Calculation of Silo Wall Pressure considering the Intermediate Stress Effect

+e reasonable determination of wall pressure is critical for the design of silo structures. In this study, the primary objective is to present four novel wall pressure coefficients based on four true triaxial strength criteria in the quasiplane strain state. +ese four strength criteria are the Drucker-Prager (D-P) criterion, the Matsuoka-Nakai (M-N) criterion, the Lade-Duncan (L-D) criterion, and the unified strength theory (UST), and they all consider the effect of the intermediate stress yet to different extent. +ese coefficients have a wide application range and are readily used to predict the distribution of wall pressure for deep and squat silos. Comprehensive comparisons are made between the predictions from the wall pressure coefficients described herein and experimental data reported in the literature as well as the results from the European, American, and Chinese silo standards or the Rankine and the modified Coulomb theories. It is found that the effect of the intermediate stress on the wall pressure is very significant for both deep and squat silos; the wall pressure of the D-P criterion is underestimated, whereas that of the MohrCoulomb (M-C) criterion is overestimated; the L-D criterion is recommended to be adopted to calculate the soil wall pressure.

e static pressure during the filling and storage constitutes the primary load acting on the silo wall.ere are several classical theories to predict static wall pressures [24][25][26].Earlier studies have shown that the wall pressure during the filling and storage can be expressed as Janssen's equation.However, a consensus with regard to the wall pressure coefficient k is not generally accepted.
For deep silos, most design standards are based on the Janssen theory, but each of them uses different wall pressure coefficients.e commonly used wall pressure coefficients are the Rankine active earth pressure coefficient k � (1 − sin φ)/(1 + sin φ) [27], the static earth pressure coefficient k � 1 − sin φ [28], the modified static earth pressure coefficient, and the wall pressure coefficient considering wall friction [29][30][31].e classical theories of wall pressure for squat silos include the Rankine theory and the modified Coulomb theory [32,33].ese two theories are based on the Mohr-Coulomb (M-C) criterion without considering the intermediate principal stress.
In fact, the materials stored in silos are in a quasiplane strain state where three-dimensional unequal stresses exist.As the influence of the intermediate principal stress on the material strength is in general significant, the M-C criterion is not appropriate to predict such strength.e silo wall pressure using the M-C criterion is then overestimated leading to a conservative silo design [34][35][36][37][38]. Consequently, choosing an appropriate true triaxial strength criterion, such as the Drucker-Prager (D-P) criterion, the Matsuoka-Nakai (M-N) criterion, the Lade-Duncan (L-D) criterion, or the unified strength theory (UST) can not only improve silo quality and durability but also generate economic benefits.
In this study, the D-P criterion, the M-N criterion, the L-D criterion, and the UST are adopted to derive four novel wall pressure coefficients in quasiplane strain to consider the intermediate stress effect.
e applicability condition for these coefficients is also provided.Furthermore, these coefficients are used to calculate the wall pressure for deep and squat silos.Finally, the calculated results are compared with several sets of experimental data and different standards or theories.

Principles.
e distribution of wall pressure depends on the silo type.According to the load characteristics, silos are usually divided into deep silos and squat silos.ese two types of silos have different calculation methods [27].
For a deep silo (H/D ≥ 1.5, where H is the silo height and D is the silo diameter), as shown in Figure 1, the wall pressure P h on the silo wall [6] is written as where C h is the correction factor related to the calculated height and can be determined by GB50077-2003 [27], c is the gravity density of bulk solids, ρ is the hydraulic radius of the net horizontal cross section, μ is the friction coefficient between the silo wall and bulk solids, k is the wall pressure coefficient, and s is the depth from the material top or from the center of cone gravity to the calculation section.For a squat silo (H/D < 1.5), the wall pressure P h [6] is expressed as ( When the Rankine theory is used, the wall pressure coefficient k is When the modified Coulomb theory is used, the wall pressure coefficient k is where φ is the internal friction angle of bulk solids and δ is the wall friction angle. It can be seen from Equations ( 1) and ( 2) that the key to calculate the wall pressure for both deep and squat silos is how to rationally determine the wall pressure coefficient k.In addition, bulk solids are similar to granular materials.erefore, the wall pressure coefficient k in the quasiplane strain state can be determined from the analogy with the lateral earth pressure for sandy soils acting on retaining walls.

Basic Assumptions
(1) e calculation of silo wall pressure can be regarded as a quasiplane strain problem [6].e intermediate stress in bulk solids could be considered approximately as the intermediate principal stress σ 2 equal to the average value of the maximum principal stress σ 1 and the minor principal stress σ 3 without significant error [34][35][36][37][38]. is correlation is written as (2) Bulk solids can be analogous to sandy soils.at is to say, the cohesion c of bulk solids is negligible, i.e., c � 0. (3) e compressive stress is assumed to be positive, and the tensile stress is thus negative.

D-P Criterion.
e D-P criterion [39,40] known as the generalized Mises criterion makes the assumption that the intermediate principal stress and the minor principal stress have an identical effect on the material strength.e D-P criterion is expressed as where J 2 is the second invariant of stress deviation and I 1 is the first invariant of stress tensor.

M-N Criterion.
e M-N criterion [41,42] is suitable for cohesionless materials and overcomes the singularity of the M-C criterion in a deviatoric plane as well as the condition of where I 2 and I 3 are the second and third invariants of stress tensor, respectively.

L-D Criterion.
e expression of the L-D criterion [43,44] is similar to the M-N criterion, but the limit locus of the former in the deviatoric plane is slightly larger than that of the latter.e L-D criterion is expressed as Substituting Equations ( 5), (8), and (12) into Equation ( 14), the wall pressure coefficient based on the L-D criterion (i.e., k LD ) is obtained as 3.4.UST.With fully considering the intermediate principal stress effect and its interval, the UST [45] covers the entire region from the lower bound to the upper bound of all convex strength criteria.e UST can thus be applied to various bulk solids with different tension-compression characteristics and is expressed as where b is the UST parameter which reflects the influence of the intermediate principal stress on the material strength, with the range 0 ≤ b ≤ 1.In addition, there is a positive correlation between the parameter b and the intermediate principal stress effect.In other words, the greater the parameter b, the higher the strength of bulk solids achieved, due to considering the intermediate principal stress effect.Also, b is a parameter for choosing different strength criteria.
For instance, the UST becomes the M-C criterion when b � 0; the twin-shear stress criterion is obtained when b � 1; a series of new strength criteria are set up when 0 < b < 1.
From Equation ( 5), it is found that the quasiplane strain condition satisfies Equation (16b) for sin(φ) ≥ 0. Substituting Equation ( 5) into Equation (16b), the wall pressure coefficient based on the UST (i.e., k UST ) is obtained as 3.5.Applicable Conditions.Substituting Equations ( 9), ( 13), (15), and ( 17) into Equations ( 1) and ( 2), four novel formulations of wall pressure corresponding to four true triaxial strength criteria are presented for deep silos and squat silos, respectively.e application of these wall pressure coefficients (and thus the corresponding wall pressure formulations) is very simple and convenient.
According to the silo wall pressure theory [6], the wall pressure coefficient should be nonnegative, and thus the applicable condition is given by In addition, the wall pressure coefficients presented herein are only associated with the internal friction angle φ of bulk solids.Accordingly, φ should fulfill Equation (18), which means that φ ≤ 42.22 °for the D-P criterion, whereas the other three strength criteria impose no restrictions on φ and therefore have a wider applicable range.4) of the modified Coulomb theory is not shown in Figure 2 in that the wall friction angle δ needs to be known.

Comparisons of Wall Pressure Coefficients
From Figure 2, it can be seen that the wall pressure coefficients all decrease with increasing the internal friction angle of bulk solids, and their relative values are (from bigger to smaller) EN1991-4, ACI313-97, GB50077-2003 � the UST (b � 0), the UST (b � 1/2), the M-N criterion, the UST (b � 1), the L-D criterion, and the D-P criterion.e European silo standard using the modified static earth pressure coefficient as 1.1 × (1 − sin φ) is the most conservative, while the American silo standard using the static earth pressure coefficient as 1 − sin φ is slightly smaller.
e Chinese silo standard using the Rankine active earth pressure coefficient Advances in Civil Engineering from the M-C criterion expressed as in Equation ( 3) is consistent with that of the UST when b � 0.
Meanwhile, the wall pressure coefficients for different strength criteria have significant differences due to the different influence of the intermediate stress.
e wall pressure coefficient based on the UST when b � 0 is relatively larger due to not considering the intermediate stress effect.On the contrary, the k DP of the D-P criterion for φ ≤ 42.22 °is the smallest due to the large influence of the intermediate stress.

Comparisons and Discussions
To demonstrate the applicability and differences of the four true triaxial strength criteria to calculate the silo wall pressure when considering different effects of the intermediate stress, the corresponding four wall pressure formulations are used for deep (H/D ≥ 1.5) and squat silos (H/D < 1.5), respectively.ese formulations are compared with several sets of experimental data as well as with the results from three standards and two theories.
Note that the experimental data involved here are all specific measured values rather than mean ones.[7], Zhang et al. [8], Ruiz et al. [11], and Munch-Andersen et al. [12] carried out model tests to measure the wall pressure distribution of deep silos.e ratio of silo height to its diameter is always greater than 1.5.

Deep Silos. Liu and Hao
e geometric data and material properties of the test silos are listed in Table 1.

Comparisons of the Results from Different Criteria.
A total of six sets of experimental data are compared with the results from six strength criteria, as shown in Figure 3. ese six strength criteria are the D-P criterion, the M-N criterion, the L-D criterion, and the UST (b � 0, 1/2 and 1).
It can be found from Figure 3 that the differences of the wall pressure for deep silos calculated by different strength criteria are obvious.e effect of the strength criterion on the wall pressure results significant.e values of the wall pressure corresponding to six strength criteria are as follows: the UST (b � 0) > the UST (b � 1/2) ≈ the M-N criterion > the UST (b � 1) > the L-D criterion > the D-P criterion.For the three groups of model tests from Liu and Hao [7], the average ratios of the P h obtained with the six strength criteria to the experimental data are (from larger to smaller) 1.21, 1.10, 1.07, 1.03, 0.97, and 0.70.For the model test from Zhang et al. [8], the ratios of the P h obtained with the six strength criteria to the experimental data are (from larger to smaller) 1.32, 1.21, 1.19, 1.15, 1.09, and 0.80.For the model test from Ruiz et al. [11], the ratios of the P h obtained with the six strength criteria to the experimental data are (from larger to smaller) 1.41, 1.26, 1.22, 1.17, 1.05, and 0.61.For the model test from Munch-Andersen et al. [12], the ratios of the P h obtained with the six strength criteria to the experimental data are (from larger to smaller) 1.53, 1.45, 1.43, 1.40, 1.29, and 0.43.
From the above analyses, we can conclude that the P h of the L-D criterion agrees best with the experimental data; the P h of the UST when b � 1/2 is similar to that of the M-N criterion.e P h using the UST when b � 0 (i.e., the M-C criterion not considering the intermediate stress effect) is the largest; on the contrary, the P h of the D-P criterion is the smallest since the intermediate stress effect is exaggeratedly considered to be the same as the minor principal stress effect.

Comparisons of the Results from the Standards.
e six sets of experimental data for deep silos are once again compared with the results from the European, American, and Chinese silo standards, and the UST (b � 0) in which the intermediate stress effect is null, as well as the D-P criterion in which the effect of the intermediate stress is the greatest, as shown in Figure 4.
Figure 4 presents the changes of the wall pressure in magnitude from three standards and two strength criteria: EN1991-4 > ACI313-97 > GB50077-2003 � the UST (b � 0) > the D-P criterion.For the three groups of model tests from Liu and Hao [7], the average ratios of the P h calculated by the three standards and the two strength criteria to the experimental data are (from larger to smaller) 1.70, 1.61, 1.21, 1.21, and 0.70.For the model test from Zhang et al. [8], the ratios of the P h calculated by the three standards and the two strength criteria to the experimental data are (from larger to smaller) 1.78, 1.70, 1.32, 1.32, and 0.80.For the model test from Ruiz et al. [11], the ratios of the P h calculated by the three standards and the two strength criteria to the experimental data are (from larger to smaller) 2.22, 2.05, 1.41, 1.41, and 0.61.For the model test from Munch-Andersen et al. [12], the ratios of the P h calculated by the three standards and the two strength criteria to the experimental data are (from larger to smaller) 1.76, 1.73, 1.53, 1.53, and 0.43.
From the above results, we find that the three standards are all more conservative than the strength criterion considering the effect of the intermediate stress (i.e., the D-P  Chen [14] carried out similar field tests to measure the wall pressure distribution of squat silos (No. 4) from Henan National Grain Reserve.In this case, the stored height h of bulk solids is considered to be the silo height H, and then h/D < 1.5.e geometric data and material properties are presented in Table 2.
Due to similar variations of the wall pressure for different stored heights, only some field experimental data from Yuan [13] (No. 4 and No. 7 silos for the first and second groups, as well as No. 8 silo for the first group) and from Chen [14] (No. 4 silo for the first group) are adopted to make the following comparisons.Experimental data with standard sand from Zhang et al. [8].(e) Experimental data with wheat from Ruiz et al. [11].(f ) Experimental data with dry sand from Munch-Andersen et al. [12].Figure 5 compares the six sets of experimental data with the results from six strength criteria for squat silos.For both cone piles and flat piles, the differences of the wall pressure for squat silos using different strength criteria are found to be significant.e values of the wall pressure corresponding to the six strength criteria are the UST (b � 0) > the UST (b � 1/2) ≈ the M-N criterion > the UST (b � 1) > the L-D criterion > the D-P criterion.is scale is the same as that for deep silos.For all six groups of field tests, the average ratios of the P h using these six strength criteria to the experimental data are (from larger to smaller) 1.21, 1.09, 1.07, 1.01, 1.00, and 0.82.It is demonstrated that the P h based on the UST when b � 0 (i.e., the M-C criterion) is the largest; the P h based on the UST when b � 1/2 is also close to that based on the M-N criterion; the P h using the UST when b � 1 is close to that using the L-D criterion.Furthermore, the P h on the basis of the L-D criterion is shown to be very consistent with the experimental data for the average ratio being 1.00.

Comparisons of the Results from the
eories.For squat silos, all standards make use of the same formulation expressed in Equation (2) to calculate the wall pressure.
ere are two classical theories, which have been adopted by different standards, to determine the wall pressure coefficient k [32,33].One is the Rankine theory.e other is the modified Coulomb theory.
Figure 6 presents the six sets of experimental data for squat silos that are now once again compared with the P h calculated by the Rankine theory, the modified Coulomb theory and the UST (b � 0) in which the intermediate stress effect is null, as well as the D-P criterion in which the intermediate stress effect is the greatest.e P h calculated by the two theories and the two strength criteria are as follows (from larger to smaller): the modified Coulomb theory > the Rankine theory � the UST (b � 0) > the D-P criterion.For the six groups of field tests, the average ratios of the P h using the two theories and the two strength criteria to the experimental data are (from larger to smaller) 1.56, 1.21, 1.21, and 0.82.
From the above results, it is demonstrated that the Rankine theory and the modified Coulomb theory are more conservative than the strength criterion considering the intermediate stress effect (i.e., the D-P criterion).e results of the UST when b � 0 (i.e., the M-C criterion) are the same as those of the Rankine theory.e experimental data are also generally located in the region between the results from and Hao [7].(d) Experimental data with standard sand from Zhang et al. [8].(e) Experimental data with wheat from Ruiz et al. [11].(f ) Experimental data with dry sand from Munch-Andersen et al. [12]. 6 Advances in Civil Engineering the UST (b � 0) and the D-P criterion.is ordering is a clear indication that the wall pressure for squat silos is not accurately calculated if the intermediate stress effect is not considered rationally.

Conclusions
rough this study, some primary conclusions can be drawn as follows: (1) Based on the principles and basic assumptions of silo wall pressure, four novel wall pressure coefficients are presented for the D-P criterion, the M-N criterion, the L-D criterion, and the UST to consider the effect of the intermediate stress.For the D-P criterion, the internal friction angle of bulk solids cannot be greater than 42.22 °, whereas the other three strength criteria are not restricted.ese four coefficients are readily used to predict the wall pressure  (3) e effect of strength criterion on the wall pressure for both deep and squat silos is found to be very significant.It is indicated that the intermediate stress effect is in fact dealt with distinctly by the different strength criteria.e wall pressure using the UST when b � 0 is overestimated since it does not consider the intermediate stress effect.On the contrary, the wall pressure based on the D-P criterion is underestimated due to the overestimation on this criterion of the effect of the intermediate stress.Overall, the wall pressure generated by the L-D criterion agrees well with the experimental data due to rational consideration that it makes use of the intermediate stress effect.Accordingly, the L-D criterion is recommended to be adopted to calculate the silo wall pressure.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.8 Advances in Civil Engineering Figure 2 illustrates the values of k DP , k MN , k LD , and k UST (b � 0, 1/2, and 1) as well as the wall pressure coefficients from the European silo standard (EN1991-4), American silo standard (ACI313-97), and Chinese silo standard (GB50077-2003) for different values of the internal friction angle φ.Equation (

Figure 2 :
Figure 2: Comparisons of the wall pressure coefficient.

Figure 3 :
Figure 3: Comparisons of the results from six strength criteria with experimental data for deep silos.(a) Experimental data with coal from Liu and Hao (b) Experimental data with wheat from Liu and Hao [7].(c) Experimental data with dry sand from Liu and Hao [7].(d)Experimental data with standard sand from Zhang et al.[8].(e) Experimental data with wheat from Ruiz et al.[11].(f ) Experimental data with dry sand from Munch-Andersen et al.[12].

Figure 4 :
Figure 4: Comparisons of the results from three standards and two strength criteria with experimental data for deep silos.(a) Experimental data with coal from Liu and Hao[7].(b) Experimental data with wheat from Liu and Hao[7].(c) Experimental data with dry sand from Liu and Hao[7].(d) Experimental data with standard sand from Zhang et al.[8].(e) Experimental data with wheat from Ruiz et al.[11].(f ) Experimental data with dry sand from Munch-Andersen et al.[12].

Figure 5 :
Figure 5: Comparisons of the results from six strength criteria with experimental data for squat silos (modified from Sun et al. [37]).(a) Experimental data of No. 4 silo for the first group from Yuan [13] (cone pile).(b) Experimental data of No. 4 silo for the second group from Yuan [13] (flat pile).(c) Experimental data of No. 7 silo for the first group from Yuan [13] (cone pile).(d) Experimental data of No. 7 silo for the second group from Yuan [13] (flat pile).(e) Experimental data of No. 8 silo for the first group from Yuan [13] (flat pile).(f ) Experimental data of No. 4 silo for the first group from Chen [14] (cone pile).

Figure 6 :
Figure6: Comparisons of the results from two theories and two strength criteria with experimental data for squat silos (modified from Sun et al.[37]).(a) Experimental data of No. 4 silo for the first group from Yuan[13] (cone pile).(b) Experimental data of No. 4 silo for the second group from Yuan[13] (flat pile).(c) Experimental data of No. 7 silo for the first group from Yuan[13] (cone pile).(d) Experimental data of No. 7 silo for the second group from Yuan[13] (flat pile).(e) Experimental data of No. 8 silo for the first group from Yuan[13] (flat pile).(f ) Experimental data of No. 4 silo for the first group from Chen[14] (cone pile).

Table 1 :
Geometrical data and material properties of deep silos.

Table 2 :
Geometrical data and material properties of squat silos.