Journal of Constructional Steel Research 96 (2014) 81–94

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Journal of Constructional Steel Research

Experimental investigations on the stability of stiffened cylindrical shellsof steel silos

Eugeniusz Hotała, Łukasz Skotny ⁎Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

⁎ Corresponding author. Wrocław University of TechnoWrocław, Poland.

E-mail address: lukasz.skotny@pwr.wroc.pl (Ł Skotny)

0143-974X/$ – see front matter © 2014 Elsevier Ltd. All rhttp://dx.doi.org/10.1016/j.jcsr.2014.01.009

a b s t r a c t

a r t i c l e i n f o

Article history:Received 25 December 2012Accepted 3 January 2014Available online 8 February 2014

Keywords:Shell stabilityStiffened shellDiscrete supportSteel siloExperimental investigations

Owing to technological reasons the silos are frequently supported in a discrete way, usually by column heads.This kind of support causes a significant accumulation of compression meridional stresses in a cylindrical shellin support regions, which may lead to local buckling. The problem of determining shell resistance over the sup-port region for different shapes of ribs constitutes a current research issue. One of highly preferred methods ofstrengthening this zone is the use of short ribs interconnected with a circumferential ring. Both the results oftests on the resistance of such shells and numerical analysis allowing to determine the course of this researchare presented in this study. It has been demonstrated that the global resistance of the stiffened shells supporteddiscretely is always much smaller from that in similar shells supported uniformly around the perimeter.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Cylindrical walls of steel silos are often supported on columns' heads(Figs. 1, 2) or on the ceiling beams in production buildings. In such casesthe silo walls are longitudinally compressed as cylindrical shells dis-cretely supported on columns along the width of s0 (Figs. 2, 3).

The vertical reaction of the single discrete support N1 (Fig. 3a) withan arbitrary width s0 causes meridional compression stress σx01 deter-mined by formula (1) at x = 0, which is much higher than the stressσx0 obtained for a shell uniformly supported around the perimeter(Eq. (2)):

σ x01 ¼ N1

s0 � tð1Þ

σ x0 ¼ n � N1

2π � r � t : ð2Þ

The meridional compression stresses σx01 for shell's column sup-ports with number n ≥ 3 are several times larger than stresses σx0

along the bearing edge (x = 0) for a uniformly supported shell. Thelocal instability of the unstiffened cylindrical shell in the discrete sup-port regions (Fig. 2a) is studied in interalia [1–4]. The strengthening ofthe supporting zone due to buckling is often executed by short ribswith the length L1 (Fig. 2b). The resistance of such shells has been testedby Komman and Pasternak [5–7].

logy, Ul. Skrzydlata 1/7, 54-129

.

ights reserved.

It is often assumed that the use of transition ring over the ribs withlength L1 N r (Fig. 2c) provides a uniform distribution of meridionalstresses σx around the entire perimeter just above the ring. One mightthink that the ribbed, bottom portion of the shell wall could act as arigid apron referred to in EN 1993-4-1:2007 + AC:2008 (p. 5.4.2(1b))[8], and that is supposed to be able to provide a state of meridionalstress, such as in the case of the shell uniformly supported on thecircumference of the silo, although the actual support is realized bymeans of columns. This provision in the standard for the role of a cir-cumferential apron in the dispersion of stress gradient from the supportreaction N1 (Fig. 3a) is not correct, which can be shown by any simplenumerical analysis of the state of stress in this zone.

The authors have done a series of experimental tests on ribbed cylin-drical shell walls supported discretely showing clearly that short ribsconnected by a transition ring (Figs. 1, 2) are not able to provide the uni-form distribution of meridional stresses σx in the cylindrical silo shell,supported discretely by columns, and that compression silo shell canbuckle in the zone placed over the short ribs.

2. Methods of the analysis of stability of cylindrical steel shell wallsin the region of discrete supports

Samuelson and Eggwertz [9] have proved clearly, that the mainstream of meridional compression stress σx01 (Fig. 3a) in the cylindricalshell supported discretely on bearingswith thewidth s1 is not dispersedat an angleα= 45°, as it is taken in planarmembers according to Saint–Venant principle. Hence, some authors [1,4] have considered that safevalue of stresses in analyzed resistance of shells is equal to the maxi-mum of meridional stresses σx = σx01 (Fig. 3a) occurring at the bottom

Fig. 1. Examples of stiffened, column supported steel silos with the use of transition ring over short ribs.

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edge of the shell. The design value of meridional stresses σx,Ed at thelevel x = 0, determined on the basis of the modified formula (Eq. (1))should be less than the value of design critical stressesσx,Rd, determinedin accordance with current standards.

Knödel andUmmenhoffer [3] have proposed a less conservativewayof determining the value of stresses σx,Ed at the level x correspondingto the half of height of the buckling wave, assuming that the angleof dispersion of the stress stream up to the half height of the wave isα = 75° (Fig. 3a). Another approach in the analysis of the resistanceof shell walls, column supported has been applied by Hotała [2], who

Fig. 2. Cylindrical shells of steel silos supported by columns: a) witho

proposed the determination of the critical reaction N1,Rd from the dis-crete support, as compared with the design value of reaction N1,Ed. Hehas also calibrated the buckling curve needed to determine the valueN1,Rd. Similarly, the analysis of the resistance of cylindrical, column sup-ported shells has been proposed by Komman [5], who determined thecritical reactionN1,Rd in the case of short ribs used over the column sup-ports (Fig. 2b). Komman and Pasternak [5–7] have clearly shown thatthe use of short ribs has a beneficial influence on the resistance ofdiscretely supported cylindrical shell, and that the basic shape of buck-ling for this system occurs as displacement of the upper edges of ribs

ut ribs, b) with short ribs, c) with short ribs and transition ring.

Fig. 3. Distribution of stress σx in a cylindrical shell over the column support with the width so and reaction N1: a) simplified model, b) stress contours σx from FEM analysis.

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towards the inside of the shell (Fig. 4b). Owing the form of buckling theuse of intermediate rings appears to be justified in order to prevent thiseffect.

The strong concentration of stress σx,Ed is observed along the axesof columns supporting the cylindrical shell. The standard EN 1993-4-1:2007 [8] provides the procedure for a reliable determination of thestress σx1,Ed (Fig. 5a) for a relatively smooth distribution of the stressaround the perimeter. The actual distribution of the stress in thediscussed above zone (Fig. 5b) differs greatly from that given in stan-dard. The difference is caused by greater discrepancy between thevalues of stresses directly placed over the support axisσx0,Ed, and stress-es σx1,Ed occurring at the distance y (Eq. (3)) from the support axis(Fig. 5b). From the standard assumptions their ratio s should satisfythe condition (Eq. (5)):

y ¼ rΔθ ¼ 4ffiffiffiffi

rtp

ð3Þ

s ¼ σ x1;Ed

σ x0;Edð4Þ

0:3 b s b 1:0: ð5Þ

On the basis of own numerical analysis of the shells discretelysupported it has been found, that for the small widths s1 of bearing(Fig. 2a), the stress ratio is usually s = 0.1 ÷ 0.2. In this situation,when condition (5) of this standard is not satisfied, the distance yshould be chosen so that s≈ 0.5. Therefore for narrowdiscrete supports

Fig. 4.Obtained from GNA analysis the shapes of buckling for a discretely supported compresseby an intermediate ring (displacements enlarged 10 times).

the actual stress distribution is not important since arbitrarily imposedstress ratio s = 0.5 must be used while distance y has no influence ondesign. This approach raises considerable doubts.

A completely different issue is the possibility to determine the resis-tance of discretely supported cylindrical shells by using numericalmethods, introduced in standards EN 1993-4-1:2007 [8] and EN 1993-1-6:2009 [10]. One should stress the importance of rapid and simplemethods for the shell's resistance evaluation to verify the correctnessof numerical analyses. Simple analytical methods are always useful ina practical design, especially when they are supported by experimentalinvestigations.

The standard EN 1993-4-1:2007 [8] presents rather complicated so-lutions of discrete shells' support, but there are no references to a rela-tively simple and very efficient constructions of supporting zones forshells' column supports, presented in Figs. 1 and 2c. The introducedtest results of cylindrical models of silo shells have shown tangible ben-efits resulting from the application of such design solutions. These testshave also revealed that even good stiffening of the supporting zonesdoes not provide a compression resistance of the shell comparable tothe resistance of shells uniformly supported around the perimeter.

3. Characteristics of tested models of cylindrical shells

Experimental investigations of the resistance of cylindrical, discrete-ly supported shells under the longitudinal compression have beencarried out on steel models (Fig. 6) with diameter D = 1000 mmand length L = 1000 mm. The cylindrical shells of models have been

d shell: a) without ribs in the supporting zone, b) with short ribs, c) with short ribs topped

Fig. 7. Measuring strip.

Fig. 5. Distribution of compression, meridional stresses over the discrete support: a) pre-sented in standard EN-1993-4-1, b) obtained from FEManalysis for shellwith r=500mm,t = 1 mm, s0 = 2 mm.

Fig. 8. Results of the analysis of convergence for the numerical model with parameters:L1 / L = 0.27, r / t = 500, s0 / 2πr = 0.0191.

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made out of steel with Young modulus E = 195400 N/mm2 and yieldstrength fy = 331 N/mm2 for the shell thickness t = 1 mm, and E =204000 N/mm2, fy = 428 N/mm2 for t = 2 mm. Both edges have beenfinished by rings 20 × 20 mm (tG = tD = 20 mm), the intermediatering and stiffeners were made from a flat bar 20 × 10 mm (tP = tZ =10 mm). Stiffeners and rings from steel S235 were welded to the shell.

Basic tests have been executed onmodelswith short ribswith lengthL1 within range 0.24≤ L1 / L≤ 1.04 and topped by an intermediate ring(Fig. 6c). A part of the comparative research has been associated withunstiffened models (Fig. 6a) and with stiffened models without the in-termediate ring (Fig. 6b). Research has been carried out for 70 cases ofloading.

Two widths of discrete support have been used: s0 = 0.0006πDand s0 = 0.0191πD. The first width corresponds to almost a spotsupporting and the second one is the same as generally used for silo col-umn heads. Three or four discrete supports have been applied. The class

Fig. 6.Geometry of tested cylindrical shells'models: a)without ribs in the region of column supports, b)with short ribs over the supports, c)with short ribs topped by an intermediate ring,d) cross-section through the shell in support axis.

Fig. 9. Schemes of individual numerical models: a) model 1, b) model 2, c) model 3.

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of workmanship of the tested shells has been high (the quality parame-ter Q = 25 according to EN 1993-1-6:2009 [10]) and evaluated on thebasis of detailed measurements of geometrical imperfections (Fig. 7).

Fig. 10. Results of elastic analysis for model 1 (L1 / L = 0.27, r / t= 500, s0 / 2πr= 0.0191) loadisplacements.

Fig. 11. Results of elastic analyses formodel 2 (L1 / L= 0.27, r / t=500, s0 / 2πr= 0.0191) loadeb) distribution of vertical displacements. Color scale matches the scale of Fig. 10.

In the described tests above the way of loading has been opposite tothe loadpattern occurring in the real silos. The upper edge of eachmodel(Fig. 6) has been supported directly on a flat non-deformable surface

ded uniformly: a) distribution of meridional compressive stress, b) distribution of vertical

d by a uniformdisplacementΔ (Fig. 9b): a) distribution ofmeridional compressive stresses,

Fig. 12. Load–displacement relationship for numerical models 1–3.

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around the entire perimeter, so the edge with the discrete supports hasbeen placed on the top. The longitudinal load has been applied to thediscrete supports by hydraulic cylinders until shell buckling. The verticaldisplacement was controlled in order to stop the process of buckling inany time, and for reading the load–displacement relationship.

4. Pre-experimental numerical analyses

Many numerical analyses of cylindrical silo shells have been carriedout before the experimental tests. The aim of this was to confirm thepredicted shape of models' collapse and to examine their resistance inthe region of the single support for the state of a single local load in com-parison with total load applied to all supports simultaneously. The nu-merical analyses of elastic shell models with perfect geometry have toconfirm that the critical reaction N1cr causing the local buckling of theshell over the single support did not depend significantly on whetheron other supports similar reactions were applied at the same time ornot. This issue is important frompractical point of view, because a singleshell model might be employed in tests 3 or 4 times throughout thelongitudinal loading applied only to the single discrete support. On thecontrary, the axial loading applied to all supports simultaneouslymight lead to the total buckling of the model in support regions and,consequently such a model could not be further used. It has been ex-pected that for the tested shell the instability has a local character, andthe load-bearing capacity attained in one part of the shell over a singlesupport does not change the resistance of the rest of support regions.This assumption might be valid for small number of supports, wherethe interaction of individual shell's support regions is very limited. In

Fig. 13.Modelwith L1 / L= 0.27, r / t=500, s0 / 2πr= 0.0191 loaded by displacement of one disin elastic range of model's behavior. Color scale matches the scale of Fig. 10.

numerical analysis of the models the 4 discrete supports have beenadopted, assuming the less favorable case.

As it has been mentioned before the test model has been reversedwith the discrete supports upwards, and the load of individual supportshas been forced by displacements Δ (Fig. 9b). Such a model has beenalso adopted for numerical studies.

GNA analysis has been employed because in real models bucklingoccurs in the elastic state, hence the inclusion of plastic properties ofthe material was unnecessary. To validate the assumptions of the anal-ysis and select the appropriate finite elements, the calculation for cylin-drical shell model with welding imperfections (of type A) has beenmade. The results have been introduced by Rotter [11], and they aremost commonly used during tests on shells' stability. For a finite ele-ment S4R the obtained results corresponds perfectly to these obtainedby Rotter. It is generally believed that finite elements S4R arewell suitedfor analysis of stability, that is why they have been chosen for the anal-ysis described here. A series of calculations have been made for a shellmodel with parameters: L1 / L = 0.27, r / t = 500, s0 / 2πr = 0.0191to optimize the sizes of the finite elements. Obtained results are pre-sented in Fig. 8. The horizontal axis shows the inverse of the numberof finite elements in themodel, which enables interpolation to the accu-rate result by comparison to the number 0 and not infinity. The verticalaxis shows the result for a given size of the finite element R in compar-ison with the result obtained from ultimately selected size of the finiteelement Rcho. Finally the elements with the length of the side 0.008 mhave been chosen, which is about 0.36(r / t)1/2, allowing a relativelyquick calculation with accuracy to 2.5%.

The first computational model (Fig. 9a) has presented a shell withparameters: L1 / L = 0.27, r / t = 500, s0 / 2πr = 0.0191. The upperring (at the unstiffened side) has been loaded by a uniform linearforce directed vertically downwards, acting on the entire circumference.Distribution of meridional, compressive stresses and the vertical dis-placements within the elastic range of the structure's behavior (beforebuckling) has been presented in Fig. 10.

The second numerical model (Fig. 9b) has the same geometry asmodel 1, but the model 2 has been loaded by a uniform displacementΔ of the upper edge. Some results of the elastic analysis of model 2have been introduced in Fig. 11 (results for displacements for whichthe sum of support reactions is equal to these obtained for the model1 presented in Fig. 10).

The third analyzed model (Fig. 9c), having the same parameters asmodels 1 and 2 has concerned the reverse situation to that adoptedfor the model 2: for all 4 discrete supports the same displacement Δhas been forced, and the uniform support has been realized on the

crete support: a) distribution ofmeridional compression stresses, b) vertical displacements

Fig. 14. Load–displacement relationships for a model with L1 / L = 0.27, r / t = 500, s0 / 2πr = 0.0191, loaded sequentially at discrete supports 1/4, 2/4, 3/4, 4/4.

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ring from the unstiffened side. This model corresponded therefore tothe manner of planned tests. The load–displacement relationship forall 3 numerical models has been presented in Fig. 12. For models 2and 3 the obtained results have been the same.

All models have lost the stability in a local manner over short ribs,although there has been a significant difference in the distribution ofmeridional compressive stresses, vertical displacements and resistancesbetween model 1 and models 2 and 3. It is obviously seen that thechange the way of loading (from uniformly applied forces to displace-ments) is essential for the numerical analysis. Usually, the load is forcedin this case by displacements produced by rigid disks of the testing de-vice. In presented tests, the models have been supported at theunstiffened side by a ring on undeformable base, whereas the controlled

Fig. 15.Modes of buckling of two models from series No. 1 (r / t = 500, L1 / r = 0.24,s0 / 2πr = 0.0191).

displacements Δ have been forced by a hydraulic press from the otherside of the model, like in Fig. 9c, but for the model inverted by 180°.

The cylindrical silo shells are quite high, so themeridional rigidity ofthe adjacent cross sections is practically infinite. Thatmeans that for anyhorizontal cross-section of the shell, the vertical displacements are thesame, while the stress might change along the shell's perimeter. Thetest models correspond to the bottom portion of the entire silo shell,therefore the uniform meridional displacements at the unstiffenedside of themodel's ring is fully correct. The employment of the inversedmodel in comparison to the Fig. 9c is pertinent. The next step of thenumerical analysis was the study of a model supported as the model 3(Fig. 9c), on the ring from the unstiffened side, but for a single supportloaded (Fig. 13). This analysis has revealed a good agreement betweencritical reactions of the analyzed model and the model loaded at foursupports — the difference did not exceed 1.2%.

The last step, very important for numerical computations, prior tojoining the experimental investigations has been the entire cycle of

Table 1Results of tests for shell models from series No. 1.

L1 / r[ – ]

r / t[ – ]

s0 / 2πr[ – ]

Test no.[ – ]

RLIM[kN]

Vertical weld[ – ]

0.24 500 0.0191 1/4 31.10 Yes1/3 36.57 No3/3 33.14 Yes4/4 33.46 No

0.0006 2/3 31.89 No2/4 32.94 No3/4 31.82 Yes

250 0.0191 1/3 130.90 No2/3 120.83 Yes3/3 121.45 No1/4 114.92 No2/4 109.86 Yes

0.0006 3/4 115.33 No4/4 102.00 Yes

Table 2Results of tests for shell models from series No. 2.

L1 / r[ – ]

r / t[ – ]

s0 / 2πr[ – ]

Test no.[ – ]

RLIM[kN]

Vertical weld[ – ]

0.54 500 0.0191 3/3 36.76 No1/4 27.38 Yes3/4 28.06 Yes4/4 30.26 No

0.0006 1/3 34.30 Yes2/3 30.62 No2/4 41.26 No

250 0.0191 1/3 152.86 No3/3 139.50 Yes1/4 136.78 No2/4 107.96 Yes4/4 120.72 Yes

0.0006 2/3 132.26 No3/4 115.54 No

Table 5Results of the additional tests of shells.

L2 / r[ – ]

r / t[ – ]

RLIM[kN]

L2 / r[ – ]

r / t[ – ]

RLIM[kN]

L2 / r[ – ]

r / t[ – ]

RLIM[kN]

0 500 20.14 0.54 500 22.64 0.24 500 18.4220.62 21.64 21.6020.16 21.74 22.5819.94 17.24 1.04 500 26.3418.70 20.92 21.2419.96 250 79.60 250 70.66

250 81.98 74.64 71.7676.14 81.76 73.7276.50 68.8881.16 72.8475.36 74.6272.38 74.10

Table 3Results of tests for shell models from series No. 3.

L1 / r r / t s0 / 2πr Test no. RLIM Vertical weld[ – ] [ – ] [ – ] [ – ] [kN] [ – ]

1.04 500 0.0191 1/3 48.92 No2/3 34.28 Yes1/4 41.30 No2/4 35.84 Yes

0.0006 3/3 49.10 No3/4 48.94 No4/4 40.64 Yes

250 0.0191 1/3 133.60 No3/3 147.22 Yes1/4 148.34 No2/4 173.34 Yes

0.0006 2/3 145.54 No3/4 142.88 No4/4 136.98 Yes

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loading and disloading executed for a single model. The analysis hasbeen based on the subsequent loading of the single support (until shell'sbuckling over this support). After the local buckling of one supportshell's region, the other, stable discrete support, has been loaded. Thelocal bucklingdeformationshave been treated asmodel's imperfections.For 4 analyzed discrete supports, 4 load–displacement relationshipshave been obtained (Fig. 14).

A very good agreement has been reached for critical reaction of all 4discrete supports. The difference between critical reactions in support2/4 and 3/4 has been 0.9%, whereas the maximum difference (betweensupport 1/4 and 4/4) has been 1.7%. Hence, it has been considered thatthe tests can be carried out similarly.

5. Experimental investigations on cylindrical shells models

During each of the loading tests for subsequent discrete support, theboth support's displacement Δ and the load R have been recorded,which allows to determine the load-bearing capacity RLIM and the R–Δrelationship as well. On this basis the maximum load-bearing capacityand corresponding displacement have been determined. Moreover,the shape of local buckling has been recorded as well as its localization.Two different modes of local instability have been observed. The basicone concerned the unstiffened part of the shell along the extension ofthe support ribs (e.g. Fig. 15), the second one appeared usually afterthe primary mode of buckling in the unstiffened part between thesupport ring and the intermediate ring next to short ribs (Fig. 18b).

Table 4Results of shells' tests from series No. 4.

s0 / 2πr[ – ]

L1 / r[ – ]

r / t[ – ]

RLIM[kN]

0.0191 0.24 500 20.4828.52

250 107.64107.68

0.54 500 29.8226.58

250 105.48111.02

1.04 500 25.2225.84

250 133.70140.38

For long ribs L1 = r there have been cases, when buckling near theshort rib appeared as the first one (Fig. 21b). The models' shells havebeen made of two pieces of hot rolled sheet welded along the meridian(see e.g. Fig. 15a). For models with 3 ribs and 3 discrete supports such aweld has occurred along the axis of one of the ribs. For models with 4ribs such weld has occurred in two opposite ribs regions. These placeshave been marked in presented results.

The experimental results are shown in Tables 1–5. Each specimen isdescribed by the number of test (the first digit) and the number of test-edmodel supports (the second digit). For instance “Test No. 2/3”meansthat in model with 3 discrete supports the second of them has beenloaded. For one value of parameter L1 / r four models have created a se-ries of tests: two of them with slenderness r / t = 500 and two other

Fig. 16. Load–displacement relationships for tested models from series No. 1 (r / t= 500,L1 / r = 0.24): a) models with support widths s0 / 2πr = 0.0191, b) models with supportwidths s0 / 2πr = 0.0006.

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with r / t=250,whereby for each slenderness r / t oneof themodels hashad 3 short ribs, and the other one 4 short ribs. Under each short rib, thediscrete support has been arranged.

Tests have been carried out in four series. In the first one (No. 1), theshells has had ribs satisfying the condition L1 / r = 0.24. In each of thetests (for models with r / t = 500 and r / t = 250) the loss of stabilityhas taken the shape of rhomboidal indentation at the unstiffenedregion, over the short rib and intermediate ring (Fig. 15). The relevantresults have been shown in Table 1, where RLIM denotes the value of re-action at the single discrete support with width s0, for which the shellhas lost the stability. In this series of investigations, the loss of stabilityhas not been noticed in the region between the intermediate ring andthe support ring, even after local buckling at the unstiffened area. Theload–displacement relationship for models of series No. 1 have beenshown in Figs. 16 and 17.

The test series No. 2 has been associated with models with L1 / r =0.54. For all models the load-bearing capacity has been accompaniedby the local instability of the unstiffened area (Fig. 18a), whereas formodels with r / t = 500, the instability has been additionally recordedat the stiffened area (between the support ring and the intermediatering), already after buckling of the unstiffened zone (Fig. 18b) — thissecondary instability has appeared after the load-bearing capacity ofthemodel have been reached. The results of tests formodels from seriesNo. 2 have been shown in Table 2 and in Figs. 19 and 20.

The third series of tests (No. 3) has been applied to models withL1 / r = 1.04. For these models, the instability has been observed atthe unstiffened area and between the rings as well. Both of the modesappeared almost for the same loads coming from discrete supports. Inshells with r / t= 500 mostly the unstiffened area has lost the stability(Fig. 21), wherein in one test the rib has buckled (test 3/3 marked as

Fig. 18. Bucklingmodes formodels from series No. 2: a) buckling at the unstiffened shell'sarea, b) additional buckling next to the ribs.

Fig. 17. Load–displacement relationship for testedmodels from series No. 1 (r / t= 250,L1 / r= 0.24): a) models with support widths s0 / 2πr= 0.0191, b) models with supportwidths s0 / 2πr = 0.0006.

bold in Table 3). It has been basically assumed that the ribs are suffi-ciently rigid and able to resist buckling along the adjacent shell. Forthis reason, the ribs have been suitable stiffened in the rest of tests.This treatment has not affected the critical values of loads RLIM, as onlythe ribs buckling length was reduced. For equal terms of comparisonwith previous models the enlarging of ribs' cross-section area hasbeen postponed.

For shells with r / t = 250 in tests 1/3 and 3/3 (marked as bold inTable 3), the rib has buckled, and the described above treatment hasbeen employed as for shells with r / t = 500. Obtained load–displacement relationships have been presented in Figs. 22 and 23.

The fourth (No. 4) series has concerned the comparative models,where models with short ribs but without an intermediate ring havebeen examined. Owing to the merely complementary and comparativenature of these tests, the range of studies has been limited to a smallnumber of models with support width s0 / 2πr = 0.0191. The ribs inthe discussedmodels have been arranged in such away that the verticalmeridional welds have not been in the vicinity of the load axis. As it hasbeen expected, the buckling has taken the shape of the rib indentationinside the shell (Fig. 24). Results from this test series have been present-ed in Table 4, where the number of each single test corresponds to theparticular load–displacement relationships shown in Fig. 25.

Unstiffenedmodelswith support rings, butwithout longitudinal ribshave been tested as well (Fig. 26a). These tests of secondary charactershould be regarded as indicative only. After the primary studies ofshell models, the strength of applied steel has been investigated onspecimens taken from portions located far from buckling deformationson testedmodels. Additional tests have been executed for shellswithout

Fig. 20. Load–displacement relationship for tested models from series No. 2 (r / t = 250,L1 / r = 0.54): a) models with support width s0 / 2πr = 0.0191, b) models with supportwidth s0 / 2πr = 0.0006.

Fig. 19. Load–displacement relationship for tested models from series No. 2 (r / t = 500,L1 / r = 0.54): a) models with support width s0 / 2πr = 0.0191, b) models with supportwidth s0 / 2πr = 0.0006.

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longitudinal ribs, but with an intermediate ring, placed at distance L2from the support ring (Fig. 26b).

6. Summary and conclusions

The study has covered discretely supported shells (Fig. 2) of steel silos.Models with short ribs topped with intermediate ring (e.g. Fig. 27c) de-scribed in the summary as “type A”, and with short ribs without an inter-mediate ring (e.g. Fig. 27b) described in the summary as “type B” havebeen predominantly studied. Additional studies have been carried outfor discretely supported shells without ribs (e.g. Fig. 27a) as well. Twosupport widths have been analyzed s0 = 0.0006πD (support type I) ands0= 0.0191πD (support type II) with different lengths of the longitudinalribs within the range: 0.24≤ L1 / L≤ 1.04. Abbreviated description of themodels is used in charts (Figs. 28 and 29): type Awith supports of type IIis described as “typeA/II”, othermodels are described in a similarmanner.

In summary charts (Figs. 28 and29) results of the individual tests areshown by marked points, whereas the averaged values (for one set ofparameters) are depicted by a continuous line. Point and line chart forone model type are described in the above explained manner. As ex-pected, the use of vertical ribs of L1 length influence the load-bearing ca-pacity RLIM of shells loaded on discrete supportswithmeridional force R.The increase of the length for ribs L1 and the use of intermediate ringover the ribs clearly affects the growth of capacity RLIM of axially com-pressed shells supported discretely (Figs. 28, 29).

The authors of presented herein study believe that the most impor-tant conclusion frompresented experimental tests is the negation of theusually formulated opinion that the use of longitudinal ribs of consider-able length (L1 / r ≥ 0.5) topped with intermediate ring in axially com-pressed cylindrical shells protects the unstiffened section, localizedabove the intermediate ring against the local instability. Presented

experimental studies with simplified numerical analyses clearly showthat the part of the shell above the intermediate ring buckles over ver-tical ribs even if used ribs are of considerable length (L1 / r ≥ 0.5). Thestrengthening of zones adjacent to supports of short ribs, and interme-diate rings increase obviously the buckling capacity of the shell, but thedifferences in the capacities of uniformly supported shell around itscircumference and shells supported on discrete supports are alwaysvery high. Experimental tests of shells supported on entire circumfer-ence have not been carried out, so GNA numerical analysis has beenperformed. In this calculation shell model with parameters: r / t =500, s0 = 0.0006πD and the number of supports n = 4 has been ana-lyzed for all four cases of geometry and support way shown in Fig. 27.The analyses have enabled to obtain a critical load for a perfect elasticshell. Resultant axial force for shell supported on entire circumference(Fig. 27d) is labeled Po. For shells supported on n= 4 discrete supports(Fig. 27a–c) the resultant critical force was determined as P= 4 × RLIM.Perfect shells have been analyzed, but it is expected that planned by au-thors GMNIA analysis of thosemodelswill lead to similar differences be-tween global capacity P and Po as in GNA analysis.

The results from numerical analysis of the perfect shell models(Fig. 30), clearly show that the use of vertical ribs in the support zoneof shells increases the global load capacity P (line b), in relation tothe unstiffened shell (line a). The use of an intermediate ring overshort ribs leads to additional increase in load capacity P (line c) and im-proves the effectiveness of longer ribs. It should be strongly emphasizedthat there is a very big difference between the global load-bearingcapacities of shells without ribs evenly supported around the cir-cumference (Fig. 30— line d), and global capacities P= n × RLIM of par-tially ribbed shells based on discrete supports (Fig. 30 — lines a, b, c).These differences are related to the phenomenon described in [9], as astrong concentration ofmeridional stressσx above the discrete supports

Fig. 21. Modes of buckling for models from series No. 3 (r / t = 500, L1 / r = 1.04): a) s0 / 2πr = 0.0191, b) s0 / 2πr = 0.0006.

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in axially compressed shells without longitudinal ribs (Fig. 3b). The lon-gitudinal ribs are also conducive to the meridional stress concentration(Figs. 10a, 11a), and in the axes of these ribs above intermediate ring themaximummeridian stress σx,max is only slightly smaller than the corre-sponding stress σx,max over the support in the shells without longitudi-nal ribs (for the same support width). This issue is often overlooked in

Fig. 22. Load–displacement relationship for tested models from series No. 3 (r / t = 500,L1 / r = 1.04): a) models with support width s0 / 2πr = 0.0191, b) models with supportwidth s0 / 2πr = 0.0006.

the design of silos, leading to incorrect assessment of ribbed cylindricalshells capacity when the traditional and simplified methods of designare applied. For a discretely supported cylindrical silo shell to be de-signed according to procedures given for uniformly supported shellsin codes [8,10] the obtained resistance would be too small even whenthe strengthening (ribs and circumferential ring) is present. The total

Fig. 23. Load displacement relationship for tested models from series No. 3 (r / t = 250,L1 / r = 1.04): a) models with support width s0 / 2πr= 0.0191, b) models with supportwidth s0 / 2πr = 0.0006.

Fig. 24. Shapes of buckling for models from series No. 4 (r / t = 500, s0 / 2πr = 0.0191): a) L1 / r = 1.04, b) L1 / r = 0.54.

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resistance of all supports P0 (lines a, b, c in Fig. 30) is always definitelysmaller than the analogical capacity P0 of the same shell supportedalong the entire circumference. The usage of solution analyzed inthis paper (line c in Fig. 30) increases significantly the capacity P0 of dis-cretely supported shell (in this case of 4 supports) in comparison to the

Fig. 25. Load–displacement relationship for models from series No. 4 (s0 / 2πr = 0.0191,L1 / r according to Table 4): a) r / t = 500, b) r / t = 250.

same shell but without ribs and circumferential ring (line a in Fig. 30),anyhow the resulting resistance is always much smaller than the onedetermined for a uniformly supported shell. The use of numerical

Fig. 26. Buckling modes for additional tests: a) r / t = 250, s0 / 2πr = 0.0006, L2 / r = 0,b) r / t = 500, s0 / 2πr = 0.0191, L2 / r = 0.54.

Fig. 27. Examples of analyzed shells with four discrete supports (a–c) and evenly supported shell (d).

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analysis GMNIA according to [10] should provide a safe evaluation of theanalytical values of capacities for steel silo shells.

The positive influence ofmaterial horizontal pressure on silo shell onglobal capacity of the shell due to stability failure is widely known. Localunsymmetrical horizontal pressure thatmay appear in shells is alsowell

Fig. 28. Summary of results of exper

Fig. 29. Summary of results of experiment

recognized [12]. This unsymmetrical pressure can appear randomly inany section of the shell and in some cases leads to local disappearanceof material horizontal pressure. Since in case of the local buckling of dis-cretely supported shells (above supports or ribs; Figs. 4, 26), it cannot beexcluded that disappearance of the horizontal pressure will take place

imental shell models of type A.

al shell models of type A and type B.

Fig. 30. Comparison of global capacity P0 of perfect shell supported on entire circumfer-ence (d) with capacities P = 4RLIM of shells supported on discrete supports: without ribs(a) and with ribs (b and c)— marking according to Fig. 27.

94 E. Hotała, Ł Skotny / Journal of Constructional Steel Research 96 (2014) 81–94

in the regionwhere bucklingwave is forming. In authors opinion in caseof discretely supported shells positive influence of horizontal bulk ma-terial pressure on shell capacity due to local stability failure should notbe taken into account. However if the constant horizontal pressurewould be provided in the axially compressed shell supported discretely,shells global capacity P0would increase in a similar degree as capacity ofan analogical shell supported uniformly. It can be assumed that Fig. 30would have a similar course but values of global capacity P0 would beproportionally higher. Capacity of shells due to buckling with regardto the normal pressure can be calculated according to standards [8,10].

Acknowledgments

Calculations have been carried out in Wroclaw Centre for Network-ing and Supercomputing (http://www.wcss.wroc.pl), grant No. 241.

Experimental and numerical studies presented in this paperwere made for PhD thesis of Łukasz Skotny “Nośność granicznaużebrowanych cylindrycznych powłok płaszczy stalowych w strefiepodpór odcinkowych” (in Polish), Prof. Hotała was a supervisor of thisdissertation. Several figures and tables presented in this paper weretaken from this thesis or are based on figures presented in it.

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