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Buckling behavior of cylindrical steel tanks with concavity of vertical weld line imperfection
Journal of Constructional Steel Research 145 (2018) 289–299
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Journal of Constructional Steel Research
Buckling behavior of cylindrical steel tanks with concavity of vertical weld line imperfection Mehdi Rastgar a,⁎, Hossein Showkati b a b
Department of Civil Engineering, Khoy Branch, Islamic Azad University, Khoy, Iran Department of Civil Engineering, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 29 December 2016 Received in revised form 6 February 2018 Accepted 22 February 2018 Available online xxxx Keywords: Field study Cylindrical tank Buckling behavior Vertical weld line imperfection External pressure
a b s t r a c t Shell structures are built using a number of welded curved panel parts. Hence, some geometrical imperfections emerge. These imperfections have a direct impact on structural behavior of shells during the external compressive loading. In this research, a field study was accomplished on the implementation of the storage tanks in a refinery site and then, the resulted imperfections were identified and categorized. The survey of imperfections revealed that the imperfection in form of concavity of vertical weld line is the most prevalent type of imperfection seen in the steel tanks. This imperfection experimentally modeled and the buckling behavior of these tanks was evaluated under uniform external pressure. Comparing obtained results of estimation, ASME code and experimental research represented a considerable difference in the amount of buckling load. Results show that the imperfections due to concavity of vertical weld line are very important in buckling of the tanks under uniform external pressure. This imperfection decreases initial, full and post buckling capacity of the tanks under uniform external pressure, significantly. Findings of this research show that for design of steel tanks under uniform external pressure load, 65% of the buckling load obtained from the ASME Code should be used. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Shell structures have curved initial shapes with the thickness less than other dimensions. In some states, radius-to-thickness ratio reaches 3000. The force field generated in these structures includes membrane and bending forces, which can change depending on shell thickness. Steel tanks designed in cylindrical form are commonly used shell structures in industrial facilities. In geometrical terms, these tanks have a very small thickness compared to their other dimensions and are categorized as thin-walled structures for buckling failure caused by the influence of uniform external pressure loading on the tank wall [1]. The buckling of tanks under uniform external pressure is usually caused by operational problems during the discharge of the liquid contents in such a way that partial vacuum is produced. These events are usually classified as accidents, and occur in individual tanks in a tank farm, rather than affecting many tanks in the same event, such as those produced by a natural disaster. The objective of this paper is identification and categorization of created imperfections in implementation of the storage tanks via a field study. By identifying the prevalent imperfection, buckling behavior
⁎ Corresponding author. E-mail addresses: [email protected] (M. Rastgar), [email protected] (H. Showkati).
https://doi.org/10.1016/j.jcsr.2018.02.028 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
and failure of these tanks is investigated under uniform external pressure loading. 2. Stability in shell structures Evaluating the stability problems in thin-walled shells is important for two reasons. First, the ratio of thickness to other dimensions is very low in these structures, which highlights the instability problem. Second, shell structures are exposed to compressive stresses and forces; however, shell buckling is caused by low thickness. One of the main properties of shells is having much higher membrane stiffness than bending stiffness. Accordingly, a shell can absorb a large amount of membrane energy without experiencing a great deformation, while large deformations and rotation in the cross-section need to absorb this energy through bending deformations [1]. 2.1. Buckling of shell structures Buckling is considered as a nonlinear phenomenon, in which the structure cannot take further load with the same geometry and changes its shape in order to find alternative equilibrium configurations. Shell buckling occurs as the structure response to the membrane forces. Membrane forces act along the component axis and tangent to the middle surface of the shell and the buckling occurs when the structure
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converts membrane strain energy into bending strain energy without any change in the applied external load [1]. There have been a number of studies addressing buckling behavior and instability, see e.g. [2]. and references therein. Below we mention several contributions relying on experimental approach. Influence of primary boundary condition on the buckling of shallow cylindrical shells was studied by Showkati and Ansourian experimentally [3]. Wang and Koizumi investigated the buckling of cylindrical shells with longitudinal joints under external pressure [4]. Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure was considered by Chen et al. [5]. Aghajari et al. conducted an experimental study on the buckling of thin cylindrical shells with two-stepwise variable thickness under external pressure [6]. Experimental and numerical investigation of composite conical shells stability subjected to dynamic loading was investigated by Jalili et al. [7]. Ghazijahani et al. studied longitudinally stiffened corrugated cylindrical shells under uniform external pressure [8]. 2.2. Geometric imperfections on shells buckling Geometric imperfections include all the deviations in the form of a structural component compared with its ideal geometric composition. In construction of shell structures, due to the large dimensions, curved plates or panels can be used. The seam between various surfaces of the main source of deviation is in the real form, these deviations or imperfections can be generated as a result of welding or appropriate incompatibility of the plates with larger dimensions than other plates. In contrast to various structures, the buckling strength of shells with no imperfections is significantly different from the buckling strength of the same shell with imperfections. In some states, geometric imperfections may strengthen the structure and increase its capacity. This feature puts shells among the structures which are called sensitive to imperfection. In Fig. 1, the buckling behavior of columns, flat plates and cylindrical shells are schematically shown. In these curves, black lines show the system with no geometric imperfections or perfect system, while red lines represent the corresponding behavior of the imperfect system. As can be seen, the flat plates and column elements are not sensitive to imperfection, while cylinders which are the examples of thin-walled structures are very sensitive to imperfections [9]. There have been a number of studies addressing the influence of imperfections on the buckling behavior of shell structures. Below we mention several contributions relying on experimental approach. Calladine studied the causes of imperfection-sensitivity in the buckling of thin shells [10]. Teng et al. investigated the geometric imperfections in full-scale welded steel silos [11]. In other work, buckling behavior of large steel cylinders with patterned welds was considered
by Hubner et al. [12]. Influence of imperfection on the buckling of thin cylindrical shell under uniform external pressure was studied experimentally by Lo Frano and Forasassi [13]. Maali et al. investigated the Buckling behavior of conical shells under weld-induced imperfections experimentally [14]. Yang et al. studied the buckling of cylindrical shells with general axisymmetric thickness imperfections under external pressure [15]. Fatemi et al. conducted experiments on imperfect cylindrical shells under uniform external pressure and observed that geometric imperfections have a more considerable impact on the behavior of shells [16]. In other work, inelastic stability of liners of cylindrical conduits with local imperfection under external pressure was studied by Khaled El-Sawy [17]. Ghazijahani et al. conducted experiments on dented cylindrical shells under peripheral pressure [18]. Thompson presented advances in shell buckling. He studied on the buckling of axially compressed cylindrical shells with arbitrary thickness imperfections theoretically and experimentally [19]. Cao et al. studied buckling of cylindrical shells with arbitrary thickness imperfections under axial compression analytically [20]. Lee et al. investigated the geometric role of precisely engineered imperfections on the critical buckling load of spherical elastic shells. Their investigation combined precision experiments, finite element modeling and numerical solutions of a reduced shell theory, all of which were found to be in excellent quantitative agreement [21]. Evkin et al. investigated the buckling of a spherical shell under external pressure and inward concentrated load [22]. In other work, Hutchinson and Thompson studied nonlinear buckling behavior of spherical shells subject to external pressure. They found that the nonlinear axisymmetric post-buckling behavior of perfect thin spherical shells and their asymmetric bifurcations are characterized providing results for a structure/loading combination with an exceptionally nonlinear buckling response [23]. Due to the mentioned studies about imperfections in shell structures, field study and vertical weld line imperfection has not been studied. In this research this imperfection has been introduced and buckling behavior of cylindrical steel tanks has been investigated under uniform external pressure. 3. Analytical equations for buckling of cylindrical shells Donnell in 1933 obtained an equation for the buckling of cylindrical shells as follow [24]: 4
D∇8 w þ
Et ∂ w R2 ∂x4
2
þ ∇4 Nx
∂ w ∂x
2
þ 2Nxy
! 2 2 ∂ w ∂ w þ Nθ 2 ¼ 0 ∂x ∂θ ∂θ
ð1Þ
This equation can be used for various loading conditions. Let us take the simple case of applied lateral pressure, where Nx = Nxy = 0 and Nθ = Pr − σcrt. Eq. (1) then becomes [24]: 4
8
D∇ w þ
Et ∂ w R2 ∂x4
2
4
þ∇
Nθ
∂ w ∂θ
2
! ¼0
ð2Þ
An expression for the deflection that satisfies the simply supported boundary condition of a cylinder can be expressed as: w ¼ wmn sin
mπx L
sin
nπy πR
ð3Þ
Substituting this expression into Eq. (2) gives the nontrivial 2
Fig. 1. Load-axial displacement graph of columns, flat plates and cylindrical shells in perfect and imperfect states [9].
2 2
ðm þβ 2 π KE solution σ cr ¼ 12ð1−υ 2 Þ ðt=LÞ where K ¼ β2 pffiffiffiffiffiffiffiffiffiffiffiffiffi L2 2 Z ¼ Rt 1−υ 2
Þ
þ
12Z2 2, π4 β2 ð1þβ2 =m2 Þ
nL β ¼ πR ,
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The number of circumferential waves of cylindrical shell formed in the body is as follows: [3] sffiffiffiffiffiffiffiffiffiffiffiffi ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi u 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 u 1−ν 6π R R 4 u n¼t ≈ 2:74 L 2 t L t R R
ð5Þ
3
Et Et mπR So that D ¼ 12ð1−υ 2 Þ, C ¼ ð1−υ2 Þ, m ¼ L
Based on the ASME, Boiler and Pressure Vessel Code Section VIII, Rules for Construction for Pressure Vessels, the allowable external pressure is calculated from: [24,26] P cr ¼
2:42E 3=4 ð1−υ2 Þ
ðt=2RÞ2:5 h i L=2R−0:45ðt=2RÞ1=2
ð6Þ
Fig. 2. A plot of K [24]
Considering m = 1 the minimum value of K becomes: K ¼ ð1þβ β2
2 2
Þ
þ
12Z2 2 π4 β 2 ð1þβ 2 Þ
For structures with large (R/t) ratios, Eq. (6) can be simplified to: [24,26] P cr ¼
A plot of K defined by above equations is shown in Fig. 2.
2:42E 3=4 ð1−υ2 Þ
ðt=2RÞ2:5 ðL=2RÞ
ð7Þ
3.1. Short cylinders For short cylinders, the curvature parameter Z can be set to zero and then K = (β + 1/β)2 3.2. Intermediate-length cylinders As the ratio L/R increases, the quantity (β + 1) can be approximated 2
by β. Hence K ¼ β2 þ π12Z 4 β6 3.3. Long cylinders For long cylinders, the buckling mode is similar to that of a circular ring and is elliptic in shape. Hence, n = 2 and β = 2L/πR and K reduces to K = β2. Considering simply supported conditions for the ends of the cylindrical shell and m = 1, and application of uniform external pressure to the roof of cylindrical tanks that causes a uniform axial force to the body of tanks, buckling load of these tanks under the simultaneous effect of uniform external pressure and axial load is obtained by the following equation: [25] 1 Pcr ¼ R
4 D
þ m4 1−υ2 C R2 2 m 2 þ n2 n2 þ 0:5m2
m2 þ n2
ð4Þ
4. Field survey In order to store oil products in a refinery site, the construction of numerous tanks has begun, which are being implemented as steel cylindrical tanks. These tanks have the height of 12 m, diameter of 23 m, and sheet thickness of 18 mm and 8 of them are under construction. For the construction of these tanks, the steel sheets with the dimensions of 6 × 1.5 m and thickness of 18 mm are used. The sheets are entered into the rolling device and the curvature operations are performed during several steps to reach the radius of the tank. Then, they are transferred to the relevant location by the special vehicles for transporting the rolled sheets and their installation operations are carried out using the crane and human force. From this step on, numerous imperfections caused by various implementation factors are made; some of them are partly modified and some others are remained in the tank even after the implementation is complete. Installation of sheets is shown in Fig. 3. According to this Figure, the rolled sheets are put in their place by the crane and then temporarily welded next to each other (tack weld). When the first stage is completed, the next layers are implemented similar to the first layer. The rolling, transportation, and installation of these sheets cause various imperfections in the implemented layer. As shown in Fig. 3, the edges of the circumferential sheets do not match each other; these edges should get close to each other by a chain and a crane until the imperfection is modified. Basically, some of
Fig. 3. Circumferential sheets installation process in the body of tank and generation of different imperfections.
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the created imperfections are removed from the body of the tanks and some others are left permanently. In this research, after attending the tank implementation location and field monitoring of the installation of circumferential sheets, various types of imperfections are observed which are classified as below: a) Non-compliance to the edges of consecutive horizontal rows that occurred outside or inside b) Vertical seam not executed properly and sharp towards inside or outside c) Non parallel vertical edges d) Nonaligned horizontal edges e) Cause by unsuitable rolling f) Vertical different layers in height g) Effect of welding temperature which usually appear in circumferential form, and h) Wind Effects causing troughs or bulge, in the peripheral of the last layer of tank. These imperfections are shown in Fig. 4. By continuing the field study on the implementation of these tanks in a 6-month period and statistical inference from the imperfections introduced in all the 8 tanks being constructed in the refinery site, the frequency distribution graph of the imperfections is drawn, as shown in Fig. 5. In this Figure, the horizontal axis shows the type of imperfection and the vertical axis is the frequency percentage of the corresponding imperfection.
Fig. 5. Frequency distribution graph of the imperfection types in all tanks.
Accordingly, it can be seen that, among all the introduced imperfections, imperfection types b) and f) occur more frequently than others, respectively. In imperfection type b), the edges of the circumferential sheets do not match each other. This Imperfection named that vertical weld line imperfection includes concavity of vertical weld line towards inside at the location of two adjacent sheets. This imperfection is shown in Fig. 6. In this research buckling behavior of steel tanks with vertical weld line imperfection is studied.
a)
b)
c)
d)
e)
f)
g)
h) Fig. 4. Types of observed imperfections in the field study of tanks.
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Fig. 6. Imperfection type b) (vertical weld line imperfection)
5. Experimental program 5.1. Geometric specifications of experimental specimens According to Fig. 5, imperfection type b) named as the vertical weld line imperfection occurs more frequently than others. For the experimental evaluation of the impact of this imperfection on the buckling behavior of tanks, three experimental specimens are made with the following characteristics: The first specimen of Spec1 has continuous circumferential plate in an integrated form without intended imperfection. Due to the limited length of the used plates which is 3 m and providing for the 3.6 m perimeter of this specimen, there are two vertical welding lines in the body of tank. CO2 welding is used in this specimen and others. The weld line is perfectly circular and has no dents or bumps.
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The second specimen of Spec2 has entirely an intentional imperfection in terms of a vertical weld line with 15 mm in depth within tank height. Similar to the previous specimen, this specimen has two vertical weld lines on the body of the tank that the imperfection is located on one of these weld lines and the tank is fully circular in another weld line and has no dents or bumps. The third specimen of Spec3 that is in an integrated form, has two intentional imperfections of the type of vertical weld line with 15 mm over the tank height and they are located on one of the tank's diameters with 180 degrees angle between them. There are two vertical weld lines in the body of this tank and the imperfections exist along these welded lines. All three specimens have the diameter of 1.15 m, height of 0.6 m, conical roof with the height of 0.2 m, body plate thickness of 1 mm and floor and roof plate thickness of 2 mm. These specimens are shown in Fig. 7. (See Fig. 8.) In order to prevent the floor and roof buckling of these experimental tanks influenced by external pressure, strengthening annular straps with the diameter of 0.3, 0.6 and 0.9 m and section of 20 × 5 mm are used. These straps could increase the bending stiffness in the roof and floor of the tanks; so, under the uniform external pressure, these two parts maintain their initial situation and hence, do not buckle; only the body of the tanks starts to move and buckles. The size of the experimental specimens is selected in accordance existing laboratory facilities. According to provided dimensions, it is observed that experimental-scale models are on the scale of 1:20 compared to real tanks. Welding on experimental models is quite different from real tanks. These specimens are welded using CO2. The shielding gas reduces generated heat and 1 mm plates are subsequently welded without any significant imperfections. A great care is taken in construction of experimental specimens and it is tried to produce only considered imperfection. Despite high accuracy in producing the experimental specimens, various local imperfections emerge in these specimens; the major one
Fig. 7. Experimental specimens.
Fig. 8. Measurement of intentional imperfection value.
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(A)
(B)
Fig. 9. (A) The three samples (B) tensile test (C) stress-strain diagram of material.
which includes imperfections caused by welding. This fact is due to low thickness of sheets used as troughs in the body, troughs in the floor and roof of the specimens caused by transportation from the workshop to laboratory, as well as movement at different places in the laboratory. The local imperfections may also affect buckling behavior of tanks. As previously noted, the experimental specimens have intentional imperfections in terms of a dent on vertical weld line on tank's body with 15 mm size. In Fig. 6, the measurement of intentional imperfection value is shown in specimens. A metal ruler is coordinated with circular body of the tank to measure considered imperfection. In imperfection location, the distance between the ruler and vertical weld line equals 15 mm, which is the same as considered imperfection. An intentional imperfection in specimen 2 and two intentional imperfections in specimen 3 are produced. But there is no intentional imperfection in specimen 1 and vertical weld line is completely adapted to the ruler. Therefore, this specimen does not have considered imperfection and is indicated as the specimen with no imperfection. By using the approximate Eqs. (4) and (6), the buckling load caused by uniform external pressure along with axial load is obtained which is equal to 23.5 and 23.3 kPa, respectively. Also number of circumferential waves formed in the body of the tank is 13 waves based on Eq. (5), In this study, the amount of the obtained buckling load is used for the evaluation and comparison of the results taken from experimental specimens. 5.2. Mechanical properties of the specimens Mechanical properties of the specimens have been achieved from tensile test. These properties including elasticity Modulus, yield stress, failure phase stress and equivalent strain of this point. Tensile test was carried out for 3 samples of the used sheets in experimental specimens. The test temperature is the same as the temperature of the laboratory and approximately 28 degrees Celsius. The material is carbon steel.
Table 1 Mechanical properties of the specimens. Average modulus of elasticity
Average yield stress
Average failure stress
Average failure strain
200 GPa
194 MPa
325 MPa
30.24%
Dimensions of tensile samples are selected based on recommended dimensions in the ASTM E8/E8M−15a code. In Fig. 9 the samples, tensile test and stress-strain diagram of material are shown. Mechanical properties of the specimens are given in Table 1. These values are the average results of three samples. 6. Testing of specimens and evaluation of results 6.1. Uniform external pressure loading In order to evaluate the buckling behavior of experimental specimens influenced by uniform external pressure, the suction device is utilized. It is an electro-pump device that discharges the air inside the tank with the constant flow of 40 m3/h to the outside. So, by discharging the air inside the tank, the atmospheric pressure is uniformly imported into the external surfaces of this tank and loading is applied as uniform external pressure to the tank. Since air discharge flow of the tank is too high, it is necessary to embed a control valve to adjust the discharge flow and thus control the external pressure. In order to measure the internal pressure in the tank, a pressure gauge is used. This device shows the moment-to-moment pressure inside the tank, which is decreased due to the discharge by the suction device. So, external pressure is obtained by subtracting atmospheric pressure and the pressure inside the tank. There are three holes in the tank floor: the first hole is connected to the suction device, through which discharge operation is done (applying uniform external pressure). The second hole is connected to the discharge valve and controls the flow of tank discharge (loading control) and the third one is connected to the pressure gauge to measure the internal pressure (loading measurement). Fig. 10 depicts the laboratory equipment including the suction pump for loading application, loading control valve, loading measurement device, deformation measurement devices, location of test specimens, the data logger, and the computer. 6.2. Measurement devices of deformation In order to take the experimental specimens deformation, measurement devices are installed in different parts of these specimens in the middle of the height radially and at the top of the conical roof of the specimens. These devices include LVDT, strain gauge and pressure sensor. The LVDT is an acronym used for Linear Variable Differential Transformer. It is a transformer which is used for measuring linear
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295
Fig. 10. A view of the laboratory equipment's and experimental specimen
Fig. 12. Schematic view of all installed sensors with location of imperfection and failure
displacement. LVDTs take the directional displacements vertically, horizontally, or radially in the locations they are installed and then, send them to the data logger. The strain gauges are electronic circuits that are attached using a special adhesive in the desired location and they measure the strain in their installed direction and send it to the computer. In this research, in all the tests, five LVDTs and two horizontal and vertical strain gauges are used to record the deformation of different parts of the experimental specimens. In Fig. 11, a sample of these measurement devices which is installed on the experimental specimens is represented and Fig. 12 shows schematic view of all installed sensors with location of imperfection and failure.
complete and, finally at the external pressure of 16.5 kPa, failure suddenly happens and the tank becomes unstable. Fig. 14(A) and (B) shows full buckling and failure of the experimental Spec2 respectively. The Spec 3, similar to previous specimens, continues gradually and in a controlled manner to the failure stage and full instability. In this specimen, by increasing the external pressure 10 waves are formed in the tank circumference and the buckling of the body becomes complete and, finally, at the external pressure of 19.5 kPa, failure suddenly happens and the tank becomes unstable. Fig. 15(A) shows full buckling and Fig. 15(B) demonstrates instability and failure of the experimental Spec3. According to these Figures, full buckling and unstable form of all three specimens is similar and up to full buckling, regular circumferential waves are seen in the tanks body but unstable location of Spec2 and 3 is different from Spec1 and failure of these specimens occurs at the location of imperfection.
6.3. Test implementation Air discharge from Spec1 starts by turning on the suction pump device and the difference between the internal and external pressure of the tank is gradually increased. As a result, external pressure is gradually applied to the tank and the initial buckling of the tank body begins. By increasing the external pressure, the number of the waves formed in the tank body is also increased and the tank moves towards full buckling. Then, after making 12 waves in the tank circumference, the buckling of the body becomes complete and, finally, at the external pressure of 21.6 kPa, the tank failure suddenly happens and the tank becomes unstable. Fig. 13(A) shows full buckling and Fig. 13(B) demonstrates instability and failure of the experimental Spec1. In the Spec2, similar to Spec1, loading continues gradually and in a controlled manner to the failure stage and full instability. After making 10 waves in the tank circumference, the buckling of the body becomes
7. Evaluation of experimental results By taking the data from the measurement devices installed at different parts of these tanks that shown in Fig. 12, radial displacement graphs of these points are drawn relative to the applied uniform external pressure. In Fig. 16 radial displacement diagrams of Spec1 at the locations of installed LVDTs are shown. Considering these diagrams it can be seen that radial displacement at the location of LVDT3 is higher with respect to other portions. Jumping in these diagrams indicates buckling wave created due to the effect of external pressure applied on it. It is observed that the first jump occurs at 5 kPa external pressure at LVDTs 3 and 8 locations with 11 and 5 mm values, respectively. This stage is called the initial
Fig. 11. Installation of measurement devices on the experimental specimen.
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(A)
(B)
Fig. 13. (A) Full buckling of Spec1 and instability threshold (B) failure of the Spec1.
Fig. 14. (A) Full buckling of Spec2 and instability threshold (B) failure of the Spec2.
Fig. 15. (A) Full buckling of Spec3 and instability threshold (B) failure of the Spec3.
Fig. 16. Radial displacement of Spec1 at the locations of installed LVDTs.
buckling of the specimen. By continuation of the loading, other buckling waves are created in the body of the tank where the jumps in the diagrams exhibit this issue. Finally at pressure loading of 19 kPa the buckling of the tank body is completed. This stage is called as the full buckling of the tank. By continuation of the pressure loading up to 22 kPa the specimen reaches its ultimate strength and then it experiences failure and collapse occurs suddenly. The stage between the full buckling and failure of the specimen is called the post-buckling stage. With respect to the diagrams, the post-buckling capacity of the specimen reached 3 kPa. The specimen failed at LVDT3 location, and the corresponding diagrams of this point are utilized in the consequent comparisons. In Fig. 17 radial displacement diagrams of SPEC2 at the locations of installed LVDTs are shown in terms of uniform external pressure. These diagrams show that radial displacement in this specimen at the location of LVDT3 differs with respect to other portions. In this specimen
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297
Fig. 19. Radial displacement in all specimens at the locations of installed LVDTs. Fig. 17. Radial displacement of SPEC2 at the locations of installed LVDTs.
with the start of loading, the radial displacement of the tank body begins at LVDT3 location. While at other LVDTs small displacement about 1 mm is observed for pressure loading up to 10 kPa, the displacement at LVDT3 is 6 mm. The reason for this issue is the imperfection in the form of concavity of vertical weld line in the tank body which is in accordance with the first mode shape of buckling. In this specimen, there is no initial buckling stage and by continuation of the loading other buckling waves occur in the tank body and the jumps in the diagrams indicate this issue. Finally at 16 kPa pressure, the buckling of different parts of the tank body is completed. By continuation of the pressure loading up to 16.5 kPa the specimen reaches its ultimate strength where it experiences damage and collapse occurs suddenly. With respect to the diagrams, the post-buckling capacity for this specimen is 0.5 kPa. Failure of this specimen occurred at LVDT3 location and the corresponding diagram of this point is used in other comparisons. In Fig. 18 radial displacement diagrams of SPEC3 at the locations of installed LVDTs are shown in terms of uniform external pressure. These diagrams show that radial displacement at the locations of LVDT3 and LVDT8 differs with respect to other portions. In this specimen with the start of loading, the radial displacement of the tank body begins at these LVDTs locations. While at other LVDTs small displacement about 1 mm is observed for pressure loading up to 14 kPa, the displacement at LVDT3 and LVDT8 is 20 and 17 mm respectively. The reason for this issue is concavity of vertical weld line imperfection in the tank body which is in accordance with the first mode shape of buckling. In this specimen, there is no initial buckling stage and by continuation of the loading other buckling waves occur in the tank body and the jumps in the diagrams indicate this issue.
Finally at 19 kPa pressure, the buckling of different parts of the tank body is completed. By continuation of the pressure loading up to 19.5 kPa the specimen reaches its ultimate strength where it experiences failure and collapse occurs suddenly. With respect to the diagrams, the post-buckling capacity for this specimen is 0.5 kPa. Failure of this specimen occurred at LVDT3 location and the corresponding diagram of this point is used in other comparisons. In Fig. 19 radial displacement of these three specimens is compared with each other. As is seen in Spec1, Point A is the start of buckling, Point B is the initial buckling, Point C is the full buckling, CD zone is the postbuckling capacity and point D is the ultimate capacity and collapse. In Spec2, Buckling starts from point O and continues to point E and here buckling is completed. EF zone is the post-buckling capacity and point F is the ultimate capacity and failure. In Spec3, Buckling starts from point O and continues to point G and here buckling is completed. GH zone is the post-buckling capacity and point H is the ultimate capacity and failure. In Table 2, numerical values of buckling loads at these points are summarized which is indicative the buckling behavior of the specimens Considering the diagrams in Fig. 19 and Table 2, it is observed that the initial buckling and full buckling load of specimens SPEC2 and SPEC3 are lower. Comparing these two specimens it is seen that SPEC3, despite having two imperfections, is stronger than SPEC2 with an imperfection. Thus it could be concluded that the concavity of vertical weld line imperfection causes reduced initial membrane strength and reduction in the full buckling and ultimate load. Whatever the amount of this imperfection is increased its impact on the buckling strength increases. The reason for this issue is compatibility between the form of imperfection and buckling mode shape of structure. So that by start of loading the location of imperfection moves and the first buckling wave is formed at the location of imperfection and ultimately damage takes place and collapse occurs at the imperfection location. It seems that increased number of concavity of vertical weld line Imperfection causes slight increase in the tank membrane strength.
Table 2 Buckling load of all experimental specimens.
Fig. 18. Radial displacement of the tanks body at the unstable location.
Spec1 Without imperfection Spec2 An imperfection Spec3 Two imperfection
Initial buckling Full buckling load load
Failure load
Post-buckling capacity
5 kPa
19 kPa
22 kPa
3 kPa
– kPa
16 kPa
16.5 kPa
0.5 kPa
– kPa
19 kPa
19.5 kPa
0.5 kPa
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Table 3 Buckling behavior of all specimens based on theory, ASME code and experiment methods.
Spec1 Without imperfection Spec2 An imperfection Spec3 An imperfection a
Buckling Load (Theory)a Eq. (4)
Buckling Load (ASME)b Eq. (6)
Initial Buckling Load (EXP)
Full Buckling Load (EXP)
Failure Load (EXP)
Post- Buckling Load (EXP)
23.5 kPa n = 13 23.5 kPa n = 13 23.5 kPa n = 13
23.3 kPa – 23.3 kPa – 23.3 kPa –
5 kPa
19 kPa
22 kPa
3 kPa
– kPa
16 kPa
16.5 kPa
0.5 kPa
– kPa
19 kPa
19.5 kPa
0.5 kPa
4
4
Pcr ¼
D¼
2 2 4 2 2 4 0:0188 2 D 2 1 ðm þ n Þ ðR2 Þ þ m ð1−υ ÞC 1 ð3 þ 13:13 Þ ð0:5752 Þ þ 3 ð1−0:3 Þ225274:72 ¼ ¼ 23:5 kPa 2 2 R ðm2 þ n2 Þ ðn2 þ 0:5m2 Þ 0:575 ð32 þ 13:132 Þ ð13:132 þ 0:5 32 Þ
Et3 205 106 0:0013 Et 205 106 0:001 mπR 1 π 57:5 ¼ ¼3 ¼ 0:0188; C ¼ ¼ 225274:72; m ¼ ¼ ¼ L 60 12ð1−υ2 Þ ð1−υ2 Þ ð1−0:32 Þ 12ð1−0:32 Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffi u 2 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi u 1−ν R R 57:5 57:5 4 6π n¼t ≈ 2:74 ¼ 2:74 ¼ 13:13: 2 L t 60 0:1 ðRL Þ ðRt Þ b
P cr ¼
ðt=2RÞ2:5
2:42E ð1−υ2 Þ
3=4
½L=2R−0:45ðt=2RÞ1=2
¼
2:42 205 106 ð1−0:32 Þ
3=4
ð1=1150Þ2:5 ½600=1150−0:45ð1=1150Þ1=2
In Table 3 the results of investigating the effect of imperfection due to concavity of vertical weld line in the experimental specimens are given and comparison is made between the theoretical relations and the relations in the ASME Code. Regarding this table the following issues could be referred to: 1- The corresponding failure load of SPEC1 is lower with respect to that of the ASME Code which is smaller about 8.5%. 2- The full buckling loads obtained in the experimental specimens SPEC2 and SPEC3 are about 10% lower with respect to that of the specimen without imperfection also it is about 27% lower with respect to that of the ASME Code. Therefore the imperfection due to concavity of vertical weld line is very important in buckling of the tanks under uniform external pressure. 3- The corresponding failure loads obtained in the experimental specimens SPEC2 and SPEC3 are lower with respect to the ASME Code and are about 16% smaller. 4- The deformations created in the tank with two imperfections is 4 times of those in the healthy tank and the tank with one imperfection is two times of those in the healthy tank. 5- The post-buckling capacity of the tanks with imperfection due to concavity of vertical weld line is negligible, and the full buckling is accounted as the threshold of collapse and failure. While in the tanks without this type of imperfection, the post-buckling capacity amounts to about 13% of the full buckling. 6- The number of buckling waves obtained from the theory is 13 waves and the number of buckling waves obtained in the experiment is 10 waves. 7- As there is a high probability of imperfection due to concavity of vertical weld line in the tanks, the effect of this imperfection should be accounted for when calculating the full buckling load in the steel cylindrical tanks. It is recommended to multiply the full buckling load obtained from the ASME Code by 0.65. In other words for the design of these tanks under uniform external pressure load it is recommended that 65% of the buckling load obtained from the ASME Code be used as the full buckling load i.e.: Pdesign ≤ 0:65 PASME 8. Conclusions In this research, a field study was accomplished on the implementation of the storage tanks in a refinery site and resulted imperfections
¼ 23:3 kPa:
were identified and categorized. The survey of imperfections revealed that the imperfection in form of concavity of vertical weld line is the most prevalent type of imperfection seen in the steel tanks. Presence of this imperfection causes reduced created waves in the body of the tanks; consequently, the lengths of these waves are increased, resulting in a reduced buckling load of the tank. Results show that the imperfections due to concavity of vertical weld line are very important in buckling of the tanks under uniform external pressure. This imperfection decreases initial, full and post buckling capacity of the tanks under uniform external pressure, significantly. Findings of this research show that for design of steel tanks under uniform external pressure load, 65% of the buckling load obtained from the ASME Code should be used. Nomenclature D bending stiffness of plate E modulus of elasticity L height of cylindrical shell m number of waves in the z-direction n number of waves in the θ-direction N number of circumferential waves in the buckling situation R radius of cylindrical shell t thickness of cylindrical shell w deflection in the z-direction Z curvature parameter Nθ axial load in the θ-direction Pcr buckling load υ Poisson's ratio σcr critical stress
References [1] M. Farshad, Stability of Structures, 2nd ed. Mc Grow Hill, New York, 1983. [2] J.G. Teng, Buckling of thin shells: recent advances and trends, Appl. Mech. Rev. 49 (1996) 263–274. [3] H. Showkati, H. Ansourian, Influence of primary boundary condition on the buckling of shallow cylindrical shells, J. Constr. Steel Res. 36 (1) (1995) 53–75. [4] J.H. Wang, A. Koizumi, Buckling of cylindrical shells with longitudinal joints under external pressure, J, Thin-Walled Struct. 48 (2010) 897–904. [5] L. Chen, J.M. Rotter, C. Doerich, Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure, J. Eng. Struct. 33 (2011) 3570–3578. [6] S. Aghajari, H. Showkati, K. Abedi, Experimental investigation on the buckling of thin cylindrical shells with two-stepwise variable thickness under external pressure, Int. J. Struct. Eng. Mech. 39 (6) (2011) 849–860.
M. Rastgar, H. Showkati / Journal of Constructional Steel Research 145 (2018) 289–299 [7] S. Jalili, J. Zamani, M. Shariyat, N. Jalili, M.A.B. Ajdari, M. Jafari, Experimental and numerical investigation of composite conical shells' stability subjected to dynamic loading, J. Struct. Eng. Mech. 49 (5) (2014) 555–568. [8] T.G. Ghazijahani, H. Jiao, D. Holloway, Longitudinally stiffened corrugated cylindrical shells under uniform external pressure, J. Constr. Steel Res. 110 (2015) 191–199. [9] J.R. Vinson, The behavior of thin-walled structures: beams, plates and shells, Mechanics of Surface Structures, 8, Kluwer Academic Publisher, 1998. [10] C.R. Calladine, Causes of imperfection-sensitivity in the buckling of thin shells, J. Thin-Walled Struct. 23 (1995) 215–235. [11] J.G. Teng, X. Lin, J.M. Rotter, X.L. Ding, Analysis of geometric imperfections in fullscale welded steel silos, J. Eng. Struct. 27 (6) (2005) 938–950. [12] A. Hubner, J.G. Teng, H. Saal, Buckling behavior of large steel cylinders with patterned welds, Int. J. Press. Vessel. Pip. 83 (2006) 13–26. [13] R. Lo Frano, G. Forasassi, Experimental evidence of imperfection influence on the buckling of thin cylindrical shell under uniform external pressure, J. Nucl. Eng. Des. 239 (2009) 193–200. [14] M. Maali, H. Showkati, S.M. Fatemi, Investigation of the buckling behavior of conical shells under weld-induced imperfections, J. Thin-Walled Struct. 57 (8) (2012) 13–24. [15] L. Yang, Z. Chen, F. Chen, W. Guo, G. Cao, Buckling of cylindrical shells with general axisymmetric thickness imperfections under external pressure, Eur. J. Mech. A/Solids 38 (2013) 90–99. [16] S.M. Fatemi, H. Showkati, M. Maali, Experiments on imperfect cylindrical shells under uniform external pressure, J. Thin-Walled Struct. 65 (2013) 14–25.
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[17] M. Khaled El-Sawy, Inelastic stability of liners of cylindrical conduits with local imperfection under external pressure, J. Tunneling Undergr. Space Technol. 33 (2013) 98–110. [18] T.G. Ghazijahani, H. Jiao, D. Holloway, Experiments on dented cylindrical shells under peripheral pressure, J. Thin-Walled Struct. 84 (2014) 14–25. [19] J.M.T. Thompson, Advances in shell buckling: theory and experiment, Int. J. Bifurcation Chaos 25 (2015) 1530001. [20] G. Cao, Z. Chen, L. Yang, H. Fan, F. Zhou, Analytical study on the buckling of cylindrical shells with arbitrary thickness imperfections under axial compression, J. Press. Vessel. Technol. 137 (2015) 1–9. [21] A. Lee, F.L. Jimenez, J. Marthelot, J.W. Hutchinson, P.M. Reis, The geometric role of precisely engineered imperfections on the critical buckling load of spherical elastic shells, J. Appl. Mech. 83 (2016) 1–11. [22] A. Evkin, M. Kolesnikov, D.A. Prikazchikov, Buckling of a spherical shell under external pressure and inward concentrated load: asymptotic solution, J. Math. Mech. Solids 1 (2016) 1–13. [23] J.W. Hutchinson, J.M.T. Thompson, Nonlinear buckling behavior of spherical shells: barriers and symmetry-breaking dimples, Philos. Trans. A Math. Phys. Eng. Sci. 375 (2093) (2017). [24] M.H. Jawad, Theory and Design of Plate and Shell Structures, 1st edition Chapman & Hall Inc, 1994. [25] D.O. Brush, B.O. Almorth, Buckling of Bears Plates and Shells, Mc Grow Hill, New York, 1975. [26] American Society of Mechanical Engineers, Boiler and pressure vessel code section VIII, rules for construction for pressure vessels, division 1. New York, ASME (2013).
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Buckling behavior of cylindrical steel tanks with concavity of vertical weld line imperfection Mehdi Rastgar a,⁎, Hossein Showkati b a b
Department of Civil Engineering, Khoy Branch, Islamic Azad University, Khoy, Iran Department of Civil Engineering, Urmia University, Urmia, Iran
a r t i c l e
i n f o
Article history: Received 29 December 2016 Received in revised form 6 February 2018 Accepted 22 February 2018 Available online xxxx Keywords: Field study Cylindrical tank Buckling behavior Vertical weld line imperfection External pressure
a b s t r a c t Shell structures are built using a number of welded curved panel parts. Hence, some geometrical imperfections emerge. These imperfections have a direct impact on structural behavior of shells during the external compressive loading. In this research, a field study was accomplished on the implementation of the storage tanks in a refinery site and then, the resulted imperfections were identified and categorized. The survey of imperfections revealed that the imperfection in form of concavity of vertical weld line is the most prevalent type of imperfection seen in the steel tanks. This imperfection experimentally modeled and the buckling behavior of these tanks was evaluated under uniform external pressure. Comparing obtained results of estimation, ASME code and experimental research represented a considerable difference in the amount of buckling load. Results show that the imperfections due to concavity of vertical weld line are very important in buckling of the tanks under uniform external pressure. This imperfection decreases initial, full and post buckling capacity of the tanks under uniform external pressure, significantly. Findings of this research show that for design of steel tanks under uniform external pressure load, 65% of the buckling load obtained from the ASME Code should be used. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Shell structures have curved initial shapes with the thickness less than other dimensions. In some states, radius-to-thickness ratio reaches 3000. The force field generated in these structures includes membrane and bending forces, which can change depending on shell thickness. Steel tanks designed in cylindrical form are commonly used shell structures in industrial facilities. In geometrical terms, these tanks have a very small thickness compared to their other dimensions and are categorized as thin-walled structures for buckling failure caused by the influence of uniform external pressure loading on the tank wall [1]. The buckling of tanks under uniform external pressure is usually caused by operational problems during the discharge of the liquid contents in such a way that partial vacuum is produced. These events are usually classified as accidents, and occur in individual tanks in a tank farm, rather than affecting many tanks in the same event, such as those produced by a natural disaster. The objective of this paper is identification and categorization of created imperfections in implementation of the storage tanks via a field study. By identifying the prevalent imperfection, buckling behavior
⁎ Corresponding author. E-mail addresses: [email protected] (M. Rastgar), [email protected] (H. Showkati).
https://doi.org/10.1016/j.jcsr.2018.02.028 0143-974X/© 2017 Elsevier Ltd. All rights reserved.
and failure of these tanks is investigated under uniform external pressure loading. 2. Stability in shell structures Evaluating the stability problems in thin-walled shells is important for two reasons. First, the ratio of thickness to other dimensions is very low in these structures, which highlights the instability problem. Second, shell structures are exposed to compressive stresses and forces; however, shell buckling is caused by low thickness. One of the main properties of shells is having much higher membrane stiffness than bending stiffness. Accordingly, a shell can absorb a large amount of membrane energy without experiencing a great deformation, while large deformations and rotation in the cross-section need to absorb this energy through bending deformations [1]. 2.1. Buckling of shell structures Buckling is considered as a nonlinear phenomenon, in which the structure cannot take further load with the same geometry and changes its shape in order to find alternative equilibrium configurations. Shell buckling occurs as the structure response to the membrane forces. Membrane forces act along the component axis and tangent to the middle surface of the shell and the buckling occurs when the structure
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converts membrane strain energy into bending strain energy without any change in the applied external load [1]. There have been a number of studies addressing buckling behavior and instability, see e.g. [2]. and references therein. Below we mention several contributions relying on experimental approach. Influence of primary boundary condition on the buckling of shallow cylindrical shells was studied by Showkati and Ansourian experimentally [3]. Wang and Koizumi investigated the buckling of cylindrical shells with longitudinal joints under external pressure [4]. Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure was considered by Chen et al. [5]. Aghajari et al. conducted an experimental study on the buckling of thin cylindrical shells with two-stepwise variable thickness under external pressure [6]. Experimental and numerical investigation of composite conical shells stability subjected to dynamic loading was investigated by Jalili et al. [7]. Ghazijahani et al. studied longitudinally stiffened corrugated cylindrical shells under uniform external pressure [8]. 2.2. Geometric imperfections on shells buckling Geometric imperfections include all the deviations in the form of a structural component compared with its ideal geometric composition. In construction of shell structures, due to the large dimensions, curved plates or panels can be used. The seam between various surfaces of the main source of deviation is in the real form, these deviations or imperfections can be generated as a result of welding or appropriate incompatibility of the plates with larger dimensions than other plates. In contrast to various structures, the buckling strength of shells with no imperfections is significantly different from the buckling strength of the same shell with imperfections. In some states, geometric imperfections may strengthen the structure and increase its capacity. This feature puts shells among the structures which are called sensitive to imperfection. In Fig. 1, the buckling behavior of columns, flat plates and cylindrical shells are schematically shown. In these curves, black lines show the system with no geometric imperfections or perfect system, while red lines represent the corresponding behavior of the imperfect system. As can be seen, the flat plates and column elements are not sensitive to imperfection, while cylinders which are the examples of thin-walled structures are very sensitive to imperfections [9]. There have been a number of studies addressing the influence of imperfections on the buckling behavior of shell structures. Below we mention several contributions relying on experimental approach. Calladine studied the causes of imperfection-sensitivity in the buckling of thin shells [10]. Teng et al. investigated the geometric imperfections in full-scale welded steel silos [11]. In other work, buckling behavior of large steel cylinders with patterned welds was considered
by Hubner et al. [12]. Influence of imperfection on the buckling of thin cylindrical shell under uniform external pressure was studied experimentally by Lo Frano and Forasassi [13]. Maali et al. investigated the Buckling behavior of conical shells under weld-induced imperfections experimentally [14]. Yang et al. studied the buckling of cylindrical shells with general axisymmetric thickness imperfections under external pressure [15]. Fatemi et al. conducted experiments on imperfect cylindrical shells under uniform external pressure and observed that geometric imperfections have a more considerable impact on the behavior of shells [16]. In other work, inelastic stability of liners of cylindrical conduits with local imperfection under external pressure was studied by Khaled El-Sawy [17]. Ghazijahani et al. conducted experiments on dented cylindrical shells under peripheral pressure [18]. Thompson presented advances in shell buckling. He studied on the buckling of axially compressed cylindrical shells with arbitrary thickness imperfections theoretically and experimentally [19]. Cao et al. studied buckling of cylindrical shells with arbitrary thickness imperfections under axial compression analytically [20]. Lee et al. investigated the geometric role of precisely engineered imperfections on the critical buckling load of spherical elastic shells. Their investigation combined precision experiments, finite element modeling and numerical solutions of a reduced shell theory, all of which were found to be in excellent quantitative agreement [21]. Evkin et al. investigated the buckling of a spherical shell under external pressure and inward concentrated load [22]. In other work, Hutchinson and Thompson studied nonlinear buckling behavior of spherical shells subject to external pressure. They found that the nonlinear axisymmetric post-buckling behavior of perfect thin spherical shells and their asymmetric bifurcations are characterized providing results for a structure/loading combination with an exceptionally nonlinear buckling response [23]. Due to the mentioned studies about imperfections in shell structures, field study and vertical weld line imperfection has not been studied. In this research this imperfection has been introduced and buckling behavior of cylindrical steel tanks has been investigated under uniform external pressure. 3. Analytical equations for buckling of cylindrical shells Donnell in 1933 obtained an equation for the buckling of cylindrical shells as follow [24]: 4
D∇8 w þ
Et ∂ w R2 ∂x4
2
þ ∇4 Nx
∂ w ∂x
2
þ 2Nxy
! 2 2 ∂ w ∂ w þ Nθ 2 ¼ 0 ∂x ∂θ ∂θ
ð1Þ
This equation can be used for various loading conditions. Let us take the simple case of applied lateral pressure, where Nx = Nxy = 0 and Nθ = Pr − σcrt. Eq. (1) then becomes [24]: 4
8
D∇ w þ
Et ∂ w R2 ∂x4
2
4
þ∇
Nθ
∂ w ∂θ
2
! ¼0
ð2Þ
An expression for the deflection that satisfies the simply supported boundary condition of a cylinder can be expressed as: w ¼ wmn sin
mπx L
sin
nπy πR
ð3Þ
Substituting this expression into Eq. (2) gives the nontrivial 2
Fig. 1. Load-axial displacement graph of columns, flat plates and cylindrical shells in perfect and imperfect states [9].
2 2
ðm þβ 2 π KE solution σ cr ¼ 12ð1−υ 2 Þ ðt=LÞ where K ¼ β2 pffiffiffiffiffiffiffiffiffiffiffiffiffi L2 2 Z ¼ Rt 1−υ 2
Þ
þ
12Z2 2, π4 β2 ð1þβ2 =m2 Þ
nL β ¼ πR ,
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291
The number of circumferential waves of cylindrical shell formed in the body is as follows: [3] sffiffiffiffiffiffiffiffiffiffiffiffi ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi u 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi2 u 1−ν 6π R R 4 u n¼t ≈ 2:74 L 2 t L t R R
ð5Þ
3
Et Et mπR So that D ¼ 12ð1−υ 2 Þ, C ¼ ð1−υ2 Þ, m ¼ L
Based on the ASME, Boiler and Pressure Vessel Code Section VIII, Rules for Construction for Pressure Vessels, the allowable external pressure is calculated from: [24,26] P cr ¼
2:42E 3=4 ð1−υ2 Þ
ðt=2RÞ2:5 h i L=2R−0:45ðt=2RÞ1=2
ð6Þ
Fig. 2. A plot of K [24]
Considering m = 1 the minimum value of K becomes: K ¼ ð1þβ β2
2 2
Þ
þ
12Z2 2 π4 β 2 ð1þβ 2 Þ
For structures with large (R/t) ratios, Eq. (6) can be simplified to: [24,26] P cr ¼
A plot of K defined by above equations is shown in Fig. 2.
2:42E 3=4 ð1−υ2 Þ
ðt=2RÞ2:5 ðL=2RÞ
ð7Þ
3.1. Short cylinders For short cylinders, the curvature parameter Z can be set to zero and then K = (β + 1/β)2 3.2. Intermediate-length cylinders As the ratio L/R increases, the quantity (β + 1) can be approximated 2
by β. Hence K ¼ β2 þ π12Z 4 β6 3.3. Long cylinders For long cylinders, the buckling mode is similar to that of a circular ring and is elliptic in shape. Hence, n = 2 and β = 2L/πR and K reduces to K = β2. Considering simply supported conditions for the ends of the cylindrical shell and m = 1, and application of uniform external pressure to the roof of cylindrical tanks that causes a uniform axial force to the body of tanks, buckling load of these tanks under the simultaneous effect of uniform external pressure and axial load is obtained by the following equation: [25] 1 Pcr ¼ R
4 D
þ m4 1−υ2 C R2 2 m 2 þ n2 n2 þ 0:5m2
m2 þ n2
ð4Þ
4. Field survey In order to store oil products in a refinery site, the construction of numerous tanks has begun, which are being implemented as steel cylindrical tanks. These tanks have the height of 12 m, diameter of 23 m, and sheet thickness of 18 mm and 8 of them are under construction. For the construction of these tanks, the steel sheets with the dimensions of 6 × 1.5 m and thickness of 18 mm are used. The sheets are entered into the rolling device and the curvature operations are performed during several steps to reach the radius of the tank. Then, they are transferred to the relevant location by the special vehicles for transporting the rolled sheets and their installation operations are carried out using the crane and human force. From this step on, numerous imperfections caused by various implementation factors are made; some of them are partly modified and some others are remained in the tank even after the implementation is complete. Installation of sheets is shown in Fig. 3. According to this Figure, the rolled sheets are put in their place by the crane and then temporarily welded next to each other (tack weld). When the first stage is completed, the next layers are implemented similar to the first layer. The rolling, transportation, and installation of these sheets cause various imperfections in the implemented layer. As shown in Fig. 3, the edges of the circumferential sheets do not match each other; these edges should get close to each other by a chain and a crane until the imperfection is modified. Basically, some of
Fig. 3. Circumferential sheets installation process in the body of tank and generation of different imperfections.
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the created imperfections are removed from the body of the tanks and some others are left permanently. In this research, after attending the tank implementation location and field monitoring of the installation of circumferential sheets, various types of imperfections are observed which are classified as below: a) Non-compliance to the edges of consecutive horizontal rows that occurred outside or inside b) Vertical seam not executed properly and sharp towards inside or outside c) Non parallel vertical edges d) Nonaligned horizontal edges e) Cause by unsuitable rolling f) Vertical different layers in height g) Effect of welding temperature which usually appear in circumferential form, and h) Wind Effects causing troughs or bulge, in the peripheral of the last layer of tank. These imperfections are shown in Fig. 4. By continuing the field study on the implementation of these tanks in a 6-month period and statistical inference from the imperfections introduced in all the 8 tanks being constructed in the refinery site, the frequency distribution graph of the imperfections is drawn, as shown in Fig. 5. In this Figure, the horizontal axis shows the type of imperfection and the vertical axis is the frequency percentage of the corresponding imperfection.
Fig. 5. Frequency distribution graph of the imperfection types in all tanks.
Accordingly, it can be seen that, among all the introduced imperfections, imperfection types b) and f) occur more frequently than others, respectively. In imperfection type b), the edges of the circumferential sheets do not match each other. This Imperfection named that vertical weld line imperfection includes concavity of vertical weld line towards inside at the location of two adjacent sheets. This imperfection is shown in Fig. 6. In this research buckling behavior of steel tanks with vertical weld line imperfection is studied.
a)
b)
c)
d)
e)
f)
g)
h) Fig. 4. Types of observed imperfections in the field study of tanks.
M. Rastgar, H. Showkati / Journal of Constructional Steel Research 145 (2018) 289–299
Fig. 6. Imperfection type b) (vertical weld line imperfection)
5. Experimental program 5.1. Geometric specifications of experimental specimens According to Fig. 5, imperfection type b) named as the vertical weld line imperfection occurs more frequently than others. For the experimental evaluation of the impact of this imperfection on the buckling behavior of tanks, three experimental specimens are made with the following characteristics: The first specimen of Spec1 has continuous circumferential plate in an integrated form without intended imperfection. Due to the limited length of the used plates which is 3 m and providing for the 3.6 m perimeter of this specimen, there are two vertical welding lines in the body of tank. CO2 welding is used in this specimen and others. The weld line is perfectly circular and has no dents or bumps.
293
The second specimen of Spec2 has entirely an intentional imperfection in terms of a vertical weld line with 15 mm in depth within tank height. Similar to the previous specimen, this specimen has two vertical weld lines on the body of the tank that the imperfection is located on one of these weld lines and the tank is fully circular in another weld line and has no dents or bumps. The third specimen of Spec3 that is in an integrated form, has two intentional imperfections of the type of vertical weld line with 15 mm over the tank height and they are located on one of the tank's diameters with 180 degrees angle between them. There are two vertical weld lines in the body of this tank and the imperfections exist along these welded lines. All three specimens have the diameter of 1.15 m, height of 0.6 m, conical roof with the height of 0.2 m, body plate thickness of 1 mm and floor and roof plate thickness of 2 mm. These specimens are shown in Fig. 7. (See Fig. 8.) In order to prevent the floor and roof buckling of these experimental tanks influenced by external pressure, strengthening annular straps with the diameter of 0.3, 0.6 and 0.9 m and section of 20 × 5 mm are used. These straps could increase the bending stiffness in the roof and floor of the tanks; so, under the uniform external pressure, these two parts maintain their initial situation and hence, do not buckle; only the body of the tanks starts to move and buckles. The size of the experimental specimens is selected in accordance existing laboratory facilities. According to provided dimensions, it is observed that experimental-scale models are on the scale of 1:20 compared to real tanks. Welding on experimental models is quite different from real tanks. These specimens are welded using CO2. The shielding gas reduces generated heat and 1 mm plates are subsequently welded without any significant imperfections. A great care is taken in construction of experimental specimens and it is tried to produce only considered imperfection. Despite high accuracy in producing the experimental specimens, various local imperfections emerge in these specimens; the major one
Fig. 7. Experimental specimens.
Fig. 8. Measurement of intentional imperfection value.
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(A)
(B)
Fig. 9. (A) The three samples (B) tensile test (C) stress-strain diagram of material.
which includes imperfections caused by welding. This fact is due to low thickness of sheets used as troughs in the body, troughs in the floor and roof of the specimens caused by transportation from the workshop to laboratory, as well as movement at different places in the laboratory. The local imperfections may also affect buckling behavior of tanks. As previously noted, the experimental specimens have intentional imperfections in terms of a dent on vertical weld line on tank's body with 15 mm size. In Fig. 6, the measurement of intentional imperfection value is shown in specimens. A metal ruler is coordinated with circular body of the tank to measure considered imperfection. In imperfection location, the distance between the ruler and vertical weld line equals 15 mm, which is the same as considered imperfection. An intentional imperfection in specimen 2 and two intentional imperfections in specimen 3 are produced. But there is no intentional imperfection in specimen 1 and vertical weld line is completely adapted to the ruler. Therefore, this specimen does not have considered imperfection and is indicated as the specimen with no imperfection. By using the approximate Eqs. (4) and (6), the buckling load caused by uniform external pressure along with axial load is obtained which is equal to 23.5 and 23.3 kPa, respectively. Also number of circumferential waves formed in the body of the tank is 13 waves based on Eq. (5), In this study, the amount of the obtained buckling load is used for the evaluation and comparison of the results taken from experimental specimens. 5.2. Mechanical properties of the specimens Mechanical properties of the specimens have been achieved from tensile test. These properties including elasticity Modulus, yield stress, failure phase stress and equivalent strain of this point. Tensile test was carried out for 3 samples of the used sheets in experimental specimens. The test temperature is the same as the temperature of the laboratory and approximately 28 degrees Celsius. The material is carbon steel.
Table 1 Mechanical properties of the specimens. Average modulus of elasticity
Average yield stress
Average failure stress
Average failure strain
200 GPa
194 MPa
325 MPa
30.24%
Dimensions of tensile samples are selected based on recommended dimensions in the ASTM E8/E8M−15a code. In Fig. 9 the samples, tensile test and stress-strain diagram of material are shown. Mechanical properties of the specimens are given in Table 1. These values are the average results of three samples. 6. Testing of specimens and evaluation of results 6.1. Uniform external pressure loading In order to evaluate the buckling behavior of experimental specimens influenced by uniform external pressure, the suction device is utilized. It is an electro-pump device that discharges the air inside the tank with the constant flow of 40 m3/h to the outside. So, by discharging the air inside the tank, the atmospheric pressure is uniformly imported into the external surfaces of this tank and loading is applied as uniform external pressure to the tank. Since air discharge flow of the tank is too high, it is necessary to embed a control valve to adjust the discharge flow and thus control the external pressure. In order to measure the internal pressure in the tank, a pressure gauge is used. This device shows the moment-to-moment pressure inside the tank, which is decreased due to the discharge by the suction device. So, external pressure is obtained by subtracting atmospheric pressure and the pressure inside the tank. There are three holes in the tank floor: the first hole is connected to the suction device, through which discharge operation is done (applying uniform external pressure). The second hole is connected to the discharge valve and controls the flow of tank discharge (loading control) and the third one is connected to the pressure gauge to measure the internal pressure (loading measurement). Fig. 10 depicts the laboratory equipment including the suction pump for loading application, loading control valve, loading measurement device, deformation measurement devices, location of test specimens, the data logger, and the computer. 6.2. Measurement devices of deformation In order to take the experimental specimens deformation, measurement devices are installed in different parts of these specimens in the middle of the height radially and at the top of the conical roof of the specimens. These devices include LVDT, strain gauge and pressure sensor. The LVDT is an acronym used for Linear Variable Differential Transformer. It is a transformer which is used for measuring linear
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295
Fig. 10. A view of the laboratory equipment's and experimental specimen
Fig. 12. Schematic view of all installed sensors with location of imperfection and failure
displacement. LVDTs take the directional displacements vertically, horizontally, or radially in the locations they are installed and then, send them to the data logger. The strain gauges are electronic circuits that are attached using a special adhesive in the desired location and they measure the strain in their installed direction and send it to the computer. In this research, in all the tests, five LVDTs and two horizontal and vertical strain gauges are used to record the deformation of different parts of the experimental specimens. In Fig. 11, a sample of these measurement devices which is installed on the experimental specimens is represented and Fig. 12 shows schematic view of all installed sensors with location of imperfection and failure.
complete and, finally at the external pressure of 16.5 kPa, failure suddenly happens and the tank becomes unstable. Fig. 14(A) and (B) shows full buckling and failure of the experimental Spec2 respectively. The Spec 3, similar to previous specimens, continues gradually and in a controlled manner to the failure stage and full instability. In this specimen, by increasing the external pressure 10 waves are formed in the tank circumference and the buckling of the body becomes complete and, finally, at the external pressure of 19.5 kPa, failure suddenly happens and the tank becomes unstable. Fig. 15(A) shows full buckling and Fig. 15(B) demonstrates instability and failure of the experimental Spec3. According to these Figures, full buckling and unstable form of all three specimens is similar and up to full buckling, regular circumferential waves are seen in the tanks body but unstable location of Spec2 and 3 is different from Spec1 and failure of these specimens occurs at the location of imperfection.
6.3. Test implementation Air discharge from Spec1 starts by turning on the suction pump device and the difference between the internal and external pressure of the tank is gradually increased. As a result, external pressure is gradually applied to the tank and the initial buckling of the tank body begins. By increasing the external pressure, the number of the waves formed in the tank body is also increased and the tank moves towards full buckling. Then, after making 12 waves in the tank circumference, the buckling of the body becomes complete and, finally, at the external pressure of 21.6 kPa, the tank failure suddenly happens and the tank becomes unstable. Fig. 13(A) shows full buckling and Fig. 13(B) demonstrates instability and failure of the experimental Spec1. In the Spec2, similar to Spec1, loading continues gradually and in a controlled manner to the failure stage and full instability. After making 10 waves in the tank circumference, the buckling of the body becomes
7. Evaluation of experimental results By taking the data from the measurement devices installed at different parts of these tanks that shown in Fig. 12, radial displacement graphs of these points are drawn relative to the applied uniform external pressure. In Fig. 16 radial displacement diagrams of Spec1 at the locations of installed LVDTs are shown. Considering these diagrams it can be seen that radial displacement at the location of LVDT3 is higher with respect to other portions. Jumping in these diagrams indicates buckling wave created due to the effect of external pressure applied on it. It is observed that the first jump occurs at 5 kPa external pressure at LVDTs 3 and 8 locations with 11 and 5 mm values, respectively. This stage is called the initial
Fig. 11. Installation of measurement devices on the experimental specimen.
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(A)
(B)
Fig. 13. (A) Full buckling of Spec1 and instability threshold (B) failure of the Spec1.
Fig. 14. (A) Full buckling of Spec2 and instability threshold (B) failure of the Spec2.
Fig. 15. (A) Full buckling of Spec3 and instability threshold (B) failure of the Spec3.
Fig. 16. Radial displacement of Spec1 at the locations of installed LVDTs.
buckling of the specimen. By continuation of the loading, other buckling waves are created in the body of the tank where the jumps in the diagrams exhibit this issue. Finally at pressure loading of 19 kPa the buckling of the tank body is completed. This stage is called as the full buckling of the tank. By continuation of the pressure loading up to 22 kPa the specimen reaches its ultimate strength and then it experiences failure and collapse occurs suddenly. The stage between the full buckling and failure of the specimen is called the post-buckling stage. With respect to the diagrams, the post-buckling capacity of the specimen reached 3 kPa. The specimen failed at LVDT3 location, and the corresponding diagrams of this point are utilized in the consequent comparisons. In Fig. 17 radial displacement diagrams of SPEC2 at the locations of installed LVDTs are shown in terms of uniform external pressure. These diagrams show that radial displacement in this specimen at the location of LVDT3 differs with respect to other portions. In this specimen
M. Rastgar, H. Showkati / Journal of Constructional Steel Research 145 (2018) 289–299
297
Fig. 19. Radial displacement in all specimens at the locations of installed LVDTs. Fig. 17. Radial displacement of SPEC2 at the locations of installed LVDTs.
with the start of loading, the radial displacement of the tank body begins at LVDT3 location. While at other LVDTs small displacement about 1 mm is observed for pressure loading up to 10 kPa, the displacement at LVDT3 is 6 mm. The reason for this issue is the imperfection in the form of concavity of vertical weld line in the tank body which is in accordance with the first mode shape of buckling. In this specimen, there is no initial buckling stage and by continuation of the loading other buckling waves occur in the tank body and the jumps in the diagrams indicate this issue. Finally at 16 kPa pressure, the buckling of different parts of the tank body is completed. By continuation of the pressure loading up to 16.5 kPa the specimen reaches its ultimate strength where it experiences damage and collapse occurs suddenly. With respect to the diagrams, the post-buckling capacity for this specimen is 0.5 kPa. Failure of this specimen occurred at LVDT3 location and the corresponding diagram of this point is used in other comparisons. In Fig. 18 radial displacement diagrams of SPEC3 at the locations of installed LVDTs are shown in terms of uniform external pressure. These diagrams show that radial displacement at the locations of LVDT3 and LVDT8 differs with respect to other portions. In this specimen with the start of loading, the radial displacement of the tank body begins at these LVDTs locations. While at other LVDTs small displacement about 1 mm is observed for pressure loading up to 14 kPa, the displacement at LVDT3 and LVDT8 is 20 and 17 mm respectively. The reason for this issue is concavity of vertical weld line imperfection in the tank body which is in accordance with the first mode shape of buckling. In this specimen, there is no initial buckling stage and by continuation of the loading other buckling waves occur in the tank body and the jumps in the diagrams indicate this issue.
Finally at 19 kPa pressure, the buckling of different parts of the tank body is completed. By continuation of the pressure loading up to 19.5 kPa the specimen reaches its ultimate strength where it experiences failure and collapse occurs suddenly. With respect to the diagrams, the post-buckling capacity for this specimen is 0.5 kPa. Failure of this specimen occurred at LVDT3 location and the corresponding diagram of this point is used in other comparisons. In Fig. 19 radial displacement of these three specimens is compared with each other. As is seen in Spec1, Point A is the start of buckling, Point B is the initial buckling, Point C is the full buckling, CD zone is the postbuckling capacity and point D is the ultimate capacity and collapse. In Spec2, Buckling starts from point O and continues to point E and here buckling is completed. EF zone is the post-buckling capacity and point F is the ultimate capacity and failure. In Spec3, Buckling starts from point O and continues to point G and here buckling is completed. GH zone is the post-buckling capacity and point H is the ultimate capacity and failure. In Table 2, numerical values of buckling loads at these points are summarized which is indicative the buckling behavior of the specimens Considering the diagrams in Fig. 19 and Table 2, it is observed that the initial buckling and full buckling load of specimens SPEC2 and SPEC3 are lower. Comparing these two specimens it is seen that SPEC3, despite having two imperfections, is stronger than SPEC2 with an imperfection. Thus it could be concluded that the concavity of vertical weld line imperfection causes reduced initial membrane strength and reduction in the full buckling and ultimate load. Whatever the amount of this imperfection is increased its impact on the buckling strength increases. The reason for this issue is compatibility between the form of imperfection and buckling mode shape of structure. So that by start of loading the location of imperfection moves and the first buckling wave is formed at the location of imperfection and ultimately damage takes place and collapse occurs at the imperfection location. It seems that increased number of concavity of vertical weld line Imperfection causes slight increase in the tank membrane strength.
Table 2 Buckling load of all experimental specimens.
Fig. 18. Radial displacement of the tanks body at the unstable location.
Spec1 Without imperfection Spec2 An imperfection Spec3 Two imperfection
Initial buckling Full buckling load load
Failure load
Post-buckling capacity
5 kPa
19 kPa
22 kPa
3 kPa
– kPa
16 kPa
16.5 kPa
0.5 kPa
– kPa
19 kPa
19.5 kPa
0.5 kPa
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Table 3 Buckling behavior of all specimens based on theory, ASME code and experiment methods.
Spec1 Without imperfection Spec2 An imperfection Spec3 An imperfection a
Buckling Load (Theory)a Eq. (4)
Buckling Load (ASME)b Eq. (6)
Initial Buckling Load (EXP)
Full Buckling Load (EXP)
Failure Load (EXP)
Post- Buckling Load (EXP)
23.5 kPa n = 13 23.5 kPa n = 13 23.5 kPa n = 13
23.3 kPa – 23.3 kPa – 23.3 kPa –
5 kPa
19 kPa
22 kPa
3 kPa
– kPa
16 kPa
16.5 kPa
0.5 kPa
– kPa
19 kPa
19.5 kPa
0.5 kPa
4
4
Pcr ¼
D¼
2 2 4 2 2 4 0:0188 2 D 2 1 ðm þ n Þ ðR2 Þ þ m ð1−υ ÞC 1 ð3 þ 13:13 Þ ð0:5752 Þ þ 3 ð1−0:3 Þ225274:72 ¼ ¼ 23:5 kPa 2 2 R ðm2 þ n2 Þ ðn2 þ 0:5m2 Þ 0:575 ð32 þ 13:132 Þ ð13:132 þ 0:5 32 Þ
Et3 205 106 0:0013 Et 205 106 0:001 mπR 1 π 57:5 ¼ ¼3 ¼ 0:0188; C ¼ ¼ 225274:72; m ¼ ¼ ¼ L 60 12ð1−υ2 Þ ð1−υ2 Þ ð1−0:32 Þ 12ð1−0:32 Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffi u 2 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi u 1−ν R R 57:5 57:5 4 6π n¼t ≈ 2:74 ¼ 2:74 ¼ 13:13: 2 L t 60 0:1 ðRL Þ ðRt Þ b
P cr ¼
ðt=2RÞ2:5
2:42E ð1−υ2 Þ
3=4
½L=2R−0:45ðt=2RÞ1=2
¼
2:42 205 106 ð1−0:32 Þ
3=4
ð1=1150Þ2:5 ½600=1150−0:45ð1=1150Þ1=2
In Table 3 the results of investigating the effect of imperfection due to concavity of vertical weld line in the experimental specimens are given and comparison is made between the theoretical relations and the relations in the ASME Code. Regarding this table the following issues could be referred to: 1- The corresponding failure load of SPEC1 is lower with respect to that of the ASME Code which is smaller about 8.5%. 2- The full buckling loads obtained in the experimental specimens SPEC2 and SPEC3 are about 10% lower with respect to that of the specimen without imperfection also it is about 27% lower with respect to that of the ASME Code. Therefore the imperfection due to concavity of vertical weld line is very important in buckling of the tanks under uniform external pressure. 3- The corresponding failure loads obtained in the experimental specimens SPEC2 and SPEC3 are lower with respect to the ASME Code and are about 16% smaller. 4- The deformations created in the tank with two imperfections is 4 times of those in the healthy tank and the tank with one imperfection is two times of those in the healthy tank. 5- The post-buckling capacity of the tanks with imperfection due to concavity of vertical weld line is negligible, and the full buckling is accounted as the threshold of collapse and failure. While in the tanks without this type of imperfection, the post-buckling capacity amounts to about 13% of the full buckling. 6- The number of buckling waves obtained from the theory is 13 waves and the number of buckling waves obtained in the experiment is 10 waves. 7- As there is a high probability of imperfection due to concavity of vertical weld line in the tanks, the effect of this imperfection should be accounted for when calculating the full buckling load in the steel cylindrical tanks. It is recommended to multiply the full buckling load obtained from the ASME Code by 0.65. In other words for the design of these tanks under uniform external pressure load it is recommended that 65% of the buckling load obtained from the ASME Code be used as the full buckling load i.e.: Pdesign ≤ 0:65 PASME 8. Conclusions In this research, a field study was accomplished on the implementation of the storage tanks in a refinery site and resulted imperfections
¼ 23:3 kPa:
were identified and categorized. The survey of imperfections revealed that the imperfection in form of concavity of vertical weld line is the most prevalent type of imperfection seen in the steel tanks. Presence of this imperfection causes reduced created waves in the body of the tanks; consequently, the lengths of these waves are increased, resulting in a reduced buckling load of the tank. Results show that the imperfections due to concavity of vertical weld line are very important in buckling of the tanks under uniform external pressure. This imperfection decreases initial, full and post buckling capacity of the tanks under uniform external pressure, significantly. Findings of this research show that for design of steel tanks under uniform external pressure load, 65% of the buckling load obtained from the ASME Code should be used. Nomenclature D bending stiffness of plate E modulus of elasticity L height of cylindrical shell m number of waves in the z-direction n number of waves in the θ-direction N number of circumferential waves in the buckling situation R radius of cylindrical shell t thickness of cylindrical shell w deflection in the z-direction Z curvature parameter Nθ axial load in the θ-direction Pcr buckling load υ Poisson's ratio σcr critical stress
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