Currently manufactured girders with sinusoidal corrugated web are available with three basic thickness values of web
Axial forces and bending moments in plate girders with corrugated webs are applied through flanges, and shear forces through the corrugated web. Tests on resistance and stiffness of plate girders with corrugated webs usually focus on problems concerning the loss of stability and shear resistance of the corrugated web [1–7]. In the case of bending resistance, lateral-torsional buckling of the compression flange of plate girders [8] and not thoroughly tested relationship between torsional buckling of the compression flange and corrugation of the web [9] are analysed. However, problems with the compensation for bending and shear resistance of girders are observed for plate girders with corrugated webs. There is also no reference to random parameters of steel used in flanges and their effect on bending resistance of plate girders. The possible effect on displacements particularly of long girders is an important parameter.
In the case of plate girders with corrugated webs, flanges are manufactured from flat sheet steel S235JRG2 or S355J2+N. Flanges made of steel S275 are available on request. The guaranteed yield strength of delivered hot rolled flat steel sheet known as the specified minimum
Flanges are connected to webs with single-sided welds at a specially prepared assembly stand (Fig. 1). So, flanges are prevented from the excessive amount of heat that can change mechanical properties of steel.
Statistical studies on strength of metallurgic products made of structural steel are described in the papers [13, 14, 15, 16]. They also include strength parameters of flat sheets. These tests are the base to determine coefficients of variation of yield strength
This paper describes statistical tests on random parameters of strength of sheet steel in flanges of plate girders with corrugated webs. The effect of strength parameters of flange steel on bending resistance and displacement of plate girders with corrugated web was analysed. The tests on strength parameters were performed on randomly collected samples of 20 girders with corrugated webs which had been already tested. Based on the tests, partial factors of yield strength
Bending resistance of plate girders with corrugated web is connected with resistance of flanges. The applicable design methods can be found in German guidelines DASt 015 [17], EC 3 [18] or the paper by Siokola [8]. These references include phenomena related to the impact of local or global buckling of compression flanges on bending resistance.
The method of calculating bending resistance specified in EN 1993-1-5 [18] is similar to German guidelines DASt-015 [17]. This method consists of designing tension and compression flanges of the girder taking into account local and global buckling. In the case of the compression flange, EC3 [18] associates resistance only with the reduction in yield strength considering the local buckling:
In the case of lateral buckling of the compression flange concerning flexural buckling, the resistance according to EC3 [19] is expressed with the following equation:
Following the purpose of Eurocode, the design bending resistance
Eurocode neglects the effect of local torsional buckling on the flange resistance for calculations made for the compression flange. In the case of the global buckling, only the flexural buckling is taken into account neglecting lateral torsional buckling.
The method presented by Siokola [8] includes both the local torsional buckling of the flange, and its impact on the compression flange.
Compression resistance related to the local buckling of the compression flange is described by Siokola [8] using the following formula:
To determ ine the impact of global bending stability of the compression flange, Siokola applied the theory of bending in-plane buckling of bar cross-sections neglecting the commonly used truss model. Hence, the bending resistance for plate girders with the corrugated web can be determined by the following formula [8]:
It should be noted, however, that this method has been directly transferred from the design for plate girders with flat webs. As Yu and Sause described in the paper [21], this method understates results for girders with the length up to 12 m. Formulas acc. to D IN correspond to test results when girder span exceeds 12m. It clearly indicates the effect of the support for the compression flange by the corrugated web as in the case of local torsional buckling. Similar conclusions were presented in the paper by Sherif [22] who related the obtained results to calculations made in accordance with EC3 [18].
In the case of erection load, Kowal proposed in 2005 [23] description of buckling.
According to recommendations specified in D IN 1880 Part 2 [20], there is the possibility of mutual interaction between local and global buckling mode of the compression flange of the girder. Then, the value of critical moment of lateral-torsional buckling has to be reduced to the value obtained from the equation:
And the ultimate limit resistance for the compression flange restrained in the lateral direction within the span cb is expressed as:
Therefore, the resistance of corrugated web girder taking into account the tension flange, the compression flange subjected to local and global buckling, should be estimated from the following formula:
According to Siokola [8], the bending resistance comes down to the following formula:
All quoted solutions refer to the effect of local and global buckling on the compression flange. However, they do not depreciate the positive effect of yield strength of flanges. They only introduce the reduction factor of yield strength when local buckling of the compression flange can be expected.
Prior to material testing of strength parameters of steel used in flanges, tests were at first conducted on resistance of girders with the corrugated web. The effect of stiffening at the support on critical shear resistance of the corrugated web has been analyzed. Thus, three different types of corrugated web girders were selected depending on stiffening at the support (Fig. 2): a) girders with a semi-rigid end stiffener; b) girders with end stiffener reinforced by tee-bar; c) girders with cantilever connected with the span through the stiffener. The tests were performed on 20 girders in total. The girders were selected to ensure that failure was determined by shear resistance, and the nominal flange resistance was greater. The difference between the reached resistance considering strength parameters of steel used in flanges and the nominal bending resistance was described in Chapter 6.
The static scheme of a simply supported beam (Figs. 2a and 2b) and a simply supported beam with a single cantilever was assumed for all girders. The program of investigations is presented in Table 1.
Program of investigations
Girder |
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Support Stiffener [mm] | Girder |
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Support Stiffener [mm] |
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
M1.11 | 500x2 | 300x15 | 25x300 | M 1.12 | 500x2 | 300x15 | 2x20x300 |
M 1.21 | 1000x2.5 | 300x15 | 25x300 | M 1.22 | 1000x2 | 300x15 | 2x25x300 |
M 1.31 | 1000x2.5 | 300x20 | 25x300 | M 1.32 | 1000x2.5 | 300x20 | 2x25x300 |
M 1.41 | 1250x2 | 300x15 | 25x300 | M 1.42 | 1250x2 | 300x15 | 2x25x300 |
M 1.51 | 1500x2 | 300x15 | 25x300 | M 1.52 | 1500x2 | 300x15 | 2x25x300 |
M 2.11 | 500x2.5 | 300x15 | 25x300 | M 2.12 | 500x2 | 300x15 | 2x20x300 |
M 2.21 | 1000x2 | 300x15 | 25x300+tee bar | M 2.22 | 1000x2 | 300x15 | 2x25x300 |
M 2.31 | 1000x2.5 | 300x15 | 25x300+tee bar | M 2.32 | 1000x2.5 | 300x15 | 2x25x300 |
M 2.41 | 1000x3 | 300x15 | 25x300+tee bar | M 2.42 | 1000x3 | 300x15 | 2x25x300 |
M 2.51 | 1500x3 | 300x15 | 25x300+tee bar | M 2.52 | 1500x2 | 300x15 | 2x25x300 |
Plate girders with the corrugated web were designed and performed in accordance with the current literature and standards [18, 24]. Girder flanges were made from hot rolled flat sheets of S275JRG2 with a thickness of 15 or 20 mm, while corrugated webs were made from hot rolled flat sheets of S235JRG2 with a thickness of 2; 2.5, and 3 mm [25]. Individual plate girders were from different batches. Certificates confirming steel grade were attached to each batch. The tested group of corrugated web girders included eighteen girders with flanges of 300 x 15 mm, and two girders with flanges of 300 x 20 mm (Table 1).
Plate girders with corrugated web were assembled from three items (Fig. 2) prepared in SIN girder production plant. Individual items of tested girders were assembled with end plate connection by means of M20 or M24 high strength prestressing bolts of class 10.9. In the case of the girders with end stiffeners (Figs 2a and b), load exerted by a pair of forces 2 x
The analysed girders (Fig. 3) were tested until the ultimate resistance was reached determined by the failure of the web and formation of plastic hinges in the span or cantilever part of the flange. The samples for material testing of steel in flanges affecting the ultimate resistance of the girders, were collected from the tested girders, undamaged parts of items subjected to failure.
Three flange samples were collected from each tested plate girder with the corrugated web. In the first stage, pieces of flanges were cut out from undamaged area of items of the girders (Fig. 4).
Then, the samples were cut out from flanges in accordance with the standards [24, 26]. The samples for material testing were cut out from the flanges providing their edges did not touch the plastic hinge formed in the flange. Moreover, the strength parameters of the flanges could be also affected by the zone of thermal effects of the web to flange weld. Thus, the samples were cut out near the flange edges. All samples were mechanically processed using a milling machine.
Random strength parameters of steel in flanges were tested in accordance with the standard [26]. The samples of “5-fold” base were cut out from the flanges (Fig. 5). Geometric dimensions of flange samples were measured using a caliper with a scale of 0.1 mm. In total, 60 samples were tested, including: 18 x 3 = 54 samples having a nominal thickness of 15 mm, and 2 x 3 = 6 samples with a nominal thickness of 20 mm. Results obtained from the tests on flange samples were affected only by the direction of hot rolling of steel sheet.
The tests on strength parameters of the flange samples were conducted using the test machine PUL 400 VEB Werkstoffprufmaschinen Leipzig. An attempt was made not to exceed the stress increment rate of 8 MPa/s throughout the testing cycle. And the measurements were performed for tensile force
Average parameters of yield strength, tensile strength and modulus of Young tests on flange samples
Girder No. of sample |
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mm | mm | mm2 | mm | mm | % | kN | kN | MPa | MPa | GPa | ||
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15.0 | 24.4 | 366.5 | 110.0 | 142.3 | 29.4 | 108.0 | 166.6 | 298.2 | 454.5 | 201.6 | 1.52 |
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15.0 | 24.4 | 365.5 | 110.0 | 142.4 | 29.4 | 110.9 | 177.4 | 303.4 | 485.5 | 213.2 | 1.60 |
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15.0 | 24.8 | 371.5 | 110.0 | 143.8 | 30.7 | 104.4 | 164.9 | 281.2 | 443.9 | 201.9 | 1.58 |
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15.0 | 24.5 | 367.0 | 110.0 | 141.7 | 28.8 | 106.8 | 165.8 | 291.1 | 451.8 | 208.3 | 1.55 |
M 2.11 | 15.0 | 24.3 | 364.0 | 110.0 | 141.5 | 28.7 | 114.0 | 176.3 | 313.3 | 484.5 | 208.4 | 1.55 |
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15.0 | 24.5 | 368.0 | 110.0 | 136.3 | 23.9 | 118.5 | 171.3 | 322.0 | 465.5 | 202.4 | 1.45 |
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15.0 | 25.1 | 376.5 | 110.0 | 137.5 | 25.0 | 123.5 | 175.6 | 328.0 | 465.6 | 204.4 | 1.42 |
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15.0 | 25.0 | 375.0 | 110.0 | 137.8 | 25.3 | 122.5 | 174.3 | 326.7 | 464.7 | 207.3 | 1.42 |
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15.0 | 24.2 | 363.0 | 110.0 | 143.8 | 30.7 | 103.9 | 154.7 | 290.3 | 432.0 | 203.5 | 1.49 |
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15.0 | 24.4 | 366.5 | 110.0 | 143.3 | 30.3 | 109.1 | 162.5 | 297.5 | 443.1 | 196.3 | 1.49 |
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15.0 | 24.5 | 367.0 | 110.0 | 144.5 | 31.4 | 105.3 | 164.3 | 287.2 | 448.0 | 199.8 | 1.56 |
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15.0 | 25.0 | 375.5 | 110.0 | 143.9 | 30.8 | 113.7 | 165.4 | 302.8 | 440.4 | 198.5 | 1.45 |
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15.0 | 24.5 | 367.5 | 110.0 | 143.0 | 30.0 | 114.5 | 163.3 | 312.5 | 445.6 | 199.9 | 1.43 |
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15.0 | 24.4 | 366.5 | 110.0 | 143.8 | 30.8 | 114.0 | 174.6 | 311.2 | 476.7 | 188.1 | 1.53 |
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15.0 | 26.6 | 398.5 | 110.0 | 142.6 | 29.6 | 128.9 | 183.9 | 323.5 | 461.4 | 233.8 | 1.43 |
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15.0 | 24.4 | 366.5 | 110.0 | 145.5 | 32.3 | 107.7 | 162.0 | 293.9 | 442.1 | 241.4 | 1.50 |
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15.0 | 26.6 | 399.5 | 110.0 | 140.4 | 27.7 | 120.4 | 175.7 | 301.5 | 439.8 | 204.7 | 1.46 |
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15.0 | 24.8 | 371.5 | 110.0 | 141.5 | 28.6 | 108.1 | 158.4 | 306.7 | 449.3 | 203.5 | 1.47 |
Average value for samples 15 mm: |
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20.0 | 28.3 | 566.7 | 140.0 | 185.0 | 32.2 | 167.6 | 244.3 | 298.9 | 435.7 | 205.5 | 1.46 |
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20.0 | 29.1 | 582.0 | 140.0 | 179.1 | 28.0 | 153.8 | 268.2 | 264.2 | 460.9 | 198.5 | 1.74 |
Average value for samples 20 mm: |
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Average value for all samples: |
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All tested samples of steel collected from flanges of corrugated web girders had a proper steel structure. Discards of steel samples from flanges were regular and unambiguously indicated homogeneity of the material (Fig. 6).
The tests conducted on tension of steel samples from the flanges were used to draw graphs illustrating relationships between stress and strain
The guaranteed yield strength of delivered flat sheets used for preparing flanges of corrugated web girders was defined by the manufacturer as the specified minimum
The parameters of normal distribution were applied for the probability density distribution of yield strength of flanges
where:
Coefficients of variation of yield strength representing the safety level were determined in the first place as true
And partial factors of yield strength
where:
The partial factor of yield strength
Figure 9 illustrates normal distributions of yield strength obtained for the samples cut out from the flanges of girders with cantilever corrugated webs and for the whole tested population of the girders. Graphs in Figure 9 present distributions of mean value
The parameters of normal distribution and determined partial yield strength factors of the flange steel samples from corrugated web girders are shown in Table 3.
The parameters of normal distribution of yield strength
Girder No. of sample |
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γ |
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298.2 | 290.2 | 283.4 | 23.68 | 4.87 | 0.016 | 0.047 | 1.024 | 0.970 |
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303.4 | 299.3 | 295.8 | 6.18 | 2.49 | 0.008 | 0.057 | 1.012 | 0.930 |
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281.2 | 269.0 | 258.5 ** | 55.78 | 7.47 | 0.027 | 0.013 | 1.040 | 1.064 |
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291.1 | 283.0 | 276.1 | 24.52 | 4.95 | 0.017 | 0.034 | 1.025 | 0.996 |
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313.3 | 305.3 | 298.5 | 23.74 | 4.87 | 0.016 | 0.075 | 1.023 | 0.921 |
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322.0 | 311.4 | 302.4 | 41.43 | 6.44 | 0.020 | 0.089 | 1.030 | 0.909 |
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328.0 | 317.0 | 307.7 | 44.76 | 6.69 | 0.020 | 0.099 | 1.030 | 0.894 |
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326.7 | 310.7 | 297.1 | 94.99 | 9.75 | 0.030 | 0.096 | 1.046 | 0.926 |
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290.3 | 280.5 | 272.1 ** | 35.7 | 5.97 | 0.021 | 0.032 | 1.031 | 1.011 |
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297.5 | 289.2 | 282.0 | 25.94 | 5.09 | 0.017 | 0.046 | 1.025 | 0.975 |
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287.2 | 275.3 | 265.0 ** | 53.13 | 7.29 | 0.025 | 0.026 | 1.039 | 1.038 |
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302.8 | 287.9 | 275.1 | 82.76 | 9.10 | 0.030 | 0.056 | 1.046 | 0.999 |
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312.5 | 301.9 | 292.9 | 41.57 | 6.45 | 0.021 | 0.073 | 1.031 | 0.939 |
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311.2 | 302.3 | 294.7 | 29.58 | 5.44 | 0.017 | 0.071 | 1.026 | 0.933 |
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323.5 | 310.3 | 299.0 | 65.04 | 8.06 | 0.025 | 0.091 | 1.038 | 0.920 |
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293.9 | 282.6 | 272.9 ** | 47.93 | 6.92 | 0.024 | 0.039 | 1.036 | 1.008 |
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301.5 | 300.0 | 298.7 | 0.86 | 0.93 | 0.003 | 0.054 | 1.004 | 0.921 |
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306.7 | 298.7 | 292.0 | 23.52 | 4.85 | 0.016 | 0.063 | 1.023 | 0.942 |
Average values 15 mm |
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298.9 | 287.6 | 278.03 | 47.11 | 6.86 | 0.023 | 0.049 | 1.035 | 0.989 |
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264.2 * | 256.6 | 250.02 ** | 21.75 | 4.66 | 0.018 | -0.025 | 1.026 | 1.100 |
Average values 20 mm |
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Average values for all samples |
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mean value of yield strength of flanges lower than the manufacturer's declaration
design value of yield strength of flanges lower than the manufacturer’s declaration
Mean coefficients of variation
Similar conclusions can be drawn by analyzing partial factors of yield strength
The partial factors
The presented chapter shows that the use of the yield strength for bending resistance of girders equal to the value of the specified minimum
The performed tests showed that the mean yield strength
The difference between bending resistance of the tested girders, which was reached due to the guaranteed yield strength
For five tested girders, the obtained results for bending resistance that was determined on the basis of the specified minimum
It should be added that tension and compression resistance of flanges of corrugated web girders does not only depend on random variability of yield strength [28], but also on the arrangement of cross-sectional area of flanges in accordance with the following equations (17 and 18):
where:
where:
Moreover, random effects of yield strength and the geometry of flanges sum up in accordance with the principles of quantile algebra.
The chapter illustrates the impact of the true modulus of Young obtained from material tests on the displacements of the tested girders containing end – plate connections. Considering the modulus of Young, it is crucial to use this quantity on the basis of its mean value determined from the tests. The total mean modulus of longitudinal deformation for all samples reached 206.1 GPa during the tests on flange samples from the girders. Thus, this result did not significantly differ from the value specified in the standard EC3 [19]. But values obtained for some samples were far from the mean value. This situation was described on the example of the normal distribution of mean values of the longitudinal deformation modulus
When the mean value of Young modulus for the whole samples population was used in the calculations, then the effect of displacements was negligible. Another situation was observed when the mean modulus determined for the given girder was used to estimate displacements. If this modulus was additionally combined with the effect of displacements resulting from the used semi-rigid connections, the allowable displacements specified in the standard [19] were expected to be exceeded.
The effect of modulus of Young on displacements was shown on two examples of the tested girders: M 1.11 (Fig. 2a) with the scheme of simply supported beam and M 1.12 (Fig. 2c) with the scheme of simply supported beam with the cantilever.
In the case of corrugated web girder M 1.11 with the scheme of simply supported beam, the overall displacement y taking account of the effect of bending, shearing and rotation of two end-plate connections with stiffness
where:
In the case of corrugated web girder M 1.12 with the scheme of simply supported beam with a single cantilever, the overall displacement
where:
Rotational stiffness of connections in the tested girders were equal to 580 MNm/rad and 110 MNm/rad respectively. Displacements were determined for the critical load obtained from the tests [25].
Table 4 presents results for displacements that depend on the modulus of longitudinal deformation and rotational stiffness of connections
Total displacements y of girders with a corrugated web depending on the modulus of Young
Girder No. of sample | y [mm] |
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M 1.11 | 24.9 | 24.3 | 24.9 | 1.02 | 1.23 | 23.3 |
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M 1.12 | 6.0 | 5.6 | 4.3 | 1.07 | 1.4 | 6.6 |
The effect of longitudinal deformation modulus
Bending resistance of corrugated web girders was affected by the quantile of the product of yield strength and random sectional area of the corrugated web.
When yield strength of ready flanges for corrugated web girders are defined, it is recommended to specify the relationships between the design yield strength and its mean value expressed as:
The design yield strength related to the specified minimum value expressed below can be alternatively used to design of the structure made of corrugated web girders, which was confirmed by the conducted tests.
Steel having similar yield strength is generally used to manufacture flanges of corrugated web girders.
Yield strength values of flange samples from the corrugated web girders from different batches were in 75% cases greater than the values declared by the manufacturer of corrugated webs, and in the remaining 25% cases these values were smaller. Thus, construction elements used at the construction site can have significantly diverged strength parameters.
Experimentally determined partial factor of yield strength
It should be mentioned that tests on metallurgical products do not include random arrangement of sectional thickness of the sample, which also affects bending resistance.
The effect of the modulus of Young