Abstract
The seismic behavior of flat-bottom silos containing grain-like material still presents strong uncertainties and current design codes tend to provide too conservative formulations for the estimation of the seismic actions. Over the years, some researchers focused on the dynamic behavior of such silos mainly through numerical investigations. Analytical formulations for the evaluation of the pressures exerted by the ensiled grain on the silo wall under seismic excitation were developed by Younan and Veletsos (J Struct Eng ASCE 124(1):62–70 1998) and, more recently, by Silvestri et al. (Bull Earthq Eng 10(5):1535–1560 2012). Experimental shaking-table tests were performed on silo specimens (Silvestri et al. EESD 2015, submitted), which showed good agreement with the Silvestri’s analytical formulations, even if some theoretical limits of validity were not satisfied. This has encouraged a complete revision and refinements of the theoretical framework, which is the object of this paper. In detail, the static and the dynamic actions exchanged between different grain portions and between the grain and the silo wall are idealised in a more physically consistent way. The analytical developments are carried out by means of simple free-body dynamic equilibrium equations. The refinements yield to a significant extension of the theoretical limits of validity and to a new set of analytical formulas for the wall pressures and for the wall shear and bending moment. A comparison of the analytical formulas with (i) the consolidated Janssen (Zeitschrift des vereines deutcher Ingenieure 39:1045–1049 1895) and Koenen (Centralblatt der Bauverwaltung 16:446–449 1896) theory for static design of silos and (ii) with the Eurocode 8 provisions for seismic design of silos and with the experimental results is also performed in order to (i) check the updated theoretical model in static conditions and (ii) verify the reliability of the different formulations in accelerated conditions, respectively. The refined theory confirms that the portion of ensiled material that interacts with the silo wall is significantly smaller than the effective mass suggested by Eurocode 8.
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Acknowledgments
The authors acknowledge the financial support received from the European Community’s Seventh Framework Program [FP7/2007-2013] under Grant Agreement N° 227887 for the SERIES Project (ASESGRAM project: “Assessment of the seismic behaviour of flat-bottom silos containing grain-like materials”).
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Appendix: Notation
Appendix: Notation
The following symbols are used in the paper:
p 0(z) | Horizontal grain-wall pressure for the Janssen and Koenen theory |
p v (z) | Vertical grain–grain pressure for the Janssen and Koenen theory |
τ 0(z) | Vertical frictional grain-wall stress for the Janssen and Koenen theory |
μ GW | Grain-wall friction coefficient |
μ GG | Grain-grain friction coefficient |
μ GB | Grain-bottom friction coefficient |
γ (N/m3) | Specific weigh of the ensiled material |
λ | Pressure ratio between horizontal and mean of the vertical pressures |
R (m) | Internal radius of the silo |
H (m) | Height of the ensiled material inside the silo |
z (m) | Distance of a generic grain layer from the top free-surface of the content |
Δp n,GW (ϑ) (N/m2) | Radial additional pressure for the Trahair formulation |
Δp ϑ,GW (ϑ) (N/m2) | Circumferential additional pressure for the Trahair formulation |
α g | Ratio of the constant horizontal acceleration to the gravity acceleration g |
g | [m/s2] Gravity acceleration |
f | Slenderness function |
α(z) | Ratio of the response acceleration of the silo at a vertical distance z form the equivalent surface of the stored content, to the acceleration of gravity |
ϑ | Latitude of the radial direction respect to x axis (clock-wise) |
A (m2) | Surface of the horizontal cross-section of the silo |
H/R | Slenderness ratio of the silo |
a eh (z) | Along-the-height profile of the horizontal acceleration |
a ev (z) | Along-the-height profile of the vertical acceleration |
a eh⊥(z, ϑ) | Portion of horizontal acceleration on the radial direction |
a eh∥(z, ϑ) | Portion of the horizontal acceleration parallel to the external vertical surface of element E |
a eh0 | Constant horizontal acceleration |
a ev0 | Constant vertical acceleration |
ν(z) | Vertical acceleration factor |
ν 0 | Constant vertical acceleration factor |
s st (z) (m) | Thickness of the material which leans of the wall in static conditions |
s(z, ϑ) (m) | Thickness of the material which leans of the wall in dynamic conditions |
p v,GG (z) (N/m2) | Vertical grain–grain pressure acting on disk D in accelerated conditions |
p h,GG (z) (N/m2) | Horizontal grain–grain pressure acting on disk D in accelerated conditions |
p h,GW (z) (N/m2) | Horizontal grain-wall pressure in accelerated conditions |
τ v,GW (z, ϑ) (N/m2) | Vertical frictional grain-wall stress in accelerated conditions |
τ h,GG (z) (N/m2) | Horizontal frictional grain–grain stress acting on disk D in accelerated conditions |
τ h,GB (z) (N/m2) | Horizontal frictional grain-bottom stress in accelerated conditions |
τ h,GW (z, ϑ) (N/m2) | Horizontal frictional grain-wall stress in accelerated conditions |
Δp h,GW (z, ϑ) (N/m2) | Horizontal grain-wall overpressure in accelerated conditions |
p h,GW,st (z) (N/m2) | Horizontal grain-wall pressure in static conditions |
C D (z) | Boundary of the disk D on the horizontal section |
T (N) | Shear action at the base of the silo wall for the Silvestri’s theory |
M (N m) | Moment at the silo wall for the Silvestri’s theory |
M Completed (N m) | Moment at the base of the silo wall accounting for vertical frictional grain-wall stress contribution for the Silvestri’s theory |
β(z, ϑ) | Dynamic factor |
β 0(ϑ) | Constant dynamic factor |
ω(z) (m) | Static function |
a | Double static function-radius ratio |
A D (z) (m2) | Surface of the horizontal cross-section of disk D |
f Iv,D (z) (N) | Vertical inertial force of the disk D |
f Ih,D (z) (N) | Horizontal inertial force of the disk D |
f Iv,E (z, ϑ) (N) | Vertical inertial force of the disk E |
f Ih,E (z, ϑ) (N) | Horizontal inertial force of the disk E |
V D (z) (m3) | Volume of a grain layer of disk D of infinitesimal thickness dz |
r(z, ϑ) (m) | Radius of the disk D in accelerated conditions |
C 1 | First constant of integration |
C 2 | First constant of integration |
O | Center of the horizontal cross-section of the silo |
O′ | Local center for each point holding to C D (z) in accelerated conditions |
R h,GG (z) (N) | Resultant of the projection of p h,GG (z) on C D (z) along the x axis |
Ω(z, ϑ) | Auxiliary latitude |
r *(z, ϑ) (m) | Auxiliary radius of the contour C D (z) |
dA D,v (z, ϑ) (m2) | Lateral vertical surface of an infinitesimal portion of the disk C D (z) |
E(z, ϑ) | Elementary portion of element E |
A E,Ext (z, ϑ) (m2) | Vertical lateral surface of E(z, ϑ) |
A E,din (z) (m2) | Surface of a horizontal cross-section of element E in contact with the wall |
A E,Int (z, ϑ) (m2) | Vertical lateral surface of element E(z, ϑ) in contact with disk D |
V E,din (z) (m3) | Volume of element E along the height |
V D,din (z) (m3) | Volume of disk D along the height |
T xx (z) (N) | Shear action at the base of the silo wall for the refined theory |
M yy,1(z) (N m) | Moment at the silo wall due to overpressure for the refined theory |
M yy,2(z) (N m) | Moment at the silo wall due to frictional vertical stresses for the refined theory |
M yy (z) (N m) | Wall bending moment for the refined theory |
\( p_{h,GW} (z)_{I^\circ } \) (N/m2) | First order approximation with Taylor’s series of the horizontal grain-wall pressure provided by Janssen and Koenen |
\( p_{h,GW} (z)_{II^\circ } \) (N/m2) | Second order approximation with Taylor’s series of the horizontal grain-wall pressure provided by Janssen and Koenen |
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Pieraccini, L., Silvestri, S. & Trombetti, T. Refinements to the Silvestri’s theory for the evaluation of the seismic actions in flat-bottom silos containing grain-like material. Bull Earthquake Eng 13, 3493–3525 (2015). https://doi.org/10.1007/s10518-015-9786-2
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DOI: https://doi.org/10.1007/s10518-015-9786-2