Available rotation capacity of wide-flange beams and beam-columns Part 2. Experimental and numerical tests

Available rotation capacity of wide-flange beams and beam-columns Part 2. Experimental and numerical tests

J. Construct.SteelRes.Vol.43, Nos.1-3,pp. 219-244, 1997 © 1997ElsevierScienceLtd.All fightsreserved Printedin GreatBritain Pn: S0143-974X(97)00045-X 0...

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J. Construct.SteelRes.Vol.43, Nos.1-3,pp. 219-244, 1997 © 1997ElsevierScienceLtd.All fightsreserved Printedin GreatBritain Pn: S0143-974X(97)00045-X 0143-974X/97 $17.00+ 0.00 ELSEVIER

Available Rotation Capacity of Wide-Flange Beams and Beam-Columns Part 2. Experimental and Numerical Tests Victor Gioncu a & Dana Petcu b apolitechnica University Timi~oara, Timi§oara, Romania OWest University Timi~oara, Timi~oara, Romania

ABSTRACT In the first part of the paper, the theoretical approaches for determining the available rotation capacity of wide-flange steel beams and beam-columns are presented. Using a plastic mechanism, the computer program DUCTROT 96 is elaborated to determine this rotation capacity. The present paper is engaged with a comparison between the experimental data and theoretical results (using FEM) and remarks on the numerical tests obtained using the DUCTROT 96 computer program. A systematic study of the influence of the main factors on the rotation capacity is presented. The major conclusion of this study is the proposal to change the concept of cross-section behavioural classes, used in codes for determining the rotation capacity, with the concept of member behavioural classes. © 1997 Elsevier Science Ltd.

1 INTRODUCTION In the companion paper [1] the available rotation capacity of wide-flange beams and beam-columns is determined based on using a local plastic mechanism, composed of plastic zones and yield lines. A computer program DUCTROT 96 (DUCTility of ROTation) was elaborated at INCERC Timi~oara by Petcu and Gioncu [2]. The analysis takes into account the main factors influencing the available rotation capacity: material properties, member dimensions, interaction between flanges and web, moment variation, etc. Nevertheless, a very important question has to be formulated: how realistic are such analyses? In order to obtain answers to this question, experimental investigations play an essential role. In the present paper a comparison between the theoretical results obtained using the D U C T R O T 96 computer program and the experimental tests related 219

220

V. Gioncu and D. Petcu

to the technical literature is performed. A comparison with the theoretical results using FEM is also presented. The comparison shows a good correspondence, conferring confidence in this method and in the computer program. In the last part of the paper, the influence of the main factors is studied. The major conclusion of this study is that the actual provisions of codes are very far from showing the real behaviour of plastic hinges.

2 COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS Because the theoretical approaches contain some important simplifying assumptions, it is absolutely necessary to compare the theoretical results with those obtained during the experimental tests.

2.1 Length of plastic mechanism Because it is a very disputed problem, special attention was paid to this aspect. Unfortunately, there are only three papers in which this problem is mentioned [3-5]. The Haaijer [3] and Lukey-Adams [4] experimental data are presented in Table 1. The /3 E values refer to the measured parameter of a buckled shape length, while the/3c are the calculated values from the relationship

0 6( tf/3/4(d/TM ~c=

" \tw/

\b/

"

(1)

One can see a very good correspondence between the experimental values and the theoretical ones, the average values of the ratio 13c//3~ being 0.999 with the coefficient of variation cv = 11.7%. Kuhlmann [5] has obtained average values /3E = 1.20, with a very good correspondence with the theoretical average values/3c = 1.176 (Fig. 1). One must mention that the correspondence between the theoretical and experimental values (Fig. 2) can be considered an excellent one, due to the difficulties in measuring the buckled lengths. Another aspect which must be mentioned is that Lay and Galambos [6] consider that a half-wavelength of the buckled plastic mechanism must be considered in the case of uniform moment, while for the moment gradient, full wavelength is required in the analysis. The experimental evidences show that even for the moment gradient the half-wavelength must be taken into account.

Rotation capacity of beams and beam-columns, part 2 TABLE

221

1

Length of Buckled Shape

No.

Source

b (ram)

0

1

1

Haaijer

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Lukey and Adams

2 101.00 83.19 101.85 102.62 73.28 101.75 88.0 51.3 36.95 43.05 47.0 48.40 50.95 36.85 42.95 46.75 44.95

P 1.4

d

ty

tw

fie

tic

(~dfle)

3 4 5 227.00 10.90 7.47 184.07 9.73 5.99 225.00 13.00 8.33 182.22 12.09 7.82 234.75 8.08 5.89 235.1 10.80 7.65 235.1 10.80 7.65 191.3 5.28 4.45 191.3 5.28 4.45 191.3 5.28 4.45 191.3 5.28 4.45 191.3 5.26 4.45 241.2 5.26 4.60 241.2 5.26 4.60 241.2 5.26 4.60 241.2 5.26 4.60 241.2 5.26 4.60 average value standard deviation coefficient of variation Cv

6 1.2 1.0 1.1 1.0 1.2 1.04 1.10 0.89 1.03 0.91 0.95 0.8t 0.82 0.93 1.04 0.95 1.30

7 0.975 1.053 1.021 0.960 1.080 0.958 0.981 0.948 1.029 0.990 0.969 0.959 0.979 1.061 1.021 1.00 1.01

8 0.813 1.053 0.928 0.960 0.900 0.919 0.894 1.095 0.998 1.088 1.020 1.184 1.194 1.141 0.982 1.053 0.777 0.999 11.71% 0.117

1.2 mean of experimental values

÷

+

+

+

+

+ .~.

1.2 4"

+

.L

+ + 4"

1.0

.L.

1.176 mean of theoretical values

0.8 0.6 +theoretical values for Kuhlmann experimental test 0.4 0.2

2'

4'

' 6 '~ 8 ' 1'0 . .12. . 14 ' 16'

' 18 ' 20

t

22

'

' 24

- test

number

Fig. 1. Length of plastic mechanism for Kuhlmann experimental tests.

222

V. Gioncu and D. Petcu

.S

1.2

/,

1.0

0.8

I

Ix I I

equality

I I + Lucky & Adams I X Haaijer

0.6

I

I

I I

0.4

I •

.

018.

i

i

1.0

i

I

1.2

i

bE

Fig. 2. Correlation theoretical-experimental values for length of plastic mechanism.

2.2 Rotation capacity Generally, the scatter area for this type of problem is very large. For the stability problems, this is due to the influence of residual stresses and out-ofstraightness. Nakashima [7] shows that in the ductility analysis, neither aspect has a major influence. In exchange, the scatter area is due to the significant influence of hardening characteristics, which are not under the control of the designer. Due to this fact, in the study on variation capacity performed by Nakashima, the coefficient of variation of experimental ductility ratios covers values between 0.03 and 0.93. All these aspects underline that in this problem type it is possible to obtain a large scatter of results, when the theoretical and experimental results are compared. In Table 2 the experimental data for standard beam SB 1, collected from the technical literature, are compared with the theoretical results obtained using the DUCTROT 96 computer program. The geometrical and mechanical properties of experimental specimens are presented. The experimental rotation capacity, RE, is compared with the calculated values, Re, for the two forms of plastic mechanism, in-plane (M1) and out-ofplane (M2). The ratio between these values is determined, and finally the average value, standard deviation and coefficient of variation are calculated. From the 81 experimental tests, six results were eliminated from the statistical analy-

Rotation capacity of beams and beam-columns, part 2

223

sis because of great differences between the other values, showing that some experimental deficiencies were produced. (i) Lukey and Adams [4]. The experimental program consists of 12 specimens organized in three series of tests on rolled wide-flange beams. The selected sections had relatively slender flanges (b/tf = 7-10). A comparison between the tested moment-rotation relationship for beam B3 (position 5 in Table 2) with the calculated one using DUCTROT 96 is shown in Fig. 3. One can see that the out-of-plane mechanism is the dominant phenomenon, in agreement with the aspect of the tested beam (see fig. 8 of Ref. [4]), where the S buckled shape is evident. (ii) Kuhlmann [5,8]. The experimental program consists of 24 specimens on built-up welded wide-flange sections. The slendernesses of flanges are within 7.5 and 12. A good agreement between experimental results for specimens 7, 8, 9 (positions 19, 20, 21 from Table 2) and DUCTROT 96 results exists (Fig. 4). (iii) Spangemacher [9]. Spangemacher tested 34 specimens of rolled wideflange beams, but only 24 of these are available for comparison because there M/Mp

1.2

1.0

out-of-~ 0.8

Specimen B3 ]

Luckey-Adams (Position 5 in table 2)

- --a-- --a-- e x p e r i m e n t a l v a l u e s

DUCTROT 96

0.6

.4_3._0_543.05

] l != 528

fyf = 371 N/mm 2

fyw = 395 N/mm 2 0.4. ¢

,~

5.28

0.2 lateral buckling m o d e

_

x

1254 I 0.01

--

I

I

I

I

0.O2

0.03

0.04

0.05

0.06

Fig. 3. Lukey-Adams experimentaltests.

=, -

0.07 0 (rad)

2 Lukey-Adams

1 1 2 3a 4 5 6 7 8 9 10 11 12 13 14 15'~ 16 17

Kuhlmann

Source

No.

3 101.75 88.00 51.30 36.95 43.05 47.00 48.40 50.95 36.85 42.95 46.75 44.45 70.5 75.0 80.0 80.0 80.0

b

4 235.1 235.1 191.3 191.3 191.3 191.3 191.3 241.2 241.2 241.2 241.2 241.2 278.0 278.0 277.0 261.0 258.0

d

5 10.8 10.8 5.28 5.28 5.28 5.28 5.26 5.26 5.26 5.26 5.26 5.26 8.0 8.0 8.5 8.0 8.0

t~

6 7.65 7.65 4.45 4.45 4.45 4.45 4.45 4.60 4.60 4.60 4.60 4.60 5.0 5.0 5.5 6.0 5.0

tw

Geometrical data (mm)

7 3480 2946 1554 1036 1254 1396 1448 1372 960 1168 1296 1280 3404 3704 4000 2540 2636

l

8 283 283 371 371 371 371 371 371 371 371 371 371 236 236 449 287 287

Li

9 308 308 395 395 395 395 395 350 350 350 350 350 217 217 217 260 252

~

Yield stresses (N/mm 2)

TABLE 2 Comparison with Experimental Data

10 11.8 13.6 2.9 10.4 6.7 3.4 3.2 4.2 13.7 8.0 4.2 6.5 8.0 7.0 1.0 12.7 8.6

RE

11 7.9 10.8 6.3 11.7 8.8 7.4 7.0 6.8 12.3 9.0 7.7 8.0 7.9 6.9 3.3 7.2 7.6

MI

Rc

12 11.8 12.2 7.2 8.0 7.6 7.4 7.3 7.2 7.9 7.3 7.0 6.9 8.0 7.8 4.9 9.9 9.3

M~

Rotation capacity

13 0.67 0.79 2.48 '~ 0.77 1.13 2.17 2.19 1.62 0.58 0.91 1.68 1.06 0.99 0.99 3.3 a 0.57 0.88

RdRE

Ix9 lxa

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Spangemacher

80.0 80.0 80.0 80.0 85.0 91.0 95.0 70.5 75.0 80.0 80.0 80.0 80.5 80.0 80.0 80.0 85.0 91.5 95.0 109.3 109.3 109.5 109.2 110.0 110.5 112.8 111.0 139.7

259.0 280.0 280.0 275.0 279.0 278.0 278.0 239.6 239.0 237.2 148.6 200.0 269.0 278.0 279.0 279.0 279.0 278.4 278.6 186.5 184.9 186.1 185.2 188.0 189.0 188.3 192.6 241.2

8.0 8.0 8.0 8.0 8.0 8.0 8.0 10.2 10.0 10.4 10.2 10.0 10.0 10.0 10.0 10.0 10.0 10.3 10.2 16.3 16.2 16.3 16.1 10.5 11.0 11.0 10.7 17.8

4.0 5.0 5.0 5.0 5.0 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 6.0 6.0 6.0 6.0 6.0 6.0 9.8 9.4 9.6 9.4 7.5 7.4 7.5 7.5 10.9

2716 1796 2196 2598 2802 3~2 3400 30~ 3200 3508 23~ 22~ 2100 2000 ~02 28~ 2406 2500 2700 35~ 3500 35~ 35~ 30~ 40~ 3000 4000 30~

287 287 287 287 236 236 236 333 333 333 333 333 333 333 333 333 333 333 333 486 486 278 278 282 282 420 420 248

252 252 252 252 217 217 217 709 709 709 709 709 709 341 349 349 349 349 349 532 532 286 286 308 308 437 437 252

4.6 13.5 11.5 7.8 5.5 8.9 7.6 5.1 3.8 3.6 10.5 9.5 6.6 12.0 8.7 7.2 10.0 6.7 5.2 6.4 7.8 18.9 19.8 12.0 9.3 2.8 1.5 34.1

9.1 13.9 9.4 7.7 8.2 6.7 5.7 4.7 3.8 3.5 17.8 10.0 5.2 14.3 11.4 9.8 10.9 10.7 9.3 9.1 9.3 11.6 15.4 7.6 6.3 5.3 3.9 19.8

8.8 14.1 10.7 9.1 10.7 10.8 10.0 6.6 6.6 6.7 13.4 10.4 8.2 11.2 9.3 8.2 9.7 10.0 9.5 10.7 10.4 18.1 17.8 15.5 13.0 10.6 8.7 24.4

1.92 1.03 0.82 0.98 1.49 0.76 0.76 0.92 1.00 0.97 1.27 1.05 0.79 0.93 1.07 1.13 0.97 1.52 1.78 1.42 1.19 0.61 0.78 0.64 0.67 1.90 2.60~ 0.58

~2

2

1 46 47 48 49 50 51 52 53 ° 54a 55 56 57 58 59 60~ 61 62 63

Boerave et al.

Source

No.

3 139.5 141.7 142.0 140.0 140.0 140.5 140.5 141.7 141.6 141.5 117.8 117.8 150.3 150.3 150.3 100.3 100.1 100.7

b

4 246.6 246.4 249.7 240.8 243.4 250.4 249.6 246.4 249.9 246.9 185.4 186.1 320.0 320.0 320.0 155.1 153.9 154.4

d

5 17.7 17.4 17.4 12.6 12.8 12.6 12.7 17.3 17.4 17.3 10.3 11.1 15.0 15.0 15.0 14.1 14.7 15.1

t:

6 10.8 11.4 11.5 8.0 7.5 9.0 9.3 11.3 11.5 11.35 7.25 7.65 10.0 10.0 10.0 8.8 9.5 9.5

tw

Geometrical data (mm)

7 4000 3000 4000 3000 4000 3000 4000 3000 3000 3000 4000 4000 3000 3000 3000 3000 3000 3000

I

TABLE 2 Continued

8 248 489 489 276 276 504 504 489 489 489 275 430 248 486 817 303 375 445

fy:

9 252 539 539 311 311 535 535 535 535 535 302 448 323 990 813 342 421 462

fyw

Yield stresses (N/mm 2)

10 20.5 9.5 8.3 19.0 6.4 6.4 4.1 0.9 2.0 10.4 10.3 2.6 16.9 2.7 0.9 16.8 12.1 10.0

RE

11 14.2 9.4 7.0 9.3 7.5 4.9 3.7 9.3 9.4 9.3 5.2 3.7 14.9 5.1 4.3 13.6 11.7 9.9

M,

Rc

12 19.2 12.6 10.1 17.1 13.3 9.7 8.0 12.3 12.3 12.3 13.7 9.0 26.7 9.1 6.3 18.6 16.2 13.4

M~

Rotation capacity

13 0.69 0.99 0.85 0.49 1.18 0.76 0.90 10.33 ° 4.70 ~ 0.90 0.51 1.44 0.88 1.88 4.78 ~ 0.81 0.97 0.99

RclRE

bo I,.o

Suzuki

Kemp

100.2 100.0 75.0 72.5 53.0 74.5 74.5 70.0 77.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0 75.0

156.6 159.5 217.8 217.4 273.9 217.9 217.1 209.5 120.3 132.0 132.0 132.0 132.0 132.0 132.0 132.0 132.0 132.0

Values eliminated from statistical analysis.

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

14.6 9.6 3000 14.9 9.4 3000 8.09 6.65 3660 10.57 6.82 3660 7.05 5.85 3660 8.56 6.78 1830 1.44 6.78 1830 10.77 6.76 1830 9.83 7.44 3660 9.0 6.0 1200 9.0 6.0 1200 9.0 6.0 1200 9.0 6.0 1200 9.0 6.0 1800 9.0 6.0 1800 9.0 6.0 1800 9.0 6.0 1800 9.0 6.0 1800 average value RclRE standard deviation Rc/RE coefficient of variation Cv

261 409 340 285 332 340 294 288 313 291 527 291 526 291 291 526 687 291

291 426 358 329 388 353 300 329 300 340 340 509 509 340 509 509 340 686

24.3 9.2 2.7 6.6 2.2 15.2 14.8 14.0 8.4 33.4 19.2 22.3 9.4 27.2 18.5 7.7 9.4 15.7

16.6 10.8 4.8 10.5 4.5 10.6 11.9 22.4 6.3 23.0 20.3 25.8 13.2 13.7 17.2 6.9 5.7 20.1

23.2 14.1 7.5 9.5 4.4 11.9 13.8 15.3 15.8 26.9 22.6 24.9 16.2 18.7 19.3 10.3 9.3 19.7 40.5% 0.379

1.065

0.68 1.17 1.70 1.44 1.98 0.70 0.80 1.09 0.74 0.69 1.06 1.11 1.40 0.50 0.93 0.90 0.61 1.25

tO tO "--1

V. Gioncu and D. Petcu

228

M/Mp1[i 1.2,

/ //"

I"

1.0

0.81 Specimen 7,8,9 Kuhlmann (Position 19,20,21 ; table 2)

0.61

0.4

. . . .

80

experimental values DUCTROT 96

_~

fyf = 287 N/mm 2 fyw = 252 N/mm 2

80

0.2 I = 1798 mm 2198 mm 2598 mm I

I

I

I

I

I

I

2

4

6

8

10

12

14

F i g . 4. K u h i m a n n

experimental

0]8p

tests.

are 10 very short beams, for which failure is produced by shear. One can see that in all cases the in-plane mechanism produces the beam collapse. The comparison of moment-rotation for specimen DO1A4M (position 37 from Table 2) (Fig. 5) shows a good correspondence between experimental values and theoretical results. (iv) Boeraeve et al. [10]. Five specimens of hot rolled section (HEB 200) were tested. The experimental moment-rotation for specimen D3 (position 63 in Table 2) is presented in Fig. 6, showing a good correspondence. (v) Kemp [11,12]. A series of 12 specimens on built-up welded I-beams was tested, but only seven specimens can be used for comparison because for the others some important data are missing. (vi) Susuki et al. [13]. The testing program contains nine specimens of builtup welded hybrid beam. In the paper only rotation capacities corresponding to the maximum moment are presented, but it is known that the ultimate rotation capacity is R ~ 2Rmax.

229

Rotation capacity of beams and beam-columns, part 2 M/Mp l 1.21

1.0

0.8 ¸

0.6

- -

I

Specimen DOIA4M

]

Spangemacher

- .o.- --o--

experimental values

~

DUCTROT 96

(Position 37, table 2)

0.I

i100,t014I ]18.3 1.86.5

fyf = 486 N/mm a fyw = 532 N/mm 2

~' J

3500

0.2; ,

2

,-....~16.3

,

4

,

6

I

8

10

I

12

01;P

Fig. 5. Spangemacherexperimental test.

2.3 Conclusions concerning the comparison with experimental data The correlation between the 75 available experimental results, RE, and the calculated rotation capacity, Re, using the DUCTROT 96 computer program is given in Fig. 7. The average value of calculated values and experimental data is 1.065, showing a very good correspondence. But the scatter area is very large, the coefficient of variation being Cv -- 0.379. This high dispersion is due to the following factors. (i) The characteristics of material in the hardening range are not presented in the analysed papers, so only average values are considered. (ii) The effect of the transverse stiffener is not considered in the analysis, because its dimensions are not specified in the studied papers. The rigidity of this stiffener can have a great influence on the shape of the plastic mechanism. (iii) In the papers used not all information requisite to the analysis is given regarding differences between theoretical and experimental data. Only if these differences are very high, in discordance with these values, can one decide

230

V. Gioncu and D. Petcu

I M (kNm)

"<'b,. t'_70~

300 ~ ^ ~

~

100.3 100.3 .

I / 100 t~/" I /

(Position 63, table 2)



, ::::~" 14,1

8.8

---o.--.-.- experimentalvalues ~ . ~ FEM simulation DUCTROT96 2.5°

,

I

7.5o

155.1 ~

I

10°

14.1

I

12.5°

I

15°

2=0

Fig. 6. Boeraeveexperimental test.

to eliminate the corresponding values from the statistical analysis, considering that some shortcomings occur during the experimental work. (iv) The calculated rotation capacity depends highly on the dimensions of the plastic mechanism, especially on the length of the plastic mechanism. The values determined using DUCTROT 96 consider average geometrical dimensions. The use of real values of plastic mechanisms, determined during the experimental work, improves considerably the ratio between theoretical and experimental results. Unfortunately, only in the paper of Lukey and Adams [4] is there information concerning the lengths of plastic mechanisms. (v) Experimental data concerning the influence of axial forces and eccentricities are lacking from the technical literature in a form to be used in comparison with the theoretical values. Taking into account this scatter of results, a value

~/m= 1.5 for the partial safety factor proposed by Spangemacher [9] must be applied to get the design values.

231

Rotation capacity of beams and beam-columns, part 2 RG

/' / 22 20

qbro•/ ~,,./ / ./

18

• equality

4//

16

44-

14

&

0

,,'7

°

12 ./

10 t,

*'$ 6 "1~ 4

4"~ ^zx

8

13 ~x /

~÷.,,, o a +°

~/

~

. a o

]

e Luckey-Adams A Kuhlmann + Spangemacher o Boeraeve • Kemp e Susuki

• 4'

A

o



,,,~ I, 2

2

4

6

8

I

10

12

14

16

I

1

1

I

1

18

20

22

24

26

RE

Fig. 7. Correlationtheoretical--experimentalvalues for rotation capacity. 3 COMPARISON WITH FEM RESULTS 3.1 Numerical evaluation of experimental data

Software has been developed by Kuhlmann [5,8], Spangemacher [9], and Boeraeve et al. [10] to calculate the rotation capacity of the tested specimens. Kuhlmann and Boeraeve used a developed computer program based on FEM to analyse the experimental results. In Spangemacher's research work, the general computer programs ABAQUS and PROFIL are used to determine the rotation capacity for tested specimens. Those results obtained using FEM are compared with the values obtained using the DUCTROT 96 computer program in Table 3. One can see that a good correspondence exists, but the scatter is also too high. 3.2 Comparison with the FEM parameter studies

With the aim of studying the influence of different geometrical and mechanical parameters, Spangemacher [9] has performed a FEM parametric study. The

232

V.

Gioncu and D. Petcu

TABLE 3 Comparison with FEM results No. a

Source

16 17 18 19 20 21 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 56 57 58 59 61 62 63 64 65

Kuhlmann

RFEM

8.7 6.7 4.2 10.5 9.2 6.5 Spangemacher 6.9 6.9 14.6 14.6 11.8 9.7 3.5 17.6 14.8 9.8 7.9 9.9 7.7 4.4 2.8 8.3 2.2 10.0 4.1 Boeraeve et al. 13.2 11.1 8.9 22.8 9.7 average values standard deviation coefficient variation

Rc

7.2 7.6 8.8 13.9 9.4 7.7 9.1 9.3 11.6 15.4 7.6 6.3 5.3 19.8 14.2 9.4 7.0 9.3 7.5 4.9 3.7 5.2 3.7 14.9 5.1 13.6 11.7 9.9 16.6 10.8

Rc]RFEM

0.83 1.13 2.09 1.32 1.02 1.18 1.34 1.32 0.79 1.05 0.64 0.65 1.51 1.13 0.96 1.04 0.89 0.94 0.97 1.11 1.32 0.63 1.68 1.49 1.24 1.03 1.05 1.11 0.73 1.11 1.11 31.2 0.281

a The test number corresponds to the experimental specimens from Table 2.

following factors are considered: the influence of beam span, flanges and web thicknesses, flange and web slendernesses and steel grades. Fig. 8 shows the comparison between Spangemacher's studies using FEM and DUCTROT 96 results, considering the variation of beam length. The same comparison for the influence of flange slenderness is presented in Fig. 9. This comparison shows a good enough correspondence. A better concurrence is noticed for the ratio b / d ~ 0.3, very likely due to the fact that in this case the shape of the

Rotation capacity of beams and beam-columns, part 2

233

R~t. 10.0,

\ 8.0

b/d --- 0.303 b/ft= 10

o~

d/tw = 33

I I

6.0

NNN 4.0

--'e'--FEM -Spangemacher a

DUI;TROT 96

2.0

I=

I i

1000

2000

3000

4000

I(mm)

Fig. 8. Comparison FEM-DUCTROT 96 values.

local plastic mechanism corresponds better to the actual buckling shape. For b/d < 0.3, the calculated values for rotation capacity are rather underestimated, while for b/d > 0.3, these values are overestimated.

4 FACTORS INFLUENCING THE ROTATION CAPACITY OF BEAMS

4.1 Shape of plastic mechanism The shape of the plastic mechanism is defined by the following parameters. (i) The length of the plastic mechanism, /m, is given by the relationship (21) in Ref. [1]. The variation of this length is given by lml = /311m-

(2)

The influence of coefficient /31 on the rotation capacity is shown in Fig. 10. One can see that the value /31 does not correspond to the minimum of rotation capacity, which is obtained for 131 ~ 0.55. But this is under the value obtained using FEM (calculated by Spangemacher [9]) and confirms the first principle of plastic mechanism (see section 6.1 of Ref. [1]) that the mechanism shape

234

V. Gioncu and D. Petcu Ra

10



o

6

o

\

4.

-

b/d = 0.189.....0.340 d/tw= 33

-

~

' ~ ' ~

~

"~ ~ "

--o.-. FEM-Spangemacher

"~

DUCTROT96

6

7

8

9

10

11

Fig. 9. Comparison FEM-DUCTROT 96 values.

R 1311m

8.0

S

6.0

1

4.0

lOO

I I

lOO

t[

\FEM

I

33ol

2.0

3000 StE 460

[.,

~1o

, 0.4

I 0.6

I 0.8

1

I I ] I

I

1.0

1.2

Fig. 10. Influence of length of plastic mechanism.

b/tf

Rotation capacity of beams and beam-columns, part 2

235

is not obtained by minimizing the rotation capacity as a function of the length of the buckled shape. (ii) The asymmetry of the plastic mechanism, la, which was observed during experimental works (see section 5.1 of Ref. [1]): l. =

abl

(3)

where a is the parameter of asymmetry. Fig. 11 shows the influence of this parameter on the rotation capacity. One can see that the effect of mechanism asymmetry, which occurs due to the presence of the stiffener under the concentrated force, is not significant, so the rotation capacity can be determined for a symmetric shape (a = 0). (iii) Parameter of buckled shape (see relationship (43) of Ref. [1]): /3bl

X- &/-

(4)

In this relationship, the only variable value is the parameter ~ which defines the height of the web buckled shape (i.e. the centre of rotation of the plastic mechanism), which does not belong to the first plastic buckling shape of flanges, so it must be used in the minimization of plastic moment (Fig. 12). It is very important to notice that the height of the web plastic mechanism increases with increasing flange width (Fig. 13), and this increase is associated with a reduction of rotation capacity.

Ra j

12.0

,rot

• 100

100

.

,-10

8.(

10 3O0

Fe 360

4.0

~--~lO

1 l

0.05

I 0.10

3000

I oGbl

'10.15

0.20

I

0.25

Fig. 11. Influence of asymmetry of plastic mechanism.

0.3"5-~'~

236

V. Gioncu and D. Petcu M Mp

1.25 J

36.85. 36.85 I 1 1 ,-'i5.26

0.75

1

I I

0.50- ,[

l

I

i

1.00

4.6 - ~

960

]

241.2

"--'--'r--i I

L

I I I I

0.25

---~5.26

I 0.15

0.20

0.235 0.25

0.30

0.35

0.40

Z

1.08

0.81

0.69 0.65

0.54

0.46

0.41

c5

Fig. 12. Influence of depth of plastic mechanism.

i

aa~

12.0 "

b

b

6o-.-~

!1,° 1,7

t.0

0.8 Ra

~

-

0.6 3.4

4"0l.~ ' 0.2 80

I

100

I

120

I

140

160

180

200 b(mm)

Fig. 13. Relation between rotation capacity and depth of plastic mechanism.

4.2 Asymmetry of flanges The influence of different dimensions of the flanges is presented in Figs 14 and 15. In the first figure, the influence of the ratio of flange depths is plotted, while in the second the effect of different thicknesses is calculated. In these figures, the movement of the position of the rotation point is also presented. One can see that the reduction of the tensile flange dimensions leads to an increase of the rotation capacity of the beam, due to the favourable position of the centre of the mechanism rotation.

0 -h

o

i

!

0

i

0

o

Oq

o

k

r

J

o

C~

o

o

o

o

o

i

i

, O

io,~-

i

0~ o

t

o

o

IL ..

I

o c3

i

m

J

\

j/

~ o

/,

ix,.) bJ

ho

238

V. Gioncu and D. Petcu

4.3 Flange slenderness Fig. 16 shows the influence of flange slenderness on the rotation capacity. This influence is very high, this factor being one of the most important, influencing decisively this rotation capacity. The limitations refer to preventing the elastic buckling (relationship (32) of Ref. [1]) or the fracture collapse (relationship (24)). In Fig. 16 the classification of cross-section behaviour after EC3 (classes 1, 2, 3) and that for member behavioural classes is given (H - high, M medium, L - low ductility, see section 8 of Ref. [1]). One can see that the classification in cross-section classes does not correspond to the actual behaviour, because the member behaviour frames the high ductility, even for class 3. So, it is very clear that the classification after cross-section must be replaced by the member behavioural classes.

4.4 Web slenderness In Fig. 17 the influence of web slenderness is plotted. The limits refer to preventing the shear plastic collapse (relationship (31) of Ref. [1]) and shear elastic buckling (relationship (28)). It is very interesting to notice that a minimum of rotation capacity occurs for d/tw ~ 40, the first part corresponding to the elasto-plastic buckling of the web, the second thus being for plastic buckling of the flange. In Fig. 17 the two classifications are also presented,

I

Ra

16.0

/

,,

12.0 limit to prevent elastic buckling "~1L 8.0 -H-- - - - - ~ - M

!l I1 TIll

4.0

I ~ " / --'

I t

-'i 100 ~100 !~t 350

',t

~.t

8 12.56

"limit to preve

fracture collapse

. 1_.0 --

~. I i,=.

6 I 8.67

I ,

-::~t ol

'l

10 I0.00

------

12 8.33

Fig. 16. Influence of flange slenderness.

tf(mm) btlf

Rotation capacity of beams and beam-columns, part 2

16

T

'1

i

I

~1 ~

I I

,

12

/

I I

1

i

100

t

- --H-- - - "-~ ~ , ~ :.limit to prevent shear collapse I I M ~ .limit to prevent elastic buckling of web . . . . .

L

100

tw

I

__

I

4 82.5

6 55.0

t ,--':.10 33O • '10

T I AJ

. _ ,. . . .

2 165.0

/ ..I'

1"

I I

239

8 41.25

I I

', 10 33.0

12 27.6

14 t,,, (mm) d/tw

Fig. 17. Influence of web slenderness.

after EC3 and after the proposed member behavioural classes. One can see that even for classes 4, the member behavioural class belongs to high ductility.

4.5 Beam length Fig. 18 shows the influence of beam length on the rotation capacity. As a function of this length, the rotation capacity varies from very high ductility for short beams to medium for long beams. One can see that this factor is one of the most important, and the missing of this parameter from the EC3 classification is the most important shortcoming of these rules.

4.6 Steel grade Fig. 19 shows the effect of steel grade on the rotation capacity for hybrid and homogeneous beams. One can see that the increase of steel grade for homogeneous beams leads to the decrease of rotation capacity. This rotation of hybrid beams with high strength steel webs and mild steel flanges is larger than that of homogeneous beams. Thus, in order to increase the rotation capacity, hybrid beams with mild strength flanges and high strength web are recommended.

V. Gioncu and D. Petcu

240 Ra

16 ¸ ,J~lO

,

• 10 12

330

i~10 i 1 0 0 1100 l h M

8 H M 4 L 2000

3000

4000

5000

6000

I (mm)

Fig. 18. Influence of b e a m length.

Ra

II

iI I

I

I

I

f# = 235 N/ram2 (Fe360)

10.0

I

I

8.0

I

%//

I

/ff'~ 1/

It

. ~ homogeneous beam '1 ~d.--"-

' I 2 f~ = 355 N/mm (Fe510)

6.0' I

I I

I

I I I

I

I

I I

I I

I I

I

I

i

4.0

T j 200

I

i

I

Fe360

Fe430

I I i

300

Fe510

,

,-:10

. 10 ..

i loo!1ool

1

400

Fig. 19. Influence of steel grade.

330 ,-;10

fyw (N/mm 2)

Rotation capacity of beams and beam-columns, part 2

241

4.7 Mechanism type There are three mechanism types which can occur in the case of a frame joint: in-plane, symmetric or asymmetric out-of-plane (Fig. 20). The parametric studies show that the rotation capacity for symmetric out-of-plane mechanisms is always higher than that corresponding to asymmetric modes. The out-ofplane mechanism type depends on the torsion rigidity of the column, because the asymmetric form involves the rotation of joints together with column. Thus, the following coupling between the two mechanism types can occur: (i) between in-plane and asymmetric out-of-plane mechanisms, where lateral rotation of joints is free. In this case, one can see that the beam collapse occurs by out-of-plane plastic buckling; (ii) between in-plane and symmetric out-of-plane mechanisms, if the joint rotation is prevented when the collapse takes place by an in-plane mechanism. The first case is characteristic for continuous beams, where both in-plane and out-of-plane buckling modes can occur, while the second is characteristic R 24

~y,,.,,~u,~. vu[-v,-plane

20

~

..1-,

.

16

I

--t' in-plane

12

80 . 80 .

I

/ j~ 8

~ ,

/'~llp~.

/

- -

asymmetric out-of-plane r 300

, - 12

5000

I

4000

Fig. 20. Influence of mechanism type.

6000

I(mm)

V. Gioncu and D. Petcu

242

for the beams which belong to a frame, where the in-plane buckling mode produces the plastic collapse, due to the rigidity to rotation of the column.

5 FACTORS INFLUENCING THE ROTATION CAPACITY OF BEAMCOLUMNS 5.1 Axial force The influence of axial force is plotted in Fig. 21. One can see an important decrease in rotation capacity in the first part of the curve, for n p < 0.1, followed by an almost horizontal plateau. The dimensions of cross-section correspond, after EC3, to the first class of ductility, while the framing in the member behavioural classes shows a low ductility capacity. 5.2 Column slenderness The influence of column slenderness is very high, as one can see from Fig. 22. A cross-section which belongs to the first class, after EC3, works under conditions of low ductility if the member behaviour is considered. The limitation of column slenderness, given in EC3 for braced frames, seems to be too generous. A limitation of values X ~ 0.7-0.8 for columns with plastic hinges is recommended with the aim of preventing a drastic decrease in rotation capacity.

Ra H

N p_.,. '

~l

----I-- ['1001-100.1,~) "~ ,0 3850 - -~" ~:~12 ~=o.6

Class1

J

nP

L

1

I

0.1

0.2 Fig. 21.

0.3

0.4

1.0 nip I

Influence o f axial force.

0.5 nP=N/Np

Rotation capacity of beams and beam-columns, part 2

Ra~

100

6

243

N=169.2 kN

i ] 100 .

\

= 2570 mm ~, = 0.4 I

"M

i

\

12 nP= 0.1

11,

I

I limitslenderr I after EC,~ I

1

0.4

I

i

0.6

I

1.o

0.8

]

i

_

1.2

Fig. 22. Influence of column slenderness.

5.3 Upper m o m e n t s The rotation capacity for a column in a frame with upper moment is presented in Fig. 23. One can see that an upper negative bending moment increases the rotation capacity, while for positive moment, the rotation capacity is reduced. Thus, it is very important to have a point of inflection on column displacements to improve the rotation ductility.

• 100

Ra

100

H

.

"~ = 0.6

rip=0.1

,d oo

N=169.2 kN

~

- ~'-I

MP

~.oM_o._~

p[ -2.0

-1.0

Mp 0

i P_.__ _ __~P

1.0

~ , l 3850mm

2.0

L

M/Nd

Fig. 23. Influence of upper bending moment.

244

V. Gioncu and D. Petcu 6 CONCLUSIONS

The use of a collapse plastic mechanism allows us to determine the available rotation capacity with an accuracy confirmed by experimental and theoretical research work. The numerical tests, in which the factors influencing the rotation capacity of beams and beam-columns are considered, show that the provisions for ductility classes, given in codes, are not sufficient for a proper design. The cross-section behaviour classes should be substituted by the concept of member behavioural classes.

REFERENCES 1. Gioncu, V. and Petcu, D., Available rotation capacity of wide-flange beams and columns, Part 1. Theoretical approaches. Journal of Constructional Steel Research, 1997, 44, JCSR 1487. 2. Petcu, D. and Gioncu, V., Plastic rotation capacity of steel beams and beamcolumns. DUCTROT 96, User Guide. INCERC, 1996. 3. Haaijer, G., Plate buckling in the strain-hardening range. Journal of the Engineering Mechanics Division, ASCE, 1957, $3(EM2), 12121-12147. 4. Lukey, A. F. and Adams, P. F., Rotation capacity of beams under moment gradient. Journal of the Structural Division, ASCE, 1969, 95(ST6), 1173-1188. 5. Kuhlmann, U., Definition of flange slenderness limits on the basis of rotation capacity values. Journal of Constructional Steel Research, 1989, 14, 21-40. 6. Lay, M. and Galambos, T. V., Inelastic beams under moment gradient. Journal of the Structural Division, ASCE, 1967, 93(ST1), 381-399. 7. Nakashima, M., Variation of ductility capacity of steel beam-columns. Journal of Structural Engineering, 1994, 120(7), 1941-1960. 8. Kuhlmann, U., Rotations kapazit~it biegenbeanspuchter I - Profile under Beriicksichtigung des plastischen Beulens. Technical Reports, Mitteilung Nr. 86-5, June 1986. 9. Spangemacher, R., Zum Rotationsnachweis von Stahlkonstruktion, die nach dem traglastverfahren berechnet werden. Dissertation, Technischen Hochshule Aachen, 1991. 10. Boeraeve, Ph., Lognard, B., Janss, J., Gerardy, J. C. and Schleich, J. B., Elastoplastic behaviour of steel frameworks. Journal of Constructional Steel Research, 1993, 27, 3-21. 11. Kemp, A. R. and Dekker, N. W., Available rotation capacity in steel and composite beams. The Structural Engineer, 1991, 69(5), 88-96. 12. Kemp, A. R., Interaction of plastic local and lateral buckling. Journal of Structural Engineering, 1985, 111(10), 2181-2196. 13. Suzuki, T., Ogawa, T. and Ikarashi, K., A study on local buckling behaviour of hybrid beams. Thin-Walled Structures, 1994, 19, 337-351.