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The present paper investigates the structural behavior of circular steel silos subject to patch loads. For reference purposes, first the patch loads are assumed to be in two squares that are in generally accordance to silo loading codes [DIN 1055. Design loads for buildings. DIN 1055 Part 6, Deutsches Institute fur Normung, Berlin; May 1987; Bases for design of structures—loads due to bulk materials. ISO11697; 1995]. The investigations reveal that the patch loads have a great effect on the stress states in the silo from the linear elastic analysis (LA). Geometrical non-linearity and primary pressures have beneficial effect. Fourier decompositions of the two square-shaped patch loads show that the effect of the shape of patch loads depends not only on the harmonic index, but also on particular stress component. For a pressure with a lower harmonic index (e.g. cos θ, cos 2θ), only limited effect was observed for all stress components. A pressure with medium-sized harmonic index (cos 4θ, cos 6θ) has a great effect on meridional compressive stress, while for higher harmonic index; the effect was significant for von Mises equivalent stress. Buckling analyses with geometrical non-linearity and material non-linearity taken into account show that the effect of patch loads could be covered by a certain percentage increase of the vertical frictions, if the patch load approach were adequate to represent the non-uniformity of wall pressures in circular flat-bottomed steel silos.

Circular steel silos have been widely used in civil engineering to store various bulk solids. Design of silos has to take two kinds of non-uniform pressures into considerations: (a) wind loads and (b) wall loads induced by bulk solids. The former is closely connected with empty silos, while the latter with fully filled silos. Structural behavior of empty silos subject to wind loads have been investigated in detail by many researchers [3], [4], [5], [6], [7], [8] and [9]. On the other hand, despite buckling failures having been re-produced in early laboratory tests [10], [11] and [12] and frequently reported of late [13] and [14], only limited researches have been carried out on the effects of non-uniform pressures caused by bulk solids [15], [16], [17], [18], [19] and [20]. The investigation of the structural behavior of steel silos is inextricably linked with the description of wall pressures. In the literature, there are dozens of models for the distribution of horizontal wall pressures in silos [21], but no accurate and commonly accepted one is available in this time. Due to the complexity of horizontal pressure distributions, investigations of non-uniform pressures caused by stored bulk solids on structural behavior of steel silos are based on simplified horizontal wall pressure models. Silo loading codes suggest that horizontal pressures consist of two components: axisymmetric (primary) pressures and locally distributed pressures. Therefore, the non-uniformity of the pressures is expressed through adding or subtracting locally distributed pressures to the primary pressures. The locally distributed pressures have different terminology in silo loading codes, for example, partial pressures in the German DIN standard [1], patch loads in International standard [2] and in Part 4 of Eurocode 1 [22], reduced and increased pressures in the Australian standard [23]. In the present paper, the patch loads is adopted for the locally distributed pressures. Rotter [15] investigated the pressure distributions and the structural behavior of silos under eccentric discharge. Linear elastic analyses indicated that non-uniform pressures produced highly non-uniform axial compressive stresses, which in turn worked as catalyst to silo wall buckling. The suggested method in [15] to assess the buckling strength of silos under non-uniform axial compressive forces was later employed for a reference silo (D=12 m and H=3D) [17]. The assessment illustrated that buckling failure might occur well above the wall of a silo. Gillie and Rotter [20] carried out parametric studies to investigate stress distributions in silos subject to patch loads. The parameters included (a) the distribution within; (b) the position of; and (c) the distribution area of the patch loads. The investigations [15], [16], [17] and [20] revealed that that the stress distributions in silos subjected to non-uniform pressures were very complex. Significant Mises stress and compressive membrane stresses arose in the wall of silos, which meant that both elastic buckling and plastic collapse were possible. However, these researches were based on the linear elastic theory, the effect of material and geometrical non-linearity was not included. Guggenberger’s research [19] involved both material and geometrical non-linearity. The distribution of the horizontal pressures was generally in line with silo loading codes [1] and [2]. The research [19] leads to the conclusions that (a) geometrical non-linearity has a significant beneficial effect and (b) material non-linearity has a weakening effect. However, Guggenberger’s research was confined to investigate the load carrying capacity of silos under non-uniform horizontal pressures and the interaction of the primary pressures and the patch loads, no vertical frictions were considered. Observed failures mainly due to the circumferential stresses though axial compressive stress was found to be important in a few cases. This kind of failure mode contradicted to the expected dominant failures caused by or connected to the axial compressive stresses as mentioned in [20] for practical silos. Based on the pioneering work [15], [16], [17], [18], [19] and [20], the present paper continued to investigate the structural behavior of circular thin-walled steel silos subject to non-uniform pressures. As the first step, a reference silo subject to two square-shaped patch loads was issued, followed by the discussion of the effects of the primary pressures on the stress states in the silo. Then, the square-shaped patch loads were decomposed into Fourier series to investigate the effect of the shape of patch loads in single harmonic term on structure behavior. Finally, the effects of the patch loads on the buckling loads were discussed. Both material and geometrical non-linearity were taken into consideration in the present paper.

A detailed investigation on the structural behavior of steel silos subject to patch loads was carried out in the present paper. The study on the distributions of the stresses (including von Mises equivalent stress) revealed that the structural behavior was very complex. Both plastic collapse and elastic buckling could be probable in silos subjected to wall loads including patch loads. Geometrical non-linearity and primary pressures were identified to have beneficial effects. Fourier decompositions of the two square-shaped patch loads and the followed numerical analyses illustrated that the form of the patch loads had great influence on the structural behavior. Non-uniform pressures expressed in an individual harmonic term had different performances when geometrical non-linearity and the primary pressures were considered simultaneously. Beneficial effects from these two factors were more significant for pressures expressed in the harmonic pressures with higher index. Comparison between the buckling loads of the silo subjected to different shapes of patch loads demonstrated that the buckling loads also varied with the shape of the patch loads. Patch loads recommended in Part 4 of Eurocode 1 [22] had little effect on the buckling loads, while the patch loads, expressed as the two squares and a cos (4θ) curve, might have somewhat great influence on the buckling loads. On the other hand, increasing the vertical frictions by certain percent, for example 10% as suggested in the German code [1], may be a feasible way to account the detrimental effects of the non-uniformity of the horizontal wall pressures on the buckling loads, if the non-uniformity could be expressed by patch loads.