Theory of bending, torsion and buckling of thin-walled members of open cross section

Journal of

The Pranklin Institute Devoted to Science and the Mechanic Vol. 239

Arts

MAY, 1945

No. 5

THEORY OF BENDING, TORSION AND BUCKLING OF THINWALLED MEMBERS OF OPEN CROSS SECTION. BY

STEPHEN

P. TIMOSHENKO,

Professor of Theoretica! and Applied Mechanics, Emeritus, Stanford University.

PART III.

8. Buckling b y F l e x u r e and Torsion u n d e r Central T h r u s t . - - L e t us consider now the general case of buckling where, under central thrust, not only torsion but also bending of the axis of the compressed bar occurs. We assume that x and y are the principal centroidal axes of the cross section of the bar and xo, y,, are the co6rdinates of the shear center. The deflections of the shear center axis in the x- and y-directions, we denote by u and v, respectively, and denote, as before, by ¢ the angles of rotation of cross sections with respect to the shear center axis. Then the deflections of the centroidal axis during buckling, as can be seen from Fig. 26, are u + yo¢

and

v -

Xo¢.

Assuming that only a thrust P is acting at the ends as in the case of simply supported bars, we find that bending moments with respect to principal axes at any cross section are M~ = P(v -

xo¢),

M~, = -

P ( u + yo¢).

The differential equations (I7) for the deflection curve of the shearcenter axis become d 2IA _

E l , , dz 2

P ( u + yo¢),

(56)

d2u E I . dz 2 -

P(v -

Xo¢).

(Note---The Franklin Institute is not responsible for the statements and opinions advanced by contributors in the JOURN-~aL.) 343

344

STEPHEN

P. T1MOSHI,ZNKO.

IJ. I;. I.

To write the equation for the angle of twist ~, we proceed as in the previous article. We take a strip of cross section tds and defined by co6rdinates x and y in the plane of the cross section. T h e components of its deflection in the x and y directions during buckling are u + (yo -

and

Y)9

v -

(Xo -

x)~o, respectively.

T a k i n g the second derivatives of these expressions with respect to z and considering an element dz of the fiber, we find, as before, t h a t the compressive forces atds acting on the slightly rotated ends of the element d z give the following forces in the x and y directions: d2 - atds ~ Eu + (yo - y ) ~o~dz and -atds ~z 2 Ev - (Xo - x ) ~o~dz. Denoting, as before, by primes, the derivatives with respect to z and making the m o m e n t s of the above forces with respect to the shear-

t

*

FIG. 26.

center axis, we obtain, as the contribution of one strip, the following torque per unit length of the bar: dmz = -

atdsEu"

+ (yo -

y)~"~(yo

-

y)

+ atds[-v" -

(xo -

x)~"-](xo

-

M a k i n g integration over the entire cross section and observing t h a t

f we obtain m , = (XoV" -- y o u " ) P

-- ro2P~o '',

where ro2 = xo2 +

yo2 +

I,: + I~,

x).

May, ~945.]

THI..'ORY OF ])JENI)ING, "]'ORSIOX A.\I) I~)UCKLIN(;.

,~45

is the square of the radius of giration of the cross section with respect to its shear center. S u b s t i t u t i n g the calculated value of mz into eq. (51 ) we obtain C , ~ , ~v -

(C -

r o ~ P ) ~ '' -

P(xoV"

-

you")

= o.

(57)

This e q u a t i o n t o g e t h e r with eqs. (56) will now be used to determine the critical value of thrust P.* It is seen t h a t the angle of rotation ~ enters in all three equations, which indicates that, in the general case, torsional buckling is accompanied b y bending of the axis, and we have a combined torsional and flexural buckling. In the particular case when Xo = yo = o, i.e. when the shear-center axis coincides with the centroidal axis, each of eqs. (56) and (57) contains only one u n k n o w n and can be treated separately. E q u a t i o n s (56) give then t w o values of the critical load, corresponding to Euler's buckling in two principal planes. E q u a t i o n (57) contains, in such case, only one u n k n o w n ~ and gives us the critical load for purely torsional buckling a l r e a d y discussed in the preceding article. From the three values of the critical load n a t u r a l l y the smallest will be taken into consideration in practical applications. R e t u r n i n g to the general case, let us assume t h a t the ends of the compressed bar cannot rotate with respect to the z-axis, Fig. ~3, b u t are free to warp and to rotate freely with respect to the x- and y-axes. In such case, the end conditions are u = v = ~ = 0'l

for

!

z = o

and

z = I.

All these conditions will be satisfied b y taking the solutions of eqs. (56) and (57) in the form u

7rZ =

Alsin

I

7rZ '

v =

A,.,sin

I

7rg '

~

=

A~sin

l

"

(58)

S u b s t i t u t i n g these expressions into eqs. (56) and (57), we obtain the following equations for calculation c o n s t a n t s A ~, A 2 and A :~: 71-2

i +PyoA3

= o,

2 -- PxoA3

= o,

71"2

PyoA 1 -

PxoA 2 +

(

-

C'1 ~T~ --

C -Jr- r o 2 P

)

(59)

A a = o.

These e q u a t i o n s are satisfied b y taking A1 = A.., = Aa = o, which cor* The system of equations equivalent to eqs. (56) and (57) was first obtained by Robert Kappus; see " J a h r b u c h der deutsehen Luftfahrtforschung," I937 and "Luftfahrtforsehung," vol. I4, p. 444, 1938.

346

STEPHEN

P.

TIMOSHENKO.

[J. F. I.

responds to the straight form of equilibrium of the compressed bar. To have a buckled form of equilibrium eqs. (59) must yield for A 1, A 2, A 3 solutions different from zero which is possible only if the determinant of eqs. (59) vanishes. To simplify the writing we introduce here the notations: z~EI~ r~EIx i( r 2) P1 12 , P2 = l~-, P3 = r° 2C + Cl-fi , (60) in which P1 and P2 are the Euler loads for buckling in two principal planes and P3 denotes the critical load for purely torsional buckling, as given by eq. (54). Then, equating to zero the determinant of eqs. (59), we obtain P

--

P1,

O, YoP,

O, P-P2, - xoP,

YoP] -xoP = o ro2(P P3)

which, after evaluation, gives the following cubic equation for calculating critical values of P : ( - - ro2 + Xo2 + Yo2)P 3 + -

E(P1 -']-P2

+ P3)ro ~ - xo2P1 - yo~P2-]P2

ro2(P,Pz + P2P3 + P 1 P 3 ) P + P1PzP3ro 2 = o.

(6I)

From this equation one important conclusion can be obtained. Assume t h a t P1 < P2, i.e. that the smaller Euler load corresponds to bending in the xz-plane, and investigate the sign of the left-hand side of eq. (61) for various values of P. If P is very small, we can neglect all terms containing P and the left-hand side of the expression reduces to P1P2P3ro ~, which is positive. Assume now that P takes the value PI; then the left-hand side of eq. (61) reduces to yo2P12(P1 - P2) and, since P1 < P2, it is negative. This indicates that eq. (61) has a root smaller than P1 and that taking into consideration the possibility of torsion during buckling we will always obtain a critical load smaller than the Euler load. To find the critical load in a particular case, we calculate, by using notations (6o), the numerical values of the coefficients of eq. (61) and then solve this cubic equation.* We will get three values for P from which the smallest will be taken in practical application. Substituting these values into eqs. (59), we find for each possible critical form the ratios A I:A 3 and A2:As. These ratios, establishing the relation between rotation and translation of cross sections, define for each critical form the position of the axis, parallel to the axis of the bar, with respect to which the cross-sections rotate during buckling. If the bar has very thin wall and short length l, P3 may become small in comparison with P1 and P2. In such case the smallest root of eq. * This solution can be greatly simplified by the use of nomogram as shown in the previously mentioned paper by Kappus.

May, ~945.1

347

TFIEORY OF BENDING, TORSION ANI) BUCKLIN(;.

(61) approaches the value Pa. Substituting this in eqs. (59), we find and A2 are small in comparison with rotational displacements, which indicates t h a t the buckled form approaches purely torsional buckling discussed in the preceding article. In the case of a thick wall and large length I, Pa is usually large in comparison with P1 and P2 and the smallest root of eq. (61) approaches the value P,. The effect of torsion on the critical load is small in such case and the usual column formula gives satisfactory results. If the cross section has an axis of symmetry, the calculation of PeT is simplified. Let the x-axis be the axis of symmetry, then yo = o and the term containing ~ in the first of eqs. (56) vanishes. Buckling of the bar in the plane of s y m m e t r y is independent of torsion and the corresponding critical load is given by Euler's formula. We have to consider only buckling perpendicular to the plane of s y m m e t r y and torsion. The corresponding equations are t h a t A1

Elxv"

=

-

C,~o TM - - ( C -

P(v

--

Xo~),

ro2P)so '' = P x o v " .

Proceeding as before and using solution (58), we obtain for calculating the critical load P -

P2,

-- xoP,

-

ro~'(P

xoP Pa)

= o

which gives ro2(P -

P~)(P

-

Pa) - xo2P2 = o.

(62)

This quadratic equation gives two solutions which, together with the Euler load for buckling in the plane of symmetry, represent the three critical values for P from which th~ smallest must be used in practical applications. Considering the left-hand side of eq. (62), we see that for very small values of P it reduces to the value r o 2 p 2 P a which is positive. We see also t h a t it is negative for P = P2 and for P = P,, since its value then reduces to - x o 2 P 2. It is negative also for all values of P between P = P2 and P = Pa, since the first term in it becomes negative and the second is always negative. From this discussion we conclude that one root of eq. (62) is smaller than either P2 or Pa while the other is larger than either. The smaller of these roots or the Euler load for buckling in the plane of s y m m e t r y gives us the required critical load. All above made conclusions are based on solution (58). W i t h o u t any complication we can take the solution in a more general form and assume ftTrg

u = A~sin~-,

nTrZ

v = A2sin--~-,

= Aasin

f/Trz

1

(63)

which corresponds .to the assumption that during buckling the bar is subdividing in n half waves. Our previous conclusions will hold also

348

STEPHEN

P.

TIMOSHENKO.

IJ. V. [.

in this case; we have only to substitute into expression (6o) the values n2rr=/l ~ instead of re2~12. The corresponding critical values of the load

will naturally be larger than those obtained for buckling in one half wave and are of practical interest only if the bar has intermediate equidistant lateral supports. If the ends of the bar are built-in, the end conditions are U t

u = v = ~0 = o,} =

Vt

=

~0 t

=

for

O,

z = o

and

z = l.

Since, during buckling, end m o m e n t s will appear, we will have, instead of eqs. (56), the following equations: E I v d2u dz 2

E

I

P(u

d2v-

+ Yo~') + EI~, k d z ~

,=o'

(64)

[ + Eix t, d z 2 "] z=o"

e(v -

These equations, together with eq. (57),* will define now the buckling forms of the bar and the corresponding critical loads. All these equations and the end conditions will be satisfied by taking U =

A1

(

I -- COS~-

=

A a

,

(

V =

( 2-z)

7)

A2

I - - COS T

I -- cos--

,

(65)

.

Substituting these expressions into eqs. (57) and (64), we obtain, for calculating critical loads, the same e% (61) ; it is only necessary to use 47r=/P instead of 7r~/l 2 in notations (60). 9. Buckling of a Bar by Torsion and Flexure in an Elastic Medium. - - L e t us consider the stability of a centrally compressed bar if, during buckling, there appear lateral reactions proportional to the deflection. \Ve assume t h a t these reactions are distributed along an axis N parallel to the axis of the bar and defined by the c06rdinates h~ and h~, Fig. 27. Denoting, as before, the components of deflection of the shear-center axis by u and v and the angle of rotation with respect to t h a t axis by ~, we find t h a t the components of deflection of the N-axis, along which the reactions are distributed, are u + (Yo -- h ~ ) ~

and

v -

(Xo -

h~)~.

T h e corresponding reactions per unit length will be -- k , E u + (yo -

hu)~0]

and

-

kyEv -

(Xo -

hx)~],

* E q u a t i o n (57) was developed from consideration of an element of.the bar between two a d j a c e n t cross sections and is not affected by changes in the end conditions.

May, 1945.[

THEOI¢Y OF BENI)IN(;, TORSION

ANI) t~UCKIAN{;.

,g4 ()

where kx and ky are constants defining the rigidity of the elastic medium. To these reactions we add lateral forces obtained from the action of initial compressive forces of the fibers on their slightly rotated cross sections, These previously discussed forces (see art. 8) give, per unit length,

-- f, GtdsEu" + (Yo- Y)~"]

and

- f, o-tds[v" - (x,,- x)¢"].

Integrating and adding the results to the above calculated elastic reactions, we finally obtain the following expressions for the intensities of lateral force distribution: q~

=

-

k~Eu +

q,, = -- k u [ v -

(yo

-

I~)¢] - P(#'

!

t!

+ )o~ ), (66)

(Xo -

h~)~] - P ( v " - x o ¢ " ) .

? [:[G. 2 7.

Substituting these into eqs. (I8), we obtain Elvu

TM

+ P(u"

+ y o ~ " ) + k ~ [ u + (yo -

h~,),p~ = o,

-

hx)¢]

(67) Elzv

TM

+ P(v"

X o ¢ " ) + kyEv -

(xo -

= o.

To these equations of the deflection curve we have to add the equation for the angle of twist ¢.. Since the reactions are distributed along the axis N which does not coincide with the shear-center axis they will contribute to the torque distributed along the bar. The magnitude of this torque per unit length is -- k,[-u + (yo -

hy)9](yo -

h , ) + kyEv -

(xo - hx)9-](xo - hx).

The elastic medium m a y resist torsion also in the case where the N-axis coincides with the shear-center axis. This action on the bar can be represented by continuously distributed torque the intensity of which is proportional to the angle of twist and can be taken in tile form - k,~, where k, is again a constant depending on the rigidity of the medium. Adding these two components of the intensity of torque distribution to

35 °

STEPHEN

P. TIMOSHENKO.

[J. [;, I,

that previously obtained from the action of initial compressive forces on rotated cross-sections of fibers (see art. 8), we obtain m , = (XoV" -

yoU")P -

ro2P~ '' -

k x [ u + (yo -

+ k,[v -

(Xo -

hu)~](yo - h~j)

hx),p](xo -

hx) -

k**.

(68)

With this value of m, we obtain, instead of eq. (57) of the preceding article, the following equation for the angle of twist: C1~ Iv -

(C -

ro2P)~ '' -

P(xov" -

you") hy)9](yo -

+ k , [ u + (yo -

ky[v-

(Xo-

by) h~)~](Xoh,) + k , ~

= o.

(69)

This equation, together with eqs. (67), defines the forms of buckling of the bar in an elastic medium and the corresponding values of the critical loads.* If the ends of the bar cannot rotate but are free to warp, we take the solution of these equatior~s in the form (63), and the calculation of critical loads will require, as before, the solution of a cubic equation. In some particular cases the calculation can be greatly simplified. Take, as a simple example, the bar with two planes of symmetry. In such a case, the shear-center axis coincides with the centroidal axis and we have xo -- yo = o. Assuming t h a t elastic reactions are distributed along the centroidal axis, we have also h, = hv = o and eqs. (67), (69) become EI~u

TM

+ Pu"

+ k x u = o,

E I , v Iv + P v " + kyv = o, Cl,p Iv -

(C -

ro2P)~ '' + k , 9

= o.

We see t h a t in this case buckling of the bar in the planes of s y m m e t r y is independent of torsion and the three forms of buckling can be discussed separately. The first two equations give the known critical loads for bending in the planes of symmetry. The last equation gives the critical load for torsional buckling. Taking the solution in the form (63), we find for the later critical load the expression n~z2

Cl W Per =

+ C+ ?'o?

12 (7 o)

In each particular case, knowing C1 and k,, we select for n the value which makes expression (70) a minimum. When k, = o, we have to take n = I and the critical load (70) becomes equal to that previously obtained from eq. (55). Consider now the case when the cross section of the bar has one axis of s y m m e t r y which we take as the x-axis. Then yo = o. We as* These equations were first obtained by V. Z. Vlasov in his book, loc. cit., J. F. I., Apr. x945, p. 256.

May, I945.] THEORY OF BENDING, TORSlON AND BUCKLING.

35I

sume also that elastic reactions are distributed along the shear-center axis. T h e n hu = o, h, = xo and eqs. (67), (69) give EI~u

TM

+ Pu"

+ k . u = o,

E I ~ v ~v + P v " + k~v C , ~ ~v -

Pxo~"

C ~ " + k ~ o + Pro2~ '' -

= o,

P x o v " = o.

We see t h a t buckling in the plane of s y m m e t r y is independent of torsion and can be treated separately. Buckling in the direction perpendicular to the plane of s y m m e t r y is connected with torsion and we have to consider the last two of the above equations simultaneously. T a k i n g the solution of these equations in the form (63), we obtain, for calculating the constants A 2 and A 3,

n47r4 A2

El,

A2Pxo-~

+

n~rr2

l-7- -

P -~

)

n2rc2

+ k,

C~ --fi- + (U -

+ Pxo-fi-

A a = o.

r J P ) --fi- + k~

A 3 = o.

The critical value of the load is obtained by equating to zero the det e r m i n a n t of these two equations. In this way we obtain a quadratic equation for P~ from which in each particular case the critical values of the load can be calculated. If k~ vanishes, we have to take n = I and obtain for P~ the equation which coincides with the previously derived eq. (62). Using eqs. (67) and (69), we can investigate buckling of the bar in the case where the axis about which cross sections are rotating during buckling is prescribed. To obtain a rigid axis of rotation, we have only to assume that k~ = k~ = ~ . T h e n the axis N in Fig. 26 remains . straight during buckling and cross sections rotate with respect to this axis. Equations (67) give in this case u +

(yo -

h~)~

=

o,

v

(xo

h~)~

=

o.

-

-

and we obtain u = -- (yo -- h~)~o,

v = (xo -

hx)~o.

We have also, from eqs. (67), k , Eu + (yo -

hu)~o-] = - E l v u

k£v

h.)~

-

(xo -

=

-

ELy

TM

-- P ( u "

~v -

P(v"

+ yo~o"), -

xo~").

Substituting into eq. (69), we obtain ~oIvE,c1 + (yo -

hu)~EI~ + (Xo + Pg"[-ro ~ -

h~)2EI,~ -

C~"

( x j + yo~) + h 2 + hv ~-] + k ~ o = o.

(7~)

352

STEPHEN

P. TIMOSHENKO.

[J. F. 1.

Taking the solution in the form (63), we can calculate in each particular case the critical load from this equation. In the particular case of a bar with two planes of symmetry, we have Xo = yo = o.

nzrz.

Substituting ~ = A3 sin ~

m eq. (7I), we obtain

n2¢2 12 ( Cx + hv2EI~ + h.2EI~) -lq- + C + n-~r2 k ,

t

e.

=

ro2 + h~2 + h~2

(72)

In each particular case we have to take for n such value as to make expression (72) a minimum.

I

Fm. 28.

If the Shear-center axis is taken as the axis of rotation, we have h~ = ,xo, hv = yo, and eq. (7I) gives •

rt27r2

C,-FPer =

When k~ vanishes, we have to the same value as eq. (55) as it Taking the axis of rotation assuming, for example, h~ = ~ ,

12

-t- C-t- n--y~2k, ro 2

(73)

take n = I, and eq. (73) gives for P . should. at infinite distance from the bar and we obtain from eq. (7I)

9IvEIv + P g " = o, which gives for P ~ the known Euler load for buckling of the bar in the xz-plane. Sometimes, instead of a fixed axis of rotation there m a y be prescribed a plane parallel to which certain fibers of the bar have to deflect during buckling. If a bar, for example, is welded to a thin sheet, Fig. 28, the fibers of the bar coinciding with the surface nn cannot deflect during buckling in the plane of the sheet but deflect in the direction perpendicular to that plane. In discussing such problems it is advantageous to take the centroidal axes x, y parallel and perpendicular to the sheet. Usually the axes will no longer be principal axes and the corresponding differential equations of the deflection curve must be changed. From

May, [945.1

TIIEORY OF BEN1)IN(;, TORSION

ANI) []U('KLIN(;.

,353

eqs. (I) and (2) we have, in the general case, M¢ = -

EI~v" -

M~ = Elyu"

Elxyu",

+ Elxyv".

Differentiating these equations twice with respect to z and using eqs. (5) and (6), we obtain q~' = EI~vXV + EI~yuIV' 1 q~ E l y u l V + ElxuviV. ,

(74)

Considering any shape of cross section, Fig. 29, assume that a fiber N

I V

l

FIG. 2 9 .

with co6rdinates hx, hu cannot deflect in the x-direction. 'Denoting, as before, by u and v deflections of shear-center axis 0 in the x- and ydirections, we find for deflections of the fiber N the following expressions: u +

~,(yo -

by),

v -

~(Xo - h 2 .

From the conditions of restraint of the fiber N we conclude t h a t '

u + ~(yo -

hy) = o.

(75)

Due to this restraint there will be reactions parallel to x-axis and continuously distributed along the fiber N. Let qo be the intensity of this distributed force. T h e n the quantities q, and qu in eqs. (74) are obtained by substituting k~ = kv = o in eqs. (66) and adding qo to the first of these equations. In this m a n n e r we obtain q~ = -

P(u"

+ y o ~ " ) + qo,

q~ =

P(v"

-

-

xw").

Substituting into eqs. (74), we obtain qo = P ( u "

+ y o g " ) + E l y u xv + E I ~ v ~v,

E I , v Iv + E I ~ v u Iv + P ( v " -- x o ~ " ) = o.

(76 )

T h e second of these equations, after substituting for u its value from

354

STEPHEN P. TIMOSHENKO.

[J. F. I.

eq. (75), gives the following e q u a t i o n containing v and ~ only: Elxv

TM

+ Pv"

-

EIx~(yo

h~)~ TM - P x o ~ "

-

= o.

(77)

T h e second e q u a t i o n for v a n d ~ is o b t a i n e d from a consideration of torsion of the bar. F o r this purpose we use eq. (69). S u b s t i t u t i n g in this e q u a t i o n k~ = k~ = o, - k , E u + (yo - h ~ ) ~ ] = qo, we obtain CI~ ~v -

(C -

r o ~ P ) ~ '' -

P(xoV"

-

you")

-

qo(yo -

h~) = o.

S u b s t i t u t i n g in this e q u a t i o n the value of u from eq. (75) and the value of qo from the first of eqs. (76), we obtain [-C~ + E I ~ ( y o

-

h~)2-}~Iv -

C~"

+ P , Y ' ( r o 2 + h~ 2 -

yo2)

I

-

EI,~(yo

-

h ~ ) v ~v -

Pxov"

(78)

= o.

E q u a t i o n s (77) and (78) will now be used for calculation of critical loads.* Assuming t h a t the end conditions are 9 = 9 " = v = o,

for

z = o

and for

z = l,

FIG. 30.

we t a k e t h e s o l u t i o n of eqs. (77) and (78) in the form (58) and, after substitution, we o b t a i n B

-W-

(y° -

h,) -

1

exo j

= o,

7r 2

+

[C1 + EI~(yo

-

h~)~-Ig + C

-

P ( r o ~ + h~' -

yo')

A ~ = o.

E q u a t i n g to zero t h e d e t e r m i n a n t of these equations we obtain, as before, a q u a d r a t i c e q u a t i o n for P from which the critical load in each particular case can be calculated. If the b a r is s y m m e t r i c a l with respect to the y-axis, as in the case of a channel, Fig. 3o, the x- and y-axes are principal axes; hence, I~u = o, * These equations were obtained by J. N. Goodier, Bull., no. 27, I94I, Cornell University Engineering Experiment Station.

May, I945.1

THEORY

OF BENDING,

TORSION

AND B U C K L I N G .

355

xo := o, and the above equations become ( EIxTr2

p)A2=o,

12 [c1 + EI~(yo -

h~)2l -g + c - P(ro ~ + h~~ - y j )

A3 = o.

From the first of these equations, we obtain the Euler load for buckling in the plane of symmetry. From the second equation we obtain the torsional critical load, the axis of rotation being in the plane of the sheet, 71-2

ECl + E l ~ s o

P~

- h~)~-I l~ + C

ro2 + h . ~ -- yo 2

,

v

{~

(t9

-~M,

1 Z

~1z

P Fit;. 3I.

This same result would be obtained from eq. (7I) by substituting 7rg

xo = hx = k~ = o, ~ = A 3 s i n

1

IO. Stability of Thin-waUed Members under Bending and Compression.--It is known that beams bent in the plane of their maximum rigidity may fail due to 'ateral buckling accompanied by torsion.* Let us consider the case when the bar is submitted to the action of central thrust P and bending couples M, and Ms applied at the ends, Fig. 31. We assume that the effect of P on bending stresses can be neglected. In * This kind of buckling in the case of beams of a narrow rectangular cross section was investigated by L. Prandtl; see his dissertation, Nfiremberg, 1899; and by A. G. M. Michell, Phil. Mag., vol. 48, 1899. The case of an I-beam was discussed by the writer in his previously mentioned paper; see J. F. I., Apr. I945, p. 254. The extension of the theory to bars of any thin-walled open section was made by V. Z. Vlasov in his previously mentioned book; see J. F. ]., Apr. I945, p. 256. Some cases are discussed also by J. N. Goodier, Bulletin no. 28, Engineering Experiment Station, CorneU University, 1942.

356

STEPHEN P. TIMOSHENKO.

[J. F. I.

such case, the normal stress at any point is independent of z and is given by the equation =

-

P A -

MO' Ix

+

M2x I,.,

,

(8o)

in which x and y are the centroidal principal axes of the cross section. T h e initial deflection of the bar due to couples M1 and M2 we consider as very small. In investigating the stability of this deflected form of equilibrium, we proceed as before and assume t h a t additional deflections u and v of the shear-center axis together with rotations ~ with respect to t h a t axis are produced, and write equations of equilibrium for this new form slightly different from the initial curved form produced by the couples M, and Ms. In writing these equations we neglect small initial deflections and proceed as in the preceding cases in which the axis of the bar was initially straight. T h e n the components of deflection of any longitudinal fiber of the bar, defined by co6rdinates x and y, are

u+

(Yo - Y)9,

v-

(Xo - x)¢.

T h e intensities of the lateral fictitious loads and distributed torque obtained from the action of initial compressive forces in the fibers on their slightly rotated cross-sections are obtained as before (see art. 8) and are given by the equations qx = -- £

a t d s [ u " + (yo - y)~o"],

q~ = - £

atds[v" -

(Xo - x)~o"-l,

m. = -- .~. . t d s [ u " + (So

y)~o"~(yo

Y)

+ j ~ atds[v" -

(xo - x)~o"-](xo-x).

Substituting expression (8o) for ~ and integrating, we obtain q~ = - P u " -

(Pyo + M 1 ) ~ " ,

qy = - P v " + (Pxo -- M 2 ) ~ " , rn. = -

(Pyo + M 1 ) u " + (Pxo - M z ) v " + (MI~I -

M2132 - Pro2)9 '',

where the following notations are introduced:

fA yadA + fA x2ydA 1 ~1 = Ix - 2yo, (8I)

[~2 =

Iv

- 2xo.

May, ~945.]

357

THEORY OF BENDIN(;, TORSION AND BUCKLING.

Equations (I8) and (5I) then give E I ~ u ~v + P u " EIxv Cl~

TM

TM

-- (C @ M I ~ I - - M 2 ~ 2 -

+ (Pyo + M 1 ) ~ "

+ Pv" -

(Pxo -

M2)¢'

= o, ~ = o, t

"(82)

ProS)~ ''

+ (Pyo + M , ) u "

-

(P.ro -

M 2 ) v " = o. )

These are the general equations of equilibrium for the buckled form of the bar. From these equat!ons the critical values of external forces can be calculated for any given end condition. Let us begin with the case of eccentric thrust. If e~ and ey denote the coSrdinates of the point of application of the thrust P, we have Ml

= -- Pc,,,

M.,. = Pe,..

Substituting in eqs. (82) we obtain E I ~ u ~v + P u "

C ~ ~v -

(C-

Pev~

-

+ P(yo -

e ~ ) ~ " = o,

E I ~ v ~v + P v " - P(.ro -

e ~ ) ~ " = o,

Pe~s

-

(83)

Pro")¢'

+ P(yo -

e~)u" -

P(a:o -

ex)v" = o.

In the case of simply supported ends, we have the following endconditions: u Vtt = v~ Ut! ~-

= @tt9 = .~. o,} O,

for

z = o

and

z = 1.

We will satisfy these conditions by taking u

=

A lsin

rrg l '

A ssin

v

7r,g l '

¢

=

A asin

7rg / .

Substituting into eqs. (83), we obtain ( E I ~ - P ) A z -

EL-fi -

P(yo -

e.u)A~ + P ( x o +

Cz-~+

-

P

P(yo-e:~)Aa•

A s+P(xo-ex)Aa

=o, = o,

(84)

ex)As C-PeyB1

-Pe~2-Pro

s

A a = o.

Equating to zero the determinant of these equations, we obtain, as before, a cubic equation for calculating Pc~.

[J.

STEPHEN P. TIMOSHENKO .

358

F. r.

In a particular case when the thrust P acts along the shear-center axis, we have ex = xo,

ey =

yo,

and eqs. (83) are greatly simplified, since each of them contains only one variable . Lateral buckling in two principal planes is independent of each other and independent of torsion . We obtain three critical values of P, two given by the Euler formula, and one corresponding to torsional buckling . If the point of application of thrust does not coincide with the shearcenter the three eqs. (83) are interconnected and flexure of the bar during buckling is connected with torsion . The problem is considerably simplified if the bar has one plane of symmetry. Assume that the yz-plane is the plane of symmetry and that the thrust P acts in that plane . Then xo = ez = o, and eqs. (84) become

2 (EI,l2 -Pl A1-P(yo-e,,)A3=o, 2

(EIxl2 -P)A2=o,

(85)

72 1 -P(yo-ey)A1+(C,12 +C-Pe yal-Py2 'A3=o.

From the second equation we see that buckling in the plane of symmetry is independent of torsion and the corresponding critical load is the same as the Euler load. Buckling in the xz-plane and torsion are interconnected and the corresponding critical loads are obtained by equating to zero the determinant of the first and third of eqs. (85). This gives

r

. 2

(EIvl2 _P~, (yo -

ey ,

- P(y. - ev),

C

(ll2

.+-C-PeyO l -Pr.2 ) .

=o

Using notations (6o) and evaluating the determinant, we obtain for calculating critical loads the following quadratic equation (P1 -

P)[P3r2 - P(r2

-[-

e,R1)] - P 2 (yo

-

ey) 2 = o.

(86)

For very small values of P, the left-hand side of this equation is positive . When P = P 1 , it is negative . Hence, there is a root of eq. (86) smaller than P1, i.e. smaller than the Euler load for buckling in the xz-plane . If e, .= yo, the thrust is applied at the shear-center, and eq. (86) gives (P1 -

P)CP3r 2 - P(r2 + eval)I = O.

May, 1945.1

THEORY

OF B E N I ) I N G , T O R S I O N

359

ANt) B U C K L I N G .

T h e two solutions of this equation are P = P1,

P3

P -

evil t I +--to 2

T h e first one corresponds to flexural buckling in the xz-plane; the second, to purely torsional buckling. Since the left-hand side of eq. (86) is positive when P is small and gradually diminishes as P increases, we conclude t h a t the smallest root of the equation will be increased b y making the parenthesis in the last term vanish; i.e. b y taking e~ = yo. H e n c e the critical load reaches its m a x i m u m value if the t h r u s t is applied at the shear center. If ey = o, eq. (86) is of the same kind as eq. (62) previously obtained for centrally applied thrust. If the cross section has t w o axes of s y m m e t r y , we have Yo = ~1 = O and eq. (86) becomes (P1 -- P)(P3

-

P ) r o ~" -

P2ey2 = o,

or

P2(ro2 -

e~ 2) -

+ P 1 ) r o 2 + P1P,~ro 2 = o.

P(P~

(87)

F r o m this equation, we find P =

P1 -{- Pa -4- ~[(Pa -

P1) 2 + 4PiP3euZ/ro 2

(88)

2(i - eu2/ro 2)

t

It is seen t h a t if eu < ro there are t w o positive roots which a p p r o a c h the values PI and P3 as ev approaches zero. If e~ > ro, there will be one root positive, a n o t h e r negative; the later indicating t h a t for larger eccentricities, the b a r m a y buckle under the action of eccentric tension. W h e n ro = ev, we see from eq. (87) t h a t one root is P=

P1P~ Pa + P3 ;

the o t h e r becomes infinity. W e discussed the case of a b a r with simply s u p p o r t e d ends. ends are built-in, the end conditions are u = Vrv == ~pP ~ == O, o,}

Ut =

for

z = o

and

z = 1.

W e will satisfy these conditions b y t a k i n g u = A1

I -- c o s ~ -

,

v = A2

I

--

COS~-

,

If the

360

IJ. t:. 1.

STEPHEN 1O. TIMOSHENKO.

Substituting in eqs. (83), we obtain, for calculation critical loads, an equation similar to that already derived for simply supported ends. The difference is only t h a t 4zc2/F appears instead of ~r2/l2 which we had before. Let us consider now the case when the bar is submitted to the action of pure bending. Substituting P = o in eqs. (82), we obtain E l y u ~v + M I ~ "

=

o,

EI~v ~v + M2~" = o,

(89)

C1~oIv - (C + M1/31 - Md~2)¢" + M ~ u " + M 2 v " = o. T a k i n g for u, v, and ~o, the trigonometric expression which we used before, the equation for calculating critical values of bending m o m e n t s can be readily derived. Of particular interest is the case where the bar has flexural rigidity in one principal plane m a n y times larger than in the other and is bent in the plane of greater rigidity. Assume, for example, t h a t the yz-plane is the plane of greater rigidity and t h a t the bar is bent in this plane by couples M1. To determine the critical value of M1 at which lateral buckling occurs, we use the first and last of eqs. (89). Assuming that the ends are simply supported, we substitute in these equations u = Alsin

l

¢,=

A3sin

1 '

Ms

o.

and obtain 71.2 EIy-fi A1-

M1At+

Cl-f+

M1A3 = o,

)

C+MI~I

A3 = o.

E q u a t i n g to zero the d e t e r m i n a n t of these equations and using notations (60), we obtain, for calculating the critical values of M,, the equation M12 - - P I D 1 M 1 -P2P3ro ~ = o, from which (M1)cr =

Pfl ± x/P'2/3'2+ --

P,P3ro 2.

(90)

4

In the particular case of a cross section with two axes of symmetry,/~, vanishes and we obtain (M,)~,. - ~

= 7

.EI~,

C~ g + C

)

.

(9I)

In the case of point symmetry, as in Fig. 32a, again Hi --- o, and eq. (9I) holds.

May, ~945.[

THEORY OF BENDING, TORSION AND [{U('KLIN(;.

,~6I

The same equation holds also in the case of cross sections with one axis of s y m m e t r y if the bending couple is acting in the plane perpendicular to the s y m m e t r y axis as in Fig. 32b, since 81 vanishes also in this case. If the bending couple M1 acts in the plane of symmetry, as in Fig. 32c, ~1 does not vanish, and eq. (9o) must be used for the calculation of (M1)cT. It was assumed in the above discussion that EL, is small in comparison with EIx. [f C and C1 are also small, buckling will occur at small values of M~, i.e. at small bending stresses in tile bar.

C

(e) FIG. ,32.

If EIx is of the same order as EI~,, lateral buckling will occur at small stresses only if C and C1 are very small. This condition may be fulfilled if the cross section is like that shown in Fig. 23. C1 vanishes in this case and C is very small if the flange thickness is small. We have discussed here bending of a bar by couples applied at the ends. Only in this ca~e are the normal stresses (8o) independent of z so that we obtain eqs. (82) with constant coefficients. If the bar is bent by lateral load, the bending stresses vary with z and, instead of eqs. (82), we will obtain a system of linear conditions with variable coefficients. The calculation of critical values of lateral load then becomes more involved. Several cases of this kind of instability have been discussed by the writer in his previously mentioned paper.* * Loc. cit., J. F. I., Apr. I945, p. 254.

See also "Theory of Elastic Stability," p. 239.