F I N I T E - E L E M E N T V O L U M E S

By Thomas G. Davis j

ABSTRACT: The finite-element-volumes method is a new earthwork volumes tech- nique quite unlike conventional methods. The algorithm provides automatic cur- vature and prismoidal correction using ordinarily available cross-section data in conjunction with horizontal baseline geometry. The cross sections are approximated as a series of rectangular elements of equal, user-specified width. As this width approaches zero, cross-sectional area 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 opposing element does not exist. The resulting volume elements are thus curvilinear wedges or frustums of wedges. Lin- ear, circular, and Cornu spiral baseline (clothoidal spline) components are accom- modated by the method. Numerical examples show excellent agreement with exact results even when the mass components are not prismoidal. A general formula for the volume of a curvilinear mass component and a new, high-precision, prismoidal curvature-correction technique are also presented.

INTRODUCTION

Traditionally, the problem of computing volumes associated with trans- portation alignments (roadways, railways, and waterways) has been accom- plished 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. When this is done, the transverse areas at the beginning and end of the mass component are nearly the same and the assumption of linearity is made more realistic. The second issue, however, cannot be ameliorated by any amount of data collection. That is, if the baseline exhibits significant horizontal curvature, the average-end-area formula cannot de-

liver 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 component 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 iden- tifying 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-of- pyramid 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, photogram- metrically 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 geom- etry. The technique takes its name from the fact that volumes are computed by the summation of finite, three-dimensional elements. The finite-element- volumes 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 s u m .

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 cur- vilinear 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 soft- ware, 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 STA- ELV PLANE

~ --OF THE OFFSET LINE

/ STA-ELV PLANE / ~ - O F THE BASELINE

OF ; ,v / / OFF. E,V

/ o / I I /'nrr FORWARD /

1+00o /

LINE BASELINE j STA-OFF PLANE

FIG. 1. Orthogonal Curvilinear Coordinate System

while those to the right are signed positive. The offset depicted in Fig. 1 is negative. Elevations correspond directly to Z values.

The STA-ELV "plane" is that curved surface extruding above and below the baseline into the Z half-spaces. When reference is made to the STA- ELV 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 STA-OFF 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 contin- uously as a horizontal alignment, the elevation data are usually collected as transverse cross sections in the OFF-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 STA-ELV 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) Per- spective 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 ~ CUT

-- -- -- TERRAIN SECTION t 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 in-

dividual 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 cross- section area is computed. Element volumes are then calculated as the prod- uct 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 ele- ments. 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 con- verges to a continuous representation as the element width approaches zero.

98

k . L l . . . l u I I I 1 . . . 1 ~ 1 I I

E L E M E N T W I D T H

- - �9 - - E L E M E N T P A T H L E N O T H

FIG. 4. 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 dif- ference in elevation between the design and terrain surfaces varies linearly

99

ELEVATION

HE,GHT

I

I

ELEVATION ~

HEIGHT x~----~~@

DESIGN -- PROFILE

TERRAIN --- PROFILE

DIFFERENCE ------ PROFILE

DATUM

(a)

DATUM

~ F I L L

~ C U T

DATU M

(b)

DATUM

STATION STATION BACK AHEAD

FIG. 5. Cavalieri's Theorem in a Plane: (a) Linear Profiles; (b) Nonlinear Profiles

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 (1)

100

l e L~

STATION L

BACK

(a)

~- OFFSET LINE

e ~- BASELINE

STATION

AHEAD

S T A T I O N ~ STATION

"~-,. r r / ' ~ OFFSET LINE ~ BASELINE

CC

(b)

STATION

BACK ~ STATION

V- \ r, O SET L, NE

o / 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 offset- line 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 offset- line length is given by

[ e 1 t 0 = t 1 -}- ~ (k 1 -~ k2) (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 pos- itive.

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 \ ' _ ........ ~ / STATION

BASELINE ~ AI~F_AD

(b)

FIG. 7.

STATION AHEAD

S T A T I ~ BACK ~ ~ . , ~ ~ OFFSET LINE

BASELINE ",d

(e)

Taper Termination: (a) Forward; (b) Backward; (c) Double

102

H > 0

HI

FIG. 8.

" L (FILL) " ~

o DAYLIGHT POINT

H < 0

~ - - 1. 2 I- L "l Element Profile Examples in STA-ELV Plane of Baseline

CUT ~ ~ ~ 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 Le .4 = + 1-12) + T 2 (H2 - H1) (k2 - k l ) (5)

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 ter- minal radii, (5) subsumes (4).

When multiple baseline objects are represented between any pair of con- secutive cross sections, the element parameters are computed on an object- by-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 = A W (6)

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

FIG. 10.

I Lo (FILL) Lo (CUT)

> O ~ H <0

o DAYLIGHT POINT

L0 t 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 Altitude Base Area Eccentricity (m) (m) (rn) (m 2) (m) Subscript (1) (2) (3) (4) (5) (6)

500 2.0 15.0 15.0 _+ 5.0 1 550 1.5 11.25 8.4375 +_ 3.75 M 600 1.0 7.5 3.75 _+2.5 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 = RIGHT HALF-SECTION ~ m

1 5m 1 5m

BASELINE

FIG. 12. Cross Section Data for Example 1

Prismoidal modeling, again without regard to baseline geometry, pro- duces

L 100 V e = ~ (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 inter- secting 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 d x + 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 infini- tesimal 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

~ BASELINE

~(x) ~

FIG. 13. 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 f inite-element-volumes method.

The second term on the right-hand side of (9) is the curvature-correction value. Curvature correction, as presented here, is "pr ismoidal ." 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 illus- trate the mechanics of prismoidal curvature correction and the behavior of the finite-element-volumes model. Fini te-element-volume results are com- puted 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 slope- stake limits.

The prismoidal curvature correction for the circular case is

_~_ 100 Vc = (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

(a)

0+600

s o+ oo\ ~ STA 0+600

(b)

STA O+500

0 + 6 0 0

(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 converges to 892.6 m 3 with an e l e m e n t wid th of

0.001 m. On the right side of the a l ignment , V = 875 - 17.578125 = 857.421875

m 3. The f in i t e - e l emen t -vo lumes 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 converges to 857.4 m 3 with an e l e m e n t wid th o f

0.001 m.

Case 2

Fig. 14(b) is the plan v iew of a spiral curve a l ignment with s tar t ing and

ending radii of rl = 500 m and r2 = 125 m, respect ive ly . T h e ins tan taneous radius of curva ture at the midd le sect ion is g iven by rM = 2/ (kl + k2) = 200 m.

The pr ismoidal cu rva tu re co r rec t ion for the spiral case is

108

TABLE 2. Fill Volume Results for Example 1

Method (1)

Average-end-area method Prismoidal modeling

Left volume (m 3)

(2)

(a) Curvature Independent

937.5 875.0

(b) Case 1

Right volume (m 3)

(3)

937.5 875.0

Prismoidal modeling with curvature correction 892.578125 857.421875 Finite-element-volumes method 892.6 857.4

Exact 892.578125 857.421875

(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

889.296875

889.3 889.34375

860.703125

860.7 860.65625

895.859375 854.140625 895.8 854.2 895.8125 854.1875

TABLE 3. Cut Section Data for Example 2

Station Cut area (m) (m 2) (1) (2)

500 100 600 50 700 0

/~ 100 [15(=5) V C = --~ ( A l e l k 1 Jr- 4AMeMkM + A2e2k2) = ~ - L 5oo

4(8 .4375) (_ 3.75) 3.75(+_ 2 .5 ) ] + 200 + 1 ~ _] = _ 14.296875 m 3 ( ] l )

For the left side of the a l ignmen t , V = Vv + Vc = 889.296875 m 3. The f in i te -e lement -volumes m e t h o d yields 890.6 m 3 with an e l em en t width of

0.1 m and converges to 889.3 m 3 with an e l e m e n t width of 0.0001 m. On the right side of the a l ignmen t , V = Ve + Vc = 860.703125 m 3. The

f in i te -e lement -volumes m e t h o d yields 861.8 m 3 with an e l em en t width of 0.1 m and converges to 860.7 m 3 with an e l e m e n t width of 0.005 m.

Case 3 Fig. 14(c) is the p lan view of a spiral curve a l i gnmen t with s tar t ing and

ending radii of rl = 125 m and r2 = 500 m, respectively. This base l ine is the reflect ion of the case 2 basel ine . T h e i n s t a n t an eous radius of curva ture at the middle sect ion is rM = 200 m as in case 2.

Prismoidal curva ture cor rec t ion is g iven by

109

" " ~ ' ~ " ~ ' ~ ' ~ STA 0+700

. lOm lOm _1

BASELINE

. , - , . .

lOOm

STA

STA 0+600

STA 0+500

STA O+500

I-

] CUT

FILL

DAYLIGHT LINE

(a)

)+600 STA 0+700

-I- lOOm "1

BASELINE

(b)

FIG. 15. Cross Section Data and Plan View for Example 2: (a) Cross Sections;

(b) Plan View

L 100 [.15(_+ 5) Vc = -~ (A le lkx + 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 ignment , V = Vp + Vc = 895.859375 m 3. The f in i te -e lement-volumes m e t h o d yields 897.1 m 3 with an e l em en t width of 0.1 m and converges to 895.8 m 3 with an e l eme n t width of 0.001 m.

On the right side of the a l ignment , V = Vp + Vc = 854.140625 m 3. The f in i te -e lement-volumes m e t h o d yields 855.4 m 3 with an e l emen t width of 0.1 m and converges to 854.2 m 3 with an e l eme n t width of 0.005 m.

Prismoidal curva ture cor rec t ion is exact if the p roduc t A e k is no more

110

•

c(x)

T

b(*)

lOre m.

(a)

I (x)

--~CUT

DAYLIGHT LINE

STA 0+600

t lOOrn t

FILL

BASELINE

STA 0+700

(b)

FIG. 16. Right Half-Section Dimensions for Example 2: (a) Cross Section; (b) Plan

View

TABLE 4. Cut Volume Results for Example 2

STA 500-600 volume STA 600-700 volume Method (m 3 ) (m s )

(1) (2) (3)

Average-end area 7,500.0 2,500.0 Prismoidal modeling 7,357.022604 1,666.666667

Finite-element volumes 6,931.5 1,931.5 Exact 6,931.471806 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-vol- ume 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 - - X)3(100 + 3X) dx = _+ 14.34375 m 3 d0

(~3)

and the exact correction value for case 3 is

l lO0

V c = +1.875(10) -"~ (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 com- ponents 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:

L A 100(50) V - ~ - - 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

L V : ~ ( 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 l O0 V = A dx = 104 In 2 - 5000 = 1.931.471806 m 3 (17)

J0

The fini te-element-volumes method yields 1,931.4 m 3 with an e lement width of 0.1 m and converges to 1,931.5 m 3 with an e lement width of 0.05 m.

In a similar fashion, the cut volume be tween stations 500 and 600 is given by V = 104 In 2 = 6,931.471806 m 3. The f ini te-element-volumes method yields 6,931.4 m 3 with an e lement width of 0.1 m and converges to 6,931.5 m 3 with an e lement width of 0.05 m.

Cut volume results for Example 2 are summarized in Table 4. In the preceding examples , exact agreement with analytical results has

been shown. The f ini te-element-volumes model is essentially a roadway design model with l inear baseline profiles. Within these constraints, the algorithm can be made to yield exact results.

CONCLUSIONS

The f ini te-element-volumes technique has been tested under a variety of circumstances and found to faithfully model volumes of revolution, curva- ture correction, pr ismoidal correct ion, and pyramid and frustum-of-pyramid calculations associated with cut-to-fill t ransit ion areas. Fur thermore , finite- element volumes correctly models superelevat ion transit ions that give rise to nonprismoidal components of volume.

When horizontal curvature or daylighting (cut-to-fill transit ion) is exten- sive, the increased accuracy can be significant. It is not, however, increased accuracy that is the pr imary benefit of using f ini te-element volumes. Rather , it is the logical accountabil i ty of a technique that does not, for instance, favor one side of a horizontal curve over the other. Considerable arbi t rat ion and litigation may be avoided by using an algori thm that both developer and contractor find equitable.

In the course of this research, a general formula for the volume of a curvilinear mass component was deve loped together with a new, high-pre- cision, prismoidal curvature-correct ion technique. It is hoped that the vol- umetric paradigm presented here will be of benefit to researchers and prac- titioners alike.

ACKNOWLEDGMENTS

The concept of f ini te-element volumes was originally suggested by C. L. Miller of eLM/Sys tems , Inc., Tampa, Fla. Many thanks are due Miller and his organization for their support and encouragement throughout the course of this research. My sincere thanks also go to Sum Lin of ESRI , 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.

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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.

A P P E N D I X II. N O T A T I O N

The following symbols are used in th& paper:

A = transverse or longi tudinal cross-sect ion area;

e = eccentr ic i ty;

H = e l emen t height ;

k = basel ine curva ture ;

L = basel ine length;

Lt~ = offset- l ine length;

r = basel ine radius;

V = vo lume; and

W = e l emen t width.

Subscripts A = ave rage -end-a rea value;

C = curva ture cor rec t ion va lue ;

M = middle value;

P = pr ismoidal va lue;

1 = initial va lue; and 2 = te rminal value.

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