Generalized beam theory—an adequate method for coupled stability problems

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Thin- Walh, d Structures 19 (1994) 161 180 ,(', 1994 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/94/$7.00

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ELSEVIER

Generalized Beam Theory An Adequate Method for Coupled Stability Problems

R. Schardt Technical Universityof Darmstadt, Institute of Statics, Alexanderstr. 7, D-64283 Darmstadt, Germany

A BS TRA C T First-order generalized beam theory describes the behaviour of prismatic structures by ordinary uncoupled d(ff'erential equations, using &~['ormation functions .for bending, torsion and distortion. In second-order theoo,, the differential equations are coupled by the effect of deviating Jorces. The basic equations .for second-order generalized beam theoo, are outlined. Solutions .for pin-ended supports are presented, demonstrating the coupling effect by modes and by loads. In the d(fferent ranges of length, the individual modes are sufficient approximations .[or the critical load. The application to a thin-walled bar with C-section under eccentric normal force demonstrates the quality of the single-mode compared to the exact solution.

NOTATION b B

C D e

f(s) f ~ , f o , t9

k( ) K I

Width of element Transverse bending stiffness Warping constant Torsion constant Eccentricity of longitudinal force Displacement function Displacement components of element Indicating mode k Plate modulus Length of column Critical length 161

162

111(s) q(x) (),. S, S I

V W .V, .V, Z

h

A O'er

uO

R. Suhardt

Transverse bending moment Load function Indicating node r or element r Local coordinates in elements Thickness of element Warping function Deformation resultant Warping moment Global coordinates Ratio of interactions Ratio of eccentricity Unit deviation forces Load factor Critical stress Euler stress Ratio of single mode solutions

1 FIRST-ORDER G E N E R A L I Z E D BEAM THEORY Thin-walled prismatic structures such as cold-rolled profiles often fail by instability before the stresses have reached the yield point. The related deformation depends upon the stress-distribution in the cross-section and upon the length of the structure. In general, the deformation is a combination of local buckling, distortion and rigid-body displacements of the cross-section. For monosymmetric sections, a sudden change from symmetric to antisymmetric deformation is even possible at special lengths. Two or more different individual deformation modes interact in such a way that the overall buckling load is lower than the lowest individual buckling load. Numerical solution methods such as the finite element method or the finite strip method can treat single cases, but cannot give information about the general mechanical behaviour of the structures. Generalized Beam Theory (GBT) describes the mechanical behaviour of prismatic structures by ordinary differential equations using deformation resultants kV(x) similar to the individual modes of buckling. GBT provides these resultants by a special eigenvalue problem. The main property of the resultants is the orthogonality between the modes for the work of the warping stresses and the work of the distortion (transverse bending) stresses. In the first-order theory, the ordinary differential equations E kC-/" V'"' - G kD. I'V" +/'B./" V = kq

( 1)

Generalizedbeam theoo'

163

for each mode k are not coupled and can be solved independently. The coefficients have the generalized meaning of the well-known section properties of the ordinary bending and torsion theory, such as the warping constant

kc =

kU2(S)dA + -~ ,4

~f2. ds , s

the torsion constant

kD = ~I ,I. k]'2(S) t3 ds .

and the 'elastic foundation', which is the transverse bending stiffness

km~(s)

kB = .L ~

ds

in which

u(s) = warping function j(s) = displacement function rn(s) = transverse bending moment K = plate constant For a C-shaped section, the deformation modes are shown in Section 5. The theory for linear problems, which provides these resultants, is developed in Ref. 1. Second-order effects may arise from normal stresses in the longitudinal or transverse direction of the structure or by shear stresses. In this paper, interest will be focused on the effect of longitudinal stresses which, for practical problems, is the most important effect.

2 SECOND-ORDER G E N E R A L I Z E D BEAM THEORY 2.1 The ~-values

The second-order effect of the longitudinal stresses ax depends upon the distribution of the stresses ax(s) in the cross-section and the elastic curvature f"(s) of the elements of the section. The second-order theory requires the satisfaction of equilibrium conditions with respect to the deformations of the structure. The system of differential equations of the GBT is found as the virtual work of stresses and applied

R. Schardl

164

loads at virtual displacements ~V -- 1. The virtual work of the second-order terms is represented below for one strip t. ds of an element dx (Fig. 1). The axial stress in the cross-section is expressed by the m stress resultants 'W and the warping functions :u

imiu(s)

cr,(s) = - _

~

(2)

i=: I

The elastic inclination of a fibre which results from a deformation / V'(x) is i.l = :'V ' . . L /. In the x-s plane ./~

In the x-,~ plane

0c' = / V ' . ( /

and the corresponding elastic curvature is In the x - s plane In the x - s plane

//.,~" = iV" • ./~i" //"'. = / V " - ! /

The deviating forces q~l and qU caused by the stresses a, and the deformations i v " ( x ) according to Fig. 2, are In the s-direction d q~II = (or,. - :.'f't~+ dory- (/~)t ds =

liuT~tds - ~ (imJvtt -~-iwt Jvt) i-C ::1

and in the ~-direction

l iu.!ftds ( i w / v " + i w ' iv') iC

dqll : i:l

The virtual work of the deviating forces at the virtual displacement k l / = 1, expressed by Aj~ and ~)"

J7-~ /

Fig. I. Deformation of a cross-section element.

Generalized beam theory

165 ~x

S.

a°dx

idql

fs

Fig. 2. Formulation of the deviating forces. qlsl " k~s + q I I . kf z _

~

© L J (iwJvr)'

-[-"~

i bl(jL k~s ~- jfkf)t ds

i=1 ~

can be introduced directly into the system of differential equations. The unified symbol for the integral is ,jk.

,j j +Jfkf)tds

= iC j

(3)

where i indicates the stress distribution due to the stress resultants (warping moments), j indicates the elastic deformation and k indicates the equilibrium condition in which the deviating forces are involved. The values have to be evaluated for the warping moments 1 < i < m, and the deformation modes 2 _
~-~ijkl~.(iw.Jvt)lzkq

.jr ~

(4)

2
i= 1 .i=2

For a systematic evaluation, the unit warping functions u(s) in each element r are expressed by ~r and fir [Fig. 3(a)]. The unit displacementfunctions f~(s) and f ( s ) are expressed by the constant values f~, fQ and O, representing the approximate element deformation [Fig. 3(b)]. This leads to ([kl~ = r=l

-brtr iC

b2]

i~lr jVfsrk L r - [ - J f Q r k f Q r ~ - ~ r j~gr kOr

1 br AriLlr -~ [JfQrkOr ~- JOrkfQr] J}

r = 1 ... n

(5)

166

R. S c h a r d t ~v

o)

bl

0 //'~\

4" •

/

-

/>

:5 \\

Fig. 3. Unit deformation values of a cross-section element r for the evaluation of the integrals ;/~t~. (a) Warping function along one cross-section element; (b) displacements in the 1' : plane.

A comparison with the conventional tbrmulation of the beam theory identifies the following i/~n-values: i

1

i

2

4

I

--1

2

2

3

4

- - 71M

1

-1 ")

--tiM!

m QM

i

-

-

l'rl~,

i-M

3

2

i=4 3

[

4 -1

2

2

3

4

[ i

i

-!

r~,

4

1

I

The matrix of the '/kts-values for warping moment i is symmetrical. Values for i >_ 4 do not appear in the conventional theory. In general, they are of less importance.

2.2 The eigenvalue problem Assume the structure to have pin-ended supports and to be loaded by a centric normal force which is constant along the x-axis. In this case, the index i of the load is 1. The differential equation (4) is reduced to

Generalized beam theory

Ekc.kv,,,, _ GkD.kV,, + kB.kV + I W £

tik~. iv,, = 0

167

(6)

i=2

The solution for k V(x), which satisfies the support conditions, is a sine function: kV(x) = kV,,, . sin rrx /

(7)

Introducing this into eqn (6), we get the following matrix eigenvalue problem:

x

iV,,,. ~ik~.

= 0

(8)

i=2

in which A is the load-factor. IWcr = A-IW0 It is presumed that, when the load reaches the critical value, the structure is undeformed.

3 C O U P L I N G BY M O D E S 3.1 The individual modes

If we only allow a single m o d e k for the buckling deformation, so that all K-values with.j :fi k are neglected, then for each mode we have the following simple formula for the critical load: •

~P(l)

'kWcr- i~-k~

(9)

with the abbreviation kp(l) = E . k c

+ G . k D + B.

(10)

The three components of kp show different characteristics concerning the length: • the warping stiffness of the system decreases by I l l 2 • the torsion stiffness is independent of the length • the distortion stiffness increases with / 2

16~

R. Scltardt

1~., where

The corresponding characteristic length found to be /~.

d,

• I+~., is minimal, is

4rE./'C 7rv aB

(I I)

For ~B equal to zero,/c is infinite. The m i n i m u m critical warping m o m e n t '~ We,- at lc is min i~W~.,- = i~aIh~ (2x/E.aC ' AB+G "/'D)

(12)

Equation (10) divided by 2 v ~ . kC. kB gives

2x/E.kC.~B

+2v/E.~C./,B

2V~k/-/)

and introducing/c leads to the dimensionless form

/'P(I)

1 (~) 2

2x/E.~C.kB

2

G/'D

1 ( ~ ) -~

(13)

+ 2x/E-~C./'B +

The normalized solution for eqn (13) is plotted in Fig. 4. In double logarithmic scale, the kb(/)-function consists of two straight parts and a transition curve in the range of the m i n i m u m value of kh (see Section 5). Numerical solutions by individual modes for a C-section are also discussed in Section 5. 2,5

~I kp(l ) '\/2~

2.0 1.5kp(l) 1,0 0,5 o

~GO -i . . . . . . . 0

1,0

0,5 --

I

1,5

IB

It Fig. 4. N o r m a l i z e d s o l u t i o n for q u a d r a l i c e q n I I 3).

2,O

169

Generalized beam theory

3.2 Two-mode interaction in buckling deformation If we allow two modes j and k for buckling interaction, we have the eigenvalue problem (JP - OJI,g. iWcr ) • J v m - ikjt%" i W c r . k v m ~- 0

(14)

(kp _ ikkt~ " iWcr) " kVm ~- 0

(15)

ijk . i W c r . j V m +

Equation (14) is divided by ~Jt~and eqn (15) by ikk. With the abbreviation iJW, which is the critical load i W¢rwhen only JV is allowed, we find

(

iJm __ i m c r ~Jkl% - ikkN " iWcr

i, ) //00/ ijjl~ " iWcr

. i k w -- tWcr

{ JVm

" ~ kVm

=

(16)

The critical v a l u e imcr is given as the solution of a quadratic equation (Det = 0) ijkt~, ikjt,;

2 ( O W - ' W c r ) ( i k w - iWcr) - ijj~;, ikk-------~' i Wcr = 0

(17)

We extract the lowest value of iJw and ikw as factors of the left side of eqn (17) and introduce the dimensionless parameters

1 qkt~ "ikjt~

/3 --

1

(18)

~jj~. ikk~

and ik W

w = ~j--~_ 1

(19)

and find

'

/3 (1 +w)[1 - ~/1 -

'mcr = q m . ~

4~

/3(1 --[-0.))2

]

= ''/"/JW

(20)

The parameters /3 and w vary between 1 and w. The lowest possible value of 7 is ½, w h e n / 3 = 1 and w = 1. The function 7(w,/3) is plotted in Fig. 5.

R. Schardt

170

1.0

O~ = 0.95

I O,g

: 0.9

5'

0.7

: 0.8 : 0.6

0.6

:0.4

0.5

:0

0.2

0

0.4

0.6

1--.,,..

0.8

1.0

u,}

Fig. 5. Interaction coefficient7-

4 C O U P L I N G BY LOADS 4.1 T w o w a r p i n g m o m e n t s as l o a d

4.1.1 One individual mode as deJormation

The load has the two components iW0 and/W0, and the buckling deformation is assumed to consist of only one mode ~V. Both load components are increased by the load-factor nA. In the eigenvalue problem [ j p _ il/~ . (ijjl% " iWo 4- ljl~ . /m0)] " il]Vm = 0

(21)

the coupled solution can be expressed by the solutions for individual loads (eqn (9))

Jp iWcr = ::-qJl~

JP and

IWcr =

Introducing the abbreviation

q -

(22)

~i%./Wo i~/~. i W o

in eqn (21), we get the following equations: ilA. iWo 1 - - -iWcr 1 4-q

and

hA. lWo 1 - lWcr 1 4 - -1 q

(23)

Generalized beam theory

171

It is obvious that, independent of the length, the interaction function (Fig. 6) is a straight line (Dunkerly line). This is a consequence of the fact that both the single load solutions and the interacting load solution have the same buckling deformation iV. 4.1.2 Two or more modes as deformation

The eigenvalue problem can be taken from eqn (8). In this case we have mixed interaction between modes and loads. Therefore, it is not easy to express the coupling effect explicitly, but according to the previous case we can state that, if the deformation vectors Okv and 0kV for the single load solution are affine, the interaction curve is also a straight line. This effect is demonstrated in the numerical examples.

5 EXAMPLES 5.1 Cross-section properties of C-section

The cross-section (Fig. 7) has 5 plane elements (main elements) and 6 main nodes. The 6 natural degrees of freedom (warping ordinates of the main nodes) are sufficient to describe the lateral buckling, lateraltorsional buckling and the buckling of the lips (simple distortion). To include also the plate-buckling of the web (higher distortion modes) the displacement of intermediate nodes is introduced as additional degrees of freedom. The arrangement in Fig. 7 with 8 intermediate nodes allows 16 modes. The higher modes (k > 10) are of no importance to our problem. The warping functions, the deformations and the transverse

"A. ~Wo ~W~

0.5 %X. iWo

0.5

1

v

iWer

Fig. 6. Linear interaction.

172

R. Schardt 10

r 6

...........

L,' ®

10 Y

@

31

4, + e

11

4,~

®

12

---1I!] 13

Fig. 7. C-section: dimensions (cm) and location of nodes 1 14.

bending moments are shown in Figs 8-10. Of these, 5 are symmetric and 5 are antisymmetric. Table 1 contains the stiffness values for symmetric and antisymmetric modes separately. Table 2 contains the to-values (eqn 5); two for normal force ~W and two for bending moment about y-axis 3W complete the section properties. Symmetric and antisymmetric modes are totally uncoupled. 5.2 The single mode solution for centric normal force To achieve a dimensionless solution, the buckling factor kb is introduced. In our case it is the buckling stress oer related to the Euler stress oe of the web °e - - 12(]---- ~2) O'cr = kb "O'e

(;)2

-~-

18980. 0.22 102 -- 7.592 kN/cm 2

(24)

(25)

Results from the single-mode solution, depending upon the length, are shown in Fig. 11 on a double logarithmic scale. It is obvious that some modes dominate special ranges of the length, where they have the minimal value. Beginning with very short lengths (about 8 cm), we have the local buckling of the web expressed by mode j = 7. The minimum value of 5.27 for kb at a length of 8-2 cm shows how much the usual assumption of kb = 4, which neglects the elastic restraint by the flanges, is on the safe side. At medium lengths (minimum at 48 cm)

Generalized beam theory

173

warping function k=3

warpln9 function k=4

warping function k=5

warping function k,,6

warping luncUon k=7

warping function k-8

2;

warping function k=9

wuzplng function k=lO

Fig. 8. Warping functions for C-section (modes k = 1-10).

174

R. Schardt

deformation k= 1

deformaUon k - 2

i___ deformation k - 4

deformation k - 3

,; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

delormatlon k - 5

i

deformation k - 6

2 ' -..... deformation k=7

de|orrnallon k - 8

/ defonnillon k,-9

delormatlon k - l O

Fig. 9. Deformations for C-section (modes k = 1-10).

Generalized beam theory

bending moment k-1

bendlng momen! k=3

L J bending momenl k-5

175

bending moment k-2

L J L P J bending moment k=4

bending moment k,,6

bending moment k=7

bendlng moment k-8

~

bending momenl k-lO

moment k ~

3 Fig. 10. Transverse bending moments for C-section (modes k = 1-10).

176

R. Schardt TABLE 1

Modal Constants kC, kD and kB for C-Section, Symmmetric Modes k

kC kD kB

1

3

5

7

9

5.000 0 0

27.082 0 0

20.742 0.09190 8.3716

0.00382 0.00245 1.6930

0.00505 0.00978 12-515

Modal Constants for Antisymmetric Modes k 2

kc kD kB

87.626 0 0

4

623.173 0.06667 0

6

0.26099 0.00121 0.26362

8

0-00655 0.01232 12-142

10

0.00549 0.01315 31-823

the distortion modes j = 5 and j = 6 (symmetric and antisymmetric buckling of the lips) have the lowest critical values (kb = 4.7 for mode 5), while column buckling begins at a length of about 90 cm. A change from symmetric lip-buckling to antisymmetric lateral-torsional buckling occurs at a length of about 90 cm. The solid curve, which shows the solution for all modes in interaction, demonstrates the good approximation by the single modes, especially at the characeristic lengths. The curves in the bottom diagram of Fig. 11 show the participation in percent o f the different modes in the interactive buckling form. 5.3 Interaction of normal force and bending moment as load

As an example of two interacting warping moments, we consider an eccentric normal force. Then we have i = 1 for the normal force and 1 = 3 for bending about the y-axis. The interaction parameter r/(eqn (22)) becomes 31it~ • 3W0

7 / - 1ij~. 1W0 The dominant deformation for lip-buckling is j = 5. Then r/is given by =

-0.1052. e. N = 0-4707- e -0.2235.N

with the eccentricity e of the normal force in the z-direction.

Generalized beam theory

177

TABLE2 K-Values~rC-Section(Fig. 7) x-Values likK(symm.) k /

3

5

3 5 7 9

-l.O000 -0.1376 -0.2283 0.1335

-0.1376 -0.2235 -0.0931 -0.0809

7 -0.2283 -0.0931 -0.2076 0-00061

9 0.1335 -0.0809 0.00061 -0.2750

K-Values likK(antisymm.) k j

2

4

6

8

2 4 6 8

-l.O00 -5.210 -0.2745 -0.3173

-5.210 -50.084 -1.812 -1.842

-0.2745 -1.812 -0.2235 -0.1135

-0.3173 -1.842 -0.1135 -0.3567

K-Values 3ikK(symm.) k j

3

5

7

9

3 5 7 9

0 0.0168 0.1116 0.0122

0.0168 -0.1052 0.0210 -0.0344

0.1116 0.0210 0.0798 -0.0470

0.0122 -0.0344 -0.0470 -0.0573

K-Values 3~K (antisymm.) k j

2

4

6

8

2 4 6 8

0 -1.000 -0.1548 -0-0541

-1.000 -13.734 -1.223 -0.7334

-0.1548 -1-223 -0.1275 -0.0177

-0.0541 -0.7334 -0.0177 -0.0533

178

R. Schardt

l

100

k -%

50

b- Oe

10 5

i I

i

10[

100%

50

10C.

0%

\

'~,.

".h,5001" h.,. 1000 I Icm]

Fig. ! 1. Buckling coefficient k b tbr single mode solutions, coupled solutions and c o m p o nents of modes in coupled solulion.

We choose node 1 (see Fig. 7) for the critical stress. For 'q = I (e = 2.125 cm) we find the critical stress for a single load at the characteristic length/c = 48 cm using eqn (25): ICrcr = 5.623 x 7.592 = 42.69 k N / c m 2 3crc,. = 8.580 × 7-592

65.14 k N / c m 2

Assuming linear interaction, we find 13crc,. =

42.69 + 65.14

= 53.9 k N / c m 2

2 The exact solution, using the modes 3.5, 7, 9, is 1~c~,. = 54-59 k N / c m 2 which confirms a nearly linear interaction, so that eqn (23) is a good a p p r o x i m a t i o n . The results for some positive eccentricities are plotted in Fig. 12. Negative eccentricities lead to web buckling (mode 7). N o d e 5 is chosen for the critical stress. The results for some negative eccentricities are shown in Fig. 13. The buckling coefficient kb is nearly independent of the eccentricity e. The c o n t r i b u t i o n s of the modes to the buckling d e f o r m a t i o n in Fig. 13 are valid for e = - 0 . 5 .

Generalized beam theory

179

100 k

Oct,1

z

50

b= T

20

f

10 5

5

10

20

50

100

200

100% - -

0% t [cm]

Fig. 12. Buckling coefficient kb for positive eccentricities e (cm) (lip buckling).

l

kb=

100

oc~s 50

20 10

e---O.5 5

2

20 100 %

50

100

200

1000

I

0% t lcm]

Fig. 13. Buckling coefficient kb for negative eccentricities e (cm) (web buckling).

I ~0

R. Schardz

The example demonstrates the advantage of the GBT formulation especially for stability problems. It is the consequence of the orthogonality of the modes, which is not only a mathematical but also a mechanical attribute. The modes are 'natural' components of the buckling deformation.

REFERENCES 1. Schardt, R., Vera/l,~emeinerte Teclmische Germany, 1989.

Bie,~etheoHe. Springer-Verlag,