Buckling of Beams, Plates and Shells

This introductory chapter starts with some basic concepts of stability theory. This is followed by a consideration of the so-called four Euler cases, which represent very basic cases of beam buckling and which can already be used to identify some important characteristics of buckling problems. The chapter concludes with an overview of the contents of this book and references for further reading.

The fundamental problem areas of stability theory and the principles that lead to their solution can already be dealt with very well using systems of rigid beams , which we will examine in this chapter. We will initially limit ourselves to systems with one degree of freedom; a corresponding generalization to systems with an arbitrary number of degrees of freedom will follow later in this chapter.

In this chapter, we consider members in the framework of second-order theory that are loaded both by compressive forces and by loads that lead to bending. We will refer to this problem area as the so-called beam-columns. We want to determine how the consideration of member deformations affects the development of internal forces and deformations. For this purpose, the governing differential equation is derived and solved for a number of elementary examples. Finally, in addition to different types of load and boundary conditions, we also investigate the influence of pre-deformations.

After  we have discussed the problem of beam-columns under bending and axial compression in the previous chapter, we now want to specialize the considerations in this chapter to beams that are free of transverse loads and only subjected to axial compressive loads. In this case, the load-bearing behavior is different. For the beam-columns considered so far, there was a clearly defined deflection of the beam for each load level, and it was shown that the state variables increase beyond all limits when the axial compressive force F approaches a certain value, namely the critical load or the bifurcational buckling load \(F_{\text {crit}}\). In the case of beams under pure compressive load, on the other hand, it can be seen that the beam remains straight at loads below the buckling load and only undergoes axial compression. When the bifurcational buckling load is reached, i.e. when \(F=F_{\text {crit}}\) applies, the bar deflects abruptly perpendicular to its longitudinal axis. We call this phenomenon of sudden deflection of the beam under compressive load buckling or flexural buckling, the latter due to the similarity of the occurring deformations to a bending deformation. This chapter is dedicated to the investigation of the buckling behavior of beams and the determination of buckling loads.

The previous explanations in this book on beam buckling were essentially based on equilibrium considerations. However, the concept of stability can be defined clearly by considering the energy of a structure, which is the subject of this chapter.

For a number of quite simple buckling problems, it is possible to derive exact analytical solutions. However, buckling analyses of beams very often lead to transcendental nonlinear buckling conditions that cannot be solved exactly for the buckling load and therefore require a numerical-iterative treatment. In such cases, approximation methods can be employed with the help of which (mostly based on energy methods) approximate solutions for the buckling loads can be determined. A selection of approximation methods is treated in this chapter.

So far, the explanations regarding the stability of slender compressed beams have been limited to flexural buckling/Euler-type buckling. However, there are other stability problems of such structures, which we denote as torsional buckling or flexural-torsional buckling, depending on the load and cross-section type. In this chapter, we want to address these stability problems, which are also extremely important in practice. We begin the discussion with a consideration of so-called torsional buckling, which is a special case of flexural-torsional buckling, before moving on to the more general case of flexural-torsional buckling of prismatic beams with arbitrary cross-sections. After the analytical treatment of elementary cases, an energetic treatment of flexural-torsional buckling is carried out and, as in the case of flexural buckling, a number of energy-based approximation methods are discussed. The chapter concludes with the investigation of beams with a fixed axis of rotation and of elastically supported beams.

A stability case closely related to flexural-torsional buckling is the so-called lateral buckling of beams under longitudinal and transverse loads, which occurs mainly in slender beams with strongly varying moments of inertia. In this chapter, we first discuss an elementary lateral buckling problem of a beam under moment load for motivation before we turn to a generalized formulation within the framework of second-order theory. The resulting differential equations are then solved in an exact analytical manner for elementary cases. The Ritz method is again considered as a suitable approximation method. This chapter concludes with a consideration of the lateral buckling of beams with a fixed axis of rotation and beams with torsionally elastic foundation.

Plates are thin-walled structures, and for this reason their buckling behavior under inplane load must be investigated in the practical design process. In this chapter, we discuss the buckling behavior of thin plates in the context of the so-called Kirchhoff plate theory and present some analytical solutions as well as approximation methods before discussing the buckling behavior of stiffened plates. The chapter is rounded off with explanations of the postbuckling behavior of plates.

Shells are curved thin-walled structures. In addition to the general strength verification, they must always be examined with regard to their stability behavior. Analogous to plate buckling, we speak of so-called shell buckling in this case. In addition to the ideal bifurcational buckling problem—i.e. the determination of the critical buckling load—the postbuckling behavior must also be considered in lightweight construction applications. In this chapter, we consider the buckling equation for the special case of circular cylindrical shells and provide closed-form analytical solutions for determining the critical buckling load. For the circular cylindrical shell, we also consider the postbuckling behavior in a closed-form analytical manner. Circular cylindrical shells not only have a fundamental and technically very important significance, e.g. in aerospace engineering, but also have some special features with regard to their stability behavior, which justifies their detailed consideration.