Bridge Scour and Stream Instability Countermeasures: Experience, Selection, and Design GuidanceThird Edition
Chapter 4
COUNTERMEASURE DESIGN CONCEPTS
4.1 COUNTERMEASURE DESIGN APPROACH
4.1.1 Investment in Countermeasures
At stream crossings, the objective of DOTs is to protect highway users and the investment in the highway facility, and to avoid causing damage to other properties, to the extent practicable. Countermeasures should be designed and installed to stabilize only a limited reach of stream and to ensure the structural integrity of highway components in an unstable stream environment. Countermeasures are often damaged or destroyed by the stream, and streambanks and beds often erode at locations where no countermeasure was installed. However, as long as the primary objectives are achieved in the shortterm as a result of countermeasure installation, the countermeasure installation can be deemed a success. Therefore, the DOT's interest in stream stability often entails longterm protection of costly structures by committing to maintenance, reconstruction, and installation of additional countermeasures as the responses of streams and rivers to natural and maninduced changes are identified.
While it is sometimes possible to predict that bank erosion will occur at or near a given location in an alluvial stream, one can frequently be in error about the exact location or magnitude of potential erosion. At some locations, unexpected lateral erosion occurs because of a large flood, a shifting thalweg, or from other actions of the stream or human activities. Where the investment in a highway crossing is not in imminent danger of being lost, it is often prudent to delay the installation of countermeasures until the magnitude and location of the problem becomes obvious.
Thus, for stream instability countermeasures, a "wait and see" attitude may constitute the most economical approach. Retrofitting can be considered sound engineering practice in many locations because the magnitude, location, and nature of potential instability problems are not always discernible at the design stage, and indeed, may take a period of several years to develop.
4.1.2 Service Life and Safety
A lifecycle concept, as applied to an erosion or scour countermeasure incorporates a host of factors into a framework for decision making considering initial design, construction, and longterm maintenance. These factors could include engineering judgment applied to design alternatives, materials availability and cost, installation equipment and practices, and maintenance assumptions (see Section 3.2.5).
When selecting a service life criterion for various types of countermeasures for transportation facilities, safety must be a primary consideration. For DOTs safety of the traveling public is the first priority when setting servicelife standards for countermeasures. Concurrent goals are protection of public and private property, protection of fish and wildlife resources, and enhancement of environmental attributes (Lagasse et al. 2006).
Thus, servicelife for a countermeasure installation should be based on the importance of the facility to the public, that is, the risk of losing the facility and how that loss may directly or indirectly affect t he traveling public, as well as the difficulty and cost of future repair or replacement. The conditions that constitute an "end of service life" for a countermeasure installation are largely dependent on the confidence one has that a degraded condition will be detected and corrected in a timely manner (e.g., during a postflood inspection). Generally, for facilities that are rarely checked or inspected a very conservative (i.e., shorter) service life would be appropriate, while a less conservative standard could be used for facilities that are inspected regularly.
Service life for a countermeasure installation can be considered a measure of the durability of the total, integrated bank, pier or abutment protection system. The durability of system components and how they function in the context of the overall design will determine the service life of an installation. The response of the system over time to typical stresses such as flow conditions (floods and droughts) or normal deterioration of system components must also be considered. Response to less typical (but plausible) stresses such as fire, vandalism, seismic activity or accidents may also affect service life. Finally, there may be opportunities for maintenance during the life cycle of the installation and, where such work does not constitute total reconstruction or replacement, maintenance should not be considered as the end of service life for the system. In fact, a lifecycle approach to maintenance may extend the service life of a countermeasure installation and reduce the total cost over the life of the project.
4.1.3 Design Approach
The design of any countermeasure for the protection of highway crossings requires the designer to be cognizant of the factors which affect stream stability and the morphology of the stream. In most cases, the installation of any countermeasure will cause the bed and banks to respond to the change in hydraulic conditions imposed by the countermeasure.
Thus, the analysis procedures outlined in HEC20 (Lagasse et al. 2001a) are a necessary prerequisite to the detailed design of specific countermeasures. The goal in any countermeasure design is to achieve a response which is beneficial to the protection of the highway crossing and to minimize adverse effects either upstream or downstream of the highway crossing. An appropriate level of hydraulic analysis must support the design of any stream instability or bridge scour countermeasure. In general, a countermeasure should be designed to provide a level of protection commensurate with the design event for the structure involved (see Section 4.3).
The bridge scour and stream instability countermeasures matrix (Table 2.1) helps define the set of specific countermeasures that are best suited to specific site conditions. The countermeasures matrix is intended, primarily, to assist with the selection of an appropriate countermeasure. Consideration of potential environmental impacts, maintenance, constructionrelated activities, and legal aspects can be used to refine the selection. The final selection criteria, and perhaps the most important, are the initial and longterm costs. The countermeasure that provides the desired level of protection at the lowest total cost may be the "best" for a particular application.
The following principles should be followed in designing and constructing stream instability and bridge scour countermeasures:
 Initial and longterm cost should not exceed the benefits to be derived. Countermeasures to make the bridge safe from scour and stream instability should be used for important bridges on main roads and where the results of failure would be intolerable. Expendable works may be used where traffic volumes are light, alternative routes are available, and the risk of failure is acceptable.
 Designs should be based on studies of channel trends and processes and on experience with comparable situations. The environmental effects of the countermeasures on the channel both up and downstream should be considered.
 Field reconnaissance by the designer is highly desirable and should include the watershed and river system up and downstream from the bridge.
 Evaluation of timesequenced aerial photography is a useful tool to detect longterm trends in river stability [see for example NCHRP Report 533 (Lagasse et al. 2004)].
 Soil and geotechnical characteristics of the site and their influence on countermeasure and filter design must be considered. This may include a geotechnical slope stability assessment, possibly under rapid drawdown conditions. This will require input from a licensed geotechnical engineer.
 The possibility of using physical model studies as a design aid should receive consideration at an early stage.
 Countermeasures must be inspected periodically after floods to check performance and modify the design, if necessary. The first design may require modification. Continuity in treatment, as opposed to sporadic attention, is advisable. The condition of the countermeasure should be documented with photographs to enable comparison of its condition from one inspection to another.
 In most cases, the countermeasure does not "cure" the instability or scour problem, and planning (funding) for continued maintenance of the countermeasure will be required.
In some cases, a combination of two or more countermeasures could be required due to sitespecific problems or as a result of changing conditions after the initial installation. The great number of possible countermeasure combinations makes it impractical to suggest design procedures for combined countermeasures. However, combined countermeasures should complement each other. That is to say, the design of one countermeasure must not adversely impact on another or the overall protection of the highway crossing. The principles of river mechanics, as discussed in HDS 6 (Richardson et al. 2001) and HEC20, coupled with sound engineering judgment should be used to design countermeasure strategies involving two or more countermeasures.
4.2 ENVIRONMENTAL PERMITTING
The environmental permitting process can have a significant effect on the planning, design and implementation of river engineering works. Often, permitting can become a lengthy process for the implementation of bridge scour and stream instability countermeasures. To expedite this process, a memorandum dated February 11, 1997, was prepared jointly by the U.S. Army Corps of Engineers (USACE) Directorate of Civil Works and the Federal Highway Administration (FHWA). The purpose of the memorandum is to facilitate timely decisions on permit applications for work associated with measures to protect bridges determined to be at risk as the result of scouring around their foundations. The USACE and FHWA consider this agreement essential to assure the safety of the traveling public while protecting the environment.
Recognizing the importance of protecting the foundations of our Nation's scour critical bridges with properly designed scour countermeasures and the need for environmentally sound projects, the FHWA and the USACE agree to work together with the bridge owners, in a cooperative effort, to plan ahead for managing projects that will need a USACE permit. A strong cooperative effort will aid in advanced planning to avoid and minimize environmental impacts, and in identifying locations where mitigation may be appropriate. If the bridge foundation has been determined to be scour critical as part of the bridge owner's scour evaluation program, the USACE will give priority to the bridge owner's request for authorization for the installation of scour countermeasures. Bridge owners must provide the FHWA and USACE Districts advance notice of the proposed countermeasure design and construction schedule. The notice must include an evaluation of the environmental impacts of the proposed scour countermeasure and appropriate mitigation of unavoidable impacts to aquatic resources, including fisheries and wetlands. This will allow appropriate and timely cooperation on project reviews. The USACE will make the maximum use possible of forms of expedited authorization, such as nationwide permits and regional permits, and Letters of Permission and the use of FHWA's Categorical Exclusion when the condition of the bridge foundation meets the criteria for codes 0 through 4 for Item 113 (FHWA 1995).
4.3 HYDRAULIC ANALYSIS
4.3.1 Overview
To be successful, the design of any countermeasure must incorporate an appropriate level of hydraulic analysis. The hydraulic principles of open channel flow and fundamentals of alluvial channel flow are summarized in HDS 6 and hydraulic factors that influence stream stability are presented in HEC20. In addition, HEC20 provides a general solution approach which includes hydrologic and hydraulic analysis steps in a multilevel analysis procedure (see Figure 1.1). Finally, HEC18 (Richardson and Davis 2001) provides references and discussion of the standard 1 and 2dimensional hydraulic computer models used for riverine and tidal analyses.
Both physical hydraulic modeling in a laboratory and numerical computer modeling are among the standard techniques available to analyze the scour problem and design countermeasures. This section introduces the use of physical modeling for the design of scour and stream instability countermeasures. Guidance is also provided for the analysis of several complex hydraulic conditions applicable to countermeasure design: scour at both transverse structures (spurs, jetties, dikes, and guide banks), and longitudinal structures (bendway revetment and vertical walls). The concept of radial stress on a bend is introduced as a method to evaluate (or predict) countermeasure performance in an eroding bendway.
4.3.2 Physical Models
The use of physical models as a tool in hydraulic design is commonly accepted. Many hydraulic phenomena which occur in nature are too complex to be described by rigorous mathematical techniques and models are used as an alternative means of obtaining the information necessary to complete an efficient and satisfactory design. Even in relatively simple situations, such as the design of spillways or water diversion structures, it is often impossible to predict the exact nature of the flow patterns without conducting a model study (Sharp 1981).
A summary of the principles of physical modeling for both rigidboundary and movablebed river models can be found in Shen (1979) who notes that hydraulic modeling has contributed significantly to design of hydraulic structures, training of rivers, and basic hydraulic research. It is a common practice to conduct hydraulic model tests to verify or modify the design of prototype structures. Hydraulic model tests are particularly useful in the study of complex flow phenomena for which no completely satisfactory theoretical analysis is available.
Some hydraulic tests are rather routine; many others are complex. For simple situations, hydraulic model tests provide accurate information that can be applied directly to prototype situations. However, for complex situations, hydraulic modeling is still more of an art than a science (Shen 1979). The following comments summarize important considerations for applying a physical model to the design of hydraulic structures, including countermeasures for bridge scour and stream instability.
 A physical model is very useful in the study of characteristics of complex flow phenomena involving significant flow variations in all three dimensions where no theoretical analysis is available.
 Dimensional analysis and physical reasoning are essential approaches to the selection of the governing similarity criteria. If more than one similarity criterion are needed, extensive knowledge of the basic process under investigation is necessary to deal with the situation.
 If the prototype is large, a distorted model may be necessary. In a distorted model the vertical model scale is usually smaller than the horizontal scale.
 A movable bed model may be necessary if a significant 3dimensional variation of sediment movement occurs in the prototype. Since movable bed model results are difficult to interpret, it can be advantageous to first investigate general flow variations in a fixed bed model.
 Verification of model results is absolutely necessary. Model results are usually verified with at least three flow conditions: high, medium, and low flow.
 In order to design a river model study correctly, one must decide the purpose of the model tests, know the principles of modeling thoroughly, and also have a thorough technical knowledge of hydraulics and river mechanics.
FHWA's "River Engineering for Highway Encroachments" (HDS 6) provides additional discussion of similitude for rigidboundary and mobilebed models.
A 1998 investigation of European practice for bridge scour and stream instability countermeasures (TRB 1999) concluded that in Europe it is much more likely that physical modeling, often in conjunction with computer modeling, will be used as an integral part of the hydraulic design process for bridge foundations and countermeasures than we are accustomed to in the United States. Government research agencies and private sector laboratories (e.g., Delft Hydraulics in the Netherlands) maintain extensive physical modeling capabilities for the following reasons: validation of computer modeling, fundamental research with respect to physical processes, and solving problems for which computers cannot presently be applied.
In a report on testing the effectiveness of scour countermeasures by physical modeling, the Federal Waterways Engineering and Research Institute (BAW) in Karlsruhe, Germany notes that the physical modeling of the scouring process at bridge piers is a proven method to get information about the size of the scour and the flow velocities generating the scour. On the basis of this information, appropriate countermeasures can be designed. The advantage of the physical model is its application on even the most complex pier geometries (Eisenhauer and Rossbach 1999).
At BAW, model tests were conducted for the piers of a new bridge over the Rhine River near Mannheim, Germany (Figure 4.1). Soon after the driving of sheet pile as a formwork for the lower part of the pier (pier width 36 ft (11 m) below mean water level and 18 ft (5.5 m) above mean water level) severe scouring of the river bed (d_{50} = 8 mm) occurred. As a consequence, the stability of the sheet pile formwork was endangered. An emergency countermeasure of placing riprap of 68 inches (1520 cm) diameter into the scour hole did not stop local scouring; however, an additional cover layer of coarser stones (diameter 8  24 inches [2060 cm]) was placed on top of the previous layer, stopping the erosion process at mean flow. A series of model tests were conducted in order to estimate the durability and stability of the emergency countermeasure for flood events. The tests proved the riprap to be stable even at flood stage while the scour was shifted away from the pier to the margin between the riprap and the sand of the natural river bed.
Figure 4.1. BAW laboratory, Karlsruhe, Germany, pier scour model of railway bridge over Rhine River near Mannheim (Eisenhauer and Rossbach 1999).
4.3.3 Scour at Transverse Structures
Several commonly used countermeasures for channel instability or scour protection project transversely into the flow (e.g., spurs, dikes, and jetties) or intercept overbank flow as it returns to the main channel (e.g., guide banks). Estimating scour at the nose of these structures is critical to successful design. Equation 4.1 is presented in HEC18 as an alternative abutment scour equation when the projecting embankment/abutment length is large in relation to flow depth (a/y_{1} > 25).
 (4.1) 
where:
y_{s} 
= 
Equilibrium depth of scour (measured from the mean bed level to the bottom of the scour hole), ft (m) 
y_{1} 
= 
Average upstream flow depth in the main channel or on the overbank outside the influence of the structure, ft (m) 
a 
= 
Structure length projecting normal to the flow, ft (m) 
F_{r} 
= 
Upstream Froude Number outside the influence of the structure 
This equation is based on field data on scour at the nose of rock spurs in the Mississippi River (obtained by the USACE) and is suggested here for estimating local scour at the nose of any transverse structure projecting into the flow.
For cases where the transverse structure length is small in comparison to flow depth (a/y_{1} ≤ 25) HDS 6 (see "Highways in the River Environment," 1990 Edition) presents the following equation for local livebed scour in sand at a stable spill slope when the flow is subcritical:
 (4.2) 
Where the variables are defined as for Equation 4.1. This equation is suggested here for estimating local scour at the nose of a transverse structure projecting into the flow when the conditions for Equation 4.1 are not met.
4.3.4 Scour at Longitudinal Structures
Variations in bed elevation during flow events or after bank hardening can result in the undermining of bank protection structures including longitudinal structures. Therefore, methods are needed for estimating maximum scour in order to design stable bank protection. The following sections provide methods for estimating scour along longitudinal countermeasures such as bulkheads and vertical walls.
Scour with Flow Parallel to a Vertical Wall. The probable mechanism causing scour along a vertical wall when the flow is parallel to the wall is an increase in boundary shear stress produced by locally increased velocity gradients that result from the reduced roughness of the vertical wall, as compared to the natural channel. It is reasonable to conclude that this scour will continue until the local flow area has increased enough to reduce the local velocity, and hence the local boundary shear stress, to values typical of the rest of the channel cross section (RCE 1994).
The magnitude of boundary shear stress around the perimeter of a channel is not constant. In channels of uniform roughness, the boundary shear stress has a maximum value near the channel centerline, and a secondary peak about onethird of the way up the sideslope. On average, the maximum on the bottom is about 0.97 times the average boundary shear stress (e.g., as defined by (γRS) for the cross section and the maximum on the side is about 0.76 times the average boundary shear stress. However, experimental data indicate a range of values, with maximum shear stresses as much as 1.6 times the average. In general, the boundary shear stress distribution is more uniform as the width to depth ratio increases.
Similar information is not available for channel cross sections of nonuniform roughness; however, reasonable conclusions can be drawn from intuitive arguments. For a straight channel with a vertical wall with smoother roughness than the rest of the channel along one side, the boundary shear stress distribution would be skewed towards the wall side of the channel. The sideslope peak value would be larger and could possibly be greater than the peak along the channel bed, which would also be shifted off the centerline location. These effects would be more pronounced in narrow channels and/or channels with steep sideslopes. As the channel gets wider, or the sideslope flattens, these effects would be diminished.
Insight on the magnitude of these effects can be obtained by considering local velocity conditions as determined by conveyance weighting concepts (see HEC18 and HEC20). The analysis assumes that the boundary roughness within the channel can be divided into two distinct regions: one region defining the roughness of the channel banks and the other defining the roughness of the channel bottom (note that this division of roughness, while logical, is not always analytically useful as it can create numerical problems leading to errors in the computation of conveyance for the entire cross section).
For purposes of illustration, a wide, shallow natural channel has a uniform roughness with a Manning's n value of 0.03, but with a concrete vertical wall the n value of the bank region is reduced by a factor of two, to 0.015. Evaluation of the distribution of discharge by conveyance weighting shows that this reduction of "n" nearly doubles the conveyance, discharge, and velocity adjacent to the bank (i.e., next to the wall). Recognizing that boundary shear stress is proportional to velocity squared, this increase in velocity increases the boundary shear stress by a factor of 4.
Based on the experimental results for a uniform roughness channel, the maximum boundary shear stress along the vertical wall could be as much as 3 times the average boundary shear stress. However, this is not totally accurate given the simplistic assumptions made and the likely changes in the distribution pattern that would result under conditions produced by a vertical wall. Nonetheless, this simplified analysis suggests that significant increases in the boundary shear stress are possible adjacent to the wall.
To apply this concept, it is appropriate to define a shear stress multiplier that can be applied to the average boundary shear stress to define the locally increased boundary shear stress adjacent to a vertical wall. Based on the above argument, a shear stress factor of 3 is suggested. Recognizing that boundary shear stress is proportional to velocity squared, the reduction in velocity necessary to lower the shear stress to an acceptable value is defined by the inverse of the square root of the shear stress multiplier (0.577) for the shear stress factor of 3. For the reduction in velocity to occur, the flow area must then be increased by the inverse of this factor (1/0.577 = 1.73). For a vertical wall, this calculation simplifies to a unit width basis and the scour depth is a multiplier of the average flow depth (0.73 y_{1}).
It is important to understand that this provides a first approximation of the potential scour along a vertical wall due to flow parallel to the wall. Using this relation, the total scour along the wall due to parallel flow can be approximated as the sum of the above relation, which results from a differential in shear stress, plus scour associated with the passage of antidunes (see HDS 6). This results in the following relationship:
 (4.3) 
where:
y_{s} 
= 
Equilibrium depth of scour (measured from the mean bed level to the bottom of the scour hole), ft (m) 
y_{1} 
= 
Average upstream flow depth in the main channel, ft (m) 
F_{r} 
= 
Upstream Froude Number 
This equation is applicable only where parallel flow can be assured (e.g., vertical walls along both banks).
Scour with Flow Impinging at an Angle on a Vertical Wall. When an obstruction such as an abutment or vertical wall projects into the flow, the depth of scour at the nose or face of the obstruction can be estimated from Equation 4.1. Considering the physical configuration of the channels for which the data on which this relation is based, this can reasonably be assumed to be the upper limit of the scour that could be expected for flow along a vertical wall when the flow impinges on the wall at an approximately 90 ° angle. The total scour along a vertical wall, thus, will vary as a proportion of that given by Equations 4.1 and 4.3. Assuming that the relative significance of the two scour mechanisms is related to the change in momentum associated with the change in flow direction from some angle θ relative to the wall, the two relations can be combined using a weighting factor based on the sine or cosine, respectively, of the angle of the flow to the wall (0 ° to 90 ° ). The resulting relationship is given by (RCE 1994):
 (4.4)

where:
θ 
= 
Angle between the impinging flow direction and the vertical wall 
Scour Along a Vertical Wall Relative to Unconstrained Valley Width. The potential scour that could occur along a vertical wall due to changes in planform as the channel evolves can be estimated by combining Equation 4.4 with the relationships for ideal meander geometry (see HEC20). Using these relationships, it can be shown that the maximum angle will vary from zero, when the width of the valley is constrained to the width of the channel, to approximately 71 ° , when the unconstrained valley width is approximately 3.5 times the width of the channel. These values are based on the assumption that the meander wavelength is 14 times the channel width. The resulting dimensionless scour depth as a function of the unconstrained valley width is plotted in Figure 4.2 for a range of Froude Numbers (F_{r}).
It is possible for the channel to impinge perpendicular to the wall due to local flow deflection or other local factors. For this case, the angle of impingement is no longer related to the valley width, and the maximum scour depth can best be estimated based strictly on Equation 4.1.
Figure 4.2. Scour along a vertical wall as a function of unconstrained valley width (RCE 1994).
In using Figure 4.2, it is important to recognize that the relationships are based on an assumed ideal meander geometry and scour relationships that, while they are the best available, are very approximate. Considering the extreme local variability that can occur in a given stream and the approximate nature of the relationships upon which these results are based, engineering judgment is critical in evaluating the reasonableness of the results for a specific problem. In particular, the potential for flow deflection and its effect on the angle of impingement on the wall should be considered and a conservatively large angle applied in Equation 4.4. If there is any reasonable possibility of flow perpendicular to the wall, an angle of 90 ° (thus, Equation 4.1) is recommended. When the results of this analysis are used to design the burial depth for a vertical wall, a safety factor of at least 1 ft (0.3 m) should be added to the predicted scour depth.
4.3.5 Scour at Protected Bendways
Deep sections at the toe of the outer bank of a bendway are the result of scour. High velocity along the outer bank is caused by secondary currents and greater outerbank depths, and together with the resultant shear stress, produce scour and cause a difference between the sediment load entering and exiting the outerbank zone. Since secondary currents transport sediment supplied, in large part, from outer bank erosion toward the inner bank of a bend, hardening of the outer bank by longitudinal bank protection structures may cause the channel cross section to narrow and deepen by preventing the recruitment of eroded outer bank sediments.
Experience is usually the most reliable means of estimating scour depth when designing a bank protection project for a particular stream. Lacking experience on a particular stream, scour depths may be estimated using physically based analytical models or empirical methods. Although scourdepth can be estimated analytically or empirically, empirical methods were generally found to provide better agreement with observed data.
Maynord (1996) provides an empirical method for determining scour depths on a typical bendway bank protection project. Although his studies are restricted to sand bed streams, the Maynord method agrees reasonably well with the limited number of gravelbed data points obtained by Thorne and Abt (1993). Nonetheless, the techniques presented by Maynord are restricted to meandering channels having naturally developed widths and depths, and cannot be applied to channels that have been confined to widths significantly less than a natural system.
Maynord's method of estimating scour depth is based on a regression analysis of 215 data points. The scour data used in developing his equation were measured at high discharges that were within the channel banks and had return intervals of 15 years. Maximum depth as defined in his bestfit equation for scour depth estimation is a function of R_{c}/W, width to depth ratio, and mean depth as follows:
 (4.5) 
where:
R_{c} 
= 
Centerline radius of the bend, ft (m) 
W 
= 
Width of the bend, ft (m) 
D_{mxb} 
= 
Maximum water depth in the bend, ft (m) 
D_{mnc} 
= 
Average water depth in the crossing upstream of the bend, ft (m) 
The terms D_{mxb} and D_{mnc} are defined in Figure 4.3.
Figure 4.3. Definition sketch of width (W) and mean water depth (D_{mnc}) at the crossing upstream of the bend and maximum water depth in the bend (D_{mxb}).
The applicability of Maynord's equation is limited to streams with R_{c}/W from 1.5 to 10 and W/D_{mnc} from 20 to 125 because of the lack of data outside these ranges. He recommends that for channels with R_{c}/W <1.5 or width to depth ratios less than 20, the scour depth for R_{c}/W = 1.5 and W/D_{mnc} = 20, respectively, be used.
In addition, Thorne and Abt (1993) suggest these methods are valid until there is significant interaction between the main channel flow and overbank flow. Therefore, Maynord (1996) recommends that application of these empirical methods to overbank flow conditions should be limited to overbank depth less than 20% of main channel depth.
4.3.6 Hydraulic Stress on a Bendway
The ratio of bend radius of curvature to flow width provides insight into the force on the meander bend margin, but this parameter does not include discharge. A quantitative technique which considers a singleevent discharge and an estimate of the radial stress on a meander bend margin was developed to evaluate the performance of alternative streambank erosion protection techniques for the U.S. Army Corps of Engineers, Vicksburg District (WET 1990). This technique could also be used by highway engineers to evaluate alternative channel instability countermeasures for a bridge located in a meander bend.
Begin (1981) defines radial stress as the centripetal force divided by the outer bank area. The centripetal force is responsible for deflecting the flow around the bend and is equal to the apparent reactive force of the flow on the bend. Based on this concept of centripetal force, the equation for the radial stress ( φ _{r}) of flow on a meander bend is:
 (4.6) 
where:
F 
= 
Centripetal force, lbs, (N) 
A_{b} 
= 
Area of outer bank, ft^{2}(m^{2}) 
ρ 
= 
Fluid density, lbs/ft^{3} (kg/m^{3}) 
Q 
= 
Discharge, ft^{3}/s (m^{3}/s) 
V 
= 
Flow velocity, ft/s (m/s) 
Y 
= 
Mean flow depth, ft (m) 
R_{c} 
= 
Radius of curvature, ft (m) 
W 
= 
Topwidth, ft (m) 
Thus, the radial stress is defined as a force per unit area (lbs/ft^{2} or N/m^{2}). Although it is not suggested that the radial stress is directly responsible for meander bend migration or failure of bank protection countermeasures, Begin did show that the radial stress is related to meander migration (Begin 1981). It is assumed that shear stress is related to radial stress because of water surface superelevation and increased nearbank velocity gradients.
Field investigations and computation of radial stress on banklines for channels in the Yazoo River basin in Mississippi clearly showed that rudimentary countermeasures, such as usedtire revetment were generally unsuccessful in bends with even low to moderate radial stress (WET 1990). The study also showed that stone structures including longitudinal stone dikes and stone spurs performed well in reaches of high radial stress. Isolated failures of stone structures did occur at locations with the highest radial stress. The 2year storm discharge was used in the computations for radial stress at these sites.
As an alternative, the increased shear force on the outside of bends can be calculated by multiplying the bed shear stress τ_{0} by a dimensionless bend coefficient K_{b}. The sharper the bend, the greater the shear stress imposed on the outer bank. The bend coefficient K_{b} is related to the ratio of the bend radius of curvature R_{c} divided by the top width of the channel T, as shown in Figure 4.4.
Figure 4.4. Shear stress multiplier K_{b} for bends (HEC15, 2005).

