1 Introduction

1.1 Background

Steel beams with web openings are modern steel structural elements compared to solid web beams. Introducing a web opening in a steel beam has various benefits, such as economic construction, since these beams allow for ease of service (cable trays and pipelines) through the web openings without increasing the beam’s height. Additionally, unlike solid web beams, they have an appealing architectural appearance [1]. From the structural viewpoint, the web opening reduces the cross-sectional area of the beam, thereby reducing the load carrying capacity of steel beams significantly. However, web openings minimize/eliminate the dead weight of steel beams reducing/eliminating useless web regions. In the early twentieth century, Redwood [2] investigated steel beams with circular web openings subjected to static load. In 1969, he deduced a design method that used an equivalent rectangular opening with adjusted proportions. After that, studies on the behavior of steel beams with a web opening were conducted, including Darwin and Lucas [3] in 1990 and Redwood and Cho [4] in 1993. They proposed a method for designing stiffened and unstiffened steel composite girders with a web opening. Hence, Chung et al. [5] explained the Vierendeel mechanism in 2001, which is considered one of the failure mechanisms for beams with web openings. This was accomplished via finite element analysis and analytical solutions. Many scholars [6,7,8,9,10] have conducted investigations using finite element theory and have performed experimental tests [11,12,13,14,15,16,17,18,19,20]. While experimenting with many shapes and configurations of web openings, they observed that four plastic zones were formed in the vicinity of the web opening due to stress concentrations generated by the Vierendeel mechanism. Recently several studies have examined the innovative use of non-standard shapes of openings [21], multiple openings [22] and loading cases [23]. Ferreira et al. [21] studied the influence of the web-post geometric parameters on shear buckling resistance of perforated steel beams with novel non-standard elliptically based web openings. To overcome transportation problem of deep section, Liu et al. [22] presented a numerical study for the bolted castellated steel beams with multiple octagonal web openings. The effect of the hogging moment on the behavior of steel–concrete composite cellular beams was presented numerically by Oliveira et al. [23].

Fatigue is considered the weakening of steel materials, and it causes the failure of steel structures. As noted by Dexter et al. [24], the investigations done by Fisher et al. in the seventies of the twentieth century laid the groundwork for developing different specifications for the design of steel bridge girders against fatigue failures, such as AASHTO [25]. Kyung et al. [26] studied the causes of fatigue cracks in steel plate girders, and consequently devised and validated a method for repairing the induced fatigue cracks. In addition, Ryu et al. [27] evaluated experimentally and numerically full scale two-span pre-stressed composite plate girders subjected to static and fatigue loads.

Crane girder fatigue life determination using SN and LEFM methods was introduced by ´Avila et al. [28]. It was concluded that the S–N method associated with FE analysis has proven to be an efficient technique for determining the most critical structure joints damage. Recently, Fatigue failure analysis of steel crane beams with variable-section supports was studied by Zhao et al. [29]. They presented the main reasons for frequent fatigue failures of steel crane beams with variable-section supports.

Since fatigue failure is defined as the propagation of cracks caused by cyclic load, steel structural elements subjected to cyclic loading were primarily studied to enable the design of these elements against seismic loading caused by earthquakes [30]. As a result, many researchers performed extensive studies on steel beam-to-column connections with solid beam web (no web openings) [31,32,33,34]. Calado [31] proposed a nonlinear analytical model for predicting the behavior of these connections under cyclic loading. Mashaly et al. [32], however, introduced a 3D finite element model to simulate the performance of these joints. Additionally, Ma et al. [33] and Kim et al. [34] experimentally studied the seismic performance of T-shaped steel connections. The experimental tests showed that, plastic hinges appeared near the column face and away from the remaining beam span.

Meanwhile, steel moment resisting connections with circular web openings in beams subjected to seismic loading were investigated experimentally and numerically [35, 36]. It was deduced that, reducing the beam section resulted in shifting the location of the plastic hinge from the column face to a more desirable location in the vicinity of the web opening. Consequently, a strong column weak beam mechanism was developed, which considerably enhanced the seismic behavior of the entire connection. Other researchers [37,38,39] continued investigating steel beam-to-column connections under seismic loading numerically and experimentally while changing the opening shape (circular, elliptical based and elongated), opening size, and column face distance and introducing closely spaced web openings. They concluded that introducing a web opening can result in increasing the ductility and dissipating seismic energy of these connections.

Furthermore, Yang et al. [40] investigated steel moment frames under cyclic loading, focusing on beam-to-column connections with circular web openings in the beam. Experimental tests and numerical models were developed to study the behavior of these connections. They realized that, introducing a web opening could change the stress distribution in the beam. As a result, it can protect the connection from brittle weld failure and increase its ductility. However, Shin et al. [41] analyzed steel frames with openings in the beam’s web that were subjected to lateral cyclic loads to understand the failure mechanisms in beams using finite element modeling in addition to experimental tests. During this research innovative non-standard shapes of openings and multiple openings were used. They concluded that changing the configuration of web openings can alter the failure mechanism. It became obvious from previous studies that researchers focused their efforts mostly on seismic loads. On the contrary, they did not give much attention to fatigue failure of steel beams.

1.2 Research Significance and Motivations

According to the previously mentioned studies, it became clear that, although the previous studies focused mostly on the structural performance of steel beams with web opening subjected to static and seismic loading, the effect of fatigue behavior was overlooked. Thus, the primary objective of this research is to study the behavior of steel beams having one web opening subjected to fatigue loading using nonlinear finite element analysis to predict the normal stress fatigue life caused by the repeated loading. Hence, synthetic SN curves for the studied beams were introduced, which are important for evaluating stress estimation in terms of fatigue damage. It is worth mentioning that application of fracture mechanics to calculate the cumulative fatigue damage or the propagation of fatigue crack length isn’t the main interest of this research.

1.3 Aims and Methodology

The study aims to: (1) Explore the structural behavior of steel beams with a single web opening subjected to fatigue load; (2) Suggest suitable fatigue categories that comply with steel beams with web opening; (3) and present the synthetic SN curves for the studied beams. Then, it can be added to the current design recommendations for steel structure sectors. The previously mentioned aims can be achieved according to the following methodology: (1) Constructing a FE model to simulate the behavior of steel beams with web opening subjected to fatigue loading. (2) Calibrating and validating the proposed FE model against previous work containing static and cyclic loading experimental and FE results, and (3) Using the proposed FE model to introduce a full parametric study to investigate the effect of geometric parameters such as opening size and opening location on the fatigue stress life of steel beams with a web opening.

2 Finite Element Model and Validation

2.1 Case of Study

The considered case of study is a simply supported single span steel beam with a web opening simulating a beam installed in a multi-story garage. The movement of cars may cause multiple loading cycles for all the structural elements, including the beam. This cyclic loading eventually leads to high cyclic fatigue loading that will result in fatigue failure of the steel beam. The existence of a web opening changes the stress distribution in the steel beam with a web opening when compared to a beam without an opening. Because the web opening reduces the section’s shear and bending moment capacity, it may induce fatigue failures due to cyclic loading. This case study was investigated through a finite element FE model analysis.

To understand the effects of web opening on stress distribution as well as the effect of fatigue loading on steel beams with web opening, FE models from previous work [5, 37] were studied and remodeled using ANSYS 17.1 [42] finite element software program. Then, the FE models were verified according to the work done by previous researchers, as well as the validation of the normal category of solid web plate girders adopted by the Eurocode 3 part 1–9 [43]. Several levels of validation are conducted, as discussed next.

2.2 Validation of the FE Model to Simulate Beams with Web Opening

Chung et al. in 2001 [5] performed an analytical and numerical investigation of steel beams with web opening subjected to static loading. They validated their FE model against test data present in the literature. Chung et al. described the modeling process and results for two test specimens referred to as 2A and 3A. Dimensions of tested beams (2A and 3A) are shown in Fig. 1.

Fig. 1
figure 1

Details and dimensions of static loading FE model [5]

Full size image

Former researchers have used solid elements to simulate beams [30] with web opening and obtained results that were similar to the results achieved when shell elements were used [8]. In previous, solid element 186 was used to simulate steel beams [44], frames, [45] and beam-to-column joints [46, 47]. A combination of solid elements 186 and 187 with three degrees of freedom per node was used to model the beams in ANSYS 17.1. These elements are suitable for plasticity, large deflections, and large strains and they both have quadratic displacement behavior. The quadrilateral brick solid element 186 has 20 nodes, whereas the tetrahedral solid element 187 has 10 nodes, making it suitable for irregular meshes due to web opening.

The mesh, loads, and boundary conditions for the models are shown in Fig. 2. To save computation time and ensure acceptable accuracy, guide solution runs were performed, and a mesh size of 20 mm was chosen for the elements as shown in Fig. 2. A mesh refinement was applied on the inner surface of the opening to help in increasing the accuracy of the simulation. Nonlinearity in the material was accounted for using a bilinear stress–strain curve with a linear strain hardening according to appendix C.6 in Eurocode 3 part 1–5 [48]. Homogenous isotropic steel with a modulus of elasticity of 200,000 MPa, a tangent modulus of elasticity of 2000 MPa and a Poisson’s ratio of 0.3 was used in modeling. Yield strength of flanges and webs on both tested beams are shown in Table 1. Large displacement was allowed, and nonlinear model was solved by the Newton–Raphson method.

Fig. 2
figure 2

Static loading finite element model (a: Meshing shape, b: Loading and boundary conditions

Full size image
Table 1 Calibration against test data studied by Chung et al. in 2001 [5]
Full size table

To simulate the boundary conditions, all nodes of the bottom flange supported by the roller were restricted to only moving in the direction of the beam span (Ux and Uy are prevented, Uz is permitted). Movement in all directions was restricted at nodes of the bottom flange that were supported by hinged support (Ux, Uy, Uz are prevented). The mid-span point was restricted from moving in the lateral direction (Ux is prevented) to the beam span to prevent lateral torsional buckling failure during loading. Details of boundary conditions are shown in Fig. 2.

Load was applied as a concentrated force at points shown in Fig. 2. Geometric nonlinearity was considered for an accurate FE analysis by applying initial geometric imperfections. These imperfections, such as web bowing commonly occur in steel beams as a result of production and transportation processes. Eigen value linear buckling analysis was used in ANSYS [42] to specify the possible buckling modes of the beam; then, the deformations and stresses were superimposed on the FE model before the actual load was applied. According to Eurocode 3 [48], the first eigen buckling mode was used with a scale factor of H/200, where H is the total girder depth. The final shape of loads and boundary conditions are presented in Fig. 2.

A comparison between the moment-deflection curves for the proposed model and the results obtained by Chung et al. [5] for beams 2A and 3A is shown in Fig. 3. The presented FE model accurately predicts the beams’ moment-deflection curves. In addition, the finite element model accurately represented failure mode of the beams by Vierendeel mechanism as discussed by Chung, et al. in 2001 [5] and a corresponding failure moment in the center of the opening as shown in Fig. 4. The obtained results from the finite element models were precise when compared to the previous results introduced by Chung, et al. in 2001 [5] using by the experimental and numerical results.

Fig. 3
figure 3

Comparison between moment-deflection curves (a: beam 2A, b: beam 3A)

Full size image
Fig. 4
figure 4

Comparison between Von-Mises stress distribution for beam (2A) (a: FE model made by Chung et al. [5], b: suggested FE model)

Full size image

2.3 Validation of FE Model to Simulate Perforated Beams under Cyclic Loads

In 2014, Tsavdaridis et al. [37] studied steel beams with web opening subjected to cyclic loading. Cantilever beam-to-column connections with various configurations of web openings were modeled with different dimensions, sections, and opening shapes. The connection A1-300 was remodeled herein to verify the simulation of a perforated beam with a circular opening subjected to cyclic loads. The connection A1-300 had a circular opening with a diameter of 115 mm and a column face distance of 300 mm as shown in Fig. 5. As suggested before a combination of solid elements (186 and 187) was used to model the specimens. Steel was modeled as a homogenous isotropic von- Mises material with a bilinear stress–strain curve with kinematic hardening according to appendix C.6 in Eurocode 3 part 1–5 [48] to account for material nonlinearity. All steel parts had a modulus of elasticity of 207,000 MPa, a tangent modulus of 1000 MPa and a Poisson’s ratio of 0.3, with yield and ultimate stresses of 305 MPa and 510 MPa, respectively, being used for all steel parts. The final shape for the model is shown in Fig. 6.

Fig. 5
figure 5

Dimensions, sections and opening shapes of modeled specimen A1-300 [37]

Full size image
Fig. 6
figure 6

Simulation of the cyclic loading specimen A1-300 of [37] using ANSYS (a: F.E. mesh, b: boundary conditions)

Full size image

Achieving acceptable precision and minimizing solving time were the governing factors for choosing the element mesh size of the model. So, a mesh size of 20 mm was chosen for the cantilever beam, whereas a mesh size of 50 mm was chosen to simulate the column based on guide solution runs. In addition, mesh refinement was applied around the opening to enhance the accuracy of the results. Final mesh shape is demonstrated in Fig. 6a. Convergent solutions were ensured by allowing the finite element software to automatically adjust the solution increment size and large displacement was allowed. The load was applied as a cyclic displacement at the center line of the cantilever at the tip according to the SAC loading protocol [49] for 32 loading cycles with increasing tip displacements.

Regarding the boundary conditions, the upper and lower ends of the vertical column were considered fixed, so they were prevented from moving in any direction (Ux, Uy and Uz are prevented), while lateral support was provided at the free end of the cantilever beam (only Ux is prevented), which was allowed to move freely in other directions. Geometric nonlinearity was considered by including an initial imperfection of the cantilever beam. An Eigen value buckling analysis was employed to study the different modes of buckling, and the deformations and stresses were applied to the FE model before applying the tip displacements. An initial imperfection was taken as tw/200 with the first eigen buckling mode shape according to Tsavdaridis et al. [37]. The details of the boundary conditions are shown in Fig. 6b.

The beam A1-300 was remodeled using the same dimensions and material properties as the original model and the results were compared. It was found that the created finite element models accurately represented the beam model (A1-300) in failure mode by the Vierendeel mechanism as shown in Fig. 7. The obtained results from the created finite element model were precise when compared to the previous results introduced by [37] as shown in Table 2.

Fig. 7
figure 7

Comparison between stress distribution and failure mechanism at cycle 27 (a: suggested FE model, b: previous work by Tsavdaridis et al. [37])

Full size image
Table 2 Comparison between the F.E model studied by Tsavdaridis et al. [37] and the presented model
Full size table

2.4 Validation of FE Model to Simulate Unperforated Beams According to Eurocode 3 [43] SN Curves

A 6000 mm span, hot-rolled steel compact section (HEB400) simply supported beam was used to simulate real-life structures such as stringer bridges and floor beams in multi-story garages, with hinge and roller supports on the left and right sides, respectively. It must be noted that validation of the unperforated beams is required before simulating the perforated beam with the same cross section, loads and boundary conditions.

As indicated before in Sect. 2.2 about the boundary conditions of simply supported beams, nodes of the bottom flange supported by the roller were restricted from moving vertically and horizontally perpendicular to the beam span (Ux and Uy are prevented) but allowed to move in the direction of the span (Uz is permitted), whereas all nodes supported by hinged support were restricted from moving vertically, in the direction of beam the span and also perpendicular to the beam span (Ux, Uy and Uz are prevented). Lateral bracing was provided at the upper flange at the mid-span point to ensure an unsupported length of 3000 mm. The lateral support was prohibited from moving perpendicular to the beam span (Ux = 0) to avoid lateral torsional buckling. The beam was examined and proven safe for lateral torsional buckling. Stiffeners were used to strengthen the support areas (hinge and roller) against the stress concentration caused by the reaction of the support, which causes unwanted damage to the lower flange and web of the beam. All dimensions and boundary conditions of the beam are shown in Fig. 8.

Fig. 8
figure 8

Beam dimensions, meshing and boundary conditions of the current finite element model

Full size image

The fatigue loading was applied to the top flange of the beam as a fully reversed, downward uniformly distributed load. The uniform load that induced first yield in the beam’s section was determined (\({W}_{y}\)) and then a small amount of it was applied to the beam (\(0.1 {W}_{y}\)), and the model was run with the applied fatigue load amplitude ranging from (0.1 \({W}_{y}\)) to (-0.1 \({W}_{y}\)). After that, the applied load was gradually increased, and the fatigue life of the simulated beam was recorded for each load step.

The beam section (flanges and web) and the stiffeners were modeled in steel S235 with an elasticity modulus of \(E=210000 N/{mm}^{2}\), a Poisson’s ratio \(\nu =0.3\), and a yield strength of 240 MPa. According to appendix C.6 in Eurocode 3 Part 1–5 [48], the bilinear stress- strain curve has a tangent modulus of elasticity of \({E}_{t}=2100 N/{mm}^{2}\).

It should be note that, fatigue stress life was calculated based on the fatigue tool which is a built-in option available in ANSYS 17.1. The fatigue tool used SN curve with the option of zero mean stress and fully reversed load to calculate fatigue stress life of investigated beams. Hence the normal stress fatigue category 160 [43] was used to define the failure criteria in the ANSYS model. The fatigue category 160 was drawn to number of cycles 1*108 to allow the designer to use any type of fatigue loading (constant or variable amplitude) without any problems.

As discussed before, solid elements 187 and 186 were used to create the current FE model. According to previous studies on steel beams with web openings, the optimal element mesh size ranged between 5 and 40 mm [9, 13, 15, 49]. A mesh convergence study was performed herein using guide runs to ensure that the optimum element size produced accurate results while consuming minimum computational time. The unperforated beam was remodeled while changing the element size (from 10 to 60 mm) and maintaining the same uniform distributed load (\(0.5 {W}_{y}\)). A 20 mm element size was used to complete the finite element analysis because it guaranteed a reasonable solving time and a satisfying accuracy of about 97.3% between the numerical stress result (\({\sigma }_{Num}\)) produced by the FE models and the theoretical stress result (\({\sigma }_{th}\)) as presented in Fig. 9.

Fig. 9
figure 9

Mesh convergence analysis for current FE model

Full size image

It is worth mentioning that, using tetrahedral elements (shown in Fig. 2) fits very well with arbitrary shaped geometries, like circular opening, with their simple computations. Although Hexahedra meshes are economic in the CPU time (or processing time), the behavior of tetrahedral elements was enhanced through a convergence study presented in Fig. 9.

The material fatigue properties were determined according to Eurocode 3 part 1–9 [43] using a fatigue category of 160 MPa for normal stress as shown in Fig. 10. This value for the fatigue category was selected because it belongs to the hot-rolled sections.

Fig. 10
figure 10

SN curve of the current FE model compared to [43]

Full size image

No initial imperfection was applied to the finite element model because the unperforated model’s normal stress fatigue life values at various stress ranges were identical to the Eurocode 3’s [43] normal stress fatigue category 160, which included initial imperfection. The comparison between normal stress SN curves of Eurocode 3 and the corresponding numerical normal stress fatigue life values of the unperforated model is shown in Fig. 10. For every cross mark shown in Fig. 10, the fully reversed uniform load was applied to the beam. Then, the fatigue stress life was calculated based on the fatigue tool which is a built-in option available in ANSYS. The fatigue tool calculates the number of loading cycles to failure corresponding to the given load. The cross marks were plotted using the normal stress ranges (equals double the stress amplitude) on the vertical axis versus the number of corresponding loading cycles to failure on the horizontal axis.

Fatigue strength values of the control beam (no opening) defined in ANSYS can represent both constant and variable amplitude loading according to Eurocode 3 part 1–9 [43]. If the designer wants to design for constant amplitude loading, then the fatigue curve is assumed horizontal after 2*106 loading cycles which is the number of loading cycles corresponding to normal stress fatigue category. If the designer wants to design for variable amplitude loading, then the fatigue curve is extended to 1*108 loading cycles then the curve is assumed to be horizontal as stated by Eurocode 3 part 1–9 [43].

2.5 Results

From previous sub-sections, it can be seen that the presented FE model accurately simulates beams with web opening and perforated beams under cyclic loads, as well as the ability of the model to predict the fatigue life of the unperforated beam with high accuracy. In general, the presented model is used with confidence in the next sections.

3 Parametric Study

3.1 Scope

Steel beams with a single web opening are examined in this section while subjected to high cyclic fatigue loading. The fatigue behavior of steel beams with different web opening configurations was investigated. The results were compared to each other and to a control beam that had a solid web without web opening. A circular web opening was chosen because it is easier to fabricate. Additionally, circular web openings reduce stress concentrations that arise from web openings with sharp edges, such as rectangular, octagonal and hexagonal openings [8, 15]. For all web opening models, the edge was assumed to be ground to remove the roughness and become smooth.

3.2 Parametric Variables

The parametric study of the FE models involved the effects of two parameters that greatly affect the fatigue behavior of steel beams with web opening. These two parameters were as follows:

  1. 1.

    Opening size parameter (α) represents the ratio between the web opening diameter (\({D}_{h}\)) and web depth (\({d}_{w}\)). Such that \(\alpha = {D}_{h}/ {d}_{w}\).

  2. 2.

    Opening location parameter (β) denotes the ratio between the location of the web opening along the beam span (\(A\)) and the span’s full length (\(L\)). Thus, \(\beta = A/L\)

where;

\(A\) is the distance between the left side of the steel beam and center of web opening, tf denotes flange thickness, H is the total depth of section, Dh is the diameter of web opening, dw is the depth of web, and L denotes the full span length of the beam, as shown in Fig. 11.

Fig. 11
figure 11

Parameters of the current FE model

Full size image

Four values of the opening size parameter were selected (0.25, 0.5, 0.75, 1.0), whereas seven different values of the opening location parameter (0.05, 0.1, 0.2, 0.25, 0.3, 0.4, 0.5) were taken into consideration in the parametric study. The previously mentioned parameters \(\left(\alpha \mathrm{and} \beta \right)\) were investigated while the steel section and span length of the beam were kept constant during the investigation. A total of twenty-nine (29) FE models were constructed including 28 models with different web opening configurations and one solid web beam (control beam) with no web opening, as shown in Table 3.

Table 3 Key parameters of the studied beams
Full size table

3.3 Finite Element Results

The results were presented in two phases. Phase 1 investigated the effect of each parameter on the fatigue behavior of steel beams with web opening. Phase 2 suggested fatigue categories for the beams under consideration numerically.

3.3.1 Effect of Opening Size (\(\boldsymbol{\alpha }\))

This subsection numerically studies the effect of the opening size (\(\alpha \)) on the fatigue life of a rolled steel beam with a web opening at different opening locations. The studied simply supported beams have the same section (HEB 400), the same span length (6000 mm) and a single circular web opening symmetric about the major axis. The fatigue strength for each beam was recorded \(({\Delta \sigma }_{c})\) as presented in Table 3, whereas the fatigue strength is defined as the stress range at which the number of loading cycles equals 2 × 106 as stated by Eurocode part 1–9 [43]. The results showed that increasing the web opening size parameter (\(\alpha \)) was inversely proportional to the fatigue strength of steel beams while the opening location (\(\beta \)) was kept constant. The SN curves that showed the effects of web opening size are shown in Fig. 12. For example, the ratio between the normal stress fatigue category of the FE model and the normal stress fatigue category of the control beam \(\left(\frac{{\Delta \sigma }_{c}}{{\Delta \sigma }_{c0}}\right)\) dropped from 87.50% to 26.88% when the opening size (\(\alpha \)) was increased from 0.25 to 1.0 while the opening location (\(\beta \)) stayed constant at 0.05. The effect of changing opening sizes was similar if the opening location (\(\beta \)) was kept constant at 0.1, 0.2, 0.25, 0.3, 0.4 or 0.5, as it resulted in reducing the normal stress fatigue category ratio as demonstrated in Fig. 13. The previously mentioned figure showed that reducing the size of web opening less than 50% of the web depth (\(\alpha \le \) 0.5) did not have a noticeable impact on plummeting the fatigue category of the studied specimen if the opening location missed the high shear region (\(\beta \ge \) 0.25), as the minimum fatigue category ratio of 98.13% belonged to the specimen of (\(\alpha \)) equals to 0.5 and (\(\beta \)) equals to 0.25.

Fig. 12
figure 12

SN curves showing the effect of web opening size parameter (a: \(\beta \)=0.05, b: \(\beta \)=0.1)

Full size image
Fig. 13
figure 13

Effect of opening size with different opening locations on normal stress fatigue category ratio

Full size image

3.3.2 Effect of Opening Location (\({\varvec{\upbeta}}\))

The effect of the web opening location parameter (\(\beta \)) was investigated using different web opening locations along the beam span (from 0.05L to 0.5L), yet the opening size parameter (\(\alpha \)) remained unchanged. Increasing the value of the opening location parameter (\(\beta \)) was directly proportional to the fatigue strength of steel beams with a single circular web opening, as it ensured a significant improvement in the normal stress fatigue life of the studied FE models. For example, in the case of raising the opening location parameter (\(\beta \)) from 0.05 to 0.5, the fatigue strength ratio \(\left(\frac{{\Delta \sigma }_{c}}{{\Delta \sigma }_{c0}}\right)\) developed from 26.88 to 76.25%, although the opening size parameter (\(\alpha \)) remained constant at 1.0. SN curves for studied FE models that represent the effect of web opening location parameter (\(\beta \)) are shown in Fig. 14.

Fig. 14
figure 14

SN curves showing the effect of web opening location parameter (a: \(\alpha \)=1.0, b: \(\alpha \) =0.75)

Full size image

It was obvious that, when the opening location was close to support (\(\beta \) = 0.05 and \(\beta \) = 0.1), the reduction in normal stress fatigue life became more severe, as shown in Fig. 15. This was due to the Vierendeel mechanism that was more critical near concentrated forces such as the vertical reaction of support. Vierendeel’s collapse was the cause of the failure when the opening was close to support. However, global bending moment induced failure when the opening was at mid span (\(\beta \) =0.5). This was valid for any opening size. The previous observation was made in several prior studies [5, 13, 50] and was supported in this study by the FE results.

Fig. 15
figure 15

Normal stress fatigue category ratios of different finite element models with various opening sizes showing the effect of changing the web opening location

Full size image

3.4 Suggested Fatigue Categories

A common question engineers confront when designing simply supported beams with a single circular web opening under fatigue load is which fatigue categories to use. In this section, fatigue strength categories were suggested to design steel beams with different web opening configurations based on the fatigue categories provided by the Eurocode 3 part 1–9 [43]. The studied FE models were divided into five groups as demonstrated in Tables 4 and 5. A suitable fatigue category was selected for each group of investigated FE models. The chosen fatigue categories were proven safe for their selected groups as illustrated in Fig. 16.

Table 4 Suggested normal stress fatigue categories
Full size table
Table 5 Suggested normal stress fatigue categories
Full size table
Fig. 16
figure 16

Suggested normal stress fatigue categories of different finite element models (a: group D, b: group E)

Full size image

All the next categories were suggested under the next assumptions:

  • The beam section is rolled with a short span.

  • The web opening is symmetric about the major axis.

  • The edge of the opening is improved by grinding to be smooth and all roughness is removed.

  • The stress–strain curve of the studied FE model adapted from Eurocode 3 part 1–9 [43] includes effects of geometrical and structural imperfections from material production and execution.

4 Synthetic SN Curves

In this section, a systematic method for constructing SN curves in the absence of test data was described. The synthetic SN curves can predict the fatigue behavior of steel beams with a single web opening of varying sizes and locations along the span of the beam and symmetric about the beam’s major axis. First, a simplified method for estimating the governing SN curve for steel structural elements is discussed based on the technical report presented by Pedersen [51], Second, synthetic SN curves for beams with the same web opening configurations as those investigated numerically in a previous section (Sect. 3.3) were determined by the general method. Then, the synthetic curve results were compared to finite element results for all studied models. It should be noted that, this analytical study has the same assumptions listed in Sect. 3.4.

4.1 Synthetic Curve Prediction Method for Beams with Web Opening

A simplified procedure for obtaining the SN curves, that can predict the fatigue behavior of any steel component, was performed by Pedersen [51] using prior research in the literature. In order to forecast the fatigue behavior SN curve for a particular steel component, the SN curves for the base steel material should be predicted first as stated by M. Pedersen [51]. Hence, the material fatigue strength of the base material \({\sigma }_{F-mat}\) should be calculated as:

$${\sigma }_{F-mat}={\alpha }_{o}\times {F}_{U}+{\beta }_{o}$$
(1)

where;

\({F}_{U}\) is the ultimate tensile strength of steel material, \({\alpha }_{o}\) and \({\beta }_{o}\) are constants for rolled steel sections as indicated in Table 6.

Table 6 Key synthetic curve factors for tested beams
Full size table

To predict synthetic SN curves for rolled steel beams, a number of corrections should be made to the synthetic SN curve of the base steel material. \({\sigma }_{F-mat}\) can be corrected to the component fatigue strength \({\sigma }_{F-comp}\).. The steel component herein represents the rolled steel beams. \({\sigma }_{F-mat}\) is calibrated considering several factors that affect the fatigue life of steel structural elements, such as size effect of steel parts, surface treatment, surface roughness, environmental effects, probability of survival and most importantly the fatigue notch effect that arises in steel beams due to any geometric change [51]. If the values for these factors are obtained, \({\sigma }_{F-comp}\) can be easily estimated as presented in Eq. (2).

$${\sigma }_{F-comp}=\frac{{K}_{reliab}\times {K}_{mean}{\times K}_{size}\times {K}_{env}\times {K}_{treat}\times {\sigma }_{F-mat}}{\sqrt{{{K}_{f}}^{2}-1+\frac{1}{{{K}_{surf}}^{2}}}}$$
(2)

where;

\({K}_{reliab}\) represents the reliability factor at 99.9% survival probability, \({K}_{mean}\) accounts for the effect of mean fatigue stress and equals 1 for cases of fully reversed loads with zero mean fatigue stress, \({K}_{size}\) stands for reduction in beam fatigue strength due to large steel thickness and equals 1 in case of steel thickness less than 25 mm, \({K}_{env}\) symbolizes the reduction in beam fatigue strength due to temperatures change or corrosion, \({K}_{treat}\) represents the increase in fatigue strength due to good surface treatment, \({K}_{surf}\) is a reduction factor based on material surface roughness and equals 0.7412 for rolled steel sections and the fatigue notch factor \({K}_{f}\) that accounts for the decrease in beam fatigue strength as a result of stress concentrations that arise in the steel beam due to changes in geometry [51].

Based on the finite element analysis of various models, it was shown that stress concentration occurs in our case due to the presence of a web opening, which is induced by the Vierendeel mechanism. Values for the previously mentioned factors are shown in Table 6.

A synthetic SN curve for any steel beam consists of two straight lines with different slopes. (\(\frac{1}{{S}_{1}}\) and\(\frac{1}{{S}_{2}}\)). They intersect at a point defined as the knee point as demonstrated in Fig. 17. \(\frac{1}{{S}_{1}}\), \(\frac{1}{{S}_{2}}\) and the number of loading cycles which causes fatigue crack initiation at the knee point (\({N}_{D}\)) can be estimated using the following equations [51].

Fig. 17
figure 17

Standard synthetic curve for base steel material and steel beam [51]

Full size image
$${S}_{1}=\frac{{\alpha }_{k}}{{{K}_{f}}^{2}-1+\frac{1}{{{K}_{surf}}^{2}}}+{\beta }_{k}$$
(3)
$${S}_{2}=2\times {S}_{1}-1$$
(4)
$${N}_{D}={10}^{{\alpha }_{N}-\frac{{\beta }_{N}}{{S}_{1}}}$$
(5)

where;

\({\alpha }_{k}\), \({\beta }_{k}\), \({\alpha }_{N}\) and \({\beta }_{N}\) are constants for steel beams with web opening. The values of \({\alpha }_{N}\) and \({\beta }_{N}\) are 6.4 and 2.5, respectively, from [51]. \({\alpha }_{k}\) and \({\beta }_{k}\) were adjusted based on the finite element results and are specified in Table 6.

It should be noted that the existence of the web opening forms four points of stress concentration (plastic hinges) in the area of the web opening due to Vierendeel mechanism [5, 50]. As a result, (\({K}_{f}\)) for the control beam with no opening is equal to 1. However, (\({K}_{f}\)) for any web opening configuration must be more than or equal to 1.

It is worth noting that, all of the factors shown in Table 6 have constant values for steel beams with a single web opening that is symmetrical about the major axis, except for \({K}_{f}\). \({K}_{f}\) has a different value for each web opening configuration as it depends on stress concentration around the web opening area. Thus, an analytical approach was used to calculate \({K}_{f}\) based on previous studies [5, 50], while involving the geometric parameters of the web opening (\(\alpha \) and \(\beta \)) into the proposed approach.

The procedure introduced herein depends on calculation of normal stresses induced in the critical T-section near the web opening \({\sigma }_{cr, T}\) on the low moment side of the opening. For simplicity, this critical T-section was assumed to be inclined by \({25}^{o}\) to the vertical axis of the web opening as stated by [5, 50] and is shown in Fig. 18. Calculating \({\sigma }_{cr, T}\) enables easy estimation of fatigue notch factor (\({K}_{f}\)) using Eq. 6.

Fig. 18
figure 18

Critical section of circular web opening

Full size image

It is worth mentioning that, if a large deformation or a fatigue crack occurred at the web opening surface, the value of \({K}_{surf}\) and \({K}_{f}\) would change significantly. \({K}_{surf}\) would be lower and \({K}_{surf}\) would be higher. This change will result in a lower value of synthetic curve slopes (\(\frac{1}{{S}_{1}}\) and \(\frac{1}{{S}_{2}}\)).

$${K}_{f}=\frac{{\sigma }_{cr, T}}{{\sigma }_{n}}$$
(6)
$${\sigma }_{cr, T}=\left(\frac{{N}_{T}}{{A}_{T }}\right)+\left(\frac{{M}_{X, T}}{{I}_{X, T }}\right){Y}_{max T}$$
(7)
$${\sigma }_{n}=\frac{{W}_{y}{L}^{2}}{8{S}_{X}}$$
(8)

where \({\sigma }_{n}\) is the maximum normal stress induced at the mid-span section of the control beam calculated in Eq. 8 based on the properties of area of the steel section. Where \({S}_{X}\) is the elastic modulus, A is section area and \({A}_{w}\) is the web area (\({d}_{w}\times {t}_{w}\)). \({Y}_{max, T}\) is the distance of the furthest point on the inclined steel section away from the inclined T section centroid. While properties of area of the inclined section \({A}_{T}\) and \({I}_{X,T}\) denote the area and moment of inertia of the inclined T-section, respectively, determined using equations 9 and 10. \({N}_{T}\) and \({M}_{X, T}\) are the normal stress and Vierendeel bending moment acting on the inclined T-section, respectively, and can be calculated through equations 11 and 12.

$${A}_{T }=0.5 (\frac{A}{\mathrm{cos}25}-\alpha {A}_{w})$$
(9)
$${I}_{X, T}=\frac{{b}_{f}\times {({t}_{f}/\mathrm{cos}25)}^{3}}{12}+{b}_{f}\times \frac{{t}_{f}}{\mathrm{cos}25}{\left({\overline{Y} }_{T}-0.55{t}_{f}\right)}^{2}+\frac{{t}_{w}\times {\left(H-2{t}_{f}-0.9\times \alpha \times {d}_{w}\right)}^{3}}{71.46}+{t}_{w}\times 0.55\left(H-2{t}_{f}-0.9\times \alpha \times {d}_{w}\right)\times {\left({\overline{Y} }_{T}-1.1{t}_{f}-0.275[H-2{t}_{f}-0.9\times \alpha \times {d}_{w}]\right)}^{2}$$
(10)

where \({\overline{Y} }_{T}\) is the distance between the top edge and the centroid of the inclined T-section.

$${N}_{T}=\frac{0.45{W}_{y}{L}^{2}\beta (1-\beta )}{{d}^{*}}+0.21{W}_{y}L(0.5-\beta )$$
(11)
$${M}_{X, T}=0.233{V}_{C}\left(\frac{H}{2}-0.9{\overline{Y} }_{T }\right)-{N}_{C}\left(0.9{\overline{Y} }_{T }-\overline{Y }\right)$$
(12)

where \({d}^{*}\) represents the distance between centroids of the vertical upper and lower T-sections at the center of web opening, \({V}_{C}, {N}_{C}\) and \({M}_{C}\) are the total global shear force, normal force and bending moment at the center of web opening, respectively.

$${V}_{C}={W}_{y}L\left(0.5-\beta \right)$$
(13)
$${N}_{C}=\frac{{M}_{C}}{{d}^{*}}$$
(14)
$${M}_{C}=\frac{{W}_{y}{L}^{2}}{2}\beta \left(1-\beta \right)$$
(15)

It is worth noting that when Eq. 6 was applied to all steel beams included in the parametric study, the value of \({K}_{f}\) was found to be less than 1 for some of the tested beams. This means that the stress induced at the inclined critical T-section (\(\theta ={25}^{o}\)) was less than the maximum stress at the steel section in the beam’s mid-span and, consequently, that the web opening had no effect on the normal stress fatigue life of the beam. As a result, the value of \({K}_{f}\) should be set to 1 for these beams. Values of fatigue notch factor (\({K}_{f}\)) obtained from Eq. 6 are demonstrated in Table 6.

4.2 Comparison between Synthetic Curves and Finite Element Results

In this section, synthetic SN curves for the tested beams with different web opening configurations were obtained from the simplified procedure presented by [51] and compared to SN curves obtained from a parametric study based on the finite element results for the same web opening configurations for tested beams. A comparison between the synthetic SN and finite element SN curves for some of the tested beams is shown in Fig. 19. Synthetic SN curves that clarify the effects of the opening size parameter (\(\alpha \)) on fatigue behavior are shown in Fig. 20 with a constant value of the opening location parameter (\(\beta =0.05\)) and several values for the opening size parameter (\(\alpha =0.25 to 1.0\)). However, the effect of the web opening location parameter (\(\beta \)) with a constant value of the opening size parameter (\(\alpha =1.0\)) and several values for the opening location parameter (\(\beta =0.05 to 0.5\)) is shown in Fig. 21.

Fig. 19
figure 19

Comparison between some synthetic SN curves and the corresponding finite element SN curves

Full size image
Fig. 20
figure 20

Synthetic SN curves showing the effect of web opening size parameter (\(\alpha \)) with constant opening location (\(\beta =0.05)\)

Full size image

Although the synthetic SN curves provided a conservative estimation of the fatigue behavior of all the tested beams with various web opening configurations, the predicted fatigue behavior had an acceptable degree of accuracy, particularly for cycle counts less than \(2*{10}^{6}\). This particular number of fatigue loading cycles (\(2*{10}^{6}\)) is of major importance as it is defined to be the number of cycles that corresponds to the normal stress fatigue category (\({\Delta \sigma }_{c}\)) as stated by Eurocode 3 part 1–9 [43]. With fatigue loading cycles of more than \(2*{10}^{6}\), the results obtained by synthetic curves become highly conservative when compared to numerical results.

It should be noted that there is a slight difference between the synthetic SN curves and finite element SN curves (Fig. 19). The authors believe that this difference is due to some reasons such as: (1) the synthetic SN was performed using a simplified procedure for obtaining the SN curves for the fatigue behavior of any steel component by predicting the SN curves for the base steel material first. Then, a number of corrections were made. (2) The analytical method to obtain the value of \({K}_{f}\) assumes that the critical section is inclined by \({25}^{o}\) to the vertical axis of the web opening which is not the case for the different values of web opening sizes and locations. While there are many factors affecting the synthetic SN curve, the fatigue notch factor (\({K}_{f}\)), it is considered to be the most effective parameter as it affects the slopes (\(\frac{1}{{S}_{1}}\) and \(\frac{1}{{S}_{2}}\)) of the curve.

It is worth mentioning that, the synthetic SN curves were extended to 1*108 loading cycles. Then, the synthetic SN curve is assumed to be horizontal for larger number of loading cycles. If the steel beam is designed for constant amplitude fatigue loading, the synthetic SN curve is assumed horizontal after 2*106 cycles. But, if the beam is designed for variable amplitude loading, the synthetic SN curve is extended to 1*108 cycles then it is assumed horizontal. This was the same case with the fatigue curve defined in ansys model (see Sect. 2.3). Thus, the synthetic SN curve is valid for both constant amplitude and variable amplitude loading.

4.3 Application of the Proposed Analytical Method

As an application to the previously mentioned analytical method some other steel beams with rolled sections other than HEB400 were investigated numerically and analytically to ensure the accuracy of the proposed method under the same fatigue loading conditions. Four steel beams with different rolled sections, spans, and web opening configurations were examined. Details of the investigated beams are shown in Table 7, while the straining actions acting on the critical inclined T-section (\({N}_{T}\) and \({M}_{X, T}\)) were calculated at vertical load of 0.2 \({W}_{y}\) for each beam.

Table 7 Details of the examined application models
Full size table

The proposed method was applied to the tested beams by calculating the properties of the critical inclined T-section. The critical normal stress around the web opening \({\sigma }_{cr, T}\) and the maximum normal stress induced at mid-span \({\sigma }_{n}\) were estimated according to eqs. 7 and 8, respectively. Fatigue notch factor \({K}_{f}\) can be calculated from Eq. 6. After determination of fatigue notch factor, the same procedure can be applied to the examined beams to determine the properties of the synthetic SN curves of the beams, such as fatigue strength \({\sigma }_{F-comp}\), the two slopes (\(\frac{1}{{S}_{1}}\) and \(\frac{1}{{S}_{2}}\)), and the number of loading cycles that cause fatigue crack initiation at the knee point \({N}_{D}\). Synthetic SN curves for the tested application beams along with the corresponding SN curves obtained from FE results are shown in Fig. 22.

Fig. 21
figure 21

Synthetic SN curves showing the effect of web opening location parameter () with constant opening size (=10)

Full size image
Fig. 22
figure 22

FE SN curves for application models along with the corresponding synthetic SN curves using the proposed analytical approach

Full size image

It was noticeable that, the value of the fatigue notch factor \({K}_{f}\) of the first application model (Model 1 of IPE 500) exceeded 1. This means that the maximum normal stress induced in this model is in the vicinity of the web opening unlike the case of the other application model (2, 3 and 4), which had a theoretical fatigue notch factor \({K}_{f}\) of value less than 1 which indicated that the maximum induced normal stress was at mid-span of the beam as verified using finite element results and is demonstrated in Fig. 23.

Fig. 23
figure 23

Normal stress distribution for the application models showing the location of maximum stress

Full size image

5 Summary and Conclusions

In this study, the numerical behavior of steel beams with a single web opening under fatigue loading was investigated. The effect of the opening size and location on the fatigue life of steel beams with a single web opening was examined. Normal stress fatigue categories available in Eurocode 3 part 1–9 [43] were proposed to design steel beams with a web opening against fatigue failures. Synthetic SN curves for steel beams with a single web opening were determined based on a simplified approach introduced by earlier researchers [51].

The following conclusions were drawn for short span steel beams with a single web opening symmetric about the major axis of the beam’s section:

  1. 1.

    Web opening size parameter (\(\alpha \)) was inversely proportional to the normal stress fatigue life of steel beams. On the contrary, the web opening location parameter (\(\beta \)) was directly proportional to the normal stress fatigue life.

  2. 2.

    The opening location parameter (\(\beta \)) has been shown to be more critical than the opening size parameter (\(\alpha \)) in terms of decreasing the normal stress fatigue life.

  3. 3.

    It was found that an opening with a size parameter of 0–0.5 and a location parameter of 0.25–0.5 had almost no effect on the normal stress fatigue life when compared to steel beam with a solid web (no opening).

  4. 4.

    It became clear that, regardless of the size of the web opening, the decrease in normal stress fatigue life caused by the Vierendeel mechanism when the opening was close to the support was greater than the decrease in normal stress fatigue life caused by the global bending moment when the web opening was near mid-span of the beam.

  5. 5.

    The synthetic SN curves obtained using the simplified procedure were proven to be safe for their corresponding finite element models for all the investigated web opening configurations.

  6. 6.

    Synthetic SN curves produced conservative results when the number of loading cycles exceeded the number of loading cycles corresponding to the normal stress fatigue category (\(2*{10}^{6}\) \(\mathrm{cycles}\)). Nonetheless, for loading cycles less than (\(2*{10}^{6}\)), the synthetic curve result values were accurate and had an acceptable accuracy.

As a result, the simplified analytical approach can be used to design steel beams with a single web opening that are subjected to fatigue loading in a conservative manner.