Fan Blade Stress: Method

Blade Loads, Stress and Frequencies

Loads

A fan blade is a rotating cantilever. Three loads act on it: the centrifugal force of its own mass, which pulls along the span and grows from zero at the tip to its full value at the root; the thrust distribution, which bends the blade out of the rotor plane (flap); and the in-plane force that reacts the torque, which bends it in the plane of rotation (lag). The thrust and in-plane loads along the span come from the same blade-element momentum model as the Ducted Fan Performance tool, so the stress analysis uses the loads the aerodynamic design implies. The centrifugal force at a station is the integral of ρ_m A Ω² r over the blade outboard of it; the bending moments are the integrals of the outboard loads times their arm.

Sections

The section at each station is modeled as an ellipse whose axes are the local chord and thickness, either solid or a thin shell of constant skin thickness. Its area and second moments are computed exactly by the site's section-geometry library (Green's theorem on the polygon), the same code that serves the Section Properties and Beam Stress tools. An ellipse has about 20 percent more area than a real airfoil of the same chord and thickness and a comparable flapwise second moment, so the stresses are a first estimate; for a detailed check, draw the real section in the Beam Stress tool with the loads reported here.

The section sits at the local pitch θ, so the flap moment (vector in the rotor plane) and the lag moment (vector along the axis) are resolved into moments about the chord and about the thickness axis before the bending stress is formed. The stress σ = CF/A + M_u v / I_u − M_v u / I_v is evaluated around the ellipse and its largest tensile value taken; the station with the largest value is the critical one, usually near the root for a tapered blade but sometimes outboard where the thickness falls faster than the moment.

Margins

The operating-point loads are limit loads. The yield margin is written at limit load times the yield factor FS_yld (default 1.15, the value several certification agencies require for no detrimental permanent set; 1.0 for a plain limit check), the ultimate margins at limit load times the factor of safety FS (default 1.5), following aircraft practice. The overspeed case multiplies the centrifugal stress by the square of the overspeed factor while holding the aerodynamic bending: a conservative bound on the stress of a runaway rotor, since the aerodynamic loads would in fact also rise.

Frequencies

The first flap and lag bending frequencies are found by Rayleigh's method with the cantilever shape ψ = 1 − cos(π(r − R_h)/(2L)), which satisfies the clamped root. At rest, ω₀² = ∫EI ψ″² dr / ∫m ψ² dr with the flapwise or lagwise second moment. Rotation stiffens the blade through the centrifugal tension: Southwell's relation gives ω_flap² = ω₀² + K Ω² with K = ∫CF ψ′² dr / (Ω² ∫m ψ² dr), and for the lag mode the in-plane centrifugal softening reduces the coefficient by one. Rayleigh's frequencies are upper bounds, typically within a few percent for the first mode.

The Campbell diagram plots both frequencies against rotor speed together with the per-rev lines 1P to 6P. A crossing near the operating speed is a resonance: 1P is excited by any inflow asymmetry (a crosswind, an inlet distortion), and the duct sees the blade-passing order. The tool flags a first mode within 10 percent of an integer per-rev at the operating speed.

Not included

Torsion and the pitch-flap coupling of a swept or skewed blade, the root attachment and its stress concentration, fatigue (the alternating 1P bending of a fan in a crosswind), composite ply-level allowables, and the aerodynamic damping of the modes.