Ji-Cheng Wang, Jianlin Yu, Shiguo Ma & Xiaonan Gong (2015): Hammer's Impact Force on Pile and Pile's
Penetration, Marine Georesources & Geotechnology, DOI: 10.1080/1064119X.2015.1016637
To link to this article: http://dx.doi.org/10.1080/1064119X.2015.1016637 Accepted author version posted online: 28 May 2015.
Hammer’s Impact Force on Pile and Pile’s Penetration
Ji-Cheng Wang
Institute of Architectural Engineering, Ningbo University of Technology, Ningbo, Zhejiang, China Jianlin Yu
Downloaded by [Jicheng WANG] at 21:22 29 May 2015 College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China Ma Shiguo
Ningbo Urban Construction Investment Holding Company Limited, Ningbo, Zhejiang, China Xiaonan Gong College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China Address correspondence to Ji-Cheng Wang, No. 201, Fenghua Road, Jiangbei District, Ningbo, Zhejiang 315211, China. E-mail: wjc1818@zju.edu.cn Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/wifa. Abstract
Methods of energy and momentum conservation, vibration theories, and model tests are employed to research into impact force of hammer on pile and pile’s penetration Analytical formula of impact force of hammer on pile is obtained. Comparison of results from the three methods finds that the analytical formula proposed by this paper conforms well to practical situations. Research results
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show that, If hammer’s impact energy remains unchanged, Penetration increases with the increase of hammer weight and cushion stiffness; Impact force the pile head receives degrades with the increase of hammer weight but increases with the increase of cushion stiffness; Impact time decreases with the increase of cushion stiffness but increases with the increase of hammer weight. Model test shows that, compared with cotton cloth cushion, elastic cushion’s advantages lie in that relatively small pile head impact force can achieve big pile penetration, and cotton cloth is gradually compacted with the increase of blow counts, hence impact force the pile head receives tends to increase gradually.
Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Keywords
INTRODUCTION Hammer piling is simple and economical, and has a great penetrability, hence is widely used. However, hammer’s impact force on pile is usually very big, thus causing damage to pile head. Besides, hammer weight selection is generally empirical. Hammer’s impact force on pile is usually calculated through pile driving formula, which are deduced from laws of energy and momentum conservation together with empirical parameters as energy utilization efficiency, coefficient of restitution, etc., such as the widely used Hiley (1925) Formula. Smith’s (1960) model used lumped mass as hammer, and used weightless spring as anvil cushion. Helmet and cap cushion were all regarded as part of the pile. Wave equation analysis was used to simulate dynamic response of pile driving (Smith 1960). Rausche, Goble, and Likins (1985) proposed a pile driving dynamic model which was more complex than Smith’s (Rausche, Goble, and Likins 1985). Deeks and Randolph (1990, 1991, 1993) built a model comprising piling hammer, anvil cushion, and helmet which was directly attached to pile top and proposed analytical solutions to this model
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considering anvil cushion’s elasticity and viscoelasticity (Randolph, Banerjee, and Buttefield 1991; Deeks and Randolph 1993, 1995). Koten (1991) researched into a simple hammering model comprising piling hammer and anvil cushion (Koten 1991). Take, Valsangkar, and Randolph (1999) analyzed interaction of piling hammer, anvil cushion, helmet, and cap cushion. The paper used lumped masses for the ram and anvil, a spring for the anvil cushion, and another spring for the cap cushion on the top of a pile. Pile and soil interaction was not considered. Take analyzed blow stress wave on pile top by using mass-spring-dashpot model and numerical computation method (Take, Valsangkar, and Randolph 1999). Runfu and Qing (2000) simplified pile to one Downloaded by [Jicheng WANG] at 21:22 29 May 2015 dimensional rod and analyzed it by using stress wave solution and proposed a stress solution of pile (Runfu and Qing 2000). Based on one dimensional stress wave theory, Shifang et al. (2003) researched into impact force and penetration by building an interaction model of hammer, pile, and soil and obtained analytical solutions of pile end displacement and velocity (Shifang, Renpeng, and Yunmin 2004). Hehua, Yongjian, and Huaizhong (2004) used different lumped mass for piling hammer and helmet, and parallel spring and dashpot for anvil cushion, and spring for cap cushion, and dashpot for pile, and established an equilibrium equation, and then used the Laplace Transform to deduce analytical solutions of hammering force (Hehua, Yongjian, and Huaizhong 2004; Yongjian et al. 2005). Liyun and Pu'an (1994) carried out a simulation piling test of small diameter steel pipe pile and concluded that disc spring could improve piling efficiency (Liyun and Pu'an 1994). Renpeng et al.(2001) monitored pile body stress in piling and obtained relationships among blow count, pile length, and pile body’s maximal stress (Renpeng et al. 2001). Peng-kong and Subrahmanyam (2003) researched into pile driving formula considering pile length factor by using one dimensional wave equation (Peng-kong and Subrahmanyam 2003). Based on Hiley Formula, So and Ng (2010) researched into impact compression behaviors of high-capacity long piles (So
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and Ng 2010). With examples, Middendorp and van Weel (1986) attested that under most conditions, simple models could provide force and velocity response curve which was accurate enough (Middendorp and van Weel 1986; Middendorp 2004). This article simplifies the system comprising hammer, anvil cushion, helmet, cap cushion, pile, and soil, and regards helmet and cap cushion as part of pile, and regards anvil cushion as weightless spring. Three methods are used to research into hammer’s impact force on pile and pile’s penetration. ANALYSES BASED ON ENERGY AND MOMENTUM Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Effect of Hammer Weight Process of piling is illustrated in Figure 1. Free fall height is H. Hammer velocity is v0
when reaching cushion. Initial kinetic energy is E0, and the weights of hammer and pile are mH and mP respectively. Cushion weight is not considered. Hammer and pile are subjected to collision. After collision, the velocities of hammer and pile are vH and vP respectively. Pile’s penetration is S. When hammer hits pile, part of the energy ELoss gets lost through sound wave, cushion warming, pile’s plastic deformation, etc. Hence we have: 12E0 mHv0 WELOSS2 (1) In which W is part of the energy that can be used in hammer’s initial kinetic energy E0. Completely Inelastic Collision If the cushion is plastic, completely inelastic collision occurs between hammer and pile, viz. hammer and pile stick together to move downward after collision. Here vH vP vc. Upon receiving kinetic energy, pile overcomes soil’s frictional resistance and sinks, then:
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1W (mHmP)vc22 (2)
According to momentum conservation, we have:
mHv0 (mHmP)vc (3)
The solution is: vc mHv0mHmP (4) Substitute Eq. (4) into Eq. (2), we have: mHEmmP1mv 0H W E02mHmPmHmPmH1mP (5) 22H0Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Let energy utilization efficiency K W, then: E0mHmPK mH1mP (6) As for completely inelastic collision, viz. cushion is plastic, the relationship between energy utilization efficiency K and mH/mp is shown in Figure 2. It can be seen from Figure 2 that energy utilization efficiency increases gradually with the increase of mH/mp. Assume that during piling, soil’s frictional resistance f for pile remains unchanged, then: 22v01mHW fSPlastic2mHmP (7)
In which SPlastic is penetration when complete plastic impact occurs between pile and hammer. From Eq. (7) we have:
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SPlasticmHmPE0 mH1fmP
(8)
122 remains It can be seen from Eq. (8) that if hammer’s kinetic energy E0 mHv0unchanged, increasing hammer weight, viz. “heavy hammer and low drop”, can achieve relatively bigger penetration. As for completely inelastic impact, relationship between penetration and mH/mp is illustrated in Figure 2. It can be seen from Figure 2 that just as energy utilization efficiency K, penetration S increases gradually with the increase of mH/mp. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Elastic Collision If cushion is elastic, elastic impact occurs between hammer and pile. According to energy and momentum conservation, we have: 111222°E0 mHv0 mHvHmPvP222®°¯mHv0 mHvHmPvP (9) Because hammer separates from pile after collision, hammer’s kinetic energy does no work 2 will get lost. during pile’s sinking process after collision, viz. hammer’s kinetic energy mHvH122 does work during pile’s sinking process. According to However, pile’s kinetic energy W mPvP12Eq. (9), we have: vP 2mHv0mHmP (10) Then energy utilization efficiency is:
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m124HmPvPmPW2K E01mv2§m·2H2H0¨1m¸P¹© (11)
Relationship between energy utilization efficiency K and mH/mp is illustrated in Figure 2. It can be seen that when mH penetration is the biggest, and SElastic E0. When mH>mp, penetration SElastic degrades gradually f3E0, be it using 4fwith the increase of mH. It can also be seen that when mH=3mp, penetration is elastic cushion or plastic cushion. From Eqs. (8) and (13), we have: SPlasticSElastic mH3mP22mHv022f(mHmP) (14) When mH!3mP (for example: pile is relatively short or pile driving just gets started), Downloaded by [Jicheng WANG] at 21:22 29 May 2015 (15) Compared with elastic impact, pile’s penetration is bigger when complete inelastic impact occurs between hammer and pile. In this case, using plastic cushion is a better choice. When mH3mP (pile is relatively longer), SPlasticSElastic (16) SPlastic!SElastic Compared with elastic impact, pile’s penetration is smaller when complete plastic impact occurs between hammer and pile. In this case, using elastic cushion is a better choice. In practical engineering, based on experience, a hammer which weight is roughly equal to or slightly bigger than pile weight is chosen. Therefore, using elastic cushion has higher energy utilization efficiency than using plastic cushion. Penetration is also bigger. Effect of Cushion Stiffness Assume elastic cushion stiffness coefficient is kH. Pile will not sink if anvil cushion’s thrust for pile is smaller than soil’s frictional resistance to pile f. This part of hammer’s energy is transferred to cushion’s compression energy, hence: 8 ELoss1f2 fx 22kH (17) In which x is compressed length of cushion under the effect of energy loss ELoss. Energy utilization efficiency f22kHf2K 1 121kmv2HH0mv2H0 (18) Energy utilization efficiency is illustrated in Figure 3. The bigger the spring’s stiffness is, the smaller its compression amount is, the more useful work the hammer will do, hence the higher Downloaded by [Jicheng WANG] at 21:22 29 May 2015 the energy utilization efficiency will be. Pile does work overcoming soil’s frictional resistance f2E0 fS2kH (19) EfS 0f2kH (20) It can be seen from Eq. (20) that, just as energy utilization efficiency, penetration S increases with the increase of cushion stiffness kH. When kHof, then Soin Figure 3. When ELoss = E0, viz. hammer’s kinetic energy is completely transferred to cushion’s compression potential energy, then 1f22mHv0 22kH (21) E0, as illustrated fWe can obtain: f2kH 2mHv0 (22) f2, hammer’s all kinetic energy is transferred to Viz. when cushion stiffness kH2mHv0cushion’s compression potential energy and will not cause pile to sink. 9 When Soil's Resistance to Pile Is Very Big When pile receives very big resistance during sinking process and penetration is very small and is negligible, foundation soil’s stiffness can be seen as kP f, then the system comprising hammer, cushion, and pile can be illustrated by Figure 4(a). Then: 112mHv0 kHx222 (23) In which x is cushion’s compression amount. It can be calculated that Downloaded by [Jicheng WANG] at 21:22 29 May 2015 x (24) Then the maximal impact force pile head receives is: Fmax,kP f kHx kHmHv0mHv0kH (25) Relationship between impact force Fmax,kP f pile head receives and cushion’s stiffness coefficient kH is illustrated in Figure 5. It can be seen that pile head tends to receive bigger impact force with the increase of cushion stiffness. When Soil's Resistance to Pile Is Very Small When pile receives very small soil resistance which is negligible during sinking process, foundation soil’s stiffness can be seen as kP = 0, then the system comprising hammer, cushion, and pile can be illustrated by Figure 4(b). Hammer receives cushion’s resistance and its velocity degrades gradually from v0. However, due to cushion’s downward impact force, pile speeds up its sinking. When hammer’s velocity vH equals to pile’s velocity vP (assumevH vP vc), cushion is compressed to the thinnest, hence cushion receives the biggest pressure. Then mHv0 (mHmP)vc (26) 10 We can obtain: vc mHv0mHmP (27) The whole system’s kinetic energy 11(mHv0)22EK (mHmP)vc 22mHmP (28) Kinetic energy degrades 11(mHv0)22'E E0Ek mHv022mHmP (29) Downloaded by [Jicheng WANG] at 21:22 29 May 2015 The degraded kinetic energy is transferred to spring’s compression potential energy 111(mHv0)222'E kHx mHv0222mHmP (30) We can obtain: mHmPv0kH(mHmP)x (31) The maximal extruding force the cushion receives is the maximal impact force the pile head receives kHmHmPv0mHmP (32) Relationship between the maximal impact force Fmax,kP 0 the pile head receives and cushion stiffness coefficient is illustrated in Figure 5. A conclusion which is similar to the conclusion drawn from Eq. (25) can be drawn from Eq. (32), viz. impact force the pile head receives will also increase with the increase of cushion stiffness. From Eqs. (25) and (32), we have: Fmax,kP 0 kHx 11 Fmax,kP fm 1H!1Fmax,kP 0mPIt can be seen from Eq. (33) that Fmax,kP f (33) !Fmax,kP 0, and the bigger mH/mp is, the bigger the difference between Fmax,kP f and Fmax,kP 0 will be. Actually, foundation soil’s stiffness kP falls between 0 andf, and the maximal impact force Fmax the pile head receives satisfies the following Eq.: Fmax,kP 0FmaxFmax,kP fDownloaded by [Jicheng WANG] at 21:22 29 May 2015 (34) ANALYSES BASED ON VIBRATION THEORIES Theoretical Analysis of Impact Force Pile Head Receives Literatures (Deeks and Randolph 1993; Take, Valsangkar, and Randolph 1999; So and Ng 2010) reveal that, compared with impact force, gravity of hammer and pile is very small and can be negligible. Compared with anvil cushion and foundation soil, hammer and pile have relatively great stiffness and can be approximately seen as rigid bodies. These literatures (Deeks and Randolph 1993; Take, Valsangkar, and Randolph 1999; So and Ng 2010) also show that hammer exerts the biggest impact force on pile at initial stage of hammering. At this stage, pile and its surrounding soil take relatively small sinking and no relative displacement occurs. Hence vertical downward sinking deformation is mainly elastic deformation. This is illustrated in Figures1(c) and 1(d). Then it can be assumed that soil’s resistance and pile’s sinkage display a direct ratio. Select the equilibrium position of hammer and pile as origin of coordinates, and select the displacements xH and xP of hammer and pile which deviate from the equilibrium position as coordinates, just as illustrated in Figure 4(c). When vibration occurs with the system, the forces 12 the hammer and the pile receive are illustrated in Figure 4(d). Establish force equations of the system: mPxP kPxPkH(xHxP)®¯mHxH kH(xHxP) (35) After transposition, we have: mPxP(kPkH)xPkHxH 0®¯mHxHkHxPkHxH 0 (36) Eqs. (36) is a second-order linear homogeneous differential equation set. To make it Downloaded by [Jicheng WANG] at 21:22 29 May 2015 simpler, let: b kPkHkk,c H,d HmPmPmH (37) Eqs. (36) can be rewritten as: xPcxH 0xPbx®¯xHdxPdxH 0 (38) Assume the solutions of Eqs. (38) are: xP Asin(ZtT)®¯xH Bsin(ZtT) (39) In which A and B are vibration amplitudes, Z is angular frequency, T is initial phase. Substitute Eqs. (39) into Eqs. (38), we have: 2°AZsin(ZtT)bAsin(ZtT)cBsin(ZtT) 0®2°¯BZsin(ZtT)dAsin(ZtT)dBsin(ZtT) 0 (40) Eliminate sin(ZtT) and reorganize the equation set, we have: (bZ2)AcB 0°®2°¯dA(dZ)B 0 (41) 13 Eqs. (41) are a linear equation set of vibration amplitudes A and B. When vibration occurs with the system, the equation set has non-zero solutions, and determinant of coefficient of the equation set must be 0, viz.: bZ2dc 02dZ (42) We can obtain: bdZ 221,2§bd·¨2¸cd©¹ (43) 2Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Substitute Eqs. (43) into Eqs. (41) respectively, we have vibration amplitudes A1 and B1 corresponding to frequency Z1, and vibration amplitudes A2 and B2 corresponding to frequency Z2. With Eqs. (41) and (42) we can attest that vibration amplitudes A and B have two sets of definite ratios, viz. corresponding to first natural frequency Z1, the ratio is: A1dZ121c J1 (44) B1bZ12dCorresponding to second natural frequency Z2, the ratio is: 2A2dZ2c1 2J2 (45) B2bZ2dJ1 in Eq. (44) and J2 in Eq. (45) are proportional constants. Substitute Eq. (43) into Eqs. (44) and (45), we can obtain: 2ªbdZb11°J1 «°cc«2°¬®2°bZ21ªbd «°J2 2cc«°¬¯2º§bd·¨2¸cd»»©¹¼2º§bd·¨2¸cd»»©¹¼ (46) Kinematic equations of vibration (first principal vibration) corresponding to first natural frequency Z1 is: 14 (1)°xP A1sin(Z1tT1)®(1)°¯xH B1sin(Z1tT1) (47) Kinematic equations of vibration (second principal vibration) corresponding to second natural frequency Z2 is: (2)°xP A2sin(Z2tT2)®(2)°¯xH B2sin(Z2tT2) (48) With theories of differential equations, general solution of free vibration differential Eq. (38) can be obtained from Eqs. (47) and (48): Downloaded by [Jicheng WANG] at 21:22 29 May 2015 xP A1sin(Z1tT1)A2sin(Z2tT2)®¯xH J1A1sin(Z1tT1)J2A2sin(Z2tT2) (49) Eqs. (49) include four undetermined constants A1, A2, T1, and T2. When t = 0, the distances of hammer and pile from equilibrium position are all 0, viz. xP= 0, xH= 0. Substitute them into Eqs. (49), we have: A1sinT1A2sinT2 0®¯J1A1sinT1J2A2sinT2 0 (50) Take the derivative of Eq. (49) with t, we have: cos(Z1tT1)Z2A2cos(Z2tT2)xP Z1A1c®¯xH Z1J1A1cos(Z1tT1)Z2J2A2cos(Z2tT2) (51) When t = 0, hammer’s velocity xH v0, pile’s velocity xP 0. Substitute them into Eqs. (51), we have: Z1A1cosT1Z2A2cosT2 0®¯Z1J1A1cosT1Z2J2A2cosT2 v0 (52) From Eqs. (50) and (52), we can obtain: T1 T2 0,A1 v0v0,A2 Z1(J1J2)Z2(J1J2) (53) Substitute Eqs. (53) into Eqs. (49), we have motion equations of hammer and pile: 15 v0v0xsinZtsinZ2t 1°PZ(JJ)Z(JJ)°112212®J1v0J2v0°x sinZ1tsinZ2tH°Z(JJ)Z(JJ)112212¯ (54) It can be noted that if soil’s frictional resistance to pile is very small and is negligible, then kPo0. From Eqs. (37) we have bokH, from Eq. (43) we have Z1o0, from Eqs. (54) we have mPxHof and xPof, showing that, because pile does not receive soil’s resistance, the system comprising hammer and pile will indefinitely sink, hence Eqs. (54) is meaningless. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Take the second derivative of the second equation of Eqs. (54), we have: xH J1v0Z1JvZsinZ1t202sinZ2tJ1J2J1J2 (55) Then the force the hammer receives is: §JvZ·JvZF mHxH mH¨101sinZ1t202sinZ2t¸J1J2©J1J2¹ (56) And pile receives a downward impact force from hammer: §JvZ·JvZF mHxH mH¨101sinZ1t202sinZ2t¸J1J2©J1J2¹ (57) Comparison of Conclusions Drawn from the Two Methods When Pile Receives a Very Great Resistance. When the pile receives a very great resistance, then the penetration is very small and can be negligible, let kPof, from Eq. (37) we have bof. Substitute it into Eq. (43), we have Z1 and Z2. Again, substitute it into Eqs. (46), we have J1 and J2. Substitute Z1, Z2, J1, and J2 into second equation of Eqs. (54), we have: 16 xH J1v0J2v0mHsinZ1tsinZ2t v0sinZ1tkHZ1(J1J2)Z2(J1J2) (58) Take the second derivative of Eq. (58), we have: mHv0Z12sinZ1tkHxH (59) Because Z12 kH, then Eq. (59) can be transformed into: mHDownloaded by [Jicheng WANG] at 21:22 29 May 2015 (60) The minus in Eq. (60) means that hammer’s acceleration is opposite to the direction of xH and hammer receives an upward thrust. Then the pile receives a downward maximal impact force: Fmax mHxHmax kHmHv0 Fmax,kP fxH kHv0sinZ1tmH (61) S22nS (n The conclusion is the same as the conclusion drawn from Eq. (25). When Z1t is nonnegative integer), Eq. (61) is true. Hammer separates from pile after collision, hence Z1t equals S2 only, then the time the pile head receives the maximal impact force is: t (62) It means that what Figure 4(a) shows is an exception of Eq. (57) when kP f. When Resistance Pile Receives Is Very Small. When the resistance pile receives is very small and can be negligible, it can be regarded that kP = 0. Substitute kP = 0 into Eqs. (37), (43), (46), and (57), we have: kHmHmPv0sinZ2tmHmPSSmH 2Z12kHF (63) The maximal value of Eq. (63) is: 17 Fmax kHmHmPv0 Fmax,kP 0mHmP (64) S22nS (n The conclusion is the same as the conclusion drawn from Eq. (32). When Z2t is nonnegative integer), Eq. (64) is true. Hammer separates from pile after collision, hence Z2t equals S2 only. Substitute kP = 0 into Eqs. (37) and (43), we have Z2 kHkH, hence the time mPmHthe pile head receives the maximal impact force is: mPmHSS 2Z22(mHmP)kH (65) Downloaded by [Jicheng WANG] at 21:22 29 May 2015 t It means that what Figure 4(b) shows is an exception of Eq. (57) when kP = 0. Examples of Cushion Stiffness's Influence on Impact Force Assume hammer weight mH = 1.04 kg, pile weight mP = 1.56 kg, hammer’s fall height is 0.82 m, foundation soil’s stiffness kP = 3.6×105 N/m. Hammer’s impact force on pile can be calculated with Eq. (57), as illustrated in Figure 6. Cushion can bear pressure rather than tensile force, hence when impact force in Figure 6 is negative, it means that hammer separates from cushion and cushion restores its original state and is not get tensiled. It can be seen from Figure 6 that hammer’s impact force on pile increases with the increase of cushion stiffness, but hammer’s impact time decreases with the increase of cushion stiffness. Examples of Different Hammer Weight's Influence on Impact Force 18 Assume mP = 1.56 kg, kH = kP = 3.6×105 N/m. Hammer’s initial energy E0 8Nmremains unchanged, Change hammer weight and fall height. Hammer’s impact force on pile can be calculated with Eq. (57), as illustrated in Figure 7. As mentioned above, negative impact force means hammer separates from cushion. It can be seen from Figure 7 that, for identical hammer impact energy, the bigger the hammer weight is, the smaller the hammer’s impact force on pile will be, and the longer the impact time will last. When mH>mP(1.56 kg), viz. hammer weight is bigger than pile weight, hammer’s impact force on Downloaded by [Jicheng WANG] at 21:22 29 May 2015 pile remains bigger than 0. ANALYSES BASED ON MODEL TEST Equipment parts of the model test are shown in Figure 8(a), and assembly drawing is illustrated in Figure 8(b). Loose uniform sand, void ratio is 0.8㸪relative density Dr = 45%㸪dry unit weight is 14.5 kN/m3, pile’s lower end just touches soil surface. Lift hammer to a certain height, and let it fall down freely and hit spring, then the pile will sink. Elastic Cushion Test Hammer’s initial energy is E0 8Nm, and mH = 1.04 kg, and mP = 1.56 kg, and spring stiffness kH = 3.6×105 N/m. An accelerometer attached to hammer is used to measure hammer’s acceleration when hitting spring. Hammer’s actual impact force on spring is indirectly measured, as illustrated in Figure 9. Because soil’s stiffness coefficient kP is unknown, here let kP = 0, 180, 360, 540, 720, 900, 1260, 1440, 2000, 4000 kN/m and infinitely great. Substitute these values into Eq. (57), hammer’s impact force on pile can be achieved, as illustrated in Figure 9. It can be seen that measured impact force is approximate to theoretical value. Especially when kP = 360 kN/m, 19 the maximal impact force measured is nearly identical with theoretical value. From Eqs. (62) and (63), it can be calculated that the times when maximal impact forces occur is 0.00204 s and 0.00262 s respectively, as illustrated in Figure 9. It can be seen that the bigger the foundation soil’s stiffness coefficient kP is, the later the maximal impact force will come, and the bigger the impact force will be. Keep hammer’s impact energy unchanged, but change hammer weight. Measure the values of hammer’s maximal impact force on pile, pile’s penetration, and maximal acceleration of soil Downloaded by [Jicheng WANG] at 21:22 29 May 2015 surface’s vertical vibration at 0.2 m horizontally from pile, as illustrated in Figure 10. Let kP = 360 kN/m, Hammer’s maximal impact force on pile calculated with Eq. (57) is also illustrated in Figure 10. It can be seen that maximal impact force measured is approximate to theoretical value, and the force degrades with the increase of hammer weight, which shows that “heavy hammer and low drop” can reduce impact force on pile head and lower probability of pile head damages. It can also be seen that with the increase of hammer weight, pile’s penetration also increases, especially when hammer weight is very small. Increasing hammer weight can significantly increase pile’s penetration, which conforms to the conclusions drawn by using energy and momentum conservation method, as illustrated in Figure 2. However, it can be seen from Figure 2 that, when mH/mp > 1, penetration will degrade with the increase of hammer weight, which is different from the conclusions drawn by using methods of vibration theories and model test. This is because energy and momentum conservation method assumes that hammer separates from pile after collision. However, the actual case is not the same as assumed, as can be seen from Figure 7. When hammer weight is bigger than pile weight, the two do not separate after collision, and impact force is bigger than 0 all along, which means that, when hammer weight is bigger than pile weight, assumption of energy and momentum conservation method is unreasonable. It can 20 also be seen from Figure 10 that vibration of soil surface from 0.2 m of pile starts to weaken with the increase of hammer weight, which means that the lost energy causing soil surface’s vibration degrades, and energy utilization efficiency is improved. Penetration of “heavy hammer and low drop” is bigger than that of “light hammer and high drop”. Hammer’s initial energy E0 8Nm, and mH = 1.04kg, and mP = 1.56kg. Change spring’s stiffness. Hammer’s maximal impact force on pile, pile’s penetration, and spring’s maximal compression are measured, as illustrated in Figure 11. Let kP=3.6×105 N/m. The maximal impact Downloaded by [Jicheng WANG] at 21:22 29 May 2015 force calculated with Eq. (57) is illustrated in Figure 11. It can be seen that with the increase of spring stiffness, hammer’s impact force on spring and pile’s penetration also increase, but spring’s maximal compression degrades. This is even more evident especially when spring has a relatively small stiffness. The conclusion conforms to the conclusion drawn from Eq. (17). It can also be seen that when spring has a relatively big stiffness, theoretical value of hammer’s maximal impact force on pile is bigger than measured value, and the bigger the spring stiffness is, the bigger the differences between these two values will be. For example, if kHof, from Eq. (57) we know that Fof, viz. if there is no cushion, impact force will be infinitely great, which does not conform to reality. This is because Eq. (57) assumes that hammer and pile are all rigid bodies while actually they are not. When spring stiffness is very big, impact force will be very big. In this case, elastic deformation of hammer and pile cannot be neglected. Plastic Cushion Test Absolute plastic materials do not exist. Moreover, materials with strong plastic properties present great difficulty in practical engineering use, hence elastic cushion or elastic-plastic cushion (such as plank) is widely used in engineering. Cotton cloth is used as cushion in this model test. 21 The cotton cloth has a thickness of 50 mm and is fastened to the tray of pile. Hammer’s initial energy is E0 8Nm. Hammer’s maximal impact force on pile and pile’s penetration, when given different hammer weights, are illustrated in Figure 10. It can be seen that, with the increase of hammer weight, the maximal impact force the pile head receives will degrade, but penetration will increase. The conclusion conforms to the conclusion when using elastic cushion. It can also be seen that the maximal impact force when using plastic cushion is bigger than that of when using elastic cushion, but penetration is smaller all along. It shows that relatively small impact force can Downloaded by [Jicheng WANG] at 21:22 29 May 2015 achieve relatively great penetration when using elastic cushion, hence energy utilization efficiency is higher. Hammer weight is 1.04 kg, other factors are the same as the factors mentioned above. Repeat hammering test. Relationship between the maximal impact force and blow counts is illustrated in Figure 12. It can be seen that, compared with using spring as cushion, pile receives a greater maximal impact force when using cotton cloth as cushion. With the increase of blow counts, the maximal impact force increases with cotton cloth being gradually compacted. The increase of impact force is especially significant at the beginning of hammering when cotton cloth is significantly compacted. However, spring takes no significant changes, and impact force remains almost unchanged. In practical engineering, with the increase of blow counts, plank as cushion is gradually compacted and becomes hardened, and its color turns black. The plank may even get scorched due to hammering, thus losing its protecting function for pile head. In this case, the plank needs to be changed or replaced with elastic cushion. CONCLUSIONS The following conclusions can be drawn from this study. 22 (1) If hammer’s impact energy remains unchanged, penetration increases with the increase of hammer weight and cushion stiffness; the impact force the pile head receives degrades with the increase of hammer weight, but increases with the increase of cushion stiffness. (2) The time the pile head receives impact force degrades with the increase of elastic cushion stiffness, but increases with the increase of hammer weight. When hammer weight increases to a certain weight, hammer will not rebound and depart from pile head. (3) Hammer’s impact force on pile increases with the increase of foundation soil’s stiffness. (4) Compared with elastic-plastic cushions such as planks, elastic cushion can achieve greater penetration with relatively small pile head impact force, hence protecting head from being damaged. Cotton cloth cushion gradually gets compacted with the increase of blow counts, and impact force the pile head receives increases gradually. (5) When foundation soil’s stiffness is 0 or infinitely great, the analytic formula of maximal impact force the pile head receives obtained by using energy and momentum conservation method is the same as that of using method of vibration theories. The maximal impact force and impact force changes over time obtained from method of vibration theories coincides well with those of using model test method. (6) Energy and momentum conservation method assumes that hammer separates from pile after collision when using elastic cushion, but the actual conditions as proved by using methods of vibration theories and model test are not the case. When hammer 23 Downloaded by [Jicheng WANG] at 21:22 29 May 2015 weight is bigger than pile weight, hammer does not separate from pile after collision. Penetration achieved by using energy and momentum conservation method is inaccurate. (7) When cushion stiffness is relatively small, theoretical value obtained by using method of vibration theories coincides well with the value measured by using model test method. However, when cushion stiffness is very big, the maximal impact force obtained by using method of vibration theories has certain difference with the value Downloaded by [Jicheng WANG] at 21:22 29 May 2015 measured by using model test method. This difference tends to increase with the increase of cushion stiffness. This is because vibration theories assume that hammer and pile are all rigid bodies, but when cushion stiffness is relatively big, impact force is also big. In this case, the deformation that occurs to hammer and pile cannot be neglected, hence differences exist with rigid body assumption. References Deeks, A. J., and M. F. Randolph. 1993. Analytical modelling of hammer impact for pile driving. International Journal for Numerical and Analytical Methods in Geomechanics 17 (5):279–302. doi:10.1002/nag.1610170502 Deeks, A. J., and M. F. Randolph. 1995. A simple model for inelastic footing response to transient loading. International Journal for Numerical and Analytical Methods in Geomechanics 19 (5):307–29. doi:10.1002/nag.1610190502 Hehua, Z., X. Yongjian, and W. Huaizhong. 2004. Analytical Solution for Pile Hammer Impact and Application of Optimum De sign Technique for Determining Cushion Parameters. Journal of Tongji University (Natural Science) 32 (7):841–45. Hiley, A. 1925. A rational pile-driving formula and its application in piling practice explained. Engineering 119:657–58. Koten, H. V. 1991. Optimal pile driving. In Proceedings of the 4th international conference on piling and deep foundation, Rotterdam. pp. 65–668. Liyun, Z., and L. Pu'an. 1994. Development of new type disc-sping cap and it's application. Chinese Journal of Geotechnical Engineering 16 (4):47–55. Middendorp, P. 2004 Thirty years of experience with the wave equation solution based on the method of characteristics. In 7th International Conference on the Application of Stress Wave Theory to Piles, 2004, Kuala Lumur, Malaysia. 24 Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Middendorp, P., and P. van Weel. 1986. Application of characteristic stress wave method in offshore practice. In Proceedings of the 3rd International Conference on Numerical Methods in Offshore Piling. pp. 21–22. Peng-kong, A., and M. S. Subrahmanyam. 2003. Pile length influences on piling formula. Chinese Journal of Geotechnical Engineering 25 (3):264–67. Randolph, M. F., P. K. Banerjee, and R. Buttefield, eds. 1991. Analysis of the dynamics of pile driving, Developments in soil mechanics IV: Advanced geotechnical analyses. Elsevier Applied Science. Rausche, F., G. G. Goble, and G. E. Likins, Jr. 1985. Dynamic determination of pile capacity. Journal of Geotechnical Engineering 111 (3):367–83. doi:10.1061/(ASCE)0733-9410(1985)111:3(367) Renpeng, C., C. Yunmin, L. Qi, and T. Jianguo. 2001 Study of pile drivability with in-stiu measurement. Chinese Journal of Geotechnical Engineering 23 (2):235–38. Runfu, W., and Z. Qing. 2000. Dynamic analysis for the shell of dish-shaped spring pile-cap. Advances in Science and Technology of Water Resources 20 (2):31–34. Shifang, W., C. Renpeng, and C. Yunmin. 2004. Study on pile drivability with a simplified method. Journal of Zhejiang University (Engineering Science) 37 (6):657–63. Smith, E. A. 1960. Pile driving analysis by the wave equation. J. Soil Mech. Found. ASCE. 86 (1):35–61. So, A. K. O., and C. W. W. Ng. 2010. Impact compression behaviors of high-capacity long piles. Canadian Geotechnical Journal 47 (12):1335–50. doi:10.1139/T10-031 Take, W. A., A. J. Valsangkar, and M. F. Randolph. 1999. Analytical solution for pile hammer impact. Computers and Geotechnics 25 (2):57–74. doi:10.1016/s0266-352x(99)00018-x Yongjian, X., Z. Hehua, W. Huaizhong, and T. Lvbin. 2005. Analytical solution for model of pile hammer impact. Chinese Journal of Rock Mechanics and Engineering 24 (1):171–76. 25 Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Figure 1. Diagram of piling process. 26 Figure 2. Relationships between penetration S, energy utilization efficiency K and mH/mp. 27 Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Figure 3. Relationship between energy utilization efficiency K, penetration S and cushion stiffness kH. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 28 Figure 4. Interaction among hammer, cushion, pile, and soil. 29 Downloaded by [Jicheng WANG] at 21:22 29 May 2015 Figure 5. Relationship between the maximal impact force the pile receives and cushion stiffness. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 30 Figure 6. Cushion stiffness’s influence on impact force. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 31 Figure 7. Influence of hammer weight on impact force. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 32 Figure 8. Model test equipment (not drawn in actual proportion). Downloaded by [Jicheng WANG] at 21:22 29 May 2015 33 Figure 9. Influence of foundation soil’s stiffness on impact force the pile head receives. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 34 Figure 10. Influence of hammer weight on impact force, penetration, and soil’s vibration. Note: The acceleration in the figure is the maximal acceleration of soil surface’s vertical vibration at 0.2 m horizontally from pile. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 35 Figure 11. The maximal impact force, penetration, spring’s maximal compression when spring has different stiffnesses. Downloaded by [Jicheng WANG] at 21:22 29 May 2015 36 因篇幅问题不能全部显示，请点此查看更多更全内容