1. Simple methods are described for applying known bending and twisting moments to pigeon bones. The ultimate bending and torsional strengths of the humerus and radio-ulna are determined.

  2. Lift distributions are calculated from a strip diagram on the assumption that local lift coefficient is constant across the span. The position of the centre of lift is calculated for (a) gliding, in which the relative air speed is entirely due to the forward motion of the bird; and (b) hovering, in which it is entirely due to rotation of the wing about the shoulder joint.

  3. Estimates of the ultimate load factor of the humerus in bending and twisting yielded 8·8 and 9·0 respectively in gliding, and 5·7 and 5·6 in hovering. Corresponding figures for the radio-ulna were 6·9 and 9·1 in gliding, and 4·0 and 5·1 in hovering.

  4. The pectoralis insertion is strong enough to apply 4·2g f in gliding and 2·9g in hovering, so the muscles would be forcibly extended before any danger could arise of the bones being broken by excessive lift.

  5. A lift coefficient of at least 3·4 is achieved during the downstroke of hovering.

In the pigeon, as in other members of the three groups of vertebrates which have developed advanced powers of flight—the birds, bats and pterosaurs—the bending and torsional strength of the proximal part of the wing lies in the humerus and radio-ulna. The humerus runs obliquely back from the shoulder joint to the elbow, and from there the radio-ulna runs obliquely forward to the wrist. In flight the radio-ulna is generally set at a large angle to the humerus, often about a right angle (Fig-8).

The elbow joint is free to bend in the plane of the wing, but transmits bending moments at right angles to this plane, and twisting moments. Because of the large angle between the radio-ulna and the humerus, bending moments applied by the lift force to the radio-ulna are transmitted to the humerus largely as twisting moments, and vice versa. At the shoulder joint, the head of the humerus is free to rotate through large angles in all of the six possible ways—up, down, forward, back, and rotation in the nose-up and nose-down senses : bending and twisting moments cannot, therefore, be transmitted to the body without the help of muscles inserting on the humerus.

In this paper the ultimate strengths of the humerus and radio-ulna under bending and twisting loads are examined. The question is then asked—How much lift would be needed to break the bones in various ways? This can be answered subject to making some assumptions about the distribution of lift over the wing, and, this done, the limitations imposed by the structural strengths of the wing bones on the pigeon’s performance can be considered.

Feral pigeons Columba livia were caught in nocturnal raids on Bristol University’s clock tower, and were kept in cages until wanted. Strength tests on bones were carried out as soon as possible after killing and dissection.

Mounting of bones

In strength tests on bones, the ends of the bones were embedded in Wood’s Metal (m.p. 70° C.) in brass cups, and the test forces were then applied to the brass cups. By this means the ends of the bone were firmly gripped, but as the Wood’s Metal mould fitted the bone closely, concentrated forces were avoided.

Bending strength

Bending moments were applied to the bones with the apparatus shown in Fig. 1. The lower brass cup was rigidly fixed, and the test force was applied horizontally to the lever arm projecting upwards from the upper one, by means of a string passing over a pully. A stream of water was directed into a bucket tied to the end of the string, and diverted as soon as the bone broke. The force F in the string causing failure was then determined by weighing the bucket of water on a spring balance graduated in steps of 100 g. wt. The moment arm D was measured from the point of attachment of the string to the lever, to the point at which the bone broke (dotted line in Fig. 1). FD was then taken as the bending moment causing failure.

The arrangement also subjected the bone to a shearing stress, but as this was generally less than 20 kg. wt./cm.2 at failure, it is unlikely to have influenced the result. This feature of the tests is therefore neglected, and results are regarded as a measure of the limiting bending moments which the bones could bear.

Torsional strength

For these tests the bone was mounted horizontally as shown in Fig. 2, the cup at one end being fixed, while that at the other was mounted on a shaft supported on two ball races, so that only torsional stresses could be applied to the bone. A lever pro-jected horizontally from this shaft, at right angles to it, and test moments were applied by hanging weights from the end of this lever.

Structure of the humerus

Although the anatomy of the pigeon’s skeleton is well known, some features of the bones under test will be pointed out here in order to enable the results of the tests to be more readily followed.

The humerus is basically a hollow tube of bone, more or less circular in section, which carries the bending and twisting moments set up by the aerodynamic forces acting on the wing (Fig. 3). It would not appear to be significantly loaded in direct tension or compression as a rule. The ends are widened and reinforced with bony trabeculae, the proximal end to form the ball of the shoulder joint and to carry the insertions of the muscles acting at this joint, and the distal end to form a joint with the radio-ulna which is strong and stiff in torsion and vertical bending, but free to bend fore-and-aft.

Fig. 4 is a view along the axis of the humerus from the proximal end, to show its orientation in life. The deltoid crest forms a shelf jutting forward on the dorsal side of the humerus, with the pectoralis muscle inserting on its ventral side. The dorsal surface of the deltoid crest is approximately flat, and parallel with the plane of the wing. This was used as a guide in setting up humeri in the bending apparatus, the bending force being applied at right angles to the flat dorsal face of the deltoid crest, and in such a direction as to bend the distal end of the bone dorsally.

Bending strength

The eight humeri of four pigeons, of average weight 393 g. wt. were broken in the bending apparatus. Failure occurred at points varying between 0·38 and 0·70 (averag-ing 0·53) of the overall length of the humerus from the proximal end, the average overall length being 4·83 cm. The bending moment at failure varied between 13·4 and 21·1 kg. wt. cm., averaging 17·0 kg. wt. cm.

Torsional strength

Six humeri were subjected to twisting moments in the apparatus of Fig. 2, and broke at moments between 11·2 and 14·1 kg. wt. cm., averaging 12·4 kg. wt.cm.

Structure of the radio-ulna

The radio-ulna of the pigeon shows the typical bird arrangement. The two bones articulate with one another at both ends, but are not fused. The ulna is the thicker bone, and is curved, bulging posteriorly, while the radius is thinner and straight. In the middle part of their length the two bones are quite separate, the radius being anterior to the ulna, and in section the line joining their centres is approximately perpendicular to the direction of the lift. Neither bone is pneumatic.

Bending strength

When subjected to a bending moment simulating the normal action of lift on the wing the two bones are side by side and their strengths are additive. The radio-ulnae of the same four pigeons, whose humeri were tested as described above, were broken in the same manner in the bending apparatus. The average overall length of the ulna was 6·06 cm. and the average point of failure was 0·33 of the overall length from the proximal end, though the variation in this measurement was rather wide, ranging from 0·17 to 0·76. The bending moment at failure varied between 7·7 and 12·4 kg. wt. cm., averaging 10·6 kg. wt. cm.

Neutral axis of torsion

Twisting the radio-ulna produces a somewhat more complicated situation than is seen in the approximately cylindrical humerus. Owing to the asymmetry of the radio-ulna, with two different-sized bones separated by a gap, at least one of the bones must be off the axis about which twisting is applied, and hence is subjected to a combination of bending and twisting moments. The choice of twisting axis therefore affects the results of torsional strength measurements, and in order to minimize bending the neutral axis of torsion was used. By definition, a force applied to this axis, and at right angles to it, produces bending without twisting.

The neutral axis of torsion was located by the method illustrated in Fig. 5. The procedure was carried out at both ends of a radio-ulna, and the position of the neutral axis of torsion, thus located, is marked in Fig. 6.

For setting up radio-ulnae in the twisting apparatus a jig was constructed by which a radio-ulna could be supported so that its neutral axis of torsion coincided with the axis of the twisting apparatus. With the specimen so held, Wood’s Metal was poured into the cups, and the jig was then removed.

Torsional strength

Five radio-ulnae, from three pigeons, were broken in the twisting apparatus. The twisting moment at failure varied between 7·12 and 8·84 kg. wt. cm., and averaged 7·66 kg. wt. cm.

In order to consider the significance of the mechanical properties of the skeleton an estimate of the lift distribution must be made. This is attempted below for two extreme conditions of flight-gliding and the downstroke in hovering.

The difference between these two conditions lies in the distribution of relative air speed over the span. In a straight, unaccelerated glide (Fig. 7,a) the wings are held motionless relative to the body, and the relative air speed is the same over the whole span—it is just the speed at which the bird is gliding along. In hovering (Fig. 7 b) the bird’s body is not moving at all, and the relative air speed at the body, including the shoulder joint, is zero. If the wing tip is distant r cm. from the shoulder joint, and the wing is flapped at w radians/sec., then the speed of the tip (and hence the relative air speed at that point) is cm./sec. The relative air speed increases linearly along the span from the shoulder joint to the tip.

In the downstroke of flapping flight other than hovering the relative airspeed at any point on the wing is the resultant of two components, one due to the forward motion of the whole bird, and the other due to the angular motion of the wing relative to the body. The velocity distribution will thus be intermediate between that for the hovering case, in which the former component is zero, and that for the gliding case, in which the latter is zero.

Estimation of lift distribution

To obtain a lift distribution the wing is divided up into 10 chordwise strips, and it is assumed that the lift on each strip is proportional to the product of its area and the square of the local relative air speed, which amounts to assuming that the local section lift coefficient is constant along the span. Readers with aeronautical knowledge may think this assumption somewhat rudimentary, but it provides a simple basis for making an approximation, in the absence of any evidence favouring some more ela-borate assumption. The analysis refers, of course, only to positive (i.e. right-way-up) loading of the outstretched wing—more complicated situations, such as the inverted loading seen in the upward ‘flick’ in the upstroke of slow flapping flight (Brown, 1948) are not considered here.

The lift force acting on each strip of wing is assumed to act through a point whose spanwise position is in the middle of the strip, and whose chordwise position is a quarter of the chord behind the leading edge. The use of this ‘quarter-chord point ‘for the centre of lift of a wing section follows from the theory of thin aerofoils, and is a good approximation for most real wing sections (Abbott & Doenhoff, 1959).

A torsion arm and a bending arm are now measured for each strip with respect to the humerus (Fig. 8) and the radio-ulna (Fig. 9). The torsion arm is the distance between the centre of lift of the strip and the neutral axis of torsion of the bone (assumed to coincide with the axis of the cylindrical part of the bone in the case of the humerus). The bending arm is the distance between the centre of lift of the strip and an axis at right angies to the torsion axis. This axis passes through the centre of rotation of the head of the humerus in Fig. 8 and through the average point of failure in bending of the radio-ulna in Fig. 9.

Gliding—location of centre of lift

If speed and local lift coefficient are constant across the span, the lift for each strip is proportional to its area, and the lift distribution is the same as the area distribution (Fig. 10 a). These conditions represent gliding flight.

If the area of the ith strip is si, and its torsion arm about the humerus ti, then the lift on the strip applies a twisting moment proportional to siti to the humerus. The total twisting moment Mt of the lift about the axis of the humerus is the sum of the contributions made by each strip, that is
formula
Mt, referred to as a ‘moment ‘actually has the dimensions of volume, since wing area has been substituted for lift force. To obtain the mean torsion arm t¯ for the whole wing, Mt is divided by the total wing area, S
formula
Similarly, if the bending arm of the ith strip is bi the total bending ‘moment’ is
formula
and the mean bending arm b¯ is
formula

In Table 1 this calculation is carried out for the humerus, and the mean bending arm is found to be 12·4 cm., while the mean torsion arm is 7·0 cm. These distances are marked relative to their respective axes in Fig. 8, giving the point marked with a hollow star. This is the centre of lift in gliding, and may be regarded as the point through which the lift appears to act.

Loading of the humerus

The bending moment Mb(ult) and the twisting moment Mt((ult) at which the humerus breaks having been found, two estimates are available for the maximum lift Lmax which the wing can develop without breaking the humerus. Lmax is obtained by dividing each maximum moment by the appropriate moment arm. The mean bending arm b¯ is the distance between the centre of lift and the centre of rotation of the head of the humerus, but the humerus failed, on average, at a point 2·5 cm. distal to this. Therefore Lmax is given by
formula
Similarly, from the twisting measurements
formula

The average weight of the pigeons used was 393 g. wt., so if the body is assumed to contribute no lift, each wing must carry half the weight, or 196 g. wt., in a straight, unaccelerated glide. The bending and torsional strengths of the humerus are thus sufficient to carry 8·8 and 9·0 times the weight respectively. The humerus may be said to have ‘ultimate load factors’ of 8-8 and 9-0 in bending and torsion respectively. The similarity of the two figures indicates that the humerus is about as likely (or unlikely) to break in bending as in torsion.

Pennycuick & Parker (1966) found that the pectoralis insertion is strong enough to apply a maximum bending moment of 10·2 kg. wt. cm. about the centre of rotation of the head of the humerus. Dividing this by b¯, the corresponding lift would be 0·82 kg. wt., or 4·2 times the half-weight. If the lift exceeded this amount, the pectoralis would presumably be forcibly extended (assuming that the muscle is not capable of breaking its own insertion), and the wings would rise, so preventing the lift from becoming great enough to break the humerus. One may conclude from this that it would be practicable, and safe, for a pigeon to develop lift equal to about four times its own weight in gliding flight, and that the ultimate strength of the humerus would stand about 214 times this. It may be said that the ‘proof load factor ‘is about four, with the ultimate load factor being about 214 times as great.

In glider design a proof load factor of 4 would be considered low, 6 being more usual. An ultimate load factor of 1·5 times the proof load factor would, however, normally be regarded as giving an adequate margin of strength, so that ultimate load factors are commonly around 9 in gliders, as in the pigeon.

An important difference is that, whereas virtually any aircraft can be broken up in flight by violent application of the controls at high speed, it appears that the pigeon would not be able to break its wings in this way. The wings would be forced up, so relieving the load, before the lift could become dangerously large.

Lift equal to 4 times the weight (4g) is as much as a prudent glider pilot would intentionally apply in flight, and the pigeon, for the reason just given, can go up to about the same limit without danger. This limits the maximum acceleration which can be applied normal to the flight path, and hence the minimum turning radius. For instance, if the pigeon is assumed to be able to develop a maximum lift coefficient of 3, which is reasonable, if not conservative, in view of the calculations set out below (pp. 231-232), it would be able to circle at a speed of 11·5 m./sec. with 75° of bank, and its minimum turning radius would be 3·4 m.

Loading of the radio-ulna

Since none of the flight feathers insert on the humerus in the pigeon, one may be reasonably confident in assuming that the moments set up by the lift are applied in toto to its distal end. In the case of the radio-ulna the lift contributed by strips 8-10 approximately (Fig. 8) acts along the length of the radio-ulna, and is not applied to its distal end. These strips represent 29·4 % of the area, and hence of the lift on current assumptions, but as can be seen from Fig. 8 their moment arms in both bending and torsion would be small with respect to the radio-ulna. Their moment contributions would therefore be small as well as doubtful, and to simplify matters they are neglected.

The moments acting on the distal end of the radio-ulna are assumed to be due to the lift on strips 1-7, as shown in Fig. 9, in which bending and torsion arms for these seven strips are drawn in. The mean bending and torsion arms for this distal part of the wing with respect to the radio-ulna are calculated as before (Table 2), and the resulting centre of lift in gliding is marked with a hollow star in Fig. 9. By the same reasoning as before, the lift acting through this point, sufficient to break the radio-ulna in bending, would be 0·95 kg. wt., while 1·26 kg. wt. would be required to break it in torsion. The lift carried by strips 1-7 would be 70·6 % of the half-weight, e. 138 g. wt., so that the ultimate load factors of the radio-ulna in bending and torsion are 6·9 and 9·1 respectively. Once again there is no danger of these values being approached before the pectoralis muscle is forcibly extended.

Hovering—centre of lift

The calculation of the position of the centre of lift in hovering proceeds as for the gliding case, except that the lift on each chordwise strip is no longer proportional to the area of the strip, because each strip is moving at a different speed. The speed of the zth strip is proportional to its flapping radius, ri, which is the spanwise distance between the centre of lift of the strip and the centre of rotation of the head of the humerus. The lift is therefore proportional to the product of the area of the strip and the square of the flapping radius—assuming as before that local section lift coefficient is constant over the whole span.

The resulting lift distribution is compared with the gliding lift distribution in Table 3 and Fig. 10, the total lift represented being the same for both curves. The lift is now concentrated on the distal part of the wing—the proximal parts, although large in area, develop little lift because they are only moving slowly. Eighty-two per cent of the lift is developed on the distal five strips, in the hovering case, as opposed to 45 % in the gliding case.

Loading of the humerus

The calculation of mean bending and torsion arms for the humerus in hovering is set out in Table 1 (the ‘moments’ now have dimensions L5), and the centre of lift in hovering thus located is marked with a solid star in Fig. 8.

It can be seen that the centre of lift is considerably further out in hovering than in flapping, and this, of course, reduces the maximum lift which can be developed without breaking the humerus. The bending strength of the humerus would limit the lift to 1·12 kg. wt. (5·7g), and the torsional strength would limit it to 1·10 kg. wt. (5·6g). The maximum lift which could be carried by the pectoralis tendon would be 580 g. wt. (2·9g). Once again it would not be possible to develop enough lift to break the humerus, though the ratio of the ultimate to the proof load factor is reduced to about 2 in hovering, instead of 214 as in the gliding case. Aerodynamical considerations (see below, pp. 231-232)) suggest that the wing would be able to develop only mar-ginally over 1g in hovering, so in fact there is a substantial reserve of strength.

Loading of the radio-ulna

The hovering lift distribution (Fig. 10 b) shows that only 3·6% of the lift would be developed on strips 8-10, as opposed to 29·4% in the gliding case, so one may assume with greater confidence than in the gliding case that the moments applied to the distal end of the radio-ulna are entirely due to the lift on strips 1-7.

The calculation set out in Table 2 shows that the radio-ulna would fail in bending if the total lift on strips 1-7 exceeded 0·75 kg. wt. (4·0g) and in torsion if it exceeded 0·97 kg. wt. (5·1g).

Lift coefficient in hovering

The lift coefficient CL is a dimensionless number which enables direct comparisons to be made between wings of different sizes flying at different speeds. It is the ratio of the lift to a standard force, the latter being the product of the wing area and the pressure which would be obtained in a blind tube pointed upstream. It is given by
formula

where L is the lift in dynes, ρ the density of the fluid in g./c.c., v the speed in cm./sec., and S the wing area in cm.2 If all the relevant quantities are expressed in some other compatible system of units (like the F.P.S. system, for instance) the same lift coefficient results.

If a pigeon is assumed to be hovering in such a way that the lift during the down-stroke just equals the weight, the lift on each chordwise strip can be obtained from Table 3, while its area is given in Table 1. The density of air is taken as 1·22 × 10−3 g./c.c., the sea level value in the I.C.A.O. standard atmosphere.

The remaining quantity required to calculate CL is the speed of the strip, which is equal to the product of the flapping radius (listed in Table 1) and the angular velocity of the wing. Pennycuick & Parker (196) found that pigeons at take-off, which approxi-mates to hovering, could flap their wings 8·9 times per second, through an angle of 142°, and that the downstroke lasts for about two thirds of the cycle. This gives an average angular velocity of about 33 radians/sec. during the downstroke.

Since the calculation of lift distribution was based on the assumption that CL does not vary across the span, Eqn. (5) can be solved using data for any strip, and the answer should be the same whichever one is chosen. For instance, using values from
formula

Higher lift coefficients than this have been obtained from aircraft wings without using active boundary-layer control devices, but this has been done at Reynolds Numbers in the region of a few millions. In the hovering pigeon the Reynolds Number varies across the span (Fig. 11), but would be between 40,000 and 50,000 for strips 1-6. It is difficult to obtain lift coefficients above about 1·5 at Reynolds numbers as low as this (Schmitz, 1960), and although the pigeon’s wing is elaborately slotted, its performance in this respect is somewhat startling.

Pigeons definitely can hover in still air (inside a pigeon loft, for instance), even if only for a few wingbeats, and can indeed generate lift slightly over their weight, sufficient to accelerate them into normal flight, so there seems no escape from the conclusion that the maximum lift coefficient, averaged over the span, must be in the region of 312. If the assumption of constant lift coefficient over the span were abandoned, it would follow that the local lift coefficient of some parts of the wing would have to be higher still.

Owing to the difference in velocity distribution it does not necessarily follow that the same maximum lift coefficient can also be achieved in gliding flight, but if it could, the minimum gliding speed would be about 5·4 m./sec. (18 ft./sec., 12 m.p.h.). This should be accessible to measurement.

The bending apparatus was built by Mr B. Knights, who also carried out the torsional strength tests on humeri. Mr W. R. H. Andrews made the torsion apparatus, and carried out the torsional strength measurements on radio-ulnae.

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