# A Study on the Impact of Low-Frequency Random Loading and Unloading on Silo Structures
## 2 Reinforced Concrete Silo Force Analysis  
### Load Calculation for Reinforced Concrete Silos  
The research object in this paper is a large-diameter reinforced concrete deep silo. According to the reinforced concrete design standard (GB50077-2017) [17], a circular silo with a height/diameter ratio \(h_{n}/d_{n}\geq 1.5\) is considered a circular deep silo. Based on Section 4.2.2 of the standard, the formula for calculating the horizontal pressure \(P_{h}\) (kPa) per unit area on the silo wall at a horizontal cross-section located at a distance s(m) below the center of gravity of the stored material's top surface (Figure 1) is as follows:  
\[P_{h}=C_{h}\cdot\gamma\cdot\rho\left(1-e^{h\cdot k\cdot s/\rho}\right)/\mu \tag{1}\]  
\[k=\tan^{2}(45^{\circ}-\phi/2) \tag{2}\]  
\[\rho=\left\{\begin{array}{l}d_{n}/4(s\leq h_{n}-h_{h})\\ d_{n}/4-\left\{2\cdot\left[s-(h_{n}-h_{h})\right]\cdot\tan\phi\right\}/4(s{>}h _{n}-h_{h})\end{array}\right. \tag{3}\]  
where \(C_{h}\) is the correction coefficient for the horizontal pressure of stored material in a deep silo; \(\gamma\) represents the gravity density of the stored material (kN/m\({}^{3}\)); \(\rho\) stands for the hydraulic radius of the net horizontal cross-section (m); \(e\) is the bottom of the natural logarithm; \(s\) denotes the distance from the top surface of the stored material and the center of gravity of the stored material cone to the calculation section; \(\mu\) is the friction coefficient of the silo wall; \(k\) is the lateral pressure coefficient; and \(\varphi\) represents the internal friction angle of the stored material (\({}^{\circ}\)).  
The formula for calculating the vertical pressure \(P_{v}\) per unit area at the top surface of the funnel is as follows [17]:  
\[P_{v}=C_{v}\cdot\gamma\cdot\rho\left(1-e^{-\mu\cdot k\cdot h_{n}/\rho}\right)/\mu \cdot k \tag{4}\]  
where \(C_{v}\) is the vertical pressure coefficient of the storage material in a deep silo; \(h_{n}\) is the calculated height of the storage material. When \(P_{v}>\gamma h_{n}\), it is taken as \(\gamma h_{n}\).  
The dimensions of the reinforced concrete silo in the calculation model selected for this research are shown in Figure 2.  
The concrete strength grade code is C35 (this concrete strength grade requires a compressive strength value of \(\geq\)35 MPa for a standard cured 28-day concrete cubic test piece), with an elastic modulus of \(E\) = 31,500 MPa (this describes the ability of a material to  
Figure 1: Schematic diagram of the pressure of the storage material in a deep silo: 1. flat top storage, 2. pitched roof storage, 3. center of gravity of the cone at the top of the storage, and 4. calculated cross-section at depth \(s\).  
Figure 2: Reinforced concrete silo’s structural dimensions.  
resist deformation under external forces, that is, the linear relationship between stress and strain within the elastic range of a material before reaching plastic deformation). The cube compressive strength \(f_{cr}=35\) MPa, the cube tensile strength \(f_{tr}=2.2\) MPa, and Poisson's ratio \(v=0.2\). Poisson's ratio is also known as the ratio of transverse strain to longitudinal strain, which is an important physical constant characterizing the deformation properties of a material under stress. The horizontal and vertical stressed reinforcements use hot-rolled ribbed bars of a grade of 400 (HRB400), with an elastic modulus \(E_{s}=200\),000 MPa, yield strength \(f_{yik}=400\) MPa (which refers to the yield limit of a metal material when it yields, which is the stress that resists plastic deformation), ultimate strength \(f_{stk}=540\) MPa (the maximum stress generated when an object is damaged by an external force, which can also be called the destructive strength or destructive stress), and Poisson's ratio \(v_{s}=0.3\).