Profiles with sections belonging to Class types 1, 2 or 3 were adopted [15-17]. Profiles with sections of Class 4 were used only for the longitudinal beams carrying on the pallets. In case of DA, other profiles replaced angular sections to guarantee the correct dissipative behaviour.

To compare the two design strategies, for the same structural elements, the same section profiles were adopted, modifying the thickness and the steel grade in relation to the design demand and the need to satisfy Eurocodes’ prescriptions.

Table 1. Definition of the design loads.

Dead Load – G1	According to the Steel Profiles Used.
Permanent/non-structural loads G2	Roof panels: 0,20 kN/m2
	Side panels: 0,25 kN/m2
Accidental loads Qk	Stored goods: each pallet is 1000 kg. The weight of each pallet is reduced by 15% basing on current practice.
	Live load on roof: 0.50 kN/m2
	Wind action	1.15 kN/m2 for the upwind side		0.69 kN/m2 for the downwind side
	Snow load: 1.2 kN/m2
Definition of the Seismic action for the design (§ 3.2.3 of NTC2008) Soil of category C.		DL limit state	LS limit state (EA)		LS limit state (DA)
	ag	0.06 g	0.163 g		0.163 g
	F0	2.507	2.449		2.449
	Tc*	0.269 s	0.284 g		0.284 g
	q	1.00	1.00		2.00

The uprights presented rectangular hollow sections (class 1), the beams adopted double channel sections (Class 1) and for the bracing system, angular (Class 3 – in case of EA) and rectangular hollow sections (Class 1 - in the case of DA) were selected. C-shaped cold formed profiles were used for the longitudinal beams carrying the pallets.

The design loads were defined in relation to the building site (Modena, Italy); Table 1 summarizes adopted actions.

Linear dynamic analyses with response spectra were performed. Behaviour factors (q) were respectively equal to 1.0 and 2.0 for EA and DA. Technical reasons justify q=2.0: for example, the need of achieving Damage Limitation (DL) actions lower with respect to Life Safety (LS) ones, adoption of medium/low ductility class to simplify connections, satisfaction of Eurocodes’ requirements etc. The pallets’ weight was evaluated referring to the full load condition neglecting the influence of the load distribution. The weight of each pallet was reduced by 15%: this last assumption, not exactly representative of the current condition of ARSW, maximized the seismic masses.

Plastic hinges were located in braces in tension, dissipative members in the structural scheme adopted. A zero-compression limit was applied to each brace to neglect its contribution when compressed.

An elastic perfectly plastic behaviour characterized plastic hinges. Fig. (6) shows the adopted force/displacement relationship; different colours and symbols are associated to the achievement of different Limit States (immediate occupancy – IO, life safety – LS and collapse prevention – CP). The model of the plastic hinge included also the first yielding (represented by a magenta square) and a point beyond CP introduced to solve convergence problems (represented by a yellow crossed circle). The degrading branch, needed by the software, was not relevant for the analysis. The acceptance criteria for each limit state are provided by EN1998-3:2005 [18], being Δ_t the elongation of the considered element, hereafter summarized.

Fig. (6). Force/displacement diagram adopted for plastic hinges in tense braces.

• Limit for Immediate Occupancy - IO (blue circle): 0,25·Δ_t.
• Limit for Life Safety - LS (light blue rhombus): 7,0·Δ_t.
• Limit for Collapse Prevention - CP (green triangle): 9,0·Δ_t.

Pushover analysis adopted two different load distributions: the 1^st group is characterized by incremental horizontal actions proportional to the shear forces at each level of the ARSW; the 2^nd group is characterized by incremental horizontal forces proportional to the masses.

Analyses were performed in displacement control; positive force is directed from left to right. Results are presented in terms of capacity curves: the control point was fixed in correspondence of the top storey of the structure.

For each dissipative element, the achievement of the different limit states of Fig. (6) was assessed. In the case of DA-ARSW, forces acting on non-dissipative members were checked to confirm the efficiency of the capacity design.

Fig. (7) shows the capacity curve obtained for EA-ARSW using the 1^st group distribution and the yielding pattern achieved at collapse limit state. As visible, considering the seismic design base shear (435 kN), the structure was still in the elastic field. The collapse condition was achieved in correspondence of several braces in the bottom part of the building for a displacement equal to about 37 cm and base shear equal to 710 kN.

Fig. (7). a) Capacity curve of ARSW designed for q=1.0 for 1st group distribution; b) yielding pattern and achievement of different limit states in correspondence of the last step of the analysis (about 37 cm of displacement).

Fig. (8) shows the capacity curve obtained for EA-ARSW using the 2^nd group distribution and the yielding pattern achieved at collapse limit state. In this case also, considering the seismic design base shear (435 kN), the structure was still in the elastic field. The collapse condition was achieved in correspondence of several braces in the bottom part of the building for a displacement equal to about 35 cm and base shear equal to 745 kN.

Fig. (8). a) Capacity curve of ARSW designed for q=1.0 for 2nd group distribution; b) yielding pattern and achievement of different limit states in correspondence of the last step of the analysis (about 35 cm of displacement).

Similar considerations are valid in the case of pushover analysis on DA-ARSW. Figs. (9 and 10) highlight, once again, that for a base shear equal to the design one (about 197 kN) the structure exhibited an elastic behaviour. The collapse condition – achieved for values of displacements and base shear respectively equal to 34 cm and 333 kN for 1^st group forces and to 35 cm and 363 kN for 2^nd group forces – corresponded to a wider distribution of plastic hinges, in correspondence of the central bottom part of the building and, in parallel, of the external uprights.

Fig. (9). a) Capacity curve of ARSW designed for q=2.0 for 1st group distribution; b) yielding pattern and achievement of different limit states in correspondence of the last step of the analysis (about 33 cm of displacement).

Fig. (10). a) Capacity curve of ARSW designed for q=2.0 for 1st group distribution; b) yielding pattern and achievement of different limit states in correspondence of the last step of the analysis (about 33 cm of displacement).

Comparing EA and DA-ARSW structures, relevant differences in the yielding pattern developed were observed (Fig. 7 ÷ Fig. 10). In EA-ARSW plastic hinges mainly developed in the central bottom part of the frame; in the case of DA-ARSW also lateral braces were involved in the developed mechanism.

Table 4 presents the values of base shear and displacement at yielding and collapse; in the table, ELS and CLS are acronyms used, respectively, to show the elastic and the collapse limit states.

Looking at Fig. (11), for the same displacement of about 120 mm (i.e. value indicating the end of the elastic behaviour of the DA-ARSW), the EA structure was in the elastic field, while the first plastic hinges developed in the lower bracing elements of the dissipative structure. It is otherwise to be noted that for the EA-ARSW, the capacity curve beyond the elastic limit is provided by a mechanical relevance only if the connections among elements are enough over-resistant respect to the bracings. This condition, currently, is not mandatory for non-dissipative structures [15-17].

Fig. (11). Comparison of the capacity curves obtained from the EA and DA-ARSW: a) 1st distribution of horizontal forces; b) 2nd distribution of horizontal forces.

Referring to the yielding patterns, the DA allowed a major diffusion of plastic hinges: the higher parts of the external lines of racks took part to the mechanism. Although a wider distribution of plastic hinges was reached, a global collapse mechanism was not achieved.

Further analyses were performed including the contribution of compressed braced for the DA-ARSW. An asymmetric tension/compression behaviour characterized each plastic hinge: for the tensile force/displacement relationship, the same limitations presented for only-tension braces were adopted. The compressive behaviour was calibrated basing on preliminary numerical simulation on single brace elements.

A simple fibre model was realized using OpenSees^® [19] following methods widely presented in the current scientific literature [20, 21, 22]. An initial imperfection, equal to 1/500 of the total length of the element, was applied to model bucking phenomena; a constitutive elastic-perfectly plastic law was adopted for the material. Pushover analysis were performed to simulate the behaviour of compressed members (Fig. 12); results in terms of displacement/base shear are presented for two different sections, according to what presented in Badalassi et al [12].

The equivalent quadri-linear force/displacement law was extracted basing on energy equivalence considerations: the continuous curve was obtained from nonlinear analyses on the fibre compressed element, the dashed one represents the approximation to use in the model.

Fig. (12). Determination of the equivalent quadri-linear relationships for: a) angular L35x35x4 profile, length 1640 mm steel grade S235; b) rectangular hollow section 30x30x2,5, length 1640 mm, steel grade S235.

The quadri-linear force/displacement law is defined as follows:

• The first branch (O-1) is a parallel to the almost linear part of the curve till the greatest force is reached.
• The second branch (1-2) is the horizontal segment till point 2 is reached (point 2 is the point of the original curve where the maximum force is reached).
• Point 4 is the point of the original curve corresponding to d₂=15·d_1, being d₁ the displacement at point 1.
• Point 3 is defined basing on the energy equivalence considerations between the two curves.

Fig. (13) shows the force/displacement relationship adopted for each plastic hinge. The different colours correspond to the different limit states. The acceptance criteria are provided by EN1998-3:2005 [18], being Δ_c the shortening of the considered element.

Fig. (13). Force/displacement law adopted for each plastic hinge.

• Limit for IO (blue circle): 0,25·Δ_c for profiles of class 1 and 2.
• Limit for LS (light blue rhombus): 4, 0·Δ_c for profiles of class 1 and 1,0·Δ_c for profiles of class 2.
• Limit for CP (green triangle): 6,0·Δ_c for profiles of class 1 and 1,00·Δ_c for profiles of class 2.

The lumped plasticity bi-dimensional model of DA-ARSW was developed with SAP2000®; the model included braces in both tension and compression. Pushover analyses were performed using the same forces distribution described in the previous paragraph. Fig. (14) shows the capacity curve for the 1^st distribution of forces. Points A, B, C and D refer to the relevant conditions achieved; Table 5 gives values of corresponding base shear and displacement. More in details:

Fig. (14). Pushover analysis on DA-ARSW including compressed members in the model: a) 1st distribution of forces; b) 2nd distribution of forces.

• Point A corresponds to a base shear of about 51 kN and a displacement of 1.4 cm. In this condition, several compressed members of the upper part of the external right shelve showed buckling phenomena.
• Point B corresponds to a base shear equal to 258 kN and a displacement of 8.1 cm. In this condition, CP limit state was reached in several compressed braces; buckling was observed in most of the braces, while tense members did not show any specific structural problem.
• Points C and D correspond to values of shear base of about 360 kN and displacement equal to 12 cm. Most of compressed braces reached CP limit state; strength resistance was achieved in several braces in tension.

Similar results were observed adopting the 2^nd forces group (Fig. 14b). In particular:

• Point A corresponds to a base shear force equal to 50 kN and displacement of 1.2 cm. In this condition, several compressed braces of the upper part of the external right shelve underwent buckling phenomena.
• Points B and C correspond to values of the base shear and displacement of about 380 kN and 10-11 cm. CP was achieved in most of the braces in compression, whilst only few tense braces reached their strength capacity.
• Point D corresponds to a displacement of about 15.5 cm. The collapse condition was reached for most of the tense braces in the bottom part of the ARSW structure.

Fig. (15) shows the distribution of achieved limit states (i.e. yielding pattern) at the end of the analysis (point D) for the DA-ARSW building and the two distributions of forces. As visible, no global dissipative mechanisms were achieved: ultimate limit states were reached only in correspondence of several elements, while most of them still exhibited – at the end of the analysis – an elastic behaviour. This situation, together with the reduced absolute displacement (lower than 15 cm), was related to the concentration of deformations in correspondence of the bottom part of the DA-ARSW. This is visible from Fig. (16a), in terms of both absolute displacement and inter-storey drift. High values of inter-storey drift (up to 2.3%) were achieved in correspondence of the 4^th-5^th storeys of the middle shelves, while in the other areas, lower and differently distributed values were observed.

Fig. (15). Yielding patterns associated to the last step (point D) of analysis on the DA-building adopting: a) 1st and b) 2nd group distribution of forces.

Several considerations shall be made concerning this situation. One is related to the analysis method: the monitoring of the roof displacement – as commonly used in pushover analyses – is not fully representative in case of ARSW, where the shelves are connected only in correspondence of the roof, being otherwise independent along the whole height. This is evident considering the completely different behaviour of middle and lateral shelves (Fig. 16b).

Besides, the development of dissipative mechanisms takes place mainly in the bottom part of the DA-ARSW, where the overstrength variation limit is satisfied (Fig. 5): this indicates that Eurocode 8 prescriptions are not able, for this particular structural typology, to provide the full exploitation of structural ductile performance as required.

Fig. (16). a) Absolute displacements and b) interstorey drifts in correspondence of the different storeys of the ARSW for the middle and the lateral shelves (2nd group of forces).