Prestressed Spun Pile Design - P-M Interaction Theory (EC2)

A prestressed spun concrete pile is a hollow, centrifugally cast tube with pretensioned tendons equally spaced on a ring. It is checked to Eurocode 2 (EN 1992-1-1) for the combination of axial force $N_{Ed}$ and bending moment $M_{Ed}$ at the ultimate limit state, and for concrete stress limits at the serviceability limit state. The method below is what this calculator implements.

1. The section - a hollow annulus

Spinning the mould throws the wet concrete against the wall, leaving a dense outer ring and a central void. With outer diameter $D$ and wall thickness $t$ (inner diameter $D-2t$), the area and second moment of area are those of an annulus:

Removing the concrete near the neutral axis - where it contributes little to bending - gives a high stiffness-to-weight ratio and a tough, low-permeability outer skin. The design integrates the compression zone over this annulus.

2. The prestress chain

The tendons are tensioned, then released onto the hardened concrete, putting the whole section into compression. The stress in a tendon immediately after tensioning is limited by §5.10.3:

The residual compressive stress left in the concrete after losses is the effective prestress $\sigma_{ce}$ (commonly supplied by the manufacturer). From it follow the effective tendon stress and the tendon pre-strain, with $A_c$ the net concrete area and $A_{ps}$ the total tendon area:

The pre-strain $\varepsilon_{pe}$ is the key: at ULS it is added to the flexural strain at every tendon, so the tendons are already stressed before any bending is applied.

Prestress losses

The jacking force never fully reaches the finished pile - it is eroded by a chain of losses, which is why $\sigma_{ce}$ (after all losses) is what governs, not the jacking stress. For a pretensioned spun pile the losses are:

• Elastic shortening - when the strands are released the concrete shortens elastically, immediately relaxing the steel by $\Delta\sigma_{el}=\alpha_e\,\sigma_{c,p}$ at the tendon level. This is an immediate loss.
• Steel relaxation - the highly-stressed strand sheds stress over time at constant strain (EN 1992-1-1 §3.3.2; Class 2 low-relaxation strand, $\rho_{1000}\approx2.5\%$).
• Concrete creep - sustained compression makes the concrete creep ($\varepsilon_{cc}=\varphi(\infty,t_0)\,\sigma_c/E_{cm}$), shortening it further and relaxing the steel.
• Shrinkage - drying shrinkage $\varepsilon_{cs}$ shortens the member independently of load.

Creep, shrinkage and relaxation are the time-dependent losses (§5.10.6). A high-strength spun-pile concrete (C60–C80) and steam curing keep them modest, so a typical residual $\sigma_{ce}$ is of the order 6–10 N/mm². This calculator takes $\sigma_{ce}$ directly (supplier value or the computed initial prestress, whichever is smaller) so the loss breakdown is upstream of the section check.

3. ULS - the P-M interaction diagram

Assuming plane sections remain plane, a linear strain profile is chosen and the internal axial force and moment are integrated. The concrete follows the parabola-rectangle law (§3.1.7) with design strength $f_{cd}=\alpha_{cc}f_{ck}/\gamma_c$; the width at depth $y$ is the annular chord - outer chord minus the inner void:

Each tendon develops a force from its total strain $\varepsilon_{p,i} = \varepsilon_{pe} + \varepsilon_{flex,i}$, capped at the design strength $f_{pd}=f_{p0.1k}/\gamma_s$. A tendon sitting inside the concrete compression block loses $\eta f_{cd}$ (the concrete it displaces is already counted):

Sweeping the strain plane from full compression (the squash load $N_{Rd,c}$) to full tension ($N_{Rd,t}$) traces the whole envelope. The moment capacity $M_{Rd}$ at the design axial load is read where the curve crosses $N=N_{Ed}$.

Member capacity ratio

Adequacy is expressed as a single utilisation: draw the ray from the origin through the load point $P(M_{Ed},N_{Ed})$ until it meets the curve at $I$. The utilisation is the ratio of the two distances, and the pile passes when it is ≤ 1:

4. Slenderness and second-order effects

A pile acting in compression is checked for slenderness with effective length $l_0=K\,L$ ($K=1.0$ vertical, ≈ 0.75 raking) and radius of gyration $i=\sqrt{I_c/A}$. A geometric-imperfection eccentricity is always included (§5.2):

If $\lambda=l_0/i > \lambda_{lim}$ (§5.8.3.1), the second-order moment is added by the nominal-stiffness method (§5.8.7), where $N_B$ is the buckling load from the nominal bending stiffness $EI$ and $\beta=\pi^{2}/8$:

A pile in pure tension needs neither imperfection nor second-order moment.

5. ULS shear (§6.2)

EN 1992-1-1 §6.2 is written for rectangular webs, so the annulus is idealised as an equivalent rectangle with $b_w=D$ and effective depth $d=R+\tfrac{2}{\pi}r_{ring}$. The axial prestress raises the members-without-links resistance via the $\sigma_{cp}$ term:

The spiral provides $V_{Rd,s}=(A_{sw}/s)\,z\,f_{ywd}\cot\theta$, limited by the strut crushing resistance $V_{Rd,\max}$.

6. SLS - concrete stress limits (§7.2)

The pile is kept uncracked and elastic under service loads. With the effective prestress added, the extreme-fibre stresses are checked for two combinations:

The compression limit is $0.6f_{ck}$ under the characteristic combination (to avoid longitudinal cracking) and $0.45f_{ck}$ under the quasi-permanentcombination (to keep concrete creep linear). The prestress $\sigma_{ce}$ relieves the tension fibre, letting the pile carry more service moment before it cracks. Each limit plots as a straight line in the N-M plane and the service load point must lie inside.

Decompression and the cracking moment

Because the section is precompressed, it only starts to crack once the applied bending overcomes the prestress at the tension fibre. The decompression moment is reached when the tension-fibre stress returns to zero, and the cracking moment $M_{cr}$ when it reaches the tensile strength $f_{ctm}$:

Keeping the service moment below $M_{cr}$ (the tension-fibre check above) means the pile stays uncracked, which is normally required for piles in aggressive ground or below the water table - the dense centrifugal outer skin plus the residual compression give spun piles their well-known durability.

7. Product context - PHC vs PC spun piles

Spun piles are classified by concrete grade and the resulting effective prestress. PHC (Pretensioned spun High-strength Concrete) piles use C80-class concrete and are the modern default; PC (Prestressed Concrete) piles use lower grades (~C60). Higher-strength concrete carries more prestress with smaller time-dependent losses, giving a larger moment capacity and crack-free service range for the same wall.

Typical outer diameters run $D = 300$–$1200\ \text{mm}$ with wall thickness $t \approx 60$–$150\ \text{mm}$ (roughly $D/6$), pretensioned with high-tensile $f_{pk}\approx1420\ \text{N/mm}^{2}$ strands spaced evenly on a ring and confined by a continuous helical spiral. The hollow core also lets the pile be driven open-ended or inspected after installation.

8. Driving and handling stresses

Beyond the in-service P-M and SLS checks, a spun pile must survive handling and driving. Lifting at the wrong pick points puts the pile into bending under self-weight; pitching it vertical is the most severe handling case. During driving, each hammer blow sends a compressive stress wave down the pile and a reflected tensile wave back up - the prestress is what stops that tensile wave from cracking the concrete. Practical limits (often expressed as fractions of $f_{ck}$ and of the prestress) cap the driving compressive and tensile stresses; they are a separate driveability/wave-equation check and are not part of the section verification this tool performs.

9. Worked example

The default case is a D900 × t130 PHC pile, C65/80, with 28 tendons of ⌀10.7 mm ($f_{pk}=1420$), supplier effective prestress $\sigma_{ce}=7.5\ \text{N/mm}^{2}$, under N_{Ed}=-2000\ \text{\,\mathrm{kN}} (compression) with end moments 127.6 / 91.1 kNm. The chain gives:

The P-M envelope peaks near $M_{Rd,peak}\approx1460\ \text{kNm}$ and gives $M_{Rd}\approx1388\ \text{kNm}$ at N_{Ed}=-2000\ \text{\,\mathrm{kN}}, so with the design moment of ~209 kNm the member-capacity utilisation is about 0.27 - comfortably inside the envelope. At SLS both the characteristic ($0.6f_{ck}$) and quasi-permanent ($0.45f_{ck}$) fibre-stress checks pass. Change any input and the diagrams, the prestress chain and every check update live.