223

EVALUATION OF DUCTILITY IN STEEL AND COMPOSITE BEAM-TO-COLUMN JOINTS: ANALYTICAL EVALUATION

Luís Simões da Silvaa, Luís Caladob, Rui Simõesa, Ana Girão Coelhoc

a Civil Engineering Department, Universidade de Coimbra, Coimbra, Portugal b Civil Engineering Department, Instituto Superior Técnico, Lisboa, Portugal

c Civil Engineering Department, Instituto Superior de Engenharia de Coimbra, Coimbra, Portugal

Abstract

The current trend towards the use of partial strength, semi-rigid joints requires that enough ductility (rotation) is available, and thus the prediction of the full (non-linear) moment-rotation response of the joint. The component method currently provides independent procedures to evaluate the strength and initial stiffness of steel and composite joints. A unified, closed-form, analytical approach is presented in this paper that gives the full non-linear moment-rotation response of steel and composite joints, and, consequently, its strength, initial stiffness and maximum rotation.

1 INTRODUCTION

The current trend towards the use of partial strength, semi-rigid joints requires that enough ductility (rotation) is available, and thus the evaluation of the full (non-linear) moment-rotation response of the joint. The component method, currently widely accepted as the practical approach at predicting the behaviour of such joints (1), provides independent procedures to evaluate the strength and initial stiffness of steel and composite joints. These procedures, already incorporated in codes of practice (2, 3), were shown to reproduce satisfactorily these properties, while maintaining a relative ease of application.

The evaluation of ductility presents two added difficulties, when compared to strength and initial stiffness:

(i) knowledge of the non-linear force-deformation response of each component; (ii) knowledge of the full (non-linear) moment-rotation response of the joint.

The first item still remains quite unexplored in the literature, most of the research effort being directed in the past towards the consistent evaluation of strength and initial stiffness of the various components (4); the second involves iterative numerical procedures, given that phenomena such as plasticity and instability are necessarily present.

Assuming that the non-linear behavior of the components is known, a unified, closed-form, analytical approach is presented in this paper that gives the full non-linear moment-rotation response of steel and composite joints, and, consequently, its strength, initial stiffness and maximum rotation. Also, the yielding sequence of the various components is identified.

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2 EVALUATION OF DUCTILITY

2.1 Component characterisation

As stated above, a key aspect to the component method relates to the characterisation of the force-deformation curves for each individual extensional spring. In practical terms, the non-linear force-deformation curve may be approximated by several idealisations (5), as shown in Figure 1. Common to all is the identification of four sets of properties, namely elastic stiffness (ke), post-limit stiffness (kp), limit load (FC=PB/2) and limit displacement ( f).

F

ACTUALBEHAVIOUR

APPROXIMATION EQUIVALENT ELASTIC MODEL

F

ke

a) Linear aproximation

F

ke

F

ke

FC ke

1 +kp

1

b) Bi-linear approximation

F

ke

kp ; PB = 2FC

F

ke

FC,1ke

1 +kp1

1FC,2

ke

1+ kp1

1+ kp2

1

c) Tri-linear approximation

F

ke

kp1;P1B=2FC,1

kp2;P2B=2FC,2

F

ke

FC

d) Non-linear approximation to the post-limit behaviour

kp = kp1+4Lkp2(1-cosQ2)+12L2kp3(1-cosQ2)2+...

F

ke

kp ; PB = 2FC

L L

Q2

Fig. 1. Various idealisations of component force-deformation curves.

Following Kuhlmann et al (6), the various components may be classified according to ductility in three main groups: (a) components with high ductility, (b) components with limited ductility and

225

(c) components with brittle failure. Components with high ductility present a nearly unlimited deformation capacity, not imposing any bounds on the overall rotation ability of the joint, and include, for example: (i) column web panel in shear, (ii) end-plate in bending and (iii) column flange in bending. Components with limited ductility are characterised by a force-deformation curve exhibiting a limit point and a subsequent softening response, comprising: (iv) column web in tension and (v) column web in compression. Finally, components with brittle failure behave linearly until collapse, with very little deformation before failure, being adequately modelled with a linear approximation, typical examples being: (vi) bolts in tension, (vii) bolts in shear and (viii) welds.

2.2 Analytical models

To overcome the numerical complexity of the evaluation of the moment-rotation response of steel and composite joints, an equivalent elastic model was developed (7), able to yield closed-form analytical expressions. With reference to Figure 2, the proposed methodology (8)comprises the following steps, here illustrated for an extended end-plate steel joint:

(i) for each bolt row in tension and shear and compression zones, association of all springs (components) in series into one single equivalent spring;

(ii) association of all resulting tensile springs in parallel into an equivalent tensile spring

(iii) application of the equivalent elastic model of Figure 2c, that yields identical results to the original elastic-plastic model of Figure 2b.

k3, 1 k4, 1 k5, 1 k10, 1

k1 k2

k3, 2 k4, 2 k5, 2 k10, 2

M

z’1

z’2

Kc

Kt

M

z’h’

Centre of rotation

a) Original component model b) Basic non-linear model

kec LcLc

kpc, PCB

LtLt

ketkpt, PTB

M

z’

c) Equivalent elastic model

Fig. 2. General substitute model for steel joints.

As shown in Figure 1, both the spring transformations (series and paralel) and the equivalent elastic model require the choice of an adequate approximation for each resulting spring. Here, four possibilities are considered:

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(i) linear (L), the corresponding elastic model being simply an elastic spring with stiffness ke;

(ii) bi-linear (BL), the post-limit stiffness being reproduced by an elastic spring with stiffness kp and pre-compression 2FC;

(iii) tri-linear (TL), with two post-limit branches characterised by stiffnesses kp1 and kp2 and corresponding pre-compressions 2FC1 and 2FC2;

(iv) non-linear (NL), where the initial elastic part is followed by a polynomial non-linear branch given by:

223

2221 cos112cos14 qkLqLkkk pppp (1)

Next, the resulting equivalent elastic models are solved in the context of a post-buckling stability analysis using an energy formulation, further details of the mathematical derivation being found in (7,8). With reference to Figure 3, two basic models are considered, for steel (Figure 3a) and composite (Figure 3b) joints, the latter case including a specific tensile row for the reinforcement (9).

kc

kt

M

kc

kt2

M

kt1

a) Steel joints b) Composite jointsFig. 3. Basic non-linear models.

For each case, several possibilities must be considered, corresponding to the various combinations of equivalent spring idealisations:

(a) Steel models (a.1) Model BL-BL: bi-linear idealisation of equivalent tensile and

compressive/shear springs; (a.2) Model TL-BL: tri-linear idealisation of equivalent tensile spring and bi-linear

idealisation of equivalent compressive/shear spring; (a.3) Model TL-NL: tri-linear idealisation of equivalent tensile spring and non-linear

idealisation of equivalent compressive/shear spring; (b) Composite models

(b.1) Model BL-BL-BL: bi-linear idealisation of reinforcement, equivalent tensile and compressive/shear springs;

(b.2) Model TL-BL-NL: tri-linear idealisation of reinforcement, bi-linear idealisation of equivalent tensile spring and non-linear idealisation of compressive/shear spring.

It is noted that all these combinations yield closed-form analytical solutions for the moment-rotation response of steel and composite joints that identify the yield rotation of all relevant levels of component deformation. Its application to typical examples of steel and composite joints is illustrated in the next section.

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3 APPLICATION TO BEAM-TO-COLUMN JOINTS

3.1 Beam-to-Column Welded Steel Joint

MIPE 300HE 140 B

7 FW

S235

M

(1) (2)

(3)

a) Connection geometry b) Mechanical model

Fig. 4. Welded steel connectionIn order to illustrate the application of the equivalent elastic models, one joint configuration was chosen from the database SERICON II (Klein 105.011) (10), corresponding to a welded beam-to-column steel joint, described in Figure 4, which was tested by Klein at the University of Innsbruck in 1985.

Component FC (kN) ke (kN/m) kp (kN/m) �y

(mm) 1 218.17 3.608 105 6.013 104 0.6052 258.30 1.803 106 4.624 103 0.1433 258.30 1.803 106 4.624 103 0.143

Table 1. Component characterisation

Figure 5 compares the experimental results with the analytical results, obtained using a bi-linear approximation for the components, the various stiffness and strength values being reproduced in Table 1.

Component Component yielding sequence Failure

Absolute displacement �i (mm) Relative displacement yii /

1 -0.607 -1.372 -4.817 1.000 2.261 7.940 7.9402 -0.121 -0.143 -38.640 0.847 1.000 269.798 269.7983 0.121 0.143 38.640 0.847 1.000 269.798 269.798 2.94 5.73 151.63 1.00 1.95 51.65 51.65 Absolute joint rotation (mrad) Joint ductility index

Table 2. Ductility indexes for welded steel joint

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The moment-rotation curve of Figure 5 shows yielding of the first component (column web panel in shear), followed by simultaneous yielding of the column web in compression and in tension, at a joint rotation of about 0.006 radian. The ductile behavior of this joint is obvious, maximum rotation of 0.151 radians being reached without failure at the end of the test. Table 2 summarises the “yield” sequence of the various components and the corresponding levels of ductility.

0.012.525.037.550.062.575.087.5

100.0112.5125.0

0.0

12.0

24.0

36.0

48.0

60.0

72.0

84.0

96.0

108.

0

120.

0

132.

0

144.

0

156.

0

(mrad)

M (kNm)

Experimental results

Analytical results

Yielding of component 'co lumn web panel inshear'Yielding of components 'co lumn web incompression' and 'co lumn web in tension'

Fig. 5. Welded steel connection: moment-rotation curve (model BL-BL)

Figure 6 illustrates the application of two alternative models, TL-BL and TL-NL, using the same value of kp1., highlighting the good adjustment of the non-linear model.

0,012,5

25,037,5

50,062,575,0

87,5100,0

112,5125,0

0,0

12,0

24,0

36,0

48,0

60,0

72,0

84,0

96,0

108,

0

120,

0

132,

0

144,

0

156,

0

(mrad)

M (kNm)

Experimental results

Analytical results (model TL-NL assuming linearkpt)cws

cwc & cwt

Analytical results (model TL-NL assumingquadratic kpt)

Fig. 6. Welded steel joint: moment-rotation curve (model TL-NL)

3.2 Extended End-Plate Bolted Beam-to-Column Steel Joint

The second example corresponds to an extended end-plate bolted steel joint tested by Humer at the University of Innsbruck (Humer 105.009), illustrated in Figure 7.

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MIPE 450HE 240 B

S275553 240 41

(2)(1)

(3,2) (10,2)(4,2) (5,2)

(3,1) (10,1)(4,1) (5,1)

M

a) Connection geometry b) Mechanical model

Fig. 7. Bolted extended end-plate steel jointTable 3 indicates the chosen values for the various components. Using model BL-BL, Figure 8 compares the experimental results with the analytical results.

Component FC (kN) ke (kN/m) kp (kN/m) �y

(mm) 1 529.33 6.363 105 7.122 104 0.8322 576.13 2.474 106 3.022 104 0.233

3.1 510.78 1.426 106 2.513 104 0.3583.2 510.78 1.426 106 2.513 104 0.3584.1 476.21 5.601 106 3.131 103 0.0854.2 476.21 5.601 106 9.131 103 0.0855.1 635.40 2.315 107 8.446 103 0.0275.2 635.40 5.571 107 8.446 103 0.01110.1 635.40 1.199 106 0.530 10.2 635.40 1.199 106 0.530

Table 3. Component characterization

Yielding starts at the compression zone (column web in shear (1) followed by the column web in compression (2)). Next, the first row of bolts of the joint becomes critical, as seen in Table 4, the following components yielding in succession: column flange in bending (4.1), column web in tension (3.1), the joint reaching 0.058 radians at the end of the test.

230

0.040.080.0

120.0160.0200.0240.0280.0320.0360.0400.0

0.0

6.0

12.0

18.0

24.0

30.0

36.0

42.0

48.0

54.0

60.0

(mrad)

M (kNm)

Experimental resultsAnalytical resultsYielding of component 'co lumn web panel in shear'Yielding of component 'co lumn web in compression'Yielding of component 'co lumn flange in bending - row 1'Yielding of component 'co lumn web in tension - row 1'

Fig. 6. Bolted extended end-plate steel connection: moment-rotation curve

Component Component yielding sequence FailureAbsolute displacement �i (mm) Relative displacement y

ii /

1 -0.832 -1.562 -5.776 -6.724 -7.212 1.000 1.878 6.943 8.083 8.669 8.669 2 -0.214 -0.233 -9.271 -11.305 -12.352 0.919 1.000 39.808 48.543 53.038 53.038

3.1 0.209 0.227 0.334 0.358 1.083 0.583 0.635 0.932 1.000 3.022 3.022 3.2 0.162 0.177 0.259 0.278 0.287 0.453 0.493 0.724 0.775 0.801 0.801 4.1 0.053 0.058 0.085 11.134 16.851 0.625 0.681 1.000 130.946 198.187 198.187 4.2 0.041 0.045 0.066 0.071 0.073 0.486 0.529 0.777 0.831 0.860 0.860 5.1 0.013 0.014 0.021 0.022 0.023 0.469 0.510 0.749 0.804 0.832 0.832 5.2 0.004 0.005 0.007 0.007 0.007 0.364 0.397 0.582 0.623 0.644 0.644

10.1 0.248 0.270 0.397 0.426 0.441 0.469 0.510 0.749 0.804 0.832 0.832 10.2 0.193 0.210 0.308 0.330 0.341 0.364 0.397 0.582 0.623 0.644 0.644

3.43 7.25 29.26 48.35 58.24 1.00 2.11 8.52 14.08 16.96 16.96 Absolute joint rotation (mrad) Joint ductility index

Table 4. Ductility indexes for bolted extend end-plate steel connection

3.3 Flush End-Plate Bolted Beam-to-Column Composite Joint

In order to illustrate the application to composite joints, a double-sided bolted flush end-plate beam-to-column joint tested in bending by Simões at the University of Coimbra in 1998 (11) was selected, shown in Figure 7.

231

(13)

M

(3) (10)(4) (5) (8)

(2) (7)

MIPE 270

S235C35/45; A400NR

IPE 270

HE 220 A

M

a) Connection geometry b) Mechanical model

Fig. 7. Bolted flush end-plate composite joint

Table 5 reproduces the adopted component properties for model TL-BL-BL.

Component FC (kN) ke (kN/m) kp (kN/m) �y

(mm) 2 1550.20 3.244 106 1.000 101 0.4783 504.00 9.404 105 1.000 101 0.5364 346.20 2.982 106 1.000 104 0.1165 293.70 2.322 106 1.000 104 0.1267 578.50 3.600 104 0.0008 462.10 0.00010 444.53 2.257 106 1.000 104 0.197

124.99 2.310 10513 477.28 6.006 105

1.200 103 0.208

Table 5. Component characterization

Figure 8 compares the experimental and analytical results, The moment-rotation curve shows yielding of the first component (reinforcement), corresponding to the cracking of concrete in tension that occurs for relatively low values of bending moment and joint rotation (55 kNm and 0.8 mrad). It is noted that current Eurocode specifications for composite joints

232

0.020.040.060.080.0

100.0120.0140.0160.0180.0200.0220.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

(mrad)

M (kNm)

Experimental results

Analyt ical solut ion

Yielding of component 'concrete intension'Yielding of component 'beam weband f lange in compression'Yielding of component 'longitudinalslab reinforcement in tension'

Fig. 8. Bolted flush end-plate composite joint: moment-rotation curve

(3) disregard the cracking moment of the joint. Next, at a rotation of 5.1 mrad, yielding of the beam web and flange in compression takes place, followed by yielding of the reinforcement. Table 6 summarises the “yield” sequence of the various components and the corresponding levels of ductility.

Component Component yielding sequence Failure

Absolute displacement �i (mm) Relative displacement yii /

2 -0.055 -0.178 -0.212 -0.213 0.116 0.373 0.443 0.445 0.4453 0.058 0.188 0.223 0.224 0.109 0.350 0.416 0.418 0.4184 0.018 0.059 0.070 0.071 0.158 0.510 0.605 0.608 0.6085 0.024 0.076 0.090 0.091 0.187 0.601 0.713 0.717 0.7177 0.000 0.000 -3.007 -3.111 0.311 1.0008 0.000 0.000 0.000 0.091 0.119 0.382 0.453 0.456 0.45610 0.024 0.078 0.093 0.000 0.123 0.397 0.471 0.474 0.47413 0.208 2.053 2.554 4.729 1.000 9.866 12.274 22.724 22.724 0.76 5.07 11.55 16.53 1.00 6.72 15.29 21.88 21.879 Absolute joint rotation (mrad) Joint ductility index

Table 6. Ductility indexes for bolted flush end-plate composite joint

4 CONCLUDING REMARKS

A simple analytical procedure for the evaluation of the moment-rotation response of steel and composite joints was presented in this paper. It allows the consistent evaluation of strength, initial stiffness and ductility. Additionally, depending on the choice of component idealisation, this methodology is able to to approximate, as closely as desired, the true moment-rotation response of the joint, further identifying all relevant changes in joint response.

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Finally, it should be noted that proper application of the component method requires the adequate prediction of the post-limit stiffness of the various components, a task yet to be done.

5 ACKNOWLEDGEMENTS Finantial support from “Ministério da Ciência e Tecnologia” - PRAXIS XXI research project PRAXIS/P/ECM/13153/1998 is acknowledged.

6 REFERENCES 1 Weynand K, Jaspart J-P, Steenhuis M, The stiffness model of revised Annex J of Eurocode

3, in R. Bjorhovde, A. Colson and R. Zandonini (eds), Connections in Steel Structures III,Proceedings of the 3rd International Workshop on Connections in Steel Structures, Trento, Italy, pp. 441-452, 1995.

2 Eurocode 3, ENV - 1993-1-1:1992/A2, Annex J, Design of Steel Structures – Joints in Building Frames. CEN, European Committee for Standardisation, Document CEN/TC 250/SC 3, Brussels, 1998.

3 Eurocode 4, Draft prEN 1994-1-1, Design of composite steel and concrete structures – Part 1.1 (Draft nº1): General rules and rules for buildings. CEN, European Committee for Standardization. Brussels, 1999.

4 Faella C, Piluso V, Rizzano G, Structural steel semirigid connections: theory, design and software. CRC Press LLC, 2000.

5 Simões da Silva L, Girão Coelho A, Mode interaction in non-linear models for steel and steel-concrete composite structural connections”, in Proceedings of CIMS 2000 - 3rd

International Conference on Coupled Instabilities in Metal Structures, Lisboa, Portugal, 21-23 September, 2000 (in print).

6 Kuhlmann U, Davison JB, Kattner M, Structural systems and rotation capacity, COST Conference on Control of the semi-rigid behaviour of civil engineering structural connections, Liège, Belgium, pp. 167-176, 1998.

7 Simões da Silva L, Girão Coelho A, Neto E, Equivalent post-buckling models for the flexural behaviour of steel connections. Comp. Struct., 2000 (in print).

8 Simões da Silva L, Girão Coelho A, A ductility model for steel connections, Journal of Constructional Steel Research, 2000 (in print).

9 Simões da Silva L, Calado L, Cruz P, Dynamic behaviour of composite structures with composite connections, in Proceedings of STESSA 2000 - 3rd International Conference on Behaviour of Steel Structures in Seismic Areas, Montreal, Canada, 21-24 August, 2000 (in print).

10 Cruz P, Simões da Silva L, Rodrigues D, Simões R, Database for the semi-rigid behaviour of beam-to-column connections in seismic regions. Journal of Constructional Steel Research 1998:46(120):1-3, 1998.

11 Simões da Silva L, Simões R, Cruz P, Behaviour of end-plate beam-to-column compositejoints under monotonical loading, Engineering Structures, 2000 (submitted for publication).