Local stability of concrete arch bridge based on Ritz method

Abstract

In order to minimize the self-weight and prevent local buckling failure of thin-walled box concrete arch bridges at the same time, the limit values of width-thickness ratios are deduced based on Ritz method and equivalent strut theory of arch bridge. A new method of determining sectional forms based on the limit values of width-thickness ratios is put forward. Based on Mupeng bridge, the theoretical results are verified by finite element software ANSYS. Results show that the limits of width-thickness ratios are related to concrete grade, equivalent calculation length and radius of gyration, the allowable minimum thickness of Mupeng bridge is 15 cm to avoid local buckling. The limit values of width-thickness ratios deduced in this paper are reasonable and this new method of determining sectional forms is simple and rational to apply in engineering. A scientific engineering calculation method on arch ring design is put forward and it can provide a theoretical basis for the design of thin-walled box concrete arch bridges constructed by cantilever pouring.

Keywords

Thin-walled box concrete arch bridge cantilever pouring Ritz method local bucking the limit of width-thickness ratios

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1. Introduction

Ritz method is an approximate method to find unknown function by the condition of functional stationary value, which can be used to analyse elastic buckling of simply supported thin plates under uniform compression. The problem of local stability of box arch rings can be transformed into that of elastic thin plates, so the theoretical analysis can be carried out by using mathematical method of Ritz method. Single-box with single-cell, double-cells and three-cells as shown in Fig. 1 are the three primary forms of thin-walled box concrete arch bridges, and for thin-walled box concrete bridges constructed by cantilever pouring, reducing thickness can reduce the tensile stress during construction and cut down costs when the width and height of arch rings are fixed [1]. However, local bulking may occur if the width-thickness ratio ( $B/t$ ) as shown in Fig. 1 is too large [2]. Some bridge accidents had been caused by local bulking in history [3, 4, 5, 6].

Figure 1.

The definition of b and h for different kinds of box section.

Peng [7] put forward the limit of width-thickness ratios of concrete thin-wall piers under different constraints by analyzing local stability to avoid buckling failure and pointed out that local buckling can be prevented by limiting the width-thickness ratios for box concrete high piers, which provided an idea for the study of local stability of box arch bridges. Liu [8] pointed out that local instability played a controlling role in the class of stable for extroverted steel box arch bridges. Ma et al. [9] claimed that local instability may occur on concrete thin-walled box arch bridges and it was worth being studied according to the investigation of the actual engineering, which reminded bridge scholars to pay attention to local stability of concrete arch bridges that was used to be ignored. Yan [10] studied on web stability of thin-walled box concrete arch bridges during construction according to the minimum potential energy principle and the safety factor was obtained, but the research results were only applicable to a specific arch bridge. Zhang [11] studied stability theory of four sides simply supported stiffening plate under unidirectional uniform compression, and the stability performance of stiffened plates was evaluated. Wang [12] studied on hysteretic behavior and failure mode of steel piers with a test under horizontal recurrent loading, and the results showed that the width-thickness ratios of flange had great influence on hysteretic behavior. It had been pointed out that the current standards of axial compression member are not consistent with experimental and numerical data on design resistances, and the triangular, rectangular and trapezoidal sections need to be optimized [13, 14]. Zhou et al. [15] studied the limit values of width-thickness ratios of different concrete-filled steel tubular structures on the basis of mechanical properties of local buckling. Bian et al. [16] proposed a method for calculating ultimate base resistance coefficient and pile shaft resistance coefficient. These can be used for reference to study on local stability of arch bridges. Some researches [17, 18, 19] pointed out that a higher economic rationality was required with regard to the section design of long span arch bridges.

To sum up, to the date, the research on local stability of steel structure bridges is relatively mature, but that of concrete bridges has been paid attention to by scholars in recent years. And the theoretical research about local bulking is scarce, and there is no specific provision about local buckling for thin-walled box concrete arch rings in the current design codes, the sectional forms and sizes of arch rings are usually determined with experiences. In this paper, the limit of width-thickness ratio of box concrete arch rings without local buckling is established and studied with Ritz method and equivalent strut theory [20].

According to the investigation of the existing concrete arch bridges, only overall stability of the structure is considered in the design checking calculation, and local stability are often met by construction requirements, and their sectional forms and sizes are mainly determined according to the empirical data, which tend to be somewhat conservative. In the construction calculation of Mupeng bridge which is constructed by cantilever pouring method, the tensile stress under self-weight is large and exceeds the allowable requirement, then the temporary pre-stressed cable had to be tensioned, which increased the cost. So the sectional sizes of this kind of bridges should be taken smaller in order to reduce the self-weight, which makes the thickness of the top and bottom plates thinner that there is a hidden danger of local instability before overall failure occurs. It is very important to find the sizes that can minimize the self-weight and meet local stability at the same time. This paper studies the sectional sizes of concrete arch bridges from the perspective of local stability for the first time. The limit values of width-thickness ratios are deduced based on Ritz method and equivalent strut theory of arch bridge, and a new method about determining sectional forms of box concrete arch rings on the basis of the limits of width-thickness ratios is established. A scientific engineering calculation method on arch ring design is put forward in this paper and the study can provide a theoretical basis for the design of thin-walled box concrete arch bridges constructed by cantilever pouring.

2. The limit of width-thickness ratio of box concrete arch ring

The limit values of width-thickness ratios without local instability is deduced by making local buckling stress higher than the allowable compression stress in this paper [21]. The results are verified by finite element software ANSYS [22]. Then a new method applying the limits of width-thickness ratios to determine the sectional forms is put forward, and its rationality and adaptability are verified by six reinforced concrete arch bridges.

2.1 Theoretical derivation

2.1.1 Ritz method

Ritz method is an approximate method to solve unknown function by stationary value condition of functional [7]. This method assumes that the function of $f(x)$ to be solved is a linear combination of the known functions of $W_{i}(x)$ [7].

$\displaystyle f(x)=\sum\limits_{i-1}^{n}a_{i}W_{i}(x)$ (1)

where, $a_{i}$ is an unknown constant coefficient. $n$ equations are obtained as the follows by the condition of stationary value of $\Phi[f(x)]$ composed of $f(x)$ .

$\displaystyle\frac{\partial\phi}{\partial a_{i}}=0(i=1,2,\cdots,n)$ (2)

where $n$ unknown constant coefficients of $a_{i}$ are solved and $f(x)$ is obtained. This theory can also be extended to multidimensional problems.

When solving the displacement of elastic body, it is assumed that the displacements of $U$ , $V$ and $W$ in the directions of $X$ , $Y$ and $Z$ are superposed by a series of known continuous functions satisfying all the displacement boundary conditions of the elastic body. According to the principle of minimum potential energy in elasticity, the total potential energy variation is zero, and the stationary condition is obtained. And then the unknown coefficients can be solved and the total displacement can be obtained.

2.1.2 The derivation of the limit of width-thickness ratio

According to the theory of elastic and thin plate, deflection formula for plates can be expressed as following [7]:

$\displaystyle D\left(\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w% }{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}\right)+p_% {x}\frac{\partial^{2}w}{\partial x^{2}}=0$

(3) $\displaystyle D=\frac{Et^{3}}{12(1-u^{2})}$

where, $D$ is flexural rigidity; $E$ and $u$ are modulus of elasticity and Poisson’s ratio respectively, and $u$ is usually taking 0.2 for concrete.

For slabs simply supported on four sides subjected to uniform pressure, with Ritz method. The critical load of elastic buckling is [7]:

$\displaystyle p_{\textit{xcr}}=\frac{\pi^{2}a^{2}Et^{3}}{12(1-v^{2})m^{2}}% \left(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{B^{2}}\right)^{2}$ (4)

where, $a$ is plate length, $B$ is plate width, $t$ is plate thickness, and $m$ and $n$ are natural numbers. In order to obtain the minimum critical load, $n$ should be taking 1. The critical stress for local elastic buckling of thin plates can be expressed as:

$\displaystyle\sigma_{\textit{cre}}=\frac{p_{\textit{xcr}}}{t}=k\frac{\pi^{2}E}% {12(1-\upsilon^{2})}\left(\frac{t}{B}\right)^{2}$ (5) $\displaystyle k=\left(\frac{mB}{B}+\frac{a}{mb}\right)^{2}$

where, $\sigma_{\textit{cre}}$ can be obtained by finding the minimum value of $k$ , and $k$ is related to the natural number of $m$ . It can be proved that the minimum value of $k$ is 4 [7]. Therefore, the critical stress of elastic buckling of simply supported thin plates on four sides can be calculated as follows:

$\displaystyle\sigma_{\textit{cre}}=\frac{4\pi^{2}E}{12(1-\upsilon^{2})}\left(% \frac{t}{B}\right)^{2}$ (6)

The critical stress of the elastic support plate is more complex. This paper studies on the critical stress of simple support which is safer than elastic support, then the elastic-plastic buckling critical stress can be calculated as the following formula:

$\displaystyle\sigma_{cr}=\frac{1}{\lambda}\frac{4\pi^{2}\sqrt{\varphi_{t}}E}{1% 2(1-\upsilon^{2})}\left(\frac{t}{B}\right)^{2}$ (7)

where $\lambda$ is buckling safety factor, which can be obtained according to experimental statistics, here advisable value is 4 [7]; $\varphi_{t}$ is the proportion of elastic plastic tangent modulus to elastic modulus, which takes 0.5 with consulting the stress-strain curve under compression; $t$ and $B$ are thickness and width, respectively.

According to the equivalent compression bar theory, arch bridges can be equivalent to compression bars, and the equivalent calculation length ( $l_{0}$ ) of hingless arch bridges is 0.36 s (s is the arc length) during service stage [23].

The compression bearing capacity of arch bridges can be calculated as Eq. (8) [24]:

$\displaystyle\gamma_{0}N\leqslant 0.9\varphi(f_{cd}A+f^{\prime}_{{sd}}A^{% \prime}_{S})$ (8)

where $N$ denotes design value of axial force; $\varphi$ is stability factor of reinforced concrete members, which is related to the equivalent calculation length ( $l_{0}$ ) and radius of gyration; $\gamma_{0}$ is importance coefficient of structures which takes 1 here; $f_{{cd}}$ and $f^{\prime}_{y}$ are respectively axial compressive strength design values of concrete and steel; $A$ and $A^{\prime}_{S}$ are respectively the sectional areas of the whole member and the steel only.

As steer bars has less effect on the internal force of arch bridges, the effect of steer bars is ignored for simplicity, and Eq. (8) can be simplified as Eq. (10),

$\displaystyle N\leqslant 0.9\varphi f_{cd}A$ (9)

Then:

$\displaystyle\sigma=\frac{N}{A}\leqslant 0.9\varphi f_{cd}$ (10)

where $\sigma$ is the allowable stress.

Combining Eqs (7) and (9), and making $\sigma_{cr}$ greater than $\sigma$ , Eq. (11) is obtained as:

$\displaystyle\frac{1}{\lambda}\frac{4\pi^{2}\sqrt{\varphi_{t}}E}{12(1-\upsilon% ^{2})}\left(\frac{t}{B}\right)^{2}\geqslant 0.9\varphi f_{cd}$ (11)

Then the limits of width-thickness ratios can be established as:

$\displaystyle\frac{B}{t}\leqslant\sqrt{\frac{4\pi^{2}\sqrt{\varphi_{t}}E}{12(1% -\upsilon^{2})\lambda\times 0.9\varphi f_{cd}}}$ (12)

Equation (12) can be simplified as the following:

$\displaystyle\frac{B}{t}\leqslant\sqrt{\frac{4\pi^{2}\sqrt{\varphi_{t}}E}{12(1% -\upsilon^{2})\lambda\times 0.9\varphi f_{cd}}}=0.82\sqrt{\frac{E}{f_{cd}% \varphi}}$ (13)

Above parameters are the same as those of Eqs (7) and (8), finally, the limits of width-thickness ratios can be calculated as following:

$\displaystyle\frac{B}{t}\leqslant 0.82\sqrt{\frac{E}{f_{cd}\varphi}}$ (14)

Equation (14) shows that the limits of width-thickness ratios is related to elastic modulus of concrete. The higher the concrete strength is, the larger the elastic modulus is, and the greater the width-thickness ratio limit is.

The limit values of width-thickness ratios with different kinds of concrete (C30, C40, C50, C55 and C60) are provided in Table 1. $E$ and $f_{cd}$ are elastic modulus and design compressive strength of concrete, which can be achieved from relevant specification [24]. It is obvious that the values are related to stability factor ( $\varphi$ ) which will be investigated in detail later.

Table 1
The limits of width-thickness ratios

Strength grade of concrete $E$ (MPa) $f_{cd}$ (MPa) $B/t$

C30 3.00 $\times$ E $+$ 04 14.3 37.6/ $\sqrt{\varphi}$

C40 3.25 $\times$ E $+$ 04 19.1 33.8/ $\sqrt{\varphi}$

C50 3.45 $\times$ E $+$ 04 23.1 31.7/ $\sqrt{\varphi}$

C55 3.55 $\times$ E $+$ 04 25.3 30.7/ $\sqrt{\varphi}$

C60 3.60 $\times$ E $+$ 04 27.5 29.7/ $\sqrt{\varphi}$

The results of $B/t$ in Table 1 are the maximum limit values meeting the requirement of local stability, if the width-thickness ratios of the actual projects exceed these limits, local buckling failure may occur.
2.2 Determination of stability factor

Strength grade of concrete	$E$ (MPa)	$f_{cd}$ (MPa)	$B/t$
C30	3.00 $\times$ E $+$ 04	14.3	37.6/ $\sqrt{\varphi}$
C40	3.25 $\times$ E $+$ 04	19.1	33.8/ $\sqrt{\varphi}$
C50	3.45 $\times$ E $+$ 04	23.1	31.7/ $\sqrt{\varphi}$
C55	3.55 $\times$ E $+$ 04	25.3	30.7/ $\sqrt{\varphi}$
C60	3.60 $\times$ E $+$ 04	27.5	29.7/ $\sqrt{\varphi}$

With EXCEL, the relationship between $\varphi$ and $\lambda$ of concrete bridges is established as Eq. (2.2), and the comparison of the results between Eq. (2.2) and standard are as shown in Table 2.

$\displaystyle\varphi=0.000019\lambda^{2}-0.010064\lambda+1.359777$

(15) $\displaystyle\lambda=\frac{l_{0}}{i}$

where $\lambda$ is slenderness; $i$ is gyration radius; $l_{0}$ denotes calculated length of equivalent compression bar that is related to the shape and length of arch axis (Fig. 2).

Table 2
The comparison between Eq. (2.2) and standard

$\lambda$ $\varphi$

The result of the specification The result of Eq. (17)

48 0.92 0.920481

55 0.87 0.863732

62 0.81 0.808845

69 0.75 0.75582

76 0.7 0.704657

83 0.65 0.655356

90 0.6 0.607917

97 0.56 0.56234

104 0.52 0.518625

111 0.48 0.476772

118 0.44 0.436781

125 0.4 0.398652

132 0.36 0.362385

139 0.32 0.32798

146 0.29 0.295437

153 0.26 0.264756

Figure 2.
The sketch of arc length calculation.

For arch with a circular axis, arc length can be calculated as following [23]:

$\displaystyle s=\theta R$ (16)

As:

$\displaystyle\tan\frac{\theta}{2}=\frac{l}{2(R-f)}$ (17)

We can get

$\displaystyle\theta=2\arctan\frac{l}{2(R-f)}$ (18) $\displaystyle s=2R\arctan\frac{l}{2(R-f)}$ (19) $\displaystyle l_{0}=0.36s=0.36\times 2R\arctan\frac{l}{2(R-f)}$ (20)

For catenary arch, arch axis formulas can be expressed as the following:

$\displaystyle y=\frac{f}{m-1}\left[ch\left(k\frac{2x}{l}\right)-1\right]$ (21) $\displaystyle k=\ln(m+\sqrt{m^{2}-1})$ (22)

According to the method of curve integral, arc length can be calculated as the following:

$\displaystyle s=\int_{L}ds=\int_{0}^{l}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx$ (23) $\displaystyle\frac{dy}{dx}=\frac{f}{m-1}sh\left(k\frac{2x}{l}\right)\frac{2k}{l}$ (24) $\displaystyle s=\int_{0}^{l}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx=\int_{0}% ^{l}\sqrt{1+\left(\frac{f}{m-1}\right)^{2}\frac{4k^{2}}{l^{2}}\left[sh\left(k% \frac{2x}{l}\right)\right]^{2}}dx$ (25) $\displaystyle l_{0}=0.36s=0.36\int_{0}^{l}\sqrt{1+\left(\frac{f}{m-1}\right)^{% 2}\frac{4k^{2}}{l^{2}}\left[sh\left(k\frac{2x}{l}\right)\right]^{2}}dx$ (26)

For parabolic arch, arch axis formula is $y=\frac{4f}{l^{2}}x^{2}$ , and arc length can be calculated as the following:

$\displaystyle s=\int_{L}ds=\int_{0}^{l}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx$ (27) $\displaystyle\frac{dy}{dx}=\frac{8f}{l^{2}}x^{2}$ (28) $\displaystyle s=\int_{0}^{l}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx=\int_{0}% ^{l}\sqrt{1+\frac{64f^{2}}{l^{4}}x^{4}}dx$ (29) $\displaystyle l_{0}=0.36s=0.36\int_{0}^{l}\sqrt{1+\frac{64f^{2}}{l^{4}}x^{4}}dx$ (30)

R is radius of circular arch bridges, $s$ is arc length, $l$ and $f$ are respectively calculation span and height of arch bridges.
2.3 Calculation example of stability coefficient of real bridge

$\lambda$	$\varphi$
48	0.92	0.920481
55	0.87	0.863732
62	0.81	0.808845
69	0.75	0.75582
76	0.7	0.704657
83	0.65	0.655356
90	0.6	0.607917
97	0.56	0.56234
104	0.52	0.518625
111	0.48	0.476772
118	0.44	0.436781
125	0.4	0.398652
132	0.36	0.362385
139	0.32	0.32798
146	0.29	0.295437
153	0.26	0.264756

Taking Guizhou Haima bridge (a circular arch) and Mupeng bridge (a catenary arch) as examples to illustrate the calculation process of stability coefficient for different kinds of arch axes.

Figure 3.

The photo of Mupeng bridge and Haima bridge.

The main arch of Guizhou Haima bridge which has a computational span of 182.06 m and a computational height of 33.1 m is made of concrete C50. Its arc radius, radius angle and gyration radius are respectively 141.71 m, 80 ${}^{\circ}$ and 0.87 m. Deriving from Eq. (20), the calculation height of equivalent pressure bar is 71.17 m, so the rotating radius is obtained as 81.1, $\varphi$ is 0.67 according to Eq. (2.2). From Table 1, the limit of width-thickness ratio of Guizhou Haima bridge is 25.9 which is larger than the actual value of 13.2. The detailed calculation is as the follows:

$\displaystyle l_{0}=0.36S=0.36\times 2R\arctan\frac{l}{2(R-f)}$ $\displaystyle\quad=0.36\times 2\times 141.71\times\arctan\left(\frac{182.06}{2% (141.71-33.1)}\right)=71.17$ $\displaystyle\lambda=\frac{l_{0}}{i}-\frac{71.17}{0.87}-81.1$ $\displaystyle\varphi=0.000019\lambda^{2}-0.010064\lambda+1.359777=0.67$

The main arch of Mupeng bridge which has a computational span of 166.8 m and a computational height of 35.4 m is also made of concrete C50, and its arch axis coefficient and rotating radius are respectively 1.988 and 1.19 m. According to Eqs (22) and (26), $k$ and $l_{0}$ are calculated to be 1.31 and 54.88, respectively, finally $\varphi$ is calculated to be 0.93 by Eq. (2.2). According to Table 1, the limit of width-thickness ratio of Mupeng bridge is 25 which is larger than the actual value of 12.9. The detailed calculation is as the follows:

$\displaystyle k=\ln(1.988+\sqrt{1.988^{2}-1})=1.31$ $\displaystyle l_{0}=0.36S=0.36\int_{0}^{166.8}\sqrt{1+\left(\frac{35.4}{1.988-% 1}\right)^{2}\frac{4\times 1.31^{2}}{166.8^{2}}\cdot\left[sh\left(1.31\times% \frac{2x}{166.8}\right)\right]^{2}}\cdot dx$ $\displaystyle\quad=54.88$ $\displaystyle\lambda=\frac{l_{0}}{i}-\frac{54.88}{1.19}-46.2$ $\displaystyle\varphi=0.000019\lambda^{2}-0.010064\lambda+1.359777=0.93$

2.4 FEM verification of width-thickness ratio limit

Mupeng bridge with a main span of 165 m and a full length of 364.6 m is located in Pingshan Township, Shiqian County, Guizhou Province, it is the supporting project. Its full width is 21.5 m, the height of the net rise is 30 m and the ratio of the net rise to span is 1/5.5. The main arch ring adopts the catenary of equal section, the arch axis coefficient is 1.988. Mupeng bridge adopts the form of single-box with double-cells with a height of 2.8 m and a width of 7.5 m. The thickness of the top and bottom plates gradually changes from 60 cm to 30 cm, the thickness of side web gradually changes from 50 cm to 35 cm and the thickness of middle web gradually changes from 50 cm to 25 cm, respectively from the foot to the joint. The main parameters of the used materials are as shown in Table 3.

Table 3
Material parameters of Mupeng bridge

Material	Modulus of elasticity (Pa)	Bulk density (N/m ${}^{3}$ )	Coefficient of linear expansion
C50	3.45 $\times$ 10 ${}^{10}$	2600	1 $\times$ 10 ${}^{5}$
Steel strand	1.95 $\times$ 10 ${}^{11}$	7850	1.2 $\times$ 10 ${}^{5}$

Figure 4.

The sizes of original design and trial design (unit: cm).

It has been found that the limits of width-thickness ratios are mainly affected by concrete grade and the height and width of arch ring, so it is assumed to be a constant for the three sections (as shown in Fig. 4).

Based on the design drawings, three FEM models of Mupeng bridge with different top and bottom thicknesses (as shown in Fig. 4) are developed with APDAL language of ANSYS program [25]. The finite element model is simplified to equal section, as shown in Fig. 5. For the boundary conditions, both ends of the arch ribs are fixed in all degrees of freedom. The calculation results of the deformation under self-weight are as shown in Fig. 6.

The stability of main arch ring under self-weight is analyzed and the results are as shown in Table 4, Figs 7 and 8 [26].

Table 4

The instability modes and width-thickness ratios of different sections

Name	The top and bottom thickness/cm	Width-thickness ratio	The limit of width- thickness ratio	Instability mode of first order
Original design	30	12.9	25	In plane instability
Trial design 1	20	19.4	25	In plane instability
Trial design 2	15	25.8	25	Local buckling

Figure 5.

The finite element model (half span).

Figure 6.

The deformation of arch ring under self-weight (half span).

Figure 7.

The first order instability of original design and trial design 1 (half span).

Figure 8.

The first order instability of trial design 2 (half span).

It is obvious from Fig. 8 and Table 4 that local buckling occurred on the bottom plate at L/4 of the arch. This is because the stress at L/4 of the arch is larger than the critical stress of local buckling. To avoid local buckling, the thickness of Mupeng Bridge should be larger than 15 cm when it adopts the form of single-box with double-cells.

Figure 9.

The sectional sizes for six arch bridges (unit: cm).

2.5 Verification of width-thickness ratio of six real bridges

Figure 9 provided the actual forms and sizes of several typical large span box concrete arch bridges, which were initially determined by experience, and the limits of width-thickness ratios of these bridges are calculated as shown in Table 5. Obviously, their actual width-thickness ratios are less than the limit values, which means that local buckling will not occur, this is accord with their actually service condition. It is proved that the limit values of width-thickness ratios deduced in this paper is reasonable.

Table 5
The calculation of width-thickness ratios for six arch bridges

Name	Arch axis form	$m$	$R/m$	Section form	$s/m$	$l_{0}/m$	The limitation of width- thickness
ratio	The width- thickness ratio
Guizhou Haima bridge	Arc	/	141.7	Single-box with three-room	197.70	71.17	24.90	13.2
Guizhou Mupeng bridge	Catenary	1.988	/	Single-box with double-room	165.89	59.72	25.00	12.9
Bashagou bridge	Catenary	1.988	/	Single-box with double-room	167.52	60.31	20.60	10.6
Wanxian Yangtze River	Catenary	1.600	/	Single-box with three-room	463.45	166.8	24.70	20.75
bridge
Beppu Ming bridge of	Catenary	3.00	/	Single-box with double-room	254.05	91.46	25.94	17.2
Japan [27]
Xing midi bridge	Catenary	1.988	/	Single-box with double-room	197.88	71.24	26.20	16.6

3. The method of determining sectional forms based on the limits of width-thickness ratios

3.1 Analysis method

Analyzing on the definition of width-thickness ratios (as shown in Fig. 1), it is easy to find that the width-thickness ratio decreases with the increase of chamber number when the width and height are unchanged. Obviously, the width-thickness ratios of single box with single-cell is the largest above the three forms. A new method of determining sectional forms based on the limits of width-thickness is advanced. The computational procedures as shown in Fig. 10 are simple to apply to engineering.

Figure 10.

The computational procedure of determining sectional forms.

Table 6

The calculation process of section form

Name	Limit of width- thickness ratio	Width-thickness ratio for single box with single-cell	Width-thickness ratio for single box with double-cells	Width-thickness ratio for single box with three-cells	Theoretical sectional form	Actual sectional form
Guizhou Haima bridge	25.9	31.2 ${>}$ 25.9	14.6 ${>}$ 25.9	13.2 ${<}$ 25.9	Single box with three-cells	Single box with three-cells
Wanxian Yangtze River bridge	24.7	40 ${>}$ 24.7	26.16 ${>}$ 24.7	20.75 ${<}$ 24.7	Single box with three-cells	Single box with three-cells
Guizhou Mupeng bridge	25	25 $=$ 25	12.9 ${<}$ 25		Single box with double-cells	Single box with double-cells
Xinmidi bridge	26.2	32 ${>}$ 26.2	16.6 ${<}$ 26.2		Single box with double-cells	Single box with double-cells
Bashagou bridge	20.6	24 ${>}$ 20.6	10.6 ${<}$ 20.6		Single box with double-cells	Single box with double-cells
Beppu Ming bridge of Japan	25.94	34.4 ${>}$ 25.94	17.2 ${<}$ 25.94		Single box with double-cells	Single box with double-cells

3.2 The examples

The sectional forms of the six existing large span reinforced concrete box arch bridges are studied (as shown in Table 6) with the above method [28], and the theoretical results are accordant with their actual forms for most of these bridges, which has proved that this new method is reasonable and applicable. The detailed calculation process about Table 6 is as the follows.

Main arch ring of Guizhou Haima bridge is 7.8 m in width and 3.2 m in height, with a top and bottom thickness of 25 cm, the actual width-thickness ratios are respectively 31.2 and 14.6 for single box with single-cell and single box with double-cells. It fails to meet the requirement of local stability with adopting the form of single box with single-cell.

Main arch ring of Chongqing Wanzhou Yangtze River bridge is 16 m in width and 7 m in height, with a top and bottom thickness of 40 cm, the actual width-thickness ratios are respectively 40 and 26.16 for single box with single-cell and single box with double-cells, they are both larger than the limit of 24.7 for this bridge. It indicates that local buckling will occur if Chongqing Wanzhou Yangtze River bridge adopts the form of single box with single-cell and single box with double-cells. The width-thickness ratio reduces to 12.33 that is less than 24.7 for single box with three-cells, so it should adopt the form of single box with three-cells in theory, which is consistent with the actual situation.

The width and height of main arch ring of Guizhou Mupeng bridge are respectively 6 m and 2.7 m, and the thicknesses of top and bottom plate are 25 cm. The width-thickness ratio is 24 which is larger than the limit of 20.6 for single box with single-cell, the width-thickness ratio reduces to 10.4 for single box with double-cells. Sichuan Bashagou bridge should adopt single box with double-cells in theory, which is consistent with the actual situation.

The width and height of main arch ring of Xinmidi bridge are respectively 9.6 m and 3.5 m, and the thicknesses of top and bottom plate are 30 cm. The width-thickness ratio is 32 which is larger than the limit of 26.2 for single box with single-cell, the width-thickness ratio reduces to 16.6 for single box with double-cells. Xinmidi bridge should adopt single box with double-cells in theory, which is consistent with the actual situation.

The width and height of main arch ring of Sichuan Bashagou bridge are respectively 6 m and 2.7 m, and the thicknesses of top and bottom plate are 25 cm. The width-thickness ratio is 24 which is larger than the limit of 20.6 for single box with single-cell, the width-thickness ratio reduces to 10.4 for single box with double-cells. Sichuan Bashagou bridge should adopt single box with double-cells in theory, which is consistent with the actual situation.

The width and height of main arch ring of Beppu Ming bridge of Japan are respectively 1.72 m and 4.5 m, and the thicknesses of top and bottom plate are 50 cm. The width-thickness ratio is 34.4 which is larger than the limit of 25.94 for single box with single-cell, the width-thickness ratio reduces to 17.2 for single box with double-cells. Beppu Ming bridge should adopt single box with double-cells in theory, which is consistent with the actual situation.

The above six reinforced concrete arch bridges in this paper meet the requirements of width-thickness ratio in design, local instability will not occur, so there is no special reinforcement structure, and the key sections are provided with transverse, longitudinal and vertical prestressed reinforcement for local reinforcement.

In fact, the sectional sizes and forms of these existing bridges were determined based on experience and construction requirements, which tend to be somewhat conservative. In this paper, a calculation method on sectional sizes is proposed from the perspective of local stability for the first time and the sectional forms according to the limits of width-thickness ratios, and the results are consistent with those of the real bridges, which illustrate that this method can be applied to practical engineering.

4. Conclusions

Based on Ritz method and equivalent strut theory of arch bridge, the limit values of width-thickness ratios of thin-walled box concrete arch bridges and the equivalent strut length of parabola arch, arc arch and catenary arch are deduced in this paper. Then the forms and sizes of box concrete arch bridges can be theoretically determined on the basis of the limit of width-thickness ratio. A scientific engineering calculation method on arch ring design is put forward and it can provide a theoretical basis for the design of thin-walled box concrete arch bridges constructed by cantilever pouring. The following conclusions can be drawn:

(1)

The limits of width-thickness ratios without local bucking are only related to concrete grade and stability factor of arch ring: they are 37.6 $\varphi$ for concrete of C30; 33.8 $\varphi$ for concrete of C40; 31.7 $\varphi$ for concrete of C50; 30.7 $\varphi$ for concrete of C55; and 29.7 $\varphi$ for concrete of C60.

(2)

The minimum thickness of top and bottom plates without local bucking can be obtained according to the deduced limits of width-thickness ratios, and it is 15 cm for Mupeng bridge with adopting the form of single-box with double-cells. A new method of determining sectional forms on the basis of the limits of width-thickness ratios is presented in this paper. This new method is simple and applicable.

(3)

It is reasonable to apply Ritz method to the study of local stability of box arch bridges, the limit of width-thickness ratio in this paper are suitable for completion stage, the maximum cantilever state is the most unfavorable state, and the local stability problem during the maximum cantilever state needs to be further studied.

(4)

The proposed method is applied for one-story box structures. Further research about multi-story box structures is needed. In addition, steel bars are neglected in the study of the limits of width-thickness ratio in this paper, more accurate results need to be further studied.

Footnotes

Acknowledgments

This research was funded by Chongqing Transportation Science and technology project: Research on key technology of design and construction of Canyon long-span through-type double-connected concrete-filled steel tube arch bridge (KJXM2021-0966).

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$\lambda$	$\varphi$
	The result of the specification	The result of Eq. (17)
48	0.92	0.920481
55	0.87	0.863732
62	0.81	0.808845
69	0.75	0.75582
76	0.7	0.704657
83	0.65	0.655356
90	0.6	0.607917
97	0.56	0.56234
104	0.52	0.518625
111	0.48	0.476772
118	0.44	0.436781
125	0.4	0.398652
132	0.36	0.362385
139	0.32	0.32798
146	0.29	0.295437
153	0.26	0.264756

Local stability of concrete arch bridge based on Ritz method

Abstract

Keywords

1. Introduction

2.1 Theoretical derivation

2.1.1 Ritz method

Table 3 Material parameters of Mupeng bridge

Table 5 The calculation of width-thickness ratios for six arch bridges

3.1 Analysis method

4. Conclusions

Footnotes

Acknowledgments

References

Table 3
Material parameters of Mupeng bridge

Table 5
The calculation of width-thickness ratios for six arch bridges