Chapter IV: Part 4

45. Saint-Venant's theory shows that the true torsional rigidity is in general less than that which would be obtained by extending Coulomb's law (G = [mu][tau]I) to sections which are not circular. For an elliptic cylinder of sectional area [omega] and moment of inertia I about its central-line the torsional rigidity is [mu][omega]^4/4[pi]^2I, and this formula is not far from being correct for a very large number of sections. For a bar of square section of side a centimetres, the torsional rigidity in C.G.S. units is (0.1406)[mu]a^4 approximately, [mu] being expressed in dynes per square centimetre. How great the defect of the true value from that given by extending Coulomb's law may be in the case of sections with projecting corners is shown by the diagrams (fig. 8 especially no. 4). In these diagrams the upper of the two numbers under each figure indicates the fraction which the true torsional rigidity corresponding to the section is of that value which would be obtained by extending Coulomb's law; and the lower of the two numbers indicates the ratio which the torsional rigidity for a bar of the corresponding section bears to that of a bar of circular section of the same material and of equal sectional area. These results have an important practical application, inasmuch as they show that strengthening ribs and projections, such as are introduced in engineering to give stiffness to beams, have the reverse of a good effect when torsional stiffness is an object, although they are of great value in increasing the resistance to bending. The theory shows further that the resistance to torsion is very seriously diminished when there is in the surface any dent approaching to a re-entrant angle. At such a place the shearing strain tends to become infinite, and some permanent set is produced by torsion. In the case of a section of any form, the strain and stress are greatest at points on the contour, and these points are in many cases the points of the contour which are nearest to the centroid of the section. The theory has also been applied to show that a longitudinal flaw near the axis of a shaft transmitting a torsional couple has little influence on the strength of the shaft, but that in the neighbourhood of a similar flaw which is much nearer to the surface than to the axis the shearing strain may be nearly doubled, and thus the possibility of such flaws is a source of weakness against which special provision ought to be made.

(1) Rectilineal square. .84346. .88326.
(2) Square with curved corners and hollow sides. .8186. .8666.
(3) Square with acute angles and hollow sides. .7783. .8276.
(4) Star with four rounded points, being a curve of the eighth degree.
.5374. .6745.
(5) Equilateral triangle. .60000. .72552.]

46. _Bending of Beams._--As a second example of the application of the general theory we take the problem of the flexure of a beam. In this case also we begin by forming a simple intuition as to the nature of the strain and the stress. On the side of the beam towards the centre of curvature the longitudinal filaments must be contracted, and on the other side they must be extended. If we assume that the cross-sections remain plane, and that the central-line is unaltered in length, we see (at once from fig. 9) that the extensions (or contractions) are given by the formula y/R, where y denotes the distance of a longitudinal filament from the plane drawn through the unstrained central-line at right-angles to the plane of bending, and R is the radius of curvature of the curve into which this line is bent (shown by the dotted line in the figure). Corresponding to this strain there must be traction acting across the cross-sections. If we assume that there is no other stress, then the magnitude of the traction in question is Ey/R, where E is Young's modulus, and it is tension on the side where the filaments are extended and pressure on the side where they are contracted. If the plane of bending contains a set of principal axes of the cross-sections at their centroids, these tractions for the whole cross-section are equivalent to a couple of moment EI/R, where I now denotes the moment of inertia of the cross-section about an axis through its centroid at right angles to the plane of bending, and the plane of the couple is the plane of bending. Thus a beam of any form of section can be held bent in a "principal plane" by terminal couples of moment M, that is to say by a "bending moment" M; the central-line will take a curvature M/EI, so that it becomes an arc of a circle of radius EI/M; and the stress at any point will be tension of amount My/I, where y denotes distance (reckoned positive towards the side remote from the centre of curvature) from that plane which initially contains the central-line and is at right angles to the plane of the couple. This plane is called the "neutral plane." The restriction that the beam is bent in a principal plane means that the plane of bending contains one set of principal axes of the cross-sections at their centroids; in the case of a beam of rectangular section the plane would bisect two opposite edges at right angles. In order that the theory may hold good the radius of curvature must be very large.

47. In this problem of the bending of a beam by terminal couples the stress is tension, determined as above, and the corresponding strain consists therefore of longitudinal extension of amount My/EI or y/R (contraction if y is negative), accompanied by lateral contraction of amount [sigma]My/EI or [sigma]y/R (extension if y is negative), [sigma] being Poisson's ratio for the material. Our intuition of the nature of the strain was imperfect, inasmuch as it took no account of these lateral strains. The necessity for introducing them was pointed out by Saint-Venant. The effect of them is a change of shape of the cross-sections in their own planes. This is shown in an exaggerated way in fig. 10, where the rectangle ABCD represents the cross-section of the unstrained beam, or a rectangular portion of this cross-section, and the curvilinear figure A'B'C'D' represents in an exaggerated fashion the cross-section (or the corresponding portion of the cross-section) of the same beam, when bent so that the centre of curvature of the central-line (which is at right angles to the plane of the figure) is on the line EF produced beyond F. The lines A'B' and C'D' are approximately circles of radii R/[sigma], when the central-line is a circle of radius R, and their centres are on the line FE produced beyond E. Thus the neutral plane, and each of the faces that is parallel to it, becomes strained into an _anticlastic surface_, whose principal curvatures are in the ratio [sigma] : 1. The general appearance of the bent beam is shown in an exaggerated fashion in fig. 11, where the traces of the surface into which the neutral plane is bent are dotted. The result that the ratio of the principal curvatures of the anticlastic surfaces, into which the top and bottom planes of the beam (of rectangular section) are bent, is Poisson's ratio [sigma], has been used for the experimental determination of [sigma]. The result that the radius of curvature of the bent central-line is EI/M is used in the experimental determination of E. The quantity EI is often called the "flexural rigidity" of the beam. There are two principal flexural rigidities corresponding to bending in the two principal planes (cf. S 62 below).

48. That this theory requires modification, when the load does not consist simply of terminal couples, can be seen most easily by considering the problem of a beam loaded at one end with a weight W, and supported in a horizontal position at its other end. The forces that are exerted at any section p, to balance the weight W, must reduce statically to a vertical force W and a couple, and these forces arise from the action of the part Ap on the part Bp (see fig. 12), i.e. from the stresses across the section at p. The couple is equal to the moment of the applied load W about an axis drawn through the centroid of the section p at right angles to the plane of bending. This moment is called the "bending moment" at the section, it is the product of the load W and the distance of the section from the loaded end, so that it varies uniformly along the length of the beam. The stress that suffices in the simpler problem gives rise to no vertical force, and it is clear that in addition to longitudinal tensions and pressures there must be tangential tractions on the cross-sections. The resultant of these tangential tractions must be a force equal to W, and directed vertically; but the direction of the traction at a point of the cross-section need not in general be vertical. The existence of tangential traction on the cross-sections implies the existence of equal tangential traction, directed parallel to the central-line, on some planes or other which are parallel to this line, the two sets of tractions forming a shearing stress. We conclude that such shearing stress is a necessary constituent of the stress-system in the beam bent by terminal transverse load. We can develop a theory of this stress-system from the assumptions (i.) that the tension at any point of the cross-section is related to the bending moment at the section by the same law as in the case of uniform bending by terminal couples; (ii.) that, in addition to this tension, there is at any point shearing stress, involving tangential tractions acting in appropriate directions upon the elements of the cross-sections. When these assumptions are made it appears that there is one and only one distribution of shearing stress by which the conditions of the problem can be satisfied. The determination of the amount and direction of this shearing stress, and of the corresponding strains and displacements, was effected by Saint-Venant and R.F.A. Clebsch for a number of forms of section by means of an analysis of the same kind as that employed in the solution of the torsion problem.

49. Let l be the length of the beam, x the distance of the section p
from the fixed end A, y the distance of any point below the horizontal
plane through the centroid of the section at A, then the bending
moment at p is W(l - x), and the longitudinal tension P or X_x at any
point on the cross-section is - W(l - x)y/I, and this is related to
the bending moment exactly as in the simpler problem.

50. The expressions for the shearing stresses depend on the shape of
the cross-section. Taking the beam to be of isotropic material and the
cross-section to be an ellipse of semiaxes a and b (fig. 13), the a
axis being vertical in the unstrained state, and drawing the axis z at
right angles to the plane of flexure, we find that the vertical
shearing stress U or X_y at any point (y, z) on any cross-section is

2W[(a^2 - y^2){2a^2(1 + [sigma]) + b^2} - z^2a^2(1 - 2[sigma])]
---------------------------------------------------------------.
[pi]a^3b(1 + [sigma])(3a^2 + b^2)

The resultant of these stresses is W, but the amount at the centroid,
which is the maximum amount, exceeds the average amount, W/[pi]ab, in
the ratio

{4a^2(1 + [sigma]) + 2b^2}/(3a^2 + b^2)(1 + [sigma]).

If [sigma] = 1/4, this ratio is 7/5 for a circle, nearly 4/3 for a flat
elliptic bar with the longest diameter vertical, nearly 8/5 for a flat
elliptic bar with the longest diameter horizontal.

In the same problem the horizontal shearing stress T or Z_x at any
point on any cross-section is of amount

4Wyz{a^2(1 + [sigma]) + b^2[sigma]}
- -----------------------------------.
[pi]a^3b(1 + [sigma])(3a^2 + b^2)

The resultant of these stresses vanishes; but, taking as before
[sigma] = 1/4, and putting for the three cases above a = b, a = 10b,
b = 10a, we find that the ratio of the maximum of this stress to the
average vertical shearing stress has the values 3/5, nearly 1/15, and
nearly 4. Thus the stress T is of considerable importance when the
beam is a plank.

As another example we may consider a circular tube of external radius
r0 and internal radius r1. Writing P, U, T for X_x, X_y, Z_x, we find

4W
P = - -----------------(l - x)y,
[pi](r0^4 - r1^4)
_
W | /
U = ------------------------------- |(3 + 2[sigma]) (r0^2 + r1^2 - y^2
2(1 + [sigma])[pi](r0^4 - r1^4) |_ \
_
r0^2r1^2 \ |
- ------------- (y^2 - z^2) ) - (1 - 2[sigma])z^2|
(y^2 + z^2)^2 / _|

W
T = - ------------------------------
(1 + [sigma])[pi](r0^4 - r1^4)
_ _
| r0^2r1^2 |
| 1 + 2[sigma] + (3 + 2[sigma]) ------------- | yz;
|_ (y^2 + z^2)^2 _|

and for a tube of radius r and small thickness t the value of P and
the maximum values of U and T reduce approximately to

P = - W(l - x)y/[pi]r^3t

U_max. = W/[pi]rt, T_max. = W/2[pi]rt.

The greatest value of U is in this case approximately twice its
average value, but it is possible that these results for the bending
of very thin tubes may be seriously at fault if the tube is not
plugged, and if the load is not applied in the manner contemplated in
the theory (cf. S 55). In such cases the extensions and contractions
of the longitudinal filaments may be practically confined to a small
part of the material near the ends of the tube, while the rest of the
tube is deformed without stretching.

51. The tangential tractions U, T on the cross-sections are necessarily accompanied by tangential tractions on the longitudinal sections, and on each such section the tangential traction is parallel to the central line; on a vertical section z = const. its amount at any point is T, and on a horizontal section y = const. its amount at any point is U.

The internal stress at any point is completely determined by the components P, U, T, but these are not principal stresses (S 7). Clebsch has given an elegant geometrical construction for determining the principal stresses at any point when the values of P, U, T are known.

From the point O (fig. 14) draw lines OP, OU, OT, to represent the
stresses P, U, T at O, on the cross-section through O, in magnitude,
direction and sense, and compound U and T into a resultant represented
by OE; the plane EOP is a principal plane of stress at O, and the
principal stress at right angles to this plane vanishes. Take M the
middle point of OP, and with centre M and radius ME describe a circle
cutting the line OP in A and B; then OA and OB represent the
magnitudes of the two remaining principal stresses. On AB describe a
rectangle ABDC so that DC passes through E; then OC is the direction
of the principal stress represented in magnitude by OA, and OD is the
direction of the principal stress represented in magnitude by OB.

52. As regards the strain in the beam, the longitudinal and lateral extensions and contractions depend on the bending moment in the same way as in the simpler problem; but, the bending moment being variable, the anticlastic curvature produced is also variable. In addition to these extensions and contractions there are shearing strains corresponding to the shearing stresses T, U. The shearing strain corresponding to T consists of a relative sliding parallel to the central-line of different longitudinal linear elements combined with a relative sliding in a transverse horizontal direction of elements of different cross-sections; the latter of these is concerned in the production of those displacements by which the variable anticlastic curvature is brought about; to see the effect of the former we may most suitably consider, for the case of an elliptic cross-section, the distortion of the shape of a rectangular portion of a plane of the material which in the natural state was horizontal; all the boundaries of such a portion become parabolas of small curvature, which is variable along the length of the beam, and the particular effect under consideration is the change of the transverse horizontal linear elements from straight lines such as HK to parabolas such as H'K' (fig. 15); the lines HL and KM are parallel to the central-line, and the figure is drawn for a plane above the neutral plane. When the cross-section is not an ellipse the character of the strain is the same, but the curves are only approximately parabolic.

The shearing strain corresponding to U is a distortion which has the effect that the straight vertical filaments become curved lines which cut the longitudinal filaments obliquely, and thus the cross-sections do not remain plane, but become curved surfaces, and the tangent plane to any one of these surfaces at the centroid cuts the central line obliquely (fig. 16). The angle between these tangent planes and the central-line is the same at all points of the line; and, if it is denoted by 1/2[pi] + s0, the value of s0 is expressible as

shearing stress at centroid
---------------------------,
rigidity of material

and it thus depends on the shape of the cross-section; for the elliptic section of S 50 its value is

4W 2a^2(1 + [sigma]) + b^2
------- -----------------------;
E[pi]ab 3a^2 + b^2

for a circle (with [sigma] = 1/4) this becomes 7W/2E[pi]a^2. The vertical filament through the centroid of any cross-section becomes a cubical parabola, as shown in fig. 16, and the contour lines of the curved surface into which any cross-section is distorted are shown in fig. 17 for a circular section.

53. The deflection of the beam is determined from the equation

curvature of central line = bending moment :- flexural rigidity,

and the special conditions at the supported end; there is no alteration of this statement on account of the shears. As regards the special condition at an end which is _encastree_, or built in, Saint-Venant proposed to assume that the central tangent plane of the cross-section at the end is vertical; with this assumption the tangent to the central line at the end is inclined downwards and makes an angle s0 with the horizontal (see fig. 18); it is, however, improbable that this condition is exactly realized in practice. In the application of the theory to the experimental determination of Young's modulus, the small angle which the central-line at the support makes with the horizontal is an unknown quantity, to be eliminated by observation of the deflection at two or more points.

54. We may suppose the displacement in a bent beam to be produced by the following operations: (1) the central-line is deflected into its curved form, (2) the cross-sections are rotated about axes through their centroids at right angles to the plane of flexure so as to make angles equal to 1/2[pi] + s0 with the central-line, (3) each cross-section is distorted in its own plane in such a way that the appropriate variable anticlastic curvature is produced, (4) the cross-sections are further distorted into curved surfaces. The contour lines of fig. 17 show the disturbance from the central tangent plane, not from the original vertical plane.

55. _Practical Application of Saint-Venant's Theory._--The theory above described is exact provided the forces applied to the loaded end, which have W for resultant, are distributed over the terminal section in a particular way, not likely to be realized in practice; and the application to practical problems depends on a principle due to Saint-Venant, to the effect that, except for comparatively small portions of the beam near to the loaded and fixed ends, the resultant only is effective, and its mode of distribution does not seriously affect the internal strain and stress. In fact, the actual stress is that due to forces with the required resultant distributed in the manner contemplated in the theory, superposed upon that due to a certain distribution of forces on each terminal section which, if applied to a rigid body, would keep it in equilibrium; according to Saint-Venant's principle, the stresses and strains due to such distributions of force are unimportant except near the ends. For this principle to be exactly applicable it is necessary that the length of the beam should be very great compared with any linear dimension of its cross-section; for the practical application it is sufficient that the length should be about ten times the greatest diameter.

56. In recent years the problem of the bending of a beam by loads distributed along its length has been much advanced. It is now practically solved for the case of a load distributed uniformly, or according to any rational algebraic law, and it is also solved for the case where the thickness is small compared with the length and depth, as in a plate girder, and the load is distributed in any way. These solutions are rather complicated and difficult to interpret. The case which has been worked out most fully is that of a transverse load distributed uniformly along the length of the beam. In this case two noteworthy results have been obtained. The first of these is that the central-line in general suffers extension. This result had been found experimentally many years before. In the case of the plate girder loaded uniformly along the top, this extension is just half as great as the extension of the central-line of the same girder when free at the ends, supported along the base, and carrying the same load along the top. The second noteworthy result is that the curvature of the strained central-line is not proportional to the bending moment. Over and above the curvature which would be found from the ordinary relation--

curvature of central-line = bending moment :- flexural rigidity,

there is an additional curvature which is the same at all the cross-sections. In ordinary cases, provided the length is large compared with any linear dimension of the cross-section, this additional curvature is small compared with that calculated from the ordinary formula, but it may become important in cases like that of suspension bridges, where a load carried along the middle of the roadway is supported by tensions in rods attached at the sides.

57. When the ordinary relation between the curvature and the bending moment is applied to the calculation of the deflection of _continuous beams_ it must not be forgotten that a correction of the kind just mentioned may possibly be requisite. In the usual method of treating the problem such corrections are not considered, and the ordinary relation is made the basis of the theory. In order to apply this relation to the calculation of the deflection, it is necessary to know the bending moment at every point; and, since the pressures of the supports are not among the data of the problem, we require a method of determining the bending moments at the supports either by calculation or in some other way. The calculation of the bending moment can be replaced by a method of graphical construction, due to Mohr, and depending on the two following theorems:--

(i.) The curve of the central-line of each span of a beam, when the bending moment M is given,[1] is identical with the catenary or funicular curve passing through the ends of the span under a (fictitious) load per unit length of the span equal to M/EI, the horizontal tension in the funicular being unity.

(ii.) The directions of the tangents to this funicular curve at the ends of the span are the same for all statically equivalent systems of (fictitious) load.

When M is known, the magnitude of the resultant shearing stress at any section is dM/dx, where x is measured along the beam.

58. Let l be the length of a span of a loaded beam (fig. 19), M1 and
M2 the bending moments at the ends, M the bending moment at a section
distant x from the end (M1), M' the bending moment at the same section
when the same span with the same load is simply supported; then M is
given by the formula

l - x x
M = M' + M1 ----- + M2 --,
l l

and thus a fictitious load statically equivalent to M/EI can be easily
found when M' has been found. If we draw a curve (fig. 20) to pass
through the ends of the span, so that its ordinate represents the
value of M'/EI, the corresponding fictitious loads are statically
equivalent to a single load, of amount represented by the area of the
curve, placed at the point of the span vertically above the centre of
gravity of this area. If PN is the ordinate of this curve, and if at
the ends of the span we erect ordinates in the proper sense to
represent M1/EI and M2/EI, the bending moment at any point is
represented by the length PQ.[2] For a uniformly distributed load the
curve of M' is a parabola M' = 1/2wx(l - x), where w is the load per
unit of length; and the statically equivalent fictitious load is
(1/12)wl^3/EI placed at the middle point G of the span; also the loads
statically equivalent to the fictitious loads M1(l - x)/lEI and
M2x/lEI are 1/2M1l/EI and 1/2M2l/EI placed at the points g, g' of
trisection of the span. The funicular polygon for the fictitious loads
can thus be drawn, and the direction of the central-line at the
supports is determined when the bending moments at the supports are
known.

59. When there is more than one span the funiculars in question may be
drawn for each of the spans, and, if the bending moments at the ends
of the extreme spans are known, the intermediate ones can be
determined. This determination depends on two considerations: (1) the
fictitious loads corresponding to the bending moment at any support
are proportional to the lengths of the spans which abut on that
support; (2) the sides of two funiculars that end at any support
coincide in direction. Fig. 21 illustrates the method for the case of
a uniform beam on three supports A, B, C, the ends A and C being
freely supported. There will be an unknown bending moment M0 at B, and
the system[3] of fictitious loads is (1/12)wAB^3/EI at G the middle
point of AB, (1/12)wBC^3/EI at G' the middle point of BC, -1/2M0AB/EI
at g and -1/2M0BC/EI at g', where g and g' are the points of
trisection nearer to B of the spans AB, BC. The centre of gravity of
the two latter is a fixed point independent of M0, and the line VK of
the figure is the vertical through this point. We draw AD and CE to
represent the loads at G and G' in magnitude; then D and E are fixed
points. We construct any triangle UVW whose sides UV, UW pass through
D, B, and whose vertices lie on the verticals gU, VK, g'W; the point F
where VW meets DB is a fixed point, and the lines EF, DK are the two
sides (2, 4) of the required funiculars which do not pass through A, B
or C. The remaining sides (1, 3, 5) can then be drawn, and the side 3
necessarily passes through B; for the triangle UVW and the triangle
whose sides are 2, 3, 4 are in perspective.

The bending moment M0 is represented in the figure by the vertical
line BH where H is on the continuation of the side 4, the scale being
given by

BH 1/2M0BC
-- = ----------- ;
CE (1/12)wBC^3

this appears from the diagrams of forces, fig. 22, in which the
oblique lines are marked to correspond to the sides of the funiculars
to which they are parallel.

In the application of the method to more complicated cases there are
two systems of fixed points corresponding to F, by means of which the
sides of the funiculars are drawn.

60. _Finite Bending of Thin Rod._--The equation

curvature = bending moment :- flexural rigidity

may also be applied to the problem of the flexure in a principal plane of a very thin rod or wire, for which the curvature need not be small. When the forces that produce the flexure are applied at the ends only, the curve into which the central-line is bent is one of a definite family of curves, to which the name _elastica_ has been given, and there is a division of the family into two species according as the external forces are applied directly to the ends or are applied to rigid arms attached to the ends; the curves of the former species are characterized by the presence of inflections at all the points at which they cut the line of action of the applied forces.

We select this case for consideration. The problem of determining the
form of the curve (cf. fig. 23) is mathematically identical with the
problem of determining the motion of a simple circular pendulum
oscillating through a finite angle, as is seen by comparing the
differential equation of the curve

d^2[phi]
EI -------- + W sin [phi] = 0
ds^2

with the equation of motion of the pendulum

d^2[phi]
l -------- + g sin [phi] = 0.
dt^2

The length L of the curve between two inflections corresponds to the
time of oscillation of the pendulum from rest to rest, and we thus
have

L [root](W/EI) = 2K,

where K is the real quarter period of elliptic functions of modulus
sin 1/2[alpha], and [alpha] is the angle at which the curve cuts the
line of action of the applied forces. Unless the length of the rod
exceeds [pi][root](EI/W) it will not bend under the force, but when
the length is great enough there may be more than two points of
inflection and more than one bay of the curve; for n bays (n + 1
inflections) the length must exceed n[pi][root](EI/W). Some of the
forms of the curve are shown in fig. 24.

For the form d, in which two bays make a figure of eight, we have

L[root](W/EI) = 4.6, [alpha] = 130 deg.

approximately. It is noteworthy that whenever the length and force
admit of a sinuous form, such as [alpha] or b, with more than two
inflections, there is also possible a crossed form, like e, with two
inflections only; the latter form is stable and the former unstable.

61. The particular case of the above for which [alpha] is very small is a curve of sines of small amplitude, and the result in this case has been applied to the problem of the buckling of struts under thrust. When the strut, of length L', is maintained upright at its lower end, and loaded at its upper end, it is simply contracted, unless L'^2W > 1/4[pi]^2EI, for the lower end corresponds to a point at which the tangent is vertical on an elastica for which the line of inflections is also vertical, and thus the length must be half of one bay (fig. 25, a). For greater lengths or loads the strut tends to bend or buckle under the load. For a very slight excess of L'^2W above 1/4[pi]^2EI, the theory on which the above discussion is founded, is not quite adequate, as it assumes the central-line of the strut to be free from extension or contraction, and it is probable that bending without extension does not take place when the length or the force exceeds the critical value but slightly. It should be noted also that the formula has no application to short struts, as the theory from which it is derived is founded on the assumption that the length is great compared with the diameter (cf. S 56).

The condition of buckling, corresponding to the above, for a long strut, of length L', when both ends are free to turn is L'^2W > [pi]^2EI; for the central-line forms a complete bay (fig. 25, b); if both ends are maintained in the same vertical line, the condition is L'^2W > 4[pi]^2EI, the central-line forming a complete bay and two half bays (fig. 25, c).

62. In our consideration of flexure it has so far been supposed that the bending takes place in a principal plane. We may remove this restriction by resolving the forces that tend to produce bending into systems of forces acting in the two principal planes. To each plane there corresponds a particular flexural rigidity, and the systems of forces in the two planes give rise to independent systems of stress, strain and displacement, which must be superposed in order to obtain the actual state. Applying this process to the problem of SS 48-54, and supposing that one principal axis of a cross-section at its centroid makes an angle [theta] with the vertical, then for any shape of section the neutral surface or locus of unextended fibres cuts the section in a line DD', which is conjugate to the vertical diameter CP with respect to any ellipse of inertia of the section. The central-line is bent into a plane curve which is not in a vertical plane, but is in a plane through the line CY which is perpendicular to DD' (fig. 26).

63. _Bending and Twisting of Thin Rods._--When a very thin rod or wire is bent and twisted by applied forces, the forces on any part of it limited by a normal section are balanced by the tractions across the section, and these tractions are statically equivalent to certain forces and couples at the centroid of the section; we shall call them the _stress-resultants_ and the _stress-couples_. The stress-couples consist of two flexural couples in the two principal planes, and the torsional couple about the tangent to the central-line. The torsional couple is the product of the torsional rigidity and the twist produced; the torsional rigidity is exactly the same as for a straight rod of the same material and section twisted without bending, as in Saint-Venant's torsion problem (S 42). The twist [tau] is connected with the deformation of the wire in this way: if we suppose a very small ring which fits the cross-section of the wire to be provided with a pointer in the direction of one principal axis of the section at its centroid, and to move along the wire with velocity v, the pointer will rotate about the central-line with angular velocity [tau]v. The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section; the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only.

The stress-resultants across any section are tangential forces in the two principal planes, and a tension or thrust along the central-line; when the stress-couples and the applied forces are known these stress-resultants are determinate. The existence, in particular, of the resultant tension or thrust parallel to the central-line does not imply sensible extension or contraction of the central filament, and the tension per unit area of the cross-section to which it would be equivalent is small compared with the tensions and pressures in longitudinal filaments not passing through the centroid of the section; the moments of the latter tensions and pressures constitute the flexural couples.

64. We consider, in particular, the case of a naturally straight spring or rod of circular section, radius c, and of homogeneous isotropic material. The torsional rigidity is 1/4E[pi]c^4/(1 + [sigma]); and the flexural rigidity, which is the same for all planes through the central-line, is 1/4E[pi]c^4; we shall denote these by C and A respectively. The rod may be held bent by suitable forces into a curve of double curvature with an amount of twist [tau], and then the torsional couple is C[tau], and the flexural couple in the osculating plane is A/[rho], where [rho] is the radius of circular curvature. Among the curves in which the rod can be held by forces and couples applied at its ends only, one is a circular helix; and then the applied forces and couples are equivalent to a wrench about the axis of the helix.

Let [alpha] be the angle and r the radius of the helix, so that [rho]
is r sec^2[alpha]; and let R and K be the force and couple of the
wrench (fig. 27).

Then the couple formed by R and an equal and opposite force at any
section and the couple K are equivalent to the torsional and flexural
couples at the section, and this gives the equations for R and K

sin [alpha] cos^3 [alpha] cos [alpha]
R = A ------------------------- - C[tau] ------------,
r^2 r

cos^3 [alpha]
K = A ------------- + C[tau] sin [alpha].
r

The thrust across any section is R sin [alpha] parallel to the tangent
to the helix, and the shearing stress-resultant is R cos [alpha] at
right angles to the osculating plane.

When the twist is such that, if the rod were simply unbent, it would
also be untwisted, [tau] is (sin [alpha] cos [alpha])/r, and then,
restoring the values of A and C, we have

E[pi]c^4 [sigma]
R = -------- ------------ sin [alpha] cos^2 [alpha],
4r^2 1 + [sigma]

E[pi]c^4 1 + [sigma] cos^2 [alpha]
K = -------- ------------------------- cos [alpha].
4r 1 + [sigma]

65. The theory of spiral springs affords an application of these
results. The stress-couples called into play when a naturally helical
spring ([alpha], r) is held in the form of a helix ([alpha]', r'), are
equal to the differences between those called into play when a
straight rod of the same material and section is held in the first
form, and those called into play when it is held in the second form.

Thus the torsional couple is

/sin [alpha]' cos [alpha]' sin [alpha] cos [alpha] \
C ( ------------------------- - ------------------------ ),
\ r' r /

and the flexural couple is

/cos^2 [alpha]' cos^2 [alpha]\
A ( -------------- - ------------ ).
\ r' r /

The wrench (R, K) along the axis by which the spring can be held in
the form ([alpha]', r') is given by the equations

sin [alpha]' /cos^2 [alpha]' cos^2 [alpha]\
R = A ------------ ( -------------- - ------------- ) -
r' \ r' r /

cos [alpha]' /sin [alpha]' cos [alpha]' sin [alpha] cos [alpha]\
C ------------- ( ------------------------- - ----------------------- ),
r' \ r' r /

/cos^2 [alpha]' cos^2 [alpha]\
K = A cos [alpha]' ( -------------- - ------------- ) +
\ r' r /

/sin [alpha]' cos [alpha]' sin [alpha] cos [alpha]\
C sin [alpha]' ( ------------------------- - ----------------------- ).
\ r' r /

When the spring is slightly extended by an axial force F, = -R, and
there is no couple, so that K vanishes, and [alpha]', r' differ very
little from [alpha], r, it follows from these equations that the axial
elongation, [delta]x, is connected with the axial length x and the
force F by the equation

E[pi]c^4 sin [alpha] [delta]x
F = -------- ------------------------- --------,
4r^2 1 + [sigma] cos^2 [alpha] x

and that the loaded end is rotated about the axis of the helix through
a small angle

4[sigma]Fxr cos [alpha]
-----------------------
E[pi]c^4

the sense of the rotation being such that the spring becomes more
tightly coiled.

66. A horizontal pointer attached to a vertical spiral spring would be made to rotate by loading the spring, and the angle through which it turns might be used to measure the load, at any rate, when the load is not too great; but a much more sensitive contrivance is the twisted strip devised by W.E. Ayrton and J. Perry. A very thin, narrow rectangular strip of metal is given a permanent twist about its longitudinal middle line, and a pointer is attached to it at right angles to this line. When the strip is subjected to longitudinal tension the pointer rotates through a considerable angle. G.H. Bryan (_Phil. Mag._, December 1890) has succeeded in constructing a theory of the action of the strip, according to which it is regarded as a strip of _plating_ in the form of a right helicoid, which, after extension of the middle line, becomes a portion of a slightly different helicoid; on account of the thinness of the strip, the change of curvature of the surface is considerable, even when the extension is small, and the pointer turns with the generators of the helicoid.

If b stands for the breadth and t for the thickness of the strip, and
[tau] for the permanent twist, the approximate formula for the angle
[theta] through which the strip is untwisted on the application of a
load W was found to be

Wb[tau](1 + [sigma])
[theta] = ---------------------------------------.
/ (1 + [sigma]) b^4[tau]^2\
2Et^3 ( 1 + ------------- - ---------- )
\ 30 t^2 /

The quantity b[tau] which occurs in the formula is the total twist in
a length of the strip equal to its breadth, and this will generally be
very small; if it is small of the same order as t/b, or a higher
order, the formula becomes 1/2Wb[tau](1+[sigma])/Et^3, with sufficient
approximation, and this result appears to be in agreement with
observations of the behaviour of such strips.

67. _Thin Plate under Pressure._--The theory of the deformation of plates, whether plane or curved, is very intricate, partly because of the complexity of the kinematical relations involved. We shall here indicate the nature of the effects produced in a thin plane plate, of isotropic material, which is slightly bent by pressure. This theory should have an application to the stress produced in a ship's plates. In the problem of the cylinder under internal pressure (S 77 below) the most important stress is the circumferential tension, counteracting the tendency of the circular filaments to expand under the pressure; but in the problem of a plane plate some of the filaments parallel to the plane of the plate are extended and others are contracted, so that the tensions and pressures along them give rise to resultant couples but not always to resultant forces. Whatever forces are applied to bend the plate, these couples are always expressible, at least approximately in terms of the principal curvatures produced in the surface which, before strain, was the middle plane of the plate. The simplest case is that of a rectangular plate, bent by a distribution of couples applied to its edges, so that the middle surface becomes a cylinder of large radius R; the requisite couple per unit of length of the straight edges is of amount C/R, where C is a certain constant; and the requisite couple per unit of length of the circular edges is of amount C[sigma]/R, the latter being required to resist the tendency to anticlastic curvature (cf. S 47). If normal sections of the plate are supposed drawn through the generators and circular sections of the cylinder, the action of the neighbouring portions on any portion so bounded involves flexural couples of the above amounts. When the plate is bent in any manner, the curvature produced at each section of the middle surface may be regarded as arising from the superposition of two cylindrical curvatures; and the flexural couples across normal sections through the lines of curvature, estimated per unit of length of those lines, are C(1/R1 + [sigma]/R2) and C(1/R2 + [sigma]/R1), where R1 and R2 are the principal radii of curvature. The value of C for a plate of small thickness 2h is (2/3)Eh^3/(1 - [sigma]^2). Exactly as in the problem of the beam (SS 48, 56), the action between neighbouring portions of the plate generally involves shearing stresses across normal sections as well as flexural couples; and the resultants of these stresses are determined by the conditions that, with the flexural couples, they balance the forces applied to bend the plate.

68. To express this theory analytically, let the middle plane of the
plate in the unstrained position be taken as the plane of (x, y), and
let normal sections at right angles to the axes of x and y be drawn
through any point. After strain let w be the displacement of this
point in the direction perpendicular to the plane, marked p in fig.
28. If the axes of x and y were parallel to the lines of curvature at
the point, the flexural couple acting across the section normal to x
(or y) would have the axis of y (or x) for its axis; but when the
lines of curvature are inclined to the axes of co-ordinates, the
flexural couple across a section normal to either axis has a component
about that axis as well as a component about the perpendicular axis.
Consider an element ABCD of the section at right angles to the axis of
x, contained between two lines near together and perpendicular to the
middle plane. The action of the portion of the plate to the right upon
the portion to the left, across the element, gives rise to a couple
about the middle line (y) of amount, estimated per unit of length of
that line, equal to

/dP^2w dP^2w \
C ( ----- + [sigma]----- ), = G1,
\dPx^2 dPy^2 /

say, and to a couple, similarly estimated, about the normal (x) of
amount

dP^2w
-C(1-[sigma]) ------, = H,
dPxdPy

say. The corresponding couples on an element of a section at right
angles to the axis of y, estimated per unit of length of the axis of
x, are of amounts

/dP^2w dP^2w\
-C( ----- + [sigma]----- ), = G2
\dPy^2 dPx^2/

say, and -H. The resultant S1 of the shearing stresses on the element
ABCD, estimated as before, is given by the equation

dPG1 dPH
S1 = ---- - ---
dPx dPy

(cf. S 57), and the corresponding resultant S2 for an element
perpendicular to the axis of y is given by the equation

dPH dPG2
S2= - --- - ----.
dPx dPy

If the plate is bent by a pressure p per unit of area, the equation of
equilibrium is

dPS1 dPS2
---- + ---- = p, or, in terms of w,
dPx dPy

dP^4w dP^4w dP^4w p
----- + ----- + 2---------- = --.
dPx^4 dPy^4 dPx^2dPy^2 C

This equation, together with the special conditions at the rim,
suffices for the determination of w, and then all the quantities here
introduced are determined. Further, the most important of the
stress-components are those which act across elements of normal
sections: the tension in direction x, at a distance z from the middle
plane measured in the direction of p, is of amount

3Cz /dP^2w dP^2w\
- ---- ( ----- + [sigma]----- ),
2h^3 \dPx^2 dPy^2/

and there is a corresponding tension in direction y; the shearing
stress consisting of traction parallel to y on planes x = const., and
traction parallel to x on planes y = const., is of amount

3C(1 - [sigma])z dP^2w
---------------- ------;
2h^3 dPxdPy

these tensions and shearing stresses are equivalent to two principal
tensions, in the directions of the lines of curvature of the surface
into which the middle plane is bent, and they give rise to the
flexural couples.

69. In the special example of a circular plate, of radius a, supported
at the rim, and held bent by a uniform pressure p, the value of w at a
point distant r from the axis is

1 p /5 + [sigma] \
-- -- (a^2 - r^2) ( ----------- a^2 - r^2),
64 C \1 + [sigma] /

and the most important of the stress components is the radial tension,
of which the amount at any point is (3/32)(3 + [sigma])pz(a^2 - r)/h^3;
the maximum radial tension is about (1/3)(a/h)^2p, and, when the
thickness is small compared with the diameter, this is a large
multiple of p.

70. _General Theorems._--Passing now from these questions of flexure and torsion, we consider some results that can be deduced from the general equations of equilibrium of an elastic solid body.

The form of the general expression for the potential energy (S 27) stored up in the strained body leads, by a general property of quadratic functions, to a reciprocal theorem relating to the effects produced in the body by two different systems of forces, viz.: The whole work done by the forces of the first system, acting over the displacements produced by the forces of the second system, is equal to the whole work done by the forces of the second system, acting over the displacements produced by the forces of the first system. By a suitable choice of the second system of forces, the average values of the component stresses and strains produced by given forces, considered as constituting the first system, can be obtained, even when the distribution of the stress and strain cannot be determined.

Taking for example the problem presented by an isotropic body of any
form[4] pressed between two parallel planes distant l apart (fig. 29),
and denoting the resultant pressure by p, we find that the diminution
of volume -[delta]v is given by the equation

-[delta]v = lp/3k,

where k is the modulus of compression, equal to (1/3)E/(1 - 2[sigma]).
Again, take the problem of the changes produced in a heavy body by
different ways of supporting it; when the body is suspended from one
or more points in a horizontal plane its volume is increased by

[delta]v = Wh/3k,

where W is the weight of the body, and h the depth of its centre of
gravity below the plane; when the body is supported by upward
vertical pressures at one or more points in a horizontal plane the
volume is diminished by

-[delta]v = Wh'/3k,

where h' is the height of the centre of gravity above the plane; if
the body is a cylinder, of length l and section A, standing with its
base on a smooth horizontal plane, its length is shortened by an
amount

-[delta]l = Wl/2EA;

if the same cylinder lies on the plane with its generators horizontal,
its length is increased by an amount

[delta]l = [sigma]Wh'/EA.

71. In recent years important results have been found by considering the effects produced in an elastic solid by forces applied at isolated points.

Taking the case of a single force F applied at a point in the
interior, we may show that the stress at a distance r from the point
consists of

(1) a radial pressure of amount

2 - [sigma] F cos [theta]
----------- ----- -----------,
1 - [sigma] 4[pi] r^2

(2) tension in all directions at right angles to the radius of amount

1 - 2[sigma] F cos [theta]
-------------- -------------,
2(1 - [sigma]) 4[pi]r^2

(3) shearing stress consisting of traction acting along the radius
dr on the surface of the cone [theta] = const. and traction acting
along the meridian d[theta] on the surface of the sphere r = const. of
amount

1 - 2[sigma] F sin [theta]
-------------- ----- -----------,
2(1 - [sigma]) 4[pi] r^2

where [theta] is the angle between the radius vector r and the line of
action of F. The line marked T in fig. 30 shows the direction of the
tangential traction on the spherical surface.

Thus the lines of stress are in and perpendicular to the meridian
plane, and the direction of one of those in the meridian plane is
inclined to the radius vector r at an angle

/2 - 4[sigma] \
1/2tan^(-1) ( ------------ tan [theta] ).
\5 - 4[sigma] /

The corresponding displacement at any point is compounded of a radial
displacement of amount

1 + [sigma] F cos [theta]
-------------- ------ -----------
2(1 - [sigma]) 4[pi]E r

and a displacement parallel to the line of action of F of amount

(3 - 4[sigma])(1 + [sigma]) F 1
--------------------------- ------ --.
2(1 - [sigma]) 4[pi]E r

The effects of forces applied at different points and in different
directions can be obtained by summation, and the effect of
continuously distributed forces can be obtained by integration.

72. The stress system considered in S 71 is equivalent, on the plane through the origin at right angles to the line of action of F, to a resultant pressure of magnitude 1/2F at the origin and a radial traction of amount

1 - 2[sigma] F
-------------- --------,
2(1 - [sigma]) 4[pi]r^2

and, by the application of this system of tractions to a solid bounded by a plane, the displacement just described would be produced. There is also another stress system for a solid so bounded which is equivalent, on the same plane, to a resultant pressure at the origin, and a radial traction proportional to 1/r^2, but these are in the ratio 2[pi]:r^(-2), instead of being in the ratio 4[pi](1 - [sigma]) : (1 - 2[sigma])r^(-2).

The second stress system (see fig. 31) consists of:

(1) radial pressure F'r^(-2),

(2) tension in the meridian plane across the radius vector of amount

F'r^(-2) cos [theta] /(1 + cos [theta]),

(3) tension across the meridian plane of amount

F'r^(-2)/(l + cos [theta]),

(4) shearing stress as in S 71 of amount

F'r^(-2) sin [theta]/(1 + cos [theta]),