Smart Beam Element Approach for LRPH Device

Abstract

LRPH (Limited Resistance Rigid Perfectly Plastic Hinge) device is a special steel device mainly usable to join beam elements of plane or spatial steel frames covered by patent n. 102017000088597 at the Italian Ministry of Economic Development and identified in the International Patent System with the number PCT/IB2018/055766. In the framework of moment (rigid) connection, the main fundamental innovation of LRPH consists in the mutual independence of its own resistance and stiffness features. The device is constituted by a sequence of three steel elements of limited length bounded by two parallel steel plates joined up with the connected structure elements. The cross-sections of the three steel elements are classical I sections with appropriate wing and web thicknesses obtained by the solution of suitable optimal design problem. Therefore, the overall device shows piecewise discrete geometric and mechanical features. In order to implement this device in a frame-oriented code for the design of both 2D and 3D frame structures, it is necessary to adopt a suitable model based on a non-uniform cross section beam element. The latter element should be able to reproduce the elastic and plastic behavior of the device. Recently, in the literature it has been proposed a new inelastic beam element, belonging to the displacement based approach and formulated for uniform beams, based on variable displacement shape functions, whose analytic expressions are prone to updating (smart) in accordance to the plastic deformation evolution in the beam element. Aim of the paper is to utilize the relevant smart displacement beam element approach and extend it to the case of non-uniform beams to evaluate the nonlinear behavior of the LRPH device. The obtained results confirm the efficacy and the feasibility of the smart displacement beam element opening the way of implementing LRPH device in a FEM code.

Appendix

In this Appendix the expressions of the functions $g_{2} \left( {x;\beta_{x,i} } \right),g_{3} \left( {x;\beta_{x,i} } \right),$ and $f_{3} \left( {x;\beta_{z,i} } \right),f_{4} \left( {x;\beta_{z,i} } \right),f_{5} \left( {x;\beta_{z,i} } \right)$, dependent on the parameters $\beta_{x,i}$ and $\beta_{z,i}$ and appearing in Eqs. (7a, 7b) of the main text, are reported:

$$ \begin{array}{*{20}c} {g_{2} \left( x \right) = - x - \mathop \sum \limits_{i = 1}^{n} \left( {\frac{{\beta_{x,i} }}{{1 - \beta_{x,i} }} - \frac{{\beta_{x,i - 1} }}{{1 - \beta_{x,i - 1} }}} \right)\left( {x - x_{i} } \right)U\left( {x - x_{i} } \right)} \\ {g_{3} \left( x \right) = - \frac{{p_{x}^{\left[ 2 \right]} \left( x \right)}}{{E_{0} A_{0} }} - \mathop \sum \limits_{i = 1}^{n} \frac{1}{{E_{0} A_{0} }}\left( {\frac{{\beta_{x,i} }}{{1 - \beta_{x,i} }} - \frac{{\beta_{x,i - 1} }}{{1 - \beta_{x,i - 1} }}} \right)\left[ {p_{x}^{\left[ 2 \right]} \left( x \right) - p_{x}^{\left[ 2 \right]} \left( {x_{i} } \right)} \right]U\left( {x - x_{i} } \right)} \\ {f_{3} \left( x \right) = x^{2} + \mathop \sum \limits_{i = 1}^{n} \left( {\frac{{\beta_{z,i} }}{{1 - \beta_{z,i} }} - \frac{{\beta_{z,i - 1} }}{{1 - \beta_{z,i - 1} }}} \right)\left( {x - x_{i} } \right)^{2} U\left( {x - x_{i} } \right)} \\ {f_{4} \left( x \right) = x^{3} + \mathop \sum \limits_{i = 1}^{n} \left( {\frac{{\beta_{z,i} }}{{1 - \beta_{z,i} }} - \frac{{\beta_{z,i - 1} }}{{1 - \beta_{z,i - 1} }}} \right)\left[ {x^{3} - 3x_{i}^{2} x + 2x_{i}^{3} } \right]U\left( {x - x_{i} } \right)} \\ {f_{5} \left( x \right) = \frac{{p_{z}^{\left[ 4 \right]} \left( x \right)}}{{E_{0} J_{0} }} + \mathop \sum \limits_{i = 1}^{n} \frac{1}{{E_{0} J_{0} }}\left( {\frac{{\beta_{z,i} }}{{1 - \beta_{z,i} }} - \frac{{\beta_{z,i - 1} }}{{1 - \beta_{z,i - 1} }}} \right)\left[ {p_{z}^{\left[ 4 \right]} \left( x \right) - p_{z}^{\left[ 4 \right]} \left( {x_{i} } \right)} \right]U\left( {x - x_{i} } \right)} \\ { - \mathop \sum \limits_{i = 1}^{n} \left( {\frac{{\beta_{z,i} }}{{1 - \beta_{z,i} }} - \frac{{\beta_{z,i - 1} }}{{1 - \beta_{z,i - 1} }}} \right)p_{z}^{\left[ 3 \right]} \left( {x_{i} } \right)\left( {x - x_{i} } \right)U\left( {x - x_{i} } \right)} \\ \end{array} $$

where $p_{x}^{\left[ k \right]} \left( x \right)$, $p_{z}^{\left[ k \right]} \left( x \right)$ indicate the k-th primitive functions of the relevant external load distributions $p_{x} \left( x \right)$, $p_{z} \left( x \right)$, respectively.