Journal of Constructional Steel Research 90 (2013) 13–28

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Journal of Constructional Steel Research

Numerical analyses of semi-rigid joints subjected to bending with andwithout axial force

Huihuan Ma ⁎, Feng Fan, Gengbo Chen, Zhenggang Cao, Shizhao ShenSchool of Civil Engineering, Harbin Institute of Technology, 202 Haihe Road, Nangang District, Harbin 150090, PR China

⁎ Corresponding author. Tel.: +86 18345141582.E-mail addresses: mahuihuan@hit.edu.cn (H. Ma), fanf

chengengbo@hit.edu.cn (G. Chen), caohit@hit.edu.cn (Z. Ca

0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rhttp://dx.doi.org/10.1016/j.jcsr.2013.07.017

a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 November 2012Accepted 11 July 2013Available online xxxx

Keywords:Socket joint systemSpatial structuresSemi-rigid jointsFEA analysesMoment–rotation relationshipParametric studyTheoretical model

Socket joint systems are typical semi-rigid joints used in space structures. This paper presents numerical analysesof semi-rigid joint systems in which the bolts are pretensioned. Eleven numerical models of socket joints arestudied: three of them are subjected to bending and the others are subjected to proportional bending and axialcompression. Moment–rotation relationships of the joints are obtained using models in which the materialand geometric nonlinearity are taken into account. The load-carrying mechanism of socket joint system is inves-tigated in detail through the numerical analysis. Through comparison of the finite element analysis (FEA) andcorresponding test measurements, it is shown that the proposed FEA models can be used effectively to describethe mechanical performance of the semi-rigid joints in spatial structures, including the initial bending stiffness,ultimate bending moment, deformation and the failure mode.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The stiffness of connections has been found to be one of the factorswhich has a great effect on the behavior of space structures, and theeffect has been studied numerically and experimentally. Observationsfrom an earlier study [1,2] confirmed that connection stiffness had aconsiderable effect on the load–displacement behavior of a structure,and considerable material saving can be achieved if the effects of theactual joint properties are considered in design. Aitziber et al. [3,4], Maet al. [5], Fan et al. [6] and Kato et al. [7] verified that the rigidity ofjoint is an important factor that influences the behavior of a single-layer latticed dome. El-Sheikh [8] and Chenaghlou & Nooshin [9]found that the overall behavior and failure mode of a spatial structureare influenced by the bending stiffness of connections. Therefore, thebending stiffness and strength of connections should be consideredwhen analyzing long span and space structures.

The prediction of themechanical behavior of joints is the first step inthe process of analyzing spatial structures with semi-rigid joints. Manyjoint systems in real spatial structures consist of forged steel nodes,sleeves or washers, high strength bolts and end cones, such as theMERO joint systems [1,2,10–12], the aluminum alloy truss connectors[13], the ORTZ joints [3,4,14], the bolt-ball joint systems [5,6,15,16],the S14, D14 and D06 joints [17], the T.U.U.-S ball joints [18] and thespace-truss connectors [19]. All these joints are semi-rigid systems,and they are usually fabricated in factories. There are many merits to

@hit.edu.cn (F. Fan),o), shensz@hit.edu.cn (S. Shen).

ights reserved.

domes with semi-rigid joint systems such as material saving, beautifulappearance, high construction speed and high fabrication accuracy.However, most of the semi-rigid joints are assumed to behave as pinjoints during the design of spatial structures, and they are currentlythought to be too weak to be used in single-layer reticulated domes.One reason for adopting this simplified assumption is that the availableinformation about the actual behavior of the semi-rigid joints in spatialstructures is limited.

The interaction between individual parts of the semi-rigid jointsystem is complex; hence, physical tests are still the main sourcefor obtaining the stiffness of the joints. Experimental studies onseveral joint systems under bending were carried out and theirmoment–rotation curves obtained [2,4,5,8,17]. Two space frameconnectors [15,16] were tested to failure to study their behavior, in-cluding strength and failuremechanism under bending, axial tensionforce and axial compressive force separately. In addition, the effectsof the axial force on the mechanical performance of the joints werestudied experimentally [12,18,20]. Mohammad [12] found that theaxial compressive force can increase the initial slope of the mo-ment–rotation curve and decrease the ultimate moment capacity ofthe connection. Ueki et al. [18] found that the rigidity of the joint in-creased under constant compression and reduced under constanttension. Another method to obtain the stiffness of the joints is byusing theoretical analysis, for example the mathematical model pro-posed for calculating moment–rotation behavior which allows foraxial force effects [12]. Feng Fan et al. [20] carried out the experimen-tal study on two semi-rigid joint systems under bending and axialcompressive force, and found that the axial compressive force hasdifferent effects on different joints.

Wcon

O

Washvex

pen

xer wx sur

ing

wr

So

with rfac

NuNu

cket

e

t

tt nod

Wco

Bol

W

e

ashncav

t

hvher ve ss

withsurfa

En

hahace

d c

a

onePipe

Fig. 1. Socket joint system.

d 1

Hig

t6

L

h st

L3

t 7t 7

1

eng

L2

r

L4

h b

End

t1

con

dt 8

t 8

olt

e

1d 2

t3

d

t2

R1

1

W

00 R

a

2

sher

R1

60

R7

s

R

t4

Soc

5

R75

100

t5

nket eod

R2 R1

Fig. 2. Configurations of the socket joints.

Table 1Geometry parameters of the socket joint systems (dimensions in mm).

Group ID Number High strength bolt Washers

L1 L2 d1 d2 t1 t2 t3

S24 4 75 95 24 32 16 13 16S27 3 65 95 27 35 18 13 16S30 4 70 100 30 41 20 13 16

(a) Test diagram

Fig. 3. Testing setup for specime

14 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

Recently, a number of powerful finite element software packagessuch as ANSYS [21] and ABAQUS [22] have become commercially avail-able; their functions and capabilities are powerful and easy to imple-ment. Software of this type can deal with a wide range of engineeringproblems efficiently and accurately, and it has become another signifi-cant resourcewithwhich to investigate themechanical behavior of con-nections. For the steel beam-to-column connections, extensive FEAmodels have been developed, such as those in [23–29]. However, forspace structures, only a few references on FEA analyses of semi-rigidjoints are available [5,6,12]. A two-dimensional finite element modelof a ball joint system was established to show the distribution of stressin bolts and sleeves under compression, tension and bending respec-tively [12]. A 3D finite element model was established in ANSYS in[5,6] to investigate the bending stiffness of bolt-ball joint systemsunder bending, and the results showed a good agreement with experi-mental results in terms of the bending stiffness and failure mode of thejoints.

End cone

t4 t5 R1 R2 t6 t7 t8 L3 L4

12 8 12 25.5 8 27 13.5 25 5011 8 13.5 27 8 26 13.5 25 5010 8 15 28.5 8 24 13.5 25 50

(b) Picture of testing site

ns under bending moment.

(a) Test diagram (b) Picture of testing site

Fig. 4. Testing setup for specimens under bending and axial compression.

Table 2Loading schemes for specimens.

Joint type Group ID Model ID σN/σM Pre-tension force (kN)

Socket joint S24 S24-A 0 50S24-B 0.17 60S24-C 0.19 50S24-D 0.35 50

S27 S27-A 0 50S27-B 0.19 50S27-C 0.45 50

S30 S30-A 0 60S30-C 0.24 50S30-D 0.37 60

15H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

Compared with the physical tests, FEA models have three mainmerits reflected in the following respects:

(i) It is easy and convenient to investigate the mechanical behaviorof various connections. In real structures, the types and details ofthe connection are numerous. Tests are necessary to get the

a) Specimen subjected to bending and shear force

b)

Fig. 5. Definition of bending

mechanical performance of some connections initially. However,the workload is great if the stiffness of all the joints are obtainedthrough experiments. Besides, experiments are costly and time-consuming compared to FEA. Hence, it is important to establishan efficient and accurate numerical model to investigate themechanical behavior of different joint systems. FEA provides anattractive option for developing a database ofmechanical charac-teristics for various kinds of connections.

(ii) It is possible to undertake large-scale parametric analyses.Many new joints need to be examined to adapt to spatial struc-tures with bigger spans and new styles. This can be realized byoptimizing an existing joint system or designing a whole newjoint system. Whichever method is used, parametric analyseshave to be carried out and then the appropriate joint systemcan be selected. The parameters which need to be analyzed in-clude the geometric details of joints, the grade of the bolts, theyield strength of the steel and the bolt pretension force, etc.Due to themultitude of geometrical andmechanical parameters,FEA provides a reliable and authoritative tool for establishing theeffect of all relevant parameters. A parametric study based on

Specimen subjected to both bending, shear force and axial force

and rotation of joints.

kiD

kiC

kiB k

uA

kuB

kuC

MuB

MuC

MuA

S24-A S24-B S24-C S24-D

(rad)

M (

kN m

)

MuD

kuD

kiA

(a) S24 joints

kiA

kiC

kuAM

uA

kuBM

uB kiB

S27-A S27-B S27-C

(b) S27 joints

M (

kN m

)0.00 0.03 0.06 0.09 0.12 0.15 0.18

0

1

2

3

4

5

1

2

3

4

5

6

7

16 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

FEA is an extremely powerful way to develop a full understand-ing of the complex mechanical behavior of connections, as hasbeen demonstrated in the study of steel beam-to-column con-nections [26,27]. In someways, FEA can investigate the joint sys-tems in greater detail than the physical tests can accommodate.

(iii) FEA can provide additional useful results about the connections.The semi-rigid joints in spatial structures consist of forged steelnodes, sleeves or washers, high strength bolts and end cones.When the joint is subjected to loads, the forces are transferredto the steel nodes through the interfaces between differentparts. The bolts are the key parts for the joint systems, however,the stress distribution is difficult to measure during testing; andthe same is true for the contact behavior in the interfaces. In FEA,the response of bolts can be simulated using a solid elementwithspecial characteristics, and the complex interactions betweensurfaces of the joints can be modeled with surface-to-surfacecontact elements. The stress distribution and the deformationin the joints can be examined during the loading process.

This paper presents a short initial description of the experimen-tal program which is used to provide verification of the FEA andmathematical model. Then, eleven numerical models of the socketjoints are established and analyzed using ANSYS. There are twoloading schemes acting on various models with three subjected tobending and the others subjected to proportional bending andaxial compression. The geometric dimensions, material character-istics and the loading schemes of the FEA models are the same asthose in the test specimens. Both geometric and material nonline-arities are taken into account in the FEA models. FEA of this kindof connection in spatial structures has hitherto not been reported.The experimental and numerical results are compared in order toverify the FEA model.

2. Previous experimental work

2.1. Specimens and instrumentation

A series of test on socket joints was conducted on the socketjoint system. The socket joint systems are composed of hollowball nodes, high strength bolts, washers, end-cones and tubularmembers as shown in Fig. 1. In this joint system, the ball nodesare manufactured as hollow spheres, and the ball nodes have oneor two openings for insertion of the bolts. The bolts go from the

B

A

ui

Mi

Muku

kiM(k

Nm

)

(rad)Φ

Fig. 6.Main characters of the socket joints.

ball node to the end of members. A pair of washers is used at eachbolted connection. A washer with a concave surface is placed out-side of the socket node and a washer with a convex surface is placedinside of the socket node. The end-cones with threaded holes arewelded at both ends of the tubes. While assembling the joints, the

(rad)

(rad)

M (

kN m

)

M

0

1

2

3

4

5

6

7

8

MuA

MuC

k

MuB

kiB

kiC

kiA

(c) S30 joints

kuC

S3 S3 S3

kuA

kuB

30-A30-B30-C

A

ABC

0.00 0.03 0.06 0.09 0.12 0.15 0.18

00.00 0.05 0.10 0.15 0.20 0.25

Fig. 7. Moment–rotation curves of the socket joints.

Table 3Main characteristics of moment–rotation curves.

Group ID Specimen ID σN/σM ki (kN m/rad) Φi 1 × 10−3(rad) Mi (kN m) Φu 1 × 10−3(rad) Mu (kN m)

S24 S24-A 0 101.55 10.2 1.04 59.8 2.83S24-B 0.17 178.94 8.2 1.39 49.4 3.10S24-C 0.20 279.04 2.8 0.99 42.0 3.20S24-D 0.35 259.98 1.8 1.16 34.8 3.30

S27 S27-A 0 121.87 8.0 1.10 125.5 4.30S27-B 0.20 129.17 13.2 1.52 104.6 5.26S27-C 0.45 144.23 17.0 2.45 – –

S30 S30-A 0 218.16 4.4 1.09 76.7 5.37S30-B 0.24 241.47 10.8 1.54 111.3 6.04S30-C 0.37 322.75 4.4 1.71 56.61 6.33

Contact surfaces

Fig. 9. Contact surfaces of joints.

17H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

bolt tension force is generated to ensure the proper bearing supportof each part of the joints. The configurations of the socket joints areshown in Fig. 2 and the corresponding parameters are listed inTable 1.

Ten socket joints in three groups (S24, S27, S30) were testedunder different loading schemes. Two experiments were designedaccording to different loading schemes to obtain the bending–rotationcurves of the joints. In the first experiment, the specimens weresubjected to bending, as shown in Fig. 3. The joint specimens inthis test were regarded as a simple beam supported on the con-crete columns. The bending at the joint can be obtained throughthe load acting at the center of the specimen. In the second exper-iment, the specimens were designed to be subjected to forces intwo directions, as shown in Fig. 4. The bending at the joint wasproduced by the horizontal hydraulic jack. The specimens in thisexperiment were about half the size of those subjected to purebending moment. Following the experimental study [20], the loadingschemes in Table 2 are examined in the FEA models. Firstly, for eachgroup of joints, the stiffness of the joint under bending is obtained.Then, several other specimens in each group are modeled under differ-ent ratios of bending and axial compressive forces. The bending mo-ment and axial compressive force are gradually increased together.σN/σM in the table means the ratio of bending stress to the axial stressat the joint, and the ratio is constant during the loading. The larger thevalue of σN/σM, the larger the axial compressive force acting on thespecimen.

FEA model

Fig. 8. The models of

2.2. Definition of the moment–rotation curves

The mechanical behavior of the connections can be represent-ed by M–Φ curves that describe the relationship between the ap-plied bending M and the corresponding rotation Φ of the joint, asshown in Fig. 5. The bending moment M, acting on the connec-tion, includes two parts. One part is equal to the product of theapplied load F and the distance between the load application

SSpecimen

the socket joints.

Table 4Contact surfaces in the model.

Number Contact surface pairs

Simulated by element TARGE170 Simulated by element CONTA174

1 Bolt shank End cone2 Bolt shank Outer washer3 Bolt shank Inner washer4 Bolt shank Socket node5 Socket node Outer washer6 Socket node Inner washer7 End cone Outer washer8 Nut Inner washer

Φ (rad)

M (

kN m

)

6

5

4

3

2

1

00.00 0.03 0.06 0.09 0.12 0.15

S27-A

S27-C

0)

0.45)

Fig. 11. Comparison of moment–rotation curves of S27-A and S27-C joints.

18 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

point and the connection point, L, and the other part M′ is in-duced by the weight of the reaction girder on top of the hydraulicjack, as shown in Fig. 5. The rotation deformation, Φ, is calculatedas follows:

Φ ¼

Xnk¼1

Φij

nð1Þ

Φij ¼ arctanδi−δ j

lij

!ð2Þ

where δi and δj are the displacements measured at the points iand j (points 1–4 or 5–8 for Fig. 3(a), and points 1–3 or 4–5 forFig. 4(a)), and lij is the distance between the two points i and j.

Strain (με)

Strain (με)

Stre

ss (

MPa

)

(a) Pipes

(c) Node balls

0

100

200

300

400

Stre

ss (

MPa

)

0

100

200

300

400

0 1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

Fig. 10. Stress–strain curves for d

2.3. Moment–rotation relationships(M–Φ curves) obtained throughexperiments

TheM–Φ curves reflect themain features of the joints, such as initialbending stiffness and bendingmoment resistance, as shown in Fig. 6. Inthe figure, ki means the initial bending stiffness of the joint; Mi is theelastic flexural resistance, which corresponds to the highest point A inthe elastic phase of the curve;Mu means the ultimate bending momentof the joint, which corresponds to the point B; the value of bending stiff-ness at point B is expressed as ku = 10%ki.

The rotation responses for the ten specimens subjected to bending,together with those subjected to bending and axial compression are

Strain (με)

Strain (με)

0

100

200

300

400

500

Stre

ss (

MPa

)

(b) End cones

(d) High strength bolts

0

200

400

600

800

1000

1200

Stre

ss (

MPa

)

0 5000 10000 15000 20000 25000

0 1000 2000 3000 4000 5000

ifferent parts of socket joints.

Table 5Values of rotation and bending moment at the key points of curves.

Specimen ID ki (kN m/rad) Step 1 (k1/ki = 95%) Step 2 (k2/ki = 50%) Step 3 (k3/ki = 25%) Step 4 (k4/ki = 10%) Step 5

Ф1 × 103

(rad)M1 (kN m) Ф2 × 103

(rad)M2 (kN m) Ф3 × 103

(rad)M3 (kN m) Ф4 × 103

(rad)M4 (kN m) Ф5 × 103

(rad)M5 (kN m)

S27-A (σN/σM = 0) 115 8.31 0.93 24.0 2.34 44.0 3.33 64.1 3.85 144.1 4.52S27-D (σN/σM = 0.45) 158 7.88 1.22 20.6 2.72 41.7 3.92 62.6 4.52 107.2 4.98

19H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

given in Fig. 7. The main characters of the joints are summarized inTable 3.

From the experimental results of the socket joints, it can be seenthat: (1) the response of the connections exhibit linear behavior inthe early loading sequence and non-linear characteristics thereaf-ter, which embodies typical elasto-plastic behavior; (2) the initialbending stiffness and ultimate bending moment of the jointincreases with the increasing axial compression. Finding out thereason of the effect caused by the axial compression is not easywith the experiments, because the stress development of somekey parts of the joints cannot be monitored during loading in theexperiments. In this paper, based on the FEA simulation, the load-carrying mechanism of the socket joint is investigated, and the effectof the axial compression on the mechanical behavior of the socketjoint is explained.

3. Finite element analyses

3.1. Element chosen

The FEA of the connections in this paper is conducted using ANSYS, afinite element software package. According to the features of the socket

(a) Joint deformation

(c) Stress distribution in inner washer

(e) Stress distrib

Fig. 12. The force transmission mechanism o

joint system, three main finite elements used in the FEA models of thesocket joint are:

• All elements of the socket node, washers, end cone, pipe and highstrength bolt are meshed by the tetrahedral solid structural ele-ment SOLID92. The element is defined by ten nodes each havingthree degrees of freedom namely: translations in the nodal x, y, andz directions. It has plasticity, creep, swelling, stress stiffening, largedeflection, and large strain capabilities.

• The important interfaces between each two parts in the FEA modelsare simulated by creating contact pairs with the two 3-D elementsTARGE170 and CONTA173. CONTA174 is used to represent contactand sliding between 3-D “target” surfaces (TARGE170) and a deform-able surface, defined by this element. The element is applicable to 3-Dstructural and coupled field contact analyses. Elements TARGE170and CONTA174 are located on the surfaces of 3-D solid or shell ele-ments without mid-side nodes, and they have the same geometriccharacteristics as the solid or shell element faces with which they areconnected. The coefficient of friction for the contact surfaces in socketjoints is taken as 0.44 (as defined in the steel joint systems [27]).

• Element PRETS179 is used to apply the pretension force in the joint.When assembling the test specimens, the pretension force is generated

(b) Force transfer

(d) Stress distribution in outer washer

ution in bolt

f socket joints under bending moment.

(a) S27-A

20 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

in the joint by tightening the bolt with wrenches [20]. In the FEAmodel, a pretension section is defined in the bolt by elementPRETS179 to simulate the pretension force in the joint. PRETS179 isused to define a 2-D or 3-D pretension sectionwithin ameshed struc-ture; it has one translation degree of freedom, UX which representsthe defined pretension direction. Loads can be applied using theSLOD command.

(b) S27-C

Fig. 14. Development of the gap in the joints.

3.2. Finite element model

Since the geometry of the socket joint under bending (Fig. 3(a))is symmetric, only half of the specimen is modeled. The 3-D finiteelement model of the socket joints under bending is the same asthat under bending and axial compressive force, as shown inFig. 8. For the socket joints under bending, the axial compressiveforce N in the model is equal to zero. The boundary condition ofthe joints under different loading schemes is the same: the socketnode is fixed and the forces act on the end of the pipe. The force Ncan rotate with the model and the direction of the axial compres-sion always points to the joint center along the centerline of thepipes; hence, the secondary moment caused by the axial force canbe eliminated during the loading, as it did in the experimental spec-imens [20]. The contact surfaces in the model are shown in Fig. 9.The surface names and the element chosen for different surfacesare listed in Table 4.

3.3. Geometric details and material properties

All the geometric details and the material properties of the connec-tions used in the FEA are the same as those in the experimental speci-mens. The stress–strain relationship for the pipes, end cones andsocket nodes are taken to be the bilinear kinematic hardening model.The yield strengths and elastic modulus of the pipes are taken as254 MPa and 2.06 × 105 MPa respectively; those for end cones being

(a) Joint deformation

(c) Stress distribution in inner washer

(e) Stress distributi

Fig. 13. The force transmission mechanism of so

405 MPa and 2.06 × 105 MPa; those for socket nodes being 328 MPaand 2.06 × 105 MPa. The stress–strain relationships of these parts inthe joint finite element model are shown in Fig. 10(a)–(c). The stress–strain relationship for the high strength bolts (including the boltheads, shanks and nuts) is taken as multi-linear kinematic hardeningmodel. The whole stress–strain curve for the bolts consists of fourstages, as shown in Fig. 10(d). Von Mises' yield criterion is adopted forall steel components and the flow rule was adopted following yielding.All of thesematerial properties are taken from test data reported by theauthors [20].

3.4. Example analyses

The moment–rotation curves of joints S27-A (just under bendingmoment) and S27-C (under bending and axil compression) are shownin Fig. 11. The Geometry and material characteristics of the two jointsare same. Just as the experimental results, the value of initial bending

(b) Force transfer

(d) Stress distribution in outer washer

on in bolt

cket joints under bending and compression.

Fig. 15. Sectional drawing of the bolt and washers.

21H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

stiffness kuc and ultimate bending moment Muc of joint S27-C is higherthan that of joint S27-A. Five steps on the curves are chosen to investi-gate the load-carrying mechanism of the socket joints, as shown in

Fig. 16. Development of the stress i

Fig. 4. Step1 means that the bending stiffness at this step is 95% of theinitial bending stiffness of the joint. The bending stiffness begins toweaken; Steps 2 and 3 mean that the bending stiffness at these stepsis 50% and 25% of the initial bending stiffness of the joint separately;Step 4means that the bending stiffness at these steps is 10% of the initialbending stiffness of the joint. The bending stiffness is small at Step 4 andthe bending moment is quite near to the ultimate bending moment;Step 5 is the last loading step. The bending moments and rotations ateach step are listed in Table 5.

The force transfermechanismof socket joint under bendingmomentis shown in Fig. 12. As shown in Fig. 12(b), the compressive force on thecompressive side of the joint caused by bending moment is transferredby the contact surface between the outer washer and end cone, and thecontact surface between the outer washer and ball node; the tensileforce on the tensile side of the joint caused by bendingmoment is trans-ferred by the friction force f between the bolt shank and end cone, thecontact surface between the inner washer and nut, and the contact

n the 1–1 cross-section drawn.

Fig. 16 (continued).

22 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

surface between the inner washer and ball node. The contact surfacesbetween the outer washer and end cone separate on the tensile sideof the joint at the beginning of loading, and the gap increases with theincreasing of bending moment. The inner washer and outer washer areboth compressionmember with eccentricity, and the stress distributionsare shown in Fig. 12(c)(d). The bolt is tensile member with eccentricity,and the stress distribution is shown in Fig. 12(e). With the increasing ofthe bending moment, the yield area of the compressive side of thewashers and of both the tensile and compressive side of bolt increase,and finally, the joint achieves the ultimate strength.

The force transfermechanismof socket joint under bendingmomentand axial compression is similar with the joint under bending moment,as shown in Fig. 13. However, the axial compression can counterbalancesome tensile force caused by the bending moment at the tensile side ofthe joint, and postpone the separate of the contact surfaces between theouter washer and end cone. Therefore, at the same bending moment,the rotation of the joint under bending moment and axial compressionis smaller than the joint under bendingmoment, and the initial bendingstiffness is enhanced. As shown in Fig. 13(c–e), compared with the jointunder bending moment, the stress in the inner washer and bolt causedby the tensile force is smaller, and the stress in the outer washer causedby the compressive force is bigger.

The deformations of joints S27-A and S27-C at the five steps areshown in Fig. 14. The deformation development of the two joints issimilar. The gap between the outer washer and end cone, and the gapbetween outer washer and ball node increase with the step number.

The cross-section draws of the bolt andwashers are chosen to inves-tigate the stress development in each part of the joints in detail, asshown in Fig. 15. The critical stress positions of bolt shank, outerwasherand inner washer are marked with characters A, B and C separately.Point A is on the tensile side of bolt shank, and points B and C are onthe compressive side of outer washer and inner washer.

Figs. 16 and 17 show the stress development in 1–1 cross-sectionand 2–2 cross-section (shown in Fig. 15) separately, and Figs. 18 and19 show the stress development in the inner washer and outer washerseparately. From the figures, it can be seen that (1) the stress develop-ment and distribution in joints S27-A and S27-C is similar; (2) at step1, the maximum stress 487 MPa in joint S27-A and 465 MPa in jointS27-C occur at point A in the tensile side of the bolt shank, as shownin Fig. 16(a); (3) at step 2, along with the load added, the stress valuein the bolt shank increases, and the stress in some area has reachedthe yield stress. The critical stress positions B and C in the compressivearea of the washers have reached the yield stress too, as shown inFig. 16(b) and Fig. 17; (3) at step 3, the bending stiffness is 25% of theinitial bending stiffness. Almost all the cross-section area in the com-pressive side of the inner washer has yielded. The yield area in the com-pressive side of the outer washer increases, as shown in Fig. 16(c). Atthis step, about 60% of the area of the 2–2 cross-section in the boltshank yield, as shown in Fig. 17; (4) at step 4, the bending stiffness is10% of the initial bending stiffness. All the cross-section area incompressive side of the inner washer has yielded and the yield area inthe compressive side of the outer washer increases, as shown inFig. 16(d). At this step, about 75% of the area of the 2–2 cross-sectionin the bolt shank yield, as shown in Fig. 17; (5) at step5, the yield areaof each part increases modestly. The stress level and distribution aresimilar with that at step 4.

From the above investigation, the joint achieves that the ultimatestrength is mostly due to two reasons: (1) most area in 2–2 cross-section of the bolt shank yield; and (2) all the cross-section area in com-pressive side of the innerwasher has yielded. That is to say, the bolt andinner washer are the key parts affecting the ultimate bending moment.When the joint subjects to bending moment and axil compression, thecompressive force is advantageous to slow the progress of the stress inbolt shank and inner washer. As shown in Fig. 20, when the bending

(a) S27-A

(b) S27-C

Fig. 17. Development of the stress in the 2–2 cross-section drawn of the bolts.

23H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

moment is 2 kN m and 3 kN m, the yield area in bolt and inner washerof S27-C is smaller than that of S27-A. Therefore, the ultimate bendingmoment of S27-C is higher than that of S27-A.

3.5. Comparison of the results from FEA and experiments

The moment–rotation curves obtained through FEA and tests ofjoints S27-A and S27-C are shown in Fig. 21. From the curves, it can beseen that the curves obtained numerically and experimentally areextremely close. Comparisons of the failure modes of the FEA modelsand the test specimens are shown in Fig. 22. The figures also show thecomparisons of the deformations of the high strength bolt, washersand end cone after loading. It can be noted that the surfaces betweensocket node, outer washer and end cone are detaching on one sidebecause of the tensile force but remain in contact on the other side dueto the pressure.

Themoment–rotation curves obtained using FEA and those from thetests for other connection specimens listed in Table 4 are shown inFig. 23.

4. Discussion

From the bending–rotation curves shown in Fig. 23, it can be seenthat the response of the connections exhibit linear behavior in theearly loading sequence and non-linear characteristics thereafter, whichembodies typical elasto-plastic behavior. For most of the connections,the curves obtained numerically and experimentally are extremely

close; although for some of the connections, the agreement betweenthe FEA and test results in the nonlinear range of the curves has minordiscrepancies. The differences between numerical and experimentalcurves may be due to a variety of reasons but is often a direct conse-quence of the simplifications introduced in the numerical modeling.Firstly, the values of the applied pretension force in the bolts of FEAmodels have somedifferences to those in the test specimens. It is difficultto induce a precise predetermined bolt force by pretension techniquesbecause the pretension force is high and the construction of the joint iscomplicated. As a consequence, this has been identified as a reason fordiscrepancies between FEA and test results. Secondly, fabrication errorsin the test specimens and imperfections originated from the assemblyof the test specimens can lead to a deviation between the physical andnumerical results.

Based on the results above, it can be concluded that the FEAmethoddescribed here can simulate the bending moment–rotation behaviorand failure mode of the connections with acceptable accuracy.

5. Conclusions

(1) FEA models for investigating the mechanical performance of thesocket joint subjected to bendingwith andwithout the axial com-pressive force have been established in this paper. From the re-sults of the FEA and tests, the following conclusions can be made:

• The results obtained through the FEAmodels considering the effectof axial compressive force and pretension force comparedwellwithtest results.

(a) S27-A

(b) S27-C

Fig. 19. Development of the stress in the outer washers.

(a) S27-A

(b) S27-C

Fig. 18. Development of the stress in the inner washers.

24 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

S27-A (N=0) S27-C

(a) M=2kN·m

S27-A (N=0) S27-C

(b) M=3kN·m

Fig. 20. Development of the stress of the section 1–1.

25H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

• The FEA model can accurately and efficiently simulate and analyzethe overall behavior of connections of this type, including the bend-ing–rotation relationships, contact phenomena during loading andthe failure mode of the joints.

• The combination of bending, axial compressive force and thepretension force in the bolt, which has not been included in theFEA of such connections before, are simulatedwell with the presentmodeling.

0.000

1

2

3

4

5

6

M (

kNm

)

Muu

0.05

ki=1

(a) S27-A (N=0)

15kN

0.10

N m

0.15

ku

TeFE

estEA

(rad)0.20

Fig. 21. Comparison of moment–ro

• The FEA allows for further parametric analyses of semi-rigidjoints to obtain comprehensive results that can be used to de-velop a database of mechanical characteristics for semi-rigid jointsystems.

(2) The load-carrying mechanism of socket joint system is investi-gated in detail through the numerical analysis. From the results,the following conclusions can be made: the axial compressioncan postpone the separate of the contact surfaces between

(rad)

m)

0.000

1

2

3

4

5

6

M

M (

kN

Mu

ku

0.02

(b) S227-C (

0.04

ki==158

0.06

8kN m

N/

m

M=0.45)

0.08

TestFEA

0.10

tA

tation curves of socket joints.

Test

Test

FEA

(b) S277-C (σN/σM=0.45)

(a) S27-A(N=0)

Test

FEA

FEA

Fig. 22. Comparison of deformation of socket joints.

26 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28

the outer washer and end cone, therefore, the initial bendingstiffness increases with the increase of axial compression;the bolt and inner washer is the key parts affecting the ultimatebending moment. The compressive force is advantageous toslow the progress of the stress in bolt shank and inner washer.Therefore, the ultimate bending moment increases with the in-crease of axial compression.

Acknowledgments

This work is supported by the Science Foundation of HeilongjiangProvince of China for Young Scholar (Grant no. QC2010109).

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moment–rotation curves.

Fig. 23 (continued).

28 H. Ma et al. / Journal of Constructional Steel Research 90 (2013) 13–28