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About This Tool
This calculator performs two related analyses on a reinforced concrete column section: moment-curvature (M-φ) analysis at a given axial load or across a range of axial loads, and axial force–bending moment (P-M) interaction analysis. Both are based on strain compatibility under uniaxial bending. It handles circular and rectangular sections. The moment-curvature analysis uses nonlinear material models — a strain-hardening curve for reinforcing steel and the Mander model for unconfined cover and confined core concrete. Analyses follow AASHTO LRFD, the AASHTO Guide Specifications for LRFD Seismic Bridge Design, and the Caltrans Seismic Design Criteria.
How the Analysis Is Performed
The cross-section is discretized into horizontal fibers perpendicular to the bending axis, refined near the extreme faces and across the core-cover boundary. Longitudinal bars are modeled as discrete areas at their actual coordinates, with the displaced concrete deducted to avoid double-counting.
Both analyses rest on strain compatibility — plane sections remain plane, so section state reduces to centroidal strain and curvature — and equilibrium: fiber stresses integrate to match the applied axial load, and the moment follows from the same integration.
Moment-curvature. Curvature is imposed incrementally. At each step the centroidal strain is solved iteratively until axial equilibrium is satisfied, then the corresponding moment is computed. Concrete follows the Mander model — confined for the core, unconfined for the cover — and steel follows an elastic-plastic curve with strain hardening. The analysis proceeds until the extreme compression fiber or the extreme tension bar reaches its governing ultimate strain.
P-M interaction. The envelope is generated by sweeping the neutral-axis position from pure compression to pure tension, computing the corresponding P-M pair at each position — either from the AASHTO equivalent rectangular stress block or by full fiber integration of the material stress-strain curves.
Understanding Moment-Curvature Analysis
Why It's Needed
Conventional design asks how much moment a section can carry. That's sufficient for gravity load. It's insufficient for seismic design, where demand is imposed as displacement, not force. The governing question becomes how far the column can deform before failing — and answering it requires the section's post-yield behavior, not just its peak strength.
What the Results Are Used For
- Deformation capacity. Ultimate curvature, integrated over the plastic hinge length, gives the column's displacement capacity — the quantity AASHTO and Caltrans SDC actually check against seismic displacement demand.
- Effective stiffness. Cracked, yielded sections are far more flexible than gross-section properties imply. E·Ieff from the M-φ curve governs periods, force distribution, and predicted displacements throughout the structural model.
- Failure mode. Compression-controlled versus tension-controlled failure identifies whether confinement detailing, bar size, or cover is the limiting parameter.
Importance of Confinement
The concrete inside the hoops or spiral behaves very differently from the unconfined cover concrete outside it. Once the concrete tries to expand laterally under compression (Poisson effect), the transverse steel restrains that expansion, putting the core in a state of triaxial compression. Per the Mander (1988) model, this confinement has two major effects on the core concrete:
- Increased compressive strength — the confined peak stress f'cc can be 20–50%+ higher than the unconfined f'c, depending on the volumetric ratio of transverse steel (ρs), its yield strength, and the confinement effectiveness coefficient (ke), which accounts for arching action between hoops/spirals and between longitudinal bars.
- Much greater ultimate compressive strain (εcu,c) — while unconfined cover concrete crushes around εcu ≈ 0.003–0.005, well-confined core concrete can sustain strains many times larger before crushing. This is the single biggest contributor to ductile column behavior: it lets the extreme compression fiber continue straining well past the point where cover concrete has already spalled off.
- Because confinement dramatically extends the usable strain range of the core, it directly controls the ultimate curvature (φu) a column can reach before failure — the value reported, and adjustable, in the Mander Confined Concrete Model panel.
Bilinear Idealization: From Curve to Design Numbers
The raw M-φ curve is smooth and awkward to use directly in design, so it's idealized as two straight lines: an elastic branch from the origin to an idealized yield point (φyi, Myi), and a plastic branch running out to the ultimate point (φu, Mu). The yield point is placed using an equal-energy method — the areas under the actual and idealized curves are matched — rather than simply reading off first yield of the extreme steel fiber, which is a poor and reinforcement-insensitive benchmark on its own. This idealization produces two quantities central to seismic design:
- Effective stiffness, E·Ieff = Myi / φyi, used in place of unrealistic gross-section stiffness for computing periods, displacements, and force distribution in a structural model.
- Curvature ductility, μφ = φu / φyi, an intermediate section-level quantity — not itself a code capacity check. AASHTO and Caltrans SDC benchmark displacement ductility and displacement capacity at the member/system level, derived by integrating plastic hinge rotation (from φu and the plastic hinge length) along the column into a rotation and then a displacement at the top of the column, compared against the imposed seismic displacement demand.
Use as Member Properties in Nonlinear Analysis
The bilinear (Myi, φyi, Mu, φu) values are not just a hand-calculation check — they are the actual input properties assigned to plastic hinge or fiber elements in nonlinear structural models, used in two complementary analysis approaches:
- Pushover (nonlinear static) analysis: the structure is loaded with a monotonically increasing lateral force or displacement pattern, and the same bilinear hinge properties trace out the column's moment-rotation response as hinges form and rotate. This directly generates the structure's global capacity (pushover) curve — base shear versus displacement — which is compared against the seismic displacement demand to establish the displacement capacity/demand ratio codes actually require. φu, translated to plastic hinge rotation capacity, sets the point on this curve where the column is deemed to have reached its displacement capacity.
- Nonlinear time history analysis: the same bilinear hinge (or fiber-based) properties are assigned to lumped-plasticity or distributed fiber elements, but now subjected to a dynamic ground motion record rather than a monotonic push. φyi and φu, converted to rotations via the plastic hinge length (θp = φp·Lp), define the hinge's yield point and rotation capacity as it cycles and degrades under the actual time-varying demand.
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