A NEW ANALYTICAL BRIDGE PIER SCOUR EQUATION

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1 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt 587 A NEW ANALYTICAL BRIDGE PIER SCOUR EQUATION Youssef I. Hafez Associate Professor Hydraulics Research Institute, El Kanater, Egypt E Mail: Astract When a ridge is uilt across an alluvial channel, the ostruction of the flow y the ridge piers induces higher velocities and vortices that cause scour of the channel ed around the piers. If this scour reaches the foundation level of the ridge piers, the ridge might collapse. Bridge pier scour is the leading cause of ridge failure. In Egypt, concerns aout ridge pier scour is one of the important reasons for limiting the increase of the current flow releases from High Aswan Dam aove the current maximum. Due to the importance of ridge pier scour, many investigators have worked on this critical suject ut most have uilt their analysis on laoratory flume data that have simplified conditions and scale effects. When applying the existing empirical equations for predicting ridge pier scour to field cases, the scour depths are over-predicted which means increased construction costs. A new analytical equation for predicting ridge pier scour is developed herein ased on an energy alance theory. The developed equation expresses equilirium ridge pier scour depth in terms of flow velocity, flow depth, ed sediment specific gravity and porosity, ed sediment angle of repose, pier width over channel width ratio, and a momentum transfer coefficient. The equation has the advantages that it explains the physics of ridge pier scour in a direct way, relates the flow hydrodynamics to scour and most of all avoids the wide pier prolem. The developed equation yielded much superior agreement with field data than any other existing empirical equation when applied to the average and maximum of 515 field data points. This was also the case in the application of the equation to ridge pier scour for two ridges near Cairo, Egypt; namely Imaa and El-Tahreer ridges. Key Words: Bridge Pier, Sediment Transport 1. Introduction: When a ridge is uilt across an alluvial channel, the ostruction of the flow y a ridge pier induces higher velocities and horseshoe vortices that cause intensive sediment transport out of the area around the pier and consequent scour of the channel ed around the pier. If this scour reaches the foundation level of a ridge pier, the ridge might collapse. Bridge pier scour is the leading cause of ridge failure. In the United States alone, ridge pier scour is the leading cause of failure among more than 487,000 ridges over watercourses (Landers and Mueller 1996). In Egypt, concerns aout ridge pier scour is one of the important reasons for limiting the increase of the current flow releases from High Aswan Dam (HAD) aove the current maximum of 270 Million m 3 /day. In

2 588 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt order to release higher flows than the current maximum, knowledge is required aout how much scour is expected around ridges uilt on the Nile River. Other scour types such as general scour, constriction scour, end scour, autment scour and ank scour are equally important, however they are not discussed at this stage. Due to the importance of ridge pier scour, many investigators have worked on this critical suject ut most have uilt their analysis on laoratory data; for example: Laursen (1956) at the University of Iowa and Shen et al. (1969) at Colorado State University. Chang (1988) reports: more than 10 different formulas have een developed for predicting local scour around ridge piers, ased on essentially laoratory data. This empirical approach suffers from its associated simplified conditions and scale effects. When applying the existing empirical equations for predicting ridge pier scour to field cases, the scour depths are over-predicted (Baaeyan-Koopaei and Valentine, 1999). This means increased construction and maintenance costs as the foundation levels are required to e deeper than it should e. In this study, application of several well-known ridge- pier-scour prediction equations is implemented in addition to testing a newly developed analytically ased equation with the ojective of predicting realistically the scour depth. The already existing equations used herein are those of: The Modified Laursen y Neill (1964), Shen et al. (1969), The Colorado State University (1975), Jain and Fischer (1979), and Modified Froehlich (1999). The new analytical equation for predicting ridge pier scour is developed y Youssef I. Hafez ased on an energy alance theory. It is the ojective of this paper to find out which of the existing formulae works for the Nile River and how well the newly developed formula performs?. Utilization is made of valuale field data aout local scour at Imaa and Tahreer ridges at Cairo, Egypt and the average and maximum of 515 field data points reported in Johnson (1995). 2. Bridge-Pier- Equations: In this section a list is made of the ridge pier scour equations used in this study. In all the formulae listed elow, it is assumed that the flow angle of attack is negligile and that the pier shape is rectangular. Chang (1988) reports that the scour depth of circular piers is 90% of that for rectangular piers and 80% of that for sharp-nosed piers. As the flow angle of attack increases the scour depth, it will e assumed that the effect of flow angle of attack and the circular shape of the pier will mutually cancel each other. This is done ecause of the lack of information aout these factors in the data The Modified Laursen y Neill (1964) Equation: Neill (1964) used Laursen and Toch s (1956) design curve to otain the following explicit formula for the scour depth: D s H = ( ) (1) Where D s is the equilirium scour depth, is the ostruction width (or pier width) and H is the approach water-depth. This equation does not include the Froude numer or in other words the velocity of the attacking stream.

3 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt Shen et al. (1969) Formula: Shen et al. (1969) used the Froude numer in their scour-depth prediction in addition to the pier width as given y: D 2 1 s H 3 3 = 3.4( Fo ) ( ) (2) Where F o is the Froude numer and the other variales are as defined efore. 2.3 The Colorado State University or CSU Formula (1975): This equation is developed as a est fit to the data (laoratory) availale at the time. The formula is given as: D s = ( ) ( Fo ) (3) H H The CSU (1975) formula is similar in form to Shen et al (1969) equation. Later on correction factors were added for effects of flow angle, pier shape and ed conditions. 2.4 Jain and Fischer (1979) Equations: Jain and Fischer (1979) developed a set of equations ased on laoratory data. For (F o -F c ) > 0.2, the formula reads as: D H Where F c is the critical Froude numer. For (F o -F c ) < 0.2, the formula is D s = 2.0( Fo Fc ) ( ) (4) H s = 1.84 ( Fo ) ( ) (5) 2.5 Modified Froelich (1999) Formula: Fischenich and Landers (1999) modified Froelich s (1988) equations for live-ed scour at ridge crossings as Ds θ = 2( ) ( ) Fo + 1 (6) H 90 H Where θ is the angle of flow attack (degrees). This equation does include a safety factor (+1.0) that accounts for contraction scour in most cases. To compare this formula with other formulae, this factor will not e considered, as only local ridge pier scour is considered herein. 2.6 Youssef I. Hafez Analytical Equation: An analytical equation is developed y Youssef Hafez using energy alance theory. The energy alance theory assumes that at the equilirium geometry of the scour hole, the work done y the attacking fluid flow upstream the ridge pier is equal to the work done in removing the volume of the scoured ed material. In other words the energy contained in the fluid flow attacking the ridge pier is converted to an energy consumed in removing or transporting the ed material, thus forming a scour hole. When all the flow

4 590 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt energy is consumed in transporting the sediment out of the scour hole, scour ceases and the scour hole ecomes stale and at its maximum scour-depth. The following assumptions are made: (1) the shape of the upstream slope of the scour hole in the stagnation vertical symmetry plane is linear, i.e. the scour hole has a triangular shape, See Fig. 1 (2) the equilirium scour hole has an upstream slope that is equal to the angle of repose of the ed material, (3) The scour hole is formed due to the conversion of the horizontal momentum of water coming to the pier to downward or vertical momentum attacking the ed surface, See Fig. 1a (4) The down flow component is responsile for transferring the momentum of the attacking fluid to the ed material particles which is raised or transported to the original ed level and carried away y the stream currents or horseshoe vortices (5) the analysis is done for a jet thickness of one sediment particle diameter which is close to working only in the stagnation symmetry plane (plane stress type analysis), (6) the moment arm of the horizontal force of the attacking flow is half the water depth plus half the scour depth, and (6) the volume of the scoured ed which is assumed triangular in shape is moved to the original ed level out of the scour hole with equivalent vertical distance equal to 1/3 of the scour depth, see Fig. 1. The work done y the fluid flow of the horizontal jet coming from upstream the ridge pier in the stagnation symmetry plane can e expressed as 2 2 ρ V X H d η H D + s (7) B Where is the fluid (water) density, V x is the longitudinal flow velocity of the jet attacking the ridge in the direction normal to the pier, H is the water depth, d is the ed material sediment diameter, is a transfer coefficient of the horizontal momentum into a vertical momentum in the downward direction, is the pier width, B is the channel width in case of one pier or the ridge span or pier centerline to centerline distance in case of multiple piers and D s is the maximum or equilirium scour depth. The work done in removing the ed material from the scour hole is equal to: 1 Ds Ds ( D s d) (1 θ ) ( γ s γ ) (8) 2 tan Φ 3 Where is the slope in of the scour hole in the symmetry plane (assumed here equal to the ed material angle of repose), θ is the ed material porosity, s is the ed material unit weight and is fluid unit weight. Under the conditions of equal work at the equilirium conditions (or maximum scour) equations 7 and 8 are equated and after some manipulation yield: 2 2 D 3 s 3 tanφ 1 V D ( ) η x s = (1+ ) H ( S 1)(1 ) G 2 (1 ) g H (9) θ H B

5 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt 591 where S G is the sediment specific gravity. Equation 9 expresses the equilirium ridge pier scour depth in terms of the local velocity, local flow depth, ed material specific gravity and porosity, ed material angle of repose, pier width over channel width ratio, and a momentum transfer coefficient. The Froude numer can e easily made to appear in Eq. 9. The equation has the advantages that it explains the physics of ridge pier scour in a direct way, relates the flow hydrodynamics to scour and most of all avoids the wide pier prolem. Equation 9 is a cuic non-linear equation. Though a closed form expression for the scour depth could e otained, a few iterations could e used instead to solve for the scour depth. Fischenich and Landers (1999) state that Though not addressed y most empirical relations, the ratio of ostruction width to channel width is proaly a etter measure of scour potential than is the ostruction width alone. Indeed such is the case in Eq. 9 where the ratio of ostruction width to channel width (/B) appears in the equation. In Equations 1-6, the ostruction width has a direct effect on the scour depth, the wider the ostruction the deeper the scour. However, for ostructions having large widths, Eqs. 1-6 would predict consideraly larger scour depths than would e practically existing which leads to the wide-pier prolem. Eq. 9, however, does not suffer from the wide-pier prolem ecause the ostruction width is not directly related to the scour depth. The comination (η V x = V z ) reflects the down flow component of the velocity caused y the pier ostruction where V z is the vertical down flow velocity in front of the pier. Indeed, the down flow is responsile for the upstream scour and should e determined from 2D or 3D numerical modeling or experimental measurements. For the sake of simplicity herein, the η factor that represents the transformation of the incoming or approach horizontal momentum to vertical downward momentum will e assumed ased on personal judgment. It could e postulated that Eq. 9 is also valid for predicting scour depths downstream ridge piers when a proper selection of the momentum transfer coefficient is made. In this case η represents the transformation of the momentum from either of the horseshoe or wake turulent eddies to the ed material particles. Indeed this is an added advantage of Eq. 9 over Eqs. 1-6 that seems to predict only upstream scour depths. Another added advantage of Eq. 9 is that the ed material characteristics appear through inclusion of the ed material specific gravity, porosity and angle of repose. Regarding sediment size effect on the scour depth, Fischenich and Landers (1999) state that sediment size may not affect the ultimate or maximum scour ut only the time it takes to reach it. Therefore, it is no surprise that Eq. 9 does not contain the sediment size. 3. Verification of the Bridge Pier Equations with Field Data: 3.1. Case of Imaa Bridge at Cairo, Egypt: Imaa ridge (Km from Aswan Dam) is located several kilometers upstream the Delta Barrages (km 946 from Aswna Dam) north of Cairo. Imaa-ridge has seven piers each with 3 m diameter except pier No. 6 that has diameter of 10 m, Fig. 2. The distance etween the piers is aout 65 m. The scour holes of Imaa-ridge have een going under detailed monitoring programs y The Hydraulics Research Institute (HRI)

6 592 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt since The monitoring programs reveal that the scour holes at Imaa-ridge and neary El-Tahreer-ridge are stale under current flow conditions after HAD (HRI 1993, HRI 1997). This is ecause the peak flood flows released from HAD are dramatically reduced after its completion in Therefore, these scour holes are elieved to e resulting from very large historic flows occurred efore HAD. It is true that the higher the flow discharges, the deeper the scour holes around ridge piers. This can e seen from Eqs. 1-6 and Eq. 9 where the Froude numer and the velocity (oth are related to the discharge) appearing in the equations are proportional to the scour depth. Therefore, the historic data are searched for maximum flow conditions that are elieved to cause scour. Historic data (Nile Research Institute 1992) show that the peak of the flood season occurs in the month of Septemer every year. The data in the period from 1923 to 1968 reveal the highest Septemer flood to occur in 1959 with a monthly total of 31.0 Billion m 3 at Aswan (Km 7.0). Converting this value to a daily value assuming uniform conditions, the corresponding maximum daily flow is aout 1033 Million m3/day (11,960 m 3 /s) at Aswan. This value is to e compared to the current maximum of 270 Million m 3 /day. It is elieved that Imaa-ridge was uilt efore 1959, therefore the 1959 flood or a similar flood with this magnitude could have caused scour at Imaa-ridge. For example, the Sept. flood in 1964 had a total volume of 27.9 Billion m 3 /month. Now it is required to find out how much of the flood discharge reached the site of Imaa ridge in 1959?. This can e answered y making use of existing records for the year 1962 at a neary gauging station, namely at El Ekhsas (Km 887 from Aswna Dam). In 1962 the Sept. total flow at Aswan was 24 Billion m 3 which amounts to nearly 800 Million m 3 /day (9259 m 3 /s) while a measured flow at El Ekhsas of 7896 m 3 /s (El Moatassem 1985) was recorded in Sept. 10, From the 1962 records the ratio of El Ekhsas flow to that of Aswan flow ecomes This ratio is assumed to hold for the 1959 flood from which the flow at El Ekhsas or Imaa is 0.85 * 11,960 m 3 /s = 10,202 m 3 /s (881.5 Million m 3 /day). The 1996 field survey cross sections at Imaa (cross section No. 9 in Hydraulic Research Institute Report 1997) show a ank level of aout 20.0 m and average ed level of 10.0 m. The ank level of 20 m is assumed to e the highest possile water level at this site as could also e seen in Fig. 3. Therefore a water depth of aout 10.0 m could e assumed corresponding to the estimated flood discharge of 10,202 m 3 /s. The 1962 data at the neary El Ekhsas show water depths in the order of 10.0 m, thus supporting the aove finding. The average channel width at Imaa-ridge could e estimated from Fig.2 or Fig. 3 as 428 m. Based on the foregoing data, the average approach flood flow velocity at Imaa ridge site is estimated as (10,202) / (428*10) = m/s. The most recent scour investigation at Imaa-ridge is that y HRI in 1997 while the oldest one might e in No data aout the dimensions of the scour holes at pre- HAD flood conditions exist. This makes it difficult for otaining the exact dimensions of scour holes that are thought to e resulting from the 1959 flood or from a different flood with nearly the same magnitude. The deposition that might have occurred upstream the piers after HAD might have caused change in the scour holes dimensions after the time of

7 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt 593 their formation. Deposition of the Nile ed load occurs ecause Imaa ridge is upstream the Delta Barrages where ack-water-effects occur. Before HAD, all gates of the Nile arrages used to e opened without ack-water-effects. Therefore, some assumptions must e made to cope with this situation. Downstream the piers, little sediment deposition can e expected and therefore it can e assumed that the scour hole dimensions resemle conditions at the time of scour formation. The scour equations are therefore assumed valid for predicting the downstream scour depths. Fig. 2 shows the contours of the scoured ed around the piers where the enclosure of the contour lines around piers 1-4 supports the close equivalence etween the upstream and downstream scour depths at Imaa-ridge. Fig.3 shows cross section No. 11 just downstream Imaa-ridge where the scour holes at piers 1,2,3 and 4 (numering start from the right ank facing the downstream flow direction) are clearly feasile. The transverse ed slope pertinent to curved river reaches appears clearly and local scour depths at piers 1-4 are superimposed on it. A large scour hole to the left enveloping piers 5, 6 and 7 is the result from flow curvature (end scour), contraction scour, local ridge pier scour and scour due to flow convergence around the upstream Island of Zamalek. As there are several factors and mechanisms affecting this large left scour hole, its formation is very complex and eyond the scope of the scour predictive equations cited herein. In addition, the scour holes near piers numer 5,6 and 7 are away from the ridge and appear to e filled with filling materials and have floating fenders that make determining their original dimensions difficult. The scour holes around piers 1,2,3 and 4 of Imaa-ridge are clearly feasile and have depths of aout 1.90 m, 3.29 m, 4.29 m, and 5.25 m, respectively as seen in Fig.3. Tale 1 shows the field data extracted from Fig. 3 and from the assumptions in the foregoing and coming discussions where a water level of 20.0 m is assumed and an average ed level of 11.0 m at the ridge. Consequently the centerline depth at the ridge ecomes 9.0 m (level 20.0 level 11.0 m). Tale 1. Data used in Hole Predictions at Imaa Bridge Pier No. Local Bed Level Local Water Local Velocity (m/s) Local Froude Numer Pier No Pier No Pier No Pier No η Due to the curvature of the flow at Imaa-ridge, the longitudinal velocity must have a transversal distriution that needs to e included in the scour calculations. From investigation y Odgaard (1982), the velocity distriution is expressed with reference to the centerline quantities as follow:

8 594 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt V H rc = (10) VC HC r As the radius of curvature of the river channel at Imaa-ridge is large (aout 2000 m) compared to the channel width (aout 400 m), it can e assumed that the velocity distriution is only proportional to the depth distriution in Eq. 10. In other words, the maximum of (r c /r) is (2000/1800) which when is raised to the ¼ power amounts to 1.027, a value that can e neglected. Eq. 10 including only the depth term on the right hand side is the asis for calculating the local velocity at each pier shown in Tale 1. The momentum transformation factor, η, can e assumed to e proportional to the ratio of (the local depth/ the centerline depth) which reflects curvature effects and sediment deposition in the transverse direction. The maximum value of η is unity, which is taken at pier 4 where the local depth exceeds the centerline depth. The sediment specific gravity is taken as 2.65, sediment porosity of 0.4, and ed material angle of repose of 30. In applying Jain and Fischer (1978) formula, a mean sediment diameter of m and D 90 of m were taken as from 1962 data at El Ekhsas (El Motassem 1985). Tale 2 shows results of applying Eqs. 1-6 and Eq. 9 to the four piers of Imaa-ridge. It is clear from the tale that the developed equation y Youssef Hafez gives etter match with the oserved field data than the other empirical formulae using the same set of data. This case is a complex case where river meandering curvature induced transverse slope due to sediment deposition and the transverse slope attained equilirium conditions Case of El Tahreer Bridge at Cairo, Egypt: El Tahreer ridge is located on the eastern Nile ranch at Zamalek Island few kilometers upstream Imaa-ridge, Fig.4. As in the case of Imaa-ridge, it seems that scour occurred due to pre-had at historic flows such as the 1959 flood. The flow in the eastern ranch reaches aout 80 % of total flow (HRI 1981). Taking the same maximum flow of 1,202 m 3 /s as at Imaa-ridge for the whole river cross-section, the estimated maximum flow in the eastern ranch is aout 8162 m3/s. Assuming that the discharge is proportional to the 5/3 power of water depth (as in the Manning s equation) yields a water depth of 8.79 m at the site of El Tahreer-ridge using the 1,202 m3/s discharge and the 10 m water depth at Imaa-ridge. The river width at the ridge is 373 m (CS No. 3, HRI 1997). With these data, an average velocity of m/s results in at El Tahreerridge. The most clear upstream scour hole at El Tahreer ridge is seen at pier No. 4 from the right ank as seen in Fig. 5 where the scour depth reaches 3.5 m (level 10.0m - level 6.5 m). The rest of the scour holes are related to flow curvature and are located downstream the ridge. The plan view of the ridge shown in Fig. 4 shows that the ridge is angled to the main flow direction with an angle of aout 33. This angle was used explicitly when applying Eq.6. The velocity component normal to the ridge ecomes m/s (2.489 times Cosine 33) and the corresponding Froude numer is The ridge span is 46 m and the pier width is 3 m. The same sediment parameters as in Imaa-ridge case were used also herein. The momentum transfer factor,, is taken as 0.75 in applying Eq.9. Kwan

9 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt 595 and Melville (1994) reported that the maximum down-flow component was measured to e 0.75 U o, where U o is the approach flow velocity at an autment. Melville (1997), and Kothyari and Ranga Raju (2001) agree that considerale similarity also exists etween the flow patterns and scour processes at a ridge pier and at a ridge autment. Tale 3 shows comparison of applying Eqs.1-6 and Eq. 9. It can e seen again that the developed equation of Youssef Hafez gives the est match with the oserved scour depth Case of the Average and Maximum of 515 Field Data: Johnson (1995) reports a summary of 515 field data where the pier width, flow depth, flow velocity and oserved scour depth are seen in Tale 4. The application of Eqs.1-6 and Eq. 9 is seen in Tale 5. It is clear from Tale 5 that at average conditions the empirical equations (Eqs. 1-6) overestimated the scour depth ut still in the same order of magnitude. At maximum conditions where wide pier conditions occur, the empirical equations yielded results that is way too much. In oth case, the developed analytical equation y Youssef Hafez yields predictions very close to the oserved scour depths. Pier Numer Tale 2. Calculation at Imaa Bridge at Cairo, Egypt Measured Youssef Hafez Modified Froelich (1988) Jain & Fischer (1978) CSU Formula (1975) Shen et al. (1969) Modified Laursen (1969) Pier No Pier No Pier No Pier No Tale 3. Calculation at El Tahreer Bridge at Cairo, Egypt Pier Numer Measured Youssef Hafez Modified Froelich (1988) Jain & Fischer (1978) CSU Formula (1975) Shen et al. (1969) Modified Laursen (1969) Pier No

10 596 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt Case Tale 4. Data used in for the Average and Maximum of 515 Field Data Points Pier Width Local Water Local Velocity (m/s) Local Froude Numer Average Maximum Tale 5. Calculation for the Average and Maximum of 515 Field Data Points Case Measured Youssef Hafez Modified Froelich (1988) Jain & Fischer (1978) CSU Formula (1975) Shen et al. (1969) Modified Laursen (1969) Average Maximum Pier Pier H H 2 D s tam φ ϕ ϕ + D s D s/3 (a) () Figure 1. Schematic Diagrams of Flow At a Bridge Pier in the Symmetry Plane, (a) Longitudinal Flow Transformation into Down Flow, () Hole Shape

11 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt 597 Figure 2. Layout of Imaa- Bridge and Contour Map of the Bed Surface and Holes at the Piers Figure 3. Cross Section No. 11 Showing Holes at Piers 1,2,3 and 4.

12 598 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt Figure 4. Layout of El-Tahreer Bridge and Contour Map of the Bed Surface and Holes Figure 5. Cross Section No. 7at El-Tahreer Bridge Showing Holes at the Bridge Piers

13 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt Conclusions and Recommendations: The developed analytical equation herein, Eq.9, y Youssef Hafez ased on an energy alance theory yields much superior agreement with field data than other existing empirical equations when applied to two ridges near Cairo, Egypt; namely Imaa and El-Tahreer ridges and to the average and maximum of 515 data points reported y Johnson (1995). The developed equation has the advantages of: (1) eing analytical that explains the physics of ridge pier scour in a direct way, (2) relating the flow hydrodynamics to the induced scour, (3) avoiding the wide pier prolem, and (4) predicting realistically ridge scour depths of field cases, i.e. no overestimation of scour depths occurs especially for wide piers. It is recommended to test the developed equation, Eq. 9, using more field data and more exploration of the momentum transfer coefficient in Eq. 9 is needed. The developed equation provides a strong theoretical and analytical framework for tackling the scour phenomena of other hydraulic structures. References: Baaeyan-Koopaei K., and Valentine, E.M. (1999), Bridge Pier in Self-Formed Laoratory Channels, The XXVIII IA HR congress August Chang, H.H. (1988), Fluvial Processes in River Engineering, John Wiley & Sons. Colorado State University, (1975) Highways in the River Environment: Hydraulic and Environmental Design Considerations, prepared for the Federal Highway Administration, U.S. Department of Transportation, May Fischenich, C., and Landers, M. (1999) Computing, EMRRP Technical Notes Collection (ERDC TN-EMRRP-SR-05), U.S. Army Engineer Research and Development Center, Vicksurg, MS. Froelich, D.C., (1988). Autment Prediction. 68 th Transportation Research Board Annual Meeting, DC. Hydraulic Research Institute (HRI), (1981). Study of Nile River Bed at the Cairo Bridges, in Araic, Cairo, Egypt. Hydraulic Research Institute (HRI), (1993). Monitoring Local in the Nile River at Imaa Bridge, in Araic, Cairo, Egypt. Hydraulic Research Institute (HRI), (1997). Monitoring Local in the Nile River at Imaa and El-Tahreer Bridges, in Araic, Cairo, Egypt. Jain, S.C. and Fischer, E. E. (1979). around circular ridge piers at high Froude numers. Rep. No.FHwa-RD , Federal Hwy. Administration (FHwa), Washington, D.C.

14 600 Eighth International Water Technology Conference, IWTC8 2004, Alexandria, Egypt Johnson, P. A., (1995) Comparison of Pier Equations Using Field Data, J. of Hydraulic Engineering, Technical Note, Vol. 121, No.8. Kothyari, U. C. and Ranga Raju K. G., (2001), around spur dikes and ridge autments, Journal of Hydraulic Research, Vol. 39, No.4. Kwan, R. T. F. and Melville B. W., (1994) Local scour and flow measurements at ridge autments, Journal of Hydraulic Research, Vol., 32, No.5. Landers M.N. and Mueller D.S., (1996) Channel at Bridges in the United States Federal High Way Administration, Report numer FHWA/RD-95/184, Laursen, E.M., and Toch, A. (1956) around ridge piers and autments, Bull. No. 4, Iowa Hwy. Res. Board, Ames, Iowa. Melville, B. W. (1997), Pier and autment scour, integrated approach, Journal of Hydr. Engrg., ASCE, 118(4): Motassem, M. (1985), Recognition of River Nile Regime at Ekhsas Flood 1962,, Nile Research Institute, Formerly Research Institute of the High Aswan Dam Side Effects, Cairo. Neill, C. R. (1964), River ed, a review for ridge engineers. Contract No. 281, Res. Council of Alerta, Calgary, Alerta, Canada. Nile Research Institute (1992), Impact of Water Resources Projects on the Nile in Egypt, Motassem, M., Editor-in-Chief, Nile Research Institute, Cairo. Odgaard, A.J., (1982) Bed Characteristics in Alluvial Channel Bends, J. Hydraul. Div. ASCE, 108 (HY11), pp , Novemer Shen, H.W., Schneider, V. R., and Karaki, S. S., (1969) Local Around Bridge Piers, J. Hydraul. Div. ASCE, 95 (HY11), pp , Novemer 1969.

Section 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).

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