• ## The Rational Method

General

The rational method is used to predict peak flows for small drainage areas which can be either natural or developed. The rational method can be used for culvert design, pavement drainage design, storm drain design, and some stormwater facility design. The greatest accuracy is obtained for areas smaller than 40 hectares (100 acres) and for developed conditions with large areas of impervious surface (e.g., pavement, roof tops, etc.). Basins up to 400 hectares (1,000 acres) may be evaluated using the rational formula; however, results for large basins often do not properly account for effects of infiltration and thus are less accurate. Designers should never perform a rational method analysis on a basin that is larger than the lower limit specified for the USGS regression equations since the USGS regression equations will yield a more accurate flow prediction for that size of basin.
The formula for the rational method is:

Hydrologic information calculated by the rational method should be submitted on DOT Form 235-009 (see Figure 2-4.1). This format contains all the required input information as well as the resulting discharge. The description of each area should be identified by name or stationing so that the reviewer may easily locate each area.

[nextpage title=âRunoff Coefficientsâ]

The runoff coefficient âCâ represents the percentage of rainfall that becomes runoff. The rational method implies that this ratio is fixed for a given drainage basin. In reality, the coefficient may vary with respect to prior wetting and seasonal conditions. The use of an average coefficient for various surface types is quite common and it is assumed to stay constant through the duration of the rain storm.

Frozen ground can cause a dramatic increase in the runoff coefficient. When this condition is coupled with heavy rainfall and, perhaps, melting snow, the runoff can be much greater than calculated values that did not account for these conditions. This condition is common for larger basins that are above 300 m (1000 ft.) in elevation and is automatically accounted for in the USGS regression equations. For small basins where the rational method is being used, the designer should increase the runoff coefficient to reflect the reduction in infiltration and resulting increased surface runoff.

In a high growth rate area, runoff factors should be projected that will be characteristic of developed conditions 20 years after construction of the project. Even though local storm water practices (where they exist) may reduce potential increases in runoff, prudent engineering should still make allowances for predictable growth patterns.

The coefficients in Figure 2-4.2 are applicable for peak storms of 10-year frequency. Less frequent, higher intensity storms will require the use of higher coefficients because infiltration and other losses have a proportionally smaller effect on runoff. Generally, when designing for a 25-year frequency, the coefficient should be increased by 10 percent; when designing for a 50-year frequency, the coefficient should be increased by 20 percent; and when designing for a 100-year frequency, the coefficient should be increased by 25 percent. The runoff coefficient should never be increased above 0.90.

Hydrologic information calculated by the rational method should be submitted on DOT Form 235-009 (see Figure 2-4.1). This format contains all the required input information as well as the resulting discharge. The description of each area should be identified by name or stationing so that the reviewer may easily locate each area.

[nextpage title=âTime of Concentrationâ]

If rainfall is applied at a constant rate over a drainage basin, it would eventually produce a constant peak rate of runoff. The amount of time that passes from the moment that the constant rainfall begins to the moment that the constant rate of runoff begins is called the time of concentration. This is the time required for the surface runoff to flow from the most hydraulically remote part of the drainage basin to the location of concern.

Actual precipitation does not fall at a constant rate. A precipitation event will begin with a small rainfall intensity then, sometimes very quickly, build to a peak intensity and eventually taper down to no rainfall. Because rainfall intensity is variable, the time of concentration is included in the rational method so that the designer can determine the proper rainfall intensity to apply across the basin. The intensity that should be used for design purposes is the highest intensity that will occur with the entire basin contributing flow to the location where the designer is interested in knowing the flow rate. It is important to note that this may be a much lower intensity than the absolute maximum intensity. The reason is that it often takes several minutes before the entire basin is contributing flow but the absolute maximum intensity lasts for a much shorter time so the rainfall intensity that creates the greatest runoff is less than the maximum by the time the entire basin is contributing flow.

Most drainage basins will consist of different types of ground covers and conveyance systems that flow must pass over or through. These are referred to as flow segments. It is common for a basin to have flow segments that are overland flow and flow segments that are open channel flow. Urban drainage basins often have flow segments that are flow through a storm drain pipe in addition to the other two types. A travel time (the amount of time required for flow to move through a flow segment) must be computed for each flow segment. The time of concentration is equal to the sum of all the flow segment travel times.

For a few drainage areas, a unique situation occurs where the time of concentration that produces the largest amount of runoff is less than the time of concentration for the entire basin. This can occur when two or more subbasins have dramatically different types of cover (i.e., different runoff coefficients). The most common case would be a large paved area together with a long narrow strip of natural area. In this case, the designer should check the runoff produced by the paved area alone to determine if this scenario would cause a greater peak runoff rate than the peak runoff rate produced when both land segments are contributing flow. The scenario that produces the greatest runoff should be used, even if the entire basin is not contributing flow to this runoff.

The procedure described below for determining the time of concentration for overland flow was developed by the United States Natural Resources Conservation Service (formerly known as the Soil Conservation Service). It is sensitive to slope, type of ground cover, and the size of channel. The designer should never use a time of concentration less than 5 minutes. The time of concentration can be calculated as follows:

[nextpage title=âRainfall Intensityâ]

After the appropriate storm frequency for the design has been determined (see Chapter 1) and the time of concentration has been calculated, the rainfall intensity can be calculated. Designers should never use a time of concentration that is less than 5 minutes for intensity calculations, even when the calculated time of concentration is less than 5 minutes. It should be noted that the rainfall intensity at any given time is the average of the most intense period enveloped by the time of concentration and is not the instantaneous rainfall. The equation for calculating rainfall intensity is:

The coefficients (m and n) have been determined for all major cities for the 2-, 5-, 10-, 25-, 50-, and 100-year mean recurrence intervals (MRI). The coefficients listed are accurate from 5-minute duration to 1,440-minute duration (24 hours). These equations were developed from the 1973 National Oceanic and Atmospheric Administration Atlas 2, Precipitation-Frequency Atlas of the Western United States, Volume IX-Washington. The designer should interpolate between the two or three nearest cities listed in the tables when working on a project that is in a location not listed on the table. If the designer must do an analysis with a storm duration greater than 1,440 minutes, the rational method should not be used.

[nextpage title=âRational Formula Exampleâ]

Compute the 25-year runoff for the Olympia watershed shown above. Three types of flow conditions exist from the highest point in the watershed to the outlet. The upper portion is 10.0 hectares of forest cover with an average slope of 0.15 m/m. The middle portion is 2.5 hectares of single family residential with a slope of 0.06 m/m and primarily lawns. The lower portion is a 2.0 hectares park with 450 mm storm sewers with a general slope of 0.01 m/m.

[nextpage title=âFiguresâ]

• ## Drainage Basin

The size of the drainage basin is one of the most important parameters regardless of which method of hydrologic analysis is used. To determine the basin area, select the best available topographic map or maps which cover the entire area contributing surface runoff to the point of interest. Outline the area on the map or maps and determine the size in square meters, acres, or square miles (as appropriate for the specific equations), either by scaling or by using a planimeter. Sometimes drainage basins areÂ small enough that they fit entirely on the CADD drawings for the project. In these cases the basin can be digitized on the CADD drawing and calculated by the computer. Any areas within the basin that are known to be non-contributing to surface runoff should be subtracted from the total drainage area.

The USGS has published two open-file reports titled, Drainage Area Data for Western Washington and Drainage Area Data for Eastern Washington. Copies of these reports can be obtained from the OSC Hydraulics Branch and the Regional Hydraulics Contacts. These reports list drainage areas for all streams in Washington where discharge measurements have been made. Drainage areas are also given for many other sites such as highway crossings, major stream confluences, and at the mouths of significant streams. These publications list a total of over 5,000 drainage areas and are a valuable time saver to the designer. The sites listed in these publications are usually medium sized and larger drainage basin areas. Small local drainage areas need to be determined from topographic maps as outlined above.

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• ## Selecting hydrology analysis method

Each of the first five methods listed above are appropriate to use for different design conditions and none of the methods will cover all situations. The first step in performing a hydrologic analysis is to determine which method is most appropriate. Generally there is no need to select more than one method.

## 1. Rational Method:

This method is used when peak discharges for small basins must be determined. It is a fairly simple and accurate method especially when the basin is primarily impervious. The rational method is appropriate for culvert design, pavement drainage design, storm drain design, and some stormwater facility designs.

## 2. SBUH Method:

This method is used when peak discharges and runoff volumes for small basins must be determined. This method is not complicated but requires a computer due to its computationally intensive nature. The SBUH method is required for many stormwater facility designs and can also be used for culvert design, pavement drainage design, and storm drain design.

## 3. Published Flow Records:

This method is used when peak discharges for large basins must be determined. This is more of a collection of data rather than a predictive analysis like the other methods listed. Some agencies (primarily the USGS) gather streamflow data on a regular basis. This collected data can be used to predict flood flows for the river and is typically more accurate than calculated flows. Published flow records are most appropriate for culvert and bridge design.

## 4. USGS Regression Equations:

This method is used when peak discharges for medium to large basins must be determined. It is a set of regression equations that were developed using data from streamflow gaging stations. The regression equations are very simple to use but lack the accuracy of published flow records. USGS regression equations are appropriate for culvert and bridge design.

## 5. Flood Reports:

This method is used when peak discharges for medium to large basins must be determined. It is basically using results from an analysis that has been conducted by another agency. Often these values are very accurate since they were developed from an in-depth analysis. Flood report data are appropriate for culvert and bridge design.

• ## General Hydrology

The Washington State Department of Transportation (WSDOT) Olympia Service Center (OSC) Hydraulics Branch uses several methods of determining runoff rates and/or volumes. Experience has shown them to be accurate, convenient, and economical. The following methods will be discussed in detail in subsequent sections of this chapter

1. The Rational Method
2. The Santa Barbara Urban Hydrograph (SBUH) Method
3. Published Flow Records
4. United States Geological Survey (USGS) Regression Equations
5. Flood Reports
Two other methods, documented testimony and high water mark observations, may be used as back-up material to confirm the results of the above statistical and empirical methods. Where calculated results vary from on-site observations, further investigation may be required. The additional two methods are:
6. Documented Testimony
Documented testimony of long-time residents should also be given serious consideration by the designer. The engineer must be aware of any bias that testifying residents may have. Independent calculations should be made to verify this type of testimony. The information that may be furnished by local residents of the area should include, but not be limited to the following:
a. Dates of past floods.
b. High water marks.
c. Amount of drift.
d. Any changes in the river channel which may be occurring (i.e., stability of streambed, is channel widening or meandering?).
e. Estimated velocity.
f. Description of flooding characteristics between normal flow to flood stage.
7. High Water Mark Observations

Sometimes the past flood stage from a drainage area may be determined by observing high water marks on existing structures or on the bank of a stream or ditch. These marks along with other data may be used to determine the discharge by methods discussed in the Open Channel Flow chapter or the Culverts chapter of this manual.
Additional hydrologic procedures are available including complex computer models which can give the designer accurate flood predictions. However, these methods, which require costly field data and large amounts of data preparation and calculation time, can rarely be justified for a single hydraulic structure. The OSC Hydraulics Branch should be contacted before a procedure not listed above is used in a hydrologic analysis. For the sake of simplicity and uniformity, the OSC Hydraulics Branch will normally require the use of one of the first five of the seven methods listed above. Exceptions will be permitted if adequate justification is provided.

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• ## Construction Proposal preparation, submittal and opening

The final steps in the contractor selection phase involve decisions about the size of the markup to be added to the net project cost, the submittal and opening of all tenders, the selection of the successful contractor and, at last, the notice to proceed, which directs the contractor to begin work.

Turning the estimate into a tender

Pilcher (1992) defines tendering as the process whereby a contractor, given the net cost, converts this to the sum that will actually be submitted to the client, together with any qualifications that are seen to be required. At this stage the principal discussions are concerned with the profit and the risk, together known as the margin or the markup.

Although we have already illustrated the addition of markup, or âprofit and contingencyâ, to net project cost in the determination of total tender price, it is important that we consider some of the issues involved in determining the markup amount. Also, the separation of the rather mechanical process of determining the net project cost from the judgment-based setting of markup is an important part of the thought process. Furthermore, while the estimating staff will perform most or all of the assembly of net project cost, the setting of markup is the responsibility of upper management. If the net project cost estimate is a relatively accurate prediction of what the project will cost, the decision about markup will âmake or breakâ the project. For those reasons, we set this section apart from the âcost estimatingâ section in this presentation.

Hendrickson and Au (1989), in their discussion of the principles of competitive bidding, state that most contractors âexercise a high degree of subjective judgmentâ in the setting of markup. The process is far from exact. In a competitive tendering situation, two opposing objectives are at work: (1) the desire to be selected as the winning tenderer and (2) the desire to make a decent profit from the project. The lower the markup, the higher will be the probability of the contractor having a sufficiently low price to be selected. On the other hand, the higher the markup, the greater will be the profit if the contractor is selected, other things being equal. It is upper managementâs responsibility to set an âoptimumâ markup that balances these two objectives for any particular tender.

The factors that may be involved in setting the markup amount have some resemblance to the factors the contractor considers in making the decision whether to spend the effort to prepare the tender, discussed earlier in this chapter. The factors are related to the contractorâs potential risk if the project is won, the degree of need and desire to win the project and the expected competition. Among all the factors a contractor might take into account are the following.

 The owner and design professional and the likelihood they will cause difficulties for theÂ contractor.

 Stipulations in the contract documents for delays in payments or retention of moneys owedÂ to the contractor.
 Disclaimer clauses that place on the contractor most or all of the risk for unknown physicalÂ conditions at the site, especially underground conditions.
 Clauses making the contractor responsible for any delay in the project, even if not caused byÂ the contractor.
 Other clauses providing for procedures the contractor may believe to be unreasonable, forÂ such matters as change orders (variations), contract claims and the rendering of bindingÂ decisions in case of disputes.
 The extent to which the contractor may be liable for any worker safety-and-health problemsÂ or labour law violations.
 The projectâs location, size and complexity.
 The amount of work to be done by the contractorâs own forces in comparison to work to beÂ done by subcontractors. In general, contractors believe that more risk is associated with doingÂ the work yourself; some apply a higher markup percentage to that work than to subcontractedÂ work.
 How âhungryâ the contractor is, based on the number of projects the contractor already hasÂ under contract and the potential for other new projects. This degree of desire to win theÂ project can be a major influence on markup.
 The expected competition, including the number of tenderers and the characteristics of each.
In general, the competition will be more intense as the number of tenderers increases and theÂ contractors will need a smaller markup to have a high chance of success. Also, if some of theÂ other contractors have reputations for offering low-priced proposals, this fact may influenceÂ our contractorâs markup decision. Of course these other tenderers are considering the sameÂ factors when pricing their proposals, including their current and future workloads, so attemptsÂ to analyse their potential competitiveness will be inexact at best.

Most of these factors are quite intangible, which is why the evaluation of these risks and the decision on markup is the responsibility of those personnel with considerable experience and judgment skills. In our two examples earlier in this chapter, we used markups of 9.6% and 10.5%, before adding taxes and bond costs. In good times, these may be reasonable percentages, but it is well known that some contractors add only a few per cent for profit and contingency when the competition is intense and they are highly desirous of obtaining the work. One other comment is in order with regard to markup. Sometimes this term is defined to include general overhead, as well as profit and contingency. In that case, of course, a higher percentage would be used. Clough et al. (2000) suggest that markup may vary from 5% to more than 20%, especially if it includes general overhead.