compvtas a smc1n?es Vol. il. pp. 349-353 Pcrgamon Press Ltd., 1980. Printed in Great Britain

FLOATING ROOF ANALYSIS AND DESIGN USING MIN~COMPUTERS~

HOWARD I. EPSTEINS Department of Civil Engineering, University of Connecticut, Storrs, CT 06268, U.S.A.

(Received 26 October 1978; received for publication 3 April 1979)

A~~-~oati~ roofs are used in oil storage tanks to reduce evaporation and hand&g losses, decrease corrosion, and reduce the fire hazard. A pontoon roof is used in this paper. It consists of a circular centrai plating attached at the edge to a compartmented; buoyant ring. The primary design Ioadings for this roof are due to accumulated rainwater or a punctured deck. The design loads typically cause the deck to deflect several hundred times its thickness and consequently the deck is usually designed and analyzed as a membrane.

The complicated nature of the loading on the membrane together with the unusual boundary conditions require numerical integration of the coupled, non-linear differential equations which govern the behavior. A skilled analyst is needed to vary the starting parameters necessary for the integration. This paper describes the procedure that should be followed in the analysis and design of the roofs, and shows why an interactive computer is so impo~nt in the understanding of the forces in this structure and in the design necessary to transmit them.

NOMENCLATURE

pontoon cross section area lengths defined in Fig. 1 central water depth Youngs modulus edge oil head edge water head oil head given by HO -H, horizontal pontoon stiffness membrane radial force/unit iength nondimension~ radial force (eqn 4) pressure on membrane nondimensional pressure (eqn 4) volume of rainwater edge oil pressure radius to centroid of pontoon radius to element of membrane nondimension membrane radius (r/d) thickness of memb~ne radial dispiacement of membrane element vertical displacement of membrane element nondimensional vertical displacement (w/t) vertical displacement of membrane at r = 0 edge rotation (see Fig. 1) horizontal edge displacement (see Fig. 1) Poissons ratio oil density rainwater density radial membrane central stress radial membrane edge stress pontoon rotation

INTROLWCTION

Minicomputer have been used in structural design offices to assist with problem setups, to check the input to programs run on large computers, and in some cases, to solve entire problems. Minicomputers and time-shared systems are both gaining in popularity because of their relatively low cost. 30th have certain advantages, and users must weigh their relative merits before investing in one of the systems. Some tirms are pu~hasing mini- computers to take part of the burden off of their time-

tpresented at the American Society of Civil Engineers Con- vention and Exposition, Chicago, Illinois, K-20 October 1978.

SAssociate Professor.

sharing facility. The role of minicomputers in structural analysis and design is changing rapidly; they are being used for problems of ever-increasing complexity If, 23. As a result, many firms are ou~owing ~uipment pur- chased only a few years ago[3].

The particular application of an interactive minicom- puter to the analysis and design of pontoon floating roofs is presented in this paper. Floating roofs are installed in oil storage tanks primarily to reduce evaporation and h~dling losses, to decrease corrosion, and to reduce the fire hazard. Boating roofs may be of the pan or double deck type as well as the pontoon roofs considered herein[4-6]. The pontoon floating roof consists of a circular central plating attached at the edge to a com- partmented buoyant ring (pontoon).

Over the years, the pontoon roof has experienced structural problems as evidenced by several roof buck- ling and sinking faihrres. These failures are usually due to either the accumulation of excessive rainwater on the deck or the product leaking through punctures of the deck or pontoon compartments. The design of these roofs is typically governed by the forces produced by one of these loadings.

In recent pubIications, the author and J. Buzek have investigated the stresses and degections in pontoon floating roofs as caused by the accumulation of rainwater [7] or due to the product leaking through punc- tures onto the deck[8]. For typical geometries, these loadings cause deflections in the central plating that are so large when compared with the plate thickness that bending is negligible and the plating is analyzed and designed as a membrane. The radial membrane forces are reacted by the pontoon ring. The ring is pulled inward and, therefore, compressed tangentially. The ring is also twisted and, hence, is subjected to bending stres- ses normal to the cross section[9]. The local and overall stability of the pontoon ring have aiso been considered [lo].

The complicated nature of the loading on the mem- brane together with the unusual boundary conditions requires numerical integration of the coupled, non-linear differential equations which govern the behavior. A skilled analyst is needed to vary the starting parameters

349

350 HOWARD I. EPSTEIN

necessary for the integration. This paper desoribes the procedure that should be followed in the analysis and design of these roofs, and shows why an interactive computer is so important in the understanding of the forces present and in the design necessary to transmit these forces.

EQUILIBRIUM EQUATIONS

The basic parameters for the membrane/pontoon problem are illustrated in Fig. 1. The overall geometry with the membrane filled with rainwater is shown in Fig. l(a). Values for HO, H, and w, are unknown until the final equilibrium position for this system is determined. Note that a negative value for H, is possible as this would represent a partially filled membrane. The mem- brane is assumed to be pinned to the pontoon at point E. The membrane force is transmitted to the pontoon at point E causing a horizontal displacement, 8, and a rotation, I$.

Membrane equations The membrane geometry is shown in Fig. l(b). At any

radius, r, the vertical downward displacement relative to the edge is defined as w, and the outward radial displace- ment as u. The radial stress in the membrane at the center (r = 0) is defined as a, and at the edge (r = d) the stress is defined as crc. The rotation of the membrane edge is given by a and the thickness of the membrane is t. The lateral pressure applied to the membrane at any point below the water surface is given by

P = (w + H&L -(w + H&x, (1)

where p,,, and p0 are the rainwater and oil densities, respectively, and where a positive pressure is downward. Equilibrium of an element of the membrane leads to two coupled, first-order, nonlinear, differential equations in terms of the vertical displacement, w, and the radial force per unit length, N,. Mitchell[lll presented the differential equations for the large deflection of a cir- cular, symmetrically loaded membrane[l2, 131 by assuming bending terms to be negligibly small. These equations are

I

Fig. 1. Geometry of membrane/pontoon configuration.

(3)

where

and where E = Youngs modulus. Solutions to the membrane equations may be accom-

plished by using the Runge-Kutta integration method[7, II]. The equations are integrated from the center (i = 0) to the edge (P = 1) in small increments, A?. In order to start the integration, the center tension, a,, the depth of water at the center, 0, and the oil head, AH, must all be spectied. Basically, the overall solution to the problem is accomplished by systematically varying these parameters until all the conditions imposed on the problem are satisfied.

The radial displacement of the membrane, u, is not necessary in the integration procedure. However, it is necessary to keep track of this displacement in order to establish the horizontal movement of the edge of the membrane, 8. The resulting equation to find 8 is[7]

6=tj)[;;($!~-(1-vz)&]di (5)

where v = Poissons ratio. This equation is solved numerically as the integration of eqns (1) and (2) is being accomplished.

The volume of water contained in the membrane, Q, may also be found numerically as the integration is proceeding. For a full membrane, this volume is given by PI

Q=2?rd2 [Dtw-w.)idr I

(6) 0

The upper limit in the integration changes when the membrane is only partially full.

Pontoon equations The pontoon geometry is shown in Fig. l(c). A hollow

trapezoidal cross section of uniform thickness, g, is used in this presentation. The interior of the ring-shaped pon- toon is divided into a number of compartments by solid radial ribs. The cross-sectional dimensions of the pon- toon are usually chosen so that buoyancy is insured even with two compartments ruptured[l4], and so that the design rainfall can be contained. These restraints can be satisfied with a variety of cross section con6gurations. However, large vertical dimensions for the pontoon would require excessive freeboard allowance. Also, it is desirable to slope the bottom of the pontoon to diiect any vapors formed under the roof away from the rim space and to slope the top of the pontoon to direct any rainfall toward the center. All these considerations naturally lead to a wide, rather shallow trapezoidal cross section.

If the pontoon is assumed not to rotate when loaded only with its own weight, the center of rotation cor- responds to the center of gravity (point 0 in the figure) and is located at a radius R from the center of the membrane. When the radial membrane force is applied to the pontoon, the tendency to rotate is resisted by the oil pressure along the bottom and the restoring moment in

Floating roof analysis and design using minicomputers

the pontoon itself. Eliminating the rotation from the vertical and rotational equilibrium equations leads to 17, 111

qc = UJ C, sin a t G cos a) + Csu. sin a cos a

c, + c5f.7. cos a (7)

where q. is the oil pressure at point E and where C, to CJ are functions of the pontoon geometry, the material properties of the pontoon and the oil density.

The horizontal component of the radial membrane tensile force per unit length is given by o,f cos a. This produces an inward displacement of the pontoon

S= u.tR2 cos a

AE (8)

where A = the cross-sectional area of the pontoon.

Possible equilibrium positions Values are required for the oil head, center water

depth, and center tension in order to start the numerical integration in the membrane. When the step-by-step in- tegration for particular starting values is carried to the edge of the membrane, o, a and q. are determined. There are three conditions that must be satisfied to insure that the solution thus obtained is the correct one: First, the horizontal movement of the edge of the mem- brane numerically found from eqn (5), must equal that of the pontoon as calculated from eqn (8). Second, the water contained in the membrane, as calculated from eqn (6), must be equal to a predetermined amount, and; third, the edge quantities, cr., a, and q. found by the in- tegration must satisfy eqn (7). Since the edge stress is contained in eqn (8), the pontoon is providing a horizon- tal stiffness to the edge of the membrane. If a stifYness, k, is defined as the force per unit circumferential length required for a unit radial displacement, eqn (8) gives

AE k=F

Thus, the matching of eqns (5) and (8) is assuring horizontal compatibility, while the matching of q* as given by eqn (7) is assuring vertical compatibility at the interface.

Effect of varying the initial parameters Each of the three parameters for starting the in-

tegration has its own effect on the final shape of the membrane and the position, slope and stress at the edge. It has been shown that a systematic variation of these three parameters can lead to the final equilibrium position[7]. For given values of D and AH, the higher the value for k, as given by eqn (9), the larger is the required center tension. Therefore, different horizontal stiffnesses can be matched by changing the center tension and hence horizontal compatibility can be assured.

The effects of varying D and AH can be seen in Figs. 2 and 3. It is fist assumed that AH is given and D is allowd to vary. Typical results are shown in Fig. 2 where the deflected shape of the membrane is plotted for a given AH and varying central water depths, D, > DZ > . . . > 4. For each depth, the central tension is adjusted to give a specific stillness at the edge. Ds (referred to as the floating depth) is the depth at which the oil pressure

I I 1 I I I 0 0.2 a4 0.6 0.6 1.0

T

Fig. 2. Typical deflected shapes for the membrane.

Fig. 3. Effects of changes in membrane parameters.

from below equals the water pressure from above. Below that depth, the net pressure is downward and therefore the curvature of the membrane is upward. The reverse is true above that depth, and, therefore, when the central depth is close to the floating depth, the deflected shape of the membrane tends to oscillate about the floating depth.

For ranges of central depth, a plot showing the edge oil pressure, q.. vs edge rotation, a, is given in Fig. 3(a). Large central depths give large q* and a. As the central depth decreases to the floating depth, the curve spirals inward to the crossed point representing the oil pressure at the floating level (zero edge rotation). A further decreasing of the depth produces another branch of the curve which spirals outward until zero central depth is reached.

If the oil head. AH, is increased, the floating depth is also increased, and the plot in Fig. 3(a) is shifted upward by a uniform amount (as partially shown by the upper dashed curve). Similarly, decreasing the oil head lowers the curve, except that sufficiently decreasing the oil head results in cases of partially filled membranes (the mem- brane extends above the water surface) and this portion of the curve cannot be obtained by simply shifting the plot downward. As D increases further, some of the partially filled cases result in membranes which project above the oil surfacl (qe CO). This condition is physic- ally not possible.

For a given AH, as D varies, a q. vs a spiral is generated as shown in Fig. 3(a). The volume of contained water, Q, also changes as D varies, as can be seen in Fig. 2. Plots of constant Q curves are shown in Fig. 3(b). For a particular volume Q, each point along the correspond-

352 HOWARD I. EPSTEIN

ing curve represents a possible equilibrium con&ration. If the center tension has been adjusted to satisfy horizontal compatibility with the pontoon, the vertical and force compatibility and, hence, the unique equili- brium position can be found by determining the configuration which also satisfies eqn (7).

SOLUTION PROCEDURE

A flow chart of the overall process which is used in the analysis and design of floating roofs is shown in Fig. 4. Seven basic steps are identified in the process.

The input necessary in Step 1 is: membrane and pon- toon geometry and material properties; the densities of the supporting and contained liquids, and; the conditions for which the system must be designed. Step 2 selects the starting parameters necessary for the numerical in- tegration. Step 3 accomplishes the numerical integration of eqns (2) and (3) as well as (5) and (6).

If, in Step 4, eqns (5) and (8) give identical results, horizontal compatibility is assured, and the solution can proceed; if not, the center tension is adjusted until the numbers converge. At this point it should be noted that some combinations of D and AH do not result in pos- sible equilibrium positions[7] and for these, the process must be restarted.

Once Step 4 is passed, one point on a constant AH spiral (see Fig. 3a) has been found. In Step 5, if the volume of contained liquid, as calculated from eqn (6), is not what it should be, more of the spiral is generated until one point on the correct constant Q curve (see Fig. 3b) is generated. If this point also satisfies eqn (7) in Step 6, vertical and force compatibility is assured, and the unique equilibrium position has been found; if not, a new AH is assumed and the process is repeated.

The final equilibrium position obtained by this process can be used in finding the forces and stresses in the membrane and pontoon[7,11]. It has been shown that typical floating roof pontoons are subjected to consider- able axial stresses considering the geometry of the cross

r eff ei

t

Fig. 4. Flow chart.

section. This introduces a final complication into the process of analysis and design of pontoon floating roofs and that is the presence of local buckling of the pontoon cross section as an active factor. If local buckling is a consideration, only a fraction of the cross-sectional area can be considered to be effective. This area is a function of the axial forces on the pontoon cross section which, in turn, is a function of the effective area.

In Step 7, the equilibrium position found in Step 6 must be examined in light of the stresses found and the local and overall buckling criteria. If the design is in- adequate, or the effective area calculated is not in ac- cordance with that which was assumed in finding the equilibrium position and associated forces, the process must be repeated. The effect of local and overall buck- ling of the pontoon and the iterative solution procedure to be followed when these factors are present have been investigated [ IO].

THE USE OF AN INTJ%RACTIVE COMfUTER It is possible to write a computer program to ac-

complish all the steps shown in the flow chart in Fig. 4. However, there are many difficulties which must be resolved in order to write such a program including, but not limited to:

0 In which directions should the starting parameters be incremented?

0 What step sixes should be used? l How and when should these step sixes be changed? 0 How to recognize inadmissible starting parameters. 0 What to do when the design is inadequate. 0 What constitutes a good design?

There are several reasons why the writing of such a program has not been undertaken and these deal mainly with the last two areas mentioned above.

The problem is ideally suited for solution on an inter- active computer. Once the effects of varying the starting parameters of the integration (as shown in Fig. 3) are understood, finding an equilibrium position for any given input data (including effective area) can be accomplished routinely. After a little experience, efficient step sixes and directions, which are needed to quickly solve the problem, can usually be found.

In going through the steps necessary to find an equili- brium position, the interactive computer gives insight into the effect of the starting parameters, the displaced conIiguration of the membrane and pontoon, and the way in which the internal forces are transmitted. This insight enables the engineer to thoughtfully answer questions such as:

0 What is the importance of the location of the attachment point between the membrane and the pon- toon?

0 Is it necessary to consider intermediate equilibrium states, or just the final position, when the decking is being filled with water or the product is leaking through punctures onto the deck?

0 Is the so-called sag-full condition (Hw = 0) a reasonable water level to use as a design condition?

0 What effect will changing the geometry of the pon- toon have on the stresses and displacements?

I ocai@~ O.K. The engineer becomes an integral part of the solution

process by determining what should be changed if the design proves inadequate and in estimating what the effects of these changes will be in the next analysis sequence. It is the interaction with the analysis process that gives the engineer the feel for the structure. This

Floating roof analysis and design using minicomputers 353

is a very important part of the process and it is the main reason that an interactive solution procedure is so vital in solving the problem.

Various pontoon floating roofs have been analyzed on an interactive time-sharing terminal and on a desk-top minicomputer. It doesnt make much difference what system is used as long as it possesses an interactive mode and solves each iteration in a reasonable time period since several hundred iterations may be necessary to analyze the roof.

REFERENCES

I. 0. 0. Storaasli, On the role of minicomputers in structural design. Comoul. Strucfures 7. 117-123 (1977).

2. J. A: Swans& Use of mini-computersfor iarge scale struc- tural analysis programs. Comput. Structures 7, 291-294 (1977).

3. II. W. Haeseker and C. S. Hodge, West coast consulting firm gets larger minicomputer to keep pace with growth. Civil Engineering-ASCE, Apr. 60-65 (1978).

4. J. de Wit, Floating roof tanks. Engng 210,55-58 (July 1970). 5. The storage of volatile liquids. Tech. Bull. No. 20. Chicago

Bridge and Iron Co., Plainfield, Illinois (1947).

6. W. B. Young, Floating roofs-their design and application. ASME Petroleum Mech. Engng Con/. Los Angeles, Califor- nia (16-20 Sept. 1973) (Preprint 73-Pet-44).

7. H. I. Epstein and J. R. Buzek, Stresses in floating roofs. J. Srrucr. Div.. ASCE 104.735-748 (1978).

8. H. 1. Epstein and J. R.Buzek, Sires& in ruptured floating roofs. .I. Pressure Vessel Technology, Trans. ASME 100, 29l-2% (1978).

9. J. F. Harvey, Pressure Vessel Design. Van Nostrand, Prin- ceton, New Jersey (1%3).

10. H. I. Epstein and J. R. Buzek, Design of pontoons for floating roofs. ASMEICSME Pressure Vessels and Piping Conf. Montreal, Canada, (25-30 June 1978 (Preprint 78-PVP- Ill).

11. G. C. Mitchell, Analysis and stability of floating roofs. J. Engng Mech. Div., ASCE 99, 1037-1052 (1973).

12. E. H. Mansfield, The Bending and Stretching of Plates. MacMillan, New York (1964).

13. T. von Karman, Festigkeitsprobleme in mashinenbrau. Encyklopadie der mathematishen Wissenschaften IV, 349 (1910).

14. Welded Steel Tanks for Oil Storage (5th Edn.). American Petroleum Institute Standard 650 (1973).

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