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Megson Example 20.3: shear flow in an idealized channel of four booms under 4.8 kN, and its shear centre

ToolOpen Section Shear Flow
SourceMegson, T. H. G. (2007). Aircraft Structures for Engineering Students, 4th ed. Butterworth-Heinemann, Oxford. Example 20.3 (idealized channel, eq. 20.6) and Section 17.2 (eq. 17.14, shear centre of Example 17.2).
LevelUndergraduate

Problem

A channel section is idealized into four booms of 300 mm² at the corners, 400 mm apart vertically and 200 mm horizontally, joined by skins that carry shear only. A vertical shear load of 4.8 kN acts through the shear centre. Find the second moment of area of the booms, the shear flow in the two flange skins and the web skin, and the position of the shear centre.

Inputs to Enter

  • Section type: booms and skins; booms (x, y, B, skin t to the next row): (200, 200, 300, 1), (0, 200, 300, 1), (0, −200, 300, 1), (200, −200, 300, –) mm, mm², mm
  • Skins carry bending: off (t_D = 0); V_x = 0, V_y = 4,800 N through the shear centre
Open the Open Section Shear Flow tool (loaded with this example)

Answers

  • I_xx = 4 × 300 × 200² = 48 × 10⁶ mm⁴ (book 48 × 10⁶), k_y = V_y / I_xx = 10⁻⁴ N/mm⁴
  • q_12 = −10⁻⁴ × 300 × 200 = −6 N/mm in the top flange skin (flowing toward the free boom 1), q_23 = −12 N/mm in the web, q_34 = −6 N/mm (book −6, −12, −6 N/mm); |q|max = 12 N/mm, τ = 12 MPa in the 1 mm web
  • Web resultant 12 N/mm × 400 mm = 4,800 N = V_y; flange resultants ±1,200 N cancel (balance check)
  • Shear centre 100 mm to the left of the web mid-line (2 × 6 × 200 × 200 / 4,800), on the axis of symmetry; the book marks S there in its Figure 20.7

Notes

The tool's default inputs are this case; press Submit. The test script test_open_section_shear_flow.php also reproduces Megson's Example 17.1 (Z-section: q at the flange-web junction 3/7 S_y/h and the web maximum 9/7 S_y/h, printed as 0.42 and 1.275 with the book's rounded coefficients), Example 17.2 (channel closed form 3b²/(h + 6b)), Problems P.17.8 (2.66 r) and P.17.9 (87.5 mm), the circular-arc closed form and the numpy dense-sampling script References/Open-Section-Shear/checks/openshear_check.py.