FINITE-ELEMENT VOLUMES
By Thomas G. Davis j
ABSTRACT: The finite-element-volumesmethod is a new earthwork volumestech-
nique quite unlike conventionalmethods. The algorithm provides automatic curvature and prismoidal correction using ordinarily available cross-section data in
conjunctionwithhorizontalbaselinegeometry. The cross sectionsare approximated
as a series of rectangular elements of equal, user-specifiedwidth. As this width
approaches zero, cross-sectionalarea and centroid location approach that of the
original cross section. Every element is assumed to transition linearly along an
offset curve concentric with the baseline to an opposing element upstation or to
terminate on a tapered offset curve when an opposingelement does not exist. The
resultingvolumeelements are thus curvilinearwedges or frustumsof wedges. Linear, circular, and Cornu spiral baseline (clothoidalspline) componentsare accommodated by the method. Numericalexamplesshow excellentagreementwith exact
results even when the mass componentsare not prismoidal. A general formula for
the volumeof a curvilinearmass componentand a new, high-precision,prismoidal
curvature-correctiontechnique are also presented.
INTRODUCTION
Traditionally, the problem of computing volumes associated with transportation alignments (roadways, railways, and waterways) has been accomplished by assuming that the required volume is given by the product of the
length (difference in baseline stationing) and half the sum of transverse
areas at the beginning and end of the mass component. This technique is
called the average-end-area method and is, by far, the most widely used
method of computing volumes. The average-end-area approach has the
advantage that is easily understood and implemented. Unfortunately, the
results are exact if, and only if, transverse area varies linearly with length
and the baseline is straight. These conditions are seldom met in practice.
The first issue, namely that transverse area does not, in general, vary
linearly with length, may be overcome by collecting very closely spaced
cross sections. W h e n this is done, the transverse areas at the beginning and
end of the mass c o m p o n e n t are nearly the same and the assumption of
linearity is made more realistic. The second issue, however, cannot be
ameliorated by any a m o u n t of data collection. That is, if the baseline exhibits
significant horizontal curvature, the average-end-area formula cannot deliver accurate results for arbitrary components of volume.
When curvature, is to be considered, a correction technique based on
averaging the eccentricities, as well as the areas, of the beginning and ending
cross sections is typically employed. The mass c o m p o n e n t is then modeled
as an average volume of revolution. While curvature correction can be made
to yield accurate results with dense cross section data, the process of identifying individual components of volume is difficult to automate.
If higher accuracy is required from sparse cross-section data, prismoidal
correction may be used. The prismoidal-correction technique assumes that
the volume components may be modeled as prismoids. The general prismoid
1Appl. Mathematician, CLM/Systems, Inc., 5601 Mariner Dr., Tampa, FL 33609.
Note. Discussion open until January 1, 1995. To extend the closing date one
month, a written request must be filed with the ASCE Manager of Journals. The
manuscript for this paper was submitted for review and possible publication on
January 13, 1994. This paper is part of the Journal of Surveying Engineering, Vol.
120, No. 3, August, 1994. 9
ISSN 0733-9453/94/0003-0094/$2.00 + $.25 per
page. Paper No. 7071.
94
is a solid figure such that transverse area varies cubically with length. This
technique is used to greatest advantage in regions of cut-to-fill transition
where average-end-area results are notoriously poor (Davis et al. 1981; Easa
1991; Moffitt and Bouchard 1987). The use of pyramid and frustum-ofpyramid formulas in transition regions is an example of prismoidal modeling
(Easa 1991). Again, the identification of individual volume components is
difficult to implement in computer code.
Another existing volumetric technique is that of computing the volume
between two digital terrain models (DTMs). While DTM differencing does
not suffer from errors associated with prismoidal correction, the design DTM
must be fairly dense in regions of horizontal curvature if accurate results
are to be obtained. Moreover, relatively sparse cross-section data are often
the only source for DTM creation. This typically leads to poor DTMs and
correspondingly poor volumetric results. Even when dense, photogrammetrically derived data are available, the proper boundary geometry and
triangulation can usually only be achieved by interactive editing of surface
discontinuity strings or break lines. Bear in mind also that this paper focuses
on transportation alignments; the cross section, for better or worse, is usually
mandated as the basis for data collection, design, construction and payment.
In the following sections, a new method of computing alignment volumes
will be presented. The finite-element-volumes method is an algorithm that
provides automatic curvature and prismoidal correction using ordinarily
available cross-section data in conjunction with horizontal baseline geometry. The technique takes its name from the fact that volumes are computed
by the summation of finite, three-dimensional elements. The finite-elementvolumes method is an exact solution of an approximate problem, while
classical techniques are more often approximate solutions of exact problems.
The method is a numerical integration technique and the result, a Riemann
sum.
The new theory is developed by first considering the geometric model in
a general fashion, without regard to the specific curvilinear components of
the baseline. Next, the baseline-dependent element geometry is presented.
Numerical examples introduce a general formula for the volume of a curvilinear mass component and a new, high-precision, curvature-correction
method with which to compare finite-element-volume results.
The errors associated with curvature correction and prismoidal correction
are well documented (Davis et al. 1981; Easa 1991; Hickerson 1964; Kahmen
and Faig 1988; Moffitt and Bouchard 1987). The numerical examples given
here illustrate extreme conditions and demonstrate the proposed method's
ability to return very accurate results with very sparse data. While curvature
and prismoidal correction are not generally available in commercial software, finite-element volumes accomplishes both, with fewer limitations and
less data.
MODEL
Fig. 1 illustrates an orthogonal curvilinear coordinate [station, offset,
elevation (STA, OFF, ELV)] system superimposed upon the more familiar
Cartesian coordinate (X, Y, Z) system. Curvilinear coordinate systems are
based on a planar curve called the baseline. Station values are determined
as path lengths along this baseline from some fixed point of zero station.
Offsets are measured in the plane of the baseline, the X - Y or STA-OFF
plane, perpendicular to the forward tangent of the baseline. Facing in the
forward direction of the baseline path, offsets to the left are signed negative,
95
Z
/
OF ; ,v
/
/
/
STA-ELV PLANE
--OF THE OFFSET LINE
~
STA-ELVPLANE
~ - O F THEBASELINE
/
o/
OFF.E,V
II
/'nrr
1+00o
FORWARD /
/
LINE BASELINE
j
STA-OFF PLANE
FIG. 1. OrthogonalCurvilinearCoordinateSystem
while those to the right are signed positive. The offset depicted in Fig. 1 is
negative. Elevations correspond directly to Z values.
The S T A - E L V "plane" is that curved surface extruding above and below
the baseline into the Z half-spaces. When reference is made to the STAELV plane of paths other than the baseline, it is meant that the named path
serves as the baseline of yet another curvilinear coordinate system.
In order to describe the geometry of masses located about the baseline,
two planar curves are defined together with their associated profiles. Fig.
2(a) illustrates the planar, offset and tapered offset curves in the S T A - O F F
plane. The offset curve is a constant, concentric offset curve with the baseline
as directrix. The tapered offset curve is a transition curve between unequal
offset values. The offset at any point on the taper varies linearly with respect
to baseline length. The tapered offset curves serve to terminate the constant
offset curves.
Now consider "elevating" or assigning continuous elevation values to the
offset curves as illustrated in Fig. 2(b). The resulting system of offset profiles,
or longitudinal cross sections, define a surface in curvilinear coordinates.
The superposition or intersection of two such surfaces defines a mass or
volume. We will refer to one of these surfaces as the terrain surface and
the other as the design surface. It is the volume, in cut and fill, between
these surfaces that we wish to compute.
While the baseline geometry is generally known completely and continuously as a horizontal alignment, the elevation data are usually collected
as transverse cross sections in the O F F - E L V plane at discrete station values.
In the finite-element-volumes model it is assumed that cross sections are
normal (orthogonal) to the baseline and that cross sections exist at every
nonlinear, taper-transition station. The baseline geometry together with
these cross sections comprise the data.
It is necessary to invent a rule by which elevation values transition along
the longitudinal cross sections from one known value to another. In the
finite-element-volumes model it is assumed that elevations transition linearly
along the offset and tapered offset profiles as viewed in the S T A - E L V plane
of the baseline. For the most part, this is consistent with assuming that the
96
E
BASELINE
(a)
TAPERED OFFSET PROFILE
OFFSET LINE
~
OFFSET LINE
BASELINE
(b)
FIG. 2. Offset and Tapered Offset Curves and Profiles: (a) Plan View; (b) Perspective View
baseline profiles (design and terrain) transition linearly from one known
value to another. The exception is superelevation on a tapered offset. In
this event, vertical curvature will arise even when the baseline profiles are
linear. For this case, and when the baseline profiles are nonlinear, more
cross sections need to be collected in order to adequately represent these
vertical curves as a series of short chords. Implicit in the assumption of
profile linearity is the existence of cross section data at every nonlinear,
superelevation transition station.
Approximate Problem
Fig. 3(a) represents a transverse cross section in cut and fill. Fig. 3(b)
represents the same cross section after approximating the cut and fill areas
as rectangles of uniform width. The height of each rectangle is given by the
difference of design and terrain elevations at the midpoint. These rectangles
are the transverse cross sections of the finite elements we will sum to obtain
volumes.
Clearly, as the element width decreases, the discrete, planar model agrees
97
--
--
--
DESIGN SECTION
TERRAIN SECTION
~
t
CUT
1
FILL
(a)
H
- ELEMENT HEIGHT
W - ELEMENT WIDTH
(b)
FIG. 3. Continuous and Discrete Transverse Cross Section: (a) Continuous Cross
Section; (b) Discrete Cross Section
more and more closely with the continuous model. Indeed, the discrete
model converges rapidly with respect to cross-sectional area since the errors
tend to be compensatory.
In the current implementation of the finite-element-volumes algorithm,
the maximum element width is user-specified. Internally, an element width
is chosen such that the region between transverse cross-section pairs is
spanned from left to right (Fig. 4).
Exact Solution of Approximate Problem
The volumetric problem is now one of computing and summing the individual element volumes. Every element is assumed to transition linearly
(in the STA-ELV plane of the baseline) to an opposing element upstation
or to terminate on a tapered offset curve when an opposing element does
not exist. The resulting volume elements are thus curvilinear wedges or
frustums of wedges. The length of an element is measured along the offset
curve corresponding to the centroid of the element's transverse cross section.
It is in the STA-ELV plane of this offset curve that the longitudinal crosssection area is computed. Element volumes are then calculated as the product of longitudinal area and uniform element width. For the approximate
problem, this is an exact solution.
This discrete, volumetric model is a Riemann sum of longitudinal elements. While computation-intensive, the model is very flexible and quite
powerful when employed on a modern, high-speed computer. As in the
OFF-ELV planes of the transverse cross sections, the discrete model converges to a continuous representation as the element width approaches zero.
98
k.Ll...lu
I
ELEMENT
--
FIG. 4.
9- -
ELEMENT
PATH
I I1...1~1
I I
WIDTH
LENOTH
Plan and Transverse Section Views of Elements
The number of elements required to produce volumetric convergence is
higher than that needed for transverse area convergence but still easily
attainable.
A further simplification of the model may be achieved by considering
Cavalieri's theorem in a plane:
If two planar areas are included between a pair of parallel lines, and
if the two segments cut off by the areas on any line parallel to the
including lines are equal in length, then the two planar areas are equal
(Eves 1991).
Accordingly, it is not necessary to consider the terrain and design surfaces
independently; only the difference in elevation between the surfaces, or
signed height, need be known [Fig. 5(a)].
The foregoing assumption of independent linearity in the profiles may be
weakened as follows. Along every offset and tapered offset line, the difference in elevation between the design and terrain surfaces varies linearly
99
DESIGN
PROFILE
TERRAIN
- - - PROFILE
DIFFERENCE
------ PROFILE
DATUM
ELEVATION
--
I
HE,GHT
DATUM
I
ELEVATION
(a)
~FILL
~
~CUT
DATUM
(b)
HEIGHT
x~----~~@
DATUM
STATION
BACK
STATION
AHEAD
FIG.5. Cavalieri'sTheoremina Plane:(a)LinearProfiles;(b)NonlinearProfiles
with respect to distance along the baseline. Thus, finite-element volumes
will produce accurate results even when the profiles contain vertical curves,
provided that the elevation difference is linear [Fig. 5(b)].
ELEMENTS
Throughout the following it is assumed that the baseline is at least once
continuously differentiable or "tangent." Continuous first derivative assures
the existence of a curvilinear coordinate representation. Uniqueness of this
representation is also an issue. Basically, cross-section lines must not cross
one another, and curve offsets must be limited toward the center of curvature
by the radius of curvature. Assuring both the existence and uniqueness of
curvilinear coordinate representations for cross-section data will eliminate
the possibility of gaps and gores in the model.
When the baseline object is a line segment [Fig. 6(a)], the offset-line
length is simply the baseline length, i.e.
Lo = L
100
(1)
le
L~
e
~- OFFSET LINE
~- BASELINE
STATION
L
STATION
AHEAD
BACK
(a)
S
T
A
T
I
O
N
"~-,. r
~
~
STATION
r / ' ~ OFFSET LINE
BASELINE
CC
(b)
STATION
BACK~
STATION
V\ r,
o/
O SET L,NE
BASELINE
(c)
FIG. 6. Offset Curve Length" (a) Linear Baseline; (b) Circular Baseline; (c) Spiral
Baseline
When the baseline object is a circular curve segment [Fig. 6(b)], the offsetline length is given by
Lo = L(1 + ek)
(2)
where e = eccentricity or plan distance from the baseline to the transverse
area centroid; and k -- 1/r is the baseline curvature. The eccentricity is
positive if the offset is away from the center of curvature and negative
otherwise. Note that the sign convention for the eccentricity differs from
the convention used to obtain curvilinear offset values. The eccentricity
shown in Fig. 6(b) is positive.
When the baseline object is a spiral curve segment [Fig. 6(c)], the offsetline length is given by
t 0 = t
[e
1 -}- ~ (k 1 -~ k2)
1
(3)
where kl = 1/rl = initial baseline curvature; and k2 = 1/r2 is the terminal
101
baseline curvature. The eccentricity is positive if the offset is away from the
evolute and negative otherwise. The eccentricity shown in Fig. 6(c) is positive.
The line, circular curve, and spiral curve segment compose the geometric
elements of a clothoidal spline (Walton and Meek 1990). If the line segment
is considered as a degenerate Cornu spiral with zero curvature at either end,
and the circular curve segment is considered as a degenerate spiral with the
same curvature at either end, then (3) subsumes (1) and (2).
If the transverse cross sections back and ahead have unequal minimum/
maximum offsets, some or all of the elements will be terminated on tapered
offset curves (Fig. 7). Modified baseline stationing and initial and terminal
element heights are computed by linear interpolation in the STA-ELV plane
of the baseline.
Any element, regardless of baseline geometry, appears as a trapezoid or
pair of triangles in the STA-ELV plane of the baseline (Fig. 8). Whenever
the initial and terminal heights differ in sign, the height curve passes through
the datum. It is computationally convenient to consider this element as
composed of two individual elements, one in cut and the other in fill. The
~
LINE
\V
.
.
.
.
.
.
.
(.)
STATION~
BACK \
"~ ~
' _ ........ ~
BASELN
IE ~
~'
/ STATION
A~
IF_AD
(b)
STATION
AHEAD
S T
BACK
A
~
T
I
~
~.,~
BASELINE
~
OFFSET
LINE
",d
(e)
FIG. 7. Taper Termination: (a) Forward; (b) Backward; (c) Double
102
H >0
H <0
"
L (FILL)
"
~
o DAYLIGHT POINT
HI
~-I-
FIG. 8.
L
Element Profile Examples in STA-ELV Plane of Baseline
~~
CUT
~
1. 2
"l
FILL
FIG, 9, Cut/Fill Regions Delineated by Daylight Line
103
point where the height equals zero is a daylight point. Here the design and
terrain elevations are equal. The locus of all such points, or daylight line,
divides the STA-OFF plane into regions of cut and fill (Fig. 9).
An element with a line or circular curve segment directrix appears as a
trapezoid or triangle in the STA-ELV plane of the offset line, as well as
the baseline. The longitudinal element area is given by
L0
A = -~-(H~ + //2)
(4)
where/41 = initial element height; and H2 = terminal element height.
For the case of an element with a spiral segment directrix, the line from
initial to terminal height as viewed in the STA-ELV plane of the baseline
maps into an inverse parabola in the STA-ELV plane of the offset line (Fig.
10). The longitudinal element area is given by
Lo
.4 =
Le
+ 1-12) + T 2 (H2 -
H1)(k2 -
(5)
kl)
The second term on the right-hand side of (5) accounts for the parabolic
segment area associated with spiral elements. If the line and circular curve
segment are considered as degenerate clothoids with equal initial and terminal radii, (5) subsumes (4).
When multiple baseline objects are represented between any pair of consecutive cross sections, the element parameters are computed on an objectby-object basis. The resulting compound element is treated as multiple
instances of single-object elements (Fig. 11).
Note well that the height profiles depicted in Figs. 8, 10, and 11 are
neither design nor terrain elevation profiles. Rather, they are elevation
difference profiles. Furthermore, the profiles illustrated in Figs. 10 and 11
are viewed in the STA-ELV plane of the offset line and do not represent
nonlinearities in baseline grade.
Finally, the element volume is given by
V
=
(6)
AW
where W = element width. The sum of all positive element volumes is the
required fill quantity. The cut quantity is given by the sum of all negative
I
>
Lo (FILL)
O
Lo (CUT)
~
H
oDAYLG
I HTPOINT
L0
<0
t
FIG. 10. Spiral Element Profile Examples in STA-ELV Plane of Offset Line
104
]'Lo (LINEAR ELEMENT)']" Lo (CIRCULAR ELEMENT)"l
Lo (CIRCULAR ELEMENT)'I'Lo (SPIRAL ELEMENT)"1
I-I
Lo (COMPOUND ELEMENT)
FIG. 11.
Compound Element Profile Examples
TABLE 1. Half-Section Data for Example 1
Station
(m)
(1)
Altitude
(m)
(2)
Base
(rn)
(3)
Area
(m2)
(4)
Eccentricity
(m)
(5)
500
550
600
2.0
1.5
1.0
15.0
11.25
7.5
15.0
8.4375
3.75
_+5.0
+_3.75
_+2.5
Subscript
(6)
1
M
2
element volumes. Eqs. (1)-(6) [or, in general, (3), (5), and (6)] are the
finite-element-volume equations.
NUMERICAL EXAMPLES
Example 1
Table 1 presents half-section parameters for the data depicted in Fig. 12.
The altitude and base at station 550 are linearly interpolated from the given
data at stations 500 and 600. Area, A of the triangular half-section is given
by one-half the product of the base and the altitude, and the eccentricity,
e is one-third the base. Subscripts indicate the first (1), second (2), or middle
(M) section.
Regardless of baseline geometry, the average end area method yields a
fill volume of
L
100
VA = ~ (A1 + A2) = T (15 + 3.75) = 937.5 m 3
(7)
for either side of the alignment. The average-end-area formula may be
derived by an application of the trapezoidal rule with one panel.
105
.
STATION 0 + 5 0 0
LEFT HALF-SECTION =
~
RIGHTHALF-SECTION
1 5m
m
1 5m
BASELINE
FIG. 12.
Cross Section Data for Example 1
Prismoidal modeling, again without regard to baseline geometry, produces
Ve
=
L
100
~ (A 1 + 4AM + A2) = y [15 + 4(8.4375) + 3.75] = 875 m 3
(8)
for either side of the alignment. The prismoidal formula may be derived by
an application of Simpson's one-third rule with two panels (Kahmen and
Faig 1988).
According to a theorem of Pappus and Guldinus:
If a planar area be revolved about an axis in its plane, but not intersecting the area, the volume of the solid of revolution so formed is
equal to the product of the area and the length of the path traced by
the centroid of the area (Eves 1991).
Through consideration of a differential element of volume (Fig. 13) and the
Pappus-Guldinus theorem, a completely general formula for the volume of
a curvilinear mass component may be derived. The general volume formula
is
V =
A(1 + e k ) d x
=
A dx +
A e k dr
(9)
where A, e, and k = functions of the instantaneous baseline length x. Note
that this "differential element of volume" is a transverse slab with infinitesimal length, while the "finite volume element" is a longitudinal strip with
user-defined width.
As special cases, (9) includes Cavalieri's and Pappus's theorems on areas
106
'~( ~ / ~
CENTROID
~~(x~)
FIG. 13.
BASELINE
Differential Element of Volume
and volumes as well as every other result given here. Eq. (9) is a difficult
prescription to follow; the application of this formula to typical alignment
volumes is not practical. The formula was developed so that exact results
could be obtained for certain well-behaved, but nontrivial, geometries. It
is presented here as an independent, analytical technique with which to
verify the correctness of the finite-element-volumes method.
The second term on the right-hand side of (9) is the curvature-correction
value. Curvature correction, as presented here, is "prismoidal." That is,
the correction quantity is developed as a Simpson's rule approximation of
the indicated integral. Elsewhere (Davis et al. 1981; Hickerson 1964), the
correction quantity is developed as a trapezoidal rule (average-end-value)
approximation. Three cases of baseline curvature are given below to illustrate the mechanics of prismoidal curvature correction and the behavior of
the finite-element-volumes model. Finite-element-volume results are computed by repeated evaluations of (3), (5), and (6).
Case l
Fig. 14(a) is the plan view of a circular curve alignment with a radius r
of 200 m. The mass components are bounded on the left and right by tapered
offset circular curves or Archimedean spirals. These lines are the slopestake limits.
The prismoidal curvature correction for the circular case is
_~_
Vc =
100
( A , e l + 4AMeM + Aze2) - 6(200) [15(-+5)
+ 4(8.4375)(___3.75) + 3.75(_+2.5)] = _ 17.578125 m 3
(10)
The correction value is positive on the left and negative on the right. The
required volume is given by V = Ve + V o
For the left side of the alignment, V = 875 + 17.578125 = 892.578215
m 3. The finite-element-volumes method yields 893.9 m 3 with an element
107
STA O+500~
T
A
0+600
(a)
s o+ oo\
~
STA 0+600
(b)
0+600
STA O+500
(c)
FIG. 14. Baseline Curvature Cases for Example 1: (a) Circular Curvature (rl =
r2); (b) Spiral Curvature (rl > r2); (c) Spiral Curvature (rl < r2)
width of 0.1 m and c o n v e r g e s to 892.6 m 3 with an e l e m e n t w i d t h of
0.001 m.
O n the right side of t h e a l i g n m e n t , V = 875 - 17.578125 = 857.421875
m 3. T h e f i n i t e - e l e m e n t - v o l u m e s m e t h o d yields 858.6 m 3 with an e l e m e n t
width of 0.1 m and c o n v e r g e s to 857.4 m 3 with an e l e m e n t w i d t h o f
0.001 m.
Case 2
Fig. 14(b) is the plan v i e w of a spiral c u r v e a l i g n m e n t with starting and
ending radii of rl = 500 m and r2 = 125 m , respectively. T h e i n s t a n t a n e o u s
radius of c u r v a t u r e at the m i d d l e section is g i v e n by rM = 2/(kl + k2) =
200 m.
The prismoidal c u r v a t u r e c o r r e c t i o n for t h e spiral case is
108
TABLE 2.
Fill Volume Results for Example 1
Left volume
(m3)
(2)
Method
(1)
Right volume
(m 3)
(3)
(a) Curvature Independent
Average-end-area method
Prismoidal modeling
937.5
875.0
937.5
875.0
892.578125
892.6
892.578125
857.421875
857.4
857.421875
889.296875
889.3
889.34375
860.703125
860.7
860.65625
895.859375
895.8
895.8125
854.140625
854.2
854.1875
(b) Case 1
Prismoidal modeling with curvature correction
Finite-element-volumesmethod
Exact
(c) Case 2
Prismoidal modeling with curvature correction
Finite-element-volumes method
Exact
(d) Case 3
Prismoidal modeling with curvature correction
Finite-element-volumes method
Exact
TABLE 3.
Station
Cut area
(m)
(1)
(m2)
(2)
500
600
700
100
50
0
/~
V C = --~ ( A l e l k 1 Jr-
+
Cut Section Data for Example 2
4AMeMkM+ A2e2k2) =
100 [15(=5)
~-
L 5oo
4 ( 8 . 4 3 7 5 ) ( _ 3.75)
3.75(+_ 2 . 5 ) ]
200
+
1~
_] = _ 14.296875 m 3
(]l)
For the left side of the a l i g n m e n t , V = Vv + Vc = 889.296875 m 3. T h e
f i n i t e - e l e m e n t - v o l u m e s m e t h o d yields 890.6 m 3 with a n e l e m e n t width of
0.1 m a n d converges to 889.3 m 3 with a n e l e m e n t width of 0.0001 m.
O n the right side of the a l i g n m e n t , V = Ve + Vc = 860.703125 m 3. T h e
f i n i t e - e l e m e n t - v o l u m e s m e t h o d yields 861.8 m 3 with a n e l e m e n t width of
0.1 m a n d converges to 860.7 m 3 with a n e l e m e n t width of 0.005 m.
Case 3
Fig. 14(c) is the p l a n view of a spiral curve a l i g n m e n t with starting a n d
e n d i n g radii of rl = 125 m a n d r2 = 500 m , respectively. This b a s e l i n e is
the reflection of the case 2 baseline. T h e i n s t a n t a n e o u s radius of c u r v a t u r e
at the middle section is rM = 200 m as in case 2.
Prismoidal c u r v a t u r e c o r r e c t i o n is given b y
109
""~'~"~'~'~
STA 0+700
STA 0+600
lOm
.
lOm
_1
STA 0+500
BASELINE
] CUT
(a)
FILL
DAYLIGHT LINE
.,-,..
STA )+600
STA O+500
I-
BASELINE
lOOm
-I-
STA 0+700
lOOm
"1
(b)
FIG. 15. Cross Section Data and Plan View for Example 2: (a) Cross Sections;
(b) Plan View
L
100 [.15(_+ 5)
Vc = -~ ( A l e l k x + 4AMeMkM + Azezk2) = ~ - L 125
+
4 ( 8 . 4 3 7 5 ) ( - 3.75)
3.75(_+ 2 . 5 ) ]
200
+
~
j = -4-20.859375 m 3
(12)
For the left side of the a l i g n m e n t , V = Vp + Vc = 895.859375 m 3. T h e
f i n i t e - e l e m e n t - v o l u m e s m e t h o d yields 897.1 m 3 with a n e l e m e n t width of
0.1 m a n d converges to 895.8 m 3 with a n e l e m e n t width of 0.001 m.
O n the right side of the a l i g n m e n t , V = Vp + Vc = 854.140625 m 3. T h e
f i n i t e - e l e m e n t - v o l u m e s m e t h o d yields 855.4 m 3 with a n e l e m e n t width of
0.1 m a n d converges to 854.2 m 3 with an e l e m e n t width of 0.005 m.
Prismoidal c u r v a t u r e c o r r e c t i o n is exact if the p r o d u c t A e k is n o m o r e
110
•
b(*)
c(x)
T
lOre
I
m.
(a)
(x)
--~CUT
DAYLIGHT LINE
FILL
BASELINE
STA 0+700
STA 0+600
t
lOOrn
t
(b)
FIG. 16. Right Half-Section Dimensions for Example 2: (a) Cross Section; (b) Plan
View
TABLE 4. Cut Volume Results for Example 2
Method
(1)
STA 500-600 volume
(m3)
(2)
STA 600-700 volume
(ms)
(3)
Average-end area
Prismoidal modeling
Finite-element volumes
Exact
7,500.0
7,357.022604
6,931.5
6,931.471806
2,500.0
1,666.666667
1,931.5
1,931.471806
than third order as a function of baseline length. For the general, finite
volume element, this product is quadratic; (1)-(6) (the finite-element-volume equations) may be derived using prismoidal curvature correction. For
the circular case 1, the function is cubic and the correction result is exact.
For the spiral cases 2 and 3, A e k is quartic. The exact correction value for
case 2 is
111
100
Vc = _+1.875(10)- H~
(200
d0
-
-
X)3(100 + 3X) d x = _+14.34375 m 3
(~3)
and the exact correction value for case 3 is
V c = +1.875(10) -"~
l
lO0
(200 - x)3(400 - 3x) dx = _+20.81250 m 3
J0
(14)
These exact results might also have been obtained using Simpson's rule with
end correction (Hornbeck 1975).
Fill volume results for Example 1 are summarized in Table 2.
The average of curvature corrections for spiral cases 2 and 3, exact or
prismoidal, is precisely the curvature correction for the circular case 1. This
is not coincidental; the spiral parameters were chosen such that the mean
radius of curvature (200 m) was that of the circle.
Cross-section dimensions were also carefully chosen for this example such
that prismoidal transitions with linear profiles were possible. The mass components in this problem are curvilinear prismoids. The following example
illustrates an instance in which the mass components are not prismoidal. As
before, finite-element-volume results are computed by repeated evaluations
of (3), (5), and (6).
Example 2
Table 3 presents cut-section parameters for the data depicted in Fig. 15(a).
Consider the cut quantity between stations 600 and 700. It is widely held
(Davis et al. 1981; Easa 1991; Moffitt and Bouchard 1987) that the volume
of this mass component is given by the pyramid formula:
V-
LA
~-
-
100(50)
3
- 1,666.666667 m 3
(i5)
Next consider the cut quantity between stations 500 and 600. Easa (1991)
suggests that this volume is given by the pyramid frustum formula
V :
L
~ (A 1 + ~
+ Az)
100
3 (100 + X/5,000 + 50) = 7,357.022604 m 3
(16)
The given cross section data cannot, however, represent sections of a
pyramid. The design plane must twist or superelevate between stations in
order to obtain the geometry shown. Between stations 500 and 600, the left
shoulder rolls up while the right shoulder rolls down, and between stations
600 and 700 the left shoulder rolls down while the right shoulder rolls up.
This superelevation transition produces curvature in the daylight line as
indicated in Fig. 15(b).
Reconsider the cut quantity between stations 600 and 700. The baseline
profile varies linearly between 0 m and 5 m while the right-shoulder profile
varies linearly between - 10 m and 0 m. The right half-section dimensions
between stations 600 and 700 illustrated in Fig. 16 are given by a = (100
- x)/10; b = 20(100 - x)/(200 - x); and c = x/20. The cut area is A =
112
ab/2 = (100 - x)2/(200 - x). The cut volume, by virtue of Cavalieri"s
theorem for space [or (9), for that matter], is
l lO0
V =
J0
A dx = 104 In 2 - 5000 = 1.931.471806 m 3
(17)
The finite-element-volumes m e t h o d yields 1,931.4 m 3 with an element width
of 0.1 m and converges to 1,931.5 m 3 with an element width of 0.05 m.
In a similar fashion, the cut volume b e t w e e n stations 500 and 600 is given
by V = 104 In 2 = 6,931.471806 m 3. T h e finite-element-volumes method
yields 6,931.4 m 3 with an e l e m e n t width of 0.1 m and converges to 6,931.5
m 3 with an element width of 0.05 m.
Cut volume results for E x a m p l e 2 are summarized in Table 4.
In the preceding examples, exact a g r e e m e n t with analytical results has
been shown. The finite-element-volumes model is essentially a roadway
design model with linear baseline profiles. Within these constraints, the
algorithm can be made to yield exact results.
CONCLUSIONS
The finite-element-volumes technique has been tested under a variety of
circumstances and found to faithfully model volumes of revolution, curvature correction, prismoidal correction, and p y r a m i d and frustum-of-pyramid
calculations associated with cut-to-fill transition areas. F u r t h e r m o r e , finiteelement volumes correctly models superelevation transitions that give rise
to nonprismoidal components of volume.
When horizontal curvature or daylighting (cut-to-fill transition) is extensive, the increased accuracy can be significant. It is not, however, increased
accuracy that is the primary benefit of using finite-element volumes. Rather,
it is the logical accountability of a technique that does not, for instance,
favor one side of a horizontal curve over the other. Considerable arbitration
and litigation may be avoided by using an algorithm that both d e v e l o p e r
and contractor find equitable.
In the course of this research, a general formula for the volume of a
curvilinear mass c o m p o n e n t was d e v e l o p e d together with a new, high-precision, prismoidal curvature-correction technique. It is h o p e d that the volumetric paradigm presented here will be of benefit to researchers and practitioners alike.
ACKNOWLEDGMENTS
The concept of finite-element volumes was originally suggested by C. L.
Miller of e L M / S y s t e m s , Inc., T a m p a , Fla. Many thanks are due Miller and
his organization for their support and e n c o u r a g e m e n t throughout the course
of this research. My sincere thanks also go to Sum Lin of E S R I , Inc. for
his many helpful suggestions.
APPENDIXI.
REFERENCES
Davis, R. E., Foote, F. S., Anderson, J. M., and Mikhail, E. M. (1981). Surveying
theory and practice. McGraw-Hill, New York, N.Y.
Easa, S. M. (1991). "Pyramid frustum formula for computing volumes of roadway
transition areas." J. Surv. Engrg., ASCE, 117(2), 98-101.
Eves, H. (1991). "Geometry." Standard mathematical tables, W. H. Beyer, ed., CRC
Press, Boca Raton, Fla., 102-114.
113
Hicke'rson, T. F. (1964). Route location and design. McGraw-Hill, New York, N.Y.
Hornbeck, R. W. (1975). Numerical methods. Prentice-Hall, Englewood Cliffs, N.J.
Kahmen, H., and Faig, W. (1988). Surveying. Walter de Gruyter, New York, N.Y.
Moffitt, F. H., and Bouchard, H. (1987). Surveying. Harper and Row, New York,
N.Y.
Walton, D. J., and Meek, D. S. (1990). "'Clothoidal splines." Comp. and Graphics,
14(1), 95-100.
APPENDIX II.
NOTATION
The following symbols are used in th& paper:
A
e
H
k
L
Lt~
r
V
W
=
=
=
=
=
=
=
=
=
transverse or longitudinal cross-section area;
eccentricity;
e l e m e n t height;
baseline c u r v a t u r e ;
baseline length;
offset-line length;
baseline radius;
v o l u m e ; and
e l e m e n t width.
Subscripts
A
C
M
P
1
2
=
=
=
=
=
=
a v e r a g e - e n d - a r e a value;
curvature correction value;
middle value;
prismoidal value;
initial value; and
t e r m i n a l value.
114