Automatic Design of Drilled Piers considering PM interaction (ACI 318-23)

A technical blog on automated reinforced concrete drilled pier design using strain compatibility and interaction diagram checking

Introduction

Drilled piers are commonly used as deep foundations for buildings, bridges, industrial structures, retaining systems, and heavy equipment foundations. In many practical projects, drilled piers are subjected not only to axial compression but also to bending moment from lateral loads, column eccentricity, seismic effects, wind, or construction tolerances.

Because of this combined action, checking only the axial capacity of the pier is not enough. A more rational design should consider the combined effect of axial force and bending moment using a P-M interaction diagram.

This is the main idea behind the automatic drilled pier design code. The program automates the design process by generating the axial-moment capacity curve of the reinforced concrete drilled pier section and checking whether the applied design demand falls safely within the interaction envelope.

Why P-M Interaction Is Important for Drilled Pier Design

A drilled pier is essentially a reinforced concrete circular column embedded in soil. Structurally, the upper part of the pier often behaves like a cantilever column resisting axial load, moment, and sometimes shear.

For a drilled pier section, the nominal axial-moment strength depends on concrete compressive strength, steel yield strength, pier diameter, longitudinal reinforcement ratio, bar arrangement, concrete cover, neutral axis depth, strain distribution, strength reduction factor, and maximum compressive capacity limit.

When axial load is high, the available moment capacity may reduce significantly. When bending moment is high, the usable axial capacity also changes. Therefore, the interaction between P and M must be considered directly.

A conventional manual design approach may check only selected points, such as pure compression, balanced condition, or pure bending. However, an automatic design tool can generate the full curve and give a much clearer view of safety.

Concept of the Automatic Design Code

The code follows the same general philosophy as reinforced concrete column design. The section is divided into concrete and steel components. For each assumed neutral axis depth, the strain profile is calculated. Then, concrete compression force and reinforcement force are computed. The internal axial force and moment are then obtained from equilibrium.

1.      Define section geometry and material properties.

2.      Generate reinforcement layout.

3.      Sweep the neutral axis depth.

4.      Compute concrete compression force.

5.      Compute steel strain and steel stress.

6.      Sum axial force.

7.      Sum moment about the section centroid.

8.      Apply capacity limit and strength reduction factor.

9.      Generate the P-M interaction curve.

10.   Compare applied Pu-Mu demand with the available capacity.

This converts drilled pier design from a manual checking problem into a systematic section-capacity evaluation problem.

Section Modeling

For a circular drilled pier, the gross concrete area is calculated as:

Ag = pi D^2 / 4

where D is the drilled pier diameter and Ag is the gross area of the concrete section.

The total longitudinal reinforcement area is:

Ast = nb Ab

where nb is the number of longitudinal bars, Ab is the area of one bar, and Ast is the total longitudinal steel area.

The longitudinal bars are arranged around the circular perimeter, usually with equal angular spacing. This allows the program to compute the exact location of each bar and evaluate its strain based on the neutral axis location.

Pure Axial Compression Capacity

One important reference point is the nominal pure axial compression capacity:

P0 = 0.85 f'c (Ag - Ast) + fy Ast

For design plotting, the maximum usable compression strength is usually capped. In the code logic, this means the computed nominal axial strength Pn should not exceed the maximum permitted compression level.

P0 = (0.85*fc*(Ag-Ast) + fy*Ast)/1000;phiP0 = 0.75*P0;Pn = min(Pn, 0.85*P0);phiPn = min(phiPn, 0.85*phiP0);

This is important because the interaction diagram should not show unrealistic compressive strength higher than the design limit. The cap should be applied consistently to both nominal and design-strength curves.

Moment Calculation About the Section Centroid

A critical point in this type of code is where the moment is calculated. For structural design comparison, the moment capacity should be calculated about the centroid of the gross section, not about the neutral axis and not about the resultant axial force location.

This is because structural analysis software usually gives the demand as Pu and Mu at the member centroidal axis. Therefore, the capacity diagram should also be expressed in the same coordinate system.

If the program sums moment about the location of the resultant compression force, the moment may become zero by definition. That is mathematically possible but not useful for design comparison. The useful design moment is the internal resisting moment about the section centroid.

Mn = sum(Fi .* yi_relative_to_centroid);

Each internal force contribution is multiplied by its lever arm from the section centroid.

Concrete Compression Contribution

For each neutral axis depth, the concrete compression block is calculated. The compression force is based on the equivalent stress block concept. For circular sections, the compressed area is not rectangular; therefore, the program must evaluate the compressed segment of the circular section depending on the neutral axis location.

Cc = 0.85 f'c Ac

where Ac is the compressed concrete area. The corresponding moment contribution is:

Mc = Cc yc

where yc is the distance from the section centroid to the centroid of the compressed concrete area. The sign of this moment must be handled carefully. If the compression block is on the opposite side of the centroid, its lever arm changes sign. This is why sign convention is one of the most important details in the code.

Note: The sum of Pn and Mn is referenced to the centroid of the section as a standard approach for PM analysis. This is to match the results obtained from the structural analysis program.

Reinforcement Contribution

Each longitudinal bar is evaluated individually. The strain at each bar is computed from the linear strain distribution:

eps_s = eps_c (c - ys) / c

where eps_s is the steel strain, eps_c is the extreme concrete compression strain, c is the neutral axis depth, and ys is the bar location measured from the compression face.

The steel stress is then limited by the yield strength:

fs = max(-fy, min(Es eps_s, fy))

The steel force is:

Fs = As fs

The total reinforcement contribution to axial force and moment is then:

Ps = sum(Fs),     Ms = sum(Fs ys)

Again, the moment is taken about the section centroid.

Strength Reduction Factor

For reinforced concrete compression members, the strength reduction factor depends on the strain condition of the section. A compression-controlled section has a lower strength reduction factor, while a tension-controlled section has a higher value.

phi = 0.75 (spiral)+ 0.15*(eps_t - 0.002)/0.003;phi = min(max(phi,0.75),0.90);

This allows the program to generate both the nominal strength curve Pn-Mn and the design strength curve phiPn-phiMn. For design, the factored demand should be checked against the design strength curve.

Demand-Capacity Check

Once the P-M interaction diagram is generated, the program checks the applied factored load combination Pu and Mu against the capacity envelope.

(Pu, Mu) <= (phi Pn, phi Mn)

In practice, because the interaction curve is a nonlinear boundary, the code may interpolate the available moment capacity at a given axial load:

phiMn_available = interp1(phiPn_curve, phiMn_curve, Pu);Mu <= phiMn_available

This is more reliable than checking only one or two simplified design equations.

Automatic Design Procedure

The automatic design code can be extended into a full design engine. The program can search through possible drilled pier designs by varying pier diameter, number of bars, bar size, reinforcement ratio, concrete strength, steel grade, clear cover, and tie or spiral configuration.

for D = D_list    for nb = nb_list        for bar = bar_list            generate_section(D, nb, bar);            compute_PM_interaction();            if demand_is_safe(Pu, Mu)                store_feasible_design();            end        end    endend

Then the program can select the most economical feasible design based on an objective function such as:

Cost = Cc Vc + Cs Ws

or more simply by minimizing diameter, steel area, or a constructability-weighted score.

Why This Is Better Than Manual Trial-and-Error

Manual drilled pier design usually requires repeated checking. The engineer may start with a diameter, select reinforcement, compute capacity, revise the design, and repeat. The automatic P-M design code improves this process in several ways:

·        It checks the full interaction behavior, not only simplified axial or moment limits.

·        It reduces human error, especially in strain compatibility and sign convention.

·        It allows rapid comparison of many design alternatives.

·        It produces clear graphical output, useful for design review and reporting.

·        It can be integrated with optimization algorithms, such as Jaya, PSO, DE, TLBO, or genetic algorithms.

This is especially useful for projects with many drilled piers under different load combinations.

Practical Output of the Program

A good drilled pier design program should produce at least the following outputs:

·        P-M interaction diagram

·        factored demand points

·        governing load combination

·        selected pier diameter

·        selected reinforcement layout

·        reinforcement ratio

·        maximum usable axial strength

·        available moment capacity at applied axial load

·        design status: OK / NG

·        exportable CSV or report file

The plot is especially important. When the demand point is inside the design curve, the section is safe. When it is outside, the design must be revised.

Key Programming Issues

1. Correct order of plot data

The P-M curve must be ordered properly before plotting. If the first point represents pure compression and the next point is not arranged correctly, the horizontal cap line may disappear or connect incorrectly. A practical fix is to move the first pure-compression point to the end of the plotting array or sort the curve carefully by moment and axial force.

2. Consistent capacity cap

The axial compression cap should be applied consistently. If the cap is applied only to the factored curve but not the nominal curve, the plotted behavior may become inconsistent.

3. Moment reference axis

Moment must be computed about the section centroid to match structural analysis results.

4. Sign convention

Concrete compression, steel compression, and steel tension must be handled with a consistent sign convention. A common source of error is reversing the moment contribution of compression blocks located on the opposite side of the centroid.

5. Multiple demand points

In real design, there may be many load combinations. The program should check all demand points. The governing design is the one with the highest demand-capacity ratio.

for i = 1:nCombo    check_PM_capacity(Pu(i), Mu(i));end

Future Development

The current code can be expanded into a more complete drilled pier design platform. Possible future features include:

·        biaxial P-M-M interaction

·        automatic load combination import

·        soil lateral spring model

·        second-order moment amplification

·        shear design

·        development length and splice checks

·        constructability checks

·        automatic optimization using metaheuristics

·        report generation in PDF or DOCX format

·        integration with ETABS, SAFE, SAP2000, or in-house FEM software

A particularly powerful extension is to combine the P-M interaction check with an optimization engine. The optimizer can search for the most economical drilled pier design while satisfying all strength, serviceability, and detailing constraints. This creates a direct path from section analysis to automatic foundation design.

Conclusion

Automatic drilled pier design considering P-M interaction is a major improvement over traditional trial-and-error design. By using strain compatibility, force equilibrium, and full interaction curve generation, the program can evaluate the real combined axial and bending capacity of the drilled pier section.

The most important concept is that the drilled pier should be treated as a reinforced concrete compression member subjected to combined axial load and bending moment. The design demand from structural analysis should be compared directly against the phiPn-phiMn interaction curve.

With proper implementation, this code becomes more than a calculation tool. It becomes the foundation of an intelligent drilled pier design system that can support automation, optimization, reporting, and future integration with structural analysis software.

References:

CRSI, Design Guide for Drilled Piers, 1st edition, 2016

ACI 318-23, Building Code Requirements for Structural Concrete and Commentary