16-Civ-A2 Elementary Structural Design · May 2018

Question 3 of 7: A3 — Stability and moment resistances of a fabricated plate section

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Question 3: A3 — Stability and moment resistances of a fabricated plate section (8 + 6 + 6 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. All plates 20 mm thick, G40.21 350W (F_y = 350 MPa). From Figure A3: a base plate 400 mm wide; two outer webs 200 mm high standing on the extreme edges of the base plate; one central web 400 mm high on the axis of symmetry. The overall depth is therefore 420 mm and y-y is the axis of symmetry.

Find. The plate-element classification of each component, and the resulting moment resistances M_rx and M_ry.

Approach. Compute the gross section properties, classify each plate against the S16 Table 2 limits for elements supported along one edge, then compute the moment resistance appropriate to the governing class about each axis — treating the two senses of bending about x-x separately, because the section is unsymmetrical about that axis.

1. Gross section properties. Taking areas of 8000 mm² (base plate at y = 10 mm), 4000 mm² each (outer webs at y = 120 mm) and 8000 mm² (central web at y = 220 mm), the total area is A = 24 000 mm² and $$\bar{y}=\frac{\sum A_{i}y_{i}}{A}=\frac{2.800\times10^{6}}{24000}=\boxed{116.7\ \text{mm from the soffit}}$$ The second moments of area follow from the parallel-axis theorem: $$I_{x}=310.1\times10^{6}\ \text{mm}^{4},\qquad I_{y}=396.0\times10^{6}\ \text{mm}^{4}$$ Note that I_y exceeds I_x: this section is wide and shallow, so y-y is the stronger axis in the elastic sense.
2. Classify the plate elements (S16 Table 2, elements supported along one edge). With F_y = 350 MPa the limits are $$\frac{145}{\sqrt{F_{y}}}=7.75\ \text{(Class 1)},\qquad\frac{170}{\sqrt{F_{y}}}=9.09\ \text{(Class 2)},\qquad\frac{200}{\sqrt{F_{y}}}=10.69\ \text{(Class 3)}$$ Outer webs: b/t = 200/20 = 10.0, between 9.09 and 10.69, so Class 3. Central web: b/t = 400/20 = 20.0, which exceeds 10.69, so the central web is Class 4 (slender). The panels of the base plate between webs are supported along two edges with b/t = 190/20 = 9.5 against a Class 1 limit of 525/√F_y = 28.1, so the base plate is never critical.
3. Bending about x-x, top in compression (sagging). The tall central web projects into the compression zone, so its Class 4 slenderness governs and S16 Cl. 13.5(c) requires an effective section. The effective outstand, measured from the supported (welded) edge, is $$b_{e}=\frac{200t}{\sqrt{F_{y}}}=\frac{200(20)}{\sqrt{350}}=213.8\ \text{mm}$$ Discarding the central web above y = 20 + 213.8 = 233.8 mm and recomputing gives A_eff = 20 276 mm², a centroid that migrates down to 78.1 mm, and I_eff = 104.5 × 10⁶ mm⁴. The extreme fibre of the effective section is 155.8 mm above that centroid, so $$S_{e}=\frac{I_{\text{eff}}}{c}=\frac{104.5\times10^{6}}{155.8}=671\times10^{3}\ \text{mm}^{3},\qquad M_{rx}=\phi S_{e}F_{y}=\boxed{211\ \text{kN}\cdot\text{m}}$$
4. Bending about x-x, bottom in compression (hogging). Now the compression zone is the base plate, an element supported along two edges at b/t = 9.5, and the webs lie almost entirely in tension where no slenderness limit applies. The compressed portions of the webs are only 66.7 mm deep, well inside Class 1. The section can therefore reach its plastic moment. Equal areas place the plastic neutral axis at y_p = 86.7 mm (base plate 8000 mm² plus 4000 mm² of web below the axis equals half of 24 000 mm²), and summing A_i|y_i − y_p| over all elements, $$Z_{x}=2.213\times10^{6}\ \text{mm}^{3},\qquad M_{rx}=\phi Z_{x}F_{y}=\boxed{697\ \text{kN}\cdot\text{m}}$$
5. Bending about y-y. The section is symmetric about y-y, so $$S_{y}=\frac{I_{y}}{c}=\frac{396.0\times10^{6}}{200}=1.980\times10^{6}\ \text{mm}^{3}$$ Under y-y bending the outer webs carry a stress that is uniform over their height (each sits at a fixed distance from the axis), so they are uniformly compressed outstands at b/t = 10.0 — Class 3, the worst element in play, because the central web lies on the neutral axis and carries no stress at all. A Class 3 section is limited to first yield: $$M_{ry}=\phi S_{y}F_{y}=0.90(1.980\times10^{6})(350)=\boxed{624\ \text{kN}\cdot\text{m}}$$

The three answers tell a coherent story. Hogging about x-x is the strongest case (697 kN·m) because all the plate area then works at full yield with a favourable lever arm; bending about y-y is nearly as strong (624 kN·m) but is held back to first yield by the Class 3 outer webs; and sagging about x-x is by far the weakest (211 kN·m) because the very element that provides the depth is too slender to be counted over its full height.