A comparison of the mechanical properties of curved box girders with corrugated steel webs to curved box girders with concrete webs

Abstract

Curved composite box girder bridges with corrugated steel webs (CSWs) have already been constructed around the world. However, limited work has been done on the comparisons of mechanical properties between curved box girders (CBGs) with CSWs and CBGs with traditional concrete webs. This can be attributed to the difficulty separating flexure, torsion, and distortion using experimental or finite element analysis methods. In this paper, first, a practical method that can solve deflections, torsional angles, distortional angles, stresses, and internal forces of simply supported and continuous CBGs with intermediate diaphragms is introduced. Then, a parametric analysis is conducted to study the difference in mechanical properties of CBGs with CSWs and CBGs with concrete webs. The results show that: (1) Compared with traditional concrete box girders, the torsional and distortional resistances of box girders with CSWs are significantly reduced. Under external loading, all deflections, torsional angles, distortional angles, flexural normal stresses, restrained torsional and distortional warping normal stresses are generally larger. (2) For the example presented in this paper, the torsional warping normal stresses can reach 30% of the flexural normal stresses, and the distortional warping normal stresses can reach 80% of the flexural normal stresses. For this reason, they cannot be neglected in the design. (3) By increasing the number of intermediate diaphragms, drawbacks of using CSWs due to the increase in distortional angles and distortional warping normal stresses can be mitigated, closing the gap between the 2 different web types. In this regard, it is suggested that at least 1 diaphragm should be installed at midspan and if the diaphragms are arranged by equal spacing, their number is suggested to be odd.

Keywords

Curved box girders corrugated steel webs flexural properties torsional properties distortional properties

Introduction

The steel–concrete composite box girder with corrugated steel webs (CSWs) is a relatively new type of bridge structure overcoming the weight problem of common concrete box girders. Compared with traditional concrete webs, the replacement of concrete webs by CSWs will greatly reduce its dead weight, thus improving the spanning capacity. CSWs can also solve potential cracking of concrete webs. Since the Cognac Bridge, the world’s first composite box girder bridge with CSWs was built in France in 1986 (Cheyrezy & Combault, 1990), this new bridge structure has been widely used in other countries. Currently, Japan is the world’s largest builder of composite box girder bridges with CSWs, with more than 220 built or under construction (CSWBA, 2021). China is the second largest with more than 80 (Shen, 2018). Apart from straight box girders (SBGs), CSWs are also widely used in curved box girders (CBGs) (see, e.g., Figure 1), for example, the Meaux viaduct in France, the Altwipfergrund viaduct in Germany, the Nakano viaduct in Japan, the Yuwotou bridge, and the No. 3 East River bridge in China.

Figure 1.

Curved composite box girder bridges with CSWs.

In this concept, the CSWs have a low longitudinal stiffness due to the accordion effect, so the CSWs mainly carry the shear force and barely carry axial force (Hamilton, 1993). Because of this characteristic, the mechanical properties of composite box girders with CSWs are quite different from traditional concrete box girders (Cheyrezy and Combault, 1990). For example, the shear deformations of CSWs become a major contributor to the total deformations (Zhou et al., 2016), and also, the replacement of the concrete webs by CSWs will reduce the torsional stiffness of the structure to about 30%–40% of that of common concrete box girders (PCTA, 2005). In the calculation example given by Jiang et al. (2015), the vertical flexural stiffness and lateral flexural stiffness of composite box girders with CSWs are 10% and 26% smaller than those of concrete box girders respectively. In addition, the restrained torsional and distortional warping normal stresses under eccentric load should not be neglected (Qiao et al., 2018). However, the above examples are SBGs with CSWs. Because CBGs must consider the flexure–torsion coupling effect, and torsion is an important effect in CBGs, it is of great importance to make a comparison of the mechanical properties of CBGs with CSWs to CBGs with concrete webs.

The mechanical properties of SBGs with CSWs have been studied by various researchers in recent years. A first research topic is related to the flexural properties. Mo et al. (2003) conducted four reduced-scale prestressed concrete box girders with CSWs and proposed an analytical model that can be used to predict the load–displacement relationship. He et al. (2014) conducted tests on full-scale specimens to investigate the deflections, strain distribution, loading capacity, and stiffness during the loading process and presented a simplified analytical model to predict the load capacity. Xu et al. (2015) introduced the spatial grid model, a new thought for the detailed design, for the mechanical analysis. Chen et al. (2015) first numerically and experimentally studied the flexural ductility of reinforced and prestressed concrete sections with CSWs. Then, they extended the sandwich beam theory to predict the full-range behavior taking into account the interaction between the girder, diaphragms, and external tendons (Chen et al., 2016). Lu and Ji (2018) introduced two methods to optimize and upgrade the traditional prestressed concrete composite box girders with CSWs. Dong et al. (2021) proposed the equations to calculate the flexural bearing capacity under positive and negative bending moments for box girders with CSWs and trusses.

A second topic is related to torsional and distortional properties. As is mentioned above, apart from the flexural tests, Mo et al. also performed torsional tests (Mo et al., 2000) and presented an analytical model to predict the torsional behavior of such girders based on these test results (Mo and Fan, 2006). Ding et al. (2012) first investigated the influence of the thickness of CSWs and the strength of the concrete on the torsional properties. Then they carried out tests on four specimens under pure torsion (Ding et al., 2013) and on two specimens under torsion combined with bending (Ding et al., 2014). The results showed that the ultimate torsional strength of these specimens was linearly related to the thickness of the CSWs and the compressive strength of the concrete. The presence of bending decreases the torsional strength, and the influence of the flexural moment on the torsional strength is more obvious for box girders with CSWs compared to concrete box girders. Ko et al. (2013) proposed a simplified torsional moment–twist angle relationship. Shen et al. experimentally, numerically, and theoretically studied the torsional behavior of single-box multi-cell box girders with CSWs under pure torsion (Shen et al., 2018a) and presented a theoretical model called the unified softened truss model for torsion (Shen et al., 2018b). Qiao et al. (2018) theoretically studied the warping torsional and distortional stresses and suggested that the warping normal stress and the additional shear stress due to warping torsion and distortion should be considered in the structural analysis. From the studies mentioned above, it is clear that a lot of work has been done related to the mechanical properties of box girders with CSWs. However, all of the mentioned work is only related to SBGs with CSWs.

While a lot of work has been done related to the mechanical properties of SBGs with CSWs, limited work has been done on CBGs. Compared to SBGs, the mechanical properties of CBGs are much more complicated because of the flexure–torsion coupling effect. Therefore, it is of great significance to make a systematic and intensive study of the mechanical properties of CBGs with CSWs for the promotion and application of this technique in curved bridges.

Unfortunately, it is difficult to separate flexure, torsion, and distortion using experimental or finite element analysis (FEA) methods. In this paper, first, a practical method that can solve the deflections, torsional angles, distortional angles, stresses, and internal forces of simply supported and continuous CBGs with intermediate diaphragms is introduced. Then, a parametric analysis is conducted to study the difference of the mechanical properties of CBGs with CSWs and CBGs with concrete webs.

Theoretical background

It is difficult to extract the pure flexural effect and the pure torsional effect of CBGs due to the flexure–torsion coupling effect from experimental results and a data obtained by FEA. In this paper, an analytical method is used to make a comparison of the mechanical properties of CBGs with CSWs to CBGs with concrete webs.

Liu et al. (2021) proposed a practical method which can solve deflections considering the shear deformations of CSWs, torsional angles, distortional angles, stresses, and internal forces of simply supported and continuous CBGs with intermediate diaphragms. The results of a series of tests performed on three CBGs with CSWs, FEA results of CBGs with CSWs, and published test results, FEA results, theoretical results of SBGs with CSWs were used to verify the correctness of the analytical method. Similarly, an analytical method that applies to traditional concrete CBGs also needs to be proposed.

For the flexure–torsion coupling effect, the analytical method proposed by Liu et al. (2021) considers both the shear deformations as well as the accordion effect of CSWs. However, the shear deformations of the webs are very small and can be ignored for traditional concrete box girders. In that case, there are only 3 displacement functions (deflection function, torsional angle function, and warping function) in the flexure–torsion governing differential equations. Just as the solving method of CBGs with CSWs, the 3 displacement functions adopt trigonometric series, and the flexure–torsion governing differential equations of traditional concrete CBGs can be solved by the Galerkin method. For the distortion effect, the analytical method proposed by Liu et al. (2021) considers the accordion effect of CSWs. Also, the distortional angle function adopts trigonometric series, and the distortion governing differential equation of traditional concrete CBGs can be solved by the Galerkin method. Therefore, a practical method that can solve the deflections, torsional angles, distortional angles, stresses and internal forces of simply supported and continuous concrete CBGs with intermediate diaphragms can also be proposed. The analytical method proposed by Liu et al. (2021) can be referenced for the detailed solution process.

The published FEA and theoretical results obtained by Guo et al. (2018), Park et al. (2002), and Heins and Oleinik (1976) are used to verify the analytical method for concrete or steel box girders which consider only one elastic modulus and one shear modulus. The geometric dimensions and material characteristics of the specimens are given in Table 1. All specimens are single-span simply supported box girders with 2 end diaphragms. The number of intermediate diaphragms n is also shown in Table 1. As is shown in Figure 2, b_t is the top flange width; b_b is the bottom flange width; t_t is the thickness of the top flange; t_b is the thickness of the bottom flange; b_t1 and b_b1 are the top flange width and bottom flange width of the closed box girder cross section; h_w is the web length equal to the clear distance between the top and bottom concrete flanges.

Table 1.

Geometric dimensions and material characteristics of the published concrete or steel box girders.

Specimen	R (mm)	n	Span L (mm)	Top flange (mm)			Bottom Flange (mm)			Web (mm)		E_c or E_s (GPa)	μ_c or μ_s
Specimen	R (mm)	n	Span L (mm)	b _t	b _t1	t _t	b _b	b _b1	t _b	t _w	h _w	E_c or E_s (GPa)	μ_c or μ_s
Guo et al. (2018)	∞	0	40000	9500	4700	220	4700	4700	340	300	1840	34	0.2
Park et al. (2002)	∞	0	40000	3000	3000	10	3000	3000	10	10	1990	205.947	0.3
Park et al. (2002)	∞	4	40000	3000	3000	10	3000	3000	10	10	1990	205.947	0.3
Heins and Oleinik (1976)	30480	7	15240	4876.8	2438.4	24.0538	2438.4	2438.4	12.7	9.525	635.6731	206.832	0.2931

Note. E_c and μ_c are the elasticity modulus and Poisson’s ratio of concrete; E_s and μ_s are the elasticity modulus and Poisson’s ratio of steel.

Figure 2.

The cross section of a box girder.

Case 1

Guo et al. (2018) theoretically studied the restrained torsional warping normal stress of concrete SBGs under eccentric load. The eccentric uniform distributed load q=100 kN/m is applied over the entire girder span. Figure 3 shows the comparison of restrained torsional warping normal stresses at the midspan and quarter span cross sections calculated by the analytical method used in this paper and obtained by Guo et al. (2018). In Figure 3, the theoretical results obtained by Guo et al. (2018) are presented between brackets ( ). It can be seen that the restrained torsional warping normal stresses calculated by the two methods are in good agreement, which means that the analytical method used in this paper is effective to calculate the restrained torsional warping normal stress.

Figure 3.

Restrained torsional warping normal stresses (unit: MPa) (Guo et al., 2018). (a) Midspan cross section (b) quarter span cross section.

Case 2

Park et al. (2002) theoretically studied the distortional warping normal stress of steel SBGs under eccentric concentrated load p = 981 kN applied at the midspan cross section. The distortional warping normal stresses of the bottom edge are 202.46 MPa and 39.64 MPa for the SBGs without intermediate diaphragms and with 4 intermediate diaphragms using the theoretical method used in this paper, and the results obtained by Park et al. (2002) are 201.11 MPa and 39.24 MPa, respectively. It can be seen that the stresses obtained by the two different methods are in good agreement, which means that the method used in this paper is effective to calculate the distortional warping normal stress.

Case 3

Heins and Oleinik (1976) theoretically studied the flexure–torsion and distortion of CBGs. Tables 2 and 3 give the displacements, internal forces, and normal stresses of a steel CBG under eccentric concentrated load (see Figure 4, p = 4×4.448 kN). It can be seen from Tables 2 and 3 that the results calculated by the method used in this paper agree well with the theoretical results obtained by Heins and Oleinik (1976) except for the restrained torsional warping normal stresses. This can be attributed to the fact that Heins and Oleinik (1976) adopted the first-order derivative of the torsion angles as the warping function for restrained torsion analysis, which is not precise. However, the theoretical method used in this paper and in Liu et al. (2021) introduces a new warping function β, which is more precise.

Table 2.

Displacements, internal forces of the steel CBG (Heins and Oleinik, 1976).

Location	Point	The method proposed by Heins and Oleinik (1976)						The method used in this paper
Location	Point	v (mm)	φ (×10⁻⁴)	γ (×10⁻⁷)	M_x (kN.m)	Q_y (kN)	T (kN.m)	v (mm)	φ (×10⁻³)	γ (×10⁻⁷)	M_x (kN.m)	Q_y (kN)	T (kN.m)
0	1	0	0	0	0	130.24	245.46	0	0	0	0	129.77	245.00
0.04 L	9	1.09	−1.15	6.43	84.30	130.24	244.59	1.09	−1.15	6.41	84.30	129.76	244.15
0.08 L	17	2.17	−2.29	7.17	168.56	130.24	242.06	2.17	−2.29	7.15	168.56	129.74	241.60
0.12 L	25	3.23	−3.40	1.00	252.75	130.24	237.85	3.22	−3.40	0.99	252.75	129.68	237.34
0.16 L	33	4.23	−4.49	−6.94	336.85	130.24	231.95	4.23	−4.49	−6.92	336.85	129.55	231.32
0.2 L	41	5.18	−5.53	−13.50	420.81	130.24	224.37	5.18	−5.53	−13.43	420.80	128.94	223.19
0.24 L	49	6.05	−6.47	−4.87	493.42	94.65	182.64	6.05	−6.47	−4.84	493.42	95.10	183.05

Note. v, φ, γ are the deflections, torsional angles, distortional angles, respectively; M_x, Q_y, T are the flexural moment, shear force, torsional moment, respectively.

Table 3.

Normal stresses of the steel CBG (Heins and Oleinik, 1976).

Location, L	Point	The method proposed by Heins and Oleinik (1976) (MPa)						The method used in this paper (MPa)
Location, L	Point	σ _m,E	σ _m,B	σ _w,E	σ _w,B	σ _dw,E	σ _dw,B	σ _m,E	σ _m,B	σ _w,E	σ _w,B	σ _dw,E	σ _dw,B
0.04	9	−1.125	3.737	0.036	0.049	0.025	−0.122	−1.124	3.733	0.044	0.059	0.025	−0.122
0.08	17	−2.250	7.471	0.072	0.098	0.031	−0.154	−2.248	7.465	0.091	0.124	0.031	−0.154
0.12	25	−3.374	11.204	0.112	0.153	0.001	−0.006	−3.371	11.194	0.148	0.202	0.001	−0.004
0.16	33	−4.496	14.931	0.181	0.246	0.000	−0.002	−4.493	14.919	0.226	0.308	0.000	−0.002
0.2	41	−5.617	18.653	0.426	0.581	−0.062	0.303	−5.613	18.637	0.343	0.468	−0.062	0.303
0.24	49	−6.586	21.872	0.457	0.623	−0.038	0.185	−6.582	21.853	0.379	0.517	−0.038	0.186

Note. σ_m,E, σ_w,E, σ_dw,E are the flexural normal stress, restrained torsional warping normal stress, distortional warping normal stress of the corner point E; σ_m,B, σ_w,B, σ_dw,B are the flexural normal stress, torsional warping stress, distortional warping stress of the corner point B.

Figure 4.

A steel CBG under eccentric concentrated load (Heins and Oleinik, 1976). (a) Longitudinal direction (b) radial direction.

From the above, we can see that the analytical methods are effective and can be used to make a comparison of the mechanical properties of CBGs with CSWs to concrete CBGs in the next section.

A comparison of the mechanical properties of CBGs with CSWs to concrete CBGs

A single-span simply supported CBG with CSWs with 0, 1, 2, 3 intermediate diaphragms with the following dimensions is adopted. The girder span L is 48m. As is shown in Figures 5 and 6, b_t=8m, b_b=5.2 m, b_t1=4.8 m, b_b1=4.8 m, t_t=t_b=300 mm, t_w=10 mm, h_w=2.4 m, the flat panel width a_w=430 mm, the horizontal projection of the inclined panel width b_w=370 mm, the inclined panel width c_w=430 mm. Material parameter: E_c=3.45×10⁴ MPa, E_s=2.1× 10⁵ MPa, μ_c=0.2, μ_s=0.3. A concentrated load and a uniformly distributed load as shown in Figure 7 are applied. In addition, a CBG with 350 mm concrete webs, which is common in actual concrete bridges, is also studied. The distance between two adjacent diaphragms is L/(n+1), where n is the number of the intermediate diaphragms.

Figure 5.

Cross section of CBGs (Unit: mm). (a) a CBG with CSWs (b) a CBG with concrete webs.

Figure 6.

Geometric dimensions of CSWs (Unit: mm).

Figure 7.

Loading cases. (a) Loading points applied in the longitudinal direction. (b) Loading points applied in the radial direction.

Displacements analysis

Table 4 gives the flexural, torsional, and distortional parameters of CBGs with 2 different web types. In Table 4, I_x is the flexural geometric moment of inertia; I_d is the torsional moment of inertia; I_ρ is the torsional parameter; K_dw is the stiffness of the box girder cross section against the distortion; I_dw is the distortional warping constant. The equations of I_x, I_d, I_ρ, K_dw, I_dw are found in Liu et al. (2021).

Table 4.

Flexural, torsional, and distortional parameters of CBGs with different webs.

Webs	I_x (m⁴)	I_d (m⁴)	I_ρ (m⁴)	K_dw (kN)	I_dw (m⁶)	Self-weight (kg)
10 mm CSWs	6.9221	4.965	7.6354	114910	5.0261	494643
350 mm concrete webs	8.1752	14.165	16.136	298540	7.6430	676800
Ratio of 350 mm concrete webs/10 mm CSWs	1.181	2.853	2.113	2.598	1.521	1.368

It can be seen from Table 4 that the flexural moment of inertia I_x of box girders with CSWs is somewhat smaller than that of concrete box girders by ignoring the small contribution of the CSWs. In addition, compared to CBGs with 350 mm concrete webs, the torsional parameters I_d and I_ρ, and the distortional parameters K_dw and I_dw of CBGs with 10 mm CSWs are significantly smaller.

Table 5 gives the deflections, torsional angles, and the additional deflections of the outer corner point B (see Figure 7) caused by torsional angles at the midspan cross section for box girders with 2 different web types. In Table 5, v₁, φ₁, and Δv_φ1 are the deflections, torsional angles, and the additional deflections of the outer corner point B caused by torsional angles of the box girders with 10 mm CSWs; v₂, φ₂, and Δv_φ2 are the deflections, torsional angles, and the additional deflections of the outer corner point B caused by torsional angles of the box girders with 350 mm concrete webs. Δv_φ, the additional deflections of the outer corner point B caused by torsional angles, can be calculated by equation (1)

Δ v_{φ} = - \frac{b_{b 1}}{2} φ

(1)

Table 5.

Deflections, torsional angles, and the additional deflections of the outer corner point B caused by torsional angles at the midspan cross section for different radii.

LoadingCase	R (m)	Box girders with 10 mm CSWs				Box girders with 350 mm concrete webs				v₁/v₂	φ₁/φ₂
LoadingCase	R (m)	v₁ (mm)	φ₁ (×10⁻⁴)	Δv_φ1 (mm)	Δv_φ1/v₁ (%)	v₂ (mm)	φ₂ (×10⁻⁴)	Δv_φ2 (mm)	Δv_φ2/v₂ (%)	v₁/v₂	φ₁/φ₂
Case 1	30	21.17	−15.03	3.61	17.0	12.04	−7.05	1.69	14.1	1.76	2.13
	45	11.91	−7.04	1.69	14.2	7.23	−3.31	0.79	11.0	1.65	2.13
	60	9.77	−4.73	1.14	11.6	6.09	−2.22	0.53	8.7	1.60	2.13
	90	8.47	−2.93	0.70	8.3	5.40	−1.37	0.33	6.1	1.57	2.14
	120	8.06	−2.14	0.51	6.4	5.17	−1.00	0.24	4.6	1.56	2.14
	150	7.87	−1.69	0.41	5.2	5.07	−0.79	0.19	3.7	1.55	2.14
	200	7.73	−1.26	0.30	3.9	5.00	−0.59	0.14	2.8	1.55	2.14
	250	7.67	−1.00	0.24	3.1	4.96	−0.47	0.11	2.3	1.54	2.13
	∞	7.55	0	0	0.0	4.90	0	0.00	0.0	1.54	—
Case 2	30	24.77	−19.23	4.62	18.6	13.73	−8.70	2.09	15.2	1.80	2.21
	45	13.61	−9.99	2.40	17.6	8.02	−4.41	1.06	13.2	1.70	2.27
	60	10.91	−7.37	1.77	16.2	6.62	−3.20	0.77	11.6	1.65	2.30
	90	9.18	−5.38	1.29	14.1	5.73	−2.27	0.54	9.5	1.60	2.37
	120	8.57	−4.53	1.09	12.7	5.42	−1.88	0.45	8.3	1.58	2.41
	150	8.28	−4.05	0.97	11.7	5.26	−1.65	0.40	7.5	1.57	2.45
	200	8.03	−3.60	0.86	10.8	5.14	−1.44	0.35	6.7	1.56	2.50
	250	7.91	−3.33	0.80	10.1	5.08	−1.32	0.32	6.2	1.56	2.52
	∞	7.55	−2.31	0.55	7.3	4.90	−0.84	0.20	4.1	1.54	2.75
Case 3	30	17.56	−10.83	2.60	14.8	10.35	−5.41	1.30	12.5	1.70	2.00
	45	10.22	−4.10	0.98	9.6	6.44	−2.20	0.53	8.2	1.59	1.86
	60	8.63	−2.10	0.50	5.8	5.56	−1.25	0.30	5.4	1.55	1.68
	90	7.77	−0.48	0.12	1.5	5.07	−0.48	0.12	2.3	1.53	1.00
	120	7.54	0.25	−0.06	−0.8	4.93	−0.13	0.03	0.6	1.53	−1.92
	150	7.47	0.67	−0.16	−2.2	4.88	0.06	−0.01	−0.3	1.53	11.17
	200	7.43	1.08	−0.26	−3.5	4.86	0.26	−0.06	−1.3	1.53	4.15
	250	7.43	1.33	−0.32	−4.3	4.85	0.38	−0.09	−1.9	1.53	3.50
	∞	7.55	2.31	−0.55	−7.3	4.90	0.84	−0.20	−4.1	1.54	2.75
Case 4	30	21.06	−15.16	3.64	17.3	12.17	−7.11	1.71	14.0	1.73	2.13
	45	11.64	−7.08	1.70	14.6	7.27	−3.32	0.80	11.0	1.60	2.13
	60	9.46	−4.75	1.14	12.1	6.11	−2.23	0.54	8.8	1.55	2.13
	90	8.14	−2.93	0.70	8.6	5.41	−1.38	0.33	6.1	1.51	2.12
	120	7.72	−2.14	0.51	6.7	5.18	−1.01	0.24	4.7	1.49	2.12
	150	7.53	−1.7	0.41	5.4	5.08	−0.79	0.19	3.7	1.48	2.15
	200	7.38	−1.26	0.30	4.1	5.00	−0.59	0.14	2.8	1.48	2.14
	250	7.32	−1	0.24	3.3	4.96	−0.47	0.11	2.3	1.47	2.13
	∞	7.20	0	0.00	0.0	4.90	0	0.00	0.0	1.47	—
Case 5	30	24.69	−18.98	4.56	18.4	13.87	−8.60	2.06	14.9	1.78	2.21
	45	13.34	−9.63	2.31	17.3	8.06	−4.26	1.02	12.7	1.65	2.26
	60	10.60	−6.99	1.68	15.8	6.65	−3.04	0.73	11.0	1.59	2.30
	90	8.84	−4.99	1.20	13.5	5.74	−2.11	0.51	8.8	1.54	2.36
	120	8.23	−4.14	0.99	12.1	5.42	−1.72	0.41	7.6	1.52	2.41
	150	7.93	−3.66	0.88	11.1	5.27	−1.49	0.36	6.8	1.51	2.46
	200	7.69	−3.2	0.77	10.0	5.14	−1.28	0.31	6.0	1.50	2.50
	250	7.56	−2.94	0.71	9.3	5.08	−1.16	0.28	5.5	1.49	2.53
	∞	7.20	−1.92	0.46	6.4	4.90	−0.68	0.16	3.3	1.47	2.82
Case 6	30	17.42	−11.34	2.72	15.6	10.46	−5.62	1.35	12.9	1.67	2.02
	45	9.94	−4.53	1.09	10.9	6.47	−2.37	0.57	8.8	1.54	1.91
	60	8.32	−2.51	0.60	7.2	5.58	−1.41	0.34	6.1	1.49	1.78
	90	7.43	−0.88	0.21	2.8	5.08	−0.64	0.15	3.0	1.46	1.38
	120	7.20	−0.15	0.04	0.5	4.94	−0.3	0.07	1.5	1.46	0.50
	150	7.12	0.27	−0.06	−0.9	4.89	−0.1	0.02	0.5	1.46	−2.70
	200	7.08	0.68	−0.16	−2.3	4.86	0.1	−0.02	−0.5	1.46	6.80
	250	7.08	0.93	−0.22	−3.2	4.85	0.21	−0.05	−1.0	1.46	4.43
	∞	7.20	1.92	−0.46	−6.4	4.90	0.68	−0.16	−3.3	1.47	2.82

Figure 8 shows v₁/v₂, the ratios of the deflections of CBGs with 10 mm CSWs to those of CBGs with 350 mm concrete webs, as a function of the curvature radius R. Figure 9 shows Δv_φ1/v₁, the ratios of the additional deflections of the outer corner point B caused by torsional angles to the deflections, as a function of the curvature radius R for CBGs with 10 mm CSWs.

Figure 8.

Values of v₁/v₂ at the midspan cross section.

Figure 9.

Δv_ϕ1/v₁ of CBGs with CSWs at the midspan cross section.

It can be seen from Table 5 and Figure 8 that: (1) Due to ignoring the very small contribution of CSWs to bending resistance and the large deflections caused by the shear deformation of CSWs, the deflections of box girders with CSWs are larger than those of concrete box girders. For the dimensions of the example, the deflections of box girders with 10 mm CSWs are 1.5∼1.8 times of those of box girders with 350 mm concrete webs. (2) Under symmetric load and outer eccentric load, v₁/v₂ decreases with the increase of R, and shows a converging trend. That is to say, the values of v₁/v₂ of CBGs are larger than those of SBGs, so one should pay more attention to the deflection control of CBGs with CSWs. (3) Under inner eccentric load, v₁/v₂ first decreases with the increase of R for R=30m∼250 m, and then increases slightly with the increase of R for R=250m∼∞. (4) If R increases towards infinity, the CBGs become SBGs, and the values of v₁/v₂ are equal for SBGs under symmetrical load, outer eccentric load and inner eccentric load. That is to say, the three curves as shown in Figure 8 converge to the same value if R increases toward infinity.

It can be seen from Table 5 and Figure 9 that: (1) The ability for torsional resistance of box girders with CSWs is smaller than that of concrete box girders. Compared with box girders with 350 mm concrete webs, the torsional angles of box girders with 10 mm CSWs are 2–3 times those under symmetric load and outer eccentric load. (2) Compared with concrete box girders, the additional deflections of the outer corner point B caused by torsional angles of box girders with 10 mm CSWs are larger. Under symmetric load and outer eccentric load, the values of Δv_φ1/v₁ decreases with the increase of R. That is to say, the values of Δv_φ1/v₁ of CBGs are larger than those of SBGs. For CBGs with CSWs with small R, the deflections of the outer corner point B may increase by 18% when considering the additional deflections caused by torsional angles, which must be considered, so some measures should be taken to control the torsional angles of CBGs with CSWs with large curvatures, such as, concrete-lined CSWs, to improve the torsional stiffness.

It is worth mentioning that the torsional effect caused by the inner eccentric load and that caused by the flexure–torsion coupling effect have opposite directions, so the torsional angles under the inner eccentric load may appear opposite in value with an increase of curvature. In this case, the torsional angles are very small, and should not be considered.

Table 6 gives the distortional angles, and the additional deflections of the outer corner point B (see Figure 7) caused by distortional angles at the midspan cross section for box girders with 2 different web types. In Table 6, γ_1,0 is the distortional angle of the box girders with 10 mm CSWs without intermediate diaphragms; γ_1,2 is the distortional angle of the box girders with 10 mm CSWs with 2 intermediate diaphragms; Δv_γ1,0 is the additional deflection of the outer corner point B caused by distortional angles of the box girders with 10 mm CSWs without intermediate diaphragms; γ_2,0 is the distortional angle of the box girders with 350 mm concrete webs without intermediate diaphragms; γ_2,2 is the distortional angle of the box girders with 350 mm concrete webs with 2 intermediate diaphragms; Δv_γ2,0 is the additional deflection of the outer corner point B caused by distortional angles of the box girders with 350 mm concrete webs without intermediate diaphragms. The additional deflections of the outer corner point B caused by distortional angles Δv_γ can be calculated by equation (2) when b_b1=b_t1

Δ v_{γ} = \frac{b_{b 1}}{4} γ

(2)

Table 6.

Distortional angles and the additional deflections of the outer corner point B caused by distortional angles at the midspan cross section for different radii.

LoadingCase	R (m)	Box girders with 10 mm CSWs					Box girders with 350 mm concrete webs					$\frac{γ_{1, 0}}{γ_{2, 0}}$	$\frac{γ_{1, 2}}{γ_{2, 2}}$
LoadingCase	R (m)	γ_1,0 (10⁻⁴)	γ_1,2 (10⁻⁴)	$\frac{γ_{1, 0}}{γ_{1, 2}}$	Δv_γ1,0 (mm)	Δv_γ1,0/v₁ (%)	γ_2,0 (10⁻⁴)	γ_2,2 (10⁻⁴)	$\frac{γ_{2, 0}}{γ_{2, 2}}$	Δv_γ2,0 (mm)	Δv_γ2,0/v₂ (%)
Case 1	30	11.44	1.83	6.25	1.37	6.5	4.50	1.10	4.09	0.54	4.5	2.54	1.66
	45	6.42	1.06	6.06	0.77	6.5	2.53	0.63	4.02	0.30	4.2	2.54	1.68
	60	4.56	0.76	6.00	0.55	5.6	1.80	0.45	4.00	0.22	3.5	2.53	1.69
	90	2.93	0.49	5.98	0.35	4.1	1.16	0.29	4.00	0.14	2.6	2.53	1.69
	120	2.17	0.36	6.03	0.26	3.2	0.86	0.22	3.91	0.10	2.0	2.52	1.64
	150	1.73	0.29	5.97	0.21	2.6	0.68	0.17	4.00	0.08	1.6	2.54	1.71
	200	1.29	0.22	5.86	0.15	2.0	0.51	0.13	3.92	0.06	1.2	2.53	1.69
	250	1.03	0.17	6.06	0.12	1.6	0.41	0.10	4.10	0.05	1.0	2.51	1.70
	∞	0	0	—	0	0	0	0	—	0	0	—	—
Case 2	30	8.80	0.47	18.72	1.06	4.3	3.31	0.32	10.34	0.40	2.9	2.66	1.47
	45	3.20	−0.39	−8.21	0.38	2.8	1.11	−0.2	−5.55	0.13	1.7	2.88	1.95
	60	1.19	−0.72	−1.65	0.14	1.3	0.31	−0.39	−0.79	0.04	0.6	3.84	1.85
	90	−0.55	−1.00	0.55	−0.07	−0.7	−0.37	−0.56	0.66	−0.04	−0.8	1.49	1.79
	120	−1.34	−1.14	1.18	−0.16	−1.9	−0.68	−0.64	1.06	−0.08	−1.5	1.97	1.78
	150	−1.80	−1.21	1.49	−0.22	−2.6	−0.87	−0.69	1.26	−0.10	−2.0	2.07	1.75
	200	−2.25	−1.29	1.74	−0.27	−3.4	−1.04	−0.73	1.42	−0.12	−2.4	2.16	1.77
	250	−2.52	−1.33	1.89	−0.30	−3.8	−1.15	−0.76	1.51	−0.14	−2.7	2.19	1.75
	∞	−3.56	−1.51	2.36	−0.43	−5.7	−1.56	−0.86	1.81	−0.19	−3.8	2.28	1.76
Case 3	30	14.08	3.19	4.41	1.69	9.6	5.70	1.87	3.05	0.68	6.6	2.47	1.71
	45	9.63	2.51	3.84	1.16	11.3	3.95	1.46	2.71	0.47	7.4	2.44	1.72
	60	7.94	2.23	3.56	0.95	11.0	3.29	1.3	2.53	0.39	7.1	2.41	1.72
	90	6.41	1.98	3.24	0.77	9.9	2.68	1.15	2.33	0.32	6.3	2.39	1.72
	120	5.68	1.86	3.05	0.68	9.0	2.4	1.08	2.22	0.29	5.8	2.37	1.72
	150	5.26	1.79	2.94	0.63	8.5	2.23	1.03	2.17	0.27	5.5	2.36	1.74
	200	4.83	1.72	2.81	0.58	7.8	2.06	0.99	2.08	0.25	5.1	2.34	1.74
	250	4.58	1.68	2.73	0.55	7.4	1.96	0.97	2.02	0.24	4.8	2.34	1.73
	∞	3.56	1.51	2.36	0.43	5.7	1.56	0.86	1.81	0.19	3.8	2.28	1.76
Case 4	30	11.23	1.43	7.85	1.35	6.4	4.36	0.88	4.95	0.52	4.3	2.58	1.63
	45	6.26	0.79	7.92	0.75	6.5	2.43	0.48	5.06	0.29	4.0	2.58	1.65
	60	4.44	0.56	7.93	0.53	5.6	1.72	0.34	5.06	0.21	3.4	2.58	1.65
	90	2.85	0.36	7.92	0.34	4.2	1.10	0.22	5.00	0.13	2.4	2.59	1.64
	120	2.11	0.26	8.12	0.25	3.3	0.82	0.16	5.13	0.10	1.9	2.57	1.63
	150	1.68	0.21	8.00	0.20	2.7	0.65	0.13	5.00	0.08	1.5	2.58	1.62
	200	1.25	0.16	7.81	0.15	2.0	0.49	0.10	4.90	0.06	1.2	2.55	1.60
	250	1	0.12	8.33	0.12	1.6	0.39	0.08	4.88	0.05	0.9	2.56	1.50
	∞	0	0	—	0	0.0	0	0	—	0	0	—	—
Case 5	30	9.79	1.42	6.89	1.17	4.8	3.83	0.85	4.51	0.46	3.3	2.56	1.67
	45	4.25	0.71	5.99	0.51	3.8	1.68	0.42	4.00	0.20	2.5	2.53	1.69
	60	2.28	0.46	4.96	0.27	2.6	0.91	0.26	3.50	0.11	1.6	2.51	1.77
	90	0.58	0.25	2.32	0.07	0.8	0.26	0.13	2.00	0.03	0.5	2.23	1.92
	120	−0.19	0.15	−1.27	−0.02	−0.3	−0.04	0.07	−0.57	0.00	−0.1	4.75	2.14
	150	−0.64	0.09	−7.11	−0.08	−1.0	−0.22	0.04	−5.50	−0.03	−0.5	2.91	2.25
	200	−1.07	0.04	−26.75	−0.13	−1.7	−0.38	0.01	−38.00	−0.05	−0.9	2.82	4.00
	250	−1.33	0.01	−133	−0.16	−2.1	−0.48	−0.01	48.00	−0.06	−1.1	2.77	−1.00
	∞	−2.34	−0.12	19.50	−0.28	−3.9	−0.88	−0.09	9.78	−0.11	−2.2	2.66	1.33
Case 6	30	12.67	1.43	8.86	1.52	8.7	4.88	0.90	5.42	0.59	5.6	2.60	1.59
	45	8.26	0.86	9.60	0.99	10.0	3.17	0.55	5.76	0.38	5.9	2.61	1.56
	60	6.6	0.65	10.15	0.79	9.5	2.53	0.42	6.02	0.30	5.4	2.61	1.55
	90	5.11	0.47	10.87	0.61	8.2	1.95	0.30	6.50	0.23	4.6	2.62	1.57
	120	4.41	0.38	11.61	0.53	7.3	1.68	0.25	6.72	0.20	4.1	2.63	1.52
	150	3.99	0.32	12.47	0.48	6.7	1.51	0.22	6.86	0.18	3.7	2.64	1.45
	200	3.58	0.27	13.26	0.43	6.1	1.35	0.19	7.11	0.16	3.3	2.65	1.42
	250	3.33	0.24	13.88	0.40	5.6	1.26	0.17	7.41	0.15	3.1	2.64	1.41
	∞	2.34	0.12	19.50	0.28	3.9	0.88	0.09	9.78	0.11	2.2	2.66	1.33

Figure 10 shows $γ_{1} / γ_{1} γ_{2} γ_{2}$ , the ratios of the distortional angles of CBGs with 10 mm CSWs to those of CBGs with 350 mm concrete webs, as a function of the curvature radius R. It can be seen from Table 6 and Figure 10 that: For box girders without intermediate diaphragms under symmetric load and inner eccentric load, the distortional angles of box girders with 10 mm CSWs are 2.3∼2.7 times of those of box girders with 350 mm concrete webs. However, for box girders with 2 intermediate diaphragms, $γ_{1} / γ_{1} γ_{2} γ_{2}$ decreases from 2.3∼2.7 times to 1.3∼1.8 times.

Figure 10.

Values of $γ_{1} / γ_{1} γ_{2} γ_{2}$ at the midspan cross section.

Figure 11 shows $γ_{i, 0} / γ_{i, 0} γ_{i, 2} γ_{i, 2} (i = 1, 2)$ , the ratios of the distortional angles of CBGs without intermediate diaphragms to those of CBGs with 2 intermediate diaphragms, as a function of the curvature radius R. It can be seen from Table 6 and Figure 11 that: Compared with box girders without intermediate diaphragms, the ability for distortional resistance of box girders with 2 intermediate diaphragms is greatly improved, but the amplification of box girders with 2 different web types is different. By comparing the values of $γ_{1, 0} / γ_{1, 0} γ_{1, 2} γ_{1, 2}$ and $γ_{2, 0} / γ_{2, 0} γ_{2, 2} γ_{2, 2}$ , it can be seen that the amplification of the ability for distortional resistance by installing intermediate diaphragms for box girders with 10 mm CSWs is larger than that for box girders with 350 mm concrete webs.

Figure 11.

A Comparison of $γ_{i, 0} / γ_{i, 0} γ_{i, 2} γ_{i, 2} (i = 1, 2)$ of CBGs with 10 mm CSWs to CBGs with 350 mm concrete webs at the midspan cross section.

Figure 12 shows $Δ v_{γ i, 0} / Δ v_{γ i, 0} v_{i} v_{i} (i = 1, 2)$ , the ratios of the additional deflection of the outer corner point B caused by distortional angles to the deflections for CBGs without intermediate diaphragms, as a function of the curvature radius R. It can be seen from Table 6 and Figure 12 that: Compared with concrete box girders, the additional deflections of the outer corner point B caused by distortional angles of box girders with 10 mm CSWs are larger. For CBGs with 10 mm CSWs, the deflections of the outer corner point B may increase by 11% when considering the additional deflections caused by distortional angles, which must be considered, so some measures should be taken to control the distortional angles of CBGs with CSWs, such as the method of installing intermediate diaphragms.

Figure 12.

A Comparison of $Δ v_{γ i, 0} / Δ v_{γ i, 0} v_{i} v_{i}$ of CBGs with 10 mm CSWs to CBGs with 350 mm concrete webs at the midspan cross section.

It is worth mentioning that the distortional effect caused by the outer eccentric load and that caused by the flexure–torsion coupling effect have the opposite direction, so the distortional angles under the outer eccentric load may appear opposite in value with the increase of curvature. In this case, the distortional angles are very small, and should not be considered.

Normal stresses analysis

Table 7 gives the normal stresses of the outer corner point B (see Figure 7) at the midspan cross section for different radii. Table 8 gives the analysis of the normal stresses. In Tables 7 and 8, σ_m1 and σ_w1 are the flexural normal stresses and restrained torsional warping normal stresses of the box girders with 10 mm CSWs respectively; σ_dw1,0 is the distortional warping normal stresses of the box girders with 10 mm CSWs without intermediate diaphragms; σ_dw1,2 is the distortional warping normal stresses of the box girders with 10 mm CSWs with 2 intermediate diaphragms; σ_m2 and σ_w2 are the flexural normal stresses and restrained torsional warping normal stresses of the box girders with 350 mm concrete webs, respectively; σ_dw2,0 is the distortional warping normal stresses of the box girders with 350 mm concrete webs without intermediate diaphragms; σ_dw2,2 is the distortional warping normal stresses of the box girders with 350 mm concrete webs with 2 intermediate diaphragms.

Table 7.

Normal stresses of the outer corner point B at the midspan cross section for different radii.

LoadingCase	R (m)	Box girders with 10 mm CSWs				Box girders with 350 mm concrete webs
LoadingCase	R (m)	σ_m1 (MPa)	σ_w1 (MPa)	σ_dw1,0 (MPa)	σ_dw1,2 (MPa)	σ_m2 (MPa)	σ_w2 (MPa)	σ_dw2,0 (MPa)	σ_dw2,2 (MPa)
Case 1	30	2.187	−0.392	−0.545	−0.845	1.747	0.053	−0.200	−0.434
	45	1.881	−0.222	−0.317	−0.484	1.502	0.030	−0.118	−0.248
	60	1.796	−0.158	−0.228	−0.347	1.434	0.022	−0.085	−0.177
	90	1.740	−0.102	−0.148	−0.224	1.390	0.014	−0.055	−0.115
	120	1.722	−0.076	−0.110	−0.166	1.375	0.010	−0.041	−0.085
	150	1.713	−0.060	−0.087	−0.132	1.369	0.008	−0.033	−0.068
	200	1.707	−0.045	−0.065	−0.099	1.364	0.006	−0.024	−0.051
	250	1.704	−0.036	−0.052	−0.079	1.361	0.005	−0.020	−0.040
	∞	1.699	0.000	0.000	0.000	1.357	0.000	0.000	0.000
Case 2	30	2.362	−0.757	0.196	0.065	1.887	0.125	0.172	−0.033
	45	1.981	−0.568	0.450	0.338	1.583	0.100	0.264	0.174
	60	1.867	−0.498	0.547	0.487	1.492	0.090	0.300	0.251
	90	1.787	−0.438	0.632	0.618	1.427	0.082	0.331	0.318
	120	1.756	−0.411	0.672	0.678	1.403	0.078	0.346	0.349
	150	1.741	−0.395	0.695	0.714	1.391	0.076	0.355	0.367
	200	1.727	−0.379	0.718	0.748	1.380	0.074	0.363	0.384
	250	1.720	−0.370	0.731	0.768	1.374	0.073	0.368	0.395
	∞	1.699	−0.333	0.784	0.848	1.357	0.068	0.388	0.436
Case 3	30	2.012	−0.027	−1.285	−1.626	1.607	−0.019	−0.572	−0.835
	45	1.780	0.123	−1.083	−1.306	1.422	−0.039	−0.499	−0.670
	60	1.724	0.181	−1.002	−1.181	1.377	−0.047	−0.469	−0.606
	90	1.694	0.234	−0.927	−1.066	1.353	−0.054	−0.442	−0.547
	120	1.687	0.259	−0.891	−1.011	1.348	−0.058	−0.428	−0.519
	150	1.686	0.274	−0.870	−0.978	1.347	−0.060	−0.420	−0.502
	200	1.686	0.289	−0.848	−0.946	1.347	−0.062	−0.412	−0.486
	250	1.688	0.298	−0.835	−0.926	1.348	−0.063	−0.407	−0.476
	∞	1.699	0.333	−0.784	−0.848	1.357	−0.068	−0.388	−0.436
Case 4	30	1.852	−0.359	−0.394	−0.701	1.480	0.047	−0.127	−0.366
	45	1.544	−0.200	−0.217	−0.388	1.234	0.026	−0.070	−0.203
	60	1.459	−0.142	−0.153	−0.275	1.165	0.019	−0.049	−0.144
	90	1.403	−0.091	−0.098	−0.176	1.121	0.012	−0.031	−0.092
	120	1.385	−0.067	−0.073	−0.130	1.106	0.009	−0.023	−0.068
	150	1.376	−0.053	−0.058	−0.104	1.099	0.007	−0.018	−0.054
	200	1.370	−0.040	−0.043	−0.077	1.094	0.005	−0.014	−0.040
	250	1.367	−0.032	−0.034	−0.062	1.092	0.004	−0.011	−0.032
	∞	1.362	0.000	0.000	0.000	1.088	0.000	0.000	0.000
Case 5	30	2.001	−0.456	−0.401	−0.663	1.598	0.060	−0.137	−0.341
	45	1.626	−0.279	−0.204	−0.315	1.299	0.037	−0.073	−0.160
	60	1.517	−0.215	−0.134	−0.191	1.212	0.028	−0.051	−0.095
	90	1.441	−0.161	−0.076	−0.086	1.151	0.021	−0.032	−0.040
	120	1.412	−0.137	−0.049	−0.038	1.128	0.018	−0.023	−0.015
	150	1.398	−0.122	−0.034	−0.011	1.117	0.016	−0.018	−0.001
	200	1.386	−0.109	−0.019	0.016	1.107	0.014	−0.013	0.013
	250	1.380	−0.100	−0.010	0.032	1.102	0.013	−0.011	0.022
	∞	1.362	−0.068	0.025	0.094	1.088	0.009	0.001	0.054
Case 6	30	1.704	−0.262	−0.388	−0.739	1.361	0.035	−0.118	−0.392
	45	1.462	−0.121	−0.230	−0.462	1.168	0.016	−0.067	−0.247
	60	1.400	−0.068	−0.172	−0.358	1.119	0.009	−0.048	−0.192
	90	1.366	−0.020	−0.120	−0.266	1.091	0.003	−0.031	−0.144
	120	1.357	0.002	−0.096	−0.222	1.084	0.000	−0.023	−0.121
	150	1.354	0.016	−0.082	−0.196	1.082	−0.002	−0.019	−0.108
	200	1.353	0.029	−0.067	−0.171	1.081	−0.004	−0.014	−0.094
	250	1.354	0.037	−0.059	−0.155	1.081	−0.005	−0.011	−0.086
	∞	1.362	0.068	−0.025	−0.094	1.088	−0.009	−0.001	−0.054

Table 8.

Analysis of normal stresses of the outer corner point B at the midspan cross section.

LoadingCase	R (m)	$\frac{Torsional warping stress}{Flexural normal stress}$		$\frac{Distortional warping stress}{Flexural normal stress}$				$\frac{Box girder with 10 mm CSWs}{Box girder with 350 mm concrete webs}$
LoadingCase	R (m)	$\frac{σ_{w 1}}{σ_{m 1}}$	$\frac{σ_{w 2}}{σ_{m 2}}$	$\frac{σ_{dw 1, 0}}{σ_{m 1}}$	$\frac{σ_{dw 2, 0}}{σ_{m 2}}$	$\frac{σ_{dw 1, 2}}{σ_{m 1}}$	$\frac{σ_{dw 2, 2}}{σ_{m 2}}$	$\frac{σ_{m 1}}{σ_{m 2}}$	$\frac{σ_{w 1}}{σ_{w 2}}$	$\frac{σ_{dw 1, 0}}{σ_{dw 2, 0}}$	$\frac{σ_{dw 1, 2}}{σ_{dw 2, 2}}$
Case 1	30	−0.18	0.03	−0.25	−0.11	−0.39	−0.25	1.25	−7.38	2.73	1.95
	45	−0.12	0.02	−0.17	−0.08	−0.26	−0.17	1.25	−7.35	2.69	1.95
	60	−0.09	0.02	−0.13	−0.06	−0.19	−0.12	1.25	−7.34	2.68	1.95
	90	−0.06	0.01	−0.08	−0.04	−0.13	−0.08	1.25	−7.33	2.68	1.95
	120	−0.04	0.01	−0.06	−0.03	−0.10	−0.06	1.25	−7.33	2.67	1.95
	150	−0.04	0.01	−0.05	−0.02	−0.08	−0.05	1.25	−7.33	2.67	1.95
	200	−0.03	0.00	−0.04	−0.02	−0.06	−0.04	1.25	−7.33	2.67	1.95
	250	−0.02	0.00	−0.03	−0.01	−0.05	−0.03	1.25	−7.33	2.67	1.95
	∞	0	0	0	0	0	0	1.25	—	—	—
Case 2	30	−0.32	0.07	0.08	0.09	0.03	−0.02	1.25	−6.04	1.14	−1.97
	45	−0.29	0.06	0.23	0.17	0.17	0.11	1.25	−5.69	1.71	1.94
	60	−0.27	0.06	0.29	0.20	0.26	0.17	1.25	−5.51	1.83	1.94
	90	−0.25	0.06	0.35	0.23	0.35	0.22	1.25	−5.33	1.91	1.94
	120	−0.23	0.06	0.38	0.25	0.39	0.25	1.25	−5.24	1.94	1.94
	150	−0.23	0.05	0.40	0.26	0.41	0.26	1.25	−5.18	1.96	1.95
	200	−0.22	0.05	0.42	0.26	0.43	0.28	1.25	−5.11	1.98	1.95
	250	−0.21	0.05	0.43	0.27	0.45	0.29	1.25	−5.07	1.99	1.95
	∞	−0.20	0.05	0.46	0.29	0.50	0.32	1.25	−4.91	2.02	1.95
Case 3	30	−0.01	−0.01	−0.64	−0.36	−0.81	−0.52	1.25	−1.44	2.25	1.95
	45	0.07	−0.03	−0.61	−0.35	−0.73	−0.47	1.25	−3.13	2.17	1.95
	60	0.11	−0.03	−0.58	−0.34	−0.68	−0.44	1.25	−3.84	2.14	1.95
	90	0.14	−0.04	−0.55	−0.33	−0.63	−0.40	1.25	−4.31	2.10	1.95
	120	0.15	−0.04	−0.53	−0.32	−0.60	−0.38	1.25	−4.49	2.08	1.95
	150	0.16	−0.04	−0.52	−0.31	−0.58	−0.37	1.25	−4.58	2.07	1.95
	200	0.17	−0.05	−0.50	−0.31	−0.56	−0.36	1.25	−4.67	2.06	1.95
	250	0.18	−0.05	−0.50	−0.30	−0.55	−0.35	1.25	−4.72	2.05	1.95
	∞	0.20	−0.05	−0.46	−0.29	−0.50	−0.32	1.25	−4.91	2.02	1.95
Case 4	30	−0.19	0.03	−0.21	−0.09	−0.38	−0.25	1.25	−7.56	3.09	1.91
	45	−0.13	0.02	−0.14	−0.06	−0.25	−0.16	1.25	−7.57	3.11	1.91
	60	−0.10	0.02	−0.11	−0.04	−0.19	−0.12	1.25	−7.57	3.12	1.91
	90	−0.06	0.01	−0.07	−0.03	−0.13	−0.08	1.25	−7.57	3.12	1.91
	120	−0.05	0.01	−0.05	−0.02	−0.09	−0.06	1.25	−7.57	3.12	1.91
	150	−0.04	0.01	−0.04	−0.02	−0.08	−0.05	1.25	−7.57	3.12	1.91
	200	−0.03	0.00	−0.03	−0.01	−0.06	−0.04	1.25	−7.57	3.13	1.91
	250	−0.02	0.00	−0.03	−0.01	−0.05	−0.03	1.25	−7.57	3.13	1.91
	∞	0.00	0.00	0	0	0	0	1.25	—	—	—
Case 5	30	−0.23	0.04	−0.20	−0.09	−0.33	−0.21	1.25	−7.59	2.92	1.94
	45	−0.17	0.03	−0.13	−0.06	−0.19	−0.12	1.25	−7.60	2.79	1.97
	60	−0.14	0.02	−0.09	−0.04	−0.13	−0.08	1.25	−7.61	2.66	2.01
	90	−0.11	0.02	−0.05	−0.03	−0.06	−0.04	1.25	−7.63	2.39	2.14
	120	−0.10	0.02	−0.03	−0.02	−0.03	−0.01	1.25	−7.64	2.12	2.53
	150	−0.09	0.01	−0.02	−0.02	−0.01	0.00	1.25	−7.65	1.84	14.39
	200	−0.08	0.01	−0.01	−0.01	0.01	0.01	1.25	−7.66	1.39	1.21
	250	−0.07	0.01	−0.01	−0.01	0.02	0.02	1.25	−7.67	0.92	1.48
	∞	−0.05	0.01	0.02	0.00	0.07	0.05	1.25	−7.71	48.61	1.74
Case 6	30	−0.15	0.03	−0.23	−0.09	−0.43	−0.29	1.25	−7.53	3.29	1.89
	45	−0.08	0.01	−0.16	−0.06	−0.32	−0.21	1.25	−7.49	3.46	1.87
	60	−0.05	0.01	−0.12	−0.04	−0.26	−0.17	1.25	−7.43	3.61	1.86
	90	−0.01	0.00	−0.09	−0.03	−0.19	−0.13	1.25	−7.12	3.87	1.84
	120	0.00	0.00	−0.07	−0.02	−0.16	−0.11	1.25	−16.60	4.13	1.83
	150	0.01	0.00	−0.06	−0.02	−0.14	−0.10	1.25	−8.24	4.38	1.82
	200	0.02	0.00	−0.05	−0.01	−0.13	−0.09	1.25	−7.92	4.78	1.81
	250	0.03	0.00	−0.04	−0.01	−0.11	−0.08	1.25	−7.84	5.17	1.80
	∞	0.05	−0.01	−0.02	0.00	−0.07	−0.05	1.25	−7.71	48.61	1.74

Figure 13 shows $σ_{w} / σ_{w} σ_{m} σ_{m}$ , the ratios of the restrained torsional warping normal stresses to the flexural normal stresses, as a function of the curvature radius R. It can be seen from Table 8 and Figure 13 that: For CBGs with 10 mm CSWs, the restrained torsional warping normal stresses can reach 30% of the flexural normal stresses. For CBGs with 350 mm concrete webs, the restrained torsional warping normal stresses can reach 7% of the flexural normal stresses. Thus it can be seen that, compared with concrete box girders, the restrained torsional warping normal stresses take up a larger proportion in the flexural normal stresses for CBGs with CSWs, and must be considered.

Figure 13.

A Comparison of $σ_{w} / σ_{w} σ_{m} σ_{m}$ of CBGs with 10 mm CSWs to CBGs with 350 mm concrete webs at the midspan cross section.

Figure 14 shows $σ_{dwi, 2} / σ_{dwi, 2} σ_{mi} σ_{mi} (i = 1, 2)$ , the ratios of the distortional warping normal stresses to the flexural normal stresses, as a function of the curvature radius R for CBGs with 2 intermediate diaphragms. It can be seen from Table 8 and Figure 14 that: When installing 2 intermediate diaphragms, the distortional warping normal stresses can reach 80% of the flexural normal stresses for CBGs with 10 mm CSWs. For CBGs with 350 mm concrete webs, it can reach 50% of the flexural normal stresses. It can be concluded that, compared with concrete CBGs, the distortional warping normal stresses take up a larger proportion of the flexural normal stresses for CBGs with CSWs, and must be considered.

Figure 14.

A Comparison of $σ_{dwi, 2} / σ_{dwi, 2} σ_{mi} σ_{mi} (i = 1, 2)$ of CBGs with 10 mm CSWs to CBGs with 350 mm concrete webs at the midspan cross section.

It can be seen from Table 8 that: Because CSWs barely carry axial force due to the accordion effect, the concrete flanges need to carry more axial force. Compared with CBGs with 350 mm concrete webs, the flexural normal stresses of CBGs with 10 mm CSWs are about 25% larger.

Figure 15 shows $σ_{w 1} / σ_{w 1} σ_{w 2} σ_{w 2}$ , the ratios of the restrained torsional warping normal stresses of CBGs with 10 mm CSWs to those of CBGs with 350 mm concrete webs, as a function of the curvature radius R. It can be seen from Table 8 and Figure 15 that: The ability for torsional resistance of box girders with CSWs is smaller than that of concrete box girders. Under symmetric load and outer eccentric load, the restrained torsional warping normal stresses of CBGs with 10 mm CSWs can reach 7 times of those of CBGs with 350 mm concrete webs.

Figure 15.

Values of $σ_{w 1} / σ_{w 1} σ_{w 2} σ_{w 2}$ at the midspan cross section.

Figure 16 shows $σ_{dw 1} / σ_{dw 1} σ_{dw 2} σ_{dw 2}$ , the ratios of the distortional warping normal stresses of CBGs with 10 mm CSWs to those of CBGs with 350 mm concrete webs, as a function of the curvature radius R. It can be seen from Table 8 and Figure 16 that: The ability for distortional resistance of box girders with CSWs is smaller than that of concrete box girders. For box girders without intermediate diaphragms under symmetric load and inner eccentric load, the distortional warping normal stresses of box girders with 10 mm CSWs are 2∼5 times of those of box girders with 350 mm concrete webs. However, installing intermediate diaphragms can decrease the gap between box girders with 2 different web types.

Figure 16.

Values of $σ_{dw 1} / σ_{dw 1} σ_{dw 2} σ_{dw 2}$ at the midspan cross section.

Influence of intermediate diaphragms on the distortional effect

In view of the observed influence of intermediate diaphragms on the distortional effect, CBGs with n=0, 1, 2, 3 intermediate diaphragms are studied in the following analysis.

It can be seen from the analysis of distortional angles and distortional warping normal stresses in the sections Displacements analysis and Normal stresses analysis that the distortional angles and distortional warping normal stresses are larger under inner eccentric concentrated load. Figures 17 and 18 show the influence of the number of diaphragms on the distortional angles and distortional warping normal stresses of CBGs with 2 different web types when R=120m under inner eccentric concentrated load (Case 3). It can be seen from Figures 17 and 18 that the distortional angles and distortional warping normal stresses of CBGs with 10 mm CSWs and those of CBGs with 350 mm concrete webs show a similar variation. The main conclusions are (1) The distortional angles and distortional warping normal stresses in the midspan cross section are maximal for CBGs without intermediate diaphragms. (2) By installing 1 diaphragm in the midspan section, both the distortional angles, and the distortional warping normal stresses in the midspan section, can be greatly reduced. (3) Compared with installing 1 diaphragm in the midspan section, installing 2 intermediate diaphragms does not reduce the distortional angles and distortional warping normal stresses greatly, but increases the distortional angles and distortional warping normal stresses in the midspan section. (4) The distortional angles and distortional warping normal stresses become very small when installing 3 intermediate diaphragms. (5) Installing intermediate diaphragms is a powerful measure to reduce distortional angles and distortional warping normal stresses. It is suggested that at least 1 diaphragm should be installed in the midspan section of CBGs with CSWs. In addition, if the diaphragms are arranged in equal spacing, the number of diaphragms is suggested to be odd.

Figure 17.

The influence of the number of diaphragms on the distortional angles of CBGs with 2 different web types when R =120m under inner eccentric concentrated load.

Figure 18.

The influence of the number of diaphragms on the distortional warping normal stresses of CBGs with 2 different web types when R=120m under inner eccentric concentrated load.

Figures 19 and 20 show a comparison of distortional angles and distortional warping normal stresses of CBGs with 2 different web types when R=120m under inner eccentric concentrated load. It can be seen from Figures 19 and 20 that, if CBGs with 2 different web types have the same intermediate diaphragms, the distortional angles and distortional warping normal stresses of CBGs with 10 mm CSWs are larger than those of CBGs with 350 mm concrete webs. With the increase of the number of the intermediate diaphragms, the difference of distortional angles and distortional warping normal stresses between CBGs with the 2 different web types is declining. Although there is still a considerable gap between the distortional angles and distortional warping normal stresses of CBGs with CSWs and CBGs with traditional concrete webs, the way of installing intermediate diaphragms is an effective method to reduce distortional angles and distortional warping normal stresses, and to close the gap between the 2 different web types.

Figure 19.

A comparison of distortional angles of CBGs with 2 different web types when R =120m under inner eccentric concentrated load.

Figure 20.

A comparison of distortional warping normal stresses of CBGs with 2 different web types when R =120m under inner eccentric concentrated load.

Conclusions

In this paper, deformations and stresses of CBGs with CSWs and CBGs with traditional concrete webs are theoretically studied and compared. The following main conclusions can be drawn:

Compared with traditional concrete box girders, the ability for torsional and distortional resistance of box girders with CSWs is significantly reduced. Under external load, the deflections, torsional angles, distortional angles, flexural normal stresses, restrained torsional and distortional warping normal stresses of CBGs with CSWs are larger than those of CBGs with concrete webs.

For the example geometry in this paper, the restrained torsional warping normal stresses can reach 30% of the flexural normal stresses, and the distortional warping normal stresses can reach 80% of the flexural normal stresses, and must be considered.

With the increase of the number of the intermediate diaphragms, the differences in distortional angles and distortional warping normal stresses between CBGs with the 2 different web types are declining. Although there remains a considerable gap in the distortional angles and distortional warping normal stresses, the way of installing intermediate diaphragms is an effective method to reduce the values, and to close the gap between the 2 different web types. In addition, it is suggested that at least 1 diaphragm should be installed in the midspan section of CBGs with CSWs. If the diaphragms are arranged in equal spacing, the number of diaphragms is suggested to be odd.

Footnotes

Acknowledgements

The study was supported by the National Natural Science Foundation of China (Grant No. 51378106), the China Scholarship Council (Grant No. 201806090108), and the Scientific Research Foundation for High-Level Talents of Jinling Institute of Technology (Grant No. jit-b-202131). The financial support is gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The study was supported by the National Natural Science Foundation of China (Grant No. 51378106), the China Scholarship Council (Grant No. 201806090108), and the Scientific Research Foundation for High-Level Talents of Jinling Institute of Technology (Grant No. jit-b-202131).

ORCID iD

Sumei Liu

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