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杭の容量を推定するためのさまざまな方法

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.

フィギュアロードメカニズム, Estimating Pile Capacity,

図 1: 杭の荷重伝達メカニズム

\( {Q}_{あなた} = {Q}_{p} + {Q}_{s} \) (1)

Qあなた : Ultimate load-carrying capacity

Qp : End-bearing load capacity

Qs : Skin-frictional resistance

Universal equations for Qp とQs


End-bearing capacity (Eq. 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 (Eq. 3).

\( {Q}_{p} = {あ}_{p} \回 {q}_{p} \) (2)

p : Pile tip area
qp : Unit point resistance

\( {q}_{p} = (c \times {N}_{c}) + (q’ \回 {N}_{q}) + (\gamma \times D \times {N}_{\gamma}) \) (3)

c : Soil cohesion at the tip of the pile
q’ : Effective vertical stress at the tip of the pile
D : Pile width
そして : Soil unit weight
Nc , Nq, : 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:

\( {Q}_{p} = {あ}_{p} \回[ (c \times {N}_{c}) + (q’ \回 {N}_{q}) ] \) (4)

 

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

\( {Q}_{s} = ∑ (p×ΔL×f) \) (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)cr usually varies from 16 に 18. この方法では, piles in the sand are assumed to have zero cohesion (c ≈ 0), and the unit point resistance should not exceed limiting point resistance (ql), which is given by Eq. 7. The bearing capacity factor (Nq) values are directly proportional to the soil friction angle of the bearing stratum (テーブル 1). Based on Meyerhof’s theory, the universal equation for Qp (Eq.4) can be simplified to:

\( {Q}_{p} = {あ}_{p} \回 (q’ \回 {N}_{q}) \leq ({あ}_{p} \回 {q}_{l}) \) (6)

\( {q}_{l} = 0.5 \回 {p}_{a} \回 {N}_{q} \times tan (\phi’) \) (7)

ql : Limiting point resistance

pa: Atmospheric pressure (≈100 kN/m2)

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

Ø
Nq
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

テーブル 1: Interpolated values of Nq (Meyerhof’s theory)

Clay Soil

方程式 4 can also calculate the end-bearing capacity of piles in clay or cohesive soils (φ ≈ 0). Since soil friction angle is neglected and the bearing capacity factor (Nc) has a constant value of 9 for cohesive soils, Eq.4 can be written as:

\( {Q}_{p} = {あ}_{p} \times c \times {N}_{c} = 9 \times c \times {あ}_{p} \) (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:

\( {Q}_{p} = {あ}_{p} \times \bar{\sigma’}_{の} \回 {N}_{\sigma} \) (9)

\(\バー{\sigma’}_{の} = frac{1 + (2 \回 {K}_{の})}{3} \times q’\) (10)

\( {K}_{の} = 1 – sin \phi’\) (11)

\( {N}_{\sigma} = frac{3 \回 {N}_{q}}{1 + (2 \回 {K}_{の})} \) (12)

\(\バー{\sigma’}_{の} \) : 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, Eq. 4 is also applicable to calculate the end-bearing capacity of piles in clay. しかしながら, the value of the bearing capacity factor (Nc) is a factor of rigidty index (私r). According to his theory of expansion of cavities, Nc そして私r can be estimated by:

\( {N}_{c} = (\フラク{4}{3}) \回 [ln({私}_{r}) + 1] + \フラク{\パイ}{2} + 1 \) (13)

\( {私}_{r} = frac{{E}_{s}}{3 \times c} \) (For φ ≈ 0)(14)

r: Rigidity index

Es: 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

\( {Q}_{p} = {あ}_{p} \回 (q’ \回 {N}_{q}) \) (15)

図 2: Variation of Nq with L/D & ファイ’ (Redrawn after Coyle & Castello, 1981)
ソース: それか, Braja. 基礎工学の原則 (7第版, p.564)

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 (K) & soil-pile friction angle, which both have varying values depending on which approach to use or to the available soil data.

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

K: Effective earth pressure coefficient

σ’: Effective vertical stress at the depth under consideration

NS: 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.

NAVFAC DM 7.2

杭タイプ 圧縮 Uplift
被駆動H杭
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
ドリブンジェットパイル
0.4-0.9
0.3-0.6
退屈な杭 (< 24″ファイ)
0.7
0.4

テーブル 2: Earth pressure coefficient, K (NAVFAC DM 7.2)

Average K Method

The earth pressure coefficient (K) can also be evaluated by taking the average of earth pressure coefficient at rest (K0), active earth pressure (Ka), and passive earth pressure (Kp), as shown from Equations 16-19.

\( K =\frac{{K}_{0} + {K}_{a} + {K}_{p}}{3} \) (16)

\( (K)_{0} =1sin \phi \) (17)

\( (K_{a} =1 – {tan}^{2}( \フラク{45 – \ファイ}{2}) \) (18)

\( (K_{p} =1 + {tan}^{2}( \フラク{45 + \ファイ}{2}) \) (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.

杭タイプ K
H-piles
1.65
Steel pipe piles
1.26
Precast concrete piles
1.5

テーブル 3: Earth pressure coefficient, K (Mansur and Hunter, 1970)

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. しかしながら, 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. 一方, NAVFAC DM7.2 uses these values to estimate the friction angle between the soil and pile:

杭タイプ NS
Steel pile
20°
Timber pile
3/4 ファイ
Concrete pile
3/4 ファイ

テーブル 4: Soil-pile friction angle (NS) (NAVFAC DM 7.2)

 

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. しかしながら, 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 そして 21. λ values changes as the depth of the penetration of pile increases. テーブル 5 shows the variation of λ with the embedment length of the pile.

\( {f}_{av} = \lambda \times [\バー{\sigma’}_{の} +( 2 \回 {c}_{あなた})] \) (20)

\({Q}_{s} = p \times L \times {f}_{av} \) (21)

\( \バー{\sigma’}_{の} \): Mean effective vertical stress for the entire embedment length

cあなた: Mean undrained shear strength

L (メートル) λ
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

テーブル 5: Variation of λ with pile embedment length (L)

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 (a). テーブル 6 shows the corresponding value of the adhesion factor with the ratio of undrained cohesion and atmospheric pressure (cあなた/pa).

\(f = \alpha \times {c}_{あなた}\) (22)

したがって, the total frictional resistance of pile in clay using this method can be re-written as:

\({Q}_{s} = sum (f \times p \times \Delta L) = sum (\alpha \times {c}_{あなた} \times p \times \Delta L)\) (23)

 

cあなた/pa a
≤ 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

pa =大気圧≈ 100 kN / m2

テーブル 6: Variation of α (テルツァーギ, ペック, とメスリ, 1996)

 

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, 用語 (K × tanδ) for unit frictional resistance of piles in sand shall be represented by β. The soil-friction angle (NS) shall be replaced by a remolded drained friction angle of the soil (ファイ’R). Thus the unit frictional resistance of piles in clay is estimated to be equal to:

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

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

Conservatively, the earth pressure coefficient (K) is equivalent to the earth pressure coefficient at rest (K0) which varies for normally consolidated clays and overconsolidated clays, as shown in the following equations:

\( K = {K}_{0} = 1 – それなし {\Phi ‘}_{R}\) (Normally consolidated clays) (26)

\( K = {K}_{0} = (1 – それなし {\Phi ‘}_{R}) \回 sqrt(OCR)\) (Overconsolidated clays) (27)

OCR: Overconsolidation ratio

 

 

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参考文献:

  • それか, B.M. (2007). 基礎工学の原則 (7第版). グローバルエンジニアリング
  • ラジャパクセ, R. (2016). 親指の杭の設計と建設のルール (2第2版). エルゼビア株式会社.
  • トムリンソン, M.J. (2004). パイルの設計と建設の実践 (4第版). E & FNスポンサー.
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