24-Bld-A2 Elementary Structural Design · December 2017
Question 6 of 7: Capacity of a triple-T reinforced concrete beam
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Examinations December 2017 — 07-Bld-A2 Elementary Structural Design, 3 hours, closed book (handbooks/textbooks permitted). Answer five: two of Questions A1–A3, two of B1–B3, and the one question C1 (this solution set, per pipeline convention, answers all seven). All loads shown in the exam are unfactored.
Reference texts: CSA S16:19, Design of Steel Structures; Salmon & Johnson, Steel Structures: Design and Behavior; CSA A23.3:19, Design of Concrete Structures; MacGregor & Bartlett, Reinforced Concrete: Mechanics and Design; CSA O86:19, Engineering Design in Wood; Canadian Wood Council, Wood Design Manual.
Check – assumptions applied throughout this solution set (the exam gives unfactored loads without separating dead/live, per Note 6, and instructs "assume any other data required"):
All specified (unfactored) loads are factored by a single combined load factor of 1.5 for ULS design, consistent with treating an undifferentiated load as governed by the live-load-dominant NBCC combination.
Material grades: structural steel plate/tie/beam – G40.21 350W (Fy=350 MPa); concrete f’c=35 MPa, reinforcement fy=400 MPa (as given for B1–B3); glulam – Douglas Fir-Larch 20f-E (fb=25.6 MPa, fc=30.2 MPa, E=12400 MPa, E05=9500 MPa, per CSA O86/Wood Design Manual).
Concrete cover/bar placement (exact stirrup and layer detail not dimensioned on the exam figures) is assumed at a standard 40–50 mm clear cover, giving effective depths stated with each question.
Question B3: Capacity of a triple-T reinforced concrete beam (12 + 8 marks)
Given. Precast triple-T section: overall flange width 2200 mm, flange thickness 150 mm, three 200 mm webs, overall depth 800 mm; top steel 8–20M, bottom steel 6–30M (2 per web); stirrups 20M @ 200 mm, typical each web; f’c=35 MPa, fy=400 MPa. Effective depth to the bottom bars, allowing for cover and the 20M stirrup, d≈726 mm.
Find. Mr and Vr of the section as fabricated.
Figure B3: triple-T RC cross-section, 2200 mm flange, three 200 mm webs, 800 mm overall depth.
Approach. This is section analysis, not design: check whether the compression block stays inside the 150 mm flange (then the full 2200 mm flange width governs, T-beam reduces to a wide rectangular calculation), then combine the three webs’ width and stirrup legs for a single global shear check.
Moment resistance. Bottom steel $$A_s=6(700)=4200\text{ mm}^2$$ (the top 8–20M sit in the flange near the compression face and are conservatively not counted as compression steel). Stress-block depth: $$a=\frac{A_sf_y}{\alpha_1f_c'b_E}=\frac{4200(400)}{0.7975(35)(2200)}=27.4\text{ mm} < h_f=150\text{ mm}\ \checkmark$$ (rectangular T-behaviour, full flange width applies) $$M_r=\phi_sA_sf_y\left(d-\frac{a}{2}\right)=0.85(4200)(400)\left(726-\frac{27.4}{2}\right)=\boxed{1017\text{ kN}\cdot\text{m}}$$
Shear resistance. Combined web width $$b_w=3(200)=600\text{ mm}$$, $$d_v=\max(0.9d,0.72h)=653\text{ mm}$$: $$V_c=\beta\lambda\phi_c\sqrt{f_c'}\,b_wd_v=\boxed{271\text{ kN}}$$ With 20M stirrups (Av=300 mm² per leg) × 2 legs × 3 webs = 1800 mm² total, at s=200 mm, θ=35°: $$V_s=\frac{\phi_sA_vf_yd_v\cot\theta}{s}=\boxed{2854\text{ kN}}$$ This nominal sum (3125 kN) exceeds the concrete-crushing ceiling: $$V_{r,max}=0.25\phi_cf_c'b_wd_v=\boxed{2228\text{ kN}}$$ so the diagonal-crushing limit governs: $$V_r=\boxed{2228\text{ kN}}$$ – the stirrup layout supplies more shear steel than the concrete strut can ever mobilize.
