UNIVERSITY OF PUERTO RICO MAYAGÜEZ CAMPUS DEPARTMENT OF CIVIL ENGINEERING A REVIEW OF WIND-TUNNEL R E S U LT S O F P R E S S U R E S O N TANK MODELS L U I S A . G O D OY – P. I . G E N O C K P O R T E L A – G. S. GENERAL OVERVIEW The Caribbean Islands are heavily exposed to hurricanes due its geographical location in the Atlantic and Caribbean Seas. Damage due to wind pressure of tanks used for storage different liquids have been observed and studied by different authors (Flores and Godoy (1998), Godoy and Mendez (2001)). Some of the cases studied were found in St. Croix, St. Thomas, and Puerto Rico during Hugo (1989), Marilyn (1995), and Georges (1998), respectively. Studies of wind pressures have taken place since early in the 20th century. During this era, air flow effects were investigated by different authors (H. L. Dryden et. al., 1930,). A common problem is the variation of wind pressures depending on tank geometry. Both external wall pressure, as external roof pressure distribution is strongly dependent of the tank geometry. Detailed studies concerning to wind loads on cylinders include wind tunnel tests performed by Maher (1966) to dome-cone and dome-cylinder tanks. The height of hemisphere and spherical dome roofs mounted on cylinders with base diameters of 12 in. or 24 in., was changed to account variation of wind on the roof and wall of the shell. Pressures were measured both in the meridian and parallel axes of the cylinders and different pressure patterns were developed. Purdy et. al. (1967), studied wind pressure distributions on flat-top cylinders. The aspect ratio (h/D) was varied from short (tanks) to long (silos) cylinders to quantify the dimensional effects in pressure distribution. The cylinder diameters were 12 and 24 in., and the heights were increased from 6 in. Numerical approximations were developed based in Fourier cosines series on the surface of the flat roof diameter and along the parallel and meridian axes of the shell. More recent studies concerning to pressure variations around the shells were accomplished by Esslinger et.al. (1971), Gretler (1978), Gorenc (1986), Greiner (1998), and Pircher (1998). Studies based on wind distribution on conical roof tanks were accomplished by Sabransky et.al (1986), and MacDonald et.al (1988). Variations in aspect ratios were considered in the experimental tests and the authors give contours of different pressure coefficients found at different tank sections. CIRCUM FERENTIAL VARIATION OF WIND PRESSURE AROUND TH E SHELL ( PARALLEL AXIS) Figure 1 presents distributions of wind pressure acquired from different experimental results, and measured from the angle of incidence of the wind (windward) to one half the parallel tank axis (leeward). The tests selected include the dome – cylinder geometries and flat roof cylinders studied by Maher (1966 and 1967), silos structures analyzed by Esslinger, Ahmed, and Schroeder (1971), and cylindrical shells (mainly tanks) tested by Gretler, and Pflügel (1978). Also distributions established by the DIN and ÖNORM European codes were plotted. In terms of the circumferential variation of wind pressure, different researchers have found similar patterns independent to the structure dimensions (aspect ratio), but the magnitudes are dependant of its aspect ratio. From the figure it is noted how the maximum positive pressure coefficient (≅ 1.0 as established by codes, and ≅ 0.7 to 0.95 as found in experimental test) is exerted in the windward direction, and at 30° to 45°, it changes to produce negative (suction) values reaching experimental pressure coefficient values up to -1.75, and code requirements up to -2.5, respectively. The maximum suction values are located 80° to 90° from windward. Figure 1. External wind pressure coefficients around the circumference of cylinders. Figure obtained from Reference It has been observed from experimental data how cosine families can represent circumferential pressures on shells. For this reason most of the formulations established to define circumferential patterns of pressure employs Fourier cosine series. Figure 2 shows how the accumulation of cosine terms approximates the real shape of the external pressure distribution ( q w ). The variable “m” defines each term of the series and an increment of the angle measured from windward direction. Cm is a constant representing the contribution of each term and the amplitude of the pressure coefficient wave. The pressure _ value at a specific height ( q ) is multiplied by the external pressure coefficient represented by the expression inside the brackets. Figure 2. Numerical approximation using Fourier cosine series. Figure obtained from Reference [&&&]. In 1986, Gorenc [4] defined the coefficient of external pressure for silos structures as: C s = −0.55 + 0.25 cos φ + 0.75 cos 2φ + 0.4 cos 3φ − 0.05 cos 5φ Where, φ is the circumferential angle from windward direction. (1) According to Greiner (1998) the circumferential distribution of pressure can be defined as: C s = −0.55 + 0.25 cos φ + 1.0 cos 2φ + 0.45 cos 3φ − 0.15 cos 4φ (2) For simplified analysis of cylindrical shells Pircher (1998) established the following distribution: C s = −0.50 + 0.40 cos φ + 0.80 cos 2φ + 0.30 cos 3φ − 0.10 cos 4φ + 0.05 cos 5φ (3) Figure 3 presents circumferential variations of wind pressure around the wall of cylinders acquired from different experimental results, and measured from the angle of wind incidence to one half of the diameter. The values presented corresponds to the studies of Gorenc (1986), Greiner (1998), and Pircher (1998) previously mentioned, in addition to distributions from the ACI-ASCE Committee 334 (1991) and Rish (1967) used by Flores and Godoy (1998) in recent investigations. Other distributions have been developed for long as well as short tanks by MacDonald et.al. (1988), however their results are in good agreement with the distributions shown in Figure 1 and Figure 3. In some circumstances when a cylindrical structure contains openings, an additional uniform negative pressure is added due to internal suction generated. The values commonly adopted are: h ≥ 2 Cs = -0.8 D h ≤ 1 Cs = -0.5, D The height of the tank is h, and D represents the diameter. Intermediate values are usually linearly interpolated. Similar behavior is possible to be found in tanks with opened roof as presented in Figure 4 (Schmidt, 1998), which sometimes are reinforced with a ring stiffener at the top. Figure 5 also presents differences between the wind pressure distributions of close and open tanks (Resinger and Greiner, 1982). 1.5 External pressure coefficient 1 0.5 0 0 20 40 60 80 100 120 140 160 -0.5 -1 -1.5 -2 Circumferential angle from windward direction [degrees] Greiner (1998) ACI-ASCE Commitee 334 (1991) Pircher (1998) Gorenc (1986) Rish (1967) Figure 3. External wind pressure coefficients around the circumference of cylinders. 180 Figure 4. Envelope of total internal and external wind pressure in open tanks Figure 5. Wind pressure distribution for close (dome roof) and open tanks Flores and Godoy (1998) studied short cylindrical tanks (h/D = 0.4) that suffered buckling on its top part during hurricane Marylin occurred in 1995. The formulation presented for the circumferential distribution of wind pressure is also a Fourier series including 7 terms and defined as: 7 p = λ ∑ Ci cos(iφ ) (3) 0 Where, Ci = Coefficient of external pressure p = External wind pressure λ = Parameter used to increase the load pressure These values are plotted in Figure 6 for both sources used in order to define the coefficients of the Fourier series. Figure 6. Wall pressure distribution used by Flores and Godoy (1998) The pressure coefficients defined by the ACI-ASCE (1991) and Rish (1967) were employed in the analysis. Table 1 presents the values obtained for each case. Ci C0 ACI-ASCE (1991) 0.2765 Rish (1967) 0.3870 C1 -0.3419 -0.3380 C2 -0.5418 -0.5330 C3 -0.3872 -0.4710 C4 -0.0525 -0.1660 C5 -0.0771 0.0660 C6 0.0039 0.0550 C7 -0.0341 N/A Table 1. Pressure coefficients used by Flores and Godoy. Other parameters will affect the wind pressure distribution along the parallel axis of the tank’s shell. For example, Megson, Harrop and Miller (1987), studied the effect produced by the Reynolds number in wind pressure variations. This effect can be observed in Figure 7, where Reynolds numbers less than a critical value of 2 x 105 reduce dramatically the wind pressure coefficients. This effect should be considered when experimental tests are conducted in wind tunnels. Figure 7. Effect of Reynolds number on pressure coefficients VARIATIO N OF WIND PRESSURE ALONG TH E HEIGHT (M ERIDIAN AXIS) During years a common practice have been to assume a constant pressure distribution along the height of the tanks, in order to simplify computations. Although, a constant distribution seems to be acceptable for some special geometries it is not true for many practical cases. On the other hand, american structural codes as ASCE-7, and UBC-97 use power models to define pressure variations. Figure 8 presents the velocity pressure coefficient distribution along the height according to these codes and assuming an exposure D (unobstructed areas exposed to wind flowing over open water for a distance of at least 1 mile). Flores and Godoy (1998) in their study used three different vertical wind pressure variations including a constant unit pressure distribution, the pattern established by the ASCE – 1995 (qh using exposure D) until a pressure coefficient value of 1.12 at roof level, and a linear pressure coefficient variation until a value of 1.03 (at a height of 4.6 m) followed by the latter distribution. Figure 9 shows the three assumed distributions. Maher (1966) found variable distributions in the vertical shell direction of dome-cones and domecylinders. The height of the hemisphere and spherical dome roofs mounted on the cylinders and cones with base diameters of 12 in. or 24 in. was changed to account for the variation of wind pressure at the wall and the roof. Purdy, Maher and Frederick (1967), studied wind pressure distributions on flat-top cylinders also varying the aspect ratio (h/D) to quantify the dimensional effects in pressure distribution. They developed numerical formulations for pressure distributions around the circumference of the walls, along the height of the wall and on the surface of flat roofs varying its roughness coefficient. Vertical Distribution of Velocity Pressure Coefficient Height above ground level [ft] 200 160 120 80 40 0 0 0.5 1 1.5 2 2.5 Velocity Pressure Coefficient ASCE-7 (1993) UBC (1997) Figure 8. Vertical distribution of velocity pressure Figure 9. Variation of wind pressure coefficients along the height of the tank The distributions found by Maher (1966) in the windward direction (shell) of cylinders with hemisphere roofs are presented in Figure 10 and Figure 11, showing certain pressure variations in height. However, unlike to the results found using flat roofs (Purdy (1967)), at this point ( φ =0), the pressures at the top level of the wall are not reduced. On the other hand, it can be observed that maximum suction (80º to 90º) would be assumed constant along the height. Figure 12 and Figure 13 present the results obtained on the wall of spherical dome roof cylinders. The maximum positive pressure is obtained in the windward direction presenting small variations up to a height of 0.6 h, approximately, where pressure values increase and subsequently decrease close to the top level. The maximum suction presents small variations in height, for which constant distributions would be a good approximation. Figure 14 and Figure 15 shows circumferential and height distributions of wind pressures for flat roof on short tanks with different roughness coefficients. The experimental tests reveal constant values of pressure (or at least poor variation) in the windward direction until a height of 0.45 h to 0.5 h, approximately (Purdy et. al. (1967)). The range in values was attributed to differences in the roughness coefficient of the roof. After this point (height) the value increase and eventually decrease again near to the top level of the tank. At the circumferential angle producing the maximum negative (suction) pressure (80º to 90º) it would be acceptable to consider a constant pressure for all heights. Although, again there is some decrease at the top level of the tank. The fact that a region close to the upper part of the tank’s wall receives greater pressure than the rest of the wall, it would evidence the buckling modes observed at top level of tanks without roof. Sabransky (1987), and MacDonald (1988) provide information regarding to variation of pressure along the height of tanks supporting conical roofs. Figure 16 shows circumferential variations for different heights of a cylinder with aspect ratio of 1, according to MacDonald. Similar results were found for lower aspect ratios (h/D = 0.5). From the graph it can be observed how both the maximum positive and maximum negative pressures occur at a vertical distance representing the 57% to 81% of the total height. PRESSURE DISTRIB UTION ON ROOFS Distribution of wind pressures on roofs was also included by Maher (1966) in his study of domecylinder and dome-cone shapes, but numerical approximations were not developed. From Figure 10 and Figure 11 it can be noted that positive and negative pressure are developed on hemisphere roofs. The positive values are in the windward region and are the same to those found in the top level of the windward wall. The maximum negative pressure is located at the center of the roof. The magnitude of the coefficient of pressure is greater to the one found in the maximum point of suction on the wall. Figure 12 and Figure 13 show the pressure distribution on spherical roofs supported by cylindrical tanks. The distribution is different to that observed on hemispherical roofs, being negative all pressure coefficients values. The maximum negative pressure occurs in the windward region being greater than the pressure found in the wall. Pressures are reduced toward the leeward direction. Figure 10. Pressure coefficient on tank supporting hemisphere roof (A.R. = 6/24) Figure 11. Pressure coefficient on tank supporting hemisphere roof (A.R. = 3/12) Figure 12. Pressure coefficient on tank supporting spherical roof Figure 13. Pressure coefficient on tank supporting spherical roof Figure 14. Pressure coefficient on tank supporting flat roof (plywood) Figure 15. Pressure coefficient on tank supporting flat roof (plexiglass) Purdy, Maher and Frederick (1967) developed formulations for pressure distributions across the diameter of flat roofs, and established numerical formulations depending on the roughness coefficient of the roof. The pressure coefficient equation developed for an aspect ratio of 0.25 is: 2 3 2 3 Cp := −.0640 + .1586 ρ − 1.3949 ρ + .7567 ρ + ( −.0105 + .5347 ρ − 3.0931 ρ + 2.1036 ρ ) cos ( .01745329252 θ ) 2 3 2 3 + ( −.0044 + .3706 ρ − 2.0012 ρ + 1.6364 ρ ) cos ( .03490658504 θ ) + ( .0043 − .0492 ρ − .1964 ρ + .2471 ρ ) cos ( .05235987758 θ ) 2 3 2 3 + ( .0042 − .0896 ρ + .1293 ρ + .0124 ρ ) cos ( .06981317008 θ ) + ( .0017 − .0634 ρ + .1446 ρ − .0265 ρ ) cos ( .08726646262 θ ) 2 3 2 3 + ( .0019 − .0964 ρ + .3493 ρ − .2735 ρ ) cos ( .1047197551 θ ) + ( .0003 − .0555 ρ + .1865 ρ − .1489 ρ ) cos ( .1221730477 θ ) 2 3 2 3 + ( −.0003 + .0083 ρ − .0398 ρ + .0369 ρ ) cos ( .1396263402 θ ) + ( .0005 − .0052 ρ + .0073 ρ − .0023 ρ ) cos ( .1570796327 θ ) (4) Figure 14 and Figure 15 show contours of the pressure distribution characterized by equation (4). As observed, only suction is developed, and the values are reduced toward the leeward direction. Figure 17 presents a diagram showing different variations of pressure coefficient across the diameter of flat roofs. In the figure are considered different aspect ratios involving tanks and silos. It is observed how for aspect ratio greater than 1.5 the results converge, while for smaller ratios exists some variations. Esslinger et.al. (1971), proposed distributions of pressure on conical roof sustained by silos, but for tanks this topic is already undefined. However, Godoy and Mendez (2001) used the distributions defined by Esslinger (1971) for analyze the buckling mode shape and pressure of a buckled tank. The results acquired seem to have good correlation with the evidence revealed in the tank after hurricane effects. Esslinger found negative pressures on the windward region of the roof and positive in the leeward region. At the center of the roof the values are zero, until a point between the center and the edge of the roof. On the other hand, Sabransky (1987) found a different distribution as depicted in Figure 18 and Figure 19 for aspect ratios of 1.16 and 0.66, respectively. The vertical axis represents the pressure coefficient and the horizontal axis represents the fraction distance from the windward point to the leeward along the roof. Contrarily to Esslinger (1971), the maximum suction is found at the windward point and at the middle region of the roof. Figure 16. Circumferential variations for different heights presented by MacDonald Figure 17. Pressure coefficients across diameter for different geometry configurations Sabransky, and Melbourne (1987) Pitch = 27°, H/D = 1.16, Re = 1 x 10^5 -1.3 -1.3 Pressure Coefficient [Cp] -1.1 -0.9 -0.7 -0.5 -0.5 -1.1 -1.1 -0.9 -0.7 -0.9 -0.7 -0.5 Along Windward Line 0.05 0.07 0.08 0.10 0.17 0.32 0.36 0.40 0.42 0.43 0.61 0.67 0.85 0.74 0.93 1.00 X _____ Dìam. Figure 18. Pressure coefficient distributions on conical roofs presented by Sabransky and Melbourne (1987). h/D = 1.16, pitch = 27º Sabransky, and Melbourne (1987) Pitch = 27°, H/D = 0.66, Re = 1.5 x 10^5 Pressure Coefficient [Cp] -1.1 -0.9 -0.7 -0.5 -1.3 -0.9 -0.7 -1.3 -1.1 -0.9 -1.1 -0.7 -0.5 -0.5 -0.5 Along Windward Line 0.00 0.05 0.07 0.10 0.34 0.37 0.41 0.43 0.45 0.61 0.63 0.67 0.80 1.00 X _____ Dìam. 0.65 Figure 19. Pressure coefficient distributions on conical roofs presented by Sabransky and Melbourne (1987). h/D = 0.66, pitch = 27º Similar results were found by MacDonald et.al. (1988) for an aspect ratio of 0.5, as shown in Figure 20. A 25º pitch was used quite similar to the 27º used by Sabransky. More data points were used to measure pressure on the tank roof especially at the middle part, where a –1.6 peak suction pressure was acquired. McDonald, Kwok, and Holmes (1988) Pitch = 25°, H/D = 0.5, 1, 2, Re = 2 x 10^5 Pressure Coefficient [Cp] -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 Along Windward Line X _____ 0.03 0.12 0.05 0.07 0.08 0.27 0.35 0.43 0.45 0.48 0.55 0.61 0.65 0.72 0.78 0.90 1.00 Dìam. 0.53 0.51 Figure 20. Pressure coefficient distributions on conical roofs presented by MacDonald, Kwok, and Holmes (1988). h/D = 0.5, 1, 2, and pitch = 25º GROUP EFFECTS Esslinger, Ahmed and Schroeder (1971) studied wind pressure distributions of two silos closely spaced. Figure 21 shows the pressure patterns found along the height and the spherical cap roofs. Three different wind directions were established as cases A, B, and C. Case A CASE A Case B CASE B Case C CASE C Figure 21. Wind pressure distribution on a group of two closely spaced silos. Shielding effects are observed in case A, where the first tank is receiving pressure distributions similar to a single tank. In case B, both tanks are experimenting considerable wind pressures, but these are differently distributed. The worst scenario is observed in case C. REFERENCES [1]. Flores F. G. and Godoy L. A. (1998), “Buckling of short tanks due to hurricanes”. Engineering Structures. Vol. 20, No. 8. [2]. Godoy L. A. and Mendez J. (2001), “Buckling of above ground storage tanks with conical roof”. Third International Conference on Thin Walled Structures. Elsevier Science Ltd., pp. 661-668. [3]. Dryden, H., and Hill, G. C. (1930), “Wind pressure on circular cylinders and chimneys”. Research Paper No. 221, National Bureau of Standards, U.S. Department of Commerce, Washington D.C. [4]. Maher, F.J. (1966), “Wind loads on dome-cylinders and dome-cone shapes”. Journal of Structural Division, ASCE, Vol. 91, No. ST3, Proc. paper 4383. [5]. Purdy D. M., Maher P. E. and Frederick D. (1967), “Model studies of wind loads on flat-top cylinders”. Journal of Structural Division, ASCE. [6]. Esslinger M., Ahmed S. and Schroeder H. (1971), “Stationary wind loads of open topped and rooftopped cylindrical silos (in German). Der Stalbau, 1-8. [7]. Gretler W. and Pflügel, M. (1978), “Wind tunnel tests of cylindrical shells at the TU Graz”. Unpublished. [8]. Gorenc, B. E., Hogan, T. J. and Rotter, J.M. (1986), “Guidelines for the assessment of loads on bulk solid containers”. Institution of engineers. Australia. [9]. Greiner R. (1998), Cylindrical shells: wind loading. Chapter 17 in: Silos (Ed. C, J. Brown & L. Nilssen), EFN Spon, London, pp. 378-399. [10]. Pircher M., Guggenberger W., Greiner R., Bridge R. (1998), “Stresses in elastic cylindrical shells under wind load”. University of Western Sydney, Nepean, pp. 663-669. [11]. Sabransky I. J., Melbourne W. H., (1987), “Design Pressure Distribution on Circular Silos with Conical Roofs”. Journal of Wind Engineering and Industrial Aerodynamics, 26, pp. 65-84. Elsevier Science Publishers B.V. Amsterdam. [12]. Macdonald P. A.,Kwok K. C. S., and Holmes J. H. (1988), “Wind loads on circular storage bins, silos and tanks: I. Point pressure measurements on isolated structures”. Journal of Wind Engineering and Industrial Aerodynamics, 31, pp. 165-188. Elsevier Science Publishers B.V. Amsterdam. [13]. ACI-ASCE Committee 334 (1991), Reinforced concrete cooling tower shells-practice and commentary, ACI 334,2R,91. American Concrete Institute, New York. [14]. Rish, R. F. (1967), “Forces in cylindrical shells due to wind”, in: Proc. Inst. Civil Engineers. Vol. 36, pp. 791-803. [15]. Scmidt H., Binder B., and Lange H. (1998) “Postbuckling strength design of open thin walled cylindrical tanks under wind load”. First International Conference on Thin Walled Structures. Elsevier Science Ltd., pp. 203-220. [16]. Resinger F., and Greiner R. (1982), “Buckling of wind-loaded cylindrical shells-application to unstiffened and ring-stiffened steel tanks”, in Buckling of shells, Ramn E. (ed.), Springer, Berlin, pp. 217281. [17]. Megson T. H. G., Harrop J. and Miller M. N. (1987), “The Stability of Large Diameter Thin Walled Steel Tanks Subjected to Wind Loading”. Proceedings of an International Colloquium on Stability of Plate and Shell Structures, Ghent University, Belgium. [18]. Uniform Building Code (1997), Structural Engineering Design Provisions. Volume 2. International Conference of Building Officials.

© Copyright 2023 Paperzz