# A Review of the Network Arch Bridge
## 4 Research on Network Arch Bridges
### Stability Analysis  
The first to address the critical in-plane buckling load of network arches was [PERSON] in ref. [64], for the case with no hanger relaxation. He derived a formula for assessing the critical elastic in-plane buckling load on network arches, with the load over the entire span of the bridge. The design formula derived was verified in some network arches already built, and the variation compared to the finite element analysis was smaller than 5%.  
The out-of-plane behavior of network arch bridges was addressed in ref. [65] as part of a method to assess the elastic buckling design of NABs. The proposed method was developed for arches with K-shaped bracings. The paper develops a set of formulas to assess the critical buckling force. The proposed formulation has a maximum error of 10% with respect to several refined FE results, in comparison to the simplified formulation in EC3 that shows large overestimation of the instability strength (up to 70% error compared to finite element analysis).  
Another group studied the post-buckling of NABs subjected to vertical loads [66]. Their study was conducted for a 100 m long bridge with a width of 10 m, a rise-to-span ratio of 0.2, and the shape of the arch a parabola. The study included three analyses: a linear elastic buckling analysis, followed by a nonlinear analysis considering the geometrical nonlinearity and, finally, a pushover analysis with the purpose of evaluating the global out-of-plane capacity of the arch. The behavior of the arch was considered under traffic loads, without seismic force. Also, the study included a scenario with cable loss. The dynamic behavior of the arch observed was similar for one or two broken cables; the main difference in the case of two broken cables was in the amplitude of the out-of-plane displacement of the arch. The dynamic response of the arch was influenced by the position of the broken cable. Such a scenario near the abutments increases the dynamic sensitivity of the arch. On the other hand, a cable loss near the center of the arch can reduce the lateral buckling capacity by up to 25% in the case of slender arch cross-sections. Another group of authorsinvestigated the behavior of NABs subjected to cable loss under moving loads [67; 68]. The method chosen was a simplified static analysis of the 2D structure without the broken cable, subjected to dead and live loads, followed by two static forces that reproduced the dynamic effect of the sudden loss. For the bridge studied, the maximum displacement of the tie in the case of sudden cable loss was found to be near the sides of the bridge, at 2/5\(\times\)L and 3/4\(\times\)L, where L is the span of the bridge.  
The most recent research into the nonlinear buckling analysis of NABs was conducted by [PERSON] et al. [69]. They compared the critical buckling factor using a linear and nonlinear buckling analysis. Nonlinear buckling analysis takes into account the nonlinear behavior of the cables. As their requirement was to find the limit point, third-order theory analysis was chosen, providing more accurate results for a slender structure in the case of large deflection, such as in the case of cables. The study was conducted for the bridge over the Vistula River in Cracow, and the vertical live load was a train load model from Eurocode, LM71. One interesting conclusion of their study, that is worth investigating in the future, is that additional tension in the hangers, above the forces resulting from self-tensioning, decrease the value of the critical load factor.