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Journal articles on the topic 'Parabolic arches'

Author: Grafiati

Published: 3 May 2022

Last updated: 30 July 2025

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1

Leitman, M. J., and P. Villaggio. "Optimal parabolic arches." International Journal of Engineering Science 48, no. 11 (2010): 1433–39. http://dx.doi.org/10.1016/j.ijengsci.2010.03.005.

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2

Shen, Fulin, and Xiaochun Song. "Internal Force Analysis of Parabolic Arch Considering Shear Effect under Gradient Temperature." IOP Conference Series: Earth and Environmental Science 631, no. 1 (2021): 012053. http://dx.doi.org/10.1088/1755-1315/631/1/012053.

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Abstract This paper theoretical analysis the internal force of the fixed parabolic arches under radient temperature gradient field incorporating shear deformations. The effective centroid of the arch-section under linear temperature gradient is derived. Based on force method and energy method, the analytical solutions of the internal force of fixed parabolic arches at pre-buckling under linear temperature gradient field are derived. A parameter study was carried out to study the influence of linear temperature gradient on the internal force of the fixed parabolic arches with different rise-spa

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3

Pan, Zhenyu, Deyuan Deng, Changsheng Feng, et al. "In-Plane Instability of Parabolic Arches under Uniformly Distributed Vertical Load Coupled with Temperature Gradient Field." Advances in Civil Engineering 2022 (February 22, 2022): 1–12. http://dx.doi.org/10.1155/2022/8973013.

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In civil engineering, arches, such as steel arch roofs and arch bridges, are often subjected to linear temperature gradient field. It is known that the in-plane instability of parabolic arches is caused by the significant axial force. The arch under the linear temperature gradient field produces complex axial force, and so the instability of arches would be affected by temperature gradient field significantly. However, the analytical solutions of in-plane instability of parabolic arches being subjected to the uniformly distributed vertical load and the temperature gradient field are not solved

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4

Jones, M. L., and S. Richmond. "An Assessment of the Fit of a Parabolic Curve to Pre- and Post-Treatment Dental Arches." British Journal of Orthodontics 16, no. 2 (1989): 85–93. http://dx.doi.org/10.1179/bjo.16.2.85.

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Study casts of the teeth are routinely used for diagnosis and to assess treatment change: the crowding of the teeth or shortage of space available within the dental arch is usually assessed visually. A full cast analysis program has been developed previously, making use of the three-dimensional Reflex Plotter linked to a computer. This study examines the validity of the fit of computer generated parabolic curves to dental arches, as performed in the measurement of crowding. Using a visual analogue method it was found that the parabola appeared to fit lower post-treatment dental arches best. Ho

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5

LEE, BYOUNG KOO, TAE EUN LEE, JONG MIN CHOI, and SANG JIN OH. "DYNAMIC OPTIMAL ARCHES WITH CONSTANT VOLUME." International Journal of Structural Stability and Dynamics 12, no. 06 (2012): 1250044. http://dx.doi.org/10.1142/s0219455412500447.

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This paper deals with dynamic optimal arches, built with a constant volume of arch material, that have the largest fundamental natural frequencies. The cross-section of each arch is a solid regular polygon with its depth varying in a functional fashion. Three shapes of arch (circular, parabolic, and sinusoidal) and three kinds of taper type (linear, parabolic, and sinusoidal) are considered. Differential equations governing free vibrations of such tapered arches are derived, in which the effect of rotatory inertia is included; these equations are numerically solved to calculate the natural fre

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6

Shi, Jun, Kangkang Yang, Kaikai Zheng, Jiyang Shen, Guangchun Zhou, and Yanxia Huang. "AN INVESTIGATION INTO WORKING BEHAVIOR CHARACTERISTICS OF PARABOLIC CFST ARCHES APPLYING STRUCTURAL STRESSING STATE THEORY." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 25, no. 3 (2019): 215–27. http://dx.doi.org/10.3846/jcem.2019.8102.

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This paper conducts the experimental and simulative analysis of stressing state characteristics for parabolic concretefilled steel tubular (CFST) arches undergoing vertical loads. The measured stain data is firstly modeled as the generalized strain energy density (GSED) to describe structural stressing state mode. Then, the normalized GSED sum Ej,norm at each load Fj derives the Ej,norm-Fj curve reflecting the stressing state characteristics of CFST arches. Furthermore, the Mann-Kendall criterion is adopted to detect the stressing state change of the CFST arch during its load-bearing process,

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7

Lee, Byoung Koo, Sang Jin Oh, Guangfan Li, and Kou Moon Choi. "Free Vibration Analysis of Parabolic Arches in Cartesian Coordinates." International Journal of Structural Stability and Dynamics 03, no. 03 (2003): 377–90. http://dx.doi.org/10.1142/s021945540300094x.

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The differential equations governing free vibrations of the elastic, parabolic arches with unsymmetric axes are derived in Cartesian coordinates rather than in polar coordinates. The formulation includes the effects of axial extension, shear deformation and rotatory inertia. Frequencies and mode shapes are computed numerically for arches with clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged ends. The convergent efficiency is highly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters are reported as func

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8

Wang, Tao, Mark A. Bradford, and R. Ian Gilbert. "Creep Buckling of Shallow Parabolic Concrete Arches." Journal of Structural Engineering 132, no. 10 (2006): 1641–49. http://dx.doi.org/10.1061/(asce)0733-9445(2006)132:10(1641).

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9

Azad, Abul K., and Hani M. M. Mohdaly. "Optimum design of parabolic steel box arches." Structural Engineering and Mechanics 9, no. 2 (2000): 169–80. http://dx.doi.org/10.12989/sem.2000.9.2.169.

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10

Fan, Jun Jian, and Kuan Tang Xi. "Out-of-Plane Elastic Stability Analysis of Parabolic and Circular Arches." Applied Mechanics and Materials 638-640 (September 2014): 190–96. http://dx.doi.org/10.4028/www.scientific.net/amm.638-640.190.

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Effects of different load modes, rise-span ratios and support conditions on out-of-plane buckling and differences between parabolic and circular arches were studied. With the increase of rise-span ratio, buckling loads of arches under vertical load uniformly distributed along the horizontal line get bigger and bigger compared with those of vertical load uniformly distributed along the axis. With increase of rise-span ratio, the buckling loads of hingeless and two-hinged arch increase after decrease, then decrease. The buckling load of three-hinged arch decreases after increases. When rise-span

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11

Al-ossmi, Laith H. M. "Beyond circular trigonometry: Parabolic functions from geometric identities." Alifmatika: Jurnal Pendidikan dan Pembelajaran Matematika 7, no. 1 (2025): 1–33. https://doi.org/10.35316/alifmatika.2025.v7i1.1-33.

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This paper presents an innovative extension of trigonometric functions to parabolic geometry, introducing the parabolic sine (sinp u) and parabolic cosine (cosp u) functions. Geometrically, sinp u and cosp u are defined via the relationship between a point on a parabola and its focus: sinp u represents the vertical displacement ratio, while cosp u corresponds to the horizontal displacement ratio, normalized by the focal distance. These functions generalize circular trigonometry to a parabolic framework, preserving key structural identities while exhibiting unique behaviors, such as fixed asymp

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12

Sabale, Aditya, and K. V. Nagendra Gopal. "Nonlinear In-Plane Stability of Deep Parabolic Arches Using Geometrically Exact Beam Theory." International Journal of Structural Stability and Dynamics 18, no. 01 (2018): 1850006. http://dx.doi.org/10.1142/s0219455418500062.

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In this paper, we investigate the in-plane stability and post-buckling response of deep parabolic arches with high slenderness ratios subjected to a concentrated load on the apex, using the finite element implementation of a geometrically exact rod model and the cylindrical version of the arc-length continuation method enabled with pivot-monitored branch-switching. The rod model used here includes geometrically exact kinematics of the cross-section, exact kinetics, and a linear elastic constitutive law; and advantageously employs quaternion parameters to treat the cross-sectional rotations and

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13

Lewis, W. J. "Mathematical model of a moment-less arch." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2190 (2016): 20160019. http://dx.doi.org/10.1098/rspa.2016.0019.

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This paper presents a mathematical model for predicting the geometrical shapes of rigid, two-pin, moment-less arches of constant cross section. The advancement of this work lies in the inclusion of arch self-weight and the ability to produce moment-less arch forms for any span/rise ratio, and any ratio of uniformly distributed load per unit span, w , to uniformly distributed arch weight per unit arch length, q . The model is used to derive the shapes of two classical ‘moment-less’ arch forms: parabolic and catenary, prior to demonstrating a general case, not restricted by the unrealistic load

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14

Moon, Jiho. "Inelastic Out-of-plane Design of Parabolic Arches." International Journal of Railway 8, no. 2 (2015): 46–49. http://dx.doi.org/10.7782/ijr.2015.8.2.046.

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15

Moon, Jiho, Ki-Yong Yoon, Tae-Hyung Lee, and Hak-Eun Lee. "In-plane strength and design of parabolic arches." Engineering Structures 31, no. 2 (2009): 444–54. http://dx.doi.org/10.1016/j.engstruct.2008.09.009.

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16

Palkowski, Szymon. "Buckling of parabolic arches with hangers and tie." Engineering Structures 44 (November 2012): 128–32. http://dx.doi.org/10.1016/j.engstruct.2012.05.028.

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17

Glisic, Branko. "New in Old: Simplified Equations for Linear-elastic Symmetric Arches and Insights on Their Behavior." Journal of the International Association for Shell and Spatial Structures 61, no. 3 (2020): 227–40. http://dx.doi.org/10.20898/j.iass.2020.006.

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Closed-form equations for determination of reactions and internal forces of linear-elastic symmetric arches with constant cross-sections are derived. The derivation of the equations was initially made for segmental, threehinged, two-hinged, and hingeless arches. Not all derived equations are simple, but still not excessively complex to apply, and they reveal several new insights into the structural behavior of arches. The first is an extremely simple approximate equation for horizontal reactions of a hingeless arch under self-weight, which could be also applied with excellent accuracy to caten

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18

Pan, W. H., and C. M. Wang. "Elastic In-Plane Buckling of Funicular Arches." International Journal of Structural Stability and Dynamics 20, no. 13 (2020): 2041014. http://dx.doi.org/10.1142/s021945542041014x.

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Buckling loads of arches could be significantly affected by the assumptions made on the load behavior during buckling. For a funicular arch whose centerline coincides with the compression line, we may consider two types of load behaviors based on how the line of load action shifts during buckling. This paper presents the governing differential equations for the elastic in-plane buckling problem of funicular circular arches under uniform radial pressure based on the two different load behavior assumptions, as well as analytical and numerical methods for analysis. For the analytical method, buck

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19

Cho, Jin-Goo, and Keun-Soo Park. "Dynamic Behavior of Plane Parabolic Arches with Initial Deflections." Journal of The Korean Society of Agricultural Engineers 46, no. 2 (2004): 67–75. http://dx.doi.org/10.5389/ksae.2004.46.2.067.

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20

Eroglu, U., and G. Ruta. "Fundamental frequencies and buckling in pre-stressed parabolic arches." Journal of Sound and Vibration 435 (November 2018): 104–18. http://dx.doi.org/10.1016/j.jsv.2018.07.038.

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21

Cai, JianGuo, Jian Feng, Yao Chen, and LiFeng Huang. "In-plane elastic stability of fixed parabolic shallow arches." Science in China Series E: Technological Sciences 52, no. 3 (2009): 596–602. http://dx.doi.org/10.1007/s11431-009-0057-9.

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22

Zewudie, Besukal Befikadu. "Nonlinear Finite Element Analysis and Comparison of In-Plane Strength of Circular and Parabolic Arched I-Section Cellular Steel Beam." Advances in Civil Engineering 2022 (August 11, 2022): 1–13. http://dx.doi.org/10.1155/2022/4879164.

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The advancement in steel fabrication technology extends the structural and constructional advantages of cellular steel beams into arched cellular steel structure members. However, less attention is given to understanding the in-plane and out-of-plane structural behavior and performance of arched cellular steel beams. This article presents a numerical study using the finite element package ABAQUS to investigate the effect of arch axis geometry (circular and parabolic) and the impact of end support types on the in-plane inelastic buckling strength and buckling mode of I-section arched cellular s

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23

Daftardar, Anand. "Economic Design of Rectangular Water Tank Walls using Parabolic Arches." International Journal for Research in Applied Science and Engineering Technology 8, no. 2 (2020): 242–52. http://dx.doi.org/10.22214/ijraset.2020.2036.

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24

Temel, Beytullah, and Ahmad Reshad Noori. "Transient analysis of laminated composite parabolic arches of uniform thickness." Mechanics Based Design of Structures and Machines 47, no. 5 (2019): 546–54. http://dx.doi.org/10.1080/15397734.2019.1572518.

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25

Cai, Jianguo, and Jian Feng. "Buckling of parabolic shallow arches when support stiffens under compression." Mechanics Research Communications 37, no. 5 (2010): 467–71. http://dx.doi.org/10.1016/j.mechrescom.2010.05.004.

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26

Moon, Jiho, Ki-Yong Yoon, Tae-Hyung Lee, and Hak-Eun Lee. "In-plane elastic buckling of pin-ended shallow parabolic arches." Engineering Structures 29, no. 10 (2007): 2611–17. http://dx.doi.org/10.1016/j.engstruct.2007.01.004.

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27

Hu, Chang-Fu, Shun-Shun Zhu, Wen-Jun Luo, Jian-Ping Zhou, Li Li, and Qi-Han Wang. "Lateral-torsional bucking of parabolic arches in Cartesian coordinate system." Structures 80 (October 2025): 109638. https://doi.org/10.1016/j.istruc.2025.109638.

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28

Zhu, Jianbei, Mario M. Attard, and David C. Kellermann. "In-Plane Nonlinear Buckling of Funicular Arches." International Journal of Structural Stability and Dynamics 15, no. 05 (2015): 1450073. http://dx.doi.org/10.1142/s0219455414500734.

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This paper presents a numerical technique to determine the full pre-buckling and post-buckling equilibrium path for elastic funicular arches. The formulation includes the effects of shear deformations and geometric nonlinearity due to large deformations. The Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are considered without approximation. The finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for the internal actions are based on a hyperelastic constitutive model. Using the differential

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29

Kairalla, Silvana Allegrini, Giuseppe Scuzzo, Tarcila Triviño, Leandro Velasco, Luca Lombardo, and Luiz Renato Paranhos. "Determining shapes and dimensions of dental arches for the use of straight-wire arches in lingual technique." Dental Press Journal of Orthodontics 19, no. 5 (2014): 116–22. http://dx.doi.org/10.1590/2176-9451.19.5.116-122.oar.

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INTRODUCTION: This study aims to determine the shape and dimension of dental arches from a lingual perspective, and determine shape and size of a straight archwire used for lingual Orthodontics. METHODS: The study sample comprised 70 Caucasian Brazilian individuals with normal occlusion and at least four of Andrew's six keys. Maxillary and mandibular dental casts were digitized (3D) and the images were analyzed by Delcam Power SHAPET 2010 software. Landmarks on the lingual surface of teeth were selected and 14 measurements were calculated to determine the shape and size of dental arches. RESUL

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30

Eroglu, Ugurcan, Achille Paolone, Giuseppe Ruta, and Ekrem Tüfekci. "Exact closed-form static solutions for parabolic arches with concentrated damage." Archive of Applied Mechanics 90, no. 4 (2019): 673–89. http://dx.doi.org/10.1007/s00419-019-01633-x.

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31

Liu, Changyong, Qing Hu, Yuyin Wang, and Sumei Zhang. "In-Plane Stability of Concrete-Filled Steel Tubular Parabolic Truss Arches." International Journal of Steel Structures 18, no. 4 (2018): 1306–17. http://dx.doi.org/10.1007/s13296-018-0122-y.

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32

Uzman, Umit, Ayse Daloglu, and M. Polat Saka. "Optimum design of parabolic and circular arches with varying cross section." Structural Engineering and Mechanics 8, no. 5 (1999): 465–76. http://dx.doi.org/10.12989/sem.1999.8.5.465.

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33

Järvenpää, Esko, Antti H. Niemi, and Matti-Esko Järvenpää. "Vaults in snow constructions." Rakenteiden Mekaniikka 57, no. 4 (2024): 138–56. https://doi.org/10.23998/rm.145956.

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The article discusses the principles of arch design as they apply to snow vaults and presents different types such as parabolic, catenary, circular and constant stress. The parabolic momentless arch requires a constant vertical load throughout the span, resulting in a decreasing snow thickness from the crown to the base. In contrast, the catenary arch is formed by an inverted hanging chain, maintaining a uniform snow thickness throughout the structure, governed by a hyperbolic cosine function. The shape of the constant stress standalone arch is determined by the unit weight and the compressive

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34

Esko, Järvenpää, and Quach Thanh Tung. "Simple innovative comparison of costs between tied-arch bridge and cable-stayed bridge." MATEC Web of Conferences 258 (2019): 02015. http://dx.doi.org/10.1051/matecconf/201925802015.

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The proposed paper compares tied-arch bridge alternatives and cable-stayed bridge alternatives based on needed load-bearing construction material amounts in the superstructure. The comparisons are prepared between four tied arch bridge solutions and four cable-stayed bridge solutions of the same span lengths. The sum of the span lengths is 300 m. The rise of arch as well as the height of pylon and cable arrangements follow optimal dimensions. The theoretic optimum rise of tied-arch for minimum material amount is higher than traditionally used for aesthetic reason. The optimum rise for minimum

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35

Bradford, Mark Andrew, Tao Wang, Yong-Lin Pi, and R. Ian Gilbert. "In-Plane Stability of Parabolic Arches with Horizontal Spring Supports. I: Theory." Journal of Structural Engineering 133, no. 8 (2007): 1130–37. http://dx.doi.org/10.1061/(asce)0733-9445(2007)133:8(1130).

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36

Wang, Tao, Mark Andrew Bradford, R. Ian Gilbert, and Yong-Lin Pi. "In-Plane Stability of Parabolic Arches with Horizontal Spring Supports. II: Experiments." Journal of Structural Engineering 133, no. 8 (2007): 1138–45. http://dx.doi.org/10.1061/(asce)0733-9445(2007)133:8(1138).

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37

Eroglu, Ugurcan, and Giuseppe Ruta. "Closed-form solutions for elastic tapered parabolic arches under uniform thermal gradients." Meccanica 55, no. 5 (2020): 1135–52. http://dx.doi.org/10.1007/s11012-020-01153-x.

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38

Chen, Yao, and Jian Feng. "Elastic stability of shallow pin-ended parabolic arches subjected to step loads." Journal of Central South University of Technology 17, no. 1 (2010): 156–62. http://dx.doi.org/10.1007/s11771-010-0025-3.

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39

Teixeira, Guilherme S. ,., and Marco D. De Campos. "Influence of Wind Angle Incidence and Architectural Elements on the External Pressure Coefficient of Hyperbolic Paraboloid Roofs." DESIGN, CONSTRUCTION, MAINTENANCE 2 (June 30, 2022): 208–16. http://dx.doi.org/10.37394/232022.2022.2.27.

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In the study of wind loads in buildings, the aerodynamics of roofs with parabolic shapes, which cause complex pressure distributions due to their sensitivity to wind, are often omitted and neglected by several codes and norms. In this way, computer simulations are a viable and reliable alternative. Here, wind action was considered in an innovative project composed of parabolic and circumferential generatrices: the Church of Saint Francis of Assisi. Designed by Brazilian architect Oscar Niemeyer in Belo Horizonte, Brazil, two paraboloid vaults and three circular arches of reinforced concrete co

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40

CAI, JIANGUO, and JIAN FENG. "EFFECT OF SUPPORT STIFFNESS ON STABILITY OF SHALLOW ARCHES." International Journal of Structural Stability and Dynamics 10, no. 05 (2010): 1099–110. http://dx.doi.org/10.1142/s0219455410003919.

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The nonlinear behavior and in-plane stability of parabolic shallow arches with elastic rotational supports are investigated. A central concentrated load is applied to create the compression in the supports. Nonlinear buckling analysis based on the virtual work formulation is carried out to obtain the critical load for both symmetric snap-through buckling and anti-symmetric bifurcation buckling. It is found that the effect of rotational stiffness of the elastic supports on the critical loads is significant. The critical load increases as either the initial stiffness coefficient α or the stiffen

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41

Sophianopoulos, Dimitris S., and George T. Michaltsos. "Analytical Treatment of In-Plane Parametrically Excited Undamped Vibrations of Simply Supported Parabolic Arches." Journal of Vibration and Acoustics 125, no. 1 (2003): 73–79. http://dx.doi.org/10.1115/1.1521952.

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The present work offers a simple and efficient analytical treatment of the in-plane undamped vibrations of simply supported parabolic arches under parametric excitation. After thoroughly dealing with the free vibration characteristics of the structure dealt with, the differential equations of the forced motion caused by a time dependent axial loading of the form P=P0+Pt cos θt are reduced to a set of Mathieu-Hill type equations. These may be thereafter tackled and the dynamic stability problem comprehensively discussed. An illustrative example based on Bolotin’s approach produces results valid

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42

Wu, Xinrong, Changyong Liu, Wei Wang, and Yuyin Wang. "In-Plane Strength and Design of Fixed Concrete-Filled Steel Tubular Parabolic Arches." Journal of Bridge Engineering 20, no. 12 (2015): 04015016. http://dx.doi.org/10.1061/(asce)be.1943-5592.0000766.

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43

Hu, Chang-Fu, and Yan-Mei Huang. "In-plane nonlinear elastic stability of pin-ended parabolic multi-span continuous arches." Engineering Structures 190 (July 2019): 435–46. http://dx.doi.org/10.1016/j.engstruct.2019.04.013.

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44

Bidnichenko, Helen. "CURVES AND SURFACES OF THE SECOND ORDER IN NATURE AND ARCHITECTURAL STRUCTURES." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 103 (December 23, 2022): 3–15. https://doi.org/10.32347/0131-579x.2022.103.3-15.

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This work is devoted to the analysis of curved lines and curvilinear surfaces of the second order and their practical use in architectural structures. The article analyzes the curves of the second order formed by conic sections: ellipse, parabola and hyperbola. Their mathematical equations are given; images are constructed in the Cartesian coordinate system. Examples of such forms in the natural environment are given. Curvilinear surfaces of the second order formed from these curves are analyzed, their mathematical description is given, and diagrams of visual images in the Cartesian coordinate

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45

Fan, Linzi, Ying Zhang, Yaroslav Zhuk, Ivan Goroshko, and Pooya Sareh. "Nonlinear in-plane buckling of shallow parabolic arches with tension cables under step loads." Archive of Applied Mechanics 92, no. 1 (2021): 335–49. http://dx.doi.org/10.1007/s00419-021-02060-7.

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46

Eroglu, U., G. Ruta, and E. Tufekci. "Natural frequencies of parabolic arches with a single crack on opposite cross-section sides." Journal of Vibration and Control 25, no. 7 (2019): 1313–25. http://dx.doi.org/10.1177/1077546319825681.

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We study natural vibration of elastic parabolic arches, modeled as plane curved beams susceptible to elongation, shear, and bending, exhibiting small concentrated cracks. The crack is simulated by springs between regular chunks, with stiffness evaluated following stress concentration in usual crack opening modes. We evaluate and compare the linear dynamic response of the undamaged and damaged arch in nondimensional form. The governing equations are turned into a system of first-order differential equations that are solved numerically by the so-called matricant. The original contribution of thi

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47

Nie, Zhihua, Chao Fu, Yongfeng Yang, and Jiepeng Zhao. "Transient Dynamic Response of Generally Shaped Arches under Interval Uncertainties." Applied Sciences 14, no. 13 (2024): 5918. http://dx.doi.org/10.3390/app14135918.

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This paper endeavors to investigate the characteristics of the transient dynamic response of a generally shaped arch when influenced by uncertain parameters while being subjected to specific external excitation. The equations of motion of the generally shaped arches are derived by the differential quadrature (DQ) method, and the deterministic dynamic responses are calculated using the Newmark-β method. By employing the Chebyshev inclusive function, an interval method based on a non-intrusive polynomial surrogate model is developed, and the uncertain dynamic responses are reckoned by enabling n

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48

Cai, Jianguo, Yixiang Xu, Jian Feng, and Jin Zhang. "In-Plane Elastic Buckling of Shallow Parabolic Arches under an External Load and Temperature Changes." Journal of Structural Engineering 138, no. 11 (2012): 1300–1309. http://dx.doi.org/10.1061/(asce)st.1943-541x.0000570.

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49

Hu, Chang-Fu, Yong-Lin Pi, Wei Gao, and Li Li. "In-plane non-linear elastic stability of parabolic arches with different rise-to-span ratios." Thin-Walled Structures 129 (August 2018): 74–84. http://dx.doi.org/10.1016/j.tws.2018.03.019.

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50

Wei, Jiangang, Qingxiong Wu, Baochun Chen, and Ton-Lo Wang. "Equivalent Beam-Column Method to Estimate In-Plane Critical Loads of Parabolic Fixed Steel Arches." Journal of Bridge Engineering 14, no. 5 (2009): 346–54. http://dx.doi.org/10.1061/(asce)1084-0702(2009)14:5(346).

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