Tools › Structures › Thin-Walled Structures › Shear Lag in Box Beams › Example
Kuhn and Chiarito's box beam of NACA Report 739 Figure 50: substitute single-stringer shear-lag analysis by the recurrence formula (Tables I to VII)
| Tool | Shear Lag in Box Beams |
| Source | Kuhn, P., and Chiarito, P. T. (1942). Shear Lag in Box Beams: Methods of Analysis and Experimental Investigations. NACA Report 739, National Advisory Committee for Aeronautics, Washington, DC. Part III, numerical examples, Figure 50 and Tables I to VII. |
| Level | Graduate |
Problem
A cantilever box beam 48 in long carries 600 lb at the tip. Its half cover is 10.875 in wide with a corner flange whose idealized area grows from 0.229 in² at the tip to 0.392 in² at the root, three stringer pitches of 3.625 in with stringer areas 0.194 to 0.318 in² (the centre stringer counted half), a skin strip of 0.018 to 0.030 in² beside the flange, a 0.015 in cover sheet with G/E = 0.40 and a depth of 6.18 in. Find the shear-lag parameter and the recurrence-formula coefficients of each 8 in bay, the X-forces and the flange and stringer stresses at every station in the first and second approximation of the substitute width, and the chordwise distribution of the stringer stresses at station 5 and at the root.
Inputs to Enter
- Half-width 10.875 in, 3 stringer pitches; stringer area 0.194 / 0.318 in² (tip / root), flange strip 0.018 / 0.030 in², flange area 0.229 / 0.392 in²; sheet 0.015 in, depth 6.18 in
- E 10 Msi, G 4 Msi (G/E 0.40); span 48 in, 6 bays, tip load 600 lb, no running load, rigid root
- Two substitute-width approximations (the report's practice), station of interest 5
Answers
- Table I: A_L 0.503 → 0.825 in², A_T 0.732 → 1.217 in², σ^P = M/(h A_T) 956 … 3,835 psi; stringer centroid b_S1 = 6.288 in (report 6.28)
- Table II (first approximation): K = 0.0757 … 0.0610 per in, p = K/tanh KL 0.1400 … 0.1347, q = K/sinh KL 0.1177 … 0.1201, γ = S A_L/(h A_T) 66.7 … 66.0 lb/in; Table III: X = 49 / 114 / 213 / 375 / 636 / 1,056 lb (report 48 / 113 / 212 / 373 / 633 / 1,054), root σ_F 6,523 psi (report 6,525), average Yb 1.56 (report 1.54), 1 − y_L/b = tanh(Yb/2)/Yb = 0.419 (report 0.420)
- Second approximation: b_S = 2 × 6.288 × 0.419 = 5.27 in, root K 0.0666 per in (report 0.0665), X_root 970 lb and σ_F 6.304 ksi, σ_L 2.653 ksi (report 987 lb, 6,355 and 2,640 psi with its miscopied coefficient p_4 = 0.1336), flange efficiency η = σ^P/σ_F = 0.607; shear-lag increase of the flange stress 64.6 % above My/I
- Station 5 chordwise distribution: Yb 1.725 from tanh(Yb)/Yb = 0.544 (report 1.760 from its figure 13), σ_CL = σ_F/cosh Yb = 1.710 ksi (report 1,673 psi), finite-stringer correction 1.119 (report 1.120), stringer stresses B / C / D = 3.325 / 2.239 / 1.914 ksi (report 3,320 / 2,203 / 1,873 psi)
- Root (Table VII station 6): flange 6.304 ksi, stringers B / C / D 3.624 / 1.932 / 1.466 ksi (report 6,355 / 3,399 / 2,020 / 1,630 psi); the centre stringer carries 62 % less than My/I. Constant-section closed form with the root properties: X = (γ/K) tanh KL = 984 lb against 970 lb from the recurrence
Notes
The tool's default inputs are this beam; press Submit. The report's Tables I to III are reproduced within 0.5 % (its slide-rule rounding); its Tables V to VII differ by 2 to 3 % on the flange stress and up to 10 % on the small centre-stringer stresses because the report read Yb from a graph and solved its second approximation with the printed coefficient p_4 = 0.1336 where 0.1386 follows from its own K and L (the test script reproduces the report's X-forces exactly from the printed coefficients). Every value is checked in test_shear_lag.php (302 checks) against the report's tables, the Report 608 Appendix B closed forms and the numpy script References/Shear-Lag/checks/shearlag_check.py.