Jozef Djubek

Equilibrium bifurcation of elastic plates subject to large deflections

Abstract In the numerical analysis of non-linear problems we often are faced with the phenomenon of the original finite deformation becoming unstable, at least in regard to the computations involved. This feature of non-linear differential equations of elastic bodies, in so far as it concerns the bifurcation of equilibrium, is dealt with in this paper. Sections 1 and 2 give a survey of notation and formulae, as well as the constitutive equations of the stable and neutral equilibrium of an elastic body. From these, the functional relations of the stable and neutral equilibrium of slender web plates are derived in Section 3. The solution of the problem using the Galerkin method is discussed in Section 4. As the non-linear operator here is potential, the Galerkin method is equivalent to the Ritz method. The numerical results, characterizing the bifurcation of equilibrium, are given for a web plate loaded in compression by increasing and decreasing load.

Large Deflections of Slender Webs Fitted with Ribs

While in Chapter 7 we investigated the post-buckled behaviour of plates by means of an orthotropic plate concept, i.e. all stiffeners being smeared over the plate surface, this chapter is going to study the post-critical performance of stiffened webs as these actually behave, i.e. as a system of web sheet and discrete stiffeners.

Large Deflections of Elasto-Plastic Webs

The webs of steel structures present unavoidable initial deviations from the ideal state and, moreover, residual stresses occur in them.

Problems of Solution of a System of Non-Linear Algebraic Equations

The solution of non-linear problems of theory of webs and shells by direct approximate methods leads to a system of non-linear algebraic equations, depending usually on a parameter. As usual, considerable difficulties are encountered in the solution of such systems. These difficulties mainly concern the initial approximation for iteration methods, the realization of the solution and also the identification of the obtained solution. Some methods of the solution to systems of non-linear algebraic equations are mentioned in the following text. It will be seen from the involved formulations that a general approach is in fact illusory. Every individual case needs to be studied separately, and then we can decide which method, or combination of methods, to apply.

Large Deflections of Elastic Isotropic Webs

The question of stability and behaviour of slender webs in the post-critical range is becoming important in connection with the production of high strength metal sheet and plates. This chapter presents a solution of the non-linear problem of the deformation of slender initially curved rectangular webs which are stiffened along their edges by elastically compressible stiffeners, flexible in the web plane. By introducing an initial deflection, we take account of certain (geometrical, structural and other) imperfections of “actual” (in comparison with ideally plane) webs.

Large Deflections of Orthotropic Webs

The problem of correct design of one-sidedly stiffened orthotropic plate elements of box-girder bridges attracted during the last years, and still attracts at the present time, plenty of attention; this being particularly due to the recent accidents on a number of large-span bridges, which highlighted the necessity to deal thoroughly with the afore-said problem.

Interaction of the Buckling of Thin-Walled Bars with the Buckling of Their Plate Elements

The design of the plate elements of compressed bars is based on the requirement that local buckling shall not occur before the bar as a whole has buckled while the effect of all initial imperfections is neglected.

Basic Assumptions of Theory of Slender Webs

Let us investigate a thin web of constant thickness, i.e. a rectangular prismatic body of which one dimension (thickness) is small in comparison with the other ones (width and depth) (Fig. 1.1). The plane which is parallel to the web surface and situated at the distance of t/2 from it will be termed middle plane. The line of intersection of the middle plane with the web edges is called boundary of the web. The middle plane is selected as plane xy.

Approximate Methods of Solution

Let us denote by R m a m-dimensional Euclid space. For every \( a = \{ {a_i}\} _{i = 1}^m,\;b = \{ {b_i}\} _{i = 1}^m \in {R^m} \) let us define