16-Civ-A2 Elementary Structural Design · December 2015

Question 3 of 7: A3 — Built-up plate section — moments of resistance

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

Notes on this paper

Paper format. National Examinations, December 2015 — 98-Civ-A2 Elementary Structural Design, 3-hour closed-book paper (textbooks and design handbooks permitted). Seven questions in three parts: Part A (A1–A3, steel), Part B (B1–B3, reinforced concrete), Part C (C1, timber). The candidate answers two from Part A, two from Part B and the one question in Part C — five solutions, all of equal value. All seven are solved here, because the set is a study resource rather than an exam script.

Reference texts.

CSA S16, Design of Steel Structures — Clauses 11 (classification), 13.3 (compression), 13.5–13.6 (flexure), 13.8 (axial + bending), 13.11–13.12 (bolts, block shear).
CISC, Handbook of Steel Construction — section property tables for W-shapes and hollow structural sections.
CSA A23.3, Design of Concrete Structures — Clauses 10 (flexure and axial), 11 (shear), 10.15 (slender columns).
CSA O86, Engineering Design in Wood — Clause 6 (modification factors), Clause 7 (sawn lumber, bending and shear).
Canadian Wood Council, Wood Design Manual — specified strengths for Beam and Stringer grades.

Check — load factoring. Note 6 on page 1 states “all loads shown are unfactored”, so every load taken from a figure is a specified load and is factored here as 1.25D + 1.5L (NBCC Table 4.1.3.2, Case 2), treating the drawn point loads as live and member self-weight as dead. Where a question states a load directly in its text without a dead/live split (B3), that load is taken as already factored; this is stated again at that question.

Question 3: A3 — Built-up plate section — moments of resistance (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.

Element (all plates 25 mm thick, $F_y = 350$ MPa)	Size and position
Closed box	600 mm wide × 1375 mm deep overall, webs at $x = \pm 287.5$ mm
Cover plate on top of the box	1000 mm wide, overhanging the box 200 mm each side
Four upstanding stiffener plates	200 mm tall, at $x = \pm 487.5$ and $\pm 200$ mm
Overall depth / overall width	1600 mm / 1000 mm

Find. $M_r$ about the horizontal centroidal axis a-a and about the vertical centroidal axis b-b.

[Figure not reproduced: Figure A3 (redrawn): all nine plates are 25 mm thick; a-a is the horizontal centroidal axis and b-b the vertical axis of symmetry. See the official exam paper.]

Check — figure interpretation. The hand-drawn figure is dimensioned but not to scale. The 1400 mm dimension is read from the underside of the box to the top surface of the cover plate, with the 200 mm stiffeners projecting above it, giving an overall depth of 1600 mm; the 500 mm dimension runs from the cover-plate edge to the axis b-b, so the cover plate is 1000 mm wide; and the 200 mm dimension at the left confirms the cover-plate overhang, consistent with a 600 mm box. All four readings close on one another, which is the check that the interpretation is right.

Approach. Locate the centroid and compute $I_x$, $I_y$; classify every plate element separately for each axis of bending; then use the plastic modulus where the section is Class 1 or 2 and an effective section modulus where any element is Class 4.

Gross section properties. Summing the nine plates with $y$ measured up from the underside of the box, $$A=141\,250\ \text{mm}^{2},\qquad \bar y=926.4\ \text{mm}$$ $$I_x=40.82\times10^{9}\ \text{mm}^{4},\qquad I_y=11.24\times10^{9}\ \text{mm}^{4}$$ The extreme fibre about a-a is the bottom of the box, 926.4 mm from the centroid, so the governing elastic modulus is $S_x = 44.06\times10^{6}\ \text{mm}^{3}$.
Element classification for bending about a-a. With $F_y = 350$ MPa the S16 Table 2 limits are $525/\sqrt{F_y}=28.1$ for a plate supported on both edges, $145/\sqrt{F_y}=7.75$ (Class 1) and $170/\sqrt{F_y}=9.09$ (Class 2) for an outstand, and $1100/\sqrt{F_y}=58.8$ for a web in flexure. Checking each element gives $400/25 = 16.0$ for the cover plate between the inner stiffeners, $575/25 = 23.0$ for the box top flange, $1325/25 = 53.0$ for the box webs — all Class 1 — and $200/25 = 8.0$ for the stiffener outstands, which falls between 7.75 and 9.09 and is therefore Class 2. The section as a whole is Class 2, so the plastic moment may still be developed.
Plastic modulus about a-a. The plastic neutral axis divides the section into equal areas, not equal first moments, and lands at $$y_p=1137.5\ \text{mm}\quad\text{versus}\quad\bar y=926.4\ \text{mm}$$ some 211 mm above the elastic centroid, because so much material is concentrated in the cover plate and stiffeners at the top. Summing $Z=\sum A_i\left|y_i-y_p\right|$ over the nine plates split at $y_p$, $$Z_x=65.82\times10^{6}\ \text{mm}^{3}$$ a shape factor of $Z_x/S_x = 1.49$, far above the 1.12 typical of a rolled I-shape, because this section carries much of its area near the neutral axis.
Moment of resistance about a-a. For a Class 2 section, $$M_{r,aa}=\phi Z_x F_y=0.90(65.82\times10^{6})(350)=\boxed{20\,730\ \text{kN}\cdot\text{m}}$$
Classification for bending about b-b is different — and severe. When the section bends about the vertical axis, each box web lies at essentially a single value of $x$ and is therefore in uniform compression over its whole 1325 mm length. As a stiffened element in uniform compression its limits are 28.1 (Class 1 and 2) and $670/\sqrt{F_y}=35.8$ (Class 3), and $$\frac{h}{w}=\frac{1325}{25}=53.0\ >\ 35.8$$ so the box webs are Class 4 and the section must be designed on effective widths. Every other element remains fully effective: the cover plate and box flanges are now “webs” in flexure at $1000/25 = 40.0$ and $600/25 = 24.0$, and the stiffeners are outstands at $8.0 < 200/\sqrt{F_y} = 10.7$.
Effective width and shifted centroid. S16 Clause 13.5(c)(iii) gives the effective width of a stiffened element as $$b_e=\frac{670\,t}{\sqrt{F_y}}=\frac{670(25)}{\sqrt{350}}=895.3\ \text{mm}$$ so only 895 mm of the 1325 mm compression-side web remains effective. Removing the balance makes the section unsymmetrical about b-b, and the centroid shifts 23.7 mm toward the tension side. Recomputing on the reduced section, $$I_{y,\text{eff}}=10.279\times10^{9}\ \text{mm}^{4},\qquad S_e=\frac{I_{y,\text{eff}}}{c_{\max}}=19.63\times10^{6}\ \text{mm}^{3}$$
Moment of resistance about b-b. $$M_{r,bb}=\phi S_e F_y=0.90(19.63\times10^{6})(350)=\boxed{6180\ \text{kN}\cdot\text{m}}$$ Using the gross elastic modulus instead would give 7080 kN·m, overstating the capacity by 15 %; using a plastic modulus would overstate it far more. The section is more than three times stronger about a-a than about b-b, which is what the 1600 mm depth against the 1000 mm width predicts.

Check. Both resistances are cross-sectional values. No laterally unbraced length is given, so lateral-torsional buckling is not evaluated; for bending about the strong axis a-a the compression flange must be braced, or $M_{r,aa}$ must be reduced by S16 Clause 13.6. The closed box gives the section a large torsional constant, so any such reduction will be modest.

Result	Value
Area / elastic centroid $\bar y$	141 250 mm² / 926.4 mm
$I_x$ / $I_y$	40.82 × 10⁹ / 11.24 × 10⁹ mm⁴
Class for a-a bending	Class 2 (stiffener outstand $b/t = 8.0$)
Plastic neutral axis / $Z_x$	1137.5 mm / 65.82 × 10⁶ mm³
$M_{r,aa}$	20 730 kN·m
Class for b-b bending	Class 4 (box web $h/w = 53.0 > 35.8$)
Effective width / $S_e$	895.3 mm / 19.63 × 10⁶ mm³
$M_{r,bb}$	6180 kN·m

Topic. Section classification of built-up plate girders and the plastic versus effective section modulus.

Key relations. $M_r = \phi Z F_y$ for Class 1 and 2, $M_r = \phi S F_y$ for Class 3, $M_r = \phi S_e F_y$ for Class 4; effective width $b_e = 670t/\sqrt{F_y}$ for stiffened elements and $200t/\sqrt{F_y}$ for outstands; the plastic neutral axis is the equal-area axis.

Why this works. A section can only reach the yield or plastic moment if none of its compression plates buckles locally first, so S16 sorts every plate element by its width-to-thickness ratio before any strength is computed. The crucial insight in this question is that classification depends on the axis of bending: the same 1325 mm box web is a lightly stressed flexural web when the section bends about a-a, but becomes a uniformly compressed stiffened element when it bends about b-b, and the limiting ratio drops from 58.8 to 35.8. That single change moves the section from Class 2 to Class 4 and forces a completely different calculation. The equal-area rule for the plastic axis is the other key idea: at full plasticity every fibre carries $F_y$, so equilibrium requires equal areas above and below, not equal first moments as in elastic analysis.

Common pitfall. Classifying the section once and reusing that class for both axes. A close second is locating the plastic neutral axis at the elastic centroid; here that error would misplace the axis by 211 mm and understate $Z_x$.

Source. CSA S16, Design of Steel Structures, Clause 11.2 with Table 2 (classification), Clause 13.5 (bending resistance, including 13.5(c)(iii) for Class 4).

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