Упруго-пластическое состояние круговой тороидальной

ISSN 1819-432X print / ISSN 1993-3495 online
ÑÓ×ÀÑÍÅ ÏÐÎÌÈÑËÎÂÅ ÒÀ ÖȲËÜÍÅ ÁÓIJÂÍÈÖÒÂÎ
ÑÎÂÐÅÌÅÍÍÎÅ ÏÐÎÌÛØËÅÍÍÎÅ È ÃÐÀÆÄÀÍÑÊÎÅ ÑÒÐÎÈÒÅËÜÑÒÂÎ
MODERN INDUSTRIAL AND CIVIL CONSTRUCTION
ÒÎÌ 3, N2, 2007, 67-77
ÓÄÊ 624.04+624.074
ÏÐÓÆÍÎ-ÏËÀÑÒÈ×ÍÈÉ ÑÒÀÍ ÊÐÓÃÎÂί ÒÎÐίÄÀËÜÍί
ÎÁÎËÎÍÊÈ Ç ÏÐßÌÎÊÓÒÍÈÌ ÎÒÂÎÐÎÌ
Â.Ï. Ìóùàíîâ, À.². Äåìèäîâ
Äîíáàñüêà íàö³îíàëüíà àêàäåì³ÿ áóä³âíèöòâà ³ àðõ³òåêòóðè,
âóë. Äåðæàâèíà, 2, ì. Ìà곿âêà, Äîíåöüêà îáëàñòü, Óêðà¿íà, 86123.
E-mail: mvf@dgasa.dn.ua
Îòðèìàíà 12 áåðåçíÿ 2007; ïðèéíÿòà 30 êâ³òíÿ 2007.
Àíîòàö³ÿ. Ñòàòòÿ ïðèñâÿ÷åíà çàñòîñóâàííþ ðàí³øå ðîçðîáëåíî¿ ìåòîäèêè ç âèçíà÷åííÿ ïðóæíî-ïëàñòè÷íîãî íàïðóæåíî-äåôîðìîâàíîãî ñòàíó îáîëîíîê äîâ³ëüíî¿ ôîðìè íà îñíîâ³ òåî𳿠ìàëèõ ïðóæíîïëàñòè÷íèõ äåôîðìàö³é. ˳íåàðèçàö³ÿ ð³øåííÿ çàäà÷ âèêîíóºòüñÿ ìåòîäîì ïðóæíèõ ð³øåíü. Ó êîæíîìó íàáëèæåíí³ çàñòîñîâóºòüñÿ âàð³àö³éíå ð³âíÿííÿ Ëàãðàíæà â ïåðåì³ùåííÿõ òî÷îê ñåðåäèííî¿ ïîâåðõí³ îáîëîíêè â ê³íöåâèõ ð³çíèöÿõ. Ïðè çàïèñ³ âàð³àö³éíîãî ð³âíÿííÿ Ëàãðàíæà çâ'ÿçîê ì³æ íàïðóæåííÿìè ³ äåôîðìàö³ÿìè ïðåäñòàâëåíèé ó ôîðì³ çàêîíó Ãóêà, àëå ç äîäàòêîâèìè ÷ëåíàìè, ùî âðàõîâóþòü ïëàñòè÷í³ äåôîðìàö³¿. Ãåîìåòðè÷í³ ð³âíÿííÿ ïðèéíÿò³ â ë³í³éí³é ïîñòàíîâö³ ó ôîðì³ ñï³ââ³äíîøåíü
Êîø³. Ìàòåð³àë õàðàêòåðèçóºòüñÿ â³äïîâ³äíîþ ä³àãðàìîþ ðîçòÿãó öèë³íäðè÷íîãî çðàçêà, ìîäóëåì íîðìàëüíî¿ ïðóæíîñò³ ³ êîåô³ö³ºíòîì Ïóàññîíà. ʳíåìàòè÷í³ ãðàíè÷í³ óìîâè çàäîâîëüíÿþòüñÿ òî÷íî, à
ñòàòè÷í³ íà â³ëüíèõ â³ä çàêð³ïëåííÿ êðàÿõ îáîëîíêè - ïðèáëèçíî. Ðîçãëÿäàºòüñÿ ïðèêëàä ðîçðàõóíêó
òîðî¿äàëüíî¿ îáîëîíêè, ð³âíÿííÿ ñåðåäèííî¿ ïîâåðõí³ ÿêî¿ çàïèñàí³ â ïàðàìåòðè÷í³é ôîðì³. Âíóòð³øí³é
³ çîâí³øí³é êðàÿ îáîëîíêè çàêð³ïëåí³ àáñîëþòíî æîðñòêî, à á³÷í³ ñòîðîíè çàêð³ïëåí³ øàðí³ðíî. Îáîëîíêà îñëàáëåíà âåëèêèì íåï³äêð³ïëåíèì ïðÿìîêóòíèì îòâîðîì ³ çàâàíòàæåíà íîðìàëüíèì ð³âíîì³ðíî
ðîçïîä³ëåíèì íàâàíòàæåííÿì. Çàäà÷à âèð³øóºòüñÿ ìåòîäîì ïðóæíèõ ð³øåíü ó ñïîëó÷åíí³ ç âàð³àö³éíîð³çíèöåâèì ìåòîäîì ó ïåðåì³ùåííÿõ òî÷îê ñåðåäèííî¿ ïîâåðõí³ îáîëîíêè. гøåííÿ çàäà÷³ äîâåäåíå äî
÷èñåëüíèõ ðåçóëüòàò³â. Ïðåäñòàâëåí³ ïîëÿ ³íòåíñèâíîñò³ äîòè÷íèõ íàïðóæåíü ïî âíóòð³øí³é ñâ³òëîâ³é
ïîâåðõí³ îáîëîíêè, ïåðåì³ùåíü, ïîçäîâæí³õ ñèë ³ çãèíàëüíèõ ìîìåíò³â ïðè ïðóæíî-ïëàñòè÷íîìó ð³øåíí³
çàäà÷³ äëÿ âñ³º¿ ñ³òêîâî¿ îáëàñò³. Ðîáîòà ïðèñâÿ÷åíà àêòóàëüíîìó ïèòàííþ áóä³âåëüíî¿ ìåõàí³êè îáîëîíîê, îñëàáëåíèõ âåëèêèìè îòâîðàìè ïðè ïðóæíî-ïëàñòè÷íîìó äåôîðìóâàíí³ ïðè ïðîñòîìó ïðîöåñ³
íàâàíòàæåííÿ.
Êëþ÷îâ³ ñëîâà: íåçàìêíóòà òîðî¿äàëüíà îáîëîíêà, âàð³àö³éíî-ð³çíèöåâèé ìåòîä, ãðàíè÷í³ óìîâè,
ìåòîä ïðóæíèõ ð³øåíü, ³íòåíñèâí³ñòü äåôîðìàö³é çðóøåííÿ, ³íòåíñèâí³ñòü äîòè÷íèõ íàïðóæåíü,
ôóíêö³ÿ ïëàñòè÷íîñò³.
ÓÏÐÓÃÎ-ÏËÀÑÒÈ×ÅÑÊÎÅ ÑÎÑÒÎßÍÈÅ ÊÐÓÃÎÂÎÉ
ÒÎÐÎÈÄÀËÜÍÎÉ ÎÁÎËÎ×ÊÈ Ñ ÏÐßÌÎÓÃÎËÜÍÛÌ
ÎÒÂÅÐÑÒÈÅÌ
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
Äîíáàññêàÿ íàöèîíàëüíàÿ àêàäåìèÿ ñòðîèòåëüñòâà è àðõèòåêòóðû,
óë. Äåðæàâèíà, 2, ã. Ìàêååâêà, Äîíåöêàÿ îáëàñòü, Óêðàèíà, 86123.
E-mail: mvf@dgasa.dn.ua
Ïîëó÷åíà 12 ìàðòà 2007; ïðèíÿòà 30 àïðåëÿ 2007.
Àííîòàöèÿ. Ñòàòüÿ ïîñâÿùåíà ïðèìåíåíèþ ðàíåå ðàçðàáîòàííîé ìåòîäèêè ïî îïðåäåëåíèþ óïðóãîïëàñòè÷åñêîãî íàïðÿæåííî-äåôîðìèðîâàííîãî ñîñòîÿíèÿ îáîëî÷åê ïðîèçâîëüíîé ôîðìû íà îñíîâå
òåîðèè ìàëûõ óïðóãî-ïëàñòè÷åñêèõ äåôîðìàöèé. Ëèíåàðèçàöèÿ ðåøåíèÿ çàäà÷ âûïîëíÿåòñÿ ìåòîäîì
óïðóãèõ ðåøåíèé. Â êàæäîì ïðèáëèæåíèè ïðèìåíÿåòñÿ âàðèàöèîííîå óðàâíåíèå Ëàãðàíæà â
68
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
ïåðåìåùåíèÿõ òî÷åê ñðåäèííîé ïîâåðõíîñòè îáîëî÷êè â êîíå÷íûõ ðàçíîñòÿõ.
Ïðè çàïèñè âàðèàöè-
îííîãî óðàâíåíèÿ Ëàãðàíæà ñâÿçü ìåæäó íàïðÿæåíèÿìè è äåôîðìàöèÿìè ïðåäñòàâëåíà â ôîðìå çàêîíà Ãóêà, íî ñ äîïîëíèòåëüíûìè ÷ëåíàìè, ó÷èòûâàþùèìè ïëàñòè÷åñêèå äåôîðìàöèè. Ãåîìåòðè÷åñêèå óðàâíåíèÿ ïðèíÿòû â ëèíåéíîé ïîñòàíîâêå â ôîðìå ñîîòíîøåíèé Êîøè. Ìàòåðèàë õàðàêòåðèçóåòñÿ ñîîòâåòñòâóþùåé äèàãðàììîé ðàñòÿæåíèÿ öèëèíäðè÷åñêîãî îáðàçöà, ìîäóëåì íîðìàëüíîé óïðóãîñòè è êîýôôèöèåíòîì Ïóàññîíà. Êèíåìàòè÷åñêèå ãðàíè÷íûå óñëîâèÿ óäîâëåòâîðÿþòñÿ òî÷íî, à
ñòàòè÷åñêèå - íà ñâîáîäíûõ îò çàêðåïëåíèÿ êðàÿõ îáîëî÷êè ïðèáëèæåííî. Ðàññìàòðèâàåòñÿ ïðèìåð
ðàñ÷åòà òîðîèäàëüíîé îáîëî÷êè, óðàâíåíèÿ ñðåäèííîé ïîâåðõíîñòè êîòîðîé çàïèñàíû â ïàðàìåòðè÷åñêîé ôîðìå. Âíóòðåííèé è íàðóæíûé êðàÿ îáîëî÷êè àáñîëþòíî æåñòêî çàäåëàíû, à áîêîâûå ñòîðîíû çàêðåïëåíû øàðíèðíî. Îáîëî÷êà îñëàáëåíà áîëüøèì íåïîäêðåïëåííûì ïðÿìîóãîëüíûì îòâåðñòèåì è çàãðóæåíà íîðìàëüíîé ðàâíîìåðíî ðàñïðåäåëåííîé íàãðóçêîé. Çàäà÷à ðåøàåòñÿ ìåòîäîì óïðóãèõ ðåøåíèé â ñî÷åòàíèè ñ âàðèàöèîííî-ðàçíîñòíûì ìåòîäîì â ïåðåìåùåíèÿõ òî÷åê ñðåäèííîé
ïîâåðõíîñòè îáîëî÷êè. Ðåøåíèå çàäà÷è äîâåäåíî äî ÷èñëåííûõ ðåçóëüòàòîâ. Ïðåäñòàâëåíû ïîëÿ èíòåíñèâíîñòè êàñàòåëüíûõ íàïðÿæåíèé ïî âíóòðåííåé ñâåòîâîé ïîâåðõíîñòè îáîëî÷êè, ïåðåìåùåíèé, ïðîäîëüíûõ ñèë è èçãèáàþùèõ ìîìåíòîâ ïðè óïðóãî-ïëàñòè÷åñêîì ðåøåíèè çàäà÷è äëÿ âñåé
ñåòî÷íîé îáëàñòè. Ðàáîòà ïîñâÿùåíà àêòóàëüíîìó âîïðîñó ñòðîèòåëüíîé ìåõàíèêè îñëàáëåííûõ îáîëî÷åê áîëüøèìè îòâåðñòèÿìè ïðè óïðóãî-ïëàñòè÷åñêîì äåôîðìèðîâàíèè ïðè ïðîñòîì ïðîöåññå íàãðóæåíèÿ.
Êëþ÷åâûå ñëîâà: íåçàìêíóòàÿ òîðîèäàëüíàÿ îáîëî÷êà, âàðèàöèîííî-ðàçíîñòíûé ìåòîä, ãðàíè÷íûå
óñëîâèÿ, ìåòîä óïðóãèõ ðåøåíèé, èíòåíñèâíîñòü äåôîðìàöèé ñäâèãà, èíòåíñèâíîñòü êàñàòåëüíûõ
íàïðÿæåíèé, ôóíêöèÿ ïëàñòè÷íîñòè.
AN ELASTIC-PLASTIC STATE OF A CIRCLE TOROIDAL SHELL WITH
A RECTANGULAR OPENING
V.F. Muschanov, A.I. Demidov
The Donbas National Academy of Civil Engineering and Architecture,
2, Derzhavin str., Makiyivka 86123, Ukraine.
E-mail: mvf@dgasa.dn.ua
Reicieved 12 March 2007; accepted 30 April 2007.
Abstract. This article is devoted to the use of the technique developed for determining a plasto-elastic stressstrain state of arbitrary curvature shells on the base of the theory of small plasto-elastic deformations.
Linearization of task solution is done by the method of elastic solutions. Lagrange's variational equation in
the form of point displacements of a shell middle surface in finite differences is used in every approximation.
When writing down Lagrange's variational equation a functional dependence between stresses and strains
is given in the form of Hooke's law but with additional members taking into account plastic deformations.
Geometrical equations are used in a linear form as Cauchy correlations. A material is characterized by a
corresponding tension diagram of a cylindrical specimen, by a normal elasticity modulus, and Poisson's
ratio. Cinematic boundary conditions are exactly met, and static ones on free loose shell edges are met
approximately. There is considered an example of calculating a toroidal shell middle surface equations of
which are written in a parametrical form. Shell internal and external edges are fastened quite rigidly, and its
side faces are hingedly fastened. A shell is weakened by a large unsupported rectangular opening and is
loaded with a normal uniformly distributed load. The task is solved by the elastic solution method and the
variation-difference method of point displacements of a shell middle surface. The solution is taken as far as
numerical results. There are given the fields of intensity of shearing stresses by a shell internal light surface,
displacements, axial forces, and bending moments at a plasto-elastic solution of the task for the whole net
area. The paper is devoted to an actual problem of structural mechanics of the shells weakened by large
openings at the plasto-elastic deformation in a simple loading.
Keywords: loose toroidal shell, variation-difference method, boundary conditions, elastic solution
method, intensity of shear strains, intensity of shearing stresses, plasticity function.
69
Óïðóãî-ïëàñòè÷åñêîå ñîñòîÿíèå êðóãîâîé òîðîèäàëüíîé îáîëî÷êè ñ ïðÿìîóãîëüíûì îòâåðñòèåì
Ôóíêöèÿ ïëàñòè÷íîñòè À.À. Èëüþøèíà [1]
Ïîñòàíîâêà çàäà÷è
 êíèãå [2] ðàçðàáîòàíà ìåòîäèêà ÷èñëåííîãî
ïðåäñòàâëÿåò òàêîå âûðàæåíèå:
ðàñ÷åòà òîíêèõ óïðóãî-ïëàñòè÷åñêèõ îáîëî÷åê
ïðè ïðîñòûõ ïðîöåññàõ íàãðóæåíèÿ íà îñíîâå ìåòîäà óïðóãèõ ðåøåíèé. Ôèçè÷åñêèå óðàâíåíèÿ ïðèíÿòû â ôîðìå îáîáùåííîãî çàêîíà
Ñåêóùèé ìîäóëü
(
]
H
P
P
]
m
ìîäóëü óïðóãîñòè,
]
H S] S]
]
ãäå
Å
]
(
]
]
H (
]
]
S]
S]
S]
G
H
H
S]
= SÃ
-1
, 2
G Å
=
H
]
H S]
>
]
]
S]
,
ùèõ ñ ïåðåìåùåíèÿìè
]
]
>
]
@
]
,
(5)
]
]
ëàáëåííóþ ïðÿìîóãîëüíûì íåïîäêðåïëåííûì
êðàÿ
êîòîðîãî
,
a1
ñîâïàäàþò
íàïðàâëåíèåì êîîðäèíàòíûõ ëèíèé
è
a2
ñ
(ðèñ. 1). Ñðåäèííóþ ïîâåðõíîñòü ïðåäñòàâèì ïà-
(2)
ðàìåòðè÷åñêèìè óðàâíåíèÿìè:
x
y
]
]
]
R R
R R
z R1
= (
= (
2
2
+
+
a1) cosa2
a1) sin a2
sin a
1
cos
1
cos
1
(6)
]
ïðèíÿ-
D u, v, w, òî÷åê ñðåäèííîé
äåâèàòîðà
äåôîðìàöèé
]
]
]
5 D D
K
D
D 5
D
@ ] D ( i = 1,2,3), à ñðåäíÿÿ äåôîðìà-
öèÿ:
˜
íóþ îáîëî÷êó ïîñòîÿííîé òîëùèíû h=3 ñì, îñ-
îòâåðñòèåì,
êíèãîé Â. Â. Íîâîæèëîâà [4].
LL
]
Ïðèìåð ðàñ÷åòà òîðîèäàëüíîé îáîëî÷êè.
ïîâåðõíîñòè, ïðèíèìàåì â ñîîòâåòñòâèè ñ
LL
(1)
òû â âèäå ëèíåéíûõ ñîîòíîøåíèé, ñâÿçûâàþ-
]
(4)
ª H] H] H] H] º
«
»
« H] H] J ] »
«¬
»¼
]
Îñíîâíûå äåôîðìàöèè
]
m)-1
H Êîìïîíåíòû
]
=
]
(1+
ñîñòîÿíèè íàõîäèì ïî ôîðìóëàì [3, 5]:
,
]
,
s
äåôîðìàöèé ñäâèãà Ã ïðè ïëîñêîì íàïðÿæåííîì
— êîýôôèöèåíò Ïóàñ-
]
íàõî-
Ðàññìîòðèì êðóãîâóþ íåçàìêíóòóþ òîðîèäàëü-
ïðåäñòàâëåíû â âèäå:
S]
G
Èíòåíñèâíîñòü êàñàòåëüíûõ íàïðÿæåíèé S è
ñîíà. Äîïîëíèòåëüíûå ÷ëåíû â óðàâíåíèÿõ (1)
2
,
,
è ìîäóëü ñäâèãà
s
ãî îáðàçöà èç ñîîòíîøåíèé:
6]
P S]
H P
G
V ,
äèì ïî äèàãðàììå ðàñòÿæåíèÿ öèëèíäðè÷åñêî-
Ãóêà, íî ñ äîïîëíèòåëíûìè ÷ëåíàìè, ó÷èòûâàþùèå ïëàñòè÷åñêèå äåôîðìàöèè.
*
*
ïðîèçâîëüíîé ôîðìû ïîñòîÿííîé òîëùèíû
,
]
,
(3)
Ðèñ. 1. Ñõåìà òîðîèäàëüíîé íåçàìêíóòîé îáîëî÷êè
ñ îòâåðñòèåì.
70
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
R1
R2
£a1£p,
£a2£p/2. Ïîëîæåíèå îòâåðñòèÿ îïðåäåëèì ïàa10=3p/10 è a20=p/7.
Íàðóæíûå êðàÿ îáîëî÷êè ïðè a = 0 è a = p ,
1
1
Ïðèìåì çíà÷åíèÿ
=1.5 ì,
Òàáëèöà 1.
= 3.0 ì, 0
0
6K
ðàìåòðàìè
î
î
æåñòêî çàäåëàíû,
6
6K
ãäå ïåðåìåùåíèÿ
â òàáëèöå 1. Ñãóùåíèå ñåòêè äàåò óòî÷íåíèå â
wZ
u=v=w=
w ïðè
;
ïðåäåëàõ 5,4 % ìàêñèìóì. Ïîýòîìó äëÿ ðåàëèçàöèè ÷èñëåííîãî ðàñ÷åòà â êàæäîì ïðèáëè-
a2 = 0 è a 2 = p/2 - øàðíèðíî çàêðåïëåíû,
ãäå ïåðåìåùåíèÿ
îñòàíîâèìñÿ íà ñåòêå ðàçìåðîì 23
u=v=w=0. Îáîëî÷êà èçãîòîâ-
Ïðèâåäåì ðåçóëüòàòû ïîëó÷åííîãî ðåøå-
äèàãðàììîé ðàñòÿæåíèÿ S= (Ã) (ðèñ. 2), ìîäó-
ôèöèåíòîì Ïóàññîíà
m=0,3.
5
íèÿ. Ðåøåíèå âûïîëíåíî ñ òî÷íîñòüþ ñõîäè-
ÌÏà, êîýô-
ìîñòè
f
f(e)
èòåðàöèîííîãî
ïðîöåññà
äî
4,5%.
Ïðîöåññ ñîøåëñÿ íà 21-ì ïðèáëèæåíèè.
Äèàãðàììó S= (Ã) ïîëó÷èì èç äèàãðàììû
ðàñòÿæåíèÿ ñòàëè s=
óçëîâ
(ðèñ.3).
ëåííàÿ èç ñòàëè 12ÕÃÍÌÔ, õàðàêòåðèçóåòñÿ
f
ëåì íîðìàëüíîé óïðóãîñòè Å = 2·10
´17
æåíèè âà ð è à ö è î í í î - ð à ç í î ñ ò í û ì ìåòîäîì
 òàáëèöå 2 ïðèâåäåíû çíà÷åíèÿ èíòåí-
íà îñíîâå ñîîòíî-
ñèâíîñòè êàñàòåëüíûõ íàïðÿæåíèé ïî ëèíèè
øåíèé:
6
,
,
Ïî ãåîìåòðèè îáîëî÷êà èìååò îäíó îñü
ñèììåòðèè. Ïîýòîìó ïðè ðàñ÷åòå ðàññìîòðèì
îäíó å¸ ïîëîâèíó. Îáîëî÷êà íàãðóæåíà íàãðóçêîé
qzz=6
ÌÏà, ïðè êîòîðîé â îòäåëüíûõ
ýëåìåíòàõ å¸ âîçíèêàþò ïëàñòè÷åñêèå äåôîð-
ìàöèè.
D
Äëÿ îïðåäåëåíèÿ íàïðÿæåííî-äåôîðìèðî-
âàííîãî ñîñòîÿíèÿ âîñïîëüçóåìñÿ ìåòîäîì
óïðóãèõ ðåøåíèé. Ñíà÷àëà äëÿ âûáîðà íåîáõî-
äèìîé ãóñòîòû ñåòêè ïðè ðåøåíèè çàäà÷è
óïðóãîñòè â êàæäîì ïðèáëèæåíèè îáîëî÷êà
´10
´ 17
áûëà ðàññ÷èòàíà íà äåéñòâèå íàãðóçêè
ñ ñåòêàìè 13
è 23
qzz=1 ÌÏà
´17 óçëîâ.
óçëîâ. Ðåçóëüòàòû
Ðèñ. 3. Ñõåìà ñåòî÷íîé îáëàñòè òîðîèäàëüíîé îáî-
ðàñ÷åòà äëÿ ñîîòâåòñòâóþùèõ óçëîâ ïîêàçàíû
6
ëî÷êè ñ îòâåðñòèåì 23
D
Ðèñ. 2. Äèàãðàììà ðàñòÿæåíèÿ ñòàëè 12ÕÃÍÌÔ.
Óïðóãî-ïëàñòè÷åñêîå ñîñòîÿíèå êðóãîâîé òîðîèäàëüíîé îáîëî÷êè ñ ïðÿìîóãîëüíûì îòâåðñòèåì
71
Òàáëèöà 2.
óçëîâ
M
6 K
6
6K
6 K
6
6K
j = 10, ïðîõîäÿùåé ÷åðåç óãëîâûå òî÷êè
çàäà÷è â óçëå 14, 8 ïðè
z = -h/2 ñîñòàâèëî 750,661
îòâåðñòèÿ (óçëû 8,10 è 16,10) ïðè óïðóãîì è
ÌÏà, à ïðè óïðóãî-ïëàñòè÷åñêîì — 426,517 ÌÏà.
óïðóãî-ïëàñòè÷åñêîì ðåøåíèè çàäà÷è ïî òîë-
Òàêèì îáðàçîì, çà ñ÷åò ïåðåðàñïðåäåëåíèÿ ïðî-
ùèíå
h
îáîëî÷êè, ÷òî ïðîèëëþñòðèðîâàíî
èçîøëî ñíèæåíèå èíòåíñèâíîñòè êàñàòåëüíûõ íà-
ñîîòâåòñòâóþùèìè ãðàôèêàìè (ðèñóíêè 4, 5).
ïðÿæåíèé âáëèçè óãëîâîé òî÷êè (14,8) íà 43,18%.
Íàèáîëüøåå çíà÷åíèå èíòåíñèâíîñòè êàñàòåëü-
Ñëåäóåò çàìåòèòü, ÷òî êîíöåíòðàöèÿ èíòåíñèâ-
íûõ íàïðÿæåíèé S ïðè óïðóãîì ðåøåíèè
íîñòè êàñàòåëüíûõ íàïðÿæåíèé S â óãëîâîé òî÷êå îòâåðñòèÿ 14,8 çíà÷èòåëüíåå, íåæåëè â òî÷êå
6,8. Ïðè ïîñòðîåíèè ãðàôèêîâ çà óçåë 0, 0 ïðè-
Òàáëèöà 4.
M
íÿò óçåë 2,2 ñåòêè (ðèñ. 3).
L
Z
Z
X
X
 òàáëèöå 3 ïðèâåäåíî ïîëå èíòåíñèâíîñòè
êàñàòåëüíûõ íàïðÿæåíèé S ïî âñåé ñåòî÷íîé îá-
ëàñòè çàäàííîé îáîëî÷êè ïðè
z=-h/2. Íà ðèñ. 6
ïîêàçàíî íàãëÿäíîå ïðåäñòàâëåíèå èçìåíåíèÿ
èíòåíñèâíîñòè êàñàòåëüíûõ íàïðÿæåíèé S ïî
âñåé ñåòî÷íîé îáëàñòè ïî âíóòðåííåé ñâåòîâîé ïîâåðõíîñòè îáîëî÷êè. Íà ýòîì ðèñóíêå
âèäíû ïëàñòè÷åñêèå îáëàñòè, ãäå èíòåíñèâíîñòü
êàñàòåëüíûõ íàïðÿæåíèé ïðåâûøàåò 242,5 ÌÏà.
u è w ïî
j = 14 äàíû â òàáëèöå 4 ïðè óïðóãîì è
Íàèáîëüøèå çíà÷åíèÿ ïåðåìåùåíèé
ðÿäó óçëîâ
è òàáëèöû 4 âèäíî, ÷òî ïðè ðåøåíèè çàäà÷è ñ
ó÷åòîì óïðî÷íåíèÿ ìàòåðèàëà îáîëî÷êè ïåðåìå-
óïðóãî-ïëàñòè÷åñêîì ðåøåíèè çàäà÷è. Èç ðèñ. 7
ùåíèÿ
u è w âîçðàñòàþò. Íàèáîëüøåå çíà÷åíèå
72
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
6
6K
6
6K
D
Ðèñ. 4. Èíòåíñèâíîñòü êàñàòåëüíûõ íàïðÿæåíèé ïðè óïðóãîì ñîñòîÿíèè òîðîèäàëüíîé îáîëî÷êè.
6
6K
6
6K
D
Ðèñ.5. Èíòåíñèâíîñòü êàñàòåëüíûõ íàïðÿæåíèé ïðè óïðóãî-ïëàñòè÷åñêîì ñîñòîÿíèè òîðîèäàëüíîé
îáîëî÷êè.
w âîçðîñëî íà 7,24 %, à ïåðåìåùåíèÿ u –
Ðåçóëüòàòû ðàñ÷åòà ïîêàçûâàþò, ÷òî ñòàòè÷åñ-
íà 9,98%. Ðåçóëüòàòû ðàñ÷åòà ïîêàçûâàþò, ÷òî
êèå ãðàíè÷íûå óñëîâèÿ íà ñâîáîäíûõ êðàÿõ îò-
êèíåìàòè÷åñêèå ãðàíè÷íûå óñëîâèÿ íà íàðóæ-
âåðñòèÿ îáîëî÷êè óäîâëåòâîðÿþòñÿ ïðèáëèæåí-
íûõ êðàÿõ îáîëî÷êè óäîâëåòâîðÿþòñÿ òî÷íî.
íî. Ìàêñèìàëüíîå çíà÷åíèå èçãèáàþùåãî ìîìåí-
ïðîãèáà
 òàáëèöàõ 5 è 6 ïðèâåäåíû çíà÷åíèÿ èçãèáàþùèõ ìîìåíòîâ Ì
1
è ïðîäîëüíûõ ñèë ïî ëèíè-
ÿì óçëîâ îò j=9 äî j=14, êîòîðûå ñîäåðæàò êðàÿ
a2
îòâåðñòèÿ, êîòîðûå ïàðàëëåëüíû êîîðäèíàòíîé
ëèíèè
îáîëî÷êè.
ïðè óïðóãîïëàñòè÷åñêîì ñîñòîÿíèè
òà Ì =-77,923 êÍì/ì â óçëå 20,14, à íà ñâîáîä-
1
íîì êðàþ åãî çíà÷åíèå ñîñòàâèëî Ì
1
-4,174
êÍì/ì â âóçëå 14,14, ÷òî ñîñòàâèëî 5,36% îò
ìàêñèìàëüíîãî çíà÷åíèÿ.
Ïðè ïðîäîëüíûõ ñèëàõ N
1
ïî íàïðàâëåíèþ
óçëîâ i (a ) ïðîñëåæèâàåòñÿ òåíäåíöèÿ ê èõ
1
73
Óïðóãî-ïëàñòè÷åñêîå ñîñòîÿíèå êðóãîâîé òîðîèäàëüíîé îáîëî÷êè ñ ïðÿìîóãîëüíûì îòâåðñòèåì
Ðèñ. 6. Èíòåíñèâíîñòü êàñàòåëüíûõ íàïðÿæåíèé S(-h/2) â óïðóãîïëàñòè÷åñêîì ñîñòîÿíèè òîðîèäàëüíîé
îáîëî÷êè.
XZ
Z
Z
X
X
D
Ðèñ. 7. Ïåðåìåùåíèÿ u, w ïðè óïðóãîì è óïðóãî-ïëàñòè÷åñêîì ñîñòîÿíèè òîðîèäàëüíîé îáîëî÷êè ñ îòâåðñòèåì ïî ëèíèè óçëîâ j=14.
74
Òàáëèöà 3.
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
75
Óïðóãî-ïëàñòè÷åñêîå ñîñòîÿíèå êðóãîâîé òîðîèäàëüíîé îáîëî÷êè ñ ïðÿìîóãîëüíûì îòâåðñòèåì
Òàáëèöà 5.
L
M
M
M
M
M
M
Òàáëèöà 6.
L
M
M
1
M
M
M
M
óáûâàíèþ ïî ìåðå ïðèáëèæåíèÿ ê íåçàêðåï-
ðîâàííîãî ñîñòîÿíèÿ òîíêîé òîðîèäàëüíîé
ëåííîìó êðàþ îòâåðñòèÿ. Èõ çíà÷åíèÿ âû÷èñ-
îáîëî÷êè ñ ïðÿìîóãîëüíûì âûðåçîì, êðàÿ êî-
ëÿþòñÿ ÷åðåç îäíîñòîðîííèå ðàçíîñòè íå íå-
òîðîãî ïàðàëëåëüíûì êîîðäèíàòíûì ëèíèÿì
ïîñðåäñòâåííî íà êðàÿõ îòâåðñòèÿ, à â óçëàõ â
è
ñåðåäèíå ÿ÷åéêè îñíîâíîé ñåòêè. Ïîýòîìó
òîäó óïðóãèõ ðåøåíèé â ñî÷åòàíèè ñ ìåòîäîì
ìîæíî ñ÷èòàòü, ÷òî è ïî ïðîäîëüíûì ñèëàì N
êîíå÷íûõ ðàçíîñòåé íà áàçå âàðèàöèîííîãî
ñòàòè÷åñêèå ãðàíè÷íûå óñëîâèÿ óäîâëåòâî-
óðàâíåíèÿ Æ. Ëàãðàíæà.
1
a
2,
a1
íà îñíîâå ðàçðàáîòàííîé ìåòîäèêè ïî ìå-
ðÿþòñÿ, õîòÿ è ïðèáëèæåííî, íî äîñòàòî÷íî
òî÷íî.
Âûâîäû
Òàêèì îáðàçîì, ìîæíî ñäåëàòü âûâîä î òîì, ÷òî
Ëèòåðàòóðà
1. Èëüþøèí À.À., Ëåíñêèé Â.Ñ. Ñîïðîòèâëåíèå
ìàòåðèàëîâ.— Ì: Ôèçìàòãèç, 1959.— 372ñ.
2. Èçáðàííûå ìåòîäû ñòðîèòåëüíîé ìåõàíèêè â
ðàñ÷åòàõ
ïîëó÷åíî ðåøåíèå çàäà÷è ïî îïðåäåëåíèþ óïðóãî-ïëàñòè÷åñêîãî íàïðÿæåííî-äåôîðìè-
ïðîñòðàíñòâåííûõ
êîíñòðóêöèé/
Ïîä îáùåé ðåä. ä.ò.í., ïðîô. Â.Ô. Ìóùàíîâà. —
Ìàêååâêà, 2006. — C.55-136.
76
Â.Ô. Ìóùàíîâ, À.È. Äåìèäîâ
0 D
M
M
M
M
M
M
Ðèñ. 8. Èçãèáàþùèå ìîìåíòû Ì1 ïî ðÿäàì óçëîâ îò j =9 äî j = 14.
1
M
M
M
M
M
M
D
Ðèñ. 9. Ïðîäîëüíûå ñèëû N
1
ïî ðÿäàì óçëîâ îò j = 9 äî j = 14
3. Ìóùàíîâ Â.Ô., Äåìèäîâ À.È. Ëèíåéíûå è íåëè-
6. Øåâ÷åíêî Þ.Í.,Ïðîõîðåíêî È.Â. Ìåòîäû ðàñ÷å-
íåéíûå çàäà÷è òåîðèè óïðóãîñòè â ðàñ÷åòàõ òîí-
òà îáîëî÷åê. Òåîðèÿ óïðóãî-ïëàñòè÷åñêèõ îáîëî-
êîñòåííûõ êîíñòðóêöèé. — Ìàêååâêà: ÐÈÑ ÎÌÑ
÷åê ïðè íåèçîòåðìè÷åñêèõ ïðîöåññàõ íàãðóæåíèÿ.-
ÄîíÃÀÑÀ, 2000, 182 ñ.
Êèåâ: Íàóê. Äóìêà, 1981. — 296 ñ.
4. Äåìèäîâ À.È. Óïðóãî-ïëàñòè÷åñêîå íàïðÿæåííî-
7. Sen S. K., Gould P. L. Critweria for finite element
äåôîðìèðîâàííîå ñîñòîÿíèå òîíêîé íåçàìêíóòîé
discretization of shells of revolution.—Int. J. for Numerical
òîðîèäàëüíîé îáîëî÷êè: — Ñá. Ñîâðåìåííîå ïðîìûøëåííîå è ãðàæäàíñêîå ñòðîèòåëüñòâî, ÄîíÍÀÑÀ ò. 2, íîìåð 4, 2006, ñ. 163–176.
5. Íîâîæèëîâ Â.Â. Òåîðèÿ òîíêèõ îáîëî÷åê. — Ëåíèíãðàä: Ñóäïðîìãèç, 1962.— 432 ñ.
Metods in Engineering, 1973, 6, 2, Ð. 265–274.
8. Stang G. Linear algebra and its applications. – New
York, Sanfracisco, London: 1976. — Ð. 456
9. Êîëàðîâ Ä., Áîí÷åâà Í. Ìåõàíèêà íà ïëàñòè÷íèòå ñðåäè.— Ñîôèÿ: Èçäàòåëñòâî íà Áúëãàðñêàòà
Àêàäåìèÿ íà íàóêèòå, 1975. — 510ñ.
Óïðóãî-ïëàñòè÷åñêîå ñîñòîÿíèå êðóãîâîé òîðîèäàëüíîé îáîëî÷êè ñ ïðÿìîóãîëüíûì îòâåðñòèåì
77
Ìóùàíîâ Âîëîäèìèð Ïèëèïîâè÷ ïðàöþº çàâ³äóâà÷åì êàôåäðè « Òåîðåòè÷íà ³ ïðèêëàäíà ìåõàí³êà », ïðîðåêòîðîì ç íàóêîâî¿ ðîáîòè Äîíáàñüêî¿ íàö³îíàëüíî¿ àêàäå쳿 áóä³âíèöòâà ³ àðõ³òåêòóðè. ×ëåí ì³æíàðîäíî¿ îðãàí³çàö³¿ „Ïî ìîñòàì ³ áóä³âåëüíèì êîíñòðóêö³ÿì” òà ì³æíàðîäíî¿ àñîö³àö³¿ “Ïðîñòîðîâ³ êîíñòðóêö³¿”, àóäèòîð ñèñòåìè ñåðòèô³êàö³¿ ÓêðÑÅÏÐÎ. Íàóêîâ³ ³íòåðåñè: òåîð³ÿ íàä³éíîñò³, ðîçðàõóíîê, ïðîåêòóâàííÿ òà òåõí³÷íà ä³àãíîñòèêà ïðîñòîðîâèõ ìåòàëåâèõ êîíñòðóêö³é.
Äåìèäîâ Îëåêñàíäð ²âàíîâè÷ ïðàöþº íà ïîñàä³ äîöåíòà êàôåäðè “Òåîðåòè÷íà ³ ïðèêëàäíà ìåõàí³êà”, ñåêö³ÿ
“Îï³ð ìàòåð³àë³â”. Íàóêîâ³ ³íòåðåñè : ô³çè÷í³ ë³í³éí³ ³ íåë³í³éí³ çàäà÷³ òåî𳿠òîíêèõ îáîëîíîê ïîñò³éíî¿ ³ çì³ííî¿
òîâùèíè.
Ìóùàíîâ Âëàäèìèð Ôèëèïïîâè÷ ðàáîòàåò
çàâåäóþùèì êàôåäðîé òåîðåòè÷åñêîé è ïðèêëàäíîé ìåõàíèêè,
ïðîðåêòîð ïî íàó÷íîé ðàáîòå. ×ëåí ìåæäóíàðîäíîé îðãàíèçàöèè “Ïî ìîñòàì è ñòðîèòåëüíûì êîíñòðóêöèÿì”
è ìåæäóíàðîäíîé àññîöèàöèè “Ïðîñòðàíñòâåííûå êîíñòðóêöèè”, àóäèòîð ñèñòåìû ñåðòèôèêàöèè ÓêðÑÅÏÐÎ. Íàó÷íûå èíòåðåñû: òåîðèÿ íàäåæíîñòè, ðàñ÷åò, ïðîåêòèðîâàíèå è òåõíè÷åñêàÿ äèàãíîñòèêà ïðîñòðàíñòâåííûõ ìåòàëëè÷åñêèõ êîíñòðóêöèé.
Äåìèäîâ Àëåêñàíäð Èâàíîâè÷ ðàáîòàåò â äîëæíîñòè äîöåíòà êàôåäðû “Òåîðåòè÷åñêàÿ è ïðèêëàäíàÿ ìåõàíèêà”, ñåêöèÿ “Ñîïðîòèâëåíèå ìàòåðèàëîâ”. Íàó÷íûå èíòåðåñû: ôèçè÷åñêè ëèíåéíûå è íåëèíåéíûå çàäà÷è òåîðèè òîíêèõ îáîëî÷åê ïîñòîÿííîé è ïåðåìåííîé òîëùèíû.
Muschanov Volodimir Pilipovich is a Principal of the Department Theoretical and Applied Mechanics, vice-rector of
Donbas National Academy of Civil Engineering and Architecture. He is a member of International Association for
Bridge and Structural Engineering, and member of International Association of Spatial Structures, auditor of certification
scheme UKRSEPRO. His research interests include the reliability theory, calculation, designing and engineering
diagnostics of spatial metal structures.
Demidov Alexander Ivanovich is an Associate Professor of the Department of Theoretical and Applied Mechanics,
Materials Resistance Section.
Research interests: Physically linear and non-linear problems of the theory of thin
coverages of a constant and variable thickness.