Methods, assumptions and derived calculations

A.1 Evidence statuses and conventions

Each figure and table carries one of four statuses. Sourced: the value was read in the cited source (Annex C). Derived: the value is Cameroon Steel’s arithmetic on sourced inputs, with the method given below. Assumed: a stated modelling or planning assumption. Illustrative: a schematic or simulated example that shows a mechanism, not data.

Currency is the CFA franc (FCFA, XAF), pegged at 655.957 FCFA to the euro. Prices are quoted as published, without inflation adjustment. Stresses are in megapascals (MPa = N/mm²); forces in kilonewtons (kN); masses in kilograms (kg) or tonnes (t). Bar sizes are nominal diameters in millimetres. Percentages of composition are by mass. Standards are cited by their reference; summaries of requirements are for orientation and the current edition of each standard governs.

A.2 Bar geometry and mass per metre

The nominal cross-sectional area of a bar of nominal diameter d (mm) is A = π d² / 4 (mm²). Its nominal mass per metre is A × 7,850 kg/m³ × 10⁻⁶ = 0.0061654 × d² kg/m, usually rounded to 0.00617 × d². This reproduces the nominal masses tabulated in the product standards (for example 0.888 kg/m for 12 mm). The mass of a 12 m bar is 12 times the mass per metre; the lightest bar allowed is the nominal mass multiplied by (1 minus the tolerance), with tolerances of 6.0% for d ≤ 8 mm and 4.5% for d > 8 mm [S31][S32]. Bars per tonne are 1,000 divided by the mass of a 12 m bar, rounded down.

The mass of a bar one millimetre undersize is computed for diameter d minus 1 mm. The loss of section is 1 minus ((d minus 1)/d)²: 16.0% for 12 mm, 19.0% for 10 mm, 12.1% for 16 mm.

A.3 Yield force, design value and the capacity ratio

The yield force of a bar is F = A × fy, with A the actual area and fy the actual yield strength. The design value assumed by the engineer under EN 1992-1-1 is Fd = A_nom × fyk / γs, with fyk = 500 MPa and γs = 1.15 [S36]. For a 12 mm bar: A_nom = 113.1 mm², Fd = 49.2 kN, characteristic yield force 56.5 kN.

The scenarios in Figure ES.1 use: typical certified production, fy = 560 MPa (the mean of a population with characteristic value 500 MPa and standard deviation 30 MPa, following the JCSS model in which the mean equals the nominal value plus two standard deviations [S29]); a bar of 11 mm (A = 95.0 mm²) at 500 MPa; a 12 mm bar at 400 MPa; and an 11 mm bar at 400 MPa. The capacity ratio of Figure 1.3 is (1 minus mass deficit) × fy / (500 / 1.15), on the basis that mass deficit equals area deficit for a bar of constant density.

These ratios concern the bar alone. The safety of a structural member also depends on load factors, concrete strength, detailing and workmanship; the calculation does not estimate the probability of failure of any structure.

A.4 Statistical evaluation of characteristic values

A characteristic value is the 5% fractile of the population. Conformity is commonly demonstrated with a one-sided statistical tolerance limit: the mean of n results minus k times their standard deviation must be at least the characteristic value. The factor k for a coverage of 95% at a confidence of 90% was computed from the non-central t distribution, k = t′(n minus 1, δ = 1.645 √n; 0.90) / √n, following the method of ISO 16269-6 [S57]. The required mean in the table assumes a standard deviation of 30 MPa [S29].

Table A.1 Tolerance factor k (95% coverage, 90% confidence) and required mean yield for a characteristic value of 500 MPa

Source: derived by Cameroon Steel (non-central t distribution); method after [S57]; standard deviation from [S29]. Certification schemes may use different confidence levels and sample rules.

A.5 Yield-strength distributions (Figure 5.1)

All three distributions are normal. Certified production: mean 560 MPa, standard deviation 30 MPa [S29]; share below 500 MPa = 2.3%. Uncontrolled spread: same mean, standard deviation 80 MPa, a coefficient of variation of about 14% as reported for market bars in Ethiopia [S28]; share below 500 MPa = 22.7%, below 400 MPa = 2.3%. Site pattern: mean 395 MPa, standard deviation 75 MPa (assumed), chosen so that about 52% of bars fall below 400 MPa as reported in [S19] (the study’s own counts give 46%, 11 of 24 bars); share below 500 MPa = 91.9%. With an assumed standard deviation of 60 or 90 MPa instead of 75, the share below 500 MPa for the site pattern would be 95.7% or 87.7%. The site pattern is an illustrative fit to one reported statistic, not a reconstruction of the study’s data.

A.6 Prices per kilogram (Table 4.2 and Figure 7.2)

Price per kg = homologated price per bar [S11] divided by the nominal mass of a 12 m bar (the list does not state the length; 12 m, the standard commercial length, is assumed). The effective price of a bar lighter than nominal by a fraction δ is the price at nominal mass divided by (1 minus δ). The Fe500 premium per kilogram is 5.9% at 8 mm, 5.4% at 10 mm and 5.5% at 12 mm. A delivery of 1,000 bars of 12 mm at nominal mass weighs 1,000 × 10.66 kg = 10.66 t.

A.7 Carbon equivalent map (Figure 2.3)

Ceq = C + Mn/6 + (Cr + Mo + V)/5 + (Ni + Cu)/15 [S31]. Residual elements were fixed at Cu + Ni = 0.35% and Cr + Mo + V = 0.15%, plausible for scrap-based electric arc furnace steel; the range 0.20% to 0.50% for Cu + Ni moves every Ceq contour by about plus or minus 0.01. The "QST window" (C 0.17% to 0.22%, Mn 0.65% to 1.05%) and the "high-carbon window" (C 0.26% to 0.31%, Mn 1.15% to 1.55%) are indicative illustrations, not data from any producer.

A.8 Chloride initiation model (Figure 8.3)

The time t for the chloride content at depth x (the cover) to reach a threshold Ccrit, with a constant surface content Cs and apparent diffusion coefficient D, follows from Fick’s second law: t = x² / (4 D z²), where z = erfinv(1 minus Ccrit/Cs). The model ignores the decrease of D with age, binding, cracking, convection by wetting and drying, and temperature, and is therefore illustrative only. Its value lies in the relative effects: t is proportional to the square of the cover and inversely proportional to D.

Table A.2 Assumptions of the illustrative chloride model, with plausible ranges and sensitivity

Source: assumptions by Cameroon Steel for illustration; sensitivity derived with the same model. Not a design method; durability design should follow EN 1992-1-1 and EN 206 [S36][S37] or a validated performance-based method.

A.9 Other data and simulations

Humidity (Figure 8.2). Monthly relative humidity at 2 m and temperature from the NASA POWER climatology API (1991 to 2020, modelled), at Douala (4.05° N, 9.70° E), Kribi (2.94° N, 9.91° E), Yaoundé (3.87° N, 11.52° E) and Garoua (9.30° N, 13.40° E) [S50]. Modelled values smooth local extremes; station data at Kribi are consistent with them [S51].

Control chart (Figure 5.2). Sixty heat means drawn from normal distributions (mean 566 MPa, standard deviation 8 MPa for heats 1 to 38; mean 556 MPa, standard deviation 4 MPa thereafter; random seed 10). Control limits are the mean of the first 30 heats plus or minus three standard deviations. The run rule flags the eighth successive heat below the mean.

Import values (Figure 9.1). UN Comtrade reporter data for Cameroon, HS 7213, 7214, 7216 and 7217, current US dollars [S52]. Later years were not yet reported at the cut-off.

Indicative costs (Table 11.1). Cameroon Steel estimates based on typical scopes of sampling, laboratory testing and survey work, for budgeting orientation only.

A.10 Worked examples for site acceptance

The two examples below show how the weighing check of Chapter 6 and the chemistry check of Chapter 7 are applied. All bar masses and lengths are invented for illustration.

Weighing. The nominal mass of a 12 mm bar is 0.888 kg/m; with the 4.5% tolerance the minimum is 0.848 kg/m (derived). Three bars are taken at random from each of two deliveries, weighed and measured.

Table A.3 Worked example of the weighing check on two deliveries of 12 mm bars (invented values)

Source: invented values; nominal mass and tolerance as in this annex [S31][S32].

The second delivery fails by a margin that no weighing error can explain: a deviation of about 15% matches a bar rolled at 11 mm instead of 12 mm (16% less section). Weighing three bars was enough to expose it.

Carbon equivalent. The specimen certificate of Figure 7.1 shows a cast analysis of 0.20% C, 0.85% Mn and 0.28% Cu. With residuals of 0.06% Cr, 0.02% Mo, 0.01% V and 0.10% Ni, which a full certificate would also report, the carbon equivalent is 0.20 + 0.85/6 + (0.06 + 0.02 + 0.01)/5 + (0.10 + 0.28)/15 = 0.385%, below the 0.50% limit for weldable steel (derived). A certificate that reports Ceq without the individual elements, or elements without Ceq, should be queried: the buyer should be able to repeat the calculation.

A.11 Limitations

The paper relies on published sources, several of them press reports, for the Cameroonian regulatory and enforcement record; it does not have access to the full text of NC 236, to ANOR’s certification files or to official investigation reports on building collapses. Published studies of market rebar use different methods, standards and sample sizes and are not directly comparable. Standards are summarised from secondary sources and certificates where the primary text is priced; the current edition of each standard governs. Project information about Cameroon Steel is limited to the public record and to stated objectives.