22-Agric-B2 Structural Design for Agricultural, Biosystems, and Food Industries · December 2017
Question 3 of 6: Timber Lintel B1 and Post P1
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — 04-Agric-B2, Structural Design of Agricultural, Biosystems and Food Industries — December 2017. 3-hour duration, open-book exam. Question 1 is mandatory; the exam asks for 4 of Questions 2–6 — all five are answered below as a complete study resource.
Reference texts: CSA O86-09, Engineering Design in Wood (attached Tables 6.3.1A/6.3.1D); CSA A23.3-04/14, Design of Concrete Structures (attached reinforcement-ratio Table 2.1); National Building Code of Canada (NBCC) Part 4, structural loads and load combinations; CSA A23.1/A23.2, Concrete Materials and Methods of Concrete Construction; Breyer et al., Design of Wood Structures — ASD/LRFD (shearwall/diaphragm design); MWPS-1, Structures and Environment Handbook (agricultural building loads and details).
Question 3: Timber Lintel B1 and Post P1 (20 marks)
8.4 m (2.4 m overhang + 3.6 m span + 2.4 m overhang)
Point loads
8 × 30 kN at 1.2 m centres (already-factored design reactions from the roof truss, see the check note)
Member
38×286 mm (2×12) SPF No.1, built up in $n$ plies
SPF No.1/No.2, Table 6.3.1A
$f_b=11.8$ MPa, $f_v=1.5$ MPa
Strength modification factors
all $K=1.0$ (given); $\phi_b=0.9$, $\phi_v=0.9$ (CSA O86)
Find. The number of 38×286 plies $n$ required for lintel B1.
Figure 3: beam B1 elevation. Two of the eight 30 kN loads land directly on posts P1 (grey) and bypass the beam; the beam itself carries the remaining six (red).
Approach. Build the shear and bending-moment diagram for the two-support, double-overhang beam under the six beam-applied loads, size the built-up section for the governing (largest-magnitude) moment, then check shear.
Reactions. By symmetry each post carries half of the beam's own six 30 kN loads:
$$R_{P1}=\frac{6(30)}{2}=\boxed{90.0\ \text{kN each post}}$$
Moment diagram. Because the two heavily loaded 2.4 m overhangs (two 30 kN loads each) dominate, the beam is in hogging (top fibre tension) over its entire length — the main span never reaches a positive (sagging) moment. The peak occurs directly over each post:
$$M_f = -30(2.4)-30(1.2) = -108.0\ \text{kN}\cdot\text{m at each post}$$
(mid-span moment is a lesser $-72.0$ kN·m by comparison, so the post section governs). Peak shear, approaching each post from the overhang side, is $V_f=30+30=60.0$ kN.
Size the built-up section for bending. A single 38×286 ply has $S=bh^2/6=38(286)^2/6=518{,}041\ \text{mm}^3$, and with $n$ plies acting together, $S_n=nS$:
$$M_r=\phi_b f_b S_n = 0.9(11.8)(n\times518{,}041)\times10^{-6}=5.50\,n\ \text{kN}\cdot\text{m}$$
Requiring $M_r\ge108.0$: $n\ge108.0/5.50=19.6\Rightarrow\boxed{n=20\ \text{plies}}$.
Check shear for the selected 20-ply section, $A_{20}=20(38)(286)=217{,}360\ \text{mm}^2$:
$$V_r=\phi_v f_v\left(\tfrac{2}{3}\right)A_{20}=0.9(1.5)\left(\tfrac{2}{3}\right)(217{,}360)\times10^{-3}=195.6\ \text{kN} \gg 60.0\ \text{kN} \ \checkmark$$
Shear does not govern; bending controls the design.
Check: the 30 kN point loads are taken as already-factored (ULS) design reactions from the truss (no separate D/L split is given for them, unlike Question 2's joint loads), consistent with the question describing them simply as "applied truss loads." A 20-ply built-up 38×286 section (760 mm overall width) is unusually large for a site-built multi-ply header — in practice a glulam or LVL header would likely be substituted — but this is the section size the given SPF No.1 material and stated loads require.
Find. Whether post P1 satisfies the CSA O86 combined axial-plus-bending interaction check.
Approach. Compute the factored compressive resistance $P_r$ with the two given modification factors, then apply the combined-loading interaction $P_f/P_r+M_f/M_r\le1.0$ using the stated 32% bending ratio.
Axial demand. Post P1 receives half of beam B1's reaction plus the one 30 kN truss load that lands directly on it (Figure 3):
$$P_f=90.0+30.0=\boxed{120.0\ \text{kN}}$$
Check: post species/grade is not explicitly stated for P1 in the source; SPF No.1 is assumed for consistency with the SPF No.1 lintel material specified in the same question. Even the bare axial ratio ($P_f/P_r=1.10$) already exceeds 1.0 before any bending is added — post P1, as sized, is not structurally adequate for the given loads; upsizing the post (e.g. to 184×184 mm or a stronger species/grade such as D Fir-L No.1) or adding an intermediate post to halve the tributary beam reaction would be required.
