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3. Stability of Bulk Solids during Ship Transport

3.1 Load Profiles

Fig. 11: Cross section of loaded cargo hold.

Generally, the bulk solid in the hold of an ore carrier is constrained on three boundaries with only the top surface free, as indicated in Fig. 11. The load profile depends on the shape of the hold, the hatch arrangement and degree of fill. Depending on the manner in which the bulk solid is loaded, the surface will normally be rilled, as in Fig. 11(a) with the surface inclined at an angle equal to or less than the angle of repose as a result of the settling process during the initial phase of the ship’s journey. Some trimming of the surface may be effected as in Fig. 11(b), and this is desirable for stability purposes.

It is assumed that the moisture content of the bulk solid is below the saturation (water holding capacity) level and, as such, the bulk solid can be described as a Coulomb frictional, cohesive bulk solid. The bulk solid will consolidate during initial filling, the consolidation continuing with time under constrained storage. Depending on the properties of the bulk solid, some chemical bonding of particles may occur over prolonged undisturbed storage leading to a further increase of bulk strength. Consolidation will be further promoted by the ship’s vibrations, those emanating from the engines and other machinery, and those of a lower frequency due to the ship’s motion. On some occasions, the latter may lead to a reduction in strength if significant downward accelerations occur, mainly during rolling and pitching. These effects may lead to some movement of the mass but, in general, this will be confined to the region adjacent to the free surface where the bulk mass is unconstrained. The bulk of the stored mass is constrained by the walls and floor where the combination of wall friction and pressures will prevent or limit relative motion with respect to the walls of the hold.

The angle of repose, θ, of the free surface is on occasions confused with the angle of internal friction. The angle of repose is simply the angle with respect to the horizontal formed when a bulk solid is placed into storage and depends on the loading rate, impact due to loading and the particle/lump size distribution. The angle of repose may be thought of as the angle of internal friction of a loosely packed unconsolidated stored mass. During filling it is common for some segregation to occur, with the fines remaining in the central region and the larger lumps rolling to the sides. Under these conditions the surface may have a range of slopes, being steeper in the central region due to cohesion of the fines and less steep at the outsides due to the more free-flowing nature of the lumps. Where the bulk mass has a more uniform size distribution, the surface will tend to slope at a constant angle with some settlement of the surface due to the motion of the ship.

The stability of the load depends on the cohesive strength of the bulk solid and whether any yield conditions can be generated in the mass during transport. The possible loading conditions are discussed in the following sections.

3.2 Stress Fields

The stress conditions at some location within the stored mass are depicted in the Mohr diagram of Fig. 12 in which σ₁ and σ₂ are the major and minor principal pressures respectively at the location considered.

Fig. 12: Stress condition in bulk mass indicating possible plane of failure.

It is assumed that an active state of stress exists and that σ₁ acts vertically being equal in magnitude to the hydrostatic pressure. That is:

(3)

where:

• ρ is the bulk density,
• g is the gravitational acceleration, and
• y is the depth below the surface.

In the Mohr diagram of Fig. 12, the point of intersection (Point A) of the yield locus with the Mohr semi-circle through σ₁ and σ₂ defines the stress condition at failure with σ_β being the normal stress and τ_β the shear stress acting on the plane of failure which is inclined at the angle β to the plane of the major principal stress.

From the geometry of Fig. 12, the following relationships can be derived:

(4)

(5)

where:

• δ is the effective angle of internal friction, and
• φ tirepresents the initial static angle of internal friction.

The angle β defining the slope of the yield plane with respect to the principal plane of σ1 is:

(6)

For a typical coal with an internal friction value of δ = 50 and φ_ti = 40°, the angle β can be calculated, resulting in a value of 65°. Additional to the information provided in Fig. 12, an equivalent internal friction angle φ_f is defined by:

(7)

Substituting Eqs. (4) and (5) into Eq. (7) yields the following expression for φ_f:

(8)

Fig. 13: Equivalent friction angles.

By way of illustration, Fig. 13 shows the variation of the equivalent friction angle φ_f as functions of the effective angle of internal friction δ for a given range of static angles of internal friction φ_ti. For comparison purposes, the angle δ is also plotted as is shown by the dotted line. It is noted that for a free flowing bulk solid material, the unconfined yield strength σ_c is generally zero and φ_f = φ_ti = δ.

If, for a given bulk solid and loading configuration, σ₁, δ and φ_ti are known, then the stresses τ_β and σ_β on the shear plane may be determined by Eqs. (4) and (5) respectively. Flow property tests enable δand φ_ti to be determined as functions of σ₁. However, some uncertainty arises in establishing σ₁. For many cases σ₁ may be estimated from the hydrostatic stress given by Eq. (3). Under these conditions, the plane of failure as defined by β will normally be quite steep. In some occasions, the cohesive shear stress τ_c is required to be expressed as a function of σ₁. Again, from the geometry previously provided in Fig. 12, it may be shown that:

(9)

with:

(10)

Alternatively, the cohesion term τc may also be expressed in terms of the unconfined yield stress σ_c as follows:

(11)

with:

(12)

Fig. 14: Stress condition in bulk mass during ship motion.

It is noted that φ_ti is the initial value of the static angle of internal friction. Following time consolidation, σ_c will increase to the time consolidation value of the unconfined yield strength σ_ct corresponding to the static angle of internal friction φ_t derived from the time yield locus. Very often the value of φ_t is very similar to φ_ti.

3.3 Influence of Ship’s Motion

When a bulk ship is pitching and rolling, the inertia effects due to the various acceleration components may influence the magnitude and direction of the major consolidation pressure σ₁. The case of rolling motion as depicted in Fig. 14 is reviewed. At the instant considered, the roll angle is θ_R and σ₁ is inclined to the ship’s axis at an angle ψ as shown. A potential plane of failure is inclined at an angle β = (π/4) + (ϕ_t/2) to the plane of the major principal pressure σ₁. With respect to the horizontal plane, the possible plane of failure will be inclined at an angle (θ_R + β – ψ). Slip will occur when the shear stress due to the body forces produced by the segment above the shear plane are just equal to the shear stress corresponding to the consolidation condition as given by Eq. (5). That is:

(13)

Fig. 15: Safe and possible failure conditions.

The actual failure mode is difficult to predict due to the uncertain consolidation condition. For a cohesive bulk solid, failure is likely to occur along a curved surface. If average values of consolidation are assumed, a rough approximation will be to consider the failure surface as a plane of slope angle (θ_R + β – ψ) with respect to the horizontal axis. Under these conditions failure is less likely to occur if the plane of failure intersects the wall below the surface as in Fig. 15(a). Failure is most likely to occur if the plane of failure is as in Fig. 15(b).

3.4 Consideration of Load Surcharge Zone

Since movement of the bulk solid, if at all, is most likely to be confined to the surface region in the surcharge zone, that portion not constrained by the ship’s walls, the condition for stability may be examined by considering slip along the sloping surface. In this case surface slip will occur along a straight inclined plane.

A loosely packed condition near the surface will result in the bulk solid forming a surface slope angle equal to the angle of repose θ. As a guide, θ may be estimated to be equal to the static angle of internal friction φ_t corresponding to zero consolidation pressures.

For the majority of bulk solids ranging from free flowing to highly cohesive, θ will range from about 30 to 40°. However, once the ship commences its journey, the vibration due to the ship’s propulsion machinery combined with any whipping, rolling and pitching motion, will result in some settlement of the bulk solid with the surface slope reducing below the natural angle of repose to its transportable surcharge angle θs.

If there is any consolidation near the surface then it is possible for the surface to be inclined at an upper bound stable angle φ_f to the horizontal, φ_f being the equivalent angle of internal friction as defined by Eq. (8) and presented in Fig. 13. For a triangular surcharge as in Fig. 14 where the surface is inclined at a settled transportable surcharge angle θ_s measured with respect to the horizontal, it is possible for no surface movement or slip to occur for roll angles given by:

(14)

Fig. 16: Maximum roll angles.

where:

• θ_a is an allowance for acceleration inertia effects.

Maximum roll angles have been determined for the coal of Figs. 1, 2 and 3 for a range of transportable surcharge angles from θ_s = 10° to 30°, the results of which are plotted as a function of the major consolidation stress σ₁ in Fig. 16. The corresponding depths below the surface are also shown. Based on Kirby [3], the plotted results assume an acceleration angle θ_a = 5°. If, for example, θ_a = 0, then the maximum roll angle would be 5° larger than those plotted in Fig. 16. The results demonstrate that there is generally a need to trim the free surface on loading in order to prevent material movement for normal roll angles.

4. Ship Load Stability - the Warren Spring Study

The report by Kirby of the Warren Spring Laboratory assumed a circular failure surface for the cohesive case, as is illustrated in Fig. 17 [3]. This assumption is consistent with the classical theory for slope stability in soil mechanics. Kirby defines the non-dimensional parameter N as:

(15)

where:

• τ_c is the cohesion,
• ρ is the bulk density,
• g is the gravitational acceleration, and
• L is the slope length.

Fig. 17: Stability of bulk cargo [3].

To illustrate the application of this method, an example based on the coal of Figs. 1,2 and 3 is presented with the following assumptions of a 40 m wide ship, a slope length L of 13.3 m, a maximum roll angle θ_R of 15° and an assumed acceleration angle θ_a = 5°. Based on the geometry of Fig. 15, for a nominated initial surcharge angle θ_s = 20° and y = 4.85 m, the major consolidation pressure σ₁ is approx. 55 kPa. From Figs. 1,2 and 3, this yields a bulk density of ρ = 1.135 t/m³, an effective internal friction angle δ = 50°, a static internal friction angle φ_t = 44° and an unconfined yield strength σ_c = 16 kPa. From Eqs. (9) to (12), τ_c yields a value of 3.1 kPa and the value of the Kirby non-dimensional factor N is 0.021.

From Fig. 46 of Kirby the maximum slope angle is η = 27° which corresponds to a combined roll angle (θ_R + θ_a) of 20° [3]. For comparison purposes, applying the method described in Section 4.4 shows that the surcharge angle θ_s (or η) is 27°, the maximum combined roll angle (θ_R + θ_a) is 18° and θ_R = 13°. This is reasonably close to the value of 15° previously calculated.

5. Concluding Remarks

As discussed in this article, the objective of safe ocean transport of bulk cargoes by large bulk ships is vitally dependent on the stability of the cargo under the influence of the rolling, pitching and yawing motion of the ship and the transmission of vibration from the ship’s engine and propulsion machinery as well as wave motion induced whipping. A brief review of the bulk material test procedures recommended by the International Maritime Organisation (IMO) has been presented. The limitations and empirical nature of these tests has been highlighted. The application of the well proven flow property tests and associated analytical and design procedures, widely accepted in the field of bulk solids handling, to bulk ship transportation has been demonstrated.