22-Agric-B2 Structural Design for Agricultural, Biosystems, and Food Industries · May 2018

Question 6 of 6: Bunker Silo Retaining Wall Design

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, May 2018. 3 hours duration, open book.

Reference texts: CSA O86-09, Engineering Design in Wood · CSA A23.3-19, Design of Concrete Structures · National Building Code of Canada (NBCC), load combinations · CSA A23.1/A23.2, concrete materials and testing · Breyer, Design of Wood Structures · MWPS-1, Structures and Environment Handbook.

Question 6: Bunker Silo Retaining Wall Design (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. Cantilever retaining wall stem 3.0 m tall (footing top to top of wall), silage pressure $L(H)=3.5+3.5H$ kPa acting over the full stem height on the silage side, backfill soil only 1.2 m deep on the grade side (Figure 6). $f'_c=35$ MPa, $f_y=400$ MPa, SLS allowable soil bearing $=150$ kPa.

Given data
Quantity	Value
Stem height, $H$	3.0 m
Silage pressure at top / base	3.5 / 14.0 kPa
Backfill depth over heel (grade side)	1.2 m
Concrete / rebar	$f'_c=35$ MPa, $f_y=400$ MPa
Allowable SLS bearing	150 kPa
Assumed unit weights	$\gamma_c=24$ kN/m$^3$, $\gamma_{soil}=18$ kN/m$^3$ (Check, not given)

Find. Footing thickness (A), heel length (B), wall thickness (C), toe length (D), horizontal wall reinforcing (E) and vertical wall reinforcing (F).

Figure 6 — retaining wall cross-section with dimensions A–D and reinforcing E, F.

Approach. Compute the total factored silage thrust and its line of action, trial a footing geometry, check global stability (bearing, overturning, sliding), then design the stem for cantilever flexure and the footing toe/heel for their own local bending, all via the supplied $K_r$–$\rho$ table (Table 2.1, $f_y=400$ MPa).

Total silage thrust. The pressure diagram is a rectangle ($3.5$ kPa) plus a triangle ($3.5H$ kPa at the base), giving, per metre of wall: $$P_{unf}=\tfrac12(3.5+14.0)(3.0)=26.25\text{ kN/m}, \qquad \bar y_{\text{above footing top}}=1.2\text{ m}$$ Treating the specified lateral pressure as a live/variable load (factor 1.5, check — the source gives no separate D/L split for this pressure): $$\boxed{P_f=1.5(26.25)=39.375\text{ kN/m}}$$
Trial geometry and stability check. Trying footing thickness $A=400$ mm, toe $D=0.8$ m, stem thickness $C=250$ mm, heel $B=1.8$ m (total width $W=2.85$ m), the unfactored vertical load (self-weight of footing, stem, and soil over the heel, using the 1.2 m grade-side backfill depth) is $N=84.2$ kN/m. Summing moments about the toe: $$\bar x = \frac{M_{stab}-P_{unf}\,\bar y_{base}}{N}, \qquad e = \frac{W}{2}-\bar x = 0.363\text{ m} < \frac{W}{6}=0.475\text{ m (middle third)}$$ so the base remains fully in compression, with $$q_{max}=\frac{N}{W}\left(1+\frac{6e}{W}\right)=52.2\text{ kPa} < 150\text{ kPa (SLS)}\quad\checkmark, \qquad q_{min}=7.0\text{ kPa}\ge0\quad\checkmark$$ $$FS_{overturning}=3.13\ge1.5\quad\checkmark, \qquad FS_{sliding}=\frac{0.5N}{P_{unf}}=1.60\ge1.5\quad\checkmark\ (\mu=0.5\text{ assumed, check})$$ All three stability checks pass with this trial geometry, so it is adopted: $$\boxed{A=400\text{ mm} \qquad B=1.8\text{ m} \qquad C=250\text{ mm} \qquad D=0.8\text{ m}}$$
Stem flexural design (vertical reinforcing, F). The full factored thrust acts on the stem alone with its resultant $1.2$ m above the footing top, giving a base moment $$M_f=P_f\,\bar y=39.375(1.2)=47.25\text{ kN}\cdot\text{m/m}$$ With cover 50 mm and 20M bars, $d=250-50-10=190$ mm: $$K_r=\frac{M_f\times10^6}{bd^2}=\frac{47.25\times10^6}{1000(190)^2}=1.31\text{ MPa}$$ Table 2.1 at $f'_c=35$ interpolates to $\rho=0.40\%$, giving $A_{s,req}=0.0040(1000)(190)=765$ mm$^2$/m — above the shrinkage/temperature floor of $0.002(1000)(250)=500$ mm$^2$/m, so flexure governs: $$\boxed{F:\ 15M\ @\ 250\text{ mm o.c.}\ (A_s=800\text{ mm}^2/\text{m} \ge 765\text{ mm}^2/\text{m}),\text{ tension face on the silage side}}$$ A stem shear check at the base ($V_f=P_f=39.4$ kN/m against $V_c=0.21\phi_c\sqrt{f'_c}\,b\,d_v=145.4$ kN/m) confirms shear is not close to governing.
Wall horizontal reinforcing (E). With no horizontal spanning action assumed (the stem cantilevers vertically; horizontal steel is shrinkage/temperature steel per CSA A23.3 Cl. 7.8.1), $A_{s,min}=0.002(1000)(250)=500$ mm$^2$/m total, split between the two faces: $$\boxed{E:\ 15M\ @\ 400\text{ mm o.c., each face}\ (500\text{ mm}^2/\text{m per face}, \text{exceeding the }250\text{ mm}^2/\text{m/face minimum})}$$
Footing toe and heel flexure. Using the bearing-pressure diagram (factored by the same 1.5 on the net soil reaction) less the footing/soil self-weight (factored 1.25), the net design moments at the stem face work out to $M_{f,toe}=19.2$ kN·m/m and $M_{f,heel}=32.8$ kN·m/m — both light enough (with $d=400-75-10=315$ mm at 75 mm cover for concrete cast near soil) that $K_r<0.5$ falls below Table 2.1's range and the minimum reinforcement ratio governs both: $$A_{s,min,footing}=0.002(1000)(400)=800\text{ mm}^2/\text{m} \;\Rightarrow\; 15M\ @\ 250\text{ mm o.c.}$$ top steel in the heel, bottom steel in the toe.

Final results — Question 6
Item	Value
A — Footing thickness	400 mm
B — Footing heel length	1.8 m
C — Wall (stem) thickness	250 mm
D — Footing toe length	0.8 m
E — Wall horizontal reinforcing	15M @ 400 mm o.c., each face
F — Wall vertical reinforcing	15M @ 250 mm o.c., silage-side face
Max SLS bearing pressure, $q_{max}$	52.2 kPa (< 150 kPa allowable)
$FS_{overturning}$ / $FS_{sliding}$	3.13 / 1.60

Topic: Cantilever retaining wall design for bunker-silo silage pressure, CSA A23.3 limit states.

Key relations: Trapezoidal thrust $P=\tfrac12(L_{top}+L_{base})H$; bearing pressure $q_{max,min}=\tfrac{N}{W}(1\pm6e/W)$; flexural design via $M_r=K_r bd^2\times10^{-6}$, $\rho=A_s/bd$ (Table 2.1).

Why this works: A cantilever retaining wall is really two coupled design problems — a rigid-body stability check on the whole footing (bearing/overturning/sliding) and a set of local flexural checks on the stem and footing "arms," each cantilevering off the same monolithic joint but loaded by different, largely independent pressure diagrams.

Common pitfall: Using the factored thrust's moment arm measured from the footing BASE (used for global stability) when computing the STEM's own base moment — the stem only sees the portion of the thrust and arm that lies above the footing top, not the whole wall-plus-footing height.

Source: CSA A23.3-19 (flexural design, Table 2.1 provided in the exam); Das, Principles of Geotechnical Engineering (retaining wall stability); MWPS-1, Structures and Environment Handbook (silo/bunker silo design).

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