Shields formula


The Shields formula is a formula for the stability calculation of granular material in running water.
The stability of granular material in flow can be determined by the Shields formula or the Izbash formula. The first is more suitable for fine grain material, while the Izbash formula is more suitable for larger stone. The Shields formula was developed by Albert F. Shields. In fact, the Shields method determines whether or not the soil material will move. The Shields parameter thus determines whether or not there is a beginning of movement.

Derivation

Movement of soil material occurs when the shear pressure exerted by the water on the soil is greater than the resistance the soil provides. This dimensionless ratio was first described by Albert Shields and reads:
where:
The shear stress that works on the bottom is:
where:
  • is the shear tension exerted by the flow on the bed;
  • is the water depth;
  • is the gradient.
It is important to realise that is the shear stress exerted by the flow and is the shear stress at which the grains move.
The shear stress velocity is often used instead of the shear stress:
The shear stress velocity has the dimension of a velocity, but is actually a representation of the shear stress. So the shear stress velocity can never be measured with a velocity meter.
By using the shear stress velocity, the Shields parameter can also be written as:
where:
  • is the dimensionless grain density
Shields found that the parameter is a function of, in which is the kinematic viscosity. This parameter is also called the granular reynolds number:
Shields has performed tests with grains of different densities, and the found value of plotted as a function of. This led to the above graph.
Van Rijn found that instead of the granular reynolds number a dimensionless grain size could be used:
Because usually the values of are quite constant, the true grain size can also be set on the horizontal axis. This means that the value of is only a function of the grain diameter and can be read directly.
.
From this follows that for grains greater than 5 mm the Shields parameter gets a constant value of 0,055.
The gradient of a river can be determined by Chézy formula:
in which = the coefficiënt of Chézy ; This is often in the order 50. For a flat bed C can be approximated with:
By introducing this into the stability formula, a critical grain size formula is found at a given flow rate:
In this form, the stability relationship is usually called the "Shields formula".

Definition of "incipient motion"

The line of Shields in the graph is the separation between "movement" and "no movement". Shields has defined as "movement" that almost all grains move on the bottom. This is a useful definition for defining the beginning of sand transport by flow. However, if one wants to protect a bed from erosion, the requirement is that grains should hardly move. To make this operational, Breusers defined 7 phases of movement in 1969:
  1. Every now and then a moving stone
  2. Frequent movement in some places
  3. Frequent movement in several places
  4. Frequent movement in many places
  5. Continuous movement at all points
  6. Transport of all grains at the bottom
These phases are shown in the figure below:
In practice, this means that for bed protections, a design value of Ψ=0.03 must be used.

Calculation Example

Question: At what speed of flow does sand of 0.2cm move at a water depth of 1m?
Question: What stone size is needed to defend this soil against a current of 2 m/s?

Restrictions

The Shields approach is based on a uniform, permanent flow with a turbulence generated by the bed roughness. In the case of a rough bed in shallow water, and in case of unusual turbulence, the Izbash's formula is therefore more recommended.