# Rotation Stiffness Investigation of Spatial Joints with End-Plate Connection

Authors: Shizhe Chen; Jianrong Pan; Zhan Wang; Chao Zhou
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

Spatial joints with end-plate connections show significant spatial coupling effects under spatial loading. Mechanical behaviour and failure modes of these spatial joints differ from those of planar joints. This study involved experiments and finite element analyses with respect to planar joints with end-plate connections under static load. The numerical results agreed well with the experimental data, and this verified the adequacy of the finite element analyses. Then, finite element models of the spatial interior joint, exterior joint, and corner joint were established to analyse the difference between the mechanical behaviour of spatial joints and planar joints. The component method was used to analyse components contributing to the initial stiffness of spatial joints. An initial rotation stiffness calculation model of spatial joints was proposed based on the deformation of joints. The findings indicated that results of the calculation models were in good agreement with those of the finite element analyses, and this proved that the calculation model proposed in this study could act as a reference method.


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## Body

## 1. Introduction

Joints are an important part of an entire structure. Some previous studies [1, 2] examining beam-column connections focused on in-plane behaviour. However, the actual loading state of the joint should be spatial. The behaviour of the frame structure joints depends on the combined effect of the major-axis and minor-axis connections. Only planar connections were considered in actual structural design, and this does not reflect the actual mechanical behaviour of joints. Thus, it is necessary to separately consider coupling effects under spatial loading.The moment-rotation behaviour of the end-plate joint with an extended stiffener was investigated by Tartaglia et al. [3, 4], and also, the design method was developed by Francavilla et al. [5]. Katalin and Miklós [6] investigated a modified component method that could be applied to the analysis of spatial joints. Cabrero and Bayo [7, 8] investigated the connection behaviour of a new spatial joint subjected to proportional spatial loading. Simoes da Silva [9] presented a conceptual design model of spatial joints. This conceptual design model was a combination of the calculation models of spatial joints based on the component method principle. Loureiro et al. [10] investigated spatial joints with end-connections on the major axis and minor axis. The results indicated that minor-axis connections offered a higher degree of improvement to the rotation stiffness of the joints when compared with that of major-axis connections. Gil et al. [11] used an experimental study and finite element analyses to investigate the interaction between major-axis and minor-axis connections of spatial joints. Zhou [12] conducted a set of experiments with respect to spatial composite joints.Researchers also developed simplified models and calculation methods of initial rotation stiffness of planar joints, such as modifying the component method [13], application of new joints [14], and new loading condition [15]. The component method in the Eurocode was the most widely used analytical method. However, a few studies investigated the initial rotation stiffness of spatial joints. Calculation models for the initial rotation stiffness of steel joints loaded along the major axis were provided by EC3 [16], which were only applied to planar joints. The interaction stiffness effect between the major axis and minor axis was not investigated before. And, this effect could not be neglected in mechanical behaviour research of spatial joints.This study investigated the rotation stiffness of the spatial joints considering the interaction stiffness effect between the major axis and minor axis. The accurate finite element modelling method was proposed by comparing the resultant curves with static experiments of planar joints. Finite element models of spatial joints were then constructed based on the verified modelling method. Differences in the initial rotation stiffness between spatial joints with end-plate connections and planar joints were evaluated. The theoretical rotational stiffness calculation method of the spatial joint was developed, and the result proved to be accurate.


## 2. Experimental Tests on the Joint Stiffness

Two kinds of specimens were designed. These joints were designed according to the code for design of steel structure [17], and the dimension was referenced from a practical engineer of an automobile repair workshop. According to the code, the column web and flange were designed to satisfy the shear deformation of the panel zone, and for the connection along the minor axis, the thickness of the column was increased to enhance the mechanical performance of the panel zone. The first specimen involved a major-axis interior joint with an extended end-plate connection that was numbered as MZ-1. The other specimen involved a minor-axis interior joint with flush end-plate connections that was numbered as MZ-2. Figure 1 shows the details of the panel zone.Figure 1
Details of the specimens. (a) MZ-1. (b) MZ-2.
(a)
(b)The dimensions of the triangular stiffener on the right beam of MZ-1 corresponded to 75 mm × 75 mm × 10 mm (Table1). Both columns and beams were composed of hot-rolled H-section steel with a strength grade of Q235B. The test mainly includes planar joints under symmetric loading and the material quality testing of the components. The properties of components are listed in Table 2. The high-strength bolt is of 10.9 grade, and the material property of the bolt was referenced from Guo [18] and listed in Table 3. Different connection types on the left and right of the joints were labelled as the left connection and right connection, respectively.Table 1
Geometric properties of the specimens.
Specimen label
Column dimension (mm)
Beam dimension (mm)
End-plate dimension (mm)
Bolt diameter (mm)
MZ-1
H244 × 175 × 7 × 11
H194 × 150 × 6 × 9
344 × 150 × 16
20 mm
MZ-2
H244 × 175 × 7 × 11
H194 × 150 × 6 × 9
194 × 150 × 16
20 mmTable 2
Material properties of steel.
Component
Thickness (mm)
Yield stressfy (MPa)
Tensile stressfu (MPa)
Elastic modulusE (GPa)
Elongation (%)
Column web
7
263.5
467.0
205
22.0
Beam flange
9
262.4
435.3
196
22.7
Stiffener
10
241.8
384.3
208
25.0
Column flange
11
263.1
445.6
206
20.2
End-plate
16
251.6
405.8
196
26.0Table 3
Material properties of the bolt.
High-strength bolt (Grade 10.9)
Yield stressfy (MPa)
Tensile stressfu (MPa)
Elasticity modulusE (GPa)
Yield strainεy
940
1040
206
45.6Figure2 presents the test setup and details of the material test. A reaction frame and a distributive beam were used to fix the top of the column, and a hydraulic jack was used to support the other end. It should be noted that it was necessary for the point of the resultant force of the hydraulic jack and the loading beam to coincide with the centroid of the column. Synchronized loads were applied to both ends of the beams. An oil distributor system was adopted to guarantee the synchronization of loading. Additionally, brackets were installed close to both ends of the beams to prevent lateral buckling.Figure 2
Test setup and the material test. (a) Joint of the interior column loaded. (b) Testing rig. (c) Material test samples. (d) Necking area.
(a)
(b)
(c)
(d)The entire loading process involved unidirectional symmetric loading, and the specific loading process is as follows: (1) The column was first subjected to a prescribed axial force (axial compressive ratio = 0.3) corresponding to 0.3fyAs, where fy and As denote the design strength of steel and the cross-section area of the column, respectively. (2) The ends of the beams were then subjected to unidirectional symmetric forces. Force control was employed prior to the yielding of the specimens. Following this, displacement control was applied at the ends of beams until there was local buckling at the upper flange of the beam, and the specimens could not continue to bear the load.Mechanical parameter results are provided in Table4. The moment-rotation response for all specimens is illustrated in Figure 3. Figure 3(a) shows that large differences existed between the left connection and right connection. These differences were attributed to the stiffener in the end-plate of the right connection. The stiffener enhanced both bearing capacity and stiffness of the joint and guaranteed the rotation capacity of the joint. The test data of the minor-axis end-plate connection are shown in Figure 3(b). This study only presented the test results of the left connection because specimen MZ-2 was bilaterally symmetrical and under symmetric loads. Although the ends of the specimen were asymmetric due to the nonsynchronization of loading positions and loading processes, they were still controllable.Table 4
Test results.
Specimen
Position
Yield moment (kN·m)
Ultimate moment (kN·m)
Yield rotation (rad)
Initial rotation stiffness (kN·m·rad−1)
Max displacement of beam end (mm)
Max load of beam end (kN)
Failure pattern
MZ-1
Left connection
73.7
110.1
0.0081
9879
64
114
Buckling at flange, large deformation of end-plate
Right connection
89.9
121.2
0.0072
13574
78
109
MZ-2
Left connection
32.0
58.7
0.0035
9769
65
60
Pull crack of bolts, large deformation of end-plate
Right connection
—
—
—
—
—
—Figure 3
Moment-rotation curves for specimens. (a) Specimen MZ-1. (b) Specimen MZ-2.
(a)
(b)Wang and Wang [19] reported that the displacement of the beam end has a nonlinear relationship with the rotation angle. The displacement of the beam end in the elastic stage consists of the joint rotation angle and deflection of the beam itself. In the plastic stage, the effect of beam plastic deformation should be considered. Therefore, the moment-rotation relationship of the connection was calculated using the following equation:(1)s=θL+ML23EI,s=θL+θ′L+ML23EI,where s is the vertical displacement of the beam end, θ is the rotation angle of the beam to the column connection, θ′ is the plastic rotation angle of the beam, M is the moment of the beam end, θL is the elastic deformation at the beam end in the elastic stage, θ′L is the plastic deformation at the beam end in the plastic stage, and EI is the bending rigidity of the beam.


## 3. Finite Element Model

A FE modelling software, ANSYS [20], was used to develop three-dimensional FE models of planar joints. The SOLID45 element was used to model the beams, columns, end-plates and bolts of the spatial joints. The element sizes of the bolts and the joint panel zone corresponded to 8 mm, and the element sizes of other parts were in the range of 20–30 mm. A surface-to-surface contact formulation was defined between the different parts of steel and between the bolts and the steel. The target faces and contact faces were modelled by the TARGE170 element and the CONTA174 element, respectively. The friction coefficient between the contact faces was 0.45.The grade of the steel used was Q235B in whichfy = 235 MPa, according to the code for design of steel structure [17]. Tables 2 and 3 provide a summary of the material properties of the H-section steel and bolts. Considering that the high-strength bolt is less ductile as compared with the H-section steel and the yield stress of the high-strength bolt is much higher, two constitutive models were provided to describe the H-section steel and the bolt, respectively. A trilinear isotropic hardening model was used as the constitutive model of the H-section steel. Grade 10.9 (M20) high-strength bolts with a yield strength of 940 MPa were used. A bilinear kinematic hardening model was used as the constitutive model of the bolt, and the value of the modulus of the hardening stage was 10% of the elasticity modulus as shown in Figure 4. For the H-section steel, fy = 260 Mpa, fu = 436 Mpa, εy = 0.001327, and εu = 0.010306. For the bolt, fy = 940 Mpa, fu = 1040 Mpa, εy = 0.004563, and εu = 0.009417.Figure 4
Constitutive model of materials. (a) Constitutive model of H-section steel. (b) Constitutive model of the bolt.
(a)
(b)Figure5 shows the FE models and mesh division. The boundary conditions of the spatial joints in the FE model included fixing the bottom of the column and constraining the horizontal movement of the top of the column. An axial force with a value that was 0.3 times the yielding bearing capacity of the column cross-section was applied on the top of the column. The loading mode corresponded to symmetric loading at the ends of beams.Figure 5
FE model and mesh division. (a) Major-axis connection. (b) Minor-axis connection.
(a)
(b)


### 3.1. Comparison of the Test Results with Finite Element Analysis

Figure6 shows the comparisons of the moment-rotation curves of specimen MZ-1 and specimen MZ-2. The left connection curves were compared to illustrate the effectiveness of the FE simulation. It can be observed that there was good agreement between the test results and the FE simulation in the elastic stage. Additionally, in the stage after yielding, there was some disparity due to the simplification of the constitutive behaviour of the materials and test errors. As shown in Figure 7, the final deformations of the test specimens and the numerical simulation were in good agreement. According to the comparison of curves and deformations, this finite element model had been proven to be sufficiently accurate for simulating these planar joints (Figure 8).Figure 6
Comparisons of the moment-rotation curves. (a) Specimen MZ-1. (b) Specimen MZ-2.
(a)
(b)Figure 7
Comparisons of the final deformation. (a) Right connection of MZ-1 in the test. (b) Right connection of MZ-1 in the FE model. (c) Left connection of MZ-2 in the test. (d) Left connection of MZ-2 in the FE model.
(a)
(b)
(c)
(d)Figure 8
A spatial joint.


### 3.2. FE Models of Spatial Joints

The joints can be classified as three types, namely, middle column joints, edge column joints, and corner column joints, as shown in Figure9. Tables 5 and 6 list the specific sizes and labels of the FE models of planar joints and spatial joints, respectively.Figure 9
FE models of the spatial joints. (a) YZ-1. (b) YZ-2. (c) YZ-3.
(a)
(b)
(c)Table 5
Sizes of the planar joints.
Label
Column section (mm)
Beam in major axis (mm)
Beam in minor axis (mm)
End-plate thickness (mm)
Joint type
Note
PZ-1
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
—
14
Planar joint
1 major-axis beam
PZ-2
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
—
14
Planar joint
2 major-axis beamTable 6
Sizes of the spatial joints.
Label
Column section (mm)
Beam in major axis (mm)
Beam in minor axis (mm)
End-plate thickness (mm)
Joint type
Note
YZ-1
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Corner column
1 major-axis beam, 1 minor-axis beam
YZ-2
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Edge column
2 major-axis beam, 1 minor-axis beam
YZ-3
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Middle column
2 major-axis beam, 2 minor-axis beam


### 3.3. Influence of Different Loading Modes

Spatial loading represents the loading mode that involved concentrated loads at the ends of the beams connected with the joint. Planar loading represents the loading mode that only included the action of unidirectional loading. In order to improve computation efficiency, the FE models of the middle column joints and the edge column joints could be reduced by half based on the structural forms and the force conditions. The FE models and the mesh division of the middle column joints, edge column joints, and corner column joints are shown in Figure9.Figure10 shows comparisons of the moment-rotation curves of the major-axis connections of the spatial joints under spatial loading and planar loading, and Table 7 lists the mechanical properties of the joints. The elastic ultimate moment could be defined by the yielding of components such as a beam flange, beam web, column flange, column web, end-plate, and high-strength bolt. And, the initial stiffness could be calculated by the ratio between moment and rotation at the elastic ultimate stage.Figure 10
Moment-rotation curves of the spatial FE model under different loading modes. (a) YZ-1. (b) YZ-2. (c) YZ-3.
(a)
(b)
(c)Table 7
Mechanical properties of the spatial joints.
Label
Elastic ultimate momentsM (kN·m)
Initial rotation stiffnessKi (kN·m·rad−1)
Plastic ultimate momentsMu (kN·m)
Planar loadingMe1
Spatial loadingMe2
Planar loadingK1
Spatial loadingK2
K
1/K2
Planar loadingMu1
Spatial loadingMu2
M
u1/Mu2
YZ-1
18.96
15.63
12837
13068
1.018
89.43
77.58
0.87
YZ-2
20.99
18.16
25520
25815
1.011
98.14
88.89
0.91
YZ-3
21.58
21.37
26764
26770
1.000
97.09
97.26
1.00As shown in Figure10 and Table 7, in the elastic stage, the initial rotation stiffness and elastic ultimate moments of the spatial joints under planar loading were almost identical to those under spatial loading. The maximum error of the initial rotation stiffness of the spatial corner column joints under different loading modes corresponded to 1.8%. However, the middle column joints and the edge column joints exhibited a greater difference in their bearing capacities under different loading modes. The joints under spatial loading yielded first, and then the plastic development was more rapid than that of the planar joints and the plastic region exceeded more than that of the planar joints. The bearing capacity of the joint under spatial loading was smaller than that under planar loading, and the maximum decrease of the corner column joints corresponded to 13%. The degradation of the bearing capacity of planar joints was slower than that of spatial joints. With respect to the middle column joints, the difference in the behaviour of the spatial load and that of the planar load was small.Furthermore, as shown in Table7, the initial stiffnesses of the middle column joints and edge column joints are larger than that of the corner column joints. With respect to the major-axis connection, the middle column joints and the edge column joints were under symmetric loading while the corner column joints were under unilateral loading. Hence, it was necessary to consider the effect of the shear deformation of the column web in the corner column joints which increased the relative rotation of the connection and reduced the rotation stiffness accordingly. The rotation stiffnesses of the middle column joints and the edge column joints were larger because it was not necessary to consider the effect of the shear deformation of the column web.


## 3.1. Comparison of the Test Results with Finite Element Analysis

Figure6 shows the comparisons of the moment-rotation curves of specimen MZ-1 and specimen MZ-2. The left connection curves were compared to illustrate the effectiveness of the FE simulation. It can be observed that there was good agreement between the test results and the FE simulation in the elastic stage. Additionally, in the stage after yielding, there was some disparity due to the simplification of the constitutive behaviour of the materials and test errors. As shown in Figure 7, the final deformations of the test specimens and the numerical simulation were in good agreement. According to the comparison of curves and deformations, this finite element model had been proven to be sufficiently accurate for simulating these planar joints (Figure 8).Figure 6
Comparisons of the moment-rotation curves. (a) Specimen MZ-1. (b) Specimen MZ-2.
(a)
(b)Figure 7
Comparisons of the final deformation. (a) Right connection of MZ-1 in the test. (b) Right connection of MZ-1 in the FE model. (c) Left connection of MZ-2 in the test. (d) Left connection of MZ-2 in the FE model.
(a)
(b)
(c)
(d)Figure 8
A spatial joint.


## 3.2. FE Models of Spatial Joints

The joints can be classified as three types, namely, middle column joints, edge column joints, and corner column joints, as shown in Figure9. Tables 5 and 6 list the specific sizes and labels of the FE models of planar joints and spatial joints, respectively.Figure 9
FE models of the spatial joints. (a) YZ-1. (b) YZ-2. (c) YZ-3.
(a)
(b)
(c)Table 5
Sizes of the planar joints.
Label
Column section (mm)
Beam in major axis (mm)
Beam in minor axis (mm)
End-plate thickness (mm)
Joint type
Note
PZ-1
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
—
14
Planar joint
1 major-axis beam
PZ-2
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
—
14
Planar joint
2 major-axis beamTable 6
Sizes of the spatial joints.
Label
Column section (mm)
Beam in major axis (mm)
Beam in minor axis (mm)
End-plate thickness (mm)
Joint type
Note
YZ-1
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Corner column
1 major-axis beam, 1 minor-axis beam
YZ-2
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Edge column
2 major-axis beam, 1 minor-axis beam
YZ-3
H160 × 160 × 8 × 12
H240 × 120 × 6 × 9
H180 × 90 × 5 × 8
14
Middle column
2 major-axis beam, 2 minor-axis beam


## 3.3. Influence of Different Loading Modes

Spatial loading represents the loading mode that involved concentrated loads at the ends of the beams connected with the joint. Planar loading represents the loading mode that only included the action of unidirectional loading. In order to improve computation efficiency, the FE models of the middle column joints and the edge column joints could be reduced by half based on the structural forms and the force conditions. The FE models and the mesh division of the middle column joints, edge column joints, and corner column joints are shown in Figure9.Figure10 shows comparisons of the moment-rotation curves of the major-axis connections of the spatial joints under spatial loading and planar loading, and Table 7 lists the mechanical properties of the joints. The elastic ultimate moment could be defined by the yielding of components such as a beam flange, beam web, column flange, column web, end-plate, and high-strength bolt. And, the initial stiffness could be calculated by the ratio between moment and rotation at the elastic ultimate stage.Figure 10
Moment-rotation curves of the spatial FE model under different loading modes. (a) YZ-1. (b) YZ-2. (c) YZ-3.
(a)
(b)
(c)Table 7
Mechanical properties of the spatial joints.
Label
Elastic ultimate momentsM (kN·m)
Initial rotation stiffnessKi (kN·m·rad−1)
Plastic ultimate momentsMu (kN·m)
Planar loadingMe1
Spatial loadingMe2
Planar loadingK1
Spatial loadingK2
K
1/K2
Planar loadingMu1
Spatial loadingMu2
M
u1/Mu2
YZ-1
18.96
15.63
12837
13068
1.018
89.43
77.58
0.87
YZ-2
20.99
18.16
25520
25815
1.011
98.14
88.89
0.91
YZ-3
21.58
21.37
26764
26770
1.000
97.09
97.26
1.00As shown in Figure10 and Table 7, in the elastic stage, the initial rotation stiffness and elastic ultimate moments of the spatial joints under planar loading were almost identical to those under spatial loading. The maximum error of the initial rotation stiffness of the spatial corner column joints under different loading modes corresponded to 1.8%. However, the middle column joints and the edge column joints exhibited a greater difference in their bearing capacities under different loading modes. The joints under spatial loading yielded first, and then the plastic development was more rapid than that of the planar joints and the plastic region exceeded more than that of the planar joints. The bearing capacity of the joint under spatial loading was smaller than that under planar loading, and the maximum decrease of the corner column joints corresponded to 13%. The degradation of the bearing capacity of planar joints was slower than that of spatial joints. With respect to the middle column joints, the difference in the behaviour of the spatial load and that of the planar load was small.Furthermore, as shown in Table7, the initial stiffnesses of the middle column joints and edge column joints are larger than that of the corner column joints. With respect to the major-axis connection, the middle column joints and the edge column joints were under symmetric loading while the corner column joints were under unilateral loading. Hence, it was necessary to consider the effect of the shear deformation of the column web in the corner column joints which increased the relative rotation of the connection and reduced the rotation stiffness accordingly. The rotation stiffnesses of the middle column joints and the edge column joints were larger because it was not necessary to consider the effect of the shear deformation of the column web.


## 4. Initial Stiffness of Spatial Joints with End-Plate Connections

### 4.1. Flexural Stiffness of the Column Flange and End-Plate

An equivalent T-stub was considered to calculate the flexural stiffness of the column flange and the end-plate [21]. A very low level of prying force is produced by bolts in the elastic stage. Prying force usually occurs in the plastic stage, and it is a significant characteristic of the ultimate state of the joint [22]. Hence, it could be considered that the tightening effort of the bolt is continuous in the elastic stage, and based on this assumption, the bolted connection was assumed as a fixed connection and the prying action could be ignored. Figure 11 shows the mechanical model of the simplified equivalent T-stub.Figure 11
Mechanical model of the simplified equivalent T-stub. (a) The transformation of T-stub. (b) Simplified equivalent T-stub.
(a)
(b)The flexural stiffness of the column flange is calculated based on the theory of structural mechanics and material mechanics:(2)kcfb=2FΔ=2×192EIcflcf3=32Eleff,cftcf3lcf3,where Icf=1/12⋅leff,cf⋅tcf3.The flexural stiffness of the end-plate is expressed as follows:(3)kepb=2FΔ=2×192EIeplep3=32Eleff,eptep3lep3,where leff,cf and leff,ep denote the effective lengths of the equivalent T-stubs of the column flange and the end-plate, respectively, which are calculated by a method proposed by Faella et al. [23]. The determined value is shown in Figure 12. Table 8 presents the values of leff,cf and leff,ep. Additionally, tcf and tep represent the thickness of the column flange and the end-plate, respectively.Figure 12
Calculation sketch ofleff.Table 8
Effective length of the equivalent T-stub.
T-stub type
Value
The effective lengths of the equivalentT-stub of end-plate leff,ep
min{2m + dm; m + e + dm/2;m + w/2 + dm/2; m + w/2 + e}
The effective lengths of the equivalentT-stub of column flange leff,cf
min{2m + dm; m + e + dm/2;m + w/2 + dm/2; m + w/2 + e}Furthermore,dm in Figure 12 denotes the effective diameter of the bolt head, and it is calculated as dm = 1.5db where db denotes the nominal diameter of the bolt. Moreover, e denotes the distance between the centre of the bolt and the outer edge of the end-plate or the column flange, m denotes the distance between the bolt axis and the weld foot of the beam flange or the minor-axis connection plate, he denotes the effective height of the welded seam, and rc denotes the radius of the web-to-flange fillet of the column.


### 4.2. Tension and Compression Stiffness of the Column Web

Partial areas between the minor-axis bolts are subject to minor tension and compression forces and involve low elastic strain in the compressive and tensile regions. These components were ignored when compared with the other components that are in high stress. Additionally, the decrease in the calculation height of the column web led to the increase in the initial stiffness of these joints. With respect to the effect of the thickness of the column web, the spread angle of the stress that is transferred from the bolt-hole edge to the centre of the column web was assumed asθ, and its values are presented in equation (4). The areas of the ignored parts of the compressive and tensile regions were then calculated, and the corresponding illustration of the calculation is shown in Figure 13. Table 9 presents the calculated results where db in the table denotes the nominal diameter of the bolt, and the definitions of other symbols can be found in Figure 13.(4a)Fortrep<2tcw,θ=45°,(4b)Fortrep>2tcw,θ=60°,where trep is the thickness of the minor-axis end-plate and tcw is the thickness of the web.Figure 13
An illustration of the calculation of ignored parts of the column web.Table 9
Calculation of the ignored area of the column web.
Tensile region of the column web
Compressive region of the column web
Condition for ignored tensile area
Ignored tensile areaSt
Condition for ignored compressive area
Ignored compressive areaSc
(d − db)tan θc
0.5{(d − db)2 tan θ + db2}
s0.5{beff,c − (d − db)tan θ}
0.5{(d − db)2 tan θ + db2}
(d − db)tan θc
0.25{c(2d − 2db − ccot θ) + 2db2 + (d − db)2 tan θ}
s0.5{beff,c − (d − db)tan θ}s0.5{beff,c + (d − db)tan θ}
0.25{(d − db)2 tan θ + db2}
s0.5{beff,c + (d − db)tan θ}
0Wherec is the transverse bolt pitch of the up flange along the minor-axis, d is the longitudinal bolt pitch of the up flange along the minor-axis, s is the distance from the bolt-hole center of the bottom flange to the bottom flange center.In the table, if the column is made of hot-rolled H steel, thenbeff,c = tbf + 2he,ep + 2tep + 2(tcf + rc). However, if the column is made of welded H steel, beff,c = tbf + 2he,ep + 2tep + 2(tcf + he,c), where he,ep and he,c denote the effective heights of welded seams of the end-plate and column, respectively. Additionally, rc denotes the radius of the web-to-flange fillet of the column. The definitions of he,ep, he,c, tcf, rc, and tbf are shown in Figure 14.Figure 14
Sizes of the column web. (a) Front view of the welded joint. (b) Top view of the welded joint. (c) Top view of the hot-rolled joint.
(a)
(b)
(c)The principle of equivalent volume states that the ignored volume of the column web equals the decreased volume of the tensile and compressive stiffness components. According to this principle, the equivalent calculation height of the column web in tension or compression is calculated as follows:(5a)hcw,t=hcw−Stbeff,t,(5b)hcw,c=hcw−Scbeff,c,where hcw denotes the calculation height of the column web that is given as follows:(6)hot‐rolled H steel column:hcw=hc−2tcf+rc,welded H steel column:hcw=hc−2tcf+he,c.The tensile and compressive stiffness of the column web can be calculated by simplifying the column web as axial tension and axial compression plates. Furthermore, it is necessary to consider the effect of different loading modes on the stiffness of the joint.The compressive stiffness of the column web is calculated as follows:(7)kcwc=Ebeff,ctcwφhcw,c,where hcw,c denotes the equivalent calculation height of the column web in compression; tcw denotes the thickness of the column web; and beff,c denotes the effective height of the column web in compression. According to Eurocode 3 [16], φ denotes the influence coefficient of load and corresponds to 0.5 for symmetric loads, 1 for antisymmetry loads, and 1 for other loads.The compressive stiffness of the column web is expressed as follows:(8)kcwt=λEbeff,ttcwφhcw,t,where hcw,t denotes the equivalent calculation height of the column web in tension. This is equal to λ(p/w)3 based on the principle that states that the distribution of the tension of the bolts depends on the stiffness of the column web and the minor-axis connection plate in conjunction with the principle that states that the column web and the minor-axis connection plate as shown in the present study have the same equivalent tensile deformation. Additionally, λ denotes the influence coefficient of bolt spacing. Each parameter is illustrated in Figure 12. The effective tensile width is shown in Figure 15, and its values are listed in Table 10.Figure 15
Calculation sketch of the effective tensile width of the column web.Table 10
Value of effective tensile width.
Criterion
Value of effective tensile width
m
c(p − dm)/2
b
eff,t2mc + dm
m
c(p − dm)/2
b
eff,t
m
ctan 45°+ (dm + p)/2


### 4.3. Tensile Stiffness of Bolts

The study ignored the effect of the prying force of the bolts in the rotation stiffness calculation. Additionally, it was necessary to modify the suggested formula of the Eurocode to calculate the tensile stiffness of the bolts. Furthermore, the coefficientγ was introduced into the calculation model for high-strength bolts based on the effect of prestress. The value of γ is based on a previous study [24] and generally corresponded to 10. The tensile stiffness of the bolts can then be derived as follows:(9a)kbt=21+γEAsLb,(9b)Lb=tep+tcf+2twh+th+tn2,where As denotes the effective area of the bolt rod and is typically (in engineering) equal to 0.8 times the nominal area of the rod. Lb denotes the calculation length of the bolt. The meaning of thickness variation t can be found in the appendix.


### 4.4. Modification of the Rotation Stiffness Calculation Model of the Spatial Joints

The deformation of the upper connection includes the deformation of the bolts, end-plates, the bending deformation of column flanges, and tensile deformation of the column web. The deformation of the lower flanges could be reflected by the compressive deformation of the column web. The connection was in accordance with the plane section assumption of the steel beam and the assumption that the rotational centre of the joint was located in the center of the lower flanges along the thickness direction. The initial rotation stiffness calculation model of the spatial middle column joint is illustrated in Figure16.Figure 16
Calculation model of the rotation stiffness of the spatial middle column joint.The total deformation of the spatial middle column joints can be divided into three parts, namely, Δt that corresponds to the tensile deformation of the column web and bolts, Δb that corresponds to the bending deformation of the end-plate and column flange, and Δc that corresponds to the compressive deformation of the column web.Based on the above assumptions, the rotation of the joint in the elastic range is given by the following expression:(10)θ=Δt+Δb+Δcheq=Fheq12kcwt+12kbt+1kcfb+1kepb+1kcwc,where heq denotes the equivalent calculation height with a value that is equal to the distance between the centre of the upper flange of the beam and the rotational centre, and it can be expressed as hep = hb − tbf where hb denotes the height of the beam.The calculation of the initial rotation stiffness of the joint is as follows:(11)Kini=Fheqθ=heq21/2kcwt+1/2kbt+1/kcfb+1/kepb+1/kcwc.


### 4.5. Verification of the Rotation Stiffness Calculation Model of the Spatial Joints

Several FE models were built to verify the accuracy of equation (11). The specific forms and sizes of joints are specified in Table 11. For spatial joints of C series, the form of the major-axis connection remains unchanged; the variable parameters along the minor axis include the thickness of the end-plate, the bolt pitch, and the bolt diameter. The minor axis connections of D series are in different forms. The end-plate was extended at both ends, while the variable parameters along the minor axis include end-plate thickness, bolt-hole diameter, and bolt pitch. Comparisons between results by the theoretical analysis and results by FEM analysis are listed in Table 12.Table 11
Geometric properties of the spatial joints.
Sizes of beam and column (mm)
The major-axis end-plate type
Joint label
The minor-axis end-plate sizes (mm)
The minor-axis end-plate type
Spacingd
Distancen
DiameterΦ
End-plate thicknesst
C-series
ColumnH200 × 200 × 8 × 12The major-axis beamH250 × 150 × 6 × 9The minor-axis beamH250 × 125 × 6 × 9The major-axis end-plate340 × 150 × 14
C01
66
40
20
12
C02
76
40
16
12
C03
76
40
24
12
C04
86
40
20
12
C05
86
40
20
20
C06
106
40
20
12
D-series
ColumnH300 × 250 × 10 × 15The major-axis beamH300 × 200 × 8 × 12The minor-axis beamH300 × 150 × 8 × 12The major-axis end-plate thickness18
D01
70
40
20
12
D02
80
40
16
12
D03
90
40
20
12
D04
90
30
20
12
D05
120
40
20
12
D06
90
40
20
22Table 12
Comparison results between the theoretical and FEM analysis.
Joint label
Initial rotation stiffness of the major-axis connection (kN·m/rad)
Theoretical result (T)
FEM result (F)
Error (T − F)/F %
C01
29859
31617
−5.56
C02
31426
31686
−0.82
C03
31479
32603
−3.45
C04
31989
32686
−2.14
C05
33586
33879
−0.87
C06
33510
33220
0.87
D01
57831
60474
−4.37
D02
59046
59662
−1.03
D03
60413
61483
−1.74
D04
54576
65031
−16.07
D05
66599
63448
4.97
D06
64676
61791
4.67As shown in Table12, the results between theoretical calculation and FEM analysis results are in good accordance for most of the spatial joints, with errors less than ±5%. While for few of the joints, the relatively larger errors occurred due to the poor design of bolt spacing, such as for joint D04, there is a large bolt spacing difference between the vertical and horizontal axis. So generally, the theoretical equation proposed in this article is of good applicability for most reasonably designed joints.


## 4.1. Flexural Stiffness of the Column Flange and End-Plate

An equivalent T-stub was considered to calculate the flexural stiffness of the column flange and the end-plate [21]. A very low level of prying force is produced by bolts in the elastic stage. Prying force usually occurs in the plastic stage, and it is a significant characteristic of the ultimate state of the joint [22]. Hence, it could be considered that the tightening effort of the bolt is continuous in the elastic stage, and based on this assumption, the bolted connection was assumed as a fixed connection and the prying action could be ignored. Figure 11 shows the mechanical model of the simplified equivalent T-stub.Figure 11
Mechanical model of the simplified equivalent T-stub. (a) The transformation of T-stub. (b) Simplified equivalent T-stub.
(a)
(b)The flexural stiffness of the column flange is calculated based on the theory of structural mechanics and material mechanics:(2)kcfb=2FΔ=2×192EIcflcf3=32Eleff,cftcf3lcf3,where Icf=1/12⋅leff,cf⋅tcf3.The flexural stiffness of the end-plate is expressed as follows:(3)kepb=2FΔ=2×192EIeplep3=32Eleff,eptep3lep3,where leff,cf and leff,ep denote the effective lengths of the equivalent T-stubs of the column flange and the end-plate, respectively, which are calculated by a method proposed by Faella et al. [23]. The determined value is shown in Figure 12. Table 8 presents the values of leff,cf and leff,ep. Additionally, tcf and tep represent the thickness of the column flange and the end-plate, respectively.Figure 12
Calculation sketch ofleff.Table 8
Effective length of the equivalent T-stub.
T-stub type
Value
The effective lengths of the equivalentT-stub of end-plate leff,ep
min{2m + dm; m + e + dm/2;m + w/2 + dm/2; m + w/2 + e}
The effective lengths of the equivalentT-stub of column flange leff,cf
min{2m + dm; m + e + dm/2;m + w/2 + dm/2; m + w/2 + e}Furthermore,dm in Figure 12 denotes the effective diameter of the bolt head, and it is calculated as dm = 1.5db where db denotes the nominal diameter of the bolt. Moreover, e denotes the distance between the centre of the bolt and the outer edge of the end-plate or the column flange, m denotes the distance between the bolt axis and the weld foot of the beam flange or the minor-axis connection plate, he denotes the effective height of the welded seam, and rc denotes the radius of the web-to-flange fillet of the column.


## 4.2. Tension and Compression Stiffness of the Column Web

Partial areas between the minor-axis bolts are subject to minor tension and compression forces and involve low elastic strain in the compressive and tensile regions. These components were ignored when compared with the other components that are in high stress. Additionally, the decrease in the calculation height of the column web led to the increase in the initial stiffness of these joints. With respect to the effect of the thickness of the column web, the spread angle of the stress that is transferred from the bolt-hole edge to the centre of the column web was assumed asθ, and its values are presented in equation (4). The areas of the ignored parts of the compressive and tensile regions were then calculated, and the corresponding illustration of the calculation is shown in Figure 13. Table 9 presents the calculated results where db in the table denotes the nominal diameter of the bolt, and the definitions of other symbols can be found in Figure 13.(4a)Fortrep<2tcw,θ=45°,(4b)Fortrep>2tcw,θ=60°,where trep is the thickness of the minor-axis end-plate and tcw is the thickness of the web.Figure 13
An illustration of the calculation of ignored parts of the column web.Table 9
Calculation of the ignored area of the column web.
Tensile region of the column web
Compressive region of the column web
Condition for ignored tensile area
Ignored tensile areaSt
Condition for ignored compressive area
Ignored compressive areaSc
(d − db)tan θc
0.5{(d − db)2 tan θ + db2}
s0.5{beff,c − (d − db)tan θ}
0.5{(d − db)2 tan θ + db2}
(d − db)tan θc
0.25{c(2d − 2db − ccot θ) + 2db2 + (d − db)2 tan θ}
s0.5{beff,c − (d − db)tan θ}s0.5{beff,c + (d − db)tan θ}
0.25{(d − db)2 tan θ + db2}
s0.5{beff,c + (d − db)tan θ}
0Wherec is the transverse bolt pitch of the up flange along the minor-axis, d is the longitudinal bolt pitch of the up flange along the minor-axis, s is the distance from the bolt-hole center of the bottom flange to the bottom flange center.In the table, if the column is made of hot-rolled H steel, thenbeff,c = tbf + 2he,ep + 2tep + 2(tcf + rc). However, if the column is made of welded H steel, beff,c = tbf + 2he,ep + 2tep + 2(tcf + he,c), where he,ep and he,c denote the effective heights of welded seams of the end-plate and column, respectively. Additionally, rc denotes the radius of the web-to-flange fillet of the column. The definitions of he,ep, he,c, tcf, rc, and tbf are shown in Figure 14.Figure 14
Sizes of the column web. (a) Front view of the welded joint. (b) Top view of the welded joint. (c) Top view of the hot-rolled joint.
(a)
(b)
(c)The principle of equivalent volume states that the ignored volume of the column web equals the decreased volume of the tensile and compressive stiffness components. According to this principle, the equivalent calculation height of the column web in tension or compression is calculated as follows:(5a)hcw,t=hcw−Stbeff,t,(5b)hcw,c=hcw−Scbeff,c,where hcw denotes the calculation height of the column web that is given as follows:(6)hot‐rolled H steel column:hcw=hc−2tcf+rc,welded H steel column:hcw=hc−2tcf+he,c.The tensile and compressive stiffness of the column web can be calculated by simplifying the column web as axial tension and axial compression plates. Furthermore, it is necessary to consider the effect of different loading modes on the stiffness of the joint.The compressive stiffness of the column web is calculated as follows:(7)kcwc=Ebeff,ctcwφhcw,c,where hcw,c denotes the equivalent calculation height of the column web in compression; tcw denotes the thickness of the column web; and beff,c denotes the effective height of the column web in compression. According to Eurocode 3 [16], φ denotes the influence coefficient of load and corresponds to 0.5 for symmetric loads, 1 for antisymmetry loads, and 1 for other loads.The compressive stiffness of the column web is expressed as follows:(8)kcwt=λEbeff,ttcwφhcw,t,where hcw,t denotes the equivalent calculation height of the column web in tension. This is equal to λ(p/w)3 based on the principle that states that the distribution of the tension of the bolts depends on the stiffness of the column web and the minor-axis connection plate in conjunction with the principle that states that the column web and the minor-axis connection plate as shown in the present study have the same equivalent tensile deformation. Additionally, λ denotes the influence coefficient of bolt spacing. Each parameter is illustrated in Figure 12. The effective tensile width is shown in Figure 15, and its values are listed in Table 10.Figure 15
Calculation sketch of the effective tensile width of the column web.Table 10
Value of effective tensile width.
Criterion
Value of effective tensile width
m
c(p − dm)/2
b
eff,t2mc + dm
m
c(p − dm)/2
b
eff,t
m
ctan 45°+ (dm + p)/2


## 4.3. Tensile Stiffness of Bolts

The study ignored the effect of the prying force of the bolts in the rotation stiffness calculation. Additionally, it was necessary to modify the suggested formula of the Eurocode to calculate the tensile stiffness of the bolts. Furthermore, the coefficientγ was introduced into the calculation model for high-strength bolts based on the effect of prestress. The value of γ is based on a previous study [24] and generally corresponded to 10. The tensile stiffness of the bolts can then be derived as follows:(9a)kbt=21+γEAsLb,(9b)Lb=tep+tcf+2twh+th+tn2,where As denotes the effective area of the bolt rod and is typically (in engineering) equal to 0.8 times the nominal area of the rod. Lb denotes the calculation length of the bolt. The meaning of thickness variation t can be found in the appendix.


## 4.4. Modification of the Rotation Stiffness Calculation Model of the Spatial Joints

The deformation of the upper connection includes the deformation of the bolts, end-plates, the bending deformation of column flanges, and tensile deformation of the column web. The deformation of the lower flanges could be reflected by the compressive deformation of the column web. The connection was in accordance with the plane section assumption of the steel beam and the assumption that the rotational centre of the joint was located in the center of the lower flanges along the thickness direction. The initial rotation stiffness calculation model of the spatial middle column joint is illustrated in Figure16.Figure 16
Calculation model of the rotation stiffness of the spatial middle column joint.The total deformation of the spatial middle column joints can be divided into three parts, namely, Δt that corresponds to the tensile deformation of the column web and bolts, Δb that corresponds to the bending deformation of the end-plate and column flange, and Δc that corresponds to the compressive deformation of the column web.Based on the above assumptions, the rotation of the joint in the elastic range is given by the following expression:(10)θ=Δt+Δb+Δcheq=Fheq12kcwt+12kbt+1kcfb+1kepb+1kcwc,where heq denotes the equivalent calculation height with a value that is equal to the distance between the centre of the upper flange of the beam and the rotational centre, and it can be expressed as hep = hb − tbf where hb denotes the height of the beam.The calculation of the initial rotation stiffness of the joint is as follows:(11)Kini=Fheqθ=heq21/2kcwt+1/2kbt+1/kcfb+1/kepb+1/kcwc.


## 4.5. Verification of the Rotation Stiffness Calculation Model of the Spatial Joints

Several FE models were built to verify the accuracy of equation (11). The specific forms and sizes of joints are specified in Table 11. For spatial joints of C series, the form of the major-axis connection remains unchanged; the variable parameters along the minor axis include the thickness of the end-plate, the bolt pitch, and the bolt diameter. The minor axis connections of D series are in different forms. The end-plate was extended at both ends, while the variable parameters along the minor axis include end-plate thickness, bolt-hole diameter, and bolt pitch. Comparisons between results by the theoretical analysis and results by FEM analysis are listed in Table 12.Table 11
Geometric properties of the spatial joints.
Sizes of beam and column (mm)
The major-axis end-plate type
Joint label
The minor-axis end-plate sizes (mm)
The minor-axis end-plate type
Spacingd
Distancen
DiameterΦ
End-plate thicknesst
C-series
ColumnH200 × 200 × 8 × 12The major-axis beamH250 × 150 × 6 × 9The minor-axis beamH250 × 125 × 6 × 9The major-axis end-plate340 × 150 × 14
C01
66
40
20
12
C02
76
40
16
12
C03
76
40
24
12
C04
86
40
20
12
C05
86
40
20
20
C06
106
40
20
12
D-series
ColumnH300 × 250 × 10 × 15The major-axis beamH300 × 200 × 8 × 12The minor-axis beamH300 × 150 × 8 × 12The major-axis end-plate thickness18
D01
70
40
20
12
D02
80
40
16
12
D03
90
40
20
12
D04
90
30
20
12
D05
120
40
20
12
D06
90
40
20
22Table 12
Comparison results between the theoretical and FEM analysis.
Joint label
Initial rotation stiffness of the major-axis connection (kN·m/rad)
Theoretical result (T)
FEM result (F)
Error (T − F)/F %
C01
29859
31617
−5.56
C02
31426
31686
−0.82
C03
31479
32603
−3.45
C04
31989
32686
−2.14
C05
33586
33879
−0.87
C06
33510
33220
0.87
D01
57831
60474
−4.37
D02
59046
59662
−1.03
D03
60413
61483
−1.74
D04
54576
65031
−16.07
D05
66599
63448
4.97
D06
64676
61791
4.67As shown in Table12, the results between theoretical calculation and FEM analysis results are in good accordance for most of the spatial joints, with errors less than ±5%. While for few of the joints, the relatively larger errors occurred due to the poor design of bolt spacing, such as for joint D04, there is a large bolt spacing difference between the vertical and horizontal axis. So generally, the theoretical equation proposed in this article is of good applicability for most reasonably designed joints.


## 5. Conclusion

This study involved analyzing the interaction between the major axis and the minor axis of spatial joints and the differences in the performances of the spatial joints and planar joints. Finally, a calculation equation for the initial rotation stiffness of spatial joints was presented based on the component method. The key conclusions of the study include the following:(i)
The rotation stiffness of spatial joints under spatial loading typically exceeded that under planar loading. However, the bending moments of the spatial joints under spatial loading were lower than those under symmetric planar loading.(ii)
The initial rotation stiffness and bearing capacity of the spatial joints were significantly higher than those of the planar joints. Mechanical performances between the spatial joints and planar joints are different. The components that contributed to the stiffness of the spatial joints were analysed to derive the stiffness calculation formula of each component based on elastic theory. Additionally, the stiffness of the specific components of the spatial joints was calculated, and compared with the FEM results, a relatively accurate formula of stiffness was presented.(iii)
The calculation model and the theoretical equation of the initial rotation stiffness of joints were rebuilt based on the sources of the deformation of the joints. Furthermore, the accuracy of the calculation formula presented in this study was verified based on calculation examples and several FE models.



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*Source: 8131052-2019-10-30.xml*