Moisés Arroyo Contreras

Bernard A Jacob
email: bernard.jacob@ifsttar.fr

Franziska Schmidt
email: franziska.schmidt@ifsttar.fr

Moises G Arroyo Contreras

Corresponding Author

Case of Normandy bridge

Keywords: Fatigue, Fatigue Lifetime, Orthotropic Deck, Traffic Loads, Abnormal Loads, Load Effects, Stresses

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Assessment in fatigue of an orthotropic steel deck bridge under traffic loads and impact of abnormal loads.

PROBLEM STATEMENT, METHODOLOGY AND CHALLENGE

There are only three bridges crossing the Seine river between the city of Rouen and Le Havre (located at the Seine's mouth), over 100 km. The suspended bridge of Tancarville, located 30 km upstream of Le Havre, with a total length of 1420 m and a main span length of 608 m, was opened in 1959. The suspension cables were replaced in 1996-99 because of the corrosion. The bridge of Brotonne, located 50 km upstream of Le Havre, is a cable-stayed bridge of 1278 m in length, with a main span of 320 m. It opened in 1977, but the truck traffic is now restricted because of heath problems. The bridge of Normandy is the latest built near to Le Havre, and opened in 1995. At that time, it was the longest span (856 m) cable-stayed bridge in the World, and its total length is 2141 m. Until now, none of these three bridges is open to abnormal loads, i.e. gross vehicle mass (GVM) exceeding the National (and European) legal limits, of 44 tons (40 tons in the European Union, EU). However, the port of Le Havre is the second busiest in France with 67 millions tons of freight and 2.6 million twenty-foot equivalent units (TEU) per year. Thus, a large number of heavy vehicles cross the Seine River to access to or to leave from the port, some of them carrying indivisible abnormal loads. Under the current regulation, these abnormal loads have to make a long detour of app. 200 km, through the city of Rouen, and to cross the city dowtown. Therefore, the prefet of Normandy asked to the Chamber of Commerce of the Seine Estuary (CCISE), concessionary of the bridge of Normandy, to investigate the feasibility and conditions of opening the bridge to abnormal loads (not divisible and exceeding the legal mass limits) up to 94-96 tons, 108 and 120 tons. The French Ministry of Transport, the conceding authority, supervises the study and the final decision of accepting or not accepting the abnormal loads. The concession contract ends in 2027, and by that time, the CCISE must return the bridge in a very good state to the French State.

The CCISE, acting as a responsible concessionary, first commissioned a consulting company (Quadric) to check the fatigue lifetime of the most sensitive and critical details, using the standardized fatigue load model (LM3) of the Eurocode EN1991-2 [START_REF]Eurocode 1: Actions on structures -Part 2: Traffic loads on bridges[END_REF]. The LM3 consists of two tandem axles. The center to center distance of the LM3 tandem axles is 7.2 m.. Also each of the four axle loads weigh 120 kN. This model was found much too conservative for the bridge of Normandy, designed before the Eurocodes, and leads to unrealistic calculated lifetimes of a few years. Therefore, advised by Michel Virlogeux, the designer of the bridge (and of many other large bridges around the World), the CCISE commissioned in 2017 a study to the French Institute for Science and Technology of Transport, Planning and Networks (IFSTTAR). The proposed methodology was first to collect load data uder the current traffic on the bridge, because only the traffic volume (truck flow by category) was known using the bridge tolling information. However, the truck and axle loads remained unknown. These data were compared to three other traffic load patterns measured on other French motorways. The next step consisted to re-assess the influence lines of the most sensitive details by FEM calculations (done by Quadric), and to compare them to those measured in 1995 during the initial study in fatigue of the bridge (2). Then a simplified fatigue assessment and lifetime calculation were carried out using a simplified load model made of a standard tridem axle of 27 tons, the maximum allowed mass on a tridem according to the European Directive 96/53/EC (3). A more realistic assessment was done using the current traffic loads measured on the bridge of Normandy and the lifetimes were compared with those calculated with the three other traffic patterns, and with the results of the intial study of 1995 [START_REF] Carracilli | Etudes en fatigue du pont de Normandie[END_REF]. Finally the superimposition of the expected abnormal loads and the current traffic loads led to new fatigue lifetime assessment to evaluate the additional damage and lifetime reduction due to the abnormal loads.

The main question was to check if two daily crossings of four defined abnormal loads would significantly reduce the expected fatigue lifetime of the bridge.

NORMANDY BRIDGE STRUCTURE AND TRAFFIC LOAD ASSESSMENT

Bridge of Normandy: a long span cable-stayed bridge with an orthotropic deck

This exceptional bridge, built from 1989 until 1995, remained the longest cable-stayed bridge in the World until 2003, even if a longer cable-stayed main span was built in 1998 in Japan. The bridge crosses the Seine estuary between Le Havre and Honfleur, and carries the four traffic lanes of the motorway A29, plus two cycling paths and two footpaths (Figure 1 The steel plate is 14 mm thick under most of the slow lanes and 12 mm thick under the fast lanes. The thickness of the stiffeners are respectively 7 and 6 mm. In the initial design, the thickness changed between the two traffic lanes, but a cycling path of 1.1 m in width was added afterwards, which shifted the left edge of the slow lane on the thiner part of the plate (Figure 8a).

Traffic load assessment using a bridge weigh-in-motion (B-WIM) system

Since the opening of the bridge of Normandy, the number of truck crossing the bridge every year raised from 200,000 to more than 1 million, for both directions. The tolling data give accurate statistics of the truck number per category (number of axles), but do not provide any information about their loading. Thus, the CCISE was highly interested to collect traffic loads using a WIM system. Howewer, it is not easy to find an appropriate location to install a road sensor WIM system.

On the steel deck, the pavement is too thin to install WIM sensors and the risk of water intrusion is too high to allow that. On the concrete deck, the longitudinal slope is above the limits proposed by the European Specification of WIM by COST323 (4). In addition, close to the toll area the trucks are accelerating or breaking, which does not comply with the WIM requirements. Therefore, it was decided to install a B-WIM system, which allows measuring axle and vehicle loads on an orthotropic bridge by instrumenting the bridge deck [START_REF] Dempsey | Orthotropic Bridge WIM for determining Axle and Gross Vehicle Weights[END_REF][START_REF] Schmidt | Experimentation of a Bridge WIM System in France and potential Applications for Bridge Monitoring and Overload Detection[END_REF][START_REF] Jacob | Bridge Weigh-in-Motion on Steel Orthotropic Decks: Millau Viaduct and Autreville Bridge[END_REF]. IFSTTAR and the Cerema (a French public technical organisation on risks, environement, mobility and planning)

have installed in June 2017 the SiWIM commercialised by Cestel, a Slovenian company [START_REF] Schmidt | Use of B-WIM for monitoring Traffic on a Specific Route in France[END_REF]. It has been installed in the south-north direction, in the 21 st segment nearby the midspan, where the slope is neglictible. Both lanes in this direction have been instrumented (Figure 3), by:  14 extensometers under all the longitudinal stiffeners (7 under the slow lane, and 7

under the fast lane), used for weighing (Figure 3b).

 2+2 extensometers, one per lane in the two adjacent sections of the segment, across the cross beams, used to detect the coming vehicles and to measure its speed and axle spacing (Figure 3b).

The SiWIM has been configurated and calibrated jointly by a calibration truck whose weights and dimensions were measured in static. The accuracy was assessed according to the COST323 WIM specifications (4), using 36 trucks (mainly 5-axle articulated T2S3) from the traffic flow, weighed in static on the toll area, over one day (August 2, 2017). The results (statistics of the relative errors, tolerances , confidence levels  and accuracy classes) are given in Table 1.

The test conditions are R4/E1 (extended reproducibility/environmental repeatability) according to [START_REF]European Draft Standard: Weigh-in-Motion of Road Vehicles[END_REF]. Four entities are in the accuracy class C(15), which is an average but acceptable accuracy.

However the single axles are only in class D(25) because of a high scattering (standard deviation) due to load transfer between axles. This rather low accuracy does not affect too much the fatigue calculations because the mean bias are low, while a 10% increase of the scattering may induce an overestimation of the fatigue damage by app. 20%, and thus the lifetime is underestimated in the safe side. The system monitored and recorded the traffic (trucks above 3. The traffic flow on Normandy bridge is quite low, comparable to the traffic on the A20 (Massay) which is a lower volume. However, it shows a rather large proportion of heavier trucks (the second mode of the GVW PDF contains 40% of the vehicles), but with a second mode 8% lower than on the other sites, but much more scattered (almost twice larger Std. Dev.).

Abnormal loads

Four abnormal vehicles were considered as representatives of the potential very heavy loads, which could cross the bridge of Normandy if allowed:

 Two cranes of 96 and 108 tons, with 8 and 9 axles, all loaded at 12 tons (Figure 5);

 Two conventional abnormal 8-axle vehicles C1 (94 tons) and C2 (120 tons) (Figure 6).

Bridge modeling: influence lines

The most sensitive details in fatigue in an orthotropic deck under traffic loads are the welds between the upper deck plate and the lateral web of the longitudinal stiffeners, with a risk of crack propagation either in the deck plate or in the stiffener web (Figure 7a). These details are in the fatigue class 71 for a full weld penetration or in class 50 for a partial penetration, according to the Eurocode 1993-1-9 [START_REF]EN 1993-1-9, Eurocode 3: Design of steel structures -Part 1-9: Fatigue, EN 1993-1•92005[END_REF]. For the study carried out in 1995 (2), 35 strain gauges have been installed at the critical locations of two bridge sections (Figure 7b) to monitor the stresses induced by the traffic loads. The influence lines (IL) were measured using a 2-axle test truck, and more precisely its second axle is equipped with twin wheels and weigh 130 kN. This truck crossed the segment at various lateral locations and very slow speed and the IL were derived by an inverse calculation from the measured strains. They are shown in Figure 10a for three of the four most sensitive details, corresponding to the stress in three stiffener webs (gauges 7, 10 and 12). The last detail is in the deck plate (gauge 8), but the IL is not shown here. The stress range (ordinate of IL's peak) depend on the position of the wheels and the thickness of the deck plate. The gauges 7, 8, 10 and 12 are located under the deck plate of 14 mm in thick, but the gauges 7, 8 and 10 are closer to the thiner plate, which explains the stress range 50 to 60% higher than for the gauge 12. The IL of the gauge 8 is rather close to the IL of the gauge 7.

The IL of the sensitive details in the stiffener webs have been calculated by Quadric using a 3D-FEM in 2016 [START_REF] Quadric | circulation de convois exceptionnels, vérification à la fatigue du platelage métallique orthotrope[END_REF]. The IL were provided at 12 locations (sections 1 to 12 in red, Figure 8a), for 6 lateral locations of the wheels (Figure 8b) as recommended by the EN1991-2 (Figure 9b), and the three types of wheel/axle configuration proposed by the Eurocode 1991-2: (A) single regular wheels, (B) twin wheels, and (C) single large tyre wheel (Figure 9). The calculated IL are given in Figure 10b for the most unfavourable transverse location (N°6) of a type B axle, and are compared to the measured IL (gauges 7, 10 and 12). According to the Figure 8a, the correspondance between the measured and calculated IL should be: gauge 7 = section 9, gauge 10 = section 8 and gauge 12 = section 7, and the wheel configuration is of type B. However, the geometry of the truck used in 1995 was slightly different from the modern truck one (Figure 9), with a slightly narrower wheel track. Moreover, the lateral location of the test truck in 1995 is not accurately known. Therefore, the measured and calculated influence lines do not exactly fit each to the other.

For the gauge 7 and section 9, the fit is excellent for the transverse location N°6 (Figure 10b). For the gauges 10 and section 8, the best fit is found for a transverse location between N°3 and 5, i.e. a right shift of 0.15 m. For the gauge 12 and section 7, the fit is found for the transverse location N°5, i.e. a right shift of 0.20 m. This comparison allows performing the fatigue assessment using either the measured or the calculated influence lines. In the following, the measured influence lines were mainly used in order to compare the results to those gathered in 1995. However, if more detailed information could be get on the real transverse location of the wheels by more advanced WIM systems, the calculated influence lines would be more adapted.

Fatigue assessment method

The fatigue assessment is made according to the EN1993-1-9 [START_REF]EN 1993-1-9, Eurocode 3: Design of steel structures -Part 1-9: Fatigue, EN 1993-1•92005[END_REF]. The stress variations under traffic loads are accounted by the rain-flow method, and then the Miner's law is used to calculate the whole lifetime of the detail, using the relevant fatigue class and the corresponding S-N (fatigue) curve. The minimum required lifetime for the large bridge of Normandy is 120 years. However, the Miner's model has quite a lot of uncertainties, and the order of magnitude of the calculated lefetimes are more relevant than the accurate values.

In the following section, the fatigue assessment is made for several loading cases:

 A simple load model, less conservative than the LM3 of the EN1991-2, which consists of a tridem axle (3 x 9 tons or 88.3 kN = 27 tons or 265 kN, i.e. the maximum permitted load by the European Directive 96/53EC); the axle spacing is 1.3 m. We assume that this tridem crosses the bridge (in one direction, on the slow lane) 600 times per day, or has the same aggressivity in fatigue than two trucks. This assumption is validated in the next section;

 The real loads measured by the SiWIM on the bridge of Normandy (from July 2017 until January 2018), and the other traffic loads measured on other sites (Table 2);  The abnormal loads (Figures 5 and6) superimposed to the real traffic of the Normandy bridge.

The results are presented for the influence lines measured in 1995 by the gauges 7, 10 and 12. The calculated influence lines are used to assess the effect of the lateral scaterring of ht loads and to open some perspectives. The fatigue classes 71 (the most relevant one according to the bridge design and construction) and 50 (quite pessimistic, just in case of poor welds) are used.

RESULTS: STRESSES AND FATIGUE LIFETIMES

Stress calculation, ranges and variations

The main parameters of the S-N curves for the classes 50 and 71 are reminded:  Class 71: fatigue limit (at 5 millions cycles) = 52.3 MPa, truncation limit (at 100 millions cycles) = 28.7 MPa;  Class 50: fatigue limit (at 5 millions cycles) = 36.8 MPa, truncation limit (at 100 millions cycles) = 20.2 MPa.

According to the Miner's model, if the maximum stress range does not exceed the fatigue limit, the lifetime is infinite (no fatigue damage). Otherwise, the lifetime is finite, and all the stress cycles above the truncation limit are taken into account. For the simple conventional load (the 265 kN tridem axle), as well as for any short series of axles or single truck, the rain-flow histogram is easy to assess manually. As shown in Figure 11 for a 5-axle articulated truck (T2S3), the first step consist to identify the longitudinal location of the load model (truck or group of axles) on the influence line, which provide a local extremum (maximum or minimum) of the stress path. In Figure 11, the arrows represent the axles with their rank, and each line or arrows corresponds to a local extremum of the stress path (the various axle loads are those measured by the SiWIM system, as explained above). After plotting these local extrema, it is easy to derive the rain-flow histogram, i.e. a few number of cycles and their amplitude.

First we assume that the influence lines are almost symmetrical, with a peak (maximum) value and two negative parts, as shown in Figures 10a and11 (but for the gauge 12). Then while the tridem is crossing the influence line, according to the axle spacing (1.3 m) and the abscissa of the influence lines, two minima are reached when the centre of the tridem is close to the minima of the influence line, and one maximum when the tridem is centred on the peak of the influence line. Assuming that the two minima are close (which is the case), the rain-flow only contains one cycle with a range of max-min. Table 3 gives the results for the tridem axle and the Equivalent Single Axle Load (ESAL)

providing the same unique stress cycle (rain-flow), and thus the same elementary fatigue damage. As explained in Figure 11, the stress cycles induced by the four abnormal trucks described in Figure 5 and 6 are simply computed by axle group, because the low spacing between axles and the spacing between groups of axles longer than the influence lines (positive part). Table 4 gives the four stress cycles induced while the cranes are crossing the influence line, and the three cycles induced by the C1 and C2 vehicles. Some of these cycles have an amplitude below the truncation limit, and therefore do not contribute to the fatigue damage. This is the case for the front axle of the vehicles C1 and C2. For these vehicles, the 5-axle group induces only one stress cycle, as the tridem because of the short spacing of the axles.

Fatigue lifetime assessment

Table 5 give the computed lifetimes for the four traffic paths (Normandy/A29, Fabregue/A9, Massay/A20 and Maulan/RN4), compared to the lifetimes resulting of the 600 daily crossings of the tridem, for both fatigue classes (50 and 71). The results are consistent. The traffics of A20 and RN4 gives lifetimes of the same order of magnitude than the traffic of A29 (bridge of Normandy), event if the RN4 traffic is a bit more aggressive. The traffic of A9 is much more aggressive, one of the most aggressive in France, but it will never occur on the bridge of Normandy. The lifetimes are also consistent with those calculated in 1995. The lifetimes for the IL10 even in class 71 are a bit short, and the calculation seems to be pessimistic. The effect of the abnormal vehicles is assessed by calculating the damage induced by two crossings per day (but Sundays and public holydays) of each of the four vehicles. This frequency was given by the CCISE as the basic assumption and leads to 300 crossing per year. The stress cycles of Table 4 allow the calculation of the elementary damage for each crossing of one abnormal vehicle, and then multiplying it by 300 gives the annual damage due to each abnormal vehicle i:

Di. The lifetime Li of the detail submitted only to the abnormal vehicle i crossings is:

Li = 1/Di (1)
The lifetimes for each abnormal vehicle and the three most sensitive details are given (in years) in the first lines of each part of Table 6 (by detail), for the fatigue classes 50 and 71. The lines below give the percentage ri of lifetime reduction due to the abnormal vehicle i, derived from the lifetime L of the detail under the current traffic of the A29 (Table 5):

L'i = 1 / (1/L + 1/Li) (2)
is the lifetime under the superimposition of the current traffic and the abnormal vehicle i, calculated as the harmonic mean of L and Li, and then:

ri = (L-L'i)/L (3)
The two last columns on the right of Table 6 gives the lifetime La of the detail under the superimposition of the four abnormal vehicles, i.e. the harmonic mean of the Li, 1≤ i ≤ 4, and below the percentage of lifetime reduction due to the four abnormal vehicles, computed with the formula (3), replacing L'i by La. The two cranes induce very close individual damage in fatigue, which is not surprising because they have the same axle loads and induce very similar stress cycles. The vehicle C1 is less aggressive than the cranes but the C2 is much more because of the 17.5 tons axles. Finally, the lifetimes of the details could be reduced by 3 to 11% if the abnormal loads are allowed and as frequent as assumed. However, the damages assessed for the four abnormal vehicles are rather overestimated, because of the very wide tyres of 0.37 m in width instead of 0.22 or 0.27 m as assumed by the EN1991-2, and for the influence lines assessment. However, the overestimation is even higher for the vehicles C1 and C2, because of the 8 wheels under five of their eight axles.

The sharing of the axle loads on more wheel and surface, all along the axle width, and thus on more longitudinal stiffeners, highly reduces the stresses in the details.

In addition, the fatigue assessment made at this stage neglects the lateral random distribution of the wheel path. If applying the statistical distribution proposed by the EN1991-2 (Figure 9b) for the whole traffic loads, the lifetimes could be significantly increased. Some preliminary calculations indicates that these lifetimes could increase by 50 to 100%. CONCLUSIONS IFSTTAR provided its expertise to install and operate a B-WIM system (the SiWIM) on the orthotropic deck of the bridge of Normandie, to measure the traffic loads over almost 11 months, and used 7 months of data to assess the lifetimes of the most sensitive details in fatigue. The influence lines measured in 1995 were double-checked by comparison with 3D-FEM calculations done by a consulting company Quadric, which was involved in the bridge design.

The effects of single vehicles or even of a tridem axle were simply calculated, which allowed assessing the potential contribution of the proposed abnormal vehicles to the fatigue damage, and thus the risk of lifetime shortening. It was shown that the lifetimes would not be shortened by more than 5 to 10%, even using rather conservative assumptions and neglecting the posiive effects of very wide tyres and 8-wheel axles on the abnormal vehicles. The methodology used may be applied to all orthotropic deck bridges, and even some other types of steel bridges.

The study could benefit of further analysis, using more detailed computed influence lines or

1 FIGURE 2

 12 FIGURE 2 Bridge of Normandy: (a) cross section of the steel orthotropic deck, (b) inside of the box.

FIGURE 3

 3 FIGURE 3 (a) Extensometers of the SiWIM fixed under each longitudinal stiffener, (b) mapping of the instrumentation.

FIGURE 4

 4 FIGURE 4 Traffic on Normandy bridge (July 2017-January 2018): (a) Truck distribution by category, (b) Gross vehicle weight PDF.

6

 6 FIGURE 5 Abnormal loads: Cranes, (a) 96 tons, (b) and (c) 108 tons

7

 7 FIGURE 7 Normandy bridge deck: (a) weld between the deck plate and a longitudinal stiffener web, and expected cracks under fatigue (named 1, 2 and 3), (b) Intrumentation of the deck with strain gauges (1995).

  FIGURE 8 (a) Sections (locations) of the calculated influence lines (1 to 12 in red), (b) Transverse locations of the (middle of) axle in the slow lane for the calculation of the influence lines.

  FIGURE 9 (a) Types of wheel/axle for load application, (b) statistical transversal distribution of the wheels in the traffic flow, both accorging to the EN1991-2.

FIGURE 11

 11 FIGURE 11 Assessment of the rain-flow histogram for a simple loading case: 5-axle articulated truck (axle loads 7, 12, 7-7-7 tons).

For a whole traffic

  record, a computer program is performing the same procedure for all the series of vehicles crossing the bridge. The local extrema are identified and stored, and then the rain-flow histogram is computed, for each influence line and traffic path. IFSTTAR uses the CASTOR/POLLUX software, which calculated the convolution of the influence lines and the measured axle loads. Therefore the input are the WIM data recorded on site, and the influence lines. The output are the rain-flow histograms and the lifetimes, depending on the fatigue class.

TABLE 1 Accuracy of the SiWIM, August 2, 2017, in-service verification, R4/E1 (4)

 1

	Entity	Number	Mean (%)	Std Dev. (%)	o (%)	Class	 (%)	 min (%)	 c (%)	 (%)	Accepted class
	Gross weight	36	-0.67	5.57	90.3	C(15)	15	11.5 11.5 97.5
	Groups of axles	41	-2.63	8.36	90.8	C(15)	18	17.7 14.7 95.2	D(25)
	Single axles	68	-0.99	14.36	92.1	D(25)	30	29.2 24.2 93.0
	Axles in a group	148	-2.36	9.41	93.4	B(10)	20	19.5	9.7 94.1	25

TABLE 2 Main characteristics of four French traffics (trucks) on highways and motorways

 2

	Site	Period Days	Trucks Nb	ADTF	Prop. 2 nd mode	Median 1 st mode (kN)	Std. Dev. 1 st mode (kN)	Median 2 nd mode (kN)	Std. Dev. 2 nd mode (kN)
	Normandy bridge	17/07-18/01	183 224,435 1226	40 %	169	54	356	53
	(A29)	18/01	29	35,710	1231	41 %	164	49	347	52
	Fabrègues (A9)	15/01-06	189 901,231 4768	41%	234	85	371	27
	Massay (A20)	2015	362 498,269 1376	20 %	241	79	384	27
	Maulan (RN4)	2015	353 755,757 2141	29%	216	69	382	33
	ADTF: Average Daily Truck Flow

TABLE 3 Stress variations, rain-flow and ESAL for a tridem axle (3 x 88.33 kN = 265 kN)

 3

		Max IL	Min IL	max	min	Cycle	ESAL
		(MPa/kN)	(MPa/kN)	(MPa)	(MPa)	(MPa)	(kN)
	IL 7	0.2720	-0.020	29.8	-1.7	31.5	108
	IL 10	0.2942	-0.031	38.1	-2.7	40.8	126
	IL 12	0.1823	-0.016	17.2	-1.4	18.6	94
	IL 8	0.2702	-0.0199	27.3	-1.8	29.1	100

TABLE 4 Stress variations and rain-flow cycles for the four abnormal loads

 4

			Cycles (MPa)	Cycles (MPa)
			Crane 96 t	Crane 108 t		C1	C2
		Tridem	36.5	34.9	single axle	16.0	18.7
	IL 7	Tandem	2 x 34.8	3 x 34.8	tandem	36.6	38.9
		single axle	32.0	-	5-axle group 39.1	53.5
		Tridem	45.3	42.4	single axle	17.3	20.2
	IL 10	Tandem	2 x 40.2	3 x 40.2	tandem	42.7	45.4
		single axle	34.6	-	5-axle group 48.2	66.0
		Tridem	22.9	21.9	single axle	10.7	12.5
	IL 12	Tandem	2 x 22.5	3 x 22.5	tandem	23.1	24.5
		single axle	21.5	-	5-axle group 24.5	33.6
		Tridem	33.0	32.3	single axle	15.9	18.6
	IL 8	Tandem	2 x 32.8	3 x 32.8	tandem	34.7	36.9
		single axle	31.8	-	5-axle group 35.7	48.7

TABLE 5 Lifetimes (in years) of the four main details under the tridem and the four traffics

 5

				Class 50					Class 71
		Tridem A29	A20	RN4	A9 Tridem	A29	A20	RN4	A9
	IL 7	61	32	29	24	7	352	254	296	274	75
	IL 10	20	11	7	6	2	96	57	40	33	10
	IL 12	∞	691	237	131	43	∞	61279	35145	13445 4426
	IL 8	92	62	46	45	10	528	586	588	573	113

TABLE 6 Contribution of the abnormal loads to the fatigue

 6

	Veh.	Crane 96 t	Crane 108 t	C1 (94 t)	C2 (120 t)	Total
	Class	50	71	50	71	50	71	50	71	50	71
	LI 7	2823 1.12%	16296 1.53% 1.15% 1.57% 0.76% 1.10% 2751 15883 4158 22923	2088 1.51%	6873 3.56% 4.40% 7.37% 696 3190
	LI 10	1606 0.68%	7248 0.78% 0.71% 0.78% 0.47% 0.63% 1540 7247 2323 8923	1161 0.94%	3567 1.57% 2.74% 3.67% 390 1496
	LI 8	3821 1.60%	22063 2.59% 1.61% 2.61% 1.06% 1.73% 3793 21900 5766 33293	2689 2.25%	10467 5.3%	934 6.22% 11.26% 4618

surfaces, and taling into account the geometry of abnormal vehicles and the lateral scattering of the wheel paths.

Finally, this study could be refined by using the real wheel imprints for each type of tyre mounted, and above all for the cranes and abnormal loads. Because these abnormal vehicles are equipped with wider tyres than the test truck used in 1995 to assess the influence lines, the stress amplitudes would be reduced for the same loads. And the fatigue lifetime would be extended. The results provided here are thus a bit conservative.

Author Contribution Statement

The


Corinne Dupuy

Abdelkader Benabou

Sylvain Shihab

O Messal

Stéphane Clenet

Patrice Peyre
email: patrice.peyre@ensam.eu

S Clénet

Influence of laser powder bed fusion process conditions and resulting microstructures on the electromagnetic properties of a 16MnCr5 steel

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Laser powder bed fusion (L-PBF) of metal parts is increasingly used in several industrial fields like medical, aeronautical or tooling. It allows building complex and precise metallic components, which are not directly achievable by conventional methods and can provide similar or better mechanical properties than casting or forging. Current limitations of the L-PBF process come from:

(1) the supporting structures, considered to be mandatory for the overhanging surfaces above a given threshold angle (usually near 45 • ) and hard to remove [START_REF] Bartsch | Benchmark parts for the evaluation of optimized support structures in Laser Powder Bed Fusion of metals[END_REF], (2) the rather deleterious surface finish (R a > 5 µm), compared to as- machined surfaces due to powder agglomerates and staircase effect [START_REF] Strano | Surface roughness analysis, modelling and prediction in selective laser melting[END_REF].

(3) the occurrence of residual porosities -usually less than 0.5% for optimized process conditions -that can impact fatigue properties [START_REF] Andreau | Texture control of 316L parts by modulation of the melt pool morphology in selective laser melting[END_REF].

A number of previous studies have shown that parts processed by L-PBF have anisotropic characteristics due to grain orientations and growth morphology as shown by Andreau et al. [START_REF] Andreau | Texture control of 316L parts by modulation of the melt pool morphology in selective laser melting[END_REF]. Due to the mechanisms of grain re-melting and epitaxial growth during the laser-induced melting-solidification procedures, the grain texture of cubic crystals are often oriented according to the crystallographic axis 〈001〉 or 〈011〉. The microstructural anisotropy is strongly affected by both the laser energy input density and the laser scan strategy as indicated by [START_REF] Zhang | Magnetic properties of in-situ synthesized FeNi3 by selective laser melting Fe-30%Ni powders[END_REF]. Also, the high cooling (10 5 -10 7 K/s) and solidification rates (0.1-1 m/s) generate a fine cellular microstructure (cell size of the order of 1 µm) and strong chemical intercellular segregation. Most of L-PBF materials exhibit, in the optimized state, 0.01-0.5% of porosities (lacks-of-fusion or blowholes) with sizes ranging between 0.5 µm and 30 µm. Yan et al. have shown that L-PBF materials also contain a high dislocation density (around 10 15 m -2 ) originating from the rapid contraction of liquid metal and localized near the intercellular spaces [START_REF] Yan | Grain structure control of additively manufactured metallic materials[END_REF], which often makes the material recrystallizable during post-L-PBF annealing as shown by Kreitcberg et al. on a nickel-based superalloy [START_REF] Kreitcberg | Effect of heat treatment and hot isostatic pressing on the microstructure and mechanical properties of Inconel 625 alloy processed by laser powder bed fusion[END_REF].

Most of the published researches on metal alloys used in AM, deal with their mechanical properties, as indicated in the 2014 ′ s review by Frazier [START_REF] Frazier | Metal additive manufacturing: a review[END_REF], but only few bibliographic data are available about their electromagnetic properties. Metallurgical parameters are expected to strongly impact the electromagnetic properties of L-PBF manufactured pieces by impeding the magnetic domain-wall movement and changing their structuration. Intrinsic parameters such as the chemical composition can modify the different magnetic energy terms or the electrical conductivity, while extrinsic parameters such as the crystallographic texture, grain sizes, defects or residual stresses can modify the magnetic domain structure or cause domain wall pinning and act as a source of coercivity. The result can have dramatic impact on the material objective of optimizing hardness distribution after case hardening. The resulting observed effect was an increase of both hardness level and depth for L-PBF + case hardening samples compared with classical manufacturing processes. This was supposed to be due to the fine grains in the L-PBF material. Therefore, our work was focused on the effects of processing conditions on the microstructure and resulting electrical and magnetic properties of 16MnCr5.

Experimental details

Material

The 16MnCr5 steel (composition given in Table 1) is a cementation steel used for parts requiring a combination of hardness and wear resistance such as gears, shafts and steering parts in mechanical engineering. Due to its ferromagnetic properties, it is also envisaged for combined magneticmechanical applications. For instance, in [START_REF] Zhao | Investigation on electromagnetic models of high-speed solenoid valve for common rail injector[END_REF], this material is employed in the study of an electromagnetic high-speed solenoid valve for common rail injector.

The material used for L-PBF experiments was a gas-atomized 16MnCr5 steel powder supplied by TLS Technik. To obtain the desired range of powder diameters, the powder was sieved to limit the maximum diameter to 63 µm. The resulting particle size distribution was d 10 = 15 µm, d 50 = 30 µm, d 90 = 48 µm, and an average size of approximately 35 µm. The particles had a spherical morphology, with a few agglomerated satellites (see Fig. 1).

L-PBF experiments

A SLM125HL machine, from SLM solutions GmbH was used to print the samples. The machine is equipped with a f420 F-Theta lens and a scanning head which ensures the movement of the beam on the surface of the powder. The used laser is a diode pumped fiber laser of IPG Photonics, with a maximum output power of 400 W, and 1070 nm wavelength. Prior to L-PBF processing, the build chamber is inerted under Argon to limit the O 2 rate to less than 500 ppm.

Two sets of parameters were used as shown in Table 2: one corresponding to the parameters recommended by the manufacturer, and the other one at a lower output volume energy density (VED) to evaluate the impact of defects such as pores on the magnetic and electric properties. As a reminder, the VED is calculated using (Eq. ( 1)). The velocity, the manufacturing strategy (number of contours, lasing strategy and scheme, focusing position) and the layer thickness remained the same. The layer thickness (= slicing distance) was 30 µm.

VED = P 0 (V 0 .hd.∆h) (1)
with VED (J/mm 3 ) = volume energy density, P 0 (W) = laser power, V 0 (mm/s) = scan speed, hd = hatch distance (mm), Δh = layer height (mm). Two dimensions of toroid, both of 4 mm height, were built to analyze magnetic properties:

(1) 38 mm inner diameter/44 mm outer diameter (small samples "S"). (2) 50 mm inner diameter/60 mm outer diameter (large samples "L").

Some samples were built parallel to the (x,y) building plane (flat performance working for example under a 50 Hz cyclic magnetic field of a sinusoidal waveform, with an increase in their magnetic losses and a decrease in their magnetic relative permeability, as indicated by Everhart et al. [START_REF] Everhart | Grain size effects in selective laser melted Fe-Co-2V[END_REF]. Some authors investigated the impact of L-PBF processes on magnetic properties. Hilzinger and Rodewald [START_REF] Hilzinger | Magnetic materials: Fundamentals, Products, Properties, Applications[END_REF] compared the composition, phase and magnetic properties of a wrought Fe-Co-2V alloy to the AM version with and without heat treatment. They both exhibit a significant improvement of magnetic properties with an ordering heat treatment (2 × 900 • C, 2 h). However, L-PBF samples had smaller grain sizes that resulted in higher yield strength and magnetic coercivity, i.e., higher iron losses. Different post-L-PBF heat treatments were also conducted by Riipinen et al. [START_REF] Riipinen | Properties of soft magnetic Fe-CoV alloy produced by laser powder bed fusion[END_REF]. One of them (pre-annealing 700 • C, 2 h then 820 • C, 10 h) improved the magnetic properties of iron-cobalt soft magnetic alloys (Fe-49Co-2V). Yakout et al. [START_REF] Yakout | Selective laser melting of soft magnetic alloys for automotive applications[END_REF] compared the density, microstructure, magnetic and electrical properties of L-PBF built motor stators produced from soft magnetic alloys (430 L, Fe5-O-Ni, Fe6-Si) with that of an original motor with laminations and isolators. Only Fe5-O-Ni and 430 L samples were tested, as cracks appeared on the Fe6-Si sample. The electrical resistance of the L-PBF motors were in line with that of the original laminated motor but their inductance was lower due to poorer magnetic properties. Recently, Li et al. [START_REF] Li | Additively manufactured Ni-15Fe-5Mo Permalloy via selective laser melting and subsequent annealing for magneticshielding structures: process, microstructural and soft-magnetic characteristics[END_REF] studied the effect of the microstructure of annealed L-PBF samples (Ni-15Fe-5Mo) on their mechanical and magnetic properties. A magnetic anisotropy was observed, the better results being obtained when the magnetic induction was parallel to the building direction. The obtained saturation magnetization value was equivalent to conventionally processed samples but the coercivity was higher.

Garibaldi et al. [START_REF] Garibaldi | Effect of annealing on the microstructure and magnetic properties of soft magnetic Fe-Si produced via laser additive manufacturing[END_REF][START_REF] Garibaldi | Relationship between laser energy input, microstructures and magnetic properties of selective laser melted Fe-6.9%wt Si soft magnets[END_REF] focused their work on the FeSi 6.9%, a material with practical importance in electrical engineering due to its lower electrical conductivity (i.e., high electrical resistivity) compared to the more common FeSi 3%, as well as lower magnetostriction coefficient. Considering ring samples, the authors investigated the effect on magnetic properties of (1) post L-PBF annealing up to 1200 • C [13], (2) scan strategy and laser energy input [START_REF] Garibaldi | Relationship between laser energy input, microstructures and magnetic properties of selective laser melted Fe-6.9%wt Si soft magnets[END_REF], and correlated the variation of magnetic properties with the grain size and presence of B2/DO3 phase and texture orientation. For the same material, Goll et al. [START_REF] Goll | Additive manufacturing of soft magnetic materials and components[END_REF] evaluated the impact of laser input energy and a 1 h -1150 • C heat treatment on magnetic losses at 50 Hz and correlated the observed variations with the porosity rate and the occurrence of cracks in the material.

Otherwise, due to the high thickness of L-PBF parts, eddy-current losses can constitute a significant part of the magnetic losses at a given working frequency. To reduce these losses, Goll [START_REF] Goll | Additive manufacturing of soft magnetic materials and components[END_REF] and Plotkowski et al. [START_REF] Plotkowski | Influence of scan pattern and geometry on the microstructure and soft-magnetic performance of additively manufactured Fe-Si[END_REF] studied the impact of more complex topological core structures on electromagnetic losses. Plotkowski et al. showed that the electromagnetic losses can be decreased by a factor 20 depending on the filling strategy. Among the tested configurations, the thin wall samples based on Hilbert filling pattern demonstrated the best performances.

Finally, building magnetic materials by L-PBF presents additional difficulties compared to mechanical parts and requires specific optimization criteria, due to clear dependence of magnetic properties on the induced microstructure. However, there is a growing interest of the industry for the L-PBF manufacturing of magnetic parts, especially when the soft magnetic piece cannot be produced by cold rolling or when the application requires non-laminated magnetic core.

In this study , the feasibility of manufacturing 16MnCr5 (1.7131) ferromagnetic components by L-PBF is investigated. 16MnCr5 is a case hardened steel mostly used in gear applications, and containing approximately 0.15% C. On the same material, Schmitt et al. [START_REF] Schmitt | Examination of the processability of 16MnCr5 by means of laser powder bed fusion[END_REF][START_REF] Schmitt | Laser-based powder bed fusion of 16MnCr5 and resulting material properties[END_REF] have recently indicated a rather good L-PBF processability with more than 99.5% density of built parts [START_REF] Schmitt | Examination of the processability of 16MnCr5 by means of laser powder bed fusion[END_REF] and up to 99.9% in their more recent work. The same authors also investigated the influence of post-L-PBF heat treatments on L-PBF gears [START_REF] Schmitt | Laser-based powder bed fusion of 16MnCr5 and resulting material properties[END_REF]. Prior to case hardening, they recommend a recrystallization annealing after L-PBF to provide a large stress relief, a better mechanical isotropy and to favour machining.

Bartels [START_REF] Bartels | Investigation on the case-hardening behavior of additively manufactured 16MnCr5[END_REF] also manufactured 16MnCr5 steel with L-PBF with the samples "F") and some tilted at 25 • with respect to the (x,y) plane (tilted samples "T") to check the effect of the grain growth direction on the material properties. Bars (5 × 3 × 30 mm 3 ) for electrical conductivity measurements were also built in two directions, parallel to the (x,y) plane (samples H) and vertically, along the z axis (samples V), as depicted in Fig. 2.

Density measurements

The Archimedes' method was used on the smallest samples (bars) to evaluate the density of as-built parts. The measurements were performed using a Mettler Toledo MS104TS hydrostatic weighing system equipped with a MS-DNY-43 solid sample weighing module.

This technique, based on Archimedes' principle enables to determine the volume of the samples, and their density. Samples were weighed in air atmosphere and in demineralized water at room temperature (RT). This procedure was carried out five times, with at least 30 min air drying between consecutive tests, in order to provide similar wettability conditions. The density of the samples was calculated using (Eq. ( 2)) from the mass obtained in each of the conditions and the density of the fluids at RT. The comparison of the results obtained with the theoretical density enables to calculate the porosity rate. The reference density ρ dense of the dense 16MnCr5 steel was set to 7.9 g/cm 3 as reported on the material datasheet. Demineralized water density ρ fluid was set (at 21. ).

In order to analyze the porosity rate of the larger toroid samples, optical micrographs (magnification ×100) were also performed, using a ZEISS Axio Imager optical microscope at × 100 and × 500 magnifications. The obtained × 500 images were then binarized using the same luminosity threshold and analyzed using ImageJ software. 20 images were taken on each cross section representing approximately 10% each of the total cross-section at × 500 magnification (2452 × 2056 pixels = 0.34 mm × 0.29 mm = 0.097 mm 2 for each image).

Magnetic measurements

In energy conversion devices, the relevant magnetic properties are the magnetic permeability represented by the non-linear B-H curve and the iron losses linked to the area of the magnetic hysteresis loop. In practice, the magnetic permeability represents the capability of the device to amplify the magnetic flux, which is directly linked to the volume energy density involved in the energy conversion process, whereas the iron losses represent the energy dissipation during this process. Therefore, performances in terms of magnetic properties are obtained when higher magnetic permeability (slope of the B-H curve) and lower iron losses are exhibited by the material.

To characterize the magnetic properties, the ring core technique

H(t) = N 1 I 1 (t) l ( 3
)
B(t) = 1 N 2 S ∫ V 2 (t)dt (4)
where l and S are respectively the average magnetic path length and the cross section of the considered ring core. Finally, the relevant physical characteristic is the B(H) magnetic hysteresis loop which represents the magnetic core properties. The measurements were performed at RT (~20 • C), for each ring core, at several magnitudes of magnetic flux density up to the magnetic saturation with controlled sinusoidal magnetic flux.

For the magnetic analyzes, we will consider first the normal B-H curves, obtained from the extrema B max (H max ) of the measured hysteresis loops (see Fig. 3-left) and second, the iron losses which are obtained from the areas of the hysteresis loops (energy dissipation during the magnetization process). From the normal curve, we can also deduce the relative magnetic permeability μ r (H) = B/(μ 0 H) (see Fig. 3-right) where μ 0 is the vacuum magnetic permeability. From the hysteresis loops, two specific points can be defined (Fig. 3-left): the coercive field H c when B = 0 and the remanent magnetic flux density when H = 0.

The classical approach for iron loss decomposition proposed by Bertotti [START_REF] Bertotti | General properties of power losses in soft ferromagnetic material[END_REF] consists in separating into three contributions the underlying physical mechanisms such that the total losses P tot can be written as:

P tot = P h + P cl + P exc [START_REF] Yan | Grain structure control of additively manufactured metallic materials[END_REF] where P h , P cl and P exc are, respectively, the hysteresis losses, classical losses (macroscopic eddy currents) and excess losses. For a sinusoidal magnetic flux density, these contributions may be expressed as:

P tot = k h fB α max + k cl f 2 B 2 max + k exc f 3/2 B 3/2 max ( 6
)
with k h , α, k cl and k exc the loss model coefficients that are usually identified from the experiment. Magnetic characterizations are performed under sinusoidal magnetic flux density.

Electrical conductivity measurements

Regarding the electrical conductivity, the four-point method is employed [START_REF] Smits | Measurement of sheet resistivities with the four-point probe[END_REF]. In the configuration with a sensor having four aligned needles, a d.c. current I is imposed between the outer pair of needles and a voltage V is measured between the inner needles separated by a distance l . From the measured resistance, the electrical conductivity σ can be deduced. In the present work, the test samples are the bars (5 × 3 × 30 mm 3 ) presented in Section 2.2. Electrical conductivity measurements are very sensitive to the physical environment, especially the temperature. Therefore, to guarantee stable conductivity measurements, i.e. less sensitive to temperature fluctuation, all measurements are performed in a temperature chamber controlled at 40 • C with an homogeneity of ± 1 • C.

Rigorously, the 4 point method requires applying a correction factor F that depends on the position of the electrical contact points and shape of the sample: (flux-metric method) is applied to the toroids presented in Section 2.2. The measurements have been realized on the MPG200D equipment from Brockhaus Measurements GmbH in accordance with the IEC 60404-4 standard [START_REF]Magnetic materials -Part 4: Methods of measurement of d.c. magnetic properties of magnetically soft materials[END_REF]. The equipment exhibits a repeatability of the measurements less than 0.1%.

To magnetize the magnetic core and measure the magnetic flux density, a primary coil and a secondary coil are wound on the magnetic core with respectively N 1 and N 2 turns. From the measure of the primary electrical current I 1 (t) and secondary induced voltage V 2 (t), the excitation magnetic field H (A/m) and magnetic flux density B (T) are respectively calculated:

σ = Il VS F (7)
with S the section of the bar crossed by the electrical current. In our case, the correction factor F is determined from an electrokinetic numerical simulation [START_REF] Jamil | Development and validation of an electrical and magnetic characterization device for massive parallelepiped specimen[END_REF] confirming that, for the considered geometry and needle distance, F ≈ 1.

Results

Estimation of porosity rate

Porosity rate was estimated with the use of either Archimedes' method (double weighing in air and liquid) or micrographs + image analysis. Archimedes' method results indicate a rather clear tendency: the porosity rate of the bars (see Table 2) manufactured with a low VED (49.6 J/mm 3 ) and a larger hatch exhibit a much higher porosity (~3.5%) than that obtained with the high VED at 67.5 J/mm 3 (~1.4%). The reduced overlap between single L-PBF tracks with low VEDlarge hatch distance logically explains such a difference. More precisely, when VED decreases (low energy condition here below), melt-pool dimensions (depth and width) -dependent on P/V -and overlap -dependent on hatch distance -both decrease, the combination of which provokes lacks-of-fusion.

Micrographs were carried out on cross section of the bars and toroid rings to provide a comparison with density measurement and analyze microstructures. Micrographs were performed on two toroids which were cut into six parts each, for cross section analysis (Fig. 5). This method does not take into account the whole sample but allows to overcome errors from Archimedes' method related to the presence of roughness-dependent air bubbles stuck at the surface, open porosities, and above all to the occurrence of entrapped powder in close porosities which are mostly removed during metallographic preparation. The mean porosity rate measured was 0.15 + /-0.05% for the toroid built with the high VED and of 1.1 + /-0.2% for the low VEDlarge hatch distance condition.

At low VED (150 W -720 mm/s and a hatch of 140 µm), most of the porosities exhibit angular contours (Fig. 6) while small spherical porosities are mainly observed at high VED (175 W -720 mm/s and a hatch distance of 120 µm). Different physical phenomena explain the different porosity shapes. Spherical pores are formed by gas occlusion unlike pores with angular contours correspond to lacks-of-fusion and come from insufficient tracks overlap at low VED -large hatch distance. The distribution of porosity diameters is shown in Fig. 7. Most of them are below 50 µm, and small pores (< 10 µm) are predominant, especially for the high VED condition.

The reason why Archimedes' method (Fig. 4) provided higher porosity rates than micrographs + image analysis may be due to:

(1) a small variation of liquid density with RT which plays a nonnegligible role on data, (2) the small volume of the samples (0.45 cm 3) results in a large specific surface (surface/volume) which enhances the influence of possible wetting issues during sample immersion, especially on high roughness L-PBF surfaces.

(3) a relatively larger porosity rate is usually shown at the L-PBF samples surface, which could explain partially why porosity is higher with Archimedes' method (Fig. 9).

A slight difference between the H-bars and V-bars porosity at high VED (Fig. 4) could be explained by the presence of pores near the overlapping area between the contour and the inskin area (six times more contour contribution with the V-bars). However, this was not confirmed at low VED.

Microstructural analysis

A chemical etching (HCl, CH 3 CO 2 H, picric acid) was applied on the tilted (T) and flat (F) built toroidal samples to analyze the microstructure and reveal the single L-PBF tracks.

A columnar microstructure is shown, oriented along the build direction (O, z). Elongated grains, of 180 ± 70 µm length, 40 ± 10 µm width, in the manufacturing direction, can be observed. These columnar grains grow by epitaxy on the grains formed during the previous fusion step (Fig. 8).

Macrographs also confirm a clear inclination of 25 • of the solidification microstructure versus the build direction for the T toroidal samples. Lacks-of-fusion between adjacent tracks are also clearly confirmed at low VED (Fig. 8, right). The difference observed on the top layers between the T and F toroidal samples corresponds to the overlapping fusion area between the upskin and the inskin zone in the case of the T samples (Fig. 10).

At a higher magnification, micrographs indicate a Bainitic to Martensitic structure (Fig. 11), in accordance with CCT diagrams, but further material analysis would be necessary to make a distinction between these two structures. Moreover, no clear difference could be evidenced between high and low VED microstructures. A coarser microstructure was also observed near the L-PBF tracks boundary (Fig. 12).

Magnetic properties

The use of magnetic materials in electro-mechanical energy conversion devices [START_REF] Zhao | Investigation on electromagnetic models of high-speed solenoid valve for common rail injector[END_REF] relies on their high magnetic permeability and low iron losses during the energy conversion process. Therefore, it is required to characterize these properties when considering the L-PBF manufacturing process, especially because these properties are dependent on the mechanical and microstructural state of the material. In the following, and given a rather thick cross section of the toroid samples, the magnetic properties have been characterized at the low frequency 10 Hz for which we expect a limited influence of the dynamic effects due to eddy currents.

a. Repeatability analysis

Magnetic properties are sensitive to mechanical and microstructural properties. Therefore a first analysis is conducted to know if there may be variations in the manufacturing process which might or may not influence the magnetic properties. Thus, all toroid samples (see Table 2) have been duplicated with the manufacturing parameters described in Section 2.2.

To evaluate the repeatability error between experimental results of two duplicated toroids (labeled 1 and 2), the relative error is considered such as:

err rel = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ∫ Bmax 0 [X 1 (B) ] 2 dB √ √ √ √ ( 8
)
where X(B) is for either the B-H curve H(B) or the loss curve P(B).

Overall results obtained for the samples having the same geometries and manufacturing parameters (6 pairs of toroids) lie within a margin of error of 1.4% for the B-H curves and 1.8% for the loss curves. This demonstrates a very good repeatability of the manufacturing process regarding the magnetic properties. As illustration, some results are given in Fig. 13 in terms of normal B-H curves for two large LT toroid samples (L for large and T for the 25 • inclination angle) and also in terms of iron loss curves for two SF toroid samples (S for small and F for flat manufactured samples).

Given these observations, we will consider in the following only one sample per geometry and manufacturing parameters (low and high VED).

b. Effect of the manufacturing parameters

The main magnetic property to be analyzed is the B-H curve given in Fig. 14 for the small toroid samples (on the left) and for the large ones (on the right).

As a first observation, the manufacturing parameters show a significant impact on the magnetic permeability (which is the slope of the B-H curve). Indeed, we have seen in the previous section that the measurement repeatability is lower than 1.4%. The observed gaps between the B (H) curves in Fig. 14 are more significant and are clearly due to the magnetic material behavior which is different from one sample to another. If we compare the toroids manufactured horizontally (F samples), the high VED configuration (175 W 720 mm/s 120 µm) gives the best magnetic performances for both toroid geometries (S and L). This analysis is supported by the porosity characterization in Section 3.1 that showed clearly a higher porosity rate for the low VED configuration. From the magnetic point of view, the porosities act as a distributed airgap within the material, which degrades the global magnetic performances, especially the B-H curves. Besides, a lower magnetic flux density is observed near saturation for the samples having higher porosity rates due to a lower magnetic volume to be magnetized. In fact,

M = B μ 0 -H (9)
Moreover, when comparing for the same VED (high VED in Fig. 14) the F toroids manufactured flat with the T toroids manufactured with a 25 • inclination angle, the magnetic performances are better for the F toroids. This result is observed for both sample geometries.

When compared to conventional 16MnCr5, the results from L-PBF 16MnCr5 are close in terms of magnetization curve. For instance, in [START_REF] Zhao | Physical Cauer circuits in nonlinear eddy-current modeling[END_REF] the B-H curve shows that a magnetic field of about 6 kA/m is required to reach a magnetic flux density of 1.5 T. In Fig. 14, we observe, for the same magnetic flux density, a magnetic field in the range 4.3-6 kA/m for the S toroids and in the range 5.6 -6.6 kA/m for the L toroids. However, our results, especially those of the S toroids, show slightly better magnetic behavior (lower required magnetic field) than the conventional 16MnCr5 in [START_REF] Zhao | Physical Cauer circuits in nonlinear eddy-current modeling[END_REF]. Indeed, in this latter reference, the authors considered the magnetic properties from [START_REF] Zhao | Investigation on electromagnetic models of high-speed solenoid valve for common rail injector[END_REF] where the steel was employed in an electromagnetic valve and has been through a case-hardening process that is known to be detrimental for the magnetic properties [START_REF] Logan | The magnetic properties of a case-hardened steel and their effect on the performance of an electromagnetic actuator[END_REF].

The evolution of iron losses versus magnetic flux density is reported in Fig. 15. Similar to the B-H curves, we observe that the small toroid samples (on the left) and the large ones (on the right) show the same behavior with regard to the manufacturing parameters. For the F samples, the high VED configuration gives also the best performances with lower iron losses.

However, in the case of large toroids, the LF and LT samples obtained from high VED exhibit very close evolutions of their iron losses, whereas for the small toroids, the ST sample losses are in-between those of SF samples obtained with respectively high and low VED.

The lower performances of the low VED samples in terms of losses are also linked to the porosity rate that is higher in that case as observed in Section 3.1. In practice, for the same global magnetic flux density (associated to the cross section of the toroid), the low VED samples will exhibit a higher effective magnetization within the material volume (porosities are not magnetic) and iron losses will increase as predicted by (Eq. ( 6)). Also, the angular contours of the lack-of-fusion porosities, observed for the low VED samples, suggest high local stress concentrations that are expected to degrade the magnetic properties.

A global observation for the previously presented magnetic results is related to the difference of behavior between the S and L samples for the same manufacturing parameters. The S samples always show better magnetic performances in terms of B-H curves and iron losses. In fact, at the considered excitation frequency (10 Hz), the dynamic effects, especially classical eddy currents, can explain this difference. To further analyze and support this hypothesis, a loss decomposition approach has been carried out in the following section.

To synthetize the effect of manufacturing parameters on the magnetic properties, Fig. 16 reports the evolutions of the coercive field (left) and relative magnetic permeability (right) for the SF samples obtained from high and low VED. These data are extracted from the hysteresis loops as explained in Section 2.4. The coercive field evolution is obviously consistent with the evolution of the iron losses (Fig. 15): higher coercive field yields higher iron losses. As mentioned earlier for the low VED samples, the angular contours of the lack-of-fusion porosities suggest high local stress concentrations. Mechanical stress is known to degrade the magnetic properties, especially the coercive field that will increase [START_REF] Stupakov | Investigation of magnetic response to plastic deformation of low-carbon steel[END_REF]. Regarding the relative permeability evolution, this latter is also consistent with the B-H curves presented in Fig. 14 that show lower magnetic properties for low VED samples.

c. Analysis of the toroid geometry factor

The B-H and loss curve evolutions between SF and LF toroids are emphasized in Fig. 17 for the low VED configuration. As previously mentioned, the dynamic effects, due to classical eddy currents in different thicknesses of toroids (3 mm for S samples and 5 mm for L samples), are suspected to lead to these different behaviors.

Starting from (Eq. ( 6)), the hysteresis losses can be extracted from the evolution of the specific energy P tot /f versus the frequency:

P tot f = k h B α max + k cl fB 2 max + k exc f 1/2 B 3/2 max ( 10
)
To have a good estimation of the hysteresis losses, the extrapolation of the specific energy to "zero" frequency requires additional data at lower frequencies. In Fig. 18, the specific energy is reported versus the frequency for a peak magnetic flux density of 1 T, for both the SF and LF samples manufactured at low VED. The typical behavior described by (Eq. ( 10)) is observed for the whole range of frequency up to 260 Hz. A zoom at the low frequency region shows that the specific energy tends towards the same value for zero frequency. Therefore, it can be concluded that the quasi-static loss term is similar for both toroid geometries. This observation is made for all other S and L toroids having the same manufacturing parameters.

Finally, it can be assumed that the loss difference observed at higher frequencies for toroid rings built with the same L-PBF parameters, is mainly due to dynamic effects, especially the classical eddy current Fig. 16. Comparison of coercive field (left) and relative magnetic permeability (right) evolutions at f = 10 Hz between high and low VED for the small toroids build flat.

the intrinsic magnetic property of the material, the magnetization M defined by (Eq. ( 9)) and that reflects the number of magnetic moments per unit volume, will also decrease. losses.

The same explanation is behind the small difference in terms of B-H curve, shown in Fig. 17. In fact, eddy currents tend to attenuate the magnetic flux density B. Consequently, to obtain the same magnetic flux density in both toroids, the magnetic field H should be higher for the L sample than for the S sample where the eddy current effect is the lowest.

d. Heat treatment effect

The L-PBF process may introduce residual stresses [START_REF] Shiomi | Residual stress within metallic model made by selective laser melting process[END_REF]. Therefore, knowing that residual stresses are likely to degrade the magnetic properties, a tempering heat treatment has been considered for stress relief in the sample exhibiting the best magnetic performances, i.e. the SF sample built with high VED. In [START_REF] Schmitt | Laser-based powder bed fusion of 16MnCr5 and resulting material properties[END_REF][START_REF] Bartels | Investigation on the case-hardening behavior of additively manufactured 16MnCr5[END_REF] the stress relief heat treatment for 16MnCr5 has been performed at 650 • C for 6 h. However, these studies also reported recrystallization and grain refinement after the heat treatment. In the present work, to avoid potential recrystallization that would also have an effect, in addition to the already mentioned stress relief effect, on the magnetic properties, we chose a lower temperature for the heat treatment. This latter was performed with a heating rate of 40 • C/min, a 2 h time maintain at 400 • C under argon atmosphere, and a air-cooling.

Results of magnetic characterization after the thermal treatment are given in Fig. 19. A significant improvement of the magnetic properties, for both the B-H curve and iron losses, is observed. For example, at 1.8 T, the required magnetic field is decreased by 33% and the losses are decreased by 15%. This noticeable result suggests to consider, after the L-PBF process, a heat treatment procedure of pieces designed for magnetic applications.

Electrical conductivity

The electrical conductivity of a material is determined by its composition but also by other material characteristics, such as the porosity, the density of dislocations, the precipitates and the crystal structure that will also influence the ability for electrons to move within the material. Therefore, it is of interest to characterize the electrical conductivity with regard to the manufacturing process parameters that have an influence on the microstructure of the material.

The average electrical conductivity obtained for 4 manufacturing process parameters (combining low/high VED with horizontal/vertical sample position) is reported in Fig. 20. First, the obtained values are in the expected range 4-6 MS/m for the 16MnCr5 material grade as reported by different manufacturers [START_REF]16MnCr5 data sheet from[END_REF] for a casted steel. Moreover, the low VED samples exhibit a slightly lower electrical conductivity (~3.95 MS/m) than the high VED condition (~4.1 MS/m). The lower electrical conductivity (-5%) may be explained by the higher porosity rate that consequently induces a lower sectional area for the electrical current. However, for a given VED, the sample building direction does not have any significant influence on the electrical conductivity.

Conclusions

The present study was focused on 16MnCr5 samples built with L-PBF additive manufacturing technique in order to investigate as-built microstructures and resulting magnetic and electric properties. Various manufacturing process conditions were considered, including the influence of volume energy density (VED), hatch distance and inclination angle, and built part size. Two levels of porosity rates were obtained for the two different series: near 1.1% for the low VEDlarge hatch, and near 0.15% for the high VEDsmall hatch distance, with a low influence of build angle. The large porosity rate of 1.1% was mainly attributed to lacks-of-fusion due to insufficient re-melting of adjacent tracks.

Regarding the magnetic properties, we observed an increase of the iron losses and a degradation of the normal B-H curve (decrease of magnetic permeability) with: (a) low VED and high porosity rate (lacksof-fusion), (b) the inclination of built parts, (c) the size of toroid rings, whereas electrical conductivity seems to be only dependent on the porosity.

Finally, the influence of a low temperature stress relief was also demonstrated, with a clear improvement of magnetic properties after heat treatment. However, additional investigation is still needed to optimize the heat treatment for recovering optimal magnetic properties.

.

  5 • C) to 0.9979 g/cm3 and air density ρ air at 0.0012 g/cm 3 ρ sample = density of the analyzed sample (g cm -3 ), M air = weight of the sample in the air (g), M fluid = weight of the samples in the fluid (g), ρ fluid = fluid density (g cm -3 ), ρ air = air density (g cm-3

Fig. 1 .

 1 Fig. 1. (a) SEM image of 16MnCr5 starting powder, (b) grain distribution.

Fig. 3 .

 3 Fig. 3. Magnetic measurements obtained on the LF toroid sample with VED at 67.5 J/mm 3 : hysteresis loops and normal B-H curve (left); relative magnetic permeability (right).

Fig. 4 .

 4 Fig. 4. Porosity rates obtained from L-PBF bars (Archimedes' method). H = Horizontal building, building angle θ = 0 • , V = Vertical building, θ = 90 • ). Error bars have been estimated over five distinct measurements on each specimen.

Fig. 5 .

 5 Fig. 5. Scheme of the cut sections of toroids.

Fig. 6 .

 6 Fig. 6. Micrographic cross-sections of T samples manufactured at high VED on the left (~0.15% porosity) and F samples manufactured at low VED on the right (~1.1% porosity) (with z = building direction), x 100 magnification.

Fig. 7 .

 7 Fig. 7. porosity size distributions (number of counted pores versus pore diameter) obtained from micrographic analysis + image analysis.

Fig. 8 .

 8 Fig. 8. Macrographs of etched cross-sections for: (a) samples T (25 • inclined) manufactured at high VED (left column) and (b) samples F (0 • building angle = parallel to the building platform) manufactured at a low VED (right column) with lacks of fusion (z axis corresponds to the building direction).

Fig. 9 .

 9 Fig. 9. Macrograph of a samples T (25 • inclined) manufactured at high VED -Occurrence of porosity in the contour-hatching transition zone.

Fig. 10 .

 10 Fig. 10. Microstructure versus toroid orientation.

Fig. 11 .

 11 Fig. 11. Bainitic-Martensitic microstructure of as-built L-PBF sample for both vertical (a) and horizontal building (b), P = 175 W, V = 720 mm/ s, h = 120 µm.

Fig. 12 .

 12 Fig. 12. Local microstructural changes versus tracks' overlaps (Nital 2% etchant). A coarser microstructure is shown near the track's contour boundaries.

Fig. 13 .

 13 Fig. 13. Repeatability analysis for (left) B-H curves of LT toroid samples and (right) iron loss curves of SF toroid samples, at f = 10 Hz.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. B-H curves for (left) small toroids samples and (right) large toroid samples at f= 10 Hz.

Fig. 17 .

 17 Fig. 17. Comparison of B-H curve and iron loss evolutions between small and large toroids manufactured with the same parameters (low VED) at f = 10 Hz.

Fig. 18 .

 18 Fig. 18. Extrapolation of specific energy to "zero" frequency for both toroid geometries.

Fig. 19 .

 19 Fig. 19. Comparison of B-H curve and iron loss evolutions (at f = 10 Hz) before and after heat treatment (400 • C 2 h @Ar) for a SF sample manufactured with high VED.

Fig. 20 .

 20 Fig. 20. Average electrical conductivity at 40 • C for different L-PBF configurations.

Table 1

 1 Chemical composition of 16MnCr5 steel (supplier data).

	Element	C	Si	Cr	Mn	S
	wt%	0.17	0.19	1.1	1.3	0.02

Table 2

 2 Manufacturing parameters.

	Samples	P	V	hatch	Spot	Volume
		(W)	(mm/	(µm)	diameter	Energy
			s)		(µm)	Density (J/ mm 3 )
	Bars: 3H, 2V	175	720	120	73	67.5
	Toroids: 2 LF, 2 LT, 2
	SF, 2 ST
	(LF = Large Flat,
	LT = Large Tilted,
	SF = Small Flat,
	ST = Small Tilted)
	Bars: 3H, 3V	150	720	140	73	49.6
	Toroids: 2 SF, 2 LF

Fig. 2. Location of samples on a building plate.

CRediT authorship contribution statement

C. Dupuy: Conceptualization and design of study, Acquisition of data, Drafting the manuscript, Approval of the version of the manuscript to be published (the names of all authors must be listed); A. Benabou: Conceptualization and design of study, Acquisition of data, Analysis and/or interpretation of data, Drafting the manuscript, revising the manuscript critically for important intellectual content, Approval of the version of the manuscript to be published (the names of all authors must be listed); S. Shihab: Acquisition of data, Analysis and/or interpretation of data, Approval of the version of the manuscript to be published (the names of all authors must be listed); O. Messal: Acquisition of data, Analysis and/or interpretation of data, revising the manuscript critically for important intellectual content, Approval of the version of the manuscript to be published (the names of all authors must be listed); S. Clénet: Conceptualization and design of study, Drafting the manuscript, revising the manuscript critically for important intellectual content, Approval of the version of the manuscript to be published (the names of all authors must be listed); P. Peyre: Conceptualization and design of study, Analysis and/or interpretation of data, Drafting the manuscript, revising the manuscript critically for important intellectual content, Approval of the version of the manuscript to be published (the names of all authors must be listed).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.