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Highway Capacity Manual 2000

24-i Chapter 24 - Freeway Weaving

CHAPTER 24

FREEWAY WEAVING

CONTENTS

I. INTRODUCTION..................................................................................................... 24-1Scope of the Methodology................................................................................ 24-1Limitations of the Methodology ........................................................................ 24-1

II. METHODOLOGY .................................................................................................... 24-1LOS .................................................................................................................. 24-2Weaving Segment Parameters ........................................................................ 24-3Determining Flow Rates ................................................................................... 24-3Weaving Segment Diagram ............................................................................. 24-3Weaving Segment Configuration...................................................................... 24-5Determining Weaving and Nonweaving Speeds.............................................. 24-5

Determining Weaving Intensity ................................................................. 24-7Constants for Computing Weaving Intensity Factors................................ 24-7

Determining Type of Operation ........................................................................ 24-7Determining Weaving Segment Speed ............................................................ 24-8Determining Density ......................................................................................... 24-9Determining Weaving Segment Capacity ........................................................ 24-9Multiple Weaving Segments........................................................................... 24-19Collector-Distributor Roadways ...................................................................... 24-19

III. APPLICATIONS .................................................................................................... 24-19Computational Steps ...................................................................................... 24-20Planning Applications ..................................................................................... 24-20Analysis Tools ................................................................................................ 24-20

IV. EXAMPLE PROBLEMS ........................................................................................ 24-22Example Problem 1 ........................................................................................ 24-23Example Problem 2 ........................................................................................ 24-26Example Problem 3 ........................................................................................ 24-29Example Problem 4 ........................................................................................ 24-32Example Problem 5 ........................................................................................ 24-37

V. REFERENCES ...................................................................................................... 24-39APPENDIX A. WORKSHEET ..................................................................................... 24-39

Freeway Weaving Worksheet

EXHIBITS

Exhibit 24-1. Freeway Weaving Methodology ............................................................ 24-2Exhibit 24-2. LOS Criteria for Weaving Segments ..................................................... 24-3Exhibit 24-3. Parameters Affecting Weaving Segment Operation ............................. 24-4Exhibit 24-4. Construction and Use of Weaving Diagrams ........................................ 24-4Exhibit 24-5. Determining Configuration Type ............................................................ 24-5Exhibit 24-6. Constants for Computation of Weaving Intensity Factors ..................... 24-6Exhibit 24-7. Criteria for Unconstrained Versus Constrained Operation of

Weaving Segments ............................................................................... 24-8Exhibit 24-8. Capacity for Various Weaving Segments............................................ 24-10Exhibit 24-9. Freeway Weaving Worksheet.............................................................. 24-21


24-1 Chapter 24 - Freeway WeavingIntroduction

I. INTRODUCTIONBackground and concepts forthis chapter are given inChapter 13, “FreewayConcepts”

SCOPE OF THE METHODOLOGY

Detailed procedures for the analysis of operations in freeway weaving segments arecontained in this chapter. Guidelines are also given for the application of theseprocedures to weaving segments on multilane highways.

A discussion of basic concepts and definitions is given in Chapter 13, “FreewayConcepts.” This section contains a complete discussion and definition of unconstrainedand constrained operations in weaving segments and of the three types of weavingconfiguration: Type A, Type B, and Type C. An understanding of these concepts anddefinitions is critical to the correct application of the methodology of this chapter and toadequately interpret the results of analysis.

The procedures of this chapter have been assembled from a variety of sources andstudies. The form of the speed prediction algorithm was developed as a result of aresearch project in the 1980s (1). Concepts of configuration type and type of operationwere developed in an earlier study (2) in the 1970s and updated in another study offreeway capacity procedures published in 1979 (3). The final weaving procedures for the1997 HCM are also documented (4). Several other documents describe the developmentof weaving segment analysis procedures (5–7).

LIMITATIONS OF THE METHODOLOGY

The methodology in this chapter does not specifically address the following subjects(without modifications by the analyst):

• Special lanes, such as high-occupancy vehicle lanes, in the weaving segment;• Ramp metering on entrance ramps forming part of the weaving segment;• Specific operating conditions when oversaturated conditions occur;• Effects of speed limits or enforcement practices on weaving segment operations;• Effects of intelligent transportation system technologies on weaving segment

operations;• Weaving segments on collector-distributor roadways;• Weaving segments on urban streets; and• Multiple weaving segments.

The last subject, which has been treated in previous editions of this manual, has beendeleted. Multiple weaving segments must now be divided into appropriate merge,diverge, and simple weaving segments for analysis.

II. METHODOLOGY

The methodology presented in this chapter has five distinct components:• Models predicting the space mean speed (average running speed) of weaving and

nonweaving vehicles in the weaving segment (models are specified for eachconfiguration type and for unconstrained and constrained operations);

• Models describing the proportional use of lanes by weaving and nonweavingvehicles, used to determine whether operations are unconstrained or constrained;

• An algorithm that converts predicted speeds to an average density within theweaving segment;

• Definition of level-of-service (LOS) criteria based on density within the weavingsegment; and

• A model for the determination of the capacity of a weaving segment.Exhibit 24-1 summarizes the methodology for freeway weaving segments.


Chapter 24 - Freeway Weaving 24-2Methodology

EXHIBIT 24-1. FREEWAY WEAVING METHODOLOGY

Volume Adjustment- Peak-hour factor- Heavy vehicles- Driver population

Input- Geometric data- Weaving and nonweaving volumes- Free-flow speed of freeway segment

before and after the weaving segment

Compute flow rates

Establish weaving segmentconfiguration type

Compute unconstrained weaving andnonweaving speeds

Check for constrained-flowoperation

If constrained

If un

cons

train

ed

Compute constrained weaving andnonweaving speeds

Compute average space mean speedwithin the weaving segment

Compute density within theweaving segment

Determine LOS

LOS

The LOS of the weaving segment is determined by comparing the computed densitywith the criteria of Exhibit 24-2. A single LOS is used to characterize total flow in theweaving segment, although it is recognized that in some situations (particularly in casesof constrained operations) nonweaving vehicles may achieve higher-quality operationsthan weaving vehicles.


24-3 Chapter 24 - Freeway WeavingMethodology

EXHIBIT 24-2. LOS CRITERIA FOR WEAVING SEGMENTS

Density (pc/mi/ln)

LOS Freeway Weaving Segment Multilane and Collector-DistributorWeaving Segments

A ≤ 10.0 ≤ 12.0B > 10.0–20.0 > 12.0–24.0C > 20.0–28.0 > 24.0–32.0D > 28.0–35.0 > 32.0–36.0E > 35.0–43.0 > 36.0–40.0F > 43.0 > 40.0

In general, these criteria allow for slightly higher densities at any given level-of-service threshold than on a comparable basic freeway segment or multilane highwaysegment. This follows the philosophy that drivers expect and will accept higher densitieson weaving segments than on basic freeway or multilane highway segments. The LOSE/F boundary does not follow this approach. Rather, it reflects densities that aresomewhat less than those identified for basic freeway or multilane highway segments.Because of the additional turbulence on weaving segments, it is believed that breakdownoccurs at somewhat lower densities than on basic freeway and multilane highwaysegments.

WEAVING SEGMENT PARAMETERS

Exhibit 24-3 illustrates and defines the variables that are used in the analysis ofweaving segments. These variables are used in the algorithms that make up themethodology.

All existing or projected roadway and traffic conditions must be specified whenapplying the methodology. Roadway conditions include length of the segment, numberof lanes, type of configuration under study, and type of terrain or grade conditions. Iffreeway free-flow speed (FFS) is not known, the characteristics of the basic freewaysegment or multilane highway must be specified to allow its determination using thealgorithms of Chapter 21 or 23.

DETERMINING FLOW RATES

All of the models and equations in this chapter are based on peak 15-min flow ratesin equivalent passenger cars per hour. Thus, hourly volumes must be converted to thisbasis using Equation 24-1.

If 15-min flow rates arespecified initially, set the PHFto 1.00 before applying thisconversion

v = VPHF * fHV * f p

(24-1)

wherev = peak 15-min flow rate in an hour (pc/h),V = hourly volume (veh/h),

fHV = heavy-vehicle adjustment factor (from basic freeway segment ormultilane highway methodology), and

fp = driver population factor (from basic freeway segment or multilanehighway methodology).

WEAVING SEGMENT DIAGRAM

After volumes have been converted to flow rates, it is useful to construct a weavingdiagram of the type shown in Exhibit 24-4. All flows are shown as flow rates inequivalent passenger cars per hour, and critical analysis variables are identified andplaced on the diagram. The diagram may now be used as a reference for all inputinformation required in applying the methodology.



EXHIBIT 24-3. PARAMETERS AFFECTING WEAVING SEGMENT OPERATION

vo1 or vo2vw1 or vw2

vw1 or vw2vo1 or vo2

Symbol Definition

L Length of weaving segment (ft)N Total number of lanes in the weaving segmentNw Number of lanes to be used by weaving vehicles if unconstrained operation is to be

achievedNw(max) Maximum number of lanes that can be used by weaving vehicles for a given

configurationNnw Number of lanes used by nonweaving vehicles

v Total flow rate in the weaving segment (pc/h)vo1 Larger of the two outer, or nonweaving, flow rates in the weaving segment (pc/h)

vo2 Smaller of the two outer, or nonweaving, flow rates in the weaving segment (pc/h)

vw1 Larger of the two weaving flow rates in the weaving segment (pc/h)

vw2 Smaller of the two weaving flow rates in the weaving segment (pc/h)

vw Total weaving flow rate in the weaving segment (pc/h) (vw = vw1 + vw2)

vnw Total nonweaving flow rate in the weaving segment (pc/h) (vnw = vo1 + vo2)

VR Volume ratio; the ratio of weaving flow rate to total flow rate in the weaving segment(VR = vw/v)

R Weaving ratio; the ratio of the smaller weaving flow rate to total weaving flow rate(R = vw2/vw)

Sw Speed of weaving vehicles in the weaving segment (mi/h)

Snw Speed of nonweaving vehicles in the weaving segment (mi/h)

S Speed of all vehicles in the weaving segment (mi/h)D Density of all vehicles in the weaving segment (pc/mi/ln)

Ww Weaving intensity factor for prediction of weaving speed

Wnw Weaving intensity factor for prediction of nonweaving speed

EXHIBIT 24-4. CONSTRUCTION AND USE OF WEAVING DIAGRAMS

1500500

300400

A

B

C

D

1500

500300

400

A

B

C

D

Segment and flows

Weaving diagram



WEAVING SEGMENT CONFIGURATIONSee Chapter 13 for diagramsand concepts of the threeweaving segmentconfigurations

Weaving segment configuration is based on the number of lane changes required ofeach weaving movement. A complete discussion of this concept is found in Chapter 13.Exhibit 24-5 may be used to establish configuration type.

EXHIBIT 24-5. DETERMINING CONFIGURATION TYPE

Number of Lane Changes Required by Movement vw2

Number of Lane ChangesRequired by Movement vw1 0 1 ≥ 2

0 Type B Type B Type C1 Type B Type A N/A

≥ 2 Type C N/A N/A

Note:N/A = not applicable; configuration is not feasible.

The three types of geometric configurations are defined as follows:• Type A—Weaving vehicles in both directions must make one lane change to

successfully complete a weaving maneuver.• Type B—Weaving vehicles in one direction may complete a weaving maneuver

without making a lane change, whereas other vehicles in the weaving segment must makeone lane change to successfully complete a weaving maneuver.

• Type C—Weaving vehicles in one direction may complete a weaving maneuverwithout making a lane change, whereas other vehicles in the weaving segment must maketwo or more lane changes to successfully complete a weaving maneuver.

DETERMINING WEAVING AND NONWEAVING SPEEDS

The heart of the weaving segment analysis procedure is the prediction of space meanspeeds of weaving and nonweaving flows within the weaving segment. They arepredicted separately because under some conditions they can be quite dissimilar, and theanalyst must be aware of this.

The algorithm for prediction of average weaving and nonweaving speeds may begenerally stated by Equation 24-2.

Si = Smin +Smax − Smin

1 +W i(24-2)

whereSi = average speed of weaving (i = w) or nonweaving (i = nw) vehicles

(mi/h),Smin = minimum speed expected in a weaving segment (mi/h),Smax = maximum speed expected in a weaving segment (mi/h), and

Wi = weaving intensity factor for weaving (i = w) and nonweaving (i = nw)flows.

For the purposes of these procedures, the minimum speed, Smin, is set at 15 mi/h.The maximum speed, Smax, is taken to be the average free-flow speed of the freeway

segments entering and leaving the weaving segment plus 5 mi/h. The addition of 5 mi/hto the free-flow speed adjusts for the tendency of the algorithm to underpredict highspeeds. Setting the minimum and maximum speeds in this way constrains the algorithmto a reasonable prediction range. With these assumptions incorporated, the speedprediction is given by Equation 24-3.

Si =15 + SFF −101 +W i

(24-3)



where SFF is the average free-flow speed of the freeway segments entering and leavingthe weaving segment (mi/h).

Initial estimates of speed are always based on the assumption of unconstrainedoperation. This assumption is later tested, and speeds are recomputed if operations turnout to be constrained.

Attributes of weavingsegments captured bythe model

The combination of Equations 24-2 and 24-3 yields sensitivities that are consistentwith observed operations of weaving segments.

• As the length of the weaving segment increases, speeds also increase, and theintensity of lane changing declines.

• As the proportion of weaving vehicles in total flow (VR) increases, speedsdecrease, reflecting the increased turbulence caused by higher proportions of weavingvehicles in the traffic stream.

• As average total flow per lane (v/N) increases, speeds decrease, reflecting moreintense demand.

• Constrained operations yield lower weaving speeds and higher nonweaving speedsthan unconstrained operations. This reflects the fact that weaving vehicles areconstrained to less space than equilibrium would require, whereas nonweaving vehicleshave correspondingly more than their equilibrium share of space. In Exhibit 24-6, this isreflected by differences in the constant a.

EXHIBIT 24-6. CONSTANTS FOR COMPUTATION OF WEAVING INTENSITY FACTORS

General Form

W =

a(1+ VR)bvN

c

Ld

Constants for Weaving Speed, Sw Constants for Nonweaving Speed, Snw

a b c d a b c d

Type A Configuration

Unconstrained 0.15 2.2 0.97 0.80 0.0035 4.0 1.3 0.75Constrained 0.35 2.2 0.97 0.80 0.0020 4.0 1.3 0.75

Type B Configuration


Type C Configuration


• Type B configurations are the most efficient for handling large weaving flows.Weaving speeds of such flows are higher than for Type A and C configurations of equallength and width.

• The sensitivity of speeds to length is greatest for Type A configurations, becauseweaving vehicles are often accelerating or decelerating as they traverse the weavingsegment.

• The sensitivity of nonweaving speeds to the volume ratio (VR) is greatest for TypeB and C configurations. Because these configurations can accommodate higherproportions of weaving vehicles and because each has a through lane for one weavingmovement, nonweaving vehicles are more likely to share lanes with weaving vehiclesthan in Type A configurations, where the opportunity to segregate is greater.

The last point is important and serves to highlight the essential difference betweenType A configurations (particularly ramp-weaves) and others (Types B and C). Becauseall weaving vehicles must cross a crown line in Type A segments, weaving vehicles tendto concentrate in the two lanes adjacent to the crown line, whereas nonweaving vehicles



gravitate to outer lanes. Thus there is substantially more segregation of weaving andnonweaving flows in Type A configurations.

This difference makes Type A segments behave somewhat differently from otherconfigurations. Speeds tend to be higher in Type A segments than in Types B or C giventhe same length, width, and demand flows. However, this does not suggest that Type Asegments always operate better than Types B or C for similar lengths, widths, and flows.Type A segments have more severe restrictions on the amount of weaving traffic that canbe accommodated than do other configurations.

Determining Weaving Intensity

The weaving intensity factors (Ww and Wnw) are a measure of the influence of

weaving activity on the average speeds of both weaving and nonweaving vehicles. Thesefactors are computed by Equation 24-4.

W i =a(1 +VR )b

vN

c

Ld (24-4)

whereWi = weaving intensity factors for weaving (i = w) and nonweaving (i = nw)

flows;VR = volume ratio;

v = total flow rate in the weaving segment (pc/h);N = total number of lanes in the weaving segment;L = length of the weaving segment (ft); and

a, b, c, d = constants of calibration.

Constants for Computing Weaving Intensity Factors

Constants for computation of weaving intensity factors (a, b, c, d) are given inExhibit 24-6. Values for these constants vary on the basis of three factors:

• Whether the average speed prediction is for weaving or nonweaving vehicles,• Configuration type (A, B, or C), and• Whether the operation is unconstrained or constrained.

DETERMINING TYPE OF OPERATION

The determination of whether a particular weaving segment is operating in anunconstrained or constrained state is based on the comparison of two variables that aredefined in Chapter 13:

Nw = number of lanes that must be used by weaving vehicles to achieveequilibrium or unconstrained operation, and

Nw(max) = maximum number of lanes that can be used by weaving vehicles for agiven configuration.

Definition of constrainedweaving segment

Fractional values for lane use requirements of weaving vehicles may occur becauseweaving and nonweaving vehicles share some lanes. Cases for which Nw < Nw(max) are

unconstrained because there are no impediments to weaving vehicles using the number oflanes required for equilibrium. If Nw ≥ Nw(max), weaving vehicles are constrained tousing Nw(max) lanes and therefore cannot occupy as much of the roadway as would be

needed to establish equilibrium operations. Exhibit 24-7 provides algorithms for thecomputation of Nw and shows the values of Nw(max), which are discussed more fully in

Chapter 13.



EXHIBIT 24-7. CRITERIA FOR UNCONSTRAINED VERSUS CONSTRAINED OPERATION OFWEAVING SEGMENTS

Configuration Number of Lanes Required for Unconstrained Operation, Nw Nw(max)

Type A 0.74(N) VR0.571L0.234/Sw0.438 1.4

Type B N[0.085 + 0.703VR + (234.8/L) – 0.018(Snw – Sw)] 3.5

Type C N[0.761 + 0.047VR – 0.00011 – 0.005(Snw – Sw)] 3.0a

Note:a. For two-sided weaving segments, all freeway lanes may be used by weaving vehicles.

The equations of Exhibit 24-7 rely on the prediction of unconstrained weaving andnonweaving speeds. The equations take these results and predict the number of lanesweaving vehicles would have to occupy to achieve unconstrained speeds. If the resultindicates that constrained operations exist, speeds must be recomputed using constrainedequations.

The limit on maximum number of weaving lanes, Nw(max), is most restrictive for

Type A segments and reflects the need for weaving vehicles to cluster in the two lanesadjacent to the crown line. The through weaving lane in Type B and C configurationsprovides for greater occupancy of lanes by weaving vehicles.

Type A segments have another unusual, but understandable, characteristic. As thelength of a Type A segment increases, constrained operation is more likely to result. Asthe length increases, the speed of weaving vehicles is also able to increase. Thus,weaving vehicles use more space as length increases, and the likelihood of requiring morethan the maximum of 1.4 lanes to achieve equilibrium also increases.

Types B and C show the opposite trend. Increasing length has less effect on weavingspeed than in Type A configurations. First, acceleration and deceleration from low-speedramps are less of an issue for Types B and C, which are, by definition, major weavingsegments. Second, the substantial mixing of weaving and nonweaving vehicles in thesame lanes makes the resulting speeds less sensitive to length. In Type B and Csegments, the proportion of lanes needed by weaving vehicles to achieve unconstrainedoperation decreases as length increases.

The analyst should note that under extreme conditions (high VR, short length), theequation for Type B segments can predict values of Nw > N. While this is not practical

and reflects portions of the research database with sparse field data, it may always betaken to indicate constrained operations.

DETERMINING WEAVING SEGMENT SPEED

Once speeds have been estimated and the type of operation determined (which maycause a recomputation of estimated speeds), the average space mean speed of all vehiclesin the segment is computed according to Equation 24-5.

S = v

vw

Sw

+ vnw

Snw

(24-5)

whereS = space mean speed of all vehicles in the weaving segment (mi/h),

Sw = space mean speed of weaving vehicles in the weaving segment (mi/h),Snw = space mean speed of nonweaving vehicles in the weaving segment

(mi/h),v = total flow rate in the weaving segment (pc/h),

vw = weaving flow rate in the weaving segment (pc/h), andvnw = nonweaving flow rate in the weaving segment (pc/h).

mdixon

Note

should be -0.00011*L



DETERMINING DENSITY

The average speed for all vehicles may be used to compute density for all vehicles inthe weaving segment as shown in Equation 24-6.

D =

vN

S(24-6)

where D is the average density for all vehicles in the weaving segment (pc/mi/ln).

DETERMINING WEAVING SEGMENT CAPACITYCapacity of a weavingsegment defined

The capacity of a weaving segment is any combination of flows that causes thedensity to reach the LOS E/F boundary condition of 43.0 pc/mi/ln for freeways or 40.0pc/mi/ln for multilane highways. Thus, capacity varies with a number of variables:configuration, number of lanes, free-flow speed of the freeway or multilane highway,length, and volume ratio. Because of the form of predictive algorithms, generation of asimple closed-form solution for capacity given the specification of the other variables isnot possible. Rather, a trial-and-error process must be used.

Capacity attributes of weavingsegments

Exhibit 24-8 shows tabulated values of weaving segment capacity for a number ofsituations. As a rough estimate, straight-line interpolation may be used for intermediatevalues. The tabulated capacities reflect some other limitations on weaving segmentoperations that reflect field observations:

• The capacity of a weaving segment may never exceed the capacity of a similarbasic freeway or multilane highway segment.

• Field studies suggest that weaving flow rates should not exceed the followingvalues: 2,800 pc/h for Type A, 4,000 pc/h for Type B, and 3,500 pc/h for Type Cconfigurations. Even though higher weaving flows have been observed, they are likely tocause failure regardless of the results of analysis using the procedures in this manual.

• Field studies indicate that there are also limitations on the proportion of weavingflow (VR) that can be accommodated by various configurations: 1.00, 0.45, 0.35, or 0.20for Type A with two, three, four, or five lanes, respectively; 0.80 for Type B; and 0.50 forType C. At higher volume ratios, stable operations may still occur, but operations will beworse than those anticipated by the methodology, and failure could occur.

• For Type C segments, the weaving ratio, R, should not exceed 0.40, with the largerweaving flow being in the direction of the through weaving lane. At higher weavingratios or where the dominant weaving flow is not in the direction of the through weavinglane, stable operations may still occur, but operations will be worse than those estimatedby the methodology. Breakdown may occur in some cases.

• The maximum length for which weaving analysis is conducted is 2,500 ft for allconfiguration types. Beyond these lengths, merge and diverge areas are consideredseparately using the methodology of Chapter 25, “Ramps and Ramp Junctions.”

As noted previously, the capacity of a weaving segment is represented by any set ofconditions that results in an average density of 43 pc/mi/ln (for freeways) or 40 pc/mi/ln(for multilane highways). Thus, capacity varies with the configuration, the length andwidth of the weaving segment, the proportion of total flow that weaves (VR), and thefree-flow speed of the freeway. For any given set of conditions, the algorithms describedherein must be solved iteratively to find capacity.



EXHIBIT 24-8. CAPACITY FOR VARIOUS WEAVING SEGMENTS

(A) Type A Weaving Segments—75-mi/h Free-Flow Speed

Length of Weaving Segment (ft)

Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,030 6,800 7,200b 7,200b 7,200b

0.20 5,460 6,230 6,680 7,010 7,200b

0.30 4,990 5,740 6,210 6,530 6,7900.40 4,620 5,340 5,480c 5,790c 6,040c

0.45d 4,460 4,840c 5,240c 5,540c 5,780c

Four-Lane Segments

0.10 8,040 9,070 9,600b 9,600b 9,600b

0.20 7,280 8,300 8,910 9,350 9,600b

0.30 6,660 7,520c 8,090c 8,520c 8,830c

0.35e 6,250c 7,120c 7,690c 8,000f 8,000f

Five-Lane Segments

0.10 10,050 11,340 12,000b 12,000b 12,000b

0.20g 9,100 10,540c 11,270c 11,790c 12,000b

(B) Type A Weaving Segments—65-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 5,570 6,230 6,620 6,890 7,050b

0.20 5,070 5,740 6,130 6,410 6,6200.30 4,670 5,320 5,720 6,000 6,2200.40 4,330 4,950 5,090c 5,360c 5,570c

0.45d 4,190 4,520c 4,870c 5,140c 5,340c

Four-Lane Segments

0.10 7,430 8,310 8,830 9,190 9,400b

0.20 6,760 7,660 8,170 8,550 8,8300.30 6,180c 6,970c 7,470c 7,830c 8,110c

0.35e 5,870c 6,620c 7,120c 7,470c 7,760c

Five-Lane Segments

0.10 9,290 10,390 11,040 11,490 11,750b

0.20g 8,450 9,700c 10,320c 10,760c 11,090c

Notes:Refer to the last page of Exhibit 24-8.



EXHIBIT 24-8 (CONTINUED). CAPACITY FOR VARIOUS WEAVING SEGMENTS

(C) Type A Weaving Segments—60-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 5,330 5,940 6,280 6,520 6,7000.20 4,870 5,480 5,850 6,100 6,2800.30 4,490 5,100 5,460 5,720 5,9200.40 4,180 4,570c 4,880c 5,140c 5,330c

0.45d 4,040 4,360c 4,680c 4,920c 5,120c

Four-Lane Segments

0.10 7,110 7,920 8,380 8,690 8,9300.20 6,500 7,310 7,800 8,140 8,420c

0.30 5,960c 6,680c 7,140c 7,470c 7,710c

0.35g 5,660c 6,370c 6,810c 7,140c 7,400c

Five-Lane Segments

0.10 8,880 9,900 10,480 10,870 11,500b

0.20f 8,120 9,270c 9,830c 10,220c 10,530c

(D) Type A Weaving Segments—55-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 5,080 5,630 5,940 6,140 6,3000.20 4,660 5,210 5,550 5,770 5,9400.30 4,300 4,850 5,190 5,430 5,6100.40 4,010 4,360c 4,670c 4,890c 5,070c

0.45d 3,880 4,180c 4,480c 4,700c 4,880c

Four-Lane Segments

0.10 6,780 7,500 7,920 8,190 8,4000.20 6,210 6,950 7,400 7,690 7,940c

0.30 5,710c 6,380c 6,780c 7,090c 7,310c

0.35e 5,440c 6,090c 6,490c 6,800c 7,030c

Five-Lane Segments

0.10 8,480 9,380 9,900 10,240 10,5100.20g 7,960c 8,800c 9,320c 9,660c 9,930c





(E) Type B Weaving Segments—75-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 7,200b 7,200b 7,200b 7,200b 7,200b

0.20 6,810 7,200b 7,200b 7,200b 7,200b

0.30 6,120 6,670 7,000 7,230 7,200b

0.40 5,540 6,090 6,430 6,650 6,8300.50 5,080 5,620 5,940 6,170 6,3600.60 4,750 5,250 5,560 5,790 5,9700.70 4,180 4,980 5,290 5,510 5,6900.80h 3,890 4,810 5,000f 5,000f 5,000f

Four-Lane Segments

0.10 9,600b 9,600b 9,600b 9,600b 9,600b

0.20 9,090 9,600b 9,600b 9,600b 9,600b

0.30 8,160 8,900 9,330 9,600b 9,600b

0.40 7,390 8,120 8,570 8,860 9,1100.50 6,660c 7,490 7,920 8,000f 8,000f

0.60 6,060c 6,670f 6,670f 6,670f 6,670f

0.70 5,570c 5,760f 5,760f 5,760f 5,760f

0.80h 5,000f 5,000f 5,000f 5,000f 5,000f

Five-Lane Segments

0.10 12,000b 12,000b 12,000b 12,000b 12,000b

0.20 11,360 12,000b 12,000b 12,000b 12,000b

0.30 10,200 11,120 11,670 12,000b 12,000b

0.40 9,250c 10,000f 10,000f 10,000f 10,000f

0.50 8,000f 8,000f 8,000f 8,000f 8,000f

0.60 6,670f 6,670f 6,670f 6,670f 6,670f

0.70 5,760f 5,760f 5,760f 5,760f 5,760f

0.80h 5,000f 5,000f 5,000f 5,000f 5,000f





(F) Type B Weaving Segments—65-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,930 7,050b 7,050b 7,050b 7,050b

0.20 6,220 6,670 6,930 7,050b 7,050b

0.30 5,610 6,090 6,360 6,560 6,7000.40 5,110 5,590 5,870 6,070 6,2200.50 4,710 5,170 5,460 5,650 5,8100.60 4,410 4,850 5,120 5,320 5,4700.70 4,190 4,620 4,880 5,070 5,2300.80h 3,650c 4,460 4,720 4,920 5,000f

Four-Lane Segments

0.10 9,240 9,400b 9,400b 9,400b 9,400b

0.20 8,300 8,900 9,240 9,400b 9,400b

0.30 7,490 8,120 8,480 8,740 8,9300.40 6,810 7,450 7,830 8,090 8,3000.50 6,180c 6,900 7,280 7,540 7,7400.60 5,640c 6,470 6,670f 6,670f 6,670f

0.70 5,210c 5,730 5,760f 5,760f 5,760f

0.80h 4,870c 5,000f 5,000f 5,000f 5,000f

Five-Lane Segments

0.10 11,550 11,750b 11,750b 11,750b 11,750b

0.20 1,0370 11,120 11,550 11,750b 11,750b

0.30 9,360 10,150 10,610 10,930 11,1700.40 8,540c 9,320 9,790 10,000f 10,000f

0.50 7,720c 8,000f 8,000f 8,000f 8,000f

0.60 6,670f 6,670f 6,670f 6,670f 6,670f

0.70 5,760f 5,760f 5,760f 5,760f 5,760f

0.80h 5,000f 5,000f 5,000f 5,000f 50,00f





(G) Type B Weaving Segments—60-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,540 6,890 6,900b 6,900b 6,900b

0.20 5,900 6,320 6,540 6,700 6,8100.30 5,340 5,780 6,040 6,210 6,3400.40 4,880 5,320 5,570 5,760 5,9000.50 4,520 4,940 5,200 5,380 5,5200.60 4,230 4,640 4,890 5,080 5,2200.70 4,020 4,420 4,670 4,850 4,9900.80h 3,520c 4,280 4,520 4,700 4,840

Four-Lane Segments

0.10 8,720 9,190 9,200b 9,200b 9,200b

0.20 7,860 8,420 8,730 8,930 9,0900.30 7,120 7,710 8,050 8,280 8,4500.40 6,510 7,090 7,430 7,690 7,8600.50 5,920c 6,590 6,930 7,180 7,3700.60 5,420c 6,190 6,520 6,670f 6,670f

0.70 5,020c 5,520c 5,760f 5,760f 5,760f

0.80h 4,700c 5,000f 5,000f 5,000f 5,000f

Five-Lane Segments

0.10 10,900 11,490 11,500b 11,500b 11,500b

0.20 9,830 10,530 10,910 11,170 11,3600.30 8,910 9,640 10,070 10,350 10,5600.40 8,170c 8,860 9,290 9,610 9,8300.50 7,400c 8,000f 8,000f 8,000f 8,000f

0.60 6,670f 6,670f 6,670f 6,670f 6,670f

0.70 5,760f 5,760f 5,760f 5,760f 5,760f

0.80h 5,000f 5,000f 5,000f 5,000f 5,000f





(H) Type B Weaving Segments—55-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,160 6,470 6,630 6,750b 5,750b

0.20 5,570 5,950 6,160 6,300 5,4000.30 5,070 5,460 5,690 5,850 5,9600.40 4,640 5,040 5,280 5,450 5,5700.50 4,310 4,700 4,940 5,100 5,2400.60 4,050 4,420 4,660 4,830 4,9500.70 3,870 4,230 4,450 4,620 5,7500.80h 3,390c 4,090 4,310 4,480 4,610

Four-Lane Segments

0.10 8,210 8,620 8,850 9,000b 9,000b

0.20 7,430 7,930 8,210 8,400 8,5400.30 6,760 7,290 7,590 7,800 7,9500.40 6,190 6,730 7,040 7,260 7,4300.50 5,660c 6,260 6,590 6,800 6,9900.60 5,210c 5,900 6,210 6,440 6,6100.70 4,830c 5,280c 5,760f 5,760f 5,760f

0.80h 4,520c 4,950c 5,000f 5,000f 5,000f

Five-Lane Segments

0.10 10,260 10,780 11,060 11,250b 11,250b

0.20 9,290 9,920 10,260 10,500 10,6700.30 8,450 9,110 9,490 9,750 9,9400.40 7,770c 8,410 8,810 9,080 9,2800.50 7,080c 7,830 8,000f 8,000f 8,000f

0.60 6,520c 6,670f 6,670f 6,670f 6,670f

0.70 5,760f 5,760f 5,760f 5,760f 5,760f

0.80h 5,000f 5,000f 5,000f 5,000f 5,000f





(I) Type C Weaving Segments—75-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 7,200b 7,200b 7,200b 7,200b 7,200b

0.20 6,580 7,200b 7,200b 7,200b 7,200b

0.30 5,880 6,530 6,920 7,190 7,200b

0.40 5,330 5,960 6,340 6,610 6,8300.50i 4,890 5,490 5,870 6,140 6,350

Four-Lane Segments

0.10 9,600b 9,600b 9,600b 9,600b 9,600b

0.20 8,780 9,600b 9,600b 9,600b 9,600b

0.30 7,850 8,710 9,220 9,590 9,600b

0.40 7,110 7,950 8,450 8,750 8,7500.50i 6,520 7,000f 7,000f 7,000f 7,000f

Five-Lane Segments

0.10 12,000b 12,000b 12,000b 12,000b 12,000b

0.20 11,510c 12,000b 12,000b 12,000b 12,000b

0.30 10,120c 11,160c 11,530 11,670f 11,670f

0.40 8,750f 8,750f 8,750f 8,750f 8,750f

0.50i 7,000f 7,000f 7,000f 7,000f 7,000f

(J) Type C Weaving Segments—65-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,740 7,050b 7,050b 7,050b 7,050b

0.20 6,010 6,570 6,870 7,050b 7,050b

0.30 5,420 5,970 6,310 6,530 6,7000.40 4,930 5,480 5,810 6,040 6,2300.50i 4,540 5,070 5,400 5,640 5,820

Four-Lane Segments

0.10 8,980 9,400b 9,400b 9,400b 9,400b

0.20 8,020 8,760 9,160 9,400b 9,400b

0.30 7,230 7,970 8,410 8,710 8,9300.40 6,570 7,310 7,750 8,060 8,3100.50i 6,060 6,760 7,000f 7,000f 7,000f

Five-Lane Segments

0.10 11,500b 11,500b 11,500b 11,500b 11,500b

0.20 10,500c 11,320c 11,460 11,500b 11,500b

0.30 9,320c 10,180c 10,520 10,890 11,1700.40 8,330c 8,750f 8,750f 8,750f 8,750f

0.50i 7,000f 7,000f 7,000f 7,000f 7,000f





(K) Type C Weaving Segments—60-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,380 6,830 6,900b 6,900b 6,900b

0.20 5,730 6,230 6,490 6,680 6,8300.30 5,170 5,690 5,990 6,190 6,3500.40 5,720 5,240 5,540 5,740 5,9100.50i 4,360 4,850 5,160 5,370 5,540

Four-Lane Segments

0.10 8,500 9,100 9,200b 9,200b 9,200b

0.20 7,640 8,310 8,660 8,910 9,1000.30 6,900 7,590 7,990 8,260 8,4700.40 6,300 6,990 7,380 7,660 7,8800.50i 5,820 6,470 6,880 7,000f 7,000f

Five-Lane Segments

0.10 11,250 11,500b 11,500b 11,500b 11,500b

0.20 9,980c 10,720c 10,820 11,140 11,3800.30 8,880c 9,680c 9,980 10,330 10,5900.40 7,980c 8,750f 8,750f 8,750f 8,750f

0.50i 7,000f 7,000f 7,000f 7,000f 7,000f





(L) Type C Weaving Segments—55-mi/h Free-Flow Speed


Volume Ratio,VR

500 1,000 1,500 2,000 2,500a

Three-Lane Segments

0.10 6,010 6,400 6,610 6,740 6,750b

0.20 5,420 5,870 6,120 6,270 6,4000.30 4,930 5,380 5,650 5,850 5,9700.40 4,510 4,980 5,250 5,450 5,5900.50i 4,180 4,630 4,900 5,100 5,250

Four-Lane Segments

0.10 8,020 8,540 8,810 8,980 9,000b

0.20 7,230 7,830 8,160 8,360 8,5400.30 6,570 7,180 7,540 7,800 7,9700.40 6,020 6,640 7,000 7,260 7,4500.50i 5,570 6,180 6,540 6,800 7,000f

Five-Lane Segments

0.10 10,560c 11,100c 11,020 11,230 11,250b

0.20 9,420c 10,090c 10,200 10,460 10,6700.30 8,430c 9,160c 9,420 9,750 9,9600.40 7,610c 8,350c 8,750f 8,750f 8,750f

0.50i 6,930c 7,000f 7,000f 7,000f 7,000f

Notes:a. Weaving segments longer than 2,500 ft are treated as isolated merge and diverge areas using the procedures of Chapter 25,“Ramps and Ramp Junctions.”b. Capacity constrained by basic freeway capacity.c. Capacity occurs under constrained operating conditions.d. Three-lane Type A segments do not operate well at volume ratios greater than 0.45. Poor operations and some localqueuing are expected in such cases.e. Four-lane Type A segments do not operate well at volume ratios greater than 0.35. Poor operations and some local queuingare expected in such cases.f. Capacity constrained by maximum allowable weaving flow rate: 2,800 pc/h (Type A), 4,000 (Type B), 3,500 (Type C).g. Five-lane Type A segments do not operate well at volume ratios greater than 0.20. Poor operations and some local queuingare expected in such cases.h. Type B weaving segments do not operate well at volume ratios greater than 0.80. Poor operations and some local queuingare expected in such cases.i. Type C weaving segments do not operate well at volume ratios greater than 0.50. Poor operations and some local queuingare expected in such cases.

It is possible to do so using a spreadsheet properly programmed for such iteration.Capacities have been determined for freeway facilities and are shown in Exhibit 24-8.These capacities represent maximum 15-min flow rates under equivalent base conditionsand are rounded to the nearest 10 pc/h. To find the capacity under a given set ofprevailing conditions Equation 24-7 is used.

c = cb * fHV * fp (24-7)

wherec = capacity under prevailing conditions stated as a flow rate for the peak

15 min of the hour (veh/h),cb = capacity under base conditions stated as a flow rate for the peak 15 min

of the hour in Exhibit 24-8 (pc/h),fHV = heavy-vehicle adjustment factor (basic freeway segments or multilane

highways), andfp = driver population factor (basic freeway segments or multilane

highways).



If a capacity in terms of an hourly volume is desired, it may be computed using Equation24-8.

ch = c * PHF (24-8)

wherech = capacity under prevailing conditions expressed as an hourly volume

(veh/h), andPHF = peak-hour factor.

MULTIPLE WEAVING SEGMENTS

When a series of closely spaced merge and diverge areas creates several sets ofweaving movements (between different merge-diverge pairs) that share the same segmentof the roadway, a multiple weaving segment is created. In previous editions of thismanual, a specific procedure for analysis of two-segment multiple weaving segments,involving two sets of overlapping weaving movements, was presented. Although itconstituted a logical approach, it did not address cases where three or more sets ofweaving movements overlapped, and it was not extensively supported by field data.

It is recommended that such cases be segregated into merge areas, diverge areas, andsimple weaving segments, as appropriate, and that each segment be analyzed accordingly.Chapter 22 contains information on this procedure.

COLLECTOR-DISTRIBUTOR ROADWAYS

A common design practice often results in weaving segments that occur on collector-distributor roadways that are part of a freeway or multilane highway interchange.Although the procedures of this chapter could be applied to such cases (using appropriatefree-flow speeds), whether the LOS criteria specified herein should apply is unclear.Because many such segments operate at low speeds and correspondingly high densities,stable operations may exist beyond the maximum densities specified herein, which areintended for freeway or multilane highway weaving.

III. APPLICATIONSGuidelines for required inputsand estimated values aregiven in Chapter 13

The methodology of this chapter can be used to analyze the capacity and LOS offreeway weaving segments. First, the analyst identifies the primary output. Primaryoutputs typically solved for in a variety of applications include LOS, number of lanesrequired (N), weaving segment length required (L), and weaving segment configurationtype (Type). Performance measures related to density and speed are also achievable butare considered secondary outputs.

Second, the analyst must identify the default values or estimated values for use in theanalysis. Basically, the analyst has three sources of input data:

1. Default values found in this manual,2. Estimates and locally derived default values developed by the analyst, and3. Values derived from field measurements and observation.

For each of the input variables, a value must be supplied to calculate the outputs, bothprimary and secondary.

A common application of the method is to compute the LOS of an existing or achanged segment in the near term or distant future. This type of application is termedoperational, and its primary output is LOS, with secondary outputs for density and speed.Another application is to check the adequacy of or to recommend the required number oflanes, weaving segment length, or weaving configuration given the volume or flow rateand LOS goal. This application is termed design since its primary outputs are geometric


Chapter 24 - Freeway Weaving 24-20Applications

attributes of the weaving segment. Other outputs from this application include speed anddensity.

Another general type of analysis is termed planning. These analyses use estimates,HCM default values, and local default values as inputs in the calculation. LOS orweaving segment attributes can be determined as outputs, along with the secondaryoutputs of density and speed. The difference between planning analysis and operationalor design analysis is that most or all of the input values in planning come from estimatesor default values, whereas the operational and design analyses use field measurements orknown values for inputs. Note that for each of the analyses, FFS of the weaving segment,either measured or estimated, is required as an input for the computation.

COMPUTATIONAL STEPS

The worksheet for freeway weaving computations is shown in Exhibit 24-9. Theworksheet is also included in Appendix A. For all applications, the analyst providesgeneral information and site information.

Operational (LOS) For operational (LOS) analysis, all required input data are entered as input. Afterconverting volumes to flow rates, the unconstrained weaving intensity factor is used toestimate weaving and nonweaving speeds. The number of lanes weaving vehicles mustoccupy to achieve unconstrained operation is determined. If this value is less than themaximum number of lanes, then unconstrained flow exists, and previously computedspeeds will apply to the analysis. If the number of lanes required for unconstrainedoperation is greater than or equal to the maximum number of lanes, then weaving andnonweaving speeds must be computed for constrained operation. Then the space meanspeed for all vehicles in the weaving segment is computed followed by density. Finally,level of service is determined using the density value for the weaving segment.

Design (N, L, Type) The objective of design (N, L, Type) analysis is to estimate the length of a weavingsegment, the number of lanes, or weaving segment configuration type given volumes andfree-flow speed. A desired level of service is stated and entered in the worksheet. Then aweaving segment length, number of lanes, and configuration type are assumed, and theprocedure for operational (LOS) analysis is performed. The level-of-service result withthe assumed parameters is then compared with the desired level of service. If the desiredlevel of service is not met, a new combination of parameter values is assumed. Theseiterations are continued until the desired level of service is achieved.

PLANNING APPLICATIONSPlanning (LOS)

Planning (N, L, Type)The two planning applications, planning (LOS) and planning (N, L, Type), directly

correspond to procedures described for operational and design analysis.The first criterion that categorizes these as planning applications is the use of

estimates, HCM default values, or local default values on the input side of the calculation.Another factor that defines a given application as planning is the use of annual averagedaily traffic (AADT) to estimate directional design-hour volume (DDHV). Guidelinesfor calculating DDHV are given in Chapter 8. The analyst typically has few, if any, ofthe input values required to perform planning applications. More information on the useof default values is contained in Chapter 13.

ANALYSIS TOOLS

The worksheet shown in Exhibit 24-9 and provided in Appendix A can be used toperform operational (LOS), design (N, L, Type), planning (LOS), and planning (N, L,Type) analyses.


24-21 Chapter 24 - Freeway WeavingApplications

EXHIBIT 24-9. FREEWAY WEAVING WORKSHEET

FREEWAY WEAVING WORKSHEET

General Information Site Information

Analyst _________________________ Freeway/Direction of Travel _______________________

Agency or Company _________________________ Weaving Segment Location _______________________

Date Performed _________________________ Jurisdiction _______________________

Analysis Time Period _________________________ Analysis Year _______________________

� Operational (LOS) � Design (N, L, Type) � Planning (LOS) � Planning (N, L, Type)

Inputs

Sketch (show lanes, L, vo1, vo2, vw1, vw2)

Conversion to pc/h Under Base Conditions

(pc/h) AADT K D V PHF % HV fHV fp(veh/day) (veh/h)

vo1vo2vw1vw2vwvnwv

Weaving and Nonweaving Speeds

Unconstrained ConstrainedWeaving (i = w) Nonweaving (i = nw) Weaving (i = w) Nonweaving (i = nw)

a (Exhibit 24-6)b (Exhibit 24-6)c (Exhibit 24-6)d (Exhibit 24-6)

Weaving intensity factor, Wi

Weaving and nonweaving speeds, Si (mi/h)

Number of lanes required for unconstrained operation, Nw (Exhibit 24-7) __________Maximum number of lanes, Nw(max) (Exhibit 24-7) __________� If Nw < Nw(max) unconstrained operation � If Nw ≥ Nw(max) constrained operation

Weaving Segment Speed, Density, Level of Service, and Capacity

Weaving segment speed, S (mi/h)

Weaving segment density, D (pc/mi/ln)

Level of service, LOS (Exhibit 24-2)Capacity for base condition, cb (pc/h)(Exhibit 24-8)Capacity as a 15-min flow rate, c (veh/h)c = cb * fHV * fpCapacity as a full-hour volume, ch (veh/h)ch = c(PHF)

Freeway free-flow speed, SFF = _________ mi/h

Weaving number of lanes, N = _________

Weaving segment length, L _________ ft

Terrain � Level � Rolling

Weaving type � Type A � Type B � Type C

Volume ratio, VR = _________

Weaving ratio, R = _________

a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =


Chapter 24 - Freeway Weaving 24-22Example Problems

IV. EXAMPLE PROBLEMS

Problem No. Description Application

1 Determine level of service of a major weaving segment Operational (LOS)2 Determine level of service of a ramp-weaving segment Operational (LOS)3 Determine level of service of a ramp-weaving segment with

constrained operationOperational (LOS)

4 Design a major weaving segment for a desired level of service Design (N, L, Type)5 Design a weaving segment using sensitivity analysis Planning (N, L, Type)


24-23 Chapter 24 - Freeway WeavingExample Problems

EXAMPLE PROBLEM 1

The Weaving Segment A major weaving segment on an urban freeway as shown onthe worksheet.

The Question What are the level of service and capacity of the weaving segment?

The Facts√ Volume (A-C) = 1,815 veh/h, √ PHF = 0.91,√ Volume (A-D) = 692 veh/h, √ Level terrain,√ Volume (B-C) = 1,037 veh/h, √ Drivers are regular commuters,√ Volume (B-D) = 1,297 veh/h, √ FFS = 65 mi/h for freeway, and√ 10 percent trucks, √ Weaving segment length = 1,500 ft.

Comments√ Use Chapter 23, “Basic Freeway Segments,” to identify fHV and fp.

Outline of Solution All input parameters are known, so no default values are required.Demand volumes are converted to flow rates, and weaving configuration type isdetermined. Weaving and nonweaving speeds are computed and used to determineweaving segment speed. The density in the weaving segment is calculated, and level ofservice is determined. Finally, capacity is determined.

Steps1. Convert volume (veh/h) to flow

rate (pc/h) (use Equation24-1).

v = V(PHF)(fHV )(fp )

v(A-C) = 1,815

(0.91)(0.952)(1.000)= 2,095 pc/h

v(A-D) = 799 pc/hv(B-C) = 1,197 pc/hv(B-D) = 1,497 pc/h

1a. Determine fp (use Chapter23).

fp = 1.000

1b. Determine fHV (use Chapter23). fHV = 1

1+ PT (ET − 1) + PR(ER − 1)

fHV = 11+ 0.10(1.5 − 1) + 0

= 0.952

2. Determine weaving segmentconfiguration type (use Exhibit24-5).

Type B (Movement A-D requires one lane change;Movement B-C requires none)

3. Compute critical variables. vw = 1,197 + 799 = 1,996 pc/hvnw = 2,095 + 1,497 = 3,592 pc/hv = 1,996 + 3,592 = 5,588 pc/h

VR = 1,9965,588

= 0.357

R = 7991,996

= 0.400



4. Compute weaving andnonweaving speeds assumingunconstrained operation (useExhibit 24-6 and Equations24-3 and 24-4).

Wi =a(1+ VR)b

vN

c

(L)d Si = 15 + SFF − 101+ Wi

Ww =0.08(1+ 0.357)2.2 5,588

4

0.70

(1,500)0.50 = 0.643

Wnw =0.0020(1+ 0.357)6.0 5,588

4

1.0

(1,500)0.50 = 0.450

Sw = 15 + 65 − 101+ 0.643

= 48.5 mi/h

Snw = 15 + 65 − 101+ 0.450

= 52.9 mi/h

5. Check type of operation (useExhibit 24-7).

Nw = 4[0.085 + 0.703(0.357) + (234.8/1,500)– 0.018(52.9 – 48.5)] = 1.65Nw(max) = 3.5, therefore unconstrained operation

6. Compute weaving segmentspeed (use Equation 24-5). S = v

vw

Sw

+ vnw

Snw

= 5,5881,99648.5

+ 3,59252.9

= 51.2 mi/h

7. Compute weaving segmentdensity (use Equation 24-6).

D =

vN

S=

5,5884

51.2= 27.3 pc/mi/ln

8. Determine level of service(use Exhibit 24-2).

LOS C

9. Determine weaving segmentcapacity (use Exhibit 24-8 andEquations 24-7 and 24-8).

cb = 8,110 pc/h [Exhibit 24-8(F)]c = cb * fHV * fp = 8,110 * 0.952 * 1.000 = 7,721 veh/hch = c * PHF = 7,721 * 0.91 = 7,026 veh/h

The Results This weaving segment will operate at LOS C during the peak hour.Weaving segment density is 27.3 pc/mi/ln. The capacity of the weaving segment isestimated as 7,720 veh/h (for the 15-min volume), or 7,030 veh/h (for the full-hourvolume).



Example Problem 1FREEWAY WEAVING WORKSHEET







Inputs




vo1vo2vw1vw2vwvnwv


















a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =

M.E.CEI8/3/99AM-Peak

X

---1999

6541,500

XX

0.357

0.400

1,815 0.91 10 0.952 1.000 2,0951,297 0.91 10 0.952 1.000 1,4971,037 0.91 10 0.952 1.000 1,197692 0.91 10 0.952 1.000 7991,729 0.91 10 0.952 1.000 1,9963,112 0.91 10 0.952 1.000 3,5924,841 0.91 10 0.952 1.000 5,588

0.08 0.00202.2 6.00.70 1.00.50 0.50

0.643 0.450

48.5 52.9

X

1.653.5

51.2

27.3

C

8,110

7,721

7,026

692

1037

1815

1297

L = 1,500 ftA C

DB



EXAMPLE PROBLEM 2

The Weaving Segment A ramp-weaving segment on a rural freeway as shown on theworksheet.


The Facts√ Flow rate (A-C) = 4,000 pc/h, √ Flow rate (A-D) = 300 pc/h,√ Flow rate (B-C) = 600 pc/h, √ Flow rate (B-D) = 100 pc/h, and√ FFS = 75 mi/h for freeway, √ Weaving segment length = 1,000 ft.

Outline of Solution All input parameters are known, so no default values are required.Demand flows are given as equivalent passenger cars per hour under base conditions.Thus, no conversion of flows is required. Weaving configuration type is determined.Weaving and nonweaving speeds are computed, followed by weaving segment speed.The density in the weaving segment is calculated, and level of service is determined.Capacity is then determined.

Steps1. Determine weaving segment

configuration type (use Exhibit24-5).

Type A (Movements A-D and B-C require one lanechange)

2. Compute critical variables. vw = 600 + 300 = 900 pc/hvnw = 4,000 + 100 = 4,100 pc/hv = 900 + 4,100 = 5,000 pc/h

VR = 9005,000

= 0.180

R = 300900

= 0.333


Wi =a(1+ VR)b

vN

c

(L)d Si = 15 + SFF − 101+ Wi

Ww =0.15(1+ 0.180)2.2 5,000

4

0.97

(1,000)0.80 = 0.867

Wnw =0.0035(1+ 0.180)4.0 5,000

4

1.3

(1,000)0.75 = 0.405

Sw = 15 + 75 − 101+ 0.867

= 49.8 mi/h

Snw = 15 + 75 − 101+ 0.405

= 61.3 mi/h


Nw = 0.74(N)VR0.571L0.234/Sw0.438

Nw = 0.74 (4) (0.1800.571) (1,0000.234 )

49.80.438 = 1.01

Nw(max) = 1.4, therefore unconstrained operation


vw

Sw

+ vnw

Snw

= 5,00090049.8

+ 4,10061.3

= 58.9 mi/h




D =

vN

S=

5,0004

58.9= 21.2 pc/mi/ln


LOS C

8. Determine weaving segmentcapacity (use Exhibit 24-8).

cb = 8,454 pc/h [Exhibit 24-8(A)]

The Results This weaving segment will operate at LOS C during the peak hour. Theweaving segment density is 21.2 pc/mi/ln, and the capacity is estimated to be 8,450 pc/h.Because neither the traffic composition nor the PHF is specified, capacities per full hourand for prevailing conditions cannot be determined.










Inputs




vo1vo2vw1vw2vwvnwv


















a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =


X

7541,000

X

0.180

0.333

4,0001006003009004,1005,000

0.15 0.00352.2 4.00.97 1.30.80 0.75

0.867 0.405

49.8 61.3

X

1.011.4

58.9

21.2

C

8,454

-

-

---1999

4000

100600

300

L = 1,000 ft

A C

B D



EXAMPLE PROBLEM 3

The Weaving Segment A ramp-weaving segment on an urban freeway as shown onthe worksheet.


The Facts√ Volume (A-C) = 975 veh/h, √ PHF = 0.85,√ Volume (A-D) = 650 veh/h, √ Rolling terrain,√ Volume (B-C) = 520 veh/h, √ Drivers are regular commuters,√ Volume (B-D) = 0 veh/h, √ FFS = 65 mi/h for freeway, and√ 15 percent trucks, √ Weaving segment length = 1,000 ft.

Comments√ Use Chapter 23, “Basic Freeway Segments,” to identify fHV and fp.

Outline of Solution All input parameters are known, so no default values arerequired. Demand volumes are converted to flow rates, and weaving configuration type isdetermined. Weaving and nonweaving speeds are computed and used to determineweaving segment speed. The density in the weaving segment is calculated, and level ofservice is determined. Capacity is then determined.

Steps1. Convert volume (veh/h) to flow

rate (pc/h) (use Equation24-1).

v = V(PHF)(fHV )(fp )

v(A-C) = 975(0.85)(0.816)(1.000)

= 1,406 pc/h

v(A-D) = 937 pc/hv(B-C) = 750 pc/h

1a. Determine fp (use Chapter23).

fp = 1.000

1b. Determine fHV (use Chapter23). fHV = 1

1+ PT (ET − 1) + PR(ER − 1)

fHV = 11+ 0.15(2.5 − 1) + 0

= 0.816

2. Determine weaving segmentconfiguration type (use Exhibit24-5).

Type A (all weaving vehicles must make one lanechange)

3. Compute critical variables. vw = 937 + 750 = 1,687 pc/hvnw = 1,406 pc/hv = 1,406 + 1,687 = 3,093 pc/h

VR = 1,6873,093

= 0.545 R = 7501,687

= 0.444




Wi =a(1+ VR)b

vN

c

(3.28L)d Si = 15 + SFF − 101+ Wi

Ww =0.15(1+ 0.545)2.2 3,093

3

0.97

(1,000)0.80 = 1.302

Wnw =0.0035(1+ 0.545)4.0 3,093

3

1.3

(1,000)0.75 = 0.927

Sw = 15 + 65 − 101+ 1.302

= 38.9 mi/h

Snw = 15 + 65 − 101+ 0.927

= 43.5 mi/h


Nw = 0.74(N)VR0.571L0.234/Sw0.438

Nw = 0.74(3)(0.5450.571)(1,000.234)/38.9.438 = 1.59Nw(max) = 1.4, therefore constrained operation

6. Repeat Step 4 for constrainedoperation.

Ww =0.35(1+ 0.545)2.2 3,093

3

0.97

(1,000)0.80 = 3.037

Wnw =0.0020(1+ 0.545)4.0 3,093

3

0.97

(1,000)0.75 = 0.530

Sw = 15 + 65 − 101+ 3.037

= 28.6 mi/h

Snw = 15 + 65 − 101+ 0.530

= 50.9 mi/h


vw

Sw

+ vnw

Snw

= 3,0931,68728.6

+ 1,40650.9

= 35.7 mi/h


D =

vN

S=

3,0933

35.7= 28.9 pc/mi/ln


LOS D

10. Determine weaving segmentcapacity (use Exhibit 24-8 andEquations 24-7 and 24-8).

cb = 4,520 pc/h [Exhibit 24-8(B)]c = cb * fHV * fp = 4,520 * 0.816 * 1.000 = 3,688 veh/hch = c * PHF = 3,688 * 0.85 = 3,135 veh/h

The Results The weaving segment will operate at LOS D during the peak hour. Theweaving segment density is 28.9 pc/mi/ln, and the capacity is estimated as 3,690 veh/h(for the 15-min flow rate), or 3,130 veh/h (for the full-hour volume). Note that the three-lane ramp-weave has a volume ratio of 0.545, which exceeds the maximumrecommended for such segments (0.45). Thus, operations may actually be worse thanpredicted.

The capacity estimates must also be carefully considered. They reflect the maximumVR that is tabulated for Type A, three-lane weaving segments (0.45). The actual value fora VR of 0.545 would be expected to be lower than the values shown.

One approach to improving operations would be to change the configuration (Type Asegments do not handle volume ratios of 0.545 efficiently) to Type B by adding a lane tothe off-ramp on Leg D. This lane, not needed for general purposes, would have to be



dropped or designed into the ramp’s other terminus. This calculation, in effect,emphasizes the importance of configuration. The Type A configuration is not appropriatefor the balance of flows presented.








Inputs




vo1vo2vw1vw2vwvnwv





Weaving and nonweaving speeds, Si (km/h)



Weaving segment speed, S (km/h)

Weaving segment density, D (pc/km/ln)









a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =


X

6531,000

X

X

0.545

0.444

975 0.85 15 0.816 1.000 1,4060 0.85 15 0.816 1.000 0650 0.85 15 0.816 1.000 937520 0.85 15 0.816 1.000 750

1,6871,4063,093

0.15 0.0035 0.35 0.00202.2 4.0 2.2 4.00.97 1.3 0.97 1.30.80 0.75 0.80 0.75

1.302 0.927 3.037 0.530

38.9 43.5 28.6 50.9

X

1.591.4

35.7

28.9

D

4,520

3,688

3,135

---1999

C

D

A

B

L = 1,000 ft

520

650

975



EXAMPLE PROBLEM 4

The Weaving Segment Alternative major weaving segments are being considered at amajor junction between two urban freeways as shown on the worksheets. Several designconstraints exist. Entry Leg A (left side) has two lanes; Entry Leg B (right side) has threelanes. Exit Leg C (left side) has three lanes; Exit Leg D (right side) has two lanes. Themaximum length of the weaving segment is 1,000 ft. The FFS of all entry and exit legs is75 mi/h.

The Question What are the required number of lanes and weaving segmentconfiguration to achieve LOS C?

The Facts√ Flow rate (A-C) = 2,000 pc/h, √ Flow rate (B-D) = 2,000 pc/h,√ Flow rate (A-D) = 1,450 pc/h, √ FFS = 75 mi/h for both freeways, and√ Flow rate (B-C) = 1,500 pc/h, √ Weaving segment length = 1,000 ft.

Outline of Solution Demand flows are given in passenger cars per hour for equivalentbase conditions. Therefore, no conversion of flows is required. Weaving number of lanesand configuration are assumed, and LOS is determined. If the LOS is below C, a betterconfiguration is assumed until LOS C or better is achieved.

Steps1. Assume lane configuration

shown on the worksheet.

2. Determine weavingsegment configurationtype (use Exhibit 24-5).

Type C (Movement B-C requires no lane change;Movement A-D requires two lane changes).

3. Compute critical variables. vw = 1,500 + 1,450 = 2,950 pc/h

vnw = 2,000 + 2,000 = 4,000 pc/h

v = 4,000 + 2,950 = 6,950 pc/h

VR = 2,9506,950

= 0.424

R = 1,4502,950

= 0.492

4. Compute weaving andnonweaving speedsassuming unconstrainedoperation (use Exhibit24-6 and Equations 24-3and 24-4).

Wi =a(1+ VR)b

vN

c

(L)d Si = 15 + SFF − 101+ Wi

Ww =0.08(1+ 0.424)2.3 6,950

5

0.80

(1,000)0.60 = 0.934

Wnw =0.0020(1+ 0.424)6.0 6,950

5

1.1

(1,000)0.60 = 0.758

Sw = 15 + 75 − 101+ 0.934

= 48.6 mi/h

Snw = 15 + 75 − 101+ 0.758

= 52.0 mi/h



5. Check type of operation(use Exhibit 24-7).

Nw = N[0.761 + 0.047VR – 0.00011L – 0.005(Snw – Sw)]

Nw = 5[0.761 + (0.047 * 0.424) – (0.00011 * 1000)– 0.005(52.0 – 48.6)] = 3.27

Nw(max) = 3.0, constrained operation

6. Repeat Step 4 forconstrained operation.

Ww =0.14(1+ 0.424)2.3 6,950

5

0.80

(1,000)0.60 = 1.635

Wnw =0.0010(1+ 0.424)6.0 6,950

5

1.1

(1,000)0.60 = 0.379

Sw = 15 + 75 − 101+ 1.635

= 39.7 mi/h

Snw = 15 + 75 − 101+ 0.379

= 62.1 mi/h

7. Compute weavingsegment speed (useEquation 24-5).

S = v

vw

Sw

+ vnw

Snw

= 6,9502,95039.7

+ 4,00062.1

= 50.1 mi/h

8. Compute weavingsegment density (useEquation 24-6). D =

vN

S=

6,9505

50.1= 27.7 pc/mi/ln


LOS C










Inputs




vo1vo2vw1vw2vwvnwv


















a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =


X

7551,000

X

0.424

0.492

2,0002,0001,5001,4502,9504,0006,950

0.08 0.0020 0.14 0.00102.3 6.0 2.3 6.00.80 1.1 0.80 1.10.60 0.60 0.60 0.60

0.934 0.758 1.635 0.379

48.6 52.0 39.7 62.1

X

3.273.0

50.1

27.7

C

---1999

A

B

C

D

L = 1,000 ft

2000

15001450

2000



The trial design just meets the design objective of LOS C. There are other troublingaspects of the results as well. The R value (0.492) exceeds the maximum recommendedfor Type C segments (0.40), and the segment might, therefore, operate worse thanexpected. The constrained operating condition produces a large difference in speedbetween weaving and nonweaving traffic streams. That is also undesirable.

There is no additional length available, since the maximum of 1,000 ft has already beenused. The width cannot be made six lanes without adding lanes to entry and exit roadwaysas well, where they are apparently not needed. The only other potential change would be toalter the configuration to a Type B design. This is best accomplished by adding a lane toLeg D. The lane will make weaving more efficient and can be dropped further downstreamon Leg D.

10. Assume lane configurationshown on the worksheet.

11. Determine weavingsegment configurationtype (use Exhibit 24-5).

Type B (Movement B-C requires no lane change;Movement A-D requires one lane change).

12. Compute weaving andnonweaving speedsassuming unconstrainedoperation (use Exhibit24-6 and Equations 24-3and 24-4).

Wi =a(1+ VR)b

vN

c

(L)d Si = 15 + SFF − 101+ Wi

Ww =0.08(1+ 0.424)2.2 6,950

5

0.70

(1,000)0.50 = 0.873

Wnw =0.0020(1+ 0.424)6.0 6,950

5

1.0

(1,000)0.50 = 0.733

Sw = 15 + 75 − 101+ 0.873

= 49.7 mi/h

Snw = 15 + 75 − 101+ 0.733

= 52.5 mi/h

13. Check type of operation(use Exhibit 24-7).

Nw = N[0.085 + 0.703VR + (234.8/L) – 0.018(Snw – Sw)]

Nw = 5[0.085 + (0.703 * 0.424) + (234.8/1,000)– 0.018(52.5 – 49.7)] = 2.84

Nw(max) = 3.5, unconstrained operation

14. Compute weavingsegment speed (useEquation 24-5).

S = v

vw

Sw

+ vnw

Snw

= 6,9502,95049.7

+ 4,00052.5

= 51.3 mi/h

15. Compute weavingsegment density (useEquation 24-6). D =

vN

S=

6,9505

51.3= 27.1 pc/mi/ln


LOS C

The Results Whereas both alternatives provide the desired LOS C, there are manyother beneficial effects of providing the Type B configuration. Unconstrained operationprevails, and the difference in speed between the weaving and nonweaving streams isreduced substantially. This calculation illustrates the advantage of a Type B configurationin handling high proportions of weaving traffic.










Inputs




vo1vo2vw1vw2vwvnwv


















a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =


X

7551,000

0.424

0.492

2,0002,0001,5001,4502,9504,0006,950

0.08 0.00202.2 6.00.70 1.00.50 0.50

0.873 0.733

49.7 52.5

X

2.843.5

51.3

27.1

C

---1999

X2000

15001450

2000

A

B

C

D

L = 1,000 ft



EXAMPLE PROBLEM 5

The Weaving Segment A major interchange is to be built to join two major freeways ina suburban area.

The Question What are the required number of lanes, length, and configuration of theweaving segment to achieve LOS C operation?

The Facts The flows analyzed are shown in the weaving diagram below. The flowrates are given in passenger cars per hour under base conditions. Since the interchangewill join two future facilities, there is considerable flexibility in both the length and the widthof the segment. The free-flow speed is 75 mi/h.

1500

1000

1000

700

A

B D

C

The Results Since the length, width, and configuration to be used are open toquestion, so is the issue of whether to have a weaving segment at all. Speed, density,and level of service will be determined from trial computations for a range of conditionscovering three, four, and five lanes, with the length ranging from 500 ft to 2,500 ft. Allthree types of weaving configuration will also be evaluated. This is a time-consumingprocess, and the use of a programmable calculator or spreadsheet is recommended. Theanalysis results are shown in the table on the next page.

A number of points should be made concerning the results and the effect on the finaldesign decision.

1. Before all the potential solutions are examined, the configuration of entry and exitlegs should be considered. To provide for minimum LOS C, two lanes are needed oneach entry and exit leg. The five-lane option is eliminated because the four-lane option isable to meet the criteria.

2. If the legs are simply connected, a four-lane Type A configuration will result, andLOS C is produced. The main drawback of this configuration is that the operation isconstrained. This indicates serious imbalance between weaving and nonweaving flows.Also, the VR of 0.405 is higher than the maximum recommended for Type A, 0.35. Trafficoperation may be worse than predicted here.

3. There is no easy way to produce a Type C configuration given the criteria forentry and exit legs.

4. A Type B configuration can be achieved by adding one lane to Leg C. Theresulting four-lane Type B segment will operate within all recommended parameters andmeet minimum required LOS C for all lengths evaluated. A three-lane Type Bconfiguration will result if a lane is merged at the entrance to and diverged at the exit fromthe segment. LOS C can also be met with three lanes if the length is 1,000 ft or longer.

The results do not yield a clear answer, but they provide the analyst with theinformation to make a judgment. Obviously, the best operation will result from a long TypeB segment with four lanes. However, economic and environmental considerations willalso affect the final decision.



No. of Lanes L (ft) S (mi/h) D (pc/mi/ln) LOS Cons Y/N

Type A Configurations

500 37.2 37.6 E N1,000 45.5 30.8 D N

3 1,500 50.6 27.7 C N2,000 49.8 28.1 D Y2,500 52.4 26.7 C Y

500 38.8 27.1 C Y1,000 46.3 22.7 C Y

4 1,500 50.9 20.6 C Y2,000 54.2 19.4 B Y2,500 56.7 18.5 B Y

500 42.0 20.0 B Y1,000 49.7 16.9 B Y

5 1,500 54.3 15.5 B Y2,000 57.5 14.6 B Y2,500 59.9 14.0 B Y

Type B Configurations

500 46.6 30.0 D N1,000 52.2 26.8 C N

3 1,500 55.3 25.3 C N2,000 57.5 24.3 C N2,500 59.1 23.7 C N

500 50.6 20.8 C N1,000 56.0 18.8 B N

4 1,500 59.0 17.8 B N2,000 61.0 17.2 B N2,500 62.4 16.8 B N

500 52.2 16.1 B Y1,000 58.8 14.3 B Y

5 1,500 61.6 13.6 B Y2,000 63.4 13.2 B Y2,500 64.8 13.0 B Y

Type C Configurations

500 44.6 31.4 D N1,000 51.4 27.2 C N

3 1,500 55.2 25.4 C N2,000 57.8 24.2 C N2,500 59.7 23.5 C N

500 49.2 21.3 C N1,000 55.7 18.9 B N

4 1,500 59.3 17.7 B N2,000 61.6 17.0 B N2,500 63.3 16.6 B N

500 51.7 16.2 B Y1,000 57.8 14.5 B Y

5 1,500 62.1 13.5 B Y2,000 64.3 13.1 B Y2,500 65.9 12.7 B Y

Note:Cons—constrained flow; Y/N—yes/no.


24-39 Chapter 24 - Freeway WeavingReferences

V. REFERENCES

1. Reilly, W. R., J. H. Kell, and P. J. Johnson. Weaving Analysis Procedures for theNew Highway Capacity Manual. Technical Report. JHK & Associates, Tucson,Ariz., 1984.

2. Pignataro, L. J., W. R. McShane, R. P. Roess, B. Lee, and K. W. Crowley.NCHRP Report 159: Weaving Areas: Design and Analysis. TRB, NationalResearch Council, Washington, D.C., 1975.

3. Roess, R. P., et al. Freeway Capacity Analysis Procedures. Final Report, ProjectDOT-FH-11-9336. Polytechnic Institute of New York, Brooklyn, 1979.

4. McShane, W. R., R. P. Roess, and E. S. Prassas. Traffic Engineering, 2nd ed.Prentice-Hall, Upper Saddle River, N.J., 1998.

5. Jack E. Leisch & Associates. Completion of Procedures for Analysis and Design ofTraffic Weaving Sections. Final Report. Federal Highway Administration, U.S.Department of Transportation, 1983.

6. Roess, R. P. Development of Weaving Area Analysis Procedures for the 1985Highway Capacity Manual. In Transportation Research Record 1112, TRB,National Research Council, Washington, D.C., 1987, pp. 17–22.

7. Fazio, J. Development and Testing of a Weaving Operational Analysis and DesignProcedure. Master’s thesis. University of Illinois at Chicago, Aug. 1985.

APPENDIX A. WORKSHEET


Chapter 24 - Freeway Weaving









Inputs




vo1vo2vw1vw2vwvnwv


















a(1 + VR)b (v/N)c

(L)dWi =

SFF – 101 + Wi

Si = 15 +

v/NS

D =

vwvvw2vw

v

+vwSw

vnwSnw

S =

( )( )

VPHF * fHV *fp

v =

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