Laplace Transform is often used in solving the free vibration problems of structural beams. In existing research, there are two types of simplified models of continuous beam placement. The first is to regard the continuous beam as a single-span beam, the middle bearing of which is replaced by the bearing reaction force; the second is to divide the continuous beam into several simply supported beams, with the bending moment of the continuous beam at the middle bearing considered as the external force. Research shows that the second simplified model is incorrect, and the frequency equation derived from the first simplified model contains multiple expressions which might not be equivalent to each other. This paper specifies the application method of Laplace Transform in solving the free vibration problems of continuous beams, having great significance in the proper use of the transform method.
The issues of local stability and ultimate resistance of a continuous beam with thin-walled box section (Class 4) were reduced to the analysis of the local buckling of bilaterally elastically restrained internal plate of the compression flange at longitudinal stress variation. Critical stress of the local buckling was determined using the so-called Critical Plate Method (CPM). In the method, the effect of the elastic restraint of the component walls of the bar section and the effect of longitudinal stress variation that results from varying distribution of bending moments were taken into account. On that basis, appropriate effective characteristics of reliable sections were determined. Additionally, ultimate resistances of those sections were estimated. The impact of longitudinal stress variation and of the degree of elastic restraint of longitudinal edges on, respectively, the local buckling of compression flanges in the span section (p) and support section (s) was analysed. The influence of the span length of the continuous beam and of the relative plate slenderness of the compression flange on the critical ultimate resistance of box sections was examined.