# Yield Behaviour of Welded I-Shaped Steel Cross-Sections
## 1 Introduction
The LRPDs represent the most recent proposal in this topic; these innovative devices can be positioned either at the ends of beams and at the bases of columns, and their peculiar design not only incites the onset of plastic strains on the selected portions of the equipped beam elements, but, furthermore, also guarantees the invariance of the bending stiffness of the overall involved elements. The first version of LRPD was referred to as limited-resistance rigid perfectly plastic flexural hinges (LRPH). Later, the design problem was enriched by taking into account the simultaneous presence of both normal and tangential stresses and adopting a special analysis of the beam elements by means of smart beam elements specialized to the design of a frame with assigned capacity curve [40], as well as performing both numerical and experimental validation. Other formulations, among which one is approximated to be useful in practical applications, have been proposed to compare the results with those achieved by different typologies of reduced beam section (RBS) connections [41]; some new formulations have also been proposed to consider the local buckling and to protect the welded connections between beams and columns [42]. The actual version, referred to as LRPD, is a beam element with multistep geometry, constituted by three subsequent I-shaped portions whose web and flange thicknesses are suitably designed. In this last version of LRPD, the inner portion with reduced dimensions shows limited strength and allows the onset of plastic strain, while the outer portions are equal to each other and show greater web and flange thicknesses to guarantee that the bending stiffness of the overall involved element is unaltered. As can be easily deduced, the LRPD design requires the solution to appropriate optimal design problems, and their production has to be performed in a factory by welding steel plates of appropriate thickness. Expanding the focus to welded I-shaped sections of any size and referring to beam elements belonging to frame structures, these sections are subjected to the simultaneous action of axial force \(N\), shear force \(T\), and bending moment \(M\). It is evident that the knowledge of both the elastic and plastic limit domains of such sections is essential to evaluate the overall structural safety. The actual international standards [43] provide approximated expressions to identify such domains, and, therefore, it is appropriate to examine this topic in depth to obtain more rigorous expressions for practical applications. In the literature, some papers have recently been proposed for either elastic [41] or plastic [44],domains but the formulations for plastic domains still present approximations, and they involve standard sections. Therefore, the aim of the present paper is to propose a rigorous procedure to identify the plastic domains of welded I-shaped sections subjected to the simultaneous action of axial force \(N\), shear force \(T\), and bending moment \(M\), assuming an elastic-perfectly plastic material behaviour and imposing that the section belongs to Class 1 of the EC3 standard [43]. The expressions of the plastic domain contours in the \((N,T,\ M)\) space, as well as those in the \((N,M)\), \((N,T)\), and \((M,T)\) planes, will be provided. Some numerical applications for the case of welded sections and a comparison with the results from the EC3 standard conclude the paper.