Issue 50

C. C. Silva et alii, Frattura ed Integrità Strutturale, 50 (2019) 264-275; DOI: 10.3221/IGF-ESIS.50.22 266 Figure 2 : Inverted “U-frame” mechanism with two beams (Oliveira [1]). According to Fan [26], the inverted "U-frame" mechanism is more adequate to represent the behavior to LDB compared to the model of a composite beam composed of a single steel profile superimposed on a concrete slab ("T" cross section) because it represents better the lateral displacement and torsional constraints imposed on the steel profile by the concrete slab and the shear connection. The "U-frame" mechanism also has a direct relation with the usual situations, once most constructions use floor systems composed of parallel steel beams equally spaced under the concrete slab. In the literature there are two types of inverted "U" mechanism, the continuous one, which has only rigid internal supports, and the discrete one, which has regularly spaced transversal stiffeners throughout the hogging moment region, which contributes to the restriction of LDB. The EN 1994-1-1:2004 [4] and ABNT NBR 8800:2008 [3] standards only present formulations considering the continuous "U-frame" mechanism for the verification of composite beams. Rotational Stiffness A fundamental parameter for the calculation of the elastic critical moment ( M cr ) is the rotational stiffness of the composite beam, k s , also known as the rotational stiffness of the inverted "U-frame" mechanism. This stiffness, considered simplified by a rotational spring located in the top flange of a steel profile, allows to reproduce the influence of the "U-frame" mechanism at the bending moment resistant to the LDB, considering the bending of the slab, the distortion of the web and the deformation of the shear connection (Fig. 3). According to Johnson [27], this stiffness is obtained by unit of length, relating the moment at point A, located in the geometric center of the top flange, caused by forces, F , applied to the bottom flanges of the parallel beams of the "U-frame" mechanism, with the corresponding rotation, θ, of these flanges. This rotation is obtained by the ratio between the lateral displacement of the bottom flange (δ) and the distance between the geometric centers of the flanges of the steel profile ( h o ). The bending moment at point A is the product between force F and distance h o . Taking one of the beams, the rotation at point A will be equal to δ/ h o , and since the moment in A is given by the product F.h o , the following general expression for the rotational stiffness is obtained: / o s o F h k h   (1) To determine precisely Eqn. (1), it is necessary to carry out experimental or numerical analyses. Alternatively, the rotational stiffness of the composite beam ( k s ) can be obtained as a result of the series association of the cracked slab bending stiffness, k 1 , the steel profile web rotational stiffness, k 2 , and shear connection bending stiffness, k 3 , as follows: 1 1 2 3 1 1 1 s k k k k           (2) For the calculation of the slab bending stiffness, k 1 , the slab is considered as a beam fixed on the profiles. This stiffness is characterized by the bending moments that arise when applying unitary rotations on the fixed ends (Fig. 4) and, generally, can be obtained as: