Tools › Structures › Beams › Beam Stress › Example
Peery Chapter 7 Example 1: Z-section stringer in unsymmetrical bending (Fig. 7.6)
| Tool | Beam Stress |
| Source | Peery, D. J. (1950). Aircraft Structures. McGraw-Hill, New York. Chapter 7, Example 1, Fig. 7.6. |
| Level | Undergraduate |
Problem
A Z-section of uniform 2 in thickness, 12 in deep and 10 in wide overall (each flange projects 4 in beyond the 2 in web, the upper flange to the left) carries bending moments M_x = 100,000 lb-in and M_y = 10,000 lb-in about its centroidal axes. Find the section properties, the principal axes and the bending stress at the critical fibers.
Inputs to Enter
- Polygon (in): (−5, 4), (−1, 4), (−1, −6), (5, −6), (5, −4), (1, −4), (1, 6), (−5, 6); grid 1 in
- Convention: aerospace fiber rule (Bruhn, Peery); restraint: free
- One load case: M_x = 100,000 lb-in, M_y = 10,000 lb-in, P = V_x = V_y = T = 0
- E = 10.5 MSI, no allowables
Answers
- I_x = 693.33 in⁴, I_y = 173.33 in⁴, I_xy = −240.00 in⁴ (book 693.3, 173.3, −240)
- Principal moments I_p = 787.17 in⁴, I_q = 79.50 in⁴, principal axis at 21.355° (book 787.1, 79.5, 21.35°)
- σ = −315.3 y − 494.3 x psi over the section (D = I_x I_y − I_xy² = 62,578 in⁸)
- Maximum tension 2,386 psi at (−1, −6) in, maximum compression −2,386 psi at (1, 6) in, neutral axis through the centroid at −57.5° from x (tool, confirmed by hand from the unsymmetrical-bending formula)
- M_x = 100,000 lb-in alone: 2,045 psi at the web-flange corner as a free beam, 865 psi uniform (M c / I) with sideways deflection restrained (tool's Restraint option)
Notes
The tool opens on a default I-beam, so draw the Z-section (or load it from My Sections where saved) and enter the two moments; only the section properties are the book's printed values, the stresses are the tool's with a hand check of the formula. Check case of test_beam_stress.php (Peery Z, I_xy −240, free 2,045.3 psi, restrained M c / I).