Eccentrically Loaded Column (Secant Formula)
A load that misses the column's center bends it, and the bend moves the load farther off center.
Example
You enter
- Axial load P (lbf) 31100
- Eccentricity e (in) 0.75
- Distance to the extreme fiber (in) 2
- Modulus E (psi) 29000000
- Yield strength Sy (psi, blank to skip) 0
- Moment of inertia I (in⁴) 8
- Cross-section area A (in²) 3.54
- Unbraced length L (in) 96
- End condition fixed-free
You get
- Maximum stress 22000
- Lateral deflection 0.939 in
- Critical load (lbf) 62100
Checked against published worked examples from: McGraw-Hill (Beer, Johnston, DeWolf).
Details, formula, and sources
ScopeThe secant formula gives the peak stress and the sideways deflection, and both grow without limit as the load nears the Euler buckling load. An 8 ft flagpole tube (I 8.0 in^4, A 3.54 in^2) carrying half its 62,113 lbf Euler load 0.75 in off center deflects 0.94 in and peaks at 21,900 psi, 2.5 times the 8,790 psi a centered load gives. Because stress is not proportional to load, enter the yield strength and it finds the load at first yield, which is where the safety factor belongs. A design aid; the engineer of record governs.
Formula or tablePcr = pi^2 E I/(K L)^2; r^2 = I/A; sigma_max = (P/A)[1 + (e c/r^2) sec((K L/2r) sqrt(P/(E A)))] = (P/A)[1 + (e c/r^2) sec((pi/2) sqrt(P/Pcr))]; y_max = e[sec((pi/2) sqrt(P/Pcr)) - 1]; with Sy entered, the load at first yield solves sigma_max(P) = Sy (bisection on 0 < P < Pcr).
Source and editionThe secant formula for eccentrically loaded columns (Beer, Johnston and DeWolf, Mechanics of Materials, Sec. 10.6; Shigley, Mechanical Engineering Design, Ch. 4), by name; it reproduces Beer and Johnston Sample Problem 10.2 (22.0 ksi, 0.939 in).
Free accessBeer and Johnston Sample Problem 10.2 is worked in course slides posted free by Benha University (feng.stafpu.bu.edu.eg, '12 Buckling.pdf', slides 10-16 to 10-18); the load, eccentricity, section properties, length, and end condition are the user's inputs.
What governsEstimate. AHJ and licensed professional govern.
API field namesload_lbf, eccentricity_in, extreme_fiber_in, modulus_psi, yield_strength_psi, moment_of_inertia_in4, area_in2, length_in, end_condition, max_stress_psi, max_deflection_in, critical_load_lbf
- Secant formula sigma_max = (P/A)[1 + (e c/r^2) sec((pi/2) sqrt(P/Pcr))] at mid-heightBeer and Johnston Sec. 10.6
- End condition K = 1.0 pinned, 2.0 fixed-free, 0.5 fixed-fixed, 0.7 fixed-pinned (theoretical), as in euler-johnson-columnShigley Ch. 4
- Safety factor on load the load at first yield, found by bisection, divided by the applied loadShigley Ch. 4