Buckling of shell structures - Plate Shell Structures of Glass

This section summarizes established knowledge about buckling of shell structures, to the extend that it is relevant for the non-linear study of plate shells. The information can for example be found in [18], [24],[37], [38] and [51].

In a thin-walled shell structure, the membrane stiffness is many times greater than the bending stiffness, if the structure is shaped and supported appropriately.

Consider a shell structure loaded in compression by a given load of increasing magnitude. Until the load reaches a certain level, the shell carries the load mainly by membrane action. Energy is then stored in the structure mainly as membrane strain energy, and the corresponding displacements are small. This is referred to as the prebuckling state. When the load level reaches a critical load, certain displacements can take place without adding any load. Buckling occurs at this load level, as membrane strain energy from compression is converted into bending strain energy, with large displacements as a con- sequence. The displacements that arise during buckling are referred to as the bifurcation buckling mode. The term true bifurcation buckling refers to a situation where the bifurcation buckling mode has zero amplitude until buckling occurs.

The structure’s state after buckling, referred to as the postbuckling state, can be either

stable or unstable1_{. The postbuckling state is stable if the load-bearing capacity of the}

1_{Note that the term “stable” here is used in a different context than in Chapter 1.}

Chapter 6 Non-linear investigations

This chapter focuses on the structural behaviour of plate shell structures when effects from geometrically non-linearity are included. This potential area of research is very large, and the study in the present chapter should be seen as an introductory investigation into the topic. The aim of the study is to promote an intuitive understanding of the non-linear behaviour of a facetted shell, in order to direct further research in the area, or (in a design situation) to help set up appropriate FE models for analysis.

Section 6.1 is a short summary of buckling of shell structures, taken from the literature. The non-linear behaviour of plate shells is addressed in Section 6.2.

6.1 Buckling of shell structures

In a thin-walled shell structure, the membrane stiffness is many times greater than the bending stiffness, if the structure is shaped and supported appropriately.

The structure’s state after buckling, referred to as the postbuckling state, can be either

stable or unstable1_{. The postbuckling state is stable if the load-bearing capacity of the}

1_{Note that the term “stable” here is used in a different context than in Chapter 1.}

Non-linear investigations 6.1 Buckling of shell structures

structure increase with increasing amplitude of the bifurcation buckling mode. An example of a structure which is stable in its postbuckling state is a simply supported plate under compression in its own plane. The postbuckling state is unstable if the load-bearing capacity decreases with increasing amplitude of the bifurcation buckling mode. If a part of the load is not removed after the buckling has taken place, the structure will collapse. Spherical shell structures belong to this category. A neutral state, where an arbitrary size of the bifurcation mode is possible at the critical load level, can be formulated as an eigenvalue problem. This is a linearized model of the elastic stability of the structure, in which it is assumed that no (or negligible) displacements take place in the prebuckling state.2

In an ideal structure, where there are no geometric or material imperfections, and where the displacements in the prebuckling state are negligible, the linearized model of the elastic stability yields the critical load with a good approximation. In any real structure true bifurcation buckling does not exist, because no real structure is free of imperfections. Imperfections include geometric imperfections (for example from construction, support settlements and geometric changes due to creep and shrinkage), material imperfections (inhomogeneities etc.), residual stresses, temperature movements and/or stresses, and other effects. Imperfections reduce the critical load, compared to the linearized bifurcation load. The difference between the bifurcation load of the perfect structure, and the critical collapse load of the imperfect structure, depends on the amplitude of the imperfection. The collapse load of a shell structure with imperfections can be determined by a geomet- rical non-linear FE-calculation, where imperfections have been assessed and implemented in the model. The load is applied in steps, each for which the displacements are calcu- lated, and equilibrium is found for the deflected structure. The deflected structure is then basis of the following load step. Generally, the stiffness of the structure decreases with increasing load. FE software which can handle geometrically non-linear analysis, apply various methods for the stepwise attaining of the equilibrium state.

A dome shell structure is unstable in the postbuckling state. Since the load is not reduced on the structure when buckling occurs, the structure will collapse. The collapse happens almost instantly. For a shell structure, which has geometric and material imperfections,

the actual load-bearing capacity can be 1/6th _{or as little as 1/10}th _{of the critical load}

corresponding to bifurcation of the perfect structure [51]. Therefore, it is essential in a design process to assess possible geometric imperfections, and carry out geometrically non-linear FE calculations on the imperfect structure. In most cases, a linear combination of the lowest linear buckling modes for a shell structure are known to produce the worst possible imperfection shapes for the structure, given a certain maximum amplitude of deformation. The effect of imperfections of other types than geometric must also be assessed.

2_{For a simply supported column, the solution to this problem is the Euler load, and the corresponding}

bifurcation mode is a single or multiple sine arches.

Non-linear investigations 6.1 Buckling of shell structures

the actual load-bearing capacity can be 1/6th _{or as little as 1/10}th _{of the critical load}

2_{For a simply supported column, the solution to this problem is the Euler load, and the corresponding}

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