Experimental and Analytical Studies of U-Shaped Thin-Walled RC Beams Under Combined Actions of Torsion, Flexure and Shear
Abstract
U-shaped thin-walled concrete bridge beams usually suffer the combined actions of flexure, shear and torsion, but no research about the behavior of U-shaped thin-walled RC beams under combined actions has been reported in literature. Three large specimens of U-shaped thin-walled RC beams were tested under different torque–bending moment ratios (T–M ratios) of 1:5, 1:1 and 1:0 to investigate the mechanical responses such as crack patterns, reinforcement strains, failure modes and ductility. The testing results showed that ductile flexural failures occurred for all three of the U-shaped thin-walled beam specimens, although the combined shear effect of circulatory torque, warping torque and shear force increased as the T–M ratio increased from 1:5 via 1:1 to 1:0, reflected by diagonal cracks and stirrup strains. More specifically, basically symmetrical flexural failure was dominated by the bending moment when the T–M ratio was 1:5; flexural failure of the loaded half of the U-shaped thin-walled section was dominated by the combined action of the bending moment and warping moment, while there were only a few cracks on the other half of the U-shaped section when the T–M ratio was 1:1; and anti-symmetrical flexural failure was dominated by the warping moment when the T–M ratio was 1:0 (pure torsion). A simple method to calculate the ultimate load of such U-shaped thin-walled RC beams under different T–M ratios was suggested, and the calculating results were corresponding well with the experimental results.
Keywords
reinforced concrete U-shaped thin-walled beam combined action of torsion and flexure warping torsion ultimate load1 Introduction
U-shaped thin-walled reinforced concrete (RC) bridge beams have been widely used in urban construction of rail viaducts in China, such as in Shanghai Subway Line 6 and Guangzhou Subway Line 2, due to the advantage of lower construction elevations, the excellent sound proof effect and attractive appearance (He 2003). Under normal service conditions, this type of member is typically subjected to the combined action of bending, shear and torsion due to eccentric traffic load (especially in multilane cases and curved structure cases) and transverse wind load. However, according to the authors’ knowledge, no research about the mechanical response of such U-shaped thin-walled RC members under combined actions has been reported in literature. The existing studies about the U-shaped thin-walled RC bridge beams are mainly concentrated on bending and shear. In China, the additional torsional effect is indirectly considered by improving the safety reserve in bending and shear design, which is unreasonable and uneconomical. Thus, the mechanical response of the U-shaped thin-walled RC beams under the combined action of bending, shear and torsion should be studied to lay the foundation for the development of a rational design provision.
The major difference in the mechanical mechanism between a member with an open thin-walled section and a member with a closed section is the torsional response. When a closed section member is under pure torsion, the warping effect is too weak to be neglected, thus only the well-known circulatory torsion, or St. Venant’s torsion, is considered. After a century of exploration, there are lots of achievements in research aimed at the torsional response of RC members with closed sections, and the widely applied theoretical model is the spatial softened truss model (Mitchell and Collins 1974; Hsu and Mo 1985; Vecchio and Collins 1986; Rahal and Collins 1995a, b, 2006; Jeng and Hsu 2009; Bernardo et al. 2012a, 2015). Basically, with specific revisions, it is applicable to all situations dominated by circulatory torsion, such as high strength concrete members (Bernardo et al. 2012b) and box section members (Bernardo et al. 2013; Jeng 2015; Wang et al. 2015). The softened truss model is also the theoretical model used to simulate the behavior of closed section concrete members under shear (planar softened truss model) (Vecchio and Collins 1981, 1986, 1988; Pang and Hsu 1995; Tadepalli et al. 2015; Liang et al. 2016) as well as under the combined actions of bending, shear and torsion (Rahal and Collins 2003; Rahal 2007; Greene and Belarbi 2009a, b).
As to torsional members with open thin-walled sections, the warping effect is not negligible. According to Vlasov’s elastic theory of the open thin-walled member (Vlasov 1961), when the warping deformation of an open thin-walled member under torsion is restrained, a new internal force called warping moment corresponding to warping normal stress will appear. When it occurs, two kinds of internal torque appears simultaneously, which are circulatory torque (the same as that in the closed section case) and warping torque. In 1961, Vlasov (1961) developed the sectorial coordinate system and derived the theoretical formula to calculate warping torque and warping moment for open thin-walled members, which became the basis for analyzing open thin-walled members under torsion. Thereafter, some research outcomes on the elastic torsional response of the open thin-walled member, especially focusing on the shear deformation induced by the warping torque, have been reported (Pavazza 2005; Erkmen and Mohareb 2006; Murín and Kutiš 2008; Aminbaghai et al. 2016). When it comes to the post cracking torsional behavior of RC members with an open thin-walled section, the above mentioned softened truss model for circulatory torsion is not accurate anymore because of the considerable warping effect (Luccioni et al. 1991). In addition the Vlasov’s elastic theory should be revised due to the cracking of concrete. Zbirohowski-Koscia (1968) first addressed issues related to the post-cracking behavior of open thin-walled RC beams under the warping moment. In 1981, Krpan and Collins tested the torsional response of a fixed–fixed U-shaped thin-walled RC beam (Krpan and Collins 1981a). The results confirmed the dominate role that the warping moment played. In the analogy to bending, based on Vlasov’s theory, the method to simulate the post cracking torsional behavior of the U-shaped thin-walled RC beam was proposed (Krpan and Collins 1981b). Then Hwang and Hsu (1983) analyzed the entire torsional behavior of the RC channel beam with a method from the Fourier series approach. In the following two decades, few research outcomes on the torsional behavior of open thin-walled RC members under torsion were reported in literature. Due to the wide application of U-shaped thin-walled RC beams in the construction of rail viaducts in recent years, their torsional behavior has again drawn research’s attention. Theoretical and experimental studies on the torsional behavior of U-shaped thin-walled RC beams have been carried out by our research group (Chen et al. 2016a, b), and based on Vlasov’s torsional theory and the nonlinear constitutive relations of materials, a nonlinear model to predict the torsional response of such U-shaped thin-walled RC beams has been proposed.
Research into the mechanical response of open thin-walled RC members under combined actions of bending, shear and torsion is quite rare. Analytical and experimental studies on the behavior of RC I-beams under combined bending, shear and torsion were conducted by Luccioni et al. (1991, 1996), and a calculation method of ultimate load based on the skew bending theory (Elfgren et al. 1974) was proposed, where the skew bending theory was modified by taking warping torque into account. In the calculation method, the effect of the warping moment was neglected because, due to the specific geometrical properties of the I-section, the effect of the warping moment was weak and ignorable compared to the effect of the bending moment. However, it is not the case for U-shaped thin-walled RC beams studied in this paper as the geometrical properties of the U-section will make the effect of the warping moment as strong as the bending moment.
Considering the strong warping effect in the U-sections, the interaction results of bending moment and warping moment, as well as the interaction results of circulatory torque, warping torque and shear force should be experimentally studied. In the current paper, considering the variation of the torque–bending moment ratio (T–M ratio) in practice, three large U-shaped thin-walled RC beam specimens will be respectively tested under different T–M ratios of 1:5, 1:1 and 1:0 to investigate the mechanical response. The crack pattern, reinforcement strain, failure mode and ductility will be analyzed. Finally, an approach of calculating the cracking load and the ultimate load of each kind of beam will be proposed, which will make meaningful contributions to developing design guidelines for U-shaped thin-walled RC beams subjected to combined bending, shear and torsion.
2 Experimental Plan
2.1 Testing Specimens
Material properties and loading conditions of beam specimens.
MEM-1:5 | MEM-1:1 | MEM-1:0 | |||
---|---|---|---|---|---|
Material properties | Longitudinal bars | Yield strength (MPa) | 353.33 | 353.33 | 353.33 |
Ultimate strength (MPa) | 573.33 | 573.33 | 573.33 | ||
Elastic modules (GPa) | 200 | 200 | 200 | ||
Diameter (mm) | 8 | 8 | 8 | ||
Stirrups | Yield strength (MPa) | 276.7 | 276.7 | 276.7 | |
Ultimate strength (MPa) | 446.7 | 446.7 | 446.7 | ||
Elastic modules (GPa) | 200 | 200 | 200 | ||
Diameter (mm) | 6 | 6 | 6 | ||
Spacing (mm) | 70 | 70 | 70 | ||
Concrete | Compressive strength of prism specimen (MPa) | 39.62 | 35.40 | 40.92 | |
Elastic modules (GPa) | 36.7 | 34.5 | 34.4 | ||
Loading conditions | Eccentricity (mm) | 166.25 (L/40) | 831.25 (L/8) | Pure torsion | |
T-BM ratio | 1:5 | 1:1 | 1:0 |
2.2 Experimental Setup
2.3 Location of Potential Critical Sections
As shown in Fig. 3b, normal stress due to warping moment M_{ω} anti-symmetrically distributes around the cross-section, while normal stress due to bending moment M_{b} symmetrically distributes; therefore they enhance each other on the loaded half U-section while counteracting on the unloaded half U-section. Thus, for MEM-1:5 and MEM-1:1, under the combined actions of bending moment and warping moment, the loaded half U-section will be more critical than the unloaded half U-section. As shown in Fig. 3b, shear stresses due to circulatory torque, warping torque and shear force distribute differently around the cross-section, and they enhance one another on the external surface of the loaded web while they do not elsewhere. Therefore, under the combined actions of circulatory torque, warping torque and shear force, the external surface of the loaded web will be more critical than other surfaces of the U-section.
2.4 Arrangement of Measurement Points and Testing Procedure
3 Test Results and Discussion
3.1 Failure Procedure
Characteristic loads and their locations of beam specimens.
MEM-1:5 | MEM-1:1 | MEM-1:0 | ||
---|---|---|---|---|
Flexural cracking | Load percentage | 13% | 11% | 12% |
Load detail | P = 27.0, e = 0.166, T = 4.5, M = 22.5 | P = 11.5, e = 0.831, T = 9.6, M = 9.6 | P = 13.0, e = 1.33, T = 17.3, M = 0 | |
Location | Top of loaded web at support | Top of loaded web at support | Top of loaded web at support and top of unloaded web at mid-span | |
Diagonal cracking | Load percentage | 49% | 33% | 24% |
Load detail | P = 103.0, e = 0.166, T = 17.1, M = 85.5 | P = 36.6, e = 0.831, T = 30.4, M = 30.4 | P = 26.8, e = 1.33, T = 35.7, M = 0 | |
Location | External surface of loaded web at 3L/8-section | External surface of loaded web at 5L/16-section | External surfaces of both webs at L/4-section | |
Steel bar yielding | Load percentage | 57% | 58% | 60% |
Load detail | P = 119.3, e = 0.166, T = 19.8, M = 99.0 | P = 64.3, e = 0.831, T = 53.4, M = 53.4 | P = 66.3, e = 1.33, T = 88.2, M = 0 | |
Location | Top of loaded web at support | Top of loaded web at support | Top of loaded web at support and top of unloaded web at mid-span | |
Stirrup yielding | Load percentage | Not yield | Not yield | 86% |
Load detail | Not yield | Not yield | P = 95.1, e = 1.33, T = 126.4, M = 0 | |
Location | Not yield | Not yield | External legs of both webs at L/4-section | |
Ultimate state | Load detail | P = 209.6, e = 0.166, T = 34.8, M = 174.0 | P = 110.7, e = 0.831, T = 92.0, M = 92.0 | P = 110.5, e = 1.33, T = 147.0, M = 0 |
Location | Ductile flexural failure at both support and mid-span sections of loaded web | Ductile flexural failure at both support and mid-span sections of loaded web | Ductile flexural failure at both support and mid-span sections of two webs |
3.2 Rotations, Deflections and Ductility
Ductility coefficient of beam specimens.
Rotational ductility coefficient | Deflection ductility coefficient | |
---|---|---|
MEM-1:5 | 9.2 | 10.3 |
MEM-1:1 | 9.3 | 9.5 |
MEM-1:0 | 8.4 | No deflection |
3.3 Crack Patterns
The crack patterns of the observed span of the three beam specimens are shown in Fig. 5, in which the numerical values are the corresponding external torques, and the bordered ones are first flexural cracking torques (the smaller ones) or first diagonal cracking torques (the larger ones). It can be seen from Fig. 5 that, for the three beam specimens, the cracks at support and mid-span are mainly vertical flexural types, displaying small shear features, which means at support and mid-span, warping moment and bending moment predominate while shear force and warping torsion had little influence. Cracks at the L/4-section are diagonal types with an inclination of about 45°. The distributions of cracks along the beam span conform well to the distributions of internal forces shown in Fig. 3a. For MEM-1:5, flexural cracks on the loaded web were more fully developed than those on the unloaded web; for MEM-1:1, flexural cracks were fully developed on the loaded web but only a few flexural cracks appeared on the unloaded web; for MEM-1:0, flexural cracks anti-symmetrically developed on two webs. The different development of flexural cracks on two webs of each beam was attributable to the interaction of the bending moment and warping moment under different T–M ratios, which will be discussed in detail in the following section. As to diagonal cracks, for MEM-1:5 and MEM-1:1, they developed most fully on the external surface of the loaded web; for MEM-1:0, they developed most fully on the external surfaces of the loaded and unloaded webs. That agrees with the distribution of shear stress resultant around the U-section (Sect. 2.3). Overall, with the increase of the T–M ratio from 1:5 via 1:1 to 1:0, diagonal cracks were more and more fully developed, which means the effect of shear stress resultant increased.
3.4 Failure Modes
It can be also seen from Fig. 12 that for the three beam specimens, longitudinal bar strains changed in a linear fashion around the U-shaped section, which agrees with the distribution of normal stresses induced by the bending and warping moment.
4 Method to Calculate the First Cracking Load and the Ultimate Load
A method to calculate the first cracking (i.e. flexural cracking) load and the ultimate load of the U-shaped thin-walled RC beams under combined actions of torsion, bending and shear will be suggested here. Considering the experimental results that, for all three beam specimens under different T–M ratios, the ductile flexural failure was dominated by the warping moment and bending moment at mid-span and at support, and the shear action (combined shear action of shear force and torque) at mid-span and at support is very weak and negligible. This was reflected by the extremely small stirrup strains at mid-span shown in Figs. 9b, 10b and 11b; thus the algorithm is measured by the normal stresses caused by the warping moment and bending moment at support and mid-span, where the shear effect is neglected.
4.1 Calculation of the Flexural Cracking Load
Comparison of test cracking torque and calculated cracking torque.
Beams | Cracking torque (kNm) | Test/calculation | |
---|---|---|---|
Test | Calculation | ||
MEM-1:5 | 4.5 | 4.7 | 0.96 |
MEM-1:1 | 9.6 | 10.2 | 0.94 |
MEM-1:0 | 17.3 | 16.9 | 1.02 |
MEM-Collins-1:0 | 23 | 18.5 | 1.24 |
Average value | 1.04 | ||
SD | 0.12 | ||
Coefficient of variation | 11.5% |
4.2 Calculation of the Ultimate Load
Considering the interaction of bending normal stress and warping normal stress, for MEM-1:5 and MEM-1:1, the flexural failure modes have confirmed that the loaded half U-section is more critical than the unloaded half U-section (see details in Sects. 2.3 and 3.4). As for MEM-1:0, the flexural failure mode was dominated by the warping moment, the loaded half U-section was also critical as the unloaded half U-section. Therefore, for U-shaped thin-walled RC beams with different T–M ratios, the ultimate bearing capacity could be determined with the loaded half U-section at support and mid-span considering the combined actions of the bending moment and warping moment.
4.2.1 Equivalent Action Acting on the Loaded Half U-Section at Mid-Span
4.2.2 Ultimate Equilibrium Equation on the Loaded Half U-Section at Mid-Span
4.2.3 Comparison of Calculating Results and Testing Results
Comparison of test ultimate torques and calculated ultimate torques.
Beams | Ultimate torque (kNm) | Test/calculation | |||||
---|---|---|---|---|---|---|---|
Test | Proposed method | Elfgren et al. (1974) | Luccioni et al. (1991) | Proposed method | Elfgren et al. (1974) | Luccioni et al. (1991) | |
MEM-1:5 | 34.8 | 35.5 | 28.9 | 38.3 | 0.98 | 1.20 | 0.91 |
MEM-1:1 | 92.0 | 88.5 | 71.8 | 111.5 | 1.04 | 1.28 | 0.83 |
MEM-1:0 | 147.0 | 139.0 | 107.3 | 198.2 | 1.06 | 1.37 | 0.74 |
MEM-Collins-1:0 | 266.0 | 248.9 | 198.7 | 369.8 | 1.07 | 1.34 | 0.72 |
Average value | 1.04 | 1.29 | 0.80 | ||||
SD | 0.04 | 0.07 | 0.07 | ||||
Coefficient of variation | 3.8% | 5.4% | 8.8% |
Also, the ultimate capacities are calculated with the method proposed by Elfgren et al. (1974) and with the method proposed by Luccioni et al. (1991). As can be seen from Table 5, according to the method proposed by Elfgren and Karlsson et al., the average ratio of test results and calculated results was 1.29, which should be attributed to the fact that the method based on the skew bending theory did not take into account the restrained torsion mechanism. For the calculation method proposed by Luccioni and Reimundin et al., as shown in Table 5, the average ratio of test results and predicted ultimate torques is 0.8. The reason for overestimating the ultimate capacities is that although the restrained torsion mechanism was considered, the effect of warping moment is underestimated.
5 Conclusions
- 1.
Ductile flexural failures occurred on the three beam specimens, which were dominated by the combined actions of the bending moment and warping moment. Specifically, for T–M ratios of 1:5, flexural failure was mainly dominated by the bending moment; for T–M ratios of 1:1, flexural failure of the loaded half U-section was dominated by the combined action of the bending moment and warping moment, while normal stress resultant on the unloaded half U-section was very small; for T–M ratios of 1:0, antisymmetric flexural failure of the two webs was dominated by the warping moment.
- 2.
As the T–M ratio increased, the combined shear stress resultant of circulatory torque, warping torque and shear force increased, reflecting more diagonal cracks and larger stirrup strains.
- 3.
The three beam specimens under different T–M ratios showed good ductile performance, with a deflection ductility coefficient larger than 9.0 and rotational ductility coefficient larger than 8.0.
- 4.
A simple method to calculate the flexural cracking torque and the ultimate torque of the U-shaped thin-walled RC beams under combined actions of torsion, bending and shear was suggested, and the calculated results were corresponding well with the testing results.
Notes
Acknowledgements
The funding from the National Natural Science Fund (NSF) of China (Grant No. 51778032 and 51278020), as well as the funding from the Texas Department of Transportation (Project No. 0-6914) are highly appreciated. Also, the scholarship funded by China Scholarship Council (CSC) with No. 201706020108 for the first author to study at the University of Houston is appreciated.
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