Experimental study on horizontal pressure of column-supported concrete group silos under earthquake force

ABSTRACT The overpressure of silo wall caused by granular materials under earthquake is one of the main reasons of silo damage. Two shaking table model tests were carried out with column supported silos and independent single silos, respectively. The seismic response of silo granular materials was analyzed first, and then the dynamic lateral pressure of silo wall under different levels of earthquake and overpressure coefficient were obtained. The results showed that the interaction between the granular materials and the silo wall in the group silos is related to the position of the individual silo. The seismic response of granular materials is linked with the spectral characteristics of seismic waves. The overpressure coefficients of the corner silo and the side silo are greater than that of the independent single silo with a 7% and 9% higher value, respectively. Apparently, the seismic response characteristics of the independent single silo are totally different from that of the group silos. It was concluded that when designing the wall of the group bin, the position of the bin should also be considered.


Introduction
Food security is a strategic issue related to national security, economic development and social stability. Meanwhile, the food storage structure is an important facility to ensure the security of national food reserves. As one of the main forms of storage structures, reinforced concrete silos have been widely used in practical engineering in recent years because of their large storage capacity, less land occupation and high degree of mechanization. The reinforced concrete silo bears considerable lateral pressure from the granular materials, especially the dynamic lateral pressure, which is the key factor for silo structure design. The lateral pressure of granular materials on the silo wall can significantly increase under earthquake or during loading and discharging. The increase of lateral pressure will cause local damage and even overall damage to the silo structure.
The numerical research on the dynamic lateral pressure distribution during loading and discharging was carried out by many researchers. Up to date, the distribution law and changing mechanism of the dynamic lateral pressure have been found. However, research on the dynamic lateral pressure of silo walls under earthquake conditions is relatively limited (Keli, Zhaoran, and Shihao 2018). There is no clear conclusion on the dynamic lateral pressure of granular materials under earthquake actions. Silvestri derived a formula for the dynamic lateral pressure of silo structure walls under earthquake action based on European code considering the horizontal shear force, whereas the formula is only applicable to the flat bottom silos (Silvestri et al. 2012). They performed a simulated seismic shaking table test on two silo-supported coal silo models and analyzed the nonlinear seismic response on the silo. Further, they derived a formula for calculating the dynamic lateral pressure of the silo under earthquake action, but the parameter C e in the formula needs to be measured by testing, and the formula is cumbersome to use in practice (Weixing and Bolong 1993). Considering the motion characteristics and the interaction between granular material and silo, the incremental constitutive relationship of the granular materials in a column supported single silo was adopted. Based on this relationship, a simplified calculation model was established under earthquake Zhanxin 1995, 1997). A shaking table test using a column-supported single-silo plexiglass model was conducted, and the interaction of the kinematic phase difference between the granular materials and the silo was investigated (Shujiang, Ligeng, and Wenge et al. 1998). A simulation study of the flow conditions of bulk particles in silos was conducted during vibration, and the dynamic pressure at the silo wall was studied (Zhao and Ji 1993). The shaking table simulation tests on silo-bearing and column supported reinforced concrete group silos were conducted. According to the test, the maximum dynamic lateral pressure of the silo wall under earthquake action appeared in the middle and upper regions of the silo wall, and the dynamic lateral pressure at the top of the silo wall was significantly higher than other parts. The finite element program SAP was used to calculate the self-vibration characteristics of 864 reinforced concrete cylinder-bearing cluster silo models (tu, Vimonsatit, and li 2018). The reinforced concrete silo was the actual object of study, and the experimental data obtained from the shaking table test were subjected to incremental dynamic analysis and numerical simulation, and the seismic susceptibility was evaluated (Kunpeng Guo 2016). The numerical model of the silo structure was established by using ABAQUS, and the correction values of the corresponding parameters were obtained by fitting the pressure change curve of the storage side according to the obtained pressure change curve, and the correction formula of the storage side pressure considering the silo-storage interaction was derived (Changdong Zhou 2015). However, only the variation of dynamic lateral pressure in a single silo has been studied in this paper, and no study of dynamic lateral pressure in group silos has been carried out to illustrate the differences between single and group silos (Xuesen and Yonggang 2021).
Currently, the dynamic lateral pressure of silos under earthquakes has been investigated by shaking table tests and numerical simulations. Whereas, most of the existing studies take a single silo as the research object, the more widely used column-supported group silos are not studied insufficiently. For columnsupported group silos, the dynamic interaction and connection constraints between each single silo make the interaction between the granular materials and silo wall in the group silos more complicated, which is no longer the same as the independent single silo. In actual engineering, the group silos are simplified as single silo during the design stage, which will bring certain hidden damage to the group silo body.
In this paper, the column-supported group silos (3 × 3 combination) and an independent single silo are treated as the research objects. The dynamic lateral pressure of different silos and an independent single silo in the group silos under the action of different seismic waves through simulated seismic shaking table tests were obtained. And the dynamic lateral pressure distribution law and overpressure coefficient of the silo wall were analysed through comparison. This study is assumed to provide a basis for improving the seismic design of the group silo structure.

Model design
Based on the performance parameters of the shaking table and the test conditions, the geometric scale of the test model was set at a ratio of 1:25 compared with real silos. And the model of column-supported cylinder group silos and an independent single silo were designed. The total height of the model is 1.62 m, of which the height of the silo is 1.28 m, and the height of the supporting column is 0.32 m. The inner diameter of each silo is 0.48 m, the outer diameter is 0.50 m, and the thickness of the silo wall is 0.01 m.
Photos of the test models are shown in Figure 1(a and b). The material of the model silo is organic glass, and its material properties are shown in Table 1. The Figure 1. Test model. granular materials in the silo are quartz sand, with particle sizes ranging from 1.18 to 2.36 mm. The density of the sand is 1340 kg/m 3 , the internal friction angle is 35°, and the surface friction coefficient of the organic glass is 0.45.

Similarity relation
The stress similarity coefficient was set as 0.08, and the acceleration similarity coefficient was set as 1.25. The rest of the similarity coefficients could be determined according to the method of dimensional analysis (Ying and Xilin 2016), and the main similarity coefficients are shown in Table 2. According to the density similarity coefficient between the model structure and the prototype structure, the additional mass of the model structure in the three storage states is determined according to the model size and the density of the organic glass and the granular materials according to the principle of the equivalent mass density of the under the artificial model.

Layout of the measuring points
The setup of the measuring points was shown in Figure 2(a, b c, and d). For a single silo, four pressure sensors were set up along the silo wall from top-tobottom in the independent single silo (Figure 2a, b). As shown in Figure 2(c), the corner silos for the model (No. 1, 3, 7 and 9 silos) with two faces solidly connected to other silos, and the boundary connection conditions of each corner silo are the same. The side silos for the model (No. 2, 4, 6 and 8 silos) are intermediate silos with three faces solidly connected to the other silos, and the boundary connection conditions for each side silos are the same. The four faces of the center silo (No. 5 silo) is fixed to other silos. According to the symmetry of the silo model and the boundary conditions of each silo, silos 1, 2, and 5 in the group silos are selected as the test objects. Similar to the single silo, the same setup of measurement is arranged to the No. 1, 2, 5 silo for the model. The same number of acceleration sensors are arranged at the same position as the pressure sensors, and the specific measurement point arrangement is shown in Figure 2(d). The main parts of the model include: the top plate, silo body, ring beam, funnel, support column and bottom plate, each part is directly bolted and bonded with chloroform (CHCl 3 ), the group silo is composed using nine single silos in 3 × 3 combination, the silo is connected to each other at the middle of the silo by three connections.

Loading
According to the Chinese Code for Seismic Design of Buildings (GB50011-2016 2016), the seismic waves with acceleration peaks of 0.0625 g, 0.125 g, 0.159 g, 0.25 g, and 0.281 g were applied to the test models   along the X direction of the shaking table. The El-Centro wave, Tangshan wave, and artificial wave were selected as the seismic simulation shaker table input waveforms. Before and after each acceleration peak seismic wave input, the model was subjected to a white noise sweep with an acceleration peak of 0.05 g. A total of 21 loading conditions were tested.

Seismic response characteristics of the stored materials
When the granular materials are particles, the seismic energy and action are reduced due to the motion and friction between the granular material and the silo wall. To study the seismic response of granular materials, the time histories curve and power spectrum of the dynamic lateral pressure of the silo wall were analysed. The time-history curve and power spectrum of the PT-2 measurement point are shown in Figure 3 (a and b).
If the dynamic lateral pressure increases from the static lateral pressure under the seismic action, it is called pressure increment, and if it decreases from the static lateral pressure, it is called pressure decrement. From Figure 3(a), it can be seen that the dynamic lateral pressure values of the silo are generally in the upper half of the horizontal coordinate and are dominated by the pressure increment. It is well known that the pressure decrement of the silo wall will not exceed its static pressure value. The pressure decrement will no longer increase with the peak of the input acceleration increase. And the pressure increment will increase proportionally presenting a significant increase trend of the lateral pressure. The dynamic lateral pressure time history curve of the stored material generally reflects the load bearing state of the silo structure under the horizontal earthquake action on the silo wall.
It can be seen from Figure 3(b) that the power spectrum of the lateral pressure in the corner, side and middle silos have prominent peaks, and the peak of the power spectrum corresponds to a frequency equal to 8.30 Hz, which is closer to its natural frequency (7.96 Hz). The power spectrum of the lateral pressure in the independent single silo shows a wider distribution band, and the peak power spectrum corresponds to a frequency of 8.20 Hz, which is very different from its natural frequency (9.38 Hz). The result shows that the overall stiffness structure is enhanced due to the connection between the walls and support columns of adjacent single silos in the group silos under earthquake action. In addition, the pressure response characteristics of the granular materials are in good agreement with those of the silo body, and the lateral pressure distribution laws of No. 1, 2, and 5 silo is similar.

Static lateral pressure distribution
When the granular materials are in a static state, the lateral pressure of the granular material on the silo body can be calculated according to the Janssen formula (Janssen 1895), as shown in Eq. (1): where P h is the horizontal pressure at depth s on the unit area of the silo wall and s is the distance from the granular material top surface to the calculated crosssection. γ is the weight density of granular materials, μ is the coefficient of friction between the wall and the granular materials, ρ is the hydraulic radius equal to F/ L, κ is tan 2 (45°-φ/2), φ is internal friction angle. The static lateral pressure was measured from the four measuring points along the silo wall and can be represented by P h01 , P h02 , P h03 , and P h04 from the top-tobottom. According to the test results, P h01 = 745 Pa, P h02 = 1348 Pa, P h03 = 1800 Pa, and P h04 = 2218 Pa. The theoretical value calculated by the Janssen formula from the top-to-bottom along the height of silo wall is P' h01 = 647 Pa, P' h02 = 1408 Pa, P' h03 = 1976 Pa, and

Distribution of the dynamic lateral pressure of individual silo in the group silos
To study the response performance of the dynamic lateral pressure on the silo walls of a group of silos under earthquake action, the dynamic lateral pressure data from the corner, side and center silo walls are analysed in Figures 5, 6 and 7.
From Figure 5 to Figure 7, the following analysis was carried out: The difference between the dynamic lateral pressure increment and decrement of corner silos is small, and the dynamic lateral pressure increment of side silos and middle silos are larger than their pressure decrements, respectively. The dynamic lateral pressure increment is positively correlated with the peak acceleration, and the greater the peak acceleration is, the larger the dynamic lateral pressure increment.
The El-Centro, Tangshan and artificial waves were selected as the loading formation. With the increase of the input peak acceleration, the deformation of the silo structure gradually increases, and the increasing trend is becoming more obvious. Therefore, the seismic wave type is significant to the seismic response of granular materials. Moreover, the sensitivity of the corner silos to the three waves is stronger than the side silos and middle silos. The magnitude of the increase in the lateral pressure is maximal under the Tangshan wave. The changing trend of the increments for dynamic lateral pressure along the height direction of the silo wall is consistent under different seismic waves.
The distributional shape for the dynamic lateral pressure increment is "small in the middle and large  at the top and bottom" along the height for the corner silo. The shape is "large at the top and small at the bottom" for the side silo, but for the center silo, the distribution shape is "large in the middle and small at the top and bottom." The lateral pressure distributions of the corner, side and center silos are obviously different from each other. The calculation results are shown in Table 3. The results show that the different positions of the silos in the group silos will cause different distribution patterns of the dynamic lateral pressure along the wall height. As for the increased percentage, the whiplash effect at the top of the group silos is significant, and this effect of corner silos and side silos is larger than that of middle silos. The corner silo, side silo and center silo have two, three and four points of constraint with other silos on the plane. Due to less constraints of the corner and side silo, the whiplash effect at the top of the silo is larger than that of the center silo. At the bottom of the silo body, the percentage increment of the corner silo is much larger than that of the side silo and the center silo, and the bottom measurement point is located far away from the connection point, which is subject to weak constraints.
Given the above results, we know that the strength of the interaction between the granular material and the silo body is related to the location of the silo. Therefore, it is invalid to calculate and design the silo wall according to the same working condition without distinguishing the difference in the position of every single silo in the group silos, and it is recommended to calculate the internal force of the silo wall according to the position of the silo in the group silos.

Comparison of dynamic lateral pressure
The dynamic lateral pressure distribution of the silo wall of the independent single silo is shown in Figure 8. To study the difference in dynamic lateral pressure between different silo under varying acceleration peaks action, the ratio of the corner, the side, and the center silo to the independent single silo in the case of the dynamic lateral pressure increment is obtained. Considering the different acceleration peak conditions, the ratios are shown to be in mean values in Figure 9.
From Figure 8, the independent silo shows a similar dynamic lateral pressure distribution shape as the central silo of the group silos, but the location where the maximum value appears is different. In detail, the maximum value of the former is closer to the top than the latter one. In terms of the independent single silo, the dynamic lateral pressure increment is generally greater than the decrease in dynamic lateral pressure. Compared with the side and corner silo, the lateral pressure of the independent single layer is less affected by the seismic wave type. The increment amplitude of dynamic lateral pressure under the action of the Tangshan wave is larger than that of the artificial wave and EL-Centro wave. However, the dynamic lateral pressure value increment along the silo wall is similar in the case of three seismic waves, which is the same as the group silos. From Figure 9, in terms of the top of the silo, the ratio for the increment dynamic lateral pressure of the corner and side silos to that of the independent single silo is greater than 1.0, and the ratio coefficient of the middle silo to the independent single silo is less than 1.0. The results show that the whiplash effect of the corner and side silos is stronger than that of the independent single silo, and the center silo is weaker than that of the independent single silo. Therefore, the amount of reinforcement should be increased appropriately at this position when designing corner and side silos. Except for the top and bottom of the corner silo, the ratio coefficient of the group silos to the independent single silo is generally less than 1.0. Because of the connection and mutual restraint between the silos in the group silos, the overall stiffness of the group silos is enhanced. As shown in the test, the shaking degree of the stored material in the group silos is less than that in the independent single silo.

Theoretical equations for the dynamic lateral pressure
The lateral pressure calculation method at the height of the storage center of mass is established based on the storage particle motion balance and the modified Jensen theory. The lateral pressure of any horizontal layer in the silo is simplified by introducing a correction coefficient based on the test results.
A single silo under earthquake action is selected for analysis as shown in Figure 10, and the absolute acceleration at the height of the storage center of mass was assumed to be S a . The lateral pressure consists of two parts: increment and reduction value. Figure 10    from (b) that P ce does not cause the acceleration of granular materials where, P ce is the static lateral pressure. While the P ee (θ) part of the lateral pressure in (c) causes the acceleration of the storage particles S a where P ee (θ) is the pressure increment generated by the particle movement to the silo wall.
Assume that the distribution of P ee (θ) satisfies the Eq. (1), as follows: According to the balance of forces, the unit height is chosen, and the calculated force of P ee (θ) in the x-direction is: From Eq. (2) and Eq. (3), Eq. (4) can be obtained: An P ee ¼ P min þ P max , Eq. (4) can be expressed as: where γ is the density of granular materials (kg/m 3 ); P min is the minimum lateral pressure (kPa); P max is the maximum lateral pressure (kPa); R is the silo radius (m); P h0 is the static lateral pressure (kPa).
In consideration of the average absolute acceleration S a of the storage layer unit at the mass point, and the lateral pressure of the granular material is distributed according to Figure 10, the friction between the upper and lower reservoirs is ignored. In addition, the acceleration response of the silo wall at the height of the mass point of the storage layer unit is obtained by the shaker test. Because granular materials have energy-consuming characteristics, the average acceleration of their unit layer is smaller than the acceleration of the silo wall, so S a in Eq. (5) is calculated with the acceleration of the silo wall instead, and the lateral pressure P ee is larger than the actual lateral pressure. Furthermore, as the upper and lower layers of the granular materials have a mutual restraining effect, the acceleration of the upper layer of the granular materials is relatively large, while the acceleration of the lower layer of the granular materials is smaller. The acceleration for the lower layer of the granular materials is the result of the combined effect of the lateral pressure of the granular materials and the friction of the upper layer of the granular materials. Therefore, Eq. (5) should be multiplied by a correction factor of less than 1.0.
In Eq. (6), C e is the lateral pressure correction factor, which can be determined according to the test results. Through the above analysis and from Figure 10 (c), Eq. (7) is obtained as follows: Eq. (7) can be solved as follows approximately (Weixing and Bolong 1993): Then, dynamic lateral pressures can be calculated as follows:

Theoretical results
Taking the acceleration peak of the 0.281 g working condition as an example, the dynamic lateral pressure of the silo under the earthquake action is calculated using the Eq. (10) and (11), and the calculated results are shown in Table 4. The dynamic lateral pressure increment calculated by theory in the case of the El-Centro wave is approximately 1.5 times of the test value; for the Tangshan wave, it is approximately 1.1 times; and for the artificial wave, it is approximately 1.2 times. This indicates that the dynamic lateral pressure results obtained according to the calculation method proposed in this paper are reasonable.

Overpressure coefficient of the dynamic lateral pressure
The overpressure on the silo wall caused by seismic action is one of the causes of the damage to the silo. To study the overpressure of silo caused by granular materials under earthquake action, the measured data of the dynamic lateral pressure under three different acceleration peak seismic waves were analysed according to the shaking table test results and theoretical calculations. The overpressure coefficient is defined as the ratio of the absolute value of the dynamic lateral pressure to the static lateral pressure at each measurement point. And the silo wall is divided into three sections, the upper (1/3), middle (1/3), and lower zone (1/3), for comparison with the silo specifications (GB50077-2017 2017). The calculated overpressure coefficients are shown in Table 5.
As seen from Table 5, the maximum values of the overpressure coefficient of the silo occur in the upper zone along the silo wall because the granular material at the top has less constraint and is easy to move, and the impact is larger. The overpressure coefficient is clearly correlated with the height position, and the tendency is enlarged gradually with increasing height.
Within the upper one-third zone, the overpressure coefficients of the corner silo and the side silo are larger than those of the independent single silo, increasing by 7% and 9%, respectively. While the overpressure coefficient of the middle silo is smaller than that of the independent single silo, decreasing by 3%. The dynamic interaction between the silos of the group silos during the earthquake causes a certain torsional effect on the top of the group silos, while the torsional effect is smaller in the center silos because of stronger constraint, so the dynamic response of the corner silos and the side silos is stronger than that of the independent single silos. And the dynamic response of the middle silos is weaker than that of the independent single silos. In the middle and lower one-third of the height direction of the silo wall, except for the bottom of the corner silo, the overpressure coefficient of the independent single silo is greater than that of the group silos, which indicates that the overall stiffness of the group silo is greater than that of the independent single silo.
The seismic response characteristics of the granular materials in the group silos are correlated with the location of the silo so the current specification of simply selecting a silo to replace all the silos is not reasonable. In the calculation of the group silo wall, structural design should be considered for different locations of the silo body. The ratio is analysed which is the maximum overpressure coefficient of the corner silo, side silo, middle silo and independent single silo to the correction coefficient adopted by existing regulations taking into account factors, such as unloading and collapse of the granular materials. In the upper onethird of the silo body, the ratio coefficients are 1.65, 1.70, 1.50 and 1.55 with different acceleration peaks; in the middle one-third of the silo body, the ratio coefficients are 0.80, 0.85, 0.85 and 0.90; in the lower onethird of the silo body, the ratio coefficients are 1.30, 0.60, 0.70 and 0.75. The ratio coefficient is less than 1.0 in the middle and lower one-third of the silo body except for the lower part of the corner silo wall, and the ratio coefficient is greater than 1.0 in the upper one-third of the silo body, indicating that the overpressure of the granular materials caused by the earthquake cannot be ignored, and the comprehensive correction coefficient adopted by existing specification is small. Moreover, the correction coefficient specified in the existing specification only considers the case of an independent single silo and does not consider the calculation of the group silos structure, which is not comprehensive and brings some blindness to the engineering design.

Conclusion
Through the analysis of experimental data, the distribution law of the dynamic lateral pressure increment along the height of the silo wall for group silos and an independent single silo under earthquake action is revealed in this paper. According to the experimental and theoretical results, the following conclusions can be drawn: (1) The distribution laws of the increment dynamic lateral pressure along the silo wall are different, due to the different positions of each silo in the group silos. The results show that the magnitude of the interaction between the granular material and the silo wall is related to the position of the silo. Therefore, the independent single silo cannot fully reflect the distribution law of the dynamic lateral pressure along the silo wall of the group silo structure under earthquake action.
(2) For group silos and independent single silo, the degree affected by seismic waves is different. The seismic response of granular materials can be affected by the seismic wave type. The sensitivity of the corner silos under the three seismic waves action is stronger than the side silos and middle silos for the independent single silo. With the enhancement of earth vibration, the difference is gradually amplified with three kinds of seismic waves loaded for the increment of dynamic lateral pressure. But the changing trend of the increments for dynamic lateral pressure along the height direction of the silo wall is consistent under the three seismic wave action.
(3) The overpressure coefficient of the silo granular materials on the silo wall under earthquake action was obtained. In the upper one-third zone of the silo wall, the overpressure coefficient of the corner silo and side silo is larger than that of the independent single silo, and the overpressure coefficient of the middle silo is smaller than that of the independent single silo. The overpressure coefficients of the group silos and the independent single silo are greater than the correction coefficient adopted by the existing code. Hence, the overpressure of granular materials caused by the earthquake effect cannot be ignored. The correction coefficient adopted by the existing code is small, so the amount of reinforcement should be added appropriately when designing the top of the silo wall.
(4) Considering the difference between the dynamic pressure distribution and the overpressure coefficient of the silo wall under earthquake action, it is unreasonable to choose a single silo to replace the group silos in existing specifications. The seismic response characteristics of the granular materials of group silos cannot be fully reflected by an independent single silo. Therefore, the calculation and structural design of wall group silos should be considered for different positions of silo bodies separately.