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Preliminary structural design of metallic silo based on the Eurocode
José Carlos Lavadinho Carneira
jose.carneira@ist.utl.pt
Instituto Superior Técnico, Universidade de Lisboa, Portugal June 2019
Abstract
The objective of this work is the pre-dimensioning of a metallic circular silo for the storage of animal feed pellets supported by four columns. Initially, an introduction is presented where some of the work in the study of silos is discussed with the classification of silos as well as their constitution. The properties that characterize the stored materials are also studied and the thematic of the flow of the material stored inside the silo is addressed, while at the same time a mass flow and a funnel flow is distinguished. Afterward, the geometrical properties of the silo are presented, including the procedure that determine certain dimensions, as is the case of the hopper. These dimensions are at the heart of the next section, where the actions in the silo are defined. For the calculation of these actions, it is used the Eurocode 1 - part 4 of 4 (EN1991-4: 2006) based on Janssen's theory. After, it is used the Eurocode 3 - part 4.1 (EN1993-4-1: 2007), which gives essential guidelines to perform the structural verification, i.e., the limit parameters to which the silo may be subject. The aspects of structural resistance and some of the stability are considered.
Keywords: metallic circular silo; discrete supports; animal feed pellets; calculation methodology;
eurocode; preliminary design.
1. Introduction
In this work, the main objective is an analysis of the procedure to be followed for the pre-dimensioning of a metallic circular silo (made of steel) according to EN1993-4-1:2007. It should be noted, however, that this is not the whole project, since only the body of the cylinder, hopper, ring girder, and columns are studied critically.
The sizing of the silo was based on the following regulations: EN1991-4:2006; Regulamento de Segurança e Acções (for wind actions); EN1993-4-1:2007.
Palma (2005) states that Janssen (1895) carried out studies to determine the pressures/actions within the silos that came to serve as the basis for international standards, as is the case adopted in this work: EN1991-4:2006.
In this document, the silo to be dimensioned is unicellular, that is to say, individual circular silo and according to EN1993-4-1:2007, its constitution is presented in Figure 1 (a) with the designations of the main elements.
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(a) (b)
Figure 1: (a) Terminology used in circular silo [Source: EN1993-4-1 (2007)]; (b) Animal feed pellets
2. Properties of particulate solids
In order to proceed to the sizing of the silo, first of all, the properties related to the material that is stored must be obtained through the Annex E (Table 1) of the EN1991-4:2006.
Table 1: Particulate solids properties [EN1991-4(2006)]
Type of particulate
solid
Unit weight 𝛾 (N/m3)
Angle of repose 𝜙𝑟
(in degrees)
Angle of internal friction 𝜙𝑖 (in degrees)
Lateral pressure
ratio Κ
Wall friction
coefficient 𝜇 Patch solid reference factor 𝐶𝑜𝑝
𝛾𝑙
(Lower) 𝛾𝑢
(Upper) 𝜙𝑖𝑚 𝑎𝜙 𝐾𝑚 𝑎𝑘
Wall type D1
𝑎𝜇 Animal feed
pellets 6500 8000 37 35 1.06 0.47 1.07 0.23 1.20 0.7
3. Flow
The flow type determines in a preponderant way the sizing of the silo. According to EN1991-4:2006, there are three types of flow: mass flow (Figure 2 (a)); funnel flow (Figure 2 (b)) and flow in mixture (Figure 2 (c)).
(a) (b) (c)
Figure 2: Types of flow: (a) Mass flow; (b) Funnel flow; (c) Flow in mixture. [Source: EN 1991-4 (2006)]
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Material flow occurs when the actuating pressures are such that the shear stresses occur without disturbing and modifying the properties of the materials. (Jenike, 1964)
The type of flow in the silo can be determined through the graph of Figure 3 (a) (conical hopper that is the case in this work), according to the angle 𝛽 (degrees) of inclination of hopper wall measured from the vertical (Figure 3 (b)), and the coefficient of friction in the hopper wall, 𝜇ℎ.
(a) (b)
Figure 3: (a) Types of flow in conical hoppers [Source: EN1991-4 (2006)]; (b) Angle of inclination of hopper wall measured from the vertical [Source: EN1991-4 (2006)]
“The main differences between the types of flow (Figure 2) are the differences in movement between the various zones of the flow. Usually, in mass flow, all solid material inside the hopper is in motion, but not necessarily at the same speed. In the funnel flow, only the material in the center of the flow above the hopper outlet is in motion while the material that is in contact with the walls remains stagnant ”(Pires, 2015)
According to De Andrade (2016), the mass flow is the flow considered ideal, being that most of the projects are done aiming this type of flow. Its main feature is that all the particles move simultaneously during the discharge, thus preventing the formation of stagnant zones.
4. Geometric properties
Figure 4 presents the main dimensions, according to the EN1991-4:2006.
Figure 4: Silo geometry [Source: EN1991-4 (2006)]
The principal dimensions of the silo according to the nomenclature of Figure 4 are: ℎ𝑐= 5 𝑚; 𝑑𝑐 = 2.5 𝑚; ℎℎ= 2,27 𝑚; ℎ𝑡𝑝= 0,94 𝑚; ℎ0= 0,31 𝑚; ℎ𝑏 = 7,27 𝑚; 𝛽 = 30 °; 𝑡𝑐= 5 𝑚𝑚; 𝑡ℎ = 10 𝑚𝑚;
𝑟 = 1.25 𝑚. It should be noted that two thickness measurements were defined, where 𝑡𝑐 corresponds
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to the thickness of the cylindrical wall and 𝑡ℎ corresponds to the thickness of the walls of the hopper.
The characteristics of the silo are presented in Table 2.
Table 2: Characteristics of the silo Structure
weight [using (SolidWorks,
2017)]
Storage volume (𝑽)
Maximum capacity
(𝑽 × 𝜸𝒖)
Number of columns
Cross-sectional area of cylindrical wall
𝑨
Internal perimeter of the cross-section of cylindrical wall 𝑼
Hydraulic radius 𝑹
(𝑹 =𝑨
𝑼)
958 𝑘𝑔 31.72 𝑚3 25896 𝑘𝑔 4 4.908 𝑚2 7.853 𝑚 0.625 𝑚
Figure 5 shows the dimensions of the silo, including the support columns.
Figure 5: Main dimensions of the silo
An aspect that deserves special attention is the sizing of the hopper since its dimensions define the type of flow of the material inside the silo (mass flow or funnel flow). Therefore, the hopper was sized in view of mass flow. For this, the value of 𝛽 = 30° was chosen for the angle that the hopper makes with the vertical, (Figure 4), since such value together with the coefficient of friction (𝜇 = 0.23) allows to verify according to EN1991-4:2006 (Figure 3) that the flow will be a mass one. There are also other models for choosing the angle of inclination of hopper wall measured from the vertical which are:
Jenike Model, Walters Model and McLean Model (De Andrade,2016).
In this case, the definition of the diameter of the discharge mouth was obtained based on the following equation (Palma, 2005),
𝐷 = 1.2 ×𝐻(𝛽) × 𝜎𝑐𝑟𝑖 𝛾𝑢
⇔ 𝐷 = 1.2 ×2.41 × 1259
8000 ⇔ 𝐷 = 0.46 𝑚 (1)
where 𝐻(𝛽)(see Figure 6) is the slope function of the hopper (for 𝛽 = 30°), 𝜎𝑐𝑟𝑖 is the unconstrained critical stress (value below which flow problems begin to occur) and 𝛾𝑢 is the highest value of unit weight (Table 1). Equation 1 is increased by factor 1.2 due to the recommendation of Jenike (1964), in order to avoid instabilities in the flow.
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Figure 6: Function H(𝛽) [Adapted from Palma (2005)]
The unconstrained critical stress 𝜎𝑐𝑟𝑖 was determined based on the values measured by Neto et al.(2009), who in their study had angles approximately equal to those of this work, namely the angle 𝛽 that the hopper does with the vertical. Accordingly, an average of the 4 values of unconstrained critical stress, present in Table 3, was performed.
Table 3: Values measured by Neto et. al. (2009) Instant after filling (t=0 h)
Animal feed pellets 𝜷 (°) 𝝈𝒄𝒓𝒊𝒕 (𝑷𝒂)
A 35 1262
B 35 1249
12 hours after filling (t= 12 h)
𝜷 (°) 𝝈𝒄𝒓𝒊𝒕 (𝑷𝒂)
A 35 1295
B 34 1231
5. Actions in silos
The loads on the vertical wall and hopper are based on EN1991-4:2006, which is governed by Janssen’s theory (1895). The loads acting on the vertical walls and hopper were calculated considering that it is a slender silo (ℎ𝑐⁄𝑑𝑐≥ 2) and class 1 (capacity less than 100 tons). For the loads resulting from the application of the wind, the provisions of the Regulamento de Segurança e Acções were followed.
5.1. Limit states
According to EN1991-4:2006, due to the variation of the properties, five limit states (LS#) are considered: three for the cylindrical wall and two for the hopper as presented in Table 4.
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Table 4: Value of properties to be used for different wall loading assessments [Source:(EN1991-4:2006, pp.26)]
Limit state Characteristic value to be adopted
Vertical wall 𝜇 Κ 𝜙𝑖
LS1 Maximum normal pressure on vertical wall Lower Upper Lower LS2 Maximum frictional traction on vertical wall Upper Upper Lower
LS3 Maximum vertical load on hopper Lower Lower Upper
Hopper wall 𝜇 Κ 𝜙𝑖
LS1 Maximum hopper pressures on filling Lower Lower Lower
LS2 Maximum hopper pressures on discharge Lower Upper Upper
5.2. Loads on the vertical walls and hopper
In Figure 7(a), 𝑝𝑣𝑓𝑡 is the vertical stress in the stored solid at the transition after filling, 𝑝𝑤𝑓 is the wall friction traction and 𝑝ℎ𝑓 is the. horizontal pressure. It should be noted that the pressures in Figure 7(a) are due to the filling. The loads resulting from the emptying is increased by the factor 𝐶𝑤 and 𝐶ℎ, respectively, wall frictional traction discharge factor and horizontal pressure discharge factor. In turn, Figure 7(b) shows the loads acting on the hopper.
(a) (b)
Figure 7: Loads [Source: EN1991-4:2006]: (a) Vertical walls (b) Hopper
The shear forces on vertical walls (Figure 8(a)) are the reason for the nonlinear pressure variation in Figure 8(b).
Note the difference between the pressures resulting from the filling [red line phf - Figure 8 (a)] which are smaller in relation to the emptying [blue line phe - Figure 8 (a)].
The attention that must be paid to the transition zone (𝑧 = 5 𝑚) is justified by the "jump" to the hopper curve that is present in Figure 8 (a). As illustrated in Figure 8(b), the loads applied to the walls of the silo or hopper are dependent on whether the material is being loaded or unloaded.
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(a) (b)
Figure 8: Pressures perpendicular to the wall: (a) LS1 for the study silo; (b) Discharge pressures in a steep hopper and vertical wall [Source: EN1991-4:2006]
5.3. Wind action
Regarding the wind action, the total elevation of the silo to soil was considered to be 10 𝑚 (Figure 5), and it was conservatively considered that the silo was in zone B (which includes the archipelagos of Açores and Madeira and the regions of the continent situated in a coastal zone with 5 km of extension or altitudes above 600 m), and that the aerodynamic roughness of the soil corresponds to type II (roughness that is attributed to rural areas and urban periphery).
6. Eurocode 3 – part 4.1
The main purpose of this work is the dimensioning of a metal circular silo. In this section, it is possible to obtain the allowable load, after applying the action defined in section 5. The calculation of these loads follows the provisions of EN1993-4-1(2007). For each dimensioning value obtained, it must always be less than the limiting value, i.e.: 𝑆 < 𝑅.
The allowable loads are calculated in 4 zones: cylindrical wall; conical hopper; ring girder and support columns.
It should be noted that stainless steel 304 was defined as the silo constituent material, and EN1993-4- 1:2007 states that for any type of steel Young’s modulus, equal to 210 𝐺𝑃𝑎. The ring girder consists of a profile C15x50 (EN10279) and the support columns a IPN180 profile (EN10024).
In the part concerning cylindrical walls, considering a symmetrical loading of the stored material and the actions defined in section 5, the value of design stress resistance 𝑓𝑒,𝑅𝑑 [EN1993-4-1(2007) pp.37]
is:
𝑓𝑒,𝑅𝑑= 𝑓𝑦
𝛾𝑀0 ⇔ 𝑓𝑒,𝑅𝑑= 276
1,00⇔ 𝑓𝑒,𝑅𝑑= 276 𝑀𝑃𝑎 (2) where 𝑓𝑦 is the yield strength of the stainless steel 304 .
In this way, the structure must verify:
𝜎𝑒,𝐸𝑑 ≤ 𝑓𝑒,𝑅𝑑= 276 𝑀𝑃𝑎 (3)
LS1
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Under the action of wind or vacuum, the silo may undergo buckling along with the whole height of the silo, or between changes in membrane thickness. The critical buckling external pressure for an isotropic wall should be found as [EN1993-4-1(2007),pp. 44]:
𝑝𝑛,𝑅𝑐𝑟𝑢= 0,92 × 𝐶𝑏× 𝐶𝑤× 𝐸 × (𝑟 𝑙) × (𝑡𝑐
𝑟)
2,5
(4) where 𝑙 is the height between stiffening rings or boundaries, which is 5700 mm (Figure 5). Substituting the respective values in the previous equation:
𝑝𝑛,𝑅𝑐𝑟𝑢= 0,92 × 1 × 1,85 × 210 × 109× (1,25
5.7) × (0,005 1,25)
2,5
⇔
⇔ 𝑝𝑛,𝑅𝑐𝑟𝑢= 79317 𝑁/𝑚2⇔ 𝑝𝑛,𝑅𝑐𝑟𝑢= 79.317 𝑘𝑁/𝑚2
(5) However, the design maximum external pressure under wind and/or partial vacuum should be assessed as [EN1993-4-1(2007),pp. 44]:
𝑝𝑛,𝑅𝑑=𝛼𝑛× 𝑝𝑛,𝑅𝑐𝑟𝑢
𝛾𝑀1
⟺ 𝑝𝑛,𝑅𝑑 =0,5 × 79,317
1,10 ⟺ 𝑝𝑛,𝑅𝑑= 36,05 𝑘𝑁/𝑚2 (6) where 𝛼𝑛 is the elastic buckling imperfection reduction factor, that EN1993-4-1:2007 (pp.45) recommends taking the value of 0.5. Therefore, it is observed that the considered pressures result from wind shall not be enough to induce buckling phenomena.
In its turn, the plastic mechanism resistance of the hopper is calculated by [EN1993-4-1 (2007), pp.
69]:
𝑛𝜙,𝑅𝑑 = 1 𝛾𝑀0
× (0,91 × 𝜇 + 0,27 𝜇 + 0,15 ) ×
(
𝑟 × 𝑡ℎ× 𝑓𝑦 𝑟 − 2,4 × √𝑟 × 𝑡ℎ
cos 𝛽 × sin 𝛽 ) ⟺
⟺ 𝑛𝜙,𝑅𝑑=1
1× (0,91 × 0,23 + 0,27 0,23 + 0,15 ) ×
(
1,25 × 0,01 × 276 × 106 1.25 − 2,4 × √1,25 × 0,01
cos 30° × sin 30°) ⟺
⟺ 𝑛𝜙,𝑅𝑑= 3.935 × 106 𝑁 𝑚⁄ ⟺ 𝑛𝜙,𝑅𝑑= 3935 𝑘 𝑁 𝑚⁄
(7)
In this way, the following condition must be verified [EN1993-4-1 (2007), pp. 70] so that there is no plastic collapse of the hopper:
𝑛𝜙,𝐸𝑑≤ 3935 𝑘𝑁/𝑚 (8)
In the part of the transition (ring girder), the mean circumferential stress in the ring should satisfy the condition [EN1993-4-1 (2007), pp. 110]:
𝑁𝜃,𝐸𝑑≤𝑓𝑦× 𝐴𝑒𝑡 𝛾𝑀0
(9) Where 𝐴𝑒𝑡 is the total effective area of the ring calculated on the basis of equation 10 [EN1993-4-1 (2007), pp. 109]:
𝐴𝑒𝑡 = 𝐴𝑝+ 0,4 × √𝑟 × {𝑡𝑐
3
2+ 𝑡ℎ
3 2
√cos(𝛽)}
(10)
where 𝐴𝑝 is the cross-sectional area of ring girder.
Finally, with respect to the support columns, each column was considered to have a load 𝑇 applied to the center of the profile. For the calculation of the load 𝑇 in each support column, the weight was combined with the maximum capacity of the silo (Table 2), both of which are multiplied by the amplifier
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1.35 (permanent load). Subsequently, this value is divided by 4, since this load is distributed by 4 support columns.
𝑇 =958 × 𝑔 × 1.35 + 25894 × 𝑔 × 1.35
4 ⟺ 𝑇 =958 × 10 × 1.35 + 25894 × 10 × 1.35
4 ⟺
⟺ 𝑇 = 90626 𝑁 ⟺ 𝑇 = 90.63 𝑘𝑁
(11)
where 𝑔 is the acceleration of gravity assumed to be 10 𝑚 𝑠⁄ 2.
Then, the allowable load for the xz plane is calculated (where the critical load is smaller, hence the critical plane) (Figure 9).
Figure 9: Silo support column with bracing: (a) Effective Length; (b) Buckling in xz plane; [Adapted from Beer et. al.. 2003] The allowable load 𝑃𝑎𝑑𝑚 is calculated by taking into account 3 parameters: effective length 𝐿𝑒, radius of gyration 𝑖𝑦 and elastic buckling imperfection reduction factor of column 𝛼 which are, respectively, 1500 𝑚𝑚, 17.1 𝑚𝑚 (Beer et. al.. 2003 – appendix C) and 0.34 (it is a laminated I profile with 𝐻 𝐵⁄ > 1.2 and buckling around the axis with smaller second area moment).
𝜆̅ = 𝐿𝑒
𝑖 𝜋 × √𝐸
𝑓𝑦
⟺ 𝜆̅ =
1500 17.1 𝜋 × √210 × 103
276
⟺ 𝜆̅ = 1.01
(12)
𝜒 = 1
[0.5 × 𝜆̅ × (𝜆̅ + 𝛼) − 0.1 × 𝛼 + 0.5] + √[0.5 × 𝜆̅ × (𝜆̅ + 𝛼) − 0.1 × 𝛼 + 0.5]2− 𝜆̅2
⟺ 𝜒 = 0.58 (13)
𝑃𝑎𝑑𝑚 =𝜒 × 𝐴 × 𝑓𝑦
𝛾𝑀1
⟺ 𝑃𝑎𝑑𝑚=0.58 × 2790 × 276
1.1 ⟺ 𝑃𝑎𝑑𝑚= 406.02 𝑘𝑁 (14) where 𝐴 is the cross-sectional area of the support column profile.
Since, 𝑇 < 𝑃𝑎𝑑𝑚, the stability of the support columns with bracing is verified.
7. Conclusions and future work
With regard to the preliminary structural design for the silo sizing, particularly in the part of the hopper, it was dimensioned considering the issue of mass flow; however, there is no consensus among authors, on whether to size a diameter minimum for the discharge port. For example, in Leite (2008) the author does not require the calculation of minimum diameter, while in Palma (2005) suggests a method for calculating it for the discharge port. In the present work, the author chose to design a diameter that the silo should have, although in an approximate and empirical way, since, for instance, the unconstrained critical stress was calculated based on an average of values of the unconstrained critical stress, present in Table 3. On the other hand, the angle of inclination of the hopper wall measured from the vertical was chosen by selecting the value of 30º (Figure 4), since it is considered acceptable by EN1991-4:2006. It is also interesting to state that the properties of the materials
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indicated in Table 1 do not include important properties such as humidity, which increases the unit weight of the material.
As a suggestion of future work, it is proposed to model and to verify the values obtained through the finite element method, as well as the extension of the study to the vibration modes, and to check in more detail the phenomenon of buckling (which is more pronounced at the moment in which the silo is empty); an addition of seismic loads as well as external loads, such as the action of pneumatic vibrators, in the latter case of loads, EN1991-4:2006 does not have a calculation methodology for these actions; consideration of temperature differences between the inside and outside of the silo;
study of bolted/welded connections at the mechanical design level; And finally, an introduction of eccentric loading/unloading.
Acknowledgements
Authors thanks to his supervisor Prof. Miguel Matos Neves (IST, Universidade de Lisboa) and to Eng.
Nuno Botelho of the VENTISEC Company.
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