INTRODUCTION A structural finite element analyses are performed on a double-deck roof structure. Initial analysis was carried out on a preliminary design of the structure. Few modifications are considered in order to satisfy the deformation requirement of the design. This report presents the deformation response of the finalized double-deck roof structure when subjected to different prescribed loading conditions. The corresponding stresses developed in the plate material are discussed in terms of equivalent von Mises stress. A. MODEL

The circular double-deck roof structure measures 71.54 m in diameter has been designed. It consists of top and bottom plates supported by concentric circular walls making up six radial sections. The outer section is further divided into several compartments by bulkheads. In this work, only 1/6th part of the structure is modeled due to symmetrical nature of the roof structure design. Two views of this symmetrical part are illustrated in FIGURE A1. In the top view, the arcs represent the circular walls while the radial lines are C-channels welded to the bottom of each top plate as reinforcements against lateral deformation. In the bottom view, rectangular grid lines represent welded lap joints of the steel plates forming the bottom floor. The dimensions of the model are as provided by the customer, and is reproduced in FIGURE A2 for reference. Cross-sectional dimensions of critical structural members are listed in Table A1. Table A1 – Size of critical members

Member Size Top plate 4 mm-thick Bottom plate (5’ x 20’) 6 mm-thick Inner ring wall 5 mm-thick Outermost ring wall 6 mm-thick C-channel, top plate 150 x 75 x 9 mm Lap joint 30 mm wide

The model is discretized into finite elements and nodes for structural stress analysis using the finite element method. Types of elements used in the analysis are:

Deckplate and ring - S4R (4-node doubly curved shell elements) Channel - B31 (2-node linear beam elements) Pipe post - C3D8R (8-node brick elements)

and C3D4 (4-node linear tetrahedron elements)

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Total number of elements = 122,661 Total number of nodes = 192,355

The following properties of a typical structural steel is used for the analysis:

Young’s modulus, E = 190 GPa Poisson’s ratio, v = 0.30 Shear modulus, G = 73 GPa Density, ρ = 7845 kg/m-3 = 7.845 x 10-9 tonnes/mm-3

The calculated total mass of model structure is 70.84 tonnes The boundary conditions imposed on the model are as follows: Symmetry of radial planes: Each radial plane at both sides of model is restrained such that:

(a) No displacement normal to the symmetry planes (b) Only in-plane rotational displacement is allowed for the symmetry planes.

Rigid body motion: The vertical displacement of a point (node) at the top plate is set to zero to prevent rigid body motion of the structure. Load case for empty tank: All displacement components for bases of pipe posts are zero to represent free-standing situation.

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(a)

(b) FIGURE A1 – Geometry of the 1/6th model of the double-deck roof structure: (a) top view and (b) bottom view of the model.

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FIGURE A2 – Major dimensions of the double-deck roof structure used in the analysis

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B. LOADING

Two different load cases considered in this work are listed in Table B1. The “Free standing roof” represents a field condition of empty oil tank so that the double-deck roof structure rests on the floor of the tank. The dead weight of the roof structure is supported by the legs /pipe posts. When the tank is filled with oil, the weight of the “Floating roof” structure is acted upon by the fluid pressure. This pressure is assumed to act uniformly on the outer side of the bottom plate.

Table B1 - Loading matrix for the analysis.

Load case ID

Free standing roof LC-FS Floating roof LC-FR

It is required that the maximum displacement of the deckplates does not exceed 500 mm in the “Floating roof” condition (LC-FR). C. RESULTS AND DISCUSSION Displacement field The displacement fields for the load case LC-FS and LC-FR are shown in FIGURE C1 and FIGURE C2, respectively. The ranges of maximum vertical displacement of the deckplates are summarized in Table C1. Table C1 – Maximum displacement ranges of the deckplates.

Load Case ID

Top plate (mm)

Bottom plate (mm)

LC-FS 0 to –23.5 -187.5 to –211.2 (-281.6*)

LC-FR -4.63 to –33.2 +252.5 to +281.1 (+309.6*)

* Maximum value noted in small area of the structure The top deckplate deforms downward (-ve) as a result of lateral bending due to the self-weight of the plate and the C-channels. The displacement of the top plate is smaller than the bottom plate due to added stiffness provided by the C-channels to the top plate. In the free-standing condition (LC-FS), the calculated displacement for the top plate is up to –23.5 mm. The displacement of the bottom plate ranges from –187.5 to –211.2 mm. However, a small radial region near the center of the roof structure experiences higher displacement up to 281.6 mm.

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When the roof structure is floating (LC-FR), the bottom plate is subjected to the reaction fluid pressure equivalent to the total weight of the structure. Consequently, the bottom plate curves upward (+ve) while the top plate remains curving downward due to the effect of self-weight. The top plate flexes downwards with a magnitude ranging from –4.63 to –33.2 mm. The displacement of the bottom plate ranges from +252.5 to +281.1 mm. The small radial region near the center of the structure shows higher displacement up to +309.6 mm.

The vertical displacement plot for both top and bottom plates along a typical radial section (middle section of the model) is compared in FIGURE C3. The displacement profile of each of the six concentric radial region is illustrated. Zero displacement of the bottom plate corresponds to location of the concentric wall of the roof structure.

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FIGURE C1 – Distribution of vertical displacement component for Load Case LC-FS (Free-standing structure)

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FIGURE C2 – Distribution of vertical displacement component for

Load Case LC-FR (Floating roof structure)

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LC-FS

LC-FR

-40

-30

-20

-10

0

10

20

30

0 5000 10000 15000 20000 25000 30000 35000Length (mm)

U2 (mm)

LC-FS

LC-FR

-400

-300

-200

-100

0

100

200

300

400

0 5000 10000 15000 20000 25000 30000 35000Length (mm)

U2 (mm)

FIGURE C3 – Displacement profile for a radial section of the roof structure.

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Stress distribution

The von Mises stress distribution in the bottom plate of the double-deck roof structure is shown in FIGURE C4 (a) and (b) for the load case LC-FS and LC-FR, respectively. The deformation of the bottom plate of the roof structure are largely within the elastic response of the material. In regions along the plate-to-ring wall connection, the stress ranges between 100 to 120 MPa. The yield strength of typical carbon steels ranges between 205 MPa (ASTM A283 Gr C) to 250 MPa (ASTM A36). Higher stress is predicted in the small region near the center of the structure and at the pole to plate connections (red color in FIGURE C4) due to geometric stress concentration effects. However the stress can be lowered by proper design of the joint using sleeves to increase the load bearing area.

The stress in the top plate is much lower than the yield strength of the steel plate

such that the material remains elastic throughout the loading.

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(a)

(a)

(b)

FIGURE C4 – von Mises stress distribution in the bottom plate of the roof deck structure: (a) LC-FS (Free-standing condition), (b) LS-FR (Floating roof structure)

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Reaction forces on pipe posts

The pipe post located nearest to the center of the roof structure experiences the largest reaction force with a magnitude around 57.2 kN (compressive) along the post. However, higher force is concentrated at locations around the base due to local geometric constraints as shown in Figure C5.

This axial force is small relative to the buckling load of the pipe post. Therefore, no buckling analysis is performed.

Pipepost A dimension Diameter = 76.2 mm Thickness = 12.7 mm Height = 3163 mm

FIGURE C5 – Reaction force along the critical pipe post

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CONCLUSIONS Load Case LC-FS (Free-standing, empty tank condition)

1. The top deckplate displaces downward up to -23.5 mm while the bottom plate curves downward with magnitudes in the maximum range of –187.8 to –211.2 mm.

2. Larger displacement up to –281.6 mm is predicted for a small circular region

close to the center of the roof structure.

3. The von Mises stress in the deckplates is largely elastic (less than the yield strength of structural steel at 205-250 MPa). Stress concentration occurs at the pole-to-plate connections.

Load Case LC-FR (Floating roof structure)

1. The top deckplate curves downward with magnitudes in the maximum range of –4.63 to –33.2 mm.

2. The bottom deckplate deforms upward due to the reaction from oil/fluid pressure.

The displacement shows a maximum range from 252.5 to 281.1 mm.

3. Larger displacement up to +309.6 mm is predicted for a small circular region close to the center of the roof structure.

4. The von Mises stress distribution in the deckplates is largely elastic, similar to

that for the Load Case LC-FS.

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