16-Civ-B2 Advanced Structural Design · December 2014

Question 3 of 7: Composite bridge cross-section

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

Notes on this paper

Paper format. National Examinations, December 2014 — 98-Civ-B2 Advanced Structural Design, 3 hours, closed book (handbooks and textbooks permitted). Seven design questions of equal value; any five constitute a complete paper.

Reference texts. CSA S16:19 Design of Steel Structures (Clause 13 members, Clause 14 plate girders, Clause 17 composite, Clause 21 plastic design); CISC Handbook of Steel Construction; CSA A23.3:19 Design of Concrete Structures (Clause 10 flexure and columns, Clause 11 shear, Clause 18 prestressed concrete); National Building Code of Canada 2020 for load combinations; Hibbeler, Structural Analysis, for the frame and continuous-beam analysis.

Design data printed on page 1. 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, $f_c' = 50\ \text{MPa}$, modular ratio $n = 6$, $f_{ult} = 1750\ \text{MPa}$, $f_y = 1450\ \text{MPa}$, $f_{initial} = 1200\ \text{MPa}$, losses $= 240\ \text{MPa}$. Design in SI. All loads shown are unfactored.

Check — load factors. Page 1 states only that the loads shown are unfactored; it gives no dead/live split. Throughout this paper the figure loads are treated as a single variable action and factored by 1.5, while self weight introduced by the solver (the reinforced-concrete frame, the prestressed girder, the bridge deck) is factored by 1.25, per NBCC 2020 Case 2. Mechanisms, section classifications and interaction ratios are unaffected by this choice — only the magnitudes are. Serviceability checks (the no-tension prestress design of Q2 and the soil bearing of Q5) use unfactored loads, as they must.

Question 3: Composite bridge cross-section (15 + 5 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
Simply supported span	18.0 m
Deck slab	275 mm thick, 11.7 m wide, $f_c' = 30$ MPa
Box girders	three, 1.3 m wide, at 3.8 m centres
Exterior girder tributary width	$2.05 + 1.90 = 3.95$ m (governs)
Design live load	20 kPa
Construction	unshored; 100 per cent interaction

Find. Plate sizes for one built-up rectangular box girder that satisfy flexure in both the construction and composite stages, and the number of headed shear studs needed to develop full interaction.

Figure 3.1 — Deck cross-section: 275 mm slab on three 1.3 m wide box girders at 3.8 m centres; the exterior girder carries the larger 3.95 m tributary width and therefore governs.

Approach. Take the exterior girder with its 3.95 m tributary width, build the factored moment, size a box girder so that the plastic tensile capacity of the steel is a little below the compressive capacity of the effective slab (which keeps the plastic neutral axis inside the slab and makes the lever arm simple), then check that the bare steel section alone carries the wet concrete during unshored construction, and finally divide the interface force by the resistance of one stud.

Part (a) — assemble the loads on one girder. Slab: $0.275(24)(3.95) = 26.07\ \text{kN/m}$. Steel girder self weight, from the trial section of step 2: $3.25\ \text{kN/m}$. Live load: $20(3.95) = 79.0\ \text{kN/m}$. Combining with 1.25 on dead and 1.5 on live, $$w_f = 1.25(29.32) + 1.5(79.0) = 155.1\ \text{kN/m}, \qquad M_f = \frac{w_fL^2}{8} = \frac{155.1(18)^2}{8} = \boxed{6283\ \text{kN}\cdot\text{m}}$$
Choose trial plates for the box. Try a box 1300 mm wide × 700 mm deep: two webs 10 mm, a top flange 1300 × 10 and a bottom flange 1300 × 12. Then $$A_s = 2(10)(678) + 1300(10) + 1300(12) = 42\,160\ \text{mm}^2, \qquad \bar{y} = 329.4\ \text{mm from the soffit}$$ Ten and twelve millimetre plate is the practical minimum for a bridge box; the section is set by that floor as much as by strength.
Find the effective slab width and the two plastic capacities. The effective width is the lesser of a quarter span and the girder spacing, $$b_{eff} = \min\left(\frac{18\,000}{4},\ 3800\right) = 3800\ \text{mm}$$ $$C_r = 0.85\phi_c f_c' b_{eff} t_s = 0.85(0.65)(30)(3800)(275) = 17\,321\ \text{kN}$$ $$T_r = \phi A_s F_y = 0.9(42\,160)(350) = 13\,280\ \text{kN}$$
Locate the plastic neutral axis. Since $T_r < C_r$ the whole steel section yields in tension and only part of the slab is needed in compression: $$a = \frac{T_r}{0.85\phi_c f_c' b_{eff}} = \frac{13\,280\times10^{3}}{0.85(0.65)(30)(3800)} = 210.9\ \text{mm} \;<\; t_s = 275\ \text{mm}$$ so the plastic neutral axis lies inside the slab, which is the efficient arrangement: every steel fibre is working at yield and no part of the girder is wasted in compression.
Compute the composite moment resistance. The compressive resultant acts at $a/2$ below the deck top, i.e. $700 + 275 - 105.4 = 869.6\ \text{mm}$ above the soffit, giving a lever arm of $869.6 - 329.4 = 540.2\ \text{mm}$: $$M_{rc} = T_r \times 0.5402 = 13\,280(0.5402) = \boxed{7174\ \text{kN}\cdot\text{m}} \;>\; M_f = 6283\ \text{kN}\cdot\text{m}$$ The utilisation is 0.88.
Check the unshored construction stage. Before the deck cures the bare steel box carries the wet concrete and its own weight alone: $$M_c = \frac{1.25(29.32)(18)^2}{8} = 1484\ \text{kN}\cdot\text{m}$$ For the bare section $I_s = 3895\times10^{6}\ \text{mm}^4$ and $S_{top} = 10.51\times10^{6}\ \text{mm}^3$, so $$M_{r,bare} = \phi S_{top} F_y = 0.9(10.51\times10^{6})(350) = 3311\ \text{kN}\cdot\text{m} \;>\; 1484\ \text{kN}\cdot\text{m}$$ A closed box has enormous torsional stiffness, so lateral-torsional buckling never governs and no temporary bracing is needed — which is exactly why a box is chosen for unshored construction.
Part (b) — find the interface force to be transferred. For full interaction the studs between the support and mid-span must carry the smaller of the two plastic capacities, $$V_h = \min(T_r,\ C_r) = 13\,280\ \text{kN}$$
Size one stud. Using 22 mm diameter headed studs, $A_{sc} = 380\ \text{mm}^2$, $E_c = 4500\sqrt{30} = 24\,648\ \text{MPa}$, $F_u = 450\ \text{MPa}$: $$q_r = \min\left(0.5\phi_{sc}A_{sc}\sqrt{f_c'E_c},\ \phi_{sc}A_{sc}F_u\right) = \min(130.7,\ 136.8) = 130.7\ \text{kN}$$ Concrete crushing at the stud shank governs, not shear of the stud steel.
Count and space the studs. $$n = \frac{13\,280}{130.7} = 101.6 \;\rightarrow\; \boxed{102 \text{ studs per half span, } 204 \text{ per girder}}$$ Placed in transverse rows of four (two over each web), that is 26 rows in 9 m, a longitudinal pitch of $$s = \frac{9000}{26} = 346\ \text{mm} \;\rightarrow\; \text{use } 340\ \text{mm}$$ which sits comfortably between the 6d = 132 mm minimum and the 600 mm maximum pitch, and the transverse spacing over the two webs exceeds the 4d = 88 mm minimum.

Result	Value
Governing tributary width (exterior girder)	3.95 m
Factored load / moment	155.1 kN/m / 6283 kN·m
Box girder	1300 × 700 mm; webs 2 × 10; top flange 1300 × 10; bottom flange 1300 × 12
Steel area / centroid	42 160 mm² / 329.4 mm above soffit
Effective slab width	3800 mm
Depth of compression block $a$	210.9 mm (PNA inside slab)
Composite moment resistance	7174 kN·m (0.88 utilised)
Construction-stage check	$M_{r,bare} = 3311 > 1484$ kN·m
Stud resistance (22 mm headed)	130.7 kN each
Studs required	102 per half span (204 per girder), 26 rows of 4 at 340 mm

Topic. Plastic flexural design of an unshored composite steel box girder and the shear connection that makes it composite.

Key relations. $b_{eff} = \min(L/4,\ s)$; slab capacity $C_r = 0.85\phi_cf_c'b_{eff}t_s$; steel capacity $T_r = \phi A_sF_y$; compression depth $a = T_r/(0.85\phi_cf_c'b_{eff})$; $M_{rc} = T_r \times$ lever arm; stud resistance $q_r = \min(0.5\phi_{sc}A_{sc}\sqrt{f_c'E_c},\ \phi_{sc}A_{sc}F_u)$; number of studs $n = V_h/q_r$ per shear span.

Why this works. Full interaction means the interface can transfer whatever force the weaker of the two components can generate, so the section reaches its plastic capacity and the elastic transformed-section machinery is never needed. Keeping the plastic neutral axis inside the slab is a design goal rather than an accident: it puts every steel fibre at yield, makes the lever arm a single subtraction, and avoids the compression-flange local buckling checks that a PNA in the steel would trigger. The number of studs then follows from equilibrium alone — the total interface force between the point of zero moment and the point of maximum moment must equal the flange force developed there — which is why studs are counted per shear span and spaced uniformly for a plastic design, not in proportion to the elastic shear flow.

Common pitfall. Forgetting that unshored construction means the bare steel section alone carries the wet slab. A composite section that is comfortable at the ultimate limit state can still fail while the concrete is being placed, and for a plate-girder deck that stage often governs the top flange. The second trap is designing the interior girder because it looks typical: here the exterior girder collects 3.95 m of deck against the interior girder's 3.80 m.

Source. CSA S16:19 Clause 17 (composite beams, including 17.9.4 for shear connectors); CSA S6 Canadian Highway Bridge Design Code Section 10.11; Kulak and Grondin, Limit States Design in Structural Steel, Ch. 9.

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