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The optimum support of horizontal pressure vessels made from reinforced plastic
The optimum support of horizontal pressure vessels made from reinforced plastic L. VARGA (Technical University of Budapest, Hungary) The deformation and stress state of horizontal pressure vessels are usually determined by the loads resulting from the support. The most favourable state can be realized only with a flexible support arch since its small flexural rigidity results in even distribution of the support pressure, while its small tensile rigidity provides for equal support forces. These facts have been proved by theoretical tests and experiments carried out on a pressure vessel made from reinforced plastic material and have resulted in the development of a structure providing for optimum support.
Key words: reinforced plastic; horizontal pressure vessel; optimum support; flexible support arch; displacement; strain; stress
Horizontal cylindrical pressure vessels, as thin-walled shell structures loaded by mass forces and internal pressure, can be efficiently used for the storage and transportation of liquids and gases. When such pressure vessels are designed, it is always difficult to determine the external forces and to calculate the deformations and stresses caused by the support reactions and liquid pressure. The designer's job is made difficult, on the one hand, by the fact that the character, magnitude and distribution of the support pressure deriving from the saddle can only be estimated and, on the other hand, that the relatively large displacements caused by the support result in considerable rearrangement of the classic stress diagram. Because of this complexity, the analysis of these effects is carried out with approximate, simplified formulae. Zick and Brownell 1'2 calculated circumferential and axial stresses according to the theory of elastic rings and bent beams, respectively. It should also be noted that the field of application of these formulae is rather limited since they can be applied only for the design of cylindrical vessels on two saddle supports and made from the traditional structural material (steel). Unfortunately, these formulae cannot be applied for the design of chemical-proof pressure vessels made from reinforced plastic because such materials are inhomogeneous and anisotropic. Their region of approximately linear elasticity and their 'elastic properties' change as a function of time and load level. The external loads can cause considerable displacements in vessels made of this material. Finally, it is also important that in the case of such materials the permissible
strains should also be taken into consideration in addition to the permissible stresses. The peculiarities and behaviour of reinforced plastics should be taken into consideration not only during the design of the pressure vessel, but also during the shaping and positioning of the support structure. The aim should be always to produce a 'plastic-like construction' matching the special state of deformation. This is particularly valid for pressure vessels and their support system when they are used as tanks in which liquid is stored or transported under no pressure, or relatively small pressure. In such cases the resultant stresses caused by the support forces are much greater than the membrane stresses resulting from the pressure of the stored material. As a consequence, the basic wall thickness of the tanks is determined not by the 'thinwalled equation' but by the support forces. These latter, however, are variables depending on the shape and position of the support structure. This plastic-like construction requires reliable and generalizable theoretical and experimental results as well as many operational experiences which, unfortunately, are very rare in the literature. The author's tests and experiments are dedicated to make up for part of this shortage of data. Experiments were carried out first on a tank of the traditional saddle support and then on a polyester tank reinforced with glass fibres and placed on flexible support arches. At the end of the experiments, b~sed on the test results and further theoretical considerations, the optimum support structure was elaborated.
0010-4361/91/030227-12(~) 1991 Butterworth-Hei nemann Ltd COMPOSITES. VOLUME 22. NUMBER 3. MAY 1991
227
the stresses resulting from the support, the cylindrical shell structure was thickened and the cross-sections above the saddles were reinforced by radial winding (stiffening ring). Four saddles were used for the support (0 = 85°). Therefore the failure came unexpectedly. However, the real surprise was the location and direction of the cracks.
FAILURE OF A PRESSURE VESSEL OF METAL-LIKE DESIGN BUT PLASTIC-LIKE BEHA VlOUR The polyester acid tank reinforced with glass fibres depicted in Fig. 1 displays the consequences of traditional design. After only a short operation, axial (1) and circumferential (2) cracks appeared on the outer surface of the tank (Fig. 2) despite the fact that the tank was properly designed. That is, in order to reduce 765
The circumferential cracks (2) at the support position, which extend to the inner surface of the shell structure, are testimonies of the presence and effect of great axial stresses and strains. The axial cracks (1) at the support span could only be caused by great circumferential stresses and strains. The major stresses were expected to be only in an axial direction in the upper and lower parts of the cylindrical shell structure, contradictory to the previous statements. The contradiction clearly derives from the fact that the reinforced plastic tank with four saddle supports reacted to the operating load in a completely different way from that calculated by the designer utilizing the traditional constructing principles. During investigation of the causes and circumstances of the unexpected failure, the traditional saddle structure finally became suspicious. Since the rigid support arch of the saddle was not able to follow the shape irregularity, deflection and expansion of the cylindrical body, there were great differences in the support forces and the distribution of the support pressure was uneven. As a consequence, such unfavourable deformations and stresses were produced that they resulted in the cracks on the surface of the cylindrical shell structure.
765
6000
4
17 .
.L
L
..
"
I
I
\
-
~1500 I
I
/ --D'-'
b
Fig. 1 mm
Sketch of the deficient 50 m 3 acid tank, all dimensions in
EXPERIMENTS ON A HORIZONTAL POLYESTER PRESSURE VESSEL REINFORCED WITH GLASS FIBRES Experiments were carried out on a pressure vessel filled with water and supported in three places (Figs 3 and 4). The vessel was made of polyester reinforced with glass fibres but without stiffening rings. The cylindrical shell structure consisted of layers of radial and angular windings (~0 = 67°) reinforced with veil and mat as shown in Fig. 5.
Fig. 2 Axial (1) and circumferential cracks (2) on the outer surface of the tank
L/=765
L=6000
H I
c
A
a
c
B" A-
a = 2340 I
Q
Q
a '
!z3_, l
]
~
s= 10
.
Q
Fig. 3 Sketch and characteristic cross-sections of the pressure vessel placed on traditional saddles (I) or flexible support arches (11),all dimensions in mm
228
C O M P O S I T E S . M A Y 1991
Fig. 6 Pressure vessel on three saddle supports prepared for measurement
Fig. 4 Pressure vessel made of polyester with glass fibre reinforcement placed on flexible support arches
values, those belonging to the characteristic crosssections A, B, C, A* and B* (Fig. 3) can be found in the following sections.
DEFORMATION OF PRESSURE VESSEL ON THREE SADDLE SUPPORTS
1
2
3/ 4
=10 R=
1500/
---4_L/
Fig. 5 Schematic layout of the cylindrical shell structure: 1--angular winding; 2--radial winding; 3--mat; 4~veil
The primary purpose of the experiments was to reveal the character, magnitude and distribution of the load passed from the support to the cylindrical shell structure as well as to determine the displacements and strains caused by the support forces and the support pressure. To achieve this goal, two completely different support structures were investigated. The vessel was first put on the traditional saddles (I) and then on the flexible support arches (II). Since the stout pressure vessel does not behave like a beam, the supports were positioned in such a way that the stiffening effect of the head could be felt and that almost equal reaction forces could be induced (Fig. 3). Among the variables characteristic of the plastic-like behaviour of the vessel, the displacements (w) of the cylindrical shell structure were recorded by displacement gauges fixed to the measuring frame and contacting the outer surface (Fig. 9), while the strains (~1, c2) were measured by strain gauges stuck on the outer (o) and inner (i) surface of the shell structure (Fig. 6). In most of the tests, experiments were concentrated on the middle support position and its surroundings where the stiffening effect of the head could not be felt. Overall testing of the shell section was achieved by axially moving the measuring frame and rotating the vessel. Among the measured displacement and strain
The pressure vessel prepared for the experiments is shown in Fig. 6. In this configuration there is a soft spacer (polyurethane plate) inserted between the rigid support arch (width: b=400 mm, support angle: 20 = 170°) and the cylindrical body. The spacer's function is to provide for a more even support pressure. (The spring constant of the spacer was measured prior to the experiments.) The major results obtained during the experiments are shown in Fig. 7. Examination of the left-hand side of the figure, which shows the displacements (WA, WB, We) of the characteristic cross-sections, reveals the magnitude and typical variation of the displacements. The distribution of the displacements produced noticeable 'flattening' of the cylindrical shell structure. Though its extent is reduced further away from the support position, the flattening is still present in the support span. Consequently, the deformations of the tested overall shell structure section are basically determined by the deformation state of the support position. Knowing the displacements of the support position and the spring constant of the spacer, the local values of the arch load (qs) can be calculated. These can be expressed in the following dimensionless form: 4s(V) -
qs('~)
(1)
qso
where the denominator is the constant arch load deriving from the equation:
Q q s o - 2Rsin0
(2)
The arch load calculated from the measured values can be seen on the right-hand side of Fig. 7. The sharp increase in load after ap=60 ° can evidently be explained COMPOSITES. MAY 1991 229
~wA ~wc
,'-1-0-8 ~ - 6 - - 4
w(mm) 10
0
-10_-20
-2
0
2
4
6
8
10
12 14
0
i
2
3
4
5 ~(¥)
-'.x 104
I O = 162 280 N
Fig. 7 Change of displacements (WA, WB, Wc)and the axial (clo)and circumferential strains (%j) measured in the characteristic crosssections of the pressure vessel on three saddle supports and status of support pressure [c/s(~P)]
by the effect of the rigid saddle arch and the inhibited displacements. The strains measured in cross-section B can also be found on the right-hand side of Fig. 7. The circumferential strains (c2i) measured on the inner surface are seen to follow the flattening of the cylinder. Even the surprising magnitude and variation of the axial strains (elo) measured on the outer surface can be explained by the flattening of the cylinder, to which the stiffening effect of the head has also contributed. In addition to the data and explanations provided, other experiences obtained during the experiment are also of interest, as follows. There were always considerable differences (almost 50%) in the support forces when the pressure vessel was removed, rotated and then replaced on the saddles. This occurred each time the process was performed, and several such processes were done. The rotation of the vessel on the saddles resulted in the rearrangement of the displacements and support pressures. The displacements and strains measured during filling showed sudden changes from time to time and in certain places. All these phenomena indicate the uncertain force balance of the pressure vessel supported with three saddles. Consequently, it was impossible to carry out reproducible strain measurements which could include the whole vessel.
DEFORMATION AND STRESS STATE OF PRESSURE VESSEL ON FLEXIBLE SUPPORT ARCHES The deformation and stress state of the support
230
COMPOSITES. M A Y 1991
position, which are relevant for the whole pressure vessel, can be improved, in principle, by reduction of the support force and support pressure; that is, by increasing the surface of the saddle arch and the number of saddles. However, in the case of more than two saddles, the uneven reaction forces should be taken into consideration. It can also happen that 'thickening' of the cylindrical shell structure would be a more economic solution. In addition, the surface of the saddle arch can be efficiently increased only to a certain limit (b/R<~l/3, 0~
R
w(mm) 2O
10
0
-10
-20
w(mm) -30
-30
-20
-10
0
10
20 ~
30
40
:20 =120"
I .
Q=162 260 N
%
Fig. 8 Change of displacements (w A, WB, WC, WA*, WB*) measured in the characteristic cross-sections of the pressure vessel placed on flexible support arches
shown in Fig. 9, the support position stiffened by the head (cross-section A* and B*) was also examined. These measured displacements (WA*, WB*)are shown on the left-hand side of the figure. Consideration of Fig. 8 shows that the character of the flattening is only slightly changed further away from the middle support position, while its degree is reduced, which leads to the cylindrical shell structure becoming barrel shaped. The circumferential (%0, E2i) and the axial (cto, Eli ) strains were measured on the outer and inner surface of the cylindrical structure and the values and variations in the characteristic cross-sections are shown in Figs 10, 11 and 12. The figures show that, like the displacements, the strains follow the flattening and the barrel shape of the shell structure. Since designers think rather in terms of stresses than of strains, the 'average stresses' relating to the measured strains can also be of interest. From the 'elastic properties' characteristic of the whole layered shell structure, these average stresses can be calculated simply from the following equations:
& O 1 -O2 -Fig. 9 Measurement of displacements (WA., WB*) induced in the vicinity of the support position stiffened by head
1--~12~21
E2 1-V12~'21
(El-b~12E2)
(3)
(E2"I-~21E 1)
(4)
The modulus of elasticity (/~1, /~2) and Poisson's ratio
COMPOSITES. M A Y 1991
231
rr~xl04
t~"
6
4" 2
I
/
+ ~ - - - - ~ , L - - ~~
""
6-2-~,-6-8.:1"0J"
.,,. '.,~'Ao
~I--~---
'~----~
',
-10-8-6-4-2
I
i
/
I
!
0
2
4
6
8
1~1
~,
x104
10 12, 14
Fig. 10 Change of the axial (e~) and circumferential strains (E2) measured on the inner (Cli , (E2i) and outer surface (clo, C2o)in crosssection A
/
/
/
~ x 10 4
~2x 104q
2
0
-2 -4 -6 ~=-8-10
~
.^^~I~-~--.^^. -10
-6
-4
-2
0
2
4
6
8 ~10
12 14
~E2c
Fig. 11 Change of the axial (El) and circumferential strains (c2) measured on the inner (eli, c2i) and outer surface (elo, C2o)in crosssection B
232
COMPOSITES. MAY 1991
/
eI
x 104
'\
Fig. 12 Change of the axial (c 1) and circumferential strains (c 2) measured on the inner (eli, ~2~)and outer surface (c~o, %0) in crosssection C
(~12, "~2~)along the major directions are determined in the following way for the pressure vessels investigated. Measured on a test specimen, the axial modulus was E 1 = 7500 MPa. Loading the vessel with an internal pressure of p=0.1 MPa, average strains of Cla=7 × 104 and c2a=3.4 × 104 were measured in the crosssections in the membrane state, corresponding to average stresses of O2a=2Ola = 15 MPa. Considering the above values and assuming -[?,2/~ = ~a2/~21 as well the relations in Equations (3) and (4), the following elastic characteristics can be calculated:/~2=30 000 MPa, ~t12=0.6 and "~21=0.15. Using the measured strains and the material properties as determined above, Equations (3) and (4) gave the circumferential (O2o, ozi) and axial (Oao, o10 average stresses in the characteristic cross-sections shown in Figs 13, 14 and 15. The middle stresses shown in the figures were determined from the equations: OIM = 0.5(Olo Jr- O1i) and OZM ~" 0.5(020 "~- O2i)
Examination of the stresses shows that the circumferential stresses are greater than the calculated axial stresses in the cylindrical shell structure, despite the fact that there were much greater strains axially. This phenomenon, which can be explained simpl~, by the 'strong anisotropy of the shell structure (E2/~1=4), justifies the requirement that as well as the permissible stresses, the designer should also take into consideration the permissible strains. The figures clearly demonstrate
that the magnitude and variation of the axial stresses is completely different from that expected based on the theory of bent beams. All these, of course, are the consequences of the flattening and barrel-like shape of the shell structure. It is interesting to compare the greatest stresses deriving from the valid requirements and those calculable from the measured strains. According to the requirements given by Zick ~, the greatest circumferential stress at the horn of the saddle (~p=60 °) is O2max=--335 MPa. Even considering the effective cross-section of the vessel, the greatest axial stress resulting from the bending moment is only Olmax= 1.12 MPa. This stress is also induced in the support position in the cross-section at an angle of xp=80 °. In contrast to the above values, stresses of o2i=-28.8 MPa and Oao= 15 MPa can be calculated from the greatest measured strains. These stresses are induced at an angle of ap---80° in the support positions in the cross-sections. Therefore it can be stated that, as far as the most dangerous cross-sections are concerned, the calculations and the measurement gave almost the same results. However, the present work has shown the situation to be considerably different. As can be seen, the greatest stress measured axially is considerably greater (13.4 times) than the calculated value. At the same time, the greatest measured circumferential stress is very much smaller (0.08 times) than the calculated value. The great differences between the measured and
COMPOSITES. MAY 1991 233
A
I
140
j
a 2 (MPa)
24 20 t 16 12
8
4
120"~
c 1 (MPa)
0 I-4 -8 -12
80 5
20= 120°;
80
/
Gli
~MT
Fig. 13 Change of the axial (0'1) and circumferential section A calculated from the measured strains
averagestresses(0"2)at the inside (0"1i,o2i) and outside surface (O"1o, 0"2(]) in cross-
calculated stresses prove the shortcomings and inaccuracies of the applied calculation model. The cylindrical shell structure of the pressure vessel is not deformed as an elastic ring but is flattened considerably as a consequence of the support force. In addition, the shell structure gradually becomes barrel-shaped as a result of the stiffening effect of the head and the thickened shell connection. All the above effects result in the complete rearrangement of the deformation and stress state which, unfortunately, cannot be followed with a simple model of calculation. However, it is very likely that these great differences appear only with stout pressure vessels (like the one in the test) and that their degree is reduced in the case of slim vessels 3. Returning to the experiments, the displacements could also be measured below the support; the resulting values are shown in Fig. 16. Though the displacements vary only slightly along the width of the support, it is still justified, in favour of the torsionless operation, to prepare the flexible support arch in the form of a pair of narrow ribbons (b*/R<~3/6).When the pressure vessel was placed several times on the support arches, there were no considerable differences in the support forces. The reaction forces on the separate support arches were Q = 155 0 0 0 - 1 7 0 000 N. Based on Equation (2) the value of the arch load proved to be almost constant at qso=62.5 N mm -1, remaining the same even after rotation of the vessel. It was also proved during the test that the flexible support arch is
234
COMPOSITES. M A Y 1991
less sensitive to the unevenness of the foundation and the accuracy of the setting. In addition, the assembly of the arches and the positioning of the vessel are very simple and there is no need for a soft spacer. Naturally, the advantages of the flexible support arch over the saddle support are more apparent in the case of a support angle of 20 = 170 °.
FORMATION OF THE OPTIMUM SUPPORT STRUCTURE The optimum support structure is the one which forces the pressure vessel to take the most favourable deformation and stress state. This state can be produced when there is an even distribution of the support pressure along the surface of the support arch. The support forces are equal, in the case of more than two supports. These conditions can be ensured only with the flexible support arch. The small flexural rigidity and the small tensile rigidity of this arch result in the even distribution of the support pressure and equal support forces, respectively. Since the flexural and tensile rigidities of a support arch with a thickness of 6 and a width of b* change in accordance with the following equations:
B=IEb*63 12
and
S=Eb*6
it is advisable to prepare the flexible support arch in the form of a pair of narrow ribbons made from a very
/
/
B
14( 120 100" a 2 (MPa)
(MPa) 24 20"16
12 8
4 ~0_ -4
-8 -4
-8 -12
0
20= 12oo.j
/
Fig. 14 Change of the axial (o,1)and circumferential average stresses (02) at the inside (oli, o2~)and outside surface (olo, o2°) in crosssection B calculated from the measured strains '\
J4J /'
~" /
//
/
//
:
C
,
z
/
i c 2 (MPa) ~
r
~ 20 16' 12 8
/ 4
0
-4'-8
-12-16--20"~
~
-
-
__ .
2e= 1 2 0 ° J /
-4
,
~ 0~, 4
G,
(MPa)
60 :' 40
%i Fig. 15 Changeof the axial (o1) and circumferential average stresses (02) at the inside (o1~,02i) and outside surface (Olo,02°) in crosssection C calculated from the measured strains
C O M P O S I T E S . M A Y 1991
235
elastic material. In the case of a pair of narrow steel ribbons and a support angle of 20= 120°, favourable support pressure and support forces were found during the experiments carried out so far. Despite the improvement, the cylindrical shell structure still showed considerable flattening and became barrelshaped. Therefore further tests are needed to reduce these effects; that is, to find the optimum solution.
81-
According to the previous experiments, the degrees of flattening and barrel shape are determined by the deformation state of the support position. However, for a certain support force and a recommended support arch, the magnitude and variation of the displacements at the support position are influenced only by changing the support surface and the support angle. Since the support position behaves like a ring, the effect of the support angle can be followed most easily on the model of calculation based on the theory of elastic rings. This model was worked out by assuming the balance of forces shown in Fig. 17. As a final result, the following equations were obtained for the internal forces.
40*
6t v
2~- 30
•
A
5
o~
10
I.
I
15
20
/25
30
I
,
x (cm) =.-
I 15 ° "2F 10"
For 0<~p~<0: t"
-,
b/2
MR -
Fig. 16 Changeof displacements measured in the vicinityof the flexiblesupport arch
.
.
.
.
.
.
2
:t
~
+ [ 1 . 5 + ( n - 0)cot 0] cosy
+ ap sinap
(5)
J
I
-
0 (a-b)
c
iB !
OB
s r. SlI MB
OBI
.QB
BI
IC
I
_ _ - - + _--_
I
i I MB OB=l/2Q(a-b )
+ ....
/ I~
....
0
2 R sin0 =/:t 2 rcpL ga = Oa Fig. 17 Model of calculation for the evaluation of the effect of the support angle 236
C O M P O S I T E S . M A Y 1991
-_----~--_ _-__-,
°;,(
/
20 = 120 °
-0.16
-0.12
-0.08
-0.04
0
,
Fig. 18
0.04
\ \ I
0,08
0
-
/
\f
1Qn t /p simp + [(n - e)cot e - e.5] cosap
2
+
sin0
O
a
- - (1 + cosap) ~x
(6)
For 0~
1 QRIe
+ ( 1 . 5 - c o t 0 ) c o s ~ + (xp-n)simp 1
L
3
(7) NR _
1 2
Q (xp - n)simp - (0.5 + 0cot 0) cosxp n
+ _b a
+ cosy)
The increase in the support angle also has a favourable effect on the flattening of the ring. It can be assumed that the degree of flattening is proportional to the characteristic displacement, which can be reduced with increasing angle 0. This is verified in Fig. 19 where the displacement factor derived from Equations (9) and (10) is shown as a function of the support angle. It can be deduced from Fig. 19 that the most favourable deformation and stress state of the ring can be realized at a support angle of 20= 180 °. This statement is also relevant to the pressure vessel since its cylindrical shell structure behaves like a ring. This fact can be proved by the similarity between the change of the stresses Oao
(8)
The characteristic displacements of the ring are: 0.2l m
1 R3 Q (0.7854ecote+
2 IE2 n Wy
-0.04
Bending m o m e n t diagram of the support position in the case of support angles of 20= 120° and 20= 180°
N R --
WX
20 = 180 °
_ 1 R 3 Q [ 1.5708 cote + 2 IE 2 n [ + 0.78540 - 1.2337/
~x
0 _1.5708 ] (9) sin0 3
0.1!
II 13
0.10
e-0.5n sine
(10)
Equations (5) and (7) prove that the bending moments induced in the ring and, consequently, also the displacements improve inversely with increasing support angle. This phenomenon can be clearly seen in Fig. 18. The left- and right-hand side of the figure show the moment factor variation for e = 6 0 ° and 0=90 °, respectively. According to Fig. 18, the greatest bending moment can be considerably reduced--almost to 25% of its original value--with increasing support angle.
0.05
I 20
I qo
60
80
90
e (°) Fig. 19 Change of displacements (wx,-wv) of the support place as a function of support angle (0)
COMPOSITES. MAY 1991 237
induce equal reaction forces on the supports even when there are more than three supports, it is advisable to prepare the flexible support arch from a pair of synthetic belts such as those used for conveyors. The flexible support arches should be positioned with equal spacing and utilize the stiffening effect of the head. A possible solution for the support structure devised on the basis of the tests and providing optimum support can be seen in Fig. 20. The described structural solution is used as an example in a registered patent.
ACKNOWLEDGEMENTS The author thanks Budaplast Company, F. Windisch, B. Frigyik and M. Vass for their significant contributions to this research work.
REFERENCES 1 Zick,L.P. 'Stresses in large horizontal cylindrical pressure vessels on two saddle supports' WeldingResearchSupplement (September 1951) p 435 2 Brownell, E. Process EquipmentDesign (John Wiley Inc, New York, USA, 1959) 3 Varga, L. 'Spannungsuntersuchung von horizontal gelagerten zylindrischen GFK-Fliissigkeitsbeh/iltern'Kunststoffe 65 No 3 (1975) p 144 Fig. 20 A possible solution forthe structure providing
optimum support
A U THOR shown in Figs 13-15 and the m o m e n t factor/~/60 displayed in Fig. 18. In order to achieve an even distribution of the support pressure and arch load in the range 60°~<0~<90° and to
238
COMPOSITES.
M A Y 1991
D r L. Varga is Professor of Mechanical Engineering and Director of the Institute for Machine Design, Technical University of Budapest, H-1111 Budapest Muegyetem rkp.3, Hungary.
Key words: reinforced plastic; horizontal pressure vessel; optimum support; flexible support arch; displacement; strain; stress
Horizontal cylindrical pressure vessels, as thin-walled shell structures loaded by mass forces and internal pressure, can be efficiently used for the storage and transportation of liquids and gases. When such pressure vessels are designed, it is always difficult to determine the external forces and to calculate the deformations and stresses caused by the support reactions and liquid pressure. The designer's job is made difficult, on the one hand, by the fact that the character, magnitude and distribution of the support pressure deriving from the saddle can only be estimated and, on the other hand, that the relatively large displacements caused by the support result in considerable rearrangement of the classic stress diagram. Because of this complexity, the analysis of these effects is carried out with approximate, simplified formulae. Zick and Brownell 1'2 calculated circumferential and axial stresses according to the theory of elastic rings and bent beams, respectively. It should also be noted that the field of application of these formulae is rather limited since they can be applied only for the design of cylindrical vessels on two saddle supports and made from the traditional structural material (steel). Unfortunately, these formulae cannot be applied for the design of chemical-proof pressure vessels made from reinforced plastic because such materials are inhomogeneous and anisotropic. Their region of approximately linear elasticity and their 'elastic properties' change as a function of time and load level. The external loads can cause considerable displacements in vessels made of this material. Finally, it is also important that in the case of such materials the permissible
strains should also be taken into consideration in addition to the permissible stresses. The peculiarities and behaviour of reinforced plastics should be taken into consideration not only during the design of the pressure vessel, but also during the shaping and positioning of the support structure. The aim should be always to produce a 'plastic-like construction' matching the special state of deformation. This is particularly valid for pressure vessels and their support system when they are used as tanks in which liquid is stored or transported under no pressure, or relatively small pressure. In such cases the resultant stresses caused by the support forces are much greater than the membrane stresses resulting from the pressure of the stored material. As a consequence, the basic wall thickness of the tanks is determined not by the 'thinwalled equation' but by the support forces. These latter, however, are variables depending on the shape and position of the support structure. This plastic-like construction requires reliable and generalizable theoretical and experimental results as well as many operational experiences which, unfortunately, are very rare in the literature. The author's tests and experiments are dedicated to make up for part of this shortage of data. Experiments were carried out first on a tank of the traditional saddle support and then on a polyester tank reinforced with glass fibres and placed on flexible support arches. At the end of the experiments, b~sed on the test results and further theoretical considerations, the optimum support structure was elaborated.
0010-4361/91/030227-12(~) 1991 Butterworth-Hei nemann Ltd COMPOSITES. VOLUME 22. NUMBER 3. MAY 1991
227
the stresses resulting from the support, the cylindrical shell structure was thickened and the cross-sections above the saddles were reinforced by radial winding (stiffening ring). Four saddles were used for the support (0 = 85°). Therefore the failure came unexpectedly. However, the real surprise was the location and direction of the cracks.
FAILURE OF A PRESSURE VESSEL OF METAL-LIKE DESIGN BUT PLASTIC-LIKE BEHA VlOUR The polyester acid tank reinforced with glass fibres depicted in Fig. 1 displays the consequences of traditional design. After only a short operation, axial (1) and circumferential (2) cracks appeared on the outer surface of the tank (Fig. 2) despite the fact that the tank was properly designed. That is, in order to reduce 765
The circumferential cracks (2) at the support position, which extend to the inner surface of the shell structure, are testimonies of the presence and effect of great axial stresses and strains. The axial cracks (1) at the support span could only be caused by great circumferential stresses and strains. The major stresses were expected to be only in an axial direction in the upper and lower parts of the cylindrical shell structure, contradictory to the previous statements. The contradiction clearly derives from the fact that the reinforced plastic tank with four saddle supports reacted to the operating load in a completely different way from that calculated by the designer utilizing the traditional constructing principles. During investigation of the causes and circumstances of the unexpected failure, the traditional saddle structure finally became suspicious. Since the rigid support arch of the saddle was not able to follow the shape irregularity, deflection and expansion of the cylindrical body, there were great differences in the support forces and the distribution of the support pressure was uneven. As a consequence, such unfavourable deformations and stresses were produced that they resulted in the cracks on the surface of the cylindrical shell structure.
765
6000
4
17 .
.L
L
..
"
I
I
\
-
~1500 I
I
/ --D'-'
b
Fig. 1 mm
Sketch of the deficient 50 m 3 acid tank, all dimensions in
EXPERIMENTS ON A HORIZONTAL POLYESTER PRESSURE VESSEL REINFORCED WITH GLASS FIBRES Experiments were carried out on a pressure vessel filled with water and supported in three places (Figs 3 and 4). The vessel was made of polyester reinforced with glass fibres but without stiffening rings. The cylindrical shell structure consisted of layers of radial and angular windings (~0 = 67°) reinforced with veil and mat as shown in Fig. 5.
Fig. 2 Axial (1) and circumferential cracks (2) on the outer surface of the tank
L/=765
L=6000
H I
c
A
a
c
B" A-
a = 2340 I
Q
Q
a '
!z3_, l
]
~
s= 10
.
Q
Fig. 3 Sketch and characteristic cross-sections of the pressure vessel placed on traditional saddles (I) or flexible support arches (11),all dimensions in mm
228
C O M P O S I T E S . M A Y 1991
Fig. 6 Pressure vessel on three saddle supports prepared for measurement
Fig. 4 Pressure vessel made of polyester with glass fibre reinforcement placed on flexible support arches
values, those belonging to the characteristic crosssections A, B, C, A* and B* (Fig. 3) can be found in the following sections.
DEFORMATION OF PRESSURE VESSEL ON THREE SADDLE SUPPORTS
1
2
3/ 4
=10 R=
1500/
---4_L/
Fig. 5 Schematic layout of the cylindrical shell structure: 1--angular winding; 2--radial winding; 3--mat; 4~veil
The primary purpose of the experiments was to reveal the character, magnitude and distribution of the load passed from the support to the cylindrical shell structure as well as to determine the displacements and strains caused by the support forces and the support pressure. To achieve this goal, two completely different support structures were investigated. The vessel was first put on the traditional saddles (I) and then on the flexible support arches (II). Since the stout pressure vessel does not behave like a beam, the supports were positioned in such a way that the stiffening effect of the head could be felt and that almost equal reaction forces could be induced (Fig. 3). Among the variables characteristic of the plastic-like behaviour of the vessel, the displacements (w) of the cylindrical shell structure were recorded by displacement gauges fixed to the measuring frame and contacting the outer surface (Fig. 9), while the strains (~1, c2) were measured by strain gauges stuck on the outer (o) and inner (i) surface of the shell structure (Fig. 6). In most of the tests, experiments were concentrated on the middle support position and its surroundings where the stiffening effect of the head could not be felt. Overall testing of the shell section was achieved by axially moving the measuring frame and rotating the vessel. Among the measured displacement and strain
The pressure vessel prepared for the experiments is shown in Fig. 6. In this configuration there is a soft spacer (polyurethane plate) inserted between the rigid support arch (width: b=400 mm, support angle: 20 = 170°) and the cylindrical body. The spacer's function is to provide for a more even support pressure. (The spring constant of the spacer was measured prior to the experiments.) The major results obtained during the experiments are shown in Fig. 7. Examination of the left-hand side of the figure, which shows the displacements (WA, WB, We) of the characteristic cross-sections, reveals the magnitude and typical variation of the displacements. The distribution of the displacements produced noticeable 'flattening' of the cylindrical shell structure. Though its extent is reduced further away from the support position, the flattening is still present in the support span. Consequently, the deformations of the tested overall shell structure section are basically determined by the deformation state of the support position. Knowing the displacements of the support position and the spring constant of the spacer, the local values of the arch load (qs) can be calculated. These can be expressed in the following dimensionless form: 4s(V) -
qs('~)
(1)
qso
where the denominator is the constant arch load deriving from the equation:
Q q s o - 2Rsin0
(2)
The arch load calculated from the measured values can be seen on the right-hand side of Fig. 7. The sharp increase in load after ap=60 ° can evidently be explained COMPOSITES. MAY 1991 229
~wA ~wc
,'-1-0-8 ~ - 6 - - 4
w(mm) 10
0
-10_-20
-2
0
2
4
6
8
10
12 14
0
i
2
3
4
5 ~(¥)
-'.x 104
I O = 162 280 N
Fig. 7 Change of displacements (WA, WB, Wc)and the axial (clo)and circumferential strains (%j) measured in the characteristic crosssections of the pressure vessel on three saddle supports and status of support pressure [c/s(~P)]
by the effect of the rigid saddle arch and the inhibited displacements. The strains measured in cross-section B can also be found on the right-hand side of Fig. 7. The circumferential strains (c2i) measured on the inner surface are seen to follow the flattening of the cylinder. Even the surprising magnitude and variation of the axial strains (elo) measured on the outer surface can be explained by the flattening of the cylinder, to which the stiffening effect of the head has also contributed. In addition to the data and explanations provided, other experiences obtained during the experiment are also of interest, as follows. There were always considerable differences (almost 50%) in the support forces when the pressure vessel was removed, rotated and then replaced on the saddles. This occurred each time the process was performed, and several such processes were done. The rotation of the vessel on the saddles resulted in the rearrangement of the displacements and support pressures. The displacements and strains measured during filling showed sudden changes from time to time and in certain places. All these phenomena indicate the uncertain force balance of the pressure vessel supported with three saddles. Consequently, it was impossible to carry out reproducible strain measurements which could include the whole vessel.
DEFORMATION AND STRESS STATE OF PRESSURE VESSEL ON FLEXIBLE SUPPORT ARCHES The deformation and stress state of the support
230
COMPOSITES. M A Y 1991
position, which are relevant for the whole pressure vessel, can be improved, in principle, by reduction of the support force and support pressure; that is, by increasing the surface of the saddle arch and the number of saddles. However, in the case of more than two saddles, the uneven reaction forces should be taken into consideration. It can also happen that 'thickening' of the cylindrical shell structure would be a more economic solution. In addition, the surface of the saddle arch can be efficiently increased only to a certain limit (b/R<~l/3, 0~
R
w(mm) 2O
10
0
-10
-20
w(mm) -30
-30
-20
-10
0
10
20 ~
30
40
:20 =120"
I .
Q=162 260 N
%
Fig. 8 Change of displacements (w A, WB, WC, WA*, WB*) measured in the characteristic cross-sections of the pressure vessel placed on flexible support arches
shown in Fig. 9, the support position stiffened by the head (cross-section A* and B*) was also examined. These measured displacements (WA*, WB*)are shown on the left-hand side of the figure. Consideration of Fig. 8 shows that the character of the flattening is only slightly changed further away from the middle support position, while its degree is reduced, which leads to the cylindrical shell structure becoming barrel shaped. The circumferential (%0, E2i) and the axial (cto, Eli ) strains were measured on the outer and inner surface of the cylindrical structure and the values and variations in the characteristic cross-sections are shown in Figs 10, 11 and 12. The figures show that, like the displacements, the strains follow the flattening and the barrel shape of the shell structure. Since designers think rather in terms of stresses than of strains, the 'average stresses' relating to the measured strains can also be of interest. From the 'elastic properties' characteristic of the whole layered shell structure, these average stresses can be calculated simply from the following equations:
& O 1 -O2 -Fig. 9 Measurement of displacements (WA., WB*) induced in the vicinity of the support position stiffened by head
1--~12~21
E2 1-V12~'21
(El-b~12E2)
(3)
(E2"I-~21E 1)
(4)
The modulus of elasticity (/~1, /~2) and Poisson's ratio
COMPOSITES. M A Y 1991
231
rr~xl04
t~"
6
4" 2
I
/
+ ~ - - - - ~ , L - - ~~
""
6-2-~,-6-8.:1"0J"
.,,. '.,~'Ao
~I--~---
'~----~
',
-10-8-6-4-2
I
i
/
I
!
0
2
4
6
8
1~1
~,
x104
10 12, 14
Fig. 10 Change of the axial (e~) and circumferential strains (E2) measured on the inner (Cli , (E2i) and outer surface (clo, C2o)in crosssection A
/
/
/
~ x 10 4
~2x 104q
2
0
-2 -4 -6 ~=-8-10
~
.^^~I~-~--.^^. -10
-6
-4
-2
0
2
4
6
8 ~10
12 14
~E2c
Fig. 11 Change of the axial (El) and circumferential strains (c2) measured on the inner (eli, c2i) and outer surface (elo, C2o)in crosssection B
232
COMPOSITES. MAY 1991
/
eI
x 104
'\
Fig. 12 Change of the axial (c 1) and circumferential strains (c 2) measured on the inner (eli, ~2~)and outer surface (c~o, %0) in crosssection C
(~12, "~2~)along the major directions are determined in the following way for the pressure vessels investigated. Measured on a test specimen, the axial modulus was E 1 = 7500 MPa. Loading the vessel with an internal pressure of p=0.1 MPa, average strains of Cla=7 × 104 and c2a=3.4 × 104 were measured in the crosssections in the membrane state, corresponding to average stresses of O2a=2Ola = 15 MPa. Considering the above values and assuming -[?,2/~ = ~a2/~21 as well the relations in Equations (3) and (4), the following elastic characteristics can be calculated:/~2=30 000 MPa, ~t12=0.6 and "~21=0.15. Using the measured strains and the material properties as determined above, Equations (3) and (4) gave the circumferential (O2o, ozi) and axial (Oao, o10 average stresses in the characteristic cross-sections shown in Figs 13, 14 and 15. The middle stresses shown in the figures were determined from the equations: OIM = 0.5(Olo Jr- O1i) and OZM ~" 0.5(020 "~- O2i)
Examination of the stresses shows that the circumferential stresses are greater than the calculated axial stresses in the cylindrical shell structure, despite the fact that there were much greater strains axially. This phenomenon, which can be explained simpl~, by the 'strong anisotropy of the shell structure (E2/~1=4), justifies the requirement that as well as the permissible stresses, the designer should also take into consideration the permissible strains. The figures clearly demonstrate
that the magnitude and variation of the axial stresses is completely different from that expected based on the theory of bent beams. All these, of course, are the consequences of the flattening and barrel-like shape of the shell structure. It is interesting to compare the greatest stresses deriving from the valid requirements and those calculable from the measured strains. According to the requirements given by Zick ~, the greatest circumferential stress at the horn of the saddle (~p=60 °) is O2max=--335 MPa. Even considering the effective cross-section of the vessel, the greatest axial stress resulting from the bending moment is only Olmax= 1.12 MPa. This stress is also induced in the support position in the cross-section at an angle of xp=80 °. In contrast to the above values, stresses of o2i=-28.8 MPa and Oao= 15 MPa can be calculated from the greatest measured strains. These stresses are induced at an angle of ap---80° in the support positions in the cross-sections. Therefore it can be stated that, as far as the most dangerous cross-sections are concerned, the calculations and the measurement gave almost the same results. However, the present work has shown the situation to be considerably different. As can be seen, the greatest stress measured axially is considerably greater (13.4 times) than the calculated value. At the same time, the greatest measured circumferential stress is very much smaller (0.08 times) than the calculated value. The great differences between the measured and
COMPOSITES. MAY 1991 233
A
I
140
j
a 2 (MPa)
24 20 t 16 12
8
4
120"~
c 1 (MPa)
0 I-4 -8 -12
80 5
20= 120°;
80
/
Gli
~MT
Fig. 13 Change of the axial (0'1) and circumferential section A calculated from the measured strains
averagestresses(0"2)at the inside (0"1i,o2i) and outside surface (O"1o, 0"2(]) in cross-
calculated stresses prove the shortcomings and inaccuracies of the applied calculation model. The cylindrical shell structure of the pressure vessel is not deformed as an elastic ring but is flattened considerably as a consequence of the support force. In addition, the shell structure gradually becomes barrel-shaped as a result of the stiffening effect of the head and the thickened shell connection. All the above effects result in the complete rearrangement of the deformation and stress state which, unfortunately, cannot be followed with a simple model of calculation. However, it is very likely that these great differences appear only with stout pressure vessels (like the one in the test) and that their degree is reduced in the case of slim vessels 3. Returning to the experiments, the displacements could also be measured below the support; the resulting values are shown in Fig. 16. Though the displacements vary only slightly along the width of the support, it is still justified, in favour of the torsionless operation, to prepare the flexible support arch in the form of a pair of narrow ribbons (b*/R<~3/6).When the pressure vessel was placed several times on the support arches, there were no considerable differences in the support forces. The reaction forces on the separate support arches were Q = 155 0 0 0 - 1 7 0 000 N. Based on Equation (2) the value of the arch load proved to be almost constant at qso=62.5 N mm -1, remaining the same even after rotation of the vessel. It was also proved during the test that the flexible support arch is
234
COMPOSITES. M A Y 1991
less sensitive to the unevenness of the foundation and the accuracy of the setting. In addition, the assembly of the arches and the positioning of the vessel are very simple and there is no need for a soft spacer. Naturally, the advantages of the flexible support arch over the saddle support are more apparent in the case of a support angle of 20 = 170 °.
FORMATION OF THE OPTIMUM SUPPORT STRUCTURE The optimum support structure is the one which forces the pressure vessel to take the most favourable deformation and stress state. This state can be produced when there is an even distribution of the support pressure along the surface of the support arch. The support forces are equal, in the case of more than two supports. These conditions can be ensured only with the flexible support arch. The small flexural rigidity and the small tensile rigidity of this arch result in the even distribution of the support pressure and equal support forces, respectively. Since the flexural and tensile rigidities of a support arch with a thickness of 6 and a width of b* change in accordance with the following equations:
B=IEb*63 12
and
S=Eb*6
it is advisable to prepare the flexible support arch in the form of a pair of narrow ribbons made from a very
/
/
B
14( 120 100" a 2 (MPa)
(MPa) 24 20"16
12 8
4 ~0_ -4
-8 -4
-8 -12
0
20= 12oo.j
/
Fig. 14 Change of the axial (o,1)and circumferential average stresses (02) at the inside (oli, o2~)and outside surface (olo, o2°) in crosssection B calculated from the measured strains '\
J4J /'
~" /
//
/
//
:
C
,
z
/
i c 2 (MPa) ~
r
~ 20 16' 12 8
/ 4
0
-4'-8
-12-16--20"~
~
-
-
__ .
2e= 1 2 0 ° J /
-4
,
~ 0~, 4
G,
(MPa)
60 :' 40
%i Fig. 15 Changeof the axial (o1) and circumferential average stresses (02) at the inside (o1~,02i) and outside surface (Olo,02°) in crosssection C calculated from the measured strains
C O M P O S I T E S . M A Y 1991
235
elastic material. In the case of a pair of narrow steel ribbons and a support angle of 20= 120°, favourable support pressure and support forces were found during the experiments carried out so far. Despite the improvement, the cylindrical shell structure still showed considerable flattening and became barrelshaped. Therefore further tests are needed to reduce these effects; that is, to find the optimum solution.
81-
According to the previous experiments, the degrees of flattening and barrel shape are determined by the deformation state of the support position. However, for a certain support force and a recommended support arch, the magnitude and variation of the displacements at the support position are influenced only by changing the support surface and the support angle. Since the support position behaves like a ring, the effect of the support angle can be followed most easily on the model of calculation based on the theory of elastic rings. This model was worked out by assuming the balance of forces shown in Fig. 17. As a final result, the following equations were obtained for the internal forces.
40*
6t v
2~- 30
•
A
5
o~
10
I.
I
15
20
/25
30
I
,
x (cm) =.-
I 15 ° "2F 10"
For 0<~p~<0: t"
-,
b/2
MR -
Fig. 16 Changeof displacements measured in the vicinityof the flexiblesupport arch
.
.
.
.
.
.
2
:t
~
+ [ 1 . 5 + ( n - 0)cot 0] cosy
+ ap sinap
(5)
J
I
-
0 (a-b)
c
iB !
OB
s r. SlI MB
OBI
.QB
BI
IC
I
_ _ - - + _--_
I
i I MB OB=l/2Q(a-b )
+ ....
/ I~
....
0
2 R sin0 =/:t 2 rcpL ga = Oa Fig. 17 Model of calculation for the evaluation of the effect of the support angle 236
C O M P O S I T E S . M A Y 1991
-_----~--_ _-__-,
°;,(
/
20 = 120 °
-0.16
-0.12
-0.08
-0.04
0
,
Fig. 18
0.04
\ \ I
0,08
0
-
/
\f
1Qn t /p simp + [(n - e)cot e - e.5] cosap
2
+
sin0
O
a
- - (1 + cosap) ~x
(6)
For 0~
1 QRIe
+ ( 1 . 5 - c o t 0 ) c o s ~ + (xp-n)simp 1
L
3
(7) NR _
1 2
Q (xp - n)simp - (0.5 + 0cot 0) cosxp n
+ _b a
+ cosy)
The increase in the support angle also has a favourable effect on the flattening of the ring. It can be assumed that the degree of flattening is proportional to the characteristic displacement, which can be reduced with increasing angle 0. This is verified in Fig. 19 where the displacement factor derived from Equations (9) and (10) is shown as a function of the support angle. It can be deduced from Fig. 19 that the most favourable deformation and stress state of the ring can be realized at a support angle of 20= 180 °. This statement is also relevant to the pressure vessel since its cylindrical shell structure behaves like a ring. This fact can be proved by the similarity between the change of the stresses Oao
(8)
The characteristic displacements of the ring are: 0.2l m
1 R3 Q (0.7854ecote+
2 IE2 n Wy
-0.04
Bending m o m e n t diagram of the support position in the case of support angles of 20= 120° and 20= 180°
N R --
WX
20 = 180 °
_ 1 R 3 Q [ 1.5708 cote + 2 IE 2 n [ + 0.78540 - 1.2337/
~x
0 _1.5708 ] (9) sin0 3
0.1!
II 13
0.10
e-0.5n sine
(10)
Equations (5) and (7) prove that the bending moments induced in the ring and, consequently, also the displacements improve inversely with increasing support angle. This phenomenon can be clearly seen in Fig. 18. The left- and right-hand side of the figure show the moment factor variation for e = 6 0 ° and 0=90 °, respectively. According to Fig. 18, the greatest bending moment can be considerably reduced--almost to 25% of its original value--with increasing support angle.
0.05
I 20
I qo
60
80
90
e (°) Fig. 19 Change of displacements (wx,-wv) of the support place as a function of support angle (0)
COMPOSITES. MAY 1991 237
induce equal reaction forces on the supports even when there are more than three supports, it is advisable to prepare the flexible support arch from a pair of synthetic belts such as those used for conveyors. The flexible support arches should be positioned with equal spacing and utilize the stiffening effect of the head. A possible solution for the support structure devised on the basis of the tests and providing optimum support can be seen in Fig. 20. The described structural solution is used as an example in a registered patent.
ACKNOWLEDGEMENTS The author thanks Budaplast Company, F. Windisch, B. Frigyik and M. Vass for their significant contributions to this research work.
REFERENCES 1 Zick,L.P. 'Stresses in large horizontal cylindrical pressure vessels on two saddle supports' WeldingResearchSupplement (September 1951) p 435 2 Brownell, E. Process EquipmentDesign (John Wiley Inc, New York, USA, 1959) 3 Varga, L. 'Spannungsuntersuchung von horizontal gelagerten zylindrischen GFK-Fliissigkeitsbeh/iltern'Kunststoffe 65 No 3 (1975) p 144 Fig. 20 A possible solution forthe structure providing
optimum support
A U THOR shown in Figs 13-15 and the m o m e n t factor/~/60 displayed in Fig. 18. In order to achieve an even distribution of the support pressure and arch load in the range 60°~<0~<90° and to
238
COMPOSITES.
M A Y 1991
D r L. Varga is Professor of Mechanical Engineering and Director of the Institute for Machine Design, Technical University of Budapest, H-1111 Budapest Muegyetem rkp.3, Hungary.