Pile tip torsional capacity

[LRJ]

I was reading a paper which stated the pile tip torsional capacity was as follows:

T_t = π.(D/2)².L.γ.(D/3).tan(δ)

Where:
T_t = Pile tip torsional capacity
π = pi (3.14...)
D = Pile diameter
L = Pile length
γ = Material unit weight (concrete in the example)
δ = Interface friction angle at pile tip

I understand where everything apart from the (D/3) part comes from. Can anyone shed some light on what the (D/3) part relates to/how it is derived?

EDIT: Essentially the above comes down to:

T_t = τ.(D/3)

Where:
τ = shear stress

I'm not sure why shear stress is multiplied by D/3.

 

[PanosDK]

I am not certain that I agree with your assessment. Shear stress times Diameter do not give moment units. This is how I see the equation: π(D/2)^2.L is the volume of the pile. Times this with γ gives the weight of the pile. Times this by tanδ gives the average frictional resistance of the base. So, the above equation appears to be the tip frictional resistance FORCE times (D/3). This result in moment units. In the above equation, the top load or side frictional resistance to the vertical loads are not accounted for. D/3 in this case appears to be the moment arm, which means that the shearing forces of the base have their resultant at R/3 on either side of the center, where R is the pile radius. This seems to be a bit small. Other things appear to be a bit unexpected as well. Can you share the reference of this equation?

http://www.qsystemsengineering.net

 

[LRJ]

Thanks for the explanation. Has a triangular distribution therefore been assumed at the base?

I can't actually remember the reference now, I'm afraid.

 

[Shu Jiang PEng SE]

This is what I understand the equation and it seems reasonable.

π.(D/2)2: the pile end area;

π.(D/2)2xL.γ: the contact vertical force between the pile end and the soil

π.(D/2)2xL.γ*tan(δ): the contact shear force between the pile end and the soil

D/3: the force arm.

 

[LRJ]

Thanks again. That makes sense to me.