ORIGINAL ARTICLE
Wall thickness variation effect on tank’s shape behaviour under critical harmonic settlement
Ahmed Shamel Fahmy
*, Amr Mohamed Khalil
Structural Engineering Department, Alexandria University, Egypt Received 20 June 2016; revised 18 July 2016; accepted 19 July 2016 Available online 28 September 2016
KEYWORDS Steel tanks;
Wall thickness;
Settlement
Abstract The purpose of this study was to investigate the effect of wall thickness variation on tank’s wall buckling mode under the effect of critical harmonic settlement for open top tanks.
The study was performed on four tanks which have the same geometric and material properties except wall thickness, for each case the tank was subjected to several settlement waves which has the same settlement amplitude, and the buckling mode and critical vertical settlement results were compared. For buckling mode, the results show that tanks with wall thickness at a close range have similar buckling mode behaviour and in case using too thick wall the buckling mode starts to change. And for the effect on critical vertical settlement, the results show that vertical settlement is sensitive to any variation in wall thickness beside that settlement value changes with the effected wave number and this variation could change the whole behaviour of the tanks. The study recom- mended that in case of performing analysis for a tank with neglecting the variation in wall thickness values, the value of chosen wall thickness should be the average of wall thickness values obtained from the designed equation.
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1. Introduction
By the expansion of oil industry, the need of storage devices has increased, and the tanks are constructed on coastal areas and on island to satisfy production needs. As a result, tanks are subjected to poor soil conditions and the tanks are sub- jected to many sort of settlement which has it effect on tanks and differential settlement [which also called Harmonic Settle- ment] has its effect, and a tiny settlement under tank wall could
cause a large displacement on the top fibre of the wall which would lead to failure especially in open top tanks[1,2].
In the stage of analytical solutions, researchers were inter- ested in the relation between harmonic settlement at the bot- tom of tank wall and circumferential displacement at the top of the tank where the floating roof and open top tanks were the main research topics. For instance, Malik et al.[3]started with assumption that the settlement under tank wall was inhar- monic, with extensional theory, the relation between harmonic settlement and radial settlement was presented in an equation which was suitable for a little wave number. After that, Kamyab and Palmer[4–7]included harmonic settlement and primary wind girder in their work, and they used the modified Donnel large deflection equation to include a wider range of wave number. With the development of computer technology,
* Corresponding author.
E-mail address:[email protected](A. Shamel Fahmy).
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many researches were conducted based on numerical method where the finite element method is considered as the most used method[2,8–17]. The linear buckling of closed top cylindrical shells under the effect of edge vertical settlement by Jonaidi and Ansourian[10]in their work uniform shells was consid- ered, they concluded that for a little wave number the shear buckling mode is noted and this mode is controlled by axial buckling which is related to the increase in wave number.
Lately, an experiment for a small tank with a flat roof was per- formed by Godoy and Sosa[12]and the tank was subjected to harmonic settlement. They displayed that the equilibrium path is nonlinear and the shell shows a stable symmetric bifurcation behaviour; the buckling behaviour of tank with fixed-roof sub- jected to harmonic settlement was studied by Cao and Zhao [13]. Three types of buckling were considered in their work:
elastic bifurcation analysis, geometrical nonlinear analysis of the perfect shell and geometrical nonlinear analysis of the imperfect shell. Furthermore, they investigated the effects of geometric imperfection, radius-to-thickness ratio, wave num- ber, and height-to-radius ratio on the buckling strength. They showed that both of the critical harmonic settlement and the buckling mode are related at a close range to the geometric parameters aforementioned. Gong et al. [14] reported the buckling behaviour of tanks with conical roof under the effect of harmonic settlement. The geometrical nonlinear behaviour was included in the analyses, in addition to, settlement–dis- placement curves, critical harmonic settlements and the buck- ling modes for various wave numbers were reported in his work.
The previous researches focus on the relation between height, radius and thickness relation, and how to achieve the optimum combination between them. After that the researches focus on the effect of imperfection and settlement effects for tanks with roofs. The relation between top stiffening ring on open top tank and differential settlement still un-investigated enough, even Gong et al.[18]research didn’t involve in this subject and it focuses on the imperfection effect for open top tanks.
As a result, this study will investigate that relation and the effect of wall thickness variation effect on open top tank under harmonic settlement in a simplified method.
2. Finite element model
For that purpose, a set of 4 tanks were modelled as a full 3D model on Sap 2000 package with the following properties: The radius and height of the shell areR= 15.12 m,h= 12.191 m, respectively. The finite element model was divided into mesh to include 27,300 nodes and 26,845 elements, shell element was defined using the four-node and quadrilateral shell element, and the full model was built and analysed. Full 3D model on Sap 2000 is shown inFig. 1.
A top stiffening ring was considered with the geometry UPN 100 web 100 * 6 mm, flange 50 * 8.5 mm as shown in Fig. 2.
The wall thickness will vary regarding the required wall thickness. The chosen wall thickness for each tank is shown inTable 1(seeFig. 3).
The wall thicknesses for tank no. 4 was calculated based on the resulting design equation according to BS 2654 (clause 7.2.2) [30]:
t¼ D
20Sð98wðH0:3Þ þpÞ þc
Figure 1 Finite element model of open top tank with top stiffening ring using SAP2000.
Figure 2 Typical stiffening-ring UPN section for tank top ring.
Table 1 Wall thickness arrangement for models.
tank no. 1 tank
no. 2 tank no. 3 tank
no. 4
h t mm t mm t mm t mm T.
L
0 - 2 m 8.92 7 45 6
2 - 4 m 8.92 7 45 6
4 - 6 m 8.92 7 45 8.5
6 - 8 m 8.92 13 105 10.5
8 - 10 m 8.92 13 105 12.5
12.191 10 -
m 8.92 13
105 15 G.
L
Figure 3 Section in tank wall.
For tank no. 2 the wall thickness was based on the average thickness of tank no. 4, tank no. 2 wall thickness was modelled on two layers only instead of 6 layers as in tank no. 4, and that will help in studying the effect of neglecting wall thickness in the finite element model.
Tank no. 4 wall was chosen with excessive thickness to show the effect of choosing tank wall with too thick plates.
The calculated wall thickness from the previous equation for the first layer was 4 mm, but as per BS 2654 clause 7.1.3 [32]
the minimum shell thickness should be 6 mm regarding tank diameter, so the shell thickness was chosen 6 mm to match the standard requirements.
2.1. Boundary condition
This model contains 456 support points at the lower fibre of the wall. The radial and circumferential constraints were applied at the bottom of tank wall, but the axial displacement at the base was defined as a ground displacement load and the values of this load will be defined later.
2.2. Material properties
The material nonlinearity for steel section was considered in the model using plasticity assuming the material yielding is defined by the Von Mosies yield function with isotropic mate- rial behaviour. The current model uses the stress strain curve relation as shown in theFig. 4.
2.3. Loading 2.3.1. Dead load
The own-weight of the shell is calculated in SAP2000 using the gravity constantg= 9.81 m/s2and the defined density of steel (P= 7.85 * 106kg/mm3).
2.3.2. Settlement
The main objective of this study was to investigate the effect of differential settlement, which is also can be defined as har- monic settlement, because it is usually simplified as the har- monic forms for various wave numbers. Fig. 5 represents settlement under tank’s wall.
Settlement values were calculated by the equationu= un cos (nu), wherenis the wave number; un is the amplitude of
the nth harmonic settlement; andu is the angle of the point at the base of the shell.
3. Results assessment
The analysis results were obtained from the top fibre of the tank wall.Fig. 6illustrates the detailed results for the models analysis. For tanks 2, 3 and 4 the average thickness of the wall was considered to obtain the values of Un/tto be able to estab- lish the relation between critical vertical settlement value and wave number.
The behaviour of tanks 2 and 4 is almost the same and the variation within 0–15%, except forn= 10 and 14. For tank no. 2n= 10 the value of Un/t= 0.817 and for tank no. 4 it was 0.08. For tank no. 2 n= 14 the value of Un/t= 2.06 and for tank no. 4 it was 5.16. The similar behaviour of tank 2 and 4 is natural result because the different in thickness between tank’s wall is very small and it is within 2.5%.
Besides, the behaviour of tank nos. 1, 2 and 4 is at close range from n= 2 tilln= 14 with a variation within 8–42%
except n= 10, 18 and 22. For tank no. 1 at n= 18 Un/t= 0.98206 but for tank nos. 2 and 4, Un/t= 3.52 and 26.56 respectively. Atn= 22 the results increased in a greater amount for tank no. 1 Un/t= 4.57, but for tank nos. 2 and 4,
-60000 -40000 -20000 0 20000 40000 60000
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
stress
strain
Stress Strain Curve of Steel
Figure 4 Stress–strain curve of steel.
Figure 5 Harmonic settlement under tank wall.
-50 0 50 100 150 200
0 5 10 15 20 25
UN / T
N
tank no.1 tank no.2
tank no.3 tank no.4
Figure 6 Wall thickness variation effect on critical harmonic settlement.
Un/t= 36.75 and 31.0256 respectively which represent an increase in the displacement, and this increase in settlement could be related to wall thickness increase which increased the loads. Tank no. 3 behaviour is completely unexpected and comparable with the other tanks’ behaviour. The results
atn= 2–4 represent a great decreasing in displacement com- paring with other tanks which could be regarded the great increase in wall thickness and that as a result would increase the weight of the wall which would limit the rotation of the wall. At n= 5, 6, 10 and 14 the settlement values start to
n tank 1 tank 2 tank 3 tank 4
2
3
4
5
6
10
14
18
22
Figure 7 Buckling mode for tanks.
increase which is the reversal behaviour of tanks 1,2 and 4. But at n= 18 and 22 the displacement decreased and the beha- viour turns to be similar to tank no. 1 with a rapid decreasing in settlement values.
4. Buckling mode
In the previous section the effect of wall thickness variation under critical harmonic settlement was investigated on 4 tanks, and in this section the effect of wall thickness variation is investigated under critical harmonic settlement illustrated on 4 tanks. Fig. 7 shows a comparison between buckling mode for all tanks, where the final deformed wall for each tank was compared for each wave number.
For tank nos. 1, 2 and 4, the behaviour is almost the same except a little variation in tank no. 2 atn= 18 and 22. For n= 2 and 3 represent rotation of the tank, the tank behaviour starts to change fromn= 4 where a reversal relation between wall buckling and notch appears. When wave number increases the notch starts to appear in the lower fibre of the wall in exchange of decrease in the buckling at tank’s wall till n= 22 where the notch dominates the lower fibre of the tank with tiny buckling at the wall.
Buckling mode of tank no. 3 could explain the behaviour which appeared inFig. 7, and forn= 2–3 the mode is almost the same of the other tanks which represent rotation but with fewer result regarding wall thickness effect, but for n= 4 a notch starts to appear in the middle of the wall in addition to wall buckling which was considered as a combined beha- viour comparing with other tanks behaviour. Fromn= 5 till n= 14 the notch starts to appear in the lower fibre in addition to wall buckling which is inverted behaviour ofn= 4. A rever- sal relation between notch and wall buckling could be noted where wall buckling dominates the behaviour and notch effect disappears tilln= 18 and 22, Wall buckling affects most of wall height with the disappearance of the notch and that explains the rapid decrease in settlement value between n= 18 and 22.
That behaviour for tank no. 3 comes with a huge increase in wall thickness which may – as a result – increase the cost of the tank from economic point of view. So it is a must to be careful with choosing the thickness for tank’s wall.
5. Conclusions
Neglecting wall thickness variation in steel tanks could affect the results especially if the chosen thickness was not at close range with the designed wall thickness from the standards.
Using too thick plates in steel tanks improves the behaviour of the wall under harmonic settlement in many wave num- bers but it comes with a huge increase in the cost of the tank.
In order to simplify the analysis and in case of neglecting wall thickness variation, it is important to keep the thick- ness of the wall within the average of the variable wall thickness.
The effect of the top ring changes between wave numbers, so it is vital to determine wave number ranges to decide the economic section to be used, and that should be used if the dominated load case is settlement.
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