Question 7 of 7: Check of a sawn timber floor beam to CSA O86
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. 07-Str-A2 Elementary Structural Design, National
Examinations May 2013, three-hour duration. The paper is printed on three pages: page 1 carries
the notes and the marking scheme, page 2 the seven questions, page 3 the five hand-drawn figures.
Part A (steel, questions A1–A3), Part B (reinforced concrete, questions B1–B3) and
Part C (timber, question C1) are answered by doing two of three from Part A, two of
three from Part B and the one question in Part C — five solutions in all, all
questions of equal value. Because this set is a study resource, all seven questions are
solved below. Note 3 of the paper fixes the design standards: steel to CSA S16, concrete
to CSA A23.3, timber to CSA O86 (latest editions). Note 6 states that all loads shown are
unfactored.
Reference texts. CSA S16, Design of Steel Structures, with the
CISC Handbook of Steel Construction (section tables, Class H HSS, fillet-weld tables);
CSA A23.3, Design of Concrete Structures; CSA O86, Engineering Design in Wood,
with the CWC Wood Design Manual; National Building Code of Canada (load combinations);
MacGregor and Bartlett, Reinforced Concrete: Mechanics and Design (Canadian edition);
Salmon, Johnson and Malhas, Steel Structures: Design and Behavior.
Check — load factors used throughout. The
paper states that all loads shown are unfactored but nowhere splits them into dead and live.
Every applied load in Figures A2, B1, B2 and B3 is therefore taken as a specified live load
and factored by 1.5, and self-weight (where the question asks for it)
by 1.25, per NBCC load combination case 2, 1.25D +
1.5L. If a grader intends a different split the factored actions scale linearly and every
design step below is unchanged. Self-weight of the steel members in A2 and A3 is neglected; for the
W610x241 of A2 it would add about 4 per cent to the moment at B.
Question C1: Check of a sawn timber floor beam to CSA O86 (10 + 6 + 4 marks)
Given. Sawn timber 241 mm x 343 mm, Douglas Fir-Larch, No. 1 grade, in
the Beam and Stringer size category (the depth exceeds the width by more than 51 mm). Simple span
5.0 m, beams at 2.5 m centres, specified dead 2.0 kPa (self-weight included) and live 2.5 kPa; dry
service, standard-term loading, untreated. Specified strengths from CSA O86 Table 5.3.1C:
$f_b = 15.8$ MPa, $f_v = 1.5$ MPa, $E = 12\,000$ MPa. Resistance factor $\phi = 0.9$.
Property
Value
Section area $A = bd$
$241 \times 343 = 82\,663$ mm2
Section modulus $S = bd^{2}/6$
$4.726\times10^{6}$ mm3
Second moment $I = bd^{3}/12$
$810.4\times10^{6}$ mm4
Tributary width
2.5 m
Specified $w_D$ / $w_L$
5.0 / 6.25 kN/m
Factored $w_f = 1.25w_D+1.5w_L$
15.625 kN/m
Find. Whether the member satisfies bending, shear and both deflection
limits.
Floor framing for C1: beams at 2.5 m centres spanning 5.0 m simply supported, so each beam carries a 2.5 m wide strip of floor.
Approach. Convert the area loads to a line load on one beam, compute the
factored moment and shear, compare them with $M_r$ and $V_r$ from CSA O86 Cl 5.5, and then check
the two serviceability deflection limits under specified loads.
Part (i) — line loads and design actions. Each beam carries a 2.5 m
strip:
$$w_D = 2.0(2.5)=5.0\ \text{kN/m},\qquad w_L = 2.5(2.5)=6.25\ \text{kN/m}$$
$$w_f = 1.25(5.0)+1.5(6.25)=15.625\ \text{kN/m}$$
$$M_f=\frac{w_f L^2}{8}=\frac{15.625(5.0)^2}{8}=48.8\ \text{kN}\cdot\text{m},\qquad
V_f=\frac{w_f L}{2}=39.1\ \text{kN}$$
Bending resistance. All modification factors are unity for dry service,
standard-term load, no treatment and no load sharing, so $F_b = f_b = 15.8$ MPa. For a Beam and
Stringer the size factor is $K_{Zb}=(305/d)^{1/9}=(305/343)^{1/9}=0.987$. Because
$d/b = 343/241 = 1.42$, which is less than 4, CSA O86 Cl 5.5.4.2 requires no lateral support and
$K_L = 1.0$. Hence
$$M_r=\phi F_b S K_{Zb} K_L = 0.9(15.8)(4.726\times10^{6})(0.987)(1.0)
=\boxed{66.3\ \text{kN}\cdot\text{m}}$$
$$M_r = 66.3 > M_f = 48.8\ \text{kN}\cdot\text{m}\quad\checkmark\ \text{(utilisation 0.74)}$$
Part (ii) — shear resistance. For sawn timber the factored shear
resistance is based on two thirds of the gross area:
$$V_r=\phi F_v\left(\frac{2A}{3}\right)K_{Zv}
=0.9(1.5)\left(\frac{2(82\,663)}{3}\right)(1.0)=\boxed{74.4\ \text{kN}}$$
$$V_r = 74.4 > V_f = 39.1\ \text{kN}\quad\checkmark\ \text{(utilisation 0.53)}$$
Part (iii) — deflections. Deflection is checked under specified, not
factored, loads with $EI = 12\,000(810.4\times10^{6})=9.725\times10^{12}$ N·mm2.
For total load $w = 11.25$ kN/m and for live load alone $w = 6.25$ kN/m,
$$\Delta=\frac{5wL^4}{384EI}:\qquad \Delta_{total}=9.4\ \text{mm},\qquad \Delta_{live}=5.2\ \text{mm}$$
against limits of $L/180 = 27.8$ mm and $L/360 = 13.9$ mm. Both are satisfied with a wide margin,
the live-load case being the tighter at 38 per cent of its limit.
Verdict. The 241 x 343 mm No.1 D.Fir-L beam is satisfactory on all
four counts. Bending governs at 74 per cent utilisation; shear and both deflection limits are
comfortable. Bearing at the supports has not been checked because no bearing length is given, and
it should be confirmed against $f_{cp} = 7.0$ MPa once the support detail is fixed.
Check — specified strengths. The values
$f_b = 15.8$ MPa, $f_v = 1.5$ MPa and $E = 12\,000$ MPa are those tabulated for D.Fir-L, No. 1
grade, Beam and Stringer sizes. Confirm them against the edition of CSA O86 current at the time of
the examination before quoting the numbers; the arithmetic and the sequence of checks are
unaffected, and the bending utilisation scales inversely with $f_b$ (the member would still pass
for any $f_b$ above 11.7 MPa).
