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Journal of Constructional Steel Research 61 (2005) 786807

www.elsevier.com/locate/jcsr

Wind pressures and buckling of cylindrical steeltanks with a conical roof

G. Portela

a

, L.A. Godoy

b,

aDepartment of General Engineering, University of Puerto Rico, Mayagez 00681-9044, Puerto Rico

bCivil Infrastructure Research Center, Department of Civil Engineering and Surveying, University of

Puerto Rico, Mayagez 00681-9041, Puerto Rico

Received 5 July 2004; accepted 19 November 2004

Abstract

Tanks with a conical roof are studied in this paper under wind load, for a roof which is supported

by rafters and columns. Buckling occurs in the form of deflections in the cylindrical shell and the

buckling mode is localized in the windward region. Both bifurcation analysis and geometrically

nonlinear analysis have been performed using finite element discretizations of the structure. The

wind pressures have been obtained from wind tunnel experiments performed as part of the research,

and have been obtained for tank geometries for which information was not previously available. The

results show high imperfection sensitivity of tanks with a conical roof, and buckling loads for wind

velocities in the same order as those expected to occur in the Caribbean region.

2004 Elsevier Ltd. All rights reserved.

Keywords: Buckling; Conical roof; Finite element analysis; Imperfections; Steel tanks; Wind pressures; Wind

tunnel

Corresponding address: Department of Civil Engineering, University of Puerto Rico, Mayagez Campus,006819041 Mayagez, Puerto Rico. Tel.: +1787 265 3815; fax: +1787 833 8260.

E-mail addresses:[email protected] (G. Portela), [email protected] (L.A. Godoy).

0143-974X/$ - see front matter 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2004.11.002
http://www.elsevier.com/locate/jcsrhttp://www.elsevier.com/locate/jcsr

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G. Portela, L.A. Godoy / Journal of Constructional Steel Research 61 (2005) 786807 787

1. Introduction

This paper addresses the problem of buckling of thin-walled steel tanks with a conical

roof, and compares the results with those for open-top tanks. The buckling of suchstructures has been considered by several authors, using simple estimates of wind pressures

[7,3]. Several limitations were present in such studies, including lack of information

about pressures due to wind for the geometric configurations considered, the type of

computational strategy used to evaluate buckling (eigenvalue analysis), and the lack of

information about imperfection sensitivity.

Other authors have considered the wind pressures on tanks, including Sabransky and

Melbourne [11] and Macdonald et al. [8] for tanks with a conical roof, and Purdy et al.

[10] for tanks with a flat roof. However, the geometries considered in those studies do

not reflect the dimensions of tanks typically found in the Caribbean and the south-eastern

coast of the United States. In such locations, tank farms typically include tanks with ashallow conical, floating flat, or dome roof, and the aspect ratio (height to diameter for the

cylindrical part) ranges between 0.25 and 0.60 [13].

This paper contributes to both the information about pressures in tanks with a conical

roof, and the nonlinear behavior of the shell leading to static buckling. In a companion

paper [9], the authors investigate the buckling of tanks with a dome roof, and this paper

will show that there are significant differences in the behavior of conical and dome roof

tanks.

The wind tunnel experiments are reported in Section 2, together with experimental

results on pressure distributions. The computational model is described in Section 3,

while computational results are reported inSection 4. Simplified forms of imperfections

are considered inSection 5by means of a trigonometric representation of imperfections.

Section 6contains computer results for tanks opened at the top, in order to compare them

with conical roof tanks. The influence of thickness reductions in the buckling load is

discussed inSection 7. Finally, some conclusions are drawn inSection 8.

2. Wind tunnel experiments and results

Wind tunnel experiments were performed using the same facility and methodology as

reported in Ref. [9], and only specific features of conical roof models are included in thissection. The dimensions of the model are shown inFig. 1(a) with diameter =269.2 mm,

height of the cylinder = 115.7 mm, and elevation of the roof = 25.4 mm, thus having

height to diameter ratio H/D = 0.43 and Hr/H = 0.22, where Hris the height of the

roof. The small scale tank constructed is shown inFig. 1(b). The cylinder was made with

a PVC tube, while the conical roof was constructed using fiber glass. Pressure taps were

located in the model along lines at perpendicular directions for each test, and then the

model was rotated at intervals of 22.5 from 0 to 67.5. The wind velocity in the wind

tunnel was 19.8 m/s at a height of 116 mm, to represent a wind speed of 64 .8 m/s at 10 m

in a real situation. Other details of the experiments are similar to what was reported for the

dome roof shells [9].Fig. 2(a) shows the pressure contours obtained for the conical roof of the tank with a

ratio H:D =0.43 and inclination angle of the roof (pitch) of 10.7. Due to the symmetry

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788 G. Portela, L.A. Godoy / Journal of Constructional Steel Research 61 (2005) 786807

Fig. 1. (a) Dimensions and instrumentation of the conical roof model. (b) Cylindrical model with a conical roof.

of the results with respect to the windward meridian, the values from both sides of the

axis were averaged. Only negative values were observed on the roof, which represent

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Fig. 2. Mean pressure coefficients in model CMT3; (a) roof, (b) cylinder.

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suctions or pressures exerted in an outward direction. The maximum values were obtained

on the external region of the roof at the entrance of the airflow. At this zone, high values

are expected due to the separation of flow induced by the abrupt change in the meridian

between the wall and the roof surface. Sabransky and Melbourne [11] found values similarto those of the present study on the windward region of the roof. However, their study

was aimed at silo structures, with aspect ratios H:D =0.66 and roof inclination angle of

27 compared to H:D = 0.43 and angle of 10.7 in this study. On the other hand, the

maximum suction in [11] developed at the center of the roof, while in the present study

the maximum suction occurs close to the windward region. A similar behavior (maximum

values at the center of the roof) was also observed in the work of Macdonald et al. [8] for

silos with ratios varying from H:D = 0.5 to H:D = 1.1 and roof inclination angle of 25.

Those results illustrate the large influence of the roof inclination angle on the pressures at

the central part of the roof.

An approximate value of Cp = 0.2 was obtained for the roof on the leeward

region. The coefficients found in this region by other authors (for silo structures) were

approximately 0.5, for shells with H:D = 1.0[8] and H:D = 1.16 [11].

The pressure patterns on the cylindrical wall for conical and dome roofs do not present

very considerable changes, and the magnitudes show differences that are not significant.

Contours of pressure coefficients were plotted along the circumference of the cylinders

and are shown inFig. 2(b). The peak positive pressures were obtained on the windward

meridian, while the peak suctions were found at an angle near to 90 from windward.

For all cases considered, similar distributions were found about the windward axis. The

maximum values were measured between 50% and 90% of the height of the tank. Inaddition, the experimental pressure distribution around the cylinder of the tank was

simplified by means of the Fourier coefficients presented in Eq. (1).

Cp = 0.2055 + 0.2943 cos + 0.4897 cos 2 + 0.2624 cos 3 0.0353cos4

0.0092 cos 5 + 0.0778 cos 6 + 0.0263 cos 7 (1)

where is the angle measured from the windward meridian.

Some variations were observed between the models of conical and dome roofs in terms

of the values of the maximum positive pressure found. The tank with a conical roof had a

high pressure coefficient ofCp = +0.90; the maximum positive values at the windwardmeridian were compared with results obtained in Refs. [8,11] and are shown inFig. 3(a).

The values obtained at this region by each author do not seem to be highly dependent on

the aspect ratio of the tank. Using the values obtained by both references and the present

study, variations of positive Cp are in the range between 0.78 and 0.92. In terms of the

maximum suction found at 90 from the windward meridian and at the leeward meridian

of the cylinder (180), the tank presented values ofCp = 0.83 and 0.20, respectively.

Notice that the pressure coefficients at the top of the cylinder in the leeward meridian are

similar to those measured in the leeward region of the roof. The maximum negative values

were also compared with the results obtained by other authors, although for this case a

clear influence of the aspect ratio of the tank occurs, as shown inFig. 3(b). Notice thatan increase in the aspect ratio also increases the magnitude of the negative pressures at

angles close to 90 from windward. This behavior is also observed in Fig. 4, where the

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Fig. 3. (a) Maximum positive mean pressure coefficients around the circumference of tanks with a conical roof.

(b) Maximum negative mean pressure coefficients around the circumference of tanks with a conical roof.

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Fig. 4. Mean pressure coefficients on tanks with a conical roof obtained for different aspect ratios.

distributions around the cylinder are presented for the different aspect ratios studied in

[8,11] and in the present study.

Results obtained by Sabransky and Melbourne [11] with a ratio H:D = 0.66, show

a peak positive pressure ofCp = 0.78, slightly smaller than those found in the present

study. However, the peak suction recorded was 1.28, which is 35% higher than the value

obtained in the present study.

3. Computational analysis

The stability response of the tanks was evaluated using the pressure distributions

obtained experimentally for the rigid models discussed in Section 2. The finite element

computer program ABAQUS [1] was used to carry out the analysis. The properties of the

model representing a tank with a conical roof are presented inFig. 5and show the tapered

thickness of the wall and the roof of the tank. The bottom part of the tank had a fixed

restraint and the plates are made of carbon steel with modulus of elasticityE = 2108 kPa

and yield strength y = 2.48 105 kPa. The cylindrical part of the model has 1408

quadratic shell elements (designated by ABAQUS as S8R5), with five degrees of freedomin each of its eight nodes; while the roof has 1916 STRI65 quadratictriangular elements

also with five degrees of freedom per node.

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Fig. 5. Plate thickness at different heights of the walls and the roof of the conical roof tank with rafters.

In a typical conical roof tank, the roof is not self-supported, and a secondary structural

system is required to carry the weight of the roof. This system has steel rafters supported

by girders and columns toward the inside of the tank and the edges of the rafters are pinned

to the walls.Fig. 6presents the details of the rafters supporting the roof of the tank. The

fluid that may be stored in a tank has not been included in the analysis because it has a

stabilizing effect for buckling. Thus, an empty tank is the worst scenario for wind load

buckling.

The pressure coefficients obtained experimentally were used to define the loading

conditions of the models established. Similar to the case for tanks with dome roofs, the

pressure values are referenced to a full-scale velocity of air Vair = 64.8 m/s ( P =

2.39 kPa).

4. Computational buckling results

The computational models were loaded with pressures that represent a given percentage

(based on experimental wind tunnel pressure coefficients) of the reference wind pressure

P = 2.39 kPa at a height of 10 m, which is associated with a wind velocity of 64.8 m/s.

The buckling load factors obtained from the analyses are identified by the multipliersc,

which scale the reference pressure with the value at the critical load in the tank, assuming

an unchanged wind profile.

The conical roof tank reinforced with rafters was first analyzed by means of an

eigenvalue buckling approach, considering wind pressures and self-weight (static pre-

loads). The first critical load factor obtained had a value of 1.04, representing a wind

pressure of 2.48 kPa and wind velocity of 66 m/s. All critical modes show displacementsonly in the cylindrical shell and have circumferential waves as shown in Fig. 7. The

modes are generated in the windward region of the cylinders having three well pronounced

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Fig. 6. Arrangement of rafters and girders on the conical roof tank.

circumferential waves. The modal displacements are very small in the regions close to the

roof and to the bottom of the tank, being comparable to sine functions. The peak values

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Fig. 7. First mode shapes obtained for model CMT3.

of deflections are observed in the windward region at 50% to 80% of the height of the

cylinder, and they are similar to the peak pressure distributions observed experimentally

(Cp = 0.90).

A geometrical nonlinear step-by-step analysis using the Riks method was employedin order to study a more realistic behavior of the structure. Preliminary studies indicated

that plasticity did not develop during the early stages of buckling, so that it was not

included in the analysis. The same loads were applied to the tank using an initial static

self-weight analysis followed by the incremental method applied to the wind pressures. The

maximum deformations were observed in the cylindrical part of the tank, as seen in Fig. 8,

and are similar to the eigenvectors obtained from the bifurcation analysis (Fig. 7). Nodes

located in the windward region of the cylinder at elevations close to 80% of the height of

the cylinder revealed experimentally the maximum displacements during the incremental

loading process. The loaddisplacement curve was computed by selecting a node on the

windward meridian and in the wind direction, in order to study the fundamental and initialpostcritical equilibrium path followed by the structure.

The path of perfect geometry inFig. 9, shows a 3.4% decrease in the initial critical

load factor ofc = 1.00, in comparison to the value obtained from the eigenvalue analysis.

In addition, the fundamental path presents an almost linear behavior, with a relative small

displacement = 7.6 mm in the order of the shell thickness. The close critical loads

obtained in the two analyses and the linear aspect of the fundamental path combined

with small displacements suggest that the eigenvalue analysis is a good indicator of the

buckling behavior presented by this tank. Besides this, it can be seen from Fig. 8that the

buckling deformation at the step when the critical load was reached is similar to the one

computed from the bifurcation analysis. In terms of the postcritical behavior of the tank,the buckling response may be described as an unstable bifurcation because of the negative

slope developed by the postbuckling path. Changes in the curvature of the path, producing

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Fig. 8. Fundamental mode shape in model CMT3 at the first critical load computed from the step-by-step analysis;

(a) side view, (b) bottom view.

additional stiffness, were observed at very large displacements (larger than 60 mm), butthe increment in load was very small and certainly would be accompanied with plasticity,

softening the stiffness of the tank.

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Fig. 9. Nonlinear response of model CMT3 with initial perfect and imperfect geometries.

The nonlinear response of the imperfect tank, and the imperfection sensitivity were

also investigated. The buckling response of tanks which are open at the top with an

imperfect geometry has been previously studied in Refs. [4,6]; however similar studies

for tanks with roof have not been reported in the literature. For these imperfection

analyses, the initial geometry of the tank was modified adding the first bifurcation

mode computed for the structure, and multiplying it by an amplitude coefficient . The

similarities between the first mode shape obtained from the eigenvalue analysis and the

deformed shape adopted by the structure after reaching the first critical load justify

the use of the first mode for the shape of the imperfection, instead of a combinationbetween modes. Imperfections due to eccentricities developed during construction are

very difficult to control and to predict. It is true that these types of imperfections

would not exhibit an identical shape to the deformation observed in the first buckling

mode, but values in the order of 3.0 t have been found in practice for some steel

shells [5].

The modal displacements were transformed to real displacements by referencing them

to a percentage of the minimum shell thickness used in the cylinder. The percentages used

were 10%, 25%, 50%, 100%, and 200%, which represent maximum initial displacements

in the order of 0.79, 1.98, 3.95, 7.9, and 15.8 mm, respectively. Positive imperfections

with the same direction observed in the eigenvectors and negative ones for the opposingdirection were used in the analysis. Numerically, both positive and negative eigenvectors

are real solutions of the eigenvalue problem.

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For imperfections smaller than the shell thickness, the shell presents an unstable

postcritical behavior, while for larger imperfections the maximum in the primary path and

the minimum in the postcritical path approach each other and coalesce in the limit, so

that they form an inflection point. For imperfections larger than those considered in Fig. 9( > 2 t) the buckling loads are not reduced but the displacement increases in a stable

form.

This suggests that the postcritical behavior depends on the magnitude of the

imperfection; however, in the explanation given by the Catastrophe Theory [12], the

imperfect paths represent a family of equilibrium paths that may be unfolded from the

bifurcation behavior. Besides this, when negative imperfections are considered, they tend

to deform to the inside of the tank (negative displacements) in the same direction that the

positive wind pressures were applied to the structure in. On the other hand, when positive

imperfections are considered, the same point in the shell is deformed in the outer direction

of the tank. Although positive displacements larger than 30 mm were not plotted, at some

displacement larger than this value the structure tends to turn from positive (outward) to

negative (inward) displacements. Notice that all nonlinear curves belonging to geometric

imperfections converge to the minimum load factor obtained for the case with a perfect

geometry. However, the study of postcritical effects far from the first critical point observed

in the structure is beyond of the scope of this research.

For an imperfection of 0.10 t, the critical load is reduced by 3% in comparison to

the perfect geometry, having a value of c = 0.97. This load factor represents a wind

velocity of 63.8 m/s, which is less than the minimum design wind speed of 67 m/s

established by ASCE [2]. A reduction in displacement = 7.34 mm, which is in theorder of 0.9 t, was obtained at the first critical load in Fig. 9. For an imperfection with the

same magnitude but in the opposite direction, a smaller critical load factor ofc = 0.94

was reached, representing a 6% reduction with respect to the perfect geometry case. This

value represents a wind speed of V = 62.8 m/s and a maximum reference pressure of

P = 2.25 kPa. On the other hand, the displacements at the first critical point were in the

order of twice the thickness ( = 15.2 mm), revealing a 100% increment compared to

the perfect geometry. In summary, the critical load factors producing instability in the tank

have been only marginally reduced (by less than 10%) for a 1% geometrical imperfection,

although a stiffness softening behavior is observed in the curve.

When a larger imperfection of 0.25 t is considered, the critical load factor is reducedby 11% (c = 0.89) and 13.3% (c = 0.87) for the positive and negative imperfections,

respectively. These values represent wind velocities at 10 m height of 61 m/s for the

positive imperfection and 60 m/s for the negative one. The first buckling mode computed

is similar to the deformed mode observed for the 1% imperfection, but with larger

displacements. The maximum displacements were observed toward the inside of the tank,

and are in the order of 2.4 t and 150% higher ( =19.1 mm) than the value obtained for

the case with perfect geometry. The maximum negative displacement was in the order of

1.43 t with a value of = 11.3 mm, representing an increase of 48% with respect to the

results for the perfect geometry.

For an imperfection in the order of 0.50 t, the first buckling load was reduced by 23%(c = 0.77) for imperfections in the negative direction, and 21% (c = 0.79) in the positive

direction. The former value is in agreement with a reference pressure P = 1.85 kPa

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and wind speed V = 57 m/s, and the latter with P = 1.90 kPa and V = 57.7 m/s.

In the inward direction of the tank (where the maximum deformation was observed),

displacements in the order of 3.11 t were computed, representing an increment in the order

of 220% with respect to the case with perfect geometry. In the other direction, a lowerincrement occurred (119%) with a displacement of = 13.2 mm.

For larger imperfections (i.e. 1.0 t and 2.0 t), the response of the structure changes to a

stable path with large deflections. These are points more difficult to identify because the

previous path is also stable. The equilibrium path in these cases is stable and has stable

critical points, the transition in the shape of the nonlinear curve is smooth, and the global

stiffness changes in the structural system are more difficult to determine.

Although these curves present stable states, a smaller load capacity of the tank

throughout the path is achieved. If an arbitrary point in the curve of = 2.0 t is

selected for a displacement of 25.4 mm, then the load factor of the structure is c

= 0.28,representing a reference wind speed of 34 m/s. Therefore, for smaller wind pressures

the structure has large displacements, which eventually may lead to the global collapse

of the structure. Fortunately, the structure continues with a stable behavior until the path

converges to the lowest postcritical load reached by the tank with perfect geometry. This

maximum loading factor, c = 0.75, is associated with a wind velocity of 56.1 m/s and

represents a 25% decrease in the load capacity of the tank. These results give an idea of the

response and load capacity that would be exhibited by a previously buckled tank under new

wind loadings, with maximum permanent displacements in the order of 2.0 t (a magnitude

commonly observed in locally buckled tanks).

5. Simplifications in the analysis of the tank with a conical roof

The shape of the imperfection was assumed to have the form of the eigenvector.

However, it would be interesting to understand the influence of more localized

imperfections on the buckling behavior of the shell. To model local imperfections the

geometry was defined in terms of trigonometric functions. Three different cosine waves

forms were assigned to the initial geometric shape of the tank, with m being the number

of cosine waves used to define the geometry. Two different levels of imperfections were

used, with = 1.0 t and = 2.0 t. From the buckling results inFig. 10, it was observedthat the number of cosine functions plays an important role in the response of the structure

and that only form = 25 does a considerable reduction occur in the load capacity of the

tank.Fig. 11presents the equilibrium path for imperfect shells with the eigenmode and the

trigonometric shape discussed here. Form = 25 the equilibrium path is well represented

by the local imperfection, but for large displacements the results begin to diverge from

those using eigenvectors as imperfections.

Additionally, the influence of simplifications in the loads was considered. First, the

pressures in the cylindrical part were assumed constant in elevation, and second, the

pressures were restricted to the windward region of the tank. A bifurcation critical load

c = 1.02 was obtained for these conditions, representing a difference less than 2% incomparison to the model with variable pressures along its height. The low sensitivity to the

distribution of pressures in height has been noted before [6] for open tanks.

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Fig. 10. Influence of the wave number in the response of tanks with imperfect geometries.

6. Buckling of an open-top tank

A tank open at the top was investigated in order to compare the sensitivity to

imperfections of the tank with a conical roof previously studied. The same diameter,

cylinder height, shell thicknesses, and external wind pressures were assumed for the model.

From the results, the critical load factors ofc = 0.48 andc = 0.46 were found by means

of the eigenvalue and nonlinear analyses, respectively. These values are less than half of the

maximum loads obtained for the tank with a conical roof. Besides this, they represent windvelocities in the order of 43.9 m/s, that is 32% less than the expected design wind speed

for Puerto Rico.Fig. 12(a) presents the pattern of the first modal deformation obtained

with the linear eigenvalue approach andFig. 12(b) shows the deformation resulting from

the nonlinear analysis at the first critical point. These modes have similarities with those

obtained in the tank with a roof only along the circumferential direction. On the other hand,

it can be seen that the maximum displacements develop at the top of the cylinder instead of

at a height of 80%, as in the tank with a roof. The displacements computed in the nonlinear

step for the first critical load are 143% higher ( = 18.5 mm) than the displacements

in the conical roof tank. This increment in displacement and reduction in maximum load

suggests that the roof contributes a considerable amount of stiffness to the tank, allowing itto sustain higher lateral loads. The path for the perfect geometry inFig. 13presents the

nonlinear behavior obtained for a selected point at the top of the windward region of the

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Fig. 11. Comparison between imperfections based on the bifurcation response and trigonometric functions.

tank where maximum displacements were obtained. The first critical point is found along

a rather linear fundamental path, followed by an unstable postcritical path.

Geometric imperfections with the shape used in the tank with a conical roof were also

used for the tank without a roof. For an imperfection of0.1 t, a 1.7% reduction occurs in

the critical load (c =0.45). If the same imperfection is applied in the opposite direction

(positive imperfection), then an increment in critical load of 3.5% is obtained (c = 0.48).

For a positive imperfection, inFig. 13the node selected on the shell begins to displacein the outward global direction of the tank and then returns back to the inward direction.

This behavior suggests that the tank is trying to overcome the initial imperfection in the

direction opposite to the one that it commonly deforms in and adjust it to its most sensitive

deformed shapes (those observed in the eigenvalue and perfect nonlinear analyses).

For an initial imperfection of 0.25 t, the equilibrium path is very similar to the case

with 0.10 t, but with lower buckling load and larger displacements at the critical state.

The load pressure factor reduces to c = 0.437, representing a difference in the order

of 5% with respect to the perfect geometry. However, for imperfections in the opposite

direction the path is different, as shown inFig. 13.

The maximum buckling load of the tank is reduced by 8.9% (c = 0.418) and 5.2%(c = 0.435) for the = 0.5 t and = +0.5 t imperfect cases. Larger imperfections

present significant reductions in the buckling capacity of the tank, with values ofc = 0.38

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Fig. 12. (a) First mode shape from the eigenvalue analysis, (b) deformation obtained at the first critical load using

a step-by-step analysis.

and 0.40 (reductions of 17% and 13%) for the positive and negative imperfections of 1.0 t,respectively. For imperfections of 2.0 t, c = 0.341 (V = 37.8 m/s) for the negative

imperfection andc = 0.357 (V = 38.7 m/s) for the positive imperfection (26% and 22%

with respect to the perfect geometry).

From the results discussed before and the sensitivity curve ofFig. 14, it seems that

the conical roof tank is more sensitive to imperfections than the open-top tank with

similar dimensions. The open tank has a lower capacity than the conical roof tank, but

it seems that local instability in the open tank is not associated with high reductions in its

initial postcritical capacity. Moreover, small imperfections generate small increments in

the buckling capacity due to changes in the geometrical shape of the tank. On the other

hand, the conical roof tank has more than twice the load capacity of the open-top tankbut suffers more notable reductions when an imperfection (regardless of its direction) is

induced in the original geometry.

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Fig. 13. Nonlinear response of unroofed model CMT3 without a roof with perfect and imperfect geometries.

Table 1

Original and reduced shell thickness

Original thickness (mm) Reduced thickness (mm)

12.70 11.10

9.50 7.90

7.94 6.34

7. Influence of thickness reductions on the buckling capacity of the tank

Most tanks in the Caribbean region are located near to the coast, to reduce the distance

between the arrival of the product from the sea and its storage. Furthermore, many tanks

were constructed in the 1960s and 1970s, and show clear signs of significant corrosion in

the buckling studies. As a simplified model, a uniform thickness reduction of 1.6 mm was

introduced in the data for the shell. Certainly, not all regions of a tank suffer the same loss

of thickness due to corrosion, but this reduction is used to estimate lower limits of capacity.

The conical roof tank (CMT3) was analyzed by means of bifurcation and nonlinear

analyses. The thicknesses of the different layers of steel in the cylinder were reduced as

presented inTable 1. A bifurcation load factor ofc = 0.592 was obtained from the linearanalysis, which represents a 43% reduction with respect to the tank without the thickness

reduction. This load factor is associated with a wind speed of 49.9 m/s. From the nonlinear

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Fig. 14. Sensitivity to imperfections on the initial geometry of the conical tank and the open-top tanks.

Table 2

Critical load factors computed for different imperfection amplitudes

(t) c Pc (kPa) Vc (m/s) Reduction inc (%)

+0.1 0.543 1.33 48 5.00

0.1 0.558 1.34 48.5 2.00

+0.25 0.491 1.18 45 14.00

0.25 0.503 1.20 46 12.00

+0.5 0.433 1.04 42.7 24.000.5 0.445 1.07 43.2 22.00

results, the load capacity is c = 0.57, that is, 3% less than the bifurcation. This load

factor is related to a wind velocity of 48.9 m/s, which represents 25% of the design value

established by ASCE [2]. The mode shapes observed were similar to those of the perfect

tank, and for this reason they are not presented again.

An imperfection sensitivity analysis was performed in model CMT3 with reduced

thickness and using the imperfection amplitudes of previous analyses (0.1 t, 0.25 t,

0.5 t, 1.0 t and 2.0 t).Fig. 15shows the nonlinear equilibrium paths computed for thetank using again a representative node located in the region where higher displacements

were computed in the analysis. The results are summarized inTable 2.

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Fig. 15. Nonlinear response of model CMT3 with reduced thickness and imperfect geometries.

The reductions in the buckling load computed in model CMT3 with the original

thickness due to imperfections seem to be quite similar to those of the same model with

reduced thickness, even when the capacity of the latter is highly reduced. In the tank opened

at the top, a reduction in load capacity and in imperfection sensitivity was detected.

8. Conclusions

This paper presented contributions in two fields associated with the wind buckling of

the steel tanks: first, wind pressures have been evaluated from wind tunnel experiments,

and second, the structural responses under such pressures have been computed to evaluate

critical loads, postbuckling behavior and imperfection sensitivity. The main conclusions of

the work may be summarized as follows:

1. The aspect ratio of a tank substantially influences the pressures on the cylindrical shell.

For short tanks, with aspect ratioH:Dfrom 0.4 to 0.5, the wall pressures are lower than

for taller tanks, with aspect ratio H:Dlarger than 0.5. Previous studies in the literaturefor conical roof models are restricted to ratios betweenH:D = 0.5 and 2. Other results

in the literature are available for flat roof models [10].

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2. For shallow conical roofs, the angle of inclination of the meridian of the roof is

responsible for an increase in pressures on the central part of the roof. Thus, great care

must be exercised to decide on the rise in conical roof tanks from the point of view of

structural behavior.3. The tanks investigated displayed an initially stable equilibrium path followed by an

unstable nonlinear response in the postcritical path.

4. The critical loads of tanks are only marginally affected by the negative pressure

distributions around the tank and by the distribution of pressures along the height of the

cylinder. For tanks with a conical roof, it seems that buckling is induced by local effects

due to positive wind pressures in the windward region. This behavior was observed in

the tank with a conical roof, with only localized positive wind pressures in the windward

region, for which the bifurcation loads were similar to the case in which the pressures

were distributed all around the tank.5. Previously buckled tanks (due to wind loads or to other effects) with levels of

imperfections higher than = 0.5 t, have a very flexible behavior with large

displacements at low load levels, driving the structure rapidly to reach its maximum

load. Therefore, sustained winds, instead of three-second gusts, begin to be of concern

based on the lower load capacity observed in the nonlinear path of tanks with large

imperfections. This is observed in the 25% reduction of the buckling load computed

from the nonlinear analysis.

6. A tank with a conical roof has a larger buckling load than a similar tank without a

roof; however, the reduction in buckling load due to the influence of small geometric

imperfections is higher in tanks with a conical roof. But, even accounting for thereduction in buckling load due to imperfections, the buckling capacity of the conical

roof tank is higher than that of the shell without a roof.

7. The loss of shell thickness reduces dramatically the buckling capacity of a tank.

With a thickness reduction in the order of 1.6 mm in a tank with a conical roof and

H:D = 0.43, the wind velocity associated with the first buckling point corresponds

to 49 m/s. These reference velocities are below the ASCE [2] requirements for Puerto

Rico.

8. The roof of a tank provides additional stiffness to the structure, so that the buckling

capacity of the tank with a conical roof is increased by a factor of two with respect to

that for a tank without a roof.

References

[1] Abaqus User Manualversion 5.6. Pawtucket (RI, USA): Hibbit, Karlsson and Sorensen; 1996.

[2] ASCE 7 Standard. Minimum design loads for buildings and other structures. Reston (VA, USA): American

Society of Civil Engineers; 2002.

[3] Briassoulis D, Pecknold DA. Behavior of empty steel grain silos under wind loading: Part 2: The stiffened

conical roof shell. Engineering Structures 1987;10:5764.

[4] Flores FG, Godoy LA. Buckling of short tanks due to hurricanes. Engineering Structures 1998;20(8):75260.

[5] Godoy LA. Thin-walled structures with structural imperfections: analysis and behavior. Oxford: Pergamon

Press; 1996.

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[6] Godoy LA, Flores FG. Imperfection sensitivity to elastic buckling of wind loaded open cylindrical tanks.

Structural Engineering and Mechanics 2002;13(5):53342.

[7] Godoy LA, Mendez JC. Buckling of aboveground storage tanks with conical roof. In: Zaras J, editor. Thin

walled-structures. Oxford: Elsevier Science; 2001. p. 6618.[8] Macdonald PA, Kwok KCS, Holmes JH. Wind loads on circular storage bins, silos and tanks: Point pressure

measurements. Research report no. R529, School of Civil and Mining Engineering, University of Sydney,

Sydney, Australia; 1988.

[9] Portela G, Godoy LA. Wind pressures and buckling of cylindrical steel tanks with dome roofs. Journal of

Constructional Steel Research [in press].

[10] Purdy DM, Maher PE, Frederick D. Model studies of wind loads on flat-top cylinders. Journal of the

Structural Division, ASCE 1967;93:37995.

[11] Sabransky IJ, Melbourne WH. Design pressure distribution on circular silos with conical roofs. Journal of

Wind Engineering and Industrial Aerodynamics 1987;26:6584.

[12] Thompson JMT. Instabilities and catastrophes in science and engineering. London: John Wiley & Sons;

1982.

[13] Virella JC. Buckling of tanks subject to earthquake loadings. Ph.D. dissertation, University of Puerto Ricoat Mayagez, Puerto Rico; 2004.

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