Stress and Deflection Analyses of Floating Roofs Based on a Load Modifying

ARTICLE IN PRESS International Journal of Pressure Vessels and Piping 85 (2008) 728– 738

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Stress and deflection analyses of floating roofs based on a load-modifying method Xiushan Sun, Yinghua Liu , Jianbin Wang, Zhangzhi Cen Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

article info

a b s t r a c t

Article history: Received 22 June 2007 Received in revised form 24 March 2008 Accepted 27 March 2008

This paper proposes a load-modifying method for the stress and deflection analyses of floating roofs used in cylindrical oil storage tanks. The formulations of loads and deformations are derived according to the equilibrium analysis of floating roofs. Based on these formulations, the load-modifying method is developed to conduct a geometrically nonlinear analysis of floating roofs with the finite element (FE) simulation. In the procedure with the load-modifying method, the analysis is carried out through a series of iterative computations until a convergence is achieved within the error tolerance. Numerical examples are given to demonstrate the validity and reliability of the proposed method, which provides an effective and practical numerical solution to the design and analysis of floating roofs. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Floating roof Rainwater load Load-modifying method Deflection Nonlinear analysis

1. Introduction Floating roofs are widely used in the middle- and large-scale cylindrical tanks for crude oil and other liquid hydrocarbon storages around the world because of their advantages such as reducing product evaporation, improving safety, overall operating economy, etc. After a history of over 80 years with continual development and improvement, modern floating roofs with larger diameters for open-top tanks can be classified usually into two common types: single-deck type and double-deck type [1–5]. The single-deck floating roof consists of characteristically a circular deck plate and a pontoon (i.e. a compartmented buoyant ring) which are both constructed with thin plates and jointed together by a connection component, e.g. an angle-iron ring. To meet the increasing capacity of oil storage tanks and to improve the performance of the traditional single-type and double-type floating roofs, a new-style floating roof with continuous beams was also developed [6]. This floating roof has more complex components, which increases somewhat the difficulty of structural analysis. In the practical operation, the floating roof is usually subjected to rainwater loading resulting from the accumulated rainfall. The rainwater loading will result in a much larger deformation (or deflection) in the deck compared with the plate thickness. In many codes for the design of floating roofs, the whole structure is

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E-mail address: [email protected] (Y. Liu). 0308-0161/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2008.03.003

required to possess good performances such as strength and stability under a standard rainfall of 250 mm over the tank [7,8], i.e. no failure modes such as fracture, buckling or sinking should occur in this rainwater loading. Accordingly, stress and deformation analyses of floating roofs under rainwater loading are practical problems to be solved. However, the floating roof is actually subjected to complex loads and deformations during the operation. The loads and deformations of floating roofs are nonlinearly coupled with each other, which results in the difficulty of analysis. Mitchell [9] investigated the problem of floating roofs with pontoon, in which the deck plate was treated as membrane and the membrane large deflection equations were solved numerically by assuming a range of starting values. But the proper selection of these values was usually difficult and, of course, important to the solution. A similar method was also used by Epstein et al. [3,10,11] to analyze deformations and stresses for different types of floating roofs, including pan floating roofs, pontoon floating roofs with accumulated rainwater loading or with punctures in the deck, in which the effects of various parameters such as tank diameter and pontoon geometry were also examined. Umeki and Ishiwata [12] improved Epstein’s solution and better computational efficiency was achieved, and they replaced the original Runge–Kutta numerical method by the Milne method. Another analytical method, i.e. the ODE-solver (ordinary differential equation solver) method, was proposed by Yuan et al. [13]. This method was used to solve the large deflection equation of floating roofs based on the bending theory rather than the membrane theory. To simplify the problem, some authors [4,14] also presented calculating formulas

ARTICLE IN PRESS X. Sun et al. / International Journal of Pressure Vessels and Piping 85 (2008) 728–738

Nomenclature

R1,R¯ 1

a c1,c2

R2,R¯ 2

C E f(r) fmax F g Gs h0 hc hs hw ha H0 H1, H2 Hg i KL, KNL M Mc Nr Na Nv pb(r) q(r) qc qs qw Q r R0

displacement vector in the FE equation deformation coefficient of the outer rim and inner rim of the pontoon, respectively ratio of increments of water and liquid heads Young’s modulus of the floating roof deflection of the deck plate maximum deflection of the deck plate restoring load vector in the FE equation 9.8 N/kg, gravitational acceleration weight of the floating roof excluding the deck plate typical rainfall equivalent deflection of the deck plate liquid head in the tank water head on the deck plate sinking depth of the floating roof due to slope of the pontoon’s bottom plate installing height of the deck plate heights of the outer rim and inner rim of the pontoon, respectively sinking depth of the floating roof due to its weight number of iteration in the load modification linear and nonlinear stiffness matrices in the FE equation, respectively total mass of the floating roof mass of the deck plate number of radial continuous beams number of annular continuous beams number of vertical ribs net pressure on the bottom plate of the pontoon net pressure on the deck plate weight of deck plate per unit area liquid pressure applied on the deck plate in the tank rainwater loads on the deck plate applied load vector in the FE equation radial coordinates of the floating roof radius of the tank

for the large deflection of the deck in floating roofs. These formulas, however, were based on a water test condition in which the loads on the deck plate distribute uniformly. In addition, with the development of computer modeling and corresponding numerical methods in modern engineering and sciences, the finite element method (FEM) was also employed in the structural analysis of floating roofs. Uchiyama et al. [15] and Yoshida [16] analyzed floating roofs under rainwater load by a nonlinear axisymmetric FEM, and special program codes for analysis of floating roofs, THANKS V-III and KOSTRAN, were, respectively, used in these two studies to compute the deformation and stress. The above methods for analysis of floating roofs were usually based on the axial symmetry theory, and the floating roof is simplified to a plane structure with this theory and the components such as bulkheads (necessary to divide the pontoon into several compartments) in the pontoon were neglected. These methods would be no more applicable when floating roofs with nonaxial symmetry or with 3-D complex structures such as the newly developed floating roof with continuous beams mentioned above, are used. Moreover, the rainwater was usually assumed to fill the whole deck plate in these methods. The rainwater, however, would fill only part of the deck plate if the floating roof has a large enough diameter. On the other hand, although some FEM solutions were used to conduct the analysis of floating roofs,

Rm Rw t t1, t2 t3, t4 V1, V2 Ve wA, wB z a d1, d2 Dh0 Dhw Dhs DH e l0 lw n y r0 r1 t; t¯ tw f

729

radius of the outer rim of the pontoon before and after deformation, respectively radius of the deck plate or the inner rim of the pontoon before and after deformation, respectively mean radius of the pontoon radius of the area of rainwater filling the deck plate thickness of the deck plate thicknesses of the outer rim and inner rim of the pontoon, respectively thicknesses of the top and bottom plates of the pontoon, respectively two parts of water volume on the deck plate due to redistribution water volume on the deck plate vertical displacements of the bottom of the outer rim and inner rim, respectively vertical coordinates of the floating roof tilt angle of the pontoon’s bottom plate radial displacements of the outer rim and inner rim of the pontoon, respectively rainfall increment water head increment liquid head increment difference between installing height of the deck plate and sinking depth of the floating roof error tolerance coefficient of determining the water distribution status on the deck plate ratio of equivalent water volumes on the deck plate Poisson’s ratio of the floating roof time of deformation progression water density, 1.0 106 kg/mm3 liquid (oil) density in the tank ratio of the inner rim’s and outer rim’s radii of the pontoon before and after deformation, respectively ratio of water distribution’s and deck plate’s radii rotation angle of the pontoon

these solutions were based on a simple axisymmetric method and only simple plane problem was dealt with. Accordingly, it is necessary to develop a general numerical method for practical analysis of floating roofs with 3-D structures in order to ease and aid implementations of structure design, analysis and optimization of floating roofs. This paper proposes a general and practical finite element (FE)based numerical method, i.e. the load-modifying method (LMM), for the 3-D structural analysis of floating roofs under rainwater load. A relationship between loads and deformations is developed firstly according to the equilibrium of the floating roof, in which two cases of rainwater distribution on the deck plate are considered, one case in which the rainwater fills only part of the deck plate and the other case in which the rainwater fills the whole deck plate. Then the FE analysis of the floating roof with this relationship is conducted based on the LMM. In the analysis procedure with the LMM, an initial condition (e.g. the condition with no deformation) is assumed to begin the nonlinear FE analysis with iterative computations, and then the load magnitudes in the current iteration are modified with computational results in the previous iteration and are ready for a new iterative analysis if necessary. Before each iterative analysis, the case of rainwater distribution on the deck plate is determined by results of the previous iteration. This analysis process is carried out

ARTICLE IN PRESS 730

X. Sun et al. / International Journal of Pressure Vessels and Piping 85 (2008) 728–738

continuously until the computational results converge to the real solutions. In numerical examples for applications of the proposed method, the present computational results are compared with those from other numerical or experimental methods. The validity and reliability of the proposed method are demonstrated.

2. Equilibrium analysis of the floating roof A typical single-deck floating roof consists of a circular plate, i.e. deck, and a compartmented buoyant ring, i.e. pontoon, and its geometry and characteristic dimensions are sketched in Fig. 1.

Fig. 2. Simplified loads on the floating roof.

2.1. Loads on the floating roof In the practical operation, the loads on the floating roof includes: weight (including that of appurtenances), rainwater loading on the deck, buoyancy and linearly distributed side pressures along the inner rim and outer rim. In general, the side pressures along the inner rim and outer rim are slight and contribute little to the deformation and stress of the floating roof, and these pressures can be neglected in order to simplify the analysis. In other words, only three typical loads, i.e. weight, rainwater loading and buoyancy, are considered in the following analysis. Additionally, the weight of the deck, buoyancy on the deck and the rainwater loading can be further simplified with a net pressure q(r) on the deck, as shown in Fig. 2. The weight of the floating roof and rainwater loading on the deck are balanced with the buoyancy produced by liquid in the tank, and the buoyancy is relevant to the liquid head (i.e. the liquid surface height, hs, see Fig. 3 or Fig. 4). As it has been pointed out, the deck plate is usually subjected to a large deflection under rainwater loading, and this deflection also has an influence on the value of buoyancy. Accordingly, the loads on the floating roof are coupled with its deformations. The proper relationship between the loads and the deformations is necessary to carry out the analysis of the floating roof. 2.2. Relationship between deformations and loads When there is a rainfall h0 on the top of the tank, the rainwater will accumulate into the deck plate and redistribute its volume. If the floating roof is too large, the rainwater will fill only part of the deck plate since the deck plate is subjected to a larger deflection (as shown in Fig. 3); but if the floating roof is not very large, the rainwater will fill the whole deck plate (as shown in Fig. 4).

Fig. 3. Deformation of the floating roof in case the rainwater fills only part of the deck plate.

Fig. 4. Deformation of the floating roof in case the rainwater fills the whole deck plate.

In these two cases, the pressures on the bottom plate of the floating roof are similar, but the load distributions on the deck plate are a little different. For the case that the rainwater fills only part of the deck plate (Fig. 3), the net pressure q(r) on the deck plate can be written with two different load distributions as ( qc þ qw þ qs ; 0prpRw qðrÞ ¼ (1a) qc þ qs ; Rw orpR2 where Rw is the radius of the area of rainwater filling the deck plate (Fig. 3); qc, qw and qs are loads resulted from the deck weight, rainwater on the deck plate and liquid pressure in the tank, respectively, and qc ¼

Fig. 1. Geometry of the floating roof.

Mc g 2

pR¯ 2

(2)

qw ¼ r0 g½f ðrÞ ðt=2Þ þ hw

(3)

qs ¼ r1 g½f ðrÞ þ ðt=2Þ þ hs

(4)

where the negative sign ‘‘’’ denotes the loading direction opposite to that of z-coordinate; Mc is the mass of the deck plate and g ¼ 9.8 N/kg is the gravitational acceleration; R¯ 2 is the radius


of the deck plate after deformation; r0 ¼ 1.0 106 kg/mm3 and r1 are the rainwater density and liquid density, respectively; hw and hs are the water head and liquid head (Fig. 3), respectively; f(r) is the deflection of the deck plate in the middle plane and always takes a positive value; and t is the thickness of the deck plate. The water head hw takes a negative value in Eq. (1a). Both the water head hw and the liquid head hs are relevant to the deflection f(r). For the case that the rainwater fills the whole deck plate (Fig. 4), the net pressure q(r) on the deck plate can be written as qðrÞ ¼ qc þ qw þ qs

(1b)

where qc, qw and qs have the same meanings as in Eq. (1a), but it should be noted that the water head hw takes a positive value in Eq. (1b). In Eq. (3), the water head hw is defined as positive value when the rainwater surface is above the installing position of the deck plate and as negative value when the rainwater surface is below this installing position (Figs. 3 and 4). It can be found that Eq. (1b) can be obtained with Eq. (1a) when Rw ¼ R¯ 2 . Accordingly, without loss of generality, Eq. (1a) is used to carry out the following derivations unless specified otherwise. The deflection of the deck plate can be treated as same along the circumference at a given position, r, and the equilibrium equation of the floating roof in the vertical direction can be written as Z

Rw

2p 0

Z qðrÞr dr þ 2p Z

þ 2p

R¯ 1 R¯ 2

R¯ 2 Rw

qðrÞr dr ðM Mc Þg

pb ðrÞr dr ¼ 0

pb ðrÞ ¼ r1 g½hs þ H0 ðR¯ 1 rÞ tanða þ fÞ

(6)

where a and f are the tilt angle of the pontoon’s bottom plate and rotation angle of the pontoon, respectively. The liquid head hs can be derived from Eq. (5) as r r t r0 2 hs ¼ 0 t2w hw þ hc 0 lw 1 tw þ 1 t¯ 2 2 r1 r1 r1 2

ðDH ha Þð1 t¯ Þ

(7a)

and

ha ¼

l0 ¼

2

R R¯ 2 0

Hg ¼

2

2

pr1 ðR¯ 1 R¯ 2 Þ

,

rf ðrÞdr

(9)

2 R¯ 2

1 þ 2t¯ ðR¯ 1 R¯ 2 Þ tanða þ fÞ 3 þ 3t¯

c2 R¯ 2 t¯ ¼ ¼t ; c1 R¯ 1

t¼

R2 ; R1

dk ck ¼ 1 ; R¯ k ¼ ck Rk Rk R Rw lw ¼ R0¯ R2 0

rf ðrÞ dr rf ðrÞ dr

(8)

tw ¼

(10) Rw R¯ 2

ðk ¼ 1; 2Þ

(11)

(12)

(14)

R20 h0

If Rw ¼ R¯ 2 , which results in tw ¼ 1 and lw ¼ 1, Eq. (7a) can be simplified to the case in which the rainwater fills the whole deck plate, which can be rewritten as r r t r0 þ 1 t¯ 2 hs ¼ 0 hw þ hc 0 1 r1 r1 2 r1 ðDH ha Þð1 t¯ 2 Þ

hw ¼ f ðRw Þ

(15a)

where Rw can be computed with the following formula of equivalent volume: Z

Rw

r½f ðrÞ f ðRw Þ dr ¼ pR20 h0

(16)

This integral equation can be solved with the numerically iterative method. For the case in which the rainwater fills the whole deck plate (l0p1), the volume Ve can be divided into two parts, one for capacity related to the water head hw and the other for capacity related to the deflection f(r), i.e. Ve ¼ V1+V2, as shown in Fig. 5. According to the equivalent volume in Fig. 5, the following equation can be obtained: hw ¼

R20 2 R¯ 2

h0 hc þ

t 2

(15b)

where t/2 is the additional term resulting from the thickness of the deck plate and usually can be neglected since it is a very small value.

(13a)

where H0 is the installing height of the deck plate (vertical distance between the deck plate and the outer rim’s bottom of the pontoon); R¯ 1 is the outer rim’s radius of the pontoon

(7b)

Eqs. (7a) and (7b) provide the relationships between the water head hw and the liquid head hs. As it is pointed out, the redistributions of the rainwater due to deflection of the deck plate are different for cases that rainwater fills whole or part of the deck plate. For the case that the rainwater fills only part of the deck plate (l041), the water head hw can be obtained as (Fig. 3)

0

M

2 R¯ 2 hc

where R0 is the radius of the tank. If l0p1, the rainwater fills the whole deck plate; otherwise, the rainwater fills only part of the deck plate. Additionally, the ratio of equivalent volumes, lw, can also be computed with the following formula: 8 1; l0 p1 > > < 2 2 (13b) lw ¼ R0 h0 þ Rw f ðRw Þ; l 41 0 > 2 > : R¯ 2 hc

2p

DH ¼ H0 Hg ;

hc ¼

after deformation; d1 and d2 denote respectively the horizontal displacement or radial displacement, of the outer rim and inner rim (Figs. 3 and 4). The equivalent deflection hc in Eq. (9) can be computed with the regular numerical integration method. In Eqs. (8) and (10), Hg is the sinking depth due to weight of the floating roof and ha is the sinking depth of the floating roof due to slope of the bottom plate of the pontoon. The case that the rainwater fills whole or part of the deck plate can be determined with the following ratio:

(5)

where M is the total mass of the floating roof; pb(r) is the pressure applied on the bottom plate of the floating roof (Fig. 2) and

731

Fig. 5. Redistribution of rainwater due to deflection.



"

R20 r0 t þ DH hc t¯ 2 þ Hg 2r 2 R¯ 1 1

2 ¯ þha ð1 t¯ Þ ðR1 rÞ tanða þ fÞ ; R¯ 2 prpR¯ 1

pb ðrÞ ¼ r1 g h0

hs ¼ h0

(19b)

R20 r0 t 2 t¯ ðDH ha Þð1 t¯ 2 Þ hc þ 2r 2 R¯ 1

(20b)

1

Fig. 6. Rotation of the pontoon.

The rotation angle of the pontoon f in Eq. (10) can be computed numerically with the following equation (Fig. 6): f¼

180 wB wA p R1 R2

(17)

where wA and wB are the vertical displacements of point A in the bottom of the outer rim and point B in the bottom of the inner rim, respectively, and they are computed numerically. It should be noted that wA and wB are positive values in the direction of z-coordinate (Fig. 6). The rotation angle f is selected as a positive value in the counterclockwise direction and it usually has a negative value in Eq. (17). The value of f in Eq. (17) includes two parts that result from the deformation and rigid rotation of the pontoon, respectively. Actually, Eq. (17) is an approximation on the rotation of the pontoon since the bottom plate of the pontoon would curve inward under the liquid pressure. This approximation, however, is reasonable because the angle f usually is a tiny value [10,11]. By substituting Eqs. (7a) and (15a) into Eqs. (1a) and (6), or by substituting Eqs. (7b) and (15b) into Eqs. (1b) and (6), the net pressure q(r) on the deck plate, the pressure pb(r) on the bottom of pontoon and the liquid head hs for two cases of load distribution (l041 or l0p1) can be obtained: (1) For the case that the rainwater fills only part of the deck plate (l041): h 8 > r0 g rr1 DH 2t ha ð1 t¯ 2 Þ f ðRw Þ þ 2t ð1 t2w t¯ 2 Þ > 0 > > >

> > > > hc t¯ 2 lw r1 þ M2 c þ f ðrÞ 1 r1 ; 0prpRw > r0 r0 < pR¯ 2 r0 h qðrÞ ¼ > > r0 g rr1 DH 2t ha ð1 t¯ 2 Þ þ f ðRw Þ þ 2t t2w t¯ 2 > 0 > > >

> > > > hc t¯ 2 lw rr1 þ M2 c f ðrÞ rr1 ; Rw orpR¯ 2 : 0 0 pR¯ r 2 0

Eqs. (18a)–(20b) show that the variables q(r), pb(r), hs and also hw in Eqs. (15a) and (15b) are all relevant to the deflection f(r). Usually, the net pressure q(r) distributes nonuniformly along the radial direction on the deck plate and is a function of the deflection f(r) when the liquid density is not equal to the water density, i.e. r16¼r0. Generally speaking, the radial deformations of the pontoon, i.e. d1 and d2, are much smaller compared with the radii R1 and R2, and the coefficients c1 and c2 can be taken as c1 ¼ c2 ¼ 1 in order to simplify the computation. That is to say, the variable values in all above equations can be computed with the initial dimensions (no deformation) of the floating roof, i.e. R¯ 1 ¼ R1 , R¯ 2 ¼ R2 and as a result that t¯ ¼ t. In addition, the rotation angle of pontoon, f, is also a tiny value. If f is small enough and contributes little to the pressure pb(r) in Eqs. (19a) and (19b) or to parameter ha in Eq. (10), it can also be neglected in order to simplify the computation. 2.3. Water test condition Now consider a special but practical condition, the so-called water test condition, when liquid in the tank is water and r1 ¼ r0. For example, for the case that the rainwater fills the whole deck plate (l0p1), substituting r1 ¼ r0 into Eq. (18b), we see the net pressure " ! # R2 t Mc qðrÞ ¼ r0 g h0 02 þ DH hc ha ð1 t¯ 2 Þ þ 2 (21) 2 pR¯ r R¯ 2

(18a) r r t r0 2 pb ðrÞ ¼ r1 g hc 0 lw 1 þ DH 0 t2w f ðRw Þ tw þ 1 t¯ 2 r1 r1 2 r1 þHg þ ha ð1 t¯ 2 Þ ðR¯ 1 rÞ tanða þ fÞ ; R¯ 2 prpR¯ 1 (19a)

(20a)

(2) For the case that the rainwater fills the whole deck plate (l0p1): ("

# r1 t qðrÞ ¼ r0 g h0 2 þ DH ha ð1 t¯ 2 Þ 2 R¯ 2 r0 ) r1 2 Mc r1 hc 1 t¯ þ 2 þ f ðrÞ 1 r0 r0 pR¯ 2 r0

hs ¼ h0

t 2 t¯ ðDH ha Þð1 t¯ 2 Þ h þ c 2 2 R¯

R20

(23)

1

The water head hw has the same formulation as that in Eq. (15b).

3. Load-modifying method

r r t r0 2 hs ¼ hc 0 lw 1 0 t2w f ðRw Þ tw þ 1 t¯ 2 2 r1 r1 r1 ðDH ha Þð1 t¯ 2 Þ

2 0

Eq. (21) demonstrates a uniform load applied on the deck plate in the water test condition when r1 ¼ r0. However, the net pressure q(r) is still relevant to the deflection f(r) since hc in Eq. (21) must be computed with f(r) and Eq. (9). The pressure pb(r) on the bottom plate of the pontoon in Eq. (19b) and the liquid head hs in Eq. (20b) can also be simplified respectively to " R2 t pb ðrÞ ¼ r0 g h0 02 hc þ DH t¯ 2 þ Hg 2 R¯ 1

2 þha ð1 t¯ Þ ðR¯ 1 rÞ tanða þ fÞ ; R¯ 2 prpR¯ 1 (22)

R20

Eqs. (18a)–(20a) or Eqs. (18b)–(20b) provide the basic formulations for analysis of the floating roof with FE simulation. This analysis is actually a geometrically nonlinear problem resulting from the deflection of the deck plate. Followed with a standard modeling and discretization in the FE analysis [17,18], the typical equation of solution can be established with the principle of virtual displacement in the total Lagrange formulation ðy K L þ y K NL Þa ¼ yþDy Q y F

(18b)

(24)

where KL and KNL denote the linear and nonlinear stiffness matrices, respectively; y+DyQ and yF are the applied and restoring y

y


load vectors, respectively; the superscript y denotes the time of deformation progression. Eq. (24) can be solved with an equilibrium iteration method, such as Newton–Raphson method [18], in which step-by-step loading (or loading with sub-steps) usually is necessary to obtain a satisfactorily converged solution. However, the loads of q(r) and pb(r) are coupled nonlinearly with the deformations of f(r) and f, and the above regular FE analysis faces difficulty. To solve this problem, an LMM is presented here to carry out the FE analysis. The concept of this method is based on an idea of iterative computation with the load–deformation relationship and the analysis consists of three basic sequential parts. Firstly, assume an initial condition, e.g. the undeformed condition when f0(r) ¼ 0 and f0 ¼ 0, and then the loads of q(r) and pb(r) are computed with these initial values according to Eqs. (18a) and (19a), or Eqs. (18b) and (19b), respectively. Secondly, a nonlinear FE analysis on the floating roof is carried out using the loads and conditions given above, and then solutions including f(r) and f are obtained. Lastly, the loads of q(r) and pb(r) are modified with new values of f(r) and f, and a new analysis is restarted with the modified q(r) and pb(r) until the difference between the current modified and previous values is within the error tolerance specified by the user. In the above analysis procedure, after the deformations, i.e. f(r) and f, are determined (obtained initially by assumption and then by computation), the case of rainwater distribution on the deck plate must be determined successively with Eq. (14), which is necessary for computation of loads q(r) and pb(r). This analysis can be summarized as follows: (1) construct the FE model according to the geometry of the floating roof; (2) compute Hg, DH and t with Eqs. (8) and (11), respectively; (3) assume an initial conditions, e.g. f0(r) ¼ 0 and f0 ¼ 0; (4) determine if parameter l041 or l0p1 (the case of rainwater distribution on the deck plate) with Eq. (14); (5) compute hc and ha with Eqs. (9) and (10), respectively, then determine the values of R¯ 1 , R¯ 2 and t¯ , and usually the simplifications that R¯ 1 ¼ R1 , R¯ 2 ¼ R2 and t¯ ¼ t are used; (6) if l041, i.e. the rainwater fills only part of the deck plate, compute Rw with Eq. (16), then compute tw with Eq. (11) and lw with Eq. (13a) or (13b); (7) compute net pressure on the deck plate q(r) with Eq. (18a) or (18b), and compute pressure on bottom plate of the pontoon pb(r) with Eq. (19a) or (19b); (8) load on the FE model and start a geometrically nonlinear solution; (9) modify the loads of q(r) and pb(r) with the solutions of f(r) and f according to the steps above (4)–(7); and (10) check whether the inequality |qi(r)/qi1(r)1|oe holds true, where qi(r) and qi1(r) respectively denote load in the current and previous iterative computation (iteration or modification i and i1) and e is the user-specified error tolerance; if it does, qi(r) (and also pib(r)) is the final solution; otherwise, return to step (8) and restart a new computation with the current modified loads until the inequality holds true. After q(r) and f(r) are obtained with this method, the water head hw and the liquid head hs also can be computed with Eq. (15a) or (15b) and Eq. (20a) or (20b), respectively. In each modification and iteration, the computation is based on the undeformation condition with the modified loads. Usually, the solution converges to the real value with a process of fluctuation between the maximum and minimum. For example, in the water test condition, the first modified load q(r) according to the initial

733

values that f0(r) ¼ 0 and f0 ¼ 0 has a maximum value qmax(r) which also results in a maximum deflection fmax(r), and the next modified load according to this maximum deflection will have a minimum value qmin(r). The convergence process is shown in Fig. 7.

4. Numerical examples Two typical floating roofs are presented in this section to demonstrate the applications of the proposed method. The first example is about a traditional single-deck floating roof with larger diameter [12,13], which is used to demonstrate the reliability of the present method for the analysis of general single-deck floating roofs. The other example is an experimental model of floating roof with continuous beams in both the top and the bottom plates of the pontoon [6]. This example is used to demonstrate the validity of the present method for analysis of floating roofs with 3-D complex structures or components. The operation condition and water test condition are considered in these two examples, respectively. Additionally, the case of rainwater filling only part of the deck plate is demonstrated in the first example and the case of rainwater filling the whole deck plate is demonstrated in the other example. 4.1. A traditional single-deck floating roof A traditional floating roof with a larger diameter in operation condition under a total 250 mm rainfall is examined in this example to demonstrate the application of the present method.

Fig. 7. Process of the solution convergence in water test condition.

Table 1 Parameters of the single-deck floating roof Parameter

Value

R0 R1 R2 H0 H1 H2 t t1 t2 t3, t4 a r1 Mc M E n

40,000 mm 39,600 mm 34,600 mm 300 mm 800 mm 450 mm 4.5 mm 8.0 mm 12.0 mm 4.5 mm 01 0.7 106 kg/mm3 152,000 kg 307,000 kg 210,000 MPa 0.3



The design parameters of this floating roof are listed in Table 1 [12,13], where ti (i ¼ 1,2,3,4) denotes the thickness of outer rim, inner rim, top and bottom plates of the pontoon, respectively. Since the floating roof is a rotationally periodic structure which usually is simplified to axisymmetic problem, one thirty-sixth of the whole structure is adopted as a compartment in the 3-D FE analysis, and the 3-D shell elements are adopted to simulate all the plate components in the FE model. The displacements along z-direction in the bottom of the outer rim are constrained in order to avoid the rigid body displacement in this direction, and the symmetric boundary conditions are applied to the circumferential sections of the concerned one thirty-sixth (1/36) part structure. The initial values for computation are f0(r) ¼ 0, f0 ¼ 0 and the

error tolerance for iterative load-modifying computation is 1%. The change of net pressure on the deck plate, q(r), with iteration i is plotted in Fig. 8. It can be found that the net pressure q(r) changes little after 16 iterations, and therefore the equilibrium solution of the floating roof is obtained. Fig. 9 plots the loads of the floating roof, q(r) and pb(r), in the equilibrium state. The radius of the area of rainwater filling the deck plate, Rw, is about 31,000 mm (oR2 ¼ 34,600 mm), which means that the rainwater fills only part of the deck plate. The numerical results of deflection and radial stress of the deck plate by the present method are compared with the results from other methods in Figs. 10 and 11, respectively. The deflections and radial stresses near and away from the bulkheads are both given

net pressure on deck plate, q(r) (x10-3MPa)

1.5 i=0 i=1 i=4 i=7 i=8 i=10 i=13 i=15 i=16

1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 0

5000

10000

15000 20000 radius r (mm)

25000

30000

Fig. 8. Procedures of the modified computation for net pressures on the deck plate.

Fig. 9. Loads of the floating roof.

35000


735

1600 present method (near bulkhead) present method (away from bulkhead)

deflection of the deck plate, f (mm)

1400

Milne method (Umeki and lshiwata, 1985) KOSTRAN (Umeki and lshiwata, 1985)

1200 1000 800 600 400 200 0 0

5000

10000

15000 20000 radiums, r (mm)

25000

30000

35000

Fig. 10. Deflection of the deck plate under rainfall h0 ¼ 250 mm.

pontoon results in an additional rotation of the deck plate, the deflections near the center area of the deck plate from the present numerical method are a little larger than those from the methods given by Umeki and Ishiwata [12] in which the pontoon is assumed as a rigidity.

radial stress of the deck plate, σr (MPa)

180 160

present method (near bulkhead) present method (away from bulkhead) Milne method (Umeki and lshiwata, 1985) KOSTRAN (Umeki and lshiwata, 1985)

140 120

4.2. A new type of floating roof with continuous beams

100 80 60 40 20 0 0

5000

10000

15000

20000

25000

30000

35000

radius, r (mm) Fig. 11. Radial stress of the deck plate under rainfall h0 ¼ 250 mm.

to demonstrate the present numerical results. The deflection of the deck plate near the bulkhead is nearly the same as that away from the bulkhead. However, there are different stress distributions near and away from the bulkheads, and a significant radial stress appears near the bulkhead. The results by the present method agree well with the results given by Umeki and Ishiwata [12], except for the stress in the edge area of the deck plate where the deck plate is jointed with the pontoon. It should be noted that the bending effect is neglected in the results by Umeki and Ishiwata [12] since only the membrane stress is concerned and the bulkhead is also not considered. It is known that the bending effect appears locally near the edge area of the deck plate, and there is usually a significant bending stress near the connections of deck plate and pontoon, especially the area near the bulkhead with considerable bending tendency. The effect of the bulkhead on the stress in the deck plate will be further demonstrated in the next example. Moreover, because the elastic deformation of the

In this example, an experimental model of floating roof with continuous beams is numerically simulated under rainwater loading in the water test condition, and the numerical results are compared with the experimental ones. This model is one fifth scale of the designed floating roof with continuous beams [6]. This floating roof has a pontoon with radial and annular continuous beams in both of its top and bottom plates, and the intersections of these beams are jointed by vertical ribs between the top and bottom plates, which results in a more complex structure (Fig. 12). The characteristic geometric properties and material properties of the floating roof model are listed in Table 2, where Nr, Na and Nv denote the number of radial, annular continuous beams (in both top and bottom plates of the pontoon) and vertical ribs (beam structure), respectively, all beams with L-section 10 mm 10 mm 1 mm. This model is also tested with the experimental method. Water is filled into a cylindrical container to lift up the model roof, which is used to simulate the floating roof in operation (Fig. 12a). A controlled-volume water pump is used to fill water onto the deck plate to simulate rainfall. The deflections and strains are measured by the staff gauges and the electrical resistance stain gauges, respectively, and then the stresses are computed with the strains obtained from the experiment. A load of rainfall h0 ¼ 50 mm is applied to the floating roof (i.e. R0 ¼ R1) to simulate a typical rainfall required in the design codes. Through the above process, we have obtained the experimental results in the water test condition [6]. In the numerical analysis, one-eighteenth of the floating roof, i.e. one compartment, is used for the FE model since the floating roof is a rotationally periodic structure consisting of 18 identical compartments, and the rotation angle of pontoon, f, is neglected



Fig. 13. FE mesh of the floating roof with continuous beams: (a) one-eighteenth structure (one compartment); and (b) pontoon with continuous beams (view without the top plate). Fig. 12. Model of the floating roof with continuous beams: (a) experimental model; and (b) pontoon with continuous beams (top—oblique view without the top plate).

Table 2 Parameters of the model of floating roof with continuous beams Parameter

Value

R1 R2 H0 H1 H2 t t1, t2, t3, t4 a Nr Na Nv r1 Mc M E n

4010 mm 3414 mm 57.5 mm 170 mm 140 mm 1.0 mm 1.0 mm 01 4 2 18 22 8 18 1.0 106 kg/mm3 286 kg 734 kg 200,000 MPa 0.334

because it is a very tiny value. In the FE model, the continuous beams, vertical ribs and all the plates are simulated with the 3-D beam elements and 3-D shell elements, respectively, and the meshes of the one-eighteenth structure and corresponding pontoon are shown in Fig. 13(a and b). It should be noted that the mesh of the pontoon’s top plate is not illustrated in Fig. 13b in order to demonstrate the inside structures of the pontoon. The displacement boundary conditions are similar to those in the first example. Table 3 shows the maximum deflection fmax, the equivalent deflection hc and the net pressure q(r) (constant in water test

Table 3 Procedures of the modified computation under rainfall h0 ¼ 50 mm Modification i

fmax (mm)

hc (mm)

q(r) ( 103 MPa)

0 1 2 3 4 5 6

0 55.032 48.881 49.659 49.561 49.574 49.572

0 30.210 26.770 27.206 27.151 27.158 27.157

0.27646 0.19499 0.20427 0.20309 0.20324 0.20322 0.20323

condition) in each modified procedure under rainfall h0 ¼ 50 mm. It can be found that the values of q(r) or hc in the current and previous iterations are nearly unchanged after about five iterative modifications, and these values can be treated as the final results. Table 4 lists the water head increment Dhw, the liquid head increment Dhs and the ratio C ¼ Dhw/Dhs ¼ (R1/R2)2 in different rainfall increments. The results from the present method, the approximate method [4] and the experimental method are compared, and good agreements are observed. The numerical results are more consistent with the theoretical values since the numerical model is more similar to the theoretical model. Fig. 14 plots the deflections of the deck plate under different rainfalls, i.e. h0 ¼ 50, 65 and 80 mm, and the present numerical results are compared with the experimental results. It can be found that the results from the present numerical method agree well with those from experiment. Fig. 15 shows comparison of the deflections of the deck plate for the roofs with and without bulkheads under h0 ¼ 50 mm. The deck plate is usually subjected to larger deflection for the roof without bulkheads since the bulkheads somewhat strengthen the rigidity of the pontoon.


737

Table 4 Comparison of coefficient C with different rainfall increments Parameter

Rainfall increments Dh0

Methods

50 mm-65 mm

65 mm-80 mm

50 mm-80 mm

Dhw (mm)

Present method Approximate method Experimental method

18.8 18.53 18.35

18.6 18.79 17.65

37.4 37.32 36.00

Dhs (mm)

Present method Approximate method Experimental method

13.7 13.43 11.50

13.4 13.63 13.75

27.1 27.06 25.25

C

Present method Approximate method Experimental method Theoretical method

1.372 1.380 1.596 1.38

1.388 1.378 1.284 1.38

100 radial stress of the deck plate, σr (MPa)

deflection of the decak plate, f (mm)

60

1.380 1.379 1.426 1.38

50

40

30

20 numerical results

10

experimental results

h0 = 50 mm h0 = 65 mm h0 = 80 mm

0

h0 = 50 mm h0 = 65 mm h0 = 80 mm

numerical results (near bulkhead) numerical results (away from bulkhead) numerical results (without bulkhead) experimental results (near bulkhead)

80

60

40

20

0

-20 0

500

1000

1500

2000

2500

3000

0

3500

500

1000

50

50

40

stress of the radial beams,σ (MPa)

deflection of the decak plate, f (mm)

60

40

30

20

the floating roof with bulkhead the floating roof without bulkhead

0 0

500

1000

1500

2000

2000

2500

3000

3500

Fig. 16. Radial stress of the deck plate under rainfall h0 ¼ 50 mm.

Fig. 14. Deflections of the deck plate under different rainfalls.

10

1500

radius, r (mm)

radius, r (mm)

2500

3000

3500

radius, r (mm)

numerical results (top beam) numerical results (bottom beam) experimental results (top beam) experimental results (bottom beam)

30

20

10

0

-10 3400

3500

3600

3700

3800

3900

4000

4100

radius, r (mm)

Fig. 15. Deflections of the deck plate for the roofs with and without bulkheads under rainfall h0 ¼ 50 mm.

Fig. 17. Stress distributions of the radial beams under rainfall h0 ¼ 50 mm.

The radial stress in the deck plate is shown in Fig. 15 with the present numerical results and experimental results. Both the numerical and experimental results indicate that there are significant stresses near the bulkhead at the edge of the deck plate, as shown in Fig. 16. These significant stresses, however, are

eliminated near the edge which is away from the bulkhead. The comparison of the radial stresses of the deck plate for the roof with and without bulkheads under h0 ¼ 50 mm is also shown in Fig. 16. These results demonstrate that the bulkhead has significant effects on the radial stress near the edge of the deck



stress near the edge of the deck plate. Bending stress usually appears near the connections of deck plate and pontoon, whereas the membrane stress dominates the stress states at locations away from the edge of the deck plate. The proposed method provides an effective and practical 3-D numerical solution to the design and analysis of floating roofs. The present solution can be further applied to structural analysis of floating roofs with more complicated components and load conditions, and a detailed analysis in a local structure or component of floating roofs can even be potentially conducted with the present method.

80 numerical results vertical stress annular stress

stress of the inner-rim, σ (MPa)

60

experimental results vertical stress annular stress

40

20

0

-20

Acknowledgments

-40 0

20

40

60

80

100

120

140

height, H (mm) Fig. 18. Stress of the inner rim under rainfall h0 ¼ 50 mm.

plate, which was not considered in the other methods forementioned since, wherein the bulkhead was usually neglected. Furthermore, at locations away from the edge of the deck plate, the membrane stress dominates the stress states, and the stress distributes relatively uniformly at these locations. Figs. 17 and 18 demonstrate the stress distributions in typical components of the pontoon under rainfall h0 ¼ 50 mm. The stress of the radial beams in the top and bottom plates of the pontoon is illustrated in Fig. 17. The results indicate that the radial beams in the pontoon are tensioned and there are significant stresses near the connections with the inner rim. Fig. 18 gives the vertical and annular stress distributions of the inner rim of the pontoon. The present numerical results are compared with the experimental results, and good agreements are observed, which demonstrates the present numerical method can simulate the structural problems of the floating roof with reasonable and reliable results.

5. Conclusions The LMM is proposed in this paper for analysis of floating roofs under rainwater loading. According to the equilibrium analysis of floating roofs in practical operation, the relationship between loads and deflections is derived, and two cases of the rainwater distribution on the deck plate are considered. The analyses of stress and deformation of floating roofs are developed with a geometrically nonlinear FE simulation based on the LMM. The loading magnitudes are modified with a series of iterative computations until the computational results meet the given accuracy requirements within the user-specified error tolerance in the analysis process. Numerical examples demonstrate that the proposed method is valid and reliable for analysis of floating roofs based on 3-D model. The numerical results indicate that the deck plate is usually subjected to larger deflection under rainwater loading and the bulkhead has significant effects on the radial

The research work in this paper is funded by the National Foundation for Excellent Doctoral Thesis of China (No. 200025) and the National ‘‘Tenth Five-Year’’ Key Technology Research Special Funds of China (No. 2001BA803B). The authors are grateful to Mr. Wang Fuan in Design Institute of China Petroleum Pipeline Engineering Co. for his generous help with the floating roof model data referred in the research.

References [1] de Wit J. Floating roof tanks. Engineering 1970;210:55–8. [2] Young WB. Floating roofs—their design and application. In: Proceedings of the ASME petroleum mechanical engineering conference, Los Angeles, CA, 73-Pet-44, 1973. [3] Epstein HI. Stresses and displacements for floating pan roofs. Comput Struct 1982;15(4):433–8. [4] Pan JH. Design of cylindrical metal tanks. Beijing: Petroleum Industry Press; 1984 (in Chinese). [5] Gallagher TA. Floating-roof technology advances with lessons learned from an 80-year history. Hydrocarbon Process 2003;82(9):63–7. [6] Wang FA, Sun XS. Research on the technology of the floating roof with continuous beams and dome frames. Technical Report, Design Institute of China Petroleum Pipeline Engineering Co., Langfang, Hebei Provice, China, 2004 (in Chinese). [7] ANSI/API STD 650-1988. Welded steel tanks for oil storage. Washington, DC, USA: American Petroleum Institute; 1988. [8] SH3046-92. Chinese standard specification for design of vertical steel welded storage tanks for the petroleum industry. Beijing, China: China Petroleum and Chemistry Co.; 1992 (in Chinese). [9] Mitchell GC. Analysis and stability of floating roofs. J Eng Mech Div 1973; 99(EM5):1037–52. [10] Epstein HI, Buzek JR. Stresses in floating roofs. J Struct Div 1978;104(ST5): 735–48. [11] Epstein HI, Buzek JR. Stresses in ruptured floating roofs. J Pressure Vessel Technol 1978;100(2):291–7. [12] Umeki T, Ishiwata M. Deflection and stress analyses of floating roofs under rainwater loading. Piping Eng 1985;27(5):64–9 (in Japanese). [13] Yuan S, Wang JL, Zhong HZ. Analysis of floating roofs by ODE-solver method. J Eng Mech Div 1998;124(10):1129–34. [14] Institute of Mechanics, CAS. Calculation formulas of strength and stability of oil tanks with floating roofs. Mech Pract 1982;4(2):36–40 (in Chinese). [15] Uchiyama S, Oka T, Oikawa T. The stress analysis of floating roof in the petroleum storage tank under rain-water load by FEM. J High Pressure Inst Japan 1981;19(2):81–6 (in Japanese). [16] Yoshida S. Geometrically nonlinear stress analysis of floating roofs by finite element method. Piping Eng 1983;25(8):51–7 (in Japanese). [17] Bathe KJ. Finite element procedures. Englewood Cliffs, NJ: Prentice-Hall; 1996. [18] Zienkiewicz OC, Taylor RL. The finite element method. 5th ed. Oxford: Butterworth Heinemann; 2000.

Stress and Deflection Analyses of Floating Roofs Based on a Load Modifying

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