Engineering Structures 30 (2008) 3114–3118

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Analysis of the steel frames with the semi-rigid beam-to-beam andbeam-to-column knee joints under bending and axial forcesAlfonsas Daniunas ∗, Kestutis UrbonasDepartment of Steel and Timber Structures, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

a r t i c l e i n f o

Article history:Received 18 January 2008Received in revised form11 April 2008Accepted 21 April 2008Available online 29 May 2008

Keywords:Steel semi-rigid jointsKnee jointsBending momentAxial forceComponent methodJoints’ characteristicsSteel frameworks analysis

a b s t r a c t

This paper presents an analysis of framework structures with semi-rigid joints. The semi-rigid end-platebolted joints are subjected to bending and tension or a compression axial force. Usually the influence ofan axial force on joint rotational stiffness is neglected. In some cases, the level of axial forces in the jointsof structures can be significant and has a profound influence on characteristics of semi-rigid joints. Oneof the most popular practical methods permitting the determination of rotational stiffness and momentresistance of a joint is the so-called component method. The extension of the component method forevaluating the influence of bending moment and axial force on the rotational stiffness and momentresistance of a joint and an application to the analysis of framework structures are presented in the paper.The numerical results of calculations of steel frameworks are presented in this paper as well. The studyshows that the estimation of axial force influence to rotation stiffness characteristics of the joint hassignificant influence on the distribution of internal forces and displacements of steel framework.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

One of the possibilities to improve the accuracy of steelframework analysis can be an evaluation of the actual behaviorof joints. For such purposes, the semi-rigid joint conception wasintroduced.

The most relevant influence on characteristics of the joint isproduced by bending moments. In this case, the behavior of a semi-rigid joint is described by moment–rotation M–Φ curve of the joint.This conception is adopted by most researchers [1–6] and designcode EC3 [7]. Joints are also influenced by axial and shear forces.These forces are not usually evaluated in calculating the rotationalstiffness and moment resistance of the joints.

There are many different types of joints in which the axialforce may be significant and it has a profound influence on jointrotational stiffness and moment resistance. The influence of anaxial force on characteristics of the joint has not been analysedwidely and it has not been completely investigated yet [8–10].

Various methods may be used for the establishment ofmoment–rotation M–Φ curves of semi-rigid joints [3]. The mostcommon method applied in practice is the so-called mechanicalcomponent method. It estimates the mechanical properties ofthe material and geometrical properties of the joint and allows

∗ Corresponding author. Tel.: +370 5 274 50 03; fax: +370 5 270 01 12.E-mail address: alfonsas.daniunas@adm.vgtu.lt (A. Daniunas).

0141-0296/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.04.027

calculating the rotational stiffness and moment resistance ofthe joint [3,7,11]. The component method is developed forevaluation of the bending moment influence on moment–rotationcharacteristics of joint.

This paper presents extension of the component method tosemi-rigid beam-to-beam end-plate bolted and beam-to-columnend-plate bolted knee joints under bending and axial force. Themain focus is made on an estimation of the axial force influence inaddition to bending moment influence on the joint’s rigidity andstrength.

Estimation of real behavior of the joints has significantinfluence to the behavior results of the steel frameworks [12–15]. In this article, the iterative procedure for the calculationof steel frameworks with semi-rigid joints is presented. In eachiteration, the steel frame is analysed using joint moment–rotationcharacteristics, which are calculated using the bending momentsand axial forces, obtained in the previous iteration.

2. The evaluation of bending moment and axial force in jointsby component method

The joints of steelworks are acted by bending moments as wellas axial and shear forces. The influence of these forces to thebehavior of joint depends mostly on the type of the joint. From thepractical point of view, the investigation of total bending momentand axial force influence on joint characteristics is importantfor end-plate beam-to-beam knee, beam-to-column knee and

A. Daniunas, K. Urbonas / Engineering Structures 30 (2008) 3114–3118 3115

Notation

E modulus of elasticityFc compression force in componentsFc,i,Rd resistance of the weakest compressed component iFc,max maximal compression force in componentsFc,max,i,Rd resistance of the weakest compressed component i

of the joint, when both sides of the joint are undercompression forces

Ft tension force in componentsFt,max maximal tension force in componentsFt,max,r,Rd effective tension resistance of bolt row r in tension

part of the joint, where the maximal tension force isacting

Ft,max minimal tension force in componentsFt,min,i,Rd resistance of the weakest component i in tension

part of the joint, where the minimal tension forceis acting

Ft,r,Rd effective tension resistance of bolt row rM bending momentMj,Rd moment resistanceN axial forceSj,ini initial rotational stiffness of the jointV shear forcehr distance from bolt row r to the centre of compres-

sionkeq equivalent stiffness coefficient of two or more bolt-

rows in tensionzc distance between the centre of the connection and

the centre of compressionzeq equivalent lever armzt distance between the centre of the connection and

the centre of tensionz′t distance between centre of connection and bolt row

in the lower part of connectionΦ rotation of joint∆t,eq equivalent tension deformation∆t,max,eq equivalent tension deformation of components

under maximal tension force∆t,min,i tension deformation of components under minimal

tension force

column-to-column joints. Some of them are presented in Fig. 1. Insome framework structures with these types of joints axial forcesmay be significant and may strongly influence joint rotationalstiffness and moment resistance.

In this paper the procedure, which evaluates the behavior ofthese joints under the bending moment and axial force is based onthe component method. The evaluation of the behavior of joint inthe component method consists of three main steps: identificationof the basic component, evaluation of the mechanical propertiesof the components, and assembling active components to onemechanical model [10]. According to the component method,the characteristics of the joint must be determined by the basiccomponents of the joint [16].

The component method allows to characterise all componentsof the joint independent from the type of loading applied to thejoint. An elastic stiffness coefficient and resistance that belongsonly to mechanical and geometrical data of the joint represent eachcomponent.

Axial force increases or decreases the forces in componentsfrom the bending moment. This property allows componentmethod suggested to evaluate joint behavior under the bendingmoment to extend for evaluation joint behavior under bendingmoment and axial force.

Mechanical model of the joint depends on values and directionsof acting forces [17,18]. There are three possible cases: (a)components are under compression or under tension whenone part of the joint is in tension and the other one is incompression Fig. 2(b); (b) no components are under tensionwhen both sides of the joint are in compression Fig. 2(c); (c)components are under tension when both sides of the joint are intension Fig. 2(d).

Tension and compression force in components, when one partof the joint is under tension and the other under compressionforce Fig. 2(b) can be expressed by the formula:

Ft =M

zeq+

(N · cos α± V · sin α) · zczeq

;

Fc =M

zeq−

(N · cos α± V · sin α) · ztzeq

,

(1)

where M is accepted as always positive;(N · cos α± V · sin α)expresses value of joint axial force which is perpendicular toend-plate; N is axial force in connected element and accepted aspositive for tension case; ‘±’ means ‘+’ or ‘−’ and according tothis proposition normal to the end-plate of shear force (±V · sin α)accepted as positive if it causes the tension of the joint.

Equivalent deformation of components under tension can befound by the formula:

∆t,eq =

Mzeq+

(N·cos α±V·sin α)·zczeq

E · keq=

M + (N · cos α± V · sin α) · zcE · keq · zeq

(2)

where keq is equivalent stiffness coefficient of two or more bolt-rows in tension and zeq is equivalent lever arm.

The rotation of the end-plate joint depends on the deformationof all tensed components i:

Φ =∆t,eq

zeq. (3)

The initial rotational stiffness of the joint is expressed by theratio between bending moment and rotation of the joint:

Sj,ini =M

Φ=

M · zeq∆t,eq

. (4)

Maximal compression force in components, when both sides ofthe joint with symmetric section elements are under compressionforces Fig. 2(c), can be expressed by:

Fc,max =M

2zc−

(N · cos α± V · sin α)

2. (5)

In such case, there are no components in tension and the jointaccording to EC3 [7] in such type of loading behaves as ideally rigid(see formulas (3) and (4)).

Maximal and minimal tension forces in components, when bothsides of the joint are under tension forces Fig. 2(d) can be expressedby:

Ft,max =M

(zt + z′t)+

(N · cos α± V · sin α) · z′t(zt + z′t)

;

Ft,min = −M

(zt + z′t)+

(N · cos α± V · sin α) · zt(zt + z′t)

.

(6)

Deformation of components under maximal and minimaltension forces can be found the following way:

∆t,max,eq =

M(zt+z

′t)+

(N·cos α±V·sin α)·z′t(zt+z

′t)

E · keq;

∆t,min,i =

−M

(zt+z′t)+

(N·cos α±V·sin α)·zt(zt+z

′t)

E · kt,i.

(7)

3116 A. Daniunas, K. Urbonas / Engineering Structures 30 (2008) 3114–3118

Fig. 1. Semi-rigid end-plate joints: (a) column-to-beam knee joint; (b) beam-to-beam knee joint; (c) beam-to-beam joint.

Fig. 2. Loading schemes and mechanical models of joint: (a) beam-to-beam kneejoint with positive direction of internal forces; (b) upper part of joint in tension andlower part in compression; (c) both sides of joint in compression; (d) both sides ofjoint in tension.

The initial rotational stiffness of the joint for this case of loadingis expressed by:

Sj,ini =M

Φ=

M(zt + z′t)

∆t,max,eq −∑

∆t,min,i. (8)

The above presented procedure for stiffness calculation is onlyapplicable in the case where the behavior of the structure is elastic.

The moment resistance of the joint loaded by bending and axialforces can be expressed as a function of axial force (the force thatis perpendicular to end-plate) and resistance of the components.When one part of the joint is under tension and the other one isunder compression force Fig. 2(b), the moment resistance can beexpressed by the formula:

Mj,Rd = min{∑

(Ft,r,Rd · hr)− (N · cos α± V · sin α) · zcFc,i,Rd · zeq + (N · cos α± V · sin α) · zt

}(9)

where hr— distance from bolt row r to the centre of compression;∑— indicate the sum according to r, where r is the number of bolt

rows in tension part of joint.When both sides of the joint are under compression forces

Fig. 2(c), the moment resistance can be expressed by the formula:Mj,Rd = zc (2Fc,max,i,Rd + (N · cos α± V · sin α)) . (10)

When both sides of the joint are in tension forces Fig. 2(d), themoment resistance can be expressed by the formula:

Mj,Rd = min{∑

(Ft,max,r,Rd(hr − zc + z′t))− (N · cos α± V · sin α)

× z′t (N · cos α± V · sin α) · zt − Ft,min,i,Rd(zt + z′t)}. (11)

where Ft,max,r,Rd—effective tension resistance of bolt row r intension part of the joint, where the maximal tension force is acting;Ft,min,i,Rd—resistance of the weakest component i in tension part ofthe joint, where the minimal tension force is acting.

Fig. 3. Flowchart of framework analysis with semi-rigid joints under bending andaxial forces.

3. Analysis of steel frames with semi-rigid joints includinginfluence of bending moments and axial forces

In this chapter, the procedure for calculation of steel frameswith semi-rigid joints suggested by the authors is presented. Thisprocedure can be used as part of an iterative design process forsteel frames. The calculation of steel frames according to thisprocedure can be performed with the software used in steelworkdesign. The evaluation of the rotational stiffness of joints is theminimum requirement for the software to be used.

When the influence of bending moment and axial forces istaken into account the characteristics of joint and magnitudeof acting forces are interdependent. In this case characteristicsof joints as well as frame forces and displacements have to becalculated by an iterative procedure until the changes of jointcharacteristics and forces in structure between two iterationsbecome very small (Fig. 3). When the influence of the axialforces is disregarded, then no iteration procedure is needed fordetermination of characteristics of the joints and internal forces ofthe frame.

A. Daniunas, K. Urbonas / Engineering Structures 30 (2008) 3114–3118 3117

Fig. 4. Portal frame.

Fig. 5. Horizontal displacement of joint “A” of the left column (Fig. 4) when theinfluence of axial forces is taken into account (M + N) and disregarded (only M).

4. Numerical examples

In order to investigate the effect of semi-rigid joints on the be-havior of framed structures the portal frame and roofing structureare examined. The joint stiffness and strength characteristics un-der bending and axial forces are obtained and analysis of framedstructures is performed according to the procedure given in previ-ous sections.

Example 1. The investigated structure is presented in Fig. 4. Thestructure was investigated for the four cases of inclination angle α(0, 15, 30 and 45 degrees). It was loaded by uniformly distributedvertical and horizontal loads and they are applied simultaneously.The steel grade of all members is S235 and Young’s modulus is210 GPa. Beams and columns in joints “A” and “B” are connectedby the bolted end-plate knee joint with four M24 8.8 classes bolts.Four cases of end-plates with the thickness of 12, 16, 20 and26 mm were analysed. The calculations were performed when thecharacteristics of the joints “A” and “B” were calculated taking intoaccount the influence of axial force or disregarding it.

The iterative procedure is applied for problem solving. Theinitial rotational stiffness of joints for different end-plate thicknessat the final iteration is presented in Table 1. Figs. 5 and 6 show theinfluence on frame displacements when the rotational stiffness ofthe joint is obtained by taking into account bending moment andaxial force and when only bending moment is taken into account.

Example 2. Investigated roofing structure is presented in Fig. 7[18]. Beams were loaded by a uniformly distributed vertical load.The steel grade of all members is S355 and Young’s modulus is210 GPa. Beams were connected by the bolted end-plate knee joint‘A’ with four M24 8.8 classes bolts. Four cases of end-plates with thethickness of 12, 16, 20 and 26 mm were analysed.

The calculations were performed when the rotational stiffnessof the joint ‘A’ is calculated by taking into account the influence

Fig. 6. Vertical displacement of rooftop joint “B” (Fig. 4) when the influence of axialforces is taken into account (M + N) and disregarded (only M).

Fig. 7. Roofing structure.

Fig. 8. Influence of the thickness of the end-plate to the bending moment in thejoint “A” (Fig. 7) when the influence of axial forces is taken into account (M+N) anddisregarded (only M).

Fig. 9. Influence of the thickness of the end-plate to the axial force in the joint“A” (Fig. 7) when the influence of axial forces is taken into account (M + N) anddisregarded (only M).

of the axial force or disregarding it. This example shows the casewhen acting tension axial force in beams decreases the rigidity ofthe joint ‘A’ and changes the behavioral results of the whole roofingstructure.

The iterative procedure is applied for problem solving. Someobtained results (Table 2, Figs. 8 and 9) of the analysed roofingstructure’s behavior have shown a significant influence of theaxial force to the internal forces of the structure. In addition,internal forces and displacements of the structure are significantlyinfluenced by the thickness of the end-plates of the joint ‘A’.

3118 A. Daniunas, K. Urbonas / Engineering Structures 30 (2008) 3114–3118

Table 1The initial stiffness SA,ini (joint “A”) and SB,ini (joint “B”) (Fig. 4) with the different thickness of end-plate and angle of inclination α (in brackets is given initial stiffness, whenaxial force is disregarded)

Initial stiffness (kNm/rad) Thickness of the end-plate (mm)26 20 16 12

α = 0◦

SA,ini 238 000 (215 000) 170 700 (153 700) 114 000 (102 800) 59 000 (53 200)SB,ini 96 700 (88 900) 69 300 (63 700) 46 350 (42 600) 24 000 (22 000)

α = 15◦

SA,ini 192 000 (161 000) 138 000 (116 000) 92 600 (77 500) 48 240 (40 150)SB,ini 112 800 (97 400) 80 830 (69 800) 54 040 (46 700) 28 040 (24 200)

α = 30◦

SA,ini 164 700 (130 000) 119 000 (93 000) 79 740 (62 000) 42 175 (32 000)SB,ini 176 960 (130 000) 126 800 (93 000) 84 300 (62 000) 42 610 (32 000)

α = 45◦

SA,ini 152 000 (110 000) 111 348 (78 600) 75 160 (52 500) 40 150 (27 200)SB,ini 4500 000 (215 000) 3200 000 (153 700) 2200 000 (102 800) 1300 000 (53 200)

Table 2The initial stiffness SA,ini of joint ‘A’ and vertical displacements uv of joint ‘A’ and point ‘B’ (Fig. 7) with the different thickness of end-plates in joint ‘A’

Thickness of the end-plate (mm)26 20 16 12‘A’ ‘B’ ‘A’ ‘B’ ‘A’ ‘B’ ‘A’ ‘B’

The influence of the axial force is estimatedSA,ini(kNm/rad) 53 661 37 449 23 905 10 871uv(mm) 6.97 12.07 6.90 12.44 6.80 12.99 6.58 14.17

The influence of the axial force is disregardedSA,ini(kNm/rad) 97 440 69 840 46 700 24 200uv(mm) 7.05 11.64 7.01 11.86 6.94 12.21 6.80 12.98

5. Conclusions

In the article, the suggested extension of the well-knowncomponent method for estimation of the total influence of axialforce and bending moment allows one to obtain more accuratestiffness and moment resistance characteristics of the variousjoints. The performed investigations have shown that axial forceinfluences the rotational stiffness and moment resistance of thejoints. Tension axial force decreases the rotational stiffness of thejoints and compression axial force increases it. Analysis of steelframeworks has shown that calculation results differ when totalinfluence of bending moment and axial force on characteristicsof the joints was estimated and when only influence of bendingmoment on characteristics of the joint was estimated.

Suggested simple iterative procedure for the analysis offramed structures with semi-rigid joints with estimation bendingmoments and axial forces allow to use the framework analysissoftware, which enable to evaluate only rotational stiffness of thejoints.

Therefore, joints in which significant axial forces occur have tobe designed by estimating the total influence of bending momentand axial force.

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