Stress distribution in two-dimensional silos

Simulations of a polydispersed two-dimensional silo were performed using molecular dynamics, with different numbers of grains reaching up to 64 000, verifying numerically the model derived by Janssen and also the main assumption that the walls carry part of the weight due to the static friction between grains with themselves and those with the silo's walls. We vary the friction coefficient, the radii dispersity, the silo width, and the size of grains. We find that the Janssen's model becomes less relevant as the the silo width increases since the behavior of the stresses becomes more hydrostatic. Likewise, we get the normal and tangential stress distribution on the walls evidencing the existence of points of maximum stress. We also obtained the stress matrix with which we observe zones of concentration of load, located always at a height around two thirds of the granular columns. Finally, we observe that the size of the grains affects the distribution of stresses, increasing the weight on the bottom and reducing the normal stress on the walls, as the grains are made smaller (for the same total mass of the granulate), giving again a more hydrostatic and therefore less Janssen-type behavior for the weight of the column.

Figure 1

Force chains in silos with 4000 grains (left) and 8000 grains (right). Here, we use $\mu =0.6$.

Figure 2

Forces on the lateral walls and base of the silo as a function of time, starting from the moment when the base starts to slide down. This is a typical behavior of forces for 1600 grains and $L=20$ cm. A zoom view of the peaks due to internal avalanches is displayed. We can observe that the peaks in the tangential force on the walls are synchronized to the normal force on the bottom, although with different sign. Notice that in general the tangential force in the left and right walls is close but different, and that the avalanches affect both sides of the silo simultaneously.

Figure 3

Vertical stress ${\sigma }_{zz}$ on the bottom for a silo of width $L=20$ cm, with different values of friction coefficient $\mu$. The points are averages over 16 runs. The lines are the fits to the Janssen's model. A perfect hydrostatic behavior can be observed when there is no friction ($\mu =0.0$). Notice, however, in Table 1 that $\alpha$ becomes clearly smaller than 1 for larger values of $\mu$. Full fit data in Table 1.

Figure 4

Vertical stress ${\sigma }_{zz}$ on the bottom for a silo of width $L=20$ cm, with $\mu =0.6$, and different values of dispersity of the radii. The points are averages over 16 runs. The lines are the fits to the Janssen's model. The granular columns for larger $\mathrm{\Delta }R$ are a bit higher than for smaller $\mathrm{\Delta }R$, although their bulk densities are similar. We obtained similar fits for larger dispersity of radii; the smallest $\mathrm{\Delta }R$ has a weight of saturation a bit higher. Full fit data in Table 1.

Figure 5

Graphic for ${\sigma }_{zz}/L$ as a function of the aspect ratio $z/L$ of the column, where ${\sigma }_{zz}$ is the vertical stress on the bottom for silos with $\mu =0.6$ and different widths. The points are averages over 16 runs. The lines are the fits to the Janssen's model. This vertical stress in the bottom does not reach saturation for large values of $L$, even for columns with aspect ratio as large as 10. In general, the behavior becomes more hydrostatic as $L$ grows. Full fit data in Table 1.

Figure 6

Normal stress ${\sigma }_{zz}$ at the bottom for silos with different bases but the same $W/L$ ratio ($W$: weight, $L$: base size). In the inset we show the behavior of the column height for growing size of the base of the silo: for larger bases, some rearrangement of the disks reduces slightly this height. The solid line is the fit of Janssen's model with $\alpha =1$, $K=0.45\pm 0.01$ and we used an average height of 98.7 cm.

Figure 7

(a) Normal and (b) tangential stress distributions on the walls, for different silo widths and keeping the aspect ratio fixed. The data were smoothed with Gnuplot using the “acsplines” option, where the standard errors of the data were used as smoothing weight. In (a), for larger silos ($L=60$ and 80 cm) notice the quasilinear behavior on the upper half of the silo, similar to hydrostatic pressures (black dashed lines obtained using ${\sigma }_n={\rho }_{b}gz$), confirming [6]. Notice in (b) the presence of negative tangential stress near the top of the column: there are related to the fact that on the bottom slides down, the walls “pull up” on the grains. Near the bottom, the weight of the column cancels this effect. In general, tangential stresses in the upper half of the silo are close to zero, confirming the observation of hydrostaticity.

Figure 8

Internal (a) normal and (b) tangential stress distributions for silos with 4000, $16000$, $36000$, and $64000$ grains keeping the aspect ratio fixed. The graphics are averages taken over 16 runs and normalized with the maximum values found in each case. Notice that for larger silos with aspect ratio fixed, the maximum normal stress appears at the center in the horizontal direction, and around a depth of $\sim 2/3z$. High tangential stress also tends to appear in chains, but small and mostly linear, forming an acute angle with the walls, similar to a herringbone pattern.

Figure 9

Histograms of internal (a) normal and (b) tangential stress distributions along the horizontal width for four silos with $z/L=10$. The first quintile corresponds to the bottom and the fifth quintile to the upper part. Notice how, as $L$ grows, the internal normal stress concentrates around the center, in particular for grains in the second and third quintiles. For large $L$ the system displays an inversion in the direction of the tangential stress since in the two lower quintiles this quantity goes from positive to negative going from left to right, but for the three higher quintiles the little variation one finds goes in the opposite direction.

Figure 10

Distribution of (a) normal and (b) tangential forces exerted at the walls for all times in simulations with a moving base. We observe the exponential decay in the frequency as $F_n$ grows, regardless of the number of the grains. For normal forces very close to zero ($F_n/\langle F_n\rangle <1$), the distribution shows a power law behavior. The tangential forces have a negative zone due to the moving base, but the negative forces have a minor range than the forces in the positive zone. The behavior is similar to the distribution of normal forces, showing an exponential decay for larger values of $|F_t|$, and a power law behavior for $|F_t/\langle F_t\rangle |<1$.

Figure 11

Vertical stress ${\sigma }_{zz}$ on the bottom for a silo of width $L=20$ cm, with $\mu =0.6$, and different values of average radii. The points are averages over 16 runs. The lines are the fits to the Janssen's model. We can note the similarity with Fig. 5. Full fit data in Table 1.

Figure 12

Normal stress distribution on the walls, for different average radii but keeping the mass. The data were smoothed with Gnuplot using the “acsplines” option, where the standard errors of the data were used as smoothing weight. The black dashed line is the hydrostatic behavior of the pressure. When $R_{\mathrm{ave}}$ decreases, the normal stress on the walls decreases.