# Evaluation of Criteria for Out-of-Plane Stability of Steel Arch Bridges in Major Design Codes by FE Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{cr}. When calculating the out-of-plane instability bearing capacity of a steel arch bridge, the Chinese code, the Japanese code, and the European code all regard the arch rib as the compressed lattice column, and the axial force when the lattice column is unstable is taken as the critical axial forces of arch rib, however, each specification considers different influencing factors.

## 2. Outline of Out-of-Plane Stability Provisions in Each Code

#### 2.1. Chinese Code

_{m}represents the angle between the quarter point on the arch axis and the horizontal plane. The critical axial force N

_{cr}is calculated by:

_{0}is calculated as follows:

_{b}represents the moment of inertia of the transverse bracing, and I

_{y}is the out-of-plane moment of inertia of the arch rib. The μ-value can be obtained using Equation (4), and μ-value must be less than 1.

_{cr}value is calculated according to Equations (2) and (3). Then the N

_{cr}value is substituted into Equation (4) to calculate the μ-value, and the N

_{cr}value is obtained through continuous iterative calculation.

**Figure 2.**Simplified Calculation Model of Arch Rib: (

**a**) Simplified Model of Arch Rib; (

**b**) Equivalent Vierendeel Truss.

#### 2.2. Japanese Code

_{g}is the mean value of the gross cross-sectional area of the members of a single arch, σ

_{ca}is the allowable axial compressive stress at the position of 1/4 span of a single arch rib. σ

_{ca}is only relevant to slenderness ratio l/r, and can be calculated according to some equations in tables of specifications when the local bucking of the arch rib is not taken into consideration. In this regard, the calculation method of the radius of rotation and the effective buckling length is shown in Equation (6).

_{y}, and β

_{z}takes the values as shown in Table 1. When the value of f/L falls between values given in Table 1, β

_{z}may be interpolated in a linear manner. Here, f is the rise of the arch. Values of φ are specified as follows: for a midheight-deck stiffened arch φ = 1, for an upper-deck stiffened arch φ = 1 + 0.45 k, and for a through stiffened arch φ = 1 − 0.35 k. Here, k is the ratio of the load shared by the hangers or shoring to the total load in the loading state shown in Figure 3. While considering the upper-deck stiffened arch, the value of k should be set at 1 if the arch and stiffening girder are not rigidly lined at the arch crow. P

_{1}, P

_{2,}and w in Figure 3 are the vehicle load, lane load, and dead load acting on the main structure, respectively.

#### 2.3. Eurocode 3

_{H}/sinα

_{k}of the hangers. h

_{H}is the length of each hanger, α

_{k}is the angle between the arch axis and the horizontal line at the springing. η in Figure 5 is the stiffness ratio between the arch rib and end brace, which can be calculated according to Equation (8), in which b is the arch rib spacing, I

_{0}is the bending moment of inertial of the first transverse brace along the horizontal axis.

## 3. Bridges Analyzed and Method of FE Analysis

#### 3.1. Bridges Analyzed and Study Parameters

#### 3.2. FE Analysis

#### 3.2.1. FE Modeling

_{y}represents the yield stress and the ε

_{y}is the yield strain, ε

_{st}and E

_{st}represent the strain at the onset of strain hardening and initial strain-hardening modulus, respectively. Equation (9) shows the calculation of strain hardening modulus E′:

^{5}MPa, σ

_{y}= 355 MPa, Poisson’s Ratio ν = 0.3, ε

_{st}= 7ε

_{y}, E

_{st}= E/30 and ξ = 0.06.

#### 3.2.2. Accuracy Verification of FE Analysis

## 4. Discussion of FE Analysis Results

_{cr}.

_{cr}by FE analysis with different sections is not comparable, the N

_{cr}of the arch rib is normalized for the purpose of discussion by the yield force N

_{y}at the full-section yield of arch rib, which is the product of yield stress and cross-sectional area of an arch rib. When N

_{cr}/N

_{y}= 1, that is, when out-of-plane instability occurs, the full section of the arch springing reaches the yield strength of the material. The influences of rise-to-span ratio, arch rib spacing, and γ-value with different stiffness ratios on normalized critical axial forces for the through-type bridge (as representative of the three types) are shown in Figure 13a–c. The influence of bridge type is studied by translating the location of the girder framework, as already explained, and the critical axial force with different stiffness ratios is shown in Figure 13d. These figures show each arch rib type in a different color with the same fill. Cases with the same lateral bracing are notated by marks with the same shape. Figure 13 shows that N

_{cr}/N

_{y}ranges from 0.33 to 0.97 under different construction parameters; in other words, when the steel arch bridge occurs out-of-plane instability, the structural parameters have a great influence on whether the arch springing section yield or yield area, especially the parameter of lateral bracing arrangement range.

## 5. Evaluation of Major Design Codes Based on Results of FE Analysis

#### 5.1. Influence of Rise-to-Span Ratio

_{r}-value increases with increasing rise-to-span ratio, which leads to a higher β-value, so as to decrease the out-of-plane critical axial force obtained by Equation (9). The Japanese code provides that the β

_{z}-value in Equation (6) increases with increasing rise-to-span ratio, which results in a higher slenderness ratio of the arch rib, l/r, so the allowable axial compressive stress σ

_{ca}decreases slightly. These changes keep the horizontal component of arch axial force almost constant. However, the increases in the rise-to-span ratio led to the increasing angle between the arch axis and the horizontal; thus, the critical axial force increases.

#### 5.2. Influence of Range of Lateral Bracing Arrangements

#### 5.3. Influence of Arch Rib Spacing

_{b}which relates to the stiffness of the lateral bracing. Therefore, when the stiffness ratio is relatively small, the dual effect of I and b results in a peak in critical axial force at certain spacing. Otherwise, spacing b is the main effect, which results in the critical axial force decreasing with increasing arch rib spacing. In the Eurocode, an increase in arch rib spacing causes the increase in η-value shown in Figure 5, which leads to a higher β-value and hence a reduced critical axial force of the arch rib. On the contrary, in the Japanese code, the integral lateral stiffness is taken into account while calculating the transverse radius of gyration, r, which means greater arch rib spacing results in greater integral lateral rigidity of the arch rib. Therefore, the critical flexure load according to the Japanese code increases with increasing arch rib spacing. However, the change is small.

#### 5.4. Influence of Stiffness Ratio

#### 5.5. Influence of Bridge Type

_{ca}. Although the φ-value has a large effect on σ

_{ca}when the arch rib is slender, the influence of the φ-value in this study is negligible since the slenderness ratio of the arch rib is comparatively small. For these reasons, the effect of bridge type on normalized critical axial force according to the Chinese and Japanese codes should be contrary to that discussed in relation to the FE results in Section 4. This can be seen in Figure 14, Figure 15 and Figure 16. In the Eurocode, the positive effect of the horizontal component of tension in the hangers and the negative effect of the horizontal component of compression in the columns on the out-of-plane buckling of the arch ribs, as shown in Figure 17, are considered. Since the hanger and column lengths are treated as positive and negative values, respectively, when calculating the h

_{r}-value for different types of arch bridges, the h

_{r}-values for through and half-through type arch bridges are little different, while the h

_{r}-value for a deck-type bridge is smaller. For this purpose, the critical axial force for a through-type arch bridge is only slightly different from that for a half-through type arch bridge. Moreover, this conclusion is quite similar to the FE analysis results when the stiffness ratio is relatively large. Although the critical axial force for a through-type bridge is larger than that for a half-through type bridge in the FE analysis results when the stiffness ratio is relatively small, the critical axial force according to the Eurocode is much larger than that by FE analysis. For these reasons, the normalized critical axial force given by the Eurocode for through and half-through type bridges is almost the same, while the critical axial force for a deck-type bridge is much smaller.

#### 5.6. Factors for Improving Code Accuracy

_{r}, φ

_{b}, and φ

_{s}are the critical axial force normalized by FE analysis, r is the stiffness ratio, R is the rise-to-span ratio, γ is the ratio of rib length provided with lateral bracing, as defined in Figure 7, and b is the arch rib spacing. The values of A, B, C, and D in Equations (10)–(12) are given in Table 4, Table 5 and Table 6, respectively. Since values of φ

_{b}for through and half-through type bridges according to the Eurocode are difficult to fit using simple equations, they are excluded from Table 5.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Timoshenko, S.; Gere, J. Theory of Elastic Stability, 2nd ed.; McGraw-Hill: New York, NY, USA, 1961. [Google Scholar]
- Demuts, E. Lateral Buckling of Circular Arches Subjected to Uniform Gravity Type Loading. Ph.D. Thesis, Ohio State University, Columbus, OH, USA, 1969. [Google Scholar]
- Lu, H.; Liu, A.; Bradford, M.A.; Pi, Y. Experimental investigation of out-of-plane buckling of circular arches under a central radial point load. Thin-Walled Struct.
**2020**, 148, 106198. [Google Scholar] [CrossRef] - Tokarz, F.J.; Sandhu, R.S. Lateral-torsional buckling of parabolic arches. J. Struct. Div.
**1972**, 98, 1161–1179. [Google Scholar] [CrossRef] - Kee, C.F. Lateral inelastic buckling of tied arches. J. Struct. Div.
**1961**, 87, 23–40. [Google Scholar] [CrossRef] - Tokarz, F.J. Experimental Study of Lateral Buckling of Arches. J. Struct. Div.
**1971**, 97, 545–559. [Google Scholar] [CrossRef] - Papangelis, J.P.; Trahair, N.S. Flexural-torsional buckling tests on arches. J. Struct. Eng.
**1987**, 113, 1433–1443. [Google Scholar] [CrossRef] - Papangelis, J.P.; Trahair, N.S. Flexural-torsional buckling of arches. J. Struct. Eng.
**1987**, 113, 889–906. [Google Scholar] [CrossRef] - Almeida, P.N. Lateral Buckling of Twin Arch Ribs with Transverse Bars. Ph.D. Thesis, Ohio State University, Columbus, OH, USA, 1970. [Google Scholar]
- Sakimoto, T.; Namita, Y. Out-of-plane buckling of solid rib arches braced with transverse bars. Proc. Jpn. Soc. Civ. Eng.
**1971**, 191, 109–116. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sakimoto, T.; Komatsu, S. A possibility of total breakdown of bridge arches due to buckling of lateral bracings. In Proceedings of the Final Report of the 2nd International Colloquium on Stability of Steel Structures, Liege, Belgium, 13–15 April 1977; pp. 299–301. [Google Scholar]
- Kuranishi, S.; Yabuki, T. Required out-of-plane rigidities of steel arch bridge with two main ribs subjected to vertical and lateral loads. Technol. Rep. Tohoku Univ.
**1977**, 46, 47–72. [Google Scholar] - Guo, Y.; Zhao, S.; Dou, C. Out-of-plane elastic buckling behavior of hinged planar truss arch with lateral bracings. J. Constr. Steel Res.
**2014**, 95, 290–299. [Google Scholar] [CrossRef] - Sakimoto, T.; Komatsu, S. Ultimate strength of arches with bracing systems. J. Struct. Div.
**1982**, 108, 1064–1076. [Google Scholar] [CrossRef] - Sakimoto, T.; Komatsu, S. Ultimate strength formula for steel arches. J. Struct. Div.
**1983**, 109, 613–627. [Google Scholar] [CrossRef] - Sakimoto, T.; Sakata, T.; Kobori, T. An effective length procedure for out-of-plane buckling strength estimation of steel arch bridges. In Proceedings of the 3rd Pacific Structural Steel Conference, Tokyo, Japan, 26–28 October 1992. [Google Scholar]
- Pi, Y.L.; Trahair, N.S. Out-of-Plane Inelastic Buckling and Ultimate Strength of Steel Arches; Research Report No. R737; University of Sydney: Camperdown, NSW, Australia, 1997. [Google Scholar]
- Lim, N.H.; Kang, Y.J. Out of plane stability of circular arches. Int. J. Mech. Sci.
**2004**, 46, 1115–1137. [Google Scholar] [CrossRef] - Pi, Y.L.; Bradford, M.A. Out-of-plane strength design of fixed steel I-section arches. J. Struct. Eng.
**2005**, 131, 560–568. [Google Scholar] [CrossRef] - Spoorenberg, R.C.; Snijder, H.H.; Hoenderkamp, J.C.D. A theoretical method for calculating the collapse load of steel circular arches. Eng. Struct.
**2012**, 38, 89–103. [Google Scholar] [CrossRef] [Green Version] - Liu, A. Experimental research on stable ultimate bearing capacity of leaning-type arch rib systems. J. Constr. Steel Res.
**2015**, 114, 281–292. [Google Scholar] [CrossRef] - Bouras, Y.; Vrcelj, Z. Out-of-plane stability of concrete-filled steel tubular arches at elevated temperatures. Int. J. Mech. Sci.
**2020**, 187, 105916. [Google Scholar] [CrossRef] - Zhong, Z.; Liu, A.; Fu, J.; Lin, P.; Jian, D.; Xie, Z. Analytical and experimental studies on out-of-plane dynamic parametric instability of a circular arch under a vertical harmonic base excitation. J. Sound Vib.
**2021**, 500, 116011. [Google Scholar] [CrossRef] - Chan, S. Non-linear behavior and design of steel structures. J. Constr. Steel Res.
**2001**, 57, 1217–1231. [Google Scholar] [CrossRef] - Standardization Administration of the People’s Republic of China. Fundamental Code for Design on Railway Bridge and Culvert; China Railway Publishing House Co., LTD.: Beijing, China, 2005. [Google Scholar]
- Japan Road Association. Specifications for Highway Bridges Part II; Marusan Publishing Division: Tokyo, Japan, 2012. [Google Scholar]
- European Committee for Standardization. Eurocode3: Design of Steel Structures-Part 2: Steel Bridge; European Committee for Standardization: Brussels, Belgium, 2003. [Google Scholar]
- American Association of State Highway and Transportation Officials. LRFD Bridge Design Specifications; American Association of State Highway and Transportation Officials: Washington, DC, USA, 2010.
- Wan, P. Research on Synthetical Three Factors Check Method for Ultimate Load Carrying Capacity of Long-Span Steel Arch Bridges. Ph.D. Thesis, Southwest Jiaotong University, Chendu, China, 2005. [Google Scholar]
- Usami, T. Guidelines for Stability Design of Steel Structures; Marusan Publishing Division: Tokyo, Japan, 2005. [Google Scholar]
- Sakimoto, T.; Sakata, T. The out-of-plane buckling strength of through-type arch bridges. J. Constr. Steel Res.
**1990**, 16, 307–318. [Google Scholar] [CrossRef] - Sakimoto, T.; Sakata, T.; Tsuruta, E. Elastic-plastic out-of-plane buckling strength of through type and half-through type arch bridges. Struct. Eng./Earthq. Eng.
**1989**, 2, 137–148. [Google Scholar]

**Figure 5.**Buckling Length Factors β: (

**a**) Arch Springing Hinged; (

**b**) Arch Springing Fixed; and (

**c**) Arch Springing with Wind Brace.

**Figure 8.**FE Model of Steel Arch Bridge: (

**a**) FE Model Meshing; (

**b**) Full-span Uniform Loading; (

**c**) Through-Type Bridge; (

**d**) Half-through Type Bridge; (

**e**) Deck-Type Bridge.

**Figure 12.**FE Analysis Result of A FE Model: (

**a**) Relation Curve between the Axial Force at the Springing and the Horizontal Displacement at Vault; (

**b**) Out-of-plane Instability Mode of Arch Rib.

**Figure 13.**Influence of Parameters on Normalized Critical Axial Forces by FE Analysis: (

**a**) Influence of Rise-to-span; (

**b**) Influence of Arch Rib Spacing; (

**c**) Influence γ-value; (

**d**) Influence of Bridge Type.

**Figure 17.**Influence of Hanger and Shoring on the Out-of-plane Buckling: (

**a**) Through-Type Arch Bridge; (

**b**) Deck-Type Arch Bridge.

**Figure 18.**Estimation Accuracy: (

**a**) Influence of Rise-to-span Ratio; (

**b**) Influence of γ-value; (

**c**) Influence of Arch Rib Spacing.

Section | Rise Ratio f/L | ||||
---|---|---|---|---|---|

0.05 | 0.10 | 0.20 | 0.30 | 0.40 | |

I_{y} = constant | 0.50 | 0.54 | 0.65 | 0.82 | 1.07 |

I_{y} = I_{y}_{,c}/cosφ_{m} | 0.50 | 0.52 | 0.59 | 0.71 | 0.86 |

Section | Height | Width | Thickness of Flange | Thickness of Web Plate |
---|---|---|---|---|

Arch rib 1 | 1752 | 1100 | 26 | 26 |

Arch rib 2 | 1200 | 1000 | 22 | 20 |

Arch rib 3 | 1000 | 900 | 20 | 16 |

Lateral brace 1 | 1000 | 900 | 18 | 15 |

Lateral brace 2 | 700 | 600 | 14 | 12 |

Lateral brace 3 | 600 | 500 | 12 | 10 |

Case | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 |
---|---|---|---|---|---|---|---|---|---|

Section of arch rib | 1 | 2 | 3 | ||||||

Section of lateral brace | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

Stiffness ratio | 3.11 | 12.10 | 23.06 | 1.47 | 5.72 | 10.89 | 0.84 | 3.26 | 6.22 |

Code | Bridge Type | A | B | C | D |
---|---|---|---|---|---|

Chinese code | Through | 1.602 | −0.008 | −0.919 | −0.009 |

Half-through | 1.751 | −0.025 | −0.778 | 0.015 | |

Deck | 1.704 | −0.036 | −1.179 | 0.044 | |

Japanese code | Through | 0.267 | 0.022 | 1.506 | 0.063 |

Half-through | 0.360 | 0.010 | 2.140 | 0.046 | |

Deck | 0.413 | 0.003 | 1.553 | −0.007 | |

Eurocode | Through | 0.751 | 0.051 | 0.712 | −0.091 |

Half-through | 0.768 | 0.034 | 1.593 | −0.178 | |

Deck | 0.929 | 0.001 | −1.436 | 0.016 |

Code | Bridge Type | A | B | C | D |
---|---|---|---|---|---|

Chinese code | Through | 0.333 | −0.012 | 1.193 | −0.020 |

Half-through | 0.530 | −0.015 | 1.212 | −0.030 | |

Deck | 0.275 | −0.008 | 1.215 | −0.034 | |

Japanese code | Through | 1.356 | 0.039 | −0.811 | 0.001 |

Half-through | 1.629 | 0.076 | −1.018 | −0.053 | |

Deck | 0.899 | 0.050 | −0.228 | −0.035 | |

Eurocode | Deck | −0.048 | −0.001 | 0.187 | 0.008 |

Code | Bridge Type | A | B | C | D |
---|---|---|---|---|---|

Chinese code | Through | 1.848 | −0.001 | −0.058 | −0.001 |

Half-through | 2.922 | −0.044 | −0.116 | 0.002 | |

Deck | 2.658 | −0.050 | −0.110 | 0.002 | |

Japanese code | Through | 0.663 | 0.015 | 0.004 | 0.002 |

Half-through | 0.778 | 0.007 | 0.006 | 0.002 | |

Deck | 0.603 | −0.002 | 0.012 | 0.002 | |

Eurocode | Through | 1.978 | −0.011 | −0.088 | 0.003 |

Half-through | 2.563 | −0.042 | −0.125 | 0.005 | |

Deck | 0.974 | −0.014 | −0.033 | 0.001 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, W.; Lin, Y.; Chen, K.
Evaluation of Criteria for Out-of-Plane Stability of Steel Arch Bridges in Major Design Codes by FE Analysis. *Appl. Sci.* **2022**, *12*, 12632.
https://doi.org/10.3390/app122412632

**AMA Style**

Wang W, Lin Y, Chen K.
Evaluation of Criteria for Out-of-Plane Stability of Steel Arch Bridges in Major Design Codes by FE Analysis. *Applied Sciences*. 2022; 12(24):12632.
https://doi.org/10.3390/app122412632

**Chicago/Turabian Style**

Wang, Wenping, Yanyu Lin, and Kangming Chen.
2022. "Evaluation of Criteria for Out-of-Plane Stability of Steel Arch Bridges in Major Design Codes by FE Analysis" *Applied Sciences* 12, no. 24: 12632.
https://doi.org/10.3390/app122412632