River bank stabilization using rock riprap falling aprons

RIVER RESEARCH AND APPLICATIONS

River. Res. Applic. 25: 1036–1050 (2009)

Published online 19 November 2008 in Wiley InterScience

RIVER BANK STABILIZATION USING ROCK RIPRAP FALLING APRONS

DAVID C. FROEHLICH*

303 Frenchmans Bluff Drive, Cary, NC 27513, USA

(www.interscience.wiley.com) DOI: 10.1002/rra.1211

ABSTRACT

A rational mathematical model is presented that provides a sensible and realistic description of the physical processes that takeplace as a horizontal rock riprap apron (generally known as a ‘falling apron’) launches in stages and then gradually covers andprotects an eroding river bank. The model is a simplified kinematic description of complex riprap cover development that doesnot reference directly the forces that lead to stream bank erosion or apron deployment. The formulation accounts for theregularly repeated processes of slope erosion followed by rock settlement that take place along a receding stream bank. Ideasused to develop the model are based on published results of small-scale experiments of falling apron behaviour, and onexperience that has influenced current design practices. The model is uncomplicated and can be applied without difficulty toevaluate the adequacy of apron thickness, the amount of stone lost during deployment, the effect of rock riprap properties, andthe lateral extent of bank erosion before complete slope coverage is established. However, application is limited to river reachesthat are not highly curved. Copyright # 2008 John Wiley & Sons, Ltd.

key words: rock riprap; falling apron; bank stabilization; revetment; riprap

Received 24 August 2008; Accepted 17 September 2008

INTRODUCTION

Holding a river in place by protecting its bank against erosion with a continuous covering of loose stones or riprap is

a common engineering practice that is used throughout the world (Task Committee on Channel Stabilization

Works, 1965) When revetting a bank this way, the armour stone needs to extend below the lowest bed level expected

at the toe of the underwater slope as the streambed scours. The facing can be installed in two general ways: (1) By

excavating the bank so that armour material can be placed directly on a prepared slope, or (2) by placing a sufficient

quantity of loose stone along the top of the bank so that it will fall gradually and cover the slope as the bank recedes

and undermines the stockpile. The first approach might be preferable, but construction could be difficult or costly,

especially when underwater excavation is needed. The second method has been used widely to stabilize the banks

of both large and small alluvial streams by placing the rock supply on existing ground surfaces or in shallow

excavations.

Placing a supply of stone that will fall or launch down a slope when undermined is a technique that was used to

resist erosion in India throughout the second half of the 19th century when the first railway bridges were being built

across the Ganges River and its tributaries (Bell, 1890). Developed from many practical applications, broad

guidelines on how to apply the approach are given by Spring (1903), and the concept is shown in Figure 1, which is

nearly identical to Spring’s (1903) illustration, which has been reproduced in many references since. When

installed around a structure, such as a bridge pier, or along the base of an embankment, the deposited stone

resembles an apron, and, for this reason, is known as a falling or launching apron. As described concisely by Gales

(1920), who relied upon the technique when building river training works to control the Ganges River at the

Hardinge Bridge, ‘the design provides that, as the river cuts away the sand from under the outer edge of this apron,

the stone falls in and automatically pitches a slope in continuation of the slope of the bank’.

Current falling apron design practice [see, e.g. Brown and Clyde, 1989; U.S. Army Corps of Engineers

(USACE), 1994; Przedwojski et al., 1995; Biedenharn et al., 1997; Garg, 2002; Richardson et al., 2001; Schiereck,

*Correspondence to: David C. Froehlich, Consulting Engineer, 303 Frenchmans Bluff Drive, Cary, NC 27513, USA.E-mail: [email protected]

Copyright # 2008 John Wiley & Sons, Ltd.

Figure 1. Typical design diagram of a falling apron placed along the toe of a revetted embankment based on guidelines presented by Spring(1903)

ROCK RIPRAP FALLING APRONS 1037

2001; CIRIA, CUR, CETMEF, 2007] scarcely differs from the general principles presented by Bell (1890) and

Spring (1903), and applied by Gales (1920, 1938). Quantities of stone needed to provide adequate protective covers

on eroding banks are calculated based on a volume conservation principal that implies at best an unclear

understanding of the physical processes that take place after stone is launched and moves down a slope. Amounts of

stone swept off the slope by river currents, or that are effectively lost by burial in underlying sands, are taken into

account indirectly by increasing the assumed thickness of the final protective layer, or by adding a margin of safety

to the computed volume of launched material. The minimum thickness of an apron that is required to cover a slope

completely also is obtained indirectly as either a multiple of the diameter of stone needed to resist movement by

river currents, or as a multiple of the bank height.

A rational mathematical model that provides a reasonable representation of the physical processes that take place

as a horizontal rock apron launches in stages and then progresses down an eroding slope is presented. The model is

based on the results of published scale-model experiments of falling apron behaviour, and on practical experience

that has influenced current falling apron design theory. Because the model includes only basic aspects involved in

stone deployment and revetment development, it is presented in an uncomplicated and understandable form that

can be applied without difficulty.

CURRENT FALLING APRON DESIGN PRACTICES

Design conventions currently used to decide on rock volume requirements and shapes of falling aprons, gradation

of the rock mixture, and other design factors are summarized. This guidance has developed largely from practical

experience over more than a century.

General considerations

The fundamental idea behind placing a volume of rock at the top of an eroding river bank is that it will be

launched or deployed by itself without direct human control when the edge of the rock supply is eroded at its base,

allowing the material to fall downward along the slope to form a continuous protective cover. The best performance

occurs when aprons are laid upon channel banks composed of cohesionless soils where deep scour is expected and

comparatively uniform rates of launching can be expected (Biedenharn et al., 1997, page 151).

Falling aprons do not function well when placed on cohesive soils where stream banks erode in the form of

slumps with steep slip faces (Richardson et al., 2001, page 6.32). Such conditions were found in February 2002

along the upstream portion of the revetment at Sara, located on the left bank of the Ganges River about 5 km

upstream from the Hardinge Bridge in Bangladesh (Figure 2). Blench (1969, page 124) notes that where cohesive

Copyright # 2008 John Wiley & Sons, Ltd. River. Res. Applic. 25: 1036–1050 (2009)

DOI: 10.1002/rra

Figure 2. Horizontal falling aprons composed of rock riprap tend not to perform well when placed on cohesive banks where erosion occurs in theform of slumps with steep slip faces. These conditions were found in February 2002 along the upstream portion of the revetment at Sara, locatedon the left bank of the Ganges River about 5 km upstream from the Hardinge Bridge in Bangladesh This figure is available in colour online at

www.interscience.wiley.com/journal/rra

1038 D. C. FROEHLICH

soils are present, launched stone will not be able to come to rest on the eroded banks, thereby leaving the slopes

unprotected. He also points out that as scour deepens, cohesive banks may also fail by sliding, carrying all or part of

an apron with them to positions where the stone will likely be useless.

Apron volume

Falling aprons formed from rock riprap are often placed horizontally along the toes of embankments that are

themselves protected from erosion by a layer of rock riprap as shown in Figure 1. To estimate the rock volume

needed for the apron, the average thickness of bank coverage Tb is usually assumed to equal some multiple of the

thickness T of the loose rock facing placed on the upper embankment. If an upper embankment does not exist, or if

the upper revetment is made from a material other than what is used in the falling apron, then for design purposes T

equals the thickness of a protective cover that would be constructed on a prepared bank slope from the apron

material. Based on this notion, the volume of material that would be needed per unit length of an apron to cover an

eroding bank completely is given by

Vb ¼ TbHffiffiffiffiffiffiffiffiffiffiffiffi1 þ z2

p(1)

where H is the height of the bank that needs to be protected, or the maximum vertical distance that stone needs to

fall and z is the ratio of the horizontal distance to the vertical distance of the eroded underwater slope covered by the

launched material.

Spring (1903) recommends Tb¼ 1.25T to calculate needed rock volumes. If apron rock is placed underwater by

dumping, the India Central Board of Irrigation and Power (CBIP) (1971) suggests that Tb¼ 1.5T. The Federal

Highway Administration (FHWA) suggests that Tb¼ 1.25D100 be used to calculate the volume of apron rock

needed to provide bank coverage (Richardson et al., 2001, pages 6–31), where Di¼ equivalent spherical diameter

of Wi stone, and Wi¼weight of a stone that is heavier than i per cent of the stones in a mixture. However, In another

FHWA publication, Brown and Clyde (1989, page 42) propose Tb¼ 1.5T. The USACE (1994, pages 3–11)

recommends that T¼max{D100, 1.5D50, 0.3 m}, and that Tb¼ 1.5T, which gives Tb¼max{1.5D100, 2.25D50,

0.45 m}. Schiereck (2001, page 282) suggests that enough stone be placed in a falling apron to provide a bank cover

that has a minimum thickness Tb¼ (2.24–3.15)�D50, based on a covered slope where z¼ 2. Blench (1969, page

125) suggests Tb¼ 2D50, noting that ‘as stone tends to launch in a single layer there is a fair reserve in an apron’.

Other than Blench’s (1969) recommendation, whether or not other suggested values for Tb include a margin of

safety to account for stone loss during deployment and design uncertainties is not clear. Except for USACE (1994,


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Table I. Falling apron volume safety factors for various stream bank conditions�

Maximum vertical fall distance of apron stone, H (m) Safety factor, fs

Above water (dry) placement Underwater placement

�5 1.25 1.505 to 15 1.50 1.75

�After USACE (1994, pages 3–11).


pages 3–11), who suggest applying the safety factors given in Table I, no other design guidelines propose increasing

apron volumes. Several experimental studies (which are described in a following section) on falling apron

behaviour that were carried out in laboratory flumes all find that a protective layer only a single stone particle thick

develops on an eroding slope. For that reason, values of Tb greater than D50 used to calculate apron rock volumes in

conventional design procedures are considered to include margins of safety that are not expressed directly.

Consequently, the volume Vb given by Equation (1) is actually the recommended apron volume Va, which includes a

margin of safety.

The USACE safety factors in Table I provide for proportionally larger apron rock volumes as fall heights

H increase, implying that once launched some of the rock becomes ineffective or is ‘lost’ as it progresses down the

eroding river bank. Lack of suggested factors for H> 15 m implying that the USACE believes the design practice

should be limited to fall heights of 15 m or less, or at least that successful application at sites where H> 15 m is not

likely.

Apron shape and thickness

The general practice as recommended by Spring (Spring, 1903; and Gales, 1938), and as reported in many texts

[e.g. Blench (1969, pages 123–125), Garg (2002, page 485), or Przedwojski et al. (1995, page 456)], is to lay an

apron over a length La¼ 1.5H, as shown in Figure 1. Spring (1903) proposes that the apron thickness at the junction

of the apron and the constructed slope be the same as that of the upper slope covering, with the apron thickness

increasing towards the riverbed. In such a case the thickness of the river end of apron will be 2.25 times the

thickness of the riprap on the slope. This idea is endorsed by Verhagen et al. (2003) who suggest that constructing

an apron in the form of a wedge-shaped layer, with more material near the river bank, rather than as a horizontal

layer will offer more protection at the beginning of the deployment process.

In contrast to the longstanding beliefs noted above, the USACE (1994, pages 3–10) finds that the ‘shape of the

stone section before launching is not critical, but thickness of the section is important because thickness controls the

rate at which rock is released in the launching process’. For normal river bends where scour occurs gradually, they

propose that Ta¼ (2.0–4.0)� T. Where rapid scour occurs in impinged flow environments or in gravel bed streams,

they advise setting Ta¼ (2.5–3.0)� T to account for larger hydrodynamic forces. Using USACE guidance for T

based on D50 given previously, Ta¼ (3.0–6.0)�D50 for gradual scour, and Ta¼ (3.75–4.5)�D50 for rapid scour.

Underwater slopes

Bell (1890), Spring (1903), and Gales (1938), all rely on a 2:1 (horizontal:vertical) underwater slope in their

designs of falling aprons used to stabilize sand bed rivers in India. Richardson et al. (2001, page 6.31) suggest that

when channels are formed in cohesionless soils, falling aprons can be designed for underwater slopes of up to 2:1,

noting that scale model tests show that these slopes are realistic for sand beds. The USACE (1994, pages 3–10)

reports that rock launched on noncohesive material in both model and prototype surveys forms a 2:1 slope, although

they also find the launch slope is less predictable if cohesive material is present, since cohesive material may fail

in large blocks. From measurements at guide banks on various rivers in India and Bangladesh, Van der Hoeven


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(2002, page 35) finds that the actual slopes of launched aprons usually range from 1.5:1 to 3:1, but in some cases

may even be flatter, with the average about a 2:1 slope.

When falling aprons are created using concrete blocks instead of rock riprap (a common construction practice for

river training works on large alluvial rivers of the Indian subcontinent), the Bureau of Indian Standards (1994) and

the Indian Roads Congress (1997) specify an underwater side slope of 1.5:1, which is considerably steeper than side

slopes recommended for loose rock. Chitale (2006) questions the use of such a steep underwater slope for concrete

block, suggesting that it should be no different than the side slope used for loose rock riprap.

Apron stone gradation

Stone used in falling aprons usually consists of ‘graded’ stone (i.e. a mixture of a wide range of stone sizes) so

that the largest particles will resist hydraulic forces, and the smaller sizes will prevent loss of bank material through

gaps between the larger stones. Although, finer particles might be washed out from the top layer of the apron, the

belief is that enough fine particles will remain to prevent underlying sediments to be eroded through the fully

developed stone covering. Widely graded riprap is recommended by the USACE (1994, pages 3–10) for launching

because of reduced rock voids that tend to prevent leaching of lower bank material through the launched riprap.

They also suggest that apron stone be graded so that D85/D15� 2.

Effect of bank curvature

Along a river bank that is curved, the underwater slope that needs to be covered by rock riprap from a falling

apron is not a plane, as has been considered previously, but takes the form of an inverted conical frustum. As a

result, the farther down the slope a rock deployment stage falls, the more it has to spread out to cover the expanding

slope. No design guidance at all mentions the effect of bank curvature on apron deployment, and no suggestions for

increasing rock volume to account for the expanding area that needs to be covered was found in the literature.

Perhaps this is a reason that rock riprap aprons often fail to protect slopes adequately.

EXPERIMENTAL STUDIES

Several experimental studies that were carried out in laboratory flumes to understand better the way in which falling

aprons function as they are undermined by eroding streambanks are reviewed briefly. The earliest, and perhaps the

most informative and enlightening, were a series of studies carried out at the Central Irrigation and Hydrodynamic

Research Station (now known as the Central Water and Power Research Station or CWPRS), located in Pune, India,

in connection with protection of the Hardinge Bridge on the Ganges River (Central Board of Irrigation, 1939, pages

92–108).

The CWPRS experiments show that launching progresses slowly, simultaneously with the gradual increase of

discharge in the channel. As discharge and stage increase, the channel deepens and widens, and stones at the apron’s

edge that are undermined begin to settle and slide down the channel bank. Settlement of previously launched rock

then takes place in stages, the stones having fallen the farthest being undermined first and settling an additional

distance, thereby exposing the portion of slope they have just vacated. The newly exposed slope is eroded at the next

stage, undermining the rock above it, which is then able to fall and cover the exposed slope below. The process is

repeated, with the extent of slope coverage increasing as scour continues at the toe. Apron stones eventually form an

arrangement on the slope one particle thick that nearly shields the underlying sand completely.

Results of the CWPRS model tests confirm the notion gained from many constructed aprons that for satisfactory

launching the bed material needs to be scoured easily and evenly. If underlying soils are cohesive clays, or if the

bank consists of alternating layers of sand and clay, eroded underwater channel banks are often nearly vertical.

Based on experimental laboratory studies, Skrebkov et al. (1991) found that when stones from the apron are

undermined, they begin to slide down the bank, wobbling and turning as they move, but they do not roll. They note

that the protected banks form with a 2:1 slope with a more or less continuous rock cover along the bank that

gradually moves from the end of the falling apron towards the toe of the slope. They also found that the final

protective cover was uniformly one particle thick.


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Recently, two related experimental studies were carried out (Thiel, 2002; Van der Hoeven, 2002) to learn more

about the launching or setting process of the apron stone, and to determine the resulting slope of the protected

underwater river bank. For a stone mixture having a narrow grading (i.e. a well-sorted material), Van der Hoeven

(2002) reports that just a single layer of rock covers the scoured slope, and that z� 2. Although the covering did not

retain the underlying sand like a properly designed filter, it did slow erosion considerably. Rock settled evenly over

the entire slope, and no large areas of sand were exposed.

In a follow-up study, Thiel (2002) carried out experiments using poorly-sorted apron stone (i.e. a mixtures having

wide gradations containing particles both much smaller and much larger than the average size.) The tests show that

z� 2 as for uniform apron material, and that the launched stone is evenly spread over the entire slope from top to

bottom with an average thickness equal to the median diameter of the stone.

Summarizing the findings of Van der Hoeven (2002) and Thiel (2002), Verhagen et al. (2003) conclude that the

resulting protective cover stays limited to a single layer of granular material, and that applying a thicker apron does

not lead to the formation of a thicker protective layer on the slope. However, they find that a thicker apron will slow

the retreat of the apron edge. Also, they note that the rock size does not influence the angle of the covered slope,

which is nearly always 2:1 (horizontal to vertical).

Experimental findings of Van Der Hoeven (2002) and Thiel (2002) support the idea that as launched stones move

downward, they cannot protect a bank completely, otherwise the covered portion of the slope would be stable, or

nearly so, and would erode no further. Therefore, gaps between stones need to exist until the extent of coverage has

reached its limit, at which time the rock particles would continue to pack themselves neatly together, thereby

restricting additional erosion significantly.

All of the studies found that stable cover formed on eroding slopes were only a single particle thick. This idea is

in contrast to most riprap design schemes that require filters beneath the rock riprap to prevent underlying fine soils

from moving through the protective layers. Verhagen et al. (2003) emphasize the finding that the rock cover formed

by falling aprons does not support the filter requirements of conventional riprap revetment design guides. Although

covers only a single particle thick might not prevent complete movement of underlying soils through the layer of

rock, they seem to slow erosion to such a small level that it can hardly be noticed.

Kinematic model of falling apron behaviour

There have been only a few experimental investigations on rock riprap falling aprons, and guidelines available

for the design of this type of river bank stabilization practice can be summarized in Figure 1, which is merely a

repetition of the notions formalized by Spring (1903) more than a century ago. However, one clear finding of all the

studies is that the resulting protective rock covers are only one particle thick and, even without underlying filter

placement, can successfully halt bank erosion, or at least slow it to an almost imperceptible rate. For this reason, the

general design approach illustrated in Figure 1 is a concept that has been largely disproven by experiment, but

which, nonetheless, may still provide adequate apron designs.

Rock riprap falling aprons continue to be used to control river bank erosion, particularly at bridge crossings of

large alluvial rivers of the Indian subcontinent where transportation infrastructure expansion and improvement has

been taking place faster in recent years. However, the success of previously constructed falling aprons used to

stabilize river banks in this region has been inconsistent (see, e.g. Macrae, 1934; and Gales, 1938). Causes of

ineffective falling aprons are likely due in large part to the long-established design practices described previously,

which do not provide clear or quantifiable descriptions of falling apron development. For this reason, a

mathematical model is needed that represents the essential aspects of deployment or launching of a rock riprap

apron. Using the model, the effect of apron thickness and horizontal extent, fall height, and rock properties can be

assessed analytically, and the river bank stabilization scheme designed with a greater degree of confidence.

Such a mathematical model is developed here to represent the essential aspects of deployment or launching of a

horizontal rock riprap apron onto the side slope of a stream channel that is eroding both downward and laterally.

The model is intended to serve as a basis for estimating the rock volume needed to provide an effective protective

covering of stone on the eroded bank that will limit the extent of erosion, or at least reduce the rate of erosion

significantly. Ideas on which the model is based were drawn primarily from the concise narratives of falling apron

processes observed during scale-model experiments carried out at the CWPRS (Central Board of Irrigation, 1939,


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pages 92–108), along with the accompanying sketches, which are similarly clear and informative. Additional

intuitive understanding of apron launching was provided by the experimental findings of Van der Hoeven (2002)

and Thiel (2002), which are summarized by Verhagen et al. (2003), and by those of Skrebkov et al. (1991).

The model is based on the idea that a falling apron launches in stages as the outer edge is undermined by an

eroding river bank. Each stage results in deployment of a single layer of stone from the outer face of the apron,

which has length b¼ Ta/sina, and a thickness D¼ average rock particle diameter of the riprap mixture. The launch

volume per unit length of the apron is then Vs¼ bD.

Movement of a launch volume down the bank slope is also considered to take place in stages, and each stage is

divided into four steps. Step A consists of lateral erosion of the exposed slope immediately below the launch

volume by a horizontal distance D/sina. During Step B, the launch volume settles onto the newly eroded slope just

below it. In Step C, a second erosion episode takes place on the exposed slope below the launch volume, which is

followed by another settlement in Step D.

A launch volume is considered to remain continuous, having an average thickness D, as it moves or settles down

a slope. If no rock is lost through settlement, the length of a launch volume is constant and equals the length of the

apron face b. However, each time a launch volume settles it is likely that a portion of the riprap becomes buried in

the underlying soil, becoming effectively lost. Other stone might be swept away by river currents, also becoming

ineffective. The fraction of a launch volume that remains after each deployment step is denoted by k, where

0� k� 1. Considering k to remain constant throughout deployment, the length of a launch volume after i stages,

each with two settlement steps, is bk2i. The initial stage of apron launching (Stage 1) is illustrated in Figure 3.

At any stage before the initial launch volume reaches the bottom of the slope, a channel bank is only partially

protected. The cover consists of a series of launch volumes extending down the slope, each of which is separated

from the subsequent stage by a gap that has a length equal to that launch volume. Bank soils within the gaps are

exposed and erode laterally at the same rates. This idea is illustrated in the second stage of apron stone deployment

shown in Figure 3.

When the initial launch volume reaches the bank toe, no further downward movement takes place if the channel

ceases to scour. The following launch volume will then fill the intervening gap at the next settlement step. This

process repeats for all the subsequent launch volumes until the final volume deploys from the apron face and the

slope is covered completely by a rock layer of average thickness D. The final stage of apron rock deployment is

shown in Figure 4.

The total length of slope covered completely by n launch stages is then found as

B ¼ bXni¼1

ki ¼ bn; for k ¼ 1

bkðkn�1Þk�1

; for 0 < k < 1

�(2)

where B¼H/sin b, and b¼ tan�1(1/z)¼ initial and final bank angle. Solving Equation (2) for n gives

n ¼x; for k ¼ 1

log 1 � 1 � k

k

� �x

� �

log k; for 0 < k < 1

8><>: (3)

where

x ¼ H sina

Ta sin b(4)

is a dimensionless fall height parameter. As the fraction of rock volume retained at each stage k decreases, the

number of launch stages needed to obtain complete coverage of a slope increases. The number of launch stages n

reaches infinity when the numerator of Equation (3) for k< 1 vanishes. For this reason, channel banks where

conditions result in values of k less than

kmin ¼ x

1 þ x(5)


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Figure 3. Stages 1 and 2 of apron launching showing the four steps (erosion, settlement, erosion, settlement)


would never be able to be covered completely by stone deployed from the apron because of losses. Therefore, k

needs to exceed kmin by a sufficient margin to make certain that a protective cover can be attained.

As rock gradually settles down a bank after launching, more of the slope will be protected from erosion.

However, because of the cyclic pattern of cover and exposure upon which the kinematic model is based, the crest of

a bank will erode laterally twice as fast at the toe until the initial launch volume makes its way to the bottom of the

Figure 4. Final state of a falling apron considered by the kinematic model of stone settlement showing intermediate positions of the erodingstream bank at various stages of deployment


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slope, which occurs after m¼ n/2 rock volume launches. Erosion will continue at the bank crest until all the gaps

between successive deployments are filled and launching ceases, which occurs after n launches. Multiplying the

total number of launch stages times the horizontal width of a launch stage volume (D/sina) gives the total lateral

extent of bank erosion xe as

xe ¼ n� D

sina(6)

As a result of the unequal rates of lateral erosion at the bottom and top of a bank, the slope will steepen through the

first m launches, and then will decrease afterwards to produce a final bank angle b. The maximum bank angle that

can be sustained is limited by the mass angle of repose a of rock riprap forming the apron. At each launching stage

i<m, the lateral distance that a bank toe erodes equals 2D/sin a, and at the bank crest equals D/sina. As a result, to

maintain a bank angle that is always less than a, n needs to be less than nmax, where

nmax ¼ 2H

D

� �sina

tan b� cosa

� �(7)

The total volume of rock deployed from an apron is found by summing the launch volumes, which are the same for

each stage if the apron thickness is constant. The ratio of total volume of deployed rock to the total volume of rock

contained in the final protective bank layer is then found as

Vd

Vb

¼log 1 � 1�k

k

� �x

x log k

(8)

where Vb ¼ DB ¼ DH= sin b: The relation given by Equation (8), which is graphed in Figure 5 for several values of

k ranging from 0.75 to 0.975, shows that for a given k, the deployed stone volume increases for larger values of x.

By increasing the apron thickness Ta, thereby decreasing x, Equation (8) shows that the amount of stone deployed to

cover a slope completely is reduced, which confirms the USACE (1994) belief that thickness of an apron is the most

important design factor because it controls the rate at which rock is released in the launching process. However, a

thicker apron increases the vertical load acting on a bank, which increases the potential of bank failure by sliding.

Mass failure of a slope is always a concern, and needs to be evaluated for every apron design, taking into account the

effect of groundwater seepage.

The effect of riverbank curvature has not been taken into account in the preceding development, which applies

only to straight or nearly straight river banks. Extension of the kinematic model of falling apron deployment to

Figure 5. Graph showing relation between the ratio of deployed apron volume Vd to the volume needed to cover the eroded bank completely Vb

as a function of the fall height parameter x and the fraction of retained stone in a launch volume after each deployment step k


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Table II. Summary of the results of falling apron experiments carried out by Van der Hoeven (2002) and Thiel (2002)

Experiment no. Ta (m) H (m) x� Vd/Vb ky

FT01z 0.185 0.17 1.2 1.1 0.92FT02z 0.125 0.200 2.1 1.2 0.90FT03z 0.120 0.100 1.1 1.3 0.80FT04z 0.136 0.150 1.4 1.2 0.87EXP1x 0.110 0.150 1.8 1.7 0.76EXP2x 0.130 0.130 1.3 1.5 0.75

�The values a¼ 358 and b¼ 26.68 were used to calculate x for all experiments.yEstimated from Equation (7) based on values of x and Vd/Vb.zFrom Van der Hoeven (2002).xFrom Thiel (2002).


include curved underwater slopes will be covered in a following publication. This development will also provide a

means of designing falling aprons around bridge piers and abutments, as well as along the curved ends of guide

banks that lead flood flows safely through bridge openings.

Rock retainage fraction

Experimental studies on the behaviour of falling aprons carried out by Van der Hoeven (2002) and Thiel (2002)

provide some factual information that can be used to assess the amount of rock lost at each settlement step. Model

apron dimensions and stone deployment volumes determined from laboratory flume cross sections measured before

and after launching are summarized in Table II, along with the calculated value of k needed to reproduce the results

according to the kinematic model.

The retained fraction of stone after each deployment step was found to be nearly constant for Van der Hoeven’s

(2002) four experiments, which were carried out using well-sorted apron material, with k ranging from 0.80 to 0.92.

However, Thiel’s (2002) two following tests using poorly-sorted apron stone yield estimates of k of about 0.75,

which is significantly less than the values found from the previous results. Larger loss of widely graded material

found by Thiel (2002) might be due to greater erosion of exposed small particles from the surface of the deployed

stone mixture, or the inclination of comparatively small stones to become buried beneath larger ones, thereby

helping to prevent underlying sands from migrating through voids in the surface layer as intended. However, Thiel

(2002) does not address this aspect of protective cover development. While the ratios of scale-model fall heightH to

apron thickness Ta, and thus the ratios x, are small for both sets of experiments in comparison to those from typical

falling apron constructions, estimates of k based on the results are probably reasonable for larger values of x as well.

However, more experiments need to be carried out to quantity stone loss rates more precisely.

Additional insight on the amount of rock lost at each deployment step can be obtained from the margins of safety

implied in commonly used falling apron design procedures. From the ratios Tb/D that can be derived from the

various conventional design criteria described previously for D¼D50, Vd/Vb ranges from 2 to 3.15. To estimate k,

we assume that a¼ 408 and b¼ 26.68, and consider the ratio H/Ta to range from 2 to 8 based on our experience. For

Vd/Vb¼ 2, k¼ 0.80 when H/Ta¼ 2, and k¼ 0.94 when H/Ta¼ 8. For Vd/Vb¼ 3.15, k¼ 0.76 when H/Ta¼ 2, and

k¼ 0.92 when H/Ta¼ 8. Therefore, based on conventional design guidelines, k can reasonably range from 0.76 to

0.94.

From the published results of experimental studies, and from common design practice, we hypothesize that

k ranges from 0.75 to 0.95. Smaller values of k apply to conditions that give rise to greater rock loss, such as uneven

launching of stone due to the presence of cohesive soils beneath the apron, rapid erosion of the bank, the tendency

for bank slides to occur, or the use of poorly sorted apron stone. Larger values of k apply where bank erosion takes

place gradually, bank slides are not likely to occur, and well-sorted stone is used in the apron.

Additional experiments are needed to establish more precise estimates of k, taking into account the

hydrodynamic factors and soil properties that control the speed of bank erosion and the rapidity of stone launching.

Also, the notion of a varying degree of rock retainage (i.e. a non-constant value of k) as an apron deploys needs to be

investigated. Rock that becomes buried in underlying soils along the upper portion of an eroding slope will likely


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reduce future losses. This physical process needs to be studied carefully in controlled small-scale laboratory

experiments.

Apron volume safety factor

While the falling apron model presented here is considered to be fairly accurate, it is not exact. When calculating

the rock quantity needed for an apron, the estimate of deployed volume found from (8) needs to be multiplied by a

safety factor fs (i.e. Va¼ fsVd) to account for uncertainty of the falling apron model, imperfections in materials, flaws

in construction, and revetment degradation over time. Appropriate safety factors depend on the accuracy of design

load calculations (i.e. calculated water velocities and depths, and the shear stresses that they produce on stream

channel beds and banks) and the channel scour depths that result, the consequences of failure, and the cost of over-

designing the apron to achieve the desired factor of safety. Skrebkov et al. (1991) suggest that fs¼ 1.5 at a

minimum. We propose that the safety factors recommended by the USACE (Biedenharn et al., 1997, page A-32;

and USACE, 1994, pages 3–11), which are presented in Table I, be used.

Design procedure

The essential features of designing a horizontal falling apron composed of loose rock to protect the bank of an

alluvial river from erosion using the kinematic model are described in the following step-by-step procedure:

Step 1. Based on hydrologic, hydrodynamic, and geomorphic calculations, estimate the maximum depth H to

which the river bed will scour at the toe of the bank.

Step 2. Based on expected bank erosion rates (rapid or gradual), the construction environment, and other factors

that affect uncertainty of design calculations, select an apron volume safety factor fs.

Step 3. Calculate the stone size D that will be needed to resist movement by river currents at any location on the

bank slope.

Step 4. Estimate or measure the mass angle of repose of the apron stone a, and the bank slope angle b.

Step 5. Calculate nmax from Equation (7) using a, b, H and D.

Step 6. Estimate k¼ fraction of retained stone at each deployment step.

Step 7. Select an initial apron thickness Ta. A good choice is Ta¼ 0.2H.

Step 8. Calculate x¼ (H sina)/(Ta sin b).

Step 9. Calculate kmin¼x/(1 + x). If k� kmin, then increase Ta and go to Step 8.

Step 10. Calculate n¼ number of launch stages needed to cover the slope completely from Equation (3) using k

and x. If n \ge nmax, then increase Ta and go to Step 8.

Step 11. Calculate Vd¼ nDTa/sina¼ total volume of stone deployed from the apron.

Step 12. Calculate Va¼ fsVd¼ volume of apron material needed per unit length of the revetment. If sufficient

space is not available at the construction site to contain Va, then increase Ta and go to Step 8.

Step 13. Finally, evaluate the potential for slope failure due to sliding, taking into account groundwater seepage

affects on slope stability. A flowchart illustrating the design procedure is provided in Figure 6.

The falling apron model developed here applies only to supplies of rock riprap placed horizontally with a

constant thickness Ta. Other configurations of rock supply, particularly those with larger initial deployment

thickness, can be analyzed using the same approach. However, solutions are not likely to take on the uncomplicated

analytical form given by Equations (2)–(4). Instead, solutions can be carried out numerically, or with the aid of

graphs developed for particular apron arrangements. Additionally, care needs to be taken when applying the design

procedure to river banks that highly curved. Along these banks, the covered slopes, which are in the shape of conical

frustums, expand with depth. For this reason, larger quantities of rock will be needed than for straight river reaches.

Falling apron deployment on curved slopes is a topic that will be dealt with in a following analysis.

Ideas contained in the outlined procedure have been used to estimate volume and thickness requirements for a

falling apron constructed along the toe of a revetted embankment whose purpose is to hold the left bank of the

Ganges River at Sara, about 5 km upstream from the Hardinge Bridge in Bangladesh (Figure 7). Built in 2002 as a

replacement for a deteriorated revetment, the rock riprap falling apron shown in the photograph had functioned well


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Figure 6. Flowchart of horizontal rock riprap falling apron design procedure


throughout two monsoon floods, during which the apron had been covered by swiftly flowing water more than 4 m

deep. Stone had deployed uniformly from the apron, with only a few minor irregularities.

Example application

An uncomplicated example is presented to illustrate how the model might be used to evaluate the suitability of a

proposed falling apron. Using previously calculated values of water velocities and depths, bank shear stresses, and

channel scour depths, a river bank of height H¼ 10 m will need to be protected by rock riprap deployed from a

horizontal apron. The bank angle b¼ 26.68 is assumed based on previous experience and general rules currently in

practice. Riprap sizing relations applied to rock having a mass angle of repose a¼ 408 give D¼ 0.3 m. Because

bank erosion will take place gradually, the assumed rock retainage fraction k¼ 0.9. Additionally, the horizontal

distance from the top of the existing bank that can be used to provide space for the falling apron cannot exceed 15 m

at the site.

Alternative A. An initial apron thickness Ta¼ 1.5 m is considered, giving x¼ (10� sin 408)/(1.5�sin 26.68)¼ 9.58. The smallest feasible value of k is then kmin¼ 9.58/(1 + 9.58)¼ 0.905. Because k< kmin, a

protective rock riprap cover will never extend completely to the toe of the bank because of losses. Consequently, the

apron design would not be satisfactory. To improve performance, we need to decrease kmin either by reducing the

height of the bank that requires protection from erosion, or by increasing the volume of rock launched at each stage.

Alternative B. By increasing the apron thickness to Ta¼ 1.8 m, we obtain x¼ (10� sin 408)/(1.8�sin 26.68)¼ 7.99 and kmin¼ 7.99/(1 + 7.99)¼ 0.889. Because k> kmin, launched riprap can spread all the way


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Figure 7. Sara revetment near the downstream end in April 2004 showing excellent performance of the falling apron since being rebuilt in 2002.Rock has deployed uniformly throughout two annual monsoon floods during which depth of water flowing over the apron had exceeded 5 m This

figure is available in colour online at www.interscience.wiley.com/journal/rra


down to the toe of the bank before the cover disintegrates due to losses. From Equation (3), the number of launch

stages needed to cover the slope completely is found as

n ¼log 1 � 1�k

k

� �x

log k

¼log 1 � 1�0:9

0:9

� �7:99

log 0:9

¼ 20:8 (9)

and the maximum allowable value of n is

nmax ¼ 2H

D

� �sina

tan b� cosa

� �¼ 2

10

0:3

� �sin 40�

tan 26:6� � cos 40��

¼ 34:7 (10)

which far exceeds n, so the likelihood of the bank becoming too steep as it erodes is small. The total volume of stone

deployed per unit length Vd¼ nDTa/sin a¼ 20.8� 0.3 m� 1.8 m/sin 408¼ 17.5 m3 m–1, and the lateral extent of

erosion xe¼ nD/sina¼ 20.8� 0.3/sin 408¼ 9.71 m, which is less than the 15 m available. The apron could be

extended at the same thickness to provide rock volume amounting to about 55% of the quantity that is expected to

be deployed, which provides a volume safety factor fs¼ 1.55.

Alternative C. With Ta¼ 1.5 m as in Alternative A, the bank is excavated so that the apron can be placed 2 m

lower, giving H¼ 8 m, x¼ (8� sin 408)/(1.5� sin 26.68)¼ 7.67, and kmin¼ 7.67/(1 + 7.67)¼ 0.885. The

maximum allowable number of stages nmax¼ 27.7. From Equation (3), n¼ 18.1, which is substantially less

than nmax, giving Vd¼ nDTa/sin a¼ 18.1� 0.3 m� 1.5 m/sin 408¼ 12.7 m3 m–1 and xe¼ nD/sin a¼ 18.1� 0.3/

sin 408¼ 8.45 m. Sufficient space exists at the site to provide a comfortable margin of safety by extending the apron.

This alternative is acceptable, and it requires only 73% of the volume needed for Alternative B.

Example summary

The kinematic model provides a well-reasoned means of evaluating the essential variables of falling apron

design. The example shows how a acceptable design can be brought about by modifying the level at which the apron

is placed, and thereby the height of the slope that needs to be protected, or by increasing the apron thickness, and as

a result increasing the volume of stone brought into effective action at each launch stage.

By varying the apron thickness and placement level from values used in Alternative A, both Alternatives B and C

provide adequate designs. The total amount of launched stone covering the bank with an average thickness of one


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rock particle diameter D would be the same for all variations of the example, but more rock is lost during

deployment for Alternative B, and bank excavation is needed in Alternative C. The expected apron volume that will

be deployed is 17.5 m3 m–1 for Alternative B, and 12.7 m3 m–1 for Alternative C, a 38% difference. Choosing

between the two feasible designs would likely depend on cost, the margins of safety that can be provided by each,

and possibly on other factors that are unique to the construction site.

SUMMARY AND CONCLUSIONS

The idea of letting rock deploy on an eroding stream bank under the action of gravity alone is used often,

particularly along large alluvial rivers. Alternative design approaches would frequently require that large amounts

of excavation, both above and below water, and underwater placement of the protective rock covers be carried out.

Placement of stone at a higher elevation reduces the volume of excavation that will be needed, and is less likely to

call for expensive dewatering measures or difficult underwater excavation. However, higher placements require

more stone because larger areas of underwater slopes will have to be protected. Placing stone closer to its final

position (i.e. at lower elevations) increases the likelihood of successful falling apron performance, particularly if

the bank contains cohesive material that is likely to retreat by means of irregular mass failure along the protected

slope rather than from comparatively uniform erosion. For these reasons, falling apron revetments are compromises

between economics and performance.

Guidelines for designing falling aprons proposed by Spring (1903) have scarcely changed in more than a century,

and are essentially the same as those proposed by the Central Board of Irrigation and Power (1971), the FHWA

(Brown and Clyde, 1989), and the USACE (1994). All of the design approaches rely on a vague notion of how bank

slope coverage develops as a rock apron deploys, and they all lack quantitative detail.

A consistent finding of all experimental studies on falling apron development reviewed (Central Board of

Irrigation, 1939; Skrebkov et al., 1991; Thiel, 2002; Van der Hoeven, 2002) is that the cover layer that forms on an

eroding slope is only a single rock particle thick. This notion is in contrast to most riprap design schemes that

require filters beneath the rock covers to prevent underlying fine soils from moving through the protective layer.

Although thin riprap covers might not prevent complete movement of underlying soils through the layer of rock,

they appear to slow erosion to such a small degree that it can hardly be noticed. When significant subsidence of a

section of an established cover layer does occur, the remaining apron supply is available to replenish the disturbed

areas with protective stone.

To provide a more precise way of designing falling rock riprap aprons, a rational mathematical model has been

formulated that is based on a kinematic description of the apron deployment process. Using the model, the complex

process is simplified and separated into its constituent parts in order to examine them, draw conclusions, and

thereby make reliable predictions. The model, which is based on the findings of several qualitative and quantitative

small-scale laboratory studies of horizontal apron deployment carried out by others, provides a sensible and

realistic quantitative description of the physical processes that take place as a horizontal rock apron launches in

stages. Along with experience gained over more than a century that has influenced current design practice, the

experimental findings are used to develop the kinematic model of apron development, and to evaluate a design

parameter that accounts for rocks that become ineffective as they settle down a slope.

Adequacy of falling apron thickness, the amount of stone lost during deployment, and the lateral extent of bank

erosion before complete slope coverage is established are factors that now can be assessed using the mathematical

formulation developed here. The model is uncomplicated and can be applied without difficulty to size rock riprap

falling aprons used to stabilize eroding river banks. However, the model is limited to river reaches that are straight,

or nearly so, because it does not take into account slope expansion with depth that occurs on a curved surface that

would form otherwise.

Additional experimental studies are needed to evaluate the rock retainage fraction k more precisely by explaining

the effects of hydrodynamic forces, soil properties, and geometric configurations that control the rapidity of apron

stone launching and the development of stable protective covers on eroding bank slopes. Detailed case studies on

the performance of constructed falling aprons over a period of many years are also needed to assess design

procedures more thoroughly.


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River bank stabilization using rock riprap falling aprons

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