THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS

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1 EUROSTEEL 2002, Coimbra, September 2002, p THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS Fernando C. T. Gomes 1 ABSTRACT The Eurocode 3 proposes a classification of beam-to-column joints by moment resistance (pinned, partial-strength and full-strength and by rotational stiffness (pinned, semi-rigid and rigid, but no classification by rotation capacity is provided. In the last years, the Eurocode 3 classification boundaries have been discussed. Different structures are used in this paper to assess the accuracy of those boundaries and to show the significant influence of the rotational stiffness of joints on deformations, stability and resistance of structures. The classification boundaries for rigid and pinned joints are discussed and alternative values are proposed. The classification of joints by rotation capacity was not yet introduced in the Eurocode. However, the definition of the required rotation capacity is necessary, in particular in the case of nominally pinned joints and partial-strength joints. Classification boundaries are proposed to classify joints by rotation capacity as class 1, class 2 and class 3. Key Words: Steel Structures, Eurocode 3, Joints, Classification. 1 INTRODUCTION The Eurocode 3 [1] proposes a classification of beam-to-column joints by moment resistance and by rotational stiffness, which is summarised in section 2. This classification has been discussed by several authors, in particular in the frame of the COST C1 action [2, 3]. Different alternatives have been proposed, depending on the used criteria. The criterion we use to classify joints as rigid is clearly identified in 2.3. The Eurocode classification boundaries are confronted with this criterion in sections 3 and 4, in order to assess the accuracy of those boundaries. Pinned joints are discussed in 5 and a classification by rotation capacity is found in 6. 1 Assistant, Civil Engineering Department, University of Coimbra, Portugal

2 2 THE EUROCODE 3 CLASSIFICATION 2.1 Classification by moment resistance The Eurocode 3 classification of joints with respect to its design moment resistance is shown in Fig. 1. While the boundary between full-strength and partial-strength is well defined by the design plastic moment resistance of the connected beam M pl. Rd, the boundary between partial-strength and pinned (0.25M pl.rd is polemical, as discussed in section 5. Fig. 1 - The Eurocode 3 classification by moment resistance 2.2 Classification by rotational stiffness The Eurocode 3 classification by rotational stiffness differs if the frame is braced or unbraced. A beam-to-column joint is considered as rigid if the rising portion of its momentrotation characteristics lies above the solid line on the appropriate diagram in Fig. 2. The bilinear boundary is given by 8 EI b! if M " 2 3 M L pl. Rd b for braced frames: M = $ 20 7 EI b! M pl.rd if 2 3 M pl.rd < M < M pl. Rd 25 EI b! if M " 2 3 M L pl. Rd b for unbraced frames: M = $ 25 EI b! M 2 pl.rd if 3 M pl.rd < M < M pl. Rd where M is the joint moment,! is the joint rotation and I b is the second moment of area of the beam I c is the second moment of area of the column is the span of the beam L c is the storey height for the column. The Eurocode boundary (2 may be used only if every storey satisfies the condition (1 (2 K b K c > 0.1 (3 in which K b is the mean value of I b for all the beams at the top of that storey, and K c is the mean value of I c L c for all the columns in that storey.

3 Fig. 2 - The Eurocode 3 classification boundaries for rigid joints The Eurocode 3 classifies a beam-to-column joint as nominally pinned if its secant rotational stiffness satisfies the condition 2.3 Criterion to classify a joint as rigid! 0.5EI b (4 The clause (1 of the Eurocode 3 defines a general criterion to classify a joint as rigid: if its deformation has no significant influence on the distribution of internal forces and moments in the structure, nor on its overall deformation. The clause (2 proposes the rule: The deformation of rigid joints should be such that they do not reduce the resistance of the structure by more than 5, i.e.! ult " 0.95! ult. (5 where! ult is the ultimate load multiplier considering the actual behaviour of joints, and! ult." is the ultimate load multiplier if nominally rigid joints are assumed as perfectly rigid. In the following sections 3 and 4, we analyse if the Eurocode boundaries (1 and (2 satisfy the condition (5. Several sub-structures of braced and unbraced frames are analysed, showing the significant influence of the joint stiffness on the overall behaviour of frames. 3 RIGID JOINTS IN BRACED FRAMES 3.1 Displacements and moments redistribution The stiffness of joints affects displacements and moment distribution, as illustrated in Fig. 3 for a simple beam, representative of an interior span of a braced frame. The behaviour of a beam with perfectly rigid joints, Fig. 3a, is compared to its actual behaviour, Fig 3b, where the joint stiffness is taken into account. The moment in the joints, for any load symmetrical with respect to the mid-span, is M j = M j.! + 2EI b (6 where M j.! is the moment if the joints are perfectly rigid, and is the joint secant stiffness.

4 Fig. 3 - Variation of displacements and redistribution of moments The displacement at mid-span of the beam under a uniformly distributed load is $! =! " S 5 4 j + 2EI b ' (7 ( where! " is the mid-span displacement if the joints are perfectly rigid ( =!. Let's define the displacement error! " and the moment error! M as! " = " " $ " $ and! M = M j." M j M j." (8 Assuming the joint stiffness = 8EI b, corresponding to the first branch of the Eurocode boundary (1, equations (8 give the following errors! " = 8EI b 2EI =80 and! M = b =20 (9 + 2EI b + 2EI b These errors may be even higher, if the moment in the joints falls between 2 3 M pl. Rd and M pl. Rd. In fact, according to the Eurocode boundary (1, the secant stiffness of a rigid joint may be less than = 8EI b. In this case, the error! " becomes greater than 80 and! M greater than 20. The magnitude of these errors justifies a correction of the Eurocode boundary values for rigid joints in braced frames. 3.2 Stability and resistance The buckling length of a column in a braced frame is commonly determined by using sub-frames, as represented in Fig. 4. The deformation of joints implies an increase of the buckling length, i.e., a decrease of the critical load, and thus a decrease of the resistance. The diagram in Fig. 4 shows the variation of the critical load N cr with the secant stiffness of the joints, for the sub-frame where the beams and the columns are supposed to have the same stiffness ratio, i.e. I b = I c L c. This particular sub-frame was chosen because it leads to an upper bound for the reduction of critical load. The critical load if the joints are assumed as perfectly rigid is given by

5 N cr.! = 2.12 " 2 EI c L c 2 (10 Fig. 4 - Braced frame. Critical load versus secant stiffness of the joints If we use the Eurocode boundary = 8EI b, the critical load is reduced to 0.87 N cr.!, i.e., an error of 13, Fig. 4. The influence of such a reduction of critical load on the reduction of resistance depends on the column slenderness. For instance, taking the curve "a" of the Eurocode 3 for the buckling resistance of columns, Fig. 5, a 13 reduction of critical load implies a 8 reduction of resistance if! =1, and a 12 reduction of resistance if! =2. Critical load N cr N pl variation Buckling resistance Curve "a" of EC3 N Rd variation N pl.rd! = ! = ! = ! = Fig. 5 - Critical load and resistance of a column The reduction of resistance may be even higher, if the moment in the joints is greater than 2 3 M pl. Rd. In fact, according to the Eurocode boundary (1, the secant stiffness may decrease until = 5EI b. In this case, the reduction of critical load reach 19, and the corresponding reduction of resistance may reach about 19 if the column is slender. These reductions of resistance are not compatible with the Eurocode criterion (5. Therefore, the Eurocode 3 boundary (1 for braced frames is not satisfactory. The boundary value 25EI b is proposed, leading to a maximum reduction of critical load of about 5, see Fig. 4.

6 4 RIGID JOINTS IN UNBRACED FRAMES 4.1 Horizontal displacements The horizontal displacements of unbraced frames, Fig. 6a, are influenced by the stiffness of the joints. The analysis of the sub-frame in Fig. 6b allows us to derive an upper bound for the inter-storey drift of unbraced frames with beams connecting each column at each storey level, like that in Fig. 6a. Frames with pinned or built-in column bases, Figs 6c and d, are particular cases of the general sub-frame in Fig. 6b. Fig. 6 - Unbraced frames For the general sub-frame in Fig. 6b, the inter-storey drift is given by! = HL 3 c 12EI c $ ' " + 3" EI b L b ( $ ' 1+ " + 6" 1+ 6 EI b L b ( EI b EI b where! and! are the ratios of rigidity defined in Fig. 6b and is the secant stiffness of the joints. The displacement error! " is shown in Fig. 7 for different! and! values, if the joint stiffness is = 25EI b. In this case, the upper bound! " =24 is found for! =0 (infinitely rigid columns. (11

7 Fig. 7 - Displacement error for nominally rigid joints From (11, the upper bound for the inter-storey drift of the general sub-frame is! ( $ '! " upper bound = 1+ 6 EI b (12 where! " = lim! is the inter-storey drift if the joints are assumed as perfectly rigid. " We notice that the Eurocode condition (3, equivalent to! >0.1, does not correspond to any limit situation in terms of displacements (! " 24 for 0! 0.1, Fig Stability and resistance The elastic critical load is affected by the deformations of the joints. An upper bound for this effect may be derived if one consider infinitely rigid columns ( I c =!, Fig. 8, $! cr."! cr ( ' upper bound =1 + 6 EI b (13 where! cr = V cr V Sd is the elastic critical load multiplier if the actual stiffness of joints is taken into account, and! cr." is the elastic critical load multiplier considering perfectly rigid joints. Fig. 8 - Mode of instability

8 For = 25EI b (Eurocode 3 boundary, the upper bound (13 yields $! cr."! cr ( ' upper bound =1.24 (14 The second-order effects may be evaluated by the amplified sway moment method, if the elastic critical multiplier is more than 4 (Eurocode 3 clause (4: the sway moments found by a first-order elastic analysis should be increased by multiplying them by the ratio 1 ( 1!1 " cr. The error in the calculation of the amplified sway moments is then! M = M " M M ( "1 ( 1 "1 $ cr. ( = 1 1 "1 $ cr 1 1"1 $ cr. Taking into account the upper bound (14, the following expression, plotted in Fig. 9, gives an upper bound for the moment error: (15! M.max = 0.24 " cr. $1.24 (16 Fig. 9 - Sway moments errors Fig. 9 shows that, if! cr." 4, the maximum error in the evaluation of the amplified sway moments is less than 9, which is an acceptable level of accuracy. However, if! cr." < 4, the deformation of joints classified as rigid can lead to a significant influence on the overall behaviour of unbraced frames. 5 PINNED JOINTS The Eurocode 3 does not permit to classify a joint as nominally pinned if its design moment resistance is greater than 0.25M pl.rd or if its secant rotational stiffness is greater than 0.5 EI b. However, it is not necessary to evaluate the stiffness or the resistance of a joint to classify it as nominally pinned. Only the rotation capacity, necessary to develop all plastic hinges under the design loads, should be checked. If the joint develops significant moments which might adversely affect members of the structure, the joint cannot be classified as nominally pinned.

9 6 CLASSIFICATION BY ROTATION CAPACITY The Eurocode 3 imposes to check the rotation capacity of pinned and partial-strength joints, and also of full-strength joints if its design moment resistance is less than 1.2 M pl. Rd, but it does not specify any classification by rotation capacity. Recently, three classes have been proposed [2, 3], without a definition of classification boundaries: Class 1 joints, which can form a plastic hinge with the rotation capacity required for plastic analysis; Class 2 joints, which can develop their plastic moment resistance, but have limited rotation capacity; Class 3 joints, in which brittle failure (or instability limits their moment resistance. A criterion to derive classification boundaries may be found if the classification of joints by rotation capacity is related to the classification of cross-sections. For instance, if the simple supported beam in Fig. 10 has a class 1 cross-section, the joints at the beam-ends should be of class 1 too. In fact, the rotation capacity of such joints should be sufficient to enable the plastic hinge to develop in the beam. From the classification of cross-sections in Fig. 10 (from [4], we may then derive the following classification boundaries for pinned joints: " $! Cd > 4! pl for class 1 joints $ 2! pl for class 2 joints where! pl is the rotation at the ends of a simple supported beam when the moment in the beam reaches M pl. Rd. (17 Fig Classification of cross-sections The behaviour of a beam with semi-rigid joints, under a uniformly distributed load, is illustrated in Fig. 11a, where the dashed line BC defines the joint rotation when the moment reaches M pl. Rd at mid-span. This rotation, graphically determined by the intersection of line BC with the M!" curve, is given by! o =! pl (1 " 0.5M / M pl. Rd. If the beam has a class 1 crosssection, the development of a plastic hinge will increase the joint rotation by 3! pl. In the case of a class 2 cross-section, the increase of joint rotation will be! pl. Thus, the required rotation of semi-rigid joints is defined by the solid lines in Fig. 11b, parallel to the dashed line BC, and writes " $! Cd >! o + 3! pl for class 1 joints $! o +! pl for class 2 joints (18

10 a b Fig a Joint rotation required to form a plastic hinge in the beam b Boundaries for the classification by rotation capacity However, because! o "! pl, the classification boundaries (17 for pinned joints may be safely used for semi-rigid joints, instead of (18. A further simplification may be introduced if we note that, in practical applications,! pl is in general close to 0.02 rad. Thus, the classification boundaries may be expressed in radians, allowing for a direct classification of a M!" curve (obtained from a test, for example without knowing the beam length: " 0.08 rad for class 1 joints! Cd > $ 0.04 rad for class 2 joints (19 7 CONCLUSIONS This paper shows that the Eurocode 3 classification boundaries are not satisfactory. It is proposed to classify a joint as rigid, in braced or unbraced frames, if its secant rotational stiffness satisfies the condition > 25EI b However, this condition shall not be applied for unbraced frames if the elastic critical load multiplier is less than 4. It is not necessary to satisfy the Eurocode condition (3. Pinned joints should be classified by rotation capacity only. Simple boundaries for the classification by rotation capacity are also proposed. 8 REFERENCES [1] Eurocode 3 - "Design of steel structures" - Part 1.1: General rules and rules for buildings", CEN, Brussels, [2] Jaspart, J-P (Editor, Recent Advances in the Field of Structural Steel Joints and their Representation in the Building Frame Analysis and Design Process, European Commission, Brussels, [3] Gomes, F.C.T., Kuhlmann, U., De Matteis, G., and Mandara, A. - "Recent developments on classification of joints", Proceedings of the International Conference on the Control of the Semi-rigid Behaviour of Civil Engineering Structural Connections, Liege, Belgium (Ed. R. Maquoi, European Commission, Brussels, 1998, [4] Eurocode 3 - "Calcul des structures en acier" et Document d'application Nationale - Partie 1.1: Règles générales et règles pour les bâtiments, AFNOR, Paris, 1992.

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