Wind Pressures and Buckling of Cylindrical Steel1

Journal of Constructional Steel Research 61 (2005) 808–824

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Wind pressures and buckling of cylindrical steeltanks with a dome roof

G. Portelaa, L.A. Godoyb,∗

aDepartment of General Engineering, University of Puerto Rico, Mayagüez 00681-9044, Puerto RicobCivil Infrastructure Research Center, Department of Civil Engineering and Surveying,

University of Puerto Rico, Mayagüez 00681-9041, Puerto Rico

Received 5 July 2004; accepted 19 November 2004

Abstract

An experimental/computational strategy is used in this paper to evaluate the buckling behaviorof steel tanks with a dome roof under exposure to wind. First, wind tunnel experiments using smallscale rigid models were carried out, from which pressure distributions due to wind on the cylindricalpart and on the roof were obtained. Second, a computational model of the structure (using thepressures obtained in the experiments) was used to evaluate buckling loads and modes and to studythe imperfection sensitivity of the tanks. The computational tools used were bifurcation bucklinganalysis (eigenvalue analysis) and geometrical nonlinear analysis (step-by-step incremental analysis).Geometric imperfections and changes in the buckling results due to reductions in the thickness werealso included in the study to investigate reductions in the buckling strength of the shell. For thegeometries considered, the results show low imperfection sensitivity of the tanks and buckling loadsassociated with wind speeds 45% higher than those specified by the ASCE 7-02 standard.© 2004 Elsevier Ltd. All rights reserved.

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

∗ Corresponding author. 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.2004.11.001

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G. Portela, L.A. Godoy / Journal of Constructional Steel Research 61 (2005) 808–824 809

1. Introduction

Short above-ground steel tanks are constructed in industrial plants to store oil,petroleum, fuel, kerosene, water, and other fluids. Because such tanks are thin walledstructures, buckling under wind loads is a major concern for the designer. This paperreports buckling results for tanks with shallow and deep dome roofs, while tanks with aconical roof are addressed in a companion paper [17].

Damage and collapse of short tanks have been reported in the Caribbean Region afterhurricanes Hugo in 1989, Marilyn in 1995, and Georges in 1998 [5,7]. Failures of tankshave many consequences, including loss of the structure, environmental problems due toloss of the contents, and economic problems because the storage capacity of the plant isreduced. Research in this field started in the 1960s with evaluations of critical loads viaeigenvalue analysis for simple tank models, open at the top and with uniform thickness[4,11,12,19]. Some authors noticed that for tanks with a roof there was a lack ofinformation about the actual pressures that should be used to represent wind. In the UnitedStates, ASCE [2] and Uniform Building Code [20] indicate pressures for the design oftank-like structures under exposure to wind, but the recommendations handle tanks in thecategory of “other structures” and provide extremely simplified wind pressures which areconstant in the circumferential direction. Other national building codes have adopted non-uniform wind pressure distributions, but they cover geometric ranges for tanks taller thanthose considered in this paper.

Research regarding wind pressure distributions in silos has been reported by Esslingeret al. [3], Gretler and Pflügel [10], and Pircher et al. [15]. For the present interest, the mainlimitation of such valuable studies is that mainly tall structures have been investigated in thepast, notably with relations between height and diameter larger than 1. Wind tunnel testson shorter structures were performed by Maher [13] on cylindrical tanks with sphericalroofs, and by Purdy et al. [18] on cylinders with a flat roof. For short tanks with heightto diameter ratio smaller than 1, as it would be in most tanks in the Caribbean region,the information on wind pressures is extremely scarce. This motivated new wind tunnelexperiments carried out by the authors, which are reported in the following sections.

The strategy in the research reported in this paper has been experimental/computational,in which wind tunnel experiments were carried out on rigid models to obtainpressure distributions, and the pressures were next used in a finite element model toevaluate buckling loads, buckling modes, and imperfection sensitivity. Studies concerningimperfection sensitivity of tanks subjected to wind loads have been reported in Refs. [6,9].Miller [ 14] investigated the influence of initial imperfections on the buckling of cylindersunder uniform external pressure and developed the basis for the ASME code.

2. Experimental set-up used to evaluate wind pressures in tank models

There have been a surprisingly small number of wind tunnel experiments performedon cantilever cylindrical shells with a dome roof, as evidenced by a review paper byGreiner [8]. The first set of studies seems to be that due to Maher [13] in the 1960s,while a second study in the following decade is due to Esslinger and co-workers [3].

810 G. Portela, L.A. Godoy / Journal of Constructional Steel Research 61 (2005) 808–824

Fig. 1. Schematic side view ofthe wind tunnel laboratory.

Both investigations were mainly interested in geometries which represent silo structures,with H/D > 1.

This motivated research to evaluate wind pressures on empty tanks using wind tunnelmodeling. The geometries of the tanks of interest were identified following field trips toindustrial plants in Puerto Rico [21]. One of the conclusions of the survey was that morethan 80% of the tanks with a dome roof have aspect ratios between 0.25 and 0.60. Onthe basis of that observation, it was decided to investigate pressures on two tank modelshaving H/D = 0.48, one with a shallow dome roof(Hr/H = 0.318) and the other onewith a deepdome roof(Hr/H = 0.475), whereHr is the rise of the roof at its center. Thedimensions considered for the small scale models in the experiments wereH = 115.5 mm,D = 238.1 mm, Hr = 36.7 mm (shallow dome), andHr = 54.9 mm (deep dome). Asa reference, Maher [13] studied models withH/D = 0.25 (Hr/H = 0.39 and 0.68)and H/D = 0.5 (hemispherical roof with Hr/H = 1.0). The study of Maher [13]for spherical and hemispherical roofs used different wind velocity profiles and differentheights of reference with respect to the present investigation, which makes it difficult tomake comparisonswith the pressures reported in this paper.

The relative dimensions chosen for the models were influenced by the typicalgeometries found in real tanks, and also by limitations of the experimental facility availableat the University of Puerto Rico at Mayagüez (UPRM), such as the dimensions of the windtunnel, longitudinal roughness length, wind profile, and the need to avoid blockage effects.

The wind tunnel laboratory at UPRM is 12 m long. The tunnel was constructed in 1985and has an open circuit, classified as a short tunnel because the experimental section issmaller than 5 m. A side view of the tunnel is shown inFig. 1. Full details of the windtunnel and the experiments are given in Ref. [16] andonly a brief summary is provided inthis section.

The dynamic pressures at different points of the models were referenced to pressurecoefficients defined as

Cp = PL

Pd(1)

where PL is the local dynamic pressure at a point, andPd is the reference dynamicpressure at an equivalent height of 10 m [16]. The wind pressure distributions on the tankswere measured assuming rigid models and simulating an atmospheric surface layer for anopen-terrain aerodynamic roughness condition based on a full-scale value of 0.02 m, asrecommended by Wieringa [22]. The wind velocity profile, shown inFig. 2, was simulated


Fig. 2. Wind velocity profile used for experiments. Logarithmic model. Experimental profile.

Fig. 3. (a) Model with a shallow dome roof. (b) Model with a deep dome roof.

using the logarithmic law

V (z) = 1

κu∗ ln

(z

z0

)(2)

whereV is the wind speed,z is the elevation from the surface terrain,z0 is theaerodynamicroughness length,κ is the von Karman constant (taken as 0.4), andu∗ is the frictionvelocity.

The models were made of PVC, and discontinuities in the surface were removed withthe application of a resin mix. The shell was rubbed, painted, and polished to get a finalsmooth external surface as shown inFig. 3. Twenty-one pressure taps were placed on theroof (five for each meridian at 90◦ and one at the center), and twenty were located on thecylinder.

The wind velocity in the wind tunnel at a height of 116 mm was 19.8 m/s, and representsa full-scale wind speed of 64.8 m/s at a height of 10 m. This is the design wind speed inPuerto Rico, the Caribbean region, and the south-eastern coast of United States, accordingto ASCE [2]. Temperature, humidity, and barometric pressure were monitored during the


Fig. 4. Angles at which the models wererotated in order to measure pressures.

tests in order to identify changes in the air density which could affect the local pressuresmeasured in the wind tunnel.

The models were instrumented with pressure taps separated at angles of 90◦. However,in order to account for local pressures on the roof and the cylinder, the tank was rotatedfrom 0◦ to 67.5◦ at intervals of 22.5◦, as shown inFig. 4. Thedata acquired per instrumentwas the average of 800 samples in a period of 10 s. All recorded values were correcteddue to environmental factors using psychometric relations. The experiments were repeatedseveral timesand three tests were selected to average the results.

3. Wind tunnel results

Results for pressure contours are shown inFig. 5for the shallow dome(Hr/D = 0.15)and inFig. 6 for the deep dome tanks(Hr/D = 0.23). Dome roofs have a variable slope,and the transition between the cylinder and the spherical cap is not so pronounced as itwould be, for example, in tanks with a conical or flat roof.

Due to the symmetry of the pressures measured on the roof with respect to the windwardmeridian, the values on either side of the axis were averaged. The highest pressures on theroof were measured at the center, and there was a 30% increase in the dome roof withrespect to the shallow dome roof. At the junction between the cylinder and the dome,the pressures were lower than the values at the center. This is similar to the pressurepattern found by Maher [13] in his study on hemispherical roofs. An approximate value ofCp = −0.15 was measured at the leeward region for both tanks. Maher [13] obtained avalue ofCp = −0.20 for hemispherical roofs withH/D = 0.5.


Fig. 5. Mean pressure coefficients in model CMT1;(a) roof, (b) cylinder.

Unlike pressure distributions found on the roof of the tanks, the cylindrical wall patternsdo not present very considerable changes between shallow and deep dome configurations,and only the magnitudes have differences. Contours with pressure coefficients wereplotted along the circumference of the cylinders and are shown inFigs. 5(b) and6(b).


Fig. 6. Mean pressure coefficients in model CMT2;(a) roof, (b) cylinder.

The peak pressures (positive values) were obtained on the windward meridian, while thepeak suctions (negative values) were found at an angle near 90◦. For both shallow anddeep cases, relatively similar distributions were found about the windward meridian. Themaximum values on the cylinder were measured at elevations between 0.5 H and 0.9 H ,referenced from the base.


Fig. 7. Plate thickness and dimensions of models CMT1 and CMT2.

The pressure coefficients found for each of the three experimental tests around the wallwere next fitted with Fourier series. Seven coefficients provided a good approximationof the distribution, and dispersion was found only near to the end of the period of thefunction, where higher participation factors begin to contribute. The coefficients proposedin this paper for approximating the pressure distribution at different angles around thecircumference for tests CMT1 and CMT2 are

Cp = −0.1925+ 0.3708 cosθ + 0.5612 cos2θ + 0.2297 cos3θ − 0.0263 cos4θ− 0.0168 cos5θ − 0.0062 cos6θ − 0.0288 cos7θ (1)

Cp = −0.1917+ 0.3161 cosθ + 0.5001 cos2θ + 0.2475 cos3θ − 0.0404 cos4θ− 0.0114 cos5θ + 0.0599 cos6θ + 0.0266 cos7θ (2)

whereθ is measured from the windward meridian.

4. Computational buckling analysis

The pressure patterns obtained experimentally from the rigid models were used in acomputational analysis. The finite element package ABAQUS [1] was used to model tankswith geometric properties similarly to the scale-reduced models tested in the wind tunnel.The geometries of the tanks are shown inFig. 7. In all computational models, the materialwas assumed as elastic and isotropic, with modulus of elasticityE = 2 × 108 kPa, yieldstrengthσy = 2.48× 105 kPa, Poisson ratioν = 0.3, and mass densityρ = 7850 kg/m3.The diameter was 20.12 m and the height of the cylinder was taken as 9.66 m in bothcases, but the shallow dome had roof heightHr = 2.41 m, while the deep dome roof wasmodeled with Hr = 4.60 m. The boundary conditions at the bottom of the tanks wereassumed fixed.

“P” and “H” type convergence analyses were carried out in order to select the typeof element and the mesh density to be used inthe static buckling analysis. Linear and


quadratic shell elements provided by ABAQUS were tested, but it was decided to usequadratic elements for a faster convergence in the solution.

The quadratic rectangular shell element designated by ABAQUS as S8R5 was used tomodel the cylinder, while the quadratic triangular element STRI65 was used to model theroof. Both elements use five degrees of freedom per node: the displacements in each of thethree spatial directions and two rotations with respect to the in-plane axis. The S8R5 is adoubly curved thin shell element with reduced integration and eight nodes and the STRI65is a six-node triangular element.

The pressures due to fluids stored were notincluded in the models because empty tanksare the worst case scenario under wind pressures. All the pressure values are referencedto the average air properties measured during the experiment. The air density used wasρair = 1.14 kg/m3 and the reference velocity of air wasVair = 64.8 m/s, correspondingto a reference pressure ofP = 2.39 kPa.

The computational models were loaded with pressures that represent a given percentageof the reference wind pressure, based on experimental wind tunnel pressure coefficients.The buckling load factors(λ) multiply the reference pressure such thatλc is a critical loadof the tank for a wind profile that is assumed constant during the load process.

5. Computational buckling results

The computer models are designated as CMT1 for the tank with a shallow dome roofand CMT2 for the tank with a deep dome roof. The lowest critical pressures obtainedfrom the eigenvalue analysis areλc = 2.83 for CMT1 andλc = 2.91 for CMT2, whichare associated with a critical wind velocity in the order of 110 m/s. Those are highvalues; however, a bifurcation analysis represents an upper limit of the load reached bya structural system. Furthermore, bifurcation buckling does not provide information aboutthe postbuckling behavior of the structure and its imperfection sensitivity. The roof ofthe tanks is rigid enough to sustain the wind pressures developed on the tank with a lowthicknessof 9.5 mm, as shown inFig. 8. From theseresults it seems that the roof shape(either shallow or deep dome) does not significantly influence the critical load of the tanks.

Figs. 9 and 10 show the nonlinear responses of the models CMT1 and CMT2, foran ideal or “perfect geometry” and for imperfect configurations. Load–displacementcurves were constructed using a node in the cylindrical shell representative of themaximum displacements experienced by the tank. The node where maximum deflectionswere computed in the wind direction was used as the degree of freedom to plotload–displacement curves. Positive displacements are related to outward directions, whilenegative displacements are directed toward the inside of the tank.

Geometric imperfections were assumed with the same shape as the buckling modeobtained in the eigenvalue analysis, witha maximumamplitude identified by the scalarparameterζ . Imperfection amplitudesζ = 0.1 t, 0.25 t, 0.5 t, 1.0 t , and 2.0 t , wereconsidered in the analysis, wheret is the smallest shell thickness in the cylinder. For thecase of the tank with perfect geometry, limit point load factors ofλc = 2.51 (CMT1) andλc = 2.48 (CMT2) were obtained, representing a difference with respect to the eigenvalueresults in the order of 12% (shallow dome) and 15% (deep dome). These load factors are


Fig. 8. Modes of deformation measured using an eigenvalue analysis; (a) CMT1 and (b) CMT2.

Fig. 9. Geometric nonlinear equilibrium path computed for perfect and imperfect geometries of model CMT1.

associated with wind velocities close toV = 102 m/s. These are high values in comparisonwith the design wind speed specified by ASCE [2].

Fig. 11 depicts the deformations of the models at the critical state. The initial bucklingmodes reached by both models show additional vertical deformations at the region betweenthe cylinder and the roof. The vertical displacements in the cylinder are ten times smallerthan the radial displacements. The modes of deformation observed in the cylinder aresimilar to those found in the eigenvalue analysis, but more circumferential waves develop.

The deformation shapes as well as the magnitudes of positive and negativeimperfections with values up to 0.25 t were practically the same for shallow anddeep domes. Small negative imperfections (−0.1 t to −0.25 t) tend to stiffen theload–displacement behavior of the tank, but the behavior is still very similar to that ofthe ideal tank. In other words, the equilibrium paths show that the tanks overcome the


Fig. 10. Geometric nonlinear equilibrium path computed for perfect and imperfect geometries of model CMT2.

Fig. 11. Deformed shape at the first critical load; (a) CMT1 and (b) CMT2.

initial negative imperfection without abruptchanges either in shape or in direction of thedisplacement. The low sensitivity to imperfections is also noted in the small reductionin the critical loads reached forζ = 0 to ζ = ±0.25 t . Table 1shows that the largestdifference is less than 2.8% for CMT1 and 2.02% for CMT2, with respect to the idealgeometry.

For higher imperfection amplitudes, the modes presented different shapes dependingon whether they were related to positive or negative imperfections. For model CMT1 andζ = −0.5 t , the reduction in load capacity was also negligible with a factorλc = 2.48,which represents 1.2% of the value in the ideal tank. But when the imperfection was appliedin the same direction as the eigenvector(ζ = +0.5 t), the loadcapacity was reduced by


Table 1Critical load values computed for different imperfection levels

Imperfection CMT1 CMT2Load factor(λc) Difference (%) Load factor(λc) Difference (%)

−0.10 t 2.54 −1.20 2.49 −0.40+0.10 t 2.49 0.80 2.47 0.40−0.25 t 2.56 −1.99 2.49 −0.40+0.25 t 2.44 2.79 2.43 2.02−0.50 t 2.48 1.20 2.44 1.61+0.50 t 2.32 7.57 2.32 6.45−1.0 t 2.18 13.15 – –+1.0 t 2.12 15.54 – –−2.0 t – – – –+2.0 t – – – –

A negative sign means that the load factor obtained using the corresponding imperfection is higher than the ideal.Blanks represent that a critical load was not identified in the analysis.

Fig. 12. Deformed shape computed in model CMT1 for an imperfectionζ = +2.0 t .

almost 8%. Larger positive imperfections (ζ = +1.0 t and ζ = +2.0 t) also presentsimilar deformed shapes, but even when the maximum buckling load is reduced by morethan15%, this value represents a wind speed 45% higher than the ASCE 7-02 design windspeed. FromFig. 12 (ζ = +2.0 t), it may be seen that for the selected region in thecylinder, the deformed shape at the windward meridian cannot be defined by a simple sinefunction. On the other hand, higher negative amplitudes of imperfect geometries producedmode shapes similar to those observed for imperfections with smaller magnitudes.

For model CMT2, the mode shapes at the first critical load reached forζ = ±0.5 werevery similar to those computed for small imperfections, but the load was reduced by 6.5%,as shown inTable 1. Critical points could not be identified for imperfections ofζ = ±1.0and ζ = ±2.0, because the load–displacement curve showed a tendency to converge


Fig. 13. Deformed shape computedin model CMT2 for imperfectionsζ = +2.0 t andζ = −2.0 t .

towards the lowest postbuckling capacity reached by the perfect tank. For imperfectionsof ζ = −1.0 t andζ = −2.0 t , positive displacements occur in the initial equilibrium pathfollowed by the tank (Fig. 10). Combinations of vertical waves in the meridian directionwereobserved for imperfections withζ = ±1.0 t and higher, especially forζ = ±2.0 t ,as shown inFig. 13.

For imperfections with maximum amplitude smaller than 1.0 t , the shell presents anunstable postcritical behavior. For larger imperfections, the maximum in the primary pathand the minimum in the postcritical path approach each other and they coalesce in the limit,so an inflection pointis obtained. For imperfections larger than those considered inFigs. 9and10 (ζ > 2 t) the buckling loads are not reducedbut the displacements increase in astableform.


Fig. 14. Imperfection sensitivity computed in models CMT1 and CMT2.

In general, it seems that the buckling load factors of models CMT1 and CMT2show little sensitivity to small imperfections (Fig. 14). This behavior is characteristicof limit points found in the equilibrium path ofa structure. Furthermore, due to thedifferences in critical load factors computed by eigenvalue and nonlinear analyses,the shape of the load–displacement paths, and the low sensitivity with respect togeometrical imperfections, it seems that this case displays an unstable limit pointbehavior.

6. Influence of thickness reduction on the buckling loads of tanks

The computational models CMT1 and CMT2 were also analyzed considering areduction in the effective thickness of the shell. For these cases, the thickness of thecylinder was reduced tot = 6.34 mm and the roof tot = 7.9 mm. For thetank CMT1with a shallow dome roof, a value ofλc = 1.59 results from a bifurcation analysis withreduced thickness,while for a nonlinear analysis the limit load wasλc = 1.43, whichis 10% lower than the bifurcation value. For the deep dome tank CMT2, the bifurcationload wasλc = 1.64 and the nonlinear maximum load wasλc = 1.39. An imperfectionsensitivity analysis was not repeated for tanks with thickness reduction, but it seems thatthe tank still has a limit point behavior.

If a lower limit capacity of the tanks is identified with the lowest load in the unstablepostcritical path (Fig. 15), then the capacity ofthe tanks is reduced toλc = 1.11 for bothmodels. This value represents a velocity ofV = 68.3 m/s at an elevation of 10 m. Suchhigh buckling capacity explains why such tanks with a dome roof located in Puerto Ricowere not damaged following hurricane Georges in 1998.

7. Conclusions

The main conclusions of this research may be summarized as follows:


Fig. 15. Equilibrium path including the effects of thickness reductions in models CMT1 and CMT2.

1. The wind pressures on the cylindrical part of a tank with a dome roof are positive onthe windward area (leading to a maximumCp = 0.87), and negative at about 90◦from the windward meridian (maximum negative pressuresCp = −0.83). Negativepressures are also detected on the leeward area, withCp = −0.15. ASCE [2] suggeststhat pressures on a cylindrical structure would be constant in the circumferentialdirection. As shown inSection 3this is clearly inadequate for tank structures, inwhich there are changes in values and even in the sign of the pressures around thecircumference.

The details of the geometrictransition between the cylindrical body and the roof arecrucial in the evaluation of pressures on the roof, since this transition changes the mainfeatures of the flow separation. Smooth transitions, such as in dome configurations, leadto lower pressure levels than abrupt transitions, such as in shallow conical roofs [17].Therefore, results from one case cannot be freely applied to another case with a differenttransition.

2. The tanks investigated in this paper displayed an initially stable equilibrium pathfollowed by an unstable nonlinear postcritical path. The postbuckling equilibriumpaths computed with both the perfect geometry of the tank and the case withinitial geometric imperfections converge for large displacements; however, for suchlarge displacements the structure may display plasticity, which was neglected in thisresearch. Tanks with dome roofs show smallimperfection sensitivity in bucklingloads.

3. The influence of the roof on the buckling response is manifested in two opposingways: on the one hand, there is an additional stiffness provided by the roof, andon the other hand, there is an additional surface of the structure on which suctionsoccurs. For the dimensions considered in the cases of studies, the tanks can sustainhigh wind pressures before buckling, and would have a stable response for the moststringent ASCE provisions [2]. Reductions in the shell thickness reduce the bucklingloads, but these loads are still for wind velocities higher than those required by ASCE.


The computational results seem to confirm the high buckling strength of tanks withdome roofs, for which buckling was not observed after the main hurricanes in theCaribbean islands.

Acknowledgements

The authors thank Dr. Raúl E. Zapata for his valuable contribution to the wind tunnelexperiments. This research was supported by grants from FEMA (PR00660-A) and fromNSF (CMS-9907440).

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Wind Pressures and Buckling of Cylindrical Steel1