Here are the Meyerhof formulas that are commonly used for calculating the ultimate bearing capacity of pile foundations based on the three types of soil data: Laboratory Data, NSPT (Standard Penetration Test), and Dutch Cone Penetrometer data. Each data type requires slightly different approaches.

1. Laboratory Data:

For laboratory data, Meyerhof's formula is often used in conjunction with soil parameters such as cohesion, unit weight, and angle of internal friction (φ). Here's how the ultimate bearing capacity is calculated.

End Bearing Capacity:

The end bearing capacity for cohesive soils is often approximated using:

qb=9×cq_b = 9 times cqb​=9×c

Where:

• qbq_bqb​ = End bearing capacity (kPa)

• ccc = Cohesion (kPa)

Skin Friction (Shaft Resistance):

The skin friction for cohesive soils is simplified using a constant factor that relates to the unit weight and the pile-soil interface.

fs=α×cf_s = alpha times cfs​=α×c

Where:

• fsf_sfs​ = Unit skin friction (kPa)

• αalphaα = Adhesion factor (0.5 - 1, empirical value)

• ccc = Cohesion (kPa)

Total Bearing Capacity (Q):

The total bearing capacity is the sum of end bearing and skin friction:

Q=A×qb+∑fs×AsQ = A times q_b + sum f_s times A_sQ=A×qb​+∑fs​×As​

Where:

• AAA = Cross-sectional area of the pile tip (m²)

• AsA_sAs​ = Surface area of the pile shaft (m²)

• fsf_sfs​ = Skin friction (kPa)

2. NSPT (Standard Penetration Test) Data:

Meyerhof's empirical relationship based on SPT (N-value) provides another method to estimate the pile's bearing capacity. This is generally used for granular soils (sands).

End Bearing Capacity:

qb=N×40q_b = N times 40qb​=N×40

Where:

• qbq_bqb​ = End bearing capacity (kPa)

• NNN = NSPT value (blows per foot)

Skin Friction (Shaft Resistance):

fs=β×Nf_s = beta times Nfs​=β×N

Where:

• fsf_sfs​ = Unit skin friction (kPa)

• βbetaβ = Empirical factor, typically around 0.5−1.00.5 - 1.00.5−1.0

Total Bearing Capacity (Q):

Q=A×qb+∑fs×AsQ = A times q_b + sum f_s times A_sQ=A×qb​+∑fs​×As​

Where:

• AAA = Cross-sectional area of the pile tip (m²)

• AsA_sAs​ = Surface area of the pile shaft (m²)

• fsf_sfs​ = Skin friction (kPa)

3. Dutch Cone Penetrometer Data:

For data derived from the Dutch Cone Penetrometer Test (CPT), the bearing capacity is calculated using cone resistance values. The cone resistance qcq_cqc​ from the test is directly related to both end bearing and shaft resistance.

End Bearing Capacity:

qb=0.6×qcq_b = 0.6 times q_cqb​=0.6×qc​

Where:

• qbq_bqb​ = End bearing capacity (kPa)

• qcq_cqc​ = Cone resistance (MPa) from the CPT test

Skin Friction (Shaft Resistance):

The shaft resistance is typically estimated as a percentage of the cone resistance:

fs=α×qcf_s = alpha times q_cfs​=α×qc​

Where:

• fsf_sfs​ = Unit skin friction (kPa)

• αalphaα = Empirical factor (often between 0.02 and 0.05)

Total Bearing Capacity (Q):

Q=A×qb+∑fs×AsQ = A times q_b + sum f_s times A_sQ=A×qb​+∑fs​×As​

Where:

• AAA = Cross-sectional area of the pile tip (m²)

• AsA_sAs​ = Surface area of the pile shaft (m²)

• fsf_sfs​ = Skin friction (kPa)

Summary of Formulas:

• Laboratory Data:

  • qb=9×cq_b = 9 times cqb​=9×c

  • fs=α×cf_s = alpha times cfs​=α×c

• NSPT Data:

  • qb=N×40q_b = N times 40qb​=N×40

  • fs=β×Nf_s = beta times Nfs​=β×N

• Dutch Cone Penetrometer Data:

  • qb=0.6×qcq_b = 0.6 times q_cqb​=0.6×qc​

  • fs=α×qcf_s = alpha times q_cfs​=α×qc​

Each of these formulas estimates either the end bearing capacity or skin friction based on the type of data (laboratory, NSPT, or Dutch Cone Penetrometer), and they are combined to provide the total bearing capacity of the pile.

Feel free to adapt these formulas to your project or application in the code you've developed!

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