Euler-Johnson Column Buckling

Shigley Ch. 4: radius of gyration r = sqrt(I/A), effective slenderness SR = K L/r, transition SR_D = pi sqrt(2 E/Sy). A long column (SR >= SR_D) buckles elastically by Euler, Pcr = pi^2 E I/(K L)^2; an intermediate one follows the J.B. Johnson parabola Pcr = A[Sy - (Sy SR/(2 pi))^2/E], which caps the short-column load at the squash load A Sy where Euler would run to infinity. K by end condition (pinned 1.0, fixed-free 2.0, fixed-fixed 0.5, fixed-pinned 0.7). A steel strut E 30e6, Sy 40 ksi, I 0.05 in^4, A 1 in^2, L 20 in, pinned is intermediate (SR 89 < 122) and carries 29,200 lbf by Johnson - Euler would overpredict 37,000; stretch it to 50 in and it flips to Euler at 5,920 lbf. Concentric load; the secant (eccentric) formula and code provisions are separate. Apply a safety factor. A design aid; Shigley and the engineer of record govern.

r = sqrt(I/A); SR = K L/r; SR_D = pi sqrt(2 E/Sy). Euler (SR >= SR_D): Pcr = pi^2 E I/(K L)^2. J.B. Johnson (SR < SR_D): Pcr = A[Sy - (Sy SR/(2 pi))^2/E]. K = 1.0/2.0/0.5/0.7 pinned/fixed-free/fixed-fixed/fixed-pinned.

The Euler critical load and the J.B. Johnson parabolic formula for intermediate columns with the transition slenderness (Shigley, Mechanical Engineering Design, Ch. 4 -- columns), by name.

The Euler and Johnson column formulas are standard published mechanics results; the modulus, yield, section properties, length, and end condition are the user's inputs.

Estimate. AHJ and licensed professional govern.

Field names used by the API: modulus_psi, yield_strength_psi, moment_of_inertia_in4, area_in2, length_in, end_condition, critical_load_lbf, transition_slenderness

• Euler / Johnson split Euler for SR >= SR_D = pi sqrt(2 E/Sy), Johnson parabola belowShigley Ch. 4
• End condition K = 1.0 pinned, 2.0 fixed-free, 0.5 fixed-fixed, 0.7 fixed-pinned (theoretical)Shigley Ch. 4
• Scope concentric load; secant/eccentric, local buckling, and code provisions are separatescope of this tile