Tools › Structures › Composites › Transverse Stresses in Curved Shape › Example
Laminate corner radius under axial load, shear and an opening moment: radial and transverse shear stress by Lekhnitskii against the simple equations
| Tool | Transverse Stresses in Curved Shape |
| Source | Lekhnitskii, S. G. (1968). Anisotropic Plates, 2nd ed. Gordon and Breach, New York (curved bar under an end force and under pure bending); Kedward, K. T., Wilson, R. S., and McLean, S. K. (1989). "Flexure of Simply Curved Composite Shapes." Composites, Vol. 20, No. 6, pp. 527–536. |
| Level | Professional |
Problem
A 90° laminate corner of outer radius 0.500 in and thickness 0.1248 in (24 plies of 0.0052 in) receives, at the tangency point, an axial load of 340.7 lb/in, a shear load of 141.9 lb/in and an opening moment of 105.2 lb-in/in per unit width (ultimate). Find the sections of largest moment and largest shear around the bend, the peak radial (interlaminar tension) and transverse shear stresses by the simple equations and by Lekhnitskii's curved-bar solution, the circumferential stress on the two surfaces, and the margins of safety against a radial strength of 6.0 ksi and a transverse shear strength of 10.0 ksi. The material values are typical of a quasi-isotropic carbon/epoxy laminate and are illustrative, not test data.
Inputs to Enter
- Circumferential modulus E_θ 7.5 Msi, radial modulus E_r 1.4 Msi, major Poisson's ratio ν_θr 0.30, transverse shear modulus G_rθ 0.55 Msi
- Radial strength F_r 6.0 ksi, transverse shear strength F_rθ 10.0 ksi
- Outer radius R_o 0.500 in, thickness t 0.1248 in, full angle φ 90°
- At the tangency point: N_o 340.7 lb/in, V_o 141.9 lb/in, M_o 105.2 lb-in/in; load level Ultimate
Answers
- R_i = 0.3752 in, R_m = 0.4376 in; maximum moment at θ = arctan(V_o/N_o) = 22.61°: N = 369.1 lb/in, V = 0, M = 117.6 lb-in/in
- The shear has no stationary point inside the bend, so its largest magnitude is at the far end, θ = 90°: N = 141.9 lb/in, V = −340.7 lb/in, M = 18.21 lb-in/in
- Constants: β = 4.404, k = 2.315, g_1 = −0.04232, g = −0.006213; moment carried by the pure-bending part M_c = M − N R_m = −43.89 lb-in/in
- Radial stress at the maximum-moment section: simple 3M/(2 t R_m) = 3.230 ksi (3.264 ksi with √(R_i R_o)); Lekhnitskii 3.229 ksi at r = 0.4286 in
- Transverse shear stress at θ = 90°: beam theory 3V/(2t) = −4.095 ksi; Lekhnitskii −4.104 ksi
- Circumferential stress at the maximum-moment section, inner / outer surface: beam theory 48.27 / −42.35 ksi; Lekhnitskii 54.68 / −39.57 ksi
- Margins of safety (ultimate): radial tension 6.000/3.229 − 1 = +0.86; transverse shear 10.00/4.104 − 1 = +1.44
Notes
The tool's default inputs are this corner; press Submit. The Lekhnitskii values are checked against an independent numerical solution in which the end-force and pure-bending stress fields are scaled until their resultants return N and M at the section (test_loads_transverse_stresses_radius.php); the case is the tool's own default, not a worked example printed in either reference. The end-force part alone would give 4.45 ksi of radial stress; the pure-bending part carrying M_c brings it to 3.229 ksi. The closing figures of the results show the stresses over the whole bend.