A pitch roof with a rectangular plan is chosen as example in order to investigate the aeroelastic behaviour variation due to corrosion. Fig. (4) illustrates the pitch roof geometry and dimensions. The plan size is 20x80 m. The minimum and maximum heights are 1 m and 4 m, respectively. The truss beam wheel base is equal to 5 m, while columns, that are not modelled in this phase, are 10 m high.

Fig. (4). Pitch roof geometry and dimensions (measures in meters).

Fig. (5a) shows the Finite Element model of the structure. Model contains 374 nodes, 1033 frame elements and 10 nodes were restrained by pinned. Structure was modelled by truss elements. Two different structures with a different value of the global stiffness are considered in order to make a comparison between roofs that are sensitive or not to wind actions. For case study n. 1, the main truss beam is made of circular pipe elements with a diameter of 150 mm and a thickness of 6 mm. For case study n. 2), the main truss beam is made of circular pipe elements with a diameter of 50 mm and thickness of 3 mm. In this study, the corrosion depth (d) is modelled as a variation of the pipe thickness. The yielding stress of steel members is equal to 275 MPa. The structures are dimensioned using permanent loads of 0.2 kN/m², snow actions of 1 kN/m² and wind actions estimated according to the CNR-DT 207 standard (2008) [64]. In this latter case, the pressure coefficients aging on the roof range from +0.8 to 0.2, while the reference wind speed is taken equal to 27 m/s.

Fig. (5). Modal analysis on the case study n. 1: (a) FEM model; (b) 1st vertical mode; (c) 1st transverse mode; (d) 1st torsion mode.

Fig. (6). Modal analyses roof with stiffness (2): (a) FEM model; (b) 1st vertical mode; (c) 1st transversal mode; (d) 1st torsional mode.

Similarly to truss roofs, light and high truss towers are very sensitive to corrosion because these structures do not have a protective screen. The examined case study is a truss tower with an equilateral triangular plan shape with sides of 4 m. The tower develops on six levels with a total height of 30 m. Fig. (7) shows the tower geometry and the FEM model used for analyses. Model contains 57 nodes, 126 frame elements and 3 nodes were restrained by pinned. Structure was modelled by beam elements.

The tower is made of columns with a variable S275 steel pipe section. From the ground to the top the cross-section varies from 700x8 mm to 300x8 mm. Perimeter beams are S275 steel pipe elements with a diameter equal to 200 mm and a thickness of 5 mm. The tower is loaded with a water tank of 300 kN at the top.

Fig. (8) shows the first ten mode shapes. Table 3 provides the first ten natural frequencies and their participating masses. The first two transverse modes have a frequency equal to 1.06 Hz with a participating mass equal to 59.5%. The first vertical mode has a frequency equal to 10.59 Hz and a participating mass equal to 48.7%. Finally, the first axial torsion mode has a frequency equal to 1.711 Hz and a participating mass equal to 57.5%.

Fig. (7). Truss tower: geometry and dimensions (measures in m) (a) and FEM model (b).

Fig. (8). Modal analysis on the inspected tower: vibration modes from 1st(a) to 10th(j).

In general, indication on the durability of metal structures affected by corrosion are contained in both the main European and Italian structural codes. In the European context [65-67], directly refers to [68], where only general principles regarding the protection of steel buildings from possible causes of damage are provided. On the contrary, in the new Italian codes an important innovation, which considers damage due to atmospheric corrosion as an action of entropic nature, has been introduced. Generally speaking, with reference to steel structures, both European and Italian codes only provide durability criteria and no reference is made to models able to estimate the corrosion depth. In particular, only information regarding the minimum thickness required for structural members and a classification of the corrosivity of atmospheres are indicated. With regard to the first issue, the adoption of increased thickness both for hollow sections and parts of the structure not directly accessible are usually required. Increased thickness is equal to 2 mm for structures with a design life up to 100 years. Instead, with reference to corrosive atmospheres, a qualitative classification was introduced by [69, 70], where the following four typologies of atmospheres are identified according to their degree of corrosiveness:

- Rural atmosphere: countryside and small cities, where corrosion proceeds rather slowly;

- Urban atmosphere: densely populated areas, with meaningful industrial activities;

- Industrial atmosphere: areas subject to pollutants released by industrial systems;

- marine atmosphere: areas directly in contact with sea water or areas strongly affected by the marine aerosol presence.

A quantitative measurement of corrosivity due to different atmosphere types is provided by the EN ISO 12944-2/2001 standard [71], which defines six corrosion categories (Table 1). The identification of these categories is based on experimental investigations carried out on specimens exposed in some areas, opportunely chosen and monitored (EN ISO 9226/1992) [67]. Controls on the environment in order to estimate the temperature of the air, the rate of humidity, the direction and the speed of the wind, the frequency of atmospheric precipitation and the incidence of solar and ultraviolet, including the analysis of specimens, radiations are also required in order to measure the influence of such factors on the corrosion phenomenon.

Generally, from a climatic examination it is possible to draw rough conclusions about the behaviour of corrosion. In fact, the corrosivity rate of an atmosphere tends to increase from rural environments to a city, a coastal area or an industrial zone. The only atmospheres that almost completely inhibit the electrochemical aggression of metals are polar climates, that have temperatures below zero, and desert climates, characterised by very low r.h. rates. Therefore, in cold or dry climates the process is slower than in moderate climate area, while it is greater in warm/humid and marine climate zones. The EN ISO 9223/1992 [68] provides a corrosivity classification system to assess the influence of various factors on the damage process. This classification considers both the level of corrosive impurities, characterised by the presence of sulphur dioxide and chloride particles, and the “Time Of Wetness” (TOW), estimated as the number of hours where the relative humidity exceeds 80% and the temperature exceeds 0°C. The atmospheric parameters determining the corrosivity classification do not include the effects of the potentially important corrosive pollutants and impurities, such as NO₂, hydrogen sulphide, chlorine gas, acid rain and other fumes that could be present in the general atmosphere or associated with specific environments.

As an alternative to the methodology proposed by the ISO standards, it is possible to classify atmosphere corrosivity using the PACER LIME algorithm. It measures the expected corrosion damage, related to various parameters opportunely evaluated. The most influential factor conditioning the corrosion phenomenon is the distance from the sea, because of hygroscopic salts. If such a distance turns out to be less than 4500 metres, the condition is much more severe, while for greater distances an estimate of the humidity influence is required. For r.h. greater than 60%, which corresponds to concentrations of 7.1∙g⋅m^-3, the atmosphere is severe or moderate depending on the percentage of different corrosive agents. Obviously, the situation improves, in spite of the same concentrations of pollutants, if the humidity value is inferior to the 80% critical level.

Damage modelling based on atmospheric corrosion is given by different literature references with dissimilar approaches. The heuristic approach is based on a regression of experimental data. Some examples are given by [72-78]. In particular, the 2^nd level models with a heuristic approach is obtai- ned from an observation and interpolation of experimental data, opportunely examined from a statistical point of view [15, 79]. All prediction models given by literature have an outline of the key assumptions (specific experimental dataset or explicit kinds of steel or different atmospheres) to compare the new proposed model.

Some of the first models were examined by [80-85] and they were based on experimental data regarding immersed corroded steel [86, 87]. modelled the degradation of paint coatings as well as the generation and the progress of the pitting points [88,89]. tested a ship hull girder subjected to corrosion and [90-93] worked on seawater ballast tank structures of ships [94, 95]. calibrated its model on environ- mental corrosion (i.e. non-immersed elements). Three models calibrated in atmosphere are given by [96, 97] and International Cooperative Programme (ICP). Differently, some examples of deterministic models are basically founded on corrosion mechanisms and are multi-scale models. Some of these models are listed in the following:

1. GILDS (Gas - Interface - Liquid - Deposition - Electrodic - Solid) [98-105].
2. MITReM (Multi-Ion Transport and Reaction Model) [106].
3. ANN (Artificial Neural Network) algorithm [107-109].

Particularly noteworthy are recent studies as for example [110, 111], that investigate In-situ monitoring of corrosion-induced expansion and mass loss of steel bar in steel and fiber reinforced concrete using a distributed fiber optic sensor. These methods may be extended to steel structures.

Based on the heuristic approach, the corrosion depth in this study was assumed variable according to the [112] curve given for mild carbon steel in marine atmosphere. The [112] curve is illustrated in Fig. (7) and shows that values were recorded for twenty years with the maximum value of depth (d) equal to about 1.4 mm.

As was previously discussed, the corrosion depth propagation in this study is modelled as a reduction of the pipe element thickness according to (Fig. 9). The modal analyses were repeated step by step by decreasing the pipe thickness in order to evaluate the natural frequencies and participating mass. As a consequence of corrosion depth, the structure mass decreases but the structural stiffness decreases faster. For this reason, the natural frequencies decrease as shown in Table 3.

Fig. (9). Pitch roof geometry and dimensions (measure in m).

The first vertical frequency of the structure with stiffness (2) is lower than (1) by 75%. According to the CNR-DT 207 standard (2008) [64], dynamic peak actions are equal to equivalent static actions if the first natural frequency is greater than 1.5 Hz. These values are greater than the first vertical, transversal and torsional natural frequencies for stiffness (2) for all values of d except the first torsional frequency for d which equals zero.

The wind-induced response was studied in terms of the critical vortex shedding speed given as Eq.1 [64]. The critical vortex shedding v_cr,i (Eq.1) was estimated using the first vertical, transversal and torsional natural frequency for wind action parallel to the smaller side of the roof plan (b is equal to 20 m).

where St is the Strouhal number which is assumed equal to 0.11 and was estimated according to the [64] for a thin rectangular shape (bigger side/smaller side equal to 5) section. n_1,y varies according to Tables 4 and 6. The mean value of the wind velocity is equal to 34.20 m/s (i.e.T_R0 = 50 yrs) and 41.26 m/s (i.e.T_R0 = 500 yrs).

As was expected, though the critical vortex shedding is greater than the wind speed, results showed that it varied from 131 to 124 m/s for vertical, from 145 to 111 m/s for transversal and from 333 to 85 m/s for torsional frequency. It is important to note that corrosion depth after 20 years reduces the critical vortex shedding value by 75%.

It is reasonable to think that structural thickness reduction also affects aerodynamic damping. According to the [64], aerodynamic damping is a function of the frequency, the pressure coefficient, the mean wind velocity and the roof geometry. When the structural frequency decreases the aerodynamic damping increases. For this reason, in this case reduced stiffness due to corrosion depth gives a positive effect under wind. It varies from 0.29% to 0.31% for vertical frequency; from 2.39% to 3.27% for transversal and from 1.19% to 4.24% for torsional frequency. Considering a structural damping equal to 2%, the estimated aerodynamic damping for the first vertical mode is about 1/6.

For the case regarding the truss tower, the instability caused by vortex shedding is more probable than for roofs. The reduced value between the maximum side size and the height increases the sensitivity to aeroelastic phenomena. In this case, the ratio is about 0.13. In order to have a measurement of the critical vortex shedding velocity, the Strouhal number is estimated at 0.13 (CNR-DT 207, 2008). The vortex shedding velocity for the horizontal modes decreases from 33 m/s to 31 m/s and the aerodynamic damping increases from 2.3 to 2.5%, considering d from 0 to 1.4 mm. This percentages are not neglecting because the structural damping is between 0.2 and 2%.

The wind-induced response will be studied also in terms of galloping and in terms of peak accelerations. The galloping critical speed is given as [64]:

where b is the crosswind dimension of the building (equal to 4 m), n_1,y is the factor of galloping instability, prudently estimated at 10, Sc is the Scruton number given as:

where ξ_i is the total damping ratio (that is the sum between structural damping and aerodynamic damping) and m_e,1 which is the generalized mass per unit length for the first mode m_e preliminary assumed as the mass for unit length at 2/3 of H [64].

For the truss tower study the Scruton number varied from 23.3 to 20.0 (-17%) whereas the critical velocity of galloping ranged between 19.9 m/s and 16.3 m/s. (-22%). The last results were significant because the galloping velocity was low with d equal to zero, however it decreased significantly for d equal to 1.4 mm.