Three-dimensional numerical and linearly distributed multi-parameter fitted analytical modeling of hybrid beam–column with partially welded flush end-plate connection

Abstract

The flexural behavior of partially welded flush end-plate connections incorporating built-up hybrid beams and columns is analytically and numerically investigated. An experimentally obeying three-dimensional finite element model is first constructed. To circumvent the laborious effort of three-dimensional simulation and experimental work, a new experimentally and numerically complying equation approach is introduced for the construction of a continuous moment–rotation (M–θ) description. For the proposed equation, two essential terms are required: the rotational stiffness, S_j,ini, obtained by employing the component method and the maximum moment, M_max, produced using the proposed linearly distributed multi-parameter fitting technique. To demonstrate the applicability of the proposed equation, a variation in the geometric configuration of connections within the practical range is considered. Excellent agreement is noted when comparing all M–θ relationships produced by the proposed equation to those by the finite element method and experiments. In addition, the stress distribution and main deformation modes are numerically obtained, where the ranking of stress criticality is offered for all structural parts. The depth, width, flange, and web thicknesses, as well as the yield stress of the beam, have a major influence on M_max, as predicted by the proposed equation. Also, bolts have been identified as the most critically stressed component.

Keywords

analytical modeling built-up beam finite element flush end-plate connection moment–rotation multi-parameter fitting

Introduction

Flush or extended end-plate beam-to-column connections constitute the main elementary component of common steel structures. It is well established that the behavior of such a system is directly influenced by the connection properties between the columns and beams (Gorgun and Yilmaz, 2012; Rajagopal and Prabavathy, 2013; Stoddart et al., 2013). When the axial and shearing deformations are comparatively small with respect to those due to moment, the moment–rotation relationship is principally the main characteristic consideration in the definition of the connection behavior.

It is thus far largely accepted that from experimental studies, the actual behavior of a beam–column connection can be obtained, therefore leading to a better understanding of the entire structural behavior. However, it is also generally accepted that it is rather impractical for the experimental work to cover all important parameters required for characterizing the detailed behavior of structures. The main shortcomings of experiments manifest themselves in the form of cost and time constraints, where it is somewhat inefficient to test all connection configurations. Experimental tests normally require a complete set of equipment, and the cost of fabrication is usually very high. Moreover, one set of specimens only represents one type of connection, which is not sufficient for building a broader understanding of the particular structure under study. In other words, the change in the connection details demands more tests.

As an alternative, change in the connection details can be modeled considerably accurately using a nowadays well-established numerical approach, such as the finite element method, without performing experimental work in a laborious manner. In validating and conducting parametric investigations, the numerical method imposes a relatively much cheaper cost. It must be stated that the moment–rotation relationship remains the best way of characterizing the beam–column system. As far as numerous configurations are concerned, the outcomes can be wide-ranging. Hence, it is reasonably justified that in a variety of cases, these varying behaviors can be modeled using a set of well-defined equations. In view of the fact that every new configuration of connection conventionally demands a new set of experimental studies, this study proposes an analytical procedure to circumvent the highly laborious nature of both experimental and numerical methods. In this regard, a simple analytical approach can be constructed on the basis of observations made from experimental and numerical outcomes in such a way that the load–deformation relationship of structures, with various geometric and material descriptions, can be produced in a general fashion.

There has been much research conducted on steel structures with end-plate connections over the past few decades, from which a great deal of effort has been devoted to establish experimental and prediction techniques to characterize beam–column connection behavior. Gibbons et al. (1993) investigated the behavior of column sub-assemblages in 10 tests and found that the moment transfers are directly affected by outweighs of connection. De Lima et al. (2004) performed measurements on extended end-plate beam-to-column joints subjected to bending and axial force to describe an experimental program for such a joint type, and in conjunction, to extend the component method philosophy. They concluded that in the presence of multiple bolt rows, some additional phenomena such as stiffness coupling increase the bending resistance for low levels of axial compression.

As far as numerical modeling is concerned, Krishnamurthy and Graddy (1976) established the first report of finite element analysis (FEA) for end-plate connection (Shi et al., 2008). Kukreti and Biswas (1997) developed a computer code and used it to analyze the moment–rotation behavior of three eight-bolt end-plate connections subjected to seismic loading. An overprediction of the moment at any rotation level of 5%–12% for the connection was observed when comparing moment–rotation hysteresis loops using the FEAs. Bursi and Jaspart (1998) claimed that a three-dimensional non-linear finite element model was more suitable for the analysis of extended end-plate moment-resisting steel connections from their overview study of the finite element (FE) method using the commercial ABAQUS program. Using the FEA software ANSYS, Shi et al. (2008) included the pretension force in the bolt and the contact regions around the end-plate in their analyses, which simulated the structural response well compared to existing modeling. Bahaari and Sherbourne (1996) concluded that for both hand-tightened and prestressed bolts, the magnitude of bolt preload does not affect the bolt force, prying action, and beam flange force distribution at the ultimate load. The bolt size, however, has a significant effect on the prying action.

A flexibility-based inelastic beam–column element utilizing a lumped plasticity model with a special procedure to account for the P–δ effect was proposed by Kaewkulchai and Bhokha (2009), in which the violation of the yield limit is avoided in obtaining the tangent stiffness matrix. In addition, a hysteresis model with pinching consideration was formulated and implemented by Nogueiro et al. (2009) using a spring element for the cyclic response of various end-plate beam–column joint configurations. Stoddart et al. (2013) carried out an investigation on the extended use of rate-dependent springs in component-based joint models using both the component method and finite element modeling. They concluded that the component-based method produces a collapse load some 20% lower than that predicted using the conventional approach, which employs a non-linear rotational spring in simulating the moment–rotation response. Using the strain energy approach, Wang et al. (2013) formulated an estimate for general semi-rigid joints.

In the realm of the prediction of the moment–rotation (M–θ) relationship, Yee and Melchers (1986) proposed analytical approaches for end-plate beam-to-column connections, taking into account the possible failure modes and the deformation characteristics of the connecting elements. Abolmaali et al. (2012) presented the development of the Ramberg–Osgood and Three-Parameter Power equations for the M–θ behavior of flush end-plate connections with one row of bolts each below tension and compression flanges, to study the deformation of different frame types.

Weld and bolt configurations in steel structural connections constitute the most important factor in determining the load-carrying capacity, deformation under the load, and the overall safety of structures (Soy, 2011). The influence of several surface conditions for numerous high-strength bolts on the slip coefficient of slip-critical joints was experimentally examined by Nah et al. (2009). In this study, uncoated surfaces exhibited a loss of clamping force of under 3% within the range of 4.71%–8.37% for the coated surfaces, for 1000 h of relaxation. Myers et al. (2009) studied the effect of welding details on the ductility of steel column and base plate connections using complete joint penetration (CJP) and partial joint penetration (PJP) weld details between the column and the base plate. They concluded that fracture was observed at the top of the weld in the heat-affected zone (HAZ) of the column flange. This was attributed to the strength and toughness at the weld root of the reinforced PJP welds, which permits the development of a fully yielded and strain-hardened column flange. The bolt load variation during loading in the beam–column connection, a pivotal matter in real structural behavior, was considered by Saravanan et al. (2009) in an advanced FE analysis for cyclic loads. Their moment–rotation and displacement responses agree very well with experimental results. Chen and Wang (2009) carried out an investigation on non-completely penetrated welds as an application in the extended end-plate connection, from which a satisfying result for the design requirements was obtained under the seismic condition. They concluded that the weld resistance deteriorated with respect to greater bolt size and thinner end-plate. On the other hand, they claimed that the connection would not fail if its parts were perfectly arranged, independent of the type of weld used. Also, they posited that it is possible to use non-completely penetrated welds for H-shaped beams and end-plate connections for both static and seismic loads.

The idea of using a built-up hybrid beam in flush end-plate connections has only recently become known. Some advantages of implementing the hybrid beam in the connection system are the reduction in the cost of the beam, weight saving, and the ease in dealing with parts, such as transferring and storage, compared to the use of standard beam sections. Yao et al. (2008) pointed out the possibility of involving the favorable strength and stiffness features to transfer moment between the hybrid beam and the circular column within structural frames for low- to medium-rise buildings. They concluded that the moment-resisting capability of this connection type can be potentially used to improve the stiffness of long-span beams and to resist lateral loads such as winds and earthquakes.

Following the research tradition of beam–column structures, this study presents an analytical formulation and numerical investigation on the flexural behavior of flush end-plate connections with a new configuration similar to that experimentally explored by Shek et al. (2012), involving a partially welded built-up hybrid beam of various types and details. Such a joint has not hitherto been analytically or numerically modeled. The remainder of the article is arranged as follows. First, the beam-to-column connection with a flush end-plate incorporating hybrid beam is numerically modeled using the FE software ABAQUS version 6.9 (ABAQUS, 2009) and validated with existing experimental results. Second, the formulation of a simple equation that represents the moment–rotation relation for the considered connection, using a newly proposed linearly distributed multi-parameter fitting approach with characteristics that conform to those of the numerical models and experiments, is carried out. Then, extra results, which are difficult to measure by physical test studies, are offered, such as the stress distribution given by the numerical simulation, from which the critical stress intensity sequence of the connection’s parts is discussed. The article ends with a brief note on the main findings.

Description of finite element model

Our numerical models are constructed by means of an FE approach using ABAQUS for four specimens of a beam–column system denoted N1, N2, N3, and N4, as investigated by Shek et al. (2012), further details of which are offered in Table 1. The structural configuration is displayed in Figure 1.

Table 1.

Types and details of test specimens.

Specimen	Hybrid beam size h × b × w/t_fb/w_t	Column size	Bolt number (top-bottom)	End-plate size	Bolt size
N1	400 × 140 × 41.13/12/5	305 × 305 × 118 UC	4 (2-2)	450 × 200 × 12	20
N2	500 × 180 × 63.59/16/5
N3	450 × 160 × 46.86/16/6		8 (4-4)
N4	600 × 200 × 85.91/16/6

Note all dimensions in millimeter.

Figure 1.

Full scale laboratory beam–column system (Shek et al., 2012): (a) test setup and b) one-bolt-row and two-bolt-row connection details for adjoining parts.

Figure 2 shows the modeled beam–column assembly. Following full specification of the experiments, a 3 m column is modeled with a fully fixed condition at the bottom; at the top, it is allowed to move freely only in the vertical direction (U_x = 0, U_y = free, and U_z = 0). A hybrid beam, which consists of various steel grades, is modeled in a cantilever manner. The steel used for the column, end-plate, and beam web sections is of grade S275. For the beam flanges, steel grade S355 is adopted. The end-plate is constructed to be in contact with the hybrid beam under a partially welded condition, and with the column using bolts, following those of the experimental setup (Figure 1(b)).

Figure 2.

Three-dimensional beam–column numerical model.

The contact between interfaces of the connection is defined and established through the creation of both master and slave surfaces, using the interaction and constraint type formulation as defined in ABAQUS. The reason behind prescribing the master and slave surfaces is so that the latter cannot penetrate the former. In common practice, the master surface is defined for the parts that are the strongest among all parts in contact, judging by the input, for example, each component’s Young’s modulus and yield stress. For interaction, the type of contact used is that of surface-to-surface (standard) with frictionless interaction. The surfaces involved using such an interaction are as follows:

All the surfaces of the bolts and contacted surfaces, such as the column and end-plate;

All interacted surfaces between the end-plate and column;

Non-welded regions between the beam web and end-plate.

For constraint, the type of contact used is a tie, to provide coupling between surfaces. The surfaces involved for this type of constraint are as follows:

Surfaces of both beam flanges, which are connected to both the beam web and end-plate;

Two surfaces of the beam web, which are connected partially to the end-plate;

Surface of the shell plate for load application, which is in contact with the beam top flange;

Surface of shell plate, which is in contact with the beam web for rotation measurement;

Surface of the shell plate, which is in contact with the column web for rotation measurement.

Table 2 categorizes the master and slave surfaces for all parts of the model. The load magnitude follows that of the joint moment obtained from experiment, with a lever arm of 1.3 m, from the column flange’s outer face to the load central point on the top flange of the beam. To discretize the structure, an eight-node brick solid continuum element with reduced integration (C3D8R) was adopted for all parts. Table 3 presents the number of elements for each part of connection for one-bolt-row and two-bolt-row configurations, where the minimum element size used for all connections’ components from the convergence study is 0.04 m. In all cases, there are three elements through the thickness direction. Also, a geometrically non-linear analysis is employed.

Table 2.

Definition of master and slave surfaces and interaction types for all parts of model.

Contact part	Master/slave definition	Interaction type
Column to bolt head	Slave to master	Surface-to-surface contact
Column to bolt arm	Slave to master	Surface-to-surface contact
Column to end-plate	Master to slave	Surface-to-surface contact
End-plate to bolt head	Slave to master	Surface-to-surface contact
End-plate to bolt arm	Slave to master	Surface-to-surface contact
End-plate to beam flange	Master to slave	Tie
End-plate to beam web	Master to slave	Tie
Beam flanges to beam web	Master to slave	Tie

Table 3.

Number of elements for each part of connection for one-bolt-row and two-bolt-row configurations.

Part of connection	One-bolt-row	Two-bolt-row
Beam top flange	148	148
Beam web	342	342
Beam bottom flange	148	148
Load plate	15	15
Beam–shell	1	1
Column–shell	1	1
Bolt	416	832
End-plate	192	192
Column	5232	5232
Total	6495	6911

Table 4 shows the material properties for all connections’ components. The modulus of elasticity, E, for steel for the different modeled parts is taken from the mean value of measurement (Shek et al., 2012); Poisson’s ratio, v, is set to 0.3 for all parts. The material behavior of all steel components is expressed using a tri-linear stress–strain relationship with yield and ultimate strengths, F_y and F_u, obtained from the mean values given in Table 4. The plastic behaviors are considered to be linear and horizontally constant after yield, as shown in Figure 3 for the beam flange steel part. ABAQUS permits the hardening rule of a material model to be described in a tabulated fashion using a set of user-prescribed data (ABAQUS, 2009).

Table 4.

Summary of tensile test results for each structural component (Shek et al., 2012).

Material property	Column flange	Column web	Beam flange	Beam web	End plate	Bolt
Modulus of elasticity, E (N/mm²)	214.33 ± 13.42	231.67 ± 36.63	245 ± 34.07	228 ± 5.72	235 ± 18.46	205
Yield stress, F_y (N/mm²)	373.21 ± 12.44	368.59 ± 20.29	373.10 ± 11.64	316.50 ± 16.41	333.66 ± 12.53	640
Tensile strength, F_u (N/mm²)	530.95 ± 7.90	458.09 ± 2.34	542.41 ± 0.94	547.76 ± 2.91	412.78 ± 9.11	660

Figure 3.

Example of application of tri-linear relationship in expressing the stress–strain behavior of the steel component (shown here is the relationship used to define the behavior of the beam flange).

Here, the classical metal plasticity, which uses the von Mises yield surface with associated plastic flow for isotropic yield, is specifically employed. In this respect, an isotropic hardening rule is thus adopted, since kinematic hardening is best applicable in the presence of a cyclic loading environment, which is beyond the scope of this study. The isotropic hardening rule is executable in ABAQUS under the script command *PLASTIC. In detail, for each structural component, the yield and ultimate strengths, F_y and F_u, as given in Table 4, are provided, along with their associated plastic strains obtained from the averaged values from the material tensile tests. Furthermore, a linearized format is selected to regulate the hardening rule, as can be clearly noted in the linear shape (in our case, three linear parts) of the stress–strain relationship illustrated in Figure 3, although other types of distribution such as those of quadratic, Voce, and Ramberg–Osgood may also be defined. No further input is required for the present model, since the plastic modulus for example is determined automatically from the tabulated plastic stresses and strains. The bolts are modeled adopting a fully treated M20 type and their hole allowance is in accordance with the Eurocode 3 (EC3, 2005) recommendation.

Validation of FE model with test results

Comparison of the ultimate failure modes, including the details of deformation from FE models and test results for all four specimens, is shown in Figure 4. In general, there are two obvious and similar deformations in all specimens. They are the bending of the end-plate at the tension side, and the occurrence of a gap between the beam web and end-plate. For all four specimens, there exists local deformation at the top of the end-plate, particularly around the top bolts in the tension side, where it is subjected to the highest tensile stress. This deformation is shown by the FE models in all the subfigures on the right column of Figure 4. The gap, the second deformation mode, occurred due to the bending of the beam web and end-plate, both of which are not welded around this particular region. It has been clearly shown that all the specimens experienced the beam web bending as predicted by the FE simulation, but such an observation only occurred in specimens N2 and N3 in tests. Note that the beam web bending is relatively small in models N1 and N4, implying that the test specimens may experience a similar deformation mode, which is not captured in the examination of photographs in Figure 4. Hence, the yielding of the top part of the end-plate and beam web buckling are the common resulting failure modes seen in such a beam–column system. There are deductively two reasons that lead to this deformation. First, the thickness of the beam web is small compared to the beam depth. Second, no weld line exists between the end-plate and the beam web along the bending length. Overall, the similarity in all main deformation modes is evident in both approaches, indicating a good fitness of the FE models in representing the tested beam–column behavior.

Figure 4.

Comparison of ultimate failure mode of all specimens of test and FE model (left: test; middle, right: FE).

Analytical approach for M–θ relation

In addition to the numerical modeling, an analytical expression for the moment–rotation relationship is now considered for the structure under investigation. From in-depth examination, it is proposed that the general equation for the moment–rotation curve be expressed using (see Figure 5 for a general presentation of this relation)

M = M_{max} (1 - e^{- [S_{j, ini} / M_{\max}] θ})

(1)

where M_max is the maximum moment of connection and S_j,ini is the initial rotational stiffness of the connection.

Figure 5.

General presentation of the moment–rotation relationship using currently proposed equation.

This equation is based on the observation that most beam–column connections behave somewhat linearly during the initial loading stage, producing the initial rotational stiffness expression, S_j,_ini, for the slope of the M–θ curve. The slope then decreases continually before a maximum is reached, the point of which is described by the maximum moment, M_max. Most existing analytical expressions, notably the component method, present the computational procedure of the two aforementioned values. Unfortunately, they are defined discretely as two separate magnitudes, not a continuous relation as currently described. The procedure of obtaining the M–θ relation is described next in terms of the determination of S_j,ini and M_max.

Initial rotation stiffness, S_j,ini

In practice, it is well known that connections behave in a semi-rigid manner with their actual behavior being expressed using the finite stiffness of the joints. Nowadays, many steel structural designs have applied guidelines offered by Eurocode, especially for works practiced in the European Union (González-Montellano et al., 2009). Based on the component method (EC3, 2005), a group of descriptive structural components is required to establish the expression for the rotational stiffness and moment capacity of connection. Table 5 shows the parameters of test specimens N1, N2, N3, and N4 for establishing the M–θ relation using the proposed equation. The main aim of collecting all the parameters in Table 5 is to include in our consideration all the components that may affect the total moment capacity of the connection under study. Descriptions of each term will be offered later, after the presentation of the equation for M_max. In accordance with the definition of EC3, the components involved in our computation are the column web panel shear (k₁), column web in compression (k₂), column web in tension (k₃), column flange in bending (k₄), end-plate in bending (k₅), and bolts in tension (k₁₀). Note that the rotational stiffnesses determined using the component method capture those measured satisfactorily. Therefore, S_j,ini is determined here by employing the component method.

Table 5.

Parameters for specimens N1, N2, N3, and N4 (Shek et al., 2012).

Variable	Unit	N1	N2	N3	N4
M_c	kNm	93.86	128.25	115.695	156.87
h	m	0.4	0.5	0.45	0.6
b	m	0.14	0.18	0.16	0.2
t_fb	m	0.012	0.016	0.012	0.016
t_wb	m	0.005	0.005	0.005	0.006
h_ep	m	0.45	0.55	0.5	0.65
f_yb	kN	274	274	548	548
l_w	m	0.313	0.389	0.353	0.428
m ₁	m	0.0345	0.0345	0.0345	0.034
m ₂	m	0.04	0.036	0.04	0.036
e	m	0.055	0.055	0.055	0.055
g	m	0.09	0.09	0.09	0.09
z	m	0.334	0.432	0.339	0.487
f_y,bw	kN/mm²	0.317	0.317	0.317	0.317
f_y,bf	kN/mm²	0.373	0.373	0.373	0.373
f_y,ep	kN/mm²	0.334	0.334	0.334	0.334
f_y,cow	kN/mm²	0.369	0.369	0.369	0.369
f_y,cof	kN/mm²	0.373	0.373	0.373	0.373

Development of equation for M_max

The same agreement offered by S_j,ini predicted using the component method cannot be observed in the prediction of the maximum moment for the M–θ relation description of the currently investigated connections. Differences of 4.3%–27% have been obtained. To replace the component method, which was found to be less accurate for M_max, a new linearly distributed multi-parameter fitting approach is proposed.

To define M_max, which can present any connection that follows the currently studied connection type with the mentioned parameters, a relationship of M_max with respect to all the considered parameters is proposed. The variation of our considered parameters is in agreement with the available data obtained from test specimens N1, N2, N3, and N4, and an additional set of specimens, IPE-O-220, IPE-O-600, IPE-T-400, and IPE-T-600, from the beam–column system that adopts the IPE beam section as defined in EN 10034:1993 (1993). IPE here denotes the European steel I-beams. The letters O and T after IPE designate one-bolt-row and two-bolt-row configurations, respectively. The numbers stated after these letters concern the depth of the beam. The proposed expression is presented as

M_{\max} = S χ

(2)

where

\begin{array}{l} χ = h^{α_{1}} b^{α_{2}} t_{f b}^{α_{3}} t_{w b}^{α_{4}} h_{e p}^{α_{5}} f_{y, b}^{α_{6}} l_{w}^{α_{7}} m_{1}^{α_{8}} m_{2}^{α_{9}} e^{α_{10}} \\ g^{α_{11}} z^{α_{12}} f_{y, b w}^{α_{13}} f_{y, b f}^{α_{14}} f_{y, e p}^{α_{15}} f_{y, c o w}^{α_{16}} f_{y, c o f}^{α_{17}} \end{array}

(3)

from which h is the beam depth, b is the beam width, t_fb is the thickness of the beam flange, t_wb is the thickness of the beam web, h_ep is the end-plate depth, f_y,b is the bolt capacity, l_w is the weld length, m₁ is the distance from bolt center to 20% distance into column root or end plate weld with beam web, m₂ is the distance from bolt center to 20% distance into end-plate weld with beam flange, e is the distance from the center to center of the radius of a bolt hole, g is the gage distance, z is the lever arm, f_y,bw is the material yield stress of beam web, f_y,bf is the material yield stress of beam flange, f_y,ep is the material yield stress of end-plate, f_y,cow is the material yield stress of column web, and f_y,cof is the material yield stress of column flange.

The units for all the parameters except for f_y,b, f_y,bw, f_y,bf, f_y,ep, f_y,cow, and f_y,cof are kN and m. Otherwise, the unit is kN/mm². In equation (3), χ depends on all the parameters of the beam–column structure, each component of which has an undetermined power, α_i, evaluated later in fitting. The fitting is carried out such that a linear relation that intercepts the origin is obtained. χ represents the slope of the fitted relation. In determining M_max, since the M–θ curve never really reaches a plateau but rather rises continually at a decreasing rate, it is obtained by setting a difference tolerance between values computed at one rotation level and the previous one. Here, the tolerance is set to be less than 1%. When such a tolerance is achieved, M_max is determined. A linear model is selected to express M_max for simplicity. Other types of model, for instance the polynomial group of order two or higher, can be employed, although the fitting effort may increase with the order of the equation. The aim of the current model is to achieve a good accuracy without sacrificing the simplicity in execution.

Table 6 shows all the geometrical properties of the IPE specimens. The inclusion of this extra set of beam sections demonstrates the applicability of the current equation when employed to describe the behavior of configurations other than those from the tests, such that the restriction of generality of our proposed M–θ relation is relaxed. The details of the other structural parts are kept the same as those used in the tests. The IPE specimens are only selected for demonstration purposes. Other sections if desired can be arbitrarily included. The beams considered here have depths in the range of 220–660 mm. Note that the determination of M_max is based on experimental data and the beams with minimum and maximum depths each from one-bolt-row and two-bolt-row configurations. Hence, not all sizes need to be included in the fitting process. The outcome of defining the ultimate moment capacity through proposed fitting technique is shown in Figure 6.

Table 6.

IPE beam geometric parameters (all dimensions in millimeter).

Designation	h	B	t_fb	t_wb	r
IPE220	220	110	9.2	5.9	12
IPE360	360	170	12.7	8	18
IPE400	400	180	13.5	8.6	21
IPE500	500	200	16	10.2	21
IPE550	550	210	17.2	11.1	24
IPE600	600	220	19	12	24

Note that IPE600 is used for both one-bolt-row and two-bolt-row configurations.

Figure 6.

Ultimate moment capacity relationship defined from proposed fitting technique.

The vertical axis represents the moment capacity obtained from specimens N1, N2, N3, and N4 from tests and IPE-O-220 and IPE-O-600 with the minimum and maximum depths, respectively, for one-bolt-row configuration as well as IPE-T-400 and IPE-T-600, respectively, for two-bolt-row configuration, from the available beam sections given in Table 6. Note that M_max for specimens other than those from the experiment was predetermined using the FE approach. To show the good performance of the proposed model, in addition to offering good agreement with experimental outcomes, a further comparison was made of the M–θ relationship for other specimens with different geometries (in this case, various IPE specimens were chosen) computed by the analytical equation, to those predicted by the numerical approach. In a situation where experimental outcomes are not available, and since it has been demonstrated that the results from numerical models agree excellently with those measured, the numerically produced M–θ relationships are (at this stage) the most reliable and justifiable method to be used for benchmarking the fitness of the proposed analytical equation. Only beam sections with the minimum and maximum depths are selected in the fitting, that is, not all IPE specimens are considered. This is important in order to ensure that the proposed formula is free from over-fit errors. A further proof of the applicability of the numerical model as a verification benchmark will be shown later, in terms of capturing the M–θ relationship produced by the experimental work. It is also important to note that the performance of the analytical model is scrutinized against a new set of specimens after the comparison with the measured behavior, although the connection type remains similar, the scope within which this study is limited.

In Figure 6, it can be seen that all data points of specimens N1, N2, N3, and N4 and those using the IPE are located on the straight line of best fit. From the fitted line, the equation $M_{\max} = 15, 395 χ$ is obtained. The goodness of fit is expressed by the correlation coefficient, R² = 0.9932, which shows a very good fit for all eight data points. The correlation coefficient is the most commonly adopted technique to measure error in an overall manner and also suits our modeling purpose, that is, ensuring simplicity in a readily effort-consuming consideration of multiple parameters. Furthermore, the mean absolute percentage deviation (MAPD)

\bar{M} = \frac{1}{n} \sum_{t = 1}^{n} | \frac{F_{t} - D_{t}}{F_{t}} |

(4)

(F_t = fitted value and D_t = data point) of all n discrete points against the fitted curve is computed as 2.96%, indicating an excellent fit of all data points. An investigation of the residual contributed by individual terms can also be performed separately; this falls beyond the scope of this study. With regard to individual residual examination and error evolution, Haldar and Mahadevan (2000) and Gardoni et al. (2002) are relevant. The resulting M_max expression in our case is given from the fitting as

M_{\max} = 15, 395 χ

(5)

where

\begin{matrix} χ = h^{0.01} b^{0.01} {t_{fb}}^{1.22} {t_{wb}}^{0.15} {h_{ep}}^{0.001} {f_{y, b}}^{0.3479} {l_{w}}^{0.00105} \\ {m_{1}}^{0.0079} {m_{2}}^{0.262} e^{0.001} g^{0.001} z^{0.001} {f_{y, bw}}^{0.001} {f_{y, bf}}^{0.001} \\ {f_{y, ep}}^{0.001} {f_{y, cow}}^{0.001} {f_{y, cof}}^{0.001} \end{matrix}

(6)

The same units used in equations (1) to (3) apply here. In addition, the inclusion of a high number of α_i reflects the relativity of the current model with all related parameters used to define the beam–column system. The main influencing parameters are those coming from the beam as perturbation in related α_i changes the fitness of the M_max–χ relationship. Also, it can be easily identified from equation (6) that those terms with higher α_i inflict a greater influence on the M_max–χ relationship. In this case, they are h, b, t_fb, t_wb, f_y,b, and m₂. Prescribing all considered parameters of connection into equations (5) and (6), M_max for a beam–column for different IPE specimens can be determined, that is, it is not applicable only for the structure with specimens IPE-O-200, IPE-O-600, IPE-T-400, and IPE-T-600. Rather, the equation is general for all beam–column systems that use the currently studied connection type, only focusing on one-bolt-row and two-bolt-row configurations.

Comparison of M–θ relation

There exist very good agreements of the equation approach with those of experiments, as well as FE for built-up hybrid beam specimens, as shown in Figure 7. Note that the terminal points of the FE curves are determined based on the maximum load magnitude displayed by the corresponding test specimens. To emphasize further the applicability of the proposed approach, Figure 8 presents graphs of comparison between the equation and FE approaches for structures with IPE specimens using one-bolt-row and two-bolt-row configurations. Since the FE models were validated well with the test results previously (section “Validation of FE model with test results”), and noting that the test specimens do not cover the use of IPE specimens, the FE models are used as a benchmark for comparing the M–θ relation for the equation method. It is obvious that, in general, for both one-bolt-row and two-bolt-row configurations, the curves agree well for the most part, except for some minor disagreement in the transition region, after the initial slope that defines the rotational stiffness. Hence, the developed equation is evidently adequate to present the connection system for the range of beams studied, including specimens from previously published tests and those additional, that is, the IPE specimens.

Figure 7.

M–θ relation from equation, FE and experimental approaches, for specimens N1, N2, N3, and N4.

Figure 8.

Comparison of M–θ relationship for structure using IPE beam section, predicted using equation and FE, for one- (IPE-O-h) and two- (IPE-T-h) bolt-row configurations.

In comparison, the agreements are generally better for the one-bolt-row configuration compared to those of the two-bolt-row. Also, the ultimate moment resistance increases corresponding to the beam depth as the maximum ultimate moment is about 157 kNm for N4 where the beam depth is 600 mm, and the minimum is around 94 kNm for N1 where the beam depth is 400 mm. Examining all α_i, it can be noticed for specimens N1, N2, N3, and N4 that an increase in the moment capacity, M_max, is the direct product of combined growth in beam depth, h, beam width, b, as well as the flange and web thicknesses of the beam, t_fb and t_wb, but a reduction in the parameter m₂. For the IPE specimens, h also plays an important role in increasing M_max. Again, all the aforementioned parameters (i.e. b, t_fb, and t_wb), which increase according to h following the increase in the number in the codename of IPE specimens, increase M_max. This phenomenon is attributed to the increase in h, which amplifies the corresponding second moment of area relating directly to M_max. One notable case is that M_max of IPE-T-500 is greater than that of IPE-O-550. This implies that the number of bolt rows has a greater effect even when there is an increment in the beam depth, although a further increase in h for the one-bolt-row case would surely exhibit a higher M_max. For the specimens chosen for the fitting, high accuracy can be achieved, even though only the maximum and minimum sections are considered. This is applicable as long as the considered sections fall within the range used for fitting, noting that the connection must be of a similar type.

Stress study

There exist additional valuable results offered by finite element modeling for the mechanical behavior of flush end-plate connections, which are difficult to measure and normally not directly available in physical tests. For demonstration purposes, Figure 9 shows the distribution of von Mises stress in various parts of the connection using specimens IPE-O-600 and IPE-T-600. In general, all specimens show approximately the same stress distribution in terms of the sequence of magnitude, ranking from small to large. This similarity can be summarized as follows. In terms of the stress distribution sequence, all the specimens behave similarly, such that the stress distribution begins from the tensile side of connection. As a result, the stress with the highest intensity is identified at the tensile side. The distribution of stress forms a crescent shape on the column web between the tensile and compressive sites of connection, due to the partial welding configuration, which commonly causes shearing in the column web in this particular region.

Figure 9.

Distribution of von Mises stress in various parts of connection for one-bolt-row (IPE-O-600) and two-bolt-row (IPE-T-600) configurations.

In general, all specimens agree on the critical parts of the connections, in the form of stretching of bolts and bending of the end-plate on the tensile side of the connections. The beam web also shows a small critically stressed area surrounding the welding length in both the tensile and compressive sides. The remaining parts are not as critical as the bolts and end-plate. However, high stresses are distributed surrounding the holes for the bolts at both column flanges, and in the column web at bolt row locations at both the tensile and compressive sides. This observation confirms that the regions with bolts and a hole remain the weakest link of the beam–column system. Looking particularly at the tensile side of connection, the bolts undergo the highest stress. Then, the second-most critically stressed part is the end-plate, having the highest stress concentrated at the tensile side. There also exists a high value of stress on the column web. Comparing the one-bolt-row and two-bolt-row configurations, the former experiences more of a symmetrical stress distribution in the tensile and compressive sides, while the latter exhibits a higher stress intensity at the tensile side, suggesting that the two-bolt-row configuration takes a greater load at the top region due to additional reinforcement there. Basically, all specimens exhibit the same distribution of stress, differing only in terms of intensity.

Table 7 shows in numeric sequence the characterization of the stress criticality between specimens’ parts, in accordance with the maximum stress experienced. In all specimens, it is obvious that the bolts are the most-stressed component. The column is ranked as the second critical part in N1, N2, N3, and N4 as well as IPE-T-400, IPE-T-500, and IPE-T-600. On the contrary, the beam is ranked as the second critical part in IPE-O-220, IPE-O-360, IPE-O-550, and IPE-O-600. The third criticality is divided between the column for all IPE specimens of one-bolt-row configuration, and the end-plate in N1 and N2 of the one-bolt-row as well as N3 and N4 of two-bolt-row configurations. For IPE-T-600, the third critical part is the beam. The fourth criticality is divided between the beam and end-plate. The beam is the fourth criticality in both the one-bolt-row and two-bolt-row configurations. For the one-bolt-row configuration, it occurs in N1 and N2. For that of the two-bolt-row, it appears in specimens N3 and N4 as well as IPE-T-400 and IPE-T-500. In order to determine whether these values have reached the failure stress, Table 8 provides a comparison between modeled and test results. It is therefore obvious that the bolts have failed since the measured failure stress has been exceeded. This implies that in these configurations, the failure of the beam-to-column system, if it occurs, initiates on the basis of considerable stress in the bolt.

Table 7.

Comparison of stress criticality of specimens’ parts.

Specimen	Criticality arrangement				Bolt row number
	1 = most critical, 4 = least critical
	1	2	3	4
N1 and N2	Bolt	Column	End-plate	Beam	One
N3 and N4	Bolt	Column	End-plate	Beam	Two
IPE-O-220 and IPE-O-360	Bolt	Beam	Column	End-plate	One
IPE-O-550 and IPE-O-600	Bolt	Beam	Column	End-plate	One
IPE-T-400 and IPE-T-500	Bolt	Column	End-plate	Beam	Two
IPE-T-600	Bolt	Column	Beam	End-plate	Two

Table 8.

Comparison of maximum stresses between FE and test (MPa).

Method	Column flange	Column web	Beam flange	Beam web	End plate	Bolts
Test	531	458	542	548	413	660
Model	385	385	385	385	320	665

Conclusion

3D numerical and linearly distributed multi-parameter fitted analytical modeling methods have been proposed and presented for the flexural behavior of built-up hybrid beams with a partially welded flush end-plate connection to columns. Good agreement of deformation modes was achieved when comparing the outcomes from both numerical and experimental works. In addition, both proposed numerical and analytical models exhibit a noteworthy capability to replicate the M–θ relationships of the studied connection type. The newly proposed analytical M–θ equation is a continual function, described principally by the rotational stiffness and the maximum moment.

The main deformation modes of the studied beam–column system are the yielding of the top side of the end-plate and beam web buckling. These are caused by the low thicknesses of the end-plate and beam web, as well as by the absence of the welding line between these two components. In terms of M–θ relationships, parametric exploration dictates that the highly influencing factors in affecting the moment capacity, M_max, are beam depth, h, beam width, b, flange and web thicknesses of the beam, t_fb and t_wb, as well as the distance from the bolt center to 20% distance into the end-plate weld with beam flange, m₂. An increase in M_max is the direct product of combined growth in h, b, t_fb, and t_wb, but a reduction in m₂. In terms of the stress criticality from the stress contour plots, all considered structures experience the highest stress in the bolts, implying that this component is the weakest link in the studied connection type.

The proposed numerical and analytical methods are applicable only to the type of connection examined here, although our procedure can be easily extended to explore other connection systems, suggesting that further work on different connection types is certainly feasible.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors thank the Malaysian Ministry of Higher Education (MOHE) and Universiti Teknologi Malaysia for research grant (R.J130000.7809.4F518) and facility.

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Three-dimensional numerical and linearly distributed multi-parameter fitted analytical modeling of hybrid beam–column with partially welded flush end-plate connection

Abstract

Keywords

Introduction

Description of finite element model

Validation of FE model with test results

Analytical approach for M–θ relation

Initial rotation stiffness, Sj,ini

Development of equation for Mmax

Comparison of M–θ relation

Stress study

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Initial rotation stiffness, S_j,ini

Development of equation for M_max