Sutural Pattern and Shell Stress in Baculites with Implications for Other Cephalopod Shell Morphologies Author(S): David K

Paleontological Society

Sutural Pattern and Shell Stress in Baculites with Implications for Other Cephalopod Shell Morphologies Author(s): David K. Jacobs Source: Paleobiology, Vol. 16, No. 3 (Summer, 1990), pp. 336-348 Published by: Paleontological Society Stable URL: http://www.jstor.org/stable/2400792 Accessed: 23/02/2010 16:21

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http://www.jstor.org Paleobiology, 16(3), 1990, pp. 336-348

Sutural pattern and shell stress in Baculites with implications for other cephalopod shell morphologies

David K. Jacobs

Abstract.-In Baculites,a straight shelled ammonite, the constructional limits on shell shape resulting from the limited strength of nacre in tension are circumvented by a system of vaults in the phragmocone. Vaults bridge between regions of the phragmocone supported by the complex ammonite septal suture, and maintain the shell wall in compression when hydrostatic load induces bending moments. To determine how these vaults interact in the phragmocone to resist hydrostatic loading, measurements were made on a suite of Baculites specimens. In Baculites there is a statistically significant inverse relationship between circumferential curvature (radius of curvature) of the vaulted shell surface and the size of vaults spanning between sutural elements supporting the phragmocone. The inverse relationship between radius of curvature and the sizes of spans in this system of vaults results in the generation of comparable reactive forces at the ends of the vault spans where adjacent vaults interact. The equivalence of these reactive forces prevents the lateral displacement of the vault ends. Consequently, compressive stresses from adjacent vaults are superimposed on, and reduce, the tensional stress component of bending. Limiting tensile stress is of utmost importance in a lightweight shell composed of a brittle material such as nacre, which is strong in compression but weak in tension. Baculites is particularly appropriate for this study because its straighlt shell is curved only in the circumferential direction, thus simplifying the problem. However, sutural patterns in coiled ammonites appear to be similarly constrained to produce vaults in the phragmocone which vary inversely in curvature and span size.

David K. Jacobs. Department of Geological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0796. Current address: Museum of Paleontology, University of California, Berkeley, California94720

Accepted: March 21, 1990

Introduction in the structure of the shell itself. This un- The structural implications of sutural com- derstanding of the hydrostatic load borne on plexity in ammonites have long been a subject the shell led to the application of membrane of scientific interest. Buckland (1836) argued stress formulae for hollow thin-walled that the complex septal sutures of ammonites spheres and cylinders to the concave portions served to support the flattened flanks of am- of nautiloid septa and the tubular siphuncle monite phragmocone against hydrostatic (e.g., Westermann 1973, 1977, 1982). In these pressure.This idea was also advocated by Pfaff thin-walled structures, hydrostatic load pro- (1911), Spath (1919), and Westermann (1956). duces a purely tensional or compressional However, conclusive evidence of the pres- stress, termed a "membrane stress." Mem- sure difference across the shell wall of ceph- brane stresses are proportional to the radius alopods was not available to these authors of curvature and the pressure difference across (for a historical review of functional analyses the concave nautiloid septum and tubular si- of shelled cephalopods see Jacobs 1990). Fi- phuncle; they are inversely proportional to nally, Denton and Gilpin-Brown (1961, 1966) the septal or siphuncular thickness. and Denton et al. (1967) unequivocally dem- The distribution and magnitude of the onstrated that modern shelled cephalopods stresses in the ammonite phragmocone are contain gas at or below atmospheric pressure less easily assessed than the stresses in con- while submerged. Consequently, modern cave septa or tubular siphuncles. There are shelled cephalopods and, by analogy, fossil two reasons for this. First, ammonite phrag- shelled cephalopods support the entire hy- mocones do not conform to simple shells of drostatic load of the overlying water column revolution for which equations of membrane

? 1990 The Paleontological Society. All rights reserved. 0094-8373/90/ 1603-0006/$1.00 SUTURAL PATTERN AND SHELL WALL STRESS 337 stress are readily derived. The radius of curvature around the whorl in the ammonite RADIUS OF CURVATURE phragmocone is not constant; consequently, one would expect differential stress generation in different regions of the phragmocone (Westermann 1971). Secondly, the phrag- Ri-Flank Radius mocone is supported at intervals by the septa. This support results in a flexural or bending R2-Dorsal Radius problem as the shell wall, bearing hydrostatic load, passes over and between the supports R3-Ventral Radius provided by the septal suture. Hewitt and Westermann (1986) attempted to model the bending stresses in spaces be- R2 tween sutural support in the relatively flat flanks of a Calliphyllocerasshell by analogy to Ri a flat plate. However, even the flanks of oxyconic ammonites have some curvature; they are rarely, if ever, exactly flat. Due to their R3 curvature, portions of the phragmocone bridging between sutural elements form a series of vaults rather than flat plates. In a tra- verse around the non-circular whorl section of a compressed ammonite morph, the radius of curvature necessarily changes (Fig. 1). In addition, spacing of sutural support of the shell wall is not uniform. The complexity of the suture and coiling geometry results in in the size of spans of shell wall variation FIGURE 1. The thick curve represents a cross-section between sutural elements in the sutural pat- through a Baculites shell. Three radii of curvature de- tern. This variation in sutural spacing appears scribe the major regions of curvature around the circum- ference of a Baculites shell. The radius of curvature of the to relate to the local curvature of the shell broad flanks of the shell, Rl, is many times greater than around the whorl. The larger radius of cur- that of the intermediate radius of curvature of the dorsal vature, "flatter," portions of the whorl have shell surface, R2, which is in turn substantially greater than the radius of curvature of the ventral R3. more closely spaced sutural support than the region, more tightly curved (smaller radius of curvature) portions of the shell (Buckland 1836; curvature and the spacing of support is ex- Spath 1919; Westermann 1956, 1971, 1975). amined to determine if it relates to the func- Due to the complex interdigitation of adja- tion of the ammonite shell as a series of vaults cent ammonoid sutures, septal support sur- supporting hydrostatic load. This will address rounds the vault spaces in the sutural pattern. the long standing question of the function of A greater proportion of the load on a doubly the ammonite suture and how the ammonite curved vault surface will be transferred in the suture may be functionally constrained. direction of the smaller radius of curvature. The Phragmocone Conceived of as a Series of The smaller radius of curvature necessarily Vaults.-Vaults combine the axial or mem- occurs around the whorl rather than around brane stresses, associated with thin-walled the coil of a planispiral shell. Consequently, pressurized spheres and cylinders, with the vaults in the phragmocone will bear more of bending stresses normally associated with the load around the whorl rather than around loaded beams or flat plates (Salvadori 1971). the coil, and it is this circumferential direc- Unlike plates, vaults produce outward thrust- tion around the whorl that is of interest. ing at their ends when loaded. This outward In this work, the relationship between shell thrusting is a consequence of bending mo- 338 DAVID K. JACOBS

INTERACTIONOF THRUSTS IN A SERIES OF VAULTS c A -t B TA TB

TA' TB'

S S

FIGURE2. In a series of uniformly loaded vaults which meet at supports, SA and SB, the outward thrusts of vault AB, TA and TB must be met by thrusts of similar magnitude, TA' and TB', generated by adjacent vaults if vault AB is to remain in compression. If TA' and TB' are not as large as TA and TB then the vault ends will displace outward and tensile stress will develop on the interior of the vault at C. ments in the vault. Unless this outward vented by equivalent reaction forces, then no thrusting is met by an opposing reaction force, dilation of the material on the interior of the bending moments in the vault will result in vault can occur. If there is no dilation then a large tensile stress component on the in- there must be no tensional stress. terior of the vault (Fig. 2). Ammonite shells In a Roman aqueduct each vault has an ad- are composed largely of nacre, a brittle ma- jacent vault available to supply an opposing terial; nacre from Nautilus is reported to be force maintaining the series of vaults in com- two to three times stronger in compression pression. Similarly, around the whorl of the than tension (Currey and Taylor 1974; Currey ammonite phragmocone there are adjacent 1976). In addition, cracks or imperfections in spaces in the sutural pattern. These spaces the material will limit tensile strength to a should function as vault structures in the greater extent than it will limit compressive phragmocone providing reactive forces to the strength (Hewitt and Westermann 1986). As vault ends in a manner similar to adjacent a consequence of these properties of nacre, vaults in an aqueduct (Fig. 2). tensile stress due to bending must be limited Hydrostatic pressure uniformly loads the to a small fraction of compressive stress in an vaults comprising the ammonite phragmo- optimally designed cephalopod shell. cone. These vaults vary in radius of curvature One strategy for reducing the tensile com- and in span length. Theoretically, thickness ponent of bending in a beam or vault is to of the shell wall (vault thickness) could also superimpose a compressive axial stress on the vary. Variation in vault thickness would have whole structure. In concrete beams this is done consequences for the stresses resulting from by application of a stress prior to loading in the reaction forces and bending moments in what is termed "prestressing." In vaults, a the vault. However, variation in shell thick- compressive stress is superimposed on a ness around the ammonite whorl is not large bending stress with the same result; tensile (Westermann 1971). In this study, no system- bending stresses are avoided in a brittle ma- atic variation around the whorl of Baculites terial weak in tension, but strong in com- was observed. A series of measurements at pression. The superimposed compressive various points around the whorl of two Bac- stress is supplied by the outward thrusting of ulites specimens reveals that the minor vari- the vault itself. To generate a compressive ation in thickness observed was not associ- stress effective in opposing tension due to ated with a particular region of the shell (Table bending, the outward thrusting of the vault 1). must be met by an equal and opposite force. In contrast to the minimal variation in shell The problem can also be expressed in terms thickness and hydrostatic loading, there is of strain or displacement in the vault; if out- considerable variation in span size and radius ward displacement of the vault ends is pre- of curvature in the series of vaults comprising SUTURAL PATTERN AND SHELL WALL STRESS 339

the ammonite phragmocone. If this system of TABLE 1. Measurements of shell thickness around the vaults is to perform optimally when loaded, whorl of two Baculites specimens. span size and radius of curvature must covary Specimen K L in a manner that produces equivalent reac- Species B. sp. B. sp. tion forces at the vault ends. Equivalence of Repository Peabody Mus. AMNH Specimen No. A361 Am 29545 reaction forces would limit the tensional Whorl height 3.4 cm 5.6 cm stresses due to bending so that the brittle na- Whorl width 1.5 cm 3.7 cm creous material could be used to advantage. Data (shell thickness mm) Testingthe Interactive Vault Hypothesis.-The K L vaults comprising the ammonite phragmo- Dor- Flank Flank Dor- cone are complex in shape, and accurate cal- Venter sum 1 2 Venter sum Flank culations of reaction forces cannot be readily 0.36 0.36 0.38 0.36 1.30 1.33 1.33 0.34 0.35 0.33 0.36 1.28 1.31 1.40 obtained by conventional engineering pro- 0.35 0.36 0.34 1.39 1.41 1.51 cedures. Consequently, an expected general 0.39 0.37 1.54 1.48 relationship characteristic of vaults is used to test the interactive vault hypothesis. Engi- neering equations can be used to calculate thrusts at the ends of vaults of uniform width diffuse end conditions provided by the com- that lack lateral support (Roark and Young plex ammonite sutural pattern make it diffi- 1975); no such equations exist for vaults of cult to determine a precise location where the the complex shape defined by the spaces be- vaults end, much less how to characterize tween the sutural elements of the ammonite them in engineering terms. Due to the com- phragmocone. Perhaps more importantly, en- plex shape of the vaults comprising the Bac- gineering equations assume precise end con- ulites phragmocone, and the difficulties in as- ditions; vaults are either pinned, that is, free sessing the exact nature of the end conditions to rotate in the vertical direction, or fixed, of these vaults, direct calculation of the re- where no end rotation is possible. Because action forces at the vault ends is not under- the septum is thin and curved where it meets taken here. the shell, sutural support of the phragmocone On a more general level, vaults combine is likely to be strongly elastic or spring-like the structural attributes of cylindrical shells, (Hewitt and Westermann 1986). In addition, where the radius of curvature is the critical the vaults end in regions of more closely variable, with a bending problem where the spaced sutural elements. The combination of length of the span is critical (Salvadori 1971). proximity of sutural support at the vault ends, This is illustrated by an examination of the and the elasticity of the septa allows septal simplest vault equation, that of a statically support of the shell to be diffused over a re- determined vault or arch of unit width; this gion of the shell wall. These regions of su- equation suggests that an inverse relation- tural support occur between, and determine ship between vault size and radius of cur- the end conditions of, the vault spans (Fig. vature would be expected if reaction forces 3). The diffuse support between vaults is nec- are to be equivalent at the vault ends. The essary to limit bending stresses over the sup- outward horizontal thrusts at the supports at port. A single point of support would con- the ends of statically determined arches or centrate stress and result in bending in the vaults of unit width can be readily calculated shell wall as it passed over the localized sup- (Salvadori 1971): port, resulting in tension on the exterior of T = M/h = 0.5w12/h (1) the shell. Such bending is observed over the simple strut-like septal support of the flank The horizontal thrust T at the vault ends is a of Nautilus (Hewitt and Westermann 1987b). function of the bending moment M, which Diffuse support limits bending and the pro- is in turn a function of the load w and the duction of tensile stress in the nacreous shell moment arm 1 (Fig. 4). The bending moment wall. Although structurally advantageous, the at the vault end will be given by the formula 340 DAVID K. JACOBS

SUTURAL PATTERN

DORSAL

CROSS NK , INTERSUTURAL

VENTRAL

FIGURE3. The sizes of the vaults spanning between septal sutures in the shell wall can be approximated by the sizes of circles inscribed between the sutures in the complex sutural pattern. Note that the largest circles are in the ventral region, the region of the shell with the smallest radius of curvature, and small circles are present in the large radius of curvature flanks of the shell. Circles of intermediate size fit within the sutural pattern in the dorsal region of the shell of intermediate curvature.

M = 0.5w12. This moment, when divided by tails of the type of vault, a general expectation the maximum height or rise of the arch, h, of an inverse relationship between a squared above the horizontal span, provides the for- span length term and radius of curvature is mula for the horizontal thrust T generated at predicted if the thrusts at the ends of the the end of the arch. Note that the length term vaults are to be comparable in magnitude. is squared in this expression as a consequence of uniform load along the length of the beam; Methods the load itself as well as the moment arm are Measurements. -An uncoiled ammonite, functions of beam length. Hydrostatic load Baculites,was chosen because the straight shell will also act horizontally on the height of the provides a variety of curvatures around the vault, but this term is a negligible contribu- whorl without the complexity associated with tion to the outward thrust of the vault. From curvature normal to the whorl section that this reaction equation it is apparent that if T, results from coiling. Three criteria were used the horizontal force at the ends of the vault, to select specimens for measurement. Only is to remain constant, the square of the half- specimens lacking lateral ornamentation in span length, 12, will have to vary in concert the form of ribs or bullae were used, since with the rise of the vault, h. The height h such ornament would complicate measure- varies inversely with radius of curvature (if ment and analysis. Only specimens that length of the vault is held constant then the showed no obvious asymmetry or deforma- height of the vault depends on its radius of tion were selected. Lastly, only specimens curvature alone); therefore, 12 is expected to with at least three chambers exposed were vary inversely with the radius of curvature used. This allowed measurement of the su- R. The static vault equation above applies only tural pattern at a sufficient number of loca- to vaults which are pinned and free to rotate tions for a statistical test. For each of the ex- at their ends and midpoint. The radius of cur- posed chambers, measurements of radius of vature and bending terms related to span curvature and span length were made for the length are also critical in calculating the out- three regions of uniform curvature (venter, ward thrusts produced by indeterminant flank, and dorsal) in ten specimens of Bacu- vaults, vaults that are fixed at their ends (Roark lites. and Young 1975). Thus regardless of the de- Measurements of radius of curvature were SUTURAL PATTERN AND SHELL WALL STRESS 341

Vault Equation

~ w

R Al21 R~~TW2h ~~~T=

FIGURE 4. The thrust T at the ends of a uniformly loaded vault is proportional to the load per unit length, w, multiplied by the square of the halfspan length 1. This thrust is inversely related to the height or rise of the vault, h; the rise of the vault is inversely related to the radius of curvature, R. It follows that the thrust at vault ends will be proportional to 12 and R. If the thrust produced by the vaults adjacent regions of the shell are to be equal, 12 must vary inversely with R.

made by molding the specimen with mod- encountered in a transect around the shell. eling clay and cutting away strips of the clay. The length of the radius of each of these cir- The strips of clay were then placed on paper cles was recorded for the ventral, flank, and and the curvature of the mold drafted on the dorsal curvatures characteristic of the shell. paper. The radius of this curvature was then These measurements were made for each determined using a compass. This procedure chamber where the sutural line was ade- was repeated for the three distinct curvatures quately exposed on the ten specimens. The of the shell, the ventral, flank, and dorsal radius of the largest circle inscribed in the curvature for every chamber measured in the sutural pattern consistently provides a rough ten specimens (Fig. 1). measure of the effective halfspan length in To obtain estimates of the vault span lengths the circumferential direction. The largest between sutural elements supporting the vaults in a region of constant shell curvature shell, the sutural pattern was drafted on my- are the vaults of interest since they will pro- lar; a compass was then used to find the larg- vide the largest outward thrusts which must est circle which would just fit within the spaces be met by thrust of vaults in adjacent regions in the sutural pattern (Fig. 3). These circles of the shell. These vaults are separated by one provide an approximation of the spacing of or several closely spaced septal elements. sutural support in the circumferential or hoop These regions where one or a number of sep- direction around the Baculites. Although lon- tal elements meet the shell in the septal su- ger transects that do not touch sutures may ture support the ends of, and provide the end be found in the circumferential direction, conditions for, adjacent vaults (Fig. 3). these transects pass through regions where The Statistical Test.-The general expecta- the shell is closely supported by adjacent su- tion of an inverse relationship between vault tures. This use of the largest inscribed circle span size between sutures, 12, and the radius provides an arbitrary but consistent means of of curvature, R, of the shell was tested using estimating the size of vaults in the circum- regression analyses. Because an exponential ferential direction given the complex and rather than a linear function was expected, a variable interdigitations of the suture pattern natural logarithm transformation of both 342 DAVID K. JACOBS variables was employed. The exponential re- formed on Baculites because it is straight and lationship in question was then tested by the lacks a secondary curvature associated with slope and significance of the regression line coiling. However, theoretical considerations produced by the two log-transformed vari- and a number of observations of the spacing ables. A slope of -1 would indicate a simple of septal elements on coiled shells suggest inverse relationship. Least squares regres- that the vault model applies to coiled am- sions were performed and subsequently con- monites as well. verted to reduced major axis regressions A straight phragmocone can be analogized through division of the regression slope by to a thin-walled cylindrical surface in the the correlation coefficient, as suggested by sense that the circumferential stress around LaBarbera(1986). The reduced major axis form the whorl will be the primary stress; on a of regression was preferred here because nei- cylindrical surface the stress is borne pref- ther variable was "independent." Regres- erentially in the circumferential rather than sions were performed for each of the ten spec- the longitudinal direction. This is not the case imens measured (Table 2). in a thin-walled sphere, where all directions of curvature are comparable and the stress Results generated in all directions is half that borne The regressions produced highly signifi- in the circumferential direction of a cylinder cant negative slopes for all ten specimens, of equal radius of curvature and thickness confirming the inverse relationship between supporting the same pressure difference across vault size and radius of curvature. The re- the wall. The cylindrical and spherical shapes duced major axis slopes ranged from -0.865 represent end points in a continuum of dou- to -1.270, with an average slope of -0.996, bly curved surfaces. In a cylinder, the radius trivially different from -1. All regression of curvature in the longitudinal direction is slopes were significant at the P < 0.0005 level, infinite and only the smaller direction of cur- confirming an excellent fit of the regression vature need be considered. If the radius of lines to the data. These results indicate that curvature around the coil greatly exceeds that in a series of vaults determined by the suture around the whorl, the situation approaches pattern of Baculites, vault area and radius of the cylindrical condition and the larger stress curvature covary in an inverse manner. The will be borne in the direction of smaller cur- conformity to a slope of negative one suggests vature. We would therefore expect that in that a simple inverse or hollow curve type of those ammonites where the ventral curvature relationship exists between the two variables. around the whorl is much smaller than around This simple inverse relationship indicates that the coil the interactive vault model would the series of spans between sutural elements apply and the spacing of sutural elements that comprise the ammonite shell, despite the would depend on the radius of curvature complexities of their shape and end condi- around the whorl. tions, interact in the predicted manner, func- In a transit around the whorl of an oxyconic tioning as a series of vaults. form such as Placenticeras, the ventral region The inverse covariation between halfspan has a small radius of curvature in the direc- length squared and radius of curvature re- tion around the whorl, whereas the radius of sults in relative equality of the reaction force curvature in the direction of coiling is much of the vaults around the phragmocone. This larger (Fig. 5, A). Assuming that the larger would minimize the tensile forces resulting direction of curvature around the coil can be from the bending inherent to vault action. neglected, Placenticeras should be analogous This minimization of tensile stress is advan- to Baculites, where a venter of small radius of tageous, and would be expected, in a shell curvature interacts with the broadly curved composed of brittle material such as nacre. flanks of the whorl. As in Baculites, vaults in the ventral region supply compressive reac- Discussion tion forces to the flanks of the ammonite. One Application of the Model to Other Shell Mor- can therefore predict that the sutural pattern phologies.-The work presented here was per- in Placenticeras should produce small vault SUTURAL PATTERN AND SHELL WALL STRESS 343

FIGUREs5. A, on the broadly curved flanks of Placenticeras, the sutures are closely spaced and therefore the vault spans are short. At the venter, where there is a smaller radius of curvature, the sutural spacing is much wider resulting in larger vault spans. Also note that on the extensive flank region of the compressed involute ammonite an increased number of sutural lobes, marked P, are required to achieve the close sutural spacing. B, in a globose pachydiscid ammonite, curvature around the venter and flanks is relatively constant. Note that the spacing between the elements of the complex suture is also relatively constant.

spans on the flattened flanks of the shell and closer spacing of support on the flanks of am- large spaces between sutures on the venter, monites and have related this close spacing similar to those produced by the spacing of of sutural elements to the need for greater sutural elements in Baculites. Examination of support against pressure in the flatter regions a Placenticeras reveals ornate closely spaced of the shell, although not to the vault model sutures on the flanks and widely spaced su- of support presented here. These authors ex- tures on the venter, thus confirming the pre- plicitly recognized a functional role in the dictions of the model (Fig. 5, A). relationship between shell curvature and In an ammonite lacking variation in radius proximity of sutural support. In addition, oth- of curvature around the shell, minimal vari- er authors have reported the related empirical ation in sutural spacing would be predicted relationship between shell curvature and the by the model. On globose ammonites, cur- number of sutural elements. Spath (1919) ob- vatures in the coiling direction and around served that increased numbers of sutural el- the whorl are uniform and nearly equal over ements were found on the flanks of phyllo- the exterior portion of the phragmocone (Fig. ceratids and ammonites that had evolved 5, B). In globose forms such as Stephanoceras broad curvature of the flank region. He sim- or Macrocephalites,sutural spacing varies min- ilarly observed that the number of lobes in- imally over the uniformly curved portions of creased in the broadly curved dorsal region the phragmocone. In scaphites there are lat- of depressed ammonites and that ammonites erally compressed forms and globose forms. with round whorl sections had a relatively Laterally compressed forms such as Hoplo- even distribution of lobes around the whorl. scaphites show close sutural spacing on the Spath also cited examples of evolutionary re- flanks whereas globose forms such as some sponse of the suture line to change in whorl Scaphitesshow even septal spacing around the shape within lineages. Checa (1986) and phragmocone. Hewitt and Westermann (1987a) also ob- Buckland (1836) and Westermann (1956, served the relationship between involute 1971, 1975) have similarly commented on the compressed morphologies and the develop- 344 DAVID K. JACOBS ment of additional lobes in the region of the and the outer shape of the shell in ammonites. suture supporting the flank. On the flat flanks Consequently, the structural relationship ex- of increasingly involute or oxyconic lineages, pressed here should be important in con- the sutures extend across the breadth of the structing phylogenies. As previously noted flank and converge toward the diminishing (Westermann 1956, 1971), heavy reliance on umbilicus (Fig. 5, A). The sutures approach the sutural morphology of an ammonite may each other as they approach the umbilicus be poor taxonomic practice in light of the much as the radial spokes of a wheel approach potential for convergent evolution. each other adjacent to the hub. Due to the The frequency of involute compressed oxy- elongation toward the umbilicus and the lat- cones with similar sutural development in eral compression of the whorl, additional su- the flank regions is an enlightening example. tural complexity on preexisting lobes of the Oxyconic homeomorphs with similar exter- suture can not readily control or contribute nal and sutural morphology occur at widely to sutural proximity in the region of the flank spaced intervals throughout the Mesozoic approaching the umbilicus. To control sutur- (Spath 1919). Repeated evolution of involute al spacing in this region in lineages devel- compressed morphologies with similar su- oping this compressed involute oxyconic tural patterns also occurs within individual morphology, new umbilicular lobes evolved. evolutionary lineages (Bayer and McGhee Workers have tended to concentrate on the 1984). This repetition of oxyconic morphol- addition of sutural elements with increasing ogy may relate to adaptive influences on shell radius of shell curvature such as in the broad shape which in turn constrain sutural pat- flanks of oxycones. A concomitant decrease tern. in proximity of sutural elements in those Hydrodynamically Efficient Shell Shapes and regions of the shell simultaneously under- Sutural Complexity.-Sutural complexity may going a reduction in radius of curvature also have evolved in response to selective forces occurs in the scaphites and in cardioceratid driving the ammonoid morphology away and macrocephalitid lineages observed by from circular whorl sections. Circular whorls Spath (1919). This suggests that selection on are structurally advantageous; they do not re- the suture operated to create a series of vaults quire vault support to avoid bending stresses. of comparable reactive force, rather than just Consequently, departure from a circular whorl adding support on the flanks as curvature section would require some alternative or changed. competing selective influence. Selection for Observations of the spacing of sutural el- hydrodynamically efficient morphology may ements and their relationship to shell cur- provide such a selective advantage favoring vature support predictions of vault spacing non-circular whorl sections and resulting in generated by the interactive vault hypothesis. selection for sutural complexity to accom- The vault hypothesis predicts sutural spacing modate these shapes. and constrains the evolution of greater su- Oxyconic morphologies confer hydrody- tural complexity to particular regions of the namic efficiency (Chamberlain 1976, 1981). shell in a wide range of coiled as well as Oxyconic shapes require the greatest varia- straight ammonites. Consequently, the exter- tion in surface curvature, from the venter to nal shape of the shell functionally and adap- the flank, of any planispiral ammonoid whorl tively controlled the evolution of sutural pat- shape. Without complex sutures dividing the tern in ammonites. flanks of an oxycone into a series of smaller Functional Constraint.-If the sutural pat- vaults, outward thrusting of the flanks could tern in ammonites is functionally constrained not be balanced by the vault force produced by the external shape of the shell, there are in the ventral region. Sutural complexity al- a number of implications for the study of am- lows the attainment of the large variation in monite evolution. This functional constraint shell curvature observed in oxycones. contributes to the frequent incidence of par- The argument for the adaptive influence of allel evolution involving both the suture line hydrodynamics on shell shape and its rela- SUTURAL PATTERN AND SHELL WALL STRESS 345 tionship to sutural pattern is supported by the advantageous use of the greater com- the shapes attained by ammonites and nau- pressive strength of nacre allowed by vault tiloids in the Mesozoic and Cenozoic. During action. Use of a series of vaults to avoid ten- the Mesozoic, ammonoids occupy, and appear sional bending stresses permits the construc- to competitively exclude nautilids from, the tion of a greater variety of shell shapes with morphospace of hydrodynamically advanta- less material. Nautiloid shells that counteract geous laterally compressed forms that mini- bending by local thickening will be more se- mize drag (Ward 1980; Chamberlain 1981). In verely limited in the shapes they can attain the Tertiary, after the demise of the ammo- than ammonites employing interactive vaults. nites, nautilids developed a certain amount The relationship between the shell thick- of sutural complexity in conjunction with ening on the flanks of nautilids and their lim- laterally compressed morphologies (e.g., Atu- ited suite of shell morphologies relative to ria). These observations suggest that evolu- ammonites was pointed out by Westermann tion of laterally compressed hydrodynamical- (1971) and shown empirically by Ward (1980). ly efficient cephalopod shell morphologies is In the work presented in this paper, an in- associated with the evolution of sutural com- teractive vault model provides a direct rela- plexity. The possibility that sutural complex- tionship between external shell shape and ity evolved as a consequence of selection for optimal sutural pattern in ammonites. This hydrodynamically efficient shell shapes mer- model does not appear to describe the struc- its further investigation. tural interactions in nautiloid shells. Limitations of the Interactive Vault Model.- The Role of Ornament.-As observed by Not all cephalopod shells or even all portions Buckland (1836), ornament can also have an of the ammonite phragmocone are equally important structural role in support of the constrained by the interactive vault model. shell. Ribbing reduces bending stress by plac- Nautilids range from globose forms such as ing material farther away from the neutral Eutrephoceraswith concave septa unlikely to axis, thereby increasing what is termed the support the phragmocone to laterally com- "moment of inertia." The material displaced pressed forms such as Aturia with a relatively away from the neutral axis resists bending in complex suture supporting the flanks of the a manner similar to the corrugations on a tin shell. Nautilus has a single septal strut sup- roof. Folding or "corrugation" of the shell porting the slightly flattened flanks. Hewitt wall produces the ribbing observed on the and Westermann (1987b) have observed that flanks of most ammonites. In contrast to ribs some bending of the shell does occur over produced by thickening and thinning of the this simple septal support. This suggests that shell, folding of the shell wall does not great- nautilid shells do not function as a series of ly increase the cross-sectional area of the shell. vaults in a manner here proposed for Bacu- By minimizing the increase in cross-sectional lites. The failure of nautilids, or at least of area the axial compressive stress produced by Nautilus, to employ a series of vaults to min- vault action can be maintained to limit ten- imize tensional shell stress in bending may sion due to bending. In addition, the ribs on be related to the relative thickening of the the flanks of compressed ammonites are ori- shell wall in the flank region to accommodate ented in a nearly radial direction: ribs on the bending (Westermann 1971). Thickening of broad ventral region of depressed ammonites the shell strengthens it in bending. However, such as Cadoceras,Stephanoceras or Macroceph- stress is inversely related to the cross-section- alites are oriented transversely across the al area acted on by a force; consequently, whorl. Ribs in these orientations will resist thickening to counteract bending will dimin- bending in the circumferential directions in ish the compressive stress resulting from vault regions of the shell where the greatest bend- thrusting. It is this compressive stress which ing stresses occur. Due to their orientation minimizes the tensile stresses resulting from and the fact that they do not greatly increase bending. Local thickening as a solution to cross-sectional area, most ribs facilitate struc- bending in the shell will tend to counteract tural support in a manner consistent with the 346 DAVID K. JACOBS

TABLE 2. Specimen information and data on vault span and radius of curvature for 10 specimens of Baculites: B. comp. = Baculites compressusSay; R = radius of curvature; 12 = halfspan length squared.

Specimen A B C D Species B. sp. B. sp. B. comp. B. comp. Repository Raup, D. Field Mus. Field Mus. Field Mus. Specimen No. none 3544-1 3602 36390-a Data (mm) R 12 R 12 R 12 R 12 5.0 18.14 18.5 2.05 27.0 14.32 20.0 0.94 5.0 16.55 8.4 5.86 13.0 19.58 7.0 1.59 6.0 16.55 4.5 14.99 9.0 30.08 3.2 3.90 7.0 14.17 20.0 1.83 31.0 12.57 21.0 0.76 7.0 15.76 8.5 6.34 18.0 14.96 6.5 1.35 7.0 14.96 4.5 19.83 8.5 36.13 3.5 3.76 7.0 14.96 19.0 2.38 30.0 11.30 18.0 0.94 7.0 8.12 7.0 9.60 16.0 15.12 6.0 1.94 8.5 8.12 4.3 17.99 11.5 26.26 3.0 4.38 9.0 8.75 21.0 1.83 32.0 13.85 18.0 0.94 10.0 8.12 6.8 7.35 15.0 21.17 5.8 1.78 10.0 9.23 3.0 14.26 6.0 40.59 2.8 3.76 11.0 8.12 21.5 1.83 34.0 7.96 16.5 0.70 35.0 7.48 7.0 7.35 19.0 12.57 5.5 1.40 36.0 2.71 4.2 14.99 9.5 31.99 3.0 3.60 37.0 2.07 23.0 1.83 17.0 0.76 39.0 2.39 7.4 7.32 5.2 2.96 42.0 2.39 4.0 13.53 3.5 3.39 41.0 2.39 22.0 1.83 45.0 2.07 6.0 6.84 4.0 13.53 Reduced major axis -0.958 -1.270 -0.865 -0.919 Regression slopes (all slopes are significant at P < 0.0005)

interactive vault model of sutural support of confirms predictions based on an interactive the phragmocone. vault model of the phragmocone. In this mod- Features of the Shell Not Subject to the Inter- el, the inverse relationship between the span active Vault Mechanism.-The umbilical re- size and radius of curvature results in equiv- gion of coiled cephalopods presents the prob- alent reaction forces at the vault ends in the lem of pressure inside rather than outside a series of vaults comprising the phragmocone. cylindrical structure; therefore deep tubular The equivalence of the reaction forces at the umbilici will generate tensional rather than vault ends provides a compressive stress that compressive stress. Avoiding a well defined is superimposed on and limits tensional stress umbilicus may be one mechanism of dealing due to bending. Limitation of tensional with this potential problem. Shell thicken- stresses is necessary if the ammonite shell is ing, plugs and wrinkle layers in the umbili- to support hydrostatic load optimally with cus may function to limit tensional stress in minimal nacreous shell material. This is a the umbilical wall. These features of the um- consequence of the greater strength of nacre bilicus and possibly other structures such as in compression than in tension. keels represent structural solutions beyond The interactive vault model is applicable to the scope of the simple vault model. coiled as well as straight ammonite shells. It provides a major constraint on sutural spac- Conclusions ing and therefore on sutural pattern and com- In Baculitesthere is a statistically significant plexity. The vault model indicates a direct inverse relationship between the size of vault relationship between external shell mor- spans defined by the sutural pattern and the phology and sutural pattern. If sutural pat- radius of curvature of these vaults which tern and shell shape are linked in a functional comprise the phragmocone. This relationship sense, parallel evolution would be expected. SUTURAL PATTERN AND SHELL WALL STRESS 347

TABLE2. Extended.

E F G H I J B. sp. B. comp. B. comp. B. comp. B. sp. B. comp. UICC Col. Field Mus. UICC Col. UICC Col. UICC Col. UICC Col. Cm3 36390-b 29-i F24 Fm3 31

R 12 R 12 R 12 R 12 R 12 R 12

4.0 12.14 20.0 1.10 4.5 18.14 3.0 5.40 5.5 7.35 5.3 17.33 13.0 5.40 7.2 2.71 9.0 9.01 2.8 9.01 27.0 2.44 10.6 6.83 31.0 1.94 4.2 5.86 24.5 2.40 5.2 5.40 5.0 9.60 34.0 3.17 4.5 12.14 21.0 1.69 3.5 26.26 20.0 1.83 10.8 5.40 5.4 25.34 12.0 5.86 6.0 2.58 8.5 14.99 3.2 7.88 31.0 1.83 10.8 9.96 30.0 2.71 4.0 5.67 29.0 2.10 5.5 6.83 5.6 9.01 32.0 4.52 4.0 18.14 19.0 0.94 8.0 9.01 19.0 1.35 10.0 4.95 6.0 21.60 10.0 8.44 6.8 2.23 30.0 1.91 3.0 10.19 35.0 1.13 11.5 9.60 30.0 2.39 3.5 4.54 6.0 6.83 4.8 10.90 33.0 4.14 17.5 1.13 22.0 1.59 8.0 5.86 5.7 3.25 3.5 11.46 31.0 1.59 3.0 6.14 6.5 5.40 18.0 1.78 20.0 2.10 5.2 3.17 3.2 10.82 3.2 5.40 6.5 7.56 20.0 1.45

-0.924 -0.875 -1.248 -0.959 -0.980 -0.965 (all slopes are significant at P < 0.0005)

Homeomorphy (especially in lineages tend- Acknowledgments ing toward compressed involute morpholo- I thank R. Hewitt, G. Westermann and J. gies where there is a tendency to generate Chamberlain for their critical reviews and R. new primary lobes on the flattened flanks of Bambach, B. Bennington, R. Cowen, N. Gil- the whorl) may be explained by the con- insky, A. Hubbard, and N. Landman for their straints imposed by this model. Recognition assistance with the manuscript. of the functional relationship between sutural pattern and shell shape should help dif- Literature Cited ferentiate between parallelism and homology BAYER,U., AND G. R. McGHEE,JR. 1984. Iterative evolution of in phylogenetic analyses of ammonites. Middle Jurassic ammonite faunas. Lethaia 17:1-16. BUCKLAND,W. 1836. Geology and mineralogy considered with Nautiloids do not appear to employ the in- reference to natural theology, chapter 15 (Proofs of Design in teractive model of shell support but locally the Fossil Remains of Molluscs) Vol. 1. Pp. 295-381. In The thicken the shell instead. This would appear Bridgewater Treatises on the Power, Wisdom and Goodness of God as Manifest in the Creation; Treatise VI. to explain the lower degree of sutural com- CHAMBERLAIN,J. A., JR. 1976. Flow patterns and drag coefficients plexity and more limited suite of shell mor- of cephalopod shells. Paleontology 19:539-563. phologies occupied by coiled nautiloids as CHAMBERLAIN,J. A., JR. 1981. Hydromechanical design of fossil cephalopods. Pp. 289-336. In House, M. R., and J. R. Senior opposed to ammonites. (eds.), The Ammonoidea. Systematics Association; London. The employment of a set of interactive CHECA,A. 1986. Interrelated structural variations in Physod- vaults in the ammonite phragmocone capi- eroceratinae (Aspidoceratidae, Ammonitina). Neues Jahrbuch fur Geologie und Palaontologie Mittheilungen 1986:16-26. talizes on the greater strength of nacre in CURREY,J.D. 1976. Furtherstudieson the mechanical properties compression than tension. The complex am- of mollusc shell material. Journal of Zoology, London 180:445- monite septal suture serves to determine ap- 453. CURREY,J. D., ANDJ. D. TAYLOR.1974. The mechanical behavior propriate span size, and provides end sup- of some molluscan hard tissues. Journal of Zoology, London port, in this series of vaults. 173:395-406. 348 DAVID K. JACOBS

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