The Dynamic Behavior of Silos with Grain-like Material during Earthquakes

Abstract

Grain security is an important guarantee for sustainable development. However, the dynamic behavior of silos containing grain-like material is not well understood. The effective mass and dynamic effects are the key parameters for the assessment of the silo–bulk material system during earthquakes. Herein, on the basis of the Janssen continuum model, it is proposed that the seismic energy is entirely dissipated by the interactions between the materials and the silo and the materials themselves. The seismic inertia forces among storage materials were introduced, and dynamic equilibrium equations considering the vibrations of storage materials were established. Theoretical solutions for the horizontal forces exerted and the effective mass of the silo–bulk material system during earthquakes are proposed. It is worth noting that the additional stress on the side wall proposed in this work is related to the depth, silo radius, storage density, internal friction coefficient, lateral pressure coefficient, and seismic acceleration. In addition, the effective mass coefficient is negatively correlated with the external friction coefficient, the lateral pressure coefficient, and horizontal seismic acceleration under a storage vibration state. A narrower silo (i.e., with a larger height–diameter ratio) has a low effective mass coefficient. The results from our method are in good agreement with those attained using experimental data, which demonstrates the accuracy and universality of the proposed formulations.

1. Introduction

As key nodes in the grain lifeline engineering system, silo structures are a basic guarantee of the security of national grain reserves. The failure rate of silos all over the world is much higher when compared to the failure rate of other industrial structures because the weight of the stored material in the silo is far greater than the structure’s own weight [1,2,3]. In particular, the silo–bulk material structural system can be seriously damaged as a result of earthquakes, which often involve casualties, production interruptions, and grain losses, as shown in Figure 1 [2]. Therefore, the reliability and stability of these special structures under seismic loads are of critical concern and should be understood by those involved in their design, implementation, and management.

During earthquakes, the particulate material oscillates inside the silo such that the lateral loads due to material flow and the lateral seismic loads must be considered simultaneously [4,5]. In addition, the material flow is not in step with the silo vibration. In fact, the interaction between the bulk solid and the silo walls dissipates energy well in earthquake scenarios [6,7,8]. Therefore, the seismic design of silos is usually performed on the basis of the identification of an effective mass, which interacts with the silo walls under seismic excitation, i.e., which horizontally pushes on the silo walls. The current Chinese and Eurocode standards consider an effective mass to equal approximately 80% of the total ensiled mass [9,10]. However, it is evident that this formulation in the standards is too conservative [11,12,13,14]. Hence, the identification of the effective mass is a contentious issue.

Via theoretical analysis and experimental research, seismic theory for silo structures has been developing rapidly since the 1960s [15]. Indian researchers carried out free vibration tests using half and full silos made of Perspex and steel containing wheat, cement, sand, and charcoal. It was reported that the effective mass coefficient of the content participating in seismic response varied from 0.22 to 0.54, and the effective mass coefficient decreased as the amount of material increased. Seismic design methods for silos in various countries are based on this theory [16,17,18,19]. In the mid-1980s, scholars conducted a series of dynamic and seismic tests on the effective mass coefficient of the stored material’s seismic response and proved that the representative value of the stored load participating in the vibration was far less than the actual gravity load [20,21]. Then, EC Harris and JDV Nad et al. carried out 30 simple harmonic horizontal vibration tests (at frequencies of 1, 3, 5, 7, and 9 Hz) under three working conditions: an empty silo, a full silo of wheat, and a full silo of sand, using two steel silos. According to the stiffness changes in the silos and storage materials, the calculated effective mass coefficient of the stored material varied from 0.58 to 0.9, demonstrating that the lower the vibration frequency, the greater the effective mass coefficient [22]. On the basis of the conservation of energy principle, the researcher showed that the kinetic energy was transformed into friction force acting on the stored materials and into deformation energy of the structure when the silos vibrate. The theoretical calculation method of the effective mass coefficient was derived, but the formula was complex and is not conducive to real-world applications [23,24].

Silvestri et al. [25,26] produced the dynamic equilibrium equation of a simple free body (particle units) by considering the side wall friction and the horizontal shear force of the stored material based on the work of Janssen [27] and Koenen [28]. With the help of theory analysis and FEM, the effective mass coefficient of the storage material and the along-the-height profile of the horizontal pressure acting on the vertical silo wall due to earthquake excitation could be calculated. Subsequently, the validity of the results was verified using the shaking table test [29,30]. The results indicated that the effective mass coefficient of the storage material was far less than the 0.8 stipulated in Eurocode-8, and the greater the friction coefficient between the storage material and the silo wall, the greater the horizontal pressure on the silo wall and the greater the seismic overturning force.

Researchers agree that the effective mass of silo contents is lower than the actual mass in the equations used to predict the dynamic responses of silos to earthquake movements. However, there are many influencing factors, such as the stiffness of the silo, the height-to-diameter ratio, the physical properties of the storage materials, the vibration frequency, the friction between the stock and the silo wall, etc. [31,32,33,34]. At present, the evaluation of the quality coefficient is not very accurate. Therefore, identifying the effective mass coefficient of silos with different stiffnesses and height-to-diameter ratios under dynamic response is a topic of increasing relevance in the study of seismic technology for silos as it can help optimize the design and save costs [35,36,37,38].

In this paper, a new analytical method for identifying the horizontal forces and the effective mass of the silo–bulk material system during earthquakes is proposed. This model was validated using experimental data. Furthermore, a sensitivity analysis was performed for use in silo design.

2. Analytical Framework

Under normal conditions, the walls of the silo are typically subjected to both normal and shear stresses (vertical and horizontal friction forces) produced by the particulate material inside the silo [39]. The particulate material generates additional pressure on the silo walls due to earthquake vibrations. Moreover, the particulate material vibration is not aligned with the silo vibration under the inertia force [40]. In addition, particulate–particulate friction and particulate–wall friction can efficiently dissipate seismic energy. As a result, the effective mass is lower than the total ensiled mass. For the above-mentioned idea, a dynamic equilibrium equation is proposed based on the Janssen continuous medium model, which considers seismic inertia forces [27]. Moreover, the pressures produced by the particulate material on the silo walls and the effective mass during the earthquake were obtained.

2.1. The Hypotheses on Particulate Material Inside the Silos during Earthquakes

(1)

The particulate material is assumed to be a continuum, incompressible, and compact as if it were composed of a number of rigid and infinitely resistant spherical balls.

(2)

The silo–bulk material system obeys Da Vinci’s friction law: the wall–particulate friction coefficient remains unchanged, which is the external friction coefficient; the particulate friction coefficient is set to be constant, which is the internal friction coefficient; the particulate horizontal friction needs to be considered.

(3)

The silo walls are assumed to be rigid. Thus, it can be assumed that the silo structure absorbs no energy from the earthquake [41].

2.2. The Hypotheses Concerning the Seismic Acceleration

As is known, seismic acceleration changes in real time. For simplification, it is assumed to be a fixed value. This simplification was widely adopted in previous works. It is clear that this assumption leads to a conservative simplification, given that it is representative of a fictitious single instant in time in which both the vertical and the horizontal accelerations achieve their peak values. The vertical (a_z) and horizontal (a_r) accelerations can be calculated as follows: a_z (t,z) = g + a_ez (t,z) g and a_r (t,z) = a_er (t,z)g, where g is the gravitational acceleration, and a_ez (t,z) and a_er (t,z) are the acceleration coefficients in the vertical and horizontal directions, respectively.

3. Analytical Mode

Taking a flat-bottom circular silo as an example, as shown in Figure 2a, the cylindrical coordination is used in the model of the idealized system, in which the vertical Z-axis is defined along the silo central depth, the r denotes the vertical distance from any point of storage to the Z-axis, θ denotes the latitude with respect to the X-axis, and R denotes the silo radius. On the basis of the aforementioned hypotheses, the vertical and horizontal accelerations, which are in the same direction as the coordinate axis, are set to be constant. The particulate material with a middle thickness d_z can be picked as a research object when the silo depth is assumed to be infinite.

The stress states of the object under seismic action are shown in Figure 2b, in which the variables are specified as follows:

• mg is the gravity of the stored material;
• σ_z is the vertical pressure caused by gravity and the actions of the upper layer. It is stipulated as a function of z;
• τ_rg is the horizontal shear stress due to the action of the upper layer. It is stipulated that τ_rg is a function related to z and independent of r and θ; the internal friction coefficient between stored materials is μ′;
• τ_zw is the vertical shear stress between the stored material and the lateral wall, and τ_zw is defined as a function related to z and θ (r = R). The external friction coefficient between the stored material and the wall is μ₀;
• σ_z + dσ_z is the supporting force from the lower layer;
• τ_rg + dτ_rg is the horizontal shear stress from the lower layer;
• σ_r is the horizontal pressure of the layer on the side wall, and σ_r is defined as a function related to z and θ (r = R);
• ma_ez(z)g is the vertical inertial force of vertical acceleration caused by seismic energy; unit: N;
• ma_er(z)g is the horizontal inertial force caused by seismic energy; unit: N.

3.1. Static Equilibrium Model

Without external forces (seismic wave input), there is no horizontal friction between storage materials, τ_rg = τ_rg + dτ_rg = 0. At the static status, the model in Figure 2 becomes an ideal axisymmetric model, as shown in Figure 3, and its vertical equilibrium equation can be written as follows:

$$\gamma \pi R^{2}dz+{{\sigma }^{\prime }}_z{R}^2\pi +{{\tau }^{\prime }}_{zw}2\pi Rdz=({{\sigma }^{\prime }}_z+d{{\sigma }^{\prime }}_z)\pi R^2$$

(1)

where R is the silo radius and γ is the storage gravity density. In this work, γ is assumed to be a constant and independent of depth and history, and τ′_zw, σ′_r, and σ′_z are the lateral wall shear stress, horizontal pressure, and vertical pressure at depth z under static load, respectively.

Particle material can bear static shear force, so complete boundary conditions are required to calculate the spatial stress of a particle system. For a nonlinear mechanical problem, a rigorous bottom boundary may cause a nonconvergence problem. Hence, for the sake of simplicity, the influence of the bottom boundary effect was omitted. The integrations of Equation (1), 0~z, generally produce an acceptable level of error near the bottom in real-world situations.

When z = 0, σ′_z = 0; r = R, τ′_rw = μ_0σ′r, one arrives at the Janssen formula. Thus, the validity of the computational model and method is verified.

$${\sigma }^{\prime }r(z)=\frac{\gamma R}{2{\mu }_0}(1-e^{-2{\mu }_{0}Kz/R})$$

(2)

3.2. Seismic Dynamic Equilibrium Model

On the basis of the simple ideal model, the horizontal input direction of an earthquake can be given arbitrarily. It can be assumed that there is an angle of θ between the input direction of a horizontal earthquake and the positive x-axis. Thus, Figure 3 is no longer an ideal axisymmetric model. Similarly, particulate material with a section angle of dθ in the middle thickness dz can be selected as a research object when the silo depth is assumed to be infinite. The stress status of the stored material in the silo is shown in Figure 4, in which the variables are specified as follows:

• τ_rw is the horizontal shear stress between the stored material and the side wall. In this paper, the moment of inertia is neglected, and τ_γw is only caused by the vertical component of horizontal seismic acceleration acting on the sector;
• P_rg is the horizontal force of the sector stock in the direction of θ₁ + π on the stock in the direction of θ₁;
• m′a_er(z)g is the horizontal inertial force of the fan-shaped storage materials under seismic action;
• θ₁ is the angle between the direction of the given horizontal acceleration and the midline of the sector;
• In Figure 4, certain other variables are not marked: the vertical shear stress between the stored material and the side wall τ_zw, gravity m′g, vertical inertia force m′a_ez(z)g, the horizontal shear stress τ_rg in the fan-shaped storage materials, the vertical pressure σ_z of the upper surface, the supporting force σ_z + dσ_z of the lower surface, and the horizontal shear stress τ_rg + dτ_rg of the lower surface.

In the cylindrical coordinate system, the arc length between the fan-shaped stored materials and the side wall is within the range of Rdθ, and the horizontal pressure and the side wall shear stress have uniform distributions. The friction increment between the upper and lower layers caused by the horizontal inertia force is balanced with the horizontal force on the side wall.

According to the above description, the vertical and horizontal equilibrium equations of the fan-shaped stored material at depth z and the circumferential force equilibrium equations between the sectoral stored material and the lateral wall can be formulated.

(1)

Equilibrium equation of sector storage in a vertical direction at a depth of z.

The fan-shaped storage material in the direction of θ₁ (upstream) and diagonal reservoir are assumed to be an isolator (Figure 5), and the vertical force balance is written as follows:

$$2m^{\prime }g+2m^{\prime }{a}_{ez}(z)g+2A_1{\sigma }_z=A_2{\tau }_{zw}(z,{\theta }_1)+A_2{\tau }_{zw}(z,{\theta }_1+\pi )+2A_1({\sigma }_z+d{\sigma }_z)$$

(3)

Here,

• A₁ is the sector area and can be expressed by 1/2·R²·dθ (the radian system);
• A₂ is the area of contact between the sector and the side wall in the range of dz and can be expressed by Rdθdz;
• m′ is the quality of the sector-shaped stored material, which can be expressed by 1/2·ρR²·dθdz, where ρ is the density of the stored material;
• In this paper, it is assumed that the limit equilibrium state between the storage material and the side wall is reached; thus, τ_zw = μ₀σ_r.

Therefore, Equation (3) can be rewritten as follows:

$$\rho Rgdz[1+a_{ez}(z)]=[{\tau }_{zw}(z,{\theta }_1)+{\tau }_{zw}(z,{\theta }_1+\pi )]dz+Rd{\sigma }_z$$

(4)

(2)

Horizontal sector equilibrium equation in θ1 direction.

The horizontal force balance of the fan-shaped material in the direction of θ₁ is as follows:

$$P_{rg}+m^{\prime }{a}_{er\perp }(z)g+A_1{\tau }_{rg}\cos {\theta }_1=-A_1({\tau }_{rg}+d{\tau }_{rg})\cos ({\theta }_1+\pi )+A_2{\sigma }_r(z,{\theta }_1)$$

(5)

Here,

• m′a_eh⊥(z)g is the component of the horizontal inertial force of the earthquake in the direction of the sector midline (perpendicular to the side wall), which can be expressed by m′a_eh(z)g·cosθ₁;
• τ_rg = μ′σ_z is the horizontal shear force between the upper layer of storage and this layer. It is assumed that under the action of earthquakes, all reservoirs reach the limit equilibrium state;
• There is no shear stress between the fan-shaped storage material and the adjacent storage material;
• P_rg is the horizontal force acting on the sector in the direction of θ₁ + π.
• Formula (5) can be rewritten as follows:

$$\frac{R^2\rho gd\theta dz}{2}{a}_{er}(z)\cos {\theta }_1=d{\tau }_{rg}\frac{R^2\cos {\theta }_{1}d\theta }{2}+R{\sigma }_r(z,{\theta }_1)d\theta dz-P_{rg}$$

(6)

Similarly, the horizontal sector equilibrium equation in the direction of θ₁ + π is

$$P_{rg}=\frac{R^2\rho gd\theta dz}{2}{a}_{er}(z)\cos {\theta }_1-d{\tau }_{rg}\frac{R^2\cos {\theta }_{1}d\theta }{2}+R{\sigma }_r(z,\theta +\pi )d\theta dz$$

(7)

In combination with Equations (6) and (7), the horizontal equilibrium equations of the fan-shaped stored material at depth z can be rewritten as

$$R^2\rho g{a}_{er}(z)\cos {\theta }_{1}d\theta dz=d{\tau }_{rg}{R}^2\cos {\theta }_{1}d\theta +R{\sigma }_r(z,{\theta }_1)d\theta dz-R{\sigma }_r(z,\theta +\pi )d\theta dz$$

(8)

(3)

Equilibrium equation of the circumferential force between the fan-shaped stored material and sidewall.

$$m^{\prime }{a}_{er//}(z)g+{\tau }_{rg}\frac{R^2\sin {\theta }_{1}d\theta }{2}=({\tau }_{rg}+d{\tau }_{rg})\frac{R^2\sin {\theta }_{1}d\theta }{2}+A_2{\tau }_{rw}$$

(9)

Here,

• m′a_eh//(z)g is the component of the horizontal inertial force perpendicular to the direction of the sector’s midline (parallel to the side wall), which can be expressed by m′a_eh(z)g·sinθ₁;
• Assuming no dislocation between stored material and sidewall.

Equation (9) can be rewritten as

$$\frac{R\rho g}{2}{a}_{er}(z)\sin {\theta }_{1}dz=\frac{R\sin {\theta }_1}{2}d{\tau }_{rg}+{\tau }_{rw}dz$$

(10)

4. Horizontal Additional Stress of Sidewall under Earthquake Excitation

The distribution of the seismic acceleration function must be clear to calculate the horizontal additional stress of the silo wall under seismic action. It is generally believed that the functions of horizontal acceleration a_eh can be expressed as a fixed value, a linear function, and a nonlinear function (

Table 1. The express form for acceleration.

	Acceleration	Horizontal Acceleration
Express Form
Vertical Acceleration		aez(z) = aez0
Linear Function		aez(z) = aez0 + aez1(H − z)
Nonlinear Function		aez(z) = aez0 + aez1(H − z) + aez2 (H − z)2

Note: a_ez0, a_ez1, a_ez2, a_er0, a_er1, and a_er2 are constant coefficients, and H is the filling height.

).

The suitable scope and conditions of the three main forms with seismic horizontal acceleration along the depth of a silo are as follows:

(1)

If the silo is almost rigid under the influence of horizontal seismic motion, the horizontal seismic acceleration can be assumed to have a constant distribution;

(2)

If the silo shows obvious shear lateral deformation characteristics, it can be assumed that the distribution of the horizontal seismic acceleration is linear;

(3)

If the silo shows the characteristics of bending lateral deformation, the horizontal seismic acceleration can be assumed to have a nonlinear (parabolic) distribution.

In the present work, the acceleration function is assumed to have a constant distribution along the depth z direction since the side wall is assumed to be free from lateral deformation.

Equations (4), (8), and (10) can be, respectively, expressed as

$$R\rho g{a}_{er0}\cos {\theta }_{1}dz=d{\tau }_{rg}R\cos {\theta }_1+{\sigma }_r(z,{\theta }_1)dz-{\sigma }_r(z,\theta +\pi )dz$$

(11)

$$\frac{R\rho g}{2}{a}_{er0}\sin {\theta }_{1}dz=\frac{R\sin {\theta }_1}{2}d{\tau }_{rg}+{\tau }_{rw}dz$$

(12)

In which, the unknown variables need to be adjusted as follows:

• Under the action of the vertical horizontal velocity a_r(z), the horizontal pressure of the fan-shaped reservoir on the lateral wall changes from σ′_r to σ_r under static load. It is assumed that there is still a limit equilibrium between the stored material and the side wall, so the friction force of the side wall changes from τ′_zw = μ₀σ′_r at static load to τ_zw = μ₀σ_r at static load;
• Setting σ_r(z,θ) = σx_r(z) + Δσ_r(z,θ), where Δσ_r(z,θ₁) is the horizontal additional stress of the silo wall under earthquake action. Under static load, σ′_r(z) = Kσ′_z(z). In addition, σ_r(z,θ₁) = Kσ′_z(z) + Δσ_r(z,θ₁) τ_zw = μ₀(σ′_r + Δσ_r) = τ′_zw + μ₀Δσ_r(z,θ₁).

Formula (4) can be rewritten as

$$d{\sigma }_z=\rho g[1+a_{ez0}]dz-\frac{{\mu }_0[{\sigma }_r(z,{\theta }_1)+{\sigma }_r(z,{\theta }_1+\pi )]}{R}dz$$

(13)

Formula (11) can be rewritten as

$$R\rho g{a}_{er0}\cos {\theta }_{1}dz={\mu }^{\prime }R\cos {\theta }_{1}d{\sigma }_z+{\sigma }_r(z,{\theta }_1)dz-{\sigma }_r(z,\theta +\pi )dz$$

(14)

By eliminating the term σ_r (z,θ + π) in Equations (13) and (14), one can write

$$\rho Rg[1+a_{ez0}]dz+{\mu }_{0}R\rho g{a}_{er0}\cos {\theta }_{1}dz=2{\mu }_0{\sigma }_r(z,{\theta }_1)dz+(R+{\mu }_0{\mu }^{\prime }R\cos {\theta }_1)d{\sigma }_z$$

(15)

It should be noted that under the assumption of this paper, the derivation of Equations (11)–(15) is completely rigorous and without redundant assumptions. Once the horizontal acceleration direction of a given earthquake is θ₁ = 0, the lateral pressure σ_r(z,0,R) in the same direction is the peak value of lateral pressure at the depth of z, and the corresponding σ_r(z,0,R) is the peak value of horizontal additional stress at the depth of z. Following Eurocode Part 4 [10], Δσ_r(z,θ₁,R) = Δσ_r(z,0,R)·cosθ1 (that is, Δσ_r(z,θ₁,R) = Δσ_r(z), _max·cosθ₁). Therefore,

$${\sigma }_r(z,\theta ,R)={{\sigma }^{\prime }}_r(z)+\Delta {\sigma }_{r,\max }(z)\cdot \cos {\theta }_1$$

(16)

If the vertical acceleration a_z(z) = 0, Formula (16) becomes

$$\Delta {\sigma }_{r,\max }(z)=\frac{\rho Rgdz}{2{\mu }_0\cos \theta }+\frac{R\rho g{a}_{er0}dz}{2}-\frac{{{\sigma }^{\prime }}_{r}dz}{\cos \theta }-\frac{(R+{\mu }_0{\mu }^{\prime }R\cos {\theta }_1)d{{\sigma }^{\prime }}_z}{2{\mu }_0\cos \theta }$$

(17)

The expressions of σ′_r and σ′_z were given before, so Formula (17) can be written as:

$$\begin{array}{ll}\Delta {\sigma }_{r,\max }(z)& =\frac{\rho Rgdz}{2{\mu }_0\cos \theta }+\frac{R\rho g{a}_{er0}dz}{2}-\frac{\gamma R}{2{\mu }_0\cos \theta }(1-e^{-2{\mu }_{0}Kz/R})dz-\frac{(R+{\mu }_0{\mu }^{\prime }R\cos {\theta }_1)\gamma R}{4K{\mu }_0^2\cos \theta }d(1-e^{-2{\mu }_{0}Kz/R})\\ & =\frac{\rho Rgdz}{2{\mu }_0\cos \theta }+\frac{R\rho g{a}_{er0}dz}{2}-\frac{\rho gR}{2{\mu }_0\cos \theta }(1-e^{-2{\mu }_{0}Kz/R})dz-\frac{(R+{\mu }_0{\mu }^{\prime }R\cos {\theta }_1)\rho g{e}^{-2{\mu }_{0}Kz/R}}{2{\mu }_0\cos \theta }dz\end{array}$$

(18)

Overall, the additional stresses at any points on the side wall of a silo caused by horizontal seismic acceleration can be expressed as

$$\Delta {\sigma }_{r,\max }(z)=\frac{R\rho g{a}_{er0}}{2}dz-\frac{{\mu }^{\prime }R\rho g{e}^{-2{\mu }_{0}Kz/R}}{2}dz$$

(19)

It is worth noting that the additional stress Δσ_r(z,θ) of the side wall proposed in this work is related to factors such as the grain depth z, silo radius R, storage density p, internal friction coefficient μ′, lateral pressure coefficient K, and horizontal acceleration coefficient a_er0, while the Δσ_r(z,θ) is a fixed value in the Chinese and European codes.

5. The Effective Mass of Stored Materials under Earthquake Excitation

5.1. Derivation of Effective Mass Coefficient of Stored Materials

The effective mass coefficient ξ of stored materials is an important parameter in the seismic resistance of silos. However, this value in the national codes is slightly conservative and its calculation lacks a theoretical basis. Hence, to address this issue, a theoretical model of the effective mass coefficient was formulated based on the analytical results of Equation (19).

When the direction of horizontal acceleration (θ) is 0, the additional stress Δσ_r(z,θ₁) of the upstream horizontal acceleration (0 < θ₁ < π/2, 3π/2 < θ₁ < 2π, see Figure 4) is positive and the downstream horizontal acceleration (π/2 < θ₁ < 3π/2) is negative. The additional resultant force ΔP_eh in the range of 0 < θ₁ < π/2 and 3π/2 < θ₁ < 2π can be equivalent to the base shear of the seismic action and the whole storage material because the stored materials only contribute to the increase in the wall stress in the upstream direction.

The additional resultant force ΔP_eh in the upstream direction can be expressed as

$$\begin{array}{ll}\Delta P_{eh}& =R{\displaystyle {\int }_0^H{\displaystyle {\int }_0^{\frac{\pi }{2}}\Delta {\sigma }_{r,\max }\cos \theta d\theta dz+}}R{\displaystyle {\int }_0^H{\displaystyle {\int }_{\frac{3\pi }{2}}^{2\pi }\Delta {\sigma }_{r,\max }\cos \theta d\theta dz}}\\ & =\rho R^{2}g[a_{er0}H-\frac{{\mu }^{\prime }R(e^{-2{\mu }_{0}KH/R}-1)}{2{\mu }_{0}K}]\end{array}$$

(20)

Then, the effective mass coefficient ξ can be calculated as follows:

$$\begin{array}{ll}\xi & =\Delta P_{eh}/a_{er0}g/m\\ & =\frac{1}{\pi }-\frac{{\mu }^{\prime }R(e^{-2{\mu }_{0}KH/R}-1)}{2{\mu }_{0}KH\pi a_{er0}}\end{array}$$

(21)

Equation (21) is the calculation method of the effective mass coefficient of the storage material under seismic action in this paper. We can see that the value of ξ is negatively correlated with height–diameter ratio H/R and horizontal acceleration a_er0, which is consistent with the results reported by Silvestri et al. [30].

Error Analysis

If a_er0g = 0, Equation (19) is changed to an equilibrium equation under static load, and if Δσ_r.max (z) = 0, Equation (19) takes the following form:

$$\Delta {\sigma }_{r,\max }(z)=-\frac{{\mu }^{\prime }R\rho g{e}^{-2{\mu }_{0}Kz/R}}{2}dz$$

(22)

where μ′ = 0, because there is no relative movement trend among the reservoirs under static load. Thus, Equation (19) is still valid at a_er0g = 0.

If the horizontal inertia force caused by seismic horizontal acceleration is not enough to overcome the resultant friction forces between the reservoir and its upper and lower layers and ma_er0g ≤ πR²dτ_rg(z), the reservoir still moves towards the a_er0g direction, which means the reservoir has no contribution to the additional stress on the side wall. Therefore, Equation (19) is not applicable at this time, which is a defect in the hypothesis in this paper. At this time, the stored material in the silo can be regarded as an aggregate of the particles and the relative movement trends between the silo bottom and the material (considering that the silo is a flat-bottomed circular silo).

If the inertia forces of the aggregate particles can overcome the friction with the silo bottom, ma_er0g > πR²τ_gf and ma_er0g ≤ πR²dτ_rg(z), where M is the total mass of the storage material in the silo and τ_gf is the horizontal shear stress between the stored material and the silo bottom, the additional stress Δσ_r(θ₁) distributed according to the fixed value will appear on the side wall of the silo. When ma_er0g ≤ πR²μ_0,gf · σ_z(H), Δσ_r = 0, which means that the storage material and the silo structure move together at this time, and the effective mass coefficient ξ = 1.

In the case of ma_er0g > πR²τ_gf, the height of the storage in Figure 2 changes from 0 to H, and the horizontal equilibrium equation of the two sector-shaped reservoirs with a symmetrical orientation of θ₁ = 0 in Figure 3 can be written as

$$R\rho g{a}_{er0}H={\tau }_{gf}R+2\Delta {\sigma }_{r,\max }(H)H$$

(23)

If the vertical acceleration (a_er0g) is 0 and the relative movement trends between the stored material and the floor exist at the bottom of the silo, the τ_gf is the product of the vertical pressure acting on the bottom of the silo and the corresponding external friction coefficient. When the external friction coefficient between the stored material and the bottom μ_0,gf is large enough, relative motion occurs between the stored material near the bottom. Otherwise, the relative motion occurs between the stored material and the bottom. Therefore,

$${\tau }_{gf}(H)=\min ({\mu }^{\prime },{\mu }_{0,gf})\cdot {{\sigma }^{\prime }}_z(H)$$

(24)

Inserting Equation (23) into Equation (24), one can write

$$\Delta {\sigma }_{r,\max }(H)=\frac{R\rho g{a}_{er0}}{2}-\frac{\min ({\mu }^{\prime },{\mu }_{0,gf})R}{2H}{{\sigma }^{\prime }}_z(H)$$

(25)

It can be seen in Equation (25) that the additional stress Δσ_r(θ₁) (Δσ_r(θ₁) = Δσ_r,max·cos θ₁) will be distributed along the depth direction according to the fixed value when πR²dτ_rg(z)/M ≥ a_er0g > πR²σ_z′(H)·min(μ′,μ_0,gf)/M. Therefore, Equation (19) can be rewritten as

$$\Delta {\sigma }_{r,\max }(z)=\left\{\begin{array}{l}\frac{R\rho g{a}_{er0}}{2}dz-\frac{{\mu }^{\prime }R\rho g{e}^{-2{\mu }_{0}Kz/R}}{2}dz,a_{er0}g>{\mu }_{0}Kg{e}^{-2{\mu }_{0}KH/R}\\ \frac{R\rho g{a}_{er0}}{2}-\frac{\min ({\mu }^{\prime },{\mu }_{0,gf})R}{2H}{{\sigma }^{\prime }}_z(H),\frac{\min ({\mu }^{\prime },{\mu }_{0,gf}){{\sigma }^{\prime }}_z(H)}{\rho }<a_{er0}g\le {\mu }_{0}Kg{e}^{-2{\mu }_{0}KH/R}\end{array}\right.$$

(26)

5.2. Test Verifications

On the basis of the theory, a silo model with organic glass was tested on a shaking table. The silo model had a 600 mm diameter, a 1800 mm height, and was made of organic glass with a 10 mm wall thickness, the steel bottom of which was supported with steel flanges (Figure 6). In addition, the bulk material used for the tests was high-quality wheat produced in Xingyang, Henan, China. The physical and mechanical parameters of the wheat were obtained through a series of tests, as listed in

Table 2. The parameters of the bulk material.

Wheat Variety	Density/(g·L−1)	Water Content/%	Internal Friction Angle/(°)	External Friction Angle/(°)
Henan wheat	804.00	8.69	25.00	21.80

. Other test parameters are listed in

Table 3. The test parameters of the silo–bulk material system.

Grain Height (mm)	Friction Coefficient μ0	Lateral Pressure Coefficients K	Horizontal Seismic Acceleration Coefficient aer0
1600	0.50	0.41	0.15; 0.285

.

On the basis of this theory and Equation (22), the theoretical values of the effective mass coefficient for the model test under different seismic accelerations were calculated. A comparison between the theoretical analysis and the test results is shown in Figure 7, with an error of 6%. This is sufficient to demonstrate the accuracy and applicability of this model and shows that the analytical formula for the effective mass coefficient of the silo–bulk material under vibration is valid and universal.

5.3. Sensitivity Analysis

In this section, the sensitivity of the friction coefficient, lateral pressure coefficients, and seismic accelerations to the effective mass coefficient of the silo–bulk material system during earthquakes is studied (Figure 8). The silo depth was 30 m, μ′ = 0.47, ρ = 800 kg·m⁻³, a_ez0 = 0, and other parameters are shown in

Table 4. The parameter values.

Comparison of Parameters	Friction Coefficient μ0	Lateral Pressure Coefficients K	Horizontal Seismic Acceleration Coefficient aer0
No. 1	0.35; 0.45; 0.55	0.40	0.30
No. 2	0.45	0.40; 0.50; 0.6	0.30
No. 3	0.45	0.40	0.30; 0.40; 0.50

.

It can be seen that with the increase in the external friction coefficient, the lateral pressure coefficient, or the horizontal seismic acceleration, the effective mass coefficient of the reservoir decreased gradually. The narrower the silo (i.e., the bigger the height–diameter ratio), the lower the effective mass coefficient.

6. Conclusions

Herein, the effective mass coefficient and dynamic effects of silos under dynamic response were systematically calculated, which can be used to optimize their design and save costs. In this paper, theoretical solutions for the horizontal forces exerted during earthquakes and the effective mass of the silo–bulk material system during earthquakes were proposed and their accuracy verified. Moreover, a sensitivity analysis of the parameters was conducted. The main conclusions of this paper are as follows:

(1)

On the basis of the Janssen continuum model, considering the seismic inertia force of stored materials and assuming that the storage itself has no energy loss, we hypothesized that the seismic energy is completely dissipated by the friction between the materials and the silo and between the materials themselves; thus, we established a dynamic equilibrium equation considering the vibration of stored materials under earthquake excitation. The dynamic equilibrium equation is a well-known Janssen formula without external force. The accuracy of the model was verified;

(2)

The vertical and horizontal equilibrium equations of any material storage unit and its circumferential equilibrium with the side wall were established, and the additional stresses at any point of the silo were calculated. The additional stress Δσ_r(z, θ) on the side wall obtained in this paper is not only related to the silo radius R, storage density ρ, internal friction coefficient μ′, lateral pressure coefficient K, and horizontal acceleration coefficient a_er0, but also to the depth of the silo. That is to say, its distribution is affected by the size of the silo, the physical parameters of the storage bulk, the seismic magnitude, and changes along the height of the silo, rather than by a constant distribution;

(3)

On the basis of the additional stress of the sidewall and Newton’s second law, the effective mass coefficient of the actual participation of the stored material in the vibration state was deduced. It is shown that under seismic action, the effective mass coefficient of the actual participation of stored materials decreases gradually with the increase in the external friction coefficient, the lateral pressure coefficient, or the horizontal seismic acceleration, and the narrower the silo (i.e., the larger the height–diameter ratio), the lower the effective mass coefficient. Compared with the experimental data and numerical calculation data, the deviation of the results was less than 6%, which proves the validity and universality of this method;

(4)

On the basis of the dynamic model of the silo under earthquake action proposed in this paper, the base shear, the moment of bending, and the moment of inertia of the silo can be calculated, which will be discussed in a future paper.