Moment–curvature
Fiber-integrated M–φ and nonlinear flexural rigidity for solid-circular, pipe, and rectangular steel sections.
A pile's bending stiffness is only constant while the section stays elastic. Push it harder and the extreme fibers yield, the section softens, and its effective EI drops. The moment–curvature tool traces that whole history — from elastic, through first yield, to a declared material or curvature endpoint — by the fiber equilibrium method: it discretizes the cross section, applies a curvature, solves axial equilibrium, and integrates the resulting fiber stresses into a moment. The live tool plots the M–φ curve and reports the elastic EI, yield/nominal response, governing endpoint, and convergence. This page explains the method and every input.
Fiber integration
The cross section is split into many thin fibers, each with an area A_i at a distance y_i from the neutral axis. The tool imposes a curvature φ (note: here φ is curvature, not the soil friction angle used in the lateral models). Plane sections are assumed to remain plane, so strain varies linearly across the depth:
The solver varies ε₀ until the sum of fiber forces equals the applied axial load. Each fiber's stress then comes from its declared law: bilinear steel, Hognestad/Todeschini concrete, or the zero-tension OpenSees Concrete01 envelope. Reinforcement can include hardening, effective prestress, and a tensile-rupture strain. Summing the fiber stresses gives both equilibrium and moment:
Tracing φ upward from zero and recording M at each step builds the full M–φ curve. At small curvature every fiber is elastic and the slope is the elastic rigidity. A perfect-plastic steel section approaches a fully-plastic asymptote. Hardening steel and concrete do not: they stop at the explicitly reported strain, rupture, compression, or requested-curvature endpoint.
Section geometry
Choose the section shape; the dimensions that appear adapt to your choice. The geometry is what sets the fiber distribution — and through it, the stiffness and the shape factor.
The cross-section type: solid circular, pipe (hollow circular), or rectangular. Round and rectangular reinforced-concrete sections are also supported for the nonlinear-EI lateral analysis. Round sections use a circular bar cage. Rectangular sections use a straight-sided perimeter layout with explicit top/bottom counts, side counts, bar area, physical bar diameter, and clear cover; they are never approximated as a circle. Round cages also require physical bar diameter so containment and steel-core clearance are checkable. Permanent casing and steel-core inputs apply to round sections, and either reinforcement layout can carry prestress. Reinforced sections explicitly select whether bar area replaces concrete (subtract) or overlaps the concrete patch (overlap, used only to reproduce OpenSees fiber definitions). Rectangular sections may declare a straight-sided confined core with its own Concrete01 envelope.
Why it matters. The shape sets how area is distributed about the neutral axis, which fixes both the elastic I and the shape factor Mp/My. A thin pipe concentrates material at the extreme fiber, giving a higher shape factor than a solid section.
Outside diameter of the solid circular section.
Why it matters. Controls both the stiffness and the plastic moment; for a solid circle I ∝ D⁴ and Mp ∝ D³.
Outside diameter of the pipe.
Why it matters. The outer fibers sit farthest from the neutral axis, so they carry the most moment and reach Fy first.
Pipe wall thickness.
Why it matters. A thinner wall has more of its area near the extreme fiber, so it yields sooner but has a higher shape factor — the gap between My and Mp is wider.
Section width, perpendicular to the bending plane.
Why it matters. Capacity scales linearly with width — all fibers shift their area proportionally but keep the same lever arms.
Section height, measured in the bending plane.
Why it matters. Height is the dominant dimension — capacity scales with h² because it stretches the lever arms of the outer fibers.
Material
Steel is modeled with a bilinear law: elastic to yield, then a post-yield branch set by the hardening ratio. A hardening curve must have an explicit strain/curvature endpoint because it has no finite plastic asymptote. Concrete may use the LPILE-compatible Hognestad law, Todeschini, or the OpenSees Concrete01 law. Concrete01 requires peak strain, terminal strain, residual stress, and zero tension. Core and cover laws remain separate; no confinement properties are invented from transverse reinforcement.
The elastic modulus of the steel — steel ≈ 200 GPa = 2×10⁸ kPa.
Why it matters. Sets the initial elastic stiffness: the starting slope of the M–φ curve is EI = E·I. It scales the whole elastic branch.
The yield stress of the steel — e.g. 345 MPa = 3.45×10⁵ kPa.
Why it matters. Sets the strain Fy/E at which fibers begin to yield, and therefore the yield and plastic moments where the section softens. A higher Fy raises both My and Mp proportionally.
The post-yield tangent ratio E_t/E.
Why it matters. 0 gives an elastic-perfectly-plastic material — the M–φ curve flattens to a horizontal plateau at Mp. A small positive value gives a gently rising post-yield branch instead of a flat asymptote, so the reported peak is meaningful only with the accompanying declared endpoint.
Steel-section requests are tension positive (enter compression as a negative value). Concrete-section requests use the LPILE convention and are compression positive. The tool labels the active convention.
Why it matters. Axial load shifts the strain required for equilibrium and therefore the neutral axis, first-yield state, and material endpoint. See Axial interaction below. Set 0 for pure bending.
Axial interaction
Bending and axial load share the same fibers. With no axial load, a symmetric homogeneous section's neutral axis sits at its centroid. In a steel section, either axial tension or compression consumes part of the symmetric yield envelope and lowers the first-yield and plastic bending capacity. In a reinforced-concrete section, moderate compression can initially increase flexural resistance before high compression drives the response back down; the solver therefore evaluates the declared fiber laws instead of applying a blanket reduction. This is the moment–axial (M–P) interaction.
For a homogeneous steel section, the reported first-yield moment follows directly from the uniform axial stress:
which reduces to the classic Fy·S at P = 0. At or beyond the homogeneous-steel squash load, |P| ≥ Fy·A, that declared model has no bending capacity and the tool fails closed. Concrete uses compression-positive axial thrust and its nonlinear equilibrium curve; requests above the section's scanned compression capacity likewise fail. In either case, use the actual service or strength load appropriate to the check rather than silently running at zero.
Reading the results
The tool reports three section capacities and the full moment–curvature curve.
Section capacities
- Elastic EI — the initial flexural rigidity E·I, the starting slope of the M–φ curve. This is also the stiffness the elastic lateral analysis uses.
- Yield moment My — the moment at which the extreme fiber first reaches Fy, accounting for the axial pre-stress (My = (Fy − |P|/A)·S). Below it the section is fully elastic.
- Plastic/peak moment — a true fully-plastic Mp only for a perfect-plastic steel law. For hardening steel or concrete it is the peak over the declared sweep and must be read with the governing limit state and limit curvature. Concrete also reports the moment at its declared cover-compression strain.
The M–φ curve
Moment is plotted against curvature, with reference lines at first yield and the applicable nominal/peak response. The secant slope at any point is the section's effective rigidity. The result also reports the actual fiber count, completed point count, axial-equilibrium residual, and named endpoint, so a truncated or nonconverged sweep is always visible.
Feed the reduced EI back into the lateral analysis
Validation and applicability
The rectangular core/cover path is checked against the official OpenSees moment–curvature example using its published 15 × 24-in geometry, Concrete01 core/cover materials, Steel01 reinforcement, and 180-kip axial load. A refined independent OpenSees run supplies frozen curve values; PileCalc's curve is required to stay within 1%, and its 1,200/2,400-fiber refinement difference below 0.25%. The same physical case is also run as an SI/US twin.
Declared section domain