Question 5 of 5: Wood in Compression and Steel Test Methods

Question 5: Wood in Compression and Steel Test Methods (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.

Part (a) — Stress–strain response of the wood specimen (11 marks)

Given. A clear wood prism of actual cross-section 0.75 in by 0.75 in and length 3.5 in, grain parallel to the length, loaded in compression parallel to the grain to failure, with the load and displacement record below.

Find. The stress–strain curve for the specimen, the modulus of elasticity from the straight portion of that curve, and the failure stress.

Approach. Convert each load to a stress by dividing by the 0.5625 in² cross-section and each displacement to a strain by dividing by the 3.5 in gauge length, plot the result, fit a straight line to the linear run, correct for the seating (toe) region by extending that line back to zero stress, and take the peak stress as the failure stress.

1. Establish the cross-sectional area and the gauge length. The specimen is a square prism loaded on its end, $$A=(0.75)(0.75)=0.5625\ \text{in}^{2},\qquad L_0=3.5\ \text{in}$$ The full 3.5 in length is the gauge length because the displacement was recorded across the whole specimen rather than over an attached extensometer.
2. Convert the record to stress and strain. Applying $$\sigma=\frac{P}{A},\qquad \varepsilon=\frac{\delta}{L_0}$$ to every line of the record gives the following pairs.

   Computed stress–strain pairs

   Load, lb	Displacement, in	Stress, psi	Strain, in/in
   0	0	0	0
   7	0.012	12	0.00343
   10	0.068	18	0.01943
   90	0.164	160	0.04686
   530	0.180	942	0.05143
   1705	0.208	3031	0.05943
   2800	0.236	4978	0.06743
   3600	0.268	6400	0.07657
   4250	0.300	7556	0.08571
   4870	0.324	8658	0.09257
   5050	0.360	8978	0.10286
   4400	0.384	7822	0.10971
   4275	0.413	7600	0.11800

3. Plot the curve and identify the three regions. The plotted record separates cleanly into a long, almost flat toe from zero to about 0.047 strain, in which the machine takes up the clearances and the loading platens bed into the end grain while the load barely rises; a straight run from about 0.051 to 0.067 strain; and a curved, work-softening region that peaks and then falls away as the fibres buckle.
   Figure 5.1 — The measured compression record for the wood prism. The dashed line is the straight-line fit through the linear run, extended back to zero stress at a strain of 0.0476 to correct for the seating (toe) region.
4. Fit the straight portion and obtain the modulus of elasticity. The three points at 530, 1705 and 2800 lb lie on a straight line to within 0.04 per cent, so the modulus is their slope: $$E=\frac{\Delta\sigma}{\Delta\varepsilon}=\frac{4978-942}{0.06743-0.05143}=\frac{4036}{0.01600}$$ $$\boxed{E\approx 2.52\times 10^{5}\ \text{psi}\ \left(252\ \text{ksi}\right)}$$
5. Apply the toe correction and state what the modulus means. Extending that line back to zero stress gives the corrected origin, $$\varepsilon_0=\varepsilon-\frac{\sigma}{E}=0.05143-\frac{942}{252\,222}=0.0476$$ so 0.0476 of the recorded strain, more than 40 per cent of the total, is seating rather than deformation of the wood. Strains measured from the corrected origin are the ones a designer would use.
6. Determine the failure stress. The maximum load carried is 5050 lb, after which the record falls away as the specimen crushes: $$\sigma_{f}=\frac{P_{max}}{A}=\frac{5050}{0.5625}$$ $$\boxed{\sigma_{f}=8978\approx 8980\ \text{psi}}$$ The post-peak points at 4400 and 4275 lb confirm that the peak has genuinely been passed and that 5050 lb is the ultimate load, not merely the last reading before the test was stopped. A crushing strength near 9000 psi parallel to the grain is at the top of the range for clear, dense, dry softwood and is entirely credible for a small clear specimen tested at low moisture content.
7. Comment on the reliability of the modulus. Handbook values of the modulus of elasticity for clear softwood parallel to the grain lie between 1.0 and 1.9 million psi, five to eight times the 252 000 psi computed here, and the discrepancy is not an arithmetic error — it is a consequence of how the displacement was measured. The record is crosshead travel over the whole 3.5 in specimen, so it contains the elastic deformation of the load train, the closing of the platen clearances and the local crushing of the end grain, all of which are far larger than the elastic shortening of the wood itself. The value obtained should therefore be reported as an apparent modulus for this test arrangement. A true modulus requires a compressometer or strain gauges attached to the specimen over a gauge length clear of the ends, which is exactly why ASTM D143 specifies a 2 in gauge deformeter for the small clear compression test. The failure stress, by contrast, depends only on the load and the area and is unaffected by the displacement measurement, so 8980 psi is a reliable result.

Part (b) — Significance and use of the three steel tests (9 marks)

i. Torsion test (ASTM E143 and A938). A cylindrical specimen or a length of wire is twisted about its axis while torque and angle of twist are recorded, and the shear stress at the surface is obtained from τ = Tr/J. Its significance is that it measures the response of the steel in pure shear, which no axial test can supply: the shear modulus G, the shear yield strength and the shear ultimate strength come directly from it, and because a torsion specimen deforms at essentially constant volume with no necking, the test remains valid to very large plastic strains where a tension test has already become unrepresentative. It is used to obtain G for the design of shafts, torsion members, helical springs and closed thin-walled sections; to qualify reinforcing wire and prestressing strand by the number of complete twists a specimen survives, which is a practical ductility screen for cold-drawn product; and, through the shape of the fracture surface, to distinguish ductile shear failure from brittle tensile failure on a 45-degree helix.

ii. Tension test (ASTM E8/E8M for the method, ASTM A370 and CSA G40.20 for structural product). A machined or full-section specimen is pulled in uniaxial tension while load and extension are recorded to fracture. Its significance is that it delivers almost the whole design basis for a structural steel in a single test: the modulus of elasticity, the proportional limit, the upper and lower yield point or the 0.2 per cent offset yield strength, the ultimate tensile strength, the percentage elongation over a specified gauge length and the reduction of area at the neck. It is used for acceptance of every structural shape, plate, bar, bolt and reinforcing bar; for mill certification against CSA G40.21 grades such as 350W; for verifying the yield-to-tensile ratio and the elongation on which capacity design and plastic analysis depend; and as the reference test to which all empirical strength correlations, including hardness conversions, are tied.

iii. Charpy V-notch impact test (ASTM E23, CSA G40.21 supplementary requirements). A 10 mm square bar with a standard 2 mm deep V notch is broken by a single blow from a swinging pendulum at a controlled temperature, and the energy absorbed is read from the height of the swing. Its significance is that it measures notch toughness — the ability of the steel to absorb energy in the presence of a stress raiser and at a high strain rate, the two conditions the smooth, slowly loaded tension specimen deliberately excludes. Repeating the test over a range of temperatures maps the ductile-to-brittle transition, and the appearance of the fracture surface (percentage shear lip against flat cleavage) and the lateral expansion confirm which regime the steel is in. It is used to specify steel for bridges, offshore and northern structures, pressure vessels and welded connections, where a minimum absorbed energy, commonly 27 J at a service-related temperature, guarantees that the transition temperature lies safely below the lowest anticipated service temperature and that a fatigue crack or a weld defect will not trigger a brittle, fast fracture.