T-Z and Q-Z Curves | Axially Loaded Piles

T-Z and Q-Z Curves for Axially Loaded Piles

Settlement prediction of single piles under axial loading is as important as the estimation of ultimate axial capacity which is the sum of ultimate total shaft resistance and end bearing resistance. Settlement control usually receives the most intention from geotechnical designers who usually have difficulties in complying with the settlement requirements as specified in many design specifications or standards if there is no thorough understanding of the mechanism involved in the settlement analysis of single piles under axial loading.

Numerical simulation of the load transfer curves such as T-Z curves for shaft resistance and Q-Z curves for end bearing resistance is a practical approach for the conditions where layered soil profiles and/or many potential load cases and trial designs need to be undertaken.

For the load transfer method, the pile settlement at the top under the axial load will be estimated with using T-Z curves to model the shaft displacement with the mobilization of the skin friction and Q-Z curves to model the response of the pile toe displacement to the mobilized end bearing resistance force.

T-Z Curve Models in PileAXL Program

T-Z Curves

There are nine different T-Z curves available for modeling the mobilization of shaft resistance with settlement:

T-Z curve – API Clay. It is based on the recommendation from API 2A-WSD (2000). Note that T-Z curve for clay is the same as the one from API RP2GEO (2011).
T-Z curve – API Sand. It is based on the recommendation from API 2A-WSD (2000).
T-Z curve – API RP2GEO Sand. It is based on the recommendation from API RP2GEO (2011).
T-Z curve – Coyle and Reese Clay (1966). It is based on the recommendation from Coyle and Reese (1966).
T-Z curve – Mosher Sand (1984). It is based on the recommendation from Mosher Sand (1984). Note that the initial modulus of side resistance is required for this T-Z curve model.
T-Z curve – Reese and O’Neill Clay (1989). It is based on the recommendation from Reese and O’Neill (1989).
T-Z curve – Reese and O’Neill Sand (1989). It is based on the recommendation from Reese and O’Neill (1989).
T-Z curve – O’Neill and Hassan Rock (1994). It is based on the recommendation from O’Neill and Hassan (1994). Note that the elastic modulus is required for this T-Z curve model.
T-Z curve – user-defined. User-defined T-Z curve can be input by the users in the PileAXL program. Currently, a maximum number of 10 points are allowed in the program. The required input parameters for user-defined data are (1) z/D: the ratio of the input settlement (z) to the pile diameter and (2) t/tmax: the ratio of the input mobilised shaft resistance (t) to the ultimate shaft resistance (tmax).

Q-Z Curve Models in PileAXL Program

Q-Z Curves

There are eight different Q-Z curves available for modeling the mobilization of end bearing resistance with settlement:

Q-Z curve – API Clay. It is based on the recommendation from API 2A-WSD (2000). Note that Q-Z curve for clay/sand is the same as the one from API RP2GEO (2011).
Q-Z curve – API Sand. It is based on the recommendation from API 2A-WSD (2000). Note that Q-Z curve for clay/sand is the same as the one from API RP2GEO (2011).
Q-Z curve – Skempton (1951). It is based on the concept of Skempton (1951) which predicts the load in end bearing of a pile in clay as a function of the pile tip movement.
Q-Z curve – Vijayvergiva (1977). It is based on the concept of Vijayvergiva(1977) which predicts the load in end bearing of a pile in sand as a function of the pile tip movement.
Q-Z curve – Reese and O’Neill Clay (1989). It is based on the recommendation from Reese and O’Neill (1989).
Q-Z curve – Reese and O’Neill Sand (1989). It is based on the recommendation from Reese and O’Neill (1989).
Q-Z curve – Elastic-Plastic Model. Note that Young’s modulus and Poisson’s ratio are required for this Q-Z curve model.
Q-Z curve – user-defined. User-defined Q-Z curve can be input by the users in the PileAXL program. Currently, a maximum number of 10 points are allowed in the program. The required input parameters for user-defined data are (1) z/D: the ratio of the input settlement (z) to the pile diameter and (2) Q/Qp: the ratio of the input mobilised end bearing resistance (Q) to the ultimate end bearing resistance (Qp).

T-Z Curve - API Clay & Sand

The T-Z curve for API Clay in the PileAXL program is based on the following relationship from API 2A-WSD (2000). Note that this relationship for clay is the same as the one from API RP2GEO (2011).

T-Z curve adopted for clay & sand – driven piles (after API 2000)

T-Z curve adopted for clay & sand – driven piles (after API 2011)

T-Z Curve - Coyle and Reese Clay

The T-Z curve for Coyle and Reese Clay is shown in the figure below and this is based on the relationship recommended by Coyle and Reese (1966).

T-Z Curve - Mosher Sand

The T-Z curve for Mosher Sand (1984) is shown in the figure below based on the following relationship:

where f is the mobilised shaft resistance, z is the pile movement, fs is the ultimate shaft resistance determined by various axial capacity calculation methods and Ef is the initial modulus of side resistance as summarised in the table below.

T-Z Curve - Reese and O'Neill Clay (1989)

T-Z curve – Reese and O’Neill Clay adopted in the PileAXL program is based on the recommendation as proposed in Reese and O’Neill (1989) as shown in the figure below. Note that the trend line is adopted in the PileAXL program to estimate the side load transfer based on the settlement ratio.

T-Z Curve - Reese and O'Neill Sand (1989)

T-Z curve – Reese and O’Neill Sand adopted in the PileAXL program is based on the recommendation as proposed in Reese and O’Neill (1989) as shown in the figure below. Note that the trend line is adopted in the PileAXL program to estimate the side load transfer based on the settlement ratio.

T-Z Curve - O'Neill and Hassan Rock (1994)

The following hyperbolic relationship for t-z curve as recommended by O’Neill and Hassan (1994) and shown in the figure below is adopted in the program to calculate the mobilised shaft resistance f based on the pile settlement z for rocks:

where f is the mobilised shaft resistance, z is the pile movement, fs is the ultimate shaft resistance determined by various axial capacity calculation methods, Em is the elastic modulus of rock mass, D is the pile diameter.

The following relationship proposed by Rowe and Armitage (1984) is adopted to calculate the elastic modulus of the rock mass (Em) based on the unconfined compressive strength of rocks:

where σc is unconfined compressive strength in MPa.

T-Z Curve - User-defined

User-defined T-Z curve can be input by the users in the PileAXL program. Currently, a maximum number of 10 points are allowed in the program. The dialog for the input of user-defined T-Z curve data is shown in the figure below.

The required input parameters for user-defined data are summarised as below:

z/D: The ratio of the input settlement (z) to the pile diameter.
t/tmax: The ratio of the input mobilised shaft resistance (t) to the ultimate shaft resistance (tmax).

Q-Z Curve - API Clay/Sand

The Q-Z curve for API Clay/Sand in the PileAXL program is based on the following relationship from API 2A-WSD (2000). Note that this relationship for clay/sand is the same as the one from API RP2GEO (2011).

Q-Z Curve - Skempton (1951)

This Q-Z curve is based on the concept of Skempton (1951) which predicts the load in end bearing of a pile in clay as a function of the pile tip movement. The following equations are adopted in the PileAXL program:

where f is mobilised end bearing resistance, D is the pile diameter, w is the pile toe movement, fb is the ultimate end bearing resistance and ε50 is the strain factor ranging from 0.005 to 0.02.

Q-Z Curve - Vijayvergiva (1977)

This Q-Z curve is based on the concept of Vijayvergiva(1977) which predicts the load in end bearing of a pile in sand as a function of the pile tip movement. The following equations are adopted in the PileAXL program:

where f is mobilised end bearing resistance, w is the pile toe movement, fb is the ultimate end bearing resistance and wc is the critical tip displacement given by Vijayvergiva (1977) which ranges from 3% to 9% of the pile diameter, D.

Q-Z Curve - Reese and O'Neill Clay (1989)

Q-Z curve – Reese and O’Neill Clay adopted in the PileAXL program is based on the recommendation as proposed in Reese and O’Neill (1989) as shown in the figure below. Note that the trend line is adopted in the PileAXL program to estimate the end bearing based on the settlement ratio.

Q-Z Curve - Reese and O'Neill Sand (1989)

Q-Z curve – Reese and O’Neill Sand adopted in the PileAXL program is based on the recommendation as proposed in Reese and O’Neill (1989) as shown in the figure below. Note that the trend line is adopted in the PileAXL program to estimate the end bearing based on the settlement ratio.

Q-Z Curve - Elastic-Plastic Model

According to Pells (1999), for massive and intact rock, the load-displacement behaviour is linear up to bearing pressures of 2 to 4 times the UCS. For jointed rock mass, the load-displacement behaviour is linear up to 0.75 to 1.25 times the UCS. Baguelin (1982) suggested using the following equation for the linear load-displacement relationship for end bearing up to a specific maximum displacement at which the ultimate bearing resistance (fb) is mobilised:

in which Eb is Young’s modulus at the pile toe; Sb is pile toe displacement; νb is Poisson’s ratio (0.25 is adopted in the program); D is the pile diameter and σb is the mobilised end bearing pressure at the pile toe. This Q-Z curve is shown in the figure for a typical analysis case.

Q-Z Curve - User-defined

User-defined Q-Z curve can be input by the users in the PileAXL program. Currently, a maximum number of 10 points are allowed in the program. The dialog for the input of user-defined Q-Z curve data is shown in the figure below.