07-Str-A2 · December 2018

Question 2 of 7: A2 — Thickness of a stiffened panel in one-way bending

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 2: A2 — Thickness of a stiffened panel in one-way bending (10 + 10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. The wall thickness $h$ required for the panel to carry the factored pressure in one-way bending, including the local-stability requirement that makes that flexural resistance available.

Approach. Convert the pressure into a line load on the 200 mm wide strip, obtain the simple-span moment, invert $M_r = \phi S F_y$ for $h$, and then test that thickness against the CSA S16 Table 2 width-to-thickness limits — because the strength demand on this panel turns out to be so small that local stability, not stress, sets the plate thickness.

1. Load on the 200 mm wide strip. The pressure acts over the plate width, so $$w=q\,b=2.0\ \text{kPa}\times0.200\ \text{m}=0.400\ \text{kN/m},\qquad w_f=1.5(0.400)=0.600\ \text{kN/m}$$
2. Factored moment. For the simply supported 2.0 m span, $$M_f=\frac{w_f L^{2}}{8}=\frac{0.600(2.0)^{2}}{8}=\boxed{0.300\ \text{kN}\cdot\text{m}}$$
3. Section modulus required by strength. Taking the conservative elastic resistance $M_r = \phi S F_y$, $$S_{\text{req}}=\frac{M_f}{\phi F_y}=\frac{0.300\times10^{6}}{0.90(350)}=952\ \text{mm}^{3}$$
4. Section properties as a function of h. Measuring $y$ downward from the top surface, the plate contributes $200h$ at $y = h/2$ and the two stems contribute $2(100h)$ at $y = h + 50$. The two areas are equal, so the elastic centroid sits at $\bar y \approx 0.75h + 25$ and the governing (bottom) fibre is $0.25h + 75$ away. Ignoring terms in $h^{2}$ and higher, $I \approx 4.17\times10^{5}h$ and $S_{\text{bot}} \approx 5.56\times10^{3}h$, so $$h_{\text{strength}}=\frac{952}{5.56\times10^{3}}\approx\boxed{0.17\ \text{mm}}$$ Solving the exact expression rather than the linearised one gives 0.171 mm — the same answer.
5. Why that answer cannot be used. At $h = 0.17$ mm the flat elements have width-to-thickness ratios of order $100/0.17 \approx 590$. CSA S16 Table 2 limits a plate supported along one edge (an outstand) to $$\frac{b}{t}\le\frac{200}{\sqrt{F_y}}=\frac{200}{\sqrt{350}}=10.69$$ and a plate supported along both edges to $670/\sqrt{F_y} = 35.81$ for Class 3. The strength-based thickness misses these by a factor of roughly sixty: the panel would buckle locally at a small fraction of the load, so $M_r = \phi S F_y$ would never be realised. Local stability governs the design, not stress.
6. Thickness required for local stability. Under the pressure as drawn (bearing down on the plate) the top plate is the compression flange. Its three elements give $$\text{outstands, }50\ \text{mm each: } h\ge\frac{50}{10.69}=4.68\ \text{mm};\qquad \text{interior, }100\ \text{mm: } h\ge\frac{100}{35.81}=2.79\ \text{mm}$$ The 50 mm outstands govern, so the panel must be built from plate at least 4.68 mm thick. Adopt $h = 5$ mm (5 mm is a standard plate thickness and the next size up from the requirement).
7. Confirm the adopted section. With $h = 5$ mm the panel has $A = 2000$ mm², $\bar y = 28.75$ mm from the top, $I = 2.214\times10^{6}$ mm⁴ and $S_{\text{bot}} = 29.0\times10^{3}$ mm³, so $$M_r=\phi S_{\text{bot}}F_y=0.90\left(29.0\times10^{3}\right)(350)=9.14\ \text{kN}\cdot\text{m}\ \gg\ M_f=0.300\ \text{kN}\cdot\text{m}$$ a flexural utilisation of only 0.033. Every element is now at worst Class 3, so the elastic resistance is genuinely available.

The engineering content of this question is the two-order-of-magnitude gap between the two answers. A fuselage panel of this size carries a trivial bending moment; what actually sizes its skin is the requirement that the flat elements not buckle before the yield stress can be reached, together with handling, fatigue and corrosion-allowance considerations that a strength calculation never sees.