1. science
2. engineering

M I N D A U G A S
K U K I S
STRENGTH AND
S TA B I L I T Y A N A LY S I S
OF CELLULAR
PRESSURE VESSELS
S U M M A R Y O F D O C T O R A L
D I S S E R T A T I O N
T E C H N O L O G I C A L
S C I E N C E S , M E C H A N I C A L
E N G I N E E R I N G ( 0 9 T )
Kaunas
2015
KAUNAS UNIVERSITY OF TECHNOLOGY
MINDAUGAS KUKIS
STRENGTH AND STABILITY
ANALYSIS OF CELLULAR
PRESSURE VESSELS
Summary of Doctoral Dissertation
Technological Sciences, Mechanical Engineering (09T)
2015, Kaunas
The dissertation was carried out in 2010 – 2014 at Kaunas University of
Technology, Faculty of Mechanical Engineering and Design, Department of
Mechanical engineering (of Solids Mechanics). The research was supported by
Research Council of Lithuania.
Scientific supervisor:
Prof. Dr. Habil. ANTANAS ŽILIUKAS, (Kaunas University of Technology,
Technological Sciences, Mechanical Engineering – 09T).
Dissertation Defense Board of Mechanical Engineering Science Field:
Prof. Dr. Habil. Vytautas OSTAŠEVIČIUS (Kaunas University of Technology,
Technological Sciences, Mechanical Engineering – 09T) – chairman;
Prof. Dr. Habil. Algimantas BUBULIS (Kaunas University of Technology,
Technological Sciences, Mechanical Engineering – 09T);
Dr. Rolanas DAUKŠEVIČIUS (Kaunas University
Technological Sciences, Mechanical Engineering – 09T);
of
Technology,
Prof. Dr. Vytautas GRIGAS (Kaunas University of Technology, Technological
Sciences, Mechanical Engineering – 09T);
Prof. Dr. Vytenis JANKAUSKAS (Aleksandras Stulginskis University,
Technological Sciences, Mechanical Engineering – 09T);
Prof. Dr. Habil. Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical
University, Technological Sciences, Mechanical Engineering – 09T).
The official defense of the dissertation will be held at 10 a.m. on 22nd of May,
2015 at the Board of Mechanical Engineering Science Field public meeting in
the Dissertation Defense Hall at the Central Building of Kaunas University of
Technology.
Address: K. Donelaičio st. 73 – 403, LT-44029, Kaunas, Lithuania,
Phone nr. (+370) 37 300042, Fax. (+370) 37 324144, e-mail:
[email protected]
The summary of dissertation was sent on 22nd April, 2015.
The dissertation is available at the library of Kaunas University of Technology
(K. Donelaičio st. 20, LT-44239, Kaunas, Lithuania).
2
KAUNO TECHNOLOGIJOS UNIVERSITETAS
MINDAUGAS KUKIS
KORĖTŲ SLĖGIO INDŲ STIPRUMO
IR STABILUMO TYRIMAS
Daktaro disertacijos santrauka
Technologijos mokslai, mechanikos inžinerija (09T)
2015, Kaunas
3
Disertacija rengta 2010-2014 metais Kauno technologijos universitete,
Mechanikos ir dizaino fakultete, Mechanikos inžinerijos (Deformuojamų kūnų
mechanikos) katedroje, remiant Lietuvos mokslo tarybai.
Mokslinis vadovas:
Prof. habil. dr. ANTANAS ŽILIUKAS, (Kauno technologijos universitetas,
technologijos mokslai, mechanikos inžinerija – 09T)
Mechanikos inžinerijos mokslo krypties daktaro disertacijos gynimo
taryba:
Prof. habil. dr. Vytautas OSTAŠEVIČIUS (Kauno technologijos universitetas,
technologijos mokslai, mechanikos inžinerija – 09T) – pirmininkas;
Prof. habil. dr. Algimantas BUBULIS (Kauno technologijos universitetas,
technologijos mokslai, mechanikos inžinerija – 09T);
Dr. Rolanas DAUKŠEVIČIUS (Kauno technologijos
technologijos mokslai, mechanikos inžinerija – 09T);
universitetas,
Prof. dr. Vytautas GRIGAS (Kauno technologijos universitetas, technologijos
mokslai, mechanikos inžinerija – 09T);
Prof. dr. Vytenis JANKAUSKAS (Aleksandro Stulginskio universitetas,
technologijos mokslai, mechanikos inžinerija – 09T);
Prof. habil. dr. Rimantas Kačianauskas (Vilniaus Gedimino technikos
universitetas, technologijos mokslai, mechanikos inžinerija – 09T).
Disertacija bus ginama viešajame Mechanikos inžinerijos mokslo krypties
tarybos posėdyje, kuris įvyks 2015 m. gegužės 22d. 10 val., Kauno technologijos
universitete, Centrinių rūmų disertacijų gynimo salėje.
Adresas: K. Donelaičio g. 73 – 403, LT-44029, Kaunas, Lietuva
Tel. (8 - 37) 300042, faksas (8 - 37) 321444, e. paštas [email protected]
Daktaro disertacijos santrauka išsiųsta 2015 m. balandžio 22 d.
Disertaciją galima peržiūrėti Kauno technologijos universiteto bibliotekoje (K.
Donelaičio g. 20, LT-44239, Kaunas, Lietuva).
4
INTRODUCTION
Development of technologies influences the origin of new materials, which
usage requires modern, science-based structural solutions. One of such
structures – sandwich plates with polymer core. Plates of this type are widely
used in construction industry and other fields, as cheap and effective alternative.
Whenever there is a need of lightweight structures or energy absorbing
properties, this type plates are inappropriate. Metal layered plates are used
instead. Cellular plate consists of two surface sheets and core. For the core layer
there are used various constructions – two dimensional porous geometry, such
as honeycomb with different forms of comb meshes, bended, corrugated metal
sheets. Also, for the structure of the core, considering application of plate and
necessary characteristics, there is used metal foam or rod construction.
Number of different studies was carried out trying to find out application
areas of these plates. Research results revealed that such structures not only
greatly sustain loads, but also are multi-functional. For example, using this type
of plate in car section, as a fire barrier between the engine compartment and the
passenger, it is obtained construction, not just limiting the spread of flame, but
also absorbing energy and sound. Over the time, there were new benefits
discovered of such plates and they became irreplaceable in many industry areas:
aviation, shipping, aerospace, the automotive industry and elsewhere.
Such plates are particularly acceptable because of multi-functionality, so
they can be used for pressure vessel shell design. Then the construction would
not only be lightweight, cheap and efficient, capable to sustain intended pressure,
but also could be of service technologically – cooling or heating content of the
vessel. Currently produced pressure vessels, for example used in food industry,
have different type of „jacket“, intended to keep, increase or reduce of the
temperature. Whereas, using cellular shell, technological processes can be
ensured – apply core insert as „jacket“, thus way to get lighter and cheaper
structure. From carried analysis of the works, it is clear that in most cases there
is investigated flat type of such structures. Investigated core structures usually
are very complex, requiring modern manufacturing technologies. In addition,
weak point of such structures – core cells attachment to the surface of the sheets.
Due to this defect cellular plates often have lower sustaining capacity, than
calculated theoretically.
Therefore, in this thesis it is investigated pressure vessel, where walls of
the cylindrical part are multilayered having cellular inserts. Simple structures are
selected for the core, which does not require sophisticated production techniques
and provides qualitative connection of elements to the surface sheets.
5
Aim of the work
To create and investigate pressure vessel, which cylindrical part of the wall
is multilayered, having cellular insert, featuring better functionality compared to
pressure vessels, having monolithic walls and simple manufacturing technology.
Out of investigated structures to determine the most rational.
Objectives of the study
1.
2.
3.
4.
Present calculating models of pressure vessels, having different cylinder and
core geometric parameters, to investigate their functional characteristics
using numerical models and compare them with each other using different
strength criteria;
To create pressure vessel, which wall of the cylindrical part of the shell is
multilayered, with various structures, of technologically simple porous
inserts;
Produce assays of new structure pressure vessel and it’s core segment and
experimentally investigate their strength and stability; verify calculating
models based on the search results;
In order to reduce the mass of the produced pressure vessels, to carry out
optimization of walls and core elements thickness, to compare optimized
structures with pressure vessels having monolithic walls.
Research methods
Work was carried out by means of theoretical and experimental research
methods. Theoretical studies were carried out applying analytical and numerical
methods (used software “ANSYS”, based on finite elements method). To verify
the results of numerical investigation there were produced specimens of the
pressure vessel and core fragment, performed experimental studies. Part of
experiments were carried out at Kaunas site industrial base of Panevėžys
installation company AB “Montuotojas”, the other part – in strength of materials
laboratory in Kaunas University of Technology.
Scientific novelty
Designed pressure vessel having multilayered walls of the cylindrical part,
which functional properties are much better than monolithic walls and
manufacturing technology is relatively simple and defectless. Using finite
elements system ANSYS for numerical analysis of strength of the designed
pressure vessel, there were programmed and used Drucker-Prager and MohrCoulomb strength criteria, that are absent in the program by default.
6
Relevance of the work
Designed new construction pressure vessels, which characterize higher
stability than vessels having monolithic walls and are more resistance to the
effects of the external pressure. In addition, these vessels are good alternative in
cases, when seeking to assure run of specific technological processes, there is a
need to adjust or maintain required temperature of the medium inside the vessel.
Using pressure vessels made of monolithic sheet, they should be covered with
particular additional „shirt“, where could circulate medium, keeping proper
temperature regime.
Defensive statements:
1.
2.
3.
Developed numerical research methodology is appropriate to apply for
estimation of strength and stability characteristics of cellular pressure
vessels;
Evaluating characteristics of cellular pressure vessels, Von Mises strength
criterion is too conservative, so in order to minimize materials expenditures,
it is necessary to use strength criteria, estimating mechanical characteristics
of the material;
Structure of cellular pressure vessels is rational alternative for vessels made
of monolithic sheet, subjected to internal and external pressure.
Practical value of the work
Created pressure vessels are more stable than the once with monolithic
walls, and more resistant to the effects of internal and external pressure.
Therefore, avoiding risk to the strength, there could be kept higher pressure of
the medium, circulating in the insert, thus ensuring more efficient adjustment
and maintenance of the medium thermal regime present in the vessel.
In addition, due to specifics and simplicity of the structure, likelihood of
vessels manufacturing defects is minimized.
Work structure
The dissertation consists of an introduction, 4 chapters, conclusions,
references and a list of scientific publications on the topic, also appendixes. The
dissertation contains 136 pages, 97 figures and 33 tables. References consist of
141 sources.
1. REVIEW OF LITERATURE
Due to depletion of metal ore, modern industry aims to use existing metal
recourses optimally. In products, where for manufacturing was used metal (ex.
pressure vessels), started to apply alternative materials – sandwich plates,
composites and etc. When there was noticed utility of sandwich plates, because
7
of their multi-functionality and economical advantages [10-13] comparing to
monolithic plates, there was started usage of metal sandwich plates where
manufacturing opportunities allow. These plates are also called cellular plates.
Cellular plates consist of two high density surface sheets with lower density core
between them (fig. 1.1), designed to keep them at a certain distance.
Fig. 1.1. Structure of cellular plate [14]
According plates’ core structure, i.e. what they are made of, plates can be
divided into several groups: plates with cores of metal foam (fig. 1.2, a), sheet
metal (fig. 1.2, b) and periodic cellular metal (fig. 1.2, c).
a
b
c
Fig. 1.2. Types of cellular plates: a – metal foam core [1]; b – sheet metal core [14];
c – periodic cellular metal [15]
Advantages of cellular plates: lightweight, multi-functionality, absorption
of sound, vibrations and impacts, isolation and transfer of heat, rigidity,
inexpensiveness, ability to change rigidity characteristic depending on the
applied load, etc. Comparing mentioned characteristics of such plates with
homogeneous sheet, it was observed, that cellular plates are much better. Some
of the properties, such as multi-functionality and vibration damping,
homogeneous sheet do not have at all.
Cellular plates are an acceptable alternative to aforementioned
characteristics, but their production requires usage of modern production
technologies and equipment. This demands a significant investment. Most of the
processes, in some cases even whole production process of certain boards,
because of complexity, are performed without human intervention. For this
reason, there occur defects, more or less affecting mechanical characteristics of
the plates. One defect, that is common for all cellular plates, which significantly
8
influences maximum bearing capacity of cellular plates – core detachment to the
surface sheets.
After review of the articles, where cellular plates are used for the shell of
pressure vessel, noted that only few theoretical analysis have been carried out,
where core structure is extremely complex, requiring sophisticated production.
Therefore, in such cellular pressure vessel, can occur core detachment to the
surface sheets, defect. Pressure vessel – a potentially dangerous device and there
are cannot be any defects. Thus, this works proposes simple core structures, not
requiring high-tech. All stages of the production are carried out by human,
therefore should not be any defects. Proposed core structures are acceptable also
in heat transfer aspect – cellular insert can be used for the execution of
technological processes, such as heating and cooling of vessel’s content.
2. CELLULAR PRESSURE VESSELS’ RESEARCH METHODOLOGY
2.1 Numerical research methodology
Although there were solved different tasks in the work, but the algorithm
of numerical solution of all the tasks was as follows:
 In preprocess part (Preprocessor) of the ANSYS model, there is
formed geometrical model in a way that would be easy to change its
geometry, materials; performed segmentation of the geometrical
model by finite elements;
 In a part of ANSYS solution (Solver), there are defined loads and
calculation performed;
 In a part of ANSYS calculation results presentation (Postprocessor),
there is performed analysis of obtained results.
2.1.1 Numerical models of pressure vessels
The strength and stability, of cellular cylinder with thick-walled blind
flanges, is analyzed in the thesis. This is done by using finite elements system
„ANSYS“.
Thus will be:
1. Designed model of cellular cylinder with different core structures,
which would be investigated under pressure load, i.e., performed
strength analysis of the vessel and obtained results compared to solid
cylinder having the same bearing capacity. The analysis is performed
using 3 strength criteria.
2. Estimated stability of the structure under vacuum, and obtained results
compared to solid cylinder, having the same bearing capacity
Principal scheme of the investigated cylinder presented in figure 2.1.
9
Fig. 2.1. Simplified scheme of a cellular cylinder
a
b
c
d
Fig. 2.2. Specimen fixing and loading analyzing: a –strength of a cellular cylinder;
b – stability of a cellular cylinder; c – strength of a solid cylinder; d – stability of a solid
cylinder
Since the investigated structure is axis symmetrical, thus to reduce
calculations duration, there is used only a tenth of cylinder, i.e. segment with one
core element. In order to ensure adequacy of the model to the real stress-strain
state, typical in case of the whole cylinder, structure is fixed indicating symmetry
conditions to corresponding surfaces (fig .2.2, a and b this is illustrated by the
arrows on the sidelong surfaces). Displacements of the structure are restricted by
fixing central point rigidly, and allowing for the all lower plane of the lower
blind flange to move only horizontally (fig. 2.2 arrow from the bottom).
Imitating the impact of the fluid inside the vessel, load, operating the structure,
consists of two components – operating pressure and hydrostatic medium
(medium inside vessel, which density corresponds to the density of water)
pressure. In order to find out how thick there should be walls of the monolithic
10
cylinder to withstand the same pressure load as investigated vessel with cellular
walls, there was made calculation model of the vessel segment quarter (fig. 2.2
c and d) with the monolithic walls having same inner diameter (scheme fig. 2.3).
Performing case calculations on its basis, when there was changed thickness of
the monolithic cylinder wall, there was determined minimum value of this
parameter, at which the strength of the vessel corresponds to the strength of
cellular wall vessel, what allowed comparing masses of both structures and
evaluate production rate.
Fig. 2.3. Scheme of solid cylinder to which, there will be compared results of cellular
cylinders
Since all the components of real investigated object (inner and outer
cylinders, core forming partitions, blind flanges and shackles) are basically of
flat type, its geometrical model is made from surfaces, while in numerical model
these surfaces are split using SHELL63 type finite elements. The grid was
regular, without thickening. These elements have 6 degrees of freedom in every
nodal point and allow calculating all the parameters, necessary to evaluate the
state of elastically and plastically deformable dimensional thin-walls structures.
Fig. 2.4 presents numerical models split into finite elements according
mentioned type. Designing numerical model, it is considered, that all
components of pressure vessel segment joined to form a monolithic solid, made
of surface elements with different thicknesses.
11
a
b
Fig. 2.4. Cylinders with flanges and blind flanges divided into finite elements:
a – cellular; b – solid
2.2 Implementation algorithms of cellular cylinders research
2.2.1 Strength research implementation algorithm
Chapter 3 presents strength calculation results of 10 cylindrical pressure
vessels with cellular walls, of the same dimensions, but with different cores of
the cylindrical wall (fig. 2.1), where inner cylinder is under operational and
hydrostatic pressure from the inside (fig. 2.2).
Each structure will be investigated in 5 different cases trying to find most
efficient structure version. First three cases – is changed thickness from 3 to 5
mm of internal cylinder tv, core element tk and external cylinder ti, while
remaining elements stay of 2 mm. Then is alternated the width of core
elements – proportionally increasing in three steps, at last step core elements
touch each other. In case of analysis, thickness of elements tv = tk = ti = 2 mm.
Finally it is investigated influence of the core height b to the bearing capacity.
Core height increased in three steps as well. In each of the cases the gap between
core cylinders increases by 10 mm. Element thickness is the same as in previous
case, tv = tk = ti = 2 mm.
In order to determine the influence of cylinder geometry change to the
maximum bearing capacity was carried out research of diameter and length
change influence. Cylinders diameter increases, as in search of effective
structure, in three steps. Cross-section was increased by 25 mm in each case. In
cylinder extension case, the length was increased by 1000 mm, as well in 3 steps.
12
Fig. 2.5. Algorithm numerical analysis of the cellular pressure vessel strength
To figure out cellular plate usage advantages or disadvantages against
monolithic ones, it was performed comparison of mass ratios, when given
maximum bearing capacity is the same. Obtained ratio compared to the results
got by optimizing wall thickness of the cellular vessel. Results are presented in
chapter 3. Strength analysis is carried out using 3 criteria of strength: von Mises,
Drucker-Prager, and Mohr-Coulomb.
2.2.2 Stability research implementation algorithm
In chapter 4 as well as in 3, presented results of 10 cellular cylinders of
standard dimensions with different cores, as shown schematically in fig. 2.1,
whereas detailed core structures presented in chapter 2.5. In stability research
there will be analyzed the same of each cylinder, 22 different cases, according
to the dimensions given in the appendixes P1-P10 of the thesis. These cylinders
will be loaded and fixed as shown in fig. 2.2.
13
Fig. 2.6. Algorithm for numerical analysis of the cellular pressure vessel stability
There will be presented pictures and graphs, when structure is under critical
external pressure, at which structure reaches ultimate buckling coeficient value,
equal to 1,0. In order to find out if cellular cylinder stumbled because of stability
loss, not as result of plastic deformations, there will be carried out non-linear
stumbling analysis of every case. There was compared masses of solid and
cellular cylinders at the same maximum bearing capacity as well as in the case
of strength analysis. Also performed optimization of cellular cylinder elements
thickness, which results presented in chaper 5.1.
2.3 Optimization parameters of cellular pressure vessels
Cellular cylinders were optimized using „Subproblem“ optimization
method. Purpose of the optimization – minimal wall thicknesses of the cellular
pressure vessel. The output parameters for optimization performance:
 Objective function (V – capacity)
n
min V   Vi ,
t1,t 2,t 3
i 1
14
(2.1)
 design variables ( t1 , t 2 , t 3 – wall thicknesses of cylinder elements, mm)
0,5  t1  2
0,5  t 2  2 ,
(2.2)
0,5  t 3  2
 state variable, when optimizing the strength (  – strains, MPa)
303    304 ,
(2.3)
 state variable, when performing optimization using Drucker-Prager and
Mohr-Coulomb strength criteria (  – strains, MPa). Minimum and
maximum values of strains define limits, where criteria conditions are met
min    max ,
(2.4)
 state variable, performing optimization of stumbling ( k – stumbling
coefficient)
1,000  k  1,001 .
(2.5)
There were used 100 iterations for optimization, from which 99 could be
improper. After optimization of strength analysis, there was verified structure
stability, and after stability – nonlinear stumbling analysis. In case of stumbling,
or case of nonlinear stumbling analysis, if structure stumbles because of plastic
deformations, results are adjusted accordingly, so the structure would be proper
according mentioned aspects.
2.4 Experimental research methodology
Calculating methodology, of cellular vessels strength and stability analysis
using numerical methods, is verified by experiments. Vessel having corrugated
core structure is chosen for verification test of strength analysis, as shown in fig.
2.8. The vessel is tested by filling it with water and generating internal pressure
with the help of compressed nitrogen cylinder. Pressure is increased to 0,45 MPa
every 0,05 MPa. In every stage there is measured displacement of inner cylinder,
which is conveyed through the rod, fixed to the cylinder, situated in the center
of wavy segment, in the middle of cylinder length. At the other end of the rod
there is a plate to which props displacements gauge.
The measurement system consists of manometer and displacements gauge.
Manometer accuracy class is 1,6, measurement accuracy ±0,01 MPa, measuring
range 0-0,6 MPa. Displacement gauge accuracy class 0,2, measurement
accuracy ±0,05 μm, and measuring range 0-100 μm.
Specimen used in the experiment, was made of 2 mm structural steel
ST3PS sheet.
15
Fig. 2.7 Cellular vessel testing-bench scheme
Fig. 2.8. Core structure of the tentative cellular pressure vessel
Stability analysis verification was performed using nature testing, during
which there was stumbled element of corrugated core, fig. 2.9. It was chosen
simplified experiment, because to produce cellular vessel, which would stumble
at lower that 0,07 MPa external pressure (maximum vacuum, which can be
obtained using vacuum pump is 0,1 MPa and it is absolute vacuum) is
impossible.
Fig. 2.9. Specimen of corrugated core
16
Fig. 2.10. Corrugated core element test scheme
Stumbling was performed with a universal testing machine „Amsler’,
which speed of vices is 2 mm/min, measurement error ±100 N
2.4 Cylinder structures of investigated cellular pressure vessels
a
b
c
Fig. 2.11. Cellular cylinder with the core of: a – „U“ shape; b – double corrugation
shape; c – „H“ shape
a)
b)
c)
Fig. 2.12. Cellular cylinder with the core of: „I“ (a), „A“ (b), „V with a wall“ (c) shape
17
a)
b)
c)
Fig. 2.13. Cellular cylinder with the core of: corrugated (a), „V“ (b), „X“ (c) shape
Fig. 2.14. Cellular cylinder with the core of „Y“ shape
2.7. Chapter summary
1. There were presented algorithm for calculating cellular pressure
vessels’ research results, using numerical methods and on their basis
operating strength and stability methodologies.
2. Analyzed strengths criteria, used for investigation of mechanical
structures strength and chosen the most suitable for evaluation of
cellular pressure vessels strength.
3. Presented experimental research methodologies for cellular vessel and
its core element, developed and produced testing-benches necessary to
perform experimental research.
4. Described measurements methodology, required for experimental
research, the number of demand, evaluation of results dispersion and
relative error calculation.
3. STRENGTH RESEARCH RESULTS OF CELLULAR CYLINDERS
Abbreviations used in the below presented graphs: Drucker-Prager (D.P),
Mohr-Coulomb (M.C.) and von Mises (V.M).
18
Fig. 3.1. Maximum bearing capacity dependence on the wall thickness tv of inner
cylinder
Fig. 3.2. Maximum bearing capacity dependence on the width La of core element
19
Internal pressure (MPa)
1,2
U (V.M.)
1,15
U (D.P.)
U (M.C.)
1,1
Doub. cor. (V.M.)
1,05
Doub. cor. (D.P.)
Doub. cor. (M.C.)
1
H (V.M.)
H (D.P.)
0,95
H (M.C.)
65
75
85
Core height b (mm)
95
Fig. 3.3. Maximum bearing capacity dependence on the height b of core element
Internal pressure (MPa)
1,15
U (V.M.)
U (D.P.)
1,05
U (M.C.)
Doub. cor. (V.M.)
0,95
Doub. cor. (D.P.)
Doub. cor. (M.C.)
0,85
H (V.M.)
H (D.P.)
0,75
500
525
550
Diameter Dv (mm)
575
H (M.C.)
Fig. 3.4. Maximum bearing capacity dependence on the inner diameter Dv
20
Internal pressure (MPa)
1,2
U (V.M.)
1,15
U (D.P.)
U (M.C.)
Doub. cor. (V.M.)
1,1
Doub. cor. (D.P.)
Doub. cor. (M.C.)
1,05
H (V.M.)
H (D.P.)
1
700
1700
2700
Length L (mm)
3700
H (M.C.)
Fig. 3.5. Maximum bearing capacity dependence on the cylinder length L
3.1 Strength optimization results of cellular pressure vessels
Cellular cylinders were optimized for minimum mass according strength
(when state variable – strains, load – internal pressure) using methodology
described in chapter 2. Their masses compared to masses of monolithic cylinders
at the same maximum bearing capacity, as shown in equation (3.1).
msant 
mkor
mmon
(3.1)
here msant – mass ratio between cylinders, mmon – monolithic cylinder mass,
mkor – cellular cylinder mass.
From the results of mass optimization of cellular cylinders under internal
pressure, presented in table 3.1, obvious, that even after optimization of
minimum mass, none of them is lighter compared to monolithic. The lightest
cellular cylinder – with the core of “I“ shape, applying Mohr-Coulomb strength
criterion. The heaviest – with the “X“ shaped core, applying von Mises strength
criterion. Comparing the lightest and the heaviest cylinders, applying different
strength criteria, obtained that in all strength criteria application cases the lightest
one is cylinder with core of “I” shape. In case of Drucker-Prager criterion
application, the heaviest cylinders are with the cores of “H” and “U” shapes, and
in case of Mohr-Coulomb – “H”.
21
Table 3.1. Optimization summary of cellular cylinders under inner pressure ( msant )
Core
von Mises
Drucker-Prager
Mohr-Coulomb
Non-opt.
Opt.
Non-opt.
Opt.
Non-opt.
Opt.
„U“
2,83
2,12
2,78
2,5
2,83
2,26
Double corr.
2,69
2,21
2,64
1,95
2,79
2,79
„H“
2,82
2,11
2,78
2,5
2,81
2,81
„I“
1,96
1,37
1,82
1,64
1,85
1,2
„A“
2,72
2,44
2,67
1,96
2,88
2,05
„V with a wall“
2,69
2,21
2,62
1,97
2,88
2,1
Corrugated
2,68
1,75
2,58
2,25
2,58
1,9
„V“
2,49
2,39
2,45
1,97
2,54
1,71
„X“
3,01
2,91
2,69
1,54
3,01
1,71
„Y“
2,69
2,47
2,3
1,39
2,58
1,33
3.2 Verification of cellular cylinders strength calculations using numerical
methods
In case to verify acceptability of made numerical research methodology
and calculating models, there was carried out experimental research (3.7) using
cellular pressure vessel, which cylindrical wall is with corrugated core (2.8). The
results were compared to the ones of analogous structure pressure vessel
calculating analysis (the experiment was repeated 20 times). In both cases there
were analyzed displacements of the central point of the inner cylinder generatrix
by increasing internal pressure operating cylinder by 0.05 MPa, from 0 to 0.45
MPa (calculating research scheme presented in fig. 2.7).
Fig. 3.6. Displacements measurement place in model of finite elements
22
Numerical model and experimental specimen were made as presented in
figures 3.6 and 3.7.
Fig. 3.7. Testing-bench of pressure vessel with corrugated core cylindrical wall
Comparing the results of calculations and experimental investigation,
obvious, that they coincide well – difference between numerical simulation and
experimental measurements results does not exceed 2,38%, variation coefficient
does not exceed 10%, thus can be said, that results, presented in chapter 3 are
correct and suitable for both, practical and scientific practice. Numerical results,
in research range, were verified by carried experiment, but it cannot be checked
pressure vessel limitary states (when density limit is reached), as well as the most
suitable criterion.
3.2 Chapter conclusions
1. After strength investigation of designed cellular cylinders, applying three
strength criteria, it was determined, that in many experimental cases,
maximum bearing capacity of pressure vessels with cellular walls, was
obtained using Drucker-Prager strength criterion. Classifying according
maximum bearing capacity, Mohr-Coulomb strength criterion was in the
second place, and von-Mises criterion was most conservative – according
it, maximum bearing capacities were the lowest. Therefore, in order to
minimize the weight and effectively use materials – should be used strength
criteria, evaluating mechanical characteristics of the material.
2. In case to find most rational cellular vessel structure there were performed
its strength (maximum bearing capacity) variant calculations, by changing
the thickness of cellular vessel elements walls. In all cases, thickness
change of the core walls and outer cylinder wall thickness did not have
significant influence on the increase of maximum bearing capacity. Highest
strength value of pressure vessel wit cellular cylindrical walls, is obtained
23
3.
4.
5.
6.
7.
8.
by manufacturing their inner cylinder from thicker sheet, but in such case
inevitably increases the mass of the pressure vessel.
Increase of the core element width La of the investigated core structures
had significant influence on the maximum bearing capacity. Influence is
positive or negative, depends on core structure and applied strength
criterion.
Increase of the core height b, contrary to core the element width La, didn’t
have observable influence on the maximum bearing capacity. Applying
von-Mises strength criterion, in most cases, maximum bearing capacity
was reducing significantly. Nature of core height influence, as well as
increasing core width, depends on the core structure and applied strength
criterion.
With increasing the inner cylinder diameter Dv of the cellular pressure
vessel, maximum bearing capacity significantly decreases, but increasing
the length L, in some cases, maximum bearing capacity of cellular cylinders
increased.
In all the cylinders, highest maximum bearing capacity, using DruckerPrager strength criterion, has cylinder with “X” shaped core, MohrCoulomb – “Y“, and von Mises – „X“. Meanwhile, the lowest: –
„I“(applying Drucker-Prager criterion), „H“ (applying Mohr-Coulomb
criterion) and „Y“ (applying von Mises criterion).
Pressure vessels of all structures (all 10 core variants of different wall
thickness and other geometrical parameters of the vessel), were tested for
buckling applying critical pressure, when density limit is reached. Analysis
was carried out to find out if structure stays stable when strains reach
density limit. After verification of this condition there was not recorder any
case of cylinder buckling.
Cellular cylinders were optimized to minimum mass in case to determine
advantages or disadvantages comparing to monolithic cylinder. After
optimization in two cases, i.e. when load operating the cylinder is internal
or external pressure (respectively optimizing according strength or
stability), was obtained that under the same bearing capacity none of the
cellular cylinders is lighter than the monolithic. In case of internal pressure
load present, the lightest of cellular cylinders is the one with “I” shaped
core, the heaviest – with “X” shaped core, while under external pressure
load, lightest cylinder is having “H” shaped core, and heaviest – “X”
shaped core.
In order to verify methodology of compiled numerical research and
acceptability of calculating models, there was carried out experiment of
pressure vessel with cylindrical wall having corrugated core, which results
were compared to calculating analysis results of analogous structure
pressure vessel. Experimental results have confirmed the ones obtained by
24
numerical methods (difference between numerical modeling and
experimental measurements results does not exceed 2,38%), however in
this case calculating methodology tested just with pressure vessel nonreaching limiting states, because it is impossible verify experimentally
which criterion is most suitable when density limit is reached.
4. STABILITY RESEARCH RESULTS OF CELLULAR CYLINDERS
Stability results are presented in the same format as strength analysis, thus
obtained graphs will not be presented in this chapter, just research verification
results. Stability results will be discussed in general in the conclusions of the
chapter.
Cellular cylinders were optimized to minimum mass according stability
(when state variable – buckling coefficient, load – external pressure) applying
methodology described in chapter 2.
Table 4.1. Optimization summary of cellular cylinders exposed to external
pressure ( msant (formula 3.1)).
Core
Non-opt.
Opt.
„U“
2,02
2,02
Double corr.
2,14
2,14
„H“
1,97
1,97
„I“
2,2
2,18
„A“
2,28
2,28
„V with a wall“
2,14
2,14
Corrugated
2,12
2,11
„V“
2,15
2,14
„X“
2,57
2,57
„Y“
2,23
2,23
According mass optimization results (presented in table 4.1) of cellular
cylinders exposed to external pressure, clear that even after optimization them
to minimum mass none is lighter than monolithic. Stability optimization
compared to the strength analysis optimization does not give significant results,
thus it can be concluded that exposed to external pressure initial thicknesses of
cellular cylinders elements are optimal and further optimization is unnecessary.
The lightest cellular cylinder – with “H” shaped core, the heaviest – with “X”.
25
4.1 Verification of cellular cylinder stability calculations, using numerical
methods
Fig. 4.1. The specimen „Amsler“ in testing machine
During the experiments, there was obtained critical force at which
specimen have stumbled – Fcr  7000 N. This force fells to one oblique plate of
the core.
The same experiment was carried out using numerical analysis package
„ANSYS“. Fig. 4.2 presents, how numerical model was loaded and fixed.
Arrows from the bottom illustrates fixings, and arrows from the top – acting
force.
Fig.4.2. Fixings and loads of the core element with oblique plates for investigation of
buckling
Using numerical analysis it was obtained critical force Fcr  7200 N. In
order to compare the results of numerical research, critical force was calculated
using analytical formula of critical buckling force:
Fcr   cr A
26
(4.1)
here Fcr – critical force,  cr – critical stumbling strain, A – cross-section area
of the plate.
After calculations obtained – Fcr  6780 N.
After calculation of relative error (i.e. inadequacy of calculation results
with numerical research data) obtained, that the value of critical buckling force
attained by different methods, coincides well – relative error of experimental
result – 2,78%, and of analytical – 5,83%. Therefore =, it can be stated, that
results presented in chapter 4 are correct and suitable for both, practical and
scientific use.
4.2 Chapter conclusions
1. After stability investigation of the designed cellular cylinders, it was
obtained, that cylinder with a core of „V shape with a wall“, has highest
maximum bearing capacity, and the smallest with „I“shape core.
2. Looking for most rational construction of cellular vessel, there were
performed variant stability calculations by modifying thicknesses of the
cylinder elements: inner cylinder, core elements and outer cylinder. In all
cases, change of the thicknesses of the core elements and the outer cylinder,
does not have significant influence on the maximum bearing capacity.
Therefore, seeking for bigger bearing capacity, cellular pressure vessel
should be manufactured with thicker walls of inner cylinder.
3. After stability calculations, during which there was investigated influence
of cellular cylinders members’ width La and height b to the bearing
capacity of pressure vessels, obtained, that core member width La, of all
investigated structures, do not have significant influence on it. Influence is
positive or negative, differs depending on core structure. Value of core
height b as well does not have substantial effect to the maximum bearing
capacity.
4. In order to find out the impact of cellular cylinders’ magnitude changes on
the maximum bearing capacity there were performed stability calculations
for different diameter and length cylinders. It was obtained, that the lager
the diameter of the cylinder, the smaller is bearing capacity (except the one,
having “I“ shape core, when it is the smallest and stays the same in all the
cases examined). When the length of the cylinder increases, in some
cellular cylinders, maximum bearing capacity decreases gradually, but in
some cases, increasing length from 700mm to 1700mm – decreases, if
length increases further – bearing capacity remains unchanged.
5. Each structure case exposed to critical pressure, when stability limit is
reached, was tested with nonlinear buckling. Analysis was carried out to
determine if structure buckles, when stability loss limit is reached, due to
structure slenderness or due to plastic deformations. After verification of
27
this condition there was not recorder any case of cylinder buckling because
of plastic deformations.
6. After the optimization, it was discovered, that optimization according
stability (cylinder under external pressure) results are not as significant as
in the case of optimization according the strength. Therefore, initial
thicknesses of cellular cylinders’ elements, under external pressure, are
optimal and further optimization is unnecessary. Although cellular
cylinders are heavier than monolithic, however they are considerably
stiffer. Adjusting monolithic cylinder for operation of technological
processes (cooling, heating), there should be formed additional “jacket”,
therefore it would become heavier.
7. In order to verify cellular pressure vessels stability calculating analysis
methodology and calculating models acceptability, there was produced one
of three cellular cylinders’ core segment – corrugated core, and
experimentally determined critical force, under which specimen have
buckled. Numerical research result was compared to analytical solution and
results of experiment, which determined critical buckling force of
analogical structure (obtained, that difference does not exceed 6%).
28
6. GENERAL CONCLUSIONS
1.
2.
3.
4.
There was designed pressure vessel, which cylindrical part is
multilayered with cellular insert. Vessel insert consists of
technologically simple partitions, characterized by simple production
technology.
Investigations for strength and stability of 10 cellular pressure vessels
with various core constructions operating under internal and external
pressure revealed that the highest maximum bearing capacity using the
Drucker-Prager strength criterion was a cylinder with an „X“ shaped
core, Mohr-Coulomb - „Y“ and von Mises - „X“. Minimum strength
using Drucker-Prager strength criterion - the vessel with the „I“ shaped
core, Mohr-Coulomb – „H“, von Mises - „Y“ shaped core. Numerical
stability study showed that the cylinders with „I“ shaped core has a
minimum bearing capacity and „V with a wall“ - the maximum.
The most rational core parameters analysis revealed that both for
strength and stability studies, the core member height b has no major
effect on the maximum bearing capacity of cellular pressure vessels.
Core element increase in width La affect the maximum bearing
capacity, however increase or decrease is determined by the core
structure. Evaluating the impact of changes in cylinder diameter and
length on the maximum bearing capacity one may note that the increase
in diameter reduces the maximum bearing capacity, and the influence
of the length of the cylinder depends on the core structure.
New design pressure vessel and it’s core segment samples were
produced and experimentally tested for their strength and stability;
experimental study results were used for verification of numerical
models. Comparison of numerical and experimental study results of
cellular cylinder with corrugated core strength confirmed the
acceptability of evaluation of strength via numerical methods, because
the difference between numerical studies and experimental
measurements does not exceed 2,38 %. Acceptability of using
numerical methods to assess cellular cylinder stability was confirmed
by calculations with relative error not exceeding 6 %.
Comparison of monolithic and cellular cylinders masses at the same
maximum bearing capacity determined that no cellular cylinder, in both
strength and stability study cases, is better than a monolithic with
regards to the mass, even optimized for minimal mass. From
multifunctionality aspect, the cellular insert can be used to ensure the
technological processes, cellular pressure vessel is preferable both
economically and technologically.
29
LITERATURE
1. Simone, A. E.; Gibson, L. J. Aluminium foams produced by liqui –state
processes. Acta mater, 1998, vol. 46, no. 9, p. 3109-3123.
2. Sriram, R.; Vaidya U. K.; Kim J. – E. Blast impact response of aluminum
foam sandwich composites. Journal of Materials Science, 2006, vol. 41,
p. 4023-4039.
3. Andrews, E.; Sanders, W.; Gibson L. J. Compressive and tensile
behaviour of aluminum foams. Material Science and Engineering, 1999,
vol. A270, p. 113-124.
4. Bart – Smith, H. et al. Compressive deformation and yielding
mechanisms in cellular Al alloys determined using X – ray tomography
and surface strain mapping. Acta Mater, 1998, vol. 46, no. 10, p. 35833592.
5. Harte, A.-M.; Fleck N. A.; Ashby, M. F. Sandwich panel design using
aluminum alloy foam. Advanced Engineering Materials, 2000, vol. 2, no.
4, p. 219-222.
6. Tianjian L. Ultralight porous metals from fundamentals to applications.
Acta Mechanica Sinica, 2002, vol. 18, no. 5, p. 457-478.
7. Staal, R. A. et al. Predicting failure loads of impact damaged honeycomb
sandwich panels. Journal of Sandwich Structures and Materials, 2009,
vol. 11, p. 213-244.
8. Liang, C.-C.; Yang, M.-F.; Wu, P.-W. Optimum design of metallic
corrugated core sandwich panels subjected to blast loads. Ocean
Engineering, 2001, vol. 28, p. 825-861.
9. Chang, W. – S. et al. Bending behavior of corrugated – core sandwich
plates. Composite Structures, 2005, vol. 70, p. 81-89.
10. Xue, Z.; Hutchinson, J. W. A comparative study of impulse – resistant
metal sandwich plates. International Journal of Impact Engineering,
2004, vol. 30, p. 1283-1305.
11. Lim, C. – H.; Jeon, I.; Kang, K. – J. A new type of sandwich panel with
periodic cellular metal cores and its mechanical performances. Materials
and design, 2009, vol. 30, p. 3082-3093.
12. Zok, F. W. et al. A protocol for characterizing the structural performance
of metallic sandwich panels: application to pyramidal truss cores.
International Journal of Solids and Structures, 2004, vol. 41, p. 62496271.
13. Kim, H.; Kang, K. – J.; Joo, J. – H. A zigzag – formed truss core and its
mechanical performances. Journal of Sandwich Structures and Materials,
2010, vol. 12, p. 351-368.
14. Hohe, J.; Librescu, L. Advances in the structural modeling of elastic
sandwich panels. Mechanics of Advanced Materials and Structures, 2004,
vol. 11, p. 395-424.
30
15. Queheillalt, D. T.; Wadley, H. N. G. Cellular metal lattices with hollow
trusses. Acta Materialia, 2005, vol. 53, p. 303-313.
LIST OF PUBLICATIONS
Articles in journals from Institute for Scientific Information (ISI) list:
1. Žiliukas, Antanas; Kukis, Mindaugas. Pressure vessel with corrugated core
numerical strength and experimental analysis // Mechanika / Kauno
technologijos universitetas, Lietuvos mokslų akademija, Vilniaus
Gedimino technikos universitetas. Kaunas : KTU. ISSN 1392–1207. 2013,
T. 19, nr. 4, p. 374–379. [Science Citation Index Expanded (Web of
Science); INSPEC; Compendex; Academic Search Complete; FLUIDEX;
Scopus]. [0,500]. [IF (E): 0,336 (2013)].
Articles in other international database list
1. Žiliukas, Antanas; Kukis, Mindaugas. Determination of non stability force
of sloping plates // Mechanika 2013 : proceedings of the 18th international
conference, 4, 5 April 2013, Kaunas University of Technology, Lithuania /
Kaunas University of Technology, Lithuanian Academy of Science,
IFTOMM National Committee of Lithuania, Baltic Association of
Mechanical Engineering. Kaunas : Technologija. ISSN 1822–2951. 2013,
p. 252–254. [Conference Proceedings Citation Index]. [0,500]
2. Žiliukas, Antanas; Kukis, Mindaugas. Determination of rational
geometrical parameters of cellular cylinders according to characteristics of
strength and stability // International Review of Mechanical Engineering
(IREME). London : Publishing Division. ISSN 1970–8734. 2014, Vol. 8,
no. 1, p. 100–110. [Academic Search Complete; IndexCopernicus;
Scopus]. [0,500]
Articles in other referred science publications
Material from conference papers
1. Žiliukas, Antanas; Kukis, Mindaugas. Application of strength criteria for
cellular pressure vessels // [ICME 2014 : International Conference on
Mechanical Engineering] : International Science Conference, May 26–27,
2014, London, United Kingdom. London : WASET, 2014. p. 1359–1361.
[0,500].
31
INFORMATION ABOUT AUTHOR OF THE DISSERTATION
Name, Surname: Mindaugas Kukis
Date and place of birth: 27 October 1984, Kaunas, Lithuania.
E-mail: [email protected]
Education and training
2010-09 – 2014-08
2007-09 – 2009-07
2003-09 – 2007-07
Doctoral student at Kaunas University of Technology
in the field of Mechanical Engineering Sciences.
Kaunas University of Technology, Master of
Sciences in Mechanical engineering, Mechanical
engineering.
Kaunas University of Technology, Bachelor of
Sciences in Mechanical engineering, Mechanical
engineering.
REZIUMĖ
Disertacijos apimtis ir struktūra
Disertaciją sudaro įvadas, 4 skyriai, išvados, literatūros sąrašas bei
mokslinių publikacijų disertacijos tema sąrašas ir priedai. Disertacijos apimtis
136 puslapiai, 97 paveikslai ir 33 lentelių. Literatūros sarašą sudaro 141 šaltinių.
Pirmame skyriuje pateikiama literatūros apžvalga, apžvelgiami atlikti tyrimai su
korėtomis plokštėmis, jų privalumai lyginant su monolitiniais lakštais, jų
gamyba ir galimi defektai, išryškinamos problemos ir uždaviniai. Korėtos
plokštės daugeliu aspektu yra pranašesnės nei monolitiniai lakštai, tačiau jų
gamyba yra itin sudėtinga, reikalaujanti didelių investicijų. Visos korėtos
plokštės turi vieną visoms bendrą defektą – korio neprivirinimą prie paviršiaus
lakštų. Slėginiai indai yra potencialiai pavojingi gaminiai, todėl toks defektas
mažinantis maksimalią laikomąją gebą yra neleistinas norint korėtą plokštę
panaudoti slėginio indo kevalui. Skyriuje pristatomas bedefektis korėtų cilindrų
gamybos, pagal kurį būtų nesudėtingai pagaminami tiriami korėti cilindrai.
Antrame skyriuje pristatomas tyrimo objektas, sudaroma skaitinio tyrimo
metodika, išvesta pasvirusių plokštelių nestabilumo jėgos nustatymo formulė,
apžvelgiami stiprumo kriterijai ir pateikiamos taikomų kriterijų
formulės - Drukerio-Pragerio, Moro-Kulono, von Mizeso. Pateikimi korėtų
slėgio indų stiprumo ir stabilumo tyrimo algoritmai. Pristatoma optimizavimo
atlikimo metodika, pateikiami korėtų slėgio indų optimizavimo parametrai,
aprašomas optimizavimui naudojamas „Subproblem“ optimizavimo operatorius.
Skaitinės stiprumo ir stabilumo tyrimo rezultatų verifikavimui pateikiami
verifikavimo metodai. Skaitiniai tyrimai atliekami su skaičiavimo baigtiniais
32
elementais paketu ANSYS. Stiprumo skaitinės analizės verifikavimui
pagaminamas korėtas slėgio indas su gofruoto korio konstrukcija. Stabilumo
tyrimų verifikacijai pasirinktas supaprastintas bandymas, kadangi pagaminti
korėtą indą, kuris sukluptų esant mažesniam išoriniam slėgiui nei 0.07 MPa
(maksimalus vakuumas, kurį galima sukelti vakuuminiu siurbliu. 0.1 MPa yra
absoliutus vakuumas) neįmanoma.
Trečiame skyriuje pateikiami stiprumo tyrimo ir optimizavimo rezultatai, o
skyriaus gale stiprumo analizės rezultatų verifikavimas. Atliktas bandymas su
korėtu slėgio indu, kurio korys gofruotas, patvirtino skaitinių tyrimų rezultatų
priimtinumą.
Ketvirtame skyriuje pateikiami stabilumo tyrimo ir optimizavimo rezultatai, o
skyriaus gale stabilumo analizės rezultatų verifikavimas. Atliktas supaprastintas
bandymas gofruoto korio segmentu patvirtino skaitinių tyrimų priimtinumą.
Darbo tikslas ir uždaviniai
Sukurti ir ištirti slėgio indą, kurio cilindrinės dalies sienelės yra
daugiasluoksnės su korėtu intarpu, pasižymintį geresnėmis funkcinėmis
savybėmis, negu slėgio indai su monolitinėmis sienelėmis, ir paprasta gamybos
technologija. Iš tiriamų korio konstrukcijų nustatyti racionaliausią.
Šiam tikslui pasiekti iškelti tokie uždaviniai:
1. sukurti slėgio indą, kurio kevalo cilindrinės dalies sienelės yra
daugiasluoksnės su įvairios konstrukcijos technologiškai nesudėtingais
korėtais intarpais;
2. sudaryti skirtingų cilindro ir korio geometrinių parametrų slėgio indo
skaičiuojamuosius modelius ir skaitiniais metodais ištirti jų funkcines
charakteristikas bei palyginti jas tarpusavyje naudojant skirtingus stiprumo
kriterijus;
3. pagaminti naujos konstrukcijos slėgio indo bei jo korio segmento
bandomuosius pavyzdžius ir eksperimentiškai ištirti jų stiprumą bei
stabilumą; tyrimo rezultatų pagrindu verifikuoti skaičiuojamuosius
modelius;
4. siekiant sumažinti sukurtų slėgio indų masę atlikti jų sienelių ir korio
elementų storio optimizavimą, optimizuotas konstrukcijas palyginti su
slėgio indais monolitinėmis sienelėmis.
Darbo naujumas
Sukurtas slėgio indas su daugiasluoksnėmis cilindrinės dalies sienelėmis,
kurio funkcinės savybės geresnės, negu slėgio indų su monolitinėmis sienelėmis,
o gamybos technologija palyginti paprasta ir bedefektė. Skaitiškai tiriant sukurto
slėgio indo stiprumą baigtinių elementų analizės sistema ANSYS suprogramuoti
33
ir panaudoti joje standartiškai nesantys Drukerio-Pragerio bei Moro-Kulono
stiprumo kriterijai.
Darbo aktualumas
Sukurtas naujos konstrukcijos slėgio indas, pasižymintis didesniu
stabilumu, negu indas su monolitine sienele, ir atsparesnis išorinio slėgio
poveikiui. Be to, toks indas yra gera alternatyva tais atvejais, kai siekiant
užtikrinti specifinių technologinių procesų vyksmą būtina reguliuoti ar palaikyti
reikiamą inde esančios terpės temperatūrą. Naudojant slėgio indus iš monolitinio
lakšto juos reikia apgaubti tam tikrais papildomais „marškiniais“, kuriuose
galėtų cirkuliuoti reikiamą temperatūros režimą palaikanti terpė.
UDK 621.772-419.5(043.3)
SL344. 2015-04-02, 2,25 leidyb. apsk. l. Tiražas 70 egz. Užsakymas 126.
Išleido leidykla „Technologija“, Studentų g. 54, 51424 Kaunas
Spausdino leidyklos „Technologija“ spaustuvė, Studentų g. 54, 51424 Kaunas
34