A NORMALIZED EQUATION OF AXIALLY LOADED PILES IN ELASTO-PLASTIC SOIL

Transcription

1 Journal of Geongineering, Vol. Yi-Chuan 4, No. 1, Chou pp. 1-7, and April Yun-Mei 009 Hsiung: A Normalized quation of Axially Loaded Piles in lasto-plasti Soil 1 A NORMALIZD QUATION OF AXIALLY LOADD PILS IN LASTO-PLASTIC SOIL Yi-Chuan Chou 1 and Yun-Mei Hsiung ABSTRACT Based on the elasto-plasti soil model, a set of normalized equations of axially loaded piles are established. It was found that the urves of normalized load-harateristi value of piles or normalized settlement-harateristi value of piles are distributed in a range and bounded by two lines. The upper bound is the urve of high end bearing pile and the lower bound is the urve of the pure frition pile. Due to the harater of axially loaded piles, the normalized equations present a simple and effetive approah to evaluate the load and settlement of piles. Key words: lasto-plasti soil, axially loaded pile, normalized equation. 1. INTRODUCTION The pile is widely used in the geotehnial engineering, espeially the axially loaded piles, beause piles will inrease the bearing apaity and derease the settlement of foundation effetively in soft layers. In general, the allowable settlement of struture ontrols the design of foundation. Hene, the loadsettlement relation of piles is most onerned by the geotehinal engineers. The relationship of shear stress and displaement along the pile of real soil is nonlinear and varies with the depth as shown in Fig. 1. Hene, the lose form solution is unavailable in the usual ase. For example, if the relation of shear stress and displaement is a hyperboli funtion, the solution must be expressed by an ellipti funtion. If the relation of shear stress and displaement of soil is onstant but linearly varies with depth, the solution must be expressed by the Bessel funtion (O Neil, 1991). Both the ellipti funtion and Bessel funtion are very ompliated. These solutions must be resort to the numerial analysis. The load transfer method (t-z urve) presented by Coyle and Reese (1966) is a very popular numerial method used in the pile engineering. Although this method is very effetive and simple, the omputer work is unavoidable. The purpose of this paper is to present a onvenient equation for the axially loaded piles in elasto-plasti soil in normalized form for geotehnial engineers. This normalized equation will provide us an easy way to predit the pile behavior. The solutions an be alulated without diffiulty using a alulator.. FORMULATION Sine it is only possible to obtain exat solutions for relatively simple problems, the elasto-plasti soil model to be on Manusript reeived June 30, 008; revised August 0, 008; aepted September 3, Leturer, Department of Constrution Management, Tungnan University, Shenkeng Township, Taipei County, Taiwan, R.O.C. ( yhou@mail.tnu.edu.tw). Professor (orresponding author), Department of Constrution Management, Tungnan University, Shenkeng Township, Taipei County, Taiwan, R.O.C. ( hsiung@mail.tnu.edu.tw). fs stress W Fig. 1 Axially loaded pile and soil model sidered is shown in Fig. 1(a). Before the soil displaement reahing the yielding displaement w *, the soil behaves elastially. In the elasti range, the shear stress inreases linearly with displaement. One the displaement of soil attains the yielding displaement w *, the shear stress keeps onstant as long as the displaement inreases. The ratio of maximum shear stress f s to the yielding displaement w * is the oeffiient of subgrade reation in shear stress k s..1 lasti Condition In this analysis, a irular solid pile of length l and outer diameter d is onsidered. The elasti modulus of pile shaft is. For the Winkler model, the surrounding soil layer and end bearing layer is assumed to have a onstant subgrade reation of shear stress k s and subgrade reation of vertial stress k t, respetively. The strength of the end bearing layer is usually equal to or higher than that of the surrounding soil layer. If a load P is applied to the pile top as shown in Fig. 1(b), the settlement of the pile (w) along the depth z is ontrolled by the following differential equation (Sott, 1981) w 0 non-linear model elasto-plasti model (a) displaement plasti zone elasti zone d w λ (1) dz z 0 P t (b) w 0

2 Journal of Geongineering, Vol. 4, No. 1, April 009 πdk λ s () A in whih A the ross area of pile and λ the harateristi length of pile. For q. (1), the general solution inludes two onstants as follows, λz 1 λz w C e + C e (3) The two boundary onditions onsidered are: at the pile top z 0, F P; at the pile tip z l, F P t Aw l k t. The variable w l is the displaement of the pile tip. Based on the general solution, the onstant C 1 and C are solved. Thus, the pile settlement at the pile top is kt 1 tanh( l) P + λ w 0 λ λa kt tanh( λ l) + λ From q. (4), the fore at the pile top is kt tanh( λ l) + P Aw λ λ 0 kt 1+ tanh( λl) λ Beause of the speial form of the equation above, we an define a funtion of hyperboli tangent of η whih equals the value of k t / λ. The variable η is the harateristi value of the end bearing apaity of the pile. k tanh( η ) t (6) λ The hyperboli tangent of η is in proportion to the subgrade reation of vertial stress k t and is inverse to λ. We will see later that tanh(η) is in terms of the ratio of the elasti modulus of soil at the pile tip to the elasti modulus of the pile shaft. Then, it is interesting to note that the relationship between the applied fore and settlement involves a hyperboli tangent funtion: P λa tanh( λ l+η ) (7) in whih λl is the harateristi value of the pile. Both λl and η are nondimensional value. In the elasti ondition, a linear relation exists between the settlement and load at the pile top. It is seen apparently in this ompated form that the ontribution of the values of η and λl to the pile is in the hyperboli tangent funtion. If the value of η equals zero, a pure frition pile, the value of λl will dominate the behavior.. lasto-plasti Condition As the load on the pile top inreases, the soil surrounding the pile begins to yield downward. As the soil yields to a depth of z 0, the pile an be divided into two parts in the analysis as shown in Fig. 1(b). In the upper part, the soil is yielding to the depth of z 0. In the lower part, the soil is still in an elasti ondition to the range of length l '. Sine the shear stress keeps onstant in the plasti zone, the differential equation is similar to q. (1) and (4) (5) may be expressed as d w π df s 0 (8) dz A In whih f s k s w *. There are two onstants to be determined in the general solution. The first ondition onsidered at the pile top is: z 0; F P or w w 0. The onneted point C provides the seond ondition. For the elasti zone, at the onneted point C, the settlement is w * and the fore is P z0. From the elasti solution in q. (7), P z0 λaw * tanh(λl ' + η). Hene, the load applied to the pile top may be expressed as ' * tanh( ) s 0 P λaw λ l +η +π df z (9) the settlement at the pile top is therefore 1 P 1 w* + w* w* tanh ( λ l +η) (10) λaw* Corresponding to the hange of l ' or (l z 0 ), qs. (9) and (10) give the load and settlement at the pile top. The ultimate load is the load as the frition along the pile is fully mobilized, i.e., l ' 0. In the derivation above, the soil of end bearing apaity is assumed in the elasti ondition. Basially, both the load and settlement of pile are ontrolled by the harateristi value λl and η as in the elasto-plasti ondition. If two different pile-soil systems onstitute the same λl and η values, they will display the same behavior. From q. (9), as the settlement at the pile top just equals w *, the orresponding load is very useful in the appliation and is denoted as the normalized load fator P P λ A w* tanh( λ l+η ) (11) 3. SOIL PARAMTRS In the elasti theory, the parameters are the soil modulus s and Poisson ratio ν. Sine the variation of the value of Poisson ratio is in a narrow range and has little influene on the alulated result, a onstant value of 0.35 or 0.4 usually will be used in the analysis. These parameters an be obtained onveniently from laboratory or field test. In the Winkler model, the soil parameters are the oeffiient of subgrade reations k s for side frition and k t for end bearing of the pile. These parameters are usually obtained from experienes or bak analyses via in-situ pile tests. Based on the elasti theory, Sott (1981) derived the relations of the oeffiient of subgrade reations k t and k s, respetively, as follows: s kt d(1 ν ) k s s 4 d(1 ν ) (1) (13) In whih d is the diameter of the pile. The value of k t is four times of k s. The value of some typial elasti modulus of soil for

3 Yi-Chuan Chou and Yun-Mei Hsiung: A Normalized quation of Axially Loaded Piles in lasto-plasti Soil 3 engineering purpose has been ompiled by Das (1999) and is listed in Table 1. The orresponding values of k t and k s using qs. (1) and (13) as diameter equal 1 m are listed in Table 1 also for referene. Although the qs. (1) and (13) are very simple, they are very useful for the preliminary estimation. For the in-situ onrete piles, the elasti modulus of pile shaft depends on the yielding strength of onrete. The value is about in the range of.0 ~ kn/m. The ratio of pile length to diameter (l / d) is in the range of 0 ~ 50. The less the ratio, the more rigid the pile. Conversely, the larger the ratio, the more ompressible the pile. Substituting q. (13) into q. (), we obtain s l λ l (1 ) d (14) ν If the influene of Poisson ratio is ignored, the value of λl will underestimate about 6%, then the approximate relation is l d s λl (15) The harateristi value of the pile is in terms of the ratio of length to diameter and the ratio of elasti modulus of soil to that of the pile shaft. Aording to the ratio suggested by Poulos and Davis (1980), the values of λl for various types of soil are alulated using q. (15) and listed in Table. Based on the value of λl, the pile an be divided into two ategories. For a rigid pile, the value of λl is equal to or less than 0.5. For the ompressible pile, the value of λl is larger than 0.5. It will be seen in an equation later, the amount of normalized settlement is in proportion to the square of λl. Substituting qs. (1) and (15) into q. (6), we obtain s b tanh( η ) (1 ν ) (1 ν ) (16) If the influene of Poisson ratio is ignored, the value of tanh(η) will underestimate about 6%. Then the approximate relation is tanh( η) b (17) The harateristi value of the end bearing apaity of the pile is in terms of the ratio of elasti modulus of soil at the pile tip to the elasti modulus of the pile shaft. The end bearing layer of the pile is usually stiff or dense soil or bed rok. Hene, the symbol b is used instead of s. Deere and Miller (1966) presented a lassifiation graph for intat rok speimens based on the ratio of the elasti modulus to the unonfined ompressive strength, together with the absolute value of the latter. For most roks, the ratio / q u lies in the range of 00 to 500. Hene, three lasses are defined as follows: For the higher lass, the ratio is larger than 500. For the medium lass, the ratio is between 00 and 500. For the lower lass, the ratio is less than 00. Some relevant values of roks and the alulated values of η are listed in Table 3. Based on the value of η, the pile an be divided into four ategories. For the pure frition pile, the value of η is zero. For Table 1 Soil parameters Soil type s (MN/m ) * k s (MN/m ) # k t (MN/m ) + Soft lay 4 ~ ~ ~ 3.94 Medium lay 1 ~ ~ ~ Stiff lay 41 ~ ~ ~ Loose sand 10 ~ 4.85 ~ ~ 7.36 Medium sand 17 ~ ~ ~ 31.9 Dense sand 35 ~ ~ ~ 6.70 * Das (1999) # By q. (1) for d 1m, ν By q. (11) for d 1m, ν Table Value of λl Soil type / s * l / d λ l Soft lay Medium lay Stiff lay Loose sand Medium sand Dense sand * Poulos (1980) Table 3 Value of η Rok type q u (kn/m ) * /q u b b / tanh (η) η Stiff lay Soft rok Medium rok Medium rok Hard rok Hard rok * kn/m the stiff lay or soft rok, the lower end bearing apaity, the value of η is between 0.01 and 0.5. For the medium end bearing apaity, the value of η is between 0.5 and 1.5. For the hard rok, the end bearing apaity, the value of η is between 1.5 and 3.0. Sine the limiting value of hyperboli tangent funtion is one and the value of tanh(3.0) is 0.99, it is reasonable to adopt η 3.0 to be the upper bound for end bearing. Regarding the yielding displaement of soil, it is hard to determine from the elasti theory. The value of the yielding displaement depends on the soil properties and pile dimensions. Vijayergiya (1977) has ompiled some information of yielding displaement from in-situ pile tests, and this information is listed in Table 4 for referene. 4. NORMALIZD QUATION OF LOAD AND STTLMNT The equation of load or settlement derived above an be written in a normalized form when the normalized load fator P (in q. (11)) and normalized displaement fator w * are used.

4 4 Journal of Geongineering, Vol. 4, No. 1, April 009 Table 4 Yielding displaement of soil, w * (Vijayergiya, 1977) In lay In sand Pile size (mm) w * (mm) Pile size (mm) w * (mm) ~ ~ ~ ~ ~ For the solution of elasti ondition, the normalized equation obtained from q. (7) is P P w 0 (18) w* It is a linear relation. It means that the settlement will be in proportion to yielding displaement as the applied load is less or equal to the normalized load fator P. For the elasto-plasti ondition, the normalized form of the qs. (9) and (10) are P ( λl λ l ) + tanh( λ l +η) P tanh( λ l+η) w w 0 * ( l l l l l ) (19) ( λ λ ) + ( λ λ )tanh( λ +η ) (0) Both the normalized load equation and normalized settlement equation are parametri funtions of λl, λ l ' and η. The parameter λl stands for the property of the pile-soil system. The parameter λ l ' and η stand for the amount of load mobilized along the pile and end bearing apaity, respetively. The value of η is related to the harateristi length λ of pile as shown in q. (6). The higher the value of η, the large the end bearing. As η 0, it is a pure frition pile. Therefore, the solution implies that the pure frition pile is a speial ase of the end bearing pile. As the shear stress along the pile is fully mobilized, i.e., l ' 0, the qs. (19) and (0) will be simplified to the following equations: ( λl) P tanh( λl) (3) 1 1 ( l) w + * λ (4) quation (3) is a transendental funtion and gives the ultimate load of pile. quation (4) is a paraboli funtion and gives the settlement of the pure frition pile. There is a single point orresponding to eah harateristi value of the pile. The points for η 0 are onneted using the solid line as shown in Figs. and 3. This urve represents the lower bound for a family of urves of the normalized load or normalized settlement. 4. Upper Bound Curve If η 3.0, as the pile is installed on the very hard bed rok, tanh(η) 1 and tanh(λl + η) 1. For this high end bearing pile, qs. (1) and () will redue to 1+λ l (5) P ( λ l) + tanh( η) P tanh( λ l+η) * ( l l ) (1) 1 1 ( ) ( )tanh( ) w + λ + λ η () These relations provide us a simple way to evaluate the ultimate load or settlement in terms of the normalized load fator P and normalized displaement fator w * as the frition is fully mobilized along the pile. For omparison, the urves of the normalized equations are plotted in Figs. and 3 with varying harateristi values of λl and η. The value of η is speially assigned, i.e., η 0.0, 0.5, 1.5 and 3.0. It is seen that these urves are more and more lose together as the value of η inreases. It is lear to see that these urves will approah a limiting value as η 3.0. Fig. Normalized load-λl urve 4.1 Lower Bound Curve For the pure frition pile, i.e., η 0, the qs. (1) and () will redue to Fig. 3 Normalized settlement-λl urve

5 Yi-Chuan Chou and Yun-Mei Hsiung: A Normalized quation of Axially Loaded Piles in lasto-plasti Soil (1 ) + +λ l (6) w * quation (5) is a linear funtion and gives the ultimate load of pile. quation (6) is a paraboli funtion and gives the settlement of pile. These two equations are plotted in Figs. and 3 using the long dashed line to represent the upper bound of the normalized load or normalized settlement urves. In the figure, there are another two urves for η 0.5 and η 1.5 bounded by the upper and lower urves. Due to the hyperboli tangent funtion existing in q. (3), it seems ompliated at first glane. However, as the value of λl inreases, the value of tanh(λl) will approah one. Therefore, the term on the right side of q. (3) will beome λl. Comparison between qs. (3) and (5) shows that the differene is one. It is not hard to see in Fig., the lines representing these funtions are parallel as λl is larger than.0. It is interesting to note that both qs. (4) and (6) are paraboli funtions. It means that the amount of the normalized settlement is in proportion to the square of λl. The differene of these two equations is λl. It is evident to see in Fig. 3 that the distane between the upper and lower bounds inreases with the inrease of λl. 5. APPLICATION Although the pile onsidered in the previous formulation belongs to in-situ onrete piles, these equations are still available for driven piles with hollow setions. For pipe piles, the area of the pile shaft in the alulation is the ross setion area instead of the total area. In pratie, the equations involve two different onditions as follows: 1. If the applied load is equal to or less than the normalized load fator P, the soil is in the elasti ondition, and q. (18) is used.. If the applied load is larger than the normalized load fator P, part of the soil has arrived to yielding ondition, and qs. (19) and (0) are used. 5.1 xample A bored pile was installed in the medium lay of length 50 m and diameter 1 m. From soil tests, the elasti modulus of soil is found to be s kn/m, and the yielding displaement of soil is w * 0.5 m. In the analysis, elasti modulus of pile shaft kn/m is used. Compute the settlement and load on the pile top as the pile is fully mobilized under the two onditions; A: For the pure frition pile, B: For the pile with end bearing apaity. Solution: From q. (13), for d 1 m and ν 0.35 then k s 8550 kn/m 3 From q. (1), for d 1 m and ν 0.35 then k t 3400 kn/m 3 For ondition A: η 0. From q. (), λ and then λl 1.95 From q. (11), P λaw tanh(λl + η) 333 kn ( λl) From q. (3),.03, P u 6563 kn P tanh( λl) 1 From q. (4), 1 ( l).9 w + * λ, w m For ondition B: From q. (6), tanh(η) 0.04 then η 0.04 From q. (11), P λaw tanh(λl + η) 343 kn ( λ l) + tanh( η) From q. (1),.07, P u 6713 kn P tanh( λ l+η) 1 From q. (), 1 (( l) ( l) tanh( )).98 w + * λ + λ η, w m It is seen that the equation obtained above provides a simple and effetive way for both the pure frition and end bearing pile. The urves in Figs. and 3 present an easy hart to obtain the answer. For ondition A in the example, it is a pure frition pile. The lower bound urve in Figs. and 3 gives the answer. For ondition B in the example, it is an end bearing pile with η The value of η is small. Then the urve of η 0 in Figs. and 3 an provide the answer with enough auray. For the other onditions whih are not inluded in Figs. and 3, an approximate answer will be obtained through interpolation. 5. Case Study A bored pile was installed in the medium silt lay and the end bearing layer is sandstone. The length of pile is 45 m, and the diameter is 1 m. In the field, the load test proedure follows the onventional stati load test. Sine it belongs to a proof test, there is no instrument on the pile. The maximum load of the load test is 6000 kn. Based on the reord of the field test shown on the paper of Hsiung and Hung (004), the elasti modulus of pile shaft is kn/m, the yielding displaement of soil w * is.6 mm, k s is 1000 kn/m 3 and k t is kn/m 3. What is the load and settlement of this pile using the normalized equations offered in this paper? Solution: From q. (), λ then λl.1 From q. (6), tanh(η) then η 0.81 The value of λl is more than 0.5, the pile belongs to ompressible piles. The value of η is more than 0.5, the pile belongs to medium end bearing piles. The first point to be onsidered is that the settlement at pile top just equal to the yielding displaement of soil. From q. (11), P 07 kn and w 0.6 mm. As the load on the top of pile inreases, the mobilized frition of pile goes gradually downward. For example, λl ' equals λl in the beginning, then λl ' equals 0.8λl, 0.6λl,, and 0.λl. The final point to be onsidered is under the ondition of frition fully mobilized, λl ' 0.0λl. From q. (1),.79 P, the applied load is P u 5773 kn. From q. (), 4.61 w, the settlement is w mm. *

6 6 Journal of Geongineering, Vol. 4, No. 1, April 009 Table 5 Calulated loads and settlements of the pile in the ase study λl P 0 / P w 0 / w * P (kn) w (mm) The load and settlement of these points are alulated using qs. (19) and (0) and listed in Table 5. For omparison, the alulated points and the field test data are plotted together in Fig. 4. Sine the elasto-plasti soil model is onsidered in the equations derived, the load-settlement relation displays nonlinear behavior when the load is more than the normalized load fator P. Atually, the ultimate load of this test pile should be more than 5773 kn. Beause the soil of end bearing apaity is assumed in the elasti ondition in the equation derived, the ultimate apaity of bearing layer is not mobilized fully. This assumption will indue that the estimated bearing apaity is on the onservative side. This is the limitation of equations developed in this presentation. However, the oinidene or the tendeny between the test reord and alulated value is quite well. It is evident that the normalized equations provide a simple and effetive way to evaluate the load or settlement of the pile.. The behavior of axially loaded piles is determined by two nondimensional variables, λl and η. The variable λl is the harateristi value of the pile and the variable η is the harateristi value of the end bearing apaity of the pile. Based on the value of λl, the pile an be divided into two ategories: the rigid and ompressible piles. Based on the value of η, the pile an be divided into four ategories: pure frition, low, medium, and high end bearing apaity. 3. Basially, the dominant parameters in the equations are the yielding displaement, oeffiient of subgrade reations of shear stress k s and vertial stress k t. These parameters are usually obtained from experienes or bak analyses via in-situ pile tests. The relations suggested by Sott (1981) are useful for the preliminary estimation. 4. The urves of the normalized load-λl and normalized settlement- λl for various value of η are distributed in a range and bounded by two urves. The upper bound is the urve for the high end bearing pile of η 3. The lower bound is the urve for pure frition pile of η 0. The mathematial funtions of these urves are simple. These urves present a hart to determine the load or settlement of pile easily and effetively. 5. Beause the soil of end bearing apaity is assumed in the elasti ondition, the ultimate apaity of bearing layer is not mobilized fully. This assumption will indue that the estimated bearing apaity of pile is on the onservative side. This is the limitation of equations developed in this presentation. 6. CONCLUSIONS From the equations derived, data analysis and omparison of ase study desribed in the paper, the following onlusions are drawn: 1. Based on the elasto-plasti soil model, a set of normalized equations of axially loaded pile have been established. If the applied load is equal to or less than the normalized load fator P, a linear relationship exists between the normalized load and settlement. If the applied load is larger than the normalized load fator P, two parametri equations of λl and η present the relationship of the normalized load and settlement, respetively. These equations provide a simple and effetive way to evaluate the load or settlement of the pile, and the solutions an be alulated using a alulator. Fig. 4 Load-settlement urve of ase study NOTATIONS The following symbols are used in this paper. A ross setion area of pile d pile diameter elasti modulus of pile shaft b elasti modulus of end bearing soil s elasti modulus of pile shaft soil I moment inertia of pile shaft f s maximum shear stress of soil k h oeffiient of subgrade reations of lateral stress k s oeffiient of subgrade reations of shear stress k t oeffiient of subgrade reations of vertial stress l pile length P pile load P normalized load fator P u ultimate load w 0 settlement at pile top w l settlement at pile tip w * yielding displaement w settlement of pile λ harateristi length λl harateristi value of pile η harateristi value of end bearing ν Poisson ratio of soil

7 Yi-Chuan Chou and Yun-Mei Hsiung: A Normalized quation of Axially Loaded Piles in lasto-plasti Soil 7 RFRNCS Coyle, H. M. and Reese, L. C. (1966). Load transfer for axially loaded piles in lay. Journal of the Soil Mehanis and Foundations Division, ASC, 9, SM, 1 6. Das, B. M. (1999). Priniples of Foundation ngineering. 4th d., PWS blishing. Deere, D. U. and Miller, R. P. (1966). ngineering lassifiation index properties for intat rok. Tehnial Report No. AFWL-TR65-116, Air Fore Weapons Lab., New Mexio. Hsiung, Y. M. and Hung, S. Y. (004). The normalized loadsettlement urve of axially loaded pile. Journal of Chinese Institute of Civil and Hydrauli ngineering, 16(1), O Neil, P. V. (1991). Advaned ngineering Mathematis, 3rd d., Wads Worth blishing Company, CA, USA. Poulos, H. G. and Davis,. H. (1980). Pile Foundation Analysis and Design. The University of Sydney. Sott, R. F. (1981). Foundation Analysis. Prentie-Hall, In., nglewood Cliffs, NJ Vijayergiya, V. N. (1977). Load-movement harateristis of piles. Symposium of Waterway, Port, Coast and Oean Division, ASC, Long Beah, CA, USA,,

8 8 Journal of Geongineering, Vol. 4, No. 1, April 009