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Comparison of the shell design methods for cylindrical liquid storage tanks
Engineering Structures 101 (2015) 621–630
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Comparison of the shell design methods for cylindrical liquid storage tanks Eyas Azzuni, Sukru Guzey ⇑ Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 April 2015 Revised 27 July 2015 Accepted 31 July 2015
The three methods for determining the shell thickness of steel cylindrical liquid storage tanks designed in conformance with API Standard 650, Welded Tanks for Oil Storage (API 650) are: (1) one-foot method (1FM), (2) variable-design-point method (VDM) and (3) linear analysis. We compared the shell designs based on these three methods for different tank properties: diameter, height and allowable stress. For linear analysis, we developed a stiffness–flexibility matrix method based on thin shell theory that gives the theoretical displacements and stresses at each shell course without any approximation or simplification. Results show that shell designs using VDM may produce overstressed shell courses for some of the large steel liquid storage tanks when VDM is permissible to use. Linear analysis would give more accurate shell designs for those cases. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Storage tank Shell design API 650 Thin shell theory One-foot method Variable-design-point method
1. Introduction API 650 is an industry standard used for the design and construction of large cylindrical storage tanks for liquid products [1–3]. API 650 storage tanks are vertical, cylindrical, closed- and open-top welded tanks with uniformly supported flat bottom. They are used to store petroleum, petroleum products, and other liquid products [1]. Recently, considerable research effort has been devoted to the analysis, design, and evaluation of the liquid storage tanks [4]. Much of the research conducted has focused on the buckling of and wind effects on the storage tanks [5–8]. Some researchers worked on dynamic effects related to earthquakes [9–11]. Chen et al. worked on developing a simple method to calculate shell stress [12]. A typical storage tank has a number of shell courses of uniform plate thickness. The thickest course is at the bottom and each shell course above is typically thinner than the previous one. See Fig. 1 for a typical storage tank shell cross-section. There are three methods allowed by API 650 to determine the required plate thickness of the shell. The first method is the one-foot-method (1FM) which is based on the ‘‘membrane theory’’. The required shell plate thickness for each shell course is
⇑ Corresponding author. E-mail addresses: (S. Guzey).
[email protected]
(E.
http://dx.doi.org/10.1016/j.engstruct.2015.07.050 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
Azzuni),
[email protected]
calculated using the circumferential stress at a point 0.3 m (1-ft) above the lower horizontal weld seam of the shell course due to hydrostatic pressure of the stored liquid. The reasoning behind this assumption is that the tank bottom plates provide restraint to reduce circumferential stress due to hydrostatic pressure at the bottom 0.3 m (1-ft) of the lowest shell course. Similarly, a shell course other than the lowest shell course, has generally thicker shell plates below. The plate below provides some restraint at the lower portion of the shell course in consideration. The 1FM is used successfully for the majority of the tanks. However, the designs based on the 1FM may become conservative and cost prohibitive for larger diameter tanks. Therefore, API 650 limits the applicability of this method to tanks up to 61 m (200-ft) in diameter. The second method to calculate the required shell plate thickness is the variable-design-point method (VDM) that is also based on the ‘‘membrane theory’’. The VDM was proposed by Zick and McGrath in 1968 [13] and later adopted by API 650 as a refined method to calculate the required shell plate thickness especially for tanks more than 61 m (200-ft) in diameter. The VDM takes into consideration the restraint provided by the tank bottom plates to the first shell course and the restraint provided by each lower shell course to the upper shell course. The VDM uses a variable distance instead of fixed distance of 0.3 m (1-ft), as used in 1FM, above the circumferential seam for each shell course to calculate the maximum stress due to hydrostatic pressure. The variable distance in VDM is a function of the shell plate thickness above and below
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Nomenclature H L h1 D r G CA Sd St t ti td t 1d t dx tt t 1t t tx tu tL
distance from the maximum product level to bottom of the shell course under consideration (m) ð500 DtÞ0:5 (mm) height of the bottom shell course (m) nominal tank diameter (m) nominal radius of the tank (m) the design specific gravity of the stored liquid corrosion allowance (mm) allowable design condition stress (MPa) allowable hydrostatic test condition stress (MPa) thickness of the shell thickness of a shell course design shell thickness (mm) design shell thickness for the first shell course (mm) design shell thickness (mm) hydrostatic test shell thickness (mm) hydrostatic test shell thickness for the first shell course (mm) hydrostatic test shell thickness (mm) corroded thickness of the upper shell course; approximated using 1FM for the first iteration (mm) thickness of the lower shell course (mm)
the seam. Most of the time designs based on VDM are more economical compared with those based on the 1FM. However, for some tank geometries the VDM may become unconservative and the tank shell thicknesses designed in accordance with VDM may be overstressed. Buzek showed that the restraint provided by the tank bottom on the tank shell produces circumferential stresses of sinusoidal nature varying with the distance from the tank bottom [14]. For certain tank diameter and height proportions, this sinusoidal varying restraining stress may add to the stress due to the hydrostatic circumferential stress and the design based on VDM may become unconservative. Therefore, API 650 limits the applicability of the VDM for the tanks with L/H ratio less than 1000/6 in SI units (refer to the nomenclature for the definition of these terms). For the storage tanks where the L/H ratio is more than 1000/6, tank shell thickness should be determined using linear analysis. The shell thickness calculation using linear analysis is the third method given in API 650. In this approach the boundary conditions for the analysis should be a plastic moment related to yielding of the plate under the shell and fully restrained radial movement at the bottom of the shell. API 650 does not describe a specific linear analysis method. In this study we developed a new method using thin shell theory to perform a linear analysis for the shell thickness calculation. In this method we are using exact stiffness–flexibility relations and exact shape functions originating from the so called ‘‘short shell’’ solution of the governing equations from the thin elastic shell theory. Therefore, we do not have any approximations or simplifications. The displacements, section forces and stresses obtained from this method are exactly matching the theoretical solution of thin shell theory. Overwhelming majority of texts employ only a single course solution of shell cylinder without extension to multiple shell courses with stepwise thicknesses. One can find very few references dealing with multiple shell courses in which only approximate solutions were obtained. In our treatment, we present an attractive and easy to implement formulation that renders analytical solution without any approximation. We shall investigate the efficiency and limitations of each method described above. The efficiency is defined in terms of
t1 t2
corroded thickness of the bottom shell course (mm) minimum design thickness of the second shell course (mm) t 2a corroded thickness of the second shell course (mm) as calculated for the upper shell courses w radial displacement of the cylindrical shell x axial length coordinate of the cylindrical shell E modulus of elasticity Ds shell bending rigidity p pressure m Poisson’s ratio b a parameter C1, . . ., C4, integration constants f(x) particular solution to the governing equation Li length of a shell course Q1, Q2 end shearing forces of a shell course M1, M2 end bending moments of a shell course w1, w2 end radial displacements of a shell course h1, h2 end rotations of a shell course
tn
t2 BOTTOM SHELL COURSE
t1
H
TANK BOTTOM D
Fig. 1. Typical tank shell cross-section.
minimum required shell thickness for each shell course and corresponding total weight of steel for the shell plates. A decrease in the weight of steel can be achieved by reducing the shell thickness, which will lead to a decrease in cost. Achieving a smaller design thickness is important for large diameter tanks because tank fabricators in North America typically order a plate thickness with 0.01 in. (0.25 mm) increments as well as commercially available 1/16 in. (1.59 mm) increments directly from a steel mill for sufficiently large weight of steel, about 20 tons. Another reason to achieve a smaller thickness may be to comply with the maximum thickness limit of 1 in. (25.4 mm) to avoid stress relieving requirement. Therefore, even a reduction of 0.01 in. (0.25 mm) in design thickness would be significant for large diameter tanks. Our main objective is to investigate the accuracy of 1FM and VDM and possibility of obtaining an economical result by using the linear analysis for the required shell thickness for storage tanks. We shall first summarize the three design methods for the storage tanks. Then we shall compare the shell designs based on the classic 1FM with those based on VDM. Furthermore, we shall focus on comparison of the VDM results with the theoretical solutions obtained from thin shell theory. Finally, we shall discuss the results and give conclusions.
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2. Design methods We conducted three different sets of studies of the available storage tank shell design methods in API 650: (1) Comparison of shell designs based on the classical 1FM with those based on VDM for the storage tank sizes where designs based on 1FM is permissible in API 650 (i.e. the tanks up to 61 m (200-ft) in diameter). (2) Comparison of the shell designs based on VDM with those based on linear analysis using thin shell theory for the larger storage tanks up to 122 m (400-ft) in diameter. (3) Study of the tank sizes where VDM is no longer permissible and compare the VDM results with those of obtained using thin shell theory. The storage tanks are typically constructed using 2.4 m (8-ft) or 3 m (10-ft) high shell courses of uniform thickness. In this study we used 2.4 m (8-ft) high shell courses because they are used more often than 3 m (10-ft) high shell courses. Material selection for a tank shell is an important task in the design process. The tank manufacturer, usually responsible for the tank design, selects a suitable material specification for each shell course based on cost, availability and brittle-fracture considerations. In this study, we used three different allowable stress representing low strength, medium strength and high strength material specifications permissible in API 650 tank designs. There are two different allowable stresses used in tank design: (1) design allowable stress and (2) hydrostatic test allowable stress. Design allowable stress is used with the specific gravity of the product that is going to be stored in the tank. The product specific gravity does not usually exceed 1.0 for the petroleum products. The second allowable stress is that for the hydrostatic test used for the hydrostatic test condition. The specific gravity for the hydrostatic test is usually 1.0 to represent the specific gravity of the test liquid. Hydrostatic test is a proof test before a tank is placed in service. During this test, the tank is filled with water and shell plates are stressed to the levels that the tank may not see again during its entire service life considering the product specific gravity is less than 1.0 in most cases. In order to be more general, we used a liquid specific gravity of 1.0 and selected the allowable stresses from the hydrostatic test allowable stresses. In the following sections we shall summarize the three shell-thickness design methods for storage tank design. 2.1. One-foot method The 1FM can be used for tanks up to 61 m (200-ft) in diameter. It has the advantage of being simple and easy to use with no iterations needed. The 1FM design takes the stress 0.3 m (1-ft) above the bottom of each shell course to determine its thickness. The thickness required is the larger of the two thicknesses calculated using Eqs. (1) and (2) from API 650:
td ¼
4:9DðH 0:3ÞG þ CA Sd
ð1Þ
tt ¼
4:9DðH 0:3Þ St
ð2Þ
Note that Eqs. (1) and (2) above are unit dependent and the equations in this work are given in the SI units. The reader is referred to API 650 for the equations in USC system of units. Eq. (1) is used to calculate the design shell thickness resulting from hydrostatic pressure of the liquid to be stored and desired corrosion allowance for the design life of the storage tank. Eq. (2) is used
to calculate the hydrostatic test shell thickness resulting from hydrostatic test pressure of the test liquid. As stated earlier, to be more general, we used Eq. (2), and corresponding hydrostatic test allowable stress in our study. 2.2. Variable-design-point method The VDM can be used for tanks with L/H ratio of up to 1000/6 in SI units (refer to nomenclature for the definition of L and H terms). In general, most storage tanks used in practice would have an L/H ratio smaller than the limiting value of 1000/6. Therefore, VDM can be used to determine the shell design thickness most of the time. The L/H limit is present because the VDM was based on the assumption that the base plastic moment has a small effect on the circumferential stresses in the lower shell course. The effect of base moment is of a sinusoidal nature: It may amplify the circumferential stress due to hydrostatic pressure for bigger tanks. The assumption used in the VDM for the base moment only works for tanks with L/H ratio less than 1000/6 [14]. The calculation of the shell courses thicknesses using VDM is more elaborate than the calculation using the 1FM. The VDM does not take a specific point for all the shell courses but rather finds the point where the stress is highest in each shell course to calculate the thickness. There are three categories of calculations: first shell course, second shell course, and upper shell courses. In accordance with API 650 [1] the following equations are used. For the first shell course from the bottom the greater of the two thicknesses obtained using Eqs. (3) and (4) is used as the shell thickness.
t1d ¼
0:0696D 1:06 H
sffiffiffiffiffiffiffi! HG 4:9HDG þ CA Sd Sd
t1t ¼
0:0696D 1:06 H
sffiffiffiffi! H 4:9HD St St
ð3Þ
ð4Þ
Note that the equations above are unit dependent and the equations in this work are given in SI units. If the calculated first shell course thickness using Eqs. (3) and (4) is more than the thickness obtained using the traditional 1FM, the thickness calculated from 1FM may be used as the first shell course thickness. Calculation of the shell thickness for the second and upper shell courses requires an iterative process. In this process the shell thickness converges to the required shell thickness in two or three iterations. The calculation procedure for the second and upper shell courses are given in Appendix A. 2.3. Linear analysis using thin shell theory Linear analysis is the required method of API 650 to calculate shell plate thickness when 1FM and VDM are not permissible. This method should be used when L/H ratio is greater than the limiting value of 1000/6 in SI units. For the storage tank geometries that is constructed in practice, it is very difficult to reach and exceed an L/H ratio of 1000/6. However, in this study we aimed to examine the linear analysis when the shell design based on VDM is still permissible and compare the results of the design based on linear analysis with those of the design based on VDM. In addition, we investigated the linear analysis and VDM when the design based on VDM is not permissible, i.e. L/H ratio is more than the limiting value of 1000/6. The API 650 document does not specify how to perform a linear analysis other than some instructions on how to choose the boundary conditions at the base of the shell cylinder. We performed the linear analysis using thin elastic shell theory that combines the
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bending deformations in addition to the membrane deformations. Because we are dealing with thin shells, the out-of-plane shear deformations are neglected. Consider a circular cylindrical shell of finite length and subject to axisymmetric, torsion-free loading. The governing equation of radial displacement, w may be written the following way if the thickness of the shell, t, is constant [15–17].
4
d w 4
dx
þ
Et p w¼ Ds r 2 Ds
ð5Þ
where E is the modulus of elasticity, r is the radius of shell, and p is the pressure. Shell bending rigidity Ds is defined as
Ds ¼
Et3 12ð1 m2 Þ
ð6Þ
where m is the Poisson’s ratio. Eq. (5) is very similar to the beam on elastic foundation equation [18]. Now we will define a parameter b where
b4 ¼
Et 3ð1 m2 Þ ¼ 2 4Ds r r2 t2
ð7Þ
Therefore, the governing equation can be written as 4
d w 4
dx
þ 4b4 w ¼
p Ds
ð8Þ
The general solution of the above fourth order ordinary differential equation of constant coefficients may be written as [16]
w ¼ ebx ðC 1 cos bx þ C 2 sin bxÞ þ ebx ðC 3 cos bx þ C 4 sin bxÞ þ f ðxÞ
ð9Þ
where C 1 ; . . . ; C 4 are the four integration constants to be determined from the boundary conditions and f ðxÞ is a particular solution. In our case membrane solution due to hydrostatic pressure of the stored liquid will be the particular solution and integration constants C 1 ; . . . ; C 4 above will be found from the combined membrane and bending solution. Note that all the other variables rotation, h, bending moment, M and shearing force, Q can be expressed in terms of radial displacement, w by the following equations
dw ; dx 2 d w M ¼ Ds 2 ; dx 3 dM d w ¼ Ds 3 : Q¼ dx dx
attached to a joint is needed for each joint. However, long shell solution may not give accurate results when semi-infinite length assumption cannot be justified [16]. Short shell solution gives more accurate results than the long shell solution because it incorporates all the integration constants into the solution. However, this approach requires more involving treatment because each joint cannot be solved independently. Simultaneous solution of entire tank joints equations needed. This problem may be calculated by approximate methods [16]. In this work we will use short shell solution and instead of solving the joint equations approximately we will solve them analytically. We will cast the shell equations for a finite length cylinder for each shell course in a matrix form using the approach suggested by Calladine [15]. Then we will enforce continuity and equilibrium of each joint in similar way to the direct stiffness or finite element method and solve the system of equations simultaneously. Now let us consider an axisymmetric circular cylindrical shell of length Li , constant thickness of t i subject to end shearing forces, Q 1 and Q 2 and bending moments M 1 and M 2 . Radial displacements and rotations at the end of the cylinder is given as w1 ; w2 and h1 ; h2 . See Fig. 2 for the geometry and sign convention for displacement, rotation, bending moment and shearing forces. Although there are many ways to relate displacement, rotation, bending moment and shearing forces at the ends of a short cylinder we will use the following stiffness–flexibility matrix suggested by Calladine [15].
2 3 2Dbb3 w1 s 6 6M 7 16 a b 6 6 17 7¼ 6 6 4 w2 5 2 6 d 3 4 2Ds b c M2 2
b
a b
bð2Ds bÞ bc dð2Ds bÞ
d 2Ds b3 bc b 2Ds b3 a b
3 3 2 Q1 7 7 6 dð2Ds bÞ 7 76 h1 7 76 7 a 7 4 b 5 Q2 5 c b
bð2Ds bÞ
h2
where the coefficients a; b; c; d are defined by the following functions
a ¼ ðSinh 2n sin 2nÞ=k b ¼ ðSinh 2n þ sin 2nÞ=k c ¼ 2ðCosh n sin n Sinh n cos nÞ=k d ¼ 2ðCosh n sin n þ Sinh n cos nÞ=k
h¼
ð10Þ
Cylindrical liquid storage tanks usually have shell courses of several different thicknesses. So for each shell course we need to solve the governing Eq. (8) and thus, find the four integration constants and particular membrane solution appeared in Eq. (9). In general there are two ways to approach the problem [16]. First one is to consider each shell course as semi-infinite length shell that one end of the shell does not see the effects of the loading at the other end. This approach is commonly named as long shell solution. The second approach is to consider each shell course as finite length shell that one end of the shell does see the effects of the loading at the other end. The second approach is commonly named as short shell solution. Long shell solution is relatively simple to obtain because the semi-infinite length assumption eliminates the integration constants C 1 and C 2 and remaining two integration constants may be found by construction of continuity of radial displacement w and rotation at each shell joint (weld seam). Each joint may be treated independently and only the information from a shell course
ð11Þ
Fig. 2. The sign convention used to develop the thin shell solution.
ð12Þ
E. Azzuni, S. Guzey / Engineering Structures 101 (2015) 621–630
625
where
n ¼ bLi k ¼ Cosh2n cos 2n
ð13Þ
Now we can form the stiffness–flexibility matrix for each shell course using Eq. (11) and assemble system matrix similar to finite element method and enforce continuity of displacement and equilibrium of forces together with membrane solution. Solution of system matrix will give us the shearing forces Q and rotation h at each joint (weld seam). The boundary conditions of the bottom of the tank were set to be pinned along the circumference of the tank. The top of the tank was free to move. Plastic moment was applied to the bottom of the tank simulating a bottom plate of 6.35 mm (0.25-in.) thick steel with a yield stress of 250 MPa (36,000 psi) along the circumference of the bottom shell course. Note that solution of the governing Eq. (8) was originally defined in terms of radial displacement w. Therefore, we need to recover w at any point along the shell to calculate stresses. While there are many ways to recover the integration constants appeared in Eq. (9), we would like to mention two of them here. First method is given for beams on elastic foundations in Hetenyi’s book and named as ‘‘method of initial conditions’’ [18]. This approach expresses displacement function w in terms of w0 ; h0 ; M 0 , and Q 0 quantities at the end x ¼ 0 of the shell course. The second method to obtain the integration constants is suggested by Gould’s book on shells [17]. Gould’s method constructs a 4 4 coefficient matrix that relates the integration constants to the end bending moments and shearing forces, M 0 , and Q 0 at both ends of a shell course at x ¼ 0 and x ¼ Li . We used both of the recovery methods and they gave identical results as expected. Here we would like to explain Hetenyi’s approach in detail. Displacement function w at any point along the shell course under consideration can be written as
1 1 wðxÞ ¼ w0 F 1 ðbxÞ h0 F 2 ðbxÞ 2 M0 F 3 ðbxÞ b b Ds
1 b3 Ds
Q 0 F 4 ðbxÞ
Fig. 3. Simply supported cylindrical shell with uniform pressure.
rhoop ¼ w
pL4i 2 sin a Sinh a 1 sin bx Sinh bx 4 64Ds a cos 2a þ Cosh 2a 2 cos aCosh a cos bxCosh bx cos 2a þ Cosh 2a
w¼ ð14Þ
F 2 ðbxÞ ¼ 12 ðCosh bx sin bx þ Sinh bx cos bxÞ
ð15Þ
Sinh bx cos bxÞ
Note that the displacement function w in Eq. (14) is the homogenous solution or bending solution of displacement of the governing equation. We need to add membrane solution associated with the hydrostatic liquid pressure to find the general solution for the displacement at each shell course. Radial displacement due to membrane solution at any point under hydrostatic liquid pressure can be expressed by the following equation
wm ¼
ð18Þ
Where
F 1 ðbxÞ ¼ Cosh bx cos bx F 3 ðbxÞ ¼ F 4 ðbxÞ ¼
ð17Þ
Before we move to the comparison studies of each tank design methods we would like show validation of our linear analysis using thin shell theory. As we mentioned earlier our method uses exact stiffness–flexibility matrix and exact shape functions to describe the radial displacement within the usual assumptions of thin shell theory. To validate this we shall use a circular cylindrical shell of length Li with a uniform internal pressure, p. For both ends of the shell pinned–pinned condition the closed form solution of radial displacement is given by Timoshenko by the following expression [16]
where
1 ðSinh bx sin bxÞ 2 1 ðCosh bx sin bx 4
E r
pr 2 Eti
ð16Þ
where hydrostatic pressure p is
p ¼ cðhi xÞ where c is the weight per unit volume of the stored liquid and hi is the height of the liquid from the bottom joint (seam) of a shell course under consideration. Once the general radial displacements, including both bending and membrane deformations, are combined the circumferential stress of the tank shell at any point can be calculated simply by the following expression.
a¼
bLi 2
In Eq. (18) origin of coordinate x is taken at the middle of the cylinder. We solved the above problem using our thin shell approach for different thicknesses and obtained identical results as the closed form solution given in Eq. (18). Fig. 3 shows the geometry and problem description of the problem. Fig. 4 shows the normalized radial displacement with respect to the membrane solution using thin shell approach and closed form solution for different thickness. Our thin shell solution matches the closed form solution exactly as expected. 3. Comparison studies We performed three different comparison studies to investigate the shell design methods. 3.1. Study 1: Comparison of 1FM and VDM To compare the results of 1FM with those of VDM, a spreadsheet was generated to compute the thicknesses of shell courses for different tank diameter and height. This comparison study was done using allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi). We started
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1.2
t = 0.01
t=0.1
t=1
1
w/w membrane
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
% x/L Fig. 4. Normalized radial displacement in a pinned–pinned cylinder subject to uniform internal pressure. The horizontal axis is the ratio of the height to the entire length, and the vertical axis is the normalized radial displacement w with respect to the membrane radial displacement.
Fig. 5a. Comparison of 1FM and VDM, ratio of volume of steel resulting from 1FM to VDM vs tank diameter for different tank heights in meters for the allowable stress of 158.6 MPa (23,000 psi).
with a tank height of 7.3 m (24-ft) with three 2.4 m (8-ft) shell courses and the diameter ranging from 3 m (10-ft) to 61 m (200-ft). Then we increased the tank height with 2.4 m (8-ft) increments up to a tank height of 19.5 m (64-ft) with eight 2.4 m (8-ft) shell courses with the diameter ranging from 3 m (10-ft) to 61 m (200-ft). 3.2. Study 2: Validation of VDM using the analytical approach when VDM is permissible to use API 650 Annex K provides tables to summarize the required shell thickness based on VDM for different tank sizes and allowable stresses. In this part of the study, the tank designs specified in Annex K for an allowable stress of 158.6 MPa (23,000 psi) were tested with the linear analysis using thin shell theory. The linear analysis provided the theoretical stresses in each course of the tank. With this study we may be able to see if the designs based on VDM are actually overstressed or not.
Fig. 5b. Comparison of 1FM and VDM, ratio of volume of steel resulting from 1FM to VDM vs tank diameter for different tank heights in meters for the allowable stress of 206.8 MPa (30,000 psi).
Fig. 5c. Comparison of 1FM and VDM, Ratio of Volume of Steel Resulting from 1FM to VDM vs Tank Diameter for Different Tank Heights in Meters for the Allowable Stress of 236.5 MPa (34,300 psi).
In total, 27 tanks were tested. Four different heights were considered: 12.2 m (40-ft), 14.6 m (48-ft), 17.1 m (56-ft), and 19.5 m (64-ft). The diameters were selected from API 650 Annex K table for an allowable stress of 158.6 MPa (23,000 psi). Each shell course was 2.4 m (8-ft) high, and the thickness of each course was also determined using the same table. We calculated the actual stresses using thin shell theory and the stress profile was then obtained. The points where the tank was overstressed were found.
3.3. Study 3: Validation of VDM using the analytical approach when VDM is not permissible to use This part of the study was performed to compare the shell designs based on VDM and linear analysis using the thin shell theory when the L/H ratio is more than 1000/6 in SI units (i.e. VDM method is no longer permissible). We investigated if the limiting L/H ratio of 1000/6 is a good limit for all diameters, heights and
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10 5 0
0
50
100
10 5 0
150
0
Stress (MPa) 20
10 5 0
50
100
5 0
50
Stress (MPa)
100
150
10 5
50
100
Stress (MPa)
150
D = 97.5 m
10 5 0
20
D = 109.7 m
0
100
50
100
150
Stress (MPa)
15
0
50
15
0
150
Height (m)
Height (m)
Height (m)
5
100
20
D = 91.4 m
20
10
50
0
Stress (MPa)
D = 103.6 m
0
5
Stress (MPa)
10
0
150
15
0
10
0
150
15
Stress (MPa) 20
100
Height (m)
D = 85.3 m
15
0
50
D = 79.2 m
15
Stress (MPa)
Height (m)
Height (m)
20
20
D = 67.1 m
15
Height (m)
D = 64.0 m
15
Height (m)
Height (m)
20
150
D = 115.8 m
15 10 5 0
0
50
100
150
Stress (MPa)
Fig. 6a. Circumferential stress profiles for different tank diameters for tank height of 12.2 m (40 ft) for allowable stress of 158.6 MPa (23,000 psi).
allowable stresses. This exercise was carried out by modeling tanks of relatively smaller height and relatively large diameters using thin shell theory with shell thicknesses calculated using the VDM. The modeled tanks were 3.7 m (12-ft), 4.9 m (16-ft), and 6.1 m (20-ft) high, with each tank having two shell courses each having the height of half the height of the tank. The tanks were modeled over a range of diameters and with the thicknesses of the shell courses calculated for the allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi). The circumferential stress results from the analytical solution were compared with the allowable stress.
4. Results and discussions 4.1. Study 1: Comparison of 1FM and VDM Comparisons of the shell designs based on 1FM and those based on VDM showed that the designs based on VDM led to more economical designs for almost all the tank sizes. For the tank diameters smaller than 15 m (50-ft) both 1FM and VDM produce similar shell thicknesses. However, for tank diameters more than 15 m (50-ft) design based on VDM are starting to be more economical. Figs. 5a–5c show the results of the comparison for the allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi), respectively. The vertical axis is the ratio of the volume of the steel required by the 1FM to the volume of the steel required by the VDM: V1FM/VVDM. If the ratio is higher than 1, then the 1FM requires more volume of steel than the VDM. If the ratio V1FM/VVDM was less than 1, then it requires less steel. The horizontal axis represents the diameter of the tank. There are six curves representing six different heights of tanks. 4.2. Study 2: Validation of VDM using the analytical approach when VDM is permissible to use The stress analysis for the API 650 tanks presented 11 out of the 27 tanks which have an overstress of more than 0.5% of the allowable stress.
Shown in Figs. 6a and 6b are the results of stress profiles for tanks designed using VDM with an allowable stress of 158.6 MPa (23,000 psi). The vertical axis represents the height from the bottom of the tank in meters, and the horizontal axis represents the circumferential stress at that height. The solid vertical line represents the allowable stress of 158.6 MPa (23,000 psi). Any point has a stress greater than that in the tank represents an overstress in the tank. Fig. 6a shows the stress profile for 9 tanks with the height of 12.2 m (40-ft). Most of the overstress happens in the upper shell courses. This is due to the upper courses approaching the L/H limit of 1000/6. For example, the tank with the height of 12.2 m (40-ft) and diameter of 115.8 m (380-ft) is facing an overstress of 1.4%. The L/H ratio for the tank as a whole is 122, but if the top two courses were considered by themselves as a separate tank the L/H ratio becomes 186, which is greater than the 1000/6 limit. Fig. 6b shows the stress profiles of 3 tanks with the height of 14.6 m (48-ft), 3 tanks with the height of 17.1 m (56-ft), and 3 tanks with the height of 19.5 m (64-ft). The overstress is more prevalent in the bottom shell courses. This is due to the way VDM is formulated. The first course’s design is formulated without the consideration of the upper shell courses. With taller tanks, the forces transferred from upper courses to the bottom course accumulate and eventually increase the overall stress experienced by the bottom shell course. 4.3. Study 3: Validation of VDM using the analytical approach when VDM is not permissible to use In this part tanks designed using VDM, when VDM is not permissible to use were analyzed using the linear analysis based on thin shell theory. The study shows the ratio of maximum stress, obtained from the solution for a tank designed using VDM in the tank increases with the increase in the L/H ratio. The results are presented in Figs. 7a–7c. The figures represent three allowable stresses: 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi) respectively. The horizontal axis shows the L/H ratio, and the vertical axis represents the ratio of the maximum hoop stress divided by the allowable stress.
E. Azzuni, S. Guzey / Engineering Structures 101 (2015) 621–630 20
D = 54.9 m
15
Height (m)
Height (m)
H = 14.6 m
20
10 5 0
0
50
100
10 5 0
150
0
50
Stress (MPa) 20
10 5 0
50
100
5 0
50
150
D = 75.3 m
10 5 0
50
5 50
Stress (MPa)
100
150
D = 64.4 m
20
10
0
150
Stress (MPa)
15
0
100
15
0
150
Height (m)
Height (m)
Height (m)
H = 19.5 m
5 100
100
D = 54.9 m
20
10
50
50
Stress (MPa)
D = 48.8 m
0
0
20
D = 73.2 m
10
0
150
15
0
5
Stress (MPa)
15
Stress (MPa)
20
10
0
150
Height (m)
D = 67.1 m
15
0
100
D = 79.2 m
15
Stress (MPa)
Height (m)
Height (m)
H = 17.1 m
20
20
D = 73.2 m
15
Height (m)
628
100
15 10 5 0
150
0
50
Stress (MPa)
100
150
Stress (MPa)
1.1
1.1
1.05
1.05
1
1 exact/ allowbale
exact/ allowbale
Fig. 6b. Circumferential stress profiles for different tank diameters and heights for allowable stress of 158.6 MPa (23,000 psi).
0.95
0.9
L/H=1000/6
0.9
0.85
0.8 160
0.95
L/H=1000/6
0.85
180
200
220
240
0.8 160
180
L/H H = 3.7 m
H = 4.9 m
200
220
240
L/H H = 6.1 m
Fig. 7a. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 158.6 MPa (23,000 psi) vs. L/H (the shell thickness is determined using VDM).
The points of interest are those at which the plots cross with unity in the vertical axis. Based on the results of our study, The L/H limit ratio of 1000/6 was found to be reasonably conservative and safe to use. The lowest L/H ratio found using our calculations was for the 6.1 m (20-ft) high tank, with a diameter of 152.4 m (500-ft). The L/H ratio at which the allowable stress is equal to the maximum experienced stress was established through interpolation. The L/H value found was 183. This value represents a tank with a diameter between 152.4 m (500-ft) and 167.6 m (550-ft) with a height of 6.1 m (20-ft).
H = 3.7 m
H = 4.9 m
H = 6.1 m
Fig. 7b. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 206.8 MPa (30,000 psi) vs. L/H (the shell thickness is determined using VDM).
5. Conclusions We developed a stiffness–flexibility method based on thin shell theory that gives the theoretical displacements and stresses at each shell course without any approximation or simplification. Based on the results of our studies, shell designs based on VDM may become up to 4% more economical than those of the 1FM: The larger the tank size either in terms of the diameter or the height, the greater the advantage of the VDM. For the tank diameters smaller than 15 m (50-ft) the designs based on 1FM and VDM are very close. Therefore, for the tank diameters smaller than
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sffiffiffiffiffiffiffi! 0:0696D HG 4:9HDG þ CA 1:06 H Sd Sd
1.1
t1d ¼ 1.05
t1t ¼
exact/ allowbale
1
L/H=1000/6
0.9
0.85
180
200
220
240
L/H H = 3.7 m
H = 4.9 m
H = 6.1 m
Fig. 7c. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 236.5 MPa (34,300 psi) vs. L/H (the shell thickness is determined using VDM).
15 m (50-ft) there is no need to use VDM. Instead 1FM should be used. Comparing shell designs based on linear analysis with those based on VDM, it is found that the linear analysis does not lead to more economical designs than VDM most of the time. However, results show that shell designs using VDM may produce overstressed shell courses for some of the large steel liquid storage tanks when VDM is permissible to use. Linear analysis would give more accurate shell designs for those cases. Considerations should be given to use linear analysis for larger thanks even if VDM is permissible to use. If L/H ratio is larger than 1000/6 in SI units VDM shall not be used based on the rules of API 650. Our linear analysis results also support this limitation on the use on VDM. Our studies have shown that to obtain economical results for shell thickness depending on the diameter and L/H ratio one should use: D 6 15 m ð50 ftÞ One-foot method
sffiffiffiffi! H 4:9HD St St
ðA2Þ
Note that the equations above are unit dependent and the equations in this work are given in SI units. If the calculated first shell course thickness using Eqs. (A1) and (A2) is more than the thickness obtained using the traditional 1FM, the thickness calculated from 1FM may be used as the first shell course thickness. The shell thickness for the upper shell courses are calculated the following way: The location of the maximum hoop stress is found first using the lowest of the values
0.95
0.8 160
0:0696D 1:06 H
ðA1Þ
D > 15 m ð50 ftÞ and L=H 6 1000=6 Variable-design-point method and linear analysis
L H
> 1000=6
Linear analysis
Acknowledgments Research by the first author was supported by the Lynn Fellowship and Purdue University. Research by the second author was supported by Purdue University. Appendix A For completeness the VDM calculation procedure are given here in SI system of units. The reader is referred to the API 650 for the further details of the VDM and equations in USC system of units [1]. In addition, Annex K of API 650 provides example calculations for VDM. For the first shell course from the bottom the greater of the two thicknesses obtained using following equations.
x1 ¼ 0:61ðrt u Þ0:5 þ 320CH x2 ¼ 1000CH x3 ¼ 1:22ðrt u Þ0:5 With C ¼ ðK 0:5 ðK 1ÞÞ=ð1 þ K 1:5 Þ and K ¼ tL =t u The thickness of the course is then found using the greater of:
tdx ¼
x 4:9DðH 1000 ÞG þ CA Sd
ttx ¼
x 4:9DðH 1000 Þ St
Second course thickness: h1 ðrt1 Þ0:5
t2 ¼ t1
6 1:375
1:375 6
h1 ðrt 1 Þ0:5
2:625 6
h1 ðrt 1 Þ0:5
< 2:625
t2 ¼ t 2a þ ðt1 t2a Þ 2:1
h1 1:25ðrt1 Þ0:5
t2 ¼ t 2a
References [1] API Standard 650. Welded tanks for oil storage. 12th ed. Washington, DC: American Petroleum Institute; 2013. [2] DiGrado BD, Thorp GA. The aboveground steel storage tank handbook. Van Nostrand Reinhold; 1995. [3] Myers PE. Aboveground storage tanks. McGraw-Hill; 1997. [4] Zingoni A. Liquid-containment shells of revolution: a review of recent studies on strength, stability and dynamics. Thin-Walled Struct 2015;87:102–14. http://dx.doi.org/10.1016/j.tws.2014.10.016. [5] Jaca RC, Godoy LA, Flores FG, Croll JG. A reduced stiffness approach for the buckling of open cylindrical tanks under wind loads. Thin-Walled Struct 2007;45(9):727–36. http://dx.doi.org/10.1016/j.tws.2007.07.001. [6] Uematsu Y, Koo C, Yasunaga J. Design wind force coefficients for open-topped oil storage tanks focusing on the wind-induced buckling. J Wind Eng Indust Aerodyn 2014;130:16–29. http://dx.doi.org/10.1016/j.jweia.2014.03.015. [7] Chen L, Rotter JM, Doerich C. Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure. Eng Struct 2011;33(12):3570–8. http://dx.doi.org/10.1016/j.engstruct.2011.07.021. [8] Zhao Y, Lin Y, Shen YB. Wind loads on large cylindrical open-topped tanks in group. Thin-Walled Struct 2014;78:108–20. http://dx.doi.org/10.1016/ j.tws.2014.01.002. [9] Nachtigall I, Gebbeken N, Urrutia-Galicia JL. On the analysis of vertical circular cylindrical tanks under earthquake excitation at its base. Eng Struct 2003;25(2):201–13. http://dx.doi.org/10.1016/S0141-0296(02)00135-9. [10] Virella JC, Godoy LA, Suárez LE. Fundamental modes of tank-liquid systems under horizontal motions. Eng Struct 2006;28(10):1450–61. http://dx.doi.org/ 10.1016/j.engstruct.2005.12.016. [11] Matsui T. Sloshing in a cylindrical liquid storage tank with a floating roof under seismic excitation. J Pressure Vessel Technol 2007;129(4):557–66. http://dx.doi.org/10.1115/1.2767333.
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[12] Chen ZP, Duan YY, Shen JM, Jiang JL. A simplified method for calculating the stress of a large storage tank wall. Proc Inst Mech Eng, Part E: J Process Mech Eng 2007;221(3):119–27. http://dx.doi.org/10.1243/09544089JPME125. [13] Zick LP, McGrath RV. Design of large diameter cylindrical shells. Proc Division Refining, Am Petrol Inst New York 1968;48:1114–40. [14] Buzek J. Hoop forces in cylindrical liquid storage tanks. Correspondence-API Subcommittee on Pressure Vessels & Tanks; 1979.
[15] Calladine CR. Theory of shell structures. Cambridge University Press; 1983. [16] Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. McGraw-Hill; 1959. [17] Gould PL. Analysis of shells and plates. Prentice Hall; 1999. [18] Hetenyi M. Beams on elastic foundation; theory with applications in the fields of civil and mechanical engineering. University of Michigan Press; 1946.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Comparison of the shell design methods for cylindrical liquid storage tanks Eyas Azzuni, Sukru Guzey ⇑ Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 April 2015 Revised 27 July 2015 Accepted 31 July 2015
The three methods for determining the shell thickness of steel cylindrical liquid storage tanks designed in conformance with API Standard 650, Welded Tanks for Oil Storage (API 650) are: (1) one-foot method (1FM), (2) variable-design-point method (VDM) and (3) linear analysis. We compared the shell designs based on these three methods for different tank properties: diameter, height and allowable stress. For linear analysis, we developed a stiffness–flexibility matrix method based on thin shell theory that gives the theoretical displacements and stresses at each shell course without any approximation or simplification. Results show that shell designs using VDM may produce overstressed shell courses for some of the large steel liquid storage tanks when VDM is permissible to use. Linear analysis would give more accurate shell designs for those cases. Ó 2015 Elsevier Ltd. All rights reserved.
Keywords: Storage tank Shell design API 650 Thin shell theory One-foot method Variable-design-point method
1. Introduction API 650 is an industry standard used for the design and construction of large cylindrical storage tanks for liquid products [1–3]. API 650 storage tanks are vertical, cylindrical, closed- and open-top welded tanks with uniformly supported flat bottom. They are used to store petroleum, petroleum products, and other liquid products [1]. Recently, considerable research effort has been devoted to the analysis, design, and evaluation of the liquid storage tanks [4]. Much of the research conducted has focused on the buckling of and wind effects on the storage tanks [5–8]. Some researchers worked on dynamic effects related to earthquakes [9–11]. Chen et al. worked on developing a simple method to calculate shell stress [12]. A typical storage tank has a number of shell courses of uniform plate thickness. The thickest course is at the bottom and each shell course above is typically thinner than the previous one. See Fig. 1 for a typical storage tank shell cross-section. There are three methods allowed by API 650 to determine the required plate thickness of the shell. The first method is the one-foot-method (1FM) which is based on the ‘‘membrane theory’’. The required shell plate thickness for each shell course is
⇑ Corresponding author. E-mail addresses: (S. Guzey).
[email protected]
(E.
http://dx.doi.org/10.1016/j.engstruct.2015.07.050 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
Azzuni),
[email protected]
calculated using the circumferential stress at a point 0.3 m (1-ft) above the lower horizontal weld seam of the shell course due to hydrostatic pressure of the stored liquid. The reasoning behind this assumption is that the tank bottom plates provide restraint to reduce circumferential stress due to hydrostatic pressure at the bottom 0.3 m (1-ft) of the lowest shell course. Similarly, a shell course other than the lowest shell course, has generally thicker shell plates below. The plate below provides some restraint at the lower portion of the shell course in consideration. The 1FM is used successfully for the majority of the tanks. However, the designs based on the 1FM may become conservative and cost prohibitive for larger diameter tanks. Therefore, API 650 limits the applicability of this method to tanks up to 61 m (200-ft) in diameter. The second method to calculate the required shell plate thickness is the variable-design-point method (VDM) that is also based on the ‘‘membrane theory’’. The VDM was proposed by Zick and McGrath in 1968 [13] and later adopted by API 650 as a refined method to calculate the required shell plate thickness especially for tanks more than 61 m (200-ft) in diameter. The VDM takes into consideration the restraint provided by the tank bottom plates to the first shell course and the restraint provided by each lower shell course to the upper shell course. The VDM uses a variable distance instead of fixed distance of 0.3 m (1-ft), as used in 1FM, above the circumferential seam for each shell course to calculate the maximum stress due to hydrostatic pressure. The variable distance in VDM is a function of the shell plate thickness above and below
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Nomenclature H L h1 D r G CA Sd St t ti td t 1d t dx tt t 1t t tx tu tL
distance from the maximum product level to bottom of the shell course under consideration (m) ð500 DtÞ0:5 (mm) height of the bottom shell course (m) nominal tank diameter (m) nominal radius of the tank (m) the design specific gravity of the stored liquid corrosion allowance (mm) allowable design condition stress (MPa) allowable hydrostatic test condition stress (MPa) thickness of the shell thickness of a shell course design shell thickness (mm) design shell thickness for the first shell course (mm) design shell thickness (mm) hydrostatic test shell thickness (mm) hydrostatic test shell thickness for the first shell course (mm) hydrostatic test shell thickness (mm) corroded thickness of the upper shell course; approximated using 1FM for the first iteration (mm) thickness of the lower shell course (mm)
the seam. Most of the time designs based on VDM are more economical compared with those based on the 1FM. However, for some tank geometries the VDM may become unconservative and the tank shell thicknesses designed in accordance with VDM may be overstressed. Buzek showed that the restraint provided by the tank bottom on the tank shell produces circumferential stresses of sinusoidal nature varying with the distance from the tank bottom [14]. For certain tank diameter and height proportions, this sinusoidal varying restraining stress may add to the stress due to the hydrostatic circumferential stress and the design based on VDM may become unconservative. Therefore, API 650 limits the applicability of the VDM for the tanks with L/H ratio less than 1000/6 in SI units (refer to the nomenclature for the definition of these terms). For the storage tanks where the L/H ratio is more than 1000/6, tank shell thickness should be determined using linear analysis. The shell thickness calculation using linear analysis is the third method given in API 650. In this approach the boundary conditions for the analysis should be a plastic moment related to yielding of the plate under the shell and fully restrained radial movement at the bottom of the shell. API 650 does not describe a specific linear analysis method. In this study we developed a new method using thin shell theory to perform a linear analysis for the shell thickness calculation. In this method we are using exact stiffness–flexibility relations and exact shape functions originating from the so called ‘‘short shell’’ solution of the governing equations from the thin elastic shell theory. Therefore, we do not have any approximations or simplifications. The displacements, section forces and stresses obtained from this method are exactly matching the theoretical solution of thin shell theory. Overwhelming majority of texts employ only a single course solution of shell cylinder without extension to multiple shell courses with stepwise thicknesses. One can find very few references dealing with multiple shell courses in which only approximate solutions were obtained. In our treatment, we present an attractive and easy to implement formulation that renders analytical solution without any approximation. We shall investigate the efficiency and limitations of each method described above. The efficiency is defined in terms of
t1 t2
corroded thickness of the bottom shell course (mm) minimum design thickness of the second shell course (mm) t 2a corroded thickness of the second shell course (mm) as calculated for the upper shell courses w radial displacement of the cylindrical shell x axial length coordinate of the cylindrical shell E modulus of elasticity Ds shell bending rigidity p pressure m Poisson’s ratio b a parameter C1, . . ., C4, integration constants f(x) particular solution to the governing equation Li length of a shell course Q1, Q2 end shearing forces of a shell course M1, M2 end bending moments of a shell course w1, w2 end radial displacements of a shell course h1, h2 end rotations of a shell course
tn
t2 BOTTOM SHELL COURSE
t1
H
TANK BOTTOM D
Fig. 1. Typical tank shell cross-section.
minimum required shell thickness for each shell course and corresponding total weight of steel for the shell plates. A decrease in the weight of steel can be achieved by reducing the shell thickness, which will lead to a decrease in cost. Achieving a smaller design thickness is important for large diameter tanks because tank fabricators in North America typically order a plate thickness with 0.01 in. (0.25 mm) increments as well as commercially available 1/16 in. (1.59 mm) increments directly from a steel mill for sufficiently large weight of steel, about 20 tons. Another reason to achieve a smaller thickness may be to comply with the maximum thickness limit of 1 in. (25.4 mm) to avoid stress relieving requirement. Therefore, even a reduction of 0.01 in. (0.25 mm) in design thickness would be significant for large diameter tanks. Our main objective is to investigate the accuracy of 1FM and VDM and possibility of obtaining an economical result by using the linear analysis for the required shell thickness for storage tanks. We shall first summarize the three design methods for the storage tanks. Then we shall compare the shell designs based on the classic 1FM with those based on VDM. Furthermore, we shall focus on comparison of the VDM results with the theoretical solutions obtained from thin shell theory. Finally, we shall discuss the results and give conclusions.
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2. Design methods We conducted three different sets of studies of the available storage tank shell design methods in API 650: (1) Comparison of shell designs based on the classical 1FM with those based on VDM for the storage tank sizes where designs based on 1FM is permissible in API 650 (i.e. the tanks up to 61 m (200-ft) in diameter). (2) Comparison of the shell designs based on VDM with those based on linear analysis using thin shell theory for the larger storage tanks up to 122 m (400-ft) in diameter. (3) Study of the tank sizes where VDM is no longer permissible and compare the VDM results with those of obtained using thin shell theory. The storage tanks are typically constructed using 2.4 m (8-ft) or 3 m (10-ft) high shell courses of uniform thickness. In this study we used 2.4 m (8-ft) high shell courses because they are used more often than 3 m (10-ft) high shell courses. Material selection for a tank shell is an important task in the design process. The tank manufacturer, usually responsible for the tank design, selects a suitable material specification for each shell course based on cost, availability and brittle-fracture considerations. In this study, we used three different allowable stress representing low strength, medium strength and high strength material specifications permissible in API 650 tank designs. There are two different allowable stresses used in tank design: (1) design allowable stress and (2) hydrostatic test allowable stress. Design allowable stress is used with the specific gravity of the product that is going to be stored in the tank. The product specific gravity does not usually exceed 1.0 for the petroleum products. The second allowable stress is that for the hydrostatic test used for the hydrostatic test condition. The specific gravity for the hydrostatic test is usually 1.0 to represent the specific gravity of the test liquid. Hydrostatic test is a proof test before a tank is placed in service. During this test, the tank is filled with water and shell plates are stressed to the levels that the tank may not see again during its entire service life considering the product specific gravity is less than 1.0 in most cases. In order to be more general, we used a liquid specific gravity of 1.0 and selected the allowable stresses from the hydrostatic test allowable stresses. In the following sections we shall summarize the three shell-thickness design methods for storage tank design. 2.1. One-foot method The 1FM can be used for tanks up to 61 m (200-ft) in diameter. It has the advantage of being simple and easy to use with no iterations needed. The 1FM design takes the stress 0.3 m (1-ft) above the bottom of each shell course to determine its thickness. The thickness required is the larger of the two thicknesses calculated using Eqs. (1) and (2) from API 650:
td ¼
4:9DðH 0:3ÞG þ CA Sd
ð1Þ
tt ¼
4:9DðH 0:3Þ St
ð2Þ
Note that Eqs. (1) and (2) above are unit dependent and the equations in this work are given in the SI units. The reader is referred to API 650 for the equations in USC system of units. Eq. (1) is used to calculate the design shell thickness resulting from hydrostatic pressure of the liquid to be stored and desired corrosion allowance for the design life of the storage tank. Eq. (2) is used
to calculate the hydrostatic test shell thickness resulting from hydrostatic test pressure of the test liquid. As stated earlier, to be more general, we used Eq. (2), and corresponding hydrostatic test allowable stress in our study. 2.2. Variable-design-point method The VDM can be used for tanks with L/H ratio of up to 1000/6 in SI units (refer to nomenclature for the definition of L and H terms). In general, most storage tanks used in practice would have an L/H ratio smaller than the limiting value of 1000/6. Therefore, VDM can be used to determine the shell design thickness most of the time. The L/H limit is present because the VDM was based on the assumption that the base plastic moment has a small effect on the circumferential stresses in the lower shell course. The effect of base moment is of a sinusoidal nature: It may amplify the circumferential stress due to hydrostatic pressure for bigger tanks. The assumption used in the VDM for the base moment only works for tanks with L/H ratio less than 1000/6 [14]. The calculation of the shell courses thicknesses using VDM is more elaborate than the calculation using the 1FM. The VDM does not take a specific point for all the shell courses but rather finds the point where the stress is highest in each shell course to calculate the thickness. There are three categories of calculations: first shell course, second shell course, and upper shell courses. In accordance with API 650 [1] the following equations are used. For the first shell course from the bottom the greater of the two thicknesses obtained using Eqs. (3) and (4) is used as the shell thickness.
t1d ¼
0:0696D 1:06 H
sffiffiffiffiffiffiffi! HG 4:9HDG þ CA Sd Sd
t1t ¼
0:0696D 1:06 H
sffiffiffiffi! H 4:9HD St St
ð3Þ
ð4Þ
Note that the equations above are unit dependent and the equations in this work are given in SI units. If the calculated first shell course thickness using Eqs. (3) and (4) is more than the thickness obtained using the traditional 1FM, the thickness calculated from 1FM may be used as the first shell course thickness. Calculation of the shell thickness for the second and upper shell courses requires an iterative process. In this process the shell thickness converges to the required shell thickness in two or three iterations. The calculation procedure for the second and upper shell courses are given in Appendix A. 2.3. Linear analysis using thin shell theory Linear analysis is the required method of API 650 to calculate shell plate thickness when 1FM and VDM are not permissible. This method should be used when L/H ratio is greater than the limiting value of 1000/6 in SI units. For the storage tank geometries that is constructed in practice, it is very difficult to reach and exceed an L/H ratio of 1000/6. However, in this study we aimed to examine the linear analysis when the shell design based on VDM is still permissible and compare the results of the design based on linear analysis with those of the design based on VDM. In addition, we investigated the linear analysis and VDM when the design based on VDM is not permissible, i.e. L/H ratio is more than the limiting value of 1000/6. The API 650 document does not specify how to perform a linear analysis other than some instructions on how to choose the boundary conditions at the base of the shell cylinder. We performed the linear analysis using thin elastic shell theory that combines the
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bending deformations in addition to the membrane deformations. Because we are dealing with thin shells, the out-of-plane shear deformations are neglected. Consider a circular cylindrical shell of finite length and subject to axisymmetric, torsion-free loading. The governing equation of radial displacement, w may be written the following way if the thickness of the shell, t, is constant [15–17].
4
d w 4
dx
þ
Et p w¼ Ds r 2 Ds
ð5Þ
where E is the modulus of elasticity, r is the radius of shell, and p is the pressure. Shell bending rigidity Ds is defined as
Ds ¼
Et3 12ð1 m2 Þ
ð6Þ
where m is the Poisson’s ratio. Eq. (5) is very similar to the beam on elastic foundation equation [18]. Now we will define a parameter b where
b4 ¼
Et 3ð1 m2 Þ ¼ 2 4Ds r r2 t2
ð7Þ
Therefore, the governing equation can be written as 4
d w 4
dx
þ 4b4 w ¼
p Ds
ð8Þ
The general solution of the above fourth order ordinary differential equation of constant coefficients may be written as [16]
w ¼ ebx ðC 1 cos bx þ C 2 sin bxÞ þ ebx ðC 3 cos bx þ C 4 sin bxÞ þ f ðxÞ
ð9Þ
where C 1 ; . . . ; C 4 are the four integration constants to be determined from the boundary conditions and f ðxÞ is a particular solution. In our case membrane solution due to hydrostatic pressure of the stored liquid will be the particular solution and integration constants C 1 ; . . . ; C 4 above will be found from the combined membrane and bending solution. Note that all the other variables rotation, h, bending moment, M and shearing force, Q can be expressed in terms of radial displacement, w by the following equations
dw ; dx 2 d w M ¼ Ds 2 ; dx 3 dM d w ¼ Ds 3 : Q¼ dx dx
attached to a joint is needed for each joint. However, long shell solution may not give accurate results when semi-infinite length assumption cannot be justified [16]. Short shell solution gives more accurate results than the long shell solution because it incorporates all the integration constants into the solution. However, this approach requires more involving treatment because each joint cannot be solved independently. Simultaneous solution of entire tank joints equations needed. This problem may be calculated by approximate methods [16]. In this work we will use short shell solution and instead of solving the joint equations approximately we will solve them analytically. We will cast the shell equations for a finite length cylinder for each shell course in a matrix form using the approach suggested by Calladine [15]. Then we will enforce continuity and equilibrium of each joint in similar way to the direct stiffness or finite element method and solve the system of equations simultaneously. Now let us consider an axisymmetric circular cylindrical shell of length Li , constant thickness of t i subject to end shearing forces, Q 1 and Q 2 and bending moments M 1 and M 2 . Radial displacements and rotations at the end of the cylinder is given as w1 ; w2 and h1 ; h2 . See Fig. 2 for the geometry and sign convention for displacement, rotation, bending moment and shearing forces. Although there are many ways to relate displacement, rotation, bending moment and shearing forces at the ends of a short cylinder we will use the following stiffness–flexibility matrix suggested by Calladine [15].
2 3 2Dbb3 w1 s 6 6M 7 16 a b 6 6 17 7¼ 6 6 4 w2 5 2 6 d 3 4 2Ds b c M2 2
b
a b
bð2Ds bÞ bc dð2Ds bÞ
d 2Ds b3 bc b 2Ds b3 a b
3 3 2 Q1 7 7 6 dð2Ds bÞ 7 76 h1 7 76 7 a 7 4 b 5 Q2 5 c b
bð2Ds bÞ
h2
where the coefficients a; b; c; d are defined by the following functions
a ¼ ðSinh 2n sin 2nÞ=k b ¼ ðSinh 2n þ sin 2nÞ=k c ¼ 2ðCosh n sin n Sinh n cos nÞ=k d ¼ 2ðCosh n sin n þ Sinh n cos nÞ=k
h¼
ð10Þ
Cylindrical liquid storage tanks usually have shell courses of several different thicknesses. So for each shell course we need to solve the governing Eq. (8) and thus, find the four integration constants and particular membrane solution appeared in Eq. (9). In general there are two ways to approach the problem [16]. First one is to consider each shell course as semi-infinite length shell that one end of the shell does not see the effects of the loading at the other end. This approach is commonly named as long shell solution. The second approach is to consider each shell course as finite length shell that one end of the shell does see the effects of the loading at the other end. The second approach is commonly named as short shell solution. Long shell solution is relatively simple to obtain because the semi-infinite length assumption eliminates the integration constants C 1 and C 2 and remaining two integration constants may be found by construction of continuity of radial displacement w and rotation at each shell joint (weld seam). Each joint may be treated independently and only the information from a shell course
ð11Þ
Fig. 2. The sign convention used to develop the thin shell solution.
ð12Þ
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625
where
n ¼ bLi k ¼ Cosh2n cos 2n
ð13Þ
Now we can form the stiffness–flexibility matrix for each shell course using Eq. (11) and assemble system matrix similar to finite element method and enforce continuity of displacement and equilibrium of forces together with membrane solution. Solution of system matrix will give us the shearing forces Q and rotation h at each joint (weld seam). The boundary conditions of the bottom of the tank were set to be pinned along the circumference of the tank. The top of the tank was free to move. Plastic moment was applied to the bottom of the tank simulating a bottom plate of 6.35 mm (0.25-in.) thick steel with a yield stress of 250 MPa (36,000 psi) along the circumference of the bottom shell course. Note that solution of the governing Eq. (8) was originally defined in terms of radial displacement w. Therefore, we need to recover w at any point along the shell to calculate stresses. While there are many ways to recover the integration constants appeared in Eq. (9), we would like to mention two of them here. First method is given for beams on elastic foundations in Hetenyi’s book and named as ‘‘method of initial conditions’’ [18]. This approach expresses displacement function w in terms of w0 ; h0 ; M 0 , and Q 0 quantities at the end x ¼ 0 of the shell course. The second method to obtain the integration constants is suggested by Gould’s book on shells [17]. Gould’s method constructs a 4 4 coefficient matrix that relates the integration constants to the end bending moments and shearing forces, M 0 , and Q 0 at both ends of a shell course at x ¼ 0 and x ¼ Li . We used both of the recovery methods and they gave identical results as expected. Here we would like to explain Hetenyi’s approach in detail. Displacement function w at any point along the shell course under consideration can be written as
1 1 wðxÞ ¼ w0 F 1 ðbxÞ h0 F 2 ðbxÞ 2 M0 F 3 ðbxÞ b b Ds
1 b3 Ds
Q 0 F 4 ðbxÞ
Fig. 3. Simply supported cylindrical shell with uniform pressure.
rhoop ¼ w
pL4i 2 sin a Sinh a 1 sin bx Sinh bx 4 64Ds a cos 2a þ Cosh 2a 2 cos aCosh a cos bxCosh bx cos 2a þ Cosh 2a
w¼ ð14Þ
F 2 ðbxÞ ¼ 12 ðCosh bx sin bx þ Sinh bx cos bxÞ
ð15Þ
Sinh bx cos bxÞ
Note that the displacement function w in Eq. (14) is the homogenous solution or bending solution of displacement of the governing equation. We need to add membrane solution associated with the hydrostatic liquid pressure to find the general solution for the displacement at each shell course. Radial displacement due to membrane solution at any point under hydrostatic liquid pressure can be expressed by the following equation
wm ¼
ð18Þ
Where
F 1 ðbxÞ ¼ Cosh bx cos bx F 3 ðbxÞ ¼ F 4 ðbxÞ ¼
ð17Þ
Before we move to the comparison studies of each tank design methods we would like show validation of our linear analysis using thin shell theory. As we mentioned earlier our method uses exact stiffness–flexibility matrix and exact shape functions to describe the radial displacement within the usual assumptions of thin shell theory. To validate this we shall use a circular cylindrical shell of length Li with a uniform internal pressure, p. For both ends of the shell pinned–pinned condition the closed form solution of radial displacement is given by Timoshenko by the following expression [16]
where
1 ðSinh bx sin bxÞ 2 1 ðCosh bx sin bx 4
E r
pr 2 Eti
ð16Þ
where hydrostatic pressure p is
p ¼ cðhi xÞ where c is the weight per unit volume of the stored liquid and hi is the height of the liquid from the bottom joint (seam) of a shell course under consideration. Once the general radial displacements, including both bending and membrane deformations, are combined the circumferential stress of the tank shell at any point can be calculated simply by the following expression.
a¼
bLi 2
In Eq. (18) origin of coordinate x is taken at the middle of the cylinder. We solved the above problem using our thin shell approach for different thicknesses and obtained identical results as the closed form solution given in Eq. (18). Fig. 3 shows the geometry and problem description of the problem. Fig. 4 shows the normalized radial displacement with respect to the membrane solution using thin shell approach and closed form solution for different thickness. Our thin shell solution matches the closed form solution exactly as expected. 3. Comparison studies We performed three different comparison studies to investigate the shell design methods. 3.1. Study 1: Comparison of 1FM and VDM To compare the results of 1FM with those of VDM, a spreadsheet was generated to compute the thicknesses of shell courses for different tank diameter and height. This comparison study was done using allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi). We started
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1.2
t = 0.01
t=0.1
t=1
1
w/w membrane
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
% x/L Fig. 4. Normalized radial displacement in a pinned–pinned cylinder subject to uniform internal pressure. The horizontal axis is the ratio of the height to the entire length, and the vertical axis is the normalized radial displacement w with respect to the membrane radial displacement.
Fig. 5a. Comparison of 1FM and VDM, ratio of volume of steel resulting from 1FM to VDM vs tank diameter for different tank heights in meters for the allowable stress of 158.6 MPa (23,000 psi).
with a tank height of 7.3 m (24-ft) with three 2.4 m (8-ft) shell courses and the diameter ranging from 3 m (10-ft) to 61 m (200-ft). Then we increased the tank height with 2.4 m (8-ft) increments up to a tank height of 19.5 m (64-ft) with eight 2.4 m (8-ft) shell courses with the diameter ranging from 3 m (10-ft) to 61 m (200-ft). 3.2. Study 2: Validation of VDM using the analytical approach when VDM is permissible to use API 650 Annex K provides tables to summarize the required shell thickness based on VDM for different tank sizes and allowable stresses. In this part of the study, the tank designs specified in Annex K for an allowable stress of 158.6 MPa (23,000 psi) were tested with the linear analysis using thin shell theory. The linear analysis provided the theoretical stresses in each course of the tank. With this study we may be able to see if the designs based on VDM are actually overstressed or not.
Fig. 5b. Comparison of 1FM and VDM, ratio of volume of steel resulting from 1FM to VDM vs tank diameter for different tank heights in meters for the allowable stress of 206.8 MPa (30,000 psi).
Fig. 5c. Comparison of 1FM and VDM, Ratio of Volume of Steel Resulting from 1FM to VDM vs Tank Diameter for Different Tank Heights in Meters for the Allowable Stress of 236.5 MPa (34,300 psi).
In total, 27 tanks were tested. Four different heights were considered: 12.2 m (40-ft), 14.6 m (48-ft), 17.1 m (56-ft), and 19.5 m (64-ft). The diameters were selected from API 650 Annex K table for an allowable stress of 158.6 MPa (23,000 psi). Each shell course was 2.4 m (8-ft) high, and the thickness of each course was also determined using the same table. We calculated the actual stresses using thin shell theory and the stress profile was then obtained. The points where the tank was overstressed were found.
3.3. Study 3: Validation of VDM using the analytical approach when VDM is not permissible to use This part of the study was performed to compare the shell designs based on VDM and linear analysis using the thin shell theory when the L/H ratio is more than 1000/6 in SI units (i.e. VDM method is no longer permissible). We investigated if the limiting L/H ratio of 1000/6 is a good limit for all diameters, heights and
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10 5 0
0
50
100
10 5 0
150
0
Stress (MPa) 20
10 5 0
50
100
5 0
50
Stress (MPa)
100
150
10 5
50
100
Stress (MPa)
150
D = 97.5 m
10 5 0
20
D = 109.7 m
0
100
50
100
150
Stress (MPa)
15
0
50
15
0
150
Height (m)
Height (m)
Height (m)
5
100
20
D = 91.4 m
20
10
50
0
Stress (MPa)
D = 103.6 m
0
5
Stress (MPa)
10
0
150
15
0
10
0
150
15
Stress (MPa) 20
100
Height (m)
D = 85.3 m
15
0
50
D = 79.2 m
15
Stress (MPa)
Height (m)
Height (m)
20
20
D = 67.1 m
15
Height (m)
D = 64.0 m
15
Height (m)
Height (m)
20
150
D = 115.8 m
15 10 5 0
0
50
100
150
Stress (MPa)
Fig. 6a. Circumferential stress profiles for different tank diameters for tank height of 12.2 m (40 ft) for allowable stress of 158.6 MPa (23,000 psi).
allowable stresses. This exercise was carried out by modeling tanks of relatively smaller height and relatively large diameters using thin shell theory with shell thicknesses calculated using the VDM. The modeled tanks were 3.7 m (12-ft), 4.9 m (16-ft), and 6.1 m (20-ft) high, with each tank having two shell courses each having the height of half the height of the tank. The tanks were modeled over a range of diameters and with the thicknesses of the shell courses calculated for the allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi). The circumferential stress results from the analytical solution were compared with the allowable stress.
4. Results and discussions 4.1. Study 1: Comparison of 1FM and VDM Comparisons of the shell designs based on 1FM and those based on VDM showed that the designs based on VDM led to more economical designs for almost all the tank sizes. For the tank diameters smaller than 15 m (50-ft) both 1FM and VDM produce similar shell thicknesses. However, for tank diameters more than 15 m (50-ft) design based on VDM are starting to be more economical. Figs. 5a–5c show the results of the comparison for the allowable stresses of 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi), respectively. The vertical axis is the ratio of the volume of the steel required by the 1FM to the volume of the steel required by the VDM: V1FM/VVDM. If the ratio is higher than 1, then the 1FM requires more volume of steel than the VDM. If the ratio V1FM/VVDM was less than 1, then it requires less steel. The horizontal axis represents the diameter of the tank. There are six curves representing six different heights of tanks. 4.2. Study 2: Validation of VDM using the analytical approach when VDM is permissible to use The stress analysis for the API 650 tanks presented 11 out of the 27 tanks which have an overstress of more than 0.5% of the allowable stress.
Shown in Figs. 6a and 6b are the results of stress profiles for tanks designed using VDM with an allowable stress of 158.6 MPa (23,000 psi). The vertical axis represents the height from the bottom of the tank in meters, and the horizontal axis represents the circumferential stress at that height. The solid vertical line represents the allowable stress of 158.6 MPa (23,000 psi). Any point has a stress greater than that in the tank represents an overstress in the tank. Fig. 6a shows the stress profile for 9 tanks with the height of 12.2 m (40-ft). Most of the overstress happens in the upper shell courses. This is due to the upper courses approaching the L/H limit of 1000/6. For example, the tank with the height of 12.2 m (40-ft) and diameter of 115.8 m (380-ft) is facing an overstress of 1.4%. The L/H ratio for the tank as a whole is 122, but if the top two courses were considered by themselves as a separate tank the L/H ratio becomes 186, which is greater than the 1000/6 limit. Fig. 6b shows the stress profiles of 3 tanks with the height of 14.6 m (48-ft), 3 tanks with the height of 17.1 m (56-ft), and 3 tanks with the height of 19.5 m (64-ft). The overstress is more prevalent in the bottom shell courses. This is due to the way VDM is formulated. The first course’s design is formulated without the consideration of the upper shell courses. With taller tanks, the forces transferred from upper courses to the bottom course accumulate and eventually increase the overall stress experienced by the bottom shell course. 4.3. Study 3: Validation of VDM using the analytical approach when VDM is not permissible to use In this part tanks designed using VDM, when VDM is not permissible to use were analyzed using the linear analysis based on thin shell theory. The study shows the ratio of maximum stress, obtained from the solution for a tank designed using VDM in the tank increases with the increase in the L/H ratio. The results are presented in Figs. 7a–7c. The figures represent three allowable stresses: 158.6 MPa (23,000 psi), 206.8 MPa (30,000 psi), and 236.5 MPa (34,300 psi) respectively. The horizontal axis shows the L/H ratio, and the vertical axis represents the ratio of the maximum hoop stress divided by the allowable stress.
E. Azzuni, S. Guzey / Engineering Structures 101 (2015) 621–630 20
D = 54.9 m
15
Height (m)
Height (m)
H = 14.6 m
20
10 5 0
0
50
100
10 5 0
150
0
50
Stress (MPa) 20
10 5 0
50
100
5 0
50
150
D = 75.3 m
10 5 0
50
5 50
Stress (MPa)
100
150
D = 64.4 m
20
10
0
150
Stress (MPa)
15
0
100
15
0
150
Height (m)
Height (m)
Height (m)
H = 19.5 m
5 100
100
D = 54.9 m
20
10
50
50
Stress (MPa)
D = 48.8 m
0
0
20
D = 73.2 m
10
0
150
15
0
5
Stress (MPa)
15
Stress (MPa)
20
10
0
150
Height (m)
D = 67.1 m
15
0
100
D = 79.2 m
15
Stress (MPa)
Height (m)
Height (m)
H = 17.1 m
20
20
D = 73.2 m
15
Height (m)
628
100
15 10 5 0
150
0
50
Stress (MPa)
100
150
Stress (MPa)
1.1
1.1
1.05
1.05
1
1 exact/ allowbale
exact/ allowbale
Fig. 6b. Circumferential stress profiles for different tank diameters and heights for allowable stress of 158.6 MPa (23,000 psi).
0.95
0.9
L/H=1000/6
0.9
0.85
0.8 160
0.95
L/H=1000/6
0.85
180
200
220
240
0.8 160
180
L/H H = 3.7 m
H = 4.9 m
200
220
240
L/H H = 6.1 m
Fig. 7a. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 158.6 MPa (23,000 psi) vs. L/H (the shell thickness is determined using VDM).
The points of interest are those at which the plots cross with unity in the vertical axis. Based on the results of our study, The L/H limit ratio of 1000/6 was found to be reasonably conservative and safe to use. The lowest L/H ratio found using our calculations was for the 6.1 m (20-ft) high tank, with a diameter of 152.4 m (500-ft). The L/H ratio at which the allowable stress is equal to the maximum experienced stress was established through interpolation. The L/H value found was 183. This value represents a tank with a diameter between 152.4 m (500-ft) and 167.6 m (550-ft) with a height of 6.1 m (20-ft).
H = 3.7 m
H = 4.9 m
H = 6.1 m
Fig. 7b. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 206.8 MPa (30,000 psi) vs. L/H (the shell thickness is determined using VDM).
5. Conclusions We developed a stiffness–flexibility method based on thin shell theory that gives the theoretical displacements and stresses at each shell course without any approximation or simplification. Based on the results of our studies, shell designs based on VDM may become up to 4% more economical than those of the 1FM: The larger the tank size either in terms of the diameter or the height, the greater the advantage of the VDM. For the tank diameters smaller than 15 m (50-ft) the designs based on 1FM and VDM are very close. Therefore, for the tank diameters smaller than
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sffiffiffiffiffiffiffi! 0:0696D HG 4:9HDG þ CA 1:06 H Sd Sd
1.1
t1d ¼ 1.05
t1t ¼
exact/ allowbale
1
L/H=1000/6
0.9
0.85
180
200
220
240
L/H H = 3.7 m
H = 4.9 m
H = 6.1 m
Fig. 7c. The ratio of maximum circumferential stress obtained from linear analysis to allowable stress of 236.5 MPa (34,300 psi) vs. L/H (the shell thickness is determined using VDM).
15 m (50-ft) there is no need to use VDM. Instead 1FM should be used. Comparing shell designs based on linear analysis with those based on VDM, it is found that the linear analysis does not lead to more economical designs than VDM most of the time. However, results show that shell designs using VDM may produce overstressed shell courses for some of the large steel liquid storage tanks when VDM is permissible to use. Linear analysis would give more accurate shell designs for those cases. Considerations should be given to use linear analysis for larger thanks even if VDM is permissible to use. If L/H ratio is larger than 1000/6 in SI units VDM shall not be used based on the rules of API 650. Our linear analysis results also support this limitation on the use on VDM. Our studies have shown that to obtain economical results for shell thickness depending on the diameter and L/H ratio one should use: D 6 15 m ð50 ftÞ One-foot method
sffiffiffiffi! H 4:9HD St St
ðA2Þ
Note that the equations above are unit dependent and the equations in this work are given in SI units. If the calculated first shell course thickness using Eqs. (A1) and (A2) is more than the thickness obtained using the traditional 1FM, the thickness calculated from 1FM may be used as the first shell course thickness. The shell thickness for the upper shell courses are calculated the following way: The location of the maximum hoop stress is found first using the lowest of the values
0.95
0.8 160
0:0696D 1:06 H
ðA1Þ
D > 15 m ð50 ftÞ and L=H 6 1000=6 Variable-design-point method and linear analysis
L H
> 1000=6
Linear analysis
Acknowledgments Research by the first author was supported by the Lynn Fellowship and Purdue University. Research by the second author was supported by Purdue University. Appendix A For completeness the VDM calculation procedure are given here in SI system of units. The reader is referred to the API 650 for the further details of the VDM and equations in USC system of units [1]. In addition, Annex K of API 650 provides example calculations for VDM. For the first shell course from the bottom the greater of the two thicknesses obtained using following equations.
x1 ¼ 0:61ðrt u Þ0:5 þ 320CH x2 ¼ 1000CH x3 ¼ 1:22ðrt u Þ0:5 With C ¼ ðK 0:5 ðK 1ÞÞ=ð1 þ K 1:5 Þ and K ¼ tL =t u The thickness of the course is then found using the greater of:
tdx ¼
x 4:9DðH 1000 ÞG þ CA Sd
ttx ¼
x 4:9DðH 1000 Þ St
Second course thickness: h1 ðrt1 Þ0:5
t2 ¼ t1
6 1:375
1:375 6
h1 ðrt 1 Þ0:5
2:625 6
h1 ðrt 1 Þ0:5
< 2:625
t2 ¼ t 2a þ ðt1 t2a Þ 2:1
h1 1:25ðrt1 Þ0:5
t2 ¼ t 2a
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