Coupled instability of thin-walled members under combined bending moment, axial and shear force

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Thin- Walled Structures 19 (1994) 285-297 © 1994 Elsevier Science Limited Printed in Great Britain. All rights reserved

e~'411

0263-8231/94/$7.00 ELSEVIER

Coupled Instability of Thin-Walled Members Under Combined Bending Moment, Axial and Shear Force M. A. Aiello, A. La Tegola & L. Ombres University of Calabria, Department of Structures, 1-87036 Arcavacata di Rende, Cosenza, Italy

ABSTRACT In this paper we analyse the behaviour of thin-walled steel members with an 1-shaped cross-section, in the presence of phenomena of coupled instability as overall-local type and combined state of stress of bending moment, axial and shear force. The analysis has been made by the 'column model method' and it has been used for bending moment-curvature-axial force diagrams, modified by the presence of local instability for web buckling of steel beams. The previous analysis allows one to define interaction diagrams MImax2-N, which can be used to check the steel members in relation to the overall slenderness and the local slenderness of the web panel.

NOTATION A b

dw el

E

fy hw H l

Surface of a cross-section Width of flanges Distance between two following stiffenings First-order eccentricity Elastic modulus Yield stress Height of the web beam Height of the beam Length of the beam

m = M/Mpl

mlmax = Mlmax/M

M

MI

Bending moment First-order bending moment 285

286 mll Mlmax Mp~ n = N/Np N Np

M. A. Aiello. A. La

Tego/a, L. Omhres

Second-order bending moment Maximum first-order bending moment Ultimate bending moment of an I cross-section

V

Axial force Ultimate axial force Thickness of flanges Thickness of the web Shear force

= Z/Zel 2 Z ZA Zel

Slenderness of the beam Curvature Curvature in the critical section Elastic limit curvature

tf tw

1 INTRODUCTION In structural engineering, mechanical high performance materials are very widespread, for example, steel ones, which allow one to build structural elements with small geometrical dimensions, especially of thickness; therefore, it is possible to make structures with low weight, but high mechanical resistence, in which the phenomena of instability become very important. Instability phenomena are emphasized by the presence of geometrical or mechanical imperfections, which characterize real structures, in which a coupling of overall and local instability can occur. The local instability, resulting from buckling of walls beams, can have critical values of load smaller than those of just the overall instability for eulerian load. The coupled overall-local instability phenomenon has been studied and several scientific works are available in the literature; ~ however, rigorous theoretical analyses have not been found as yet to utilize for the design problem because of analytical difficulties, which can be overcome by using numerical methods, while technical codes provide formulae based on approximate models that are easy to apply. So, the necessity for investigating new models, which are able to represent the physical phenomenon and to give effective means about the design, is clear. For this reason, in the present paper, we analyse the problem of the interaction between overall and local instability in structural steel members under the combined state of stress of bending moment, shear and axial force: for example, this is the case for steel beams that form

Coupled instability of thin-walled members under combinedforces

287

columns of frames situated in seismic areas, in which the local buckling causes a notable reduction of the critical load. The analysis has been made by the 'column model method', based on using bending moment-curvature diagrams, found in relation to the slenderness of the wall web beam between two stiffeners, for the values of stress of the bending moment, shear and axial force corresponding to the local critical state of the beam. So, it is possible to evaluate the overall critical load of the beam regarding the overall slenderness and the local slenderness of the stiffened wall web. The method, developed on the basis of the elastoplastic behaviour of the material, allows one to have interaction diagrams Mlmax-N-~.for easy use in the design and these make it possible to value the influence of the geometrical imperfections on the behaviour of the critical state of the beam.

2 PROBLEM F O R M U L A T I O N The analysis of the overall-local instability, regarding a steel I-shaped cross-section beam, has been made under the following hypotheses: - - t h e behaviour of the material is of linear elastic-perfectly plastic type; - - t h e plasticity Von Mises criterion is adopted. The reference structural model is performed by a constant section cantilever beam, stiffened along the web height. The analytical method is based on the construction of bending momentcurvature diagrams of the section in relation to the slenderness of the web wall between two following stiffenings; the diagrams, defined by the evolution of the stress and strain state, are limited by the web buckling. The analytical relationships that describe the bending moment-curvature laws are obtained by mechanisms of collapse of an I-shaped cross. "~3 section;-' the evaluation of the critical stress state regarding the web buckling has been made in reference to the plastic behaviour of the steel, 4 known as the slenderness. 2.1 Bending moment-curvature laws The collapse mechanisms regarding the steel I-shaped cross-section are shown in Fig. 1. The analytical relationships, that can be used to determine the bending moment-curvature diagrams at the changing of shear and axial force about each mechanism of collapse, are the following:

M. A. Aiello, A. La Tegola, L. Ombres

288 I-

f~

~1

B

h/2

fy (b)

(c)

(d) :d

12

h/2

12 I#

(e)

(f)

D (g)

(h)

(i)

(j)

Fig. 1. Mechanisms of collapse of (a) I-shaped cross-section; (b) Mechanism 1; (c) Mechanism 2; (d) Mechanism 3; (e), (f) and (g) Mechanism 4; (h), (i) and (j) Mechanism 5

Elastic limit

Zei

--

EH N i-

Ibtr 2 +~tw(n-2tf)(n-tf)

+

+

2

]

w h e r e A = 2btf + tw(H - 2tr)

Mechanism 1 ( H - tf>d>tr; H - 2 t f > h > 0 ) N =j~tw(H-

h - 2d) /4 2

M = fy tr (H - tO(B - tw) + tw ~ - -twd(h-

H) + Hh ~ l

Coupledinstability of thin-walledmembersundercombinedforces

-/f)

Mechan&m 2 (0 < d H H-h d---~2

289

N h 2fy Bh - (B - t w ) ( n - 2tf)

M = fyB { [ Hd + -2-Hh-

d 2 -

h23

hd

(H -21f)316hJ + ~tw (H - 2tf) 3}

Mechanism 3 (If < d < H - tr; tr < d + h < H) N = f y {2Bd+ B(2tf - 2 H + h) + t w ( H - 2tr) B - tw +----if-- [d 2 + 2d(tf - H) + H 2 + t 2 - 2Htf]}

M=fyB

Hd+ T

[

H3

d2

+fy(B-tw) H2tf---g-+H 2 Mechanism 4

(tf < d <

3Ht2 " 2t~

2 + T +t~d-Hdt

H - tf)

[ tw d)2 N = f y 2tr(B - tw) + Htw - ~ (H -

B-tw ] -h tf(2H - 2 d - lf)

M = f y I f f - - ~ (~12 +t2d+ H2tr+ H t f d - - ~3 Ht 0

Mechanism 5 (0 < d < tf; d + h > / 4 ) N = f y 2tf(B - tw) + Htw +

B(H

-

h M=-~

+8

r]

B - tw

d)2]

J

[ ( B - t w ) ( ~ t3 - 2Ht 2

+T-Y

(H 2 + 4dtf - 2Htf - 2Hd)

290

M. A. Aiello, A. La Tegola, L. Ombres

If N is known in each examined case, X is assigned equal to 2Jy/Eh, and d and h are known, it is possible to determine the value for M. 2.2 Local critical stress state

The critical stress state, regarding the local instability of the web wall between two following stiffenings, is found by the previously described method, 4 evaluating the critical shear stress zcr, which, for known values of normal stresses, causes the web wall buckling. From an analytical viewpoint the problem is solved by the definition of the stress distribution in correspondence with the edges of the wall, which is expressed by following relations: 4 a0')=2

y

r(y)=~

1-4

in elastic field, and = L

= 0

in plastic field. The structural model is a rectangular plate with b and hw dimensions, simply supported on the flanges of the beam and transversal stiffening. The critical shear stress is evaluated by the energy method based on the Dirichlet principle, assuming as critical deflection of the plate such solutions as w(x,y) =

±± //1= l

11=

Win,,sin ~ )'sin -i~- ¥

]

The stress "Ccrcorresponds to the values of geometrical dimensions d and h, which define the extension of the yield field inside the wall; it can be used to determine, on the bending moment-curvature diagram, the point corresponding to the local critical state. The critical stress state can be obtained also, by a normal stress associated with the bending moment and axial force. By this method it is possible, in relation to the slenderness of the web wall ~ = hw/b, to limit the extension of bending moment-curvature diagrams to the local critical state and then to evaluate the critical load for the overall instability.

3 D E T E R M I N A T I O N OF O V E R A L L C R I T I C A L LOAD: C O L U M N M O D E L M E T H O D With the known bending moment-curvature diagrams, modified by the local critical state, the evaluation of the overall critical load allows one to

Coupled instability of thin-walled members under combinedforces

291

define the interaction between the local and overall instability, varying the overall slenderness of the beam and the local slenderness of the web wall, where the buckling occurs. The method used for the evaluation of the overall critical load is the column model method 5'6 applicable to isostatic elements (cantilever beams) with constant cross-section. Let ZA be the value of the curvature in the clamped section, considered as a critical section, where all the deformability of the beam is concentrated, the bending moment in this section is M = MI + M I I where Mli =- 0-4ZAN2 2 is the second-order moment, and MI = Ne~ is the first-order moment with el the first-order eccentricity, due to the presence of geometrical imperfections or the simultaneous action of the axial force and bending moment. In the critical section the condition of equilibrium, which allows the equality between the external moment M = Ml + MH and the internal moment, is found by bending moment-curvature diagrams, defined for an axial force N. Therefore, the cases that can appear are drawn by 'a', 'b' and 'c' straight lines on the M-Z plane (Fig. 2). In the case 'a', points A and B are the conditions of the stable and unstable equilibrium, respectively; in the case 'b', point C is the only one of equilibrium, even if unstable; in the case 'c', there is no equilibrium. By this method it is also possible to evaluate the greatest first-order bending m o m e n t Mlmax, which the beam can bear; in fact, knowing the second-

I

/

J(b)

: .

Fig. 2. Moment-curvature diagram.

(a)

2~

292

M. A. Aiello, A. La Tegola, L. Ombres

order moment M . , the value Mlmax is the distance between the straight line M . and that one corresponding to the total value of the bending moment M, parallel to MII and tangent to the M-z-N curve (Fig. 3). The value of MImax varies with N, because M-)~ diagrams are defined for N, and in relation to the slenderness of the beam, which is a term of the expression of Mn. Knowing the geometrical cross-section of a beam, it is possible to determine the Mlmax-), N diagrams and by that we can verify the structural element. Interaction laws, so defined, that coincide for 2 = 0 with ones regarding the plastic collapse of the section, are conditioned by the presence of the local buckling phenomenon in this case; therefore they can be found for each value of the slenderness of the buckled web wall.

4 N U M E R I C A L APPLICATIONS The above method has been used to evaluate the mimax-2 N diagrams regarding a beam, made by a commercial profile IPE 300 (type A) and by a profile (type B) in which the ratio hw/twis very high. Figures 4, 5 and 6, with reference to the profile IPE 300, show the nondimensional bending moment-curvature diagram of the critical section, defined until the local critical stress state, varying axial force N for the values ~ -- 0.75, 1 and 2 of the slenderness of the web wall, where the buckling occurs. M-Z diagrams, interpolated by a polynomial second-order expression,

X Fig. 3. Determination of Mlmax.

Coupled instability of thin-walled members under combined forces

293

nl a=0.75

n=O.1 n=l)'2

~i

1).3 .4

o

i

I

I

2

I

3

Fig. 4. IPE 300 diagram of m-n-~. (~ = 0.75.)

from an analytical viewpoint, allow one to value, varying 2, the greatest first-order moment Mlmax. Figure 7 shows the interaction diagrams mlmax-2-n for the IPE 300 profile. m

c~=l

~i

n=0.1 =0"2 1).3 .4

o

i

i

i

1

2

3

Fig. 5. IPE 300 diagram of m-n-¢. (~ = 1.00.)

4

294

M. A. Aiello, A. La Tegola, L. Ombres m

c~=2

/

1

n=O. 1 n=0.2 n=0.3

£ _

-

()

7

?

i

i

i

I

2

3

4

Fig. 6. IPE 300 diagram ofm-n--¢. (~ = 2.00.)

Figure 8 shows the interaction diagrams mlmax 2-n, assuming the slenderness ~ = 2 of the buckled web wall, compared to the m - n curve carried out for 2 = 0 (a) 2'3 and the m - n curve carried out with just the local buckling (b). 4 Figure 9 shows M - Z diagrams of type B profile in which H = 940 mm, 1,() ]

m ~=2

(),60 4~

k - - ! 6 0 f ~ =

I

S

Q

k = ~

(),2

0,0 (), I

,

,

,

0,2

1),3

(L4

n (L5

0,6

Fig. 7. Interaction curves mlmax 2--/'1for IPE 300.

Coupled instability of thin-walled members under combined forces

295

m 1,0 ~

o~=2 a

0,

0,2.

0,0 0,1

,

,

,

0,2

0,3

0,4

. 0,5

. 0,6

.

.

0,7

. 0,8

il

. 0,9

1,0

Fig. 8. Diagrams of mlmax-R-n for profile IPE 300 with m-n curves: curve 'a' (2 = 0); curve 'b' (local buckling).

b = 300mm, tf = 2 0 m m and tw = 10mm; in this case the critical state of the stress is produced by the shear stress r. Figure 10 shows mlmax-2-n curves. Numerical results show, in the case of type A profile (IPE 300), that the m

1,0 ot=l n=0.1

0,8 n=0.2

0,6

(1,4

O,2

0,0 0

i

i

!

!

|

1

2

3

4

5

6

Fig. 9. Diagram of m-n-~ (I cross-section: H = 940mm, b = 300 mm, tw = 10 mm).

tf

-----

20 mm,

296

M. A. Aiello, A. La Tegola, L. Omhres m 1,0 rwith total buckling

((~- 1 )

0,8 £=20(O/ 0,6 •

0,4

0,2

0,0 (),1

n ,

0,2

(1,3

,

(1,4

0,5

0,6

Fig. 10. C o m p a r i s o n between diagrams of m-n.

local instability is produced by the normal stress associated with the bending moment and axial force, while for type B profile the local instability is produced by the shear stress; therefore, concerning the mlmax 2-n relationships, in the first case the overall instability is predominant as regards the local instability while in the second case the local instability is predominant as regards the overall instability. Furthermore we can keep in mind that, in the presence of the local instability, the value of the axial force is narrow, since the dimension h of the buckled web is almost equal to the height of the web beam hw.4

5 CONCLUSIONS On the basis of numerical applications we can draw the following conclusions: • The presence of the local instability (web buckling) produces a remarkable reduction of the resistance capacity of steel members subjected to combined bending moment, axial and shear force. This reduction is more relevant both if the critical state of stress is due to a shear stress and in the presence of high values of the axial force N. • The behaviour of steel beams, with regard to instability phenomena, is influenced by the shear force. In fact, for the elements in which the web buckling is produced by a shear force the local instability is predominant over the overall one; on the other hand, in the presence

Coupled instability of thin-walled members under combinedforces

297

of low value of the shear force, the overall instability is predominant with regard to the local instability. The utilization of the column model method allows the determination of interaction diagrams mimax-2-n, which by varying the crosssection of the beam, the local slenderness and the shear force V, allows one to carry out the check on beams. The proposed model of analysis can be easily used to evaluate the influence, both structural and mechanical, of imperfections in the resistance capacity of the steel beam in the presence of a coupled overall-local instability, since they can be taken into account in the determination of moment-curvature diagrams.

REFERENCES 1. Gioncu, V., Coupled Instabilities in Thin-Walled Members: Phenomenon, Theory and Practice. Building Research Institute, Timisoara, Romania, Nov. 1989. 2. La Tegola, A., Ombres, L., Pasca, M. & Pignataro, M., Limit analysis of steel members under axial force, shear and bending moment. In Proc. 5th Conf. on Metal Structures, Timisoara, Romania, 22-24 Sept. 1988. 3. La Tegola, A., Ombres, L., Pasca, M. & Pignataro, M., Plastic analysis of I beams under combined bending moment, axial and shear force. Report no. 118, Department of Structures, University of Calabria, Cosenza, Italy, Dec. 1989. 4. La Tegola, A. & Ombres, L., Influence of the local buckling on the plastic analysis of I steel beams. In Proc. 6th Conf. on Steel Structures, Timisoara, Romania, 10-12 Oct. 1991. 5. La Tegola, A., Sul Carico di Collasso di Aste Snelle in C.A. Atti del Convegno in memoria di R. Baldacci e M. Capurso, Roma, Oct. 1989. 6. Migliacci, A. & Mola, F., Progetto agli Stati Limite delle Strutture in C.A. Masson, Milano, 1988.