Reassessing Janssen's equation for cave stress estimation in block cave mining

Abstract

Cave mines operating at greater depths are faced with higher draw column heights, typically ranging from several hundred metres to over one kilometre. This results in significantly higher cave stress on the production level, making reliable stress estimation critical for assessing long-term stability. The current approach to estimating cave stresses in the draw column relies on Janssen's equation, developed initially using the Method of Differential Slices (MDS) and based on laboratory observations of corn and bulk solids in storage silos. This article first reviews the MDS and the assumptions used to derive Janssen's equation. The article also discusses how the limitations of the analogues (e.g. corn material and full-to-empty silo condition) used to develop the equation do not represent conditions equivalent to those observed in cave mining (e.g. fragmented rock blocks undergoing fragmentation and no-column to full-column condition as cave propagates). These mechanistic differences may lead to erroneous estimations of cave stresses. While Janssen's equation is a valuable initial tool for estimating cave stresses, practitioners must carefully assess its assumptions and limitations before applying it to engineering designs.

Introduction

Janssen's equation (Janssen, 1895), developed by German engineer H.A. Janssen in 1895, was initially used to estimate the vertical pressure exerted by bulk solids on the bottom of a silo for storage silo design. Physical experiments involving the filling of corn and other bulk materials into silos revealed that the vertical pressure at the silo base increases exponentially with the material weight but eventually stabilises. This phenomenon, known as Janssen's effect, is attributed to the shear stresses at the silo walls and the stress arching of bulk solids, reducing the vertical pressure at the silo bottom compared to the total material weight. Notably, Janssen's equation assumes fully isolated silo walls which diverge from the interacting ore columns in cave mining.

Janssen's equation is derived using the Method of Differential Slices (MDS), which applies force equilibrium to an infinitesimally thin horizontal slice of material to estimate the mean vertical stress across the silo cross-section. In recent decades, Janssen's equation has been adapted to estimating the average vertical stresses at the base of a draw column (Figure 1) for various applications of cave mine design (Castro, 2007; Pierce, 2010, 2019). Pierce (2010) applied Janssen's equation to model stresses around the Isolated Movement Zone (IMZ) in REBOP. The same author and Leonardo (2016) also proposed to use Janssen's equation combined with the Rankine's coefficient of earth pressure K to estimate normal stress on the shear band of IMZ for secondary fragmentation estimation. Gómez et al. (2017) used Janssen's equation to estimate average vertical stress as the input to the cave comminution model. Pierce (2019) employed Janssen's equation to estimate the average vertical stress at the base of the cave and analyse the stabilities of drifts in the production level. While Janssen's equation provides a practical starting point for stress estimation in cave mining, the initial assumptions of MDS and its inherent limitations are often overlooked.

Figure 1.

Illustrations of fragmentation processes inside the cave and vertical pressure on column base (modified after Elmo et al., 2015).

This paper critically reviews the MDS approach and the assumptions underlying Janssen's equation to examine their applicability and limitations in the context of cave stress estimation. Sensitivity analyses are conducted to investigate the influences of parameters in Janssen's equation on the vertical stress estimations. The study shows that relying on empiricism and expert judgment to estimate these parameters can lead to significant discrepancies in calculated cave stresses. Additionally, we need to consider the imperfect physical analogy between storage silos and underground caves. This article advocates for the cautious use of Janssen's equation in cave stress estimation. To refine current methods for estimating cave stresses, the industry will need to consider modifying Janssen's equation and developing laboratory experiments using analogues that are more representative of cave conditions.

Janssen's equation for cave stress estimation

The method of differential slices

Figure 2 illustrates the schematic of an elemental slice at depth z from the surface of a volume of bulk solids inside a container, along with the stress components acting on the slice. According to MDS theory, the elemental slice at depth z has infinitesimal thickness $dz$ . The slice is assumed to have either a circular or square shape as shown in Figure 2.

Figure 2.

Schematic of the method of differential slices.

MDS theory assumes that the sum of vertical forces on the elemental slice is considered to be zero (i.e. $\sum F_z=0$ ). The vertical forces along the slice consist of three components:

vertical force ( $A{\sigma }_{zz}$ ) on the upper element face at the depth z;

vertical reaction force ( $A{\sigma }_{zz}+Ad{\sigma }_{zz}$ ) on the lower element face at the depth $z+dz$ ; and

wall shear forces ( ${\tau }_{w}Pdz$ ) on the perimeter of the slice.

Therefore, the force equilibrium on an elemental slice can be expressed as:

$$A({\sigma }_{zz}+d{\sigma }_{zz}-{\sigma }_{zz}-\gamma dz)+{\tau }_{w}Pdz=0$$

(1)

where $\gamma$ is the material unit weight; A is the cross-sectional area of the slice; P is the perimeter of the slice; and ${\tau }_w$ is the shear stress on the silo wall.

The shear stress on the silo wall ( ${\tau }_w$ ) can be expressed as a function of horizontal stress ( ${\sigma }_{rr}$ ) and wall friction coefficient ( $\mu$ ):

$${\tau }_w={\sigma }_{rr}{\mu }_w$$

(2)

Substituting equation (2) into equation (1) results in:

$$d{\sigma }_{zz}=\gamma dz-{\displaystyle \frac{{\mu }_w{\sigma }_{rr}Pdz}{A}}$$

(3)

Janssen (1895) introduced a constant K, defined as a product of horizontal to vertical stress ratio ( ${\sigma }_{rr}/{\sigma }_{zz}$ ) and wall friction coefficient ( ${\mu }_w$ ) between material and silo walls:

$$K={\displaystyle \frac{{\sigma }_{rr}}{{\sigma }_{zz}}{\mu }_w}$$

(4)

Substituting equation (4) into equation (3) gives:

$${\displaystyle \frac{d{\sigma }_{zz}}{\gamma \left(1-{\displaystyle \frac{K{\sigma }_{zz}P}{A\gamma }}\right)}=dz}$$

(5)

Equation (5) is a first-order differential equation. Solving for ${\sigma }_{zz}$ we obtain:

$${\sigma }_{zz}={\displaystyle \frac{A\gamma }{KP}\left(1-e^{-\frac{KP(z+z_o)}{A}}\right)}$$

(6)

Assuming no surcharge on the surface of the bulk solid (i.e. ${\sigma }_{zz}=0$ at $z=0$ ), solving for the constant $z_o$ gives:

$${\sigma }_{zz}={\displaystyle \frac{A\gamma }{KP}\left(1-e^{-\frac{KPz}{A}}\right)}$$

(7)

Janssen (1895) also considered a square-shaped slice (Figure 2) where $A=L^2$ and $P=4L$ , with L being the side length of the slice. Therefore, the original Janssen's equation can be modified as follows:

$${\sigma }_{zz}={\displaystyle \frac{L\gamma }{4K}\left(1-e^{-\frac{4Kz}{L}}\right)}$$

(8)

Equation (8) represents the original Janssen's equation. A modified version was later proposed by Nedderman (1992), who derived the equation for vertical stresses while considering different silo shapes. Starting from equation (1) and assuming a circular slice:

$$A={\displaystyle \frac{\pi D^2}{4}}$$

(9)

Substituting equation (9) into equation (1) gives:

$${\displaystyle \frac{\pi D^2}{4}({\sigma }_{zz}+\gamma dz)={\displaystyle \frac{\pi D^2}{4}({\sigma }_{zz}+d{\sigma }_{zz})+{\tau }_w\pi Ddz}}$$

(10)

In Nedderman's version, the constant K was redefined as the ratio of horizontal to vertical stresses, ignoring the impact of the wall friction coefficient ( ${\mu }_w$ ):

$$K={\displaystyle \frac{{\sigma }_{rr}}{{\sigma }_{zz}}}$$

(11)

With a surface surcharge $Q_o$ at the surface of the bulk solids (i.e. ${\sigma }_{zz}=Q_o$ at $z=0$ ), the solution for ${\sigma }_{zz}$ is:

$${\sigma }_{zz}={\displaystyle \frac{\gamma D}{4{\mu }_{w}K}\left[1-\exp \left(-{\displaystyle \frac{4{\mu }_{w}Kz}{D}}\right)\right]+Q_o\exp \left(-{\displaystyle \frac{4{\mu }_{w}Kz}{D}}\right)}$$

(12)

The hydraulic radius $R_H$ as a ratio of area and perimeter of the elemental slice is given by:

$$R_H={\displaystyle \frac{A}{P}={\displaystyle \frac{\pi D^2}{4\pi D}={\displaystyle \frac{D}{4}}}}$$

(13)

This definition also applies to square slices, where $R_H=\frac{L^2}{4L}=\frac{L}{4}$ . Substituting equation (13) into equation (12) for a circular slice, we obtain:

$${\sigma }_{zz}={\displaystyle \frac{\gamma R_H}{{\mu }_{w}K}\left[1-\exp \left(-{\displaystyle \frac{{\mu }_{w}Kz}{R_H}}\right)\right]+Q_o\exp \left(-{\displaystyle \frac{{\mu }_{w}Kz}{R_H}}\right)}$$

(14)

This equation was later modified to account for the bulk density ( ${\rho }_b$ ) of broken rock fragments for cave stress estimation:

$${\sigma }_{zz}={\displaystyle \frac{R_H{\rho }_{b}g}{{\mu }_{w}K}\left[1-\exp \left(-{\displaystyle \frac{{\mu }_{w}Kz}{R_H}}\right)\right]+Q_o\exp \left(-{\displaystyle \frac{{\mu }_{w}Kz}{R_H}}\right)}$$

(15)

$${\sigma }_{zz}={\displaystyle \frac{R_H{\rho }_{b}g}{{\mu }_{w}K}\left[1-\exp \left(-{\displaystyle \frac{{\mu }_{w}Kz}{R_H}}\right)\right]\mathrm{for}{Q}_o=0}$$

(16)

Equations (15) and (16) are widely used for cave stress estimation (Castro, 2007; Castro et al., 2014, 2020; Pierce, 2010, 2019), assuming the cave walls acts as the walls of an isolated silo, and that fragmented rock blocks in a cave are equivalent to bulk solids.

Nedderman (1992) assumed the constant K to be identical to Rankine's coefficient of earth pressure:

$$K={\displaystyle \frac{1+\kappa \sin (\varphi )}{1-\kappa \sin (\varphi )}}$$

(17)

where $\varphi$ is the material interfriction angle; $\kappa =-1$ is the active case and $\kappa =1$ is the passive case. However, many practitioners have simplified the definition of K to a constant at-rest earth pressure for cave stress estimation:

$$K=1-\sin (\varphi )$$

(18)

Schulze (2021) recommends equation (18) be applied only to particle beds with infinite horizontal dimensions where silo wall effects are negligible. As such, its application to cave problems requires caution.

Assumptions of Janssen's equation

The derivations in the method of differential slices section highlight two key assumptions underlying Janssen's equation:

Stresses are uniform along the elemental slice.

The vertical and horizontal stresses on the elemental slice are principal stresses.

However, neither assumption is mechanistically correct. Numerous authors (e.g. Nedderman, 1992; Schulze, 2021; Shamlou, 2013; Walker, 1966) have presented laboratory measurements showing that the stresses are not uniform along the base of the elemental slice. Specifically, vertical stresses near the silo walls tend to be lower than those at the centre of the element due to the presence of shear stresses (Figure 2). These nonzero shear stresses at the walls further indicate that the second assumption of Janssen's equation is only valid along the central axis of the vertical section.

To address these limitations, Walker (1966) proposed a stress distribution factor ( $\mathcal{D}$ ) which is a ratio of vertical wall stress ( ${\sigma }_{zz,w}$ ) and average vertical stress on the elemental slice ( ${\overline{\sigma }}_{zz}$ ). The expression for $\mathcal{D}$ is given as:

$$\mathcal{D}={\displaystyle \frac{{\sigma }_{\mathrm{zz},\mathrm{w}}}{{\overline{\sigma }}_{zz}}={\displaystyle \frac{1+{\sin }^2\varphi -2\kappa sin\varphi \sqrt{1-{\displaystyle \frac{{\tan }^2{\varphi }_w}{{\tan }^2\varphi }}}}{1+{\sin }^2\varphi -{\displaystyle \frac{4{\tan }^2\varphi }{3{\tan }^2{\varphi }_w}\kappa sin\varphi \left(1-{\left(1-{\displaystyle \frac{{\tan }^2{\varphi }_w}{{\tan }^2\varphi }}\right)}^{3/2}\right)}}}}$$

(19)

Janssen's equation can be modified to incorporate $\mathcal{D}$ to estimate the average vertical stress ( ${\overline{\sigma }}_{zz}$ ) along the elemental slice:

$${\overline{\sigma }}_{zz}={\displaystyle \frac{{\rho }_b{R}_H}{{\mu }_{w}K\mathcal{D}}[1-\exp \left(-{\displaystyle \frac{{\mu }_w{K}_{w}z\mathcal{D}}{R_H}}\right)+Q_o\exp \left(-{\displaystyle \frac{{\mu }_w{K}_{w}z\mathcal{D}}{R_H}}\right)]}$$

(20)

Despite Walker's (1966) improvement, this adjustment has been largely ignored in practice due to the mathematical complexity of $\mathcal{D}$ .

Based on the derivations in the method of differential slices section, the following assumptions and conditions are necessary for the application of the MDS to cave problems:

Material size: The material particles must be small relative to the silo dimensions so that the elemental slice can be treated as a continuum.

Element shape: The elemental slice is assumed to have a perfect circular or square cross-section.

Wall geometry: The silo walls are assumed to be perfectly vertical and parallel to ensure vertical shear stress at walls are consistent. Furthermore, stress arching is assumed to be uniform in all directions, from the centre of the horizontal slice to the silo walls.

Surface topography: The surface of the bulk solids is assumed to be flat.

Static condition: The method is limited to static conditions and does not account for dynamic processes (i.e. flow).

These conditions significantly limit the applicability of Janssen's equation to cave stress estimation. The following section examines these limitations, as well as the critical parameters that affect the degree of stress arching and the caution required in using Janssen's equation for cave stress estimation.

Stress arching and cave stress estimation

Janssen's effect has been interpreted to be the result of stress arching within granular material and between the material and silo walls (Castro, 2007; Pierce, 2010). Specifically, a higher degree of stress arching causes a greater portion of the material's weight to be supported by the silo walls, which results in reduced pressure transmitted to the silo bottom. While Janssen's equation does not explicitly quantify stress arching as a parameter, it is indirectly represented by factors such as the hydraulic radius ( $R_H$ ) and the horizontal-to-vertical stress ratio ( $K$ ). This section examines how these parameters, as well as the size of rock fragments, influence the degree of stress arching (i.e. the percentage of pressure transferred to the draw column base) in cave mine design. Additionally, the limitations of the corn and silo analogues are discussed.

Influence of hydraulic radius ( ${\mathit{R}}_{\mathit{H}}$ )

$R_H$ was initially defined as the ratio of cross-sectional flow area and wetted perimeter in fluid mechanics. This initial definition implies that flow capacity increases with increasing $R_H$ . In mining engineering, $R_H$ is generalised as the ratio of the cross-sectional area to the perimeter. $R_H$ increases when measured with respect to the opening and activation of new drawbells in a cave mine, but it remains constant when looking at stope dimensions or the cross-sectional area of an ore column above a drawpoint. $R_H$ plays a critical role in determining the rate of vertical stress ${\sigma }_{zz}$ increment with depth, its steady state and its eventual steady state. Therefore, it becomes important to understand the impact that these two different interpretations of the $R_H$ (active foot print area in cave mines vs. cross-sectional area of a single ore column/silo) may have in the calculations of ${\sigma }_{zz}$ .

To analyse this, a sensitivity study was performed using the widely applied version of Janssen's equation (i.e. Equation 16) to investigate the influence of $R_H$ on ${\sigma }_{zz}$ as a function of the depth from the surface (in cave mining this would correspond to the draw column height). The analysis assumes a rock fragment inter-friction angle ( $\varphi$ ) of 38°, an intact rock density ( $\rho$ ) of 2700 kg/m³ and a bulk porosity ( $n$ ) of 0.34. The rock bulk density ( ${\rho }_b=\rho (1-n)$ ) is calculated as 2015 kg/m³. The at-rest earth pressure constant in Equation (18) is used to calculate the value of the constant K (= 0.38).

Figure 3 illustrates the impact of varying $R_H$ values on the curvature of the ${\sigma }_{zz}$ increment curve using the assumed parameters. The results show that ${\sigma }_{zz}$ increases exponentially with depth (draw column height) before reaching a steady state. The steady-state value of ${\sigma }_{zz}$ increases with a larger $R_H$ . For infinite $R_H$ , ${\sigma }_{zz}$ increases linearly with depth, and no steady state is achieved. Figure 4 illustrates the percentage of pressure transferred to the column base (account for stress arching using Janssen's equation) compared to that without stress arching, plotted as a function of depth from the muck pile surface for varying $R_H$ values. The results indicate that lower $R_H$ values correspond to higher degrees of stress arching, leading to less pressure being transmitted to the base of the column.

Figure 3.

Influence of $R_H$ on vertical stress increment with depth from the surface.

Figure 4.

Percentage pressure transferred to the bottom of the column plotted against the depth from the material surface (draw column height).

The function of vertical stress increment with depth when $R_H\to \mathrm{\infty }$ can be expressed as:

$${\sigma }_{zz,R_H\to \mathrm{\infty }}={\rho }_{b}gz$$

(21)

According to Equation (21), the muckpile acts as a continuum media, which is unrealistic in practice. For finite $R_H$ , the steady-state value of ${\sigma }_{zz}$ ( ${\sigma }_{zz,max}$ ) is given by:

$${\sigma }_{zz,max}={\displaystyle \frac{R_H{\rho }_{b}g}{{\mu }_{w}K}}$$

(22)

It can be seen from Equation (22) that the value of $R_H$ dominates the steady state of ${\sigma }_{zz}$ . Underestimating $R_H$ results in a steady state ${\sigma }_{zz}$ occurring at a relatively low draw column height and prevents ${\sigma }_{zz}$ from increasing as the cave propagates and the column height grows. This negatively impacts the mine design, as it misrepresents the true stress regime governing arching behaviour. With increasing column height, stresses are expected to amplify at the cave base, leading to over-stressed arches. Sudden collapse of the arch (due to slip or breakage of rock fragments) could result in dynamic loading on infrastructures, such as draw bells, drawpoints, and production drifts (Bustos et al., 2023, 2024), increasing the risks of localised failures such as spalling, rockbursts, and damage to support systems. Furthermore, the degree of secondary fragmentation would be underestimated, as reduced confining stresses (due to premature arching) limit particle breakage, leading to coarser-than-expected fragmentation profiles which could exacerbate hang-up risks.

According to Laubscher’s (2000) caveability chart, the maximum $R_H$ achieved under operational conditions is 80 m. This indicates that the vertical stress would reach the maximum value for supercave conditions where the draw column height extends to over one kilometre. For practical caving applications, Castro (2007) proposed estimating $R_H$ based on the shape of the column base using the following equation:

$$R_H={\displaystyle \frac{ab}{2(a+b)}}$$

(23)

where a and b are the dimensions of the elemental slice shown in Figure 5.

Figure 5.

Schematic of a rectangular elemental slice (after Castro, 2007).

Equation (23) deviates from the original assumptions of the MDS framework discussed in the method of differential slices section. The original equation assumed a circular or square cross-section. Therefore, equation (23) suggests that $\frac{D}{4}$ or $\frac{L}{4}$ can be replaced by $R_H$ (see equations 8 and 13).

The initial definition of $R_H$ suggests that the degree of stress arching is the same along both major and minor axes of the cross-sectional slice. When calculating the $R_H$ , equation (23) should be used with caution for irregular-shaped column bases with high aspect ratios, as the degrees of stress arching vary along the major and minor axes due to the finite size of rock fragments. The effect of column size and shape relative to the fragment size in relation to the degree of stress arching will be discussed later.

Some authors have linked $R_H$ to the production rate (i.e. tonnes per day) of a group of drawpoints (Castro et al., 2016; Pierce, 2019). However, practitioners must remain mindful of the original assumption behind Janssen's equation: the reduction of pressures transferred to the column base arises from shear stresses along the silo walls. Therefore, the estimation of $R_H$ should strictly depend on the area and perimeter of the cross-sectional slice in contact with the cave walls.

Influence of fragment and column sizes

In Janssen's equation, the relative size of the silo and the material is neglected. The assumption is that the material size is small enough compared to the silo dimensions. This implies that the material along the slice (i.e. the elemental slice) can be treated as a continuum, focusing solely on silo size (i.e. $R_H$ ). However, this assumption is invalid for cave mines, where the ratio of the size of the draw column to fragment size changes as the cave propagates and advances.

Figure 6 schematically illustrates how different sizes and shapes of rock fragments influence the degree of stress arching in a draw column of the same size. Results of numerical simulations in Figure 7 shows numerical simulation results using various sizes of granular materials. In Figure 7, the force chain magnitude (representing stress arching) decreases with decreasing particle sizes. It becomes apparent that a higher degree of stress arching is expected for larger, more angular blocks with higher aspect ratios (e.g. d/D = 10). The degree of stress arching decreases with decreasing the material size (i.e. force chain magnitude decreases at the upper part of the model in d/D − 100). Therefore $R_H$ of a silo alone without consideration of material size is insufficient to represent the degree of stress arching.

Figure 6.

Degree of stress arching with different sizes of fragments.

Figure 7.

Force chain of numerical simulation using various sizes of granular materials.

The degree of stress arching would also vary in drawing columns with high aspect ratios. Figure 8 compares the stress arching in a column with an aspect ratio of two in major and minor axes of vertical directions. The degree of stress arching along the minor axis is hypothesised to be higher than along the major axis. Janssen's equation does not account for this variation, as it assumes circular or square slices where the degree of stress arching is uniform along all axes.

Figure 8.

Degree of stress arching with different column widths.

Rock fragments in the draw column are typically smaller near the base of the ore column than at the top of the muck pile (Figure 9). As the cave propagates and the draw column height increases, rock fragments near the base experience breakage due to increased vertical loads. This breakage reduces the degree of stress arching with depth. Janssen's equation, developed for silos with relatively small heights (metres) and unbreakable materials (e.g. corn or bulk solids), does not account for the effects of breakable rock fragments over the much taller draw columns (several hundred metres to one kilometre) developed in cave mines.

Figure 9.

Degree of stress arching with increasing column height.

Influence of horizontal to vertical stress ratio ( $\mathit{K}$ )

For the design of storage silos, the horizontal to vertical stress ratio K in equation (11) has traditionally been equated to Rankine's coefficient of earth pressure in equation (17). In cave design, this constant is typically assumed to be the at-rest earth pressure constant (equation 18).

The interfriction angle ( $\varphi$ ) of rock fragments (rockfills) in equations (17) and (18) can be estimated using Barton and Kjærnsli (1981)'s empirical approach:

$$\varphi =Rlog\left({\displaystyle \frac{RBS}{{\sigma }_n}}\right)+{\varphi }_b$$

(24)

where ${\varphi }_b$ is the material basic friction angle (with an average value of 30°); R is the equivalent roughness of the rock fragment, and $RBS$ is the rock block strength modified by Laubscher and Jakubec (2001) and Pierce (2010).

Figure 10 illustrates the influence of K on the vertical stress increment from the top of the muck pile. Higher values of K result in a higher degree of stress arching along the horizontal cross-sectional slice. It is typically accepted that $K<1$ for the active case (i.e. ${\sigma }_{rr}<{\sigma }_{zz}$ ) and at-rest conditions, and $K>1$ for the passive condition (i.e. ${\sigma }_{rr}>{\sigma }_{zz}$ ). However, assuming at-rest conditions when using equation (18) can lead to inaccuracies due to the complex flow and ground conditions in caves.

Figure 10.

Influence of K on vertical stress increment with depth from surface when $R_H=200$ m.

Influence of draw column shape

Janssen's equation assumes that silo walls are perfectly upright, which is achievable in artificially designed storage silos. However, in cave mines, draw column shapes are much more complex. Factors such as the advance direction, pre-conditioning volume, cave rate, and operational blast control can create varying degrees of wall inclination, which significantly affect the degree of stress arching.

Figure 11 shows two examples of different column shapes with the same footprint area. In Cave 1, column walls are inclined outwards, whereas Cave 2 the walls are inclined inwards. Caves 1 and 2 are typical scenarios of shallow (Vyazmensky et al., 2010) and deeper caves (e.g. Elmo et al., 2015; Shapka-Fels and Elmo, 2022), respectively. Cave breakthrough to surface eventually happens in both scenarios. Because of the different cave geometry, the walls in Cave 1 support more material weight, leading to a higher degree of stress arching, whereas the walls in Cave 2 provide minimal support, resulting in a much lower degree of stress arching. Ignoring the effects of the shape of the cave can lead to unreasonable estimations of cave stress. At the same time, the shape of the cave can only be delineated once the caving is initiated and develops. In principle, discrete and hybrid numerical models can simulate caving mechanisms, including breakage and material flow above the undercut level. However, the caving industry prefers to use continuum models in which the caving and flow of material are implicitly simulated using methods that rely on Jansen's equation. This may create a paradox whereby the cave shape is determined using methods in which the stress arching and draw column shapes are not independent variables.

Figure 11.

Conceptual description of how different degrees of stress arching could develop because of different draw column shapes.

Discussion and conclusions

This paper has reviewed the MDS and Janssen's equation developed over a century ago and highlights their initial assumptions and limitations. Despite its mechanistically erroneous assumptions, Janssen's equation has been widely used for designing storage silos for corn/bulk solids. From a historical perspective, Janssen's original paper experienced a significant revival over the past three decades, coinciding with the growing interest in studying granular material flow in cave mining operations. However, this raises important questions concerning the validity of the corn/rock and silo/ore columns analogues applied to stress estimation in cave mine designs.

Janssen's equation assumes that bulk solids are relatively small, uniform, and well-graded in shape. It also presumes that the silo is a perfect cylinder or rectangular prism with smooth, upright walls, resulting in consistent stress arching along the major and minor axes of horizontal cross-sectional slices. However, these assumptions do not apply to cave mining. In cave systems, rock fragments are expected to be more angular, and their sizes are non-uniform at different depths due to breakage. In addition, draw columns deviate from ideal cylindrical or rectangular shapes and lack smooth, upright walls.

Another critical factor is the horizontal-to-vertical stress ratio (K). Estimations of K must consider the actual flow and cave conditions, including the advance rate and draw sequence, cave wall deformability, and material flow kinematics. As the cave advances, walls are expected to provide minimal support, suggesting an active stress state for the muckpile. Conversely, rock fragments flowing towards the drawpoint may induce a passive stress state. Simplification using the at-rest earth pressure constant may lead to an erroneous estimation of cave stresses.

While one may argue in favour of the use of Jansen's equation in cave mining as a practical starting point, its initial assumptions and inherent limitations are overlooked. The imperfect use of a model like Jansen's equation is not an isolated case. Indeed, empiricism and expert judgment are at the core of many methods we use in rock engineering design, from classification systems to failure criteria. It is important to acknowledge that model acceptance should not be viewed as synonymous with model validation. Even flawed (Ptolemy's geocentric model) and imperfect (Copernicus's heliocentric model) astronomical models can successfully predict planetary motions while being mechanistically incorrect. But we do not rely on them to plot the course of probes and spacecraft trajectories.

The truth is that in rock and mining engineering, the scientific validity of a method often takes a secondary role for several reasons:

It is impossible to physically recreate and study the same problem we are trying to design. For example, we cannot set up a laboratory experiment that simulates the mechanisms (e.g. flow, fragmentation) encountered in ore columns in cave mining.

It is difficult to extend the application of a model across different sites due to the natural variability of structural geology, rock properties, and stress conditions. There are no universal models in rock engineering.

Many numerical parameters used in rock engineering design are qualitatively established and, therefore, include human-induced variability. At the same time, quantitative measurements (e.g. uniaxial compressive strength) are also characterised by large coefficients of variation. Artificial and natural variability are typically addressed by adopting very conservative design limits.

There is a growing trend to rely on computer simulations and machine learning tools without critically examining their underlying principles and the role of empirical and often subjective input in the outcomes of those complex simulations.

The degree of conservativeness adopted in rock engineering design inevitably masks the imperfections and flaws of our models. This may lead to conflating empirical justifications with rigorous scientific validation.

This article did not propose an alternative to Janssen's equation, and we understand some readers may find this disappointing. However, as Elmo (2025) argues, the fundamental methodological flaws in the derivation and application of Janssen's equation in cave mining are epistemologically independent of the existence of alternative design methods. The absence of an alternative method does not mitigate the critical importance of identifying significant theoretical limitations. Acknowledging theoretical limitations and identifying methodological flaws represents a crucial first step in the scientific process of developing more robust analytical frameworks, even before alternative methods are formally proposed.

While rock engineering design may often be perceived as practical, it should not be overlooked that the scientific method provides a rigorous framework for systematically understanding mechanisms that drive what we observe in the field. The scientific method enables us to develop theories and explanations grounded in empirical evidence, thus demanding continuous critical evaluation of existing engineering methods.

Rock engineering knowledge cannot advance without the willingness to challenge established practices and beliefs. Practitioners have a professional and intellectual obligation to critically examine the limits and flaws of current accepted methods and hypotheses. A flawed method remains fundamentally flawed, regardless of its current applications in the mining industry or the immediate availability of alternatives.