Locally flattened or dented domes under external pressure

Thin-Walled Structures 97 (2015) 44–52

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Locally flattened or dented domes under external pressure J. Błachut The University of Liverpool, School of Engineering, Liverpool L69 3GH, UK

art ic l e i nf o

a b s t r a c t

Article history: Received 23 July 2015 Accepted 25 August 2015 Available online 22 September 2015

The paper provides comparisons of sensitivity of buckling pressures to initial shape imperfections, for the case of externally pressurised steel domes. A priori defined deviations from perfect shape include: Legendre polynomials, increased-radius patch, and localised inward dimple. The latter is created by a concentrated force acting radially. In this case the effects of spring back and/or annealing of the dented patch and the surrounding area on the load carrying capacity are assessed. Some of the elastic solutions related to the lower-bound approach are compared with the available experimental data. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Buckling Geometrical imperfections Increased-radius flattening Inward dents Elastic Elastic–plastic

1. Introduction The quest for oil/gas and other natural resources has pushed the underwater exploration to deeper and deeper environment. Fig. 1 depicts the range of current and planned exploration as the percentage access to the ocean floor bed. Plans for exploration for up to 3 km (zone ‘C’ in Fig. 1) are currently being actively pursuit, Refs. [1–4]. Several manned vehicles capable of exploring deep-sea (6 km), have been developed within the last few decades and they are indicated in Fig. 1. Freshly added to this list is JIAOLONG vehicle rated at 7 km depth, Ref. [5]. Pulse-waves generated by controlled implosion of buoyancy units are also actively studied as a part of new warfare (assault and/or protection of existing subsea assets), Refs. [6–8]. One of the bottle-neck in wider underwater activities is the availability of buoyancy units. This is especially true at greater ocean depth. There are two approaches to buoyancy units. In the first one, the buoyancy is provided by foams. It appears that by-and-large the necessary foam-based buoyancy units have been obtained. This can be seen, for example, in the case of DeepSea Challenger where the foam was successfully used as buoyancy during the full ocean depth dive in 2012 (11 km), Ref. [9]. The second type of buoyancy unit is of ‘the-vessel-type’ frequently referred to as pressure hull. As yet this type of buoyancy unit is not readily available. Geometry of the hull can take different shapes as this depends on applications. Typically these could be cylinders capped by domed closures – see Refs. [10–13], spheres – see Refs. [14,15], or closed toroids, Refs. [16,17]. One of critical issues when designing externally pressurised cylinders and/or doubly curved shells is the sensitivity of their buckling pressure to initial geometric imperfections. Search for sensitivity of buckling loads to initial geometrical imperfections has resulted in an enormous research effort for many of recent decades. Some results http://dx.doi.org/10.1016/j.tws.2015.08.022 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

related to imperfect domes subjected to external pressure can be found in Refs. [13,18–21]. In order to ascertain sensitivity of buckling pressure to initial deviations from perfect shape one has to decide what these shape deviations are, i.e., how they are defined, where they are positioned, what the maximum amplitude of imperfection is, etc. The above questions still remain open ones. Over the years a number of approaches to modelling shape of initial geometric imperfections have been tried. The shape which reduces the buckling strength the most has always been sought as this would allow the designer to plan for the worst possible scenario. The imperfections in fabricated domes will be distributed randomly and will normally consist of dimples and increased-radius flat spots of various sizes. The analysis is much simpler if the imperfections can be assumed to be axisymmetric. The actual localised initial imperfections may be at the apex (axisymmetric) or away from the pole. In the latter case, it still seems reasonable to assume axisymmetric behaviour if the imperfection is not too near the clamped edge (hemisphere) or not too near the spherical cap/ knuckle junction in torispherical shell. The most frequently adopted forms of initial geometric imperfections include shapes: (i) affine to eigenmode, and (ii) local flattening associated with the increased radius. Another possibility would be an inward dimple created by a concentrated force (Force-Induced-Dent, FID). This approach has been vigorously pursuit for axially compressed cylindrical shells manufactured by filament winding of carbon fibre (with epoxy resin being the binding agent), Refs. [22,23]. Inward dent, representing imperfection, was created by force acting radially and was placed in the middle of cylinder's length. Detailed and comprehensive experimental data was obtained for cylinder's geometry given by D/ t¼500 and L/R¼ 2. The aim of this paper is to explore possible advantages of this

J. Błachut / Thin-Walled Structures 97 (2015) 44–52

Fig. 1. Current and planned activities at sea bottom (Fig. 1a, A≡North Sea; B≡Gulf of Mexico, Nigeria, Brazil; C≡2020 Horizon research). Also, milestones in depth access by humans.

approach when used in externally pressurised domed ends. Previously used methods for ascertaining imperfection sensitivity are reviewed/expanded, and response of buckling pressures to ForceInduced-Dimples is studied in detail. The next sections discuss these issues for the case of externally pressurised torispherical, hemispherical and/or spherical cap end closures made from mild steel.

2. Preliminaries – buckling of geometrically perfect domes This section briefly outlines buckling of externally pressurised, and geometrically perfect domes. The domes are to be taken as torispherical or hemispherical shells – both frequently found as closures of the cylindrical portion of underwater vehicles, buoyancy units, liquid oxygen/hydrogen tanks in aerospace, as well as bulkheads in the second stage LOX tanks. In all of the ensuing computations the following two codes were used: ABAQUS, and BOSOR5, Refs. [24,25]. As an illustration, consider a torispherical head, see Fig. 2a, with its geometry given by the diameter-to-thickness ratio, D/ t¼ 1000, the knuckle radius-to-diameter ratio, r/D ¼0.10, the spherical-radius-to-diameter ratio, Rs/D¼ 1.0, and subjected to uniform external pressure, p. Let the torisphere be manufactured from steel with E ¼210 GPa, ν ¼ 0.3, and the yield point of material, syp ¼ 350 MPa. Assume that the dome is fully clamped at its equatorial edge. Pre-buckling shape of this externally pressurised torisphere is shown in Fig. 3a. This shell is able to support external pressure for up to a certain magnitude at which the axisymmetric

45

deformation, seen in Fig. 3a, suddenly changes its shape. This pressure, corresponding to an eigenvalue, is also known as bifurcation pressure, pbif. Its magnitude in the current case is, pbif ¼0.126 MPa. Fig. 3b depicts eigenshape, i.e., the shape at pressure equal to bifurcation, and which has n¼ 17 circumferential waves. The corresponding FE models are shown in Fig. 3c and d. Fig. 3c shows axisymmetric shape just prior to buckling whilst Fig. 3d illustrates the eigenshape at buckling pressure. The influence of the (D/t)-, (Rs/D)-, and (r/D)-ratios on buckling performance of steel torispheres was addressed in Ref. [26], where results of a wide parametric study are given. A number of design rules for externally pressurised domed ends have been developed – available, for example, in Refs. [27–29]. They make various provisions for initial geometric imperfections. But there is still no convincing argument which approach is the best. The current paper is the extended and revised version of conference contribution, Ref. [30]. The next section outlines some possibilities, including the increased-radius local flattening as a possible worst scenario approach to design.

3. Buckling of geometrically imperfect domes – axisymmetric models 3.1. Increased-radius patch One of possible imperfection profiles studied in the past is the increased-radius flattening patch – as sketched in Fig. 2b, see Refs. [19,20]. It is characterised by arc length over which radial imperfections are measured, simp, and the radius of imperfect portion of torispherical or hemispherical shell, Rimp. The relation between the amplitude of the imperfection at the pole, δ0, and the radius of curvature of imperfection, Rimp, is given by:

Rs ⎞⎛ Rs ⎞ δ0 1 ⎛ ⎟⎟⎜ ⎟ = α 2⎜⎜ 1 − t 2 ⎝ R imp ⎠⎝ t ⎠

(1)

For a given magnitude of imperfection, δ0/t, there is an infinite number of geometries defined by the angle, α (or radius Rimp) – see Fig. 2b. Only one of them will lead to the weakest dome, i.e., will have the lowest buckling pressure. As an illustration consider hemispherical dome, clamped at its equatorial plane. A series of computations were carried out in order to find the buckling strength of externally pressurised imperfect steel hemisphere. Results are shown in Fig. 4. It is seen in Fig. 4 that for different magnitudes of the angle, α, one obtains different magnitudes of buckling strength. The festooned curve to these response-curves represents the so called lower-bound curve. It is seen in Fig. 4a and

Fig. 2. Geometries of: (a) perfect torispherical dome (Fig. 2a), imperfect one (Fig. 2b), and view of torisphere with flat patch (Fig. 2c, black area, δ0/t ¼1.0).

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J. Błachut / Thin-Walled Structures 97 (2015) 44–52

Fig. 3. Pre-buckling (Fig. 3c) and buckling (Fig. 3b) shapes of the generator. Views of corresponding FE models in Fig. 3c and d, respectively.

Fig. 4. Illustration of numerical derivation of lower-bound curve for externally pressurised steel hemispheres (E¼ 207 GPa, ν ¼ 0.3).

b that for the yield point syp ¼ 310 MPa the lower-bound curve tends to become horizontal at δ0/t¼ 2.0. The above concept of lower-bound has been validated by experimental data carried on selected sets of geometrically imperfect torispheres subjected to vacuum. Nominal dimensions of torispheres were defined by: Rs/ D¼ 1, r/D ¼0.15, and D/t¼ 200. The flat patch, instead of being placed at the pole, was moved away from the pole by 12° in order to check claims that such position represents the weakest spot in externally pressurised hemispheres and/or torispheres – see Ref. [19] for full details. The following magnitudes of imperfection, δ0/t, were chosen for experimentation: δ0/t ¼0.10, 0.31, 0.61, and 1.02. Firstly, a batch of geometrically ‘near perfect’ models were manufactured, followed by manufacturing of imperfect torispheres. Thirty ‘perfect’ torispheres and thirty with build in imperfection for each of four magnitudes, δ0/t, were injection moulded using ABS material. Fig. 2c shows a moulded torisphere with a flattened patch characterised by an increased-radius associated with, δ0/ t¼0.10 (black area). At the same time flat coupons were also moulded and then tested under uni-directional tensile conditions. Average material properties found in a number of tensile tests for

this material are: E ¼2.52 GPa, the yield point of material, s0.2 ¼48 MPa, and ν ¼ 0.3 (with full details available in Ref. [19]). Each dome was tested under vacuum condition by evacuating air from inside of the shell. This allowed to capture the moment of buckling – which was in all cases sudden. Shells would shatter unless there was an internal mandrel able to support buckled shell. Figs. 5b and 6a depict buckled torisphere with small amplitude of build-in imperfection (δ0/t¼0.10). It is seen that the dome fails by bifurcation at the same level of pressure as geometrically perfect shell, and with n ¼ 6 circumferential waves. Observation made here is clear. Small, localised imperfections can be tolerated by the shell and they do not affect the buckling strength. It is seen in Fig. 5a that pre-buckling deformation is axisymmetric and there is no visible effect of small-size imperfection on the deformation pattern. Bifurcation is of global nature and the flattening is not affecting the regular eigenshape seen in Fig. 5b. The situation changes once the magnitude of imperfection is larger than the threshold value (δ0/t¼0.16, in this case). There is no bifurcation at failure pressure. The shell collapses by developing large, locally axisymmetric, deformation at the

J. Błachut / Thin-Walled Structures 97 (2015) 44–52

47

Fig. 5. Pre-buckling and buckling of small amplitude case (δ0/t ¼0.10, Fig. 5a and b). View of initial and collapsed case for δ0/t¼ 0.61 shown in Fig. 5c and d. Not to scale.

Fig. 6. View of failed torispherical shells: perfect (Fig. 6a), imperfect (Fig. 6b–d).

flattened area. The prediction of the failure mode seen in Fig. 5d (δ0/t¼ 0.61) corresponds well to the failure mode seen in Fig. 6c (δ0/t¼ 0.61). It is worth noting here that the ratio of experimental failure pressure to the FE predictions, pexptl/pFE, varied between 0.98 and 1.02 for the set of δ0/t¼0.0, 0.10, 0.31, 0.61, and 1.02 – again, details can be found in Ref. [19]. Fig. 7 shows experimental data plotted on the lower bound curve obtained for overall average geometry of moulded torispheres. It is also seen here that buckling strength of imperfect torisphere continuously decreases once the magnitude of imperfection exceeds the threshold value. This figure also illustrates how the lower-bound curve has been computed.

assuming

ν ¼0.3):

⎛ t ⎞2 pcr = 1.21E⎜ ⎟ ⎝ Rs ⎠

(2)

Associated with this pressure are many linearly independent buckling modes involving spherical surface harmonics of degree n. The degree n is the integer which most closely satisfies the relation: 1/2⎛ R ⎞ n(n + 1) = ⎡⎣ 12(1 − υ2)⎤⎦ ⎜ s ⎟ ⎝ t ⎠

(3)

3.2. b. Imperfections defined by Legendre polynomials The critical external pressure for the elastic buckling of a perfect complete spherical shell is given as follows (Ref. [31];

For axisymmetric buckling, the relevant spherical harmonic is the Legendre polynomial of degree n, i.e., Pn(cos φ). The value of n varies with the (Rs/t)-ratio. For n ¼9, 17, and 25 the variation of the

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J. Błachut / Thin-Walled Structures 97 (2015) 44–52

3.3. Local inward dimple of arbitrary shape One particular form of initial shape imperfection used in Ref. [18] had the following variation of radial deviation, w, from perfect shape:

w=

Fig. 7. Comparison of buckling load sensitivity in externally pressurised torisphere to lower-bound and to FID-dimple types of geometrical imperfections.

eigenmodes is illustrated in Fig. 8b. The elastic and plastic collapse pressures were obtained for clamped, imperfect steel hemispheres with the (Rs/t)-ratio of 100. The plastic collapse pressures were obtained for the yield point of material, syp ¼207 MPa, 310 MPa, 414 MPa, and the relevant value of n. The results are shown on Fig. 9 together with the corresponding results for the increasedradius imperfections. As may be seen, the imperfections in the form of Legendre polynomials give the lower collapse pressures but there is not a great deal of difference between them. It is worth noting that the curves related to the Legendre polynomials give the impression that the collapse pressure depends only on the amplitude of the imperfection. However, with the imperfection in the form of Legendre polynomials, the degree of the polynomial, n, varies with the (Rs/t)-ratio and the half-wavelength of the central dimple is:

simp = 2.6 Rst

(4)

Thus, the extent of the imperfection comes into the calculations as well.

⎛ δ0 ⎞ ⎜ ⎟ 1 − x2 ⎝ t ⎠

(

3

)

(5)

where δ0 is the amplitude of the imperfection at the apex, and x¼ s/simp is the meridional extent of the imperfection. The dimple was placed at the apex in a spherical cap clamped at the base. Geometry of the cap was defined by the parameter, λ ( λ = 2[3(1 − ν 2)]0.25 (H /t )0.5, with H being the height of the cap). Past work shows that the reduction in the buckling pressure also depends on the meridional extent of the imperfection, simp. Using a finer variation in simp produces many response curves. Drawing the envelope to all of these curves gives the lower-bound sensitivity of buckling pressure to shape imperfections. Fig. 10 (adapted from Ref. [18]) shows the relative reduction in the elastic (E) and elastic–plastic (E–P) buckling pressure in the range 0r δ0/t r2.0 obtained for the inward dimple defined by Eq. (5). Results obtained for increased-radius imperfections are superimposed in this figure, and it is seen that there is very little difference between both types of lower-bound curves. In both cases the following material data was assumed: E¼ 207.0 GPa, ν ¼ 0.3 and syp ¼414.0 MPa with elastic-perfectly plastic modelling of stressstrain curve. No noticeable difference was recorded between elastic and elastic-perfectly plastic sets of results. Next, computing has been carried out for the cases of imperfect torisphere and spherical cap in which axisymmetric inward dimple were generated at the apex by concentrated force, F, acting vertically (see Fig. 8d). 3.4. Localised inward dimple created by radial load It has been recently suggested, Refs. [22,23], that the worst profile of initial geometric imperfection, in axially compressed cylinders, is that associated with an inward dimple created by a

Fig. 8. Possible types of other axisymmetric imperfections including Force Induced Dimple (FID) – Fig. 8d.

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49

* ¼ 20.0 MPa). Fig. 9. A comparison of increased-radius and Legendre polynomials imperfections in externally pressurised hemispheres (Rs/t 100, E ¼207.0 GPa, pcr

Fig. 10. Lower-bound curves versus FID-curves for the elastic/elastic–plastic buckling pressures of spherical cap.

concentrated force acting radially. This idea was pursuit here for an inward dimple created at the apex of: (i) a torispherical shell, and (ii) a spherical cap. The geometry of the torisphere was given by D/t¼200, Rs/ t¼ 1.0, and r/D ¼0.15. Its material was ABS plastic (as described earlier). The shell was fully clamped at the base. The computation started with application of concentrated force, F, acting vertically at the apex (see Fig. 8d). Its magnitude was such that the resulting dent’s depth was δ0/t¼0.5. Whilst the magnitude of the force, F, remained constant, the uniform external pressure was applied next in an incremental way for up-to-collapse (via RIKS method). The resulting load deflection curve under the action of constant

Fig. 11. Load-deflection curves for imperfect torispheres. Comparison between increased-radius and Force-Induced-Dimple imperfections.

force, F, and incremental pressure, p, is plotted in Fig. 11. The load deflection curve for the case of increased-radius lower-bound imperfection, with the same magnitude of δ0/t¼ 0.5, is added in the figure. It is seen that there is very little difference between curves both on the pre-collapse and post-collapse parts. Table 1 provides values of collapse pressures for both increased-radius and FID-dimple models of the imperfection as obtained from Abaqus and from Bosor5 codes for two magnitudes of indentation, i.e., for δ0/t¼0.5 and 1.0. The percentage error between predictions of collapse given by both codes is [0.8%, 2.3%, 0%, 5.5%]. In three cases Abaqus predictions are higher than those given by Bosor5. Fig. 12 depicts deformed shapes of dome's generator at the collapse (one for increased-radius and one for FID-indentation). Again, the deformed shapes at the collapse are fairly similar. The next set of computations was carried for steel spherical cap. The geometry of the cap was given by λ ¼10.0 and Rs/t¼ 378.2. The cap was fully clamped at the bottom and it was made from mild steel with E ¼207.0 GPa and ν ¼0.3. Several types of analyses were carried out, i.e., elastic and elastic perfectly-plastic with various loading and unloading scenario. In the latter case the yield point of material was assumed to be, syp ¼414.0 MPa. The first sequence of computing was as follows: The force, F, of given magnitude was applied first. The resulting magnitude of indentation at the apex, δ0, was recorded – see the curve ‘A’ in Fig. 13. When loaded incrementally by concentrated force, F, the cup collapses at Fcoll ¼799.9 N. For a given indentation, δ0, the external pressure, p, was applied next in an incremental way until collapse of the cap was achieved. This means that the stress state generated by the force, including any plasticity, was taken up when pressure loading commenced. Results are plotted in Fig. 10. They show that the Force Induced Dimples lead to larger reduction of load carrying capacity than either the increased-radius or arbitrarily assumed smooth dimple given by the Eq. (5). Whilst these differences are noticeable for δ0/to1.0, they level off for larger amplitudes of indentation (1.0r δ0/t r2.0). It is interesting to see that the profiles of sensitivities remain by-and-large comparable despite the fact that the lower-bound solution is based on the shape deviations only whilst FID-indentation apart from shape deviation also includes embedded stresses. The above sequence of computations was repeated for purely elastic behaviour of the cap. The relevant results are also plotted in Fig. 10. Similar behaviour is noted as in the case of elastic perfectly-plastic analysis. It might be argued that simultaneous loading by a denting force of constant amplitude and by incremental pressure is very unlikely to be seen in practice. But this approach, from a numerical point of view, is computationally more efficient than the lower-bound approach based on the increased-radius imperfection (i.e., single

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J. Błachut / Thin-Walled Structures 97 (2015) 44–52

Table 1 Comparison of collapse pressures obtained for two different, axisymmetric models of initial shape imperfection at the apex of ABS-torisphere (D/t¼ 200, Rs/D¼ 1.0, r/ D ¼0.15). δo t

0.5

Imperfection type

Increased-radius (lower-bound)

pcoll (MPa)

Bosor5 0.0242

1.5 FID-indentation

Abaqus 0.0244

Bosor5 0.0213

analysis versus many analyses for a constant (δ0/t)-amplitude). In view of the above the following two approaches were investigated further. In the first one, for a given magnitude of indentation – the elastic off-loading (spring back) was carried out. The elastic spring back was recorded and then the external pressure was applied in an incremental way for up to collapse. The curve ‘B’ in Fig. 13 depicts the residual magnitude of indentation after the elastic unloading. Sensitivity of buckling pressure to this kind of initial geometric imperfection is shown by the dotted curve in Fig. 14. It is this case which is likely to exist in real life situations. If after the manufacture steel dome does not undergo annealing then any measured shape imperfections are those which exist after the elastic spring back plus any embedded (residual) stresses in the wall. The second approach assumed that the initial geometric imperfection was simply the deformed shape generated by the force, F. Once a given depth of indentation, δ0/t, was achieved, the shape was ‘frozen’. Next, any embedded stresses were removed. This would be equivalent to annealing in the real shell. Only then the external pressure was applied in an incremental way for up-to the collapse. The load-deflection curve under the assumption of fully elastic behaviour is plotted in Fig. 13 as curve ‘C’. The response of buckling pressure to different magnitudes of initial shape deviations (elastic analyses, E), is shown in Fig. 14 (as ‘Dented and then Annealed’ curve). It is seen that much larger reductions in buckling strength have been recorded than in the case of elastic spring back (doted curve). But the increasedradius approach, as seen in Fig. 10, leads to even larger drops in the buckling strength. The next section examines the use of FID-imperfections in a compound shell, e.g., in torispherical vessel closure. There are two segments in any of torispherical domes – spherical cap and toroidal (also known as knuckle).

Increased-radius (lower-bound) Abaqus 0.0218

Bosor5 0.0108

FID-indentation

Abaqus 0.0108

Bosor5 0.0800

Abaqus 0.0844

Fig. 13. Load deflection curves generated by vertical force, F, applied at the apex of steel cap.

Fig. 14. Sensitivity of buckling pressure for various analyses of Force Induced Dimples.

created by a concentrated force, F (Force-Induced-Dimple, FID). 4. Buckling of geometrically imperfect domes – non-axisymmetric cases

4.1. Eigenmode imperfections

Consider torispherical shell as discussed earlier (D/t¼1000, Rs/ D¼ 1.0, r/D ¼0.1). Assume that its material data is given by E ¼210 GPa, syp ¼350 MPa, and ν ¼0.30. In the next two sections two types of geometric imperfections are to be considered: (a) affine to the eigenmode, and (b) localised, inward dimple

The shell considered here buckles at pbif ¼0.126 MPa and the eigenmode has n ¼17 circumferential waves as illustrated in Fig. 3a (pre-buckling) and in Fig. 3b (bifurcation). It is customary to use the eigenshape as the possible shape of initial geometric imperfection and scale it as a function of the wall thickness. The FE

Fig. 12. Deformed shapes at collapse: (i) increased-radius imperfection (Fig. 12a) and (ii) FID-dimple imperfection (Fig. 12b).

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Table 2 Comparison of inward deflection, δ0/t, at the apex for three magnitudes of force, F, in steel torisphere (D/t¼ 1000, Rs/D¼ 1.0, r/D¼ 0.1). F (N)

100.0 200.0 300.0

ABAQUS

BOSOR5

STRI65/S8R

SAX2

0.221 0.510 0.990

0.221 0.512 1.00

0.230 0.546 1.09

Table 3 Maximum magnitude of radial deflection, δ0/t, at various positions along the shell generator in steel torisphere. Fig. 15. Imperfection sensitivity to eigenmode and to Force-Induced-Dimple (FID) shape deviations.

code Abaqus has this procedure implemented in it. Hence, calculations have been carried out for the (δ0/t)-imperfection-ratio being: 0.0 r δ0/tr 2.0. A given shape perturbation has been added to the initial geometry and then the Riks method has been used for tracing the collapse. Results are shown in Fig. 15. Collapsed shell, corresponding to δ0/t¼ 0.5 is depicted in Fig. 16a. It is seen here that the collapsed shape is through the growth of displacements at all of 17 dimples associated with the adopted eigenmode-type imperfection. After a quick drop in load carrying capacity, the magnitude of collapse load remains by-and-large unchanged for δ0/tZ0.3. Similar trend was also noted in previously studied torispherical shell with geometry given by D/t¼200, Rs/D¼1.0, r/ D ¼0.15 – see Ref. [20], Fig. 8. 4.2. Force Induced Dimple, FID, imperfections In this case the imperfect patch was initially localised at the apex and then positioned arbitrarily along the meridian. It was assumed that concentrated force, F, generating the inward dimple of magnitude, δ0/t, stays in place when incremental external pressure is applied. Force, F, is always acting in radial direction. Positioning the FID at the apex allows comparisons to be made between predictions given by axisymmetric modelling in the FE (SAX2 elements in ABAQUS library, and BOSOR5) with the 2D modelling using shell elements (STRI and S8R elements). Table 2, for example, provides comparisons between the maximum inward deflections, δ0/t, obtained for different modelling, and for the magnitudes of the indentation force being: F¼100 N, 200 N, and 300 N. It is seen here that there is very little difference between results obtained using different approaches (i.e., SAX2, STRI & S8R,). All results obtained from Bosor5, on the other hand, are higher than those obtained from the FE Abaqus code. This is likely due to modelling of concentrated force which in Bosor5 relies on the use of a small patch rather than explicit application of force at

Denting Force, F (N)

s/stot 1.0 (apex) 0.85 0.69 0.55 0.27(1) 0.19(2) 0.15 0.10

100.0

200.0

300.0

δ0/t 0.22 0.22 0.22 0.22 0.22 0.089 0.046 0.043

0.51 0.51 0.51 0.51 0.52 0.187 0.093 0.087

0.99 0.98 0.99 0.98 0.96 0.294 0.143 0.133

Notes: 0.73(1) Force, F, is applied at the maximum radial deflection at buckling; 0.81(2) Force, F, is applied at spherical cap – knuckle junction.

a single node. Table 3 gives values of, δ0/t, for imperfection patches positioned in the spherical part of the dome. The magnitude of inward deflections caused by the force, F¼100 N, 200 N, and 300 N, and measured under the application point, are significantly smaller when the force is applied within the knuckle (the last two rows of Table 3). This clearly is the effect of stiffening caused by clamped boundary condition at the base. For all of these FID-imperfections, the sensitivity of buckling pressure to the magnitude of, δ0/t, was computed. Typical responses are plotted in Fig. 17 for selected positions of the dimple along the meridional arc length (s/ stot ¼0.15, 0.19, 0.27, 1.0). View of collapsed shell with the (FID)imperfection, placed at the same place where the eigenmode has the largest inward displacement (s/stot ¼0.27), is plotted in Fig. 16a.

5. Conclusions The study reveals complicated nature of imperfection sensitivity of buckling pressure to the initial deviations from perfect shape, in externally pressurised domed end closures. Out of several possible imperfection profiles, the lower-bound approach based on the increased-radius flattened patch offers nearly always

Fig. 16. View of collapsed torispheres: FID imperfection positioned at s/stot ¼ 0.27 (Fig. 16a), eigenmode affine (Fig. 16b). In both cases δ0/t ¼0.5.

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[2]

[3]

[4]

[5] [6]

Fig. 17. Imperfection sensitivity of buckling pressure to inward dimples (FID) being positioned at different meridional locations (D/t ¼1000, Rs/D ¼ 1.0, r/D ¼ 0.10).

[7]

the largest reduction in buckling strength, i.e., it provides designers with a safe design at the preliminary stage (providing the worst possible scenario). But a local, inward dimple created by a concentrated force was found to be also a dangerous imperfection profile. Computations show that in this case the pressure reduction is comparable to the lower-bound approach. From efficiency of computing the Force Induced Dimple (FID), is far superior than lower-bound approach. Per given amplitude of the imperfection one needs to perform only a single analysis for the FID whilst several analyses are required to secure the lower-bound answer. There is however a difference in physics between both approaches. The lower-bound is truly based on the initial shape deviations from perfect shape. Once the imperfect geometry is described the buckling strength is evaluated. In the FID approach the applied force can remain active during the buckling strength analysis driven by external pressure. After denting one can remove the force and allow the elastic spring back. Any plastic straining would remain in the subsequent buckling strength analysis. This scenario is probably the closest to the real situations where damaged shells do not undergo any heat treatment and the wall contains residual stresses. Yet, another possibility is associated with the assumption of annealing the wall after the force application. In all of the latter two scenarios the reduction in buckling strength is not as big as in the previous two. One additional feature of FID-approach is worth mentioning. In compound shells (e.g., in torispheres) this method is capable of correctly tracing failure mechanisms and the corresponding magnitudes of loading. When imperfection occurs at the spherical portion it does not lower the buckling strength until it is sufficiently large (verified experimentally). Equally when placed at the junction between two segments (spherical cap – knuckle) it reduces the buckling strength significantly. Here the use of eigenmode related shape imperfections appears to be incorrect on both counts, i.e., at the spherical portion and at spherical cap – knuckle junction. The lower-bound approach has been verified by available experimental data but in view of a number of modelling possibilities of FID it would be interesting to see carefully conducted experiments in order to check validity of the FE predictions.

[8]

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