SELECTION OF SUITABLE EQUATIONS IN FINAL ESTIMATION OF LOCAL SCOUR (SECOND EDITION)

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 6, Issue 4, April (2015), pp. 13-27 © IAEME
13
SELECTION OF SUITABLE EQUATIONS IN FINAL
ESTIMATION OF LOCAL SCOUR
(SECOND EDITION)
Mr. Ravindra A Oak
Professor, Deptt. Of Civil Engineering; Bharati Vidyapeeth Deemed University,
College Of Engineering
Mr.Hossein Mortazavi
Post Graduate (M.Tech. Hydraulic Engineering) Student at Bharati Vidyapeeth Deemed
University, College Of Engineering, Pune-(411043)
ABSTRACT
local scour is one of the most significant causes of bridges damaging and destruction,
particularly on the flood time. hence determination of local scour around piers of bridges play
an important role in design of bridge. In addition to the loss of lives and properties, damaged
bridges lose their functions; and, lead to extra costs of maintenance. with the help of regression
analysis Some equations have been obtained by different researchers using experimental
observations and field data .The goal has been to obtain an equation with fewer gaps between
observed values and predicted values. In this article , local scour and its parameters are introduced
and then a comparison between scour equations, has shown .using dimensional analysis a general
relation is obtained. From this relation, observed data and applying linear multiple regression
analysis and polygon multiple regression analysis, 2 new formulas are obtained for the areas
Keywords: Local Scour, Comparison, Parameters, Regression Analysis
INTRODUCTION
Erosive action from flowing water in rivers called scours and leads to removing bed material
around piers and abutments. This natural phenomenon causes a lot of loss of lives and manmade
construction such as bridges and roads. Total scour has 3 components. Degradation and aggradation
which are natural scour. Unnatural scour which are contraction scour and finally local scour. many
parameters involve in bridge- scour such as velocity of flow, bed material characteristics, flow depth,
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 6, Issue 4, April (2015), pp. 13-27
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14
sediment size and distribution, characteristics of bridge pier such as the shape, width, and length of
the pier .many efforts have been carried out on different parts of the world by researchers and
scientists to obtain equations for forecasting the depth of scour and recognizing it’s
mechanism(Laursen and Toch, Shen et al., Breusers et al.Jain and Fischer,Melville and Sutherland,
Froehlich, Richardson, Lim, and Heza et al. Haque, Tafarojnoruz et al. and others). Pier scour is a
type of local scour which seems as a hole around a pier which affected by sediment transport. When
water flowing around a pier, the velocity of flow on the upstream face of the bridge pier becomes
zero and adverse pressure gradient is developing, consequently a horseshoe vortex is formed and
moved downward the upstream surface of the pier bridge and it tends to eradicate the bed materials
from around of pier. As the depth of scour increases, the power of horseshoe vortex will be
decreased and the level of scour from the bed is reduced, after a period of time equilibrium will be
happened. In live bed local scour, when the inflow bed material is equal to the outflow then
equilibrium will be obtained. In clear water scour scouring ends when the shear stress from
horseshoe vortex is equal to the critical shear of the sediment particles at the bottom of the scour
hole.
Following table show the equations used for anticipating scour in this article.
equation year
1- inglis ݀௦=1.7b(
௤
మ
య
௕
)଴.଻଼
–y 1949
2- laras ݀௦=1.05ܾ଴.଻ହ 1963
3-aronachelam
ௗೞ
௬
=1.95(
௕
௬
)
భ
ల -1
1965
4-coleman
ܸ
√2݃݀௦
= 0.6(
ܸ
ඥܾ݃
)଴.ଽ
1971
5-norman ݀௦=3.ܾ଴.ଷ
1975
6-Froehlich
ௗೞ
௕
=0.32Φ(
௕′
௕
)଴.଺ଶ
.(
௬
௕
)଴.ସ଺
.‫ݎܨ‬଴.ଶ
. (
௕
ௗఱబ
)଴.଴଼
1988
݀௦: Maximum scour depth
y:upstream flow depth
b:pier width
Fr: upstream Froude number
Φ: pier shape factor
݀ହ଴: Median sediment size
ܾ′
: Effective pier width
.most important elements have considered for a general equation are:
݀௦ = ݂ଵ ( ݇௦ , ݇Ө , b , V , y , g,ρ,ߤ, ‫ܦ‬ହ଴ , ߪ )

15
That݀௦, ݇௦ , ݇Ө , b , V , y , g, ρ ,ߤ, ‫ܦ‬ହ଴ and ߪ are defined maximum local scour depth, pier
shape coefficient, factor for angles between approach flow and pier axial, pier width, flow velocity
,upstream flow depth, gravity acceleration ,fluid dynamic viscosity ,fluid density ,median sediment
size and standard deviation of bed material, sequentially. Using Buckingham π theory and ignoring
ρ.௏.௬
ఓ
parameter ultimate form of general equation is:
ୢ౩
ୠ
= fହ(kୱ , kӨ,
୷
ୠ
,
ୠ
ୈఱబ
,
୴
ඥ୥୷
, σ୥)
Notation 1: Reynolds number (
ρ.௏.௬
ఓ
) in turbulent flow could be ignored since the effect of viscosity
force in comparison is negligible as compare to inertial force.
Notation 2: if there are n variables (independent and dependent variables) in physical phenomenon
and if these variables contain m fundamental dimensions (M,L,T) then the variables arranged into
(n-m) dimensionless terms each term called π term.
FIELD DATA
Data and records in different state of United States have been considered here. Values
collected by Froehlich (for comparison of classic equations) and values from New Hampshire field
areas (combined with previous values for derivation of new formulas and reliability test as compare
to observed values).
New Hampshire’s field data
row Name of river location
Discharge
(
௙௧య
௦
)
Width of river (ft)
1 (Kenduskeg) (Bangor) 6620 111
2 (kennbec) (Gardiner) 80000 890
3 (Androscoggin) Bethel 29300 340
4 (penobscot) (Lincoln) 43000 1058
5 (Aroostook) (Ashaland) 22700 332
6 (St. John) (Van Buren) 111000 682
7 (Austin) (Bingham) 2750 145
PREDICTION ACCURACY AND CONTROL
Three Statistical methods are used for evaluating of predicted values. Mean square error
(MSE), mean Absolute Deviation (MAD) and Tracking signal (TS).Scatter plot help for better
understanding of distribution of predicted values versus observed data.
MSE =
∑(஽೟ିி೟)మ
௡ିଵ
MAD =
∑|஽೟ିி೟|
௡
Smaller values of MSE or MAD, show more accurate prediction.

16
Tracking signal =
∑(஽೟ିி೟)
ெ஺஽
=
ா
ெ஺஽
In this research negative tracking signal means the summation of scour depth values in
observed site are smaller than the summation of forecast values concerned to the equation and Vice
Versa.
t = period number
‫ܦ‬௧= demand (real data) in period (t)
‫ܨ‬௧ = forecast (prediction) for period (t)
n = total number of periods
RESULTS OF COMPARISON BETWEEN CLASSIC FORMULAS
ds: observed values
ds(ft) inglis laras froehlich norman coleman arunachlam
1 14.11 16.23 10.64 7.71 15.46 2.3 33.09
2 9.84 16.26 10.64 7.82 15.46 2.35 31.77
3 5.71 6.69 5.62 3.91 11.97 1.57 11.35
4 25.59 20.92 17.15 7.31 18.7 2.76 27.51
5 0.98 3.24 4.72 1.06 11.16 1.24 4.05
6 0.98 3.62 4.72 1.25 11.16 1.29 4.7
7 0.98 3.24 4.72 1.19 11.16 1.31 4.05
8 2.49 1.98 4.72 0.69 11.16 1.17 2.21
9 4 2.21 4.72 0.65 11.16 1.1 2.51
10 2 4.21 4.72 1.48 11.16 1.39 5.84
11 2 4.12 4.72 1.46 11.16 1.39 5.66
12 2.99 4.82 4.72 1.7 11.16 1.39 7.29
13 4 4.92 4.72 1.82 11.16 1.42 7.58
14 4.49 4.82 4.72 1.74 11.16 1.41 7.29
15 3.51 4.58 4.72 1.7 11.16 1.43 6.68
16 6 4.82 4.72 1.75 11.16 1.41 7.29
17 1.51 2.78 4.72 0.99 11.16 1.32 3.33
18 2 2.78 4.72 1 11.16 1.33 3.33
19 1.51 3.94 4.72 1.31 11.16 1.34 5.29
20 2 4.45 4.72 1.47 11.16 1.35 6.35
21 2.49 4.37 4.72 1.44 11.16 1.35 6.19
22 1.51 3.62 4.72 1.24 11.16 1.35 4.7
23 2.49 4.12 4.72 1.33 11.16 1.33 5.66
24 5.91 5.13 4.72 4.29 11.16 1.44 8.27
25 6.89 5.13 4.72 4.32 11.16 1.44 8.27
26 5.91 5.37 4.72 4.6 11.16 1.43 9.38
27 7.87 5.31 4.72 5 11.16 1.41 9.02
28 5.91 10 7.95 3.22 13.75 2 15.61
29 2.95 2.97 3.39 0.93 9.78 1.14 4.34
30 12.14 15.07 16.69 4.87 18.5 2.63 18.08
31 14.11 14.02 16.69 4.91 18.5 2.7 16.53

17
32 23.95 16.55 23.58 7.71 21.25 3.29 18.34
33 22.31 14.31 22.21 8.87 20.74 3.23 15.69
34 27.89 19.47 23.58 10.03 21.25 3.55 22.28
35 34.12 35.24 31.97 16 23.99 4.02 42.1
36 9.19 8.7 9.12 3.26 14.53 1.89 11.28
37 4.27 4.83 4.67 1.51 11.12 1.44 7.4
38 4.27 4.78 4.67 1.53 11.12 1.46 7.26
39 2.64 4.48 4.67 1.42 11.12 1.45 6.49
40 2.95 3.47 4.67 1.16 11.12 1.46 4.47
41 2.95 3.35 4.67 1.11 11.12 1.44 4.25
42 1.31 3.35 4.67 1.13 11.12 1.46 4.25
43 1.31 2.92 4.67 1.01 11.12 1.44 3.56
44 1.64 2.76 4.67 0.96 11.12 1.44 3.32
45 1.31 2.76 4.67 0.98 11.12 1.45 3.32
46 1.31 2.4 4.67 0.87 11.12 1.43 2.79
47 2.1 3.67 3.95 1.32 10.4 1.21 5.4
48 1.31 1.9 3.95 0.73 10.4 1.15 2.2
49 4 3.78 3.95 1.47 10.4 1.26 5.69
50 2 2.18 3.95 0.79 10.4 1.15 2.6
51 2.49 5.49 4.72 3.17 11.16 1.42 10.65
52 2 5.26 4.72 2.94 11.16 1.46 8.78
53 2 5.01 4.72 5.22 11.16 1.43 7.87
54 2 5.47 4.72 2.97 11.16 1.49 10.15
55 2 5.47 4.72 3.09 11.16 1.48 11.4
56 2 5.46 4.72 1.17 11.16 1.51 10.04
57 2.69 6.37 5.35 1.09 11.74 1.63 11.17
58 14.11 24.78 19.08 0.98 19.52 2.98 32.94
59 26.9 26.51 19.08 1.23 19.52 3.01 36.33
60 15.09 27.2 19.08 1.32 19.52 3.1 37.76
Selected equations
equation year TS MAD MSE
1- laras 1963 -30.52 2.43 8.51
2- inglis-poona 1949 -27.73 2.40 11.91
3-Froehlich 1988 56.15 3.88 46.71
4-arunachlam 1965 -52.03 4.92 47.78
5-norman 1975 -50.67 7.57 62.07
6-coleman 1971 59.40 4.78 72.65
Comparison between relations demonstrates very wide range of changes in predicted values.
And so on an engineer cannot rely on one equation and consequently field data of observed scour
depth would be necessary. For example: under predicted scour may lead to destruction of bridge and
from other side, over prediction can lead to extra cost of operation. The reason for this wide range of
difference in prediction is that equations have been obtained in laboratory condition and in this
situation many parameters are not simultaneously considered.

18
REANALYZING OF EQUATIONS WITHOUT CONSIDERING DISTANT SCATTERED
VALUES
Observing scatter graphs show that some of points in graphs are scattered significantly .here
absolute values of differences between selected equations with respect to observed data are
calculated. in some stations the absolute value of differences between observed values and predicted
values are greater than 3.5 in all selected classic equations. These stations are deleted from next trial
to recognize its consequences.
Results of first and second trial
equation condition TS MAD MSE
1- laras
Whole
values
-30.52 2.43 8.51
2- laras No scattered values -29.89 2.33 7.92
2- inglis-poona
Whole
values
-27.73 2.40 11.91
1- inglis-poona No scattered values -26.37 1.98 6.72
3-Froehlich
Whole
values
56.15 3.88 46.71
4-Froehlich No scattered values 52.47 3. 30 37.22
4-arunachlam
Whole
values
-52.03 4.92 47.78
3-arunachlam No scattered values -50.57 4.35 34.50
5-norman
Whole
values
-50.67 7.57 62.07
5-norman No scattered values -49.53 7.68 63.71
6-coleman
Whole
values
59.40 4.78 72.65
6-coleman No scattered values -49.53 7.68 63.71
1-Except norman equation, MSE values in all equations are improved. also MAD values display
improvement in the top four equation of table. And absolute values of TS in whole equations show
reduction in the quantity.
2-priority of suitable equations are changed in the table as ‘inglis –poona’ demonstrates the best
results without considering scattered values, and arunachlam equation is replaced with Froehlich
equation as the third suitable equation.
New equations for the areas
With the help of regression analysis and mini tab software, 2 new equations are obtained
from general formula; results of all stages are as below:

19
EQUATION BY LINEAR MULTIPLE REGRESSION ANALYSIS
Notation 1: P-value determines the appropriateness of rejecting the null hypothesis in a hypothesis
test .here the less the p values between two parameters the more correlation is between them.
Notation 2: R-square Used in regression analysis to indicate how well the model predicts responses
for new observations, whereas ܴଶ
indicates how well the model fits data. Higher percentage of
ܴଶ
indicate the better modeling.
1- Predict dependent variable (
ୢ౩
ୠ
) based on independent variables ( ‫ܓ‬‫ܛ‬ , ‫ܓ‬Ө,
‫ܡ‬
‫܊‬
,
‫܊‬
۲૞૙
,
‫ܞ‬
ඥ܏‫ܡ‬
, ો܏):
Results
k_s k_Ө σ b/D50 v/√gy y/b
k_Ө 0.087
0.401
Σ -0.843 0.682
0.000 0.000
b/D50 -0.340 0.367 0.923
0.001 0.000 0.000
v/√gy -0.083 -0.056 -0.497 -0.313
0.431 0.598 0.010 0.002
y/b -0.131 0.111 -0.147 -0.104 -0.000
0.204 0.281 0.446 0.315 0.997
ds/b 0.420 0.431 -0.007 -0.032 -0.130 0.533
0.000 0.000 0.971 0.761 0.219 0.000
Cell Contents: Pearson correlation ---------- first row of each parameter
P-Value--------------------------- second row of each parameter
The results show correlation between ds/b andσ is -0.007 which is almost zero and p value is
0.971, and certainly there is no significant correlation between them so in next trial σ is removed
from analysis.
2- Regression Analysis: (
ୢ౩
ୠ
) versus ( ‫ܓ‬‫ܛ‬ , ‫ܓ‬Ө,
‫ܡ‬
‫܊‬
,
‫܊‬
۲૞૙
,
‫ܞ‬
ඥ܏‫ܡ‬
):
Results
The primary regression equation is ds/b = - 1.81 + 2.09 k_s + 0.0163 k_Ө + 0.000124 b/D50 - 0.065
v/√gy + 0.220 y/b
Predictor Coef SE Coef T P
Constant -1.8061 0.3275 -5.51 0.000
k_s 2.0921 0.3062 6.83 0.000
k_Ө 0.016329 0.003685 4.43 0.000
b/D50 0.0001241 0.0001339 0.93 0.357
v/√gy -0.0646 0.1424 -0.45 0.651
y/b 0.21961 0.02461 8.92 0.000
S = 0.292385 R-Sq = 66.4% R-Sq(adj) = 64.4%

20
Analysis of Variance
Source DF SS MS F P
Regression 5 14.3600 2.8720 33.59 0.000
Residual
Error
85 7.2666 0.0855
Total 90 21.6266
Source DF Seq ss
k_s 1 3.7124
k_Ө 1 3.5525
b/D50 1 0.0881
v/√gy 1 0.1982
y/b 1 6.8089
Unusual Observations
Obs k_s ds/b Fit SE Fit Residual St Resid
69 1.20 2.1034 1.5004 0.0816 0.6031 2.15R
72 1.20 2.0000 1.2395 0.0844 0.7605 2.72R
75 1.20 2.2759 2.3014 0.1382 -0.0256 -0.10 X
87 1.00 0.3617 1.1404 0.0481 -0.7787 -2.70R
88 1.00 0.3286 0.9321 0.0441 -0.6035 -2.09R
94 0.80 2.0000 0.9680 0.0901 1.0320 3.71R
95 0.80 0.0761 0.3153 0.1377 -0.2392 -0.93 X
96 0.80 0.1087 0.1691 0.1498 -0.0604 -0.24 X
R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large leverage.
From results tables
b
D50
,
v
ඥgy
display 0.357 and 0.651 quantity for p value and so they are not a
significant predictor. so in the next trial both parameters are not considered.
3-Regression Analysis: (
ୢ౩
ୠ
) versus ( ‫ܓ‬‫ܛ‬ ,‫ܓ‬Ө,
‫ܡ‬
‫܊‬
):
Results
The primary regression equation is ds/b = - 1.67 + 1.97 k_s + 0.0175 k_Ө + 0.206 y/b
Constant -1.6693 0.2608 -6.40 0.000
k_s 1.9671 0.2621 7.50 0.000
k_Ө 0.017474 0.003263 5.36 0.000
y/b 0.20591 0.02328 8.84 0.000
S = 0.294314 R-Sq = 64.5% R-Sq(adj) = 63.3%

21
Source DF SS MS F P
Regression 3 14.3128 4.7709 55.08 0.000
Residual
Error
91 7.8825 0.0866
Total 94 22.1953
Source DF Seq SS
k_s 1 3.9164
k_Ө 1 3.6227
y/b 1 6.7737
69 1.20 2.1034 1.4930 0.0785 0.6105 2.15R
72 1.20 2.0000 1.2587 0.0747 0.7413 2.60R
75 1.20 2.2759 2.2385 0.1318 0.0373 0.14 X
82 0.80 1.8846 1.3502 0.1076 0.5344 1.95 X
85 1.00 0.2048 0.7951 0.0354 -0.5903 -2.02R
87 1.00 0.3617 1.1407 0.0481 -0.7790 -2.68R
88 1.00 0.3286 0.9524 0.0369 -0.6238 -2.14R
94 0.80 2.0000 0.9614 0.0903 1.0386 3.71R
All the values of p values are equal to zero so Final equation obtained by linear multiple regression
is:
ds
b
= - 1.67 + 1.97 ks + 0.0175 kӨ + 0.206
y
b
ksWith 1.97 quantities as coefficient is the most important parameter in the formula.
R-square in formula is 64.5%
A comparison between predicted values obtains by new formula versus observed values
demonstrates that in a few rare stations, new formula show negative values. so a Constraint condition
is added:
൞
݂݅
ds
b
≥ 0 ‫ݐ‬ℎ݁݊
ds
b
=
ds
b
݂݅
ds
b
< 0 ‫ݐ‬ℎ݁݊
ds
b
= 0
4-Regression Analysis: ds/b versus (
‫ܡ‬
‫܊‬
,‫ܓ‬Ө, ‫ܓ‬‫ܛ‬ )
Results
The primary regression equation is ds/b = - 1.67 + 0.206 y/b + 0.0175 k_Ө + 1.97 k_s

22
Constant -1.6693 0.2608 -6.40 0.000
y/b 0.20591 0.02328 8.84 0.000
k_Ө 0.017474 0.003263 5.36 0.000
k_s 1.9671 0.2621 7.50 0.000
S = 0.294314 R-Sq = 64.5% R-Sq(adj) = 63.3%
Rearranging the Sequence of parameters in regression analysis show same results as compare
to previous analysing .hence confidently important parameters in equation are ‫ܓ‬‫ܛ‬,
‫ܡ‬
‫܊‬
and ‫ܓ‬Ө
respectively.
EQUATION BY POLYNOMIAL MULTIPLE REGRESSION ANALYSIS
Here parameters are manually squared and cubed and be analysed to check R-sq.
1-Correlations: ‫ܓ‬‫ܛ‬; (‫ܓ‬‫ܛ‬)૛
; (‫ܓ‬‫ܛ‬)૜
; ‫ܓ‬Ө; (‫ܓ‬Ө)૛
; (‫ܓ‬Ө)૜
; (
‫ܡ‬
‫܊‬
); ቀ
‫ܡ‬
‫܊‬
ቁ
૛
; ቀ
‫ܡ‬
‫܊‬
ቁ
૜
;
‫܌‬‫ܛ‬
‫܊‬
Results
k_s k_s^2 k_s^3 k_Ө k_Ө^2 k_Ө^3 y/b y/b^2 y/b^3
k_s^2 0.997
0.000
k_s^3 0.988 0.997
0.000 0.000
k_Ө 0.087 0.105 0.124
0.401 0.307 0.230
k_Ө^2 0.031 0.036 0.042 0.933
0.767 0.726 0.685 0.000
k_Ө^3 0.011 0.009 0.007 0.849 0.981
0.914 0.929 0.946 0.000 0.000
y/b -0.131 -0.104 -0.075 0.111 0.082 0.080
0.204 0.314 0.465 0.281 0.428 0.440
y/b^2 -0.156 -0.131 -0.104 0.063 0.022 0.013 0.945
0.129 0.204 0.314 0.539 0.835 0.902 0.000
y/b^3 -0.139 -0.116 -0.091 0.043 -0.009 -0.024 0.852 0.974
0.177 0.262 0.379 0.679 0.933 0.814 0.000 0.000
ds/b 0.420 0.453 0.484 0.431 0.365 0.320 0.533 0.524 0.508
0.000 0.000 0.000 0.000 0.000 0.002 0.000 0.000 0.000
In which: Pearson correlation ---------- first row of each parameter
P-Value--------------------------- second row of each parameter

23
ds/b correlation with y/b is 0.533 which shows the highest correlation value. All parameters have a
zero p-value as compare to ds/b.so all parameters are remained in formula.
Take regression analysis we have:
2-Regression Analysis:
‫܌‬‫ܛ‬
‫܊‬
versus: ‫ܓ‬‫ܛ‬; (‫ܓ‬‫ܛ‬)૛
;(‫ܓ‬‫ܛ‬)૜
; ‫ܓ‬Ө; (‫ܓ‬Ө)૛
; (‫ܓ‬Ө)૜
; (
‫ܡ‬
‫܊‬
); ቀ
‫ܡ‬
‫܊‬
ቁ
૛
;ቀ
‫ܡ‬
‫܊‬
ቁ
૜
Results
The primary regression equation is ds/b = 4.28 - 10.4 k_s + 6.44 k_s^2 - 0.0105 k_Ө + 0.00171
k_Ө^2 - 0.000025 k_Ө^3 + 0.174 y/b - 0.0346 y/b^2 + 0.00718 y/b^3
Constant 4.284 1.927 2.22 0.029
k_s -10.371 3.972 -2.61 0.011
k_s^2 6.443 2.060 3.13 0.002
k_Ө -0.01048 0.02478 -0.42 0.674
k_Ө^2 0.001706 0.002310 0.74 0.462
k_Ө^3 -0.00002525 0.00004825 -0.52 0.602
y/b 0.1736 0.1755 0.99 0.325
y/b^2 -0.03459 0.07164 -0.48 0.630
y/b^3 0.007184 0.007901 0.91 0.366
S = 0.273467 R-Sq = 71.0% R-Sq(adj) = 68.3%
Source DF SS MS F P
Regression 8 15.7638 1.9705 26.35 0.000
Residual
Error
86 6.4314 0.0748
Total 94 22.1953
Source DF Seq SS
k_s 1 3.9164
k_s^2 1 4.3436
k_Ө 1 2.1294
k_Ө^2 1 0.2188
k_Ө^3 1 0.2693
y/b 1 4.3645
y/b^2 1 0.4601
y/b^3 1 0.0618
24 1.00 1.2039 0.6184 0.0545 0.5855 2.18R
55 1.20 0.6489 1.3913 0.1205 -0.7423 -3.02R
71 1.20 2.0000 1.4622 0.1010 0.5378 2.12R
74 1.20 2.2759 2.7401 0.1939 -0.4642 -2.41RX
81 0.80 1.8846 1.5942 0.1830 0.2904 1.43 X
93 0.80 2.0000 1.0615 0.1016 0.9385 3.70R

24
R-sq is 71 per cent which is improved as compare to linear equation. The parameters with a
p-value more than 0.5 are removed from next trial to improve the correlation.
3-Regression Analysis: versus: ‫ܓ‬‫ܛ‬; (‫ܓ‬‫ܛ‬)૛
;(‫ܓ‬‫ܛ‬)૜
; (‫ܓ‬Ө)૛
; (
‫ܡ‬
‫܊‬
);ቀ
‫ܡ‬
‫܊‬
ቁ
૜
Results
The primary regression equation is ds/b = 4.82 - 11.4 k_s + 6.98 k_s^2 + 0.000562 k_Ө^2 + 0.0828
y/b + 0.00356 y/b^3
Constant 4.816 1.607 3.00 0.004
k_s -11.385 3.327 -3.42 0.001
k_s^2 6.976 1.721 4.05 0.000
k_Ө^2 0.0005616 0.0001019 5.51 0.000
y/b 0.08276 0.04128 2.00 0.048
y/b^3 0.003556 0.001276 2.79 0.007
S = 0.269732 R-Sq = 70.8% R-Sq(adj) = 69.2%
Source DF SS MS F P
Regression 5 15.7201 3.1440 43.21 0.000
Residual
Error
89 6.4752 0.0728
Total 94 22.1953
Source DF Seq SS
k_s 1 3.9164
k_s^2 1 4.3436
k_Ө^2 1 2.3476
y/b 1 4.5474
y/b^3 1 0.5651
Unusual Observations:
Observation k_s ds/b Fit SE Fit Residual
24 1.00 1.2039 0.5971 0.0439 0.6069
36 1.00 1.1842 1.3471 0.1191 -0.1629
37 1.00 1.3816 1.3471 0.1191 0.0345
38 1.00 1.1842 1.4434 0.1218 -0.2591
39 1.00 1.5789 1.3193 0.1184 0.2597
55 1.20 0.6489 1.4023 0.1004 -0.7533
71 1.20 2.0000 1.4595 0.0991 0.5405
74 1.20 2.2759 2.7062 0.1846 -0.4303
81 0.80 1.8846 1.5755 0.1590 0.3091
93 0.80 2.0000 1.0778 0.0871 0.9222

25
In this stage all the p-values of parameters are less than 0.05 and R-sq is 70 per cent which is
higher than linear regression. so final equation obtained by polynomial multiple regression analysis
is:
‫܌‬‫ܛ‬
‫܊‬
= 4.82-11.4(‫ܓ‬‫ܛ‬)+6.98(‫ܓ‬‫ܛ‬)૛
+0.000562(‫ܓ‬Ө)૛
+ 0.0828(
‫ܡ‬
‫܊‬
)+ 0.00356 ቀ
‫ܡ‬
‫܊‬
ቁ
૜
RESULTS OF COMPARISON BETWEEN ALL FORMULAS
equation year TS MAD MSE
1- New polynomial
equation
2014 20.32 0.16 0.05
2-New linear equation 2014 -0.51 0.18 0.05
3- laras 1963 -30.52 2.43 8.51
4- inglis-poona 1949 -27.73 2.40 11.91
5-Froehlich 1988 56.15 3.88 46.71
6-arunachlam 1965 -52.03 4.92 47.78
7-norman 1975 -50.67 7.57 62.07
8-coleman 1971 59.40 4.78 72.65
Statistical results display a very close prediction for new formulas in these areas. Absolute
number obtained from linear equation is considerably less than polynomial equation, which is clearly
visible in scatter plots.
Equations gained by regression analysis are more accurate and reliable. So deriving equation
of prediction for areas which they are concerned to constructions is more reliable as compare to
previous equations.
2.52.01.51.00.50.0
2.5
2.0
1.5
1.0
0.5
0.0
newlinear equation ds/b
observedds/b
Scatterplot of ds/b vs newlinear equation

26
2.01.51.00.50.0
2.5
2.0
1.5
1.0
0.5
0.0
Newpolynomial equation ds/b
observedds/b
Scatterplot of ds/b vs Newpolynomial equation
SUGGESTIONS
1 Areas are concerned to scour phenomenon need a vast range of observed data .and these values
should be gathered in different parts of rivers.
2 As it is shown, every equation has a high reliability on the fields or river basin of its own.
There are many geographical and geological characters which are specific for every area.
Driving a specific equation for every type of river basin is suggested.
3 There are some main reasons of pier local scour which have to be recognized and listed as the
main reasons
4 Considering that hydraulics structures such as bridges and culverts demand a high quality of
construction.
5 Study of local scour mechanism for every area.
REFERENCES
1 Garcia, Marcelo H.,2008:“Sedimentation Engineering: Processes, Measurements, Modeling,
and Practice”. ASCE Publications,
2 Wardhana K and Hadipriono F.C.:“Analysis of recent bridge failures in the United States",
Journal of Performance of Constructed Facilities.
3 Mohammed, T. A., MegatMohd Noor, M. J., Ghazali, A.H., Yusuf, B. and Saed, K.:
"Physical Modeling of Local Scouring around Bridge Piers in Erodable Bed", Journal of King
Saud University.
4 Johnson, Peggy A.: “Advancing Bridge – pier Scour Engineering”, ASCE.
5 Rahman Md. Munsur and Haque M. Anisul,:“Local scour estimation at bridge site:
Modification and application of Lacey formula", International Journal of Sediment Research.
6 Khwairakpam, Padmini, Ray, Soumendu Sinha, Das, Subhasish, Das, Rajib; Mazumdar,
Asis,:“Scour hole characteristics around a vertical pier under clear water scour condition”,
Journal of Engineering & Applied Sciences.
7 Tafarojnoruz, A., Gaudio, R., and Dey, S. “Flow-Altering Countermeasures against Scour at
Bridge Piers: a Review”, Journal of Hydraulic Research.

27
8 Lee, Seung Oh “Physical modeling of local scour around complex bridge piers”, PhD thesis,
Institute of Technology, Georgia, 2006.
9 Arneson, L.A., Zevenbergen, L.W., Lagasse, P.F., and Clopper, P.E.," Evaluating scour at
bridges", U.S. Federal Highway Administration, Hydraulic Engineering Circular no. 18,
2012.
10 sturm T.W, 2001. : “open channel hydraulics” New York,USA,McGraw-hill.
11 Mohamed, T.A.,M.J.M.M.Noor and A.H.Ghazali,2005.: “Validation of some bridge pier
scour formulae using field and laboratory data.” American J. Environ.
12 Froehlich .D.C, 1988:“Analysis of onsite measurements of scour at piers. in American society
of civil engineers national conference of hydraulic engineering: Colorado spring, co
American society of civil engineers.
13 Bruce W. Melville, Stephen E. Coleman 2000:–“Water Resources Publication, Technology &
Engineering” chapter.6.6 other design equation.
14 Dr.R.K.Bansal: “a textbook of fluid mechanics and hydraulic machines” chapter 12-
dimensional and model analysis.
15. Mr.Ravindra A Oak and Mr.Hossein Mortazavi, “Selection of Suitable Equations In
Estimation of Local Scour” International journal of Computer Engineering & Technology
(IJCET), Volume 5, Issue 12, 2014, pp. 277 - 281, ISSN Print: 0976 – 6367, ISSN Online:
0976 – 6375.
16. Ch.Sudha Rani and K.Mallikarjuna Rao, “Statistical Evaluation of Compression Index
Equations” International journal of Computer Engineering & Technology (IJCET), Volume
4, Issue 2, 2013, pp. 104 - 117, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.

SELECTION OF SUITABLE EQUATIONS IN FINAL ESTIMATION OF LOCAL SCOUR (SECOND EDITION)

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