Assessment of fatigue damage in welded aluminum joints subjected to multiaxial stress state

Abstract

The aluminum has mechanical properties enable to a better finishing in welded region. However, it fails eventually when a dynamic load is applied. This study estimated the damage by multiaxial fatigue present in welded joints of bicycle frames, the methodologies of analysis are based on Findley's and Dang Van's Methods. Seven experiments were performed on two pipes with welded joints. These pipes were choosen according to Bike S/A data, numerical simulations in bicycle frame and previous analysis of specialized literature. Therefore, lifespan result of frame, in function the distance, is $3\times 10^4$ km - $5.5\times 10^4$ km.

1. Introduction

In bicycles, there are various welded joints that often are subjects to dynamics loads, that produce multiaxial stresses in your joints. As a consequence of this, the bikes are susceptible to multiaxial fatigue, hence it's necessary to carry out life fatigue prediction.

The first publications of loads measurement applied to bicycles occurred in the late 1960s. Those first studies searched to define the locations of failures by means of extensometry in steel frames [1]. Already in the decade of 80 [2], performed experimental testing in aluminum frames, simulating with statics loads the forces acting on the crank arm.

The most recent publications have searched to estimate the fatigue life of bicycle frames by means of finite elements methods. In this context, the work of [1,3] stands out. The first work estimated the failure in the frame used multiaxial fatigue criterion. The second performed various simulations in bike frames, with the goal of to check the most likely location of failure. It is clear, however, that the published papers limited to estimate the loads to perform the damage prediction in the welded joints.

At that, based on the theory of fatigue for welded joints, it proposed to perform dynamics cycling tests, with the goal of measurement the strains generated by loads actuate in the bicycle during its use, hence to perform a assessment more accurate multiaxial fatigue damage at welded joints. For the correct damage prediction, methodologies based at the main methods of multiaxial fatigue were proposed and validated with cases studies performed by numerical simulation.

This research aims objective the evaluation of the damage by multiaxial fatigue at welded joints and the local of bike frame with real data collected by Strain Gauges Rosettes on the local failure concordant with Ansys's analysis of the bike frame experimented, occurred by aleatory solicitations of the dynamic pedaling experiment, using software.

2. Materials and methods

2.1. Methodologies for multiaxial fatigue

In this paper, after the experiments realization, it was realized that the loading's acting on the bicycle did not change significantly the direction of the main stresses, i.e., the loads are proportional. Due to this, was used the Findley and Dang Van methods adapted for that loading type.

2.1.1. Dang Van and Findley

Findley (1959) [4] supposes the crack is born for fatigue in the critical plane of critical point, where the damage parameter, shear stress amplitude (${\tau }_a$) with some contribution normal stress (${\sigma }_n$), is maximized and exceed the fatigue endurance shear stress of the Findley (${\tau }_{N,Fd}$) [5,6]. Thus, the Findley's equivalent stress (${\tau }_{Fd}$) can be found as follow:

$${\tau }_{Fd}={\tau }_a+k_{Fd}{\sigma }_n\text{,}$$ (1)

where $k_{Fd}$ is the normal stress sensitivity factor, which represents the influence of maximum normal stress in maximum shear stress amplitude [7]. According to Ref. [8], the $k_{Fd}$ is expressed by:

$$k_{Fd}=\left(\frac{1}{2}\right)\sqrt{-\frac{{\left({\sigma }_{F\left(0\right)}^2-{\sigma }_{F\left(-1\right)}^2\right)}^2}{\left({\sigma }_{F\left(0\right)}{\sigma }_{F\left(-1\right)}\right)\cdot \left(2{\sigma }_{F\left(0\right)}^2-5{\sigma }_{F\left(0\right)}{\sigma }_{F\left(-1\right)}+2{\sigma }_{F\left(-1\right)}^2\right)}}\text{.}$$ (2)

where ${\sigma }_{F\left(0\right)}$ means the limit fatigue stress for pulsing loading and ${\sigma }_{F\left(-1\right)}$ is the limit fatigue stress for totally reverse loading

Based on this formulation of Findley's method and in analysis of references [7,9,10], it is proposed the methodology, show in Fig. 1, for experimental assessment of fatigue multiaxial damage, in structures subject to random proportional loading. Those methodologies, the stress ${\sigma }_i$ and ${\sigma }_{ii}$, represented at Fig. 1, are calculated through the strains obtained at the strain gauges placed in the structure tested. A algorithm is used to transform the stresses ${\sigma }_i$ and ${\sigma }_{ii}$ in main stresses ${\sigma }_1$, ${\sigma }_2$ e ${\sigma }_3$ at the plane stress state, making null at least one of three main stresses in every instant of time. After obtaining the main stresses, is calculated the Findley's equivalent stress, and then used the Rainflow's method for obtaining the alternating stress and the medium stress for the Findley's equivalent stress. Finally, the Palmgren-Miner's ruler is utilized for estimate the damage of the structure tested.

Fig. 1.

Application methodology of the Findley's method for fatigue experimental analysis (prepared by the author).

According to Papuga e Halama (2019) [11], Dang Van proposed that the mesoscopic shear stress, equation (5), (${\tau }_{meso}$) is the responsible for nucleation of the crack along the slip bands in the grain region and a mesoscopic hydrostatic stress, equation (6), (${\sigma }_{meso,h}$) influence on the crack opening process. Van (2003) [12] defined that the linear combination these parameter create the Dang Van's equivalent shear stress (${\tau }_{DV}$), expressed by:

$${\tau }_{DV}={\tau }_{meso}+{\alpha }_{DV}{\sigma }_{meso,h}$$ (3)

where ${\alpha }_{DV}$ is a parameter that represent the influence of hydrostatic stress in the study point, and is found by means of system of linear equations shown in Ref. [12], which takes into account the fatigue limits under fully reversed ($R=-1$) and pulsating ($R=0$), thus, ${\alpha }_{DV}$ is found by the following equation:

$${\alpha }_{DV}=\left(\frac{3}{2}\right)\cdot \frac{{\sigma }_{F\left(-1\right)}-{\sigma }_{F\left(0\right)}}{2\cdot {\sigma }_{F\left(0\right)}-{\sigma }_{F\left(-1\right)}}\text{,}$$ (4)

The mesoscopic hydrostatic stress is the same hydrostatic stress at macro scale, and mesoscopic shear stress is equal maxima shear stress for proportional loading. Thus, these stresses can be calculated by:

$${\tau }_{meso}={\tau }_{\mathit{max}}=\frac{{\sigma }_1-{\sigma }_3}{2}$$ (5)

$${\sigma }_{meso,h}=\frac{{\sigma }_1+{\sigma }_2+{\sigma }_3}{3}$$ (6)

Then, this paper introduce the methodology shown in Fig. 2 for the damage assessment in aleatory proportional loading, with the requests being obtained from strain gauges. It can be noticed that the methodology it is the similar the proposal for Findley, changing only the equivalent stress.

Fig. 2.

Application methodology of the Dang Van's method for fatigue experimental analysis (prepared by the author).

2.2. Fatigue resistance of aluminum

The object of study, bike frame, was manufactured with aluminum pipes 6061-T6. Thus, was used the curve SN these material, available on the [13], because that author realize the fully reverse bending fatigue tests at proof bodies to type butt joints, with yield strength (${\sigma }_y$), Young's modulus ($E$) and ultimate tensile stress (${\sigma }_{ut}$) of $262MPa$, $69GPa$ and $290MPa$, respectively. Through of analysis these paper reached the value of $49MPa$ for the fatigue resistance these alloy at $10^9$ cycles. This data were used in the multiaxial fatigue methods for estimate the damage suffered by the above dynamic tests.

2.3. Numerical analysis

In this paper was carried out two transient numerical analysis, with its foundations the boundary conditions supplied by company Bike S $/$ A (case 1) and the Standard [14] (case 2). These analysis were used to validate the Findley and Dang Van methodologies, since the authors have the available results of the above mentioned cases. As a result of the failure to occur in the formed union for down tube, Saddle tube and central tube, the analyzes were restricted to this meeting.

Firstly, it was considered a transient analysis using an excitation frequency of $2.0Hz$ in both cases mentioned above. To estimate vibration modes dynamic effects was utilized the module Modal do Software Ansys 19, reaching a minimum frequency, among six degrees of freedom, of $86.5Hz$. Thus, adopted frequency is less than 1/40 of this value, the transient regime can be expressed as a punctual static one, analysing only critical points at time of greatest stress.

Furthermore, the finite element was constructed using hexahedral and tetrahedral elements with a configuration Hard for the software to follow the geometry constraints. Thus, the directional tube was used as a fixed support, where its displacement is zero; and the displacements in the hooks are zero for the YZ axis, occurring only in the X direction.

The fatigue testing, which were performed by the manufacturer (Bike S/A), using the own machine, consisted in submitting the frame to pedaling forces, only in direction y, in the case 1, then, the maximum loads on the pedals were synthesized in Fig. 3. Furthermore, the case 2 is based in the Standard ISO 4210-2 [14] suggests that fatigue testing with pulsating-cyclic loads was performed using a $1200N$ descent force at YZ-plan inclined about $7,5^{\circ }\left(\pm 0,5^{\circ }\right)$ from the y-axis, applied repeatedly at pedals. According to Fig. 3(a) and (b), it is trivial that acting force occurs alternately, each pedal at once.

Fig. 3.

Loading conditions according to a) Bike S/A, b) Standard ISO 4210-2 (prepared by the author).

2.4. Collection data by strain gauge

The measurement of data by rosettes are made at places of greatest failure, as: welded joins between down pipe, central pipe and saddle pipe according to Refs. [2,3]. Following [15,16] rosettes been associated dummys located out stress zone for effects by temperature, they are associated by Wheastone Bridge with resistance over $350\pm \mathrm{0,1}\%$ ohms. The strain gauge 1 and 2 based at $4mm$ and $14mm$ from welded joint of down tube and saddle tube. However all locations are indicated at Fig. 4(a) to (d).Where, sg1 and sg2 means Strain Gauge Rosette 1 and Rosette 2, respectively.

Fig. 4.

Locations for a)Strain Gauge - Sandle Tube, b)Strain Gauge - Down Tube, c)Dummy - Sandle Tube and d)Dummy - Down Tube (prepared by the author). Where, sg1 and sg2 means Strain Gauge Rosette 1 and Rosette 2, respectively.

2.5. Dynamic cycling tests

The bicycle was submitted to the dynamic testing at four types of different surfaces, as illustrated in Fig. 5, where it can be to note the paving surfaces, asphalt and sandy, respectively in Fig. 5(a), (b) and 5(c). Also, Fig. 5(d) show the step the one that was submitted to bike, in this test the same was put up end down a 20 cm high step.

Fig. 5.

Surfaces a)Pavement, b)Asphalt, c)Sandy and d)Step (prepared by the author).

In addition to the surfaces already shown, the test also consisted in test the bike at different pedaling position. Fig. 6 show the two pedaling positions, to which the bicycle was submitted, where the position 1 show the seated cyclist and at position 2, the pedal cyclist standing.

Fig. 6.

Cyclist positons Garmin.

Thus, the bike was submitted two test (seated cyclist and pedal cyclist standing) at the surfaces shown in Fig. 4 and 4 and 4; and only a test at surface illustrated at Fig. 4. In that the cyclist cycled on the Saddle. Therefore, were performed 7 dynamic tests (experiments) in the bicycle, that are:

• •Experiment 1: Pavement, riding on the saddle;
• •Experiment 2: Pavement, pedaling standing;
• •Experiment 3: Asphalt, riding on the saddle;
• •Experiment 4: Asphalt, pedaling standing;
• •Experiment 5: Sandy, riding on the saddle;
• •Experiment 6: Sandy, pedaling standing;
• •Experiment 7: Going up and down a step.

Some others relevant informations about these test condition: ambient temperature of $17^{\circ }C$, a cyclist of $173cm$ height, $73kg$ weight and $8.5m/s$ speed. Furthermore, was collected over $5500$ points for $50s$ to each.

3. Results

3.1. Numerical results

According to methodologies about multiaxial fatigue, linear damage law and materials properties, the damage and cycles to failure was estimated for greatest main stress at $4mm$ of weld bead, as it shows at Table 1.

Table 1.

Damage by multiaxial fatigue to cases 1 and 2 (prepared by the author).

Methodologies	Case 1		Case 2
	Damage	Cycle	Damage	Cycle
Findley	1.245 × 10−4	8034.1	1.06 × 10−5	9.4 × 104
Dang Van - Met 1	8.752 × 10−5	11426	7.42 × 10−6	1.35 × 105
Dang Van - Met 2	8.752 × 10−5	11426	7.42 × 10−6	1.35 × 105

There are two different Dang Van methodologies. One of them presented by Lee, Barkey e Kang (2012) and another one by Ref. [9], both reach the same outputs. So, the second one will be adopted.

It was created a distribution weibull to results obtained by Bike S $/$ A, estimated failure for the case 1, over $1.13\times 10^4$ - $3.14\times 10^4$ cycles noticing 10% for failures for minimum number cycles, according Fig. 7(a). However, it was used tubular joint "T" as approximation for the bike joint, presenting failure by fatigue for the case 2, about $10^5$ - $10^6$ cycles, represented by Fig. 7(b), showing that methodologies adopted for multiaxial fatigue are powerful for life analysis to propocionals loads at weld bead.

Fig. 7.

Estimated failure for fatigue a)Distribution Weibull (prepared by the author) and b)Obtained Results to Tubular Joints T (Adapted [3]).

3.2. Fatigue analysis

3.2.1. Main stresses

As a consequence of the dynamics tests performed in the bicycle were calculated the acting stresses, starting of measurement of strain of rosettes. Fig. 9 present the mains stresses obtained in the experiments, after the passage through a moving averages filter. It is noted that the main stresses in the strain gauges close to the weld bead (Strain Gauge Rosette 1) are bigger, prove the theory of stresses in welded joints, shown in Ref. [17].

Fig. 9.

Main stresses in the experiments (prepared by the author).

It is observed that on all graphs, in Fig. 9(a) to 9(n), occurred an initial peak of stress, that is due to the initial pedaling movement, i.e., break inertia to the movement. After that action, the stresses become stable with smaller amplitudes.

In addition, that among the various tests, the experiment where was imposed the bicycle up and down a step had signs of stress more unstable, being that the main stress peaks were perceptible, as shown in Fig. 9(m) and (n). It should be noted that the analyzed bike it has fork with cushioning, consequently at the moment of entering and exiting the step the dissipation of a portion of the forces occurred, causing a likely reduction of stress in frame.

3.2.2. Equivalent stresses

To calculate the equivalents stresses of Findley and Dang Van, it was used equations (1), (3)), being that the material constants $k_{Fd}$ and ${\alpha }_{DV}$ are worth 0.191 and 0.116, respectively. Fig. 10(a) to 10(n) show the calculated stresses for the experiments. These results are obtained through Goodman mean stress correction, according Fig. 8.

Fig. 10.

Equivalent stresses in the experiments (prepared by the author).

Fig. 8.

σN curve correction for Findley and Dang van methods (prepared by the author).

Then $k_{Fd}$ (equation (2)) was calculated using ${\sigma }_{F\left(0\right)}=40.38MPa$ and ${\sigma }_{F\left(-1\right)}=48.49MPa$ founded at Fig. 8; ${\alpha }_{DV}$ (equation (4)) was determinate by the same way, where ${\sigma }_{F\left(0\right)}=44.36MPa$ and ${\sigma }_{F\left(-1\right)}=48.49MPa$.

It is noted that due to formulation of both methods (Findley and Dang Van), the stresses are ever positive. It is also noted that, the graphics have the same behavior, with small variations, this occurs because of proportionality of the principal stresses.

3.2.3. Cycle counting

Finally, the Racetrack was used as a filter of equivalents stresses of Findley and Dang Van, and after it was used the Rainflow cycle counting method, since the load can be considered proportional. The latter defined the cycles, amplitude and Average stresses, grouping them on a histogram, as shown in Fig. 11.

Fig. 11.

Rainflow cycle count histogram of the experiments (prepared by the author).

The histograms of Fig. 11(a) to (n) provide a convenient way for to define the damage caused cycle to cycle per overlap of average stress the alternating stress (amplitude). Soon, the quadrants that have a larger number of cycles do not cause significant damage, i.e., closer to the amplitude and zero mean. Meanwhile, when the average and alternating stress increases, same with a shorter number of cycles, tend to damage more the joint welded.

3.3. Damage assessment

The evaluation of the damage in the experiments it was made individually, that is, the tubes was analyzed separately in the experiments. Table 2 exhibit the accumulated damage in the tests. It is evident that the damage was bigger in the experiments where the cyclist "pedaled standing". That confirmation proved that the practiced loads over the analyzed joint comes of pedaling load. It is still noticeable that the calculated damage starting of the Findley's equivalent stresses is bigger than Dang Van, corroborating with the results found in the paper [9].

Table 2.

Accumulated damage per experiment (prepared by the author).

Experiment	Down tube			Saddle tube
	Findley	Dang Van	DF/DDV	Findley	Dang Van	DF/DDV
Exp.1	3.9 × 10−7	2.2 × 10−7	1.77	1.0 × 10−7	0.64 × 10−7	1.56
Exp.2	67 × 10−7	36 × 10−7	1.86	7.4 × 10−7	5.6 × 10−7	1.32
Exp.3	0.69 × 10−7	0.36 × 10−7	1.92	0.85 × 10−7	0.59 × 10−7	1.44
Exp.4	62 × 10−7	33 × 10−7	1.88	5.8 × 10−7	3.7 × 10−7	1.57
Exp.5	4.5 × 10−7	2.6 × 10−7	1.73	1.3 × 10−7	0.86 × 10−7	1.51
Exp.6	38 × 10−7	21 × 10−7	1.81	4.1 × 10−7	2.8 × 10−7	1.46
Exp.7	14 × 10−7	8.0 × 10−7	1.75	5.5 × 10−7	3.4 × 10−7	1.62
Mean (DF/DDV)			1.82			1.50

For the evaluation of the damage in 2 h, estimated time of the daily use by maker and was experimented during 40 s, is perceived that the down tube in the exp. 2 had a greater accumulated damage, as shown in Table 3. Showing that poorly paved streets contribute of significantly for the bike failure.

Table 3.

Accumulated daily damage per experiment (prepared by the author).

Experiment	Down tube		Saddle tube
	Findley	Dang Van	Findley	Dang Van
Exp.1	0.71 × 10−4	0.41 × 10−4	0.18 × 10−4	0.11 × 10−4
Exp.2	12 × 10−4	6.5 × 10−4	1.3 × 10−4	1.0 × 10−4
Exp.3	0.12 × 10−4	0.06 × 10−4	0.15 × 10−4	0.1 × 10−4
Exp.4	11 × 10−4	5.8 × 10−4	1.1 × 10−4	0.67 × 10−4
Exp.5	0.8 × 10−4	0.46 × 10−4	0.24 × 10−4	0.15 × 10−4
Exp.6	6.7 × 10−4	3.4 × 10−4	0.72 × 10−4	0.49 × 10−4
Exp.7	3.8 × 10−4	2.2 × 10−4	1.7 × 10−4	1.0 × 10−4

Nevertheless, these experiments were susceptible to non-ideal test regions for variable control, the acquisition system has a capture of only 40 s and its noise. Even so, was obtained the bike life, in relation to the result obtained for the down tube in exp. 2, in mean of 833.3 h for Findley and 1538.5 h for Dang Van, using maximum damage of 0.5, as presented in Ref. [18]. Finally, considering typically 10 m/s as speed for pedaling, it is concluded that, the bicycle will support approximately between $3\times 10^4$ km to $5.5\times 10^4$ km [2]. found $3.3\times 10^4$ km in their static laboratory tests, thus proving the results found in this work. It's important make sure that the exp. 2 it's a extreme case, because the bicycle was submitted to poor pavement and the cyclist pedaled standing.

4. Conclusion

The methodologies developed were validated by means of two cases studies. Those it was estimated the life prevision for the analyzed joint with the proposed methodologies, and posteriorly that estimated were compared with results obtained in practical tests. Both techniques obtained satisfactory results, being that the methodology based on the Findley method had results more conservative than that established of Dang Van method, where the relation mean between the damage by the method of Findley and Dang Van was 1.82 for down tube and 1.5 for saddle tube.

For the experimental analysis, were realized 7 tests in the down tube and of saddle tube, which were measured deformations at 4 different locations near the welding region, as measurements were made using rosettes. Then, in the total, were collected 84 curves of main stresses. After the treatment of these deformations, it was concluded that less uniform surfaces (pavement and sandy) generated relatively higher stresses. And, when the cyclist pedaled, i.e., without its weight exercised the saddle, the stresses were even larger. Soon, it was found that the forces exerted on the pedals generate stresses more significant than the other forces.

As a consequence of these stresses, the calculated damage for the experiments, whose cyclist pedaled standing was on average approximately 21.3 times greater than the damage caused when the cyclist pedaled sitting on the saddle. It was also evident that more rough terrains, reduce the welded joint life. In these the damage was in mean 1.7 times greater than the caused damage for asphalt surface.

The proposed methodologies in this paper have proved effective for to realize the fatigue study in bicycle frames, because averages the results obtained in this methodologies had relative errors less than 30%, when compared with the works found in literature. However, when was compared only to methodologies that uses the Dang Van method, the error verified was only 9%. Soon, companies in the sector can use the results as base for the development of new projects and products.

Finally, the bike model supplied by the company Bike Norte S/A was resistant to damage caused by fatigue, for the load modes analyzed in this study, since, the same (bike model) obtained a life expectancy between 833.3 h (Findley method) and 1538.5 h (Dang Van method), similar to the results found in the literature.

Author contribution statement

Ricardo Cardoso Soares: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Edison da Rosa: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data.

Allan Icaro Ferreira Sousa: Conceived and designed the experiments; Analyzed and interpreted the data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data will be made available on request.