PRE-DESIGN OF SEMI-RIGID JOINTS IN STEEL FRAMES

Transcription

1 PRE-DESIGN OF SEMI-RIGID JOINTS IN STEEL FRAMES Martin Steenhuis ( ), Nol Gresnigt ( ), Klaus Weynand ( 3 ) Keywords: steel, joints, frames, semi-rigid, pre-design, stiffness, classification Abstract The response of steel frames is influenced by the mechanical properties of the joints (strength, stiffness, rotation capacity). In practice, the joints are usually considered as either rigid or pinned. Research has shown that frames with semi-rigid joints can be more economical than frames with rigid or pinned joints. Computer programs are available to determine the joint properties. These programs require a full description of the geometry of the joint. In a pre-design, these data are not yet available. Therefore, for the pre-design stage there is a need for simple rules to estimate the mechanical properties of the joints. This paper presents such simple rules. Comparisons to the Eurocode 3 are made. This paper also includes design examples.. Introduction In various European countries, traditionally, two parties are responsible for the design of steel frames: the engineer designs the beams and columns and the steel fabricator designs the connections. In this design practice the engineer specifies the mechanical requirements of joints. The steel fabricator designs the joints to fulfil these requirements. The fabricator also considers manufacturing aspects. Elastic global frame analysis is commonly used in Europe. In this analysis, pinned and rigid design of joints (see figure ) leads to a limited data flow between engineer and steel fabricator. In case of pinned joints the fabricator needs to design the joints like pins. In case of rigid joints, the fabricator needs to design the joints sufficiently rigid. In both cases, the joints should be capable of transmitting the forces determined in the global frame analysis. Pinned and rigid joints, however, are not necessarily economical []. On the contrary, the more economical semi-rigid design leads to intensive data flow between engineer and steel fabricator. The global frame analysis (task of engineer) requires the stiffnesses of the joints, which should be based on the actual geometrical layout of the joint (task of fabricator). This intensive communication (and the consequent mixture of responsibilities) between engineer and fabricator hinders the acceptance of semi-rigid design. Pinned Joint Rigid Joint Semi Rigid Joint figure : Examples of pinned, rigid and semi-rigid joints Published in Proceedings of the Second State of the Art Workshop on Semi-Rigid Behaviour of Civil Engineering Structural Connections, ed. by F. Wald, COST C, Prague, Czech Republic, 994, pp ( ) TNO Building and Construction Research, P.O. Box 49, 600 AA Delft, The Netherlands ( ) TU Delft, P.O. Box 5048, 600 AA Delft, The Netherlands ( 3 ) RWTH Aachen, Mies-van-der-Rohe Straße, 5074 Aachen, Germany

2 This paper presents in chapter. a procedure for frame design with semi-rigid joints, which respects the existing share of responsibilities between engineer and steel fabricator. The advantages of this procedure are that: ) semi-rigid design can be applied without increase of communication between designer and fabricator compared to the traditional situation, ) the fabricator has a certain freedom to design the joints in accordance with the available production technologies and materials in stock. Of course, this procedure is also applicable when the engineer designs both frame and joints. Chapter 3. of this paper concerns the determination of the stiffness of joints in the pre-design phase. Chapter 4 treats the check of the stiffness requirements during the design of the joints. Chapters 5 and 6 show some worked examples.. Design procedure Traditionally, the design process of a steel frame consists of 7 logical steps (see table ). In the modelling phase (step ), the engineer models the joints as pinned or rigid. Pinned joints should be capable of transmitting the forces calculated in design, without developing significant moments which might adversely affect the beams or columns in the frame. In elastic frame analysis, a rigid joint has no influence on the distribution of internal forces and moments in the frame, nor on its overall deformation. table : Design process of a steel frame with elastic global frame analysis, with rigid or pinned joints. Mechanical modelling of the frame in the building including modelling joints as pinned or rigid.. Estimation of loads. 3. Pre-design of beams and columns. 4. Determination of forces and displacements in the frame. 5. Check of beams and columns in limit state conditions. 6. If required, adjustment of beams and columns (continue with step 4). 7. Design of joints based on strength ( pinned or rigid). The loads and the stiffness of beams and columns are input for the frame analysis (step 4). The deflections of the frame and the force distributions are output. The member sizes and the forces which should be transmitted by the joints are the starting point for the design of joints (step 7). The purpose of this design is to find a layout capable of transmitting the forces between the beam and the column. Additionally, in case of pinned joints the fabricator should verify that no significant moments develop in the joints. In case of rigid joints, the fabricator should verify that the joint is sufficiently rigid. Computer programs exist which support the design of joints [, 3] (step 7) according to Eurocode 3 [4]. These programs are essential in the design process because they enlightens the task of the fabricator dramatically. They require the geometrical layout of the joint as input and give the strength, stiffness and rotational capacity as output. A user of a program designs the joints in an interactive way by trial and error. For example, the user first tries a simple solution. If this solution doesn't satisfy the strength criteria, the user will improve the design by adjusting the lay out of the joint. The process ends when the design is satisfactory. Some programs are also capable to check whether a joint is rigid or not. This paper proposes two modifications to make the procedure of table suitable for semi-rigid design. This leads to a process as given in table. When this paper speaks about stiffness of a joint, it is meant the elastic stiffness

3 table : Design process of a steel frame with elastic global frame analysis with semi-rigid joints. Mechanical modelling of the frame in the building.. Estimation of loads. 3. Pre-design of beams and columns and assessment of stiffness of joints. 4. Determination of forces and displacements in the frame (stiffness of the joint included in the analysis) 5. Check of beams and columns in limit state conditions. 6. If required, adjustment of beams and columns (continue with step 4). 7. Design of semi-rigid joints based on required strength and stiffness. I Inclusion of stiffness in the frame analysis Cunningham and Taylor proposed to assume a certain stiffness for the joints in the frame analysis rather than to take the joints as either rigid or pinned [6]. This stiffness is taken as a function of the beam stiffness and the recommended fixity factor (the fixity factor is defined as the relation between the stiffness of the joint and the beam). This stiffness is an estimate of the actual stiffness of the joints to be designed in step 7. This paper basically proposes, as an extension to the work of Cunningham and Taylor, to assess the stiffness of the joints relating to the beam and column properties and the type of the joints in the pre-design phase. This is explained more in detail in chapter 3. These stiffnesses can be used in the global frame analysis (step 4). II Verification of stiffness in the design of joints. It should be verified in step 7 that the stiffness of the joint designed by the fabricator is in reasonable agreement with the stiffness included in the global frame analysis. This replaces the verifications for rigid and pinned joints in traditional design. Chapter 4 gives some rules to carry out this verification. These rules are based on the same philosophy as the classification diagrams of Eurocode 3. In combination with a computer program, these rules can easily be applied. 3. Prediction of the stiffness of joints in the pre-design stage In the pre-design phase of a structure, it is difficult to assess the stiffnesses of the (semi-rigid) joints, because the joints have not been designed yet. To overcome this problem, some simplified formulae have been derived based on Eurocode 3 Annex J (revised) [5]. With help of these formulae, a designer can determine the stiffness of a joint by selecting the configuration. These formulae assume some fixed choices for the connection design. These are for endplated connections: the connection has two bolt rows in the tension zone. the bolt diameter is approximately.5 times the thickness of the column flange; the location of the bolt is as close as possible to the root radius of the column flange, the beam web and flange (about.5 times the thickness of the column flange); the end-plate thickness is similar to the column flange thickness; For European I and H sections the following rules are valid: the root radius is about the same size as flange thickness; the web is about 0.6 times the flange thickness; the clear depth of the web is about 5 times the flange thickness. As an example, a simplified formula is derived for an un-stiffened extended end-plate connection in a single sided joint configuration. In figure, first the stiffness factors according to Annex J (revised) are calculated. 3

4 Column web in shear: h c For simplicity we take equal to 0.8. This is a reasonable assumption in un-braced frames. h t A v,c 0.8 h c t w,c h t h t 0.8 h c 0.6 t f,c 0.8 t h f,c t Column web in compression and tension: We assume that in an un-stiffened joint with an extended end-plate, the deformation of the column web in tension is similar to the deformation of the column web in compression. The effective width b eff is approximately: b eff = t f,b + t e + 5 (r c + t f,c ) t f,c 0.7 b eff t w,c t f,c 0.6 t f,c t d 5 t f,c f,c Column flange and end-plate in bending h t l eff t 3 f,c 0.85 m Two bolt rows in tension t f,c t 3 f,c.5 3 t t f,c f,c t f,c h c tp d n t f,b m A 0.75 π/4.5 t b,s f,c t l b 3 t f,c f,c The stiffness of the joint is: E h t t f,c E h S j t t f,c 3,0.4 In this example, the column web in shear contributes most to the flexibility of the joint. figure : Derivation of a simplified stiffness formulae for an extended end plate connection Table 3 contains formulae for different configurations. These formulae can be derived similar to the previous example. Stiffening plates have a great influence on the stiffness. The formulae in table 3 contain only two parameters: h t and t f,c. Parameter h t is the distance between the point of compression and the centre of the tension zone. In an extended end-plate connection with two bolt rows, this distance is approximately equal to the beam depth. For the same joint with haunch, h t is equal to the sum of the beam depth and the haunch height. t f,c is the thickness of the column flange. 4

5 table 3: Formulae for the approximation of the stiffness for beam-to-column configurations Configuration S j Extended end-plate, single sided and un-stiffened 3,0 Extended end-plates, double sided, un-stiffened and symmetrically 7,4 Extended end-plate, single sided, stiffened in tension and compression 8,3 Extended end-plates, double sided and stiffened in tension and compression, symmetrically,7 Extended end-plate, single sided and Morris stiffener,7 Flush end-plate, single sided and cover plate,5 Flush end-plates, double sided, cover plate and symmetrically 6, Welded joint, single sided and unstiffened,6 Welded joints, double sided unstiffened and symmetrically 6,0 Welded joint, single sided, stiffened in tension and compression 5,6 Welded joints, double sided stiffened in tension, compression and symmetrically 5

6 4. Required stiffness Eurocode 3 gives two diagrams to classify joints according to their stiffness (pinned, semirigid, rigid): one for braced and one for un-braced frames. For braced frames Eurocode 3 says that a joint may be regarded as rigid if: S j 8 E I b l. The background of this rule for rigid joints is that the bearing capacity of the frame doesn't drop with more than 5% due to the difference between the assumed joint stiffness in the frame analysis (S j = ) and the 'actual' stiffness [7]. In the context of this paper, the 'actual' stiffness is the best value a designer can obtain for the stiffness of a particular joint. This is, for example, a value obtained from a test or based on Eurocode 3. If a difference between assumed (S j = ) and 'actual' stiffness has a limited effect on the frame behaviour, then it is not required to perform a second frame analysis with the 'actual' stiffness of the joint. The check whether a joint is rigid needs to be done in three steps, see figure 3. Step a) shows the inclusion of the joint stiffness in the frame analysis (in step 3 of table ). Step b) shows the range in which the 'actual' stiffness should be (in step 7 of table ). Step c) shows the check that the 'actual' stiffness is in this range (in step 7 of table ). M S j,app M M S j,act Frame analysis based on the assumption S j,app = (the joint is rigid) φ φ Stiffness range for rigid joints based on S = j,app see Eurocode 3 φ Check if 'actual' stiffness S j,act is in range Step a) : engineer Step b): fabricator Step c): fabricator figure 3: Check of stiffness requirement for a rigid joint This concept can be generalized to a check whether a difference between assumed and 'actual' stiffness of semi-rigid joints has a significant influence to the frame behaviour, see figure 4. The corresponding formulae for the variance between the assumption used in the frame analysis and the 'actual' stiffness of the joint are given in table 4. These criteria may be used to check whether a difference between assumed joint stiffness in the frame analysis and 'actual' stiffness have the above mentioned limited effect (5%) on the frame behaviour. M S j,app M upper boundary M S j,act lower boundary φ φ φ Frame analysis based on the Stiffness range for semi-rigid joints Check if 'actual' stiffness S j,act assumption S based on S is in range j,app j,app Step a): engineer Step b): fabricator Step c): fabricator figure 4: Check of stiffness requirement of a semi-rigid joint 6

7 table 4: Boundaries for variance between actual an approximated stiffness Frame Lower boundary Upper boundary Braced 8 S j,app E I b 8 E I b S j,act If S 0 E I b + S j,app l j,app then l 0 S j,app E I b S j,act 8 E I b - S j,app l Un-braced 4 S j,app E I b S j,act 30 E I b + S j,app l else S j,act 4 E I b If S j,app then l * 30 S j,app E I b S j,act 4 E I b - S j,app l else S j,act in which: S j,app = the assumed stiffness adopted in the frame analysis (this is an approximation of the 'actual' stiffness) S j,act = the 'actual' stiffness of a joint E = youngs modules l = beam length I b = moment of inertia of the beam * For reasons of simplicity in the formulae for un-braced frames, the Eurocode 3 rigidity boundary is rounded off from S j 5 E I b 4 E I b to S l j l 5. Comparison to the stiffness model of Annex J (revised) The stiffness predictions with table 3 are compared to the stiffness according to Annex J (revised) (this is seen as the 'actual' stiffness) for 6 different joints. Wald & Steenhuis [8] give a full description of the geometry of these 6 joints. Tables 5 and 6 give for these 6 joints successively: the approximation for the stiffness according to table 3 (step a) the range in which the Eurocode 3 prediction should lie, if the frame analysis is performed with the approximation for the stiffness according to table 3 (step b) the 'actual' stiffness assuming this is predicted by Annex J (revised) and a check if the 'actual' stiffness is in the range (step c). These tables show good agreement between prediction based on table 3 and Annex J (revised). 7

8 table 5: Comparison of stiffnesses for a braced frame, beam span l = 0 h b Joint Approximation for the stiffness, Range (step b) 'Actual' stiffness according to Check (step c) (step a) see table 3 Lower boundary Upper boundary Annex J (revised) knm/mrad * knm/mrad * knm/mrad * knm/mrad * 83# Ok 83#0 6 ** 4 8 Ok 83# Ok 9# 59 ** 3 59 Ok 9# 59 ** 3 60 Ok 9#3 8 ** Ok * Kilonewton meter per milliradian ** According to Eurocode 3 this joint is rigid table 6: Comparison of stiffnesses for an un-braced frame, beam span l = 5 h b Joint Approximation for the stiffness, (step a) see table 3 Range (step b) Lower boundary Upper boundary 'actual' stiffness according to Annex J (revised) Check (step c) knm/mrad knm/mrad knm/mrad knm/mrad 83# Ok 83# Ok 83# Not Ok * 9# Ok 9# Ok 9# Ok * In this case, the 'actual' stiffness should be introduced in the frame analysis 6. Design example The example in figure 5 shows the effect of the stiffness of a joint on the deformations and the force distribution of an un-braced frame. First order elastic analysis is used. The span of the beam is 6 meters. The column height is 4 meters. Both columns and beams are IPE 360 sections. Loads consist of a horizontal load (F = 5 kn) and an uniformly distributed vertical load (q = 40 kn/m) F = 5 kn q = 40 knm δ h M A M B M C EI c EI b j 4 m δ v G t figure 5: 6 m Deformations Single storey single bay frame M Table 7 gives internal forces and deformations in the frame for different joints which may be considered as rigid according to Eurocode 3. A stiffness of S j = 40 knm/mrad is the lowest value to be regarded as rigid. Compared to a theoretical rigid joint (S j = ), vertical deflections vary 0% and horizontal deflections vary 0%. The drop in bearing capacity is not more than 5%. Table 8 shows the frame forces and the deformations for a semi-rigid joint with a stiffness S j = 60 knm/mrad. This table also gives deformations and forces for stiffnesses S j = 35 knm/mrad and S j = 30 knm/mrad. These are the lower and upper boundaries for stiffness according to table 4 when in the frame analysis a stiffness of S j,app = 60 knm/mrad is adopted. When comparing the case S j = 60 knm/mrad to S j = 35 knm/mrad or S j = 30 knm/mrad, variations in deflections occur of 0% in vertical deflections and 0% in horizontal deflections. 8

9 The variations in deflections between different rigid joints in table 7 are in close agreement to the variations between the different semi-rigid joints in table 8. table 7: Comparison between frames with different rigid joints rigid joints S j knm/ 40 * mrad lower approximation boundary δ v mm δ h mm M A knm M B knm M C knm * In accordance with Eurocode 3 Annex J (revised) half S j is adopted in the frame analysis table 8: Comparison between frames with different rigidity for semi-rigid joints semi-rigid joints S j knm/ 35 * 60 * 30 * mrad lower approximation upper boundary boundary δ v mm δ h mm M A knm M B knm M C knm * In accordance with Eurocode 3 Annex J (revised) half S j is adopted in the frame analysis stiffened web panel unstiffened web panel 7. Conclusions This paper shows that semi-rigid design of joints can fit in standard design practice by adopting an adjusted traditional design approach. In this approach, the share of responsibilities between engineer and steel fabricator are similar to those in traditional frame design with pinned or rigid joints. In the adjusted approach, a first approximation of the joint stiffness should be included in the frame analysis. Simplified formulae based on Eurocode 3 Annex J (revised) help to make this approximation. The philosophy to check whether a joint is sufficiently stiff to be regarded as a rigid joint in frame analysis can easily be extended to the application of semi-rigid joints. In this application the difference between approximation and actual stiffness should fit within certain limits. These limits are dependent on the type of frame, the beam span, the beam stiffness and the joint stiffness. Examples show that the adjusted approach is feasible for application in practice. We intend to compare the simplified formulae from table 3 to tests form the Aachen data bank in the near future. As a possible result of this comparison, the check of the rigidity as described in figure 4 could be omitted for certain joint geometry's, especially in braced frames. 9

10 References [] DOL C. & STEENHUIS C.M. Bolted end-plate connections (in Dutch), Staalbouwinstituut Rotterdam, Bouwen met Staal nr 03, 99. [] BROZZETTI, J., Design of Connections in the EUREKA "CIMSTEEL" project, Proceedings of the Second International Workshop on Connections in Steel Structures, edited by Bjorhovde R., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 99; [3] STEENHUIS, C.M., DOL C. & VAN GORP, L. Computerised calculation of force distributions in bolted end-plate connections according to Eurocode 3, Journal of Constructional Steel Research Vol. 3, 994; [4] EUROCODE 3, ENV , Design of Steel Structures, Commission of the European Communities, European Prenorm, Brussels, Belgium, April 99; [5] EUROCODE 3, ENV , Revised annex J, Design of Steel Structures, CEN, European Committee for Standardization, Document CEN / TC 50 / SC 3 - N 49 E, Brussels, June 994. [6] CUNNINGHAM R. & TAYLOR C., Practical design allowing for semi-rigid connections, Proceedings of the Second International Workshop on Connections in Steel Structures, edited by Bjorhovde R., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 99; [7] BIJLAARD F.S.K, STEENHUIS C.M. Prediction of the Influence of Connection Behaviour on the Strength, Deformations and Stability of frames, by Classification of Connections, Proceedings of the Second International Workshop on Connections in Steel Structures, edited by Bjorhovde R., Colson A., Haaijer G. and Stark J., AISC, Chicago, Illinois, USA, 99; [8] WALD F., STEENHUIS C.M., The Beam-to-Column Bolted Joint Stiffness according to Eurocode 3, proceedings of the first COST C workshop, Strasbourgh

11 ANNEX: COMPARISON WITH SPRINT DESING TABLES Introduction In the proceedings of the second state of the art workshop on Semi Rigid Behaviour of Civil Engineering Structural Connections COST C, held in Prague 6-8 October 994, Martin Steenhuis, Nol Gresnigt and Klaus Weynand have presented a paper concerning pre-design of semi-rigid connections. They give simple formula to assess the stiffness of beam-to-column connections in the pre-design stage. This assessment should in a second step be verified against the 'actual' stiffness of the joint to be realised in the structure. It should -of course- be in reasonable agreement with the 'actual' stiffness. The formula to assess the stiffness in the pre-design stage is as follows: S E z t = k x f.c j.app where: E is the Youngs modules t f.c is the column flange thickness z is the distance between centre of compression and tension k x is a factor dependent from the type of joint (e.g. 3 for extended unstiffened beam to column joints) S j.app the 'good guess' for the initial stiffness To see whether the estimation of the stiffness is in reasonable agreement with the 'actual' stiffness of the joint, in the COST proceedings the following formula are given: For braced frames: S 8 S 0 E I + S E I j.app b j.act b j.app l but S 0 S 8 E I E I - S j.app b j.act 3 b j.app l in case S 8 E I l b j.app 4 where: I is moment of inertia S j.app is the approximation of the initial stiffness S j.act is the actual initial stiffness S j.low is the allowable lower bound for the initial stiffness S j.upp is the allowable upper bound for the initial stiffness l is the beam length For unbraced frames:

12 S 4 S 30 E I + S E I j.app b j.act = S j. low 5 b j.app l but S 30 S j 4 E I E I - S j.app b j.act = S j. upp 6 b.app l in case S 4 E I l b j.app 7 This document provides further comparisons between predictions based on the 'good guess' formula with more accurate stiffness predictions based on Eurocode 3 Annex J, as given in the "SPRINT Design Manual". For this purpose, 6 series of beam-to-column connections have been analysed. It concerned: - two series with extended unstiffened end plate connections (series "spr_ex.dat" and "spr_ex.dat"). In this case, distance z is calculated as follows: z = h b - t f.b - two series with flush end plate connections, with an extended part in the bottom flange (series "spr_fl.dat" and "spr_fl.dat"). In this case, distance z is calculated as follows: z = h b t f.b - u where u is the distance between the top of the beam to the centre of the upper bolt row - two series with flush end plate connections, with a non-extended part in the bottom flange (series "spr_lfl.dat" and "spr_lfl.dat"). In this case, distance z is calculated as follows: z = h b - t f.b - u The differences between two series within one type concerns bolt spacing and bolt grade. In all calculations, it is assumed that the beam span is 5 times the beam height. This is taken as a lower bound of the beam span in practical structures. Higher values could also be adopted, but in that case, the frame is less sensitive for the stiffness variations of the joints. It is assumed that the extended end plated joints will be adopted in unbraced frames, and the flush end plated joints in braced frames. Example calculation As an example, a flush end plate connection of chapter 3b of the "SPRINT design manual" is calculated. Column: HEB40B, Beam: IPE0 z = h b t f.b - u = * = 55.4 mm

13 S E z t = k x = = knm/rad f.c j.ini 8 S 8 S 0 E I + S E I j.app b j.act = b j.app l = knm/rad * 0 = 6 S 0 S 8 E I E I - S j.app b j.act = 5978 b j.app l =785.5 knm/rad * 0 = 6 Conclusion: The approximation is within the limits and can be used in the frame analysis. Results The figures herafter give the results to all 6 series in a graphical form on 6 different sheets. Each sheet shows four lines: - This line represents the 'actual' initial stiffness divided by the 'actual' initial stiffness (S j.act / S j.act ) - SJAPPSJ This line represents the approximated initial stiffness divided by the 'actual' initial stiffness (S j.app / S j.act ). - SJLOWSJ This line represents the lower bound of the stiffness divided by the 'actual' stiffness (S j.low / S j.act ). - SJUPPSJ This line represents the upper bound of the stiffness divided by the 'actual' stiffness (S j.upp / S j.act ). It appears in all cases, that non of the upper or lower boundaries is crossing the line '', so all S j.ini according to the SPRINT manual are within the predicted ranges based on the 'good guess' whenever the beam span is 5 * h b or more. Acknowledgement This comparison was made in the frame of the ECSC project 'Design manual'. Thanks are given to the group preparing the 'SPRINT Design Manual' for making their results available for this comparison. 0 3

14 spr_ex "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 4

15 spr_ex "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 5

16 spr_fl "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 6

17 spr_fl "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 7

18 spr_lfl "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 8

19 spr_lfl "SJAPPSJ" "SJLOWSJ" "SJUPPSJ" joint number 9