Multiaxial fatigue assessment of welded connections in railway steel bridge under constant and variable amplitude loading

Abstract

Fatigue assessment based on the critical plane approach is generally accepted to be more accurate for multiaxial non-proportional loading. Among different critical plane fatigue criteria, the modified Carpinteri-Spagnoli (C-S) criterion is rather a simple approach in which the critical plane orientation can be determined a priori. The aim of this paper is to investigate the applicability and reliability of the critical plane approach-based C-S criterion that can be employed to calculate the fatigue damage of bridge connections under railway traffic loading. A typical half through steel bridge subjected to railway traffic is analysed using Finite Element Modelling in ANSYS 17.2. The averaged principal stress directions determined through appropriate weight functions are used to predict the critical plane orientation. Then, the multiaxial fatigue damage is assessed through an equivalent stress expressed by a nonlinear combination of the normal and the shear stress components acting on the critical plane. Applicability of the C-S criterion is investigated by estimating the fatigue damage of critical welded connections and is compared with the fatigue damage computed according to Eurocode EN1993-1-9 under both constant and variable amplitude loading.

Keywords

Fatigue assessment multiaxial loading welded joint variable amplitude steel bridge

1 Introduction

About 80 to 90% of structural failures in steel bridges are due to fatigue and fracture [1]. Fatigue limits the service life of railway steel bridges and is, therefore, considered a crucial factor in design of steel bridges. The fatigue assessment of bridge connections is generally based on the concept of uniaxial S-N curves provided in the codes of practice. However, local stresses developed at welded connections under railway traffic are always multiaxial. In addition, directions of principal stresses change due to stochastically varying stress components defining the variable amplitude non-proportional stress histories. Multiaxial fatigue assessment can be done with the help of several models such as stress, strain, energy-based or fracture mechanics models. However, there is still no consensus on a method which can correctly account for non-proportional loading.

In this paper, a preliminary fatigue assessment of welded connections based on nominal stress approach as per Eurocode EN 1993 1-9 is performed to identify the most critical locations that are then subjected to a subsequent more refined non-proportional multiaxial fatigue damage assessment.

2 Literature review

There are mainly three classical methods of fatigue assessment, namely stress-based, strain-based and fracture mechanics-based approaches. Other methods such as fatigue analysis using field measurement and probabilistic methods are also briefed.

2.1 Stress-based approach

Stress life approach is most suitable for high-cycle fatigue where the stresses/strains are elastic (plastic strains are negligible) except at the fatigue crack tips. This is due to scatter in fatigue data at low-cycle fatigue. It is also known as total life approach, in which there is no distinction between the crack nucleation and propagation, instead gives us life of structural component to failure.

Basquin O. H. [2] represented the stress-life relationship of the SN curve: $N \cdot S^{m} = A$ (1)

or $log N = - m log S + log A$ (2)

Where m = slope and A = logN (x – axis) intercept are material constants.

Stress based approach can be further classified into many categories depending on the stress analysis in structures. The most commonly used approach for estimating the fatigue damage of steel bridges is the nominal stress method also known as the conventional S-N curve method. Here, the relationship between the stress range, S, and the number of cycles to failure N_f is determined by appropriate constant amplitude experimentally tested S-N curves. Each stress range is categorised as a separate S-N (Wohler’s) curve and each curve is associated with a constructional detail. In Eurocode, EN 1993-1-9 [3], there are set of 14 equally spaced S-N curves. In order to determine the fatigue damage, the nominal stress in the detail is compared to the S-N curve provided in the code. Cumulative damage method using the Palmgren-Miner linear damage assumption, also known as the Miner’s rule, extends nominal stress approach to variable - amplitude loadings.

Hotspot stress/geometric and effective notch stress method are alternate to nominal stress method for more complicated welded connections in bridge. Eurocode 3 provides stet of 7 detail categories for hotspot stress method which is more reliable and accurate than nominal stress method. However, it has not provided any recommendations for the computation of the hotspot stress. Niemi et al. [4] suggested detailed recommendations regarding hotspot stress determination for the fatigue analysis of welded components. Radaj [5] confirmed that hot spot stress could be determined either by surface extrapolation or by linearization through the plate thickness. Dong P [6] also recommended an alternate geometric/hot spot stress computation method combining the surface extrapolation and through thickness methods. Chan et al. [7] also put an effort in extending the hot spot stress approach to fatigue damage evaluation of the welded plate joint of steel structures, particularly for cable stayed bridge fatigue evaluation. A comparative study on five frequently used welded bridge details is conducted by Aygül et al. [8] to study the accuracy of nominal stress, hot spot stress, and effective notch stress method. He observed that there was no significant improvement in estimation of the fatigue strength using the effective notch stress method.

2.2 Strain-based approach

Strain life approach is most suitable for low-cycle fatigue in which the stresses induced due to plastic deformation on the macro scale are also considered. Since most of the fatigue damage in railway steel bridges are due to high-cycle fatigue, no significant research performed using strain based approach.

2.3 Fracture mechanics-based approach

The fracture mechanics approach is generally used to calculate the propagation life of crack in weldment from an initial crack or defect.

The Paris law [9] gives us the relationship between growth rate of initial crack length, a, to the number of cycles to failure, N: $\frac{da}{dN} = C Δ K^{m}$ (3) where C and m are material-related parameters and ΔK is the stress intensity factor range.

This method is commonly used to predict the bridge inspection intervals for specific increase in crack length with different assumed values of initial crack length. Fisher [10] investigated about more than twenty welded details in steel bridge using fracture mechanics approach.

2.4 Fatigue assessment using field measurement

Despite the highly developed finite element software, fatigue analysis using numerical simulation cannot capture the exact stress variations in structural members. Moreover, it is very difficult to account all possible fatigue parameters. Therefore it plays a crucial role to get field measurement data for more accurate derivation of all fatigue parameters and their corresponding statistical properties in fatigue assessment.

Chang et al. [7] considered the non-destructive methods as “local health monitoring” methods since they are used to monitor localized damage once damage is confirmed by the structural health monitoring (global methods) without locating it.

Field data can be measured by two ways,

Short term monitoring: Fatigue data can be measured either using load controlled diagnostic testing or by short term in-service monitoring using Non Destructive Evaluation [11].

Long term Structural Health Monitoring (SHM).

DeWolf et al. [12] assessed the fatigue life of various bridges using portable strain gauge data acquisition system, which has been used in Connecticut to renew the old bridges and to identify the cost-effective maintenance, repair, and replacement strategies.

2.5 Probabilistic fatigue assessment

Due to inevitable uncertainties and stochastic nature of fatigue process and measured data, the probabilistic fatigue assessment of steel bridges prevails over other classical methods. Generally, the log-normal distribution and the Weibull distribution is used to consider variable traffic loading. By considering this technique, logical strategies on inspection intervals and maintenance can be achieved using reliability indices.

Imam et al. [13] evaluated many short-span riveted railway bridges in UK using probabilistic fatigue assessment method.

3 Description of railway steel bridge

A typical single span half through steel bridge carrying single railway track is considered for fatigue analysis. It is particularly well suited for medium span bridges, where there is shallow depth between the trafficked surface and the clearance level underneath. Therefore, half through railway steel bridges are widely preferred rather than deck type plate girder bridges. The structure is simply supported and they are resting on elastomeric bearings.

The bridge is composed of a pair of plate girders spanning 27.8 m between abutments, 20 mm deck plate and inverted T cross girders. Cross girders span between main girders spaced at 0.67 m as is shown in the Fig. 2. Transverse stiffeners are provided at a spacing of 1.33 m (Fig. 1). Lateral torsional buckling is restrained by U-frame action. All structural components are made up of non-alloy European standard (EN 10025-2) structural steel S355, whose yield strength is 355 MPa and ultimate strength is 611 MPa. Rotations are allowed in all directions and translations are allowed only along the direction to accommodate longitudinal and transverse movements arising due to temperature, breaking-traction, wind, nosing, centrifugal actions, etc.

Fig.1

3D view, Front view and Side view of global numerical model of the bridge.

Fig.2

Cross section of inverted T beam cross girders.

4 Fatigue load models for railway bridges

4.1 Fatigue load model for nominal stress method

Fatigue load model LM71 represents the static effect of railway traffic loads calibrated using the International Union of Railways (UIC) and European railway traffic data. This load model shall be used for fatigue verification of railway bridges excluding ultimate strength design adjustment factor, (i.e., α= 1.0). The axle load arrangements and the characteristic values for LM 71 are given in EN 1991-2:2003 clause 6.3.2 as follows (see Fig. 3).

Fig.3

Fatigue load model LM71.

4.2 Fatigue load model for cumulative damage method

The fatigue damage of a detail depends on the traffic spectra that passes over the bridge during its service life and its detail category. According to EN 1991-2:2003, Annex D the fatigue design stress spectra is categorised as “standard”, “heavy” and “lightweight” traffic mixes. This depends on whether the railway line predominantly carries passenger cars or freight cars. Wherever the traffic mix doesn’t represent the real traffic, an alternative traffic spectra should be specified. In this paper the fatigue analysis is performed based on the heavy traffic mix. Heavy traffic mix is composed of different combination of 4 fatigue trains as is shown in Fig. 4. The axle load configuration is reported in Fig. 5.

Fig.4

Axle load and axle configuration of train types 5, 6, 11 and 12 in heavy traffic mix.

Fig.5

Tables taken from EN 1991-2:2003, Annex D of train composition of standard, heavy and light traffic mix.

5 Numerical modelling of bridge

Stress analysis of the railway bridge is performed by numerical finite element approach using ANSYS 17.2 Workbench. Global numerical model of the bridge is created using hexahedron 8 noded solid 185 type elements. Materials are assumed to be linear elastic and isotropic with modulus of elasticity 200 MPa and poison’s ratio 0.3. The contact between structural members is modelled using surface to surface bonded option. In specific, the CONTA174 and TARGE170 elements are used to model the contact and target surfaces forming a contact pair. With this global numerical model a prior identification of critical welded detail is performed using the nominal stress (S N curve) approaches.

The recommended Fatigue load models in Eurocode are the set of point loads. But in actual, these point loads are transferred through rails, sleepers and ballast and get distributed longitudinally and transversely over the certain area of deck plate. The easiest way to mimic this phenomenon is to apply the axle loads on the deck plate as a patch load. According to DIN EN 1991-2, cl. 6.3.6.3, angle of longitudinal and transverse dispersion of axle load is 14° (i.e. slope 4V:1H). In static analysis, for λ – coefficient method, LM71 is converted entirely into uniformly distributed load of 156.25 kN/m at the centre span of 6.4 m and distributed as area load over the width of 2.7 m on the bridge deck. In transient dynamic analysis, for cumulative damage method and critical plane approach, a moving patch load is considered as shown in Fig. 6. The stress distribution is more complex in real and is affected by several factors such as the angle of internal friction and weight of the material. Since the focus in this project was to generate a model that should be convenient and rational to use in practice, the stress is assumed uniform within the patch area.

Fig.6

Schematic drawing of normal stress distribution in ballast in a) longitudinal and b) transverse direction of the bridge. c) Plan view showing longitudinal, transverse normal stress distribution in ballast and patch load.

In order to get local stress at the weldments, a detailed local finite element model is created using sub modelling technique in ANSYS. It is substantially based on St. Venant’s principle. According to it, – “If an actual load distribution is substituted by a statically equivalent load system, the stress or strain distribution is altered only in the vicinity of load application.” In other words, simplifying the minor details in global model does not affect the total response of features say - cope hole, weld beads are shown in the refined submodel. The critical welded detail is then simulated by interpolating the displacement histories as a boundary condition from the global model (at the cut section) to that of local model. By this, a highly precise local stress state is obtained at a very vicinity to weld toe to consider the multiaxial notch fatigue using Taylor’s Critical Distance Point method as shown in Fig. 7.

Fig.7

Local model showing application of Taylor’s Critical Distance theory to critical welded detail.

6 Fatigue damage assessment

In order to mitigate the high computational cost, a preliminary fatigue assessment based on the nominal stress method is performed to locate the potential critical weld. The overview of the adopted methodology is shown in Fig. 8.

Fig.8

Overview of methodology.

6.1 Nominal stress method

6.6.1.1 6.1.1 λ – coefficient method

In Eurocode, the nominal stress approach is referred as theλ-coefficient method. It is based on the concept of variable amplitude loading generated by railway traffic, and is simplified to equivalent constant amplitude loading by usingλ-factors. The nominal stresses obtained from ANSYS using load model LM71 as per EN 1991-2:2003 are modified by fourλ-factors and expressed as an equivalent stress range corresponding to 2×10⁶cycles, Δσ_E,2. As a result, fatigue assessment is reduced to a simple comparison between the equivalent stress range at 2×10⁶ cycles and the detail category (fatigue strength). $γ_{Ff} λ φ_{2} Δ σ_{71} ⩽ \frac{Δ σ_{c}}{γ_{Mf}}$ (4) As per EN1993-2:2006 clause 9.5.3(9), fatigue damage equivalent factorλ related to 2×10⁶ cycles for railway bridges is calculated by using the following expression: $λ = λ_{1} \cdot λ_{2} \cdot λ_{3} \cdot λ_{4} ⩽ λ_{max}$ (5)

λ₁ - factor for the damage effect of traffic

λ₂ - factor for the traffic volume

λ₃ - factor for the design life

λ₄ - factor for the structural element loaded by more than one track

λ_max - 1.4

γ_Ff - partial safety factor for fatigue loading

γ_Mf - partial safety factor for fatigue resistance

λ - fatigue damage equivalent factor related to 2×10⁶ cycles

Φ₂ - dynamic factor calculated as per Annex D, section D.1 in EN 1991-2:2003(E)

Δσ₇₁ - stress range due to the fatigue load model

Δσ_C - detail category of welded detail

6.1.2 Cumulative damage method

In order to reflect the most onerous and real traffic spectra, fatigue analysis is performed by considering heavy traffic mix. The following steps are performed to determine the fatigue damage:

Stress history is obtained by moving each train type in heavy traffic mix on global finite element model for the time step of 0.01 sec.

Stress histogram (see Fig. 9) is constructed by splitting individual stress range spectrum in stress history through rainflow technique (ASTM E1049 – 85 [14]) method.

Endurance limit (Ni) of each stress block is calculated by the following relationship:

$N_{i} = N_{c} {[\frac{Δ σc / - γMf}{γ_{Ff} Δ σ_{i}}]}^{m}$ (6) where Nc = endurance limit of detail category Δσ_C, Δσ_i= calculated stress range

Total damage, D is calculated by summing up the damage caused by individual blocks in the stress histogram.

Fatigue life is calculated using the condition D≤1 as is given below: $D = \sum_{i} D_{i} = \sum_{i} \frac{n_{i}}{N_{i}}$ (7)

where n_i = number of cycles of individual stress block.

Fig.9

Cumulative damage method: Stress histogram of all train types in heavy traffic mix.

6.2 Identification of critical welded detail

From the preliminary fatigue assessment it is pointed out that the welded connection between cross girder and main girder at the centre of span is the most critical welded detail, as is shown in Fig. 10. Since the nominal stress approach is based on the static load model it does not give clear picture about damage rate due to each train passage. Therefore a detailed multiaxial fatigue assessment is needed to get damage rate due to each train pass.

Fig.10

(a) Critical welded connection between cross girder and main girder at the centre of span. (b) Stress concentration at flange to flange connection in critical welded detail.

6.3 Multiaxial fatigue assessment using critical plane approach

The traditional fatigue assessment of bridge connections are based on uniaxial S-N curves. However, it has been found that these approaches might give non-conservative results for the non-proportional loading. A possible explanation of these results might be due to the fact that the change in principle stress direction activates plastic strains along different slip systems leading to more damage when compared to proportional loading. Therefore the correct fatigue damage is achieved by considering multiaxial non proportional loading.

Generic critical plane models require scanning over all planes intersecting the surface either orthogonally or at some inclination for maximum value of damage parameter. Also the stress analysis has to be performed for each time step considered in the simulation of the train crossing. This cumbrous and tedious task is simplified in the C-S criterion by applying some weight functions and linking the critical plane with mean principle stress direction. The critical plane is the plane where the fatigue damage parameter is maximum. According to the C – S criterion, the damage parameter is defined as - “Effective stress Δσ_eq which is a nonlinear combination of normal and shear stress ranges acting on critical plane”

$Δ σ_{C - S} = \sqrt{Δ σ_{eq}^{2} + k \cdot Δ τ^{2}}$ (8) where k = material constant, Δσ_eq= equivalent normal stress given by, $Δ σ_{eq} = Δ σ + Δ σ_{R, - 1} \frac{σ_{m}}{R_{m}}$ (9) where Δσ, Δτ= normal and shear stress range respectively Δσ_R,-1= fatigue strength for fully reversed normal stress, σ_m= mean stress, tensile strength, R_m = 611 MPa.

Fig.11

Multiaxial fatigue assessment using C-S criterion: a) Normal and shear stress histories; b) Range mean matrix and c) Shear stress vector path on the critical plane for train type 11.

The procedure of fatigue damage assessment using the C– S criterion can be illustratively outlined as follows (see also Fig. 11).

Multiaxial notch fatigue using Point Method: C-S criterion for welded details is applied in the presence of notches by using the critical distance theory (critical point method) i.e., by taking stress tensor at a material dependent distance 0.5 L from the notch tip. Where, critical distance L is given by,

L = \frac{1}{π} {(\frac{Δ K_{th}}{σ_{af}})}^{2}

(10) where ΔK_th (threshold stress intensity) = stress intensity below which cracks will not grow; σ_af= fatigue limit under fully reversed normal.

Averaged principal stress directions $\hat{1} \hat{2}$ and $\hat{3}$ are determined from the averaged values of the principle Euler angles $\ddot{Φ}$ $\hat{θ}$ and $\hat{Ψ}$ (Fig. 10) by averaging the instantaneous Euler angles φ (t) θ (t) and ψ (t) as proposed by Carpinteri et al. [15 –17] using the weight function method as follows:

$\begin{matrix} \hat{φ} = \frac{1}{W} \int_{0}^{T} φ (t) W (t) dt, \hat{θ} = \frac{1}{W} \int_{0}^{T} θ (t) W (t) dt, \\ \hat{ψ} = \frac{1}{W} \int_{0}^{T} ψ (t) W (t) dt \end{matrix}$ (11) $W (t) = {\begin{matrix} 0 & if & σ_{1} (t) < c σ_{af} \\ {(\frac{2 σ_{1} (t)}{σ_{af}})}^{- 1 / - m} & if & σ_{1} (t) ⩾ c σ_{af} \end{matrix}$ (12) where σ₁ (t)= maximum principle stress for a given train run.

The orientation of the critical plane is linked with averaged principal directions (see Fig. 12) and can be determined by the following expression,

δ = \frac{3 π}{8} [1 - {(\frac{τ_{af, - 1}}{σ_{af, - 1}})}^{2}]

(13)

Fig.12

Averaged principal stress directions $\hat{1} \hat{2}$ and $\hat{3}$ described through the averaged principle Euler angles $\hat{Φ}$ $\hat{θ}$ and $\hat{ψ}$ .

where δ= off angle (angle between normal to critical plane $\hat{w}$ in the plane $\hat{1} \hat{3}$ and averaged principle stress σ₁ direction) σ_af,-1, τ_af,-1= fatigue strength under fully reversed normal and shear stress respectively.

The mean value and the amplitude of the shear stress acting on the critical plane are calculated by using the minimum circumscribed circle approach proposed by Papadopoulos [18].

The fatigue life estimation is performed using equivalent normal stress (N’_a,eq) and the shear stress (Ca) amplitudes acting on critical plane using material parameters taken from the literature (Table 1).

Table 1

Material parameters of steel S355

Sl.no	Material parameters	Units	Value
1	Ultimate tensile strength of steel, σ_u	MPa	611^*
2	Fatigue strength under fully reversed normal stress, σ_af,-1	–	276.58^*
3	Slope of the S–N curve, m	–	–0.15^*
4	Fatigue strength under fully reversed shear stress, τ_af,-1	MPa	183.7^*
5	Slope m of the S–N curve, m^*	–	–0.09^*
6	Threshold stress intensity, ΔKth	(Mpa√m)	7.5^**

^*Ronchei et al. [19], ^**Vukojević et al. [20].

$\begin{matrix} \sqrt{{(N_{a, eq}^{'})}^{2} + {(\frac{σ_{af, - 1}}{τ_{af, - 1}})}^{2} {(\frac{N_{f}}{N_{0}})}^{2 m} {(\frac{N_{0}}{N_{f}})}^{2 m *} C_{a}^{2}} \\ = σ_{af, - 1} {(\frac{N_{f}}{N_{0}})}^{m} \end{matrix}$ (14) where N_o = reference loading cycles, 2×10⁶, N_f = required fatigue life and N’_a,eq is calculated as below, $N_{a, eq}^{'} = N_{a} + σ_{af, - 1} {(\frac{N_{f}}{N_{0}})}^{m} (\frac{N_{m}}{σ_{u}})$ (15) where N_a = normal stress amplitude.

Based on the experimental evidences it is assumed that the mean shear stress doesn’t affect the fatigue damage.

7 Results

The multiaxial fatigue damage caused by one train run on the critical detail is calculated by applying the rainflow technique (ASTM E1049 – 85 [14]) as is shown in Table 2. Then, based on the heavy mix traffic spectra, total damage is determined. Furthermore, despite the lesser mass, damage caused by train type 11 is compared to others.

Table 2
Multiaxial fatigue damage on critical detail of each train run based on the C-S criterion^*

Train type Damage/train No. of trains/day Damage

Type 5 2.01E-06 6 0.44

Type 6 1.32E-06 13 0.63

Type 11 1.22E-06 16 0.71

Type 12 1.07E-06 16 0.62

Total damage, D 2.41

Train type	Damage/train	No. of trains/day	Damage
Type 5	2.01E-06	6	0.44
Type 6	1.32E-06	13	0.63
Type 11	1.22E-06	16	0.71
Type 12	1.07E-06	16	0.62
Total damage, D			2.41

^*Cross girder flange to main girder flange welded connection (see Fig. 13a).

The fatigue damage based on the three approaches and its demerits are given in Table 3.

Table 3

Fatigue damage using three different approaches

Method	Damage	Demerits
λ - coefficient	1.51	No dynamic analysis. Instead, Static load model (LM 71) is considered
		Neither variable amplitude nor non-proportionality loading is accounted
		No mean stress correction
		No stress gradient effect
Cumulative Damage (Miner rule)	1.96	Dynamic analysis using four standardised freight trains as per EN 1993-2
		Variable amplitude is accounted but not non-proportional loading
		No mean stress correction
		No stress gradient effect
C-S Criterion	2.41	Four standardized heavy freight trains are considered. But, in real there would be different traffic mix.
		Shear amplitude could be accurately determined using other state of the art methods

Fig.13

(a) Critical detail: Cross girder flange to main girder flange welded connection and (b) Alternate detail: Cross girder web to main girder web welded connection.

It is noticed that the fatigue damage of flange to flange welded connection is too high, and therefore, change of connection/geometry is needed. The improvement can be achieved by connecting the cross girder only to web part of the main girder thus avoiding flange to flange connection. The fatigue damage is also calculated for alternate detail (see Fig. 13b) as is shown in Table 4.

Table 4

Multiaxial fatigue damage on alternate detail of each train run based on the C-S criterion^*

Train type	Damage/train	No. of trains/day	Damage
Type 5	7.08E-08	6	0.02
Type 6	2.35E-07	13	0.11
Type 11	2.19E-07	16	0.13
Type 12	1.62E-07	16	0.09
Total Damage, D			0.35

^*Cross girder web to main girder web welded connection (see Fig. 13b).

The fatigue damage on the alternate detail based on the λ – coefficient and cumulative damage (Miner rule) method is 0.11.

8 Conclusions

It is observed that there are differences in the results obtained by different methods in the prediction of fatigue damage. The multiaxial fatigue assessment using the critical plane approach has led to a higher damage, and the SN curve approach is the method which calculates lesser damage in the critical weld. The reason of this outcome might be due to the fact that the conventional SN curve approach for non-proportional variable amplitude railway traffic loading gives single stress cycle for each train pass. This could disregard other damaging cycles resulting non-conservative values, but further investigation would be needed to strengthen this conclusion. In addition to the demerits mentioned in the Table 3, Eurocode recommends a Miner value of 1 which is the highest possible. The slopes of the Wohler’s curves are less steep for higher stress cycles N, as a consequence, lower stress amplitudes are missed out. Furthermore, in the C-S criterion fatigue accumulation due to all small amplitude cycles is accounted. Multiaxial notch fatigue is considered by using the point method.

The conclusion is that, the results from the C-S criterion are consistent and reliable. And the criterion is applicable to the welded connections in the bridges.

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Multiaxial fatigue assessment of welded connections in railway steel bridge under constant and variable amplitude loading

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Stress-based approach

2.3 Fracture mechanics-based approach

2.5 Probabilistic fatigue assessment

3 Description of railway steel bridge

4.1 Fatigue load model for nominal stress method

6.6.1.1 6.1.1 λ – coefficient method

Table 2 Multiaxial fatigue damage on critical detail of each train run based on the C-S criterion* Train type Damage/train No. of trains/day Damage Type 5 2.01E-06 6 0.44 Type 6 1.32E-06 13 0.63 Type 11 1.22E-06 16 0.71 Type 12 1.07E-06 16 0.62 Total damage, D 2.41

References

Table 2
Multiaxial fatigue damage on critical detail of each train run based on the C-S criterion^*

Train type Damage/train No. of trains/day Damage

Type 5 2.01E-06 6 0.44

Type 6 1.32E-06 13 0.63

Type 11 1.22E-06 16 0.71

Type 12 1.07E-06 16 0.62

Total damage, D 2.41