buk1-a unified solution for various boundary conditions

8/11/2019 Buk1-A Unified Solution for Various Boundary Conditions

1/12

A UNIFIED SOLUTION FOR VARIOUS BOUNDARY CONDITIONSFOR THE COUPLED FLEXURAL-TORSIONAL INSTABILITY

OF CLOSED THIN-WALLED BEAM-COLUMNS

ASHOK JOSHI and S. SURYANARAYANDepartment of Aeronautical Engineering. IndianInstitute of Technology. Bombay. India

Received 27 December 1982)

AbstrIct- The problem of coupled ftexural-torsional instability of closed thin-walled beam-columns underthe combined action of axial loads and equal end moments is studied numerically. The characteristicequations for stability are obtained and the load interaction curvesand mode shapes are plolted for variousboundary conditions. From a study of the numerical results and the chllrolcteristic equations. equivalent loadand geometry JIllrameters arc defined. These yield a single solution common ror all boundaryconditions forthe ioad interaction curve and the mode shape. The effect of various boundary conditions and modenumbers is manifest only in the value of an eigen coefficient independent of the loads which is used indefining the equivalent parameters.The case of lateral instability within the clastic limit is also examinedand it is shown that even when tensile axial loads are present elastic lateral instability may occur.

NOTATIONA area of the cross-section of beam

A A B . BIC C2, D . Dz arbitrary constants in eqn (9)

D differential operator, did:

b width of the cross-sectionb bid, size ratio of the cross-sectiond depth of the cross-sectionE Young s modulus of elasticityf (= AlV S)/P S), eqn (10)

G shear modulus of rigidityI... moment of inertia about x-axisI moment of inertia about y-axislp polar moment of inertiaJ St. Venant torsional constantL span of beam

L, (= L/a), effective span of beamM ~ . moment about y-axisM dimensionless moment defined by eqn (2)M (= Mia), effective moment parameterM equivalent moment parameter defined by eqn (24)

Me critical lateral instabilitymoment for simply-supported ends (eqn 2m mode numberP axial load

Pe , critical Euler buckling load for simply-supported ends (eqn 2)p ~ , I P ~ , ) ,dimensionless load parameterP, (Pia), effective load parameter5 slenderness parameterdefined by eqn (Sa)5, (S/a l ), effective slenderness parameter

I thickness of the wall of the beamVn nexural disglacement in I -directionilo (1J(J(1,JArl ), d i m e n s i o n l c ~ sdisplacement

x. y.: Cartesian coordinates,see Fig. I

( = l L), dimensinless spanwise coordinatea eigen value coefficientI = - v J9), location of centre of rotation of the cross-section

I, defined by eqns (20) and (23). .. defined by eqn (25)

Etimi( linear elastic limit strainlzz axial stress9 torsional displacement

I. INTRODUCTIONThe problem of coupled flexural-torsional instability of beam-columns (also referred to inliterature as lateral instability) under the combined action of axial load and end moment has

167


2/12

168 A JOSHI and S SURYANARAYAN

been a subject of study for nearly halfa century. The earlierst comprehensive review on TheLateral Buckling of Beams is that of Timoshenko[l]. Since then numerous studies consideringvarious types of loads, geometries and end conditions have appeared in the literature.Johnston[2] has studied the eccentrically loaded I-column for lateral instabilityfor two cases of

simply supported and clamped ends. He has also given the limiting values of the slenderness ofthe beam for which elastic lateral instability precedesyielding. Goodier[3] has consideredsingly-symmetric open section beams but restricted to simply-supported end conditions.Salvadori[4] has considered the lateral instability of beams subjected to unequal end momentsand axial thrust and by using the concept of an effective span obtained a single interactionequation for simply-supported as well as clamped ends. Many other investigators[5-7] havetackled the problem of lateral instability for various problem parameters like biaxial moment,boundary conditions etc. Howevertheir studies usually give separate solutions for differentcases as represented by the boundary condition, slenderness of the beam and combination ofloads. Only for the simple case of Euler buckling of beams, a completely unified solution forvarious homogeneous boundary conditions has been obtained using the concept of effectivewave length . That such a unification is possible to account for various homogeneous boundaryconditions and slenderness of the beam even for the case of coupled flexural-torsionalinstability of closed thin-walled beams under the combined action of axial loads and momentconstant along the span does not seem to have been realized. The authors have shown in anearlier work [8] that a unified solution can be obtained for various slendernesses of the beam forthe coupled flexural-torsional vibration and stability of simply-supported narrowrectangularbeams. The present paper gives a unified solution for various boundary conditions for thecoupled flexural-torsional instabilityof closed thin-walled beams subjected to the combinedaction of axial load and moment constant over the span. t is assumed that the effect of axialconstraint stresses arising from the torsion of the cross-section can be ignored. The analysis isalso applicable to open cross-sections for which this assumption is valid and is exact forwarpless cross-sections, i.e. sections which do not warp under torsion.

Firstly, solutions are obtainedfor various homogeneous boundary conditions. From an observation of these solutionsand the values of theeigen coefficients which represent the effect ofvarious boundaryconditions and mode numbers, the various dimensionlessload and geometryparameters are redefined to yield a nearly unified solution for the load interaction curve and themode shape. Further. by incorporating the effect of the slenderness of the stream into thedefinition of the equivalent moment and the mode shape parameters. the ~ o l u t o nis soobtained that (i) the load interaction curve and the mode shape in terms of the equivalent loadparameters is independent of the boundary conditions, mode numbers and the slenderness of

the beam and (ii) the characteristic equation for the eigen coefficients is entirely in terms of theboundary conditions and mode numbers and independent of the applied loads and theslenderness of the beam. The load interaction equation in terms of equivalent axijll loads andmoment parameters is similar to the one for simply-supported ends already obtainedinliterature. The characteristic equations for the eigen coefficient are the same as those obtained forthe Euler buckling of columns.

Further to the unified solution. the extent of validity of linear stability analysisas given bythe unified solution is also examined for typical geometries of closedand open warplesscross-sections. This is done by dividing the load and geometry planes into elastic and inelasticdomains by constant limit strain curves. From these the combinations of load and geometryparameters for which the linear elastic analysis is valid are obtained.

2 FORMULATION AND SOLUTIONFigure 1 shows a uniform beam of warpless cross section subjected to an axial load P and

an end Momemt M).) The governing equations for the coupled flexural y) and torsional (6)motion of the beam are,

1)


3/12

A unified solution for v a r i o u ~boundary c o n d i t i o n ~ 69

Fig. \. A typical thin-walled beam of warpless cross-section.

where A is the area of cross-section; J , J are the second moments of area about x and axes

respectively; J is the polar moment of inertia, ] is the torsional constant; E and G are materialconstants and o and 8 are the infinitesimal flexural (y) and torsional displacements of thecross-section. Using the following transformations

z = fL P = PP n M = MMcr

where

we get the governing equations as

Simplification of eqns 3) yields

Defining a slenderness parameter S as

and the dimensionlesstransverse displacement as

and substituting into the eqn 4) we get

- ~ D + P D 2

MV S D 2

where D is the differential operator did .

2)

(3)

4)

Sa)

(5b)

(6)


4/12

17

From eqn 6) we get

7)

where

8)

The general solution for o and 8 can be expressed as;

vo


5/12

A unilicu \olution for variou\ boundary ~ o n u i t i o n 7

iv) Clamped-pinned ends. The characteristic equation is

ITa = tan ITa 15a)

and the mode shape is

Vo = sin lTai lTai ITa cos lTai \); 6 = fvo. 15b)

v) Pinned-free entls. The first mode for this case represents the rigid hody rotation of thebeam and for higher modes the characteristic equation for a and normalized mode arerespectively,

sin 1Ta =0; a = m 16a)

and

V = sin lTai; 0 = fVII I6b)

The second mode for this case is the same as the first mode for pinned-pinnedends.vi) Free free ends. For this case also the first mode is the rigid body motion of the beam

and higher modes are given by eqns 16a) and t6b).

3 RESULTS FOR VARIOUS BOUNDARY CONDITIONSTable 1 gives the values of the eigen coefficient a for v a r i o u ~boundary conditionsand mode

numbers. Figure 2 shows the load interaction curves for various boundary conditionsfor aninfinitely slender beam and a very short beam of dlL = 0.25. Though the analysis here isrestricted to only Euler-Bernoulli beams, to show the limited extent of the effect of dlL on theresults an extreme value of 0.25 is chosen. Curves marked Q) correspond to the first mode forsimply-supported endsand the second mode for both simply-supported free and free-free ends.Figure 3 shows the spanwise distribution of lateral displacements Vo and torsional displacement6 It can be seen that both have the same spanwise distribution which is independent of theapplied loads. However Tj = vol8 which gives the location of the centre of rotation of thecross-section. does depend on the applied loads as shown in Fig. 4. The variation of Tj with Mis shown for two boundary conditions and two typical values of the slenderness. These showthat for dlL = .25 the effect of boundary conditionon the value of Tj is noticeable but as thebeam becomes slender dlL = 0.1) this effect diminishes.

4. DERIVATION OFTHE UNIFIED SOLUTIONThe load interaction curves in Fig. 2 and the eqn 8) suggest that it is possible to collapse

them into a single curve. By defining equivalent axial load, and moment and slendernessparameters P 1 M and S . using the following transformation relations;

M,= MIa P. = Pla 2 and S, = S/a 2 17)Table I. Values of the coefficient a for various boundary

conditions and mode numbers

Boundary Mode NumberCondition 2 3 4

pinned- 1.0000 2.0000 3.0000 4.0000pinnedpinned- 0.0000 1.0000 2.0000 3.0000freefree- 0.0000 1.0000 2.0000 3.0000freeclamped- 1.4303 2.4590 3.4709 4.4774pinnedclamped- 2.0000 2.8606 4.0000 4.9180clampedc1amped- 0.5000 1.5000 2.5000 3.5000free


6/12

-.10.

3

A. JOSHI and S. S U R Y N R Y ~

D CLAMPED - fREE CANTILEVER) 1)VPINNED-PINNED 1);PINNED-FREE 2 ) ,

FREE -FREE 2 )

d L = 0 2 5

4 ~ ~ ~_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Fig. 2. Interaction curves for various end conditions and two typical values of the slenderness.

PI NNE 0 - PINNED ENDS 1 )

O ~ ~< J 2 4 0 6 0 8 1 0CLAMPED - CLAMPED ENDS 1 )~ I~ 1CLAMPED-PINNED ENDS 1)

~CLAMPED - FREE ENDS 1)

t l l l l l ~2 0 2 0 4 06 0 8 1 0

Fig. 3. Normalized spanwisedistribution of displacements v and 8.

and substituting in eqn 8), the interaction equation may be rewritten as,

1 P lSp. = I. 18)

Figure 5 shows the above relation for three typical valuesof the effective slenderness of the beam.It can be seen that for slender beam the interaction equation 18) reduces to

- 2 -M. - p. 1 19)

aRd that for dl L. = 0.1 where L. = la the deviation in M. even for large tensile axial load


7/12

A unified solution for various boundary condition

16

I?

1 0

oI> 8

t:< 6

4

a

\. J CLAMPED-FREE < < = 0 5 )

~ D CLAMPED -CLAMPED


8/12

174

f 2k'0:-

OJ0:-

o

A JOSHI and S. SURYANARAYAN

S q C L e / d l2

0 d L O O

t ~

f i\- \\\ .V ' \ \ .

\

Fig. 6 Nearly unified mode shape showing the effect of effective s l e n d e r n e ~ , .

is desired a completely unified solution accounting for the effect of finite slenderness can also bearrived at as given by

22)

23)

It can be seen that the load interaction curve is a parabola and the variation of the centre ofrotation with the applied moment a hyperbola see Fig. 7). The parameters Mt and ~ can beobtained from eqns 18), (20), (22) and (23) as solutions of the following equations.

M ~ 4 / S .+ ~ 2 1 - l / S . )==M/ (24)

TIt == ./Y(1 + M ~ 2- I /S,. . 25)

4

3 M;)

1 e : 1 . M.

t 2'

M; -0

(-2 )2 M . 5 .

M : 2M; - 1 + 5 IQ

0

UNSTABLE

I

Fill. 7. Unified linear stability solution.


9/12

A unified slIlution for variou\ ollundary condition, 175

Figures 8 and 9 show the variations of M and 1/, with M It can be seen that M M and1/, 1/. as dl L,. ~ O For finite values of dl L M differs (rom Mr and 1/. from 1/. by onlya small amount which is of the order of 112-1 .

Thus a unified solution for the coupled flexural-torsional instability for all boundary

conditions, mode numbers, slenderness of the beam and the load ratio is obtained. This is insuch a form that the (i) load interaction curve is independent of the boundary condition andmode number and (ii) the characteristic equation for the eigen coefficient a is independent ofthe loads. Further it may be observed that (i) the load interaction equationin terms of M andp. is similar to that obtained for the case of simply-supported ends and (ii) the characteristicequations for a are the same as those for the simple case of Euler Buckling.

5 LIMITS OF ELASTIC ANALYSISThe load interaction curves show that the lateral instability is possible even when tensile

axial loads are present. The assumption of linear elasticity have been implicit in the present

formulation and therefore the interaction curves necessarily represent the linear elastic instability. It would be relevant to examine whether linear elastic instabilityis at all possible whentensile axial loads are present and if so what are the limits on the loads and the geometries. This

o 01

~


10/12

A. JOSHI and S SURY ANARA YAN

has been done for the thin-walIed beams of typical cross-sections considering various combinations of load and geometry parameters and calculating the maximum strain attained. Figure10 shows the constant limit strain curves for a simply-supported narrow rectangular beam forsome typical combinations of instabilityload. The straightlinecorresponds to the case of Euler

buckling. Thecurve correspondingto lateral instability underpure moments is a parabola. Limitstrain curves for two typical cases of lateral instability under the tensile axial loads are alsoplotted. These constant limit strain lines divide the geometry plane defined by L,/d) and tid)into the elastic zone and inelastic zone. Results show that for a limit strain equal to 0.001 theEuler buckling is elastic for L,/t values above 30. Elastic lateral instability underpure momentsis possible for L,Jd) value greater than 20 for tid = 0.1 and greater than 80 for tid = 0.2.Similarly elastic lateral instability under tensileaxial load t =0.5 is possible for L,Jd) valuegreater than 25 for tid = 0.1 and greater than 55 for tid = 0.15. It is clear from these results thatelastic lateral instabilityis possible for practically realizable geometries evenin the presence oftensile axial loads. In Fig. 11 the limit strain curves for simply supported narrow rectangular

15

t 10

J

tIQ.

3

2

1..1 L - Id ~T

LINEAR ON LINEARELASTIC (PLASTIC)ZONE ZONE

LIMIT = 0 -001

E J L i

L . / d . S

{ OJ L e i d 2 0 01 1LINEAR ONLINEARELASTIC (PLASTIC)ZONE ZONE

690

174

_ I L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Fig. 11. Constant limit strain curves superimposedon the unified stability solution in load plane.


11/12

A unified solution for variou\ bound,try condition,

2000l iNEAR NON LINEARE L A ~ I VPLASTIC)ZONY ZONE

CORRESPONOINGPOINTS OR

CIRCULAR THIN-

WALLED BEAM

Fig. 12 Constant limit strain curves for a simply-supported warplessrectangular box beam for various loadcombinations.

177

beam are superimposedon the unified stability solution in the load plane for two typical valuesof the slenderness of the beam. These show that the constant strain lines are straight in the loadplane and are near vertical for LJd = 0. Figure 12 shows the constant limit strain curves on asemi-log scale for a simply-supported warpless rectangularbox section. The results show thatelastic lateral instability is possible even for closed sections of sufficiently deep geometries. ForbId) value of 0.1 elastic lateral instability underpure moments is possible for LJd) value

greater than 40. The value of ( L J ~ )for elastic lateral instability undertensile axial load P. = I)for bId) valueof 0.1 is 57. Howeverfor higher values of bId) this limiting valueof LJd) increasessharply. For bId) = 0.25 the correspondingvalues of LJd) are 215 and 320 and for hId) = 0.5 thecorrespondingvalues are 620 and 880. For hid = 1.0 square box) elastic lateral instability underpure moments is possible for L,/d) value greater than 1200. The correspondingvalue for P, = 1 sabout 1690. The correspondingvalues of LJd) for a circularsection are 690 and 980 respectively.

Though this situation may not normally be encounteredin practice theoreticallyit is possible to getlinear elastic lateral instability even for circular sections.

6. CONCLUSIONSThe paper presents a unified solution for various boundary conditionsfor the problem of

coupled flexural-torsional instability of beam-columns of warpless section subjected to axialloads and end moments. By suitable transformations i) the load interaction equation is obtainedindependent of the boundary conditionin a form similar to that for simply supported ends, (ii)the location of centre of rotation of the cross-section is obtained as a single equation and (iii)

the characteristic equations for eigen coefficient a are obtained independentof the loads whichare the same as those obtained forthe simple case of Euler buckling. Theresults show that for allpractical purposes theresults for an infinitely slender beam can give sufficiently accurate resultsfor all slenderness and the correction for finite slenderness is only marginal. The results alsoshow that elastic lateral instability is possible even in the presence of tensile axial loads if thebeam is sufficiently slender.

A c k n o ~ l t d R t l l l t n l s -The work presented in this paper forms a part of a project sponsored by the Aeronautics Researchand Development Board, Directorate of Aeronautics, Ministry of Defence. Government ofIndia. The financial support,received is gratefully acknowledged.


12/12

178 A JOSHI and S SURY ANARAY AN

REFERENCESI S. Timoshenko. Theory of Elastic Stability. p. 279. McGraw Hill. New York (1963).2 B. Johnston, Lateral buckling of I section column with eccentric loads in the plane of web. 1. Appl. Mech. B A-176

(1941).3 J. N. Goodier. Torsional and flexural buckling of hars of thin-walled open ~ e c t i o nunder c o m p r e ~ s i v eand bending l o a d ~ .

1 Appl. Mech. 9. A-103 (1942).4. M. G. Salvadori. Lateral bucklin of [-beams. Tran.l. ASCE 120. 1165 (1955).S J. W. Clark, Lateral buckling of b e a m ~'mc. ASCE, l. StTllcl//T/J1 Viv. 86 ( ~ T - 7 ) .175 (1%(1).6. A. Chajes. Torsional-flexuralbuckling of thin-walled members.Proc. ASCE l Structural Div. 9 (ST-4 , 103 (1965).7. T. Tarnai. Variational methods for analysis of lateral buckling of beams hung at both ends. Int. 1 Meeh. Sci. 21, 6, 329

(1979).B A. Joshi and S. Suryanarayan.The erreet of static axial loads and end moment on the vibration and stability of narrow

rectangular beams. Department Tech. Rep. AETR B1121. Dec. 1981. Dept. of Aero. Engng. I.I.T. Bombay (Alsopresented at the 26th Congress of the Indian Society of Theoretical and Applied Mechanics held at Coimbatore, Indiafrom 28-31 Dec. 1981).

buk1-a unified solution for various boundary conditions

Documents