07-Str-A5 · May 2015

Question 4 of 7: Composite steel–concrete pedestrian bridge

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

Notes on this paper

Paper format. National Exams, May 2015 — 07-Str-A5 Advanced Structural Design. Three hours, "closed book" (textbooks and design handbooks permitted, no notes). Seven questions; any five constitute a complete paper and all are of equal value, the printed mark split being 20 marks each. All loads shown on Figures 1–4 are unfactored. All seven questions are answered here.

Design data printed on page 1 and used throughout. Concrete $f'_c = 30\ \text{MPa}$; structural steel $F_y = 350\ \text{MPa}$; reinforcing steel $f_y = 400\ \text{MPa}$; prestressed concrete $f'_{ci} = 35\ \text{MPa}$ at transfer and $f'_c = 50\ \text{MPa}$ in service, $n = 6$; $f_{pu} = 1750\ \text{MPa}$, $f_{py} = 1450\ \text{MPa}$, $f_{p,\text{initial}} = 1200\ \text{MPa}$, losses $= 240\ \text{MPa}$, hence $f_{pe} = 960\ \text{MPa}$ and the loss ratio $\eta = 0.80$.

Check: load factors. The paper says only that "all loads shown are unfactored" and gives no load classification. Every printed load is therefore treated as a specified live load and factored at 1.5; member self weight is treated as dead and factored at 1.25 (NBCC 2020 Case 2, $1.25D + 1.5L$). Resistance factors are $\phi = 0.90$ (steel), $\phi_w = 0.67$ and the extra 0.67 of CSA S16 Cl. 13.13.2.2 (welds), $\phi_{sc} = 0.80$ (shear studs), $\phi_c = 0.65$, $\phi_s = 0.85$, $\phi_p = 0.90$ (CSA A23.3). In an examination you would state exactly this and proceed.

Reference texts for 07-Str-A5.

CSA S16:19, Design of Steel Structures, with the CISC Handbook of Steel Construction (11th ed.) — Cl. 13.3 (compression), 13.5–13.6 (flexure and lateral–torsional buckling), 13.7 (bracing at plastic hinges), 13.8 (beam–columns), 13.13 (welds), 14.3–14.6 (plate girders), Cl. 17 (composite beams).
CSA A23.3:19, Design of Concrete Structures, with the CAC Concrete Design Handbook (4th ed.) — Cl. 10 (flexure and columns), Cl. 11 (shear), Cl. 18 (prestressed concrete).
Salmon, Johnson and Malhas, Steel Structures: Design and Behavior, 5th ed. — plastic analysis (Ch. 10), plate girders (Ch. 11), composite construction (Ch. 16).
MacGregor and Bartlett, Reinforced Concrete: Mechanics and Design, Canadian edition — frames, beam–columns, detailing.
Collins and Mitchell, Prestressed Concrete Structures — cable zone, transfer and service stress checks, harped profiles.
National Building Code of Canada 2020 for load combinations; CSA S6 (CHBDC) for the pedestrian-bridge serviceability and vibration criteria.

Question 4: Composite steel–concrete pedestrian bridge (14 + 6 = 20 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.

Quantity	Value
Span / deck width / beam spacing	20 m / 5.0 m / 4.0 m (0.5 m overhang each side)
Deck slab	220 mm reinforced concrete, $f'_c = 30\ \text{MPa}$, 24 kN/m³
Pedestrian live load	14 kPa over the full deck
Interaction	100 % (full shear connection)
Materials	$F_y = 350$, $E_s = 200\,000$, $E_c = 4500\sqrt{30} = 24\,648\ \text{MPa}$

Find. Plate sizes for the two girders, the composite moment and shear resistances, the construction-stage and serviceability checks, and the number and spacing of headed shear studs.

Cross-section of the pedestrian bridge: 5 m deck on two girders at 4 m centres, each with a 2.5 m tributary width.

Approach. Take one girder with its 2.5 m tributary strip, factor the loads, place the plastic neutral axis (it will fall in the slab because the steel is light relative to a 2.5 m × 220 mm slab), take moments about the steel centroid for $M_r$, check the bare steel under wet concrete, then govern the design by pedestrian serviceability — deflection and footfall vibration — before counting studs from the full horizontal shear.

Tributary and effective widths. Each girder takes $4.0/2 + 0.5 = 2.5\ \text{m}$. The effective flange width is the lesser of the tributary width and $L/4 = 5.0\ \text{m}$, so $b_{\text{eff}} = 2500\ \text{mm}$.
Loads on one girder. Trying flanges 2–280 × 20 with a 900 × 12 web ($d = 940\ \text{mm}$, $A = 22\,000\ \text{mm}^2$, 172.7 kg/m):
$$w_D = 0.220(24)(2.5) + 1.694 = 14.89\ \text{kN/m} \qquad w_L = 14(2.5) = 35.0\ \text{kN/m}$$ $$w_f = 1.25(14.89) + 1.5(35.0) = 71.12\ \text{kN/m} \qquad M_f = \frac{w_fL^2}{8} = 3556\ \text{kN}\cdot\text{m} \qquad V_f = 711\ \text{kN}$$
Locate the plastic neutral axis. Compare the two full capacities:
$$C_r = 0.85\phi_c f'_c b_{\text{eff}} t_s = 0.85(0.65)(30)(2500)(220)/10^3 = 9116\ \text{kN}$$ $$T_r = \phi A_s F_y = 0.90(22\,000)(350)/10^3 = 6930\ \text{kN}\ <\ C_r$$
so the neutral axis lies inside the slab and the entire steel section is in tension.
Composite moment resistance.
$$a = \frac{T_r}{0.85\phi_cf'_cb_{\text{eff}}} = \frac{6930\times10^3}{41\,438} = 167.2\ \text{mm} \qquad e = \frac{d}{2} + t_s - \frac{a}{2} = 470 + 220 - 83.6 = 606.4\ \text{mm}$$ $$\boxed{M_r = T_r e = 6930(0.6064) = 4202\ \text{kN}\cdot\text{m} \ \ge\ M_f = 3556\ \text{kN}\cdot\text{m}}$$
Utilisation 0.846 — deliberately loose, because serviceability governs at step 7.
Construction stage (unshored). Before the slab cures, the bare girder carries the wet concrete alone: $M_f = 1.25(14.89)(400)/8 = 931\ \text{kN}\cdot\text{m}$ against $\phi ZF_y = 2388\ \text{kN}\cdot\text{m}$. With flange $b/t = 6.70$ and $h/w = 75.0$ the bare section is Class 2, and temporary bracing at 5 m centres keeps the compression flange stable during the pour.
Shear. With $h/w = 75.0$ between $502\sqrt{k_v/F_y} = 62.0$ and $621\sqrt{k_v/F_y} = 76.7$, inelastic buckling governs and $F_s = 290\sqrt{F_yk_v}/(h/w) = 167.2\ \text{MPa}$, giving $V_r = 0.90(900)(12)(167.2)/10^3 = 1625\ \text{kN} \ge 711\ \text{kN}$.
Serviceability — this is what sizes the girder. Transform the slab with $n = E_s/E_c = 8.11$: $b_{tr} = 308\ \text{mm}$, the composite neutral axis sits 908 mm above the soffit and $I_{\text{comp}} = 8960\times10^6\ \text{mm}^4$.
$$\delta_{L} = \frac{5w_LL^4}{384E_sI_{\text{comp}}} = 40.7\ \text{mm} = \frac{L}{492}\ \le\ \frac{L}{400} = 50\ \text{mm}$$
The dead load acting on the bare steel deflects it 50.1 mm, so specify 50 mm of camber. For footfall vibration the dead load acts on the composite section, giving 17.3 mm and
$$f_1 \approx \frac{17.8}{\sqrt{\delta_D}} = \frac{17.8}{\sqrt{17.3}} = 4.28\ \text{Hz} \ >\ 3\ \text{Hz}$$
which clears the usual pedestrian comfort threshold. A shallower girder would satisfy flexure but fail both of these.
Part (b): shear connectors. For full interaction the horizontal shear to be transferred between a support and midspan is the smaller of the two capacities, $V_h = 6930\ \text{kN}$. Using 19 mm × 100 mm headed studs ($A_{sc} = 283.5\ \text{mm}^2$), CSA S16 Cl. 17.7.2 gives
$$q_r = 0.5\phi_{sc}A_{sc}\sqrt{f'_cE_c} = 0.5(0.80)(283.5)\sqrt{30(24\,648)}/10^3 = 97.5\ \text{kN}$$ $$\le \phi_{sc}A_{sc}F_u = 102.1\ \text{kN}\quad\text{(so } q_r = 97.5\ \text{kN)}$$ $$n = \frac{6930}{97.5} = 71.1 \ \rightarrow\ \boxed{72\ \text{studs per half span, 144 per girder, 288 in total}}$$
Arranged in pairs, 36 rows per half span at 278 mm centres, which sits between the minimum $4d = 76\ \text{mm}$ and the maximum of $\min(8t_s, 600) = 600\ \text{mm}$. Transverse spacing 100 mm on the 280 mm flange gives 90 mm edge distance.

Bridge girder: welded I, flanges 2–280 × 20, web 900 × 12, d = 940 mm.

Result	Value
Girder section (two required)	flanges 2–280 × 20, web 900 × 12, $d = 940$ mm, 172.7 kg/m
Effective flange width	2500 mm
Factored load / moment / shear per girder	71.1 kN/m / 3556 kN·m / 711 kN
Composite $M_r$ (PNA in slab, $a = 167$ mm)	4202 kN·m (0.846)
$V_r$	1625 kN (0.438)
Live-load deflection	40.7 mm = $L/492 < L/400$
Camber for unshored dead load	50 mm
First natural frequency	4.28 Hz (> 3 Hz)
Shear studs	19 mm × 100 mm, $q_r = 97.5$ kN; 144 per girder, 288 total, 36 pairs per half span at 278 mm

Topic. Full-interaction composite beam design, and why a pedestrian bridge is a serviceability problem rather than a strength problem.

Key relations. $b_{\text{eff}} = \min(\text{tributary}, L/4)$; $C_r = 0.85\phi_cf'_cb_{\text{eff}}t_s$ and $T_r = \phi A_sF_y$, the smaller fixing $V_h$; $a = T_r/(0.85\phi_cf'_cb_{\text{eff}})$; $M_r = T_r(d/2 + t_s - a/2)$; stud resistance $q_r = 0.5\phi_{sc}A_{sc}\sqrt{f'_cE_c} \le \phi_{sc}A_{sc}F_u$ (Cl. 17.7.2); $\delta = 5wL^4/384EI$; $f_1 \approx 17.8/\sqrt{\delta_D}$.

Why this works. With the neutral axis in the slab the composite couple has a lever arm of roughly half the steel depth plus the slab thickness, which is why a 940 mm girder develops 4200 kN·m while the bare steel manages only 2390. Full interaction is an assumption about the connectors, not about the concrete: it holds only if enough studs are provided to transfer the whole 6930 kN, which is why the stud count follows directly from the flexural capacity rather than from the shear diagram. Pedestrians are a light load but an unusually rhythmic one, so the 3 Hz frequency floor and the $L/400$ deflection limit bite long before the strength limit does — hence the 0.85 flexural utilisation, which is a design outcome and not slack.

Common pitfall. Taking the tributary width as the 4 m beam spacing and forgetting the 0.5 m overhangs (or, worse, taking half the 5 m deck for each of two beams, which happens to give the same 2.5 m here but fails as soon as the overhangs are unequal). The second pitfall is checking flexure, declaring the girder adequate, and never computing the deflection: a section chosen for $M_f/M_r \approx 0.97$ would deflect about 55 mm and vibrate near 2.5 Hz, both unacceptable.

Source. CSA S16:19 Cl. 17 (composite beams), especially 17.4, 17.7 and 17.9; CSA S6 (CHBDC) pedestrian-bridge deflection and vibration criteria; Salmon, Johnson and Malhas, Steel Structures, Ch. 16.

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Result	Value
Girder section (two required)	flanges 2–280 × 20, web 900 × 12, \(d = 940\) mm, 172.7 kg/m
Effective flange width	2500 mm
Factored load / moment / shear per girder	71.1 kN/m / 3556 kN·m / 711 kN
Composite \(M_r\) (PNA in slab, \(a = 167\) mm)	4202 kN·m (0.846)
\(V_r\)	1625 kN (0.438)
Live-load deflection	40.7 mm = \(L/492 < L/400\)
Camber for unshored dead load	50 mm
First natural frequency	4.28 Hz (> 3 Hz)
Shear studs	19 mm × 100 mm, \(q_r = 97.5\) kN; 144 per girder, 288 total, 36 pairs per half span at 278 mm