A screening check for RC pier columns: slenderness by kL/r, single-curvature moment magnification with EI = EcIg/2.5, and a strain-compatibility P–M interaction diagram with the variable resistance factor of AASHTO 5.5.4.2. Circular columns assume a 12-bar ring; rectangular columns assume steel split equally between the two bending faces.
A screening check for RC pier columns: slenderness by kL/r, single-curvature moment magnification with EI = EcIg/2.5, and a strain-compatibility P–M interaction diagram with the variable resistance factor of AASHTO 5.5.4.2. Circular columns assume a 12-bar ring; rectangular columns assume steel split equally between the two bending faces.
Bars are placed for real: cover + tie + half-bar sets the ring radius (circular, evenly spaced ring) or the two face rows (rectangular, half the bars per bending face). ρ and the effective bar-spread ratio are computed from that geometry, not assumed.
Magnification uses Cm = 1.0 and φK = 0.75; sustained-load creep refinement (βd) is not applied — treat results as a screen, not a final design.
| Po pure compression (nominal) | – |
| φPn,max cap | – |
| Euler Pe (EI = EcIg/2.5) | – |
| φMn at Pu | – |
| β1 | – |
Treat this as the two-minute sanity check before the real column run. The stiffness assumption EI = EcIg/2.5 is the blunt end of AASHTO’s two options and usually conservative for lightly loaded piers; if δ comes out above roughly 1.4, the column wants either a bigger section or a proper second-order analysis, not a bigger magnifier. Two modelling honesty notes: the circular section here carries a 12-bar ring and the rectangle puts all steel on the two bending faces — a real 4-face cage gives a slightly fatter diagram near pure bending, so this tool errs safe. And remember the interaction check is one axis at a time; bridge piers almost always see biaxial demand from braking plus wind or seismic, so run both directions and combine per 5.6.4.5 before calling it done. Seismic detailing, plastic-hinge confinement and P-Δ at the seismic limit state live entirely outside this screen.
A pier column carries the superstructure reactions with the moments that come from eccentric live load, braking, wind, temperature movement of the deck and, in seismic regions, the plastic hinge demands. The preliminary check has three parts: slenderness, magnified moments and the interaction diagram.
The effective length kL depends on the fixity: a cantilever column free at the top (single column, elastomeric bearings) has k ≈ 2.0; a column in a multi-column frame bent about the frame axis k ≈ 1.2–1.5; a column fixed top and bottom k ≈ 0.7–0.8 in the braced direction. With r = 0.25 D for circles and 0.29 h for rectangles, slenderness may be ignored below kL/r = 22 (unbraced) or 34 − 12 M1/M2 (braced); above 100 a refined P-Δ analysis is required.
The magnified moment Mc = δb M2b + δs M2s is then checked on the interaction diagram, in both directions when the column bends biaxially (5.6.4.5).
For a given section and bar layout, each neutral-axis position gives one point (Pn, Mn): the concrete stress block 0.85 f′c over β1c and the bar forces from the linear strain profile with εcu = 0.003. The pure-axial capacity is Po = 0.85 f′c (Ag − Ast) + fy Ast, limited to 0.85 Po (spirals) or 0.80 Po (ties). The resistance factor follows the strain in the extreme tension steel: 0.75 for εt ≤ 0.002, 0.90 for εt ≥ 0.005, linear between. The load point is safe when it lies inside the φPn–φMn curve; the radial distance to the curve is the demand-to-capacity ratio.
Ø1.8 m column, f′c 35 MPa, 24 × #32 (ρ 1.0 %), height 9 m, cantilever longitudinally (k = 2.0): kL/r = 18 / 0.45 = 40 → slender. Pu = 14,000 kN, Mu = 6,000 kN·m; EI = Ec Ig / 2.5 = 29,900 × 0.515 / 2.5 ≈ 6.2 × 106 kN·m², Pe = π² × 6.2 × 106 / 18² ≈ 189,000 kN, δb = 1 / (1 − 14,000 / (0.75 × 189,000)) = 1.11 → Mc ≈ 6,650 kN·m. The point (14,000, 6,650) is then plotted on the φ-curve of the section — the calculator draws it.
AASHTO LRFD 5.6.4.3: for members not braced against sidesway when kL/r < 22, and for braced members when kL/r < 34 − 12 (M1/M2). Most single-column or wall piers of moderate height are unbraced in the longitudinal direction and slender above about 22.
What is the moment magnification method?The approximate method of 4.5.3.2.2b: the factored moment is multiplied by δb = Cm / (1 − Pu / (φ Pe)) with Pe = π² EI / (kL)², using EI = Ec Ig / 2.5 (or (Ec Ig / 5 + Es Is) / (1 + βd)) to represent the cracked, creeping column.
How is the P–M interaction diagram built?By strain compatibility: for each neutral-axis depth the concrete stress block and the strain in every bar give a resultant axial force and moment; the resistance factor φ varies from 0.75 (compression-controlled, εt ≤ 0.002) to 0.90 (tension-controlled, εt ≥ 0.005). The column passes when the factored (Pu, Mu) point lies inside the φ-curve.
What reinforcement ratio is typical for pier columns?AASHTO 5.6.4.2 limits the longitudinal steel to 1–8 % of the gross area (0.135 f′c/fy minimum ratio when the column is oversized); practical piers use 1–2 %, with spirals or ties per 5.10.4.