Pressure profiles exerted by grain-like material on the silo walls in accelerated conditions

Transcription

1 Pressure proiles exerted by grain-like material on the silo walls in accelerated conditions T. Trombetti, S. Silvestri, G. Gasparini Department DICAM, University o Bologna, Italy D. Foti Department DICA, Technical University o Bari, Italy S. Ivorra Chorro Universidad de Alicante, Apartado 99, Alicante, Spain SUMMARY: This paper presents analytical developments devoted to the evaluation o the eective behaviour o grain in latbottom silos during an earthquake. Based on the results achieved rom this study, the paper proposes an innovative methodology or the seismic design o lat-bottom silos containing granular and grain-like materials, which allow or substantial reductions in the design seismic actions or silos characteried by squat geometrical coniguration. The analyses are developed by simulating the earthquake ground motion with constant vertical and horiontal accelerations and lead to the subdivision o the ensiled material into three dierent portions by means o plain dynamic equilibrium considerations that take into consideration the speciic mutual actions developing in the ensiled grain. The indings indicate that, in the case o squat silos the portion o grain mass that interacts with the silo walls turns out to be noticeably lower than the total mass o the grain in the silo. Keywords: lat-bottom silos, grain-like material, dynamic equilibrium, riction coeicient, seismic orces. INTRODUCTION In the general issue o the action o grain-like materials, during earthquake ground motions, on the walls o lat-bottom silos, the assessment o the horiontal interaction is o particular interest. This interest is based on the possibility o providing more appropriate design rules closer to the eective seismic behaviour o silos. A careul evaluation o the orces produced by the material in the silos makes it possible to saely design silos in a seismic area without waste o material and excessive redundancy. The walls o the silos are typically subjected to both normal and tangential stresses (normal pressures and vertical and horiontal riction orces) produced by the material inside the silo. The magnitude and distribution o both shear and normal pressures over the heit o the wall depend on the properties o the stored material and whether the silo is being illed or discharged. As it is the case or most structures, the eect o lateral loads can be signiicant especially on larger silos containing heavier material since the magnitude o the horiontal seismic load is directly proportional to the weit o the silo (Dogangun et al. 2009). For the material inside the silos, the horiontal actions are usually evaluated using the ollowing hypotheses (urocode 8 part 4):. sti behaviour o the silo and its contents (which means that the silo and its contents are subjected to the input ground accelerations; no ampliication is taken into account); 2. the grain mass corresponding to the whole content o the silo with the exception o the base cone with an inclination equal to the internal riction angle o the grain is balanced by the horiontal actions provided by the walls (supposing that the seismic orce coming rom the base cone is balanced by riction and thereore does not push against the walls). Althou there are many papers on the behaviour o liquid silos under earthquake ground motion (amdan 2000, Nachtigall 2003), there are only ew examples o scientiic investigation into the dynamic behaviour o lat-bottom grain silos under earthquake ground motion.

2 The main goal o this paper is to present an analysis o the eective behaviour o lat-bottom silos containing grain when subjected to constant horiontal and vertical accelerations, which are here taken to simulate earthquake ground motion (time-history dynamic analyses are not carried out). In more detail, the developments presented here, keeping the validity o the hypothesis, is aimed at assessing, on the basis o plain dynamic equilibrium considerations, the eect o the horiontal actions that rise on the silo walls due to the applied accelerations. The results obtained show how these horiontal actions may be ar lower with respect to those that can be obtained by adopting the urocode approach (i.e. using also the hypothesis 2), especially or silos characteried by squat geometrical coniguration. To better understand the physical meaning o the results obtained, a three-dimensional representation o the results in terms o those portions o the grain mass that act (in terms o horiontal actions) upon the silo walls is also provided, in addition to the analytical expression o the horiontal actions. The results obtained are then used to ormulate a procedure or the seismic design o silos. 2. PROBLM FORMULATION A silo with radius R and illed with grain-like material up to the heit, as represented in Fig. 2., is considered. The ree surace o the grain is assumed to be horiontal. With the objective o obtaining an approximated estimation o the pressures which the grain-like material produces on the silo walls due to the earthquake acceleration, an idealised system is studied in idealised conditions. y y x x R (a) (b) Figure 2.. Geometry o a lat-bottom grain silo and reerence system adopted. (a) vertical view; (b) plan view. The ollowing idealised system is considered as representative o the lat-bottom silo illed with grainlike material: the silo walls are assumed to be ininitely sti with respect to the ensiled grain; the grain-like material is assumed to be incompressible and compact, without voids, as it were be composed by a number o ininitely sti and ininitely resistant spherical balls, as depicted in Fig R grain wall wall V V V V V V V V grain base V V GB V V GB V GB GB base (a) (b) Figure 2.2. (a) Idealised system. (b) Mutual orces which are exchanged between a grain and another one, between the grain and the silo wall, and between the grain and the silo base.

3 It is well known that the grain provides orces onto the silo walls (Poati an Ceccoli 972). Fig. 2.2 represents the mutual orces o the schematic idealisation adopted: V,, is the vertical normal orce, perpendicular to the grain surace, which is exchanged between a single grain and another one;,, is the horiontal normal orce, perpendicular to the grain surace, which is exchanged between a single grain and another one;,, is the horiontal tangential orce, parallel to the grain surace, which is exchanged between a single grain and another one; V,, is the vertical tangential orce, parallel to the grain surace, which is exchanged between a single grain and another one;,, is the horiontal normal orce, perpendicular to the grain surace, which is exchanged between the grain and the silo wall; V,, is the vertical tangential orce, parallel to the grain surace, which is exchanged between a single grain and the silo wall;,, GB is the horiontal tangential orce, parallel to the grain surace, which is exchanged between the grain and the silo base; V,, GB is the vertical normal orce, perpendicular to the grain surace, which is exchanged between the grain and the silo base. It is here assumed that: the normal orces ( V,,,,,,,, and V,, GB) do not have limitations; the tangential orces are limited by the riction law o the contact surace considered, i.e.:,, V,, V,,,,,, GB GB V,, GB V,,,, where, GB and are the riction coeicients o the contact suraces grain-grain, grain-base and grain-wall, respectively. Note that, implicitly, urocode 8 assumes,, 0, which implies, under earthquake input, a lateral sliding behaviour o each grain layer upon the one below, so that the lateral equilibrium is ensured by the reaction o the walls which take all weit o the grain. In order to perorm an integral evaluation o the global orces that the grain produces on the silo walls, the grainlike material is treated as a set o overlapped layers o ininitesimal heit d (passage rom discrete treatment o grains to continuous treatment o grain-like material), and the above-mentioned concentrated orces becomes distributed normal pressures p and tangential stresses t, respectively: becomes v, becomes v, becomes V,, V,,,, GB p ;,, ;,, ; hgb, V,, GB becomes h, becomes h, becomes p. p ;,, p ; V,, vgb, becomes h, becomes ; ; According to Janssen (895) and Koenen (896), or the vertical translational equilibrium o a grain portion at a generic heit, the vertical pressures, p, at the base o this portion are equally distributed over the whole surace. owever, it is reasonable to assume that the vertical pressures tend to diminish rom the core o the grain portion towards the silo walls where their value is equal to ero. A limiting schematisation (that will be useul or the assessment o the actions induced on the silo walls by the applied accelerations and that lead to conservative results) is the one where each grain layer is divided into two equivalent portions composed o (i) grain completely leaning against the layers below (central portion) and (ii) grain completely sustained by the walls (and thereore characterised by a null vertical pressure between one grain and another). This schematisation requires that it exists a speciic distance s rom v, v,

4 the silo wall, in correspondence o which v, =0 and such that p, distant less than s rom the silo wall (Fig. 2.3). v =0 or all points which are s h, p h, p h, p h, p h, p v, d v, v, h, Figure 2.3. Subdivision o each grain layer: grain completely leaning against the layers below (internal disk) and grain completely sustained by the walls (external torus). Thereore, with reerence to a horiontal layer o grain characterised by heit d and placed at a generic distance measured rom the ree surace, it can be divided into two portions: an internal disk with a diameter o 2r (corresponding to the grain leaning on the layers below), hilited in Fig. 5 with blue hatching; an external torus with a unknown thickness s (corresponding to the grain sustained by the walls), hilited in Fig. 5 with red hatching. The dimensions o the internal disk and the external torus vary with the distance measured rom the ree surace o the grain, as the thickness s o the external torus varies with. r R s and is placed at a depth The internal disk D is characterised by heit d and radius measured rom the ree surace o the grain, namely at heit h = - rom the ground. The external torus is subdivided into sectors (o circular annulus), each one o them, with central angle d, is identiied by the central angle measured clockwise rom the negative semi-axis o x, as indicated in Fig. 5. ach sector o the external torus, which will be reerred herein to as element, is characterised by heit d and thickness s. A system o auxiliary coordinates (O, on the horiontal plane is also deined, where represents the radial direction (perpendicular to the lateral surace o the silo) and represents the direction perpendicular to, as indicated in Fig d y x h d y r R x s (a) (b) Figure 2.4. xternal torus (red hatching) and internal disk (blue hatching) o the grain. The above-described idealised system is studied in the ollowing idealised conditions. The earthquake ground motion is simulated with in time constant vertical and horiontal accelerations. Provided that the silo is assumed to be ininitely sti, no ampliication is here considered and thus spectral accelerations coincide with ground accelerations. Accelerated conditions with both in time constant vertical ( agv g) and horiontal ( a g ) additional accelerations will be thus studied in detail (Fig. 6). ere, the expression additional means

5 additional with respect to the acceleration o gravity. Note that both a gv and a are expressed as ractions o g. The objective o the ollowing sections is to determine the value o the thickness o the external torus in accelerated conditions, by means o simple plain equilibrium equations, in order to quantitatively identiy the portion o the grain which leans on the layers below and the one which pushes on the walls. Clearly, as intermediate results, inormation regarding normal pressures and tangential stresses will be also achieved. 3. ACCLRATD CONDITIONS The ollowing assumptions are considered: presence o in-time constant vertical additional acceleration a gv towards (positive upwards); presence o in-time constant horiontal additional acceleration a towards x (positive to the rit); the inertial orces acting on internal disk D, due to the horiontal additional acceleration, are completely balanced by the resultant o the shear (tangential) stresses developing on the lower surace o the disk (absence o horiontal sliding o the grain); the eventual negative variation (depression) o the horiontal pressure between element and the silo wall, due to the eects o the horiontal additional acceleration, is so that it cannot completely annihilate the horiontal pressure, p, between element and the silo wall (hypothesis o not annihilating the pressure). The direction o the additional acceleration (towards x) is rotated by an angle on the horiontal plane compared to the direction (towards )) perpendicular to the external vertical surace o element. The analyses are not carried out here with reerence to Janssen and Koenen s hypothesis (Poati and Ceccoli 972), but to the subdivision o the grain into disks and elements, which implies the generation o additional orces between the silo wall and the grain, in the presence o horiontal and vertical additional accelerations. With reerence to Figures 3. and 3.2, the unknown quantities o the problem are: p = vertical pressures acting on disk D;. v, 2. h, 3. h, 4. v, 5. s = thickness o element ; 6. p h, p = horiontal pressures which are exchanged between disk D and element ; = horiontal tangential stresses acting on the suraces o disk D; = vertical tangential stresses acting on the silo wall. h, 7. h, = horiontal pressures which are exchanged between element and the silo wall; = horiontal tangential stresses acting on the silo wall. The mutual actions exchanged between the grain and the silo walls are assessed here (as is usually done in seismic analyses where the eects o horiontal accelerations are evaluated), by the study o ree-body diagrams which are representative o the dynamic equilibrium conditions. d s h (a)

6 ree surace o grain s s p v, p h, v, v, V p h,..a p h, V. p h, d h, h, V.. D a ( +d) V. D p h, p h, p h, V..a V. v, v, p h, d p v, ( +d) V..a gv V..a D gv V..a gv wall element internal disk D element wall (b) Figure 3.. Vertical longitudinal section: (a) schematic trend o s ; (b) vertical and horiontal actions operating on disk D and symmetrical elements. h, p h, d a. V. s p h, y 2 x R a. V. s 2 a. V. a a. V.. V. Figure 3.2. oriontal cross-section: horiontal actions operating on symmetrical elements. Figures 3. and 3.2 show the mutual actions that disk D, elements and the external walls o the silo exchange. It must be noticed that, in addition to the normal pressures and the shear stresses which are exchanged between the portions o grain and between the grain and the silo wall, there are the vertical and additional horiontal orces reported hereater: V D = sel-weit o disk D and acting towards due to the eect o the gravity acceleration ( is the density o the grain-like material); agv VD = inertial orce coming rom the centre o the mass o disk D and acting towards due to the eect o the vertical acceleration a gv (the inertial orce is downward, as the acceleration a gv has been assumed positive upwards); a VD = inertial orce coming rom the centre o the mass o disk D and acting towards x, due to the eect o the horiontal acceleration a (inertial orce to the let as the acceleration a has been assumed to be positive to the rit); V = sel-weit o element and acting towards due to the eect o the gravity acceleration ( is the density o the grain-like material); agv V = inertial orce coming rom the centre o the mass o element and acting towards due to the eect o the vertical acceleration a gv (the inertial orce is downward, as the acceleration a gv has been assumed positive upwards);

7 a V = inertial orce coming rom the centre o the mass o element and acting towards x, due to the eect o the horiontal acceleration a (inertial orce to the let as the acceleration a has been assumed to be positive to the rit). Vertical translational equilibrium o disk D provides: gv p a (3.) v, I is the pressure ratio o the grain-like material, the ollowing relationship holds between vertical and horiontal pressures inside the grain: gv p a (3.2) h, oriontal (radial) translational equilibrium o disk D provides: h, a (3.3) I is the riction coeicient o the contact surace grain-wall, the ollowing relationship holds between the normal pressures and the vertical shear stresses along the contact surace between the grain o element and the silo wall: p (3.4) v, h, Vertical and horiontal translational equilibrium equations o element are coupled in the ollowing system o equations:, V agv v A p A a V p A h, h, Ater some calculations, this system o equations provides the closed-orm expressions o and s : p h, a cos 2 2 s R R R a cos p, (3.5) (3.6) (3.7) oriontal (tangential) translational equilibrium o element provides: h, a sin a cos (3.8) 4. FUNDAMNTAL RSULTS q. (3.6) represents the irst undamental result o this work: it provides, according to the assumptions o previous sections, the horiontal pressures which are exchanged between the grain, schematised as

8 element, and the silo wall, in seismic conditions. In the case o absence o the additional horiontal acceleration ( a 0 ), q. (3.6) simpliies as ollows: ph0 agv (4.) Then, it can be interesting to deine the overpressure (or depression), due to the eects o the horiontal accidental acceleration, between the grain and the silo wall, as ollows:,, p p p (4.2) h h, h0 q. (3.7) represents the second undamental result o this work: s, represents the thickness o the portion o grain which, according to the assumptions o previous sections, actually pushes on the silo walls in seismic conditions. In order to give physical insit into this result and to acilitate the understanding o the mutual actions which develop between the grain and the silo walls under accelerated conditions, in this section three-dimensional graphical representations are provided o the volumes o the portions o grain which basically (i) are supported by the grain below and (ii) are sustained by the silo walls. Fig. 4. provide these graphical representations or a silo characterised by same parameters reported in the previous section. Portion A is the amount o grain insisting on the lower portion o the material all the way to the silo oundation. This portion o the grain does not interact with the silo walls. From a geometrical point o view, it coincides with the vertical axis truncated cone (overturned) solid (Fig. 4., blue colour), with, as the minor base, the one obtained by drawing the curve r, R s, or on the plane (thereore at the silo oundation) and, as the major base, the silo circumerence (placed at the top o the grain, i.e. 0 ). Portion A2 is the amount o grain that is completely sustained by the lateral walls o the silo. This portion o the grain interacts with the silo walls. From a geometrical point o view, it coincides with the vertical axis cylindrical annulus (Fig. 4., red colour) and thickness s,, which is variable according to the heit and to the angle on the horiontal plane. (a) (b) Figure 4.. Three-dimensional views o portion A (in blue) and o portion A2 (in red) o the lat-bottom grain silo: (a) overview and (b) sectioned view. 5. MTODOLOGY PROPOSD FOR T ASSSSMNT OF T SISMIC ACTION ON FLAT-BOTTOM GRAIN SILOS It can be reasonably assumed that structures characteried by hi values o the vertical and horiontal stinesses, such as silos, do not ampliy or diminish the acceleration induced by the earthquake ground motion at their base. They are thereore subjected to the stresses caused by the inertial orces

9 that arise due to the accelerations at the base that, during the seismic events, vary continuously in time. It is clear that the hiest stresses induced by the seismic action are those that derive rom the peak ground acceleration. Thereore, the seismic design o silos can be based on the assumption that the action induced by earthquake ground motion is modeled as a couple o horiontal and vertical additional accelerations, a g and agv g, representative o the maximum shaking o the ground. By approximating the seismic action on silos as two additional, constant, horiontal and vertical accelerations equal to a g and agv g, their eect can be evaluated with reerence to the results obtained throu the analytical elaborations described in the previous sections. It must be hilited that the actions developing on the silo walls are not axial-symmetrical. The base shear (at the bottom o the silo walls) is given by the integral, on the lateral surace o the grain, o the projection o the additional pressures towards x (namely, along the horiontal additional acceleration). The bending moment at the bottom o the silo walls is given by the integral, on the lateral surace o the silo, o the projection o the additional pressures towards x (namely, along the direction o the horiontal additional acceleration) multiplied by their heit rom the silo oundation Skipping all analytical passages, the base shear and the base bending moment is given by: a T a R M a R a (5.) urocode 8 provisions would lead to the ollowing actions at the base o silo walls: R 6 2 TC8 a R 2 2 R 2 R M a C (5.2) For the immediate assessment o the beneits that the methodology here presented gives with respect to calculation according to urocode 8, it is appropriate to deine the ollowing ratios between: (i) the shear obtained rom the ormulation presented and the one obtained rom urocode calculations, (ii) the bending moment rom the ormulation presented and the one obtained rom urocode calculations (where /2R is the slenderness ratio) T a a T T R C8 R M 2 a 4 a M 2 M C8 3 2 R 3 R (5.3) (5.4) For illustrative purposes, Fig. 3 reports the graphs o and ratios as unctions o the slenderness ratio, or some speciic values o the parameters: GB 0.40, 0.50, a 0.5,0.25,0.35, agv 0.0, and 0.25,0.40,0.50,0.60. Careul examination and inspection o the plots reported in Fig. 5. indicates that the dierence between the results o the here proposed methodology and those o the urocode 8 procedure is maximum (i.e. low values o the ratios) or squat silos, characterised by lower values o slenderness ratio. Also:

10 the advantages o the theory here presented are larger or the bending moment, rather than or the shear; among the parameters which govern the behaviour o silos, the most important one seems to be the grain-wall riction coeicient : the beneicial eect in terms o base shear and base bending moment reduction, provided by the portion o grain which leans on the layers below and does not push on the silo walls, increases i decreases; again, the beneicial eect in terms o base shear and base bending moment reduction increases (i.e. the and ratios decrease) with decreasing values o ; the values o the additional accelerations (both horiontal and vertical) do not inluence so much the trend o the and ratios. GB =0.40 =0.50 a =0.5 a gv =0.0 GB =0.40 =0.50 a =0.5 a gv = T 0.5 M = = = 0.50 = = /(2R) 0.3 = = = 0.50 = = /(2R) Figure 5.. and ratios as unction o the silo slenderness ratio 2R or speciic parameters. 6. CONCLUSIONS In this paper, the actions o grain on the walls o lat-bottom grain silos during earthquake ground motions have been studied analytically. First, the mutual actions exchanged between the grain and the silo walls under static and accelerated conditions (constant vertical acceleration and constant horiontal acceleration) have been assessed. Then, the analytical results obtained in the irst part have been used to ormulate suggestions or assessing the seismic eects on the walls o lat-bottom grain silos. The results indicate that, in case o squat silos (characteried by low but common heit/diameter slenderness ratios), the portion o grain mass interacting with the silo walls proves to be noticeably lower than the total grain mass contained in the silo. RFRNCS Dogangun, A., Karaca, Z., Durmus, A., Seen,. (2009). Cause o Damage and Failures in Silo Structures, Journal o Perormance o Constructed Facilities ASC, Vol. March-April. P N 998-4, 2006 N 99-4, (2004). urocode 8. Design o structures or earthquake resistance, Part 4 Silos, tanks and pipelines, urocode 8. CN, Brussels. amdan, F.. (2000). Seismic behaviour o cylindrical steel liquid storage tanks, Journal o Constructional Steel Research, Vol. 53, p Koenen, M. (896). Berechnung des Seiten und Bodendrucks in Siloellen, Centralblatt der Bauverwaltung, Vol. 6, 896, p Nachtigall, I., Gebbeken, N., Urrutia-Galicia, J.L. (2003). On the analysis o vertical circular cylindrical tanks under earthquake excitation at its base. ngineering Structures, Vol. 25, p Poati, P. & Ceccoli, C. (972). Teoria e Tecnica delle Strutture, Volume Primo, Preliminari e Fondamenti. UTT, Torino.