Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:419 https://doi.org/10.1007/s40430-019-1937-y TECHNICAL PAPER Shell instability analysis by using mixed interpolation M. Rezaiee‑Pajand 1  · Amir R. Masoodi 1 Received: 20 August 2018 / Accepted: 3 September 2019 / Published online: 14 September 2019 © The Brazilian Society of Mechanical Sciences and Engineering 2019 Abstract In this paper, buckling and post-buckling behavior of plates and shells are investigated. An efcient triangular shell element having six nodes is employed in this study. To consider large defection and rotation, total Lagrangian formulation is utilized. The authors employ mixed interpolation for strain felds to deviate shear and membrane locking phenomena. The efects of diferent types of boundary conditions and geometry properties of plates and shells are studied. An almost complete litera- ture review is also provided to show the progress of researches about the current topics. Various states of buckling including bifurcation and snap-through are assessed in diferent problems. In this study, the popular benchmarks of shell structures are analyzed to show the accuracy and capability of element. In addition, some new benchmarks which can be used by other researchers are also introduced and solved. Further, a comparison study between the proposed element and NLDKGQ ele- ment in OpenSees software is implemented for some complex curved shell structures. Keywords Buckling and post-buckling · Bifurcation · Snap-through · Total Lagrangian · Triangular shell element · Mixed interpolation of strains 1 Introduction Shell is one of the most applicable structures used in civil engineering area. These structures can be constructed in dif- ferent shapes. On the other hand, thin shells are the most practical ones, because of their high strength-weight ratio. It should be noted that the critical behavior of shell struc- tures is not material strength most of the times. Based on the obtained results, the buckling loads of these structures create more problems than the yielding criteria. Therefore, it is essential to employ an efcient method to study the buckling of shell structures. Besides, the behavior of shells after reaching buckling load should be also investigated. It is found that two types of buckling behavior can happen in shell structures. One of them is bifurcation, and the other is snap-through. The frst one almost happens in shell or structures subjected to in-plane loads while the second one occurs in structures under transverse loads. In both cases, the post-buckling behavior can be studied using the struc- tural equilibrium paths. One of the most applicable methods which can be employed in this area is fnite element method. It is worth mentioning that geometric nonlinear analysis should be considered to achieve the equilibrium path and predict the post-buckling behavior of structure. Therefore, researchers have always been looking for an efcient method to achieve complete equilibrium path. Some of them devel- oped the nonlinear solution method and others formulated more efcient shell element. Several researches were implemented about buckling analysis of structures. Most of them were based on the ana- lytical solution. So a closed-form solution was available to reach the buckling load [1–3]. One of the old experimen- tal investigations about the cylindrical shell buckling was performed by Babcock. He investigated the efect of test machine on the buckling load [4]. On the other hand, the stability analysis of thin-walled cylinders was done by Har- ris under compression load. This researcher incorporated internal pressure efects in his study [5]. Stability analysis of homogenous and nonhomogeneous beams using closed- form solution was investigated theoretically [6]. Another old investigation about shell buckling, considering diferent Technical Editor: Paulo de Tarso Rocha de Mendonça, Ph.D. * M. Rezaiee-Pajand rezaiee@um.ac.ir Amir R. Masoodi amirreza.masoodi@mail.um.ac.ir 1 Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran