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 fully plastic — by the fiber method: it discretizes the cross section, applies a curvature, and integrates the resulting fiber stresses into a moment. The live tool plots the M–φ curve and reports the elastic EI, yield moment, and plastic moment. 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:
Each fiber's stress comes from the steel stress–strain law — elastic at slope E until the strain reaches Fy/E, then plastic, with an optional strain-hardening branch of slope E_t = (E_t/E)·E. Summing the fiber stresses over the area gives the axial force and the bending moment carried at that curvature:
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 EI = E·I. As the outer fibers yield the curve bends over, approaching the fully-plastic moment as its asymptote.
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.
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
The section is steel, modeled with a bilinear stress–strain law: elastic to yield, then a post-yield branch set by the hardening ratio. Inputs are in consistent units — modulus and stress in kPa, axial load in kN.
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.
Axial compression carried by the section (compression positive).
Why it matters. Axial load pre-stresses every fiber, so they reach Fy at a lower bending demand — it lowers the moment capacity. See Axial interaction below. Set 0 for pure bending.
Axial interaction
Bending and axial load share the same fibers, so they compete. With no axial load the neutral axis sits at the centroid and the section is in pure bending. Add axial compression and every fiber picks up a uniform compressive pre-stress P/A; the compression-side fibers are now closer to Fy before any bending is applied, so they yield at a smaller curvature. The net effect is that axial load lowers the moment capacity — this is the moment–axial (M–P) interaction.
Practically, run the tool at the axial load the pile actually carries, not at zero, so the M–φ curve and the reported My and Mp reflect the real demand. The reduction grows with the axial ratio — a lightly loaded pile sees little change, while one near its squash load loses a large fraction of its bending capacity.
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. Below it the section is fully elastic.
- Plastic moment Mp — the fully-plastic capacity, the asymptote of the M–φ curve and the ultimate bending capacity for design. The ratio Mp/My is the section's shape factor, which depends on geometry — higher for a thin pipe than for a solid section.
The M–φ curve
Moment is plotted against curvature, with horizontal reference lines drawn at My and Mp. The initial straight portion has slope EI; the knee at My marks first yield; the plateau toward Mp is the fully-plastic limit. The secant slope at any point is the section's effective (reduced) rigidity at that curvature.
Feed the reduced EI back into the lateral analysis