📈 Moment Curvature

A fibre section divides a cross-section into many small areas, gives each one a uniaxial material, and integrates their response. It is how a section's real non-linear behaviour — concrete crushing, reinforcement yielding, the loss of stiffness after cracking — comes out of the material laws rather than being assumed.

The Moment Curvature component then pushes such a section's curvature under a held axial force and returns the full curve, to its ultimate state; Interaction Domain gives what the section carries at the ultimate limit state, for every combination of N, My and Mz.

Note

This tab is marked βeta in Grasshopper (MomentCurvature_βeta). It works as a section-level tool. Fibre sections cannot yet be assigned to a ForceBeamColumn, which takes an elastic section.

Components in this tab

Materials — non-linear uniaxial laws, for fibres.

Component	Nickname	Models
Concrete01	Concrete	Concrete with zero tensile strength.
Steel01	Steel01	Bilinear steel with kinematic hardening.
ReinforcingSteel	ReinforcingSteel	Reinforcing bar, with a strain-hardening branch.

Fibres — the pieces a section is made of.

Component	Nickname	Builds
Fiber Point	Fiber Point	One fibre: a point, an area, a material.
Layer Fiber	Layer Fiber	A row of equally spaced fibres along a curve.
Patch	Patch	An area filled with fibres, one per mesh face.
Fiber Section	FiberSection	Collects points, layers and patches into a section.

Analysis

Component	Nickname	Does
Moment Curvature	MC	Runs the section analysis, in any direction, to its ultimate state.
Moment Curvature Diagram	M-κ	Draws the curves, with first yield and the ultimate state.
Fiber Stress Strain	FBS	Splits the per-fibre history into fibres, stresses and strains.

Resistance domain — what the section carries at the ultimate limit state.

Component	Nickname	Does
Interaction Domain	Interaction Domain	The N–My–Mz domain of the section.
Domain Check	Domain Check	Safety factors of design forces against it.
Domain 3D	Domain 3D	Draws the domain in three dimensions.
N-M Diagram	N-M	Draws its cut through the N axis at a moment direction.
My-Mz Diagram	My-Mz	Draws its levels at chosen axial forces.

Local axes and units

Fibre geometry is drawn in the world XY plane and read as the section's local yz plane:

Drawn in	Is the section's
World X	Local y — the green arrow
World Y	Local z — the blue arrow

This is the order OpenSees writes a fibre in. Fiber Section draws both arrows so you can see which is which, at the area centroid of the fibres, which the moments are taken about.

My and Mz are the moments about local y and z, by the right-hand rule with the beam's axis x coming out of the drawing: positive My stretches the +z side, positive Mz the −y side.

Everything is in metres and kN: coordinates in m, fibre areas in m², stresses in kN/m², curvature in rad/m, moments in kN·m. Strains are dimensionless.

A typical definition

Concrete01 ──► Patch  ─┐
                       ├──► Fiber Section ──► Moment Curvature ──► M, κ
ReinforcingSteel ──► Layer Fiber ─┘                            └──► Fiber Stress Strain

1. Mesh the concrete outline and give it to a Patch with a Concrete01 material — the mesh density is the fibre density.
2. Draw the bar positions as a curve and give it to a Layer Fiber with a ReinforcingSteel material.
3. Collect both in a Fiber Section.
4. Push it with Moment Curvature.
5. Or sweep its Interaction Domain, check design forces against it with Domain Check, and draw it with the views.