The Relationship Between the Buckling Load Factor and the Fundamental Frequency of a Structure

Abstract

In structural dynamics, the problem of finding oscillation natural frequencies of a structure can be solved using the mass and elastic stiffness matrices of the structure. On the other hand, the problem of structural instability of a structure and the critical buckling load is a problem, which can be solved using the elastic and geometric stiffness matrices of the structure. By including the geometric stiffness or in other words by taking into account stiffness conditions with respect to the deformed structure, a new more realistic representation of stiffness results. This influences the oscillation frequencies of the structure. This is why in general structural cases there will be a relationship between the vibration and buckling behavior. Both of these phenomena are sensitive to end boundary conditions and so if it is possible to make natural frequency measurements under different loads on a structure which take account of end conditions it may be possible to predict the critical buckling load. Stress analysis and structural analysis are the key requirements for the determination and improvement of the strength and stiffness of structures. The important problem of stability and, in particular, buckling behavior must also be considered and is necessary to complete the analysis. In stability, the importance of experimental studies in many categories of problems is evident. It has been found in many cases that the biggest problem is in defining the boundary conditions to which the buckling load is sensitive. This introduction aims to provide details of the buckling characteristics of various structural elements and to present the different analytical methods used in the solution of stability problems. To predict the critical load of a structure and then to find the relationship between the critical buckling load and the fundamental frequency of a structure, it is necessary to investigate the stability of the structure acting as a single unit. The problem of calculating the natural frequencies and corresponding mode shapes of all the kinds of variations of common structural components requires the development for each element in the model structure of a mass matrix which will represent the effect of the dynamic loading which is set up during vibration. Structural instability problems are geometrically non-linear, and a basic case of such problems is the problem of the lateral buckling under axial compression of a perfect pin-ended column or the classical Euler analysis of buckling load. In common with the vibration problem, the linear stability problem is all eigen value problem and the eigen values are the critical values of loading magnitude at which buckling occurs. Usually, only the lowest (fundamental) of these is of practical interest. A number of design examples show how the formulas can be used in practical applications. Also, these theoretical and practical examples help to understand the significance of the subject of this work and the checking of the safety in every kind of load that is applied. Also, the comments made are useful to the designers and engineers when trying to create safe structures. It also worth's noting that the engineer must design a structure, which will satisfy the demands for a functional performance, aesthetic appeal and the critical element of design.