** ⦽᧲ݡ⦺Ʊ Õᖅ⪹ĞŖ⦺ŝ ᕾᔍŝᱶ ([email protected])
**** ⦽᧲ݡ⦺Ʊ Õᖅ⪹ĞŖ⦺ŝ ၶᔍŝᱶ ([email protected]) Received March 11 2013, Revised April 8 2013, Accepted April 23 2013
Copyright ⵑ 2013 by the Korean Society of Civil Engineers
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0)
ǣǣȀȀǤǤȀͳͲǤͳʹͷʹȀ ǤʹͲͳ͵Ǥ͵͵Ǥ͵Ǥͺͷ Ǥ ǤǤ
ᢦ′ᇎ❊Ἲ#ኞᷢ㬚#㣊ⴲ㫂ⴖ#⹊ጲ⮎#ኾ㬚#⮮ጪ
ౖܛ ȵճઽఛ ȵ֫০ ȵଲஂট
Choi, Dong-Ho*, Ko, Young-Chan**, Gwon, Sun-Gil***, Lee, Joung-Sun****
Buckling Analysis of Pipelines with Reduced Cross Section
ABSTRACT
This paper proposes a theoretical solution of elastic critical buckling load of infinitely long pipelines with non-uniform thickness under external pressure. The non-uniform cross section of pipelines can be considered as corroded or stiffened pipelines so that this paper can be a fundamental research of pipelines that are essential technology for offshore industries. The theoretical solution of pipelines with non-uniform thickness is derived with an assumption that a cylindrical shell under external pressure can be considered as a simple ring.
The eigenfunctions are derived to obtain the critical buckling load. The reduced thickness and the reduced range are considered as variables in parametric analysis. The finite element analysis is performed to verify the theoretical solutions and the results of the analytic method and the finite element method are in good agreement.
Key words : Pipelines, Elastic buckling load, Critical buckling load
Ⅹಾ
ᅙᩑǍᨱᕽ۵ᇡᯕӹᅕvᰍaŁಅࡽእɁᯝ⦽ࢱ̹ෝaḡ۵❭ᯕ⥥ᯙᨱݡ⦹ᩍᯝᱶ⦽ᦶᮥၼᮥভ᮹┥ᖒ᳭Ǖ⦹ᵲᮥᯕುᱢᮝಽ
ᔑᱶ⦹ᩡ݅. ʙᯕaๅᬑʕᬱ☖⩶ᚹǍ᳑ྜྷᯙ❭ᯕ⥥ᯙᮥ݉ᙽ⦽ยǍ᳑ྜྷಽaᱶ⦹ᩡŁ, Łᮁ⧉ᙹෝᮁࠥ⦹ᩍ᳭Ǖᯥĥ⦹ᵲᮥᔑᱶ⦹ᩡ
݅. ੱ⦽, ࢱ̹ᄡ⪵᪡ࢱ̹aqᗭࡽǍe᮹ჵ᭥ᨱෙᄡᙹ⧕ᕾᮥᙹ⧪⦹ᩡ݅. ᯕುᨱ᮹⧕ᔑᱶࡽ᳭Ǖᯥĥ⦹ᵲđŝෝᮁ⦽ᗭ⧕ᕾđ ŝ᪡እƱ⦹ᩍá᷾⦹ᩡŁ, ࢱđŝ۵ᯝ⊹⧉ᮥᙹᯩᨩ݅.
áᔪᨕ ❭ᯕ⥥ᯙ, ┥ᖒ᳭Ǖ, ᳭Ǖᯥĥ⦹ᵲ
1. ᕽು
ݡℕ ᨱթḡ }ၽᯕ ⪽ၽ⧉ᨱࠥ ᇩǍ⦹Ł ᕾᮁᔑᨦ᮹ ᙹ۵ ᩍᱥ⯩ ׳݅. ᕾᮁ ၰ ᩑ aᜅ ŲǍ ┱ᔪ ၰ }ၽᯕ ᷾a ⇵ᖙᨱ
ᯩŁᦿᮝಽࠥᯕ్⦽⇵ᖙ۵ḡᗮࢁäᮝಽᅕᯙ݅. ⧕ᱡ❭ᯕ⥥ᯙᮡᘂᮁš᮹Ğᬑᨱอ(2012֥⩥ᰍ) ᱥᖙĥ᧞175,000kma
ᖅ⊹ࡹᨕᯩŁ⧕ᱡ↽ݡ᧞3,000 mʭḡᖅ⊹ࡹᨕᯩ݅. ᯕ్⦽ᝍ⧕ᱡ⪹Ğᨱᕽ᮹ᯱᬱ᮹ᔾᔑၰᬕᘂ᮹ᩎ⧁ᮥᬱ⪽⯩ᙹ⧪⦹ʑ
᭥⧕ ❭ᯕ⥥ᯙ᮹ ݉໕ᮡ ᱢᱩ⦽ vᖒŝ ݉໕ᱢᮥ wŁ ᬕᩢ ᮹ ᦶಆ ၰ ❭௲ᨱ ᮹⦽ ⦝ಽ, ᪉ࠥ, ḡḥ ॒ᨱ ᮹⦽ ⦹ᵲᮥ čॽ
ᙹ ᯩᨕ ⦽݅. ✚⯩ ⧕ᙹᨱ ᮹⦽ ᇡ ၰ ׳ᮡ ᦶಆᨱ י⇽ࡹᨕ ᯩᮝ໕ᕽ ᝍ⧕ᱡ۵ ⪹Ğᱢ ✚ᖒ ভྙᨱ ᅕᙹ ੱ⦽ ᨕಖ݅.
❭ᯕ⥥ᯙŝ šಉ⦹ᩍ, ᬱ☖ ᚹ Ǎ᳑ྜྷ᮹ ᳭Ǖ Ñ࠺ᨱ ݡ⧕ᕽ۵ ฯᮡ ᩑǍa ḥ⧪ࡹᨩ݅. Kyriakides (1986)۵ ᯝᱶ⦽ ᦶಆᨱ
ĵܓėॡ
Fig. 1. Ring under uniform external pressure ݡ⦹ᩍᬱ☖⩶ᚹᨱݡ⦽᳭Ǖ⪶ᔑᮥᝅ⨹ᮥ☖⦹ᩍȽ⦹ᩡŁ,
əᨱ ݡ⦽ ᳭Ǖ ⦹ᵲᮥá☁⦹ᩡᮝ໑, Jensen (1988)ᮡ ᬱ☖⩶
ᚹᨱݡ⦽᳭Ǖᦶಆᮥᩩ⊂⧁ᙹᯩ۵ᯕುᱢᯙႊჶᮥᱽ⦹ᩡ݅.
Tian et al. (1999)ᮡʙᯕႊ⨆ᮝಽᯥ᮹᮹᭥⊹ᨱย⩶┽᮹ᅕvᰍ aᖅ⊹ࡽᬱ☖⩶ᚹǍ᳑ෝaḡŁ݅᧲⦽᳦ඹ᮹ᦶಆᨱݡ⦹ᩍ
┥ᖒ ᳭Ǖ⧕ᕾᮥ ᙹ⧪⦹ᩡᮝ໑, Lopatin᪡ Morozov (2012)۵
ᯝᱶ⦽ᦶಆᮥၼ۵ᅖ⧊⋵❙౩ქᬱ☖⩶ᚹᨱݡ⦽᳭Ǖ⧕ᕾᮥ
ᯕುᱢᮝಽ ᙹ⧪⦹ᩡ݅.
ੱ⦽ ᯝᱶ⦽ ᦶᮥ ၼ۵ ྕ⦽⯩ ʕ ᬱ☖ ᚹ Ǎ᳑ྜྷ᮹ ᳭Ǖ
Ñ࠺ᨱ ݡ⧕ᕽࠥ ฯᮡ ᩑǍa ḥ⧪ࡹᨩ݅. ᯕ్⦽ ᳭Ǖ ᩑǍ۵
ʙᯕaๅᬑʕᬱ☖⩶ᝅฑӹ❭ᯕ⥥ᯙ᮹❭ƕෝᩩ⊂⦹۵
ႊჶᮥᱽ⦹ᩡ݅. Timoshenko᪡Gere (1961)۵ᯝᱶ⦽ᦶᮥ
ၼ۵ࢱ̹aᯝᱶ⦹Łʙᯕaๅᬑʕᬱ☖⩶ᚹǍ᳑ྜྷ᮹᳭Ǖ⦹ᵲ
ᮥᔑᱶ⦹۵ᯕುᮥᱽ⦹ᩡ݅. ʙᯕaྕ⦽⦹ʑভྙᨱᝅฑ
a⠪໕ᄡ⩶ᔢ┽᪡zᮝအಽᬱ⪹Ǎ᳑ಽaᱶ⦹ᩍ᳭Ǖ⦹ᵲᮥ
ᔑᱶ⦹ᩡ݅. ᯕ్⦽aᱶᮡ❭ᯕ⥥ʙᯕ᪡ḢĞ᮹እL/Da25ᯕᔢ ᯝভa܆⦹໑L/Da25ၙอᯝভ۵ḡḡᱱĞĥ᳑Õᨱ᮹⧕
vࠥᅕv⬉ŝaᔾʑŁ᳭ǕÑ࠺ᨱᩢ⨆ᮥၙ⊹íࡽ݅.
ᕽᯕভ۵ᮁ⦽ʙᯕ᮹ᝅฑಽeᵝ⧕⦽݅. Xue (2012)۵
ᯝᱶ⦽ᦶᮥၼ۵ʙᯕaๅᬑʕᬱ☖⩶ᚹᨱݡ⧕Donnell shell ᯕುᮥᱢᬊ⦹ᩍ᮲ಆ⧉ᙹᮥᮁࠥ⦹ᩡ݅. ᯕḡ႑ႊᱶᨱ
Ritz methodෝᯕᬊ⦹ᩍ⩶ᔢᄡ⪵ᨱෙ⦹ᵲᄡ⪵ෝᩩ⊂⦹ᩡ݅.
ə్ӹᝅᱽᔍᬊ⪹Ğᨱᖁ❭ᯕ⥥ᯙ᮹ʙᯕႊ⨆ᮝಽʙí
ᅕvᰍaᖅ⊹ࡹᨕᯩÑӹᯝᇡǍeᨱᕽᇡᯕၽᔾ⦹ᩍࢱ̹ෝ
᪉ᱥ⯩Łಅ⦹ḡ༜⧁ᙹᯩ݅. Hoo Fatt (1999)۵ᇡࡽ❭ᯕ⥥
ᯙ᮹ǎᇡ᳭Ǖŝ┥ᗭᖒÑ࠺ᨱݡ⧕ᩑǍ⦹ᩡ݅. ੱ⦽, Xue᪡
Hoo Fatt (2002)۵ ࢱ̹a ᯝᱶ⦹ḡ ᦫᮡ ʕ ᬱ☖⩶ ᚹᮥ ⦽
Ǎeᨱᕽࢱ̹aqᗭࡽᬱ⪹Ǎ᳑ಽaᱶ⦹ᩍ᳭Ǖ⦹ᵲᨱݡ⦽
Łᮁ⧉ᙹෝᮁࠥ⦹ᩡŁ, Newton-Raphson methodෝᯕᬊ⦹ᩍ
Łᮁ⧉ᙹෝอ᳒⦹۵᳭Ǖ⦹ᵲᮥᔑᱶ⦹ᩡ݅. ੱ⦽, Xue᪡Hoo Fatt (2005)۵ ᇡࡽ ❭ᯕ⥥ᯙᨱ ݡ⦽ ݡ⋎ ၰ ᩎݡ⋎ ᳭Ǖ
⪶ᔑ༉ऽᨱݡ⦽ᯕುᱢ⧕ᕾᮥvᗭᖒ⧕ᕾᮥ☖⦹ᩍᙹ⧪⦹ᩍ
ᮁ⦽ᗭ⧕ᕾđŝ᪡እƱá☁⦹ᩡ݅. ə్ӹᝅᱽ⪹Ğᨱᕽᇡ
ၽᔾᯕӹᅕvᰍᖅ⊹۵❭ᯕ⥥᮹ᩍ్ᇡᇥᨱᅖ⧊ᱢᮝಽ᯲ᬊ⧁
ᙹᯩʑভྙᨱᩍ్ᇡᇥᨱᕽࢱ̹aqᗭࡽ⩶┽᮹❭ᯕ⥥ᯙ᮹
᳭Ǖ⦹ᵲᮡ Łಅ⦹ḡ ༜⦹۵ ⦽ĥa ᯩ݅.
ᅙᩑǍ۵ᯝᇡǍeᨱᕽࢱ̹aqᗭࡽ⩶┽᮹ྕ⦽⯩ʕ❭ᯕ⥥
ᯙ᮹┥ᖒ᳭Ǖ⦹ᵲᔑᱶᮥ༊ᱢᮝಽ⦹Łᯩ݅. əႊჶᮝಽ
❭ᯕ⥥ᯙ᮹ʙᯕaๅᬑʙᨕʙᯕႊ⨆ᮝಽ⠪໕ᄡ⩶ශᔢ┽ᨱ
ᯩ݅Łaᱶ⦹Łᬱ☖ᚹǍ᳑ྜྷᮥ2₉ᬱᬱ⪹Ǎ᳑ಽ⊹⪹⦹ᩍ
ᮥᮁࠥ⧁ᙹᯩ݅. ຝᱡᯝᇡǍeᨱᕽࢱ̹aqᗭࡽ⩶┽᮹
ᬱ⪹ Ǎ᳑᮹ bࠥ ᄥಽ ݡ⋎ (Symmetric) ᳭Ǖ༉ऽ ၰ ᩎݡ⋎
(Anti-symmetric) ᳭Ǖ༉ऽ⩶┽ᨱݡ⦽ᄡ᭥⧉ᙹෝᮁࠥ⦹Ł
Ğĥ᳑Õᮥᯕᬊ⦹ᩍၙᇥႊᱶ᮹⧕ෝǍ⦽݅. əญŁၙᇥႊᱶ
᮹ĥᙹe᮹šĥෝ☖⧕Łᮁ⧉ᙹෝǍ⦹ŁNewton-Raphson methodෝ☖⧕Łᮁ⧉ᙹ᮹⧕ෝǍ⦹ᩍ᳭Ǖ⦹ᵲᮥᔑᱶ⦹íࡽ݅.
ษḡสᮝಽࢱ ᳭Ǖ༉ऽᨱ ݡ⧕ࢱ̹a qᗭࡽ Ǎe᮹bࠥ ၰ
ࢱ̹qᗭపᨱݡ⦽ᄡᙹ⧕ᕾᮥᙹ⧪⦹ᩡŁ, ABAQUSෝᯕᬊ⦽
FEM ⧕ᕾᮥ ☖⧕ Ǎ⦽ ᳭Ǖ⦹ᵲŝ ᮁࠥࡽ ᨱ ᮹⧕ ĥᔑࡽ
᳭Ǖ⦹ᵲᮥ እƱ⧕ ᯕುᮥ á᷾⦹ᩡ݅.
2. ❭ᯕ⥥Ǎ᳑ྜྷ᮹᳭Ǖ⦹ᵲᔑᱶʑᅙᯕು
ʙᯕaๅᬑʕ❭ᯕ⥥Ǎ᳑ྜྷᮡe݉⦽ยǍ᳑ྜྷಽeᵝ⦹ᩍ
᳭Ǖ⦹ᵲᮥᔑᱶ⧁ᙹᯩ݅. ᯕᨱݡ⦽ḡ႑ႊᱶᮁࠥෝ᭥⦽
ᯝၹᱢᯙaᱶᔍ⧎ᮡ݅ᮭŝz݅(Timoshenko᪡Woinowsky -Krieger, 1959; Kyriakides᪡ Corona, 2007).
a) ❭ᯕ⥥ ʙᯕ᪡ ḢĞ᮹ እ, L/D۵ 25ෝ չ۵݅. ᯕౕ Ğᬑ, ḡḡᱱ᮹ ᩢ⨆ᮥ ྕ⧁ ᙹ ᯩŁ ʙᯕ ႊ⨆ᨱ ݡ⦽ ᄡ᭥a
0ᯙ ⠪໕ᄡ⩶ ᔢ┽ಽ aᱶ⦹ᩍ 3₉ᬱ ❭ᯕ⥥ Ǎ᳑ෝ 2₉ᬱ
ยᮝಽ eᵝ⧁ ᙹ ᯩ݅.
b) ⠪Ɂ ၹḡᨱ እ⧕ ࢱ̹ t۵ ๅᬑ ᯲݅.
c) ย Ǎ᳑ྜྷ᮹ ᄡ⩶ ⩶ᔢᮡ Ⅹʑ łශ໕ ᭥ᨱ ᯩ݅.
d) ย Ǎ᳑ྜྷ᮹ ⅾ ʙᯕ ᄡ⪵۵ ྕ⧁ ᙹ ᯩ݅.
ʙᯕaๅᬑʙŁᯝᱶ⦽ᦶᮥၼ۵❭ᯕ⥥۵Fig. 1ŝzᯕ
2₉ᬱยᮝಽӹ┡ԝᙹᯩ݅. Fig. 1ᨱᕽ«ᮡย᮹ၹḡᯕŁ, ƒ۵ย᮹ࢱ̹, ©۵ย᮹ᵲᝍᮥ⨆⧕ᯝᱶ⦹í᯲ᬊ⦹۵ᇡ
ᦶಆᯕ໑, ľ۵ᬱᵝႊ⨆ᨱᕽ᮹ᯥ᮹᮹᭥⊹ෝӹ┡ԕ۵bࠥ,ƕ۵
ḡ ႊ⨆ᮝಽ᮹ ᄡ᭥ෝ ӹ┡ԙ݅.
ḡႊ⨆ᄡ᭥ƕᨱݡ⦽ၙᇥႊᱶŝ┥ᖒ᳭Ǖ⦹ᵲ©ƃ۵
bbEq. (1) ၰEq. (2)᪡z݅(Timoshenko᪡Gere, 1961).
Fig. 2. Ring with one thickness-reduced region ćƂľÏ
ƂÏƕ âƕ áà ćƒÐ
ÎÏÞÎ à ŃÏß©«Ðƕ (1)
©ƃá ćÑÞÎ à Ń ÏßÞć«ƒ ßÐ (2)
ᩍʑᕽ۵ ┥ᖒĥᙹᯕŁŃ۵ ⡍ᦥᘂ እᯕ݅.
3. ࢱ̹aᯝᱶ⦹ḡᦫᮡยǍ᳑ྜྷ᮹᳭Ǖ⦹ᵲᔑᱶ
❭ᯕ⥥ᯙ᮹ᇡᮥᰍഭᱢ✚ᖒ᮹ᄡ⪵aᦥܭʑ⦹⦺ᱢ⩶ᔢ ᮹ᄡ⪵ಽᕽeᵝ⦹ᩡŁ, Eq. (1)᮹ḡ႑ႊᱶᮥᯕᬊ⧕ࢱ̹a
ᯝᱶ⦹ḡᦫᮡยǍ᳑ྜྷ᮹ḡ႑ႊᱶᮥᮁࠥ⦹ᩡ݅. ⦽Ǎeᨱᕽ
ࢱ̹aqᗭࡽ⩶┽᪡ࢱǍeᨱᕽࢱ̹aqᗭࡽ⩶┽᮹❭ᯕ⥥ᨱ
ݡ⧕ ᯕುᮥ ᮁࠥ⦹ᩡŁ, b ⩶┽ᨱ ݡ⧕ᖁ ݡ⋎ŝ ᩎݡ⋎᮹
ࢱ}᮹᳭Ǖ༉ऽᨱݡ⧕á☁⦹ᩡ݅. b᳭Ǖ༉ऽᨱᕽ۵༉ऽ⩶ᔢ ᨱෙᕽಽ݅ෙᱽ⦽᳑Õᮥᱢᬊ⦹ᩡŁ, ੱ⦽ࢱ̹aᕽಽ݅ෙ
Ǎe᮹Ğĥᱱᨱᕽ᮹ᩑᗮ᳑Õᨱݡ⦽Ğĥ᳑Õᮥᱢᬊ⦹ᩡ݅.
ᯕෝ☖⧕ᄡ᭥ᨱݡ⦽ᮥᮁࠥ⦹ᩡŁ, ĥᙹॅe᮹šĥෝᯕᬊ⧕
Łᮁ⧉ᙹෝᮁࠥ⦹ᩡ݅. ษḡสᮝಽŁᮁ⧉ᙹ᮹⧕ෝ☖⧕᳭Ǖ⦹
ᵲᮥ ᔑᱶ⦹ᩡ݅.
3.1 ֜ԩছܪ״ԧԮীܤࠫ֜ࢄ
⦽ Ǎeᨱᕽ ࢱ̹a qᗭࡽ ⩶┽᮹ ❭ᯕ⥥ Ǎ᳑۵ Fig. 2᪡
z݅. Fig. 2ᨱᕽĢƒ۵qᗭࡽࢱ̹᮹ʙᯕᯕ໑ยᦩŝၵˆᨱ
Ɂᯝ⦹í ӹڹᨕ qᗭࡽ݅Ł aᱶ⦽݅. ĸ۵ľႊ⨆ᮝಽ ࢱ̹a
qᗭࡽǍe᮹bࠥᯕ໑, ⦽Ǎeᨱᕽࢱ̹aqᗭࡽ⩶┽᮹ࢱ̹
qᗭ Ǎe᮹ bࠥ۵ ⅾÏĸa ࡽ݅.
ࢱ̹aqᗭࡽǍe1 (Region 1)᮹ḡ႑ႊᱶŝࢱ̹aqᗭࡹ
ḡᦫᮡǍe2 (Region 2)᮹ḡ႑ႊᱶᮡbbEq. (3)ŝEq.
(4)᪡ z݅ (Xue᪡ Hoo Fatt, 2002).
ćƂľÏ
ƂÏƕ âƕ áà ćÞƒà ĢƒßÐ ÎÏÞÎ à ŃÏß©«Ð
ƕ Þà ĸ = ľ = ĸß (3)
ćƂľÏ
ƂÏƕ âƕ áà ćƒÐ
ÎÏÞÎ à ŃÏß©«Ðƕ Þĸ = ľ = Ïņ à ĸß (4)
Eqs. (3) and (4)۵ Eqs. (5) and (6)ᮝಽ ӹ┡ԝ ᙹ ᯩ݅.
ćƂľÏ
ƂÏƕ âƉÎÏƕ á × Þà ĸ = ľ = ĸß (5)
ćƂľÏ
ƂÏƕ âƉÏÏƕ á × Þĸ = ľ = Ïņ à ĸß (6)
ᩍʑᕽ
ƉÎÏá Î â ćÞƒà ĢƒßÐ
ÎÏÞÎ à ŃÏß©«Ð (7)
ƉÏÏá Î â ćƒÐ
ÎÏÞÎ à ŃÏß©«Ð (8)
ḡႊ⨆ᄡ᭥ƕᨱݡ⦽ၙᇥႊᱶᯙEqs. (5) and (6)ᮥ
⣡໕, əᨱ ݡ⦽ ⧕۵ Eq. (9)᪡ zᯕ ᱶญ⧁ ᙹ ᯩ݅.
ƕ áåƉÎľ â ƉÎľ Þà ĸ = ľ = ĸß
ƉÏľ âƉÏľ Þĸ = ľ = Ïņ à ĸß (9)
ၙᇥႊᱶ᮹ၙḡĥᙹෝǍ⦹ʑ᭥⧕Ğĥ᳑Õᮥᯕᬊ⦽݅.
ࢱ̹aqᗭࡽǍeŝࢱ̹aqᗭࡹḡᦫᮡǍeᯕอӹ۵ᱱᨱᕽ
᳭Ǖ༉ऽ⩶ᔢᨱෙᱽ⦽᳑Õᮥᱢᬊ⦽݅. ݡ⋎᳭Ǖ༉ऽ᪡ᩎݡ
⋎᳭Ǖ༉ऽᨱݡ⧕ᖁbbEq. (10)ŝEq. (11)᮹ᱽ⦽᳑Õᮥ
ᯕᬊ⦽݅.
ຝᱡݡ⋎᳭Ǖ༉ऽᨱݡ⧕ᕽ, ᯕᨱ⧕ݚ⦹۵ᱽ⦽᳑ÕᯙEq.
(10)ᮥEq. (9)ᨱᱢᬊ⦹໕, Eq. (12)᪡zᯕĥᙹA, CෝᗭÑ⧁
ᙹ ᯩ݅.
Fig. 3. Ring with two thickness-reduced regions ƕ á
Ē ē Ĕ
ĕĕ
ƉÎľ Þà ĸ = ľ = ĸß
ćƉÏņ
ƉÏľÞņ à ĸß
Þĸ = ľ = Ïņ à ĸß (12)
ੱ⦽,ľ á ĸᨱᕽ᮹ᄡ᭥᪡ʑᬙʑᨱݡ⦽ᩑᗮ᳑ÕᮥEq. (12)ᨱ
ᱢᬊ⦹໕ Eqs. (13) and (14)ಽ ᱶญ⧁ ᙹ ᯩ݅.
ƕÞĸßá ƉÎĸ á ćƉÏņ
ƉÏÞņ à ĸß
(13)
ƉÏÞņ à ĸß
(14)
᭥ࢱᮡĥᙹB, Dᨱݡ⦽ᩑพႊᱶᯕအಽ, Eq. (15)᪡zᯕ
⧪ಽ ӹ┡ԝ ᙹ ᯩ݅.
ƙ
Ɯ
ƚ
ƉƉÎƉÎĸÎĸ à ćƉÏćƉƉƉÏÞņ àĸßÏÏņņ ƛƝƞ
ƉÏÞņ àĸß Ė
Ę
ė
ęěĚ
á × (15)ĥᙹB, Daᯱ⦽⧕ෝwḡᦫʑ᭥⧕Eq. (16)ŝzᯕ⧪ᮡ
0ᯕ ࡹᨕ ⦽݅ (Xue and Hoo Fatt, 2002).
á ƉÎƉÎĸ âƉÏƉÏÞņ àĸßá × (16)
Eq. (16)ᮡ⦽Ǎeᨱᕽࢱ̹aqᗭࡽ❭ᯕ⥥Ǎ᳑ྜྷ᮹᳭Ǖ⦹
ᵲᔑᱶᮥ᭥⦽Łᮁ⧉ᙹᯕ݅. Eq. (16)ᨱᕽƉÎᮥᗭÑ⦹ʑ᭥⦹ᩍ, Eqs. (7) and (8)ಽᇡ░ᮁࠥࡹ۵ƉÎŝƉÏ᮹šĥᯙEq. (17)ᮥ
ᯕᬊ⧁ ᙹ ᯩ݅.
ƉÎá
ö
ćÎ â ćÞÎ àĢƒîƒßƉÏÏà Î Ð (17)Eq. (17)ᮥEq. (16)ᨱݡ᯦⦹໕ƉÎᯕᗭÑࡹŁƉÏaǍ⧕ḥ݅.
ᯕভ, ᯕ ႊᱶᮡƉÏᨱ ݡ⦽ እᖁ⩶ ႊᱶᮝಽᕽ, Newton- Raphson Method᪡zᮡእᖁ⩶⣡ᯕჶᮥᯕᬊ⧕⦽݅. ษḡส ᮝಽ, Eq. (8)ᮥ ᦶ ⩶┽ಽ ᱶญ⦽ Eq. (18)ᨱ ᔑᱶࡽƉÏෝ
ݡ᯦⦹໕ ┥ᖒ ᳭Ǖ⦹ᵲᯕ ᔑᱶࡽ݅.
©ƁƐá ćÎÏÞƉÞÏÏÎ à Ńà ÎßÏßÞć«ƒ ßÐ (18)
↽᳦ᱢᮝಽ, ᰍഭᱢᯙ✚ᖒᯙ ၰŃ۵┥ᖒ᳭Ǖ⦹ᵲEq. (18)ᨱอ
ᩢ⨆ᮥ ၙ⊹໑, ᳭Ǖ༉ऽ ⩶ᔢᯙƕ۵ ʑ⦹⦺ᱢᯙ ⩶ᔢ ᯙᯱᯙ
ĸ, ƒ, ၰĢƒᨱอ ᩢ⨆ᮥ ၼ۵݅۵ äᮥ ᙹ ᯩ݅.
ᩎݡ⋎᳭Ǖ༉ऽᨱݡ⧕ᕽࠥษ₍aḡಽ, Eq. (9)ᨱᯕᨱ⧕ݚ⦹
۵ᱽ⦽᳑ÕᯙEq. (11)ᮥᱢᬊ⦹ᩍEq. (19)᪡zᯕᱶญ⧁ᙹ
ᯩ݅.
ƕ á Ē ē Ĕ
ĕ ĕ
ƉÎľ Þà ĸ = ľ = ĸß
ćƉÏņ
ƉÏľÞņ à ĸß
Þĸ = ľ = Ïņ àĸß (19)
ੱ⦽,ľ á ĸᨱᕽ᮹ᄡ᭥᪡ʑᬙʑᨱݡ⦽ᩑᗮ᳑ÕᮥEq. (19)ᨱ
ᱢᬊ⦹Ł, ĥᙹ A, Da ᯱ⦽ ⧕ෝ wḡ ᦫࠥಾ อॅ໕, Eq.
(20)ŝ zᮡ Łᮁ⧉ᙹa ᮁࠥࡽ݅ (Xue and Hoo Fatt, 2002).
á ƉÎƉÏÞņ à ĸßâƉÏƉÎĸ á × (20)
ษḡสᮝಽ, Łᮁ⧉ᙹಽᇡ░᳭Ǖᯥĥ⦹ᵲᮥǍ⦹ʑ᭥⧕ݡ⋎
༉ऽ᪡ ษ₍aḡಽ Eqs. (17) and (18)ᮥ ᯕᬊ⦽݅.
3.2 ܪ֜ԩছܪ״ԧԮীܤࠫ֜ࢄ
ᔢ, ⦹݉ᨱ݉໕qᗭaၽᔾ⦽❭ᯕ⥥Ǎ᳑ྜྷ᮹᳭Ǖ⦹ᵲᔑᱶᮥ
᭥⧕Fig. 3ŝzᮡ༉ߙᮥᯕᬊ⦹ᩡ݅. ࢱ̹aqᗭࡽ⦽Ǎe᮹
bࠥa 2ĸᯕအಽ, ࢱ Ǎeᨱ ⧕ݚ⦹۵ ⅾ bࠥ۵ 4ĸa ࡽ݅.
Fig. 3ᨱ⧕ݚ⦹۵4}᮹Ǎeᨱݡ⦽ḡ႑ႊᱶᮡEqs. (21) and (22)ಽ ӹ┡ԝ ᙹ ᯩ݅.
ćƂľÏ
ƂÏƕ âƉÎÏƕ á × Þà ĸ = ľ = ĸì ņ à ĸ = ľ = ņ âĸß (21)
Fig. 4. Buckling loads of ring with one thickness-reduced region (G: symmetric mode, G: antisymmetric mode) ćƂľÏ
ƂÏƕ âƉÏÏƕ á × Þĸ = ľ = ņ à ĸì ņ â ĸ = ľ =à ĸß (22)
ḡႊ⨆ᄡ᭥ƕᨱݡ⦽ၙᇥႊᱶᯙEqs. (21) and (22)ෝ
⣡໕, əᨱ ݡ⦽ ⧕۵ Eq. (23)ŝ zᯕ ᱶญ⧁ ᙹ ᯩ݅.
ƕ á Ē ē Ĕ
ĕĕ
ƉÎľ â ƉÎľ Þà ĸ = ľ = ĸß
ƉÏľ âƉÏľ Þĸ = ľ = ņ à ĸß
ƉÎľ â ƉÎľ Þņ àĸ = ľ = ņ âĸß ƉÏľ â¡ƉÏľ Þņ âĸ = ľ =à ĸß
(23)
⦽Ǎeᨱᕽࢱ̹aqᗭࡽĞᬑ᪡ษ₍aḡಽ, ၙᇥႊᱶ᮹
ၙḡĥᙹᔑᱶᮥ᭥⧕, ༉ऽ⩶ᔢᄥᱽ⦽᳑ÕၰǍeĞĥᱱᨱᕽ᮹
ᄡ᭥᪡ ʑᬙʑ᮹ ᩑᗮ᳑Õŝ zᮡ Ğĥ᳑Õᮥ ᯕᬊ⦽݅. ຝᱡ, ݡ⋎༉ऽ᮹Ğᬑ, ᯕᨱ⧕ݚ⦹۵ᱽ⦽᳑ÕᯙEq. (10)ᮥᱢᬊ⦹໕, Eq. (23)ᨱᕽĥᙹA ၰEaᗭÑࡽ݅. əญŁľ á ĸᨱᕽ᮹ᩑᗮ᳑
Õᮝಽᇡ░ĥᙹC᪡Dෝ, ľ áà ĸᨱᕽ᮹ᩑᗮ᳑Õᮝಽᇡ░ĥᙹ
G᪡ Hෝ, bb Bᨱ š⦽ ᮝಽ ӹ┡ԝ ᙹ ᯩ݅. ษḡสᮝಽ, ľ á ņ à ĸᨱᕽ᮹ᩑᗮ᳑Õᮥᯕᬊ⦹໕, B᪡Fᨱš⦽ᩑพႊᱶ
ᯕ ᮁࠥࡽ݅. ᯕ ĥᙹ B᪡ Fa ᯱ⦽ ⧕ෝ wḡ ᦫࠥಾ, Eq.
(15)᪡zᮡႊჶᮝಽ⧪ᯕ0ᯕࡹࠥಾอॅ໕, ࢱǍeᨱᕽ
ࢱ̹aqᗭࡽĞᬑ᮹ݡ⋎᳭Ǖ༉ऽᨱݡ⦽Łᮁ⧉ᙹᯙEq. (24)a
ᮁࠥࡽ݅ (Ko, 2013).
á ÏƉÎƉÏÏƉÎĸ àåƉÎÏà ƉÏÏàÞƉÎÏâƉÏÏßÏƉÎĸæ
ƉÏÞņ à Ïĸßá ×
(24)
ᩎݡ⋎༉ऽࠥݡ⋎༉ऽ᮹ႊჶŝษ₍aḡಽ, ຝᱡᩎݡ⋎
༉ऽᨱ⧕ݚ⦹۵ᱽ⦽᳑ÕᯙEq. (11)ᮥᱢᬊ⦹໕, Eq. (23)ᨱᕽ
ĥᙹB ၰEaᗭÑࡽ݅. əญŁľ á ĸᨱᕽ᮹ᩑᗮ᳑Õᮝಽᇡ░
ĥᙹC᪡Dෝ, ľ áà ĸᨱᕽ᮹ᩑᗮ᳑Õᮝಽᇡ░ĥᙹG᪡Hෝ, bbAᨱš⦽ᮝಽӹ┡ԝᙹᯩ݅. ษḡสᮝಽ,ľ á ņ à ĸᨱᕽ ᮹ᩑᗮ᳑Õᮥᯕᬊ⦹໕, A᪡Fᨱš⦽ᩑพႊᱶᯕᮁࠥࡽ݅.
ᯕĥᙹA᪡Faᯱ⦽⧕ෝwḡᦫࠥಾอॅ໕, ࢱǍeᨱᕽ
ࢱ̹aqᗭࡽĞᬑ᮹ᩎݡ⋎᳭Ǖ༉ऽᨱ ݡ⦽Łᮁ⧉ᙹᯙEq.
(25)a ᮁࠥࡽ݅ (Ko, 2013).
á ÏƉÎƉÏÏƉÎĸ âåƉÎÏà ƉÏÏâÞƉÎÏâƉÏÏßÏƉÎĸæ
ƉÏÞņ à Ïĸßá × (25)
ࢱǍeᨱᕽࢱ̹aqᗭࡽ⩶┽᮹❭ᯕ⥥Ǎ᳑ྜྷ᮹᳭Ǖ⦹ᵲ
ᔑᱶᮥ᭥⧕, Eq. (17)ᮥEqs. (24) and (25)ᨱݡ᯦⦹Łᯕಽᇡ░
ĥᔑࡽ ĥᙹƉÏ᪡ Eq. (18)ᮥ ᯕᬊ⦽݅.
3.3 ࣡ැজ
ᄡᙹ⧕ᕾᮥ᭥⦽ᄡᙹྕ₉ᬱ⪵ෝ᭥⦹ᩍ, ᳭Ǖ⦹ᵲᨱݡ⦹ᩍ
Fig. 5. Buckling loads of ring with two thickness-reduced regions (G: symmetric mode, G: antisymmetric mode)
(a) Symmetric mode (a) Antisymmetric mode
Fig. 6. Buckling loads with respect to reduced region angle (one thickness-reduced region) Eq. (26)ŝzᯕࢱ̹aqᗭࡽǍ᳑᮹᳭Ǖ⦹ᵲᮥࢱ̹aqᗭࡹḡ
ᦫᮡǍ᳑᮹᳭Ǖ⦹ᵲᮝಽӹ٥ᨕӹ┡ԕᨩ݅. ੱ⦽, qᗭࡽࢱ̹ෝ
Ⅹʑ ࢱ̹ಽ ӹ٥ᨕ ྕ₉ᬱ⪵⦹ᩡ݅.
ć©ƃ
©ƁƐ
á ćÞƉÏÏÐà Îß
(26)
ࢱ̹aqᗭࡽǍe᮹bࠥĸ ၰྕ₉ᬱ⪵ࡽࢱ̹᮹ᄡ⪵ᨱ
ෙྕ₉ᬱ⪵ࡽ᳭Ǖ⦹ᵲ᮹ᄡ⪵ෝə௹⥥ಽӹ┡ԕᨩ݅. Fig.
4۵⦽Ǎeᨱᕽࢱ̹aqᗭࡽĞᬑᯕŁ, Fig. 5۵ࢱǍeᨱᕽ
ࢱ̹a qᗭࡽ Ğᬑ᮹ ᳭Ǖ⦹ᵲᮥ ӹ┡ԙ݅.
ࢱ̹a ᯝᱶ⦹ḡ ᦫᮡ ❭ᯕ⥥᮹ ᳭Ǖ༉ऽ۵ qᗭࡽ ࢱ̹ ၰ
qᗭǍe᮹ bࠥ᪡ zᮡ ʑ⦹⦺ᱢ ⩶ᔢᨱ ݍḥ݅. ᝅᱽ
❭ᯕ⥥᮹᳭Ǖ⦹ᵲᮡݡ⋎, ᩎݡ⋎᮹ࢱ᳭Ǖ༉ऽᵲ᯲ᮡ᳭Ǖ⦹
ᵲᮥ w۵ ༉ऽa ḡ႑⦹í ࡽ݅. ੱ⦽, ࢱ̹a ᯝᱶ⦽ Ğᬑ᮹
qᗭǍe᮹ bࠥ ᄡ⪵ᨱ ෙ ᳭Ǖ ⦹ᵲ᮹ ᄡ⪵۵, ⦽ Ǎe ၰ
ࢱǍeqᗭᨱݡ⦹ᩍbbFig. 6 ၰFig. 7ಽӹ┡ԝᙹᯩ݅.
Fig. 6ŝ Fig. 7ෝ ᅕ໕, ⦽ Ǎe ၰ ࢱ Ǎe᮹ qᗭǍeᮥ
aḡ۵ĞᬑəญŁݡ⋎ၰᩎݡ⋎༉ऽ᮹Ğᬑ༉ࢱᨱݡ⦹ᩍ, qᗭǍe᮹bࠥa᷾a⧉ᨱ᳭Ǖ⦹ᵲ᮹Ⓧʑaqᗭ⦹໑,
(a) Symmetric mode (a) Antisymmetric mode Fig. 7. Buckling loads with respect to reduced region angle (two thickness-reduced regions)
Table 1. Properties of analysis model
Radius, « (mm) 304.8
Thickness, ƒ (mm) 31.9
Elastic modulus, (GPa) 206.8
Poisson ratio, Ń 0.3
Yielding stress, ňƗ (MPa) 410
Fig. 8. Finite element model for a pipeline qᗭࡽࢱ̹a᷾a⧁ᙹಾ᳭Ǖ⦹ᵲᯕɪĊ⦹íqᗭ⧉ᮥ
ᙹ ᯩ݅.
4. ᮁ⦽ᗭ⧕ᕾᮥ☖⦽ᯕುá᷾
ᅙ ᩑǍᨱᕽ ᱽᦩ⦽ ⦽ Ǎe ၰ ࢱ Ǎeᨱᕽ ࢱ̹a qᗭࡽ
❭ᯕ⥥Ǎ᳑᮹᳭Ǖ⦹ᵲ ᔑᱶᮥá᷾⦹ʑ᭥⧕ᮁ⦽ᗭ⧕ᕾ
đŝ᪡እƱ⦹ᩡ݅. ᮁ⦽ᗭ⧕ᕾᨱᕽ۵Blue Stream ⥥ಽ᱾✙ಽ
ᇩญ۵్ᦥDzhudga᪡░┅Samsunᮥᯨ۵⧕ᱡaᜅᬕᘂš ᨱᔍᬊࡽ❭ᯕ⥥ᯙ᮹ᱽᬱᮥᯕᬊ⦹ᩡ݅(Caruso et al., 2003).
2003֥ᨱ᪥Ŗࡽ❭ᯕ⥥ᯙᮝಽᕽ, እƱᱢᔍᬊʑeᯕ᪅௹ࡹᨕ
ᇡ॒ᮝಽ݉໕qᗭaၽᔾ⧁a܆ᖒᯕᔢݡᱢᮝಽ׳ᮥäᯕ
Ł❱݉ࡽ݅. ❭ᯕ⥥ᯙᮡǍe774km, ↽ݡᙹᝍ2,150mᯕ໑
ᮁ⦽ᗭ⧕ᕾ༉ߙ᮹ ᱽᬱᮡ Table 1ŝ z݅.
Ŗ⋎ࢱ̹᪡qᗭࡽࢱ̹᮹እĢƒîƒ۵0ᇡ░0.8ʭḡ0.2݉᭥ಽ
⧕ᕾᮥᙹ⧪⦹ᩡ݅. qᗭࡽǍe᮹ⅾbࠥ۵, ⦽Ǎeᨱᕽࢱ̹ෝ
ᵥᩡᮥĞᬑᨵ30, 60, 90, 120, 150ⰺᨱݡ⧕, ࢱǍeᨱᕽࢱ̹ෝ
ᵥᩡᮥĞᬑᨵ15, 30, 45, 60, 75ⰺᨱݡ⧕⧕ᕾᮥᙹ⧪⦹ᩡ݅.
ੱ⦽, ABAQUS᮹ᮁ⦽ᗭ༉ߙᨱ۵ᖁ⩶┥ᖒᰍഭෝaḡ໑
ᱥ݉ᄡ⩶ᮥྕ⧁ᙹᯩ۵᪅ᯝ్-ᄁ٥ᯕᅕෝʑၹᮝಽ⦹۵
2ᱩᱱ ᗭᯙ B23 ᗭa ᔍᬊࡹᨩŁ, ᬱᵝႊ⨆ᮝಽ ⅾ 72}
ᗭಽӹ٥ᨕ⠪໕ᄡ⩶ᮉᔢ┽ᨱᕽ⧕ᕾᮥᙹ⧪⦹ᩡ݅. ABAQUS
༉ߙ᮹Ğᬑࠥ, ʙᯕaྕ⦽⯩ʙĞᬑ⠪໕ᄡ⩶ᮉᔢ┽᮹ยǍ᳑ಽ
Łಅa܆⦹݅۵ᯕುᱢᯙ⧕ᕾᨱᕽ᮹aᱶᔍ⧎ᮥəݡಽᱢᬊ
⍽, ᬱ☖⩶Ǎ᳑ෝᚹᗭಽ༉ߙย⦹ḡᦫŁᅕᗭෝᯕᬊ⦽
ย Ǎ᳑ಽ ༉ߙย⦹ᩡ݅.
Fig. 8ᮡABAQUSෝᯕᬊ⦽❭ᯕ⥥Ǎ᳑᮹ᮁ⦽ᗭ༉ߙᮥ
ӹ┡ԕŁᯩŁ, Fig. 9۵ᯕᨱݡ⦽᳭Ǖ⧕ᕾđŝಽᕽݡ⋎ၰ
ᩎݡ⋎༉ऽ⩶ᔢᮥӹ┡ԙ݅. Tables 2 and 3ᨱ۵⦽Ǎeᨱᕽ
ࢱ̹aqᗭࡽ❭ᯕ⥥Ǎ᳑᮹ᯕುᨱ᮹⦽᳭Ǖ⦹ᵲŝᮁ⦽ᗭ
⧕ᕾᨱ᮹⦽ ᳭Ǖ⦹ᵲᮥ ݡ⋎༉ऽ᪡ ᩎݡ⋎ ༉ऽᨱݡ⧕ bb
እƱ⦹ᩡ݅.
Tables 4 and 5ᨱ۵ࢱǍeᨱᕽࢱ̹aqᗭࡽ❭ᯕ⥥Ǎ᳑᮹
ᯕುᨱ᮹⦽᳭Ǖ⦹ᵲŝᮁ⦽ᗭ⧕ᕾᨱ᮹⦽᳭Ǖ⦹ᵲᮥݡ⋎
༉ऽ᪡ ᩎݡ⋎ ༉ऽᨱ ݡ⧕ bb እƱ⦹ᩡ݅.
Tables 2, 3, 4, and 5ෝᔕ⠕ᅕ໕, ݡ⋎༉ऽᨱᕽࢱ̹qᗭa
aᰆⓑĞᬑᨱ10% ᯕᔢ᮹᪅₉aၽᔾ⧩ḡอ, ᯕෝᱽ⦹໕
(a) Symmetric mode
(b) Antisymmetric mode Fig. 9. Mode shape of buckling
Table 2. Comparison of buckling loads for ring with one thickness-
Ïĸ Comparison Ģƒîƒ
0 0.2 0.4 0.6 0.8
30쥩
©ƁƐ (Proposed) 65.13 55.51 46.17 33.88 6.21
©ƁƐ (FEM) 65.13 55.84 47.21 35.11 7.03 Error (%) 0.00 0.59 2.20 3.50 11.66 60쥩
©ƁƐ (Proposed) 65.13 53.20 34.24 12.86 1.83
©ƁƐ (FEM) 65.13 53.39 34.45 13.18 1.96 Error (%) 0.00 0.36 0.61 2.43 6.63 90쥩
©ƁƐ (Proposed) 65.13 43.40 22.50 8.29 1.30
©ƁƐ (FEM) 65.13 43.57 23.10 8.78 1.40 Error (%) 0.00 0.39 2.60 5.58 7.14 120쥩
©ƁƐ (Proposed) 65.13 39.60 20.83 7.81 1.10
©ƁƐ (FEM) 65.13 40.00 21.84 8.61 1.22 Error (%) 0.00 1.00 4.62 9.29 9.84 150쥩
©ƁƐ (Proposed) 65.13 38.47 18.31 5.82 0.75
©ƁƐ (FEM) 65.13 38.73 18.61 5.96 0.77 Error (%) 0.00 0.67 1.61 2.35 2.60
Table 3. Comparison of buckling loads for ring with one MPa)
Ïĸ Comparison Ģƒîƒ
0 0.2 0.4 0.6 0.8
30쥩
©ƁƐ (Proposed) 65.13 58.83 43.17 21.07 6.46
©ƁƐ (FEM) 65.13 58.85 43.35 21.29 6.43 Error (%) 0.00 0.03 0.42 1.03 0.47 60쥩
©ƁƐ (Proposed) 65.13 47.85 31.41 17.28 5.56
©ƁƐ (FEM) 65.13 47.96 31.57 17.86 5.85 Error (%) 0.00 0.23 0.51 3.25 4.96 90쥩
©ƁƐ (Proposed) 65.13 46.73 28.32 10.35 1.51
©ƁƐ (FEM) 65.13 46.82 28.48 10.47 1.60 Error (%) 0.00 0.19 0.56 1.15 5.63 120쥩
©ƁƐ (Proposed) 65.13 41.23 19.81 6.45 0.93
©ƁƐ (FEM) 65.13 41.35 19.95 6.30 0.98 Error (%) 0.00 0.29 0.70 2.38 5.10 150쥩
©ƁƐ (Proposed) 65.13 34.88 15.08 4.52 0.57
©ƁƐ (FEM) 65.13 35.02 15.20 4.57 0.57 Error (%) 0.00 0.40 0.79 1.09 0.00
Table 4. Comparison of buckling loads for ring with two thickness-
Ñĸ Comparison Ģƒîƒ
0 0.2 0.4 0.6 0.8
15쥩
©ƁƐ (Proposed) 65.13 51.11 34.40 19.07 8.01
©ƁƐ (FEM) 65.13 51.70 37.25 21.11 9.03 Error (%) 0.00 1.14 7.65 9.66 11.30 30쥩
©ƁƐ (Proposed) 65.13 47.17 31.39 18.24 4.44
©ƁƐ (FEM) 65.13 48.27 33.41 20.10 5.08 Error (%) 0.00 2.28 6.05 9.25 12.60 45쥩
©ƁƐ (Proposed) 65.13 46.87 30.41 13.78 3.01
©ƁƐ (FEM) 65.13 48.00 32.31 14.84 3.40 Error (%) 0.00 2.35 5.88 7.14 11.47 60쥩
©ƁƐ (Proposed) 65.13 44.76 24.87 8.82 1.18
©ƁƐ (FEM) 65.13 45.42 25.95 9.36 1.26 Error (%) 0.00 1.45 4.16 5.77 6.35 75쥩
©ƁƐ (Proposed) 65.13 39.29 18.60 5.87 0.75
©ƁƐ (FEM) 65.13 39.56 18.90 6.01 0.77 Error (%) 0.00 0.68 1.59 2.33 2.60
༉ुʑ⦹⦺ᱢ⩶ᔢၰ᳭Ǖ༉ऽᨱݡ⧕ᕽᯕುᱢᯙđŝ᪡ᮁ⦽
ᗭ⧕ᕾ đŝa Ñ᮹ zᮡ sᮥ ӹ┡ԕᨩ݅.
Table 5. Comparison of buckling loads for ring with two thickness-
Ñĸ Comparison Ģƒîƒ
0 0.2 0.4 0.6 0.8
15쥩
©ƁƐ (Proposed) 65.13 63.29 57.72 36.67 6.21
©ƁƐ (FEM) 65.13 63.24 57.68 37.00 6.39 Error (%) 0.00 0.08 0.07 0.89 2.82 30쥩
©ƁƐ (Proposed) 65.13 54.03 34.26 12.77 1.73
©ƁƐ (FEM) 65.13 54.07 35.38 12.91 1.76 Error (%) 0.00 0.07 0.87 1.08 1.70 45쥩
©ƁƐ (Proposed) 65.13 43.28 21.59 6.96 0.89
©ƁƐ (FEM) 65.13 43.40 21.73 7.03 0.91 Error (%) 0.00 0.28 0.64 1.00 2.20 60쥩
©ƁƐ (Proposed) 65.13 36.62 16.31 4.97 0.63
©ƁƐ (FEM) 65.13 36.76 16.44 5.02 0.63 Error (%) 0.00 0.38 0.79 1.00 0.00 75쥩
©ƁƐ (Proposed) 65.13 33.81 14.38 4.28 0.54
©ƁƐ (FEM) 65.13 33.95 14.49 4.32 0.54 Error (%) 0.00 0.41 0.76 0.93 0.00
5. đು
ᅙםྙᨱᕽ۵ʙᯕaๅᬑʕᬱ☖⩶ᚹǍ᳑ྜྷᯙ❭ᯕ⥥ෝ
ยᮝಽaᱶ⦹ᩍ᳭Ǖ⦹ᵲᔑᱶᮥ᭥⦽ᯕುᮥᮁࠥ⦹ᩡ݅. ❭ᯕ
⥥۵ᇡ᮹᭥⨹ᯕⓍʑভྙᨱᯕෝŁಅ⦹ʑ᭥⧕ยǍ᳑᮹
ࢱ̹ෝᵥᯥᮝಽ៉ᇡᮥŁಅ⦹ᩡ݅. ݅᧲⦽ᇡ⩶┽ෝŁಅ⦹
ʑ ᭥⧕ ย᮹ ⦽ Ǎe ၰ ࢱ Ǎeᨱᕽ ࢱ̹ෝ ᄡ⪵⎑Ł, ᯕ
⩶┽᮹᳭Ǖ⦹ᵲᔑᱶᮥ᭥⦽ᯕುᮥᮁࠥ⦹ᩡ݅. ᯕುᨱ᮹⦽
᳭Ǖ⦹ᵲŝ ᮁ⦽ᗭ⧕ᕾᨱ ᮹⦽ ᳭Ǖ⦹ᵲᮥ እƱá᷾⦹ᩡŁ, ᅙ ᩑǍᨱ ݡ⦽ đುᮡ ݅ᮭŝ z݅.
(1) ࢱ̹aᯝᱶ⦹ḡᦫᮡยǍ᳑᮹᳭Ǖ⦹ᵲᨱݡ⦹ᩍ, ᯕುᨱ
᮹⦽đŝ᪡ᮁ⦽ᗭ⧕ᕾᨱ᮹⦽đŝෝእƱ⧕ᅕ໕, ݡ⋎༉
ऽᨱᕽࢱ̹qᗭaaᰆⓑĞᬑᨱ10% ᯕᔢ᮹᪅₉aၽᔾ⧩
ḡอ, ᯕෝᱽ⦹໕༉ुʑ⦹⦺ᱢ⩶ᔢၰ᳭Ǖ༉ऽᨱݡ⧕ᕽ
Ñ᮹ zᮡ sᮥ ӹ┡ԕᨩ݅.
(2) ࢱ̹aᯝᱶ⦹ḡᦫᮡย᮹᳭Ǖ༉ऽ⩶ᔢᮡᰍഭᱢ✚ᖒᨱ۵
ྕš⦹໑ࢱ̹aqᗭࡽǍe᮹bࠥၰࢱ̹qᗭపŝzᮡ
ʑ⦹⦺ᱢ⩶ᔢᨱʑᯙ⦹Ł, ᰍഭᱢ✚ᖒᮡย᮹᳭Ǖ⦹ᵲᮥ
đᱶ⦹۵ äᮝಽ ❱݉ࡽ݅.
ᯕುᱢᮝಽᔑᱶࡽࢱ̹aᯝᱶ⦹ḡᦫᮡ❭ᯕ⥥Ǎ᳑᮹᳭Ǖ⦹
ᵲᯕᮁ⦽ᗭ⧕ᕾᨱ᮹⦽đŝ᪡Ñ᮹ᯝ⊹⦹အಽ, ᅙםྙᨱᕽ
ᮁࠥ⦽ᯕುᮥᇡᮥŁಅ⦽❭ᯕ⥥᮹⧕ᕾၰᖅĥᨱᔍᬊࢁ
ᙹ ᯩᮥ äᮝಽ ❱݉ࡽ݅. ੱ⦽, ┥ᖒ ᳭Ǖ⦹ᵲᮡ ᦶᮥ ၼ۵
❭ᯕ⥥᮹ᖅĥӹ⧕ᕾᨱᯩᨕaᰆᅕᙹᱢᯙsᯕʑভྙᨱ❭ᯕ⥥
ᖅĥⅩʑ݉ĥᨱᕽ}ఖᖅĥෝ᭥⧕ᔍᬊࢁᙹᯩŁ, ᅙםྙᨱᕽ
ᱽ⦽ᯕುᮥ☖⧕݅᧲⦽ᄡᙹᨱݡ⦽⧕ᕾᯕa܆⧁äᮝಽ
ᅕᯙ݅. ੱ⦽, ݉ᙽ⯩⦽ǍeၰࢱǍe᮹ࢱ̹ෝᵥᯥᮝಽ៉
ᇡᮥŁಅ⦹۵ᅙםྙ᮹ḡෝၽᱥ⍽, ʙᯕႊ⨆ᮝಽᩍ్
aḡ⩶┽᮹ᅕvᰍaᯩ۵❭ᯕ⥥Ǎ᳑ྜྷᨱݡ⧕ᕽ᳭ࠥǕÑ࠺ᮥ
ᩩ⊂⧁ ᙹ ᯩᮥ äᮝಽ ❱݉ࡽ݅.
qᔍ᮹ɡ
ᅙᩑǍ۵2012֥ࠥ⡍ᜅ⎵(POSCO) ݡ⦺ᬱᔾםྙḡᬱᔍᨦᨱ
᮹⧕ ᙹ⧪ࡹᨩܩ݅. ᩑǍእ ḡᬱᨱ qᔍऽพܩ݅.
References
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Timoshenko, S. P. and Gere, J. M. (1961). Theory of elastic sta- bility, McGraw Hill, Singapore.
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