A M . ZOOLOGIST, 9:81-89 (1969).
Elementary Physics and Spider Webs
RUDOLPH M. LANGER
The J.R.M. Bege Company, Arlington, Massachusetts 02174
SYNOPSIS. Static forces in spider webs are discussed in terms of the changes they
produce in the position and shape of the web filaments. Material properties of the
silks are related to the structure and functions of the spider webs. Gravity and wind
forces are considered, but localized forces are particularly interesting. They may be
used to measure by simple optical methods the forces and energies involved in
different routine operations and emergencies.
Rapid disturbances of the filaments create two kinds of signals which circulate
separately through the web. Their speeds are given and their time development
describes the nature and location of the disturbing source.
Certain questions in the Earth Sciences
can be studied best from positions exterior
to the earth where we have only recently
gained access. There is enough to learn by
looking inward from outside the atmosphere to engage us indefinitely. Current
trends in the Space Reseaich Program are
in that direction. The study of Earth
resources from orbiting satellites is likely
to grow in importance, and permanent
platforms in nearby space may become
necessary.
The construction and use of large extraterrestrial stations would presumably require human beings to leave their protective enclosures at times to assemble, inspect, and repair the structures. These critical and essentially hazardous functions are
called Extra-Vehicular Activities (EVA).
The situations to be encountered are unfamiliar and unnatural for humans, but in
some respects they may well resemble,
more than anything else, routines in the
life of an orb-weaving spider. It behooves
us before we go too far in space to learn
what we can from this creature which
survives with its life constantly hanging on
a thread.
The astronaut in EVA, like the spider,
will prudently maintain a lifeline to a substantial body nearby when he can. His operating pressure is above that of the environment so he must be encased in an impervious articulated shell, partially flexible
and bulbous, partially rigid and tubular.
The structure to be built, being free of
gravitational and atmospheric restrictions
as to size, could well span kilometers in
space. To be economical in mass and fixed
in shape it would use networks of fine
filamentary tension members as far as possible. These are, except for size, the specifications of a spider web, and some antenna
structures already contemplated actually
resemble spider webs. Men might propel
themselves from place to place on such
web-like structures, transmit signals, and
even dispatch equipment safely, provided
they had enough understanding of the materials and the structural dynamics of the
system. Little is known about the difficult
dynamical problem of highly flexible structures and our experience with the nearest
terrestrial counterpart of the space station—namely, deep-ocean ship mooring
systems—suggests that there is much to
learn about the seemingly simple problem
of station keeping before it can be considered safe or economical.
It follows that the physical properties
and engineering behavior of spider webs
and silks and of the spiders themselves are
topics in need of further study, not only on
their own account for scientific purposes,
but for space and other applications as
well. This idea motivates the present report, an outcome of a research project partially funded by the Office of Advanced
Research and Technology of the National
Aeronautics and Space Administration
81
RUDOLI'H M. LANCER
Fig. I
(Contract No. NASW-1389)."- The main
purpose is to show how elementary methods can lead to new and useful insights
regarding flexible structures—specifically
spider webs. The methods were first applied to design and operational problems
on large mooring systems. These are unfortunately submerged structures, expensive
and difficult to observe. The spider web,
on the other hand, provides an outstandingly convenient and possibly unique display of quantitative information on highly
flexible structures which are good engineering models of certain space and ocean
systems and at the same time promise to be
useful in the study of animal and astronaut behavior.
Witt, and his collaborators (see Wilt
and Reed, 1965), have already demonstrated in many papers how a sophisticated
analysis of the simple geometry of webs
can yield valuable data, not only about
spiders but also on entirely different subjects. Eberhard's paper in this volume has
opened still another path in the study of
web-building behavior by combining geometric observations with computer-programming techniques. The importance of
web geometry can, therefore, be taken
for granted. This paper will consider forces and displacements in webs. These topics also depend on geometrical observations for elucidation, but these will be supported by data on masses, motions, elasticity, and electromagnetic forces.
l The final report on Contract NASW-1389,
hereafter designated (I), contains the material in
the present paper together with derivations of the
equations used. The equations attributed in this
manuscript to Appendix A or B [e.g., (A. 13) or
(B-6)] will always refer to the appendices in (I).
FILAMENTS UNDER TENSION
The force exerted by a spider and the
energy spent in installing a web-member
are reflected in the tension and stretch of
the member at the moment of installation.
Subsequent events bring about energy
changes which in turn are reflected in the
tension and stretch changes. Elastic filaments are, in fact, such good force-sensors
and energy storages that they are used in
many instruments to measure electric,
magnetic, gravitational, centrifugal, and
other forces. It is especially convenient for
spider or man when a flexible line serves
both as a working medium and as its own
indicator of performance.
The force of gravity is primary because
it is always present. Usually it is overshadowed or counterbalanced by other forces,
but in spider webs it may play a part because of the relatively very open spans
compared with the close weave in textiles.
A common human activity analogous to
web building is illustrated in Fig. 1 to give
a physical feeling for the problem under
consideration. A suitable analytical treatment is presented in Appendix A..
Suppose that the rope is 25 meters long
and that it weighs 0.1 kg per meter of
length (about 1 cm2 in section) and that
the man exerts a force comparable with his
own weight so that the tension in the line
is 100 kg. The sag in the line is then 8
cm—measurable, with care, but ordinarily
barely discernible.2 The corresponding sit2 This result is obtained from
Appendix A. The sag is taken as
height between the mid point of
end points which are at equal
whole line,
equation (A. 13)
the difference in
the line and the
heights. For the
83
PHYSICS AND SPIDER WEBS
s.. — st = 25 m: y. — y^ = 0; w = 0.1 kg/m:
T — 100 kg.
R (s) =
sec 6.
w
For the midpoint as S! and with the same w and
This form shows that the length of line R
T,
in which a change of direction of one radis2 — Sj = 12.5 m; sin 6\ = 0.
an takes place is at least T/w. It is smallest
Then compute,
for a horizontal line {i.e., R = T/w when
sec
6=1) and little change in direction
y2 — y, = 0.08 m = 8 cm.
occurs
(i.e., R—>oo) when the line is aluation for a spider weighing 10 mg or 10~7 most vertical (i.e., sec TT/2 = oo).
times the man's weight and in proportion
If the spider silk and nylon can be asexerting a force:
sumed equally elastic with a breaking
strength Sb = 100 kg/mm2, and a breaking
T B P = 10-7 T m = 0.01 g
stretch £ = 20% (Appendix B offers juson a silk thread 3 ^ in diameter so that, tification for these assumptions in certain
approximately at least,
instances) then both the spider's thread
and man's rope would have stretched
Wsp = 10-7 W m _ 1 0 - 8 k g / m _ 10-7 g / c m .
about 1% of their length under the tension
If the length spanned is 1% as long as for loads mentioned (namely 0.01 g and 100
the man; (s2—sx)m = 25 m.
kg, respectively). Either one could stretch
2
20
times as much before parting. On the
(s2—s^gp = 10~ (s 2 —s^ = 25 cm.
other hand a 1% stretch (e.g., 2.5 mm in 25
The sag (for a nearly horizontal segment) cm) is easy to measure with accuracy.
now turns out to be about 8 /j.. Again this The formula for stretch corrections merely
would be measurable, but only under the recognizes that the fractional stretch
special circumstances where a span of 25 equals the ratio of actual tension to breakcm could be examined and a small dis- ing tension and that total weight, w (s2—Si),
placement of 8 fi could be measured. This remains constant.
would require a precision comparator with
a 10-inch screw protected from air curT
Aw
As
rents and vibrations. A photograph ten inw
ches across would have to show a resoluThe rope in Fig. 1 can get wet and pick
tion and freedom from distortion of a few
up water amounting to several times its
microns overall.
dry weight. This increases the sag and deEquations (A.12), (A. 13), and (A.14), creases the radius of curvature for a given
of (I) show that the catenary described by tension. Droplets along a horizontal spider
equations (A.8), (A.9), and (A.10) re- web member can well increase its effective
duces, when tension is great compared w several hundred fold and curvatures
with total line weight, to a segment of a would be easy to see. Indeed a load of rain
parabola (or for short segments to a sim- or dew droplets can break a thread. It can
ple circular arc). This is made clear by be seen in the catenary equations (A.8),
means of the definition of radius of curva- (A.9), and (A. 10) that if the supports
ture in equation (A. 15). A better way to remained rigid so that (x —x ) remained
2
±
describe a taut line is to give the radius of fixed and if the filament failed
to stretch
curvature R (s) at any point s. This is so that the catenary shape remained undone in equation (A.19)
changed in spite of the increased w, the
line tension would have to increase in proportion to w. Compliance of the end
A<9
fi
points and line stretch permit a change in
which in the case of gravity reduces to shape so that the tension can be limited to
a value comparable with the weight of the
(A.12) or (A.16)
84
RUDOLPH M. LANGER
droplets supported. The analysis of this
situation is too long to include here but
some immediate applications to webs will
be listed later on.
Another important hazard for spider
webs is the wind load which acts chiefly on
the members normal to the wind (see
Appendix A). I favor equation (A.29) to
express the force per unit length fi normal to the filament
Since there is an upper limit to the fiber
stress, S, tolerated in a filament under tension, there are upper limits to the value of
R that can be tolerated regardless of the
diameter of the filament.
IMPLICATIONS FOR SPIDER WEBS
The aim up to this point has been to
cultivate a feeling for the forces on a flexible line in terms of the appearance of the
line. The remarks made about a man exert/ T / vi \2 I di \
ing a force of 100 kg on a nylon rope 1
fids- / —
I (
wds
2
cm
in cross section can be translated into
y N ^ »e ' W /
statements about the web of a nondescript
where v± is the component normal to the spider 10 mg in weight (10~7 times that of
filament of the wind velocity v. The sym- the man) exerting a force equal to its own
bol ve2 stands for
weight on a silk filament also 10~7 times
smaller than the rope in cross section so
7T
Pspg cm
that its diameter would be about 3 ^. A
load of 10 mg is likely to be easier on a 10
c
pa sec
mg
spider than 100 kg on an average man
if there are N adjacent filaments having a
but
the stress on the 3 /A spider silk would
mean diameter d (in microns). As indi2
cated in formula (A.30), when N = 4 and be the same as on the 1 cm nylon rope.
Each would be loaded to about 1% of its
d = 1^
breaking tension and each would be
ve — 12.4 cm/sec = 0.4 ft/sec = 0.25 knot
stretched about 1% of its breaking strain
= 0.3 mi/hr = 0.4 km/hr
(presumed here about 20%). Thus, a 10
cm length of spider silk would be stretched
and then
about 0.2 mm (0.2%) to 10.02 cm-a measurable amount.
Examples are listed here of the kinds of
conclusions about spider webs that can be
provided vi is expressed in cm/sec and d, is draw from the text and the appendices.
taken as 1 p. The normal force then
1. Filaments in a spider web do not
comes out in the same units as w, e.g., show sag due to gravity if a moderate teng/cm. Since the velocity, ve = 12.4 cm/sec, sion equal to the spider's weight acts in
is low (Appendix A), the ratio, vi/ve, can them.
attain large values in a storm. Thus wind2. A horizontal web-member uniformly
loading can be large compared with gravi- loaded by many small dew droplets would
ty-loading of w. This still applies when sag perceptibly in a length of 10 cm if the
droplets are attached to the web because weight of the droplets per cm of length
the normal force increases with diameter was 100 times that of the web filament.
as w/r.
The silk would stretch, and the angle of
The tension in a flexible filament can be inclination at the ends of the filament
determined when the normal force per unit would increase. A material that could not
length, fi, is known and R is observable: stretch and sag would be subject to excessive tension. A thread 3 ju in diameter
could support a tension of about 1 gm
For a nearly horizontal line, fi — w and (weight). The tension in an orginally hoR = As/Ad, so the measurement is easy. rizontal filament loaded by dew droplets is
\ 16 /
PHYSTCS AND SPIDF.R WEBS
85
greater than the weight of the droplets by radius of curvature regardless of length.
a factor inversely proportional to the size The total end-to-end difference in angle
of the angle of inclination at the ends of (A6?) will be proportional to length.
Only after taking a bend well over 90° in
the filament.
direction
will a member (with fixed end
3. The more inclined the web-member,
points)
have
stretched the 20% commonly
the less it shows sag and tension. The tension in a vertical line due to a droplet is obtainable in spider silks (and nylons).
just equal to the weight of the droplet. The longest member will be bent more
Moreover, a droplet is least likely to strike than a j4 circle and will part first, leaving the inner portions intact until higher
and to stick to a vertical filament.
loads act. Thus, there is considerable lati4. A web of given area, oriented near a tude in initial tension and final shape unvertical plane, will be safer against gravity der load. Tautness in the original line
loads if the horizontal members are shorter avoids sudden onset of high loads such as
than the vertical. The web of some spi- may happen in the cracking of a whip or
ders might, therefore, tend to be elongated the flapping of a flag, sail, or tent. Sudden
along the vertical axis, and the angular loads are destructive for materials and
spacing between radii could well be smal- structures. The high elongation found in
ler near the vertical axis than near the spider silks protects them against excessive
horizontal, so that the horizontal spiral overloads.
spans would tend to be smaller. There are
7. Air currents normal to web-elements
presumably optimal ratios but to predict
them would require a knowledge of the exert forces which increase with the square
parameters to be optimized. Witt and of the wind velocity. The force on a vertiReed (1964) have pointed to these asym- cal filament 1 ^ in diameter due to a gentle
metries in Diadematus. The nearly hori- breeze of 1 m/sec (2 mph) would be a
zontal webs of Uloborus div. do not seem hundred times as large as the force of gravto favor one axis more than another ity. The relative importance of weight
grows in proportion to the diameter. The
(Eberhard, unpublished).
5. A weight localized at a point in a response of the spider to this situation has
web, loads the supporting filament above it been to build webs in locations and orienand not the portion below. This reinfor- tations that reduce exposure to wind even
ces the presumption that there would be though this compromises the yield of insect
more radii more closely spaced in the upper prey. Large webs with heavy filaments are
portion of the web, and it also suggests often located near a wind-breaking object
that the hub in nearly vertical webs might and they are likely to be oriented so that
well be positioned above the geometrical the plane of the web is parallel to the
center of the web. This point is less clear prevailing local wind direction. Uloborus
than the others and is worthy of more webs often found near the nests of pack
detailed study. If the displacement of the rats in Arizona are, according to Eberhard,
hub does prove to be sensitive to gravity, usually oriented along the local nest conthen the redundancy of gravitational pro- tour so as to minimize wind forces.
tection might indicate that the silk has not
8. The more a filament can stretch withalways been of such high tenacity as it is out breaking, the greater the wind force it
at the present time. Perhaps the silk of can tolerate for a given breaking strength.
more primitive spiders tends to be lower in Spider silks have unusually high breaking
strength. Perhaps webs less exposed to elongations. A typical elongation of
gravity-loads reflect this in their geometry e = 20% would permit a filament to survive
as well as in the tensile strength of the silk. in 50-knot winds. The point of attachment
6. If all horizontal members of a web would then make an angle of over 50°
are uniformly loaded with droplets along with the downwind point which is normal
their length, then all will have the same to the wind on a filament 20 cm long.
86
Webs which have been stretched more
than a few percent would show yield according to the curves in Figure (B-l) of (1).
They would not return to their original
shape and would remain slack. A slack web
is subject to large excursions and relatively high accelerations even in mild gusts.
Further, as can be shown from Appendix
B, the signal-transmitting properties of
such a web would be highly abnormal.
9. An artifact caught in a web subjected
to wind acts only on upwind filaments to
give them the configuration of longer filaments. The equivalent added length is the
length between the actual member and the
virtual down-wind point that would have
the same total wind-drag as does the artifact.
10. The breaking tenacity, S^/p, and the
breaking elongation, c(l), are both exceptionally high for spider silks. Their product, Sbe/p, is consequently exceptional
also. This product is roughly equal to twice
the limiting energy density for substances
which are fairly elastic. The limiting energy density is the work that must be expanded per gram of silk (no matter what
the diameter of the filament) in order to
stretch it to breaking. The stretch energy in
a normal web must be small compared with
the stretch energy to break all the filaments.
Otherwise it would not be elastic and useful. The energy required to build a web
•with n radii of average length, r, could be
approximated by nmspgr where msp is the
spider's mass and g is the acceleration of
gravity. This product represents the energy
of lifting the spider's weight to a height
approximately equal to the full length of
the web. It, too, is small compared with
the energy required to stretch all filaments
to breaking elongations. On the other
hand, the stretching energy is small compared with the heat of combustion of the
web. It follows that web building is an
economical procedure energetically unless
the edible mass of prey is small compared
with the mass of web used in its capture. It
was thus advantageous for spiders to develop superior web materials and web designs rather than simply to lay out even
M. LANCER
larger webs. The superiority of spider silks
over other materials is already clear with
regard to tenacity and uptake energy. To
judge superiority of web design requires
deeper analysis.
The NASA report (I) treats these topics
in greater detail. Two of the results, not
made plausible in this brief paper are
presented for comment here. 1. Spider
webs are subject to destruction in exceptionally high winds. 2. A force equal to
the spider's weight (a natural unit in a
web) is ample to stretch a web element
straight and taut.
The first statement is derived from an
analysis of the bowing of a filament in a
cross wind. It was found in Appendix A of
(I) that the anchored ends of a web element could be deflected over 50° from
their initial direction before stretching the
silk by 20%. When the wind velocity
reaches about 100 knots the filaments are
subject to rupture. Winds or gusts over 100
knots do occur occasionally and so the
match between silk strength and web dimensions implied by the first statement is
significant. Species with silk of inferior
strength would be restricted to sheltered
positions for their webs. Another consideration will be adduced later on to support
this point of view.
The properties of silk are not crucial for
the second statement so long as the filaments are very fine. The thought here is
that the straight line structure of a web
gives the spider a means for observing directions, distances, and events within the
web. Such capability goes a long way to
make the web-spider system viable on the
basis of the tactile sense alone. There may
be little need for more specialized senses of
sight, smell, and hearing. Thus the very
success and versatility of tactile perception
in the rectilinear web environment may
stultify the development of other senses.
These unspectacular statements concerning commonplace physical properties do
apparently have some biological significance. They arose from an examination of
the effects of distributed forces preliminary
to a discussion of localized forces which is
87
1'HYSICS AND SI-IDER WEBS
Fig. 2
NOTATION
the primary purpose of this paper. The
preliminary study showed that gravity
would not bow the web elements perceptibly and could be disregarded in the discussion of localized forces but that winds or
air currents are strong disturbing influences. It would ordinarily be necessary to
shield against them in studying concentrated forces in webs.
fs°
f53
F=cos0 P I f(s)cos#sds-|-sin0K S iI f(s)sin0sds.
The coordinates can be chosen so that the
resultant vector is parallel to the Y-axis so
that cos0P = 0 and then
F = I f(
f(s)sin0sds.
Si
FILAMENTS UNDER CONCENTRATED LOADS
The notation in Figure 2 was devised
especially for flexible lines under gravity.
It serves readily for all kinds of forces and
bodies whether rigid or not, and is particularly convenient when the problem concerns the total change in slope angle
(62—#i = A0) between the points of suspension, Sj and s2. It is then possible to treat
the loading as concentrated at one or more
discrete points and the algebra becomes
very simple. It will suffice to begin with a
load distributed between s2 and Sj and then
let s2 and Sj approach a single common
point. The distributed vector force has a
resultant magnitude F describable by the
integral
The object is in equilibrium under the
action of the external forces F, T 2 , and
—Tj. The equilibrium conditions are
Fx—Txcos^ -|- T2cos#2 = 0
Fy—T1 sinOi + T2sin#2 = 0
The X components of the tension T x must
balance at the ends s2 and st since the
resultant X component Fx vanishes here.
A very useful equation for the resultant
force F emerges
F = —Fy = Tx(tan02— tan0a)
The tensions in the lines supporting the
body are thus
88
RUDOLPH M. LANCER
APPLICATION AND EXTENSION
Fcos02
Ta = -
sin (02—
There are here essentially three fundamental equations among the five quantities, tan 02, tan 0,, Tv T 2 , and F. The key
point in this whole paper is that when
three of these basic quantities are known
the other two may be found directly. This
holds also and is especially powerful for
concentrated loads because the resultant
force F is then merely the load at the point
where the slope suddenly changes. It does
not matter how sudden the change is so
long as the suspending lines are able to
hold the total load. It is typically possible
to observe the angles Qx and 02 and often
there are means for determining something
about the forces. A calibrated weight may
be used as the force F and then the line
tensions T j and T 2 become know. This is
the basis for studying the physical properties of spider silk and of web structure.
A particularly simple situation arises
when 0j = —Q.z because then
qn
1 — ±2
-c
'
2 sin 02
and the line tension is given as precisely as
the sine of the bending angle and the
load F can be measured. If the elastic
properties of the silk are known it becomes
possible to measure forces exerted by spiders in many of their activities. Web filaments, properly illuminated or photographed, permit measurements of small angles
accurate to ±0.1° or better. Sometimes it
is possible to draw useful conclusions in
spite of angular errors over ten times as
large.
Instruments constructed on these principles are now in use to study material properties of spider silks and other filamentous
substances. Phenomena observed in silk
from functioning webs often lie outside
elementary physics. A problem under consideration now is the persistence of the
effects of stress after a stress has subsided.
Little is known about the mechanical
properties of spider silks beyond a few
values of breaking strength and stretch.
Appendix B of (I) summarizes recent
findings of the U. S. Army Materials and
Mechanics Research Center on certain
North American species. Lucas, Shaw, and
Smith (1955) give additional data on
several fibroins mentioning interaction
with water and recovery after loading
(memory). Both of these investigations
dealt with raw silk and left out of account
the changes brought about by processing.
A web is a sophisticated structure whose
design reflects these subtleties. Spider behavior also must be understood in terms of
properties of silk already processed and
normally installed in a web.
Appendix B of (I) treats two signal velocities on a filament, a longitudinal velocity of about 2 km/sec and a transverse
velocity which is ordinarily much slower. A
disturbance at any point spreads out with
these velocities according to its nature. A
torsional signal (motion) is also possible.
All of these velocities are subject to significant change due to violent motions of prey
or wind. There the non-linear effects play
a part related to the analysis by Douglas
(1967) of materials under extreme tensions. It would be presumptuous to imagine that a spider is as insensitive to these
matters as we are. The spider, in reacting
lo visible flexible lines, in fact gives us a
chance to appreciate certain aspects of its
behavior which in other animals (including man) might be even more subtle. The
convenience of observing concentrated
loading in flexible systems is emphasized in
this article in order to encourage work
along these lines. It will take a long time
and many active groups to cover this field
including high loading, high rates of
strain, and effects of memory.
The moderate forces, quick motions, and
limited displacements typical of a spider
are usually subject to analysis by means
of the formulae already presented. Even
qualitative observation with these results
PHYSICS AND SPIDER WEBS
in mind can give much that is not discussed in the literature. The wide spread,
zig-zag, preliminary spiral of Nephila clavipes shows immediately the direction of
motion of the spider and also the ambidextrous character of its operations as it
goes back and forth building or repairing
the spiral. Moreover the surprising equality (more or less) of tension in radial and
spiral (preliminary) members is apparent,
as is the relatively low tension in the
close-spaced spirals. The structure shifts
during the process so that the tension
relaxes either in the radial strands or the
preliminary spiral strands in different places on the periphery of the web. A spider
incapacitated on the right or left side
would probably take much more time and
effort to repair a web. Further time and
motion studies of Nephila would be rewarding.
Some more general problems of web design are touched upon in (I) where typical orb-webs (denoted trigonal webs) are
shown to be such that a tug in one element
results in a proportional response in every
other element of the web. The time-delay
in response, as indicated in the comment
on signal velocity for spicier filaments, is of
the order of milliseconds. It is thus plausible
to suppose that a spider can sense or at
least utilize intervals of the order to ten
microseconds just as the human pinna
(Batteau, 1967) can utilize time-delays of
89
that order in the localization of sources of
sound.
More fundamental still is the discussion
of web design in (I) where the conclusion
is reached that a web is designed to capture available prey from a maximum area
and in order to do so the cross section of
the filament is made as small as is feasible.
It lies close at hand to suppose that spider
silk would become so gossamer thin that
webs would begin to be vulnerable to environmental limitations—specifically to exceptionally high winds. This is regarded as giving increased significance to the result already mentioned that webs of normal dimensions are subject to rupture in (100
knot) winds that can and do occur in their
environment.
Batteau, W. D. 1967. The role of the pinna in
human localization. Proc. Roy. Soc. London, 13,
158:158-180.
Douglas, W. J. 1967. Natural vibrations of finitely
deformed structures. AIAA Journal 5:2248-2253.
Langer, R. M. 1959. Catenary problems associated
with deep ocean mooring operations. Final report, Navy Contract XObs-74310.
Langer, R. M. 1969. Extra-vehicular activity. Final
report, NASA Contract NASW-1389. This report
is referred to herein as (1).
Lucas, F., J. T. B. Shaw, and S. G. Smith. 1955.
The chemical constitution of some silk fibroins
and its bearing on their physical properties. J.
Textile Inst. 46:440-452.
Witt, P. N. and C. F. Reed. 1965. Spider-web
building. Science 149:1190-1197.