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16-Civ-B2 Advanced Structural Design · 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. 98-Civ-B2 Advanced Structural Design, National Exams May 2015 — 3 hours, closed book (textbooks and design handbooks permitted, no notes). Seven design questions of 20 marks each; any five constitute a complete paper, so all seven are solved here. All loads shown on the figures are unfactored. Page 1 supplies the design data used throughout: concrete f′c = 30 MPa, structural steel Fy = 350 MPa, reinforcing steel fy = 400 MPa; for the prestressed member f′ci = 35 MPa, f′c = 50 MPa, n = 6, fpu = 1750 MPa, fpy = 1450 MPa, initial stress 1200 MPa and total losses 240 MPa.

Reference texts. CSA S16:19 Design of Steel Structures; CISC Handbook of Steel Construction, 12th ed.; CSA A23.3:19 Design of Concrete Structures; CSA S6:19 Canadian Highway Bridge Design Code; NBCC 2020 Part 4; Hibbeler, Structural Analysis, 10th ed. (plastic analysis, Ch. 18).

Check — load factors. Page 1 states only that “all loads shown are unfactored” and gives no dead/live split. Throughout this paper the applied loads on the figures are factored by 1.5, and any self weight introduced by the solution itself (the plate girder, the deck, the prestressed beam, the reinforced-concrete frame) by 1.25, consistent with NBCC 2020 Table 4.1.3.2. The mechanisms, section classifications and interaction equations are independent of that choice — only the magnitudes change.

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. A simply supported deck 20 m long and 5 m wide on two steel beams at 4 m centres (0.5 m overhang each side), acting compositely with a 220 mm reinforced-concrete slab.

QuantityValue
Span / deck width / beam spacing20 m / 5 m / 4 m
Slab thickness220 mm, f′c = 30 MPa
Live load14 kPa over the full deck
Superimposed dead load (surfacing, railings)1.0 kPa assumed
Structural steelFy = 350 MPa
Shear connectors19 mm dia. headed studs, Fu = 450 MPa

Find. The steel section for each beam, verified for strength, deflection and pedestrian comfort, and the number and spacing of shear connectors for full interaction.

220 mm r.c. slab — effective width 2500 mmW840×1762 – 19 mm studs @ 250 mm220beams at 4 m centres, deck 5 m wideplastic neutral axis 168 mm into the slab, full interaction
Composite cross-section — W840×176 acting with a 2500 mm effective width of the 220 mm slab through 19 mm headed studs.

Approach. Build the load per beam on its 2.5 m tributary width, factor it, compute the plastic composite moment resistance with the neutral axis in the slab, check the unshored construction stage, then transform the section for deflection and natural frequency, and finally proportion the studs from the total horizontal shear.

  1. Part (a) — loads per beam. Each beam takes a 2.5 m tributary width (half of the 4 m bay plus the 0.5 m overhang): $$w_{slab} = 0.220(24)(2.5) = 13.20,\quad w_{sdl} = 1.0(2.5) = 2.50,\quad w_{steel} = 1.73\ \text{kN/m}$$ $$w_D = 17.43\ \text{kN/m},\qquad w_L = 14(2.5) = 35.0\ \text{kN/m}$$ $$w_f = 1.25(17.43) + 1.5(35.0) = 74.29\ \text{kN/m}$$ $$M_f = \frac{w_fL^2}{8} = \frac{74.29(20)^2}{8} = 3714\ \text{kN}\cdot\text{m},\qquad V_f = 743\ \text{kN}$$
  2. Effective slab width. CSA S16 Cl 17.4 limits the effective width to the lesser of a quarter of the span and half the distance to each adjacent beam; here $$b_{eff} = \min\left(\frac{20\,000}{4},\ \frac{4000}{2} + 500\right) = 2500\ \text{mm}$$
  3. Trial beam and plastic composite resistance. Try W840×176 (835 × 292, flange 18.8, web 14.0), which from its nominal plate dimensions gives $A = 22\,143\ \text{mm}^2$. Assuming full interaction, the whole steel section yields in tension: $$T = \phi A_sF_y = 0.9(22\,143)(350)/10^3 = 6975\ \text{kN}$$ $$a = \frac{T}{0.85\phi_cf'_cb_{eff}} = \frac{6975\times10^3}{0.85(0.65)(30)(2500)} = 168\ \text{mm} < 220\ \text{mm}$$ so the plastic neutral axis lies inside the slab and the whole steel section is in tension: $$M_r = T\left(\frac{d}{2} + t_s - \frac{a}{2}\right) = 6975(417.5 + 220 - 84.2)/10^3 = \boxed{3860\ \text{kN}\cdot\text{m}} \ge 3714$$ a utilisation of 0.96.
  4. Construction stage. Unshored, the bare steel beam carries the wet concrete and its own weight: $$M_f = 1.25(13.20+1.73)\frac{20^2}{8} = 933\ \text{kN}\cdot\text{m}$$ against $M_r = \phi Z_xF_y = 0.9(350)(6.706\times10^6)/10^6 = 2112$ kN·m with the formwork providing lateral restraint, so the erection condition is not critical.
  5. Web shear. With $h/w = 57.0$ just above $439\sqrt{k_v/F_y} = 54.2$, $$F_s = \frac{290\sqrt{k_vF_y}}{h/w} = 220\ \text{MPa},\qquad V_r = \phi d w F_s = 0.9(835)(14)(220)/10^3 = 2316\ \text{kN} \gg 743\ \text{kN}$$
  6. Transformed section and live-load deflection. With $E_c = 4500\sqrt{30} = 24\,648$ MPa the modular ratio is $n = 8.11$, so the slab transforms to $2500/8.11 = 308$ mm wide. Taking first moments about the soffit gives $\bar{y} = 815$ mm and $$I_{tr} = 7.338\times10^9\ \text{mm}^4,\qquad \delta_L = \frac{5w_LL^4}{384EI_{tr}} = 49.7\ \text{mm} = \frac{L}{403}$$ which satisfies the usual L/360 serviceability limit.
  7. Pedestrian comfort. A footbridge is governed as much by vibration as by deflection. With the dead-load mass $m = 17\,430/9.81 = 1777$ kg/m, $$f_1 = \frac{\pi}{2L^2}\sqrt{\frac{EI_{tr}}{m}} = \frac{\pi}{2(20)^2} \sqrt{\frac{1.468\times10^9}{1777}} = \boxed{3.57\ \text{Hz} > 3\ \text{Hz}}$$ so the deck is clear of the 1.6–2.4 Hz pacing range and its first harmonic, and satisfies the CSA S6 frequency criterion without needing the more onerous L/800 deflection limit.
  8. Part (b) — total horizontal shear. For full interaction the connectors between the support and mid-span must transfer the smaller of the slab and beam capacities: $$V_h = \min\left(0.85f'_cb_{eff}t_s,\ A_sF_y\right) = \min(14\,025,\ 7750) = 7750\ \text{kN}$$
  9. Resistance of one stud. For a 19 mm headed stud ($A_{sc} = 283.5\ \text{mm}^2$, $\phi_{sc} = 0.80$), $$q_r = 0.5\phi_{sc}A_{sc}\sqrt{f'_cE_c} \le \phi_{sc}A_{sc}F_u = \min(97.5,\ 102.1) = 97.5\ \text{kN}$$
  10. Number and layout. $$N = \frac{V_h}{q_r} = \frac{7750}{97.5} = 79.5 \rightarrow 80\ \text{studs per shear span}$$ Placed in pairs, that is 40 rows of 2 – 19 mm × 100 mm studs in each half span, uniformly spaced at $10\,000/40 = 250$ mm. The spacing satisfies the CSA S16 limits (minimum $6d = 114$ mm along the beam, $4d = 76$ mm across it; maximum $8t_s = 1760$ mm), and the 100 mm height clears the $4d = 76$ mm minimum while leaving 120 mm of cover to the slab top. In total 160 studs per beam, 320 for the bridge.
ResultValue
Factored moment / shear per beam3714 kN·m / 743 kN
Steel section (each of two beams)W840×176
Effective slab width2500 mm
Composite moment resistance3860 kN·m (ratio 0.96)
Live-load deflection49.7 mm = L/403
Fundamental frequency3.57 Hz > 3 Hz
Total horizontal shear per shear span7750 kN
Shear connectors2 – 19 mm × 100 mm studs @ 250 mm, 160 per beam