The ultimate limit state is caused by a single one-off event, such as a severe wheel flat or derailment accident, which generates an extremely high load, capable of causing the concrete sleeper to fail. This severe event can cause severe cracking at the railseat or at the midspan, and a failure from this event would fit with the ultimate limit state failure definition.

Fatigue limit state is a time-dependent limit state where a single concrete sleeper accumulates damage progressively over years until it reaches a critical state, where it is considered to have reached failure. Such failures could come from progressive damage (such as abrasion or cracking) from repeated wheel–rail interaction forces over its lifetime.

Serviceability limit state is a limit state where concrete sleepers are beginning to impose some restrictions or tolerances on the operational capacity of the track, such as sleeper deformations, track stiffness changing, and rail displacement. The serviceability limit state failure of a single sleeper is rarely, if ever, a cause of a speed restriction or a line closure. However, when a cluster of sleepers fail, an operational restriction is usually applied until the problem is rectified.

In the past, a lot of countries have investigated concrete sleepers using numerical models, field studies, and experimental test methods. Researchers from Australia, Canada, Japan, and Sweden have also carried out numerical and experimental studies into the responses of prestressed concrete sleepers under impact load, in order to develop the limit states design approach (Wang, 1996; Wakui and Okuda, 1997; Kaewunruen, 2007). Fatigue properties of materials, such as concrete, reinforced steel, and prestressed steel, under fatigue loading are well established (Hanson et al., 1974; CEB-FIP, 1988; American Association of State Highway and Transportation Officials, 1989; CEB-FIP M 90, 1993; EC2, 2005; AS3600, 2009; Japan Society of Civil Engineers, 2010). A lot of fatigue characteristics of railway bridges are also studied (Li et al., 2001; Olsson and Pettersson, 2010). Recently, some fatigue experiment tests of concrete sleepers have been done (Koh et al., 2016; Parvez and Foster, 2017). However, there was relatively little work carried out on fatigue life assessment of prestressed concrete sleepers.

The main objectives of this paper are to review the reporting on the main characteristics of fatigue failure and fatigue load of prestressed concrete and present the fatigue resistance of prestressed concrete. Then, this article will provide a method to evaluate the service performance and predict the fatigue life of the concrete sleeper, which will be useful for the concrete sleeper design and the rail track maintenance in practice.

It is well know that dynamic load is used commonly in the design process of railway tracks and railway applications. The dynamic load considers the dynamic wheel–rail interaction in normal situations; it does not include the extreme impact factor (IF) in railways. Dynamic load is usually related to the train speed and is used for the permissible stress design method. The dynamic wheel load is generally expressed as a function of the static wheel load (Eq. 1) (Doyle, 1980): (1)Pd=∅Ps where P_d is the dynamic wheel load, ∅ is the dynamic wheel load factor (∅ > 1), and P_s is the static wheel load.

Railway engineers around the world have recommended several formulas and parameters for calculating the dynamic wheel load factor. Based on field data, the expression used for the calculation of the dynamic factor ∅ is typically developed empirically and is always expressed in terms of train speed (Doyle, 1980; Van Dyk et al., 2017).

Talbot (1953), Clarke (1957), and the South African Railways (1974) proposed an equation to calculate the dynamic factor that incorporates train speed and the wheel diameter (Doyle, 1980; Hay, 1982). Indian Railways (1974) proposed the dynamic factor by incorporating train speed and track modulus (Srinivasan, 1969). Eisenmann (1972) used a statistical approach to propose a dynamic factor that incorporates train speed and the condition of the track (Esveld, 2001). UIC (1965) developed the most comprehensive dynamic factor, which incorporates the train speed, track geometry, track age, curve radius, superelevation, cant deficiency, vehicle suspension, and center of gravity of the vehicle (Birmann, 1965). The Deutsche Bahn (Germany Railways, 1943) began using an equation (when the train speed is no more than 200 km/h) to calculate the dynamic factor only using train speed (Schramm, 1961). British Railways developed a dynamic factor in the 1970s, which related to a vehicle’s unsprung mass, track stiffness, and train speed (Doyle, 1980). In 1968, the Washington Metropolitan Area Transit Authority prepared an equation where the dynamic factor only depended on the train speed, using subsequent recommended standards for transit track work (Prause et al., 1974). The American Railway Engineering and Maintenance-of-Way Association defines the speed factor for concrete sleeper design in chapter 30. When checking the strength of a railway track in China, the dynamic factor, the speed coefficient, and the wheel load shift coefficient in curves are all taken into account. When calculating the load of sleepers at a railseat, a lateral force coefficient should be also considered (Xiufang Chen, 2017). Table 3 shows a summary of the main recommendations for the dynamic wheel load factor.

With some reasonable assumptions and accordingly simplified parts of the factor, the design dynamic factors, increasing with speed, are displayed in Figure 5 (Doyle, 1980; Van Dyk et al., 2017).

Brandon J. Van and J. Riley Edwards used field test data on Amtrak’s Northeast Corridor to evaluate the dynamic factor (Van Dyk et al., 2017). After removing the abnormal data or some recoded in error, the relationship between the dynamic factors and the train speeds is evaluated as follows (Van Dyk et al., 2017): (2)Φ=1.099+0.00621V.

It is not unusual that railway tracks suffer from impact loads, which are caused by the wheel–rail interactions associated with the abnormalities in the wheels or rails (Kaewunruen and Remennikov, 2006, 2009a,b, 2007) The magnitude of the impact loads on the railseat of concrete sleepers greatly exceeds the static wheel load. The value of the impact load is not only related to the train speed, but is also dependent on causes of such loads (e.g., wheel flats, out-of round wheels, wheel corrugation, short and long wavelength rail corrugation, dipped welds and joints, pitting, and shelling) (Kaewunruen and Remennikov, 2013). It should be noted that impact capacity of concrete sleepers on the railseat can be reduced due to the modification for add-on fixture in practice and surface abrasions (Ngamkhanong et al., 2017a,b,c,d).

Wakui and Okuda (1997) measured the axle-box acceleration of a wheel-set axle-box to study the characteristics of impact loads. They found that the impact load can be simplified as a shock pulse acting after the static wheel load is removed; the duration of impact wheel loading ranges widely from 1 to 10 ms.

Brandon J. Van Dyk and J. Riley Edwards used the wheel impact load detector (WILD) data to evaluate the IF of the North American railway (Van Dyk et al., 2017). The sites of WILD are typically constructed on well-maintained tangent railway track structures with concrete sleepers. The actual wheel loading at Union Pacific Rail Road’s Gothenburg, Nebraska, shows that the IF is related to the axle load of the train. Moreover, part of the impact of the wheel load is very large and even exceeds the static wheel load five times (Table 4).

Queensland University of Technology (2004), Australia also did a comprehensive investigation of actual impact loads. Nearly six million passing wheels on different heavy haul mineral lines were measured in the investigation. The static axle load of the test train ranged from 26 to 28 t, including full-load and no-load conditions. Figure 6 shows the statistical result of field test wheel loads obtained from the rail line in North Queensland of Australia (Leong, 2007).

Based on the return periods and consequences of changing operational characteristics and the statistical data, the impact force factor (k_i) is calculated for the sites as below: (3)ki=0.00278R+0.029V−0.73 where R is the return period (years); V, velocity.

The relationships between the return periods (R) and the impact forces I (kN) could be expressed as: (4)1R=10−0.0191I+5.92.

In current practice, the structural design process for concrete sleepers consists of the following steps (Grassie, 1984):

Analysis of dynamic load coefficient;

Countries or regions with rapid development of railway transportation have promulgated the design method of concrete sleepers for their own railway tracks. American Railway Engineering and Maintenance-of-Way Association (2012), Euro Norm (CEN, 2009), International Union of Railways (UIC, 2004), Australian Standard (AS1085.14, 2003), and Chinese Railway (2015) all provide formal recommendations on the analysis portion of the concrete design process. Table 5 provides a comparison of different analysis methods mentioned above.

Fatigue of concrete is a progressive process of microcrack initiation and propagation toward to the point at which failure occurs. The mechanical properties of concrete will change under repeated cyclic loading, such as permanently increasing strain on the concrete, causing the stiffness to decrease. Cyclic loading may also cause a concentration of stress at the prestressed wires’ surface, which can lead to sudden fracture (Mallet, 1991; Parvez and Foster, 2017).

In the 1970s, a lot of research regarding concrete fatigue analysis was done for the construction of offshore structures in Scandinavia, which suffered severe repeated loads due to the tide Da Mata Bilé and Student¹. The fatigue performance of concrete is commonly represented on graphs, with stress ranges plotted against the failure cycles number (Keynia, 1998). To assess fatigue performance, the Goodman diagram has been developed (Figure 7) (Keynia, 1998; Sadeghi and Fathali, 2007). fc′ is the allowable compressive stress of concrete, N_f is the number of allowable load cycles, σ_t is the maximum stress caused by a dead load, and σ_h is the maximum stress caused by both live and dead loads.

The fatigue reference compressive strength f_ck,fat of concrete (fib Model Code 2010) is calculated based on the characteristic compressive strength f_ck as follows: (5)fck,fat=βcc(t)βc,sust,t0fck1−fck400 (6)βcc(t)=exps ⌈1−(28t)0.5⌉.

In Eq. 5 and βc,sus(t,t0) is a coefficient which depends on the time under high sustained loads t − t₀ (days). The expression of β_cc(t) from Eq. 6 describes the strength of the concrete, which is developing with time, in which s depends on the cement strength class and t equals to the age of concrete age (in days) (Lantsoght et al., 2016).

The typical S–N curves of concrete under pure compression are shown in Figure 8. N is the number of cycles causing fatigue failure in plain concrete (7)Sc,max=σc,max∕fck,fat, (8)Sc,min=σc,min∕fck,fat.

In the fib Model Code 2010, the characteristic fatigue strength function for prestressed steel consists of two different slope segments (see Figure 9). The parameters of the prestressed steel’s S–N curve are shown in Table 6. ΔσN∗ is the stress range obtained from a characteristic fatigue strength function.

Extending Miner’s rule, the damage accumulation method is used in many design codes (BS5400, 1980; EC2, 2005). In this method, the cumulative damage accumulation under variable amplitude loading for a particular sample of stress history is calculated as an equivalent fatigue stress with constant amplitude (Rao and Talukdar, 2003) (9)DI=∑kbj=1 ΔDj=∑kbj=1 njNj.

In the above equation D_I is the cumulative damage index, D_j is the incremental damage, n_j is the number of cycles at stress range Sr_j, N_j is the number of cycles at constant stress range Sr_j to cause fatigue failure, k_b is the stress range number blocks in the histogram over which summation is carried out. The process of evaluating the cumulative damage index has been repeated for all the simulated response samples and the average value is used to find the fatigue life L_f using: (10)Lf=1DI.

The Damage Equivalent Stress Method, presented in Annex NN of EN 1992-2:2005, is a simplified method compared to the Cumulative Damage Method. This method is used to evaluate the fatigue safety of structures, based on an equivalent fatigue stress with constant amplitude. This assessment method is applicable to reinforced and prestressed steel for road and railway bridges.

The obtained stresses are multiplied by damage equivalent factors, which are dependent on different infected factors. The expression of damage equivalent factor is given as follows: (11)λs=φfat⋅λs,1⋅λs,2⋅λs,3⋅λs,4.

In the above equation, φ_fat is the damage equivalent IF, which is dependent on the surface roughness. λ_s,1 is a factor related to damaging effect of traffic and dependent on the critical length of the influence line or area. λ_s,2 is a factor that is controlled by the traffic volume. λ_s,3 is a factor that is dependent on the design life of the bridge. λ_s,4 is a factor used for bridges that are loaded by more than one lane.

All the methods above are commonly used for the fatigue life assessment of bridges. Because of the periodic impact load, the fatigue assessment process for concrete sleepers is different from bridges. At present, there is not a suitable fatigue assessment method for concrete sleepers.

Based on field tests and laboratory test results, Wakui and Okuda (1997) proposed a method for evaluating the fatigue life of a prestressed concrete sleeper, according to a prestressed steel rupture, due to impact load, by using the following procedure:

Construct a cumulative frequency distribution curve of wheel loading.

Kaewunruen and Remennikov (2009a,b) did a lot of research regarding the response of prestressed concrete sleepers under impact loads. By using the concept of damage accumulation, they presented a method for predicting the residual life of concrete sleepers under repeated impact loads, using the relationships between the cumulative damage of sleepers and given impact loads.

Based on laboratory tests, the relationships between the cumulative damage index and the impact loads in different track environments are shown in Figure 10. The damage index in the figure is the ratio between the maximum crack length c_max caused by the bending moment and the total depth (d) of the sleeper. This ratio is used to quantify and classify the damage states of concrete sleepers subjected to impact loads. Based on the test results, the damage index can also be used to predict the residual life of concrete sleepers.

The tested concrete sleepers comply with the requirements of AS 1085.14. Each sleeper consisted of 18 prestressed wires with 26.4 kN prestressed force per wire, loss of prestress is calculated as 12%. The geometry details are given in Table 7, and the railseat section can be seen in Figure 11.

The fatigue test almost followed the set up that was specified in AS 1085.14 (Figure 12). In order to make sure the sleeper failed below three million cycles, the upper limit of fatigue load was increased to 240 kN (36% higher than specified standard value). The fatigue test load protocol can be seen in Figure 13.

The properties of materials can be seen in Table 8.