Analysis of Hollow Section “Y” Connections with the Application of Non-Linear Material Modeling

Abstract

The use of welded structural hollow sections in civil engineering is relatively new. Design of connections in steel structures with welded structural hollow sections requires a more specified approach, compared to traditional connections achieved with gusset plates. Stress and local deformations at the connection area are non-linear and very complex. This paper presents the application of non-linear numerical analysis for the determination of the ultimate bearing capacity of “Y” connections. In the analysis, a non-linear material model was applied. In order to validate the applied method of steel modelling, comparative analysis of the analytically and numerically determined ultimate bearing capacity with the experimentally obtained results has been conducted.

1. Introduction

Constructing and dimensioning connections in steel trusses with welded structural hollow sections require a more specified approach compared to traditional connections achieved by gusset plates. This approach, unlike in the cases of typical steel connections, takes into consideration local deformations as a limiting condition which is the dominant design criterion in most cases in reality. Stresses and local deformations at the connection nodes are non-linear and very complex, as typical in welded joints [1,2], which makes the analysis of the behaviour of these connections under the ultimate limit state and the serviceability limit state with only analytical postulates impossible. Because of that, during the last 40 years a large number of both experimental and numerical studies have been conducted in order to determine theoretical basics and calculation models for determining the bearing capacity of these connections [3,4,5,6,7,8].

However, the current European standard for design of joints in steel structures, EN 1993-1-8 [9], shows some limitations in the field of analysis of connections between hollow section profiles. The proposed equations are limited to the specific geometrical configurations and element size ratios. Moreover, the application of the proposed design equations excludes the application of class 3 and 4 hollow section profiles. In order to overcome this problem, advanced numerical methods can be applied together with extensive experimental research.

Being a powerful tool for computational analysis of hollow section connections under different loading conditions, the application of the finite element method (FEM) reduced the necessity of development of specific analytical solutions for different types of connections [10,11,12]. In early stages of the FEM application, joint members have been modelled by shell elements, since shell models are simple for modelling and do not require large computational efforts [13,14]. With the increase of computing speed and capabilities, the application of solid elements for FEM analysis of hollow section joints became more popular [15,16]. The advantage of the application of solid elements is mainly connected to the simpler and more accurate modelling of welds. Simplification of FEM modelling is of great importance for engineering practice since it results in reduction of computational time and cost. Hence, in recent years much research has concentrated on simplifying the modelling procedure. Some new approaches were proposed, such as component-based finite element modelling [17].

Once verified by experimental results, numerical models can be used for parametric analysis as well as for deriving analytical equations for geometrical solutions that are not included in the available standards but can be used for increasing bearing capacity of structural elements or connections. In [18,19], the authors have numerically analysed the ultimate strength, initial stiffness, and failure mechanisms of tubular T/Y-joints reinforced with doubler plates under axially compressive load. The authors found that the ultimate strength of a doubler-plate-reinforced T/Y-joint can be up to 295% of the strength of the corresponding unreinforced joint. Furthermore, the doubler-plate-reinforcement method can considerably increase initial stiffness and improve the failure modes of tubular T/Y-joints subjected to axially compressive load. Based on the parametric investigation, the authors derived a new formula for design of such connections.

Another very important aspect of FEM modelling is the selection of adequate material parameters. In order to speed up the calculation, often a simplified rather than exact stress–strain relationship is applied. Usually, a bilinear model with isotropic hardening is applied for defining the stress–strain relationship in FEM analysis. Some limitations of this type of material model are presented in [20].

The results that can be obtained based on the limit states theory show that the behaviour of these connections at the state of fracture depend not only on geometrical characteristics of the connected element and base material quality, but also on configuration—the shape of the connection and type and level of the load in a chord.

This paper presents the application of non-linear numerical analysis of “Y” connections, in comparison with the results of experimental tests under quasi-static loading [21], considering material non-linearity. The so-called Total Lagrange’s formulation [22] with the use of a mixed incrementally iterative method, where total load is divided into few steps (increments) and in each step the iteration is conducted in order to balance residual load, is used as basic formulation for solving non-linear problems within numerical methods [23].

Comparative analysis of results of the analytical solution and obtained numerical and experimental results has been conducted in order to verify the given method of modelling under the limit states of these connections [21].

Numerical non-linear analysis has been conducted using software package ANSYS R14.5 [24].

2. Experimental Research

Experimental research was carried out on “Y” connections made of square hollow sections produced from cold-formed S235JR steel profiles. The physical model represents the connection between the truss chord and diagonal brace element. The angle between the brace and the chord was 45°. The total length of the chord was 400 mm and the length of the brace at the crown toe side was 200 mm. The two elements were welded using a fillet weld of 3 mm thickness.

Three samples were tested quasi-statically. The static load of the connection was applied with the help of a hydraulic piston on a hydraulic press with a load capacity of 100 tons. The test load was applied within the prescribed limits, according to EN ISO 6892-1 [25].

The test was aiming to determine the ultimate loading capacity of the joint according to the deformation limit criterion of reaching the local deformation of 3% of the chord width or height. The experimental setup is shown in Figure 1b. The chord was positioned inclined at an angle of 45 degrees from the horizontal plane, putting the brace in the vertical position. This allowed the vertical load to be applied to the brace element. In order to avoid any load eccentricity, the load was applied through the sphere joint element placed between the hydraulic press and the brace. The load intensity was measured using a 50 kN load cell, placed between the hydraulic press and the sphere joint (Figure 1). Simultaneously, the local deformations (displacements) were measured in three spots: at the crown toe side weld (in middle of the chord top flange), and on the two side surfaces of the chord (10 mm from the chord top surface). The displacements were measured using a HBM linear variable displacement transducer with the measuring range of ±2 mm. Thus, the largest value of the displacement was limited to 4 mm. The data recording was performed using data acquisition system SPIDER8. The sampling frequency was 10 Hz.

2.1. Material Parameters of Joined Elements

The researched “Y” connections (Figure 1) were produced from the cold-formed square hollow-section steel profiles, with determined mechanical- and partially determined chemical characteristics (

Table 1. Chemical structure of chord material (in %).

Batch	C 10−2	Si 10−3	Mn 10−2	P 10−3	S 10−3	N 10−3	Cr 10−2	Ni 10−2
4657908	5	10	33	10	11	-	-	-

and

Table 2. Mechanical characteristics of chord material.

ReH [MPa]	RM [MPa]	A [%]
282	381	43

). The connection was made according to corresponding EN provisions for bearing steel constructions [26,27].

The chord element was produced from the RHS 80 × 80 × 3, CFR(C) HS Longitudinally black steel welded profiles made of S235JR steel, with the chemical structure and mechanical characteristics shown in Table 1 and Table 2 respectively.

Brace elements were made of RHS 50 × 50 × 4, CFR(C) HS Longitudinally black steel welded pipes, made of S235JR steel with the chemical structure and mechanical characteristics shown in

Table 3. Chemical structure of brace material (in %).

Batch	C 10−2	Si 10−3	Mn 10−2	P 10−3	S 10−3	N 10−3	Cr 10−2	Ni 10−2
36503/1	7	1	38	22	21	6	3	4

and

Table 4. Mechanical characteristics of chord material.

ReH [MPa]	RM [MPa]	A [%]
255	355	34

respectively.

3. FEM Modelling

Numerical analysis was performed in the FEM-based commercial software ANSYS R14.5.

The joint was modelled as an assembly of two solid bodies. The first body included the diagonal brace together with the fillet weld. The second body was the chord element. In order to simplify the model, the weld was modelled only at the straight parts of the brace. The boundary conditions were carefully chosen to adequately represent the experimental conditions and are given in

Table 5. Boundary conditions.

Notation Figure 2	Type of Element	Fixed Displacements
A	Surface	Z = 0
B	Line	X = Y = Z = 0
C	Surface	Y = 0
D	Line (both sides of the chord)	X = 0
E	Node (middle of the top surface of the solid plate)	Load as nodal displacement Y = Z = 6 mm

and Figure 2 and Figure 3. The surfaces that were in contact with the fixed elements of the experimental rig were set to have zero displacement in the direction perpendicular to the contact surface. The contact between the brace/fillet weld and the chord, was created by defining a bonded contact pair between surface body faces.

The mesh was generated using the ANSYS Mechanical built-in automatic mesh generator. The element size was controlled using the body sizing command. In the high-stress zones, at the contact between brace and chord, automatic mesh refinement was performed on the surface level. It was set to one half of the body element size. In such a way, it was made easier to perform the mesh convergence study only by changing the element size on a body level. The results were calculated for the element size equal to 15, 10, and 5 mm. The meshing was performed using SOLID187 elements, a higher-order 3-D, 10-node element with a quadratic displacement behaviour, which is suited to modelling irregular meshes. The results of the mesh convergence studyfor the total strain development with the load increase, indicated that the calculations performed on the two finer-meshed models gave no significant difference. Thus, both meshes could be considered as adequate.

Basic input data for the material, taking geometrical and material non-linearity into consideration within the numerical analysis, are applied as follows:

The yield strength and ultimate strength of the steel material are modelled as given in Section 2.1, i.e., for the chord element, the applied yield strength was f_y = 282 MPa and the ultimate strength was f_u = 381 MPa; for the brace element, the applied yield strength was f_y = 255 MPa and the ultimate strength was f_u = 355 MPa:

• modulus of elasticity E = 2.1·10⁵ MPa;
• tangent modulus of elasticity was calculated according to [22], and equal to

  $$E_T=\frac{1}{12.5}·E=0.08·2.1·{10}^5=0.168·{10}^5\mathrm{MPa};$$
• The applied Poisson’s ratio value was 0.3;
• Classic bilinear kinematic hardening, BKIN, was applied for modelling steel nonlinearity according to recommendations given in [23]. This model is suitable for the application in the case of small strains and materials that follow the von Mises yield criteria. This option assumes that total stress range is equal to double the yield stress, which leads to including Bauschinger’s effect, that makes this model relevant for analysis of the presented type of the connection;
• Applied yield criteria: von Mises/Hill.

4. Analysis of the Results

Comparative analysis of the analytical solution and experimental results with results achieved by applying non-linear numeric analysis for the researched type “Y” connection has been conducted through the qualitative analysis of stress fields and both qualitative and quantitative analysis of local deformations at the connection area. The intensity of the ultimate bearing capacity of the connection was determined by:

• EC 3 [9], applying nominal yield strength;
• EC 3 [9], applying experimentally determined yield strength;
• Analytical model, considering the effect of welds;
• Experimentally determined load bearing capacity of connections.

In the preliminary research, during the mesh convergence analysis, it was observed that the von Mises stresses in the stress concentration regions significantly exceeded the ultimate strength of the material. The reason for this lies in the FEM computational method. Namely, the extrapolation of the solution results from the integration points to the nodes is accurate only in the elastic regions. When the plasticity is involved, this could lead to inaccurate results, since the linear extrapolation does not follow the material behaviour within one element. Errors could be reduced by decreasing the element size, i.e., reducing the distance between integration points and nodes. The extrapolation could be also controlled using the ERESX command [24], applied through the ANSYS APDL-snippet. In this way, it could be specified whether the results are as would be extrapolated or copied from integration points to nodes. In the von Mises stress distribution with allowed extrapolation, at the stage at which a large part of the chord’s top side was in the plastic region, an unrealistic increase of the stresses, far beyond the ultimate stress, could be clearly seen at the transition from the elastic to the plastic region. Reduction of the element size could result in obtaining more realistic stresses, since it decreases the difference between the extrapolated value and the value at the integration point. However, in order to “catch” the spreading of the yielded zone, a very fine mesh in a relatively large zone is required. Another solution is to disable the extrapolation or enable it partially. In the first case, the values are copied from the integration points to nodes. The second case involves extrapolation only in elements which have not yielded. However, in this case, the occurrence of the stresses above the ultimate strength is observed as well. The final calculations, as well as the mesh convergence study, were performed with extrapolation totally disabled.

The analysis results or characteristic output data are presented graphically using a contour plot of stress field and deformation. “MX” and “MN” labels represent maximum and minimal values of corresponding output data, respectively.

In Figure 3, Figure 4, Figure 5 and Figure 6, local deformations of “Y” connection (y-direction components) for different load values are given. The applied load level was equal to the bearing capacities of the researched connections that are determined using the approaches listed above. Comparative displays of ultimate bearing capacities of connections and maximum local deformations that are obtained by non-linear analysis are given in

Table 6. Results of ultimate bearing capacities and local deformations.

Ultimate Bearing Capacity Determining Procedure	Ultimate Bearing Capacity of Connection [daN]	Max Local Deformation NNA [mm]	Max Local Deformation Experimental [21] [mm]
EC 3 with nominal yield strain	3532.1	0.326	0.22
EC 3 with experimentally determined yield strain	4037.0	0.393	0.28
Analytical calculation model considering the welds effect	6952.0	1.217	0.72
Experimentally determined value	8881.0	2.470	2.40

.

In Figure 3 and Figure 4, it can be seen that the maximum deformation occurs near the crown toe contact between the brace and chord, due to the excessive local plastification of the flange surface of the chord element. The deformation at the crown heel side of the joint is noticeably smaller compared to the crown toe side. The values of the maximum local deformations in the direction of the “Y” axis are 0.326 mm and 0.393 mm, respectively. When the influence of the seams in the analytical model is taken into account, the obtained value of the local deformation is 1.217 mm, which is much closer to the experimentally determined deformation (2.400 mm). The bearing capacity of the connection is higher by 1.97% compared to the bearing capacity determined with the nominal value of the yield strength. This difference occurs due to the fact that, in the analytical model in which the influence of the seams is taken into account, the theoretical assumption of rigid-plastic behaviour of the material at the point of connection is made, while the conducted research shows that the connection shows elasto-plastic behaviour with pronounced elastic deformation. The local deformation obtained numerically for the load equal to the experimentally determined bearing capacity is 2.470 mm, which is in good agreement with the experimentally measured deformation.

Characteristic displays of this stress are given in Figure 7, Figure 8, Figure 9 and Figure 10, with the corresponding legend of the numerical values. The maximum value of equivalent stress in the base material at the connection zone is roughly similar to the yield strength. It is noticed that the field of equivalent stresses at fillet welds as well as in certain zones on the side vertical walls of chord element has an extremely non-uniform stress distribution, which can be clearly recognised by sudden peaks on the stress diagram. The maximum value of equivalent stress at welds is at some spots even above the yield strength of the weld material. At the side walls of the chord element, the stress reaches a value which is close to the ultimate strength of the material. All of these lead to the conclusion that the examined connection model arrived at a certain point of fracture, which has not happened during experimental research of the connection. These phenomena [23] happen at a very small area, which means that their impact is strictly local. Other than that, with stress this high, local plastification occurs and changes the modulus of elasticity to a significantly lower value, which leads to significantly lower stresses. This phenomenon has a local character, since the failure that corresponds to the local deformation criterion occurs at a load level which is not high enough to cause high stress values in larger areas.

The results of the bearing capacities and deformations determined according to EC3 calculation procedures and the nonlinear numerical analysis, as well as the experimentally determined values of bearing capacity that correspond to limit displacement, were compared. It was observed that the ratio between the maximum local deformation determined by the application of non-linear numerical analysis and the experimentally determined maximum deformation was in the range of 1.03–1.5. In all analysed cases, higher values of local deformation were obtained using nonlinear numerical analysis, which was on the safe side. The best match with nonlinear numerical analysis, with a difference of only 3%, was observed in the case of experimentally determined values. It should be noted that the exhaustion of the bearing capacity of the experimentally tested connections occurred due to local plastification of the flange of the chord element. The theoretical analysis of the behaviour of these connections is based on the assumption of ideal rigid-plastic behaviour of the material at the point of connection. The experimental tests carried out in this work show that the connection exhibits elasto-plastic behaviour with pronounced elastic deformation. The theoretical considerations are based on the application of the plasfification line model. The values obtained by applying this model represent the upper limit of the solution, and are on the safe side. In order to obtain the lowest value, it is possible to analyse different mechanisms of plastic fracture lines. The values obtained in this way would be non-conservative.

5. Conclusions

Based on comparative analysis of results obtained by analytical solution, numerical analysis, and experimental research, the following conclusions for defining limit states of “Y” connections can be derived:

• Local plastification of chord flange led to exhaustion of load capacity of the experimentally examined connections which is in complete agreement with the means and methods of calculation in this paper;
• Analytical models for the analysis of the behaviour of these connections are based on the assumption that there is ideal rigid-plastic material behaviour at the connection node. The experimental and numerical results of the presented study showed that the connection has elastic-plastic behaviour with significant elastic deformation. This implies that there is a reserve in the bearing capacity connected to the stress state of the elements. However, in order to increase the bearing capacity of the connection, local deformations have to be reduced;
• Stresses and local deformations at the connection area are non-linear and very complex. In order to limit local deformations at the connection and prevent rotation/deformation capacity reduction, certain limitations of the connection geometry, such as the ratio of the connection element sizes, have to be introduced. The main reason for this is to fulfil the conditions of the serviceability limit state;
• Elastic connection behaviour or elastic local deformation of the chord element at local deformation that is about 1.0% of the element width is noticeable;
• Ultimate bearing capacity of the connection is calculated using the EC3 rule with nominal yield strain and it corresponds to local deformation of the chord which amounts 0.25% of the chord width;
• Experimentally determined values of the ultimate bearing capacity are 2.5 and 2.2 times higher than the value calculated according to EC 3 applying the nominal yield strength and experimentally determined yield strength, respectively. This fact points out the necessity of the revision of standards for the design of connections between hollow section profiles;
• The experimentally determined value of the ultimate bearing capacity is around 25% higher than the numerical result with consideration of the weld effects;
• Calculated ultimate bearing capacities of experimentally examined connections are significantly conservative due to the existence of membrane effects and post-elastic rigidity of material due to plastic deformations;
• For the analysed types of “Y” connections by non-linear FEM, analysis shows good agreement with the experimental results.