Second Moment of Cross Section of a Unit Pleat as a Function of the Half Angle of a Pleat's Opening with Its Cross Section Being Vertically and Horizontally Oriented

Abstract

Formulae have been given for the second moment of cross section I of a unit pleat as the functions of the half angle of a pleat's opening θ, the half width of a unit pleat l and the thickness of cloth t, assuming that the surface of the cloth was flat. These formulae could be presented as educational tools to make a comparison of the bending between the canti lever beam of clothing material and a unit pleat.
For the vertical setting of a unit pleat under the condition of tan θ≥_t/l, the values given by the derived equation I_∧ =lt (l² cos, ²θ+t²/sin²θ) /6 were inputed into an Excel program written by VBA. As the half angle θ reached zero, the value of I_∧ diverged to infinity. For a half angle of θ more than about 5° both with l=2.5 and 4 cm, the value of I_∧ for t= 0.06 cm was approximately comparable to two times that for t = 0.03 cm, as expected, since I_∧ had a proportional relation to the thickness t for t>>l. In addition, the decreasing tendency of I_∧ for t= 0.06 cm to that for t = 0.03 cm were almost the same within the range of θ=15-55° for l=4 cm.
For the horizontal setting of a unit pleat under the condition of tan θ≥t/l the values of I_< given by the equation I_<=2l³t sin² θ/3 had a proportional relation to the thickness t, increased rapidly with l, and showed the same dependence on the half angle θ.
It has been considered that the divergence of I_∧ arose near θ=0° due to a lack of free opening of a unit pleat for the condition of tan θ≥t/l. After the deletion of one bent top and two side edges, the equation for vertical setting was written as I_∧, _deleted =lt (l² COS ² θ t ²sin² θ) /6, yielding a definite value at θ=0°.Furthermore the, difference between this and the former equation for the vertical setting was lt³ (sin^-2θ-sin²θ) /6 with the values of 547lt³ atθ=1°, 21.9lt3 at θ=5° and 5.5lt³ at θ=10°. With these decreasing tendencies and the adding condition of t>> l, the effect of the deletions has not been observed for the not so small angle of θ.
Moreover, for the horizontal setting of this model without one left bend and two edges each at the upper and bottom position, the equation was written as I_<, _deleted = (2l^3& t sin² θ + 6l² t² cos θ sin θ +2lt³ cos²θ) /3, reaching the equation given by I_<=2l³t sin² θ/3 for t>>l.