1. Introduction
Structures must provide strength, stability, and stiffness to buildings and at the same time be efficient, ensuring that they can withstand extreme events. Connections are fundamental elements of steel structures because they are responsible for transferring loads between different steel members in the structure. Furthermore, connections are important because they can have a significant impact on the overall cost and construction of the steel structure. The design of the connections must be optimized to minimize the amount of steel used, while ensuring that the connections are strong and reliable. This requires a careful analysis and design to ensure that the connection is the most efficient and economical solution.
The bolted connections are based on classic solutions of the traditional ribbed construction. Particularly, bolted lateral connections attach steel profiles with plates arranged as joint covers. Their use is especially indicated in those cases where the frontal solutions are not usable due to geometrical constraints, or because they do not allow some of the degrees of freedom of movement required for the joint [
1]. On the one hand, bolted lateral connections avoid tensile stresses perpendicular to the rolling plane of the plate, an indispensable condition for joining the earlier steels, which were very susceptible to sheet defects. On the other hand, this solution is typically selected for long-span beams that would require special transportation, which would increase its final cost significantly. Therefore, a joint is introduced in the beam that works well under shear forces, although it is true that this type is not as effective against bending moments. Thus, when selecting the joint location, it must be placed at approximately one fifth of the span, where the bending moment envelope is closest to zero. Possible joint failures include breakage of the bolt and breakage of the sheet metal, either by tearing or buckling. Given the location of the joint in the structure, in a section subjected to a certain shear loading and with low bending moment loading, failure of the steel profile would occur in the web.
Moreover, circumstances may arise on construction sites that may constrain the availability of materials or represent an opportunity to reduce costs, e.g., in relation to the length of the steel profiles, thicknesses of the cover plate, etc. If changes are made to the original design, they must be code-compliant and ensure safety.
2. Background
Limit states are those situations for which, if exceeded, the building may be considered as not meeting any of the structural requirements for which it was designed. In the case of the ultimate limit states, they are those that in the case of being exceeded, constitute a risk for people, because they produce the rupture of some structural elements, and with it the total or partial collapse of the building [
2].
The finite element method (FEM) is a widespread numerical method for solving structural analysis models, dividing the system into smaller, simpler elements, and modeling each element with equations that describe its mechanical behavior. The calibration process involves comparing the output of the simulation with experimental data obtained from physical testing, and adjusting the model parameters until a satisfactory match is achieved. Current software suites provide calibrated FEM models that reproduce structural performance of steel connections as well as the interaction between elements in a realistic manner [
3,
4].
Research has recently been conducted on evaluating the performance of bolted connection steel plates applying the FEM and the discontinuity layout optimization (DLO) [
5]. The interactions between all the structural parts involved in a bolted connection are easier to model when using the FEM. In addition, the FEM provides information on displacements and stresses distribution prior to collapse. In turn, two commonly used possibilities to analyze the structural behavior of steel joints are solid and shell FEMs. For example, solid FEMs require more computational effort while providing more accurate modeling of bending moment resistance in steel joints [
6]. Thanks to the use of the FEM, it is possible to optimize structural design as well as to develop reliable structural solutions for frames and connections between members [
7]. Different types of joints have been tested, particularly T-joints and plate connections have been extensively studied using FEMs [
3,
4,
8]. Shell elements have been used to simulate laminates in 2D FEM single-lap joints, improving the simulation of the stiffness of bolts and laminates as well as the model’s ability to simulate the secondary bending [
9].
The FEM provides valuable information about the interaction between the elements that make up the joint. A literature review shows that traditional design methods, which assume a rectangular stress block in plates, may inaccurately represent stress distribution under base plates, especially with thicker plates, leading to potentially non-conservative designs [
10]. An adequate arrangement of the bolts improves stress distribution in connection plates [
11]. Thin plates can benefit from the clamping force generated during the rivet installation process, which creates a localized compressive stress that helps to prevent the initiation and propagation of cracks. However, this positive effect diminishes as the thickness of the material increases [
12].
Modeling of composite bolted joints recently incorporated friction, bolt-hole clearance, bearing damage, and joint failure [
13]. Composite bolted joints can be modeled through a collection of components replicating the elastic properties of the bolt as demonstrated by Belardi et al. [
14]. A beam model simulated the structural behavior of the bolt shaft while a nonlinear spring was employed to represent the frictional force between the plates, accounting for bolt-hole clearance, allowing the calculation of the displacement caused by the contact between the bolt shaft and the plate holes. The deformation of the plate holes was modeled by considering a beam resting on an elastic foundation. Additionally, a second spring component was utilized to account for the rotational stiffness linking the bolt head and shaft, as well as the elastic support provided by the plates to the bolt head.
The regulatory framework for steel connections in Europe is defined by a structural standard for steel and composite connections, known as Eurocodes, prescriptions that have been met for the definition of the case studies developed in this study.
3. Materials and Methods
The joint case proposed in this study connects two IPE240 identical profiles of S275 steel by means of two plates on both sides of the web and twelve bolts (
Figure 1). The plate dimensions are conditioned by the geometry of the IPE240 profile. As shown in
Figure 1a, the meeting of the flange and the web in the profile includes a circular root, so that the available web height to accommodate the plate is 190 mm. To mount the bolts inside the plate spaced at similar distances horizontally and vertically, a length of 320 mm is selected. Furthermore, these dimensions allow us to modify the position and arrangement of the bolts, whose effect is to be studied. In addition, the parameters associated with strength, dimensions, and bolt tightening are factors that are maintained for all case studies. Particularly, prestressed connecting elements included high-strength bolts. According to the standard EN ISO 4014/4022 [
15], which specifies the characteristics of hexagon head bolts, an M12 C8.8 bolt was selected. Its yield strength is 640 N/mm
2 and its tensile strength is 800 N/mm
2. As contact surfaces do not need to be prepared in a special manner, the joint studied is grade A.
The failure modes of this connection include tearing resistance for bolt groups, local bending, crushing, and block tearing.
IDEA StatiCa v.20 software was used to compute the resistance and the stress state of the joint elements. The program works with the component-based finite element method (CBFEM), which is the combination of two methods: the FEM and the component method (CM) [
16]. The CM is implemented in the current Eurocodes [
17,
18,
19] and applied in the majority of software for structural steel used in Europe. The component model of connections begins and stems from the decomposition of a joint to components. The constitutive equations of the material model the deformation behavior in relation to the normal and shear forces. The connection components are grouped to examine joint moment–rotational behavior and classification representation in a spring-shear model. Interactions between components incorporate boundary conditions to simulate the influence between the behavior of the connection elements for their consideration in the aggregate analysis of the connection structural performance [
20]. The IDEA StatiCa software calculation procedure is supported by an extensive experimental validation campaign [
16].
The CM is based on standard procedures that evaluate the internal forces in the geometric layout of the connection and their verification, involving the prediction of resistance, stiffness, and deformation capacity. It permits predicting the 3D performance of steel joints under arbitrary loading despite the complex phenomena (nonlinearities, residual stresses, geometrical configurations, etc.) affecting the behavior of structural steel [
21]. The weakness of the standard CM lies in the analysis of internal forces and stress in a joint.
To overcome this limitation, the CBFEM method replaces the specific analysis of the internal forces in the joint with the general FEM. Check that methods of specific components like bolts, which are modeled as nonlinear springs, are performed according to the standard CM (Eurocode). In turn, special FEM components permit modeling the bolts behavior in a joint. All parts of one-dimensional members and all additional plates are modelled as plates. As elements are made of steel and their mechanical behavior is significantly nonlinear, the real stress–strain diagram of steel is replaced by the ideal plastic material for design purposes in building practice. The advantage of ideal plastic material is that only the yield strength and modulus of elasticity must be known to describe the material curve. As these elements are considered as an ideal elastic plastic material, their internal forces can be retrieved for evaluation. Although the granted ductility of construction steel is 15%, the real usable value of limit plastic strain is 5% for an ordinary design [
22]. The stress in steel cannot exceed the yield strength when using the ideal elastic–plastic stress–strain diagram.
The CBFEM method aims to model the real state precisely. Meshes of plates from different components are not merged, and no intersections are generated between them. Instead, a mesh of finite elements is generated on each individual plate independently on mesh of other plates. Between the meshes, special massless force interpolation constraints are added. They ensure the connection between the edge of one plate and the surface or edge of the other plate. End forces on members are applied as loads on segment ends, with internal forces from theoretical nodes transferred to segment ends, maintaining force values while adjusting moments due to force actions on corresponding arms, without connecting inner ends of segments [
16]. The model incorporates shear stress transmission by crushing and tensile–shear interaction. This unique calculation model provides excellent results, both from the point of view of precision and a speed analysis.
Figure 2 illustrates the joint model as displayed by IDEA StatiCa.
Circular holes are arranged both in the IPE240 profile and in the plates. As stipulated by Eurocode 3 Part 1–8 [
17] for M12 and M14 bolts, the standard diameter of the holes equals the diameter of the screw shank plus 1 mm (
Figure 3). Spacing between bolts and between bolts and the plate border is regulated by Eurocode 3 Part 1–8.
The case studies assume a small bending moment value in relation to the shear stress. This fact determines the placement of joint covers only in the web zone of the structural section. A total of 80 case studies were analyzed with the same static loading hypothesis, including the weighting coefficients of actions, resulting in a bending moment stress of 10 kN-m and a shear stress of 75 kN. These values constitute the boundary conditions, with no additional constraints considered. As a result, the model is in equilibrium after computation if the internal forces are added at the ends of related members.
Table 1 contains model and mesh CBFEM parameters as well as calculation parameters as introduced in IDEA StatiCa settings. A multiple of the height of the section is used to determine the default length of a standard element. The ratio of the plate edge length to the number of edge elements determines the number of elements in the largest flange or web. Each calculation was conducted with 25 iterations with up to three divergence steps for its evaluation. The finite element mesh is composed of linear elements. To select the size of the finite elements, a pilot sensitivity analysis was conducted. Element sizes of 2 to 8 mm, 5 to 20 mm, and 10 to 40 mm were set to analyze the effect of the mesh size on the target parameters of this study. It was concluded that reducing the mesh size beyond the latter values did not provide significant additional precision while the computational time was greatly increased. As a result, the maximum element size was set at 40 mm and the minimum element size was set at 10 mm since it provided a sufficient fineness to obtain precise enough results without delaying the computational calculation, while the mesh layout provided sufficient adaptation to the shape of the connection components.
The results of joint designs have been further checked and validated using the spreadsheets for lateral bolted joints prepared by Ortiz et al. [
1] These are based on the abovementioned EuroCode 3 specifications.
5. Conclusions
As fundamental parts of steel structures, connections are responsible for transferring loads between different steel components and have a considerable impact on the overall cost and construction of the steel structure.
This study highlighted the possibility of optimizing steel connection components within the requirements of codes under the restriction of available materials or designs. Different combinations of plate thickness, bolt spacing, and geometric arrangements were tested to observe their effects. The CB FEM was employed to assess the stress state of the junction plate, plastic deformation, and bolt shear. This is a powerful and precise numerical method used to simulate the behavior of structures under various conditions.
The nonlinear behavior of the elements that make up the connection is evident from the results. First, the evaluation of the stress distribution within the junction plate helped in identifying areas of high stress, which varied with the joint design layout with no predictable pattern other than by the finite element analysis. Second, the plastic deformation values showed that the connection components deform plastically under the applied loads increasing more than proportionally for extreme values of the design parameters. Similarly, the examination of the shear forces experienced by the bolts in the lateral connections encountered a disproportionate rise for design parameters at the extreme values.
The results showed that placing the bolts close to the edge of the plate affects the bolts more than the stress distribution of the plate itself when plate thicknesses are in the lower end of the range. In addition, it is not advisable to introduce an offset between the central bolt row and the outer ones since it worsens the stress distribution and the structural behavior of the components. In summary, this text highlights the intricate nature of connection behavior in structural systems, emphasizing the need for comprehensive analyses, careful design consideration, and ongoing optimization efforts to ensure the reliability and efficiency of engineered structures within the constraints of available materials. These effects should be borne in mind by practitioners to optimize the joint design. The findings of this research can significantly contribute to the design and construction of efficient and safe steel bolted joints for building structures.