An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure
При проектировании оболочечных конструкций важно учитывать их упругое схлопывание, поскольку оно может привести к разрушению конструкций. В частности, для обеспечения надежности проектирования следует учитывать изменение уровня нагрузки потери устойчивости вследствие уменьшения толщины стенок констр...
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| Дата: | 2012 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2012
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| Назва видання: | Проблемы прочности |
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| Цитувати: | An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure / M. Ishinabe, K. Hayashi // Проблемы прочности. — 2012. — № 2. — С. 125-134. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-960802016-03-11T03:02:37Z An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure Ishinabe, M. Hayashi, K. Научно-технический раздел При проектировании оболочечных конструкций важно учитывать их упругое схлопывание, поскольку оно может привести к разрушению конструкций. В частности, для обеспечения надежности проектирования следует учитывать изменение уровня нагрузки потери устойчивости вследствие уменьшения толщины стенок конструкции. Для решения подобных задач используется конечноэлементный метод длины дуги. Однако с помощью этого метода не всегда можно оценить траекторию изменения нагрузки схлопывания. Метод длины дуги был использован при упругом схлопывании пологого фрагмента сферической оболочки. Предложен новый алгоритм в явной конечноэлементной постановке для оценки минимальной прочности тонкостенных конструкций, согласно которому начальная деформация задается путем прижатия горизонтальной жесткой плиты к верхней точке схлопывающейся конструкции и ее перемещения по вертикали. Предполагается, что метод обеспечит хорошее инженерное решение для оценки минимальной нагрузки, при которой происходит частичное упругое схлопывание оболочечных конструкций общего вида под давлением. При проектуванні оболонкових конструкцій важливо враховувати їх пружне схлопування, оскільки воно може призвести до руйнування конструкцій. Зокрема, для забезпечення надійності проектування необхідно враховувати зміни рівня навантаження втрати стійкості внаслідок зменшення товщини стінок конструкції. Для розв’язання подібних задач використовується скінченноелементний метод довжини дуги. Однак за допомогою цього методу не завжди можливо оцінити траєкторію зміни навантаження схлопування. Метод довжини дуги використовували при пружному схлопуванні пологого фрагменту сферичної оболонки. Запропоновано новий алгоритм у явній скінченноелементній постановці для оцінки мінімальної міцності тонкостінних конструкцій, згідно з яким початкова деформація задається шляхом притиснення горизонтальної жорсткої плити до верхньої точки конструкції, що схлопується, та її переміщення по вертикалі. Припускається, що метод забезпечить добрий інженерний розв’язок для оцінки мінімального навантаження, за якого відбувається часткове пружне схлопування оболонкових конструкцій загального вигляду під тиском. 2012 Article An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure / M. Ishinabe, K. Hayashi // Проблемы прочности. — 2012. — № 2. — С. 125-134. — Бібліогр.: 11 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/96080 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Ishinabe, M. Hayashi, K. An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure Проблемы прочности |
| description |
При проектировании оболочечных конструкций важно учитывать их упругое схлопывание, поскольку оно может привести к разрушению конструкций. В частности, для обеспечения надежности проектирования следует учитывать изменение уровня нагрузки потери устойчивости вследствие уменьшения толщины стенок конструкции. Для решения подобных задач используется конечноэлементный метод длины дуги. Однако с помощью этого метода не всегда можно оценить траекторию изменения нагрузки схлопывания. Метод длины дуги был использован при упругом схлопывании пологого фрагмента сферической оболочки. Предложен новый алгоритм в явной конечноэлементной постановке для оценки минимальной прочности тонкостенных конструкций, согласно которому начальная деформация задается путем прижатия горизонтальной жесткой плиты к верхней точке схлопывающейся конструкции и ее перемещения по вертикали. Предполагается, что метод обеспечит хорошее инженерное решение для оценки минимальной нагрузки, при которой происходит частичное упругое схлопывание оболочечных конструкций общего вида под давлением. |
| format |
Article |
| author |
Ishinabe, M. Hayashi, K. |
| author_facet |
Ishinabe, M. Hayashi, K. |
| author_sort |
Ishinabe, M. |
| title |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure |
| title_short |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure |
| title_full |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure |
| title_fullStr |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure |
| title_full_unstemmed |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure |
| title_sort |
algorithm for estimating minimum strength of thin-walled structures to resist elastic buckling under pressure |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2012 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/96080 |
| citation_txt |
An Algorithm for Estimating Minimum Strength of Thin-Walled Structures to Resist Elastic Buckling under Pressure / M. Ishinabe, K. Hayashi // Проблемы прочности. — 2012. — № 2. — С. 125-134. — Бібліогр.: 11 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-07T03:15:34Z |
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2025-07-07T03:15:34Z |
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1836956395594842112 |
| fulltext |
UDC 539.4
An Algorithm for Estimating Minimum Strength of Thin-Walled
Structures to Resist Elastic Buckling under Pressure
M. Ishinabe
a,1
and K. Hayashi
b,2
a Chubu University, Kasugai, Japan
b JSOL Corporation, Nagoya, Japan
1 ishinabe@isc.chubu.ac.jp
2 hayashi.kimihiro@jsol.co.jp
ÓÄÊ 539.4
Àëãîðèòì îöåíêè ìèíèìàëüíîé ïðî÷íîñòè òîíêîñòåííûõ
êîíñòðóêöèé ïîä äàâëåíèåì ïî êðèòåðèþ óïðóãîãî ñõëîïûâàíèÿ
Ì. Èøèíàáå
à,1
, Ê. Õàÿøè
á,2
à Óíèâåðñèòåò ×óáó, Êàñóãàè, ßïîíèÿ
á Êîðïîðàöèÿ JSOL, Íàãîÿ, ßïîíèÿ
Ïðè ïðîåêòèðîâàíèè îáîëî÷å÷íûõ êîíñòðóêöèé âàæíî ó÷èòûâàòü èõ óïðóãîå ñõëîïûâàíèå,
ïîñêîëüêó îíî ìîæåò ïðèâåñòè ê ðàçðóøåíèþ êîíñòðóêöèé.  ÷àñòíîñòè, äëÿ îáåñïå÷åíèÿ
íàäåæíîñòè ïðîåêòèðîâàíèÿ ñëåäóåò ó÷èòûâàòü èçìåíåíèå óðîâíÿ íàãðóçêè ïîòåðè óñòîé-
÷èâîñòè âñëåäñòâèå óìåíüøåíèÿ òîëùèíû ñòåíîê êîíñòðóêöèè. Äëÿ ðåøåíèÿ ïîäîáíûõ çàäà÷
èñïîëüçóåòñÿ êîíå÷íîýëåìåíòíûé ìåòîä äëèíû äóãè. Îäíàêî ñ ïîìîùüþ ýòîãî ìåòîäà íå
âñåãäà ìîæíî îöåíèòü òðàåêòîðèþ èçìåíåíèÿ íàãðóçêè ñõëîïûâàíèÿ. Ìåòîä äëèíû äóãè áûë
èñïîëüçîâàí ïðè óïðóãîì ñõëîïûâàíèè ïîëîãîãî ôðàãìåíòà ñôåðè÷åñêîé îáîëî÷êè. Ïðåäëîæåí
íîâûé àëãîðèòì â ÿâíîé êîíå÷íîýëåìåíòíîé ïîñòàíîâêå äëÿ îöåíêè ìèíèìàëüíîé ïðî÷íîñòè
òîíêîñòåííûõ êîíñòðóêöèé, ñîãëàñíî êîòîðîìó íà÷àëüíàÿ äåôîðìàöèÿ çàäàåòñÿ ïóòåì ïðè-
æàòèÿ ãîðèçîíòàëüíîé æåñòêîé ïëèòû ê âåðõíåé òî÷êå ñõëîïûâàþùåéñÿ êîíñòðóêöèè è åå
ïåðåìåùåíèÿ ïî âåðòèêàëè. Ïðåäïîëàãàåòñÿ, ÷òî ìåòîä îáåñïå÷èò õîðîøåå èíæåíåðíîå
ðåøåíèå äëÿ îöåíêè ìèíèìàëüíîé íàãðóçêè, ïðè êîòîðîé ïðîèñõîäèò ÷àñòè÷íîå óïðóãîå
ñõëîïûâàíèå îáîëî÷å÷íûõ êîíñòðóêöèé îáùåãî âèäà ïîä äàâëåíèåì.
Êëþ÷åâûå ñëîâà: òîíêîñòåííàÿ êîíñòðóêöèÿ, êîðïóñ êîðàáëÿ, íàïðÿæåíèå
óïðóãîãî ñõëîïûâàíèÿ, ìåòîä äëèíû äóãè, ìåòîä êîíå÷íûõ ýëåìåíòîâ, ïîëî-
ãàÿ ñôåðè÷åñêàÿ îáîëî÷êà.
Introduction. Elastic buckling, such as snap-through, that occurs in thin-
walled structures subject to pressure is an important issue in the design of
thin-walled structures. Radha and Rajagopalan [1] studied and identified failures in
shell structures that occur because of elastic buckling. Huang [2] and Thurston [3]
pointed out that dispersions in buckling strength caused by reducing the wall
thickness of a structure is especially important in terms of design safety.
In general, the arc-length method, which is a finite element method (FEM) of
analysis, is applied to solve this problem. We confirmed the effectiveness of the
© M. ISHINABE, K. HAYASHI, 2012
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2 125
arc-length method in solving the elastic snap-through buckling problem in a partial
shallow spherical shell. We focused on the dispersion of the buckling strength and
found that the arc-length method is effective in calculating the lowest value of
buckling mode in most cases. However, we also confirmed there were some cases
in which the calculation cannot be solved. Cerini and Falzon [4] studied and
confirmed the reliability of the method. Therefore, we formulated a new algorithm
to estimate the minimum buckling load in a shallow partial spherical shell and
report it below. This method is expected to be used to estimate the minimum load
on the partial elastic snap-through buckling load of general thin-shell structures
under pressure.
Figure 1 shows the specific buckling problem studied in this paper. The main
aim of this study is to evaluate this problem, which exerts pressure on the upper
side of a shallow spherical shell. This subject has already been reported by Huang
[2] and Thurston [3]. As Fig. 2 shows, experimental buckling loads are significantly
smaller than the numerical solutions given in classical buckling theory. It also
shows the dispersion of the experimental results. In other words, this means the
experimental results are unstable and the strength of the structure is smaller
compared with the theoretical solution, as the wall thickness decreases.
Judging from our experience, it may lead to a misinterpretation that these
differences are considered to be a special case because we cannot say the lower
buckling mode never triggers a dynamic collapse deformation at large due to the
disruption of the balance of whole structures by the partial elastic snap-through.
In this study, we show that in most cases, the arc-length method is effective in
estimating the lower limit of buckling loads as stated earlier. We have derived a
solution for this problem by applying the arc-length method (implicit LS-DYNA
method) that is easier for designers to use and obtained the same results as those
proposed by Thurston [3] regarding converged cases. However, we could not
obtain a solution in the case of less thickness and deeper profile.
Figure 3 shows elastic buckling with snap-through. Point A shows the upper
limit strength for buckling, point E shows the minimum buckling strength, which is
the focus of this study, and point F shows imperfection effects, which have been
studied by many researchers [4–7].
Fig. 1. Geometry of clamped shallow spherical shells.
126 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2
M. Ishinabe and K. Hayashi
We have been recognized from our experience that the buckling calculation
with changing shape as initial imperfection must be careful.
In this paper, we propose a new algorithm using initial deformation to
calculate the minimum buckling strength without changing the initial shape of
structures. This method is applied to a case in which pressure is exerted on upper
side of a partial spherical shell, as shown in Fig. 1. The aim of this study is also to
reduce the number of procedures required for calculating correct bifurcation
buckling as marked by points C or D in Fig. 3. This algorithm assumes a rigid wall
in the center of a partial spherical shell that slightly deforms the center in the
downward direction as a result of the contact to give an effective initial deformation.
Furthermore, this method causes deformations such as snap-through, which are
generated by changing the amount of indentation on a rigid wall by the FEM
method. That is, we propose an algorithm to calculate the path of F in Fig. 3.
1. Buckling with Snap-Through in a Clamped Partial Spherical Shell.
Huang [2] and Thurston [3] studied and compared theoretical solutions of classical
buckling theory and buckling experimental results. They define a, r , t, H , �
(angle), and P (distributed load) in Fig. 1.
Value of � (horizontal axis in Fig. 2) is a standardized shape parameter of
Eq. (1) and each variable correspond to the ones in Fig. 1. Value of � is Poisson’s
ratio and 0.3 is the value under consideration. The increase of � corresponds to a
reduction in the wall thickness of the partial spherical shell. The actual load value
decreases remarkably in comparison with the theoretical value given by the
axisymmetric theory and also varies greatly.
Fig. 2. Calculated solutions for classical theory and experimental results for clamped shallow
spherical shells.
Fig. 3. Pressure–deflection curve of clamped shallow spherical shells.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2 127
An Algorithm for Estimating Minimum Strength ...
Consider Eqs. (1)–(3):
� �� �
�
�
�
�2 3 1 2 0 25
0 5
[ ( )] ,.
.
H
t
(1)
P
q
qcr
ex
�
0
, (2)
q
E t
r0
2
2
2
3 1
�
�
�
�
�
�
( )
,
�
(3)
where Pcr on the vertical axis is the standardized load parameter defined in Eq. (2),
qex is an experimental value, q0 is a numerical solution for the classical buckling
theory [7], and E is the Young modulus (210 GPa). The other variables also
correspond to the letters in Fig. 1.
The reduction in wall thickness corresponds to the increase of � on the
horizontal axis, as shown in Fig. 2. As � increases, the minimum buckling load of
the actual structure becomes notably smaller. This tendency is generally common
to the thin-wall structures. As shown in Fig. 3, elastic buckling is triggered when
the relative thickness of the structure is decreased, even if structures are made in
the same way. The load rarely reaches A as shown in Fig. 3, but snap-through
buckling occurs when the load nears C in the figure before reaching A. The path
depends on the influence of the initial imperfection, which generates various
results. In Fig. 3, F is an example of the path.
2. Calculation of Minimum Buckling Strength by the Arc-Length Method.
The calculation result is shown below by using � in Eq. (1), which is affected by
shape and thickness parameters in Fig. 1.
Consider Eqs. (4) and (5):
x
p
q
�
0
, (4)
y
u
t
�
�
�
�
�. (5)
The objects of the examination are cases in which � varies from 3.8 to 9.2.
We show the results of a study in which it is assumed that r is 200 mm, � is 0.31
rad, p in Eq. (4) is pressure (MPa), X x� 16523. is a standardized load
parameter corresponding to Eq. (2), u in Eq. (5) is the downward displacement
(mm) of the arc center, and y is a standardized displacement parameter. Thurston
calculated the problem under these conditions and reported the results that are
shown in Fig. 4. He checked the minimum buckling load value against Eqs. (1)–
(3), coordinated the relation to Fig. 2, and showed the results when � is 6.7 and
8.9 in Fig. 4. However, the scale of the horizontal axis in the study of Thurston is
squared �.
128 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2
M. Ishinabe and K. Hayashi
Next, we show the result of applying the arc-length method. Figure 5 shows
the convergence results when �� 3.8 and 4.6, and the result that did not converge
when �� 6.5. As far as we determined, the calculation converges when �
4 6.
and does not converge when ��5 3. . The double underlined numbers in the figure
are minimum buckling load values. Figure 4 shows these results as a whole. The
square symbols in Fig. 4 are the results of a calculation using the arc-length
method. Solutions calculated by Thurston in Fig. 4 and the convergence solution
obtained by the arc-length method are in good agreement. This is the reason we
decide that it is effective to apply the arc-length method to calculate the minimum
snap-through buckling load in Fig. 5 when the calculation converges.
Therefore, we investigated whether there is an effective method for a case in
which ��5 3. . As we described, an analysis of snap-through buckling by applying
an explicit method has been studied and we can find many reports regarding these
studies. However, these studies focus on the deformation that occurs when load A
Fig. 4. Calculated lowest buckling pressure for an axisymmetric mode by Thurston and the arc-length
solution with the experimental results.
Fig. 5. Calculated critical snap-through arc-length solution results for clamped shallow spherical
shells.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2 129
An Algorithm for Estimating Minimum Strength ...
jumps or after the load A jumps, and they do not show a method to calculate the
minimum snap-through buckling load that is shown as C in Fig. 3.
We have elaborated a new method to calculate the effective minimum
snap-through buckling value in the case that the arc-length method does not
converge. We present this method in the next section.
3. Proposed New Algorithm and Evaluation. As described previously, a
calculation performed by using the arc-length method does not converge when
��5 3. . This means it is impossible to calculate the minimum value that is shown in
Fig. 3 as an evaluation value of the buckling load. Here, we propose a novel
method, instead of using the arc-length method, which overcomes this challenge
and calculates the lower value as described in next section.
3.1. Algorithm.
(i) Calculate the 1st eigenmode under the pressure loading to obtain a point of
maximum displacement and its unit vector. (This procedure is not included in this
study.)
(ii) A rigid wall (Fig. 6) is placed in contact with the maximum displacement
point, which was obtained in (i). Match the movement vector with the unit vector
and move the rigid wall slightly. The movement velocity is given incrementally in
one second, that is, it should be a quasi-static loading. This calculation is
conducted using the explicit method. A specific value is decided as the movement
distance, which is from almost 1 to 6 times the wall thickness.
(iii) A pressure load is given incrementally in one second, that is, it should be
a quasi-static loading.
(iv) Plot the results of (iii) against the maximum displacement obtained in (i).
The load value when snap-through buckling occurs is decided as the evaluation
load.
(v) Change the movement distance and then iterate from (ii) to (iv). This
calculation omits the procedure to obtain the 1st eigenmode because it is clear that
the mode by which the center of Fig. 1 moves the furthest downward is the 1st
eigenmode.
3.2. Evaluation of Cases Applicable for the Arc-Length Method. Here, we
show that the minimum snap-through buckling strength agrees with the solution in
Fig. 4 as Thurston calculated [3] and the results calculated by the arc-length
method. Figure 7 shows the results of a calculation in which �� 3.8, which agrees
with the results calculated by Thurston and the results calculated by the arc-length
method. Value of t in Fig. 7 indicates the pushed distance described in Section 3.1.
Fig. 6. Scheme of the proposed method.
130 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2
M. Ishinabe and K. Hayashi
The point of which X �0 in Fig. 7 is non-displacement of the center point
when the rigid wall force into the structure with the given distance. However the
transverse value of the figure does not correspond to pushed distance because
elastic buckling has already occurred at this time. Figure 8 shows a deformed state.
The maximum displacement of the center in Fig. 8 corresponds to y in Fig. 7.
Fig. 7. Calculated critical snap-through arc-length solution results for � � 3.8 by present solutions
(1.3t, 1.5t, 1.7t, and 2t), explicit solution (t), and arc-length solution for clamped shallow spherical
shells.
Fig. 8. Buckling shape of the top of a spherical shell by the procedure outlined in step 1 of the
proposed method for the solution (2t).
Fig. 9. Calculated critical snap-through arc-length solution results for � � 4.6 by the proposed
procedure [(1.5–2.5)t] and the arc-length solution for clamped shallow spherical shells.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2 131
An Algorithm for Estimating Minimum Strength ...
Figure 9 shows the results for �� 4.6. They are the same results as when
�� 3.8. As described above, we can show the same results as those calculated by
the arc-length method when �� 3.8 and 4.6, and therefore, not only the arc-length
method but also the proposed method can evaluate the minimum snap-through
buckling strength.
3.3. Evaluation for Cases Not Applicable for Arc-Length Method. Next we
will show that this calculation method can be applied to the case that cannot be
solved by using the arc-length method. Figure 10 shows the calculation results for
�� 9.2. This shows that this calculation method can be applied in a case where the
wall thickness is very low. As described above, the shell thickness is considerably
low when �� 4.6. Figure 11 combines the results from both Figs. 4 and 10.
In such cases, the current method of estimating the minimum strength of
elastic snap-through buckling is inadequate and the proposed method is effective as
an analytical method.
Fig. 10. Calculated critical snap-through arc-length solution results for � � 6.5 by the proposed
procedure [(1.0–6.0)t] and arc-length solutions (nonconvergence) for clamped shallow spherical
shells.
Fig. 11. Calculated lowest buckling of solutions, and arc-length proposed method and experimental
results.
132 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2
M. Ishinabe and K. Hayashi
Conclusions. We proposed a method to estimate the minimum buckling load
of the partial snap-through that occurs in thin-wall structures such as marine
structures that are subject to external pressure. We investigated the method by
solving the minimum buckling strength of elastic snap-through that occurs in
clamped thin partial spherical shell structures under external pressure. In addition,
we presented a study of current theoretical analysis, the experimental results and
the relation between FEM analysis by the arc-length method and theoretical
analysis and related experiments to show the effectiveness and limitations of the
arc-length method. This study shows that the proposed method can be applied even
when the arc-length method does not converge and allows one to reduce the wall
thickness to make light products. We expect the proposed method to be applied as
a practical method.
Ð å ç þ ì å
Ïðè ïðîåêòóâàíí³ îáîëîíêîâèõ êîíñòðóêö³é âàæëèâî âðàõîâóâàòè ¿õ ïðóæíå
ñõëîïóâàííÿ, îñê³ëüêè âîíî ìîæå ïðèçâåñòè äî ðóéíóâàííÿ êîíñòðóêö³é.
Çîêðåìà, äëÿ çàáåçïå÷åííÿ íàä³éíîñò³ ïðîåêòóâàííÿ íåîáõ³äíî âðàõîâóâàòè
çì³íè ð³âíÿ íàâàíòàæåííÿ âòðàòè ñò³éêîñò³ âíàñë³äîê çìåíøåííÿ òîâùèíè
ñò³íîê êîíñòðóêö³¿. Äëÿ ðîçâ’ÿçàííÿ ïîä³áíèõ çàäà÷ âèêîðèñòîâóºòüñÿ ñê³í-
÷åííîåëåìåíòíèé ìåòîä äîâæèíè äóãè. Îäíàê çà äîïîìîãîþ öüîãî ìåòîäó íå
çàâæäè ìîæëèâî îö³íèòè òðàºêòîð³þ çì³íè íàâàíòàæåííÿ ñõëîïóâàííÿ. Ìå-
òîä äîâæèíè äóãè âèêîðèñòîâóâàëè ïðè ïðóæíîìó ñõëîïóâàíí³ ïîëîãîãî
ôðàãìåíòó ñôåðè÷íî¿ îáîëîíêè. Çàïðîïîíîâàíî íîâèé àëãîðèòì ó ÿâí³é
ñê³í÷åííîåëåìåíòí³é ïîñòàíîâö³ äëÿ îö³íêè ì³í³ìàëüíî¿ ì³öíîñò³ òîíêîñò³í-
íèõ êîíñòðóêö³é, çã³äíî ç ÿêèì ïî÷àòêîâà äåôîðìàö³ÿ çàäàºòüñÿ øëÿõîì
ïðèòèñíåííÿ ãîðèçîíòàëüíî¿ æîðñòêî¿ ïëèòè äî âåðõíüî¿ òî÷êè êîíñòðóêö³¿,
ùî ñõëîïóºòüñÿ, òà ¿¿ ïåðåì³ùåííÿ ïî âåðòèêàë³. Ïðèïóñêàºòüñÿ, ùî ìåòîä
çàáåçïå÷èòü äîáðèé ³íæåíåðíèé ðîçâ’ÿçîê äëÿ îö³íêè ì³í³ìàëüíîãî íàâàíòà-
æåííÿ, çà ÿêîãî â³äáóâàºòüñÿ ÷àñòêîâå ïðóæíå ñõëîïóâàííÿ îáîëîíêîâèõ
êîíñòðóêö³é çàãàëüíîãî âèãëÿäó ï³ä òèñêîì.
1. P. Radha and K. Rajagopalan, “Ultimate strength of submarine pressure hulls
with failure governed by inelastic buckling,” Thin-Walled Struct., 44, 309–
313 (2006).
2. N. C. Huang, “Unsymmetrical buckling of thin shallow spherical shells,” J.
Appl. Mech., 31, 447–457 (1964).
3. G. A. Thurston, “A numerical solution of the nonlinear equations for
axisymmetric bending of shallow spherical shells,” J. Appl. Mech., 28, 557–
561 (1961).
4. M. Cerini and B. G. Falzon, “The reliability of the arc-length method in the
analysis of mode-jumping problems,”in: Proc. 44th Conf. on Structures,
Structural Dynamics, and Materials, Norfolk, VA, AIAA (2003), pp. 1977–
1986.
5. R. Kao, “Large deformation elasitic-plastic buckling analysis of spherical caps
with initial imperfections,” Comp. Struct., 11, 609–619 (1980).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2 133
An Algorithm for Estimating Minimum Strength ...
6. T. Y. Yang and D. G. Liaw, “Elastic-plastic dynamic buckling of thin-shell
finite elements with asymmetric imperfections,” AIAA J., 26, No. 4, 479–486
(1988).
7. T. Koga and N. J. Hoff, “The axisymmetic buckling of initially imperfection
complete sphrical shells,” Int. J. Solids Struct., 5, 679–697 (1969).
8. A. Kaplan and Y. C. Fung, A Nonlinear Theory of Bending and Buckling of
Thin Elastic Shallow Spherical Shells, NACA Technical Note 3212, August
(1954).
9. R. H. Homewood, A. C. Brine, and A. E. Johnson, Jr., “Experimental
investigation of the buckling instability of monocoque shells,” in: Proc. of the
Society for Experimental Stress Analysis, Vol. 18 (1961), p. 88.
10. B. Budiansky, “Buckling of clamped shallow spherical shells,” in: Proc. of the
Symposium on the Theory of Thin Elastic Shells (Delft), North Holland
Publishing Co., Amsterdam (1959), p. 64.
11. M. A. Crisfield, “Snap-through and snap-back response in concrete structures
and the dangers of under-integration,” Int. J. Num. Meth. Eng., 22, 751–767
(1986).
Received 12. 05. 2011
134 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2012, ¹ 2
M. Ishinabe and K. Hayashi
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|