In presently manufactured SIN girders [16] with corrugated (undulated) webs, sinusoidal webs have three elementary thicknesses of
Statistical investigations into the strength of structural steel properties were conducted by Sowa, Murzewski, and Mendera [9, 10, 11]. The
Based on the investigations,
Because of the shape and thickness of the corrugated web in SIN girders, numerous phenomena occur that are related to web shear resistance. They are reported in studies [1, 2, 5, 6, 7]. In those phenomena, an important role is played by
The span of the SIN girder is limited by the shear resistance of the corrugated web. Shear resistance of the thin-walled web depends on
Yield strength of the corrugated web results from the individual stages of the manufacturing process:
in the casting process at the steel mill, the so-called slab, i.e. plate, 220 mm in thickness is produced;
the second stage involves rough hot rolling;
in the third stage, sheet metal is hot-rolled to the required thicknesses of 2; 2.5, and 3 mm;
the fourth stage consists in the cooling of flat sheet metal and winding it onto a coil, the internal diameter of which is 740–760 mm. The consecutive stages include:
decoiling
cold straightening
longitudinal slitting
the last stage involves by cold-formed corrugation (Fig. 1).
Due to cold process of thin sheet folding, stress state along the wave occurs, which is shown in Fig. 1.
In European standards [17, 20], the assessment of random strength of the material is based on the method of load coefficients and load bearing capacity coefficients. Partial factor for a material property, also accounting model uncertainties and dimensional variations γ
where:
This study presents the results of statistical investigations into random parameters of strength properties of steel from corrugated webs 2, 2.5 and 3 mm in thickness. Investigations were performed for samples randomly collected from twenty SIN girders that were earlier subjected to testing. Based on the investigations, variation coefficients of yield strength
The first stage of investigations covered twenty research models of girders with corrugated web, the load diagram of which was that of a simply supported beam, or a simply supported beam with a single cantilever. Three types of beams geometry were distinguished (Table 1): a) girders with a flexible end stiffener (Fig. 2a); b) girders with a rigid end stiffener (Fig. 2b); c) cantilever girders (Fig. 2c).
Program of investigations
Girder |
|
End Stiffener [mm] |
|
Girder [mm] |
|
End Stiffener [mm] |
|
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
M 1.11 | 500x2 | 300x25 | 7825 | M 1.12 | 500x2 | 300x25 | 6000 |
M 1.21 | 1000x2.5 | 300x25 | 7825 | M 1.22 | 1000x2 | 300x25 | 6000 |
M 1.31 | 1000x2.5 | 300x25 | 7825 | M 1.32 | 1000x2.5 | 300x25 | 6000 |
M 1.41 | 1250x2 | 300x25 | 7825 | M 1.42 | 1250x2 | 300x25 | 6000 |
M 1.51 | 1500x2 | 300x25 | 7825 | M 1.52 | 1500x2 | 300x25 | 6000 |
M 2.11 | 500x2.5 | 300x25 | 5825 | M 2.12 | 500x2 | 300x25 | 3750 |
M 2.21 | 1000x2 | 300x25+tee bar | 5825 | M 2.22 | 1000x2 | 300x25 | 3750 |
M 2.31 | 1000x2.5 | 300x25+tee bar | 5825 | M 2.32 | 1000x2,5 | 300x25 | 3750 |
M 2.41 | 1000x3 | 300x25+tee bar | 5825 | M 2.42 | 1000x3 | 300x25 | 3750 |
M 2.51 | 1500x3 | 300x25+tee bar | 5825 | M 2.52 | 1500x2 | 300x25 | 3750 |
The girder group under consideration comprised twelve WTA models with the web basic thicknesses of 2 mm, six WTB models having the web thickness of 2.5 mm, and two WTC models with the web thicknesses of 3 mm.
Research models of girders with corrugated webs were designed and produced in accordance with literature data and standards [19, 21]. Corrugated webs of girders were made from flat S235JRG2 steel sheets that were hot-rolled to the thickness of 2; 2.5 and 3 mm, whereas flanges were made from S275JRG2 steel [15]. It should be added that individual girders came from different batches. Each batch was provided with mill certificates indicating the grade of steel products.
Girders were assembled from items (Fig. 2) prefabricated at the plant. The items of the girders were buttconnected using High Strength Friction Grip (HSFG) bolts. Girders were tested (Fig. 3) until ultimate resistance was reached, which resulted from the condition for web failure in the span or cantilever part of the girder. The load, considered as a concentrated force P, is transferred from frame (R) by means of the actuator (1) via a dynamometer (2) or a plate (3), to the end plate of the span or cantilever part of the girder (4).
Samples were collected from examined girders to conduct materials tests on web steel, which limits the ultimate resistance of girders.
Six samples were collected from the web of each examined girder. The samples came from the areas with the lowest stress intensity level. In the first stage, the corrugated web part was cut out of undamaged area (box-marked in Fig. 4).
Then, standard samples were cut out of the plate along the fold (Fig. 6) and machined using a milling machine. Samples were cut out in such a way so that the contour of the samples was kept away from the edge on which permanent deformations and dislocations occurred.
Web strength parameters change in accordance with a change in the sinusoidal geometry of the web. Yield strength increment is altered, which occurs at the sinusoidal wave crests. At “zero” wave inflection points, yield strength remains almost unaltered. The samples were cut out at the sites near connections to flanges, where the corrugated sheet was under maximum stress (Fig. 5), which was found on the basis of computed yield strength. That was done to obtain more reliable strength estimates.
Investigations into strength properties of steel samples from webs and flanges were carried out acc. standard [24]. Samples were cut out of webs with “tenfold” base (Fig. 6). Geometric dimensions of the samples were measured using Vernier Callipers with 0.1 mm graduation. Altogether, 120 samples were examined, including: 12 × 6 = 72 samples with nominal thickness of 2 mm, 6 × 6 = 36 samples nominal thickness of 2.5 mm and 2 × 6 = 12 samples with nominal thickness of 3 mm. The results of investigations were influenced by the following: the direction of sheet hot-rolling, the direction of sheet folding, and tensile tests on samples along the fold direction.
Investigations into strength properties were conducted using PUL 400 VEB Werkstoffprufmaschinen Leipzig testing machine. In the tests, it was tried not to exceed the stress increment rate of 8 MPa/s, which corresponded to the tensile force increment level of 0.4 kN/s. In the tests, tensile force
Parameters of yield strength and tensile strength tests on web samples
Girder number |
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mm | mm | mm2 | mm | mm | % | kN | kN | MPa | MPa | ||
1 | 2 | 2 | 4 | 5 | 6 | 7 | 8 | 9 | 12 | 13 | 14 |
|
2.0 | 20.0 | 40.0 | 70.0 | 85.0 | 21.5 | 12.2 | 16.1 | 304.4 | 403.5 | 1.33 |
|
2.0 | 20.0 | 40.0 | 70.0 | 85.1 | 21.6 | 12.7 | 17.4 | 317.8 | 434.3 | 1.37 |
|
2.0 | 20.0 | 40.0 | 70.0 | 86.1 | 23.0 | 9.9 | 15.0 | 247.2 | 375.5 | 1.52 |
|
2.0 | 19.7 | 39.4 | 70.0 | 83.5 | 19.2 | 13.2 | 17.2 | 334.3 | 437.5 | 1.31 |
|
2.0 | 20.0 | 40.0 | 70.0 | 86.2 | 23.1 | 11.6 | 15.1 | 289.3 | 376.8 | 1.30 |
|
2.0 | 20.0 | 40.0 | 70.0 | 85.5 | 22.2 | 13.6 | 17.4 | 334.7 | 430.6 | 1.29 |
|
2.0 | 20.0 | 40.0 | 70.0 | 83.9 | 19.9 | 10.7 | 14.4 | 312.5 | 453.9 | 1.45 |
|
2.0 | 20.0 | 40.0 | 70.0 | 86.3 | 23.3 | 10.0 | 15.2 | 267.2 | 360.9 | 1.35 |
|
2.0 | 20.0 | 40.0 | 70.0 | 86.3 | 23.3 | 10.0 | 15.2 | 299.1 | 380.5 | 1.27 |
|
2.0 | 20.0 | 40.0 | 70.0 | 84.8 | 20.8 | 13.5 | 17.2 | 337.9 | 429.1 | 1.27 |
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2.0 | 20.0 | 40.0 | 70.0 | 85.9 | 22.5 | 13.5 | 17.1 | 336.3 | 427.3 | 1.27 |
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2.0 | 20.0 | 40.0 | 70.0 | 87.2 | 24.6 | 11.2 | 15.0 | 281.0 | 375.5 | 1.34 |
Mean value for 2 mm samples: |
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2.5 | 19.8 | 49.5 | 80.0 | 96.5 | 20.7 | 13.7 | 20.6 | 275.9 | 416.0 | 1.51 |
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2.6 | 19.7 | 51.2 | 80.0 | 96.6 | 20.7 | 13.3 | 20.6 | 260.4 | 403.0 | 1.55 |
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2.6 | 20.0 | 52.0 | 80.0 | 98.2 | 22.8 | 13.8 | 20.3 | 266.2 | 390.0 | 1.47 |
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2.5 | 20.0 | 50.3 | 80.0 | 92.0 | 15.0 | 17.3 | 24.7 | 340.8 | 487.7 | 1.43 |
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2.6 | 20.0 | 52.0 | 80.0 | 96.2 | 20.2 | 16.3 | 23.6 | 339.4 | 435.4 | 1.28 |
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2.6 | 20.0 | 52.0 | 80.0 | 95.4 | 19.2 | 16.9 | 23.5 | 324.6 | 451.5 | 1.39 |
Mean value for ∼2.5mm samples: |
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3.0 | 20.0 | 60.0 | 90.0 | 107.9 | 19.9 | 19.5 | 25.3 | 325.5 | 422.0 | 422.0 |
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3.0 | 20.0 | 60.0 | 90.0 | 103.2 | 14.6 | 26.8 | 32.6 | 445.9 | 544.1 | 1.22 |
Mean value for 3 mm samples: |
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Mean value for all samples: |
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samples without clear yield plateau
On the basis of investigations, graphs (Figs. 8 and 9) of the dependence
As regards the graphs of the static tensile test for samples that did not show discontinuities clearly indicating the material yield strength, the conventional yield strength was identified as the one corresponding to 0.2% of the permanent elongation of the samples.
Sheet corrugation produced changes in the material structure, which resulted in the shortening of the plastic flow zone and obscuration of a clear yield strength in
The guaranteed yield strength of the supplied flat sheets, from which corrugated webs are made is presented as the minimum value specified by the mill on the basis of relevant standards [20, 21] and it is given by the manufacturer as
On the basis of structural analysis, partial factors for material properties γ
For the determined parameters of normal distribution of yield strength
where:
Partial factors for material properties γ
where: γ
To illustrate the level of structural reliability, coefficients of variations of yield strength were estimated. First, the actual coefficients
Then, coefficients of yield strength γ
Parameters of normal distribution and partial factors of yield strength that were obtained are shown in Table 3.
Parameters of normal distribution of yield strength Re of steel samples of corrugated webs
Girder number | E( |
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
M 1.11 | 304.4 | 299.15 | 10.13 | 3.1826 | 294.69 | 0.0105 | 0.1390 | 0.1791 | 1.0151 | 0.7974 | 0.7296 |
M 1.41 | 317.8 | 307.57 | 38.76 | 6.2258 | 298.85 | 0.0196 | 0.1589 | 0.1972 | 1.0292 | 0.7863 | 0.7194 |
M 1.51* | 247.2 | 243.12 | 6.04 | 2.4576 | 239.68 | 0.0099 | 0.0301 | 0.0794 | 1.0144 | 0.9805 | 0.8970 |
M 2.21 | 334.3 | 321.48 | 61.41 | 7.8365 | 310.50 | 0.0234 | 0.1811 | 0.2176 | 1.0354 | 0.7568 | 0.6924 |
M 2.51 | 289.3 | 278.64 | 42.39 | 6.5108 | 269.53 | 0.0225 | 0.1144 | 0.1566 | 1.0338 | 0.8719 | 0.7977 |
M 1.12 | 334.7 | 328.38 | 15.00 | 3.8730 | 322.96 | 0.0116 | 0.1816 | 0.2181 | 1.0168 | 0.7276 | 0.6657 |
M 1.22 | 339.4 | 334.82 | 7.85 | 2.8018 | 330.90 | 0.0083 | 0.1876 | 0.2235 | 1.0118 | 0.7102 | 0.6497 |
M 1.42* | 267.2 | 257.21 | 36.78 | 6.0647 | 248.72 | 0.0227 | 0.0735 | 0.1191 | 1.0341 | 0.9448 | 0.8644 |
M 1.52 | 299.1 | 291.99 | 18.65 | 4.3186 | 285.95 | 0.0144 | 0.1307 | 0.1714 | 1.0211 | 0.8218 | 0.7519 |
M 2.12 | 337.9 | 333.88 | 6.03 | 2.4556 | 330.44 | 0.0073 | 0.1857 | 0.2218 | 1.0104 | 0.7112 | 0.6506 |
M 2.22 | 336.3 | 325.33 | 44.91 | 6.7015 | 315.95 | 0.0199 | 0.1837 | 0.2199 | 1.0297 | 0.7438 | 0.6805 |
M 2.52 | 281 | 273.17 | 23.02 | 4.7979 | 266.45 | 0.0171 | 0.0998 | 0.1432 | 1.0252 | 0.8820 | 0.8069 |
Average values 2 mm |
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M 1.21* | 275.9 | 266.26 | 34.65 | 5.8864 | 258.02 | 0.0213 | 0.0904 | 0.1346 | 1.0319 | 0.9108 | 0.8333 |
M 1.31* | 260.4 | 250.70 | 34.91 | 5.9085 | 242.43 | 0.0227 | 0.0595 | 0.1063 | 1.0341 | 0.9694 | 0.8869 |
M 2.11 | 266.2 | 260.28 | 13.13 | 3.6235 | 255.21 | 0.0136 | 0.0715 | 0.1173 | 1.0199 | 0.9208 | 0.8424 |
M 2.31 | 340.8 | 326.04 | 81.16 | 9.0089 | 313.43 | 0.0264 | 0.1893 | 0.2251 | 1.0402 | 0.7498 | 0.6860 |
M 1.32* | 312.5 | 303.33 | 30.95 | 5.5633 | 295.54 | 0.0178 | 0.1512 | 0.1902 | 1.0264 | 0.7952 | 0.7275 |
M 2.32 | 324.6 | 310.44 | 74.97 | 8.6585 | 298.32 | 0.0267 | 0.1683 | 0.2059 | 1.0406 | 0.7877 | 0.7207 |
Average values 2.5 mm |
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M 2.41 | 325.5 | 312.86 | 59.11 | 7.6883 | 302.09 | 0.0236 | 0.1695 | 0.2070 | 1.0357 | 0.7779 | 0.7117 |
M 2.42 | 445.9 | 422.88 | 197.6 | 14.0570 | 403.20 | 0.0315 | 0.1884 | 0.3157 | 1.0488 | 0.5828 | 0.5332 |
Average values 3 mm | 385.7 | 367.9 | 128.36 | 11.3296 | 352.70 | 0.0294 | 0.2290 | 0.2614 | 1.0422 | 0.6804 | 0.6225 |
Average values for all samples |
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Figure 10 shows exemplary normal distributions of yield strength obtained for samples of corrugated webs of M 2.12 and M 1.31models. Mean
According to investigations, mean coefficients of variation
Coefficients of variation mentioned above indicate great caution exercised by both the steel mills and manufacturers of corrugated webs, who adopt reduced mean yield strength E(
That is confirmed by the obtained values of partial factors of yield strength γ
The obtained values of coefficients γ
The obtained values of coefficients γ
Corrugated web yield strength for the convex part of the wave, where increase in yield strength of cold-formed elements is enhanced, is also given in standard [18] on the basis of formula (6):
where:
When corrugated web is treated as an element with one inflection point, for the yield strength of the starting material 235 MPa, the design yield strength is obtained depending on the web thickness given in Table 4.
Steel yield strength that accounts for the impact of cold rolling acc. (6) [18]:
web thickness 2 mm | web thickness 2.5 mm | web thickness 2 mm | |
---|---|---|---|
Design yield strength |
254.7 | 259.6 | 264.5 |
However, as many as 25% of the results for samples did not reach the values resulting from formula (6) [18].
However, it should be emphasised that a change in the value of yield strength acc. formula (6) for cold-formed elements is an additional source of uncertainty related to unknown level of confidence with respect to the modification of the original value. As regards the estimation of yield strength of the elements of concern, a simplified probabilistic method acc. PN-EN 1990 [20] provides a better solution.
It should be added that the resistance of elements made from corrugated webs depends not only on the scatter of yield strength, but also on the scatter of the size of cross-sectional area in accordance with formula (7):
where:
Random impacts of geometry and yield strength sum up in accordance with the principles of quantile algebra. The impact of random cross-sectional area and yield strength on the cumulative result is expressed by formula (8):
where:
Table 5 shows parameters of normal distribution of geometric dimensions of steel samples from corrugated webs, and of member resistance.
Parameters of normal distribution of geometric dimensions of steel samples from corrugated webs and of resistance obtained according to formulae (7) and (8)
Model |
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mm | mm | mm2 | |||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 9 |
Mean value for 2 mm samples | 2.0 | 19.98 | 39.95 | 0.030 | 0.170 | 0.004 | 0.047 | 0.217 | 0.018 |
Mean value for 2.5 mm samples | 2.57 | 19.92 | 51.18 | 1.111 | 1.054 | 0.021 | 0.209 | 0.457 | 0.030 |
Mean value for 3 mm samples | 3.0 | 20.0 | 60.0 | 0.000 | 0.000 | 0.000 | 0.462 | 0.680 | 0.029 |
Estimated coefficient of variation
On the other hand, resistance variation factor
The investigations conducted for the study indicate that when dimensioning structures made of girders with corrugated web, it is possible to apply the design yield strength
General conclusion:
The span of plate girders with corrugated web is limited by web shear resistance. The latter is affected by the
With respect to yield strength, the following should be stated:
The investigations indicate that when dimensioning girders with corrugated web, it is
For the manufacture of corrugated webs, steels that have
Yield strengths of prefabricated corrugated webs, separately from each girder, are
Yield strengths put together from different batches show considerable scatter of results and have values that are higher than those given by manufacturers of corrugated webs. The values are also higher than the recommended ones. That may result in a situation in which structural members delivered to the construction site
The analysis of variation coefficients
Due to the fact that the variation in resistance of girders with corrugated web is substantially affected by yield strength of steel of corrugated webs, the manufactured structural elements should be provided with the results of the materials tests for the web. They would show actual distributions of yield strength in steel webs and prove the reliability of structural members.
The direction of hot rolling, coiling and cold rolling of steel sheet intended for corrugated web of SIN girders is always perpendicular to the wave being formed. That is significant for the corrugated web dimensioning.
When yield strength of finished corrugated webs is given, it is recommended that the relation should be specified between the design yield strength and its mean value having the form:
Concerningthe cross-section, the following should be stated:
In the tests conducted by mills, random scatter of the thickness of sample section is not accounted for.
Estimated coefficient of variations
Regardless of the comments above, it should be noted that