Engineering Structures 31 (2009) 444–454 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct In-plane strength and design of parabolic arches Jiho Moon a , Ki-Yong Yoon b , Tae-Hyung Lee c , Hak-Eun Lee a,∗ a Civil, Environmental & Architectural Engineering, Korea University, 5-1, Anam-dong, Sungbuk-gu, Seoul, 136-701, South Korea b Department of Civil Engineering, Sunmoon University, Kalsan-ri, Tangjeong-myeon, Asan-si, Chungnam 336-708, South Korea c Department of Civil Engineering, Konkuk University, Seoul 143-701, South Korea article info Article history: Received 24 May 2008 Received in revised form 5 September 2008 Accepted 9 September 2008 Available online 10 October 2008 Keywords: In-plane strength Arches Buckling Design criteria Reduction factor C m abstract In the current AASHTO LRFD, the arch design formula is based on the bilinear interaction relationship between two extreme cases of the axial and the flexural strength. However, this method is not suitable for the design of the shallow arch which may buckle in a symmetric snap-through mode. Also, the use of the constant reduction factor for the design of arches leads to a conservative design. This paper investigates the in-plane buckling strength and design of parabolic arches. Firstly, the thresholds for the different buckling modes of shallow parabolic arches are summarized and boundaries for the deep and shallow arches are reported. The inelastic strengths of parabolic deep arches based on the finite element analyses are then compared with those presented in AASHTO LRFD. From the results, it is found that AASHTO LRFD provides good predictions of buckling strengths for the parabolic arches under only axial compression, while the bilinear interaction relationship provides conservative values for the in-plane strength of parabolic arches due to the use of constant reduction factors that can be applied regardless of loading and boundary conditions. The modified formulas for reduction factors are proposed for various loading and boundary conditions in this study. It is found that modified formulas for reduction factors show good match with the results obtained from finite element analyses. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction When the lateral displacement and twist rotations are fully restrained, the arches under axial compression may buckle in an asymmetric bifurcation or in a symmetric snap-through buckling, as shown in Fig. 1. For the deep arch that may buckle in an asymmetric bifurcation buckling, the classical buckling theory is adopted in order to design the arches where it is assumed that the pre-buckling behavior is linear and its effect on the buckling is ignored. According to the classical buckling theory, a deep arch that has the arc length S can be replaced by the equivalent column that has the length k (S /2), as shown in Fig. 2 where h is the rise of the arch; l is the span of the arch; S is the arc length of the arch; and k is the effective length factor of the arches that depends on the shape of the arch and boundary conditions. For the shallow arches, several researchers have investigated the geometric nonlinear behavior of the arches [1–5]. They reported that the classical buckling theory may overestimate the buckling load of shallow arches because the ratio of the pre-buckling deformation to the initial height of the arch is too large to be ignored [1–4]. Therefore, it is important to classify the buckling modes to determine the buckling strength of the arches under axial compression. ∗ Corresponding author. Tel.: +82 2 3290 3315; fax: +82 2 928 5217. E-mail address: helee@korea.ac.kr (H.-E. Lee). Subsequent to determining the buckling strength of the arch under the axial compression, the in-plane strength of the arches subjected to a transverse loading can be calculated by using the interaction formula. A number of researchers [6–8] have attempted to develop the interaction formula for the arches. Pi and Trahair [6] developed the in-plane design equation for the 2-hinged circular arches. Pi and Bradford [7] studied the in-plane behavior of fixed circular arches under general loading and proposed the interaction formula. Verstappen et al. [8] proposed the in-plane buckling design strength of deep circular arches using the linear interaction formula. For parabolic arches, Kuranish and Yabuki [9] proposed design criteria for the parabolic arches that were expressed in terms of axial force and bending moment at the quarter point of the span. However, their results are only valid for the slenderness ratio of 0.5 S /r , in the range of 50–150, and for the rise to span ratio of 0.1–0.3. A few design codes such as AASHTO LRFD [10] provide design equations for the arch. This design code also adopts interaction formula between two extreme cases of the axial strength of centrally loaded columns and the flexural strength of the beam. This beam–column analogy approach is only applicable to deep arches. However, the classifying the deep and shallow arches or buckling modes is not mentioned in AASHTO LRFD [10]. The objective of this paper is to investigate the in-plane buckling strength of parabolic arches and to focus on the 0141-0296/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2008.09.009