Buckling design of axially compressed steel silos on discrete supports

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1 Articles Arne Jansseune* Jan Belis Wouter De Corte DOI: /stco Buckling design of axially compressed steel silos on discrete supports Comparison between FE-calculated buckling strengths and Eurocode interaction curves This paper compares the buckling strengths derived from a validated shell-based FEM model of discretely supported axially compressed silos with the interaction curves given in EN for axial compression. The buckling strengths were numerically determined for a wide range of geometries and yield strengths, considering two configuration types of discretely supported steel silos: with engaged columns and partial-height U-shaped longitudinal stiffeners. For these types, remarkably good correspondence was found between the numerical results of discretely supported steel silos and the interaction curves for uniformly compressed cylindrical steel silos, although unconservative in many cases. 1 Introduction Generally, the governing design state of discretely supported axially compressed steel silos is elasto-plastic buckling (i.e. interaction of instability and yielding). To estimate the elasto-plastic buckling load, the relevant Eurocode (EN ) gives three design methods: 1) the stress design approach, 2) the MNA/LBA approach, and 3) the GMNIA approach [1]; each involves varying levels of calculation effort and accuracy. The choice for one or other of these methods depends on the size (expressed in tonnes) and the complexity of the silo. As discretely supported silos exhibit zones of high stress gradients and stress concentrations, e.g. in the silo wall above the supports, excessively conservative predictions are obtained when the failure load is estimated on the basis of a stress design approach. As a result, the first method is less suitable. In the case of the second method, two relatively simple numerical shell calculations (MNA and LBA) must be performed, from which the elasto-plastic strength can be estimated on the basis of interaction (or buckling) parameters. However, these interaction parameters for meridional compression are currently missing in the normative document for discretely supported stiffened cylindrical steel silos [1]. Instead, the Eurocode proposes that designers 1) make their own appropriate conservative choice for the interaction parameters (by comparing the current problem with similar buckling problems), 2) switch to the more complex GMNIA approach (i.e. the third method), or 3) use the default interaction parameters given * Corresponding author: Arne.Jansseune@UGent.be in the Eurocode for uniformly compressed unstiffened cylindrical steel silos [1], [2]. In other words, none of the proposed alternatives seems to provide a satisfactory answer to the lack of interaction parameters. Finally, the third and last method requires a more complex and accurate shell calculation (GMNIA = Geometric and Material Non-Linear Analysis with Imperfections). However, this method is more demanding and requires much more knowledge than both the other types in order to choose an appropriate imperfection shape, interpret the failure behaviour correctly and determine the elasto-plastic failure load (different failure criteria) of the imperfect structure. 2 Material and methods 2.1 Scope Silo configurations In this contribution, two types of discretely supported cylindrical steel silo were considered. For lighter silos, a possible solution is to engage the supporting columns over a short distance along the cylindrical barrel, combined with a light transition ring (Fig. 1a). For medium to heavy silos, a possible solution consists of terminating the supporting columns at the transition junction and adding a partial-height U-shaped longitudinal stiffener above each supporting column plus upper and lower ring stiffeners (Fig. 1b). Fig. 1. Supporting and stiffening arrangements for discretely supported silos 10 Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin Steel Construction 11 (2018), No. 1

2 Fig. 3. Cross-section of an engaged column Fig. 2. The cylindrical barrel Geometry The radius-to-thickness ratio R/t of the cylindrical barrel was varied between 100 and 1000, while the silo height h was either 2 or 10 times the cylinder radius R, representing either a short or a long barrel respectively (Fig. 2). The supporting configuration of a discretely supported silo typically has a small number of separate supports. In this study the number of supporting columns n sup was chosen as equal to four or six, equally spaced over the complete circumference and positioned either eccentrically (engaged columns) or concentrically (U-shaped stiffeners) relative to the silo wall (Fig. 3 and Fig. 4). The circumferential width d stif was varied between 0.05 and 0.30 times the cylinder radius R, corresponding to a supported proportion of the circumference m sup of between 3.2 and 19.1 % (for n sup = 4). The ratio of the radial width to the circumferential width w stif /d stif was equal to 100 or 200 % for the engaged columns and 25 % for the U-shaped longitudinal stiffeners. Stiffener thickness t stif was varied from a maximum of silo wall thickness t and 1/40 of the circumferential width d stif to a minimum of 5 times the silo wall thickness t and 1/10 of the circumferential width d stif. The sup ratio of attached height to cylinder radius h stif /R was varied between 0.5 and 2.0 in steps of 0.5, whereas a constant value of 4.0 was adopted for the unattached sup column height h stif /R. Both ring stiffeners had a rectangular cross-section and their thickness was chosen as equal to that of the silo wall. The radial width was equal to 0.10 or 0.20 times the cylinder radius R for the upper and the lower ring respectively. For a full discussion of the choice of the above scope of geometries, the reader is referred to the doctoral dissertation of the first author [3]. 2.2 Numerical model Fig. 4. Cross-section of a U-shaped stiffener All numerical simulations were performed with the Abaqus finite element package [4] using a shell-based numerical model (Fig. 5), which was validated against experimental results obtained from a small number of well-chosen destructive experiments on scale models [3], [5]. A uniform edge load q was applied to the upper edge of the barrel, generating axial compression in the shell wall (Fig. 2). Quadrilateral shell elements (S8R5) were used for the entire model. Boundary conditions restricting the out-ofroundness deformations were imposed on the upper edge and the non-supported lower edge of the barrel, representing the influence of the roof and hopper respectively (which were not modelled). In addition, the model was reduced to 1/8 (n sup = 4) or 1/12 (n sup = 6) of the entire circumference by exploiting symmetry (see Fig. 5). As a consequence, symmetrical boundary conditions were applied to all free edges created by the symmetry transformation. The discrete supports were considered to be fully clamped. Depending on the shell calculation (LBA, MNA, GNA, GNIA, GMNA, or GMNIA), material non-linearity (M), geometric non-linearity (G) and imperfections (I) were taken into account. The material behaviour was either perfectly elastic or ideal elasto-plastic, with a Young s modulus E = 210 GPa and a Poisson s ratio n = 0.3. In order to be able to deduce capacity or buckling curves for silos, the silo slenderness l was varied by changing the yield stress s y from very small values (e.g. 5 MPa) to very large values (e.g MPa) [6], [7]. Although these values may seem unrealistic in practice, they can help with the calculation of the curves. In this way, the overall slenderness can be varied without influencing the imperfection sensitivity. Based on an imperfection sensitivity study [3], an inward weld depression type A was chosen as imperfection shape, with the half-wavelength equal to the linear elastic bending half-wavelength, equivalent amplitudes according to [1], [8], [9] and located in the circumferential direction above the top of the engaged column or the U-shaped longitudinal stiffener (see Fig. 5). Steel Construction 11 (2018), No. 1 11

3 Fig. 5. Numerical model 3 Results and discussion 3.1 Numerical results points, and is compared with the interaction curves of Eurocode 3 determined on the basis of the meridional buckling parameters mentioned in section D.1.2 (Annex D) of EN [1]. It is important to note that these interaction parameters were originally formulated for unstiffened cylindrical steel silos subjected to uniform meridional compression [2]. According to EN , these parameters are rather conservative and may be used for the design of other types of axially compressed silo. Whether or not this assumption is correct for discretely supported stiffened cylindrical shells, where the axial stresses exhibit a clear non-uniform distribution in the circumferential direction, will be checked below. Fig. 7 to Fig. 14 show the data points (Abaqus) and the characteristic capacity curves (Eurocode 3) in χ l plots for silos with radius-to-thickness ratios R/t of 200 and and for a perfect silo wall as well as for imperfect silos with various fabrication tolerance quality classes (i.e. A, B and C). Surprisingly, the lower bounds of the data points are relatively close to the interaction curves. For perfect silos (all values of R/t) and for relatively thickwalled imperfect silos (R/t < 500; l < 1), a large number of In the first instance, capacity curves were determined for a wide range of discretely supported steel silos (1104 in total) and different imperfection quality classes (perfect, A, B and C) by varying the yield stress s y of the ideal elasto-plastic material behaviour. Fig. 6 depicts the capacity curves for one of the geometries in a dimensionless strength χ versus relative slenderness l plot. The upper curve represents the silo without imperfections; the other curves represent the imperfect silo with a weld depression in the critical region (see Fig. 5). Clearly, the dimensionless strength χ decreases when the fabrication quality (i.e. larger equivalent imperfection amplitude) deteriorates. 3.2 Comparison with the existing interaction curves In this section, the large dataset of the above numerical results is reduced to only those combinations of strength χ and slenderness l which correspond to the steel grades used in practice: 235, 275, 355, 460, 690 and 960 MPa. This reduced dataset still contains approx data Fig. 7. Comparison of the numerical results, the existing and the proposed interaction curves for discretely supported silos with R/t = 200 and no imperfections Fig. 6. Example of a capacity curve Fig. 8. Comparison of the numerical results, the existing and the proposed interaction curves for discretely supported silos with R/t = 200 and quality class A 12 Steel Construction 11 (2018), No. 1

4 Fig. 9. Comparison of the numerical results, the existing and the proposed interaction curves for discretely supported silos with R/t = 200 and quality class B Fig. 12. Comparison of the numerical results, the existing silos with rad R/t = a quality class A Fig. 10. Comparison of the numerical results, the existing silos with R/t = 200 and quality class C Fig. 13. Comparison of the numerical results, the existing silos with R/t = and quality class B Fig. 11. Comparison of the numerical results, the existing silos with R/t = and no imperfections Fig. 14. Comparison of the numerical results, the existing silos with R/t = and quality class C Steel Construction 11 (2018), No. 1 13

5 the data points are located below the characteristic interaction curve (20.7% in total), which means that the calculated strength χ Abq is smaller than the lower bound estimated strength χ EC, which is unsafe. In contrast, the design is safe for relatively thin-walled imperfect silos (R/t 500; l 1) where the calculated strengths χ Abq are equal to or larger than the lower bound estimates of the strength χ EC. However, this unconservative situation of the first category of data points is largely nullified by the presence of a partial coefficient applied to the resistance g M1 in design. Indeed, the characteristic value of the dimensionless strength χ is divided by 1.10 [1] to obtain the design value of the strength χ, which corresponds to a reduction of a reduction of approx. 9 %. 3.3 Proposal for adapted interaction curves In the previous section it was found that 20.7 % of the total points in the dataset were located below the Eurocode 3 interaction curves, which corresponds to an unwanted, unsafe situation. Therefore, this section attempts to find interaction curves that fit better to the lower bound of the dataset (Abaqus) and are safe for almost the complete dataset. The search for these buckling curves was carried out by varying the value of all buckling parameters (i.e. a x, b x, l 0x and h x ) within a realistic range with a small step size. The plastic range factor b x and the squash limit load l 0x are constant values, while the interaction exponent h x can be either constant or can vary linearly with the slenderness in the elasto-plastic part of the capacity curve [6], [10]. In addition, two different methods were used for determining the (im)perfect elastic reduction factor a x, (see next paragraphs). The two approaches for both h x and a x result in four different methods. To determine the meridional elastic reduction factor a x, Eq. (1) is used for the first method. Clearly, this formula is very similar to the current expression in [1], except that the constants are replaced by three variable coefficients a, b and c. The ratio of imperfection amplitude to silo thickness Dw k /t is done in the same way as in [1] and depends on the radius-to-thickness ratio R/t and the quality class. a x = 1 + b ( w /t) The second method assumes, for each imperfection quality class, a constant value for the meridional elastic reduction factor a x which is independent of the radius-to-thickness ratio R/t. In Eqs. (2) to (5) the coefficients d, e, f and g are the unknown variables and the second symbol in the subscript of the meridional elastic reduction factor a x represents the imperfection quality class. = d x,0 x,a = e x,0 x,b = f x,0 k c (1) (2) (3) (4) After comparing all combinations of buckling parameters and the consequent buckling curves, only one set of buckling parameters was selected for each method the set for which the corresponding buckling curves fits best to the lower bound of the dataset (Abaqus). An overview of these interaction parameters is given in Table 1. The best-fit lower bound interaction curves (and the Eurocode 3 curve) are plotted in Fig. 7 to Fig. 14 for silos with radius-to-thickness ratios R/t = 200 and and for a per fect silo wall as well as for imperfect silos with various fabrication tolerance quality classes (i.e. A, B and C). From these figures it was found that all new proposed buckling curves fit better to the lower bound of the dataset compared with the existing Eurocode interaction curve. Furthermore, the allowable percentage of points located below the interaction curves is limited to 2.7 % to avoid excessively unconservative design. Furthermore, the proposed interaction curves are located below and are more conservative than the existing Eurocode interaction curve for the entire range of slendernesses in the more thickwalled silos: R/t < 500 (quality class A), R/t < 333 (class B) and R/t < 250 (class C). When the radius-to-thickness ratio R/tincreases, the Eurocode interaction curve shifts downwards and the Eurocode estimate gradually decreases, as a result of which the Eurocode estimate is located below all proposals for larger slendernesses: R/t < 500 (quality class A), R/t < 333 (class B) and R/t < 250 (class C). The best correspondence (i.e. with the smallest value of dev in Table 1) was found for the second method with a variable interaction exponent h x. Since the results presented above are derived as lower bound interaction curves to the scattered dataset, it is important to mention that, for most cases, safe, but also rather conservative, outcomes are obtained when the corresponding interaction parameters are applied in design. This is the main disadvantage of the approach of fitting a lower bound. 4 Conclusions In this paper, FE-calculated buckling strengths for a wide range of axially compressed discretely supported cylindrical steel silos were compared with the relevant Eurocode interaction curves. The main conclusions drawn from this comparison are: 1. In general, surprisingly, relatively good agreement was found between the current buckling curves for meridional compression given in Eurocode 3 (based on the buckling parameters for unstiffened uniformly compressed cylindrical steel silos) and the lower bound of the dataset derived from numerical results (discretely supported stiffened cylindrical steel silos with a marked non-uniform axial stress distribution in the circumferential direction). 2. A number of sets of buckling parameters was proposed in which the corresponding buckling curves fit better to the lower bound of the scattered set of numerical results. x,c = g x,0 (5) 14 Steel Construction 11 (2018), No. 1

6 Table 1. Overview of the parameters of the best-fit lower bounds to the dataset Parameter Eurocode (constant h x ) A. Jansseune/J. Belis/W. De Corte Buckling design of axially compressed steel silos on discrete supports Method 1 (constant h x ) Method 1 (variable h x ) Method 2 (constant h x ) Method 2 (variable h x ) a [ ] b [ ] c [ ] a x [ ] d [ ] e [ ] f [ ] g [ ] b x [ ] l x,0 [ ] l x,p [ ] = f (a x ; b x ) h x [ ] h x,0 [ ] h x,p [ ] under curve [%] dev [ ] References [1] EN Eurocode 3: Design of Steel Structures Part 1.6: Strength and Stability of Shell Structures, CEN, Brussels, [2] ECCS: Buckling of Steel Shells: European Design Recommendations, 5th ed., European Convention for Constructional Steelwork, Brussels, [3] Jansseune, A.: Optimisation of the Stiffening Configuration of Axially Compressed Steel Silos on Local Supports. PhD thesis, Dept. of Industrial Technology & Construction, Faculty of Engineering & Architecture, Ghent University, [4] ABAQUS version 6.9 documentation. Dassault Systèmes Simulia Corp., Providence, RI (USA), [5] Jansseune, A.; Belis, J.; De Corte, W.: Experimental validation of a numerical model for locally supported steel silos. Proc. of Intl. Assoc. for Shell & Spatial Structures (IASS), Amsterdam, [6] Doerich, C.; Rotter J. M.: Generalised capacity curves for stability and plasticity: Application and limitations. Thin- Walled Structures, 49(2011), No. 9, pp [7] Rotter, J. M.: Buckling of shallow conical shell roofs for small diameter tanks and silos. In Proceedings of the International Conference on Design, Inspection and Maintenance of Cylindrical Steel Tanks and Pipelines, Prague, Czech Republic: , [8] Rotter, J. M.; Teng, J. G.: Elastic stability of cylindrical shells with weld depressions. Journal of Structural Engineering, 115(1989), No. 5, pp [9] Song, C. Y.; Teng, J. G.; Rotter, J. M.: Imperfection sensitivity of thin elastic cylindrical shells subject to partial axial compression. International Journal of Solids and Structures, 41(2004), No , pp [10] Doerich, C.: Strength and stability of locally supported cylinders. PhD thesis, Institute for Infrastructure & Environment, The School of Engineering & Electronics. University of Edinburgh, Keywords: cylinder; discretely supported; buckling parameters; design rule Authors Dr.-Ir. Arne Jansseune Ghent University Department of Industrial Technology & Construction Valentin Vaerwyckweg Ghent, Belgium Arne.Jansseune@UGent.be Prof. Dr.-Ir. Jan Belis Ghent University Department of Structural Engineering Technologiepark-Zwijnaarde Ghent, Belgium Jan.Belis@UGent.be Prof. Dr.-Ir. Wouter De Corte Ghent University Department of Industrial Technology & Construction Valentin Vaerwyckweg Ghent, Belgium Wouter.DeCorte@UGent.be Steel Construction 11 (2018), No. 1 15