The Brazier Effect.pdf

104

On

theFlexure of Thin Cylindrical Shells and other “ Thin" Sections. By L. G. B razier , B.Sc.

(Communicated by E. V. Southwell, F.R.S.—Received May 28, 1927.) I t is a generally appreciated deduction from St. Venant’s solution of the flexure problem that a beam in which the material is disposed at a distance from the neutral axis is superior to the solid section in economy of material. St. Venant’s solution, however, suggests th at this advantage increases without limit as the thickness of the material is reduced and the distance from the neutral axis is increased. I t has, of course, been generally realised th at this conclusion is not supported by ordinary engineering practice, and recent experience in the use of high tensile steels and problems of aircraft structure have emphasised the desirability of a further examination of the flexure problem. St. Venant’s solutions are obtained when the equations of equilibrium of an isotropic elastic solid are made linear by the neglect of terms of higher orders than the first: and by Kirchhoff’s theorem of determinancy these solutions may then be considered unique and stable. To attack problems of stability it is necessary, as is shown by R. V. Southwell* in his *General Theory of Elastic Stability,’ to include some of the second order effects. I t is, in fact, only when these become considerable that Kirchhoff’s theorem fails and instability becomes possible. By this general treatment various classes of instability are obtained or indicated, but the only ones susceptible to analysis or of practical interest (on account of the “ elastic limit ” which is a feature of all practical materials) are those in which at the moment of instability the strains are still small. Bryantf has shown that this will only occur, as in the case of thin rods and shells, when one dimension of the body is small compared with others. In a similar way we may expect that when one dimension of the cross section of the body is small compared with others it will be necessary to include second order terms in problems purely of flexural equilibrium. This is really evident in the case of the flexure problem from St. Venant’s explicit description of the stress at any element of the cross section as a function of the initial position of the element. The longitudinal stress, for example, is prescribed as directly proportional to the initial distance of the element from the line of centroids or * ‘ Phil. Trans. Roy. Soc.,’ A, vol. 213, p. 187. f ‘ Camb. Phil. Soc. Proc.,’ vol. 6, p. 199 (1888).

Flexure of Thin Cylindrical Shells.

105

neutral axis. If, however, one dimension of the cross section is small compared with others, then even while the strains remain everywhere small, large displace­ ments over the cross section may occur. I t is then clearly inaccurate to assume th at the stress in the element is a linear function of its initial position. The accurate description of the stress in the element as a function of its resultant position after including displacements due to strain, corresponds to the intro­ duction in the problem of higher terms than the first. The present paper illustrates problems of this class. I t indicates the necessary corrections to St. Yenant’s theory of flexure for cases in which some dimensions of the cross section are small compared with others. The problem is not attacked by a direct introduction of higher terms in the equations of equili­ brium, but by the variational method and the general dynamical theorem th at a position of equilibrium will be a position of minimum energy. The body is supposed strained in the manner described by St. Venant. I t is then allowed to undergo a system of displacements, and the system determined by the condition th at the final potential energy is a minimum. The system of dis­ placements is directed so th at the applied forces do no work, and the condition then is that the strain energy of the body is a minimum. I t is interesting to find th a t this treatm ent suggests a form of instability under flexure which does not appear to have been treated before. St. Venant’s solu­ tion gives a linear relation between the bending moment exerted on the beam and the resulting curvature of its central line. The effect of second or higher order terms will necessarily be to depress the bending moment progressively, below the value given by St. Venant’s relation, in the manner shown in fig. 1.

ST. VENANTS solution

Cu r v a t u r e

F ig. 1.

106

L. G. Brazier.

The solution obtained in this paper shows a point A a t which the bending moment passes through a maximum. If the bending moment is increased above this value the beam must collapse. That is to say th at the point A is a point of instability. This form of instability is characterised in comparison with the generally accepted types of instability by absence of a point of bifurcation at which an extensional and an inextensional system of displacements under the given load system become possible.* In this instability form there is no inextensional system and no bifurcation point, but a progressive reduction of the appropriate elastic rigidity through a point at which the external load system must be a maximum. In the present paper terms only of the second order have been introduced so that it will not in general give an exact expression for the maximum bending moment, but it does indicate the existence of this maximum and should give it quantitatively for sufficiently “ thin ” sections. TheThin Circular Tube. The method can conveniently be illustrated by the case of a thin circular tube subjected to flexure. To eliminate the difficulties of end effects we suppose th at a long tube is bent into a circle of large radius and joined on itself. We then consider the relation between the bending moment transmitted by the tube at any section and the curvature of the tube. We consider, therefore, a cross section of a toroidal shell. The median radius of the cross section is r, and the thickness t. We describe a moving system of co­ ordinates x, and of displacements u, v, w, as shown in fig. 2, the origin moving round the median circumference, and the axis of z remaining directed towards the centre and making an angle 0 with the F ig. 2. plane of symmetry of the toroid. The axis oi y remains in the cross section. The plane 0 = 0 contains the centre of the toroid, and the curvature of the toroid is c. * Curves similar to that of fig. 1 were given by Southwell in his general remarks on the collapse of struts formed of materials having finite limits of elasticity; but his curves do not pass through the origin since they contemplated initial curvature due to inaccuracies of workmanship or loading.

Flexure o f Thin Cylindrical Shells.

107

According to St. Y enant’s solution of tlie flexure problem the displacements in the plane of the cross section will then be :— w0 = - err2cos 0 2

(1 )

- o r 2sin 0.

( 2)

A

By expressing c in term s of Young’s modulus, the maximum stress allowable and the radius we may note th a t for ordinary materials, including wood and metals, the maximum displacement will be of the order of

w'

A

1000 We may conveniently imagine th a t when the tube was straight it was filled w ith a solidifying compound. When set this compound has an intelligent par­ tiality for St. Y enant’s solutions, and while presenting no opposition to dis­ placements given by his solutions, offers a complete rigidity against all other displacements. When the tube is bent to the toroid the cross section will therefore take up the form given by (1) and (2). If the compound is now melted the cross section will be free to take up a further system of displacements v' w'.We determine v' th a t the tube will pass to a position of minimum strain energy. As usual in problems of this type we suppose th a t the system v' w' is inextensional.* As w0 v0 have been seen to be very small the condition th a t i f are inextensional is t h a t :—

(3)

The total displacements are w =

+ w0

and

V

= v' + v0,

(4)

ca

(5)

and using (3) we obtain dv0 _ dQ "

dv

1

^ 1
of

= —

r2 cos 0

car2cos 0.

(6)

* Vide Basset, “ Extension and Flexure of Cylindrical and Spherical Thin Elastic Shells,” 4 Phil. Trans.,’ A, vol. 190, p. 433 (1890), and Rayleigh, * Roy. Soc. Proo.,’ vol. 45, p. 105 (1888). The justification for this assumption is essentially that the energy absorbed by any extensional displacement (compared with a flexural displacement) is mathematically large and therefore precluded.

L. G. Brazier.

108

The change of lateral curvature (i.e., in the plane cross section is 1

^ = vAl¥ + W ’

and using (6) we obtain

/ \ _ 11 fdzv , /7ai d*A r2W62 ^ dQJ

= 0) at a point on the ld 2w',
(7> (8)

The longitudinal strain at an element is proportional to the resultant distance from the neutral axis. This is d = (r — w) cos 0 — v sin 0. (9) From (9) and (8) we obtain (after neglecting products and squares of small quantities) for the total strain energy per unit length of the toroid the expres­ sion idzv , dv'\2 E H 2’ __________ ts o (1 - o 2)12r3\d& ^~ d + c2rt | r2— 2

r-f- r 2co cos 0^| cos2 0 — rv sin 20d0 . ( 10)

If this is to be a minimum then according to the calculus of variations v must satisfy the equation d% , „ d^v_ . d2v 18cV (1 — c2) sin 20. ( 11) dW ^ d ¥ ^ W ~ t2 The solution of this is d2v (A + B0) cos 0 -f- (H + E0) sin 0 ----- sin 20, ( 12) . dP 9 where N = 18c2r5 ( 1 a 2) . (13) —

Considerations of symmetry and of continuity of the circumference require that the constants, B, E and H, and the first of the two arising in the integration of (12) vanish. The second of these two and the constant A represent a rigid body displacement which can be disregarded. We then have v = — sin 20,

and

dv N d 9 = 18COs2eetc., cos 20 — 1 - oc r2cos 0.

(14)

(15)

Flexure o f Thin Cylindrical Shells.

109

I t will be observed th a t the term containing <7 is very small. I t represents th e St. Venant displacements (1) (2) plus a rigid body motion which is im­ material. If these expressions are substituted in (10) and the integrations effected, we obtain u = | TC^ {

i - | r V -(l-r ^

},

as)

and the couple transm itted a t a section of the toroid is given by _ dU M= —- =_ E —~rH 12c _— W dc 2 1 This is a maximum when

q - a 2))

7

(17) •

t
(18)

a t which point the couple is — M3

2a/ 2 0

’ mrt2 E *2% V i

(19)

If (18) is substituted in (14) we obtain for the maximum radial deflection at this instant A 2 ( 20 ) W= 9 f' so th a t the approximations based on the smallness of w and v are justified to this extent. The form of the cross section a t this point is shown a t fig. 3. This

F ig .

3.—Deformation of thin tube immediately before buckling.

figure makes it clear what is happening. The longitudinal compression on the inner side of a beam and the longitudinal tension on the outer side, both have a component directed towards the centre of the tube and tending to flatten it, in much the same way as the earth pressure tends to flatten a circular tunnel. These pressures are actually flattening the circular cross-section into the quasi­ oval form shown in fig. 3. That this is the physical interpretation of the equations

110

L. G. Brazier.

is clear from the following argument. The components of the longitudinal ten­ sions and compressions directed towards the median circumference of the toroid give rise at all points of the cross-section to a pressure per unit area of amount p = Ec2r£ cos 0.

(21)

And if the equilibrium of an infinitely long cylindrical shell under a vertical pressure px —Ec2r£ cos 0 is examined, it is found th at the cylinder is distorted in exactly the manner defined by equations (14) and (15) without the final term of (15) which is due to St. Venant’s flexure distortion. The analysis has dealt with an endless beam strained in such a way th at every section must undergo the same distortion. But we may reasonably expect the results to apply to the ordinary tubular beam provided it is sufficiently long to minimise the end effects. Then the relation between the terminal couples and the curvature of the beam will be equation (17) and not St. Venant’s equation M = E tt^ c.

(22)

So long as the couple is below the value (19) there is an equilibrium position, though not a unique one. If the couple exceeds (19) the beam must necessarily collapse, even with a material of infinite elastic proportionality. Terms of the second order only have been introduced, and it is necessary to consider if higher order terms are likely to be important. We have found th at all elements of the tube are effectively subjected to a pressure tending to flatten the tube to a quasi-oval section. Elements at the top of the compression side, therefore, are in a similar condition to elements of a cylindrical shell subjected to end compression and hydraulic pressure. Southwell* deals with the problem as an example of his general theory, and finds th at the tube may collapse into lobed forms of distortion which have been experimentally reproduced. In a corre­ sponding way, therefore, a lobed deformation may occur on the compression side of the bent tube and approximate calculation suggests th at this may occur before the instability point (19) is reached. This has been experimentally confirmed. The relation between terminal couples and average curvature over the length was observed for long thin tubular beams made of celluloid and subjected to pure terminal couples. These relations are shown in fig. 4. In these experiments as the terminal couple was increased the cross section of the tube was observed to take up the quasi-oval * “ On the General Theory of Elastic Stability,” ‘ Phil. Trans.,’ A, vol. 213, pp. 187-244.

flexure o f Thin Cylindrical Shells.

Ill

form, and finally collapse occurred when a lobed deformation formed on the compression side. In one case the lobe occurred just as the tube was passing C a lc u la te d in sta b ility m om en t ------— T a n g en t fo r _ c a lc u la tio n o f E T h e o r e tic a l c u r v e ~ fo r tu b e . W ith n o e n d c o n s tr a in ts

F ig. 4.—Curves showing moment-curvature characteristics of celluloid tubes. AM = Ar/4-87 — 0*005 a 8. A c = 1 * 8 85.10“ 4 AS. c = mean curvature over length of tube.

. through the critical point A of fig. 1. fig. 5.

A photograph of a lobed failure is shown at

F ig. 5.

L. G. Brazier.

112

Thin Sections other than Circular. Geometrical sections such as the ellipse can clearly be treated by the same method that has been used for the circular section. Considerable interest, however, attaches to irregular sections such as those used by aircraft engineers A typical cross-section is shown in fig. 6. The variational method can be applied to examine the equilibrium of the flange* when the beam is bent by terminal couples. Keferring to fig. 7 we

F ig . 6.

describe a moving system of co-ordinates y, z, and of displacements u, v, w. The origin moves along the median plane of the flange. The axis of x remains parallel to the axis of the spar and the axis of z is normal to the flange at the origin. We define t as the thickness of the flange, d as the initial distance of the origin from the neutral axis, the angle of the tangent at the origin to the horizontal, and p the initial radius of curvature of the section (in the plane x — Flange is used throughout in the sense that it is used by the beam engineer. is to say, a beam has essentially two components, a “ flange ” and a “ web.”

That

Flexure of Thin Cylindrical Shells.

113

const.) a t the origin. The independent variable s is measured along the flange. (See fig. 7.) We suppose th at the spar is subjected to a constant axial curvature c and th a t the stress at any point is proportional to the distance from the neutral axis. As before, we then allow the flange to take up an inextensible system of deforma­ tion w, v. The condition of inextension is dv_w (23) ds p Using (23) we obtain for the change of curvature at a point Act, = pe"' + 2pV ' +

+ !)V -X » . p/ p2

V

(24)

where the dashes denote the operation of differentiation with respect to s. For differentiations above the third order Roman numerals will be used. The resultant distance from the neutral axis is d — w cos 4* + ®sin 4*, (25) and we thus obtain after neglecting squares and products of the small quantities v and w with respect to d for the total strain energy between sx and s2 per unit length along the spar the expression Et3 j l A 2 ds jp v '" + 2 p V '+ (p' .12 (1 p2vj JSj, —

+ Etc2 [ {d2 + 2

cos(26) 4} ds .

dvsin 4 — 2

As before if this is to be a minimum, v must satisfy a sixth order linear differential equation of which the coefficients are the following known functions of the independent variable :— ao — P2> ai = 6pp', a2 = 10pp" + 2 + 4p'2,

a3 = lOpp"' + 6p'p", cc4 = 5pp* - 3

P

a (6_») being the coefficient VOL. CXVI.— A .

Si + 4p'p"'+ 9 + i , Ap

9

<£v dsn ‘ I

114

Flexure of Thin Cylindrical Shells.

The second member of the equation is the known function of the independent variable —— —- {d sin

+ dp cos

}.

(28)

In the solution of this equation the integration constants can be determined either from the limit terms given by the calculus of variations for the free ends of the flange, or as the physical significance of the equations and terms is known by inserting the conditions th at the shear and bending moments at these points are zero. The two methods give identical results. This equation, however, will not ordinarily be conveniently integrable. We observe, however, that if we imagine all portions of the flange to be subjected to a vertical pressure directed towards the neutral plane of amount p

—Ec2 dt,

(29)

and investigate the equilibrium of the flange under this pressure we arrive at exactly the same equation. Hence, although the equation cannot be solved, the engineer may form a conception of the stresses to which the flange is sub­ jected by supposing it subjected to the vertical pressure (29). In some cases this will lead to definite results. Thus, when there is only a single web attached to the centre of the flange, each half of the flange will have to resist the load system (29) as a cantilever. The moments arising from (29) can be obtained by graphic integration and the flange stresses accurately determined. Again, the integral of (29) taken over the flange has to be resisted by the web acting as a s tr u t: a form of load for which it is not normally designed. The magnitude of the load is easily obtained and the suitability of the web examined. In other cases sufficient information may be obtained from tests on rubber models subjected to the load system (29) instead of a more complicated full scale test on the actual spar. The author s acknowledgments are due to the Air Ministry for permission to publish the results contained in this paper, to the Steel Wing Company for the loan of fig. 6, to Dr. A. A. Griffith of the Royal Aircraft Establishment, in whose laboratory the experimental work was carried out, and to R. V. Southwell, F.R.S., for his detailed criticisms and advice.