**4. Discussion**

Analyzing the experimental results summarized in Table 6 and moment-rotation curves (M–θ) shown in Figures 9–11, we note the following important findings.


• For the assemblies containing B or C beams with a section height of 100 mm or 110 mm, respectively, and containing a five-tab connector, the design moment MRd and rotational sti ffness k m of the joints increase due to the upright sections having a thickness of 1.5 mm or 1.75 mm. For the assemblies containing the upright sections with a thickness of 2.00 mm, the rotational sti ffness km still increases, but design moment MRd of the connection decreases. In the assemblies with a 5-tab connector, the capable design moment MRd corresponding to the upright III having the thickness of 2.00 mm, is lower than the capable moment MRd corresponding to the upright I with a thickness of 1.75 mm upright regions in case of all types of beams.

In practice, the values of the design moment MRd and rotational sti ffness of the connection experimentally obtained and given in Table 6 are very useful in order to check if the beam-connectorupright assembly works safely from both strength and sti ffness points-of-view.

We considered a typical case for storage pallet racking systems for which two beams having the length of 2.7 m must support the weight of three wooden pallets (Figure 19a). The maximum total mass for one pallet loaded with goods is 500 kg per pallet, which means that the total weight is approximately 15,000 N for three pallets. This weight applied to one shelf is supported by two beams, which means approximately 7500N applied to the beam length of 2700 mm, which leads to a distributed force of 2.78 N/mm by assuming that the weight is uniformly distributed on the beam (Figure 19b). We considered that this beam is connected with upright regions by one type of semi-rigid connections tested for which the experimental results regarding the design moment MRd and rotational sti ffness of the connection are synthesized in Table 6.

**Figure 19.** Scheme of loading: (**a**) loading for two beams, and (**b**) equivalent scheme of loading for calculus (k m represents the rotational sti ffness of the connections).

We aim to comparatively analyze the safety coe fficient c for all beam-connector-upright assemblies tested to show in this way the reserve of each connection so that it works in the safety range. For this purpose, the moment developed at the end of the beam (at the level of beam end connection) is computed, which is denoted by Mend by using Equation (15) for the uniformly distributed force q of 2.78 N/mm applied to the beam with the length L of 2700 mm.

The safety coe fficient c corresponding to each beam-connector-upright assembly involved in this study is computed by using Equation (17). Lastly, the results are synthesized and analyzed comparatively in Table 7 for all assemblies involved in this research. In Table 7, for each class of assemblies corresponding to one type of beam, the smallest and the highest value of the safety coe fficient c is highlighted by using superscript symbols for those values (see the footnotes of Table 7).

In Table 7, we may remark that, for the assemblies containing types A or C beams, the best assemblies are based on the upright of type II and beam-end connector with four tabs (A-II-4L and C-II-4L in Table 7). Just in case of the beam of type B, the highest value of the safety coe fficient c is for the assembly containing the upright of type III, which has the greatest thickness of 2 mm, and the beam-end connector with four tabs (B-III-4L in Table 7).

For each class of assemblies corresponding to a certain type of beam, the smallest safety coe fficient c corresponds to the assembly for which the beam is connected with the upright of type III by using the beam-end connector with five tabs (A-III-5L, B-III-5L, and C-III-5L in Table 7).


**Table 7.** Results for the safety coe fficient c of the connection for the maximum deflection in case of the assemblies tested for the case of the real loading in the pallet racking storage system.

\* The greatest safety coe fficient of the connection for the group assemblies corresponding to one type of beam. \*\* The smallest safety coe fficient of the connection for the group assemblies corresponding to one type of beam. \*\*\* The allowable deflection vall is 13.5 mm for the beam length of 2700 mm, according to EN15512 [9].

It was very interesting to remark that the ratio between the highest safety coe fficient and the smallest safety coe fficient is equal to 1.68 (i.e., 4.41/ 2.63) for the assemblies containing the beam of type A, 1.69 (i.e., 4.57/ 2.70) for the assemblies containing the beam of type B, and 1.73 (i.e., 5.94/ 3.42) for the assemblies containing the beam of type C. The conclusion is that, for any type of beam involved in this research, there is a reserve of approximately 70% regarding the safety coe fficient, which depends on the type of the upright and end-beam connector used.

On the other hand, the beam used in racking storage systems must obey another requirement: the allowable deflection vall is equal to a maximum of 5% from the beam length, according to EN15512 [9]. This is called a sti ffness condition. This means that the allowable deflection vall must be 13.5 mm for the beam length of 2700 mm.

For the practical example considered, the maximum deflection vmax of the beam having two semi-rigid connections at both ends is computed by using Equation (18) for all assemblies involved in this research and the results are shown in Table 7. In order to check the sti ffness condition, the ratio between the allowable deflection vall of 13.5 mm and the maximum deflection vmax for the beam with the same semi-rigid connection at both ends is given for each type of beam-connector-upright assembly involved. It is remarked that all values computed for the vall/vmax ratio are greater than 1 and this means that the sti ffness condition is obeyed for all assemblies. As expected, the ratio vall/vmax increases as the thickness for the beam or for the upright increases. In Table 7, it is observed that the best assemblies from the sti ffness point-of-view (i.e., the greatest values of the ratio vall/vmax) do not correspond to the best values for the safety coe fficient c.

For better understanding the importance of the experimental determination of the connection's rotational sti ffness, a comparative study is presented for the same practical study case corresponding to the beam of type B for the following beam models.

(i) beam B shown in Figure 20a, for which the rotational sti ffness of the connections is equal to 0 that corresponds to hinged beam ends,


**Figure 20.** Bending moment diagram in case of different boundary conditions for beam B: (**a**) hinged beam ends, (**b**) semi-rigid connector with four tabs, (**c**) semi-rigid connector with five tabs, and (**d**) rigid connection (km represents the rotational stiffness of both end connections).

Furthermore, using RFEM software (Dlubal software Gmbh, Tiefenbach, Germany), it investigates the bending moment Mmid developed at the level of the cross section of the beam located at the midpoint of the beam, the bending moment Mend developed at the level of the connection, and the maximum deflection. Beam finite elements with two nodes were used in a numerical model of the beam loaded, as shown in Figure 19b. For finite element analysis (FEA) by using RFEM software, the beam corresponding to each case (Figure 20) was divided in 10 segments. Tridimensional support conditions were used in FEA at the beam ends in order to define the following boundary conditions for all cases: (i) beam hinged at both ends – all degrees of freedom are set to zero at supports, except the rotation with respect to the axis perpendicular to the loading plane (Figure 20a), (ii) beam with 4-tab connections at both ends – all degree of freedom are set to zero and the rotational stiffness km is set to 35.9 kN·<sup>m</sup>/rad at both end supports (Figure 20b), (iii) beam with 5-tab connections at both ends—all degrees of freedom are set to zero and the rotational stiffness km is set to 35.9 kN·<sup>m</sup>/rad at both end supports (Figure 20c), (iv) rigid connections (embedded) at both ends—all degrees of freedom are set to zero (Figure 20d). It is mentioned that all results are reported in case of the assemblies corresponding to the type B beam. In case of the beam models with semi-rigid connections at both ends (Figure 20b,c), only the cases corresponding to the extreme values of the rotational stiffness km are taken into account.

For better visual comparisons, the bending moment diagrams and deflection diagrams are graphically shown in Figure 21 and in Figure 22, respectively, for all cases shown in Figure 20. In the same manner, the distribution of the equivalent stress by Von Mises failure theory is comparatively plotted in Figure 23.

The results shown in Figures 21–23, show that it is important to determine the stiffness of such semi-rigid connections because the deflection, the bending moment, and stresses are very much influenced by the boundary conditions.

**Figure 21.** Comparison of the bending moment diagrams for different boundary conditions in case of the type B beam.

**Figure 22.** Comparison of the maximum deflection vmax for different boundary conditions in case of the type B beam.

**Figure 23.** Comparison of the distribution of the equivalent stresses (von Mises) for different boundary conditions in case of the type B beam.
