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Twin arch bridges are often designed with a bracing member system that can significantly enhance the out-of-plane stability. This study presents an exact matrix stiffness method (MSM) for the out-of-plane buckling analysis of funicular twin arch bridges and investigates the optimal bracing system design. A novel 14 × 14 s-order element stiffness matrix of three-dimensional beam–columns was used in the MSM and the out-of-plane buckling analysis was performed after assembling the exact global structural stability stiffness matrix. The influences of the bracing location, number, length, and flexural rigidity of both the transverse bracing and X-bracing systems on the buckling capacity of the twin arch system were investigated. The in-plane stiffness of the bracing system (i.e., the horizontal flexural rigidity in a transverse bracing system) was found to be the critical factor in suppressing the out-of-plane buckling of the twin arch system. In addition, the twin arch system achieved the maximum out-of-plane buckling capacity with the bracing location at S /4, where S is the total arc length. For optimal bracing system design of funicular twin arches, it is recommended that one uses the more economical X-bracing system or a transverse bracing system with adequately rigid bracing members. • A method for exact out-of-plane buckling analysis of twin arches is established. • The optimal bracing location is found to be S /4 of arch rib (of length S) for maximum lateral buckling capacity. • The in-plane stiffness of bracing is critical for lateral stability of twin arches. • The X-bracing system is highly recommended for enhanced out-of-plane stability of twin arch bridges. [ABSTRACT FROM AUTHOR]

Highlights • Applied the Hencky bar-chain model (HBM) for frame buckling analysis considering different configurations. • Used easy-to-code stiffness matrices with patterns to simplify the expression of HBM formulations. • Demonstrated the advantage of localized model modification of HBM in considering a local damage/stiffening. Abstract This paper presents the development of the Hencky bar-chain model (HBM) for the buckling analysis of non-symmetric portal frames under vertical compressive axial loads on the columns. Generic stiffness and geometric matrices are derived for the frame buckling analysis. Based on these matrices, a simple computer code is developed to determine the buckling loads. Illustrative frame buckling problems are solved to demonstrate the ease and simplicity of the HBM for analysis of different and complex design conditions imposed on the frames. The advantages of consistent formulation, localized model modifications, and convenience in obtaining the buckling modes are further noted for the HBM. [ABSTRACT FROM AUTHOR]

• The least weight design of steel BRBs against global buckling is pinpointed. • Global buckling prevention criterion of BRB is expressed as stiffness and strength requirements. • Stiffness–strength relationship curve of BRB restraining member is derived. • Feasible designs of restraining member may be overdesigned in weight significantly. This paper is concerned with the optimal (or least weight) design of steel buckling-restrained braces against global buckling. First, the global buckling prevention criterion of steel buckling-restrained braces (BRB) is reconstituted to take on a symmetric expression, i.e., (P cr / P u −1) M r /(v 0 P u) −1] ≥ 1, so as to provide clarity on the equal importance of the stiffness and strength requirements of the restraining member, where P cr is the elastic buckling load, M r the yield moment resistance, and v 0 the initial imperfection of the restraining member whereas P u is the ultimate compressive load of the core under cyclic axial loads. It is shown herein that the optimal design of the restraining member satisfies (1) the boundary of the global buckling prevention criterion, and (2) the relationship between the stiffness term P cr / P u and the strength term M r /(v 0 P u) in view of the maximum feasible cross-sectional height-to-thickness ratio as specified by the adopted design code recommendation. The satisfaction of these two conditions furnishes a unique optimal design of the restraining member. The optimal design can be expressed as an explicit equation in terms of the second moment of area of the restraining member. A design example is presented to illustrate the material (or weight) efficiency of all-steel BRB designs based on the proposed design equation. [ABSTRACT FROM AUTHOR]

• Developed easy-to-code Hencky bar-chain model (HBM) formulations for buckling analyses of general multi-bay and multi-storey frames. • Proposed a new connection model to handle joints with multiple members connecting to nodal zones by flexible rotational springs. • Demonstrated the advantage of HBM matrix method over traditional matrix method in handling non-uniform cross-sectional members. • Presented an easy-to-use computer program—HBM solver for buckling analyses of frames. Comprising rigid segments connected by hinges with elastic rotational springs, the Hencky bar-chain model (HBM) has some advantageous features over traditional structural models in that it can handle local structural changes easily and allows easy coding for analyses. The development of the HBM was previously rather limited to structural members and simple frames. Continuing along the development of an HBM for bifurcation in-plane buckling analysis of frames, a new connection model is proposed herein to handle joints receiving multiple members that are encountered in multi-bay-multi-storey frames. The connection stiffness matrices associated with the elastic strain energies of the deformed multi-member connection are formulated. The connection stiffness matrices together with the elastic stiffness and geometric stiffness matrices for members are readily coded in a computer program—HBM solver for elastic bifurcation in-plane buckling analysis of frames, and the MATLAB code is given in the Appendix. The HBM with the new connection model is verified by comparing its results with frame buckling loads and mode shapes obtained from the traditional matrix stiffness method involving stability functions. The advantages of the HBM over traditional structural models are also demonstrated, including its simplicity in handling non-uniform cross-sectional members. Image, graphical abstract [ABSTRACT FROM AUTHOR]

This paper presents a simple methodology for determining the optimal design of an elastic funicular arch under point loads and selfweight for maximum in-plane buckling capacity. This class of optimization problem has hitherto not been investigated. The cross-section of the arch is assumed to be geometrically similar and constant through its entire length. The arch is assumed to be inextensible. The supporting points of the arch may be either pinned or fixed. The methodology for determining the optimal arch shape involves first finding the funicular shape of the arch under point loads and selfweight. Next, the arch is modelled by the Hencky bar chain model (HBM) for the buckling analysis. The HBM is adopted because it eliminates the requirement for formulating the governing equation for buckling and it is a straightforward model to comprehend and implement in a computer code. Each funicular arch shape is specified by a horizontal force value, and it has to be optimally selected to maximise the buckling capacity of the arch. As it is a single parameter optimization problem, the Golden Section search was used to determine the optimal horizontal force, and hence the optimal funicular arch shape. The methodology is illustrated by solving two example problems: (1) an arch under a point load and selfweight and (2) a symmetric arch under two equal point loads and selfweight. By solving the self-buckling problem of a catenary arch, the maximum spanning capacity of the catenary arch can be obtained as shown herein. • Paper outlines a method for designing elastic funicular arches to optimize buckling capacity. • Arch supportings can be pinned or fixed, offering design flexibility. • Method finds funicular arch shape under seflweight and point loads for analysis. • Paper uses Hencky bar chain model for buckling analysis, simplifying governing equations. • Method demonstrates with two examples: an arch under point load and selfweight, and a symmetric arch under two equal point loads and selfweight. [ABSTRACT FROM AUTHOR]

• Develop Hencky bar-net model for circular and annular single layer graphene sheets. • Calibrate Eringen's small length scale coefficients for nonlocal circular and annular plates. • Calculate real buckling loads of circular and annular graphene sheets with clamped and simply supported restraints. This paper is focused on the modeling of circular and annular graphene sheets via Hencky bar-net model (HBM 1 1 HBM: Hencky bar-chain model.) and calibrating the Eringen's small length scale coefficient e 0 in Eringen's nonlocal theory. The buckling solutions of circular and annular graphene sheets based on Eringen's nonlocal continuum plate theory are first obtained. On the other hand, HBM is developed to model the same structure from the discrete view. HBM is a grid system comprising rigid bars and arcs connected by frictionless hinges with elastic rotational and torsional springs. By regarding the length of straight segments in HBM equal to the characteristic length of Eringen's nonlocal model (ENM 2 2 ENM: Eringen's nonlocal model.) and matching their solutions, the Eringen's small length scale coefficient e 0 is calibrated. It is found that for circular graphene sheet, e 0 = 0.258 for clamped edge and e 0 = 0.300 for simply supported edge. For annular graphene sheet, e 0 is dependent on the inner to outer radius ratio χ and boundary conditions. The scale coefficient e 0 takes 0.307–0.367 for clamped edges while 0.219–0.290 for simply supported edges with χ varying from 0.2 to 0.8. Another finding is that the graphene sheet will buckle with a very small load when its dimension is large, regardless of models adopted. However for small dimensions, ENM and HBM predict lower buckling loads than the classical local model because the scale effect is more obvious. [ABSTRACT FROM AUTHOR]