FINITE ELEMENT ANALYSIS OF THE ROTATION CAPACITY OF BEAM-TO-COLUMN END-PLATE BOLTED JOINT

EUROSTEEL 2014, September 10-12, 2014, Naples, Italy FINITE ELEMENT ANALYSIS OF THE ROTATION CAPACITY OF BEAM-TO-COLUMN END-PLATE BOLTED JOINT Krzysztof Ostrowski Design Office MTA Engineering Ltd., Rzeszów, Poland krzysztof.ostrowski@mta-online.net INTRODUCTION End-plate bolted joints are commonly used in steel frame structures as a very efficient connection of beam to column and beam splice. To estimate joint flexibility parameters, usually the component method is used. In accordance with analytical procedures of the component method included in EN 1993-1-8 [1] a quantitative estimation of joint resistance and stiffness is possible however, there are no procedures which can be used with regard to joint rotation capacity. The rules given in [1] are limited only to the calculation of end-plate joint thickness, at which the connection has enough rotational capacity. A precise determination of joint rotation capacity is required in the global analysis of the structure. A determination of the rotation capacity of the joints is problematic due to different failure modes, which in turn, depend on the component stiffness. The beyond the elastic limit joint components take part in the redistribution of internal forces, which results in a changeable behaviour of the components in particular phases of loading. The study of the rotation capacity was carried out by Frey and Morris [2], Yee and Melchers [3], Faella and Piluso [4], Kuhlmann [5], [6] and Beg [7]. The analytical determination of end-plate thickness and bolt forces was described by Murray and Kukreti [8] with the application of regression analysis, but this approach was related to isolated components of the end-plate joint. The cooperation of all components was not taken under consideration in the case of the bending moment distribution. In this paper, a polynomial function was proposed to describe the rotation of beam-to-column endplate bolted joint [9], [10], [11]. The determination of the rotation angle function was carried out with the use of FEM analysis. To achieve the demanded function the following variables were used: - end-plate thickness "t p ", - spacing between the centres of bolts "w", - distance from the top surface of upper flange to the axis of bolts c g1 (Fig. 1). The results of the analysis were obtained with the use of the nonlinear regression method. 1 DESCRIPTION OF NUMERICAL CALCULATIONS The determination of the rotation angle function of beam-to-column end-plate bolted joint was carried out with the use of Finite Element Method analysis. The improvement of the electronic systems made the use of the FEM possible in the current research as well as the design activity to support laboratory tests which are usually very expensive. The interpretation of the numerical calculations results is also much easier, so that laboratory experiments are often replaced by FEM analysis. In the finite element model of beam-to-column end-plate bolted joint, high strength bolts, grade 10.9 (according to standard DIN 6914 [12]) were used. The following properties of steel were considered: grade S 235, elasticity modulus 210 GPa and Poissons's ratio 0,3. A material nonlinear analysis was applied with the use of ANSYS software. End-plate components were created using eight layers of finite elements. In the stress concentration areas the mesh density was increased. To increase the calculation efficiency, the analysis of symmetrical model was carried out. The washers were made as separate elements, whereas the bolt head and shank were connected to the nut.

Fig. 1. Configuration of beam-to-column end-plate bolted joint In the region of contact between the nut and the thread, a mesh density is used with the element size of 0,3 mm. Due to the prestressing of the fasteners the thread flexibility was neglected. The bolt prestressing to the force F p,c = 175 kn was executed by the application of the compression force to the lateral surface of the shank. Fig. 2. Numerical models used in the FEM analysis of beam-to-column end-plate bolted joint: a) bolt; b) washer The mechanical properties of the materials were assumed on the basis of the laboratory tests. In Fig. 3 the stress strain diagram for bolts grade 10.9 and the steel grade S 235 used in the analysis is presented. The diagram includes curves for true and engineering stresses. Fig.3. Stress-strain curves for particular elements

The curves marked by line 1a and 2a present the stress strain distribution related to the initial crosssection A 0 area, whereas curves marked by line 1b and 2b depict the stress strain distribution related to instantaneous cross-section area A. The application of stress-strain curves as a logarithmic function of strain enables an appropriate representation of the joint component strain in the full scope of the strain state. The contacts between particular joint elements were created as nonlinear with the friction factor assumed for the surface in natural condition with the value of =0.2. Each numerical model was loaded by a bending moment, at which the joint component reached the allowable strain determined by the basis of the axial tensile testing. The following conditions were assumed up to the point of the element failure: 6 % strain for fasteners and 25 % strain for steel components. The bending moment was executed as the result of the concentrated load F, which was applied in 22 steps at the end of the cantilever (Fig. 1). The first two steps were assigned for the prestressing stage. The remaining load was divided into the next twenty steps equally. In summary, the 3D numerical model of the beam-to-column end-plate bolted joint was discretized by 292 557 hexagonal finite elements. The size of the finite elements was optimized to obtain convergent results. The difference in the results between the numerical model discretized by 292 557 finite elements and the model discretized by 1 030 074 finite elements was only 2,6 %. 2 PLAN OF NUMERICAL EXPERIMENT The central composite design experiment plan was applied in the executed numerical experiment, which is integrally connected to the approximation of the research object function as a polynomial function. One of the most frequently used polynomial function was applied in the following form: C (K M ) C (K M ) C (K M ) (1) 3 5 1 2 3 K Ct w c (2) A1 A2 A3 p g1 where: C 1, C 2, C 3, are the constants of the polynomial, C, A1, A2, A3 are the coefficients related to the joint geometry, t p is the thickness of the end-plate [mm], w is the spacing between the centres of bolts [mm], c g1 is the distance from the top edge of upper flange to the axis of bolt [mm], M is the value of the bending moment [kn m]. Table 1. Plan of Experiment - Central Composite Design DP i t pi w i c g1.i DP i t pi w i c g1.i DP i t pi w i c g1.i DP 1 12,5 150 65 DP 11 12,5 150 57,5 DP 21 13,75 165 57,5 DP 2 10 150 65 DP 12 12,5 150 80 DP 22 10 120 80 DP 3 11,25 150 65 DP 13 12,5 150 72,5 DP 23 11,25 135 72,5 DP 4 15 150 65 DP 14 10 120 50 DP 24 15 120 80 DP 5 13,75 150 65 DP 15 11,25 135 57,5 DP 25 13,75 135 72,5 DP 6 12,5 120 65 DP 16 15 120 50 DP 26 10 180 80 DP 7 12,5 135 65 DP 17 13,75 135 57,5 DP 27 11,25 165 72,5 DP 8 12,5 180 65 DP 18 10 180 50 DP 28 15 180 80 DP 9 12,5 165 65 DP 19 11,25 165 57,5 DP 29 13,75 165 72,5 DP 10 12,5 150 50 DP 20 15 180 50 - - - - where: DP i t pi, w i, c gi is the number of the design point are variables

The form of the polynomial function of the rotation capacity depends on the number of input values. Very often trivalent plans (-1, 0, 1) proposed by Hartley or Benken Box [13] were used. However, in this paper, due to specific characteristics of the analyzed problem, the central composite design plan was applied. This approach allowed the use of pentavalent (-1.0, -0.5,0,0.5, 1.0) divisions of the analyzed variable factors. The experiment plan in relation to the variable factors: t p, w and c g1 was presented in Table 1. a) b) c) d) Fig. 4. Graphical representation of the experiment plan: a) general view; b) t p - thickness of the end plate; c) w - spacing between the centres of fasteners; d) c g1 - distance from upper flange to the centre of fastener The experiment plan, which is based on 29 design points (Fig. 4), includes a variable factors combination in the following ranges: t p = 10 15 mm, w = 120 180 mm and c g1 = 50 80 mm. The solution for all the configurations of this plan enabled to create a response surface of the demanded values based on 580 measurement points. 3 RESULTS OF THE ANALYSIS FEM analysis was followed by a calculation of the polynomial factors. These factors were determined with the use of the nonlinear regression method. The polynomial factors for the determination of the rotation angle was presented in Table 2. Table 2. Coefficients of the polynomial function X i C C 1 C 2 C 3 A 1 A 2 A 3 0,00211 0,52886 0,36887 0,09749-1,53220 0,46556 0,96395 In the joints, at some parameter configurations, the process of an additional bi-directional bending of bolts may occur. The values of the bending moment M yz and M xy may be determined on the basis of the reaction moment measurement in the contact point between the washer and the end plate surface. The total value of the bending moment at the bolt head is not equal to the value of the bending moment acting at the nut. This results from the varied stiffness of the bolt along its length. There is a stiffness reduction in the thread segment of the fastener due to a lower value of the crosssection. The additional disturbance is a result of a varied strain of the adjacent elements. In the analyzed experiment plan the end plate is always thinner than the column flange. The development

of the plastic zone in the bolt which is subjected to an axial tension and bidirectional bending was presented in Fig. 5. a) b) c) Fig. 5. Development of plastic zone in the bolt resulting from axial tension and bidirectional bending. Stress maps were shown in logarithmical scale. As recommendations for the determination of the rotation capacity of beam-to-column end-plate bolted joints the following algorithm can be formulated: a) Assusmption of joint geometrical characteristics: t p, w, c g1. b) Definition of the limit strain for fasteners and steel: bolt,max, steel,max. In the analysed case the following values were considered: : bolt,max = 0,06 [mm/mm], steel,max = 0,25[mm/mm]. c) Determination of the maximum value of the bending moment M i,max which can be applied to the analyzed joint with the use of formulae (3) and (4). Equations (3) and (4) were developed using the results of FEM analysis from the response surface of the experiment design. The nonlinear estimation method was applied for the determination of equations. M 393 t w c (3) 0,294 1,987 0,553 0,732 steel,max steel,max p g1 M 1360 t w c (4) 0,463 0,663 0,298 0,375 bolt,max bolt,max p g1 where: M steel,max is the value of the bending moment which is reached for the maximum assumed strain steel,max, M bolt,max is the value of the bending moment which is reached for the maximum assumed strain bolt,max, t p, w, c g1 are the values as in formula (1), bolt,max, steel,max are the maximum acceptable values of strain. d) The determination of the rotation capacity of the joint as an angle on the basis of formula (1) with the application of the polynomial factors collected in Table 2. The value of the joint rotation angle is determined with the use of the lower value of the bending moments obtained from the formulae (3) and (4). 4 CONCLUSIONS The results of the analysis with the use of FEM method applied by the author reveals new possibilities of identification of the relation between joint components and their behaviour in postlimit stiffness range. The elaborated results posses huge applicability limitations due to the application of only two profiles in beam-to-column joint. The analysis also did not include all the variable factors which influence the joint rotation. To achieve the universal analytical formulae, which include changing geometrical characteristics of the connection components, it will be necessary to elaborate an experiment plan for the configuration containing all the variable factors.

The allowable strain limit in a fastener bolt,max is frequently reached due to the bidirectional bolt bending. In such case, the bolt does not reach the maximal force which could be transferred through the tensile stress area A s, in case of pure axial tension. The dominant parameter that has influence on the rotation of beam-to-column joint is the end plate thickness. For example: the rotation angle obtained with the application of the value of bending moment M=80 knm for parameters w=150mm, c g1 =65 mm, decidedly increases in the case of plate thickness t p =10 mm and is almost 10 times bigger than in the case of plate thickness t p = 15 mm. Finite Element Method analysis enables to obtain many additional results, e.g. force in bolts or strain values of particular joint elements and points. REFERENCES [1] EN 1993-1-8 Design of steel structures. Part 1-8: Design of joints. CEN, 2006. [2] Frye M. J., Morris G. A., 1975. Analysis of Flexibly-Connected Steel Frames., Canadian Journal of Civil Engineering, No2. [3] Yee Y. L., Melchers R. E., 1986. Moment Rotation Curves for Bolted Connections., Journal of Structural Division. ASCE. Vol. 112, ST3. [4] Faella C., Piluso V., Rizzano G., 1996. A New Method to Design Extended End Plate Connections and Semirigid Braced Frames. Journal of Constructinal Steel Research. Vol. 41, No. 1, pp. 61-91. [5] Kuhlmann U., Fürch A., 1997. Rotation Capacity of Steel Joints., COST Project C1 Meeting. [6] Kuhlmann U., Kühnemund F., 2004. Rotationskapazität nachgiebieger Stahlknoten., Stahlbau. H. 9 [7] Beg D., Zupančič E., Vayas I., 2004. On the rotation capacity of moment connections., Journal of Constructinal Steel Research. Vol. 60. [8] Murray T., Kukreti A., 1988. Design of 8-bolt Stiffened Moment End Plates, Engineering Journal, Second Quarter, American Institute of Steel Construction. [9] Sherbourne A., Bahaari M. R., 1997. Finite element prediction of end plate bolted connection behavior. I: Parametric study. Journal of Structural Engineering. [10] Bahaari M. R., Sherbourne A., 1997. Finite element prediction of end plate bolted connection behavior. II: Analitic formulation. Journal of Structural Engineering. [11] Concepcion D., Pascual M., Mariano V., Osvaldo M., 2011. Review on the modelling of joint behavior in steel frames. Journal of Constructional Steel Research. 67(2011) 741-758 [12] DIN 6914 Sechskantschrauben mit großen Schlüsselweiten, HV-Schrauben in Stahlkonstruktionen. [13] Polański Z., 1995. Experiment design. (In Polish), PWN, Warsaw.

The central composite design experiment plan was applied in the executed numerical experiment, which is integrally connected to the approximation of the research object function as a polynomial function. One of the most frequently used polynomial function was applied in the following form: C (K M ) C (K M ) C (K M ) (1) 3 5 1 2 3 K C t w c (2) A1 A2 A3 p g1 where: C 1, C 2, C 3, are the constants of the polynomial, C, A1, A2, A3 are coefficients related to the joint geometry t p is the thickness of the end-plate [mm], w is the spacing between the centres of fasteners [mm], c g1 is the distance from the top surface of upper flange to the axis of bolts [mm], M is the value of the bending moment [kn m]. FEM analysis was followed by a calculation of the polynomial factors. These factors were determined with the use of the nonlinear regression method. The polynomial factors for the determination of the rotation angle was presented in Table 1. Table 1. Coefficients of the polynomial function X i C C 1 C 2 C 3 A 1 A 2 A 3 0,00211 0,52886 0,36887 0,09749-1,53220 0,46556 0,96395 The experiment plan, which is based on 29 design points includes a variable factors combination in the following ranges: t p = 10 15 mm, w = 120 180 mm and c g1 = 50 80 mm. The solution for all the configurations of this plan enabled to create a response surface of the demanded values based on 580 measurement points. CONCLUSIONS The results of the analysis with the use of FEM method applied by the author reveals new possibilities of identification of the relation between joint components and their behaviour in postlimit stiffness range. The elaborated results posses huge applicability limitations due to the application of only two profiles in beam-to-column joint. The allowable strain limit in a fastener bolt,max is frequently reached due to the bidirectional bolt bending. In such case, the bolt does not reach the maximal force which could be transferred through the tensile stress area A s, in case of pure axial tension. The dominant parameter that has influence on the rotation of beam-to-column joint is the end plate thickness. Finite Element Method analysis enables to obtain many additional results, e.g. force in bolts or strain values of particular joint elements and points. REFERENCES [1] EN 1993-1-8 Design of steel structures. Part 1-8: Design of joints. CEN, 2006. [2] Frye M. J., Morris G. A., 1975. Analysis of Flexibily-Connected Steel Frames., Canadian Journal of Civil Engineering, No. 2. [3] DIN 6914 Sechskantschrauben mit großen Schlüsselweiten, HV-Schrauben in Stahlkonstruktionen.