On the resistance of arbitrarily ring‐stiffened welded bins subject to axial compression

Ring‐stiffened flat bottom tanks are widely used in many industries for the storage of liquids, such as oil, fertilizers, water, sewage and sludge. Material efficiency is achieved by combining thin plates with ring‐stiffeners. When the bin is covered by a dome or cone roof, high meridional forces due to live or snow loads may trigger axial buckling of the shell. Ring stiffeners are beneficial, not just against buckling due to wind, but to reduce the risk of stability failure of tanks and silos caused by meridional compression as well.


Introduction
Ring-stiffened shells are widely used as tanks or silos to store a variety of goods, such as water, oil or grain and other solids. The stiffeners are typically intended to strengthen the structure against wind and external pressure.
Covered bins may suffer meridional compression due to snow and roof live loads as well as heavy machinery placed on top of the roof. However, it is still possible to achieve an economical plate thickness when the reduced imperfection sensitivity, which is one of the benefits when using stiffening rings, is appropriately considered. It was shown in [1] that especially closely spaced ring-stiffeners at a distance of up to one buckling wave (BuW, lbuW = 3.46√ ) may allow for huge strength gain compared to unstiffened shells. Considerably larger capacity is achieved with weak stiffeners (10 % of the connected wall cross section) while heavy stiffeners (100% of connection wall cross section) allows to reach about 90% of the classical critical load. An extension to arbitrarily stiffened shells has been provided in [2] and [3] while the most recent and complete overview has been published with [4].
The previous research has been conducted employing parametrical studies using an axis-symmetric eigenform-affine imperfection. This kind of imperfection is known as being deleterious but not practically relevant. Furthermore, it does not allow to take into account a special localized placement of an imperfection.
While for unstiffened shells a weld type A shape deviation according to [5] leads to similar results as the axis-symmetric eigenform-affine imperfection, for ring-stiffened shells no specific studies are available on ring-stiffened cylinders that allow a comparison. Besides [6] there is no study available specifically dealing with the placement of a weld in relation to the stiffeners and the influence of this particular configuration on the resistance against axial compression.
For this paper the results of [6] have been extended for two stiffening ratios kst (eq. 1) to gain an overview for practically relevant parameter combinations. Effects of certain parameters are discussed briefly and a calculation procedure is proposed. For that, equations for the buckling curve parameters were extracted from the modified capacity curve [7] for all combinations of parameters. It is shown that interpolation between few reference parameter combinations allow a safe-sided design procedure. where Ast = cross-sectional area of the stiffener, ast = stiffener spacing and t = thickness of the connected shell

Analysis procedure and parameters
The analysis was carried out using the commercial FEM software Sofistik®, which is widely used in structural engineering practice and has been approved for many finite element calculations [8]. The "QUAD" element, a four node shell element using linear shape functions to calculate the deformations, was used. The reference steel grade S 355 was used without hardening (fy = fu -1 = 355 MPa) for all calculations. The mesh convergence study carried out for [6] showed that a meridional length of 0.24√ yields satisfactory results when the circumferential length does not exceed 0.96√ .
The length l of the shell was kept approximately at a length to radius (l/r) ratio of two if the stiffener configuration allowed it. Depending on the spacing of the stiffeners ast the length had to be adjusted so that in any case three rings were present. Only a quarter of the cylinder was modelled due to geometrical and load symmetry. Appropriate boundary conditions were applied on the vertical edges. The horizontal edges were chosen to represent continuity with very weak radial springs of 1 kN/m stiffness to ensure numerical stability. The edge was clamped around its tangential direction. The bottom had restrained vertical movement while the top was free to move vertically. The load was applied as a uniform line load at the top edge.
The geometrically and materially nonlinear analysis (GMNA; GMNIA with weld type A imperfection [5] ) was carried out employing the Newton-Raphson iteration scheme. Sofistik uses the modified approach in combination with a line-search. However, in some cases the full Newton-Raphson algorithm allowed for better convergence. For comparison, a dynamically stabilized quasi-static analysis was used. The increments were chosen as one percent of the critical load. The resistance was determined by carrying out an eigenvalue check for every four increments along the load deflection path. When the critical load factor switched from above one to below one between two load increments, the limit load was determined by linear interpolation between the adjacent load increments.
The radius over thickness (r/t) ratio is not directly varied but instead the yield strength of the material is changed to eliminate the effect of geometrical nonlinearity. The reference radius r is kept constant at 1000 mm with a thickness t of 1 mm.

Unstiffened shells
It was shown in [6] that the numerical results derived during the study align well with findings published in the literature. It is important to note that the weld type imperfection creates a length dependence of the capacity [10], which cannot be observed when an eigenform-affine imperfection is chosen.
The proposed modification of the formulation of the elastic buckling reduction factor α (eq. 2) was derived from [9] with an optimization in the range of imperfection depths Δw/t up to 2, which is approximately equal r/t = 2500 for fabrication tolerance class B (FQC B), and r/t ≈ 1000 for FQC C. Very slender cylinders in any case achieve a buckling capacity of about 12% of the critical load [11]. With the current codified formulation for α this effect may not be accounted for. Hence, for this paper different equations for α, β and η were used to more economically describe the behavior.

Figure 2
Buckling curves and experiments from [11] in the slender range Taking a closer look at the buckling parameters, which are proposed in [12], in comparison to the experiments gathered by [13], it is observed that the capacity for FQC A (χA) and C (χC) is determined quite unconservatively in the stocky range ( fig. 1). The extended plastic plateau yields an over estimation of up to 50% for FQC C at λ = 0.45. Using the set of parameters, which are proposed in [4], a lower bound curve on the tests can be produced.
While the current code [12] underestimates the true resistance of unstiffened shells in the slender range, using the proposal of [4] yields more economical yet safe results ( fig. 2).

4
Parameter study of ring-stiffened shells

Chosen parameters
The stiffener spacing was varied by multiples of the bending halfwave length (BeHW, lbeHW = 2.44√ ). It was anticipated that less heavy rings need to be placed closer to achieve a remarkable effect while heavy rings stiffen the shell even if they are more distant to each other. Hence, for the ring parameter kst = 0.40 the values for ast / lbeHW were chosen to 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0 and for kst = 1.00 the parameter ast / lbeHW was set to 2, 3, 4, 5, 6, 8 and 10.
The imperfection was considered to be in the line of the stiffener (aimp = 0) and away from the ring at aimp = ast / 8, ast / 4, 3 ast / 8, ast / 2.

Influence on elastic imperfection reduction factor
The variation of capacity with increasing imperfection depth Δw/t is depicted in fig. 3 for several positions of the weld imperfection. It is observed that close and heavy stiffeners reduce the imperfection sensitivity efficiently, especially when the shape deviation is in the vicinity of the stiffening ring. When an imperfection is in the line of the stiffener, circumferential compression due to the inward deformation, which results form the imperfection and meridional compression, is beared mostly by the strong ring. Since the shell suffers only minor circumferential compression, a chequer-board pattern is   unlikely to develop. Buckling can be triggered by meridional bending only. As the distance of the ring beam to the weld grows, this stiffening effect vanishes. Interestingly, between aimp = 1/8 … 1/4 ast the trend is reversed and larger capacity is achieved with higher distance of weld to stiffener. This may be due to the afore mentioned length effects coming with the weld imperfection. Several analysis runs with different mesh configurations were made to make sure that no numerical problem was present. None of the attempts significantly altered the trend shown in fig. 3. For a design procedure, this effect shall be neglected and a clear trend with descending resistance when the distance of weld to stiffener is increased should be assumed. Even at the most unfavorable position of the weld, midway between two stiffeners, the shell still is about twice as strong as an unstiffened cylinder.
The best placement for a weld is close to the stiffener. This is demonstrated in fig. 4. Closely spaced rings allow for high capacity even at deep imperfections. However, the reduction of resistance does not drop below around αx,rst ≈ 0.5 even for stiffening rings, which are so far away from each other that no stiffening effect was anticipated. The capacity is in each case is higher than for the unstiffened shell.
A criterion should be specified in the design code, to define at which spacing a stiffened cylinder becomes an unstiffened one and for which parameters further weld depressions or similar imperfections need to be taken into account to make sure that the determined capacity is sufficiently safe. In practice, standard format plates or coils may already set up this criterion for large tanks. For smaller bins however, the linear bending half-wave may be quite small so that special care needs to be taken to make sure that the assumptions of a structural analysis and the erection on site align.
As expected, the stiffened shells' resistance is close to an unstiffened cylinder as can be observed in fig. 5. As the distance between two rings gets larger, the tendency for increased strength with higher imperfection amplitude gets more pronounced. Again, several modifications in the model and the analysis procedure have been attempted to alter the results. However, the derived capacities remained approximately the same in each case. While it is well known that imperfection may enhance the capacity rather than lowering it, in the case of ring-stiffened shells, this effect may have a mechanical background rather than numerical reasons. As the imperfection gets deeper, the meridional bending moment increases, which leads to increased circumferentially alternating tension and compression membrane forces. As the stiffener keeps the shell circular, buckling can only be triggered away from the stiffeners. Each circumferential compression area is elastically supported by adjacent circumferential tension. The load at which circumferential buckling triggers the collapse may therefore increase because the tension force's support overweighs the negative influence of the deeper imperfection.
( / ) = 1000 /355 Comparing figs. 6 and 7, the dependence of the elastic imperfection reduction factorαx,rst from the effective r/t ratio is illustrated. Like unstiffened shells, weakly stiffened shells show only few to no strength gain when the r/t ratio is increased ( fig. 7). For the present case this is true for fy ≥ 89 MPa (eq. r/t ≈ 240) up to Δw/t = 2. A deeper imperfection causes a huger drop of capacity due to yielding prior to buckling when the shell is thick. Material strength lesser effects the resistance with decreasing shell thickness. Buckling then becomes elastic as it is usually expected for unstiffened shells for a large range of r/t ratios. It can be observed in fig. 7 that exceeding the yield strength of 533 MPa (eq. r/t ≈ 1500) does not result in any further gain in capacity, which means that all thinner shells buckle elastically even at very deep imperfections of 5 times the shell thickness.
When heavy rings at a close distance are employed, even at very deep imperfections, high capacities may be achieved when thinner shells are employed ( fig. 6). In the case of fy = 3550 MPa (eq. r/t = 10000), even the classical bifurcation stress can be achieved while for thicker cylinders very high elastic imperfection reduction factors were calculated. Exceeding α = 1 is only possible due to numerical effects, especially at deeper imperfections.
Ring stiffeners efficiently reduce the imperfection sensitivity, which is lowest for stocky, thick-walled shells and very high for very thinwalled cylinders. Ring stiffeners may reduce the imperfection sensitivity so much that even for r/t = 10000 the yield strength may alter the resistance of the shell [4].
Hence, it is clear that accounting for the elastic imperfection reduction factor may alone is not sufficient to determine the capacity of the shell and considering the interaction of elastic and plastic buckling is much more important for heavily stiffened shells as in the case of unstiffened cylinders. The range in which plasticity is involved in the buckling process may be evaluated using the plastic limit relative slenderness ̅ . determined by eq. 6. The plastic range parameter βx,rst needs to be explored more in detail. This is best achieved by plotting the numerical results in the modified capacity curve.

Modified capacity curve
The plastic range parameter cannot be directly calculated from the numerical results but can be determined by evaluating numerical values plotted in the modified capacity curve.
In many cases, clear trends of β can be determined, as can be seen in fig. 8. With increasing imperfection depth smaller values of 1-β are found. At a constant value of α, decreasing 1-β results in an extended elastic plastic interaction range. Since α and β usually decline as the imperfection gets deeper, only small variation of ̅ is determined, which means that the influence of the elastic imperfection reduction factor governs the capacity in a wider range.
The plastic limit relative slenderness remains almost constant for the parameters chosen in figs. 8 and 9. The trends of the curves of αx,rst and βx,rst quotient converge to be parallel ( fig. 9).
No clear trend can be deduced from the modified capacity curve for the heavily stiffened shell, which is depicted in fig. 10. An interpolation of the buckling curve parameters is anyway possible (fig. 11). While the curve for αx,rst shows almost no reduction with imperfection depth, a clear descending path of the curve of 1βx,rst can be observed. This means that with increasing imperfection depth the range of elastic-plastic interaction during the buckling process extends. Since by code more slender shells are deemed to have deeper imperfections consequently these shells profit more from heavy stiffening as the plasticity is involved in the buckling process even for very thin cylinders.
Obviously, the benefit of ring-stiffeners is most pronounced for very thin shells. However, to reach high ring parameters such as kst = 1, quite heavy rings are necessary, e.g.

The effect of the weld position
The influence of the weld position is depicted in fig. 12 as a relative value to αx,ref, which was chosen as for kst = 1, ast = 2 BeHW, S 355, aimp = 0. As a further reference, the unstiffened shell has been included as aimp = oo. It can be expected that the loss of resistance is most detrimental for the heaviest stiffened shell with closest stiffener spacing. For comparison the curves are plotted for a stiffener spacing of ast = 5 BeHW in fig 13. Clearly, the loss of capacity is less detrimental for this case. The curve shape has been interpolated using eq. 7, which lead to the parameters in tab. 1.
, / , , = + Δ / (7) No interpolation was attempted for βx,rst. For a safe-sided approach, adopting the equation for the plastic range factor of the unstiffened shell is sufficient. The buckling curve exponent ηx,rst does not alter the resistance as much as αx,rst or βx,rst so that adopting the equation of the unstiffened shell is satisfactory for a lower bound estimate.

Influence of the ring parameter
The ring parameter reduces imperfection sensitivity. Equivalent natural imperfection amplitudes should capture this effect. At the example of fig. 5 the relation given with eq. 8 can be deduced, which approximately allows to account for reduced imperfection sensitivity depending on the ring parameter.
Furthermore, weaker stiffeners do not yield the same effect as heavy stiffeners. However, even small rings may enhance the resistance considerably. This effect may be approximately accounted for by reducing the part of the elastic imperfection reduction factor, which exceeds αx of the unstiffened shell according to eq. 9.

Influence of the stiffener spacing
It can be taken from figs. 3 and 4 that at ast = 3 BeHW a minimum of αx,rst is determined for the group of distances of rings from 3 to 8 BeHW. It is hence assumed for the simplified design procedure that a straight line interpolation between 2 and 3 BeHW as well as between 8 and the unstiffened shell (10 BeHW) yields satisfactory results. The elastic imperfection reduction factor is constant between 3 and 8 BeHW.
Proposal for a simplified design procedure The procedure presented in this section is valid for ring parameters up to kst = 1 and a r/t ratio ≤ 10000. The minimum considered ring spacing is 2 BeHW.
To determine the characteristic capacity of a ring-stiffened shell with arbitrarily placed welds between the rings, the followings steps have to be undertaken.

Check of the design procedure
The design procedure has been checked against the outcome of the FEM study for the ring parameters kst = 0.40 and 1.00 (figs. 13, 14) using FQT C (Q = 16).
Less scatter is achieved when linear interpolation between all the parameter combinations is used. The required coefficients determined from the modified capacity curve are available upon request.

Weld in line of stiffener
In the stocky slenderness range a major part of the results is unconservative. This is due to the consideration of all imperfection depths. Typically, thick-walled cylinders have only little characteristic imperfections so that in fact the appropriate calculated design results do not overestimate the FEM calculations.
As the slenderness grows, results align in a band between the values χx,rst,FEM / χx,rst,cal = 0.90 and 1.3 ( fig. 13), respectively 0.90 and 1.60 for the medium stiffened shell. The design procedure hence is safe and, taking into account the complexity, allows quite accurate predctions of the numerically derived resistance.

All parameter combinations considered
When all parameter combinations are considered the scatter is larger than for the situation of welds in the line of stiffener. The scatter reaches from about 0.8 up to 5 in th case of the heavily stiffened shell ( fig. 15) while for the medium stiffened cylinder the scatter is slightly reduced (fig. 16). This outcome is partially due to increasing resistance with increasing imperfection depth in the FEM analysis, which cannot be captured with a hand calculation procedure that assumes descending capacity with increasing imperfection depth. Furthermore, if only the appropriate characteristic imperfection depths are considered, the scatter is reduced.
Less scatter may be achieved with variations in the proposed formulae. However, to capture the behavior of the about 3250 data points acquired by numerical calculation with simple algebraic formulae is a quite ambitious undertaking, especially as all the considered parameters depend non-linearly from each other.
The more promising and more economical approach would be to use the equations determined using the modified capacity curve and programming the formulae in a spread sheet. The resistances may then easily be determined using linear interpolation of the buckling reduction factors between adjacent parameter sets.

Conclusion
A finite element study concerned with the axial buckling resistance of arbitrarily ring-stiffened welded shells has been conducted with a special focus on the effect of the distance of the imperfection to the ring stiffener. The influence of the weld position, the ring parameter and the stiffener spacing on the capacity has been briefly described. A simplified design procedure has been worked out, which uses equations for the buckling curve parameters of two parameter sets of the stiffened shell and the unstiffened cylinder as reference resistances. Linear interpolation between the three cases allows to safely determine the enhanced buckling capacity of arbitrarily ringstiffened shells with randomly placed welds using a hand calculation procedure.