SIMPLIFIED FORMULAS FOR ASSESSMENT OF STEEL JOINT FLEXIBILITY CHARACTERISTICS

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1 SIMPLIFIED FORMULAS FOR ASSESSMENT OF STEEL JOINT FLEXIBILITY CHARACTERISTICS Aleksander Kozłowski; Lucjan Ślęczka Rzeszów University of Technology, Poland ABSTRACT Code includes full design procedures, based on component method, allowing to asses design resistance M j,rd and initial stiffness S j,ini. of the most often used steel joints, as end-plate bolted connections. This method, despite of many simplifying assumptions, is still burdensome and time consuming and is addressed to final check of designed structure. In the pre-design phase, when designer is forced to change input data many times, there is a need to use design tools, like tables, graphs or simplified formulas, to quick asses joint properties for global analysis of the frame. Such simplified formulae for assessment of design moment resistance M j,rd and initial stiffness S j,ini. Of steel joint has been presented in the paper. Comparison to results obtained using available software shows that simplified formulas present a good agreement (ca. 0 %). 1. INTRODUCTION Current steel structures design codes, including EN , requires taking into account during global analysis actual properties of joints and connections. Joint main characteristics, as M-ф curve, can be taken from experimental tests or analytical models. Code EN includes full design procedures, based on component method, allowing to asses design resistance M j,rd and initial stiffness S j,ini. of the most often used steel joints, as end-plate bolted connections. This method, despite of many simplifying assumptions, is still burdensome and time consuming. It requires to input many geometrical and structural properties of design joints, which are not known to designers on this stage of design. It is enough to say, that hand calculation of multi-row bolted end-plate connections takes a dozen or so pages (Kozlowski, 009). Computer software in such a situation were widely developed to support design work. There are few computers programs allowing to obtain moment resistance and stiffness of steel joints: (CoP, CRSJAE, module connections in AutodeskRO- BOT). Application of these programs are reasonable in the last stage of design, where structure design is nearly finished. In the pre-design phase, when designer is forced to change input data many times, there is a need to use design tools, like tables, graphs or simplified formulas, to quick asses joint properties for global analysis of the frame. The aim of the paper is to propose such simplified formula for steel joint.. ANALYSIS OF EFFECTIVE LENGTH AND FAILURE MODES OF T-STUBS Components of steel joint can be divided as component of: - shell behavior. - plate behavior,

2 The first group contains: column web in shear, column web in compression and column web in tension. In the second, there are: column flange in bending, end plate in bending, angle in tension. These components are modeled by T-stub. Resistance of plate components is calculated for three failure modes of T-stub (table 6. of EN ). Effective lengths for each component, for circular and noncircular modes are presented in Tables 6.4, 6.5 and 6.6 of EN Code requires to analyze all effective length and choose the smallest, what is the main reason of huge time consuming during calculation. In the frame of diploma work (Kowal-Gaska, 011) wide analysis of all possible effective lengths and failure modes were conducted, to eliminate this ones which are never or seldom possible. Conclusions from this work is as follows: for unstiffened column flange, circular failure modes are possible only when distance e (Figure 1) is,8 times bigger than bolt diameter, what is practically impossible, so as reliable non-circular effective length: l eff,nc 4m + 1,5e should be taken, for bolt row in end plate above beam flange, always the smallest is non-circular length: l eff,nc 5 b p, for bolt row in end plate below beam flange decisive is effective length: l eff,nc α. After analysis of many case, it was concluded that α 5,8 can be safely taken, for most of design cases, decisive are non-circular failure modes and final resistance of components should be calculated using: for column flange: mode of failure, for end plate: mode 1 for row above beam flange and mode for rows below beam flange.. ANALYSIS OF RESISTANCE AND STIFFNESS OF BASIC JOINT COMPO- NENTS.1 Assumptions Simplified formulas for assessing bending resistance and rotational stiffness requires some assumptions about geometry of analysed joints: As most often case in design, it was assumed cross section of beams made of IPE and cross section of columns made of HEB, The beam depth was limited to the range between 40 and 450 mm, The column depth was limited to the range between 140 and 00 mm, There were assumed following relations between geometrical dimensions of beam, column and end plate cross sections (Table 1): Table 1. Assumed geometrical properties of analyzed joints Column b c h c 11 r c 1,5 t t wc 6 t r c 1, t End plate t p t b p b fb 10 t e min 1,5 d 0 1,65d t d 1,5 t a p 7 t fb n e min t Beam z h b b fb b fb 10 t t fb 8 t a w 7 t wb 6 t wc t wb 9 t wc where: h c, b c, t, t wc, r c depth, width, thickness of flange, thickness of web and the root radius of column cross section, h b, b fb, t fb, t wb - depth, width, thickness of flange, thickness of web and the root radius of beam cross section,

3 d, d 0 nominal bolt diameter, the hole diameter for a bolt, a w, a p thickness of the weld between beam web and end plate, thickness of the weld between beam flange and end plate, (Figure 1). Figure 1. Geometrical configuration of beam-to-column joint. Resistance of basic components Considering the case of single sided (external) joint, the transformation parameter β is equal to β1,0. The values of partial safety factors according to Polish National Annex (EN ) were established as γ MO 1,0 and γ M 1,5. Column web panel in shear (eq. (6.7) in EN ): 0 9fy A vc Vwp Rd,, γm0 It can be assumed that ( h + 4t ) t ( 1, 5t + 4t ) 6t 10 5t A, vc c wc 9 fy 1 5t Vwp, Rd 5, 5t (1) Column web in transverse compression (eq. (6.9) in EN ) ωk wc beff, c, wc t wc fy Fc, wc, Rd γm0 beff, c, wc t fb + ap + 5( t + rc ) + tp t fb + 7t fb + 5( t + 1t, tc ) + t t fb + 1t 8t + 1t 15, 4t

4 Average value of ω is equal to ω 7 and k wc 1,0. Then: 7 10, 15, 4t t wc fy Fc, wc, Rd 7 15, 4t 6t 6, 5t () 10, Column flange in transverse bending (Figure 1) bc ec t wc 8rc 1, 5t ec 6t 8 1t, m For the most popular bolts M0: d0 and d 0 e 15, d 165d, 165, 15t,, 1 min 0 t e c emin + 175t,, 1t + 175t,, 85t (see Figure 1). Then m 5, 5t ec 5, 5t, 85t 165, t l eff 4m + 15e, c 4 165t, + 15,, 85t 114, t Considering mode of failure (Table 6. in EN ): 5, 5t ec Mpl,, Rd + n Ft, Rd FT, Rd m + n fy M pl,, Rd 5leff t 5 114t, t f y, 85t f y γm0 In case of bolts M16, M0 and M4: πd A s 78A 78 61d 4 k fub As 9 fub 61d 9 fub 61( 15t, ) Ft, Rd 17, t fub γ M 15, 15, n emin t < 15m, 15, 165t,, 1t Mpl,, Rd + n Ft, Rd, 85t f y + t 17, t f ub FT, Rd m + n 165t, + t 156t, f y + 75t f ub In case of steel grade S5 and bolt class 10.9: f ub 4,6f y and then Ft,, Rd FT, Rd 156t, f y + 75t f ub 156t, f y + 75t 4, 6f y 4, 8t f y In case of steel grade S55 and bolt class 10.9: f ub,8f y and then Ft,, Rd FT, Rd 156t, f y + 75t f ub 156t, f y + 75t, 8f y, 7t f y () End-plate in bending, (Figure 1) - Bolt-row outside tension flange of beam: b p bfb 10t ; l eff 5bp 5t m mx 15d, 15, 15t, 19, t Considering mode 1 of failure (Table 6.1 in EN ): fy M pl, 1, Rd 5leff tp 5 5t t f y 15t, f y γm0 4Mpl, 1, Rd 4 15t, f y F t, ep, Rd Ft, 1, Rd, 6t f y (4) m 19t,

5 - First bolt-row below tension flange of beam: bp e t wb 8 aw m e emin, 1t 10t, 1t 9t wc 8 6t wc m, 9 t 11t, wc, 9t 11, 6t, t α śr 5, 8 ; l eff αm 5, 8, t 1, 9t fy M pl,, Rd 5leff tp 5 1, 9t t f y, t f y γm0 n emin, 1t Mpl,, Rd + n Ft, Rd, t f y +, 1t 17, t f ub FT, Rd m + n, t +, 1t 149t, f y + 67t f ub In case of steel grade S5 and bolt class 10.9: f ub 4,6f y and then Ft, ep, Rd FT, Rd 149t, f y + 67t f ub 149t, f y + 67t 4, 6f y 4, t f y In case of steel grade S55 and bolt class 10.9: f ub,8f y and then Ft, ep, Rd FT, Rd 149t, f y + 67t f ub 149t, f y + 67t, 8f y, 4t f y (5) - Other inner bolt-rows: l eff 4m + 15e, 4, t + 15,, 1t 115, t fy M pl,, Rd 5 leff tp 5 115t, t f y, 88t f y γm0 Mpl,, Rd + n Ft, Rd, 88t f y +, 1t 17, t f ub FT, Rd m + n, t +, 1t 1t, f y + 67t f ub In case of steel grade S5 and bolt class 10.9: f ub 4,6f y and then Ft, ep, Rd FT, Rd 1t, f y + 67t f ub 1t, f y + 67t 4, 6f y 4, t f y In case of steel grade S55 and bolt class 10.9: f ub,8f y and then Ft, ep, Rd FT, Rd 1t, f y + 67t f ub 1t, f y + 67t, 8f y, t f y (6) Column web in transverse tension (eq. (6.15) in EN ) ωleff t wc fy Ft, wc, Rd 7 114t, 6t 4, 8t f y (7) γm0 Beam flange and web in compression (eq. (6.1) in EN ) Mc, Rd bfbtfb fy ( h tfb ) Fc, fb, Rd bfbtfb fy 10t 8t 8, 0t f y (8) h tfb ( h tfb ) γm0

6 Beam web in tension (eq. (6.) in EN ) beff, t, wb t wb fy Ft, wb, Rd beff, t, wb 9t wc beff γ M0 beff, t, wb 54t, t, wb 9 6t - In case of first bolt-row below tension flange of beam b eff,t,wb 1,9t F t, wb, Rd 1, 9t 54t 7, 0tfy - In case of next bolt-rows b eff,t,wb 11,9t F t, wb, Rd 115t, 54t 6, tfy (9) Above calculations were related to single sided joints. In double sided joints, the transformation parameter β was assumed as β 0. Reduction factor ω is equal to ω1,0. So, for some components, values of resistance are changing: V wp, Rd, F c, wc, Rd 9, t f y and F t, wc, Rd 6, 8 t f y.. Stiffness analysis of basic components The stiffness coefficients (related to single sided joint) can be calculated as below: Column web panel in shear (Table 6.11 in EN ): 0 8A vc k1, β z A vc 1 5t ; z hb b fb 10t 0t 8 10t k 1 19t (10) 1 0t (In case of double sided joint, when β 0 component is inactive). Column web in transverse compression (Table 6.11 in EN ): 7beff, c, wc twc k dc d c hc ( t + r) 1, 5t ( t + 1t, ) 9, 1t 7 15, 4t 6t k 71t (11) 9, 1t Column web in transverse tension (Table 6.11 in EN ): 7beff, t, wc t wc 7 11, 4 t 6t k 5t (1) dc 9, 1t Column flange in transverse bending (Table 6.11 in EN ): 9leff tp 9 114t, t k 4, 8t (1) m ( 165t, ) End-plate in bending (Table 6.11 in EN ): 9leff tp k5 m

7 Table. Simplified formulas to assess resistance and stiffness coefficients in basic components Component Column web panel in shear Column web in transverse compression Beam flange and web in compression Column web in transverse tension Column flange in transverse bending End-plate in bending Beam web in tension Bolt-row Effective width/ length Symbol Resistance Formula Stiffness coefficients Single-sided Double sided Symbol Formula - - V wp,rd 5,5 t f y k 1 19 t - 15,4 t F c,wc,rd 6,5 t f y 9, t f y k 71 t - - F c,fb,rd 8,0 t f y ,4 t F t,wc,rd 4,8 t f y 6,8 t f y k 5 t outside tension flange below tension flange other below tension flange 11,4 t F t,,rd 5 t 1,9 t 11,5 t 1,9 t F t,ep,rd F t,wb,rd 4,8 t f y (,7 t f y ),6 t f y 4, t f y (,4 t f y ) 4, t f y (, t f y ) k 4 k 5,8 t 66 t 1,06 t 95 t 7,0 t f y - - other 11,5 t 6, t f y - - Bolts in tension k t Remarks: t min (t ; t p ), Resistance of basic components is related to steel grade S5 (In parenthesis values for S55) 9 5t t Bolt-row outside tension flange of beam: k 5 ( 19t, ) 66t (14) 9 1, 9t t First bolt-row below tension flange of beam: k 5 (, t ) 106, t (15) 9 115t, t Other inner bolt-rows: k 5 95t (, t ) (16) Bolts in tension (Table 6.11 in EN ): k 10 16A, L b s

8 πd A s 78A 78 61d 61( 15t, ) 95t 4 L t + t t b 16, 95t t (17) t k, Table gives the review of simplified formulas, which can be used to assess resistance and stiffness coefficients of steel joints basic components. 4. MOMENT RESISTANCE AND ROTATIONAL STIFFNESS OF STEEL JOINTS Mechanical models representing different types of joints with distinguished components are shown in figures: - single sided joints (Figure ), - double sided joints (Figure ), - beam splices with bolted end-plates (Figure 4). Each from above-mentioned figures show first view of joint (figures marked a ) and next show mechanical model with calculated values of each component (figures marked b ). Such system in next step is replaced by simpler model, with only one component in every line, representing the smallest value of the tension (compression) resistance for an individual row (figures marked c ). Figure. Single sided joint with extended end plate; a) general view, b) mechanical model, c) effective design resistance of each bolt-rows, d) final model with reduced values according to resistance of compression (shear) zone

9 Figure. Double sided joint with extended end plate; a) general view, b) mechanical model, c) effective design resistance of each bolt-rows, d) final model with reduced values according to resistance of compression (shear) zone Figure 4. Beam splice with extended end plate; a) general view, b) mechanical model, c) effective design resistance of each bolt-rows, d) final model with reduced values according to resistance of compression (shear) zone Final models (figures marked as d ) show effective design tension resistances, reduced according to the resistance of compression zone (the sum of tension resistances of bolt-row cannot exceed the resistance of compression zone). The final models shown in Figure d, d and 4d can be easily used to assess design moment resistance, taking into account only the distances from bolt-rows to the centre of compression, thickness of column flange (or end plate) and yield strength. Initial stiffness single sided joint with flush end plate can be calculated, assuming z9h b, as follows:

10 ( 9h ) E b S j, ini 1 k The only one bolt-row was taken into account (first below tension flange): , k 19t 71t 5t, 8t 106t, 51t t ( 9hb ) 1 hb t S j, ini (19) 119, t In above formula input values (the depth of the beam h b and thickness of column flange t ) should be put on in [mm], then the result is calculated in [knm/rad]. Initial stiffness of double sided joint with flush end plate: , k 71t 5t, 8t 106t, 51t t ( 9hb ) t 1 hb t S j, ini (0) 6, For extended end-plate connections, simplified method was applied. Instead of calculation concerning two bolt-rows (one in the extended part of the end-plate and one between the flanges of the beam) there was used modified value for a single bolt-row in the extended part of the end-plate, taken as twice to the corresponding value. The influence of the third bolt-row was neglected. In case of single sided joint: , k 19t 71t 5t, 8t 106, 51t t ( 105h, b ) t 1 hb t S j, ini (1) 9, And in case of double sided joint with external end plate: , k 71t 5t, 8t t t ( 105h, b ) t 1 hb t S j, ini () 4, Full list of developed formulas is presented w Table ACCURACY OF DEVELOPED FORMULAS Accuracy of developed formulas (Table 4) was checked by confronting obtained values of structural properties of steel joints with accurate ones, from component method. There were used results of calculations included in (Bródka et al, 009, Kozłowski et al, 009, Kozłowski et al 01 GiŜejowski & Ziółko, 01 Sprint). Additionally calculation of a few joints were performed by Cop software (Cop). Results of comparison applying to the bending resistance is shown in Figure 5, results of rotational stiffness comparison is shown in Figure 6.

11 Table 4. List of simplified formulas to assess design moment resistance and initial rotational stiffness of steel joints Type of joint Single sided with flush end plate Double sided with flush end plate Single sided with extended end plate Double sided with extended end plate Beam splice with bolted flush end-plates Beam splice with bolted extended endplates Steel grade S5 S55 S5 S55 S5 S55 S5 S55 S5 S55 S5 S55 Design moment resistance M j,rd 1h, ( 4, h + ), 1h (, 4h + ), 7h ( 4, h + ), h (, 4h + ) 1, 9h (, 6h + ) 1 + 4, h 11h, (, 6h + ) 1 +, 4h, 0h (, 6h + ), 7h ( 4, h + ), h (, 4h + ) 1 + 4, h 11h, (, 6h + ) 1 +, 4h, 0h (, 6h + ) Initial rotational stiffness S j,ini hb t 70 hb t 9 hb t 40 hb t 18 hb t hb t 11 Remarks: h 1, h, h the distances from bolt-rows (accordingly from one outside tension flange, below tension flange and other) to the centre of compression, (Figure 1), tmin(t, t p ) Design moment resistance is calculated in the same units, as input values, while rotational stiffness is calculated in [knm/rad], on condition that input values are in [mm]. Figure 5. Results of comparison applying to the bending resistance Figure 6. Results of comparison applying to the rotational stiffness

12 6. CONCLUSIONS With reference to the prediction of the bending resistance by component method, the use of simplified formulas shows good agreement. The range of error varies between -8% do +7% in case of single sided joints with extended end plate, -7% do +9% in case of double sided joints with flush end plate and -0% do +11% in case of single sided joints with flush end plate. Rotational stiffness results obtained from simplified method differ from exact results from 0% to +40%. Proposed formulas are addressed only to predesign stage and should help designers to establish structural properties of steel joints in easy way, to use them in global analysis of structure. Sensivity analysis of frames show limited influence of joint rotational stiffness on global behaviour of frames. Even 50% change in rotational stiffness does not significantly change values of internal forces in steel frames (Steenhuis et al, 1996 and Kozłowski 1998). So proposed formulas enables practitioners to assess structural properties of joint without complicated calculation. It should be noted that full verification of the joint have to be carried out after the structure has been designed. REFERENCES Autodesk Robot Structural Analysis Professional, Bródka J., Kozłowski A., Ligocki I., Łaguna J., Ślęczka L., (009) Design and calculations of steel joints and connections (in Polish), vol., Polskie Wydawnictwo Techniczne, Rzeszów. CoP, The Connection Program, CRSJAE (1998) Computer program for calculation of resistance and stiffness of joints according to Eurocodes, Rzeszow University of Technology. EN European Committee for Standardization CEN. Eurocode : Design of steel structures - Part 1-1: General rules and rules for buildings. Brussels. EN European Committee for Standardization CEN. Eurocode : Design of steel structures - Part 1-8: Design of joints. Brussels. GiŜejowski M., Ziółko J. (ed.), (010) Building engineering, vol. 5. Steel building structures. Design according to Eurocodes with worked examples (in Polish). Arkady, Warszawa. Kowal-Gąska G., (011) Analysis of the influence of joint geometric parameters on the behaviour of steel frame structures (in Polish). Diploma thesis. Rzeszów University of technology, Rzeszów. Kozłowski A., (1999), Shaping of the steel and composite skeletons with semi-rigid joints (in Polish). Publishing House of Rzeszów University of Technology. Rzeszów. Kozłowski A., Pisarek Z., Wierzbicki S., (009), Design of end-plate connections according to PN-EN and PN-EN (in Polish), InŜynieria i Budownictwo, nr 4. Sprint. L Eurocode et les assemblages en acier. Aides de calcul pour assemblages rigides et semi-rigidies. Liege. Steenhuis M., Evers H., Gresnigt N., (1996) Conceptual design of joints in braced steel frames. IABSE Semi-Rigid Structural Connections Colloquium, Istanbul, pp 7-6.