Lateral Load Analysis of Single Piles

Lateral Load Analysis of Single Piles

ECI 281(a)Term Project

Instructor: Boris Jeremic

Yung-Tsang Chen University of California, Davis 2004

Purpose The purpose of this report is to provide knowledge of deep foundation analyses, and provide commonly used method for structure engineers. Piles are structural members made of steel, concrete, or timber, but many structure engineers are not familiar with the pile’s behavior. Therefore, in order to ensure structural safety, it is very essential to know how to analyze the soil pile behavior. Introduction Most structures are subject to lateral loads as a result of wind, earthquake, impact, waves, and lateral earth pressure. If these structures are supported on deep foundations, the foundations have to be designed for lateral loads. Laterally loaded piles should be safe against geotechnical failure, structure failure, and excessive deflections. In general geotechnical failure is reached only at very large displacements. Therefore, what we concerns about is mainly on the prediction of deflections and maximum bending moments in long piles. The problem of a deep foundation subjected to lateral loading involves the interaction of soil and structure. Therefore, the solution to the problem usually requires the use of iterative techniques because soil response is a nonlinear function of the deflection of the foundation. However, most practical engineers are interested in how to obtain the deflection and bending moment in the deep foundation. Hence, in the process of analysis, it is crucial to obtain these values. Analyses in which the interaction between the soil and the pile is modeled using concepts of subgrade reaction have been developed by Hetenyi(1946), and a number of other investigators. These analyses are based on the assumption that the soil reaction p is proportional to the deflection of the pile y. The soil reaction divided by the deflection is called the soil modulus Es. Solutions have been developed for Es constant with depth (Hetenyi 1946), for Es varying with linearly with depth (Reese and Matlock 1956), for Es varying nonlinearly with depth (Matlock and Reese 1960). The p-y method, devised by McClelland and Focht(1958), appears to be the most practically useful procedure for the design of deep foundations. The reaction of the soil against the pile is related to the deflection of the pile by means of nonlinear p-y curves. Methods for estimating the shapes of p-y curves for various types of soil and loadi9ng conditions have been developed by Matlock (1970) for soft clay, Reese et al. (1975) for stiff clay below the water table, Reese et al.(1974) for sand.

Because p-y analyses are capable of representing a wide variety of soil and loading conditions in a realistic manner, and because the results of p-y analyses have been found to be in reasonable agreement with results of field loading tests in many cases, these analyses provide good results in pile design. However, in practical engineering design, how to reduce the analyzing time becomes very important. When engineers use p-y analyses, they have to spend a lot of time on developing the input data and performing the computer analyses. Thus, a simplified method on analyzing the pile subjected to lateral load is needed. Evans and Ducan (1992) proposed a simplified analyzing method, called Characteristic Load Method. (CLM) In the following reports, I would like to introduce 3 commonly used methods, including Elastic solution for single piles, P-y method of nonlinear behavior for lateral load analysis, and Characteristic load method. Elastic solution for single piles Consider a pile of length L subjected to a lateral force Q and a moment M at the ground surface=0), the soil reaction in the direction opposite to the pile deflection can be written as EpI p

d4y =p dx 4

Figure 1. Lateral load pile(a)before deformation (b)after

(1)

According to Winkler’s model, an elastic medium can be replaced by a series of infinitely close independent elastic springs; therefore, we can assume: p = − ky

(2)

Where p=pressure on soil k=modulus of subgrade reaction y=deflection The subgrade modulus for granular soil at a depth x is defined as k x = nx x

(3)

Where nx=constant of modulus of horizontal subgrade reaction. Combining the above equations, the soil reaction can be written as

EpI p

d4y d4y + k y = E I + n x xy = 0 y p p dx 4 dx 4

(4)

By solving the above equations, pile deflection at any depth [y(x)] can be obtain

y ( x) = Ay

QT 3 MT 2 + By EpI p EpI p

(5)

Slope of pile at any depth [θ(x)] can be obtain

θ ( x) = Aθ

QT 3 MT 2 + Bθ EpI p EpI p

(6)

Moment of pile at any depth [m(x)] can be obtain

m( x) = AmQT + Bm M

(7)

E I Where Ay , B y , Aθ , Bθ , Am , Bm are coefficients, T = 5 p p is the characteristic length ny of the soil-pile system. When L ≧5T, the pile is considered to be a long pile. For L ≦2T, the pile is considered to be a rigid pile. Table 1. gives some representative values of nh. Table 2. gives the values of the coefficients for long piles. Note that, in the first column of Table 2, Z is the non-dimensional depth.

Table 1. Representative values of nh

Table 2. Coefficient for long piles(L≧5T)

P-y method of nonlinear behavior for lateral load analysis

When a deep pile is subjected to a lateral load, the equation can be written as EpI p

d4y d2y + P =p x dx 4 dx 2

Figure 2. Lateral load pile (a)deformation (b)modeled as independent elastic According to Winkler’s model, an elastic medium can be replaced by a series of infinitely close independent elastic springs; therefore, we can assume: p = − Es y

(8)

Where p=pressure on soil Es=soil modulus y=deflection However, soil modulus may not be a constant in all depth, and it may changes with depth and with p-y curves. Therefore, if we can predict a set of p-y curves at all depths, deflection, pile rotation, shear, soil reaction can be readily solved. Figure 3 shows the p-y cures under different depth. From the figure, we can observe that soil modulus is not a constant or a straight line. Soil will perform nonlinear behavior under big soil deflection. Hence, how to get a set of p-y curves under different depth has become an important thing. We can obtain the results from the experiments, and predict some formulas. Matlock (1970) proposed some procedures to obtain the p-y curves for soft clay; Reese (1975) proposed some steps and procedures for stiff clay to get p-y curves; Cox (1974) and Reese (1974) also proposed procedures to find the p-y curves.

Figure 3. (a) soil modulus varies with depth (b) p-y curves under different depth ( Prakash S., and Sharma, H. D. 1990)

The researchers mentioned above all proposed similar procedures for different soils. Reese in 1975 proposed his procedures for making a set of p-y curves, and the steps are as follows Step1: Obtain the best possible estimate of the variation of shear strength c and average effective unit weight γ with depth, and the value of ε50, the strain corresponding to one-half the maximum principal stress difference. Step2: Compute the ultimate soil resistance per unit length of pile, pu, use the smaller of the following values. x (9)  γx pu =  3 + + 0.5 cb c b  pu = 9cb

(10)

Step3: Compute the deflection, y50, at one-half the ultimate soil resistance y50 = 2.5bε 50

(11)

Step4: Points describing the p-y curve may be computed by following equation 1

 y 4 p  = 0.5 pu  y50 

(12)

The applications of p-y curves requires the use of computer programs, such as COM622 (Reese 1977), COM624 (Reese 1984), LPILE (Reese 1985), LTBASE (Borden and Gabr 1987). These programs can help us to solve the deflection, rotation, shear, moment, and soil pressure of piles under lateral loads. The advantages of using p-y analyses are that p-y curves are the p-y curves are capable of representing a wide variety of soil and loading conditions in a realistic manner, p-y curves have been found to be in reasonable agreement with results of field loading tests in many cases, and p-y analyses consider the nonlinear behavior of soils. The disadvantages of using p-y analyses are that p-y curves separate the soil into discrete elements; therefore, the soil pressure is converted to point loads. Another disadvantage is that the amount of time required developing input and performing the detailed computer analyses in engineering practice. Thus, we may need a more simplified method to be used on engineering practice. Characteristic Load Method (CLM)

In order to reduce the time of analyses, a simplified method may be needed in engineering design. Evans and Ducan (1992) proposed CLM method, which closely approximates the results of nonlinear p-y analyses, and this method obtains results more quickly. This method can be used to determine ground-line deflections due to lateral load, ground-line deflections due to moments applied at the ground line, maximum moments, and the location of maximum moments. In using the method, first, we have to find the characteristic load and moment, which are For clay

 S Pc = 7.34 D ( E p RI ) u E R  p I 2

For sand

   

0.68

 γ ' Dφ ' K p   Pc = 1.57 D ( E p RI )  E R  p I   2

(13-1)

0.68

(13-2)

For clay

For sand

 S M c = 3.86 D ( E p RI ) u E R  p I 3

   

0.46

 γ ' Dφ ' K p   M c = 1.33D ( E p RI )  E R  p I  

(14-1)

0.4

3

(14-2)

Where Pc= characteristic load, Mc=characteristic moment D= pile width or diameter, Ep=pile modulus of elasticity RI= moment of inertia ratio=ratio of moment of inertia of the pile to the moment of inertia on a solid circular cross section r’= effective unit weight of sand, Su= undrained shear strength of clay ψ’= effective stress friction angle for sand Kp= Rankine coefficient of passive earth pressure=tan2(45+ ψ’/2) Deflection due to loads applied at ground line

To estimate the ground line deflection using Fig.4, calculate the value of Pc using (13-1) for clay or (13-2) for sand. Divide the ground line load by Pc to determine the value of Pt/Pc. Using the appropriate curve in Fig. 2, determine the value of yt/D, and multiply this value by D to determine the ground line deflection yt . Fig.4 can also be used to determine the load corresponding to a given ground line deflection, by determine the load corresponding to a given ground line deflection, by entering the chart on the horizontal scale at yt/D, and determining the corresponding value of Pt/Pc from the appropriate curve. Deflection due to moments applied at ground line

To estimate the ground line deflection using Fig.5 calculate the value of Mc using (14-1) for clay or (14-2) for sand. Divide the ground line moment by Mc to determine the value of Mt/Mc. Using the appropriate curve in Fig. 5, determine the value of yt/D, and multiply this value by D to determine the ground line deflection yt.

Figure 4. Load-Deflection Curves(a) Clay(b) Sand

Figure 5. Moment-Deflection Curves(a) Clay(b) Sand

Comparison of characteristic load and p-y analyses

The CLM was derived from the results of p-y analyses, and it would be expected that these two types of analyses would agree fairly closely when used to analyze the same conditions. However, because some simplifications and approximations were made in the process of reducing the numerical p-y results to the simplified dimensionless from of the CLM, there are some differences between the results calculated by the two methods. From table 3, the results for clay calculated using the CLM are in close agreement with static p-y results for the soft clay and the stiff clay above water p-y formulations, and the CLM gives a conservative approximation for the static analyses using the stiff clay below water p-y formulation. The results of the CLM also agree fairly well with the cyclic results calculated using the soft clay formulation. Table 3. Comparison of characteristic load and p-y methods of analyses

Summary and Conclusion Analyses of laterally loaded piles can be simplified by an elastic beam, and by using elastic beam theory and Winkler’s model, we can easily solve the problem. For most cases, we can use p-y curves and computer programs to obtain the results, while most important parameters involved in the prediction of p-y curves are believed to have been considered. However, the prediction methods for p-y curves for clay or sand are based on a small amount of experimental data; therefore, for more exactly analyses, we should use continuum mechanics of lateral soil-pile interaction to analyze the problem in 3-D condition. In engineering practice, we can use CLM method to quickly obtain the approximate results, but we have to be very careful until additional data allow the method to be validated. Reference: 1. Matlock, H., and Reese, L. C. (1960). “Generalized solutions for laterally loaded piles.” Journal of Soil Mechanics and Foundation Division, ASCE, 86(5), 63-91. 2. McClelland, B., and Focht, J. A. Jr. (1958). “Soil modulus for lateral loaded piles.” Trans., ASCE, 123 1046-1063. 3. Reese, L. C., and Welch, R. C. (1975). “Lateral loading of deep foundations in stiff clay.” Journal of the Geotechnical Engineering Division, ASCE, 101(7), 633-649. 4. Das, B. M. Principal of foundation engineering.” PWS-KENT publishing. 5. Ducan, J. M. “Lateral load analysis of single piles and drilled shafts.” 6. Abedzadeh, F. and Pak, Y. S. (2004) “Continuum mechanics of lateral soil-pile interaction.” Journal of Engineering Mechanics, 130(11), 1309-1318.