Coupled Bending and Twisting of a Timoshenko Beam

Journal of Sound and Vibration (1977) 50(4), 469-477

COUPLED BENDING AND TWISTING OF A TIMOSHENKO BEAM

R. E. D. BISHOP AND W. G. F%ICE Department of Mechanical Engineering,

University College London, London WC 1 E 7JE, England

(Received 29 March 1976)

Allowance is made for shear deflection and for rotary inertia of a non-uniform beam that executes coupled bending and twisting vibration. Principal modes are found, orthogonality conditions established and modal equations of forced motion derived.

1. INTRODUCTION

When a beam vibrates in flexure, it may have to be treated as a Timoshenko beam; that is to say corrections may have to be made to the familiar Bernoulli-Euler theory to allow for the effects of shear deformation and rotary inertia [l]. Unless the beam is very thin (so the distance between adjacent nodal points is much greater than the depth of the beam) the error incurred if the corrections are not made may be substantial-particularly as regards natural frequencies. The theory is, in particular, of great importance in the analysis of sym- metric vibration of a ship hull.

The purpose of this paper is to show how the correction can also be made in a discussion of coupled bending and torsion. While, to be sure, this objective is essentially that of filling a comparatively minor gap in the existing theory of linear vibration, the matter is very far from trivial. It is well known that the antisymmetric response of ships, to waves and to propeller excitation, cannot be analysed with much confidence and, in that context, the theory to be given is very much to the point.

2. EQUATIONS OF ANTISYMMETRIC MOTION

Figure 1 shows a slice of a beam of open section in which C is the centre of mass of the slice and S is the shear centre. The shear force is represented by V while A4 is the bending

Y(x,

470 R. E. D. BISHOP AND W. G. PRICE

moment. The axes OXYZ are stationary, the beam occupying the region 0 < x 6 I when it is in its equilibrium condition with its plane of symmetry coincident with the plane OXZ. For the sake of definiteness we shall assume that the ends of the beam are both free. Other ideal end conditions may be used, as can readily be established.

The equation governing motion parallel to 0 Y is

p(x) &(x, t) = V(x, t) + Y(x, t),

where p(x) is the mass per unit length, u,(x, r) is the deflection of C and Y(x, t) is the applied force per unit length. A prime signifies differentiation with respect to x and an overdot means differentiation with respect to t. Since the deflection of S is

0, r) = %(X9 t) + Z(x) 4(x, t),

where i(x) is the distance by which S is below C and 4(x, t) is the angle through which the slice rotates, this equation may be written

p(x) [ti(x, t) - Z(x) 4(x, t)] = V(x, t) + Y(x, t). (1)

Rotation of the slice about the vertical axis through C is such that

Z,(x) 0(x, t) = M(x, t) + V(x, t). (2)

Here Z=(x) is the moment of inertia per unit length and 0(x, t) is that contribution to the slope which is due to bending. Deflection of the beam in the 0 Y direction is due to bending and to shear deformation. It is such that

u(x, t) = 0(x, r) + Y(X, I), (3)

where y(x, t) is the transverse shear strain. These three equations govern transverse bending of the Timoshenko beam. We have

now to supplement them with suitable expressions for V and M. Let

V(x, r) = kAG(x) [y(x, t) + a(x) $J(x, [)I,

Mb, t) = Mx) [wx, t) + B(x) Nx, t>l, (4) where kAG(x) is the shear rigidity and El(x) is the flexural rigidity, while a(x) and j?(x) are distributions of shear and bending damping, respectively.

In general, antisymmetric bending is associated with twisting of the beam. The equations governing the twisting motion are

Z,(x) 6;(x, t) - p(x)f(x) ii(x, t) = T - f(x) Y(x, t) + K(x, t), (5)

where Z,(x) is the moment of inertia per unit length given by

Z,(x) = &(x) + P(X) [ax)lz,

Z, being the moment of inertia about the longitudinal axis through C. The quantity K(x, t) is the applied moment per unit length and

T = C(x) $(x, t) - {C,(x) @(cc, t)} + f(x) &(x, t). (6)

In this expression, C(x) is the torsional stiffness, C,(x) is the torsional stiffness due to warping of the cross-section (see reference [2]) and T(x) represents damping of torsional motion.

COUPLED BJZNJXNG AND TWISTING OF A BEAM 471

3. PRINCIPAL MODES OF ANTISYMMETRIC MOTION

When the beam oscillates in I)UCUU with no applied actions and with no damping admitted, the equations governing its bending motion reduce to

/.l(ii - 2;) = V, 0 = e + y = e + V/kAc, Z*B=M+ V, M = EZe.

(We shall take up the rotation equations separately, later on.) Assume that motion in the rth principal mode is such that

u(x, t) = v,(x) sin 0, t, e(x, t) = e,(x) sin W, t, 4(x, t) = &(x) sin 0, t,

M(x, t) = Mr(x) sin 0, t, V(x, t) = V,(x) sin 0, t.

Then the equations of motion give

-/uuf v, + /.L202 & = If:,

u; = e, + V,/kAG,

v, = -44; - z, of e,,

Iu, = Eze;.

(7)

(8)

(9)

(10)

By eliminating M, from the last two equations one finds that

v, = -(Eze;) - z, 0f e,, so

-PO: 0, + pa: 4r = -[(Eze;) + z, ~0: e,].

Multiplying this result by vS(x) and integrating with respect to x gives

0 0 0

and, when the term on the right hand side is evaluated by parts, it gives

The integrated term is nil because the contents of the square brackets are -V, which vanishes at the boundaries x = 0 and x = 1. The integral may again be evaluated by parts to give

and again the integrated term vanishes since the bracketed term is M, which is zero at the two extremities. In other words,

-u:j~v,v.dx+~~~~~,v,bi=-~Eze~:~+~~~z~e~~;~. 0 0 0 0

This result must still hold if the suffices r and s are interchanged, so that

(11)


If the last two equations are subtracted, it is now found that

(w,z - o:,I /H.J, v, dx - 05 /&,v,dx+ CU; j&&v,dx= j EZ(B:v: - 0:v:)dx 0 0 0 0

I 1

-- of J *I,8,v:dx+w:j I, 8, v:dx.

0 0

(13) If equations (Q (9) and (10) are combined it is found that

v: = 0, - [((EZ&) + Z, w: B,}/kAG]. (14)

When this result is differentiated with respect to x, multiplied by EZei and then integrated with respect to x, it gives

I I s EZO; 0; dx = I EZe: 8:dx - 0 0

If the last integration in this equation is performed by parts, it is found to give

I[ (Ezey + z, 0: 8, (Eze;) + z, of 8, kAG kAG 1 (Eze;) dx, of which the integrated term vanishes since M, = 0 when x = 0 and x = I. Therefore

1 I

j - ,!%I; v; dx = j EZe: e: dx + s (zzze:y (Eze;y dx + w2 0 0 kAG I 0 0 When the suffices r and s in this last equation are interchanged it is found that

I 1

I EZe: v: dx = I EZe; 8: dx + s wze:) wze;) dx + o2

LAG s

z, e,(Eze:y dx, 0 0 ?^

kAG 0 0

Subtraction now reveals that

I

I Ez(e: v: - e: f$) dx = w; j 4 wm dx _ o2 4 wm) dx,

0 kAG I

s kAG 0 0

(15)

(16)

(17)

Return now to equation (14), multiply it throughout by 05 Z, 8, and integrate with respect to x. This gives

1 1 O~jzze,v:dx=w: [zzeresdx-of s '(Ezex z, 8, dx w 2 OL I

1: 6 6 _ I s

d x .

(18) 0 i,

kAG kAG 0 0

Interchanging r and s gives

1 1

0: ! zze,v:dx=w: 5 zze,e,dx-a: (19) 0 0

0 0

COUPLED BENDING AND TWISTING OF A BEAM 473

When these last two equations are subtracted they give

s (EZ&)Z, 8, dx kAG 0

If equations (17) and (20) are now added together, they give

jEl(e:.:-s:.:)~+o:jz~s.jdx-o:jl,8,v:dr=(~~-~~)~z~B.B.dr. (21) 0 0 0 0

Comparison of this result with that of equation (13) reveals that I

(O:-~:)jlw.u,dx-o:jIri~~v.~+~jZAv,br=(w:-~~~II,B,B,dx. (22)

0 0 0 0

We next go back to equations (5) and (6) governing rotation. For undamped free vibration in the rth principal mode,

&l, &P - @Jr) = - [C& - (C, &:)I = -Ti (say). (23)

If this equation is multiplied throughout by $J~ and integrated it becomes

w4j(~~,~~-~*b.)dx=-jr:hdx 0 0

and so, by interchanging r and s, one finds that

(24)

0: (~~~~,-~v*~~)dx=-iT~~,~. I 0 0 Subtraction now shows that

(m:-or:)jZ~~,Adr-o:jlliv.~.dx+w:jrriv,9.dr=-jT~),dx+iC~,dr. 0 0 0 0 0

When they are evaluated by parts the integrals on the right-hand side of this last equation become

-ILAlb+j G&b+ I~~&+~&dx. 0 0

The integrated terms vanish because T, and T, are both zero when x = 0 and x = 1. Further- more

~(T,9:-T,~;)br=jfC~:-(C,6:))~~dx-~(C~~-(C~~~)~~~ 0 0 0


This is because c#$, 45 vanish at a free end of the beam since warping is unrestrained. It follows that

(W:-w:)kl,$,8,dx_w:ipiv,~,dx+w:jpiv,b,dx=O. (25) 0 0 0

Equations (22) and (25) may be added together to give

We therefore reach the orthogonality condition 1

I bw Us + 44~4, + h&f4 - P%$J, + Us 4r)ldx = ~shs, (26)

0

where 6,, is the Kronecker delta function. Notice, however, that a special case arises when o,=o=o

A seconiorthogonality condition can now be found. Let

yr = VJkAG = U: - 8,.

It can now be shown by substituting back into equation (26) from equations (11) and (15), and also from equation (24) noting that

that I

I [EZe; e; + kAGy, ys + T, &] dx = O: ars 6,, = c,, d,,. 0

(27)

Explicit expressions for the generalised mass and generalised stiffness corresponding to pS can now be written down. If r = sin the orthogonality equations, these quantities are found to be

and

respectively.

c,, = co, a,, = I (EZei2 + kAGyf + T, c$:) dx, (29) 0

4. RIGID BODY MODES

Although they arise only from the particular choice of end conditions that we adopted, the rigid body modes are of special interest. If the x co-ordinate of the centre of mass of the beam is 2, they are

00(x) = 1, e,(x) = 0 = ye(x) = 4o(x) (for r = 0),

ul(x) = l-G, e,(x) = 0 = y,(x) = c$~(x) (for r = l), X

u2w=o=e2(4=Y2w, 42(x) = 1 (for r = 2). (30)

COUPLED BBNDING AN0 TWISTINGOFABEAM 475

All are scaled to have unit deflection at x = 0. The modal shaper&) does not satisfy equation (8) but the reason is obvious and trivial--equation (8) effectively excludes deflection in just such a mode as this and the exclusion can readily be remedied. The modes corresponding to r = 0 and r = 1 satisfy the orthogonality condition (26) and are therefore mutually orthogonal. But

I

rro2=- [pzdx=ll20#0 (for r,s = 0,2), d

I

al2 = - 5( 1 fl l-: &=a21#0 (for r,s = 1,2). 0 zi (31) In other words the Kronecker delta function in equation (26) must be ignored for these combinations of the modal indices.

5. MODAL ANALYSIS OF FORCED ANTISYMMETRIC OSCILLATION

Antisymmetric distortion of the beam may be expressed as the sum of distortions in the antisymmetric principal modes. That is, we may write

(32)

wherep,(r) is the rth principal co-ordinate. Equation (1) now becomes

If this equation is multiplied throughout by u, and then integrated over the length of the beam it is found that

- %) 0, dx - ,zo j [kAG(y,p, + ay,d,)l us dr = j YU, dx. - 0 0

Integrated by parts, the second integral gives the term

and this vanishes because the quantity in square brackets is the shearing force which vanishes at the ends of the beam. That is to say

We now transfer our attention to equation (2) and treat it in a similar manner, this time multiplying by 6,. Thus


By the same reasoning as before the integrated term can be rejected when the second integral is evaluated by parts-the bending moment is nil at the boundaries-and this leaves

-$of4/ crkAGy,O,dx = 0. 0

Next we turn to equation (5), multiplying it by & and integrating to give

rzo &/(I,& - MaJ&dx -% /]C&P, - (C, 4:>pr + r4:rirlAdx = - 0 0

I I

-

s 2& Y dx + J +S Kdx.

0 0

(34)

If the second integral is evaluated by parts, the integrated term vanishes since the term [C&p, - (C,&)p, + f&J] is equal to zero at both boundaries. We are thus left with the equation

1 I

- jZ& Ydx+ +,Kdx. 5 0 0

(35)

Adding equations (33), (34), and (35) together now gives

%b.j] ~,a, + I,& 4s + I,& 6 - PM a, + d, v,>l dx + 0

+ $ d, / bkAGy,( us - e,) + pzze: e: + rg $:I dx + k0

0

+?gop, 1 [EIe: 0: + kAGy, (v: - 0,) + Tr &I dx = I[ Y(vs - 34 + @sl dx 0 0

and this is true for s = 0, 1, 2, . . That is to say

for s = 0, 1, 2, . . ., where I

a,, = I

akAGy, ys dx, r,, = j l-4; c$; dx. 0 0 0

If the rigid body modes of equations (30) are adopted this gives

a,,& + uZojj2 = Ydx s 0

(for s = 0),

(37)

(for s = l),

C!OUF'LELlBENDINGANDTWISTINGOFABEAM 477

u&o -t a& + a&, = j (K - Z Y) dx (for s = 2), 0

1

4d%+ 2 c&s + B,,+rJI4+ c**p*= I 1 Y(v* - W + ~44 dx (for s > 2). (38) r-3 0

6. CONCLUSIONS

The simple Bernoulli-Euler theory of beam flexure may be adapted to the theory of coupled bending and twisting vibrations. We have shown that, in this more complex motion, the bending distortions may be those admitted in the theory of a Timoshenko beam. In practice, the effect of this change is often fairly minor so far as the shapes of those principal modes of lowest order are concerned. But the effect on the corresponding natural frequencies is often substantial. (Notice that the Timoshenko beam necessarily has lower natural frequencies than the corresponding Bernoulli-Euler beam since its bending stiffness is reduced by the allowance for shear deflection and its inertia is increased by the rotary inertia.)

If the cross-section of the beam is box-like, the foregoing theory takes a degenerate form. The points C and Sin Figure 1 then coincide, and so Z = 0. But if i = 0, equations (1) and (5) are independent of each other. Variations of v&t), e&t) and r(x,t) are independent of those of 4(x,t). Thus the last of equations (32) should now be replaced by some such expression as

4(x, 0 = z 41(r) 4(x), I-0

since equations (36) and (38) separate out into two distinct sets of equations. At first sight it may seem surprising that, despite the added complications of shearing

deflection and rotary inertia, the theory produces exactly the same equations as those arrived at by using the Bernoulli-Euler approach. Thus generalised masses, damping coefficients, stiffnesses, displacements and forces at the principal co-ordinates all perform the same functions as before (though of course the expressions for them are different). In fact, however, it would be surprising if this were not so; for both theories conform to the general pattern of linear vibration theory which may be developed by the use of Lagranges equation [3].

1.

2.

3.

REFERENCES

S. P. TIMOSHENKO, D. H. YOUNG and W. WEAVER 1974 Vibration Problems in Engineering. New York: Wiley, fourth edition. S. P. TIMOSHENKO 1945 Journal of the Frunklin Institute 239,201-219,249-268,343-361. Theory of bending, torsion and buckling of thin-walled members of open cross-section. R. E. D. BISHOP and D. C. JOHNSON 1960 27re Mechanics of Vibration. Cambridge University Press.

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Coupled Bending and Twisting of a Timoshenko Beam