A2 — Moments of resistance of a welded box section: 16-Civ-A2 Elementary Structural Design, December 2016, Q2

Question 2: A2 — Moments of resistance of a welded box section (8 + 12 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. A welded box built from 20 mm plate of G40.21-M350W steel ($F_y=350$ MPa): two 500 × 20 flange plates at top and bottom, and two 20 mm webs whose centrelines lie 190 mm each side of the vertical axis (dimension string 50 + 20 + 180 = 250 mm from the plate edge to the centreline). Overall 500 mm wide × 500 mm deep.

Find. M_rx and M_ry, the factored moments of resistance of the cross-section about the two centroidal axes.

Figure A2 — welded box from 20 mm plate. The webs are set in from the plate edges, leaving 50 mm flange outstands.

Approach. Compute the second moments of area, classify every plate element to S16 Table 2, and — because everything proves to be Class 1 — take the moments of resistance as $\phi Z F_y$ about each axis.

Area. Clear web height $h_w=500-2(20)=460$ mm, so $$A=2(500\times20)+2(20\times460)=20\,000+18\,400=38\,400\ \text{mm}^2$$
Second moment about x-x (flanges by the parallel-axis theorem, webs about their own centroids): $$I_x=2\left[\frac{500(20)^3}{12}+10\,000(240)^2\right]+2\left[\frac{20(460)^3}{12}\right]=1477\times10^{6}\ \text{mm}^4$$
Second moment about y-y (now the flanges are the wide plates and the webs sit at ±190 mm): $$I_y=2\left[\frac{20(500)^3}{12}\right]+2\left[\frac{460(20)^3}{12}+9200(190)^2\right]=1082\times10^{6}\ \text{mm}^4$$ so $S_x=I_x/250=5908\times10^{3}$ mm³ and $S_y=I_y/250=4326\times10^{3}$ mm³.

Because the section is doubly symmetric the plastic neutral axis coincides with the centroidal axis in both directions, so each plastic modulus is simply the first moment of the whole area about that axis.

Plastic moduli. About x-x the flanges act at 240 mm and each web half at 115 mm: $$Z_x=2(10\,000)(240)+2\Bigl[2(4600)(115)\Bigr]=6916\times10^{3}\ \text{mm}^3$$ About y-y each flange contributes two half-plates acting at 125 mm and each web acts as a whole at 190 mm: $$Z_y=2\Bigl[2(5000)(125)\Bigr]+2(9200)(190)=5996\times10^{3}\ \text{mm}^3$$
Classify the elements (S16 Table 2, $\sqrt{F_y}=18.71$). Flange plate between the webs, supported along both edges: $b/t=360/20=18.0$ against the Class 1 limit $525/\sqrt{F_y}=28.1$. Flange outstand beyond a web: $50/20=2.5$ against $145/\sqrt{F_y}=7.75$. Web in flexure: $h/w=460/20=23.0$ against $1100/\sqrt{F_y}=58.8$. Every element is Class 1 about both axes.
Moments of resistance (S16 Clause 13.5(a), $\phi=0.90$): $$M_{rx}=\phi Z_x F_y=0.90(6916\times10^{3})(350)=\boxed{2179\ \text{kN}\cdot\text{m}}$$ $$M_{ry}=\phi Z_y F_y=0.90(5996\times10^{3})(350)=\boxed{1889\ \text{kN}\cdot\text{m}}$$

The shape factors are $Z_x/S_x=1.171$ and $Z_y/S_y=1.386$; the larger value about y-y reflects how much of the area (the two flange plates) lies close to that axis and only reaches yield late in the elastic range. Note also that the two webs close the section into a box, giving it a very large St Venant torsion constant, so no lateral-torsional buckling reduction applies to $M_{rx}$: these are true cross-section resistances, valid for any unbraced length.

Property	About x-x	About y-y
Second moment of area, I	1477 × 10⁶ mm⁴	1082 × 10⁶ mm⁴
Elastic modulus, S	5908 × 10³ mm³	4326 × 10³ mm³
Plastic modulus, Z	6916 × 10³ mm³	5996 × 10³ mm³
Radius of gyration, r	196.1 mm	167.8 mm
Classification	Class 1	Class 1
Moment of resistance	M_rx = 2179 kN·m	M_ry = 1889 kN·m

Topic. Section classification and plastic moment capacity of a welded box girder.

Key relations. $Z=\sum A_i|y_i-y_p|$ with the plastic axis placed by equal areas; Class 1 limits $145/\sqrt{F_y}$ (outstand), $525/\sqrt{F_y}$ (plate supported both edges), $1100/\sqrt{F_y}$ (web in flexure); $M_r=\phi Z F_y$ for Class 1 and 2.

Why this works. Class 1 means every plate can reach yield in compression and then rotate without local buckling, so the whole cross-section can be taken to the fully plastic stress block. Only then is $Z$, rather than $S$, the correct modulus.

Common pitfall. Assuming that the deeper-looking axis is automatically the stronger one. Here the section is square, so $I_x>I_y$ only because the flange plates are the full 500 mm wide while the webs are set in; move the webs out to the plate edges and the two capacities almost swap. Always compute both.

Source. CSA S16:19 Clause 13.5 and Table 2; CISC Handbook of Steel Construction, 11th ed., Part 4.

Question 2 of 7: A2 — Moments of resistance of a welded box section

Question 2: A2 — Moments of resistance of a welded box section (8 + 12 marks)