16-Civ-A2 Elementary Structural Design · December 2015

Question 6 of 7: B3 — Slender reinforced concrete column

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

Notes on this paper

Paper format. National Examinations, December 2015 — 98-Civ-A2 Elementary Structural Design, 3-hour closed-book paper (textbooks and design handbooks permitted). Seven questions in three parts: Part A (A1–A3, steel), Part B (B1–B3, reinforced concrete), Part C (C1, timber). The candidate answers two from Part A, two from Part B and the one question in Part C — five solutions, all of equal value. All seven are solved here, because the set is a study resource rather than an exam script.

Reference texts.

CSA S16, Design of Steel Structures — Clauses 11 (classification), 13.3 (compression), 13.5–13.6 (flexure), 13.8 (axial + bending), 13.11–13.12 (bolts, block shear).
CISC, Handbook of Steel Construction — section property tables for W-shapes and hollow structural sections.
CSA A23.3, Design of Concrete Structures — Clauses 10 (flexure and axial), 11 (shear), 10.15 (slender columns).
CSA O86, Engineering Design in Wood — Clause 6 (modification factors), Clause 7 (sawn lumber, bending and shear).
Canadian Wood Council, Wood Design Manual — specified strengths for Beam and Stringer grades.

Check — load factoring. Note 6 on page 1 states “all loads shown are unfactored”, so every load taken from a figure is a specified load and is factored here as 1.25D + 1.5L (NBCC Table 4.1.3.2, Case 2), treating the drawn point loads as live and member self-weight as dead. Where a question states a load directly in its text without a dead/live split (B3), that load is taken as already factored; this is stated again at that question.

Question 6: B3 — Slender reinforced concrete column (10 + 5 + 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. Square column, unsupported length $l_u = 7$ m, fixed at both ends; $P_f = 1000$ kN at $e = 200$ mm, giving a first-order moment $M_f = 200\ \text{kN}\cdot\text{m}$; $f_c' = 35$ MPa, $f_y = 400$ MPa.

Find. A square section and its longitudinal and tie reinforcement.

Column elevation and the adopted section. The load is eccentric along the horizontal axis of symmetry, so bending is uniaxial.

Check — load status. The question states the 1000 kN load in the text with no dead/live split, so it is taken as already factored. If instead it were a specified load of, say, 40 % dead and 60 % live, the factored value would rise to 1400 kN and a 500 mm square section would be needed. The column is also assumed to be part of a braced (non-sway) storey, which is what “fixed at both ends” implies.

Approach. Test the slenderness against the A23.3 non-sway threshold, magnify the moment if the column is slender, then verify the trial section against its own $P$–$M$ interaction diagram by strain compatibility.

Trial section and slenderness. Try 400 × 400 mm. For a rectangular section $r = 0.3h = 120$ mm, and for a braced member fixed at both ends the design effective-length factor is $k = 0.65$, so $$\frac{kl_u}{r}=\frac{0.65(7000)}{120}=37.9$$ The load is applied at the same eccentricity at both ends, so the column is in single curvature, $M_1/M_2 = +1$ and the A23.3 Clause 10.15.2 threshold is $$34-12\left(\frac{M_1}{M_2}\right)=34-12=22\ <\ 37.9$$ The column is therefore slender and the moment magnifier is mandatory.
Flexural rigidity and critical load. With $E_c=4500\sqrt{f_c'}=26\,622$ MPa and $I_g=h^{4}/12=2.133\times10^{9}\ \text{mm}^{4}$, taking $\beta_d = 0$, $$EI=\frac{0.4E_cI_g}{1+\beta_d}=2.272\times10^{13}\ \text{N}\cdot\text{mm}^{2},\qquad P_c=\frac{\pi^{2}EI}{(kl_u)^{2}}=10\,830\ \text{kN}$$
Magnified moment. With $C_m = 0.6+0.4(M_1/M_2) = 1.0$ for single curvature and $\phi_m = 0.75$, $$\delta_b=\frac{C_m}{1-\dfrac{P_f}{\phi_mP_c}}=\frac{1.0}{1-\dfrac{1000}{0.75(10\,830)}}=1.140$$ $$M_c=\delta_bM_f=1.140(200)=\boxed{228.1\ \text{kN}\cdot\text{m}}$$ The second-order effect adds 14 % to the moment — modest, but not negligible, and it is the difference between a passing and a failing check.
Trial reinforcement. Provide 4–30M, one bar in each corner: $$A_{st}=4(700)=2800\ \text{mm}^{2},\qquad \rho=\frac{2800}{160\,000}=1.75\ \%$$ which sits between the A23.3 Clause 10.9.1 limits of 1 % and 8 %. With 40 mm cover, 10M ties and 30M bars, the bar centroid is 66.2 mm from each face.
Interaction check by strain compatibility. Because the effective eccentricity $M_c/P_f = 228$ mm exceeds half the section depth, the column is flexure-dominated and must be checked on the interaction diagram — the pure-axial formula $P_{r,\max}$ is meaningless here. Searching for the neutral-axis depth that produces the required eccentricity gives $$c=185\ \text{mm}\;\Longrightarrow\;P_r=1158\ \text{kN},\qquad M_r=264.1\ \text{kN}\cdot\text{m}$$ Both exceed the demands, at a utilisation of $$\frac{P_f}{P_r}=\frac{1000}{1158}=\boxed{0.86\ \checkmark}$$ For scale, $P_{r,\max}$ for this section is 4381 kN — more than four times the resistance actually available at this eccentricity, which is exactly why the interaction check cannot be skipped.
Ties. Tie spacing must not exceed the least of 16 longitudinal bar diameters (478 mm), 48 tie diameters (542 mm) and the least column dimension (400 mm). Adopt 10M ties at 400 mm, with the first tie 200 mm from the floor and closer spacing through the lap zones.
Final design. A 400 × 400 mm column with 4–30M vertical bars and 10M ties at 400 mm, cast in 35 MPa concrete with 400 MPa steel, satisfies strength, slenderness and detailing with a 14 % reserve.

Item	Value
Section adopted	400 × 400 mm
Slenderness $kl_u/r$ vs threshold	37.9 vs 22 — slender
Critical load $P_c$ / magnifier $\delta_b$	10 830 kN / 1.140
Magnified moment $M_c$	228.1 kN·m
Longitudinal steel	4–30M, $\rho = 1.75$ %
Resistances at $e = 228$ mm	$P_r = 1158$ kN, $M_r = 264.1$ kN·m
Utilisation	0.86
Ties	10M @ 400 mm

Topic. Slender reinforced concrete columns under combined axial load and uniaxial bending.

Key relations. Slenderness threshold $kl_u/r < 34-12(M_1/M_2)$ for a braced column; $P_c=\pi^{2}EI/(kl_u)^{2}$ with $EI=0.4E_cI_g/(1+\beta_d)$; magnifier $\delta_b=C_m/(1-P_f/\phi_mP_c)$; and the $P$–$M$ interaction obtained by strain compatibility at $\varepsilon_{cu} = 0.0035$.

Why this works. An eccentrically loaded column deflects laterally, and the axial load acting through that deflection generates additional moment which in turn increases the deflection. A23.3 captures this with the moment magnifier, a closed-form amplification derived from the elastic buckling load, so a second-order analysis is not required. The check itself must then be made on the interaction diagram rather than against a single axial capacity, because a column's resistance falls steeply as eccentricity grows: here 200 mm of eccentricity reduces the usable axial resistance from 4381 kN to 1158 kN. Working at the actual eccentricity by strain compatibility gives the true point on the curve.

Common pitfall. Designing with $P_{r,\max}=0.80[\alpha_1\phi_cf_c'(A_g-A_{st})+\phi_sf_yA_{st}]$, which applies only to columns with negligible eccentricity; at $e = 200$ mm it overstates the resistance nearly fourfold. The second pitfall is skipping the slenderness check because 7 m “does not seem slender”; the threshold for single curvature is only 22, and this column is well past it.

Source. CSA A23.3, Design of Concrete Structures, Clauses 10.9 (reinforcement limits), 10.10 (ties), 10.15.2–10.15.3 (slenderness and moment magnification).

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