Traction on Rods within Cylinders Containing Grains: An Analogy with the Upward Movement of Trees in Tornadoes

Abstract

In this work, the frictional traction forces developing in the annular space between two concentric vertical cylinders consisting of the outer surface of a cylindrical rod and the inner sidewall of a wider circular cylinder will be analyzed. The experiments carried out for this study allowed us to measure the traction on the rod for several filling heights, H. For the rod, it is possible to find a linear relation between the theoretically computed traction $T_{rod}$ and the traction measured experimentally, $T_{rodM}$. Based on these results, it is possible to understand the fascinating phenomenon of the lifting, by the rod, of the weights of the mass of grains and of the outer cylinder. Finally, a physical analogy between this problem and the upward movement of trees in tornadoes can be identified.

1. Introduction

Around the year 1495, Leonardo da Vinci inquired into the possible traction force of water on the sidewalls of a vessel. At that time, the mechanical properties of water were unknown and due to this, Leonardo, in the Codex Madrid I, folio 150 recto [1], wrote “I want to see how much more force water exerts on its bottom than on its banks. And I will do it as follows…” A mobile base is joined to the walls of a vessel by means of a strip of leather and is then suspended from the arm of a balance. In Leonardo’s words, “The weight that counterbalances water in the vessel is the weight of water that rests on the bottom. All the weight that doesn’t rest on the base falls on the walls of the container”. This passage suggests that Leonardo never performed the experiment; otherwise, he would probably have realized that the weight resting on the bottom is actually the entire weight of water. On the other hand, due to its frictional nature, granular matter stored in silos imposes traction on the walls. In this work, we shall characterize some properties of the traction force due to grains stored between a rod and a wider cylinder, both kept vertically standing.

In general, dry cohesionless granular materials (whose grains interact mainly by contact), even in a resting state, have unusual properties. For instance, in 1895, H. A. Janssen [2,3], from simple experiments with corn, showed the saturation of pressure with height when this granular medium was stored. More specifically, the Janssen effect predicts that, due to the friction between the grains and the sidewalls of the reservoirs of a constant cross-section (circular, squared, rectangular, etc.), the bottom of a bin does not experience the full weight of the grains stored above some critical height. This phenomenon results in the walls carrying some of the weight of the material. Assessment of the internal friction coefficient ($\mu$) and the coefficient of friction between grains and wall (${\mu }_w$) is essential in the use of the Janssen’s theory.

In his work, Janssen recognized that a publication on experiments intended to determine the pressure distribution on wheat silo cells was completed by Roberts in 1883 [4]. Despite this, Hagen, in his 1852 paper [5,6], measured this effect earlier—although not for the first time. As Hagen himself commented, Huber-Burnand in 1829 [7,8] studied the force transmission in granular materials through an experiment in which an egg covered with a thick layer of sand was able to support a heavy block of iron without breaking. Hagen attempted to predict saturation pressure with respect to height (of sand) through a quadratic law (with some cutoff) for the pressure, instead of the exponential form found by Janssen more than 40 years later.

A variation of the Janssen effect was analyzed by Lord Rayleigh in 1906 [9] for the propulsion of a horizontal column of dry sand, filling a tube, subject to forces pushing at the opposite end. In this case, when the sand is moving, the tangential force at the wall is reckoned as ${\mu }_w$ times the normal force. Rayleigh found that the application of a considerable pressure is unable to overcome a very feeble force acting in the opposite direction at a section many diameters away. Incidentally, later on it will be noted that the ordinary differential equation for the Janssen effect, with the absence of gravity, is the same as for the Rayleigh problem.

The Janssen effect is also associated with the problem of the static pressure distribution and the traction force on the sidewalls when a hollow vertical cylinder or a solid cylindrical rod (generically, both bodies will be referred to as a rod) are symmetrically aligned within a larger cylindrical reservoir, and it is later filled with a dry, non-cohesive granular material [10,11,12]. Under gravity, the traction force accounts for the portion of the weight of the grains that is supported by the sidewalls due to friction. In the cases when flow occurs, from an orifice at the bottom of a silo, measurements indicate that horizontal pressure changes significantly and the Janssen method fails to predict it [10]. Similarly, experiments on the slow pullout of a glass rod immersed in dry sand showed that the friction force is very sensitive to changes in the packing close to the rod [11]. Consequently, this study is only focused on the resting state and the dynamic part of these events was left aside.

In his study, Janssen assumed that, everywhere on the walls, the friction force has reached its maximum allowed value—given by the law of da Vinci–Amontons–Coulomb [13]: $\tau ={\mu }_w{p}_r={\mu }_{w}K{p}_z$, where K is a phenomenological coefficient termed the Janssen parameter, $\tau$ is the vertical friction stress on the sidewall of the silo, $p_r$ is the radial pressure on the cylinder inner side and $p_z$ is the vertical pressure that is constant over a horizontal plane. For the contact with the wall, it is entirely arbitrary to assume full mobilization of the friction (i.e., that the law of da Vinci–Amontons–Coulomb, $\tau ={\mu }_w{p}_r$, is identically satisfied), as in the previous formula. In fact, any value $\tau /p_r$ below the threshold would be acceptable. In this same context, the constant K reflects the pressure anisotropy of the granular medium, and this will be discussed in Section 4.

In practice, Janssen’s method (with suitable safety factors) forms the basis of most codes of practice for silo design [14,15,16,17]. By contrast, the forces on obstacles immersed in a flowing granular material have received little attention. Though not common, such obstacles are by no means unknown in industry, the silo bin activator being a well known example. Our attention has also been drawn to a grain drier with a fixed bed silo–dryer–aerator with vertical pipe-like static grain-drying cells [18].

In this work, a review of Janssen’s problem is presented, first for a cylindrical silo and then for a system of two concentric vertical cylinders. Experiments for the later case will be shown. Also, it will be argued that the latest problem has some analogies with the pure vertical pullout loading condition of a tree due to tornadoes [19]. The windthrow vulnerability assessment for synoptic weather systems is commonly conducted through hybrid-mechanistic and empirical models using both deterministic and stochastic approaches. The bases of a number of these methods are mechanistic models of tree stability based on static tree-pulling field tests, and these ultimately provide critical wind speeds for windthrow and stem break failures. Surprisingly, some of the results of the current study have a direct analogy with the mechanistic models of tree stability based on static tree-pulling field tests.

The paper is organized as follows. In next Section, a brief discussion of the vertical and radial pressures and the traction force is given. In Section 3, the same theoretical approach is applied to compute the traction in the system consisting of two concentric cylinders. In Section 4, experiments of traction on rods will be reported, while, in Section 5 an analysis of the lifting of the rod–sand–reservoir system is discussed theoretically and experimentally and it will be related to the balance of force equation for the upward movement of trees in tornado events. Finally, in Section 6, the main conclusions and limitations of this work will be presented.

2. Pressures and Traction Force in a Circular Cylinder

Pressure distribution in quiescent liquids and in non-cohesive granular materials at rest are quite different from each other. Above all, a brief review of Janssen’s theory is necessary. Consider a cylinder or cylindrical silo (with a flat bottom) of radius $r_0$ and thickness e, filled with a dry cohesionless granular material up to a certain height H, measured from the flat free surface of the granular material; meanwhile the overall cylinder has a height L, see Figure 1. Now, taking a horizontal section of the silo at a depth z in the granular material, according to Janssen’s model, the force exerted, per unit area, on this section by the material above it on the material below, is the vertical pressure $p_z\left(z\right)$. This pressure is not equal to the horizontal pressure $p_r\left(z\right)$ on the sidewall of the silo at the same depth, but the two pressures are linearly related [2,14,20,21,22,23] as

$$p_r\left(z\right)=K{p}_z\left(z\right),$$

(1)

where K is the Janssen parameter, which is, in a first approximation, a constant independent of z that characterizes the conversion of vertical stress into horizontal stress within the granulate.

The horizontal pressure acting on the sidewall of the silo causes a vertical friction stress $\tau$ between the wall and the granular material. A simple balance of the vertical forces, per unit length, acting on a slab of the granular material gives

$$\pi r_0^2{\displaystyle \frac{d{p}_z}{dz}}=\pi r_0^2\rho g-2\pi r_0\tau ,$$

(2)

where $\rho$ is the bulk density of the granular material, assumed to be independent of z, and g is the acceleration due to gravity. In Janssen’s model, assuming the full mobilization of the friction, the vertical friction stress is given by the law, $\tau ={\mu }_w{p}_r$, with a constant wall friction coefficient ${\mu }_w$. Notice that if gravity is absent Equation (2) gives the simple equation $d{p}_z/dz=-\left(2{\mu }_{w}K/r_0\right)p_z$ whose solution, when $p_z\left(0\right)=p_0$, is $p_z=p_0{e}^{-z/\lambda }$, where $\lambda =r_0/2{\mu }_{w}K$. The latter is Rayleigh’s problem of the propulsion of a horizontal column of dry sand in a pipe of radius $r_0$ [9]. The solution shows that a significant pressure at $z=0$ is unable to overcome a very feeble one acting in the opposite direction at a section many diameters away.

Continuing with Janssen’s theory, the solution of Equation (2) with the condition $p_z\left(0\right)=0$ is then [2,20,21,22,23,24]

$$p_z\left(z\right)=\rho g\lambda \left[1-exp\left(-{\displaystyle \frac{z}{\lambda }}\right)\right]\mathrm{where}\lambda ={\displaystyle \frac{r_0}{2{\mu }_{w}K}}.$$

(3)

Equation (3) shows that the vertical pressure increases linearly with z, as $p_z\left(z\right)\approx \rho gz$, for $z/\lambda \ll 1$ and tends to the limiting value $\rho g\lambda$ for $z/\lambda \gg 1$. The length $\lambda$ is the characteristic size of the region where the pressure undergoes this transition. In columns recreated in a laboratory $\lambda \approx 0.1$ m [21]; then, it follows that the sidewall of the container supports most of the weight of the grains when $H\gg 0.1$ m.

The vertical shear stress $\tau \left(z\right)$ yields an overall traction force $T\left(H\right)$, acting on the sidewall of the cylinder, filled up to a height H, that is

$$T\left(H\right)=2\pi r_0{\int }_0^H\tau \left(z\right)dz,$$

(4)

or explicitly

$$T\left(H\right)=\pi r_0^2\rho g\left\{H-\lambda \left[1-exp\left(-{\displaystyle \frac{H}{\lambda }}\right)\right]\right\}.$$

(5)

The function $T\left(H\right)$ increases quadratically with H, as $T\approx \pi r_0\rho g{\mu }_{w}K{H}^2$, for $H\ll \lambda$, and linearly, as $T\approx \pi r_0^2\rho gH$, for $H\gg \lambda$, when most of the weight of the granular material is supported by the wall of the silo. In a conventional silo standing on its base, this maximum stress results in a compression of the lower part of the wall, which makes buckling a critical factor in the design of tall silos.

3. Traction Forces on the Rod and on the Outer Cylinder

Now, to quantify the vertical pressure acting on the granules and the partial weight that rests on the rod and on the sidewall of the outer cylinder, consider a rod of radius $r_1$, concentrically aligned to the larger cylinder of radius $r_0$, and raised at a small height $\Delta H$ above its bottom. The rod can be held, for instance, by a force sensor, see Figure 2. The latter condition is important in experimental set-ups given that when sand is pouring into the annular space, the vertical shear stress exerted by the grains pulls the rod down, towards the small, otherwise empty space below the rod, and thus the measurements of traction are direct. Then, it is important to notice that the level of the sand with respect to bottom is H; meanwhile, with respect to the level $\Delta H$ is $H^{\prime }$, i.e., $H=\Delta H+H^{\prime }$, see Figure 2. Such considerations will be used in the calculation of the different tractions and weights and in the respective experimental measurements.

By alluding to Equation (2), the balance of forces is now

$$\pi \left(r_0^2-r_1^2\right){\displaystyle \frac{d{p}_z}{dz}}=\pi \left(r_0^2-r_1^2\right)\rho g-2\pi \left(r_0\tau +r_1{\tau }_1\right),$$

(6)

here the vertical shear stress on the rod is ${\tau }_1={\mu }_{w_1}{p}_r={\mu }_{w_1}{K}_1{p}_z$ where ${\mu }_{w_1}$ is the wall friction on the rod and, as mentioned previously, $\tau$ is the shear stress on the inner sidewall of the large cylinder. Close to $r=r_0$: $p_r\left(z\right)=K{p}_z\left(z\right)$ and similarly, close to $r=r_1$: $p_r\left(z\right)=K_1{p}_z\left(z\right)$, where $K_1$ is the Janssen parameter close to the rod [10,24]. Using all these relations in Equation (6), allows us to find that

$$\pi \left(r_0^2-r_1^2\right){\displaystyle \frac{d{p}_z}{dz}}=\pi \left(r_0^2-r_1^2\right)\rho g-2\pi \left(r_0{\mu }_w+r_1{\mu }_{w_1}\kappa \right)K{p}_z,$$

(7)

where $\kappa =K_1/K$. The latest differential equation under the condition $p_z\left(0\right)=0$, leads to the solution

$$p_z\left(z\right)={\displaystyle \frac{\left(r_0^2-r_1^2\right)\rho g}{2K\left(r_0{\mu }_w+r_1{\mu }_{w_1}\kappa \right)}}\left\{1-exp\left(-{\displaystyle \frac{2K\left(r_0{\mu }_w+r_1{\mu }_{w_1}\kappa \right)z}{\left(r_0^2-r_1^2\right)}}\right)\right\},$$

(8)

or

$$p_z\left(z\right)=\rho g\lambda \left\{1-exp\left[-{\displaystyle \frac{z}{\lambda }}\right]\right\},$$

(9)

where the screening parameter is now

$$\lambda ={\displaystyle \frac{\left(r_0^2-r_1^2\right)}{2K\left(r_0{\mu }_w+r_1{\mu }_{w_1}\kappa \right)}},$$

(10)

and if the inner rod is absent, $r_1=0$, the Janssen solution (3) is immediately retrieved.

In agreement with Figure 2, the traction force on the rod is

$$T_{rod}\left(H^{\prime }\right)=2\pi r_1{\int }_0^{H^{\prime }}{\tau }_1\left(z\right)dz,$$

(11)

and thus

$$T_{rod}\left(H^{\prime }\right)=2\pi {\mu }_{w_1}{K}_1\rho g\lambda r_1\left\{H^{\prime }-\lambda \left[1-exp\left(-{\displaystyle \frac{H^{\prime }}{\lambda }}\right)\right]\right\}.$$

(12)

For short rods, Equation (12) gives

$$T_{rod}\left(H^{\prime }\right)=\pi {\mu }_{w_1}{K}_1\rho g{r}_1{H}^{\prime },\mathrm{for}{H}^{\prime }\ll \lambda ,$$

(13)

and for long rods

$$T_{rod}\left(H^{\prime }\right)=2\pi {\mu }_{w_1}{K}_1\rho g\lambda r_1{H}^{\prime },\mathrm{for}{H}^{\prime }\gg \lambda .$$

(14)

The traction on the sidewall of the outer cylinder is

$$T_{wall}\left(H\right)=2\pi r_0{\int }_0^H{\tau }_0\left(z\right)dz,$$

(15)

or explicitly

$$T_{wall}=\pi r_0\rho g{\mu }_{w}K\Delta H^2+2\pi {\mu }_{w}K\rho g\lambda r_0\left\{H^{\prime }-\lambda \left[1-exp\left(-{\displaystyle \frac{H^{\prime }}{\lambda }}\right)\right]\right\},$$

(16)

where it has been taken into account that the rod is above $\Delta H$, consequently; for short silos the traction on the cylinder wall is

$$T_{wall}\simeq \pi r_0\rho g{\mu }_{w}K\left(\Delta H^2+H^{\prime 2}\right),\mathrm{for}{H}^{\prime }\ll \lambda ,$$

(17)

and for long silos

$$T_{wall}\simeq \pi r_0\rho g{\mu }_{w}K\left(\Delta H^2+2\lambda H^{\prime }\right),\mathrm{for}{H}^{\prime }\gg \lambda .$$

(18)

4. Computation and Measurements of Forces

In order to compare the theoretical predictions with the corresponding experimental measurements, the dimensions of the inner (rod) and outer concentric circular cylinders and the properties of Ottawa sand are provided below. For these parameters, experiments will be performed to measure the traction force on the concentric rod within the cylinder. The rod is placed $\Delta H=0.10$ m above the silo floor, as seen in Figure 2. Both the inner (subscript 1) and outer cylinders (subscript 0) were fabricated in acrylic and their nominal radii, lengths, and weights are given in

Table 1. Dimensions of the acrylic cylinders used in experiments.

${\mathit{r}}_0$ (${10}^{-3}$ m)	${\mathit{L}}_0$ (m)	${\mathit{W}}_{\mathit{rod}}$ (N)	${\mathit{r}}_1$ (${10}^{-3}$ m)	${\mathit{L}}_1$ (m)	${\mathit{W}}_{\mathit{res}}$ (N)
$2.8$	$1.80$	$6.18$	$1.6$	$1.90$	$12.94$

; also, both of them are $e=0.35\times {10}^{-3}$ m thick, with a nominal density of ${\rho }_{rod}=1.17\times {10}^3$ Kg/m³.

Ottawa sand, which was used in these experiments, consists of round grains, has a bulk density of $\rho =1.60\times {10}^3$ Kg/m³, a mean value of friction angle of $\theta ={31}^{\circ }$, a friction coefficient of ${\mu }_{sand}=tan\theta =0.51$ and a mean diameter of $d=0.05\times {10}^{-3}$ m; the wall friction coefficient between the sand and the acrylic is ${\mu }_w={\mu }_{w_1}=0.26$

Taking into account those data, the Janssen parameter can be estimated, for example, with the Eurocode (European Standard EN 1991-4, 2006), the European bulk storage building code [15,23], where $K=K_1=1.1\left(1-sin\theta \right)=0.54$, or with the Rankine coefficient [25] for the active state, $K=1-sin\theta /(1+sin\theta )=0.32$, because most design manuals assume that a close approximation to the active state is achieved on filling [24]. Consequently, in our calculations we shall use the Rankine coefficient.

In the active state, K is the active pressure ratio (the granular material exerts pressure on the wall), contrary to the passive pressure ratio (the wall exerts pressure on the bulk solid). The lack of consensus exists because K is not a fundamental, innate physical property of bulk solids; it is an emergent result of equilibrium, arising from the “fluid” nature of granular materials. The value of K is not realistically expected to be constant within a storage structure at all. The simplifying assumption is made because Janssen’s equation works well within a narrow column of material, but the variation of K through a large volume of bulk material and/or with the grain shapes has not been well studied experimentally. It is difficult to correctly include a varying K analytically. Janssen [2] and Rankine [25] made the simplifying assumption of constant $0<K<1$.

4.1. Computation of the Forces

Using the previous data (for the active state), from Equation (10) it is immediately found that the screening parameter for the concentric cylinders is $\lambda =7.45\times {10}^{-3}$ m, whilst from Equation (3) $\lambda =0.17$ m for the case where only the outer cylinder is present. In Figure 3, the typical theoretical plots of the forces F involved in the filling of the annular space are shown: these forces are the sand weight, $W_{sand}=\pi g\rho H^{\prime }\left(r_0^2-r_1^2\right)+\pi r_0^{2}g\rho \Delta H$, where the first term represents the weight in the annular space and the second term accounts for the weight below the rod which is near a small cylinder of height $\Delta H=0.10$ m and radius $r_0=2.8\times {10}^{-3}$ m. The plot of $W_{sand}$ as a function of H (black dashed curve) is given in comparison to the relative magnitude of the computed tractions. Similarly, the traction force on the rod, $T_{rod}$, was computed with Equation (12) (red dashed curve). Finally, the traction on the cylinder sidewall $T_{wall}$, computed with Equation (16), is given by the blue dashed curve.

Through the computed plots of Figure 3, it can be appreciated how part of the sand weight is transferred to both the rod and to the outer cylinder, as was mentioned early. In agreement with these results, the computed net weight at the bottom must be $W_{sand}^{\prime }=W_{sand}-T_{rod}-T_{wall}$, which is only a small portion of the overall weight.

An indication of the validity of the current computations will be obtained through the execution of experiments to measure the weights and the traction on the rod, at specific filling heights H, which are described in detail in the next subsection.

4.2. Force Measurements

In carrying out the experiments, a force sensor, model Pasco CI-6537, with range $\pm 50$ N and a resolution of 0.03 N was employed. The section of the laboratory in which the experiments were carried out was climate controlled ($25\pm 1$ °C and 28% R.H.). In Figure 4, plots of $T_{rodM}\left(H^{\prime }\right)$, the measured traction on the rod, and $W_{sandM}\left(H^{\prime }\right)$, the sand weight in the annular space, for several levels of filling $H^{\prime }$, above $\Delta H$, are reported. In the execution of experiments, sand masses of known weights $W_{sandM}$ were gently poured through a small funnel into the annular space between the cylinders. Upon the aggregation of a given weight $W_{sandM}$, a certain height $H^{\prime }$ was achieved. Then, the traction force on the rod, $T_{rodM}\left(H^{\prime }\right)$,was measured. In experiments, both the rod and the outer cylinder were grounded to avoid electrification due to granule–wall and granule–granule collision or friction [26]. Electrification causes the generation of electrostatic charges and the occurrence of granule clustering and granule adhesion to the wall. It has been reported that electrostatic force becomes significant near the wall boundary, where electrostatic force has the same order of magnitude as drag force.

Measurements of each $H^{\prime }$ and $T_{rodM}$, for a given weight $W_{sandM}$, were repeated seven times and averaged. Measurements of the traction on the wall of the outer pipe, $T_{wallM},$ were not carried out since the outer pipe was kept at rest on the floor. A mean maximum height of $H_{max}^{\prime }=0.52\pm 0.01$ m was determined through measurements, which is equivalent to a sand weight of $W_{sand}=14.71\pm 0.03$ N.

In Figure 5, the plot of the theoretical rod traction, $T_{rod}$, versus the measured rod traction, $T_{rodM}$, is depicted; the best fit for the set of data was obtained through the least squares method. The linear relation found is of the type $T_{rod}\sim A{T}_{rodM}$, where in this case, using the Rankine coefficient $K=0.32$ given above, we obtained that $A=1.77$; in contrast, if we use the Janssen constant, given by the Eurocode ($K=0.54$), then $A=2.47$. Consequently, the best fit was obtained by using the Rankine coefficient which confirms that, on filling, this coefficient gives a fine approximation to the experiment [24], of order of magnitude unity, which is noticeable taking into account that in the Janssen model the bulk density was assumed as constant and that K does not depend on the material depth or on the radius of the silo. With respect to the bulk density, in a granular medium filling a channel or pipe with rigid walls, there is in general an increase in the porosity as one approaches the walls. This is because the solid particles are unable to pack together as efficiently as thy can elsewhere, given the presence of the wall [27]. Experiments have shown that the porosity is a damped oscillatory function of the distance from the wall, varying from a value near unity at the wall to nearly core value at about five grain diameters from the wall [27]. In the same sense, experimental evidence was found that K may be significantly lower in the middle of the bin than the wall [28], and that K depends on the material depth [29]. These latest considerations are out of the reach of this study.

5. Analysis of the Lifting of the Rod–Sand–Reservoir System

5.1. The Mechanical Issue

An interesting effect of the traction force occurs when, once a system has been filled up to a height H, an attempt is made to pull out the rod. In experiments, the pullout force on the cylinder, $F_{up}$, is large enough to simultaneously hold the weights of the rod, the sand, and the outer cylinder, meaning that there is a rigid motion of the rod–sand–outer cylinder system; this can be noted from Figure 6, where it can also be observed that the phenomenon takes place in systems of various different sizes. It was also observed that in some events the rod slid out for a small distance before engaging in the rigid elevation of the other elements of the system. It is possible that this occurs while the initial loose packing evolves into a higher packing [11]. In order to explain the holding-up phenomenon, a simple model based on the previous traction analysis is now proposed.

Clearly, due to the enduring contacts of the grains of sand the force $F_{up}$ holds up the weights of the rod, the sand and the outer cylinder. The numerous acting forces, and their directions, occurring in the force balance are depicted in Figure 7. Then, the minimum force required to hold up the rod–sand–outer cylinder system is

$$F_{up}=T_{rod}\left(H\right)-T_{wall}\left(H\right)+W_{rod}+W_{res}+W_{sand}^{\prime },$$

(19)

if a force $F_{up}$, equal or larger than that given by Equation (19), is applied, the hold-up effect must also occur.

In Equation (19), each term expresses physically different effects, as seen in Figure 7: the traction force on the rod $T_{rod}\left(H^{\prime }\right)$ (Equation (12)) frictionally opposes the sand being moved up; meanwhile, on the sidewall of the outer cylinder the traction force $T_{wall}\left(H\right)$ opposes, by friction, the drop down of the reservoir itself. In Figure 7, both of these tangential stresses (${\tau }_1$ and ${\tau }_0$) are depicted. Other contributions are the weight of the rod, $W_{rod}$ and that of the cylinder, i.e., the weight of the reservoir $W_{res}$, since the reservoir itself may comprise the outer cylinder and its bottom; finally, as was determined in Section 4.1 $W_{sand}^{\prime }=W_{sand}-(T_{rod}+T_{wall}$), is the effective weight of the sand allowed by the traction. The introduction of the previous expression for $W_{sand}^{\prime }$ into Equation (19) gives the minimum force

$$F_{up}=W_{sand}+W_{rod}+W_{res}-2T_{wall},$$

(20)

the formula for $T_{wall}$ is given by Equation (16). In Equation (20), it is noticed that if sand, the rod, and the reservoir were a single rigid solid, then $T_{wall}=0$, and thus

$$F_{up}=W_{sand}+W_{rod}.+W_{res},$$

(21)

which are the net forces that $F_{up}$ must hold up to pull up the rigid rod–sand–reservoir system. In the current case $W_{sand}=\pi g\rho H^{\prime }\left(r_0^2-r_1^2\right)+\pi r_0^{2}g\rho \Delta H$, given that the sand weight below the rod, of height $\Delta H=0.10$ m also contributes to the overall weight of the sand; the weights $W_{rod}$ and $W_{res}$ are fixed in experiments.

5.2. Experiments on the Hold-Up Effect with a Cylinder

An expedient series of experiments was carried out to prove the validity of the previous theoretical results; so, in this case the same system of cylinders used in the experiments of traction and the same procedure of filling the annular space, were maintained to avoid the introduction of additional changes. In experiments, it was found that $W_{rod}=5.10\pm 0.03$ N, $W_{res}=10.54\pm 0.03$ N and $W_{sand}=17.61\pm 0.03$ N. Substituting in Equation (20) the previous weight values and the computed wall traction $T_{wall}=9.43$ N (see Figure 3), it is found that the minimum force required to hold up the rod–sand–outer cylinder system is $F_{up}=17.87\pm 0.03$ N. However, if it is assumed that the experimental value of the wall traction is $T_{wall,exp t}=T_{wall,theo}/1.77$ where $T_{wall,theo}$ is the theoretical wall traction computed with Equation (16); then, it can be found that $F_{up}=26.07\pm 0.03$ N.

In Figure 8, we show the schematic of the experimental array to measure first the traction on the rod (a) and (b) the force $F_{up}$ that rigidly holds up the weights of the sand, the rod, and the outer cylindrical pipe.

Figure 8. Schematic of (a) the stage of the rod traction temporal measurement and (b) the stage of the holding up of the sand–rod–outer cylinder system and consequently of the measurement of $F_{up}$, after a short dead time. A typical plot of these events is given in Figure 9.

Figure 8. Schematic of (a) the stage of the rod traction temporal measurement and (b) the stage of the holding up of the sand–rod–outer cylinder system and consequently of the measurement of $F_{up}$, after a short dead time. A typical plot of these events is given in Figure 9.

Figure 9. Temporal measurement of the traction on the rod $T_{rodM}$ and, after a small delay (dead time), measurement of the pull up force required to rigidly hold up the rod–sand–outer cylinder system. Notice that the maximum traction on the rod close to $T_{rodM}\sim 2$ N and that the time-averaged pull up force is $<F_{up}>=33.25\pm 0.03$ N.

Figure 9. Temporal measurement of the traction on the rod $T_{rodM}$ and, after a small delay (dead time), measurement of the pull up force required to rigidly hold up the rod–sand–outer cylinder system. Notice that the maximum traction on the rod close to $T_{rodM}\sim 2$ N and that the time-averaged pull up force is $<F_{up}>=33.25\pm 0.03$ N.

As a final point in this Section, it is worth commenting that in some cases we attempted to break the mechanical stability of the system (to release the rod) by hand pulling the rod and the outer cylinder in opposite directions, but it was soon discovered that this bond was very difficult to break. Physically, this is equivalent, in Equation (21), to increasing $W_{res}$ on the one hand, and $F_{up}$ on the other. If the increments are similar to one another, then the balance in Equation (21) is maintained, as is the mechanical stability of the system.

5.3. Analogy with the Upward Movement of Trees during Tornadoes

In a general context, an analogy is a partial similarity between two things for the purpose of explanation or clarification. By using this definition, we will highlight the similarities between the upward movement of trees during the passage of a tornado and the problem of the lifting of the rod–sand–reservoir system. During tornadoes, there are different types of tree failures, this has led to the identification of relevant variables in order to explain the occurrence of wind related forest damage. The upward movement of a tree (without overturning) during a tornado event is expected to be accompanied by the separation of a percentage of the outer root length from the surrounding soil. This separation can be attributed to frictional loss at the soil–root interface and the decrease in the number of roots involved in resisting the loads at the outer boundaries of the equivalent root-plate [30].

The pure pullout load ($P_u$) of the tree can be estimated, in agreement with Figure 10, using the equation [19,30,31].

$$P_u=W_{sh}+W_{rp}+Q_s,$$

(22)

which is similar to Equation (21). In Equation (22) $W_{sh}$ is the shoot weight (stem + crown) that is equivalent to $W_{rod}$, $W_{rp}$ is the root-plate weight, equivalent to $W_{sand}$, and $Q_s$ which is the pullout contribution of the exposed root length (which represents approximately 10% of the total pullout resistance) and this force plays the role of $W_{res}$. The previous comparison of Equations (21) and (22) shows a similarity between both phenomena. It is important to highlight that trees are held by soils whose granulometry, humidity (cohesion), and other factors, are complex to analyze and, consequently, these contributing factors must be taken into account in hypothetical laboratory experiments.

6. Conclusions

In this work, the Janssen effect, i.e., the saturation of the horizontal and vertical pressures beyond a critical depth $\lambda$, was described first for a cylinder and later for a system composed of a rod concentric to a larger cylinder. This effect is present also in the computation of the traction force T, as a function of the height of filling in the silo, H, on the cylinder’s inner side, and in the computation of the two traction forces acting on the rod and on the sidewall of the outer cylinder. A good agreement between the theory and experimental measurements was attained by assuming the existence of an active state, for which we used the Rankine coefficient. This latest condition and the assumption of the full mobilization of friction leads to an approximate linear correlation between the theoretical and the experimental tractions of the form $T_{rod}\sim A{T}_{rodM}$, which allows us to obtain a realistic estimation of the traction on the rod ($T_{rod}$ is of the same order of magnitude as $T_{rodM}$). Based on all these results, it is possible to shed light on another interesting phenomenon related to the frictional lifting of the combined weight of the sand, the rod, and the outer cylinder. Experimental evidence of this peculiar behavior was shown in detail. A simple relation that describes the holding phenomenon was proposed and discussed. Finally, this formula is closely related to that reported in the description of the pure upward movement of a tree in the presence of a tornado; in this sense experimental realizations with rods and elements simulating roots, immersed in cohesive and non-cohesive granular soils, could be of practical interest.