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# 估算桩容量的各种方法

## Estimating Pile Capacity

Estimating the Pile load-carrying capacity is necessary to determine the ultimate axial load that the pile can carry. The ultimate load capacity of the pile (Qu) is equivalent to the sum of end-bearing capacity (Qp) and frictional resistance (Qs), represented by Fig. 1 and Eq. 1. Numerous published studies and practices determine the pile’s end-bearing capacity and frictional resistance. This article focuses on various methods to estimate the ultimate pile capacity. $${问}_{ü} = {问}_{p} + {问}_{s}$$ (1)

s : Skin-frictional resistance

## Universal equations for Qp 其余的将被桩所承受的土壤抵抗s

End-bearing capacity (情商. 2) is the ultimate resistance per unit area developed at the tip of the pile. The unit point resistance at pile tip (qp) can be expressed similarly to the general bearing capacity equation for shallow foundations proposed by Terzaghi (情商. 3).

$${问}_{p} = {一个}_{p} \次 {q}_{p}$$ (2)

qp : Unit point resistance

$${q}_{p} = (c \times {ñ}_{C}) + (q’ \次 {ñ}_{q}) + (\gamma \times D \times {ñ}_{\gamma})$$ (3)

C : Soil cohesion at the tip of the pile
q’ : Effective vertical stress at the tip of the pile
d : Pile width

ñC , ñq, : Bearing capacity factors
Since the width of the pile is relatively small compared to shallow foundations the third term of Eq. 3 can be neglected, thus Eq. 2 can be re-written as:

$${问}_{p} = {一个}_{p} \次[ (c \times {ñ}_{C}) + (q’ \次 {ñ}_{q}) ]$$ (4)

The total frictional resistance of the pile, which is developed along its length, can be calculated using this equation:

$${问}_{s} = ∑ (p × ΔL × f)$$ (5)

p: Perimeter of the pile

ΔL: Incremental pile length over which p and f are taken

F: Unit frictional resistance at any depth

## Methods for Estimating Qp

### Meyerhof’s Method

#### Sandy Soil

According to Meyerhof, the unit point resistance (qp) of piles in sand generally increases with the embedment length until it reaches its maximum value when the embedment ratio (L/D) reaches a critical value. Critical embedment ratio (L/D) usually varies from 16 至 18. 用这种方法, piles in the sand are assumed to have zero cohesion (c ≈ 0), and the unit point resistance should not exceed limiting point resistance (q), which is given by Eq. 7. The bearing capacity factor (Nq) values are directly proportional to the soil friction angle of the bearing stratum (桌子 1). Based on Meyerhof’s theory, the universal equation for Qp (Eq.4) can be simplified to:

$${问}_{p} = {一个}_{p} \次 (q’ \次 {ñ}_{q}) \列克 ({一个}_{p} \次 {q}_{升})$$ (6)

$${q}_{升} = 0.5 \次 {p}_{一个} \次 {ñ}_{q} \times tan (\菲')$$ (7)

q : Limiting point resistance

p一个: Atmospheric pressure (≈100 kN/m2)

$$\phi’$$: Effective soil friction angle at the tip of the pile

Ø
ñq
20
12.4
21
13.8
22
15.5
23
17.9
24
21.4
25
26
26
29.5
27
34
28
39.7
29
46.5
30
56.7
31
68.2
32
81
33
96
34
115
35
143
36
168
37
194
38
231
39
276
40
346
41
420
42
525
43
650
44
780
45
930

#### Clay Soil

$${问}_{p} = {一个}_{p} \times c \times {ñ}_{C} = 9 \times c \times {一个}_{p}$$ (8)

### Vesic’s Method

Vesic’s method of calculating end-bearing capacity on sandy or clayey soils is based on his theory of the expansion of cavities.

#### Sandy Soil

Based on his theory, end-bearing capacity of piles in sand can be estimated using the following equations:

$${问}_{p} = {一个}_{p} \times \bar{\sigma’}_{的} \次 {ñ}_{\sigma}$$ (9)

$$\酒吧{\sigma’}_{的} = frac{1 + (2 \次 {ķ}_{的})}{3} \times q’$$ (10)

$${ķ}_{的} = 1 – sin \phi’$$ (11)

$${ñ}_{\sigma} = frac{3 \次 {ñ}_{q}}{1 + (2 \次 {ķ}_{的})}$$ (12)

$$\酒吧{\sigma’}_{的}$$ : Mean effective normal ground stress at the level of the pile point

Ko: Earth pressure coefficient at rest

: Bearing capacity factor

#### Clay Soil

Same with Meyerhof’s method, 情商. 4 is also applicable to calculate the end-bearing capacity of piles in clay. 然而, the value of the bearing capacity factorC) is a factor of rigidty index (一世[R). According to his theory of expansion of cavities, ñC 和我[R can be estimated by:

$${ñ}_{C} = (\压裂{4}{3}) \次 [输入({一世}_{[R}) + 1] + \压裂{\pi}{2} + 1$$ (13)

$${一世}_{[R} = frac{{Ë}_{s}}{3 \times c}$$ (For φ ≈ 0)(14)

Ës: Modulus of elasticity of soil

### Coyle and Castello’s Method (Sandy Soil)

Based on 24 large-scale field load tests of driven piles in sand, Coyle and Castello suggested that the end-bearing capacity of piles can be calculated using Eq.15. The values of the bearing capacity factor (Nq) is a factor of both embedment ratio (L/D) and the soil friction angle (φ’), as shown in Fig. 2

$${问}_{p} = {一个}_{p} \次 (q’ \次 {ñ}_{q})$$ (15) ## Methods for Estimating Qs

### Frictional Resistance of Piles in Sand

The unit frictional resistance of piles in sand, as shown in Eq. 5, considers multiple factors which are quite difficult to calculate. It includes the earth pressure coefficient (ķ) & soil-pile friction angle, which both have varying values depending on which approach to use or to the available soil data.

$$f = K\times {\sigma}_{的}’ \times tan (\delta)$$ (15)

ķ: Effective earth pressure coefficient

σ’: Effective vertical stress at the depth under consideration

d: Soil-pile friction angle

The following are the different ways to estimate the effective earth pressure coefficient and soil-friction angle values. These variables are a factor of soil frictional angle (φ’) or pile type.

### Effective earth pressure coefficient

The soil exerts lateral earth pressure to the pile surface. It is necessary to account for this pressure on the design or analysis for stability. The following are the different ways to determine the earth pressure coefficients to calculate the unit frictional resistance of piles in sand.

#### 海军陆战队DM 7.2

0.5-1.0
0.3-0.5
Round/Square Driven displacement piles
1.0-1.5
0.6-1.0
Tapered Driven displacement piles
1.5-2.0
1.0-1.3

0.4-0.9
0.3-0.6

0.7
0.4

#### Average K Method

The earth pressure coefficient (ķ) can also be evaluated by taking the average of earth pressure coefficient at rest0), active earth pressure一个), and passive earth pressurep), as shown from Equations 16-19.

$$K =\frac{{ķ}_{0} + {ķ}_{一个} + {ķ}_{p}}{3}$$ (16)

$$(ķ)_{0} =1sin \phi$$ (17)

$$(K_{一个} =1 – {tan}^{2}( \压裂{45 – \φ}{2})$$ (18)

$$(K_{p} =1 + {tan}^{2}( \压裂{45 + \φ}{2})$$ (19)

#### Mansur and Hunter (1970)

Based on different field load test results, Mansur and Hunter concluded the values of earth pressure coefficient with the corresponding pile types.

H-piles
1.65
Steel pipe piles
1.26
Precast concrete piles
1.5

### Soil-pile Friction Angle

The friction angle between the soil and the surface of the pile is an essential aspect of foundation design. Practically, many engineers approximate this value as equal to 2/3 of the internal friction angle of the soil. 然而, based on the study of Coyle and Castello in 1981, the soil-pile friction angle is approximately equivalent to 80% of the internal friction angle of the soil. 另一方面, NAVFAC DM7.2 uses these values to estimate the friction angle between the soil and pile:

Steel pile
20°
Timber pile
3/4 披
Concrete pile
3/4 披

### Frictional Resistance of piles in Clay

Calculating the frictional resistance of piles in clayey soils can be as challenging as the one in sandy soils due to the introduction of new variables, which are also not as easy to determine. 然而, there are several available methods to obtain the values of these variables.

#### λ Method

Based on the study of Vijayvergiya and Focht in 1972, the total frictional resistance of piles in clay can be estimated by determining the average unit frictional resistance of the pile, as shown by Equations 20 和 21. λ values changes as the depth of the penetration of pile increases. 桌子 5 shows the variation of λ with the embedment length of the pile.

$${F}_{av} = \lambda \times [\酒吧{\sigma’}_{的} +( 2 \次 {C}_{ü})]$$ (20)

$${问}_{s} = p \times L \times {F}_{av}$$ (21)

$$\酒吧{\sigma’}_{的}$$: Mean effective vertical stress for the entire embedment length

Cü: Mean undrained shear strength

0
0.5
5
0.336
10
0.245
15
0.200
20
0.173
25
0.150
30
0.136
35
0.132
40
0.127
50
0.118
60
0.113
70
0.110
80
0.110
90
0.110

#### α Method

The α method suggests that unit frictional resistance of piles is equivalent to the product of the undrained cohesion of the soil layer and its corresponding empirical adhesion factor (一种). 桌子 6 shows the corresponding value of the adhesion factor with the ratio of undrained cohesion and atmospheric pressure (Cü/p一个).

$$f = \alpha \times {C}_{ü}$$ (22)

$${问}_{s} = 总和 (f \times p \times \Delta L) = 总和 (\alpha \times {C}_{ü} \times p \times \Delta L)$$ (23)

Cü/p一个 一种
≤ 0.1
1.0
0.2
0.92
0.3
0.82
0.4
0.74
0.6
0.62
0.8
0.54
1.0
0.48
1.2
0.42
1.4
0.40
1.6
0.38
1.8
0.36
2.0
0.35
2.4
0.34
2.8
0.34

p一个 = 大气压 ≈ 100 千牛/米2

#### β Method

Pore water pressure around the pile increases when the pile is driven into saturated clays. This method, based on effective stress analysis, is suited for long-term (drained) analyses of the pile load capacity as it considers the gradual dissipation of the excess pore water pressure over time. According to Tomlinson (1971), piles driven in soft clays assume that failures occur in the remolded soil close to the pile surface. Based on Eq. 15, 期限 (K × tanδ) for unit frictional resistance of piles in sand shall be represented by β. The soil-friction angle (d) shall be replaced by a remolded drained friction angle of the soil (披’[R). Thus the unit frictional resistance of piles in clay is estimated to be equal to:

$$f = \beta \times {\sigma’}_{的}$$ (24)

$$\beta = K \times tan {\Phi ‘}_{[R}$$ (25)

Conservatively, the earth pressure coefficient (ķ) is equivalent to the earth pressure coefficient at rest0) which varies for normally consolidated clays and overconsolidated clays, as shown in the following equations:

$$ķ = {ķ}_{0} = 1 – 没有 {\Phi ‘}_{[R}$$ (Normally consolidated clays) (26)

$$ķ = {ķ}_{0} = (1 – 没有 {\Phi ‘}_{[R}) \次 sqrt(字符识别)$$ (Overconsolidated clays) (27)

### 参考资料:

• 计算极限承载能力, 计算极限承载能力. (2007). 计算极限承载能力 (7第版). 计算极限承载能力
• 计算极限承载能力, [R. (2016). 计算极限承载能力 (2nd版). 计算极限承载能力.
• 计算极限承载能力, 计算极限承载能力. (2004). 计算极限承载能力 (4第版). Ë & 计算极限承载能力.