castigliano’s principle of minimum...

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Castigliano’s Principle of Minimum Strain-EnergyBy R. V. So u t h w e l l , F.R.S.

{Received October 30, 1935)

In t r o d u c t io n a n d Sum m ary

1—Two important theorems can be established regarding the equilibrium of a body which obeys Hooke’s law. In the first we assume that specified displacements are imposed (by an application of suitable forces) on particular points of the body, and we compare configurations which all satisfy this condition, but of which all but one would require additional forces (some applied at other points) to maintain them in equilibrium.* (In all the strains are “ compatible ” ,— they are expressible in termsof single-valued displacements u, v, w; so the comparison is between configurations which could be obtained one from another by imposing single-valued displacements at all points except those of which the displacements are specified. Mathematically, we vary u, v, independently (except at these points) without imposing the requirement that every part of the body must be in equilibrium.) It can be shown that U, the total elastic strain-energy, has its smallest value in the equilibrium configuration: this is the First Theorem of Minimum Strain-Energy.f

As a special example of maintained displacements we have the “ self- strained ” body. Self-straining may arise from various causes,—e.g. stresses produced by casting, rolling or other manufacturing processes, or by differences in temperature at different parts of the same body; but it can always (in imagination) be relieved by making suitable cuts in the body, and conversely, the original state of self-strain can be restored by applying such forces to the cut body as are required to neutralize the “ gaps ” produced by cutting. Let A and B be two points, on opposite sides of a gap, which are brought into coincidence when the gap is neutralized: then the forces entailed at A and B may be grouped together as a generalized “ force ” , and the relative displacement of A and B may be regarded as the “ corresponding displacement. Since the gaps

* It is not, of course, postulated that equilibrium is in fact maintained. The configurations may, for example, be instantaneous configurations o f a vibrating body.

t Cf. Love, “ Mathematical Theory of Elasticity ”, § 119.$ Cf. Rayleigh, “ Theory of Sound”, vol. 1, § 74. The argument of this paragraph

is elaborated in §§ 83-85 of my “ Introduction to the Theory o f Elasticity ” (Clarendon Press, 1936).

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Castigliano’s Principle o f Minimum Strain-Energy 5

have definite magnitudes, definite displacements are required to close them : so, by a generalization of the notions of force and displacement, we may regard a self-strained body as resulting from the imposition of definite displacements upon a body which initially was in a state of ease. Compatible strains in the “ cut ” body entail non-compatible strains in the resulting (self-strained) body.

2— In the second theorem we compare configurations which all satisfythe conditions of equilibrium for a specified system of external forces, but which differ in that only one has resulted from an initial state of ease. (That is to say, we no longer require, as in the first theorem, that the strains shall be “ compatible ” , but we do, on the other hand, require that the stresses shall be “ balanced ” . Mathematically, in the second theorem the stresses are independently varied, subject to the restrictions that the equations of equilibrium must be satisfied at every point and that neither the surface tractions nor the body forces—which are the specified force- system—may be varied.) It can be shown that U takes its least value in this configuration: that is to say, the stress-distribution resulting from given forces, applied to a body initially in a state o f ease, can be deduceafrom the conditions o f equilibrium combined with the conditions for a minimum value ofU.

This second theorem is commonly known as Castigliano’s “ Theorem of Least Work ” . The title seems misleading (since it is clear that no limit is imposed upon the amount of work which could be done on a body by given forces), and the theorem is often stated in a needlessly restricted form. Accordingly I have ventured to give the title “ Castigliano’s Principle of Minimum Strain-Energy ” to the theorem as stated here.

3— Both theorems can be based on the assertion that Ui is a positive quantity, independent ofU 0, in the expression

u = u0 + ulf (1)where U 0 stands for the strain-energy which results from the imposition of specified displacements (with self-straining included as a special case), Ux stands for the addition made to U 0 by the imposition of specified forces, and U, as before, stands for the total elastic strain-energy. A mathematical proof of this assertion can be formulated,* but the physical argument is equally cogent:—The Principle of Superposition (an immediate consequence of Hooke’s law) asserts that a given system of forces, slowly appliedf to a given elastic body, will entail the same

* Cf., e.g., my “ Introduction ”, §§ 81-4.t So that equilibrium is maintained at every stage, and vibrations are not excited.



6 R. V. Southwell

definite displacement of every point, and so will do the same positive* amount of work, whether the body in question be initially strained or in a state of ease. Conservation of energy requires that the work so done shall be stored in the body as an addition (Ux) to the elastic strain-energy, and thus leads to the conclusion stated.

When the initial strain is due to the imposition of definite displacements at particular points, we must imagine that constraints are operative at the points concerned and impose whatever forces are required to maintain the displacements. Then, when the given system of forces is applied, its effects will be modified by the circumstance that certain points are prevented from moving further; but the Principle of Superposition still permits us to assert that Uj will be independent of the amount of the initial strain-energy U 0, and hence, when U 0 is due to self-straining, it has no influence on the magnitude ofDj.

Since U 0 and Ux are essentially positive quantities, we can infer from (1), on the one hand, that

U > U 0, )and on the other, that r (2)

U > Ux. )

The first of (2) states that, U 0 being given, U has its least value when Ux is zero: this is the First Theorem of Minimum Strain-Energy. The second of (2) states that, Ux being given, U has its least value when U 0 is zero : this is the second theorem.

4—Thus a simple argument from physical principles permits a very important generalization. Both theorems hold in respect of any body which obeys Hooke’s law, without restriction on shape or size; so we may regard it as a general property of materials which obey Hooke’s law, that these tend always to assume (within the imposed conditions) a configuration of minimum strain-energy.

Twelve years agof I indicated the utility of this concept by showing that it provided justification for a principle formerly based on intuition,— namely, Saint-Venant’s well-known “ principle of the elastic equivalence of statically equipollent systems of load ” 4 I showed, further, that it gives additional significance to the exact solutions for flexure, shear and torsion in a uniform prism which Saint-Venant obtained by his cele

* Mechanically unstable materials are of course excluded.t ‘ Phil. Mag.’, vol. 45, p. 193 (1923), and ‘ Aero. Res. Cttee. R. and M.’, No. 821

(1922).t Cf. Love, “ Mathematical Theory of Elasticity ”, Introduction p. 21 and §89;

also § 12 of this paper.



brated “ semi-inverse” method of attack:* those solutions are “ least- energy solutions ” for the actions considered, and as such they may be regarded as standard solutions to which all particular solutions must approximate, since all are governed by the requirement of minimum energy storage.

This paper develops the same ideas, starting from the principle stated (in italics) in § 2. Part I deduces from that principle the well-known “ conditions of compatibility for strain ” , which hitherto (so far as I am aware) have been obtained by purely kinematical reasoning.! They are the conditions which must be satisfied by the six strain-components in order that these may be expressible in terms of three single-valued components of displacement u, v, w : as such they are the conditions for zero self-strain, and hence (according to the principle) they must be obtainable as the conditions for a minimum value of U, when the variations of the stress-components are restricted by the stress-equations of equilibrium. In Part I they are actually obtained in this way.

In Part II it is shown that the same principle enables us to replace the somewhat tentative approach of Saint-VenantJ by a more direct attack on the problems which he solved,—and on other problems of equilibrium. Three examples are treated, namely, Saint-Venant’s problems of uniform flexure and torsion and a slightly harder problem in plane stress—the uniform flexure of a flat circular ring: in each the known expressions for the stress-components are derived as conditions which must be satisfied in order that U may take its least value consistent with the maintenance of equilibrium throughout the body in question. The same method can (in theory) be applied to all problems of equilibrium, although it will only occasionally provide the shortest route to a solution. It reduces them, in effect, to problems in the Calculus of Variations.

P art I—T he C o n d itio n s of C om patibility for St r a in D ed u c ed as C onsequences of the P r in c iple of M inim um St r a in -E n er g y

5—In the theory of strain, six apparently independent components of strain are introduced, but these can be expressed as differential coefficients of only three independent quantities,—the three perpendicular com-

* Love, op. cit., p. 19.t Cf. Love, op. cit., § 17 and footnote; also Todhunter and Pearson, “ History o f

Elasticity ”, vol. 2, §§ 112 and 190 (c).J “ The semi-inverse method o f solution consists in imposing a restriction on the

generality of the stress within the solid in accordance with a result based on some theory not derived from a solution of the general equations.” (Love, op. cit., 1st ed., §82.)

Castigliano's Principle o f Minimum Strain-Energy 7



8 R. V. Southwell

ponents of displacement. Therefore it is evident that the strain-components are related, and by elimination of the displacements it can be shown that the relations in question are:

d2evy +d2ezz _ d2eyz

dz2 dy2 dycz

d2ezz+ d2Cxx d2eV c zx

dx2 dz2 dz dx ’

U c X X +d2evv _ d2exv

dy2 dx2 dxdy

2 d2exx _ d ,dy dz dx

f df'yz{ dx

, de*v\ )^ dz r

2 d2evv _ _5_ 1 deyz _ Zezx d&xv\ > (3)dz dx dy \ dx dz ) ’

2 d2ezz d l deVZ | fezx _ dexy \dxdy dz \ ax dz i* ’

Now according to the concept of “ minimum energy storage ” developed in § 2, these relations, being the conditions of zero self-straining, should be the conditions for a minimum value of the total strain-energy U, as applied to stress-distributions which already satisfy the conditions of equilibrium. We shall test this prediction by expressing the variation 8U in terms of the variations of the stress-components, ensuring satisfaction of the conditions of equilibrium by expressing the stress-components in terms of the stress-functions introduced by J. C. Maxwell and by G. Morera.* (In this way we ensure that the stress-system is “ balanced” (without alteration of the body forces if operative), but we do not require the associated strain-system to be “ compatible ” (§ 1): the conditions that it shall be compatible will appear as the conditions for a stationary value of U.)

6—Without restriction to isotropic or homogeneous materials, we may assert that the strain-energy per unit volume is a homogeneous quadratic function of the six stress-components X*, ..., Y„ ..., etc. So the variation SU is expressed by

SXj. ... etc. J dx dy ,

= + ... + eyz$Yz -j- ...] dy dz, (4)

by Castigliano’s first theorem, since exx, ... (the strains involved by our stress-system) are the “ displacements corresponding w ith ” Xx, ..., etc.f This expression is replaced by

... | dx dy dz, (5)

* Cf. Love, op. cit., § 56 (e).t tt is at this point that appeal is made to Hooke’s law. The work equation

d\Jldexx = X x would be satisfied by any elastic material, but the converse relation dU jdXx — exx (Castigliano’s first theorem) is satisfied only when the relation between

X x and exx is one of direct proportionality.




when we substitute for the three stress-components in terms of Maxwell’s stress-functions xi, X2, Xs 5 and by

8U = i 2 ( exa a2H i , „ s2H 2 ^ „ 92H 3\" » cyy

eVz

dydz

d_dx dx

dzdx

+

+ e zz dxdy)

. 08^2 I ^ H ^dy dz

d /084*i 08^2 0841ZX -- ’dy \ dx dy +

_ d/SS^ 8S<];2xv d z \ dx + dy

dx dy dz,

when we use M orera’s stress-functions ^1, 42> s*7—Applying Green’s transformation to the integral in (5), we obtain

•SU = [ f f - ° V ) +J J J L * \ dz2dy2 dydz!

+ i LI dy ' (2 me

dydz/

+ (similar terms in 8x2, SX3)

- nevz) + (2 ne — meyz) +

dx dy dz +

+ h l [ m ^ + n ^ 2m - In ^ ) | +

+ (similar terms in Sx2, 8x3) dS, (7)

and from the volume integral (since each of the variations Sxi, 8x2, 8x3 can take any value at any point in the volume of the solid body) we deduce that the strain-components, for minimum strain-energy (SU = 0), must satisfy the relations

d2emi cPe = d2evz I dz2 dy2 dydz’ l (8)

. . . etc. J

These are the first three of the conditions of compatibility (3).From (6), in the same way, we obtain

•SU H i 2 dydzA (_dx \ dx

I dfzx 1 1+ ^ 7 + dz)> +

+ (similar terms in 8^2, H 3 dx dy dz +



10 R. V. Southwell

and from the volume integral in this expression (since §<K, &<]z2» H a can have any values at any point in the volume of the solid body) we deduce that the strain-components, for minimum strain-energy, must satisfy the further relations

&exxdydz — (■dx \

0 . 0 , 0 —x ev z + f y e*x + e*v

. . . etc.

These are the last three of the conditions of compatibility (3).8—At first sight it would appear that further restrictions on the strain-

components can be deduced from the surface integrals in (7) and (9). But when (8) and (10) are satisfied the strain-components exx, ..., etc. can (as the usual demonstration shows) be expressed in terms of three singlevalued functions u, v, w * and on this understanding no further condition is imposed by the surface integrals in (7) or (9). In that of (7) for example, the terms of the integrand which involve may be written in the form

+ 2v r 0 0 im — n r 0 01m — /z —dz dz dzasxi0y (i>

* A direct proof may be given as follow s: such that ■du

dx

we can always find forms for u, v, w,dv _ 8w

eT ~ ***’ If, ~eyv>and equations (3), when these substitutions are made, require that

IZ dy dw dz dy

dw du „a? + Tz - F”

du Sv ^ + Tx ~ F”(ii>

where Fl5 F 2, F 3 have the forms f x(x, y) + / 2 (z, x), f 3 (y, z) + / 4 (x, y ) , f 5 (z, x) + fe (y, z) respectively, and

A = A* dA ^ dAdx dy ’ dy dz ’ dz dx '

By adding to u, v, w terms of the forms/ (y, z ) , f ( z , x ) , f (x, y) respectively, we can make F1? F 2, F 3 zero in (ii) without affecting the expressions (i). Hence, if the con-



Castigliano’s Principle o f Minimum Strain-Energy

and the terms in the first line of this equation vanish on integration, by Stokes’s theorem.* Treating similarly the terms of the integrand which involve Sx2, 0X3, we can now express the surface integral of (7) in the form

f f L r -0 01 dh i _ w 0Ml _ — dM __ dh i |J J r l dz dyi dz dz 0 J 3y +

- f - w

d- u

d , 0- n dx' 0 0 ' l Y y - m Tx-

0SXsdx

d dn dx dz_

dh

l a*m—CX-

dzdh s

-2 +

dx}</S, (ii)

and this vanishes in view of the requirement that equilibrium must be maintained in varying U. For then at every point on the boundary we must havet o = 0XV = MX, + m0Xy + \

O = 0Y„ = MY, + m%Yy + (11)O = 0Z „ = MZ, -f- d- J

or, if we substitute for 0X„ ..., etc. in terms of Maxwell’s stress-functions,■ a 0-. dz n dx.

0 , 0m ~dx

- 0 0 '

« — m —- dy dz J

00Xdz5SXdx

^ x

-2 +

-3 +

' 0 0 ' T y - m - x l -3 = o,

dz t y -

30X0Z

= 0, (iii)

0 j dU dx dz.

9sXsdx

Therefore the co-factors of u, v and w, in (ii), vanish severally.9—The surface integral in (9) can be treated similarly. Thus, when

we substitute for the strain-components in terms of u, v, w, the terms of the integrand which involve may be written in the form

\ 0 0 -

- dz U dx_du * . , 0§4'i +

+ / 0 0 "1 /0M , 00^1 0S 11 \ I, ^ - m T x \ { T z ^ + w 7 ? - u ^ t ) +

d- u

— v

dy

0T dx.|

dz dz dx.r , 0 0 i r , 0 01L dz dx\ dx _ dy 0xJ

0 H i)0j

00^1 0X ’ (i)

ditions (3) are satisfied, the six strain-components can be expressed in the usual way in terms of some three functions u,v,w: this result is all that is needed in the subsequent argument.

* Cf. Love, op. cit., § 15 and footnote, or Lamb, “ Hydrodynamics ”, § 32. t Cf. Love, op. cit., § 47, equations (5).



12 R. V. Southwell

and the first two lines of this expression vanish on integration over the whole boundary, by Stokes’s theorem. The remaining terms, combined with the corresponding terms which involve Ha* H a , give

l L ay dx.

+

as<Kdz + ,a a

_ dz n dxas 41!

am — — ndz

JT0y_

dya s ^ 2 _|_ +dy '

+ (corresponding terms involving v and w).

These vanish at the boundary in view of the requirement that SX, SZ,, must be zero, since we have from (11)

sx„ = S2Hdydz

i _ a / a s ^ 0H2 _ a s^ ;dz \dx dy dz

8S<K dy \ dx0n — I as _|_ aa^;

dy dz /_

and two similar equations, when we substitute for SXx, SXy, 8XZ in termsof s 1? s 42? a 4*3-

P art II— Some P ar tic u la r P roblems Solved by the P r in c iple of

M inim um St r a in -E n er g y

Saint-VenanfsDiscussion of Uniform Flexure and Torsion

10—We shall here consider two of the problems treated by Saint-Venant on the basis of his celebrated “ semi-inverse ” method,—namely, the uniform bending and torsion of an isotropic cylinder.

Saint-Venant’s approach to these problems may be summarized as follows*:—The exact distribution of stress will depend upon the exact distribution of the external forces which constitute the resultant action (flexure or torsion); but if these forces are applied at points close to the ends of the cylinder it is evident that the stresses, excepting in regions very close to the ends, will be determined almost entirely by their resultant action, and hardly at all by their distribution.! Therefore we need not (initially) define that distribution, but may leave it to be determined, subsequently, from our solution; and in seeking for this solution of the

* Cf. Love, op. cit., Introduction p. 19, and § 89; also Todhunter and Pearson, op. cit., chap. X.

t Appeal is here made to Saint-Venant’s principle (c/. § 4). This appears to have been propounded by Saint-Venant without proof, as a truth which is intuitively evident. Cf. Todhunter and Pearson, op. cit., chap. X, § 8.



general equations of elasticity we may, without sensibly reducing the practical value of our results, restrict its exact application by introducing (tentatively) any simplifying assumption which is physically plausible.

Such simplifying assumptions a re :#

(1) that the stresses are propagated, in an axial direction, withoutchange in magnitude or distribution;

(2) that no lateral pressures act between adjacent longitudinal “ fibres ” , and the shearing stresses acting on such fibres have an axial direction.


In symbols, if Oz be drawn in an axial direction, these assumptions may be stated in the form

y _ y — Y1 y ^y

(Z*, z„, — 0.

The three stress-equations of equilibrium, when simplified with the aid of these relations, reduce to one only, viz.

dZx , dZ dx dy

= 0, (0

and the conditions (3) of compatibility, when expressed in terms of stress- components, impose the further restrictions

02Z„ 02Z, 02 Z,dx29Z*

dy2dzy_dx

dx dy

const.

= 0,

The first of (ii), combined with (12), shows that Zz is a linear function of x and y only: hence we deduce the familiar solution for uniform flexure. From the second of (ii), combined with (i), we obtain the solution for uniform torsion. In each instance an exact solution is obtained, provided that the external tractions have a particular distribution over the areas of the terminal cross-sections.

11—It will be observed that Saint-Venant’s approach is tentative in two respects,—namely, (1) in the appeal to intuition which is implied in his enunciation of the principle associated with his name, and (2) in the introduction of simplifying assumptions for which nothing more than plausibility can be claimed a priori, so that their validity must be left for later verification. We now develop, in contrast, the more positive



argument which can be based upon the principle of minimum strain- energy.

The Argument from Strain-Energy

12— First, we may use that principle to provide a theoretical argumentfor Saint-Venanfs principle of the elastic equivalence of statically equipollent systems of load.* The conditions of equilibrium require that the stresses shall adjust themselves to be in equilibrium with the applied tractions, and so demand a different solution for each particular mode of applying a given resultant action; but if the tractions are applied near the ends of the cylinder, its central portion (with boundaries free from stress) has merely to transmit this resultant action, and the conditions for minimum strain-energy in this portion are accordingly invariant. Thus the actual stress-distribution (since it must entail minimum strain-energy in the cylinder as a whole) will in any particular instance be a compromise between demands which call for a particular distribution close to the ends and demands which make for standardization in the central region. Evidently the compromise will be such that the stresses approach more and more nearly to some standard distribution as we leave the (terminal) regions in which forces are applied; and so, at parts remote from those regions, differences in the manner of applying a given resultant action will entail departures from this standard distribution which are negligibly small in relation to the “ standard ” stresses .f

The standard distribution is what we have to determine, since it alone has claims to adoption as a general formula for practical use. And the principle of minimum strain-energy gives us guidance in regard to its determination:—The central region of the cylinder has to transmit the same resultant action at every cross-section, and every cross-section has the same size and shape; therefore in this region the demands of minimum energy are the same at every section, and we can assert of the standard distribution that the stresses do not vary in a longitudinal direction. We thus find theoretical justification for the last three of Saint-Venant’s simplifying assumptions (12), in which z is directed along the axis of the cylinder.

13— We can now proceed further with the aid of the same principle. Having shown that in the “ standard ” solution the stresses have the same distribution at every section, we can determine that distribution by making U a minimum, subject to the conditions that the stresses are to

* Cf. the paper cited in § 4, or my “ Introduction ”, §§ 92-95.t Some such wording as this might perhaps be substituted, with advantage in

respect of precision, for the somewhat vague phrasing of Saint-Venant (cf. Todhunter and Pearson, loc. cit.).

14 R. V. Southwell




be independent of z and that the resultant action transmitted from section to section is not to be varied. In relation to isotropic material the total strain-energy may be expressed as

U = i \ j j {2 (1 + s) (Y*2 + Z,2 + X /) + X / + Y„2 +

— 2(j (Y vZz + ZzXx + X ^Y J} dy, (13)

where L is the length of the cylinder and the integration is to be taken over the whole area of the cross-section; and then it is clear that the principle of minimum energy requires X z and Y z to vanish everywhere, unless finite values are demanded by the conditions of equilibrium. Now X z, Y z do not contribute to the flexural moment, and they are not related with X x, Y v, Zz,Xv by the conditions of equilibrium, either at the boundary or in the interior of the cross-section; for according to the conclusions of § 12 we have (suppressing differentials with respect to z)

a x , ax, _ 0 dY. a Y ,_ 0dx0y ’ 0x ^ dy 9

dZx dZvdx' 0y ~ U> (14)

at all points in the cross-section, and

lX x + mXy = 0, IYX + mYy 0, IZX + mZy 0, (15)

a t all points on the boundary, where /, m stand for the cosines of the angles made with x and y by v, the normal to the (cylindrical) boundary drawn outwards. Hence we may assert ( instead o f assuming) that Xz, Y z vanish in the flexural solution.

In the torsion problem, similarly, we may assert that X*, Yy, Zz, X v vanish everywhere. They do not contribute to the torsional action (which comes from X z, Yz); they are not related with X2, by the conditions of equilibrium; and they would make, if finite, an independent contribution (necessarily positive: cf. § 3) to the total strain- energy.

Uniform Flexure

14—Hereafter we shall treat the two problems separately. In the problem of flexure we have shown that Xz, Y z vanish everywhere, so that the expression (13) is equivalent to

u = i | j ] {2 (1 + (X„2 - X, . Y„) + (Xx + Yv)2 + Z 2 -

— {Xx + Yv) Zz) dx dy. (i)



16 R. V. Southwell

This equation can now be simplified further. For the first and second of (14) permit us to write (as in the theory of plane strain)

v _ 82x v _ _ s2x y = i!x v ~ 9 v ax2’

and then we have

( 16)

f j (X,s - XxY v) dx dy a r&i a2x \ _ _ 1 / ax a xax v ax dy2. dx dy

= - f j H lX , + mXv)ds,

= 0, by the first of (15).

Accordingly (i) reduces, in the case of uniform flexure, to

L

| dx dy,

or tou = i i J j {(X* + Y»)2 + z * ~ 2” (X* + Y,) Zi} dx dy’

U = i | j f { (V .w + Z f - 2ctVx2 X • z ,)d x dy,

a2 a2where Vi2 denotes the operator — + j - - , ,y (i7>

if we substitute for X*, Y„ in terms of x-0y 0y

15—The boundary conditions (15) require that shall each have a

constant value at the boundary. Since they can be satisfied by a term of the form Ax + By (A and B constant) which, regarded as a component of X) has no effect on the stress-components (16), we shall lose no generality if we say that (15) impose the conditions

0yX = — 0, on the boundary.

dv(18>

On this understanding we have to make U as given by (17) a minimum, subject to the restriction that the resultant action is specified, so that

J j Z z dxdy, j jxZ2 dx dy, j JyZ2 dx dy

have fixed values. Hence we have, according to the rules of the calculus of variations,

SU jj{SZ*(Z, P —Qx — Ry — ctVi2x)

+ v x2 §x (Vi2x — <?Z2)} dx dy



(P, Q and R being arbitrary constants), for all permissible variationsSZ„ Sx-

Using (18), we deduce that

Z, = P + Qx + Ry + aV A ,a V ? Z z = V A ,

and hence, eliminating Zz, we have

Vx4x = 0.

This condition, combined with (18), shows that Vi2x, and therefore X, Xr, Y y, X v, must vanish everywhere: hence, by the first of (i), we deduce that Z2 has the form (P + Qx + Ry). We have thus obtained by positive arguments all the essential features of Saint-Venant’s solution.


Uniform Torsion

16—We shall here confine attention to cylinders of solid cross-section, but the same method can be applied to hollow sections.

We showed in § 13 that X„ Yz will be the only stress-components having finite values in the “ least-energy ” solution for uniform torsion, so that the expression (13) reduces in this instance to

U = (1 + <7) | f f (X,* + Y f) dx dy. (19)

The conditions of equilibrium are

a x , , 0Y. = 0

at all points in the cross-section, and

-}- rnYz = 0

at all points on the boundary.We can satisfy the first of (i) by writing

Y, = — 0T*dx ’ z dy ’

and then the second of (i) requires that W/ 0, or

T* = T c (a constant),

\

(0

/

(ii)

(20)

VOL. CLIV.— A. C



18 R. V. Southwell

at all points on the boundary. The expression (19) takes the form

u = (1 + o ) t M f H £ ) > ^and the torque is given by

T = j j (xY, - yX J dx dy = — j j ( x ? f - + y ^ f d x dy, '

= 2 j j ( ’F - T

(21)

by Green’s transformation combined with the boundary condition (20). It can be shown that (20) restricts the resultant action of Xz, Y z to a pure couple having its axis in the direction Oz.

17—The function T must be such as will give a minimum value for U within the restrictions imposed by (20) and by the requirement that T, as given in (21), has a specified value. Applying the rules of the calculus of variations, we deduce that the quantity

iff WS ■ ¥ + £ ■ i t )+A <»* - ■ *•> I * *(i)

(A being an arbitrary constant) must vanish for any variation S T such that S T has a constant value S T c on the boundary.

The integral in (i), by Green’s transformation, is equivalent to

j I (S T (A - 2VX2T) - A S T J dx dy + 2 j S T - | i

= j j (ST - S T C) (A - 2V12T) dx dy,

(ii)

(iii)

since the line integral in (ii)

= 2ST cj ^ J s = 2S T C j j Vj2 T dx dy.

Hence, since SU must vanish for every variation ST, we deduce the condition

VX2T = constant, at all points in the cross-section, which, with the boundary condition

T = T c (a constant),serves to determine T uniquely. This is the known solution.

Uniform Bending of a Thin Circular Bar {Plane Stress)18—When the central line of a beam is curved, transverse (radial)

stresses are necessary to the maintenance of equilibrium under flexural



19Castigliano's Principle o f Minimum Strain-Energy

moments, and therefore stresses analogous to X*, Yw, X v (in the straight beam directed along O z)must be expected. For most shapes of cross- section the problem of flexure is difficult, but a solution based upon the argument from minimum strain-energy can be obtained without difficulty in the particular case where the beam forms part of a circular ring cut from a flat plate of small thickness, and so presents a problem in plane stress. Reasoning on the lines of § 12 we shall seek a “ least-energy solution ” on the understanding that the resultant action is a pure couple M having an axis directed along Oz, the axis of the ring. We may postulate that Xz, Y„ Zz are zero everywhere, on the ground that these stress-components must vanish at the faces of the plate, which we take to be thin.

19—As in § 12 we can assert that the demands of least energy, and hence the stresses, will be the same (in this standard solution) at every section of the ring: so, using cylindrical coordinates 0, we have to contemplate stress-components rr, 06, r0, all independent of 0. The conditions of equilibrium are*

andrr = r0 = 0, when r — a and when r = b,

! ( ' • " > - ■ 9 e >

3 r ■ * > - < > >

(23)

(24)

J

for all values of r between the values a and b, when a and b stand for the inner and outer radii of the ring. Taken in conjunction, the first of (23)

0, while the second of (23) and (24)and (24) require that J 00 . dr

require that rO shall be zero everywhere. Thus there is no resultant tension and no shear stress on any axial cross-section, so that rr, 00 are principal stresses; and the strain-energy stored in the ring is given by

U = I {rr {rr — a 00) + 00 (00 — cr rr)} rdr, 2E J a

= J r ~~ ^ arr' ® 6 62) (25)

* Cf. Love, op.cit., § 59 (i). The three conditions o f equilibrium are reduced to two by our assumption that there is no stress on planes perpendicular to and these two are further simplified by the requirement that all stresses are independent of 0.



20 R. V. Southwell

where a is the angle subtended at the axis of the ring by its terminal sections, and t stands for the (small) thickness of the ring.

20—We have from the first of (24)

2 | r .7r . dr = [ ~ {r .Ja Jadr= 0 by the first of (23); and

j r .002 dr = j J- (r . rr) . jr (r . /r) j dr,

= — ( . ~ (r ~ J dr,]a dr dr J

when we integrate by parts and make use of (23). Hence we find that (25) may be written in the form

(r . rr) dr,

(26)

if we integrate by parts with the aid of (23) again.The flexural couple

M = t[ r . Qd dr — tI r ( r . by the first of (24). Ja Ja dr

= i t j {r2 j r rr + j r(r2 • rr)} dr,

= \t( r2 rr dr, (27)J a dr

by (23) again. We have to find that distribution of rr which will make U a minimum for all variations Srr, subject to the restrictions imposed by the boundary condition (23) and by the requirement that M, as given by (27), must have a fixed value.

21—Varying U as before, we have from these conditions (writing p , for brevity, in place of rr)

A being an arbitrary constant,

= b p { 2 A r- ^ % h (28>




by the boundary condition, when we integrate by parts. Since SU must vanish for every variation Bp in the range ( « < / * < b), we deduce that in this range p must satisfy the condition

whence we have

and

dp _ A 2Bdr r r3

^ Brr = p = A log r + — + C, (29)

B and C being constants of integration.

Making this expression satisfy the boundary condition (23), we have a unique form for rr and hence, by the first of (24), a unique form for 00. The complete solution is given by the expressions

rr = 4^ ( log ~a + a2log “ + 2 log ,

00 = 4& ( — log ~ log ~ log ^ + ),

and

(30)

where, from (27),

M = kt

These are known results.*

(Z>2 — a2)2 — 4a2b2 (log ^ '

* Cy. S. P. Timoshenko, “ Theory o f Elasticity ”, § 23, where the solution is attributed to H. Golovin (‘ Trans. Inst. Tech. St. Petersburg’, 1881).



castigliano’s principle of minimum...

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