Estimation of Uplift Pressure Equation at Key Points under Floor of Hydraulic Structures

Abstract Most of the hydraulic structures rest on an impervious foundation to reserve water at the upstream side. The water heads difference leads to water movement from the higher to the lower head through the porous soil layer beneath the foundation, generating an uplift pressure under the structure floor. In this study, a new method is presented to estimate the uplift pressures at key points by performing sub-surface flow analysis using the Analysis SYStem (ANSYS) software. Then a statistical analysis to validate the proposed equations is conducted using the SPSS software. The case study for this research is a barrage in Kufa city-Iraq. The used data to implement this study was water levels, soil permeability, and length of imperious foundation. The obtained results show good outcomes from using the proposed method to develop uplift pressure equations. The comparison of the current study results with Khosla’s equation showed good agreement where the coefficient of determination (R 2) and the standard error of estimation (SEE) for the equations were between (99.9–97.8) and (0.024–0.11), respectively.


Introduction
Most hydraulic structures reserved water upstream of structures. To achieve structures' equilibrium, water transmission (seepage) will occur from maximum to minimum head passing through the soil, generating three types of forces: uplift pressure under improvise floor, seepage discharge, and exit gradient. Uplift pressure reduces the share resistance between soil and foundation, causing a decrease in structures' stability against sliding or overturning. Increasing the seepage Mohammed Hamid Rasool ABOUT THE AUTHOR Mohammed H. Rasool is a civil engineer and holds an M.Sc. in hydraulic structures from the University of Kufa\Iraq since 2012. He worked as an engineer at the consulting engineering bureau of Kufa University. He is currently a lecturer at the College of Water Resources Engineering\AL-Qasim Green University. His general interest is seepage phenomenal, water quality and analysis of hydraulic by finite element. He had many published papers in his field.

PUBLIC INTEREST STATEMENT
Uplift pressure is an upward vertical pressure created due to penetration of the water into the porous material at the dam basement. When the depth of the excavation is deeper, the greater the upward pressure applied by the water. It is crucial to calculate this pressure and study its possible effects to take the needed precautions. One of the most used techniques to reduce the uplift pressure U.P. is the sheet piles. There are few formulas to calculate the U.P. so introducing new one will help to reduce the calculation time and give accurate results. discharge at the end of the foundation causes soil particles' movement and accelerates piping and soil erosion. The exit gradient is a criterion for designing hydraulic structures to determine their safety against the piping phenomenal (Khalili Shayan & Amiri-Tokaldany, 2015).
To solve this problem, a sheet pile is installed under the hydraulic structures with embedded vertical length. The uplift pressure under hydraulic structures with one or more vertical sheet piles was investigated using many scholars' approaches like Khosla (1936) (AN Khosla et al., 1936), Harr (1962) , Leliavsky (1955), Karl et al. (1967), and others. However, limited literature is available for seepage through a previous medium beneath a hydraulic structure with an inclined sheet pile. Bligh (1910) and E. W. Lane (1935) were the first who studied the seepage effect by estimating the length of creep for the flow passing under hydraulic structures (creep length is the line which touches the structural floor) and computed that path by summation of the vertical and horizontal distances. In the end, Bligh and Lane provided a coefficient to calculate the minimum length of the creep path depending on the soil type and water depth. However, this way is minimal to design the foundation of hydraulic structures (Bligh, 1910;E. W. Lane, 1935). Khosla (1999) presented a method to estimate the uplift pressure under hydraulic structures based on solving the Laplace equation as a predominant equation in steady conditions (Ashok Khosla, 1999).
Even though this equation can solve complex problems, it is hard to apply when the seepage flow is changed by a cut-off wall and will result in complicated integrals (Harr, 1962). Scholarstudied two cases of finite-depth seepage: with and without cut-off to introduce analytical solutions (Harr, 1962;Leliavsky, 1955;Polubarinova-Koch, 2015). Fil 'chakov, (1959) analytically studied the finite-depth seepage using weirs with cut-offs for several schemes (Fil'chakov, 1959). Abedi Koupaei (1991) estimated the uplift pressure distribution using four different methods and compared the results to find that the uplift pressures estimated by using Bligh and Lane equations are less than both Khosla and finite difference methods (FDMs) (Abedi Koupaei, 1991). Griffiths & Fenton (1997) modeled the 3D steady seepage using a combination of the finite element method (FEM) (Emmanuel, Oladipo, & Olabode) and random fields generating techniques (Griffiths & Fenton, 1997). Sedghi-Asl et al. (2005) studied the cut-off wall position effect on seepage and the flow velocity under hydraulic structures. According to their results, the best cut-off positions were at the upstream and downstream ends, respectively (Říha, 2020;SedghiSedghi-Asl et al., 2005). Ahmed and Bazaraa (2009) used FEM to investigate the 3D seepage path under and around hydraulic structures. They aimed to reduce the seepage losses and design a stable hydraulic structure by comparing 3D with 2D analyses for calculating the exit gradient (Ahmed & Bazaraa, 2009). Hillo (1993) used FEM to analyze seepage under hydraulic structures for different models to obtain pressure distribution under foundation and exit gradient variations along the downstream bed. Finite element results were obtained for seepage around a single sheet pile and two sheet piles by Hillo and Lane (Hillo, 1993 Lane, 1935). Ahmed & Elleboudy (2010) used the finite element to study the effect of increasing the length of the sheet pile more than the length of the hydraulic structures on uplift pressure and exit gradient. The authors found that increasing the length has no significant influence on uplift pressure and exit gradient. Furthermore, the sheet pile reduces the exit gradient and increases the uplift pressure (Ahmed & Elleboudy, 2010).
Obead (2013) used a computational method to simulate the seepage phenomenon and estimate the uplift pressure at a key point. He studied sheet piles location and inclination angle effect on seepage phenomena under the dam's impervious floor (Obead, 2013). Novak (2014) developed an equation (Khosla equations) to measure the uplift pressure at the key point. However, using that equation must impose no slope of the floor, one sheet pile, and neglect its thickness. Then, they suggested correction factors for thickness, slope, and influence mutual interference (Novak et al., 2014). Nassralla et al. (2016) studied the effect of using two layers of soil on seepage properties with sheet pile experimentally and discussed with a numerical computer program (Geo -Studio SEEP/W model). The results showed that the uplift pressures decrease if the upstream pile was less than half the soil layer depth. A comparison with numerical results yielded an excellent agreement (Nassralla, Rabea, & Technology, 2016). Jamel (2017) used three parameters upstream sheet piles, downstream sheet piles, and permeability of two layers to study their effect on seepage properties (uplift pressure and exit gradient) by using a computer program (Geo-Studio SEEP/W model). He then proposed empirical equations to calculate seepage discharge, uplift pressures, and exit gradient by a statistics software program (SPSS) (Jamel, 2017). Rasool (2018) studied the possibility of using the FEM to simulate uplift pressure under hydraulic structures. The other used several sheet piles for many depths and locations to study their effect on uplift pressure. The results showed good agreement between FEM and experimental results, as shown in Figure 1 (Rasool, 2018).
In this study, equations will be developed based on the finite element analysis ANSYS results to calculate the required uplift pressure at key points of a hydraulic structure foundation. The    pressure at key points of hydraulic structures. Also, verification of these results is performed using both Khosla's method and finite element results.

Analysis System (ANSYS)
ANSYS (version 15.0) is a finite element analysis software used in engineering simulation. The software creates simulated computer models of structures, machine components to simulate strength, toughness, temperature distribution, fluid flow, and other attributes. ANSYS has many elements to model structures and analyzes them for suitable loads Kamanbedast & Delvari, 2012;Stolarski et al., 2018). 2D Element (plan 77) (eight-node-one degree of a free dome) was adopted to simulate the soil layers under structures and the size of elements used to achieve the best results (Dekhn, 2008;Rasool, 2018). The permeability coefficient of soil is hydraulic conductivity (K xx = K yy ) homogenize (Hosseinzadeh Asl et al., 2020;Singh et al., 2019). Finally, the boundary conditions are the head of water upstream and downstream.

Statistical Package for the Social Sciences (SPSS)
The SPSS is a popular package of computer software. Nonlinear regression is one of the SPSS software methods, which has been used to develop the equations (Yin et al., 2019). This method is used to estimate the uplift pressure equations at key points under hydraulic structures.
The coefficient of determination (R 2 ) (Eq. 2), standard error of estimation (SEE) (Eq.3), and the mean absolute percentage error (MAPE) are the used criteria to show the percentage error for the suggested equations. They are calculated by Eqs. 2-4 as follows: where • y = observed values = sum ofsquared deviations.
• N = number of measurements  Figure 3 shows selection for a sample group when L = 25 m, which is the same producer for other groups by changing the value of (L = 30, 40, 50, 60, 70, 80, 100, 120, 140 and 160) m, respectively (Table 1).  Figure 3 shows the relationship between the head of water (H) m and the percentage of pressure at key points under hydraulic structures (ØE 1 , ØC 1 , ØD 1 , ØE 2 , ØD 2 , ØC 2 , ØE 3 , ØD 3 , ØC 3 ). The value of ØE 1 is the same as the value of pressure at the structure beginning due to its location at the first point upstream. Also, ØC 3 value is equal to the pressure downstream of the structure because of its location at the end. But the other point values are affected by the proportion of the water head; if the water head increased (50%), the average percentage of pressure at key points is reduced by 10.55%.

Relations between the variables
The soil permeability (hydraulic conductivity) causes seepage phenomena, so that we used isotropic homogenous soil K. Figure 4 shows the relationship between the uplift pressure at the key points with soil permeability. This factor is small compared to the value of pressure because it is directly on the seepage amount.
The percentage of uplift pressure is affected by the impervious foundation length L. Figure 5 shows the result of U.P. for pile 1 is increased 31% by increasing L from 40 to 160 m. Also, at 7.5% ration the difference between the result of U.P. for pile 2 with increase in L. But the values of U.P. for pile 3 is reduced by 45% with an increase in L.
Additionally, the uplift pressure is reduced by 33% with an increase depth of pile (d 1 ). But the increased depth of pile (d 2 ) leads to an increase in the pressure at upstream 12% because the flux will be specific between the wall of the sheet pile and reduces the pressure downstream. The depth of pile (d 3 ) increases the uplift pressure by 34%; furthermore, this case effects on exit gradient (Rasool, 2018).

Equations development
The results presented from ANSYS analysis for uplift pressure at the key points were used to develop equations for calculating the pressure percentage. Statistical program (SPSS) helps estimate the equation parameters.

The suggested form of the equation is
) (a) The percentage of uplift pressure at point C 1 was calculated from Eq. (5). R 2 and SEE for the proposed equation were 0.999 and 0.038, respectively. Figure 6 compares the actual values and predicted values.   Table 2.

Validation of the suggested equations
Data from a further six measurements were used to verify equations' accuracy (5-11) calibration with Novak 2014. Figure 13 shows the pressure for points under hydraulic structures. These values represent the results of Khosla's method, the suggested equations, and ANSYS analysis. Figure 14 presents the difference values for two statistical criteria (SEE and) for Eqs. 3 and 4, respectively, and the coefficient of determination (R 2 ) for Eq. 2 is presented in Figure 15.
Also, the average difference between values is less than 5% for points 1-6, so the developed equations can be used to estimate the percentage pressure at key points for hydraulic structures.

Conclusion
In this study, the upstream blanket and the sheet pile effects on reducing uplift pressure were investigated using FEM data. It is found that the FEM can be used as a useful tool for estimating the uplift pressure for a wide range of conditions (Rasool, 2018). Moreover, data analysis revealed that the sheet pile's best position is at the upstream end to reduce uplift pressure. The results agree with Sedghi-Asl et al. (2005), Khalili Shayan & Amiri-Tokaldany (2015), and AL-Musawi et al. (2006), who also found the best location of the pile to reduce the uplift pressure at the upstream end. At the same time, utilizing SPSS for analysis can be helpful in the suggested equations for calculating the U.P. at key points of each pile as follows: • The pressure at the upstream sheet pile increased by increasing d 2 , d 3 , H, and L, while reduced by increasing d 1 .
• The pressure at point E 2 was increased by increasing d 2 , d 3 , and H and reduced by reducing L, X, and d 1 due to the mutual effect between piles 1 and 2.
• The pressure at point D 2 was increased by increasing d 2 , d 3 , and H and reduced by reducing X and d 1 .
• The pressure at point C 2 was increased by increasing d 3 , L, and H and reduced by increasing d 2 , X, and d 1 .
• The pressures at points E 3 and D 3 were increased by increasing d 3 and H and reduced by increasing d 2 , L, X, and d 1 .
It may be evident that the effect of variables x (1.44 and 1.79%) and L (0.43%) were small for all sheet piles and can be neglected.
Finally, despite the progress made in this paper, however, there are limitations to using general applications, so further research is needed. Also, only one layer of soil material was selected as a porous media under the foundation of the hydraulic structures, while a wide range of materials may be available in prototype situations.