Third Chinese-German Joint Symposium on Coastal and Ocean Engineering
National Cheng Kung University, Tainan
November 8-16, 2006
Classification of Shape
and Underwater Motion Properties of Rock
Timm Stückrath*, Gerhild Völker** and Jian-hua Meng***
Fachgebiet Konstruktiver Wasserbau, Institut für Bauingenieurwesen
Technische Universität Berlin, Berlin
*[email protected]
**[email protected]
***[email protected]
Abstract
During research work focusing on rock placement in coastal engineering and
port construction it was found that an objective shape classification of rocks was
needed.
The introduced method utilizes three independent parameters which are described.
Underwater falling properties of rocks were object of systematic experimental
examination. These properties are associated to the introduced parameter space.
The digital visualization of the examined rocks and the experimental results on
properties and motion are presented in a PC-based interactive information system
RockDataBase. The user can query information on any rock by using
RockDataBase as a modern form of comparison table to acquire shape properties,
hydraulic properties and a visualization of its under water motion.
1 Introduction
Research at the Institute of Hydraulic Engineering and Water Resources
Management IWAWI of Technische Universität Berlin focused on the field of rock
placement in coastal engineering and port construction. Research on the quality of
rock dumping with seaborne methods (e. g. bulk placement by barge, fall-pipe
placement), it was found, that rock shape and roundness contribute significantly to
the hydraulic falling properties of settling velocity and lateral deviation.
The settling velocity can become crucial parameter when constructing protection
layers to damageable structures such as underwater pipelines, cables or geotextile
layers. The settling velocity is determined by the coefficient of drag cD. Lateral
deviation is crucial to the placement accuracy of falling rocks. Both characteristics
are determined by the rock’s overall shape.
It was found that a viable shape description and classification of rocks was needed.
The existing methods in engineering and geology fail to fulfil all of the required
criteria simultaneously. The introduced method utilizes three parameters: size,
aspect ratio and roundness for rock classification. Furthermore, the method
associates the shape classification to a rock’s properties of falling in water.
2 Method of shape classification
The developed method of shape classification was designed to simultaneously fulfil
the following criteria:
A.
The results must be objective. They must be unambiguous and
reproducible.
B.
The parameters are to be independent.
C. The results have to be invariant of an object’s spatial position and scale.
D.
The method and the results have to be descriptive and thus easily
applicable.
The chosen parameters were:
z
size,
z
aspect ratio, i.e. the spatial range,
z
roundness, i.e. the overall characteristic of the object’s circumference and
z
roughness, i.e. the surface’s coarseness (preliminarily omitted in further
discussion, since no significant contribution to hydraulic behaviour).
These parameters have to be defined to fulfill the demand of being independent.
They are represented by the overall term of “shape”, defined as the sum of an
object’s geometric properties.
A descriptive explanation is shown in fig. 1:
shape
size
aspect ratio
roundness
roughness
distinction of objects distinction of tabular,
of different size
equant, prolate und
bladed objects
distinction of edged
und rounded objects
distinction of objects with
plane or rough surface
classification
by volume
classification e. g.
according to
Diepenbroek or
according to the
equivalent ellipsoid of
inertia
classification e. g.
according to equivalent
sand grain roughness
classification e. g.
according to Zingg
Figure 1 Characteristics of an object’s geometrical overall shape
properties.
The overall shape -i.e. the spatial geometry- is the determining factor of an object’s
hydrodynamic properties of falling freely in water.
Size
The object’s size can be easily and unambiguously described by its volume. The
description by volume is completely independent of other geometric properties
(Huller, 1985) and can be easily determined by displacement measurement.
Aspect ratio
A variety of methods of shape classification or aspect ratio determination are
available. Some of the better known are Corey Shape Factor CSF (Komar and
Reimers, 1978) and Shape Factor E (Baba and Komar, 1981). Many are not
unambiguous: for different shapes identical factors are determined. Many methods
are not independent to the objects roundness, making them not suited for hydraulic
matters.
The here introduced method is based on the method by Zingg. The swiss road
engineer Zingg 1935 proposed to determine the aspect ratio of the enclosing
rectangular box. The three determined axes lengths of s (shortest), l (longest) and i
(intermediate) are completely independent from roundness and, due to the ratio, of
the scale. These three axes are the basis of Zingg’s diagram:
The classification by aspect ratio is completely invariant of the outer contour of an
object: e. g. a sphere has the identical aspect ratio as a cube! This simple example
shows, that the parameter of ‘aspect ratio’ does not describe ‘roundness’.
Furthermore, by using the ratio rather than absolute length, the parameter ‘aspect
ratio’ remains scale-independent.
The original method introduced by Zingg was modified:
(a) The origin of the cartesian coordinate system is defined at the object’s centre
of mass with its axes according to the main axes of inertia.
(b) The three axes are not measured at the rock itself, but at the ellipsoid, whose
momentum of inertia is identical to the rocks (equivalent ellipsoid of inertia). Thus,
irregular contours do not interfere with the measured length, making the
determination independent of the rocks’ roundness.
Figure 2 Classification of body shape in four classes according to Zingg
(Tucker, 1985). A: tabular, B: equant, C: bladed, D: prolate.
Figure 3 Determination of the longest, shortest and intermediate axis
length by the enclosing box (McLane, 1995).
Roundness
Roundness is commonly determined on a two-dimensional cross section or
projection area. Significant contributions were developed by Wentworth (1919) or
Wadell (1932). For simplicity, Roundness can be determined by comparison tables.
In these tables, the object is compared to a two-dimensional image, giving a quite
rough idea about a (however defined ‘degree of roundness’). A list of the existing
roundness-comparison tables are given by Köster (1964).
These methods do not necessarily lead reproducible results: it is up to the
individual, which cross section or projection area he may apply a mathematical
scheme to, or which he may compare to a range of images of different roundness.
For a reproducible determination of roundness, the method of contour-analysis by
determination of Fourier coefficients, as proposed by Diepenbroek (1993) was
applied. The amplitude spectrum of an objects contour line contains all information
on the objects shape.
With this method, the exact, reproducible and unambiguous degree of roundness
can be determined for a plane image (again: cross section or projection area).
When applied to the three orthogonal cross sections as defined by the main axes of
inertia, a three dimensional degree of roundness is obtained.
An alternative method was introduced by the authors 2004: the degree of
roundness as deduced by the objects equivalent ellipsoid of inertia. The basic idea
is, to compare the original object to the ellipsoid, which has the identical inertia
properties with this equivalent ellipsoid of inertia defined as ‘perfectly round’.
This method is completely independent from the objects aspect ratio and includes
the complete outer contour (not only a number of representative cross sections).
Figure 4 Contour line of a cross section in polar coordinates
(Diepenbroek, 1993).R0 is the zero-frequency and represents the mean
radius.
Figure 5 Image of a natural rock (roundness P = 0,30) and its equivalent
ellipsoid of inertia (roundness P = 1,0).
Roughness
Roughness stands for the state of an objects surface structure. The experiments at
TU Berlin confirm that roughness plays no role for the hydraulic falling behaviour of
edged objects. For precisely rounded objects however, there is an interrelation.
The responsible fluid mechanical effects and the resulting falling behaviour will be
discussed in a further publication.
Classification
For descriptive classification based on Zingg’s scheme (as displayed in fig. 6), the
original scheme was extended to a 3D parameter space (fig. 7): thus the
parameters of aspect ratio ‘flatness’ and ‘elongation’ plus the degree of roundness
point out a single, unambiguous position in the parameter space.
Figure 6 Base of the 3D parameter space with information on shape
properties. As an example, the corresponding cuboids are displayed at
their position according to their shape (excluding roundness).
Figure 7 3D parameter space visualizing the shape properties form (base)
and roundness (vertical axis). A body is represented by a position
determined by its flatness s/i, its elongation i/l and its roundness P.
3 Hydraulic Motion Properties
Rocks falling freely in water show a variety of characteristic motion patterns, which
are determined the object’s shape. The most clearly distinguishable motion
patterns can be observed on cuboids:
spatial rotation:
the object rotates around a single axis, which continuously
shifts its spatial position.
single-axis rotation: the object rotates around the longest axis, which mainly
remains perpendicular to the direction of flow.
single-axis oscillation: the object oscillates around the longest axis in a pendulum
motion characteristic, which mainly remains perpendicular to
the direction of flow (i.e. direction of falling).
spatial oscillation: the object oscillates alternately around the longest and median
axis, which mainly remain perpendicular to the direction of
flow.
These motion patterns not always occur in pure appearance. Quite often, there are
transitional patterns of motion, in which the one or the other may dominate.
The occurrence of the above mentioned motion patterns of cuboids can be
well-displayed in Zingg’s scheme:
Figure 8 Underwater motion types of cuboids associated to their shape
properties. I: spatial rotation, II: single-axis rotation, III: single-axis
pendulum motion, IV: spatial pendulum motion, M: mixed motion with no
predominant motion type.
Figure 9 Underwater motion types of natural rocks associated to their
shape properties. The rocks are classified into four classes of roundness
(angular, sub angular, rounded, well rounded). Every diagram represents
a class of roundness and displays the rock’s motion type.
|: spatial rotation,: Δ single-axis rotation, ¨: single-axis pendulum
motion, ∇ : spatial pendulum motion.
Some cuboids show an additional spiral trajectory. This spiral trajectory then
superimposes the other motion patterns. It can be observed on cuboids of all form
classes.
The experimentally tested natural rocks displayed the same four motion patterns as
described for cuboids. Also, spiral trajectories could be observed on natural rocks.
Zingg’s scheme was found not only to classify in a highly descriptive way, but also
proved to be a means of showing the effective conformity of falling properties and
shape classification. This can be seen in fig. 9.
The coefficient of drag is the central parameter for the determination of falling
velocity, which again is the key parameter for engineering purposes e.g. when
estimating grain segregation or bottom impact velocity.
Figure 10 Drag coefficient cD [-] of natural rocks, associated to their
shape properties. Each diagram represents a class of roundness.
Since coefficient of drag of free-falling, irregularly shaped objects is rarely available
in literature, it therefore became the central focus of our experimental objective.
The measured coefficient of drag cD for irregular rocks falling freely in water is
shown in fig 10.
4 Information System RockDataBase
RockDataBase is an information system, which displays the results of
experimentally tested rocks. The tested rocks represent natural rocks of common
convex shapes and degrees of roundness. These rocks were tested in the
hydraulic laboratory of TU Berlin.
The RockDataBase information system is not only the result report of the row
testing, but can also be used as a form of “digital comparison table”. Since the
determination of properties which are not mere easily measurable lengths such as
the equivalent ellipsoid of inertia and/or the main axes of inertia may be quite
troublesome in practice, the RockDataBase was designed to be used as a digital
version of traditional comparison table.
Among the available information is the coefficient of drag cD, the settling velocity,
the duration of acceleration, the motion pattern video and the geometric shape
parameters. The RockDataBase contains a tool for the interactive prediction of the
user’s own rock on the basis of the tested experimental data.
The user can easily query information on any rock to acquire shape properties,
hydraulic properties and a visualization of it’s under water motion.
5 References
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Numbers, Jour. Sed. Petrology, Vol. 51, No. 1, pp. 121-128, 1981.
Diepenbroek, M. Die Beschreibung der Korngestalt mit Hilfe der Fourier-analyse:
Parametrisierung der Morphologischen Eigenschaften von Sedimentpartikeln,
Mitteilungen des Alfred-wegener-institutes für Polar-und Meeresforschung,
Bremerhaven, 1993.
Huller, D. Quantitative Formanalyse von Partikeln, Fortschrittsberichte VDI, Reihe
3: Verfahrenstechnik, Nr. 101, VDI-verlag, Düsseldorf, 1985.
Komar, P.D. and C.E. Reimers. Grain Shape Effects on Settling Rates, Jour. Geol.,
Vol. 86, pp. 193-209, 1978.
Köster, E. Granulometrische und Morphometrische Meßmethoden
Mineralkörnern, Steinen und Sonstigen Stoffen, Enke, Stuttgart, 1964.
an
McLane, M. Sedimentology, Oxford University Press, New York, Oxford, 1995.
Stückrath, T., G. Völker and J. Meng. Die Gestalt von Natürlichen Steinen und ihr
Fallverhalten in Wasser, Mitteilung 137 des Instituts für Wasserbau und
WasserwirtscHaft der Technischen Universität Berlin, 2004.
Tucker, M.E. Einführung in die Sedimentpetrologie, Ferdinand Enke Verlag,
Stuttgart, 1985.
Wadell, H. Volume, Shape and Roundness of Rock Particles, Jour. Geol., Bd., Vol.
40, pp. 443-451, 1932.
Wadell, H. Shape Determinations of Large Sedimental Rock Fragments,
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Wentworth, C.K. A Laboratory and Field Study of Cobble Abrasion, Jour. Geol., Bd.,
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Zingg, Th. Beitrag Zur Schotteranalyse, Schweiz. Mineralog. und Petrog. Mitt.,
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