Chapter 7 — Deep Foundations: Axial Capacity¶
7.1 The capacity equation¶
The ultimate axial capacity of a single pile is the sum of shaft and base resistance:
where \(Q_s\) is the shaft (skin friction) resistance, \(Q_p\) the end-bearing (toe) resistance, and \(W\) the pile weight (often neglected). The allowable capacity is \(Q_{ult}\) divided by a factor of safety, and must also satisfy the settlement criterion.
7.2 Shaft resistance in clay¶
In clay, shaft resistance is usually computed with the total-stress (α) method:
where \(c_u\) is the undrained shear strength, \(A_s\) the shaft area, and \(\alpha\) an adhesion factor (typically ~0.5, decreasing with \(c_u\)). The effective-stress (β) method is also used, especially for long-term conditions.
7.3 Shaft resistance in sand¶
In sand, shaft resistance is computed with the effective-stress (β) method:
where \(\sigma'_v\) is the effective vertical stress, \(K\) the lateral earth-pressure coefficient, and \(\delta\) the pile–soil friction angle. The λ and α methods are alternatives for specific conditions.
7.4 End bearing¶
End bearing depends on whether the pile tip rests on rock, dense sand, or clay:
- In clay: \(Q_p = 9 \, c_u \, A_p\) (with \(c_u\) at the base).
- In sand: \(Q_p = q'N_q A_p\) (or correlations with SPT/CPT), capped to account for the limited depth over which the bearing pressure mobilizes.
- On rock: capacity from the rock's unconfined compressive strength with reduction factors.
7.5 Capacity from in-situ tests¶
Static analysis is checked and often calibrated against field tests:
- SPT and CPT correlations — capacity estimated from penetration resistance.
- Static pile load tests — the definitive measurement, with the load applied in increments and settlement recorded.
- Dynamic testing (PDA) — high-strain dynamic tests during driving give capacity and integrity.
- Statnamic and rapid load tests — alternatives where static testing is impractical.
7.6 Instrumented load tests¶
The most informative load tests are instrumented: strain gages or sister bars along the shaft and telltales or extensometers measure the load distribution between shaft and toe, separating the two components directly (see the FHWA reference, Chapter 8, and Appendix D examples).
7.7 Factors of safety and LRFD¶
Allowable-stress design uses a factor of safety on the ultimate capacity (commonly 2.5–4 for working loads). LRFD instead applies resistance factors to the shaft and base components separately, reflecting their different reliability.
7.8 Key takeaways¶
- \(Q_{ult} = Q_s + Q_p\) — shaft friction plus end bearing.
- Use the α/β/λ methods for shaft, and 9\(c_u\) / \(q'N_q\) for base.
- Calibrate static analysis with load tests, SPT/CPT correlations, and dynamic testing.
- Instrumented load tests separate shaft from toe — the gold standard.
- Allowable-stress uses a factor of safety; LRFD uses resistance factors.