ì‡ð˚
„ ÚÓ‰Óð ı
ÓÈÎ ˙∏
. ( 1=μ ) „ ˝
Ê ‡‚̇.
aai
l 3.833
3.0
2501
min
=⋅
⋅=
μ=λ
Œ‰ÓÓ Ó„ÚÎÓÎ˚Ì ‡ ı˝ÏÊ ̋ ˝„ ÓÎ ˙∏.
5.422.3
63
22.3 ≅==
Aa ÒÏ
“ ̋ „‚ ̋Î 2.1855.4
3.833==λ ·ÓÎÌÓ.
9.1-ð ı øÒÌ˝„Ú˝˝Ò 180=λ ø‰ → 19.0=ϕ
190=λ ø‰ → 17.0=ϕ ‡‚ ˙ˇ.
»ÌÚÂðÔÓÎ ˇˆ˜ËÎ ˙ˇ:
1796.02.510
17.019.019.01 =⋅
−−=ϕ′
“ÓÚ‚ÓðÊÎ˚Ì Ìº ı ̂ ºÎ ·ËÂÎÊ ·ÛÈ ˝Ò˝ı ËÈ„ ¯‡Î„‡ ˇ.7.281601796.0
=⋅=⋅ϕ R Ãœ‡
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî180
4.60063.0
10400 3
=⋅
==σ A
N Ãœ‡
“ÓÓˆÓÓÌ˚ ı ø˜‰ ̋ Î Áº‚¯ººðº„‰º ı ı˝ÏÊ ̋ ˝Ì˝˝Ò
2.121009.28
4.67.28 =⋅− %> 5% ·‡È̇.
’Ó∏ ð‰Û„‡‡ð ‰º ı øøÎ ̋ÎÚ. ϕ -ËÈÌ ‰Û̉‡Ê ÛÚ„ ̊ „ ÓÎ ˙∏.
29.02
1796.04.0
211
2 =+
=ϕ′+ϕ
=ϕ
Œ„ÚÎÓÎ˚Ì Ú‡Î·‡È„ ÓÎ ˙∏ 0086.01016029.0
104006
3
2 =⋅⋅
⋅= A ÒÏ2
Œ„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝„ ÓÎ ˙∏. 2.522.3
86
22.3 === A
a ÒÏ
“ ̋ „‚ ̋ΠχÚÂðˇÎ˚Ì Û ˇÌ ˜‡Ì‡ð
3.1602.53.0
2501
min
=⋅
⋅=
μ=λ
i
l ·ÓÎÌÓ.
9.1-ð ı øÒÌ˝„Ú˝˝Ò λ = 160 ø‰ → 24.0=ϕ
λ = 170 ø‰ → 21.0=ϕ ‡‚‡‚.
»ÌÚÂðÔÓÎ ˇˆ˜ËÎ ˙ˇ:
24.03.010
21.024.024.02 =⋅
−−=ϕ′
ÇÚÂðˇÎ˚Ì ·‡Ú ·º ı ËÈ„ ¯‡Î„‡ ˇ.
R A
N ⋅ϕ≤=σ 4.385.46;16024.0
0086.0
10400 3
≤⋅≤⋅
Áºðøø„ ÓÎ ˙∏.−
⋅ = >46.5 38.4
100 17.4 546.5
%
’ø˜‰ ̋ ÎËÈÌ Áºðøø ÓÈðÚÒÓÌ ı ̋‰ËÈ ˜ Áº‚¯ººðº„‰ÒºÌ ı˝ÏÊ ̋ ˝Ì˝˝Ò Ë ı Û˜ð‡‡Ò ‰‡ ı ËÌ ‰º ı øøÎ ̋ÎÚ ı ËËÂ.
√ Ûð‡‚‰Û„‡‡ð ‰º ı øøÎ ̋ÎÚ ϕ -ËÈÌ ‰Û̉‡Ê ÛÚ„ ̊ „ ÓÎ ˙∏.
27.02
24.029.0
222
3 =+
=ϕ′+ϕ
=ϕ
ÇÚÂðˇÎ˚Ì ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È„ ÓÎ ˙∏
0095.01016027.0
104006
3
3 =⋅⋅
⋅= A Ï2=95ÒÏ2
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
181
Œ„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝„ ÓÎ ˙∏. 4.522.3
95
22.3 ===
Aa ÒÏ
“ ̋ „‚ ̋ΠχÚÂðˇÎ˚Ì Û ˇÌ ˜‡Ì‡ð
3.1544.53.0
2501
min=⋅
⋅=
μ=λ i
l ·ÓÎÌÓ.
9.1 ı øÒÌ˝„Ú˝˝Ò 150=λ ø‰ → 27.0=ϕ
160=λ ø‰ → 24.0=ϕ ‡‚‡‚.
»ÌÚÂðÔÓÎ ˇˆ˜ËÎ ˙ˇ:
26.03.410
24.027.027.02 =⋅
−−=ϕ′
“Ó„Ú‚ÓðÊÎ˚„ ¯‡Î„‡ ˇ.
R A
N ⋅ϕ≤=σ 2.431.42;1016027.095
0086.0
10400 63
≤⋅⋅≤⋅
Áºðøø„ ÓÎ ˙∏.−
⋅ = <42.1 43.2
100 2.6 542.1
%
’ø˜‰ ̋ ÎËÈÌ Áºðøø 5 ı Û‚ËÈÌ ‰ÓÚÓ𠷇Ȅ‡‡ Û˜Ëð ÚÓ„Ú‚ÓðÊÎ˚Ì Ìº ı ̂ ºÎ ·ËÂÎÊ·‡È̇. 95= A ÒÏ2 ڇη‡ÈÚ‡È Ï‡ÚÂðˇΠÒÓÌ„Ó‚.
2. ƒ‡„ÛÛ- ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ˚Ì ÚÓÓˆÓÓ ı ËÈÂ.
9.2 ñð ÁÛð‡„› ı ν ̋‰ Áº‚ ı ºÌ ı ºÌ‰ÎºÌ ‡˜‡‡ÎÎ˚Ì (9.2ñð ÁÛð‡„) øÈΘΠ̋Ή ·‡È„‡‡‡ ‰‡ÏÌÛÛð˚Ì – ˆ ̋ „ ‰ ̋ ı ¯ËÎÊËÎÚËÈ„ ( C v ) ÓÎÌÓ. ØøÌËÈ ÚÛΉ
‡. —‡͈ ¿, ¬ -ËÈ„ ÓÎ ˙∏.
∑ =⋅−⋅−⋅= 033
20
l F l F l R M A B
105.2
3/5.2105.23/2103/3/2=
⋅+⋅⋅=
⋅+⋅⋅= l
l F l F R A ÍÕ
M 0 , ÍÕÏ
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî182
10033
20 ==⋅+
⋅⋅+⋅−=∑ B B A R
l F
l F l R M ÍÕ ·ÓÎÌÓ.
· ) C v Ú‡ ı ËÈÎÚ˚„ ‡Ì ı Ì˚ Ô‡ð‡ÏÂÚðËÈÌ ‡ð„‡‡ð ÓÎ ˙∏. ÿËÎÊËÎÚËÈÌ
Ú˝„¯ËÚ„ ̋ ÎËÈÌ ÂðºÌ ı ËÈ ı ̋η˝ðËÈ„ ·Ë˜¸Â.
6
3
2
6
3
6
33
3
00
⎟ ⎠
⎞⎜⎝
⎛ −
−⎟ ⎠
⎞⎜⎝
⎛ −
−+⋅θ+=l z
F
l z
F z
R z EI IEIv EIv A
“Û΄ÛÛð˚Ì Ìº ı ̂ ºÎ ‡¯Ë„Î‡Ê 00 , θv Ô‡ð‡ÏÂÚðËÈ„ ÓÎ ˙∏.
1ñð ̺ ı ̂ ºÎ
⎭⎬⎫
==
==
0
00
vl z
v z 2ñð ̺ ı ̂ ºÎ
1ñð ̺ ı ̂ ºÎËÈ„ Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛη‡Î 00 =v ·ÓÎÌÓ.
2ñð ̺ ı ̂ ºÎËÈ„ Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛÎÊ
=
⎟ ⎠
⎞⎜⎝
⎛
−
⎟ ⎠
⎞⎜⎝
⎛
−+⋅θ+=6
310
6
3
2
106
1000
33
3
0
l l
l l IE
162
6.1510
162
12510
6
6.15105.20 −−+⋅θ= IE
›Ì‰ ̋ ˝Ò:
( ) =⋅−−=⋅
⋅++−=θ IE IE
33
0 1093.65.2
1096.072.726
411
3
10814.31020238.0
1801093.6 −⋅−=⋅⋅⋅
⋅⋅−= ð‡‰ ·ÓÎÌÓ.
2384.528.028.0 44 =⋅== a I ÒÏ4=0.0238Ï4
11102 ⋅= E œ‡√ ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ (9.2· ñð ÁÛð)·‡È„ÛÛÎ ˙ˇ.
1ñ1 Ó„ÚÎÓÎ: ( )83.0030 11 ≤≤≤≤ z
l
z 11
z R M A z ⋅= 01 = z ø‰ 01 = z M
83.01 = z ø‰ 3.81 =′ z M ÍÕ⋅Ï
2ñ2 Ó„ÚÎÓÎ: ( )66.183.03
23 12 ≤≤≤≤ z
l z
l
⎟ ⎠
⎞⎜⎝
⎛ −−⋅=
3222
l z F z R M A z
83.02 = z ø‰ 3.82 =′ z M ÍÕ⋅Ï
66.12 = z ø‰ 3.82 =′′ z M ÍÕ⋅Ï
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
183
3ñ3 Ó„ÚÎÓÎ: ( )83.003
0 13 ≤≤≤≤ z l
z
33 z R M B z ⋅=
03 = z ø‰ 03 =′ z M ÍÕ⋅Ï
83.03 = z ø‰ 3.83 = z M ÍÕ⋅Ï
ŒÎÒÓÌ EI
v3
001093.6
;0 ⋅−
=θ= ı Óηӄ‰Î˚„ Ú˝„¯ËÚ„ ̋ Ή ÓðÎÛÛ·Ì,
25.1= z „ ̋ Ò˝Ì ÛÚ„‡‰ ¯‡ ı ‡„‰Ò‡Ì ¯Ëη˝ÌËÈ ı ºÌ‰ÎºÌ „ÛÎÁ‡ÈÎÚ ( C v ):
( )=
−−+⋅⎟
⎠
⎞⎜⎝
⎛ −⋅=
6
83.025.1
6
25.11025.1
93.60
33
EI EI EIvC
5.512.03.325.193.6 −=−+⋅−=
811
33
105.111020238.0
105.5105.5 −⋅−=⋅⋅
⋅−=⋅−= EI
vC Ï ·ÓÎÌÓ.
ðËÚËÍ ı ø˜ËÈ„ ÓÎ ˙∏. (9.8)
( )( )
( )7
2
44112
2min
2
108.175.21
104.528.010214.3⋅=
⋅
⋅⋅⋅⋅=
⋅⋅=
−
l
I E F cr
μ
π Õ
ÿËη˝ÌËÈ ·øÚ˝Ì „ÛÎÁ‡ÈÎÚ˚„ (9.10) ÓÎ ˙∏.
88
max 107.11
178000000
4000001
105.11
1
−−
⋅−=
−
⋅−=
−
==
k
C C
F
N
vvv Ï
ÿËη˝ÌËÈ ·º ı ·‡Ú˚„ ¯‡Î„‡ ˇ. (9.7)
=⋅
++=σ x
C
x W
v F
W
M
A
N 0
3.11610112
107.111010
10112
103.8
1095
104006
83
6
3
4
3
=⋅
⋅⋅⋅+
⋅
⋅+
⋅
⋅=
−
−
−−Ãœ‡<160Ãœ‡
·º ı ·‡Ú˚Ì Ìº ı ̂ ºÎ ·ËÂÎ ̋Ê ·‡È̇.
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî184
ƒ¿¿À√ ¿¬¿— π8. ƒ‡‡Î„‡‚‡ð ı Ó∏ð ·Ó‰ÎÓ„ÓÓÒ ÚÓ„ÚÓÌÓ.
1.
–Ú‡ÚËÍËÈÌ ÚÓ‰Óð ı
ÓÈ ·Ë¯ ·Ó‰ÎÓ„Ó
ª„º„‰ÒºÌ ÒÚ‡ÚËÍËÈÌ ÚÓ‰Óð ı ÓÈ ·Ë¯ ‰‡ÏÌÛÛð˚Ì (9.3ñð ÁÛð‡„) ı ºÌ‰ÎºÌ ı ø˜,„ÛÎÁ‡È΄‡ ı ÏÓÏÂÌÚ˚Ì ˝Ô˛ðËÈ„ 9.2ñð ı øÒÌ˝„ÚËÈÌ º„º„‰Îººð ·‡È„ÛÛÎ.
9.2ñð ı øÒÌ˝„Ú
π a
Ïq
ÍÕ/Ï
F ÍÕ
M ÍÕ⋅Ï π
a
Ïq
ÍÕ/Ï
F ÍÕ
M ÍÕ⋅Ï
1 2 20 30 18 11 2 40 32 362 1.2 28 28 20 12 1 38 50 22
3 1.4 26 32 24 13 1.2 36 42 24
4 1.6 24 40 30 14 1.4 34 40 26
5 1.8 30 42 32 15 1.6 32 46 28
6 1 32 36 24 16 1.8 42 34 30
7 1.2 34 48 20 17 2 46 30 32
8 1.4 28 42 26 18 1 48 50 20
9 1.6 20 36 32 19 1.2 32 28 24
10 1.8 30 28 34 20 1.4 30 26 30
9.3ñð ÁÛð‡„
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
185
9.3ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî186
9.3ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
2. ÿ‡ ı ‡„‰Ò‡Ì ¯Ëη˝ÌËÈ ÚÓ„Ú‚Óð
400= N ÍÕ ı ø˜ÌËÈ øÈΘΠ̋ν˝ð ¯‡ ı ‡„‰‡Ê ·‡È„‡‡ l ÛðÚÚ‡È ¯Ëη˝ÌËÈ(9.4ñð ÁÛð‡„) ÚÓÓˆÓÓÌ˚ ˝Ò˝ð„øøˆ ̋ Î 210= R Ãœ‡, Áº‚¯ººðº„‰º ı ı ø˜‰ ̋ Î
160=σ Ãœ‡
1. ‰‡„ÛÛ „ÛÎÁ‡ÈÎÚ˚Ì ϕ ÍÓ˝ÙÙˈËÂÌÚ˚„ ‡¯Ë„Î‡Ì ·‡„‡Ì‡Ì˚
ı ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì ı˝ÏÊ ̋ ˝„ ÒÓÌ„Ó.2. ı ºÌ‰ÎºÌ ‰‡„ÛÛ „ÛÎÁ‡ÈÎÚ˚Ì ÚÓÓˆÓÓ ı ËÈ.
ª„º„‰ÎËÈ„ 9.3ñð ı øÒÌ˝„Ú˝˝Ò ‡‚ .
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ÿ“ ≈—»¿À¤Õ ›–›—√ ØØ÷›À
187
9.4ñð ÁÛð‡„
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ì√ð‡ÙËÍ ÚÓÓˆÓÓÌ˚ ‡ÊËÎî188
9.4ñð ÁÛð‡„ (øð„ ̋ ÎÊÎ ̋Î)
9.3ñð ı øÒÌ˝„Ú
π F ,ÍÕ
l
Ïπ
F ,ÍÕ
l
Ï
1 160 3.5 11 186 3.0
2 216 3.0 12 150 3.23 176 3.4 13 180 3.4
4 124 3.6 14 224 4.0
5 110 4.0 15 282 3.4
6 210 3.2 16 144 2.8
7 172 2.8 17 110 2.6
8 140 3.6 18 220 2.4
9 100 4.0 19 280 3.4
10 200 3.5 20 140 3.2
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188 ·‚· ı χÚÂðˇÎ
’¿¬–—¿À“
Œ ̨ÛÚ‡Ì . . . . . . . . . “.ƒÓðÊ (ÍÓ‰ . . . . . )
”·‡Ì·‡‡Ú‡ð 2002
¡‡„¯ . . . . . . . . . . . .Ã.¡‡Ú (ÍÓ‰ . . . . . . )
5
5
ÏÓÏÂÌÚ˚„ ÚÓ‰Óð ı ÓÈÎÓ ı ÕËÈÎÏ ̋Î ‰øðÒËÈÌ ËÌÂðˆËÈÌ
ƒ‡‡Î„‡‚‡ð π . . .
¬‡ðˇÌÚ . . .
– ̋‰ ̋ ‚:
20
ÿËÌÊνı ” ı ‡‡Ì “ ı ÌÓÎÓ„ËÈÌ » ı –Ûð„ÛÛθÇÚÂχÚËÍËÈÌ –Ûð„ÛÛθ 5
“ ı ÌËÍËÈÌÏ ı ‡ÌËÍËÈÌ
҇η‡ð
’‡‚Ú‡ÒÌ˚ Á‡„‚‡ð
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À‡‚· ı χÚÂðˇΠ189
ÿ“ ≈—»¿À¤Õ ‘»«» ñÃ≈ ’¿Õ» »…Õ Ø«ØØÀ›À“Ø؃
1ñð ı øÒÌ˝„Ú
ÇÚÂðˇÎ
” ˇÌ ı ‡ðËÏ ı ‡ÈÌÏÓ‰Ûθ E ,
Ãœ‡
ÿËÎÊËÎÚËÈÌ ÏÓ‰Ûθ G,Ãœ‡
œÛ‡ÒÒÓÌ˚
ÍÓ˝ÙÙˈËÂÌÚν
ÿÛ„‡Ï‡Ì
Ú˝ ν ÎÚËÈÌÍÓ˝ÙÙˈËÂÌÚ
,106⋅α C 0/1
Õ ˇ„Ú ρ ,Í√ /Ï3
√ ‡Ì 5102 ⋅ 0.8∑105 0.30 12 7850
ÿËð˝Ï 5101 ⋅ 0.4∑105 0.25 10 7200
’ºÌ„ºÌ-ˆ‡„‡‡Ì
5107.0 ⋅ 0.3∑105 0.30 23 2700
«˝Ò 5101 ⋅ 0.4∑105 0.32 16 6500
ÃÓ‰(̇ðÒ)
5101.0 ⋅ - - - 550
‘»«» ’›Ã ∆ »√ ƒ›’ØØÕØ؃»…Õ Õ›√∆ »…Õ ’ŒÀ¡ŒŒ
2ñð ı øÒÌ˝„Ú
’ ˝ÏÊË„‰ ˝ı øøÌËÈÌ˝ð Õ˝„ÊËÈÌ Õ˝„ÊËÈÌ ı ‡Ï‡‡ð‡ÎÌ˝ð Ú˝Ï‰ ̋ „ν„ ̋ ˝
’ø˜, ‡˜‡‡, ÊËÌ Ì¸˛ÚÓÌ Õ 1Õ = 0.1Í„1Õ/Ï = 0.1Í„/Ï1ÍÕ/Ï = 0.1ÚÒ/Ï1œ‡ = 0.1Í„Ò/Ï2 1Íœ‡ = 0.1ÚÒ/Ï2 1Ãœ‡ = 10Í„Ò/ÒÏ2
1Õ⋅Ï = 0.1Í„Ò⋅Ï
1ÍÕ⋅Ï = 0.1ÚÒ⋅Ï
1 ∆ = 0.1Í„Ò⋅Ï
1¬Ú = 0.1 Í„Ò⋅Ï/Ò
“‡ð ı χΠ‡˜‡‡Î‡Î ̸˛ÚÓÌ/ÏÂÚð Õ/Ï
√ ‡‰‡ð„ÛÛ„‡‡ðÚ‡ð ı Ò‡Ì ‡˜‡‡Î‡Î,Ï ı ‡ÌËÍ ı ø˜‰ ̋ Î, Û ˇÌ ı ‡ðËÏ ı ‡ÈÌ ÏÓ‰Ûθ
Ô‡Ò͇θ(̸˛ÚÓÌ ⋅ ÏÂÚð2 )
œ‡
’ø˜ÌËÈ ÏÓÏÂÌÚ Ì¸˛ÚÓÌ ⋅ ÏÂÚð ÕÏ
¿ÊËÎ ( ̋ ÌÂð„Ë) ÊÓÛθ ∆
◊‡‰‡Î ‚‡ÚÚ ¬Ú
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À‡‚·ı χÚÂðˇÎ190
’ªÕƒÀªÕ Œ√ “ÀŒÀ¤Õ √≈ŒÃ≈ “— ’¿—¿ “ ≈—»–“ ≈ Ø؃»…ÕÕ›√∆ »…Õ ’ŒÀ¡ŒŒ
3ñð ı øÒÌ˝„Ú
π ’ ̋ ÏÊË„‰ ̋ ı øøÌ “ ˝Ï‰ ˝„ ν„ ̋ ˝ Õ˝„Êøø‰ËÈÌ ı ‡Ï‡‡ð‡Î
1 ’ºÌ‰ÎºÌ Ó„ÚÎÓÎ˚Ì Ú‡Î·‡È A 1ÒÏ2 4101 −
⋅= Ï2; 1Ï2 4101 ⋅= ÒÏ2
2 “‡Î·‡ÈÌ ÒÚ‡ÚËÍ ÏÓÏÂÌÚ , x S y S 1ÒÏ3 6101 −
⋅= Ï3; 1Ï3 6101 ⋅= ÒÏ3
3 ›Ò˝
ð„øøˆÎËÈÌ ÏÓÏÂÌÚ
,, y x W W
pW 1ÒÏ3 6
101 −
⋅=
Ï3
; 1Ï3 6
101⋅=
ÒÏ3
4 »ÌÂðˆËÈÌ ÏÓÏÂÌÚ p xy
y x
I D
I I
,
,, 1ÒÏ4 8101 −
⋅= Ï4; 1Ï4 8101 ⋅= ÒÏ4
5√ ‡Ì„ËÈÌ Û ˇÌ ı ‡ðËÏ ı ‡ÈÌÏÓ‰Ûθ E
6102 ⋅ Í„Ò/ÒÏ2 11102 ⋅= Õ/Ï2 = 8102 ⋅= ÍÕ/Ï2 5102 ⋅= Ãœ‡
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« ¿ — » Ã ’
¿ ¬ “ √ ¿ … ƒ Ø — – Ø Ø ƒ » …
Õ “ ¿ À ¡ ¿ … , ’ Ø Õ ƒ » … Õ “ ª ¬ » … Õ Œ Œ — ƒ » Õ ¿ “ ” ” ƒ
,
“ › Õ ’
À › √ Ø Ø ƒ “ › … ’ ¿ — ‹ ÷ ¿ Õ √ ”
… › – › — √ Ø Ø ÷ À » … Õ ¡ ¿ »
Õ ≈ — ÷ » … Õ Ã Œ à ≈ Õ “ ” ” ƒ
,
» Õ
≈ — ÷ » … Õ — ¿ ƒ » ” – ” ” ƒ
4 ñ ð
ı ø Ò Ì ˝ „ Ú
’ º Ì ‰ Î º Ì Ó „ Ú Î Ó Î ˚ Ì
ı ˝ Î · ˝ ð ø ø ‰
Œ „ Ú Î Ó Î ˚ Ì
Ú ‡ Î · ‡ È , Ò Ï 2
» Ì Â ð ˆ Ë
È Ì
Ï Ó Ï Â Ì Ú Û Û ‰ , Ò Ï 4
› Ò ˝ ð „ ø ø ˆ Î Ë
È Ì
Ï Ó Ï Â Ì Ú Û Û ‰ ,
Ò Ï 3
» Ì Â ð ˆ Ë È Ì ð ‡ ‰ Ë Û
Ò Û Û ‰ , Ò Ï
1
2
3
4
5
‚ ‡ ‰ ð ‡ Ú
b
b O
Y
X
2 b
1 2 4 b
I
I
y
x
=
=
6 3 b
W
W
y
x
=
=
b
b
i
i
y
x
2 8 9
. 0
1 2
=
=
=
’ º Ì ‰ Ë È Í ‚ ‡ ‰ ð ‡ Ú
2 1
2
b
b −
1 2
4 1
4
b
b I
I
y
x
−
=
=
b b
b
W
W
y
x
6
4 1
4
−
=
=
2 1
2
2 8 9
. 0
1 2
b
b
b
i
i
y
x
+
=
=
=
À‡‚· ı χÚÂðˇΠ191
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
»
ð Ï ˝ „ ˝ ˝ ð Ì ¸ · Ó Ò „ Ó Ò Ó Ì
Í ‚ ‡ ‰ ð ‡ Ú
2 b
1 2 4 b
I
I
y
x
=
=
3
3
1 1 8
. 0
2
6
b b
W
W
y
x =
=
=
=
b
i
i
y
x
2
8 9
. 0 =
=
“ ˝ „ ¯ º Ì ˆ º „ Ú
b h
1
2
1
2 3 3
h b
I
b h
I y x = =
6 6 2 2 h b
W
b h
W y x = =
b
b
i
h
h
i y x
2 8 9 . 0
1 2
2 8 9
. 0
1 2
=
=
=
=
192 À‡‚· ı χÚÂðˇÎ
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø
ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
’ º Ì ‰ Ë È Ú ˝ „ ¯ º Ì ˆ º „ Ú
1
1 h b
b h −
1
2
1
2
3 1 1
3
3 1
1
3
b h
h b
I
h b
b h
I y x
−
=
−
=
b
b
h
h b
W
h
h
b
b h
W y x
6 6
3 1
1
3
3 1
1
3 −
=
−
=
(
)
(
)
1
1 3 1
1
3
1
1 3 1
1
3
1 2
1 2
h b
b h
b h
h b
i
h b
b h
h b
b h
i y x
− −
=
− −
=
√ Û ð ‚ ‡ Î Ê Ë Ì
3 h
u =
b h 2 1
1 2 4
3 6 3 3 3
1
b h
I
b h
I
b h
I u u x =
= =
√ Û ð ‚ ‡ Î Ê Ì ̊ Ó
ð Ó È
‰ ˝ ˝ ð ı ø ˜ ‰ ˝ Î Ú Ó
Ó ˆ Ó Ê
· Û È ø Â ‰ : 2 4 2 b h
W x =
– Û Û ð Ë È Ì ˆ ˝ „
‰ ˝ ˝ ð
ı ø ˜ ‰ Î Ë È „ Ú Ó Ó ˆ Ó Ê · Û È
ø Â ‰ : 1 2 2 b h
W x =
h
h
i x
2 3 6
. 0 2 3
=
=
À‡‚· ı χÚÂðˇΠ193
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø
ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
¿ ‰
Ë Î ı ‡ Ê Û Û Ú „ Û ð ‚ ‡ Î Ê Ë Ì
b h 2 1
4
8 3
h b
I x =
2 4 2 h b
W x =
b
b
i x
2 0 4
.
0 2 3
6
=
=
“ ð ‡ Ô Â ˆ
1 1
0
2
3
b
b
b
b h
v
+ +
⋅
=
h b
b 2
1
+
(
)
1
2 1
1
2
3
3 6
4 b
b
b
b b
b h
I x
+
+
+
=
’ ø ˜ ‰ Î Ë È „ ‰ ˝ ˝ ‰
Ò Û Û ð Ë È Ì ˆ ˝ „ ‰ ˝ ˝ ð
Ú Ó Ó ˆ Ó Ê · Û È ø Â ‰ :
(
)
(
)
1
2 1
1
2
2
2
1 2
4
b
b
b
b b
b h
W x
+
+
+
=
’ ø ˜ ‰ Î Ë È „ ‰ Ó Ó ‰
Ò Û Û ð Ë È Ì ˆ ˝ „ ‰ ˝ ˝ ð
Ú Ó Ó ˆ Ó Ê · Û È ø Â ‰ :
(
)
(
)
1
2 1
1
2
2
2
1 2
4
b
b
b
b b
b h
W x
+
+
+
=
(
)
(
)
2 1
1
2
1 4
2 6
b
b b
b
b
b h
i x
+
+
×
×
+
=
194 À‡‚· ı χÚÂðˇÎ
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø
ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
“ ð ‡ Ô Â ˆ
h b
b 2
1
+
1 4 1
4
4 8
b
b
b
b h
I x
− −
⋅
=
1
2
4 1
4
2 4
b b
b
b
b
h
W x
− −
⋅
=
2 4
2 1
2 b b
i x
+
=
“ ˝ „ ¯ º Ì ˆ º „ Ú ø ø ‰ ˝ ˝ Ò
· ø ð ‰ Ò ˝ Ì Ú ˝ „ ¯ ı ˝ Ï Ú
‰ ‚ Û Ú ‡ ‚ ð
h b
B h
1
1
2
+
1 2
2
3
3
b h
B H
I x
−
=
;
1
2 2
3
1
3 1
B h
h b
I y
+
=
H
b h
B H
W x
6 2
3
3 −
=
;
B
B
h
h b
W y
6 2
3
1
3 1 +
=
A I
i
x
y =
;
A I
i
y
y =
À‡‚· ı χÚÂðˇΠ195
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø
ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
ƒ Û „ Û È
4 2 d π
;
4
6 4
4
4
r
d
I
I
y
x
⋅
=
=
=
π
π
4
0 5 .
0
d
I
I
y
x
≈
=
;
4
3 2
3
3
r
d
W
W
y
x
⋅
=
=
=
π
π
4
1 . 0
d
W
W
y
x
≈
=
2
4
r
d i
i
y
x
=
=
=
÷ ‡ „ Ë ð ‡ „
d d
c
1
=
(
)
4
2 1
2
d
d
−
π
(
=
−
=
=
6 4
2 1
2
d
d
I
I
y
x
π (
) ;
1
6 4
4
4
c
d
−
=
π
(
) ;
1
4
4
4
c
r
I
I
y
x
−
⋅
=
=
π
(
)
4
4 1
0 5 . 0
c
d
I
I
y
x
−
≈
=
(
) ;
1
3 2
4
3
c
d
W
W
y
x
−
=
=
π
(
) ;
1
1 . 0
4
3
c
d
W
W
y
x
−
=
=
2 1
2
4 1
d
d
i
i
y
x
+
=
=
196 À‡‚· ı χÚÂðˇÎ
7/21/2019 Hand Book.pdf
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4 ñ ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø ð „ ˝ Î Ê Î ˝ Î
1
2
3
4
5
“ ‡ Î ‰ Û „ Û È
r
r
v
4 2 4
. 0
3 4
0
≈
=
π
8 2 d ⋅
π
;
1 1 0
. 0
0 0 6 8 6
. 0
4
4
r
d
I x
≈
=
4
4
0 2 5
. 0
1 2 8
d
d
I y
≈
⋅
=
π
;
0 2 3 9
. 0
3
d
W x =
3
4
0 5 . 0
6 4
d
d
W y
≈
⋅
=
π
;
1 3 2
.
0
m i n
d
i
i x
≈
=
4 d i y =
« º ‚ Á Û ð „ ‡ ‡ Ì º Ì ˆ º „ Ú
2
5 9 8
. 2
a
A =
· Û ˛ Û
2
8 6 6
. 0
h
A =
4
0 6 . 0
h
I
I
y
x
=
=
· Û ˛
Û
4
5 4 1 . 0
a
I
I
y
x
=
=
3
3
6 2 5
. 0
1 2 . 0
a
h
W x
=
=
3
5 4 1
. 0
a
W y =
=
=
=
a
i
i
y
x
4 5 6 5
. 0
h
2 5 7 . 0 =
À‡‚· ı χÚÂðˇΠ197
7/21/2019 Hand Book.pdf
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À‡‚·ı χÚÂðˇÎ198
«¿—»Ã ’ fl À¡¿— ƒØ—–»…Õ “¿À¡¿… ’ØÕƒ»…Õ “ª¬
5ñð ı øÒÌ˝„Ú
ƒ‡ÏÌÛÛð˚̇˜‡‡ÎÎ˚Ì Ò ı ÂÏ √ ÛÎÁ‡È΄‡ ı à ÏÓÏÂÌÚ˚Ì ˝Ô˛ð ›Ô˛ðËÈÌڇη‡È, w ’ø̉ËÈÌ Úº‚ ı
øðÚνı
Á‡ÈC z C z l −
hl 2
l l
2
1
2
hl l
3
2 l
3
1
3
hl l
4
3 l
4
1
4
hl l
5
4 l
5
1
2
)( 21 l hh +
)( 3
)2(
21
21
hh
l hh
+
+ )( 3
)2(
21
21
hh
l hh
+
+
2
hl
3
l a +
3
l b +
hl 3
2 l
2
1 l
2
1
3
hl
16
5l
16
3l
)]( 3[6
2 l aal ql
++
)33( 4
386322
322
l al t a
l al l d
++
++
)33( 4
46322
322
l al l a
l al l a
++
++
7/21/2019 Hand Book.pdf
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À‡‚· ı χÚÂðˇΠ199
i M , k M ›œfi—Ø؃»…Õ Ø— ∆ ¬›—
6ñð ı øÒÌ˝„Ú
i M ˝Ô˛ð k M ˝Ô˛ð
321 h sh ⋅ )2(
6 341 hh
sh+ )2(
6 341 cc
sh−
621 h sh ⋅ )2(
6 431 hh
sh+ )2(
6 341 cc
sh−
)2( 6 562
hh
sh+
++ )2( [
6435 hhh
s
)]2( 346 hhh ++
+− )2( [
6345 cch
s
)]2( 346 cch −+
)2( 6 12
2 cc sh
− −+ )( [
6 432 hhc s
)]2( 431 hhc +−
−− )2( [6 342 ccc s
)]2( 341 ccc −−
122h f s ⋅⋅ )3(
12 43 hh f s
+ )3( 12 34 cc f s
−
42
h f s ⋅⋅
)3( 12 34 hh f s
+ )3( 12 34 cc f s
−
32h f s ⋅⋅ )(
3 43 hh f s
+ )( 3 34 cc f s
−
42h f s ⋅⋅ )35(
12 43 hh f s
+ )53( 12 34 cc f s
−
12
2h f s ⋅⋅ )35(
12 34 hh
f s+ )35(
12 34 cc
f s−
6
)'2( 2 λ +h sh )'4(
6 43 λ hehkh s
++ )'4( 6 34 kcehc s
+−λ
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À‡‚·ı χÚÂðˇÎ200
ÿ“ ≈—»¿À””ƒ¤Õ ƒ¿ √ ”” √ ”À«¿…À“¤Õ Œ›‘‘÷»≈Õ“¤Õ ”“ √ ¿
7ñð ı øÒÌ˝„Ú
λ
√ ‡Ì ÿËð˝Ï ÃÓ‰(ÚºðκºÒ
ı ‡Ï‡‡ð-„øÈ„ ̋ ˝ð)
¡ÂÚÓÌ–Ú10
–Ú2–Ú3–Ú4
–Ú5
ÕÀ-1
ÕÀ-1 –◊ 15-30
–◊ 15-15–◊ 15-36–◊ 41-40
–◊ 21-14
–◊ 28-48
’ø̉ ’ºÌ„ºÌ
0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
10 0.99 0.98 0.98 0.97 0.95 0.99 1.00 1.00
20 0.97 0.95 0.95 0.91 0.87 0.97 0.96 0.96
30 0.95 0.93 0.93 0.81 0.75 0.92 0.90 0.86
40 0.92 0.90 0.90 0.69 0.60 0.87 0.84 0.73
50 0.89 0.84 0.83 0.57 0.43 0.80 0.76 0.68
60 0.86 0.80 0.78 0.44 0.32 0.71 0.70 0.59
70 0.81 0.74 0.71 0.34 0.23 0.61 0.63 0.52
80 0.75 0.66 0.63 0.26 0.18 0.49 0.57 0.46
90 0.69 0.59 0.54 0.20 0.14 0.38 0.51 ñ
100 0.60 0.50 0.45 0.16 0.12 0.31 0.45 ñ
110 0.52 0.43 0.39 ñ ñ 0.26 ñ ñ
120 0.45 0.38 0.33 ñ ñ 0.22 ñ ñ
130 0.40 0.32 0.29 ñ ñ 0.18 ñ ñ
140 0.36 0.28 0.26 ñ ñ 0.15 ñ ñ
150 0.32 0.27 0.23 ñ ñ 0.14 ñ ñ
160 0.29 0.24 0.21 ñ ñ 0.12 ñ ñ
170 0.26 0.21 0.19 ñ ñ 0.11 ñ ñ
180 0.23 0.19 0.17 ñ ñ 0.10 ñ ñ
190 0.21 0.17 0.15 ñ ñ ñ ñ ñ
200 0.19 0.16 0.14 ñ ñ ñ ñ ñ
7/21/2019 Hand Book.pdf
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¹
’ ˝ Ï Ê ˝ ˝ , Ï Ï
Œ „ Ú Î Ó Î ˚ Ì
∆ Ë Ì
X −
V
V −
U
U −
1
1
X
X
−
0
y
b
d
R
r
Ú ‡ Î · ‡
È ,
2
Ò Ï
1 Ï , Í „
x I , Ò Ï 4
x
i
, Ò Ï
m a x
, v I
, Ò Ï 4
m a x
, v i
, Ò Ï
m i n
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I
, Ò Ï 4
m i n
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, Ò Ï
1 x I
, Ò Ï 4
Ò Ï
2
2 0
3
3 . 5
1 . 2
1 . 1 3
0 . 8 9
0 . 4 0
0
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4
1 . 4 6
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0
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b
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b
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r
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r
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Û Ú · Û Î ‡ Ì ( √ Œ – “ 8 5 0
9 - 7
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Ú
8 ñ ð ı ø Ò Ì ˝ „ Ú
À‡‚· ı χÚÂðˇΠ201
7/21/2019 Hand Book.pdf
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À‡‚· ı χÚÂðˇΠ202
8 - ð ı ø Ò Ì ˝ „ Ú Ë È Ì ø
ð „ ˝ Î Ê Î ˝ Î
¹
’ ˝ Ï Ê ˝ ˝ , Ï Ï
Œ „ Ú Î Ó Î ˚ Ì
∆ Ë Ì
X −
V
V −
U
U −
1
1
X
X
−
0
y
b
d
R
r
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2
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1 Ï , Í „
x I , Ò Ï 4
x
i
, Ò Ï
m a x
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, Ò Ï 4
m a x
, v i
, Ò Ï
m i n
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I
, Ò Ï 4
m i n
, u i
, Ò Ï
1 x I
, Ò Ï 4
Ò Ï
3
2 . 6 5
2 . 0 8
5 . 1 3
1
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8 . 1 3
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2 . 1 2
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À‡‚· ı χÚÂðˇΠ208
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’˝ÏÊ˝˝, ÏÏŒ„ÚÎÓÎ˚Ì ∆ËÌ
X X −
h b d t R r ڇη‡È, 2ÒÏ 1Ï, Í„
x I , 4ÒÏ x
W , 3ÒÏ xi , ÒÏ x
S , 3ÒÏ y I
10 100 55 4.5 7.2 7.0 2.5 12.0 9.46 198 39.7 4.06 23.0 112 120 64 4.8 7.3 7.5 3.0 14.7 11.50 350 58.4 4.88 33.7 2
14 140 73 4.9 7.5 8.0 3.0 17.4 13.70 572 81.7 5.73 46.8 4
16 160 81 5.0 7.8 8.5 3.5 20.2 15.90 873 109.0 6.57 62.3 58
18 180 90 5.1 8.1 9.0 3.5 23.4 18.40 1290 143.0 7.42 81.4 8
18a 180 100 5.1 8.3 9.0 3.5 25.4 19.90 1430 159.0 7.51 89.8 11
20 200 100 5.2 8.4 9.5 4.0 26.8 21.00 1840 184.0 8.28 104.0 11
20a 200 110 5.2 8.6 9.5 4.0 28.9 22.70 2030 203.0 8.37 114.0 15
22 220 110 5.4 8.7 10.0 4.0 30.6 24.00 2550 232.0 9.13 131.0 15
24 240 115 5.6 9.5 10.5 4.0 34.8 27.30 3460 289.0 9.97 163.0 19
24a 240 125 5.6 9.8 10.5 4.0 37.5 29.40 3800 317.0 10.10 178.0 26
27 270 125 6.0 9.8 11.0 4.5 40.2 31.50 5010 371.0 11.20 210.0 26
27a 270 135 6.0 10.2 11.0 4.5 43.2 33.90 5500 407.0 11.30 229.0 33
ƒ‚ÛÚ‡‚ð (√ Œ–“ 8239-72)
h
b
t
(b-d)/4
R rd
O
Y
X
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¹ ’˝ÏÊ˝˝, ÏÏ
Œ„ÚÎÓÎ˚Ì ∆ËÌ X X −
h b d t R r ڇη‡È,2
ÒÏ 1Ï, Í„ x I , 4ÒÏ xW , 3ÒÏ xi , ÒÏ xS , 3ÒÏ y I
30 300 135 6.5 10.2 12.0 5.0 46.5 36.50 7080 472.0 12.30 268.0 3
30a 300 145 6.5 10.7 12.0 5.0 49.9 39.20 7780 518.0 12.50 292.0 4
33 330 140 7.0 11.2 13.0 5.0 53.8 42.20 9840 597.0 13.50 339.0 4
36 360 145 7.5 12.3 14.0 6.0 61.9 48.60 13380 743.0 14.70 423.0 5
40 400 155 8.3 13.0 15.0 6.0 72.6 57.00 19062 953.0 16.20 545.0 6
45 450 160 9.0 14.2 16.0 7.0 84.7 66.50 27696 1231.0 18.10 708.0 8
50 500 170 10.0 15.2 17.0 7.0 100.0 78.50 39727 1589.0 19.90 919.0 10
55 550 180 11.0 16.5 18.0 7.0 118.0 92.60 55962 2035.0 21.80 1181.0 1360 600 190 12.0 17.8 20.0 8.0 138.0 108.0 76806 2560.0 23.60 1491.0 17
65 650 200 11.7 19.2 22.0 9.0 151.0 119.0 100840 3100.0 25.80 1790.0 2
70 700 210 12.7 20.8 24.0 10.0 174.0 137.0 133890 3830.0 27.70 2220.0 27
70a 700 210 15.0 24.0 24.0 10.0 202.0 158.0 152700 4360.0 27.50 2550.0 32
70· 700 210 17.5 28.2 24.0 10.0 234.0 184.0 175370 5010.0 27.40 2940.0 39
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¹
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ÿ‚  ΠΠ ð ( √ Œ – “ 8 2 4 0 - 7 2 )
b
h
R
d
O t
r (
b - d ) / 2
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0
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2 1 1 À ‡ ‚ Î ‡ ı Ï ‡ Ú Â ð Ë ‡ Î
7/21/2019 Hand Book.pdf
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¹ ’˝ÏÊ˝˝, ÏÏ
Œ„ÚÎÓÎ˚Ì ∆ËÌ X X − Y
h b d t R r ڇη‡È,2
ÒÏ 1Ï, Í„ x I , 4ÒÏ xW , 3ÒÏ xi , ÒÏ xS , 3ÒÏ y I , 4ÒÏ W
20a 200 80 5.2 9.7 9.5 4.0 25.20 19.80 1670 167.0 8.15 95.90 139.00 2
22 220 82 5.4 9.5 10.0 4.0 26.70 21.00 2110 192.0 8.89 110.0 151.00 2
22a 220 87 5.4 10.2 10.0 4.0 28.80 22.60 2330 212.0 8.99 121.00 187.00 3
24 240 90 5.6 10.0 10.5 4.0 30.60 24.00 2900 242.0 9.73 139.00 208.00 3
24a 240 95 5.6 10.7 10.5 4.0 32.90 25.80 3180 265.0 9.84 151.00 254.00 3
27 270 95 6.0 10.5 11.0 4.5 35.20 27.70 4160 308.0 10.90 178.00 262.00 3
30 300 100 6.5 11.0 12.0 5.0 40.50 31.80 5810 387.0 12.00 224.0 327.00 4
33 330 105 7.0 11.7 13.0 5.0 46.50 36.50 7980 484.0 13.10 281.00 410.00 5
36 360 110 7.5 12.6 14.0 6.0 53.40 41.90 10820 601.0 14.20 350.00 513.00 6
40 400 115 8.0 13.5 15.0 6.0 61.50 48.30 15220 761.0 15.70 444.00 642.00 7
7/21/2019 Hand Book.pdf
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√ ‡ð˜Ë„
ªÏ̺ ı ø„ 3
Õ›√ ƒØ √ ››— ¡ØÀ›√ . –ÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ 5
1.1. –Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ¯Ëη˝ÌËÈ ÒÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚ ÖÖÖÖÖ.ÖÖ.Ö. 5
1.2. –Ú‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ¯Ëη˝ÌËÈ ÒÛÌ„‡ÎÚ, ¯‡ ı ‡ÎÚÖÖÖÖÖÖÖÖÖ
13
’Œ®—ƒ” √ ¿¿— ¡ØÀ›√ . ÷ ̋ „ËÈÌ ı ø˜‰ ̋ ÎÚ ÚºÎº‚
2.1. ÷ ̋ „ËÈÌ ı ø˜‰ ̋ ÎÚ ÚºÎº‚ ÖÖÖÖÖÖÖÖÖÖÖÖÖ.ÖÖÖÖ.Ö. 18
ƒ‡‡Î„‡‚‡ð π1 ÖÖÖÖÖÖÖÖÖÖÖ.ÖÖÖÖ.Ö.Ö.ÖÖÖÖ.Ö. 24
√ ”—¿¬ƒ” √ ¿¿— ¡ØÀ›√ . Œ„ÚÎÓÎÚ ı Ó΄ÓÎÚ ̊ Ì ÚÓÓˆÓÓ
3.1. ¡ÓÓÎÚÓÌ ı ÓηÓÓÒ˚Ì ÚÓÓˆÓÓ ÖÖÖÖÖÖÖÖÖÖÖ.ÖÖÖÖ.Ö. 32
3.2. ÿËð˝ ̋ÎÚ˝Ì ı ÓηÓÓÒ˚Ì ÚÓÓˆÓÓÖÖÖÖÖÖÖÖÖÖ.ÖÖÖÖ.Ö. 35
ƒ‡‡Î„‡‚‡ð π2 ÖÖÖÖÖÖÖÖÖÖ...ÖÖÖÖÖÖÖÖÖÖÖÖÖ... 37
ƒª—ª¬ƒØ √ ››— ¡ØÀ›√ . ’‡‚Ú„‡È ‰øðÒËÈÌ „ÂÓÏÂÚð ¯ËÌÊ
4.1. ’‡‚Ú„‡È ‰øðÒËÈÌ „ÂÓÏÂÚð ¯ËÌÊÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö 43
ƒ‡‡Î„‡‚‡ð π3 ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 52
“¿¬ƒ” √ ¿¿— ¡ØÀ›√ . ƒÛ„ÛÈ Ó„ÚÎÓÎÚÓÈ ¯Ëη˝ÌËÈ ÏÛ¯„Ëð‡ÎÚ
5.1. ÃÛ¯„Ëð‡ÎÚ˚Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ó‰ÎÓ„Ó..Ö..ÖÖÖÖÖ..Ö..ÖÖ
58
5.2. ÃÛ¯„Ëð‡ÎÚ˚Ì ÒÚ‡ÚËÍ ÚÓ‰Óð ı ÓÈ ·Ë¯ ·Ó‰ÎÓ„ÓÖÖÖÖÖ..Ö..ÖÖ 66
5.3. ¡‡„‡ ‡Î ı
‡ÏÚ‡È ˆËÎË̉ð Ôøð¯ÌËÈ ÚÓÓˆÓÓ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 73
ƒ‡‡Î„‡‚‡ð π4 ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 78
«”—√ ¿¿ƒ” √ ¿¿— ¡ØÀ›√ . ƒ‡ÏÌÛÛð˚Ì „ÛÎÁ‡ÈÎÚ
6.1. ƒ‡ÏÌÛÛð˚Ì ‰ÓÚÓÓ‰ ı ø˜ÌËÈ˝Ô˛ðÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ
86
ƒ‡‡Î„‡‚‡ð π5ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 96
6.2. ƒ‡ÏÌÛÛð˚Ì ·‡Ú ·˝ı ËÈÌ ·øð˝Ì 115
7/21/2019 Hand Book.pdf
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.
ÚÓÓˆÓÓÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ
6.3. ƒ‡ÏÌÛÛð˚Ì ı º¯øøÌ ˜‡Ì‡ð˚Ì ÚÓÓˆÓÓ Ö..Ö..ÖÖÖÖÖ..Ö..ÖÖ 125
ƒ‡‡Î„‡‚‡ð π6ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 134
ƒŒÀŒŒƒ” √ ¿¿— ¡ØÀ›√ . ÕËÈÎÏ ̋Î ˝Ò˝ð„øøˆ ̋ Î
7.1. ∆ ˯øø „ÛÎÁ‡ÈÎÚ ÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 135
7.2. “º‚ËÈÌ ·Ë¯, ¯‡ ı ‡ÎÚ ÒÛÌ„‡ÎÚÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ
136
7.3. ÃÛ¯„Ëð‡ÎÚڇȄÛÎÁ‡ÈÎÚÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖÖ
137
ƒ‡‡Î„‡‚‡ð π7ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 156
Õ¿…Ô √ ¿¿— ¡ØÀ›√ . √ Ûð‚‡Ì ÏÓÏÂÌÚ ̊ Ì Ú ̋ „¯ËÚ„ ̋ Î
8.1.√ Ûð‚‡Ì ÏÓÏÂÌÚ˚Ì Ú˝„¯ËÚ„ ̋ Î 164
≈–ƒØ √ ››— ¡ØÀ›√ . ÿ‡ ı ‡„‰Ò‡Ì ¯Ëη˝ÌËÈ ÚÓ„Ú‚ÓðÊËÎ
9.1. ÿ‡ ı ‡„‰Ò‡Ì ¯Ëη˝ÌËÈ ÚÓ„Ú‚ÓðÊËÎ 174
ƒ‡‡Î„‡‚‡ð π8ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖ 182
’‡‚Òð‡ÎÚ ÖÖÖÖÖÖÖÖ..ÖÖÖÖÖ..Ö..ÖÖÖÖÖ..Ö..ÖÖÖÖ 188
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¿ÿ»√ À¿–¿Õ ÿ“ ≈—»¿À
1. ¿ÎÂÍ҇̉ðÓ‚.¿.¬, ¬.ƒ.œÓÚ‡ÔÓ‚, ƒÂðʇ‚ËÌ.¡.œ
–ÓÔðÓÚË‚ÎÂÌË χÚÂðˇÎÓ‚ÃÓÒÍ‚‡ , ì¬˚Ò¯‡ ˇ ¯ÍÓ·î 1995
2. ÿ.¡‡ÎÊËÌÌ ˇÏ, ¿.√ ºÎ„ºº, —.Շȉ‡Ì‰ÓðÊ, Õ.“ºÏºð·‡‡Ú‡ð, ∆ .÷ ̋ ̉Òøð˝Ì
ÇÚÂðˇÎ˚Ì ˝Ò˝ð„øøˆ ̋ Î ”·‡Ì·‡‡Ú‡ð 1996
3 ÿ ¡ ¿ √ — Õ È Õ “